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Method and device to investigate the behavior of large rotors under continuously adjustable foundation stiffness
Vibration problems have been observed after the installation of large rotating machines, such as electric machines and generators and paper machine rolls. One possible cause can be differences
between the foundation stiffness of the installation location and the testing platform where the machine is balanced and optimized. Foundation stiffness exerts a significant effect on the behavior of
a rotating system, and the above-mentioned differences can cause major unexpected changes at natural frequencies, and thus resonance. The problem is typical for large machines due to their large
mass, which leads to low natural frequencies. This induces situations where these natural frequencies coincide with rotor excitations and cause excessive vibration. This study presents a novel method
and a device for adjusting the foundation stiffness of a large rotor system, consequently enabling the investigation of the effect of foundation stiffness on rotor behavior. However, this
investigation is restricted to the horizontal axis. The characteristics of the device were analyzed together with a rotor behavior measurement that consisted of versatile measurements of
acceleration, force and displacement in different locations inside the rotating system. The device in the presented form is best applied in R&D laboratories and factory acceptance test cells, in
which it can be used to predict the behavior of various rotors on different foundations. With the dynamic rotor behavior measurement performed with the device, the natural frequencies and their
harmonic components can be presented as a function of foundation stiffness. This information can be used both to optimize rotor behavior in an installation location and also to improve the rotor
system behavior in the design phase. The method and device presented in this study can be considered effective and successful, since the natural frequencies of the first two rotor modes could be
manipulated freely at a range of 50-100 % by changing the stiffness.
• A novel method and device to adjust foundation stiffness of large-scale rotors are presented.
• The behavior of a flexible industrial rotor is investigated with different foundation stiffnesses.
• Proposed concept can be applied the best in R&D laboratories, acceptance test cells and as a semi-active actuator to optimize rotor behavior at installation location.
1. Introduction
The designing of any rotating system is complicated if foundation stiffness is unknown. Frequently, rotating machines or rotors are delivered without an accompanying installation service, with the
responsibility for a proper foundation resting with the customer. Sometimes this causes problems after the installation of a rotor when the foundation differs from that expected by the manufacturer.
In particular, foundation stiffness has a significant effect on the dynamic behavior of a rotor. An unexpected difference between estimated and actual foundation stiffness can shift the natural
frequency of the rotating system and cause major vibration problems at operating speed. The problem is difficult to solve because it appears only after the first run on the final machine bed. Such
problems are more common when operating with large machines: natural frequencies are lower due to larger masses, and thus the gap between the operating speed and natural frequency of the system is
smaller. Typically, vibration problems appear also below the natural frequency at subcritical resonance frequencies. These problems occur at fractions, such as 1/2 and 1/3 times, of a natural
frequency. The most common excitation sources for subcritical resonances are bending stiffness variation and various bearing excitations, such as errors in inner ring roundness. The present study
focuses especially on subcritical vibrations.
In general, the vibration behavior of rotating systems can be solved analytically and with the finite element method (FEM). Kang et al. [1] simulated the effects of foundation parameters on rotor
vibration, and their analysis found strong correlations between the parameters and natural frequencies. However, a simulation model cannot provide sufficiently accurate results for vibration analysis
if the foundation parameters are unknown and only estimated. Particularly in precision applications, where displacements are measured in micrometers, even a minor difference between the rotating
system and the model can cause a significant error. Therefore, machine testing, verification and performance measurements are still everyday chores in industry, especially in the case of large rotor
systems that are not mass-produced.
Vibration problems caused by uncertainty over the foundation parameters were investigated in the present study by integrating an adjustable stiffness into the test bed. The foundations of rotating
machines are usually modeled as objects that have mass, stiffness and damping [2]. Ultimately, the ideal test bed should adjust these parameters across wide ranges in six axes to simulate the large
variety of machine beds. This kind of test bed would enable the investigation of the vibration behavior of different rotating machines on a single bed located in the test laboratory.
In the present study, the first steps towards this ideal test bed were taken, and the most effective feature, stiffness, was implemented on the horizontal axis of the test bed with adjustability.
Stiffness has a significant impact on the natural frequencies of rotor systems. The adjustability of mass and damping were beyond the scope of the present study. Stiffness adjustment was achieved by
changing the length of a cantilever steel beam that was directly attached to the foundation of the rotor below the bearing housing. This principle is simple, and its adjustability is based on
classical beam theory. In addition to its contribution towards an ideal test bed, the novelty of the designed bed can be exploited in the elimination of vibration, since harmful natural frequencies
can be avoided by varying the foundation stiffness.
Other solutions are available, using various methods, for changing stiffness [3-16]. These methods implemented using springs, memory alloy metals, magnets, mechanical solutions, piezoelectric and
magnetorheological materials, have also been used in rotor vibration elimination. Winthrop et al. [17] compared these different methods and organized them by their effectiveness for stiffness
variation, and thus also their ability to change the behavior of rotors. However, the new principle developed for the present study was chosen because of its simplicity and feasibility. The method is
able to change stiffness in wide range without significantly affecting other foundation parameters, such as damping. In addition, this design allowed the results of a previous study to be exploited
This study investigates the effects of varying stiffness on dynamic rotor behavior using the developed device. The effects on dynamic response are measured by the displacement of the rotor middle
cross-section, the acceleration of the bearing housings and radial bearing forces. The results demonstrate that the method and device had a considerable impact on the behavior of the rotor and its
natural frequencies. The stiffness range of the device was approximately 2-18 MN/m, which was sufficient to freely manipulate the natural frequencies of the first two modes at a range of 50-100 %.
Thus, as the results indicate, this method for the stiffness adjustment of large rotor foundations seems promising. In particular, R&D laboratories and factory acceptance test cells can exploit the
method to optimize rotors for certain foundation stiffnesses. Because foundation stiffness can be actively controlled with a servo drive, it is applicable also in semi-active vibration control,
enabling the user or automatic control system to adjust the system stiffness characteristics to avoid resonance vibration.
2. Materials and methods
2.1. Device for adjustable foundation stiffness
In this design, the rotor was mounted on a horizontally free and vertically and axially rigid bed, illustrated in Fig. 1. This was achieved by supporting the rotor bed cradle with plate springs
mounted to the rest of the body from the other end. Thus, the cradle was suspended, and buckling could be prevented. The plate springs enabled almost free horizontal movement while holding the rotor
rigidly in respect to the axial and vertical directions. However, as the horizontal movement of the cradle increases, gravity exerts a larger effect. Nevertheless, the movement of the cradle was
small, and thus the effect of gravity could be considered negligible.
Fig. 1Horizontally, the rotor bed is supported in a very flexible manner. The cradle hangs on thin plates, which provide support mainly in the vertical and axial directions
The horizontally free rotor bed could be converted to offer adjustable stiffness by attaching an external device to control the horizontal stiffness of the cradle (rotor bed). This external device is
presented in Fig. 2. The principle of the device is simple: the position of the beam support (horizontal stiffness adjuster, HSA) can be varied, thus changing the stiffness of the beam end.
Fig. 2Adjustable stiffness device: a) the appearance of the device and b) the supporting structure inside the device. The beam, which determines the horizontal stiffness of the cradle, is colored
To enable continuous, in-process stiffness adjustment, the HSA position was controlled with a ball screw and a servo motor. The HSA included several different parts, including the frame, bearings,
axles and supports. It was tightened into one stiff structure with screws, which eliminated clearances from the system. Finally, the external device was added to the horizontally free rotor bed. The
cradle was directly mounted to the end of the beam with a connection bar, and thus the stiffness of the system was determined by the external device. The complete device with sensors is presented in
Fig. 3.
Fig. 3The rotor bed with horizontally adjustable stiffness. The length of the beam can be changed by moving the HSA device along the beam. In addition, the positions of the force and acceleration
sensors can be seen
2.2. Stiffness characterization of the system
2.2.1. Analytical stiffness characterization
The stiffness of the device in a horizontal direction can be determined with an analytical solution. Bedford and Liechti [19] have extensively presented the effects of loads on beam deflection and
have demonstrated different strategies for solving various beam configurations. Their calculations are based on Euler-Bernoulli beam theory, which assumes beam deflections to be small. The theory is
a simplification of the linear theory of elasticity, which has been widely used in the field of engineering. The theory derives a static relationship between the distributed load $q$ and the
deflection $v$, which can be now exploited to solve the stiffness of the device:
where $E$ is the elastic modulus and $I$ is the second moment of area. Both variables are in this case constant. Because in the equation the deflection is in the form of the fourth derivative, the
final deflection of the beam can be solved by integrals:
$\frac{dv\left(x\right)}{dx}=\theta ,$
$v\left(x\right)=\delta ,$
where $Q$ is shear force, $M$ is moment, $\theta$ is the slope of a beam and $\delta$ is the deflection of a beam.
Frequently, a tabulated expression is available for the required beam configuration. However, in this case, the beam deflection and stiffness must be derived using the Euler-Bernoulli equations
presented above. Before the equations can be applied, the configuration of the beam must be determined. The free body diagram in Fig. 4 illustrates the forces, their relations and distances in a
device where the beam is supported rigidly from its lower end, the movement in $y$ direction is supported by intermediate beam support (HSA) and the other end is connected to the rotor foundation.
The connection to the rotor foundation can be now modeled with force $F$.
Fig. 4The structure of the device converted into a free body diagram: a) original beam and its support, b) free body diagram, c) free body diagram with point loads in relation to force F
As the free body diagram indicates, only point loads are acting in the system. In this study, the point loads are derived for the shear forces of the beam, and then the shear forces can be directly
applied with the Eq. (2). The shear forces form two different sections on the beam; thus, two equations must be created according to these sections:
The final equation for the deflection can now be solved by integrating these equations three times:
The integration constants can be solved using boundary conditions:
Finally, the stiffness of the beam can be solved when the applied force and the deflection of the beam end is known:
By substituting the structure and material parameters from the actual device with Eq. (11), the deflection of the beam end can be calculated as a function of the intermediate support position ($R$).
This value can be divided by the applied force $F$ as in Eq. (12), which leads to the developed stiffness of the beam [N/m]. The parameters used in the calculations are presented in Table 1. The
force used can be arbitrarily decided when solving the stiffness. The other parameters are determined by the structure and materials of the device.
Table 1Structure and material parameters used in the calculations
Parameter Deflection position $x$ [mm] Force $F$ [N] Elastic modulus $E$ [GPa] Second moment of area $I$ [mm^4] Length of the beam $L$ [mm]
Value 491.75 (=$L$) 2000 210 202667 491.75
2.2.2. Simulated stiffness characterization
The stiffness of the device can also be determined with a simulation utilizing the Finite Element Method (FEM). This is achieved by loading the beam as in the analytical solution and then observing
the beam end movement in relation to the force used. Only the essential parts of the structure were left for static FEM analysis that virtually meant the analysis of the beam. Siemens NX was used for
the modelling and analysis.
Fig. 5Simulation process. On the left-hand side there is meshed beam with constraints and force. The beam end has all 6 DoF (Degrees of Freedom) constrained and different HSA positions have only the
horizontal one. On the right-hand side there is a result of the first loading case in which the force F was applied, and HSA was in position 0 mm. The result describes the absolute displacement of
the beam end in mm. h= 85 mm and H= 385 mm
The beam was first divided into suitable sections to enable correct positioning of the intermediate support and hence also analysis of the corresponding stiffness range. Then a 3D element meshes with
a maximum element size of 5 mm was created according to the actual beam. The static FEM analysis also demanded the determination of the forces and constraints of the model. The beam was constrained
rigidly from the lower end and laterally from the HSA positions. The movement of the HSA in the simulation corresponded the HSA movement in the actual device. The Fig. 5 illustrates the simulation
process and the used HSA positions. The beam was loaded from the free end with a force of 2000 N. The parameters used in the simulation are presented in Table 2.
Table 2Structure and material parameters used in the simulation
Parameter Element size [mm] Deflection position $x$ [mm] Force $F$ [N] Elastic modulus $E$ [GPa] Second moment of area $I$ [mm^4] Length of the beam $L$ [mm]
Value 5 491.75 ($=L$) 2000 206.94 Not a constant 491.75
2.2.3. Experimentally measured stiffness characterization
The stiffness of the rotor-mounting cradle in a horizontal direction could be determined by measuring the displacement of the cradle under a known force. Because the stiffness of the complete system
must be known, the measurement setup was arranged in the following way. The cradle was connected to the stiffness-adjusting beam, which was displaced with a screw through a force sensor. The
displacement of the cradle and the force that developed the displacement were measured. The measurement setup is presented in Fig. 6. A non-linear change in stiffness was expected during the movement
of the HSA; thus, it was necessary to repeat the measurement several times at different points so that the full stiffness range could be properly determined. The utilized sensors and their
specifications are introduced in Table 3.
Table 3Specifications of sensors
Measured unit Sensor Resolution Repeatability
Force HBM S9M 1 N (accuracy class 0.02) 1 N (accuracy class 0.02)
Displacement Sylvac S229 1 μm 2 μm
Fig. 6Measurement setup for the stiffness characterization. The displacement (force) on the beam was produced with a screw through the force sensor and cradle. The displacement of the cradle was
measured with the dial gauge. The blue color indicates the parts that acted with the force between the sensor and the beam
The experimental measurement was conducted as follows:
1) The horizontal stiffness adjuster (HSA) was set its highest position.
2) A force of 250 N was applied to the system.
3) 100 corresponding displacement and force samples were taken.
4) The force was increased by increments of 250 N.
5) Steps 3 and 4 were repeated until the force was 2500 N.
6) The HSA was moved downwards in increments of 30 mm.
7) Steps 2 to 6 were repeated until the full movement (300 mm) was reached and measured.
Thus, the stiffness of the system was measured in 11 different HSA locations. The measurement described above was performed for both devices at both ends of the rotor. The averaged results of the
measurement points were employed to solve the stiffness of the devices.
2.3. Subcritical rotor behavior measurement
The device presented in the previous sections was mounted at the both ends of the test rotor. Changes in subcritical vibration behavior were monitored when the horizontal stiffness of the foundation
was decreased. Monitoring was conducted by measuring the response (center point movement) at the middle cross-section of the rotor, acceleration in the bearing housings, and radial bearing forces.
The complete test setup is presented in Fig. 7.
Fig. 7Complete test setup. The accelerometers and force sensors were placed at the rotor ends, and displacement measurement was performed with laser sensors attached to the yellow arc
2.3.1. Test setup
The test setup was constructed on a rigid grinding machine bed, with the test rotor fixed to the device. The test rotor was a 735 kg paper machine roll, and its main dimensions are presented in Fig.
8. The rotor speed and position of the HSA were controlled with a CNC. The test rotor was connected to the motor drive through a gearbox with a universal joint.
Fig. 8Main dimensions of the test rotor. All dimensions are in mm
Fig. 3 shows the measurement points of the acceleration and force sensors and Fig. 7 the displacement measurement. The specifications of the sensors are presented in Table 4. The accelerometers were
attached to the bearing housings, and only the vertical and horizontal directions were measured. Measurement of the radial bearing forces was performed by integrating the force sensors into the
device. The principle of radial bearing force measurement is straightforward and has been presented in a previous study [18]. In the present study, the center point movement was measured by
exploiting the four-point method developed by Kuosmanen and Väänänen [20] and further investigated by Viitala et al. [21]. The method is a combination of two different methods: the Ozono three-point
method [22], developed for roundness measurements, and the two-point method, which is a straightforward method to measure the diameter variation of the rotor. The method was applied with four
reflective laser sensors that were arranged around the rotor at certain angles. The four-point method enabled the separation of the center point movement and the roundness of the rotor in the dynamic
Table 4Sensors used in rotor behavior measurement
Measured unit Sensor Sensitivity Range
Acceleration Brüel & Kjær type 4381 10.0 pC/ms^-2 0.1 to 4800 Hz
Force (horizontal) Kistler 901A –4.3 pC/N 0 to 7.5 kN
Force (vertical) HBM PaceLine CFW –4.3 pC/N 0 to 100 kN
Displacement Matsusita NAIS LM 300 1 V/mm 27 to 33 mm
2.3.2. Data acquisition
To ensure as simultaneous sampling as possible, all the measurement signals were obtained with a single National Instruments PCI-6259 data acquisition card (DAQ) using an external trigger. The card
applied multiplexing during the measurement, and thus the samples could not be acquired at exactly the same time. However, since the card had a sampling frequency of 1 MS/s in multichannel mode, the
problem was considered negligible.
A rotary encoder connected directly to the rotor shaft triggered the measurement and operated as an external sample clock. Thus, the sampling frequency changed as a function of rotor speed. The
encoder had 1024 pulses/rev, which consequently led to 1024 measurement samples per rotor revolution. The external phase-locked sample clock enabled a time synchronous averaging (TSA) method, which
is presented in Section 2.3.4. The method facilitates the identification of harmonic frequencies from the signal and reduces noise and other non-periodic signals.
2.3.3. Measurement procedure
The measurement was conducted simultaneously with all sensors, thus ensuring comparability between the results. The measurements were performed at a constant speed to enable averaging and FFT
analysis, which is presented in the next section. The measurement procedure was completed as follows:
1) The support of the beam was set at its stiffest point.
2) The rotor was accelerated to its starting frequency, 4 Hz.
3) 100 revolutions were measured.
4) The rotating frequency was increased by increments of 0.05 Hz.
5) 100 revolutions were measured.
6) Steps 4 and 5 were repeated until the rotating frequency was at its final value, limited by the safety margin (no crossing of the first mode natural frequency).
7) The rotating frequency was decreased to its starting frequency of 4 Hz.
8) The support of the beam was moved 10 mm downwards to decrease the horizontal stiffness.
9) Steps 3-8 were repeated until the HSA was at its lowest point.
This measurement procedure produced data measured at 31 stiffness points.
2.3.4. Signal analysis
In the present study, the signals were investigated using the time synchronous averaging (TSA) method and Fast Fourier Transform (FFT) algorithm. They both offer advantages when studying periodic
signals from vibration and are therefore widely utilized in vibration analysis.
Time synchronous averaging [23, 24] is a signal processing technique that can be considered a filtering process for periodic signals, which are extracted from non-periodic signals. Therefore,
significant information can be also lost if the focus is not exclusively periodic (harmonic) signals. The method is based on a clock signal that is phase locked with the angular position of a
rotating object. Originally, McFadden and Toozhy [23] applied the method to the investigation of bearing damage, but it is also utilized in the vibration measurement of rotors, as Widmaier [25] and
Viitala [26] have shown.
FFT was originally developed in 1964 by Cooley and Tukey [27]. It is based on the assumption that every periodic signal can be represented by combining an infinite series of trigonometric functions.
The algorithm can be applied to discrete signals, such as measurement data, and it can then present the signal in the frequency domain. In the frequency domain, the signal takes the form of complex
numbers that contain information about the absolute amplitude and phase of a signal.
3. Results and discussion
3.1. Beam stiffness characterization
Beam stiffness was determined with analytical, simulation and experimental methods. The analytically determined deflection and stiffness of the beam end can be seen below in Fig. 9 as a function of
the HSA position. In the calculations, a force of 2000 N was used, which is the same force as in the measurements. The results were obtained from the range of 0-491.75 mm, which corresponds to the
total length of the beam. The HSA can move in a range of 106.75-406.75 mm.
Fig. 9Analytic results for the deflection and stiffness of the beam. 0 mm corresponds to the upper end of the beam
As can be observed from the stiffness diagram, theoretically stiffness increases towards infinity when the rigid beam support approaches the beam upper end (position 0 mm). However, in reality, the
rigid support at the beam end and HSA (Fig. 4) are not completely rigid, which limits the stiffness.
The beam end deflection and stiffness derived by the simulation are presented below in Fig. 10. The results were obtained from 11 different positions from a range of 106.75-406.75 mm measured from
the beam upper end in which the HSA was able to move.
The results of the experimental measurement are presented in Fig. 11. The stiffness was measured with the complete assembly, in which the force was conducted through the cradle, instead of measuring
only the stiffness of the beam. This is likely to be a more accurate method for describing the real stiffness of the existing system. The interpretation of the results is straightforward except for
the slope that can be seen in the tending-end results. The slope is a result of the small clearance between the frame and the cradle; the cradle collided with its frame when using high loads.
However, this was not a problem when the device was used in rotor measurements, since excessive forces were not produced. Moreover, the problem could be eliminated by increasing the clearance between
the frame and the cradle. The results indicate that the clearance between the cradle and the frame was approximately 0.8 mm at the tending end when the collision occurred. At the driving end, the
maximal movement of the cradle was approximately 1 mm when a 2000 N force was applied.
Fig. 10Simulated results for the deflection and stiffness of the beam. HSA position 0 mm corresponds to the upper position of the HSA and 300 mm to the lower position of the HSA
Fig. 11The horizontal displacement of the cradle at a constant 2000 N force in different HSA positions. HSA position 0 mm corresponds to the upper position of the HSA and 300 mm to the lower position
of the HAS
The final stiffnesses of the devices were averages calculated from the 10 measuring points shown in Fig. 12. At each point, the measured force is divided by the corresponding displacement, which
provides the stiffness at that point. This procedure was repeated for 11 different HSA positions. However, it was necessary to eliminate some points from the averaging process due to collision and to
non-linearity when using low forces. After these issues were controlled for, the stiffnesses were linear in the force range used for each HSA position. The averaged stiffnesses of both HSA devices
and their combined average are presented in Table 5. The stiffness average of both HSA devices was used as the horizontal stiffness of the complete system.
Table 5Averaged stiffnesses in different HSA positions
Position of HSA [mm] Averaged stiffness in the driving end device [MN/m] Averaged stiffness in the tending end device [MN/m] Averaged stiffness of the system [MN/m]
0 17.86 18.77 18.32
30 12.77 12.88 12.83
60 9.18 9.55 9.37
90 6.79 7.76 7.28
120 5.60 5.90 5.75
150 4.55 4.56 4.56
180 3.64 3.68 3.67
210 3.11 3.12 3.12
240 2.74 2.61 2.68
270 2.35 2.36 2.36
300 2.01 2.06 2.04
Fig. 12Stiffness measurements under different loads in 11 different HSA positions
A regression curve was fitted to the averaged stiffness points to achieve a continuous stiffness curve that acts as a function of the HSA position. This enables accurate control of the horizontal
stiffness between the minimum and the maximum. The derived regression curve is presented together with the analytical and simulation stiffness curves in Fig. 13.
Fig. 13Measured, analytical and simulated stiffness curves as a function of HSA position. HSA position 0 mm corresponds to the upper position of the HSA and 300 mm to the lower position of the HSA
As can be observed from the Fig. 13, the measured results differ from the other results, particularly at the stiff end. According to the analytical results, the stiffness of the device varies between
1.72 and 37.44 MN/m, depending on the HSA position, whereas the simulated stiffness varies between 1.52 and 26.18 MN/m and the measured stiffness between 2.04 and 18.32 MN/m. The difference is mainly
due to errors in the analytical and simulation models. The models assume that the beam supports are completely rigid. Therefore, the stiffness curve increases exponentially when the beam support
approaches the beam end. These models could be further developed by replacing the rigid supports with spring supports.
The analytical and simulated solutions developed for this study are applicable for the dimensioning of the beam to achieve a certain stiffness range and thus also desired foundation stiffness in the
horizontal direction. This can be achieved by changing the beam material ($E$), length ($L$) or beam profile ($I$).
3.2. Rotor behavior measurement
The results were obtained by measuring the acceleration of the bearing housings, the radial bearing forces and the displacement of the rotor in the middle cross-section. The horizontal results form
the main part of the study, since the effects of varied stiffness were expected only on that axis. The effect of unbalance and eccentricity (the first harmonic component) was excluded from the
The behavior of the rotor was analyzed in two different ways utilizing the data gathered. The first approach presents the peak values of the amplitudes at a certain stiffness and rotor rotation
frequency. These results are separated into horizontal and vertical components, since different natural frequencies in these directions were expected. Figs. 14-16 illustrate the horizontal results
and Fig. 17 presents the vertical results.
The second approach presents the subcritical harmonic components of the vibration at a certain stiffness and rotating frequency after the FFT analysis. Only the horizontal results are presented in
Fig. 18, since the vibration behavior of the rotor in the vertical direction did not change according to the results presented in Fig. 17.
Fig. 14Peak values of the displacement measurement (center point movement) at a certain rotating frequency and stiffness. Different modes are color-coded, and each ridge denotes a harmonic component
Fig. 15Peak values of the radial bearing force measurement at a certain rotating frequency and stiffness. Different modes are color-coded, and each ridge denotes a harmonic component
The results presented in Figs. 14-16 reveal large changes in the horizontal natural frequencies of the rotor system at different stiffnesses. No natural frequencies were crossed, since the study
focused on subcritical vibrations.
The radial bearing force measurement in Fig. 15 allows the first three horizontal modes of the rotor to be distinguished from the vibration. The identification of the subcritical resonance curves was
straightforward, since they appear at fractions of their natural frequency, such as 1/2 times the natural frequency (2$H$) and 1/3 times the natural frequency (3$H$). This is related to harmonic
excitations that can occur multiple times per revolution, and therefore coincidence between the excitation frequency and the natural frequency is possible below the critical speed. In the results,
the harmonic frequencies were given as a function of the rotating frequency, and thus 2$H$ subcritical resonance was visible when the rotating speed was half of the corresponding critical speed.
Fig. 16Peak values of the acceleration measurement at a certain rotating frequency and stiffness. Different modes are color-coded, and each ridge denotes a harmonic component
Fig. 17Demonstration of the effect of varied stiffness on the vertical natural frequencies. The vertical harmonic components of the displacement measurement are presented. The vertical mode is
illustrated as red lines and each ridge denotes a harmonic component
However, as the results show, the measurements contain major differences in their ability to distinguish different modes. For instance, in the displacement measurement displayed in Fig. 14, the
second mode cannot be seen at all. This is a result of the test setup, as the displacement measurement was performed at the middle cross-section of the rotor. As already shown, the node of the second
mode is in the middle of the rotor, which makes detection of the second mode impossible. However, this could be avoided by selecting another measurement point along the rotor. By contrast, the
acceleration measurement presented in Fig. 16 detects all the horizontal modes, but it is also mixed with vertical modes: the first and second harmonic components can be clearly seen in the results.
Furthermore, its ability to separate the horizontal harmonic components was low compared to the other measurements.
The results indicate the significance of low harmonic components. For example, the fifth harmonic component of the first mode produces high amplitudes at low frequencies in the displacement and force
diagrams, Fig. 14 and Fig. 15 respectively. Furthermore, very low harmonic frequencies still exert a significant effect on the behavior of the rotor, as demonstrated by the eighth harmonic component
of the third mode in Fig. 14 and Fig. 16. In turn, Fig. 17 shows that the stiffness variation in the horizontal direction did not affect the vertical natural frequency as expected.
Fig. 18The effect of stiffness variation on the harmonic components. Stiffness decreases when moving downwards. In each measurement, the last presented harmonic component is the last significant one
a) Center point movement (Middle cross-section of rotor)
b) Radial bearing force (Bearing housing, tending end)
c) Acceleration (Bearing housing, tending end)
The second analysis was performed by applying the FFT algorithm to the measured data. This method allowed the harmonic components to be investigated more effectively. Fig. 18 presents the results in
the frequency domain, where each harmonic component can be separately observed. Only three different stiffness points are shown; the remainder are compressed and are available in gifs (appendix in
the electronic form of this article). The diagrams display the same natural frequencies and their decrease as in the previous representation. Utilizing this method, the higher harmonic components
could be distinguished. In the diagrams below, the first harmonic component is removed, and the highest presented harmonic component is the last that has significant amplitude.
As the measurement results show, the natural frequencies of the modes responded differently to varying levels of stiffness. Here, the frequencies of the harmonic components changed in an identical
manner, as they are fractions of the natural frequency. Thus, the natural frequencies could be determined and calculated by multiplying the frequencies of the harmonic components presented in the
results. Fig. 19 demonstrates the effect of varying stiffness on the natural frequencies and their harmonic components.
Fig. 19The effect of stiffness on natural frequencies
As the diagram indicates, the largest change was to the natural frequency of the second mode, where the natural frequency decreased by 60.7 % from 34.59 Hz to 13.61 Hz when the stiffness was
decreased 88.9 % from 18.32 MN/m to 2.04 MN/m. The natural frequencies of the first and third modes decreased 53.6 % and 21.1 % respectively. The effects of varying stiffness on the natural
frequencies can be seen in Table 6.
Table 6The effects of varying stiffness on natural frequencies.
Stiffness [MN/m] Change [%] 1st mode [Hz] Change [%] 2nd mode [Hz] Change [%] 3rd mode [Hz] Change [%]
18.32 0 20.98 0 34.59 0 63.87 0
14.31 –21.9 19.77 –5.8 31.79 –8.1 61.37 –3.9
10.36 –43.4 18.19 –13.3 28.22 –18.4 58.14 –9.0
6.18 –66.3 15.20 –27.6 22.45 –35.1 54.17 –15.2
2.04 –88.9 9.73 –53.6 13.61 –60.7 50.38 –21.1
4. Conclusions
In this study, a novel method and device for adjustable stiffness was developed and applied to the investigation of rotor behavior. The results suggest that the method and device for evaluating the
foundation stiffness effect was successful. The range of possible stiffnesses was wide, and varying the stiffness produced large changes in the natural frequencies of the rotor system. For example,
the device enabled the natural frequencies of the first two modes of the rotor to be halved. A test bed that can simulate and mimic other machine beds by adjusting its stiffness can be considered a
universal test bed. Since this study offered a successful proof-of-concept for one axis, a similar two-axis concept, which would also include the vertical axis, can be considered a promising novel
method and tool for final machine testing and balancing. It would significantly increase the reliability of the delivery of large machines where problems arise from differences between the
foundations of a test bed and an installation location. Moreover, there are surely other applications for this adjustable stiffness method than minimizing the vibration of a rotating system by
identifying the ideal level of stiffness.
In industry, major vibration problems typically originate from the first two harmonic components, and thus higher harmonic components (>2$H$) are ignored or, in the worst case, completely filtered
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rotor, and they should be taken into consideration both in vibration elimination as well as in solving problematic situations and achieving improved vibration levels.
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About this article
Mechanical vibrations and applications
adjustable stiffness
controllable stiffness
foundation stiffness
rotor dynamics
subcritical vibration
support stiffness
variable stiffness
This work was a part of the Digital Twin of Rotor System project (TwinRotor, Grant Number 313675), which was supported by Academy of Finland.
Copyright © 2020 Risto Viitala, et al.
This is an open access article distributed under the
Creative Commons Attribution License
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6.1.3A Multiplication & Division
Multiply and divide decimals, fractions and mixed numbers; solve real-world and mathematical problems using arithmetic with positive rational numbers.
Subject: Math
Strand: Number & Operation
Benchmark: 6.1.3.1 Multiplication & Division Procedures
Multiply and divide decimals and fractions, using efficient and generalizable procedures, including standard algorithms.
Benchmark: 6.1.3.2 Making Sense of Procedures for Multiplying & Dividing Fractions
Use the meanings of fractions, multiplication, division and the inverse relationship between multiplication and division to make sense of procedures for multiplying and dividing fractions.
For example: Just as $\frac{12}{4}$ = 3 means $12=3\times 4$, $\frac{2}{3}\div\frac{4}{5}=\frac{5}{6}$ means $\frac{5}{6}\times\frac{4}{5}=\frac{2}{3}$.
Big Ideas and Essential Understandings
Students at this level model multiplication and division of fractions and connect these models to procedures for multiplying and dividing fractions. Place-value patterns are used to multiply and
divide finite decimals by powers of 10. The relationship between decimals and fractions, as well as the relationship between finite decimals (i.e., a finite decimal multiplied by an appropriate
power of 10 is a whole number), is used to understand and explain the procedures for multiplying and dividing decimals. Common procedures are used to multiply and divide fractions and decimals
efficiently and accurately. Problem solving with positive rational numbers is extended to include arithmetic with decimals, fractions, and mixed numbers. Students build on understanding that percents
are ratios per 100 to solve problems in various contexts that require finding percent of a number or what percent one number is of another.
All Standard Benchmarks
6.1.3.1 Multiply and divide decimals and fractions using efficient and generalizable procedures, including standard algorithms.
6.1.3.2 Use the meanings of fractions, multiplication, division and the inverse relationship between multiplication and division to make sense of procedures for multiplying and dividing fractions.
6.1.3.3 Calculate the percent of a number and determine what percent one number is of another number to solve problems in various contexts.
6.1.3.4 Solve real-world and mathematical problems requiring arithmetic with decimals, fractions and mixed numbers.
6.1.3.5 Estimate solutions to problems with whole numbers, fractions and decimals and use the estimates to assess the reasonableness of results in the context of the problem.
Benchmark Cluster
Benchmark Group A
6.1.3.1 Multiply and divide decimals and fractions using efficient and generalizable procedures, including standard algorithms.
6.1.3.2 Use the meanings of fractions, multiplication, division and the inverse relationship between multiplication and division to make sense of procedures for multiplying and dividing fractions.
What students should know and be able to do [at a mastery level] related to these benchmarks
• Model multiplication with fractions and connect models of multiplication with fractions to procedures for multiplying fractions;
• Model division with fractions and connect models of division of fractions to procedures for dividing fractions;
• Use fractions, mixed numbers, and decimals to represent quotients in division of whole numbers;
• Recognize and use the place-value patterns in multiplying and dividing finite decimals by powers of 10;
• Use place value and their understanding of multiplication of fractions to justify procedures for multiplying finite decimals;
• Use place value and their understanding of representing quotients as fractions to justify procedures for dividing decimals;
• Recognize fractions, decimals, and percents as ways of representing rational numbers;
• Convert among fractions, decimals, and percents;
• Develop efficient, accurate, and generalizable methods for multiplying and dividing fractions and decimals;
• Estimate product and quotients of problems involving decimals, fractions, mixed numbers, and improper fractions;
• Solve problems requiring arithmetic with decimals, fractions, and mixed numbers.
Work from previous grades that supports this new learning includes:
• Multiply multi-digit numbers, using efficient and generalizable procedures, based on knowledge of place value, including standard algorithms;
• Divide multi-digit numbers, using efficient and generalizable procedures, based on knowledge of place value, including standard algorithms. Recognize that quotients can be represented in a
variety of ways, including a whole number with a remainder, a fraction or a mixed number, or a decimal;
• Estimate products and quotients of multi-digit whole numbers by rounding, using benchmarks, and place value to assess the reasonableness of results;
• Consider the context in which a problem is situated to select the most useful form of the quotient for the solution and use the context to interpret the quotient appropriately;
• Estimate solutions to arithmetic problems to assess the reasonableness of results;
• Solve real-world and mathematical problems requiring addition, subtraction, multiplication and division of multi-digit whole numbers. Use various strategies, including the inverse relationships
between operations, the use of technology, and the context of the problem to assess the reasonableness of results;
• Read and write decimals using place value to describe decimals in terms of groups from millionths to millions;
• Recognize and generate equivalent decimals, fractions, mixed numbers and improper fractions in various contexts;
• Round numbers to the nearest 0.1, 0.01, and 0.001;
• Locate fractions on a number line, including mixed numbers and improper fractions;
• Represent equivalent fractions using fractions models such as parts of a set, fraction circles, fraction strips, number lines and other manipulatives. Use the models to determine equivalent
• Apply the commutative, associative and distributive properties and order of operations to generate equivalent numerical expressions and to solve problems involving whole numbers.
NCTM Standards:
Work flexibly with fractions, mixed numbers, and decimals to solve problems
● Build on prior knowledge from previous grade levels and everyday life;
● Use of decimals and fractions includes measurements and comparisons;
● Ensure solid understanding of context when deciding among differing representations for equivalency and moving flexibly between them;
Understand and use the associative, commutative, and distributive properties to simplify computations with integers, fractions, and decimals
● Use mathematical properties to simplify many computations involving fractions and decimals;
● Use common procedures to multiply and divide fractions and decimals efficiently and accurately;
Develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use
● Develop own methods of computation and sharing results with class;
● Explain why methods chosen and subsequent solutions are reasonable;
● Compare and evaluate personal method with traditional algorithms;
Understand and use the inverse relationship of multiplication and division to simplify and solve problems
• Be mindful of the decision of when to multiply or divide when working with fractions, mixed numbers, or decimals;
• Increase understanding of division as being repeated subtraction rather than just a rote procedure of invert and multiply;
• Select appropriate methods and tools for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the
situation, and apply the selected methods
• Learn when an exact answer or an estimate is needed;
• Identify which computational method should be chosen;
• Evaluate reasonableness of solution;
• Increase mental computation and estimation.
Common Core State Standards (CCSS):
5BTB (Number And Operations In Base Ten) Perform operations with multi-digit whole numbers and with decimals to hundredths.
5NTB.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between
addition and subtraction; relate the strategy to a written method and explain the reasoning used.
5NF (Number And Operations--Fractions) Use equivalent fractions as a strategy to add and subtract fractions.
5NF.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations
to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.
5NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
5NF.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
5NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1
5NF.7.c Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations
to represent the problem.
6NS (Number System) Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
6NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the
6NS.3 Fluently multiply and divide multi-digit decimals using the standard algorithm for each operation.
6RP (Ratios And Proportional Relationships) Understand ratio concepts and use ratio reasoning to solve problems.
6RP3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
6RP3.c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
7NS (Number System) Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
7NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers.
Student Misconceptions
Student Misconceptions and Common Errors
• Students may believe that multiplication always results in a larger number, while division always results in a smaller number.
• Students struggle to make meaning of problems involving fractions, such as $\frac{1}{2}\times 1\frac{3}{4}$ and $5 \div\frac{1}{4}$, making it difficult to estimate solutions and assess
reasonableness of results.
• When comparing two rectangular area models, students may not recognize equivalencies. For example, the multiplication of $\frac{1}{2}$ and $\frac{3}{4}$ can be shown in different ways.
Although both show the result is $\frac{3}{8}$, students may believe the second model results in a larger number since it has 6 parts shaded compared to 3 in the first model.
• Students are sometimes confused when finding a fraction of a number. It helps to use the words "$\frac{1}{3}$ of 12" rather than "$\frac{1}{3}\times$ 12."
• Students who lack sufficient experience with grid or area models involving multiplication and division of fractions may misapply "the invert and multiply" rule by not inverting the second
fraction, or inverting the first fraction, or inverting both fractions.
• When using the standard algorithm for division, students may ignore 0s in problems involving multi-digit dividends where the 0 is in the middle. For example, students may treat 40.2 ÷ 6 as 42 ÷
• When using the standard algorithm for multiplying decimals, students may determine the number of decimal places in the answer by counting the decimal places to the left of the decimal point
instead of the right. For example, students may believe that 18.6 x 5.9 = 10.974, thinking that 3 decimal places are needed rather than 2.
• When placing the decimal point using the standard algorithm, students may begin counting from the left side of the product instead of the right. For example, students may understand that four
decimal places are needed, but believe that 7.91 x 0.72 = 5695.2.
In the Classroom
In the following vignette, students show the use of fraction bars and a number line to solve a problem requiring division of a mixed number.
Problem: You have $7\frac{2}{3}$ pounds of peanuts. You want to put them into 3 bags, putting the same amount into each bag. How many pounds of peanuts should you put into each bag?
Teacher: Student A, I see you used fractions bars to solve this problem. Can you show your solution?
Student A: Sure. I know that I have to share $7\frac{2}{3}$ pounds equally among 3 groups. I think of one fraction bar as one pound, so I laid out 7 pounds and 2 one-third pounds.
It's easy to see I can share 2 pounds with each of the three groups, but I have $1\frac{2}{3}$ pounds left. I broke the one pound into 3 equal one-third pound pieces.
Then I put 1 one-third pound in each group, and I also have 2 one-third pieces left to split up.
Now I can break each one-third into 3 equal pieces. Since one-third equals 3 one-ninths, then 2 one-third pounds equals 6 one-ninth pounds.
Now I can share 2 one-ninth pounds with each group.
Teacher: How much is in one group?
Student A: I see that I have 2 pounds plus $\frac{1}{3}$ plus two $\frac{1}{9}$ pounds. I use one-ninth fraction bars to help show that since 1 one-third is equal to 3 one-ninths, I have 5 one-ninths
in all. I have 2 pounds and 5 one-ninth pounds in each group.
$7\frac{2}{3}\div 3=2\frac{5}{9}$ and $2\frac{5}{9}$ pounds of peanuts should go in each bag.
Teacher: Very interesting. You used fraction bars to show how to divide $7\frac{2}{3}$ into 3 equal groups. Thank you for sharing. Student B used a number-line model with fraction bars. Will you
show us your solution?
Student B: OK. To find the number in each of 3 groups, we need to divide the distance on the number line from 0 to $7\frac{2}{3}$ into 3 equal sections and find out the size of each section. I
started by drawing fraction bars on the number line to show $7\frac{2}{3}$.
As I looked at the number line, I was wondering what intervals I could use to break up the distance from 0 to $7\frac{2}{3}$ that would give me a total number of parts divisible by 3. First I tried
breaking each one-unit interval into 3 one-third intervals. Here's what I did.
But 23, the number of intervals, cannot be divided evenly by 3, so using interval lengths
of one-third will not work. Next I tried sixths, since $\frac{1}{3}=\frac{2}{6}$.
$\frac{23}{3}=\frac{23}{3} \times \frac{2}{2}=\frac{46}{6}$.
But 46 cannot be divided evenly by 3 either, so sixths will not work. So then I tried intervals of length one-ninth.
$\frac{23}{3}=\frac{23}{3} \times \frac{3}{3}=\frac{69}{9}$.
Bingo! 69 can be divided by 3, so this will work. Then I divided my number line into intervals of length one-ninth. There will be 69 intervals between 0 and $7\frac{2}{3}$.
Now because 69 ÷ 3 = 23, I put marks to show the 3 groups of 23 intervals.
Teacher: So how much do you find for each group?
Student B: $\frac{23}{9}$ is the same as $2\frac{5}{9}$, so I agree with Student A that each group gets $2\frac{5}{9}$ pounds.
Student C: I have another strategy, and I didn't use fraction bars or a number line. May I share it?
Teacher: Of course.
Student C: I knew that $7\frac{2}{3}$ needed to be divided equally into 3 groups, so the problem can be written as $7\frac{2}{3}\div 3$.
$7\frac{2}{3}\div 3=\frac{23}{3}\div\frac{3}{1}=\frac{23}{3}\div(\frac{3}{1}\times\frac{3}{3})=\frac{23}{3}\div\frac{9}{3}=\frac{23\div 9}{3\div 3}=\frac{23\div 9}{1}=23\div 9=2\frac{5}{9}$.
Teacher: You solved the problem without any diagrams, but by using symbols. I noticed that you found a common denominator for your fractions before dividing. The interesting thing about dividing
fractions with common denominators is that the division always results in a denominator of 1. When dividing fractions with common denominators, the answer is actually determined by the division of
the numerators. You can see that in our example. The final step is 23 ÷ 9, a division of the numerators.
Student B: You mean that always happens when you use common denominators?
Teacher: Let's try it with another problem and see what happens. How about $\frac{5}{6}\div\frac{3}{4}$. What's a common denominator?
Student B: 24. I don't know if it's the lowest common denominator, but I know it's a common denominator because I just multiplied the denominators. 6 x 4 = 24.
Teacher: You're right. Common denominators are common multiples, so let's use 24 as our common denominator.
$\frac{5}{6}\div \frac{3}{4}=\frac{5\times 4}{6\times 4}\div\frac{3\times 6}{4\times 6}=\frac{20}{24}\div \frac{18}{24}=\frac{20\div 18}{24\div 24}=\frac{20\div 18}{1}=20\div 18=1\frac{2}{18}=1\frac
Student D: Wow! I like that strategy, but I used the "invert the divisor and multiply" strategy.
Teacher: How would you use that strategy to solve $\frac{5}{6}\div\frac{3}{4}$?
Student D: Like this: $\frac{5}{6}\div \frac{3}{4}=\frac{5}{6}\times \frac{4}{3}=\frac{20}{18}=1\frac{2}{18}=1\frac{1}{9}$
I got the same answer.
Teacher: Why does that strategy work?
Student D: I don't know. It just does.
Teacher: Then let's see if we can understand why. First let's start with easier numbers. We know that 12 ÷ 4 = 3. One way to think of 12 ÷ 4 is that if 12 is 4 groups, then the quotient represents
how many are in one group.
Since there are 3 in one group, then 12 ÷ 4 = 3. Using the same idea, you can think of $\frac{5}{6}\div\frac{3}{4}$ like this: if $\frac{5}{6}$ is $\frac{3}{4}$ of a group, then the quotient is
how many in one group.
Student C: OK. I understand that. Then I predict that the answer to $\frac{5}{6}\div\frac{3}{4}$ is probably a little more than 1.
Teacher: Why do you say that?
Student: Because if $\frac{5}{6}$ represents only $\frac{3}{4}$ of one group, then the size of one group must be larger then $\frac{5}{6}$. Since $\frac{5}{6}$ is pretty close to 1, I think adding
another $\frac{1}{4}$ of a group will give you just a little more than 1.
Teacher: Good reasoning. If $\frac{5}{6}$ is $\frac{3}{4}$ of a group, then the quotient of $\frac{5}{6}\div\frac{3}{4}$ is the how many in one group.
Since $\frac{5}{6}$ represents $\frac{3}{4}$ or 3 parts out of 4 of one group, we can divide $\frac{5}{6}$ by 3 to find the size of one part. Then we can multiply the size of one part by 4 to find
the size of one entire group. How can we divide $\frac{5}{6}$ by 3 to find the size of one part?
Student: Dividing $\frac{5}{6}$ by 3 is the same as finding $\frac{1}{3}$ of $\frac{5}{6}$, or $\frac{1}{3}\times\frac{5}{6}=\frac{5}{18}$. That tells us the size of each of those 3 parts is $\frac
Teacher: I can check that the size of one part is $\frac{5}{18}$ by multiplying it by 3 to see if the result is $\frac{5}{6}$. $\frac{5}{18}\times 3=\frac{5}{18}\times \frac{3}{1}=\frac{5\times 3}{18
\times 1}=\frac{15}{18}=\frac{5}{6}$. Yes, the size of one part is $\frac{5}{18}$.
But remember, there are 4 total parts in one group.
Student A: So we need to multiply by 4 to find the size of the whole group.
Teacher: Exactly. $\frac{5}{18}\times 4=\frac{5}{18}\times \frac{4}{1}=\frac{5\times 4}{18\times 1}=\frac{20}{18}=1\frac{2}{18}=1\frac{1}{9}$.
Student D: But how does that connect to "invert and multiply?'
Teacher: Let's look more closely. What we actually did was $\frac{1}{3}\times \frac{5}{6}\times \frac{4}{1}$. Using the commutative and associative properties, I'm going to rewrite that as $\frac{5}
{6}\times (\frac{1}{3}\times \frac{4}{1})$, which is $\frac{5}{6}\times \frac{4}{3}$.
Student D: There it is! $\frac{4}{3}$ is the reciprocal of $\frac{3}{4}$. Dividing is the same as multiplying by the reciprocal of the divisor!
Teacher: Terrific! You've got it!
Instructional Notes
• From prior experiences with whole numbers, students may have developed the misconception that multiplication always results in a larger number. Using concrete models and pictorial representations
to visualize multiplication as "repeated addition" helps students understand that multiplication is a scalar relationship, where one factor is multiplied by another. Scaling by a factor greater
than 1 (repeating the addition more than 1 time) results in an increased number. Scaling by a factor less than 1 (repeating the addition a fractional number of times) results in a smaller number.
The example 4 x 3 = 12 and 4 x $\frac{1}{2}$ = 2, where 4 is multiplied by scale factors 3 and $\frac{1}{2}$, can be used to illustrate this. Once students understand that multiplication does not
always result in an increased number, the relationship between multiplication and division as inverse operations that "do" and "undo" each other can be used to show that division does not always
result in a decreased number.
• It is essential to build on students' understanding of multiplication and division and connect previous experiences with whole numbers to fractions and decimals. Using simpler problems involving
whole numbers is an effective strategy to help students make meaning of problems involving computation with fractions. The example below shows how the factors 4 and 3 can be used as a foundation
for multiplying the factors 4 and $\frac{2}{3}$.
• Division has two common interpretations: measurement (or quotitive) and sharing (or partitive). Each interpretation has its associated language. Patterns in language used with dividing whole
numbers can help develop understanding of division examples that involve fractions. The chart below describes different division examples, first using the quotitive interpretation and then using
the partitive interpretation. A number line is used to represent each example. The number line can be a very efficient model for representing division. If students have used the number line to
model fraction multiplication, they will have the experience necessary to connect the number line model of division of fractions back to the number line model of multiplication of fractions.
• Since division has different interpretations, students may use different types of models to represent different situations. Also the same model may be used in a different way depending on the
context of the problem. In other words, a student might use fractions bars to model both a quotitive and a partitive division problem but the model will look different. As with multiplication, it
is important to help students connect their prior understanding of division with whole numbers to models for representing division that can lead to formal symbolic procedures for dividing
• It is important to give appropriate context to problems and not teach computational skills in isolation. One possible context for $\frac{1}{2}\times 1\frac{3}{4}$ is the need to determine how
much flour is needed to make $\frac{1}{2}$ batch of cookies that calls for $1\frac{3}{4}$ cup flour. Determining the number of people that will receive a portion of 5 candy bars that have been
broken into $\frac{1}{4}$s is a possible context for 5 ÷ $\frac{1}{4}$.
• When using rectangular area models, the same result may look different. For example, the multiplication of $\frac{1}{2}$ and $\frac{3}{4}$ can be represented in two ways.
The second model shown appears to have a larger area. Remind students that fractions represent $\frac{part}{whole}$ relationships. Since the $\frac{part}{whole}$ relationship shown in the second
model is $\frac{6}{16}=\frac{3}{8}$, both models have the same result.
• Teachers' personal algorithmic knowledge of "invert and multiply" when dividing by fractions can interfere with the construction of a more complete understanding of the concept. It is essential
for students to have multiple experiences with area and number line models connecting division of whole numbers to division of fractions that can lead to formal symbolic procedures for dividing
• Students should be introduced to strategies other "invert and multiply" when dividing by fractions. Some strategies include:
The Number Line Model for $3\frac{1}{2}\div \frac{1}{2}$ can be used to show that there are 7 groups of $\frac{1}{2}$ in $3\frac{1}{2}$ Therefore, $3\frac{1}{2}\div \frac{1}{2}=7$.
The Rectangular Area Model uses pictures to show how the division works. In the following example, an area model is used to show that there are 4 eighths in $\frac{1}{2}$. Therefore, $\frac{1}{2}\div
The Common Denominator Model works by finding a common denominator for both fractions. This results in a whole number division problem involving only the numerators. For example: $\frac{3}{4}\div\
• A focused discussion of multiplication and division of decimals should begin with an understanding of the basic idea of equivalence between fractions and decimals. To expand their understanding
of this equivalence relationship, students must also revisit their understanding of division:
• As students learn to divide in situations where the answer is not a whole number, they often encounter situations such as the following:
It is reasonable to take this computational work and say, "5 goes into 17 three times with a remainder of 2." However, one should not write 17 ÷ 5 = 3 r 2, since no meaning for the equality sign
makes this statement true. Instead, the information given by the computation is 17 = 5 x 3 + 2. We can divide both sides of the equation by 5 and get:
$17\div 5=\frac{17}{5}=\frac{5\times 3+2}{5}=3+\frac{2}{5}=3\frac{2}{5}=3\frac{4}{10}=3.4$;
• When the result of whole-number division is not a whole number, the answer can be expressed in three ways:
• As students learn about different representations of quotients, they realize that when a whole number is divided by a whole number, three results are possible:
Sometimes the remainder is 0 and sometimes it is not.
If the answer is not a whole number, then sometimes the quotient is less than 1 and sometimes it is not (i.e., it is a "mixed" number).
If the answer is not a whole number, sometimes the quotient is a fraction in "lowest terms" and sometimes it is not.
• An important skill that helps students to understand how to multiply and divide decimals is the ability to apply place-value patterns when multiplying or dividing by powers of 10:
When multiplying by greater and greater powers of 10, the digits move to greater and greater place-value positions. When dividing by greater and greater powers of 10, the digits move to lesser and
lesser place-value positions.
• When multiplying decimals, students will begin to see the relationship between the number of decimal places in the factors and the number of decimal places in the product through a variety of
carefully crafted examples:
• When dividing a finite decimal number by another finite decimal number, the following guidelines reduce the work to dividing one whole number into another whole number:
If the divisor (denominator) is not a whole number, obtain an equivalent problem by multiplying both the dividend (numerator) and divisor (denominator) by the same power of 10, chosen so that the
new divisor is a whole number.
$3.55\div 0.25=\frac{3.55}{0.25}=\frac{3.55}{0.25}\times\frac{100}{100}=\frac{355}{25}=355\div 25$.
Divide as you would normally divide if the divisor was a whole number, being careful to use understanding of place value to align the digits in the quotient with the appropriate place value
in the dividend.
Place the decimal point in the quotient directly above the decimal point in the dividend.
• It is important to ask students to estimate products and quotients before performing calculations with fractions and decimals and then assessing those results for reasonableness. This strategy
often helps students recognize errors and provides opportunities to address misconceptions. For example, students that ignore 0s in multi-digit dividends when using the standard algorithm for
division can recognize that their estimate differs by a power of 10. Estimation will also help students who incorrectly position the decimal point recognize their errors.
• Although students at this grade level can use estimation and reasonableness to justify the process for finding decimal products and quotients, teachers should understand that the equivalence
relationship between decimals and fractions is the basis of any efficient, generalizable procedure. The examples below show how the equivalence relationship between fractions and decimals can be
used to verify standard procedures for multiplying and dividing decimals:
• Students who struggle to position the decimal point correctly when multiplying decimals may benefit from writing the decimals as fractions first, and then multiplying.
• Multiplying and dividing decimals and fractions can be difficult for students who are still struggling with basic multiplication facts. Consider allowing these students to use a multiplication
chart to aid them through the problems, while encouraging them to master the facts.
Instructional Resources
Feeding Frenzy
In this activity, students multiply and divide a recipe to feed groups of various sizes. Students use unit rates or proportions and think critically about real world applications of a baking problem.
Multiplication of Fractions
Students are able to multiply and manipulate various fractions to create area models that represent any fractions needed to be compared.
Additional Instructional Resources
Literature connections
Students read and discuss "Beasts of Burden" in The Man Who Counted: A Collection of Mathematical Adventures by Malba Tahan. In this story, three brothers must divide their father's camels.
Multiplication and Division of Decimals
This interactive lesson teaches methods for multiplying and dividing decimals using whole-number divisors.
New Vocabulary
reciprocal: the multiplicative inverse of a number; in other words, a reciprocal is a number that you multiply by so the resulting product equals 1. Example: The reciprocal of $\frac{3}{5}$ is $\
frac{5}{3}$ because $\frac{3}{5}\times\frac{5}{3}=1$.
Professional Learning Communities
Reflection - Critical Questions regarding the teaching and learning of these benchmarks:
• What previous models and understanding of multiplication and division do students bring to my classroom?
• How can I use students' understanding of multiplication and division of whole numbers as a foundation for experiences with fractions and decimals?
• What strategies can be used to model multiplication and division of fractions and decimals?
• What strategies, other than standard algorithms, are students able to demonstrate for multiplying and dividing fractions and decimals?
• What evidence do I have that my students understand the relationship between decimals and fractions and recognize equivalencies?
• What evidence exists to show that students understand standard algorithms for multiplying and dividing fractions and decimals?
Materials - suggested articles and books
Unpacking a Conceptual Lesson: The Case of Dividing Fractions
This article uses pattern blocks and pictorial representations to demonstrate addition, subtraction, multiplication, and division of fractions. It offers a good description of the concepts behind
"inverting and multiplying."
What do Students Need to Learn about Division of Fractions?
This article discusses the various ways fractions are divided and why students need to learn about fractions through dividing them. Several problems are presented.
NCTM A Research Companion to Principles and Standards for School Mathematics (Details about this resource can be found in the References section.)
Chapter 8, Conclusion to facts and algorithms as products of students' own mathematical ability, pp. 120-121.
Kaput, J. (1989). Linking representations in the symbol system of algebra. In Kieran, C. & Wagner, S. (Eds.). A research agenda for the learning and teaching of algebra. Hillsdale, NJ: Lawrence
Kilpatrick, J., Martin, W., & Schifter, D. (Eds.). (2003). A research companion to principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
South Carolina State Department of Education. Math Curriculum Standards, N.p., n.d. Web. 1 Apr. 2011. <http://ed.sc.gov/agency/Standards-and-Learning/Academic-Standards/old/cso/standards/math/>.
Minnesota's K-12 Mathematics Frameworks. (1998). St. Paul, MN: SciMathMN.
National Council of Teachers of Mathematics. (2010). Focus in grade 6 teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
NJ Mathematics Curriculum Framework. The official web site for the state of New Jersey. N.p., n.d. Web. 1 Apr. 2011. <http://www.state.nj.us/education/frameworks/math/>.
(DOK Level 1)
1. Multiply: 0.14 x 1.6
a) 2.240 b) 0.224 c) 0.0224 d) 0.00224
Answer: b
(DOK Level 1)
2. Divide: $3\frac{1}{2}\div \frac{1}{3}$.
Answer: $10\frac{1}{2}$
(DOK Level 2)
3. The highest mountain on the moon is Mount Huygens. It is 5.5 about kilometers in height. Mount Everest, the highest mountain on earth, is about 8.8 kilometers in height. How many times taller
Mount Everest than Mount Huygens?
a. 4.84 times b. 3.3 times c. 1.6 times d. 0.625 times
Answer: c
(DOK Level 2)
4. Andy wants to buy $3\frac{1}{3}$ cups of cashews. There are $\frac{5}{6}$ cup of cashews in each package. How many packages of cashews should Francis buy?
Answer: 4 packages
(DOK Level 3)
5. Describe and correct the error in the solution. Explain your reasoning.
Sample Answer:
The decimal point is in the wrong place. I estimated the product to be 20 (5 x 4). Since each factor has 1 decimal place, the product will have 2 decimal places. The correct answer is 18.62.
(DOK Level 4)
6. Create a model to prove that $2\frac{1}{2}\div\frac{1}{2}=5$.
Sample Answers:
Emergent Learners
This unit uses the set model to support students who struggle with basic fraction concepts and facilitates work with comparing and ordering and working with equivalency.
This unit uses the area model to support students who struggle with basic fraction concepts and facilitates work with comparing and ordering and working with equivalency.
This applet allows students to individually practice working with relationships among fractions and shares ways of combining fractions.
When students work with physical manipulatives, one major challenge is that the manipulation of multiple pieces, representing fractions, can be confusing for the student. This can cause the students
to lose sight of the intended mathematical concept of the lesson. Having too many manipulatives prevents struggling students from connecting the mathematical concepts with their concrete
representations (Kaput, 1989.) Using virtual manipulatives on a computer can assist with student understanding as they can bridge the physical to the abstract.
Multiplication of Fractions
This website allows students to select fractions and show a pictorial representation for the algorithm on the right. Both the horizontal and vertical axes are easily manipulated.
Provide a multiplication chart to assist students who are struggling with basic multiplication facts, while encouraging them to master the facts.
English Language Learners
• Provide a graphic organizer to show the different ways division problems can be expressed.
• The language of division is especially tricky for English Language Learners, since it signals different situations. "How many in each group" signals partitive division, while "how many groups"
signals measurement division. Provide a graphic organizer that shows models for partitive and measurement division and connects division of whole numbers to division with fractions.
• Provide a graphic organizer that show models connecting multiplication of whole numbers to multiplication of fractions.
• Provide a graphic organizer that connects multiplication of decimals and fractions.
• Provide a graphic organizer that connects division of decimals and fractions.
• Use graphic organizers such as the Frayer model shown below, for vocabulary development.
Extending the Learning
In this lesson, students develop a deep conceptual understanding of the relationship between remainders and the decimal part of quotients.
Score by correctly multiplying or dividing fractions. Students need to reduce to simplest form to be correct. There are three levels and a Super Brain level that uses all 4 operations.
Classroom Observation
Administrative/Peer Classroom Observation
│ Students are: (descriptive list) │ Teachers are: (descriptive list) │
│using a variety of models, such as number lines and rectangular area models, to explore and make │connecting students' previous experiences with multiplying and dividing whole numbers to │
│sense of processes for multiplying and dividing fractions. │experiences with multiplying and dividing fractions. │
│representing multiplication and division of fractions with a variety of models. │modeling a variety of strategies to multiply and divide fractions. │
│multiplying and dividing fractions using a variety of strategies and assessing results for │asking students to estimate products and quotients of problems involving fractions and decimals │
│reasonableness. │and justify their results. │
│understanding that the commutative property applies to multiplication of decimals and fractions, │using models to demonstrate that multiplication is commutative for positive rational numbers, but │
│but not to division. │division is not. │
│understanding that division can have different meanings: "how many in each group," (partitive) and│using real-life applications to model both partitive and measurement division. │
│"how many groups" (quotitive). │ │
│exploring place-value patterns that result when decimals are multiplied and divided by powers of │making place-value patterns that result when decimals are multiplied and divided by powers of 10, │
│10. │explicit to students. │
│recognizing the relationship between the number of decimal places in the factors and the number of│carefully crafting examples that support students to learn the standard algorithm for multiplying │
│decimal places in the product when multiplying decimals. │decimals. │
│writing decimal quotients and their fractional representations side-by-side so that they begin to │using the equivalence between fractions and decimals to justify the standard algorithm for │
│see patterns. │dividing decimals. │
│having multiple opportunities to build meaning for generalizable procedures for multiplying and │allowing ample time for students to make sense of computational processes so that they can │
│dividing with fractions and decimals. │eventually apply them effectively in problem-solving situations. │
│communicating their reasoning in writing, drawings, and conversation. │asking students to explain their reasoning in a variety of formats. │
Parent Resources
• Models for the Multiplication of Fractions
This website uses visual area models to multiply fractions.
• ModelsfortheDivisionofFractions
This website uses visual area models to divide fractions.
• MultiplyingDecimals
This video demonstrates the standard algorithm for multiplying decimals.
• DividingDecimals
This video demonstrates the standard algorithm for dividing decimals. | {"url":"https://stemtc.scimathmn.org/frameworks/613a-multiplication-division","timestamp":"2024-11-03T14:10:23Z","content_type":"text/html","content_length":"97999","record_id":"<urn:uuid:8d7cb5dc-bd0b-49a4-97f1-70ffb1ea2ee5>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00521.warc.gz"} |
Nominal Logistic
You can enter values for estimated coefficients for several scenarios. For example, you may want to provide starting estimates so that the algorithm converges to a solution, or you may want to
validate a model with an independent sample. For more information, go to
Entering initial values for estimated coefficients
Starting estimates for algorithm
Enter the column containing the initial values for model parameters. Specify initial values for model parameters or parameter estimates for a validation model.
Estimates for validation model
Enter the column containing the estimated model parameters. Minitab will then fit the validation model.
In Maximum number of iterations, enter the maximum number of iterations that Minitab performs to reach convergence. The default value is 20. Minitab's logistic regression commands obtain maximum
likelihood estimates through an iterative process. If Minitab reaches the maximum number of iterations before convergence, the command terminates. | {"url":"https://support.minitab.com/en-us/minitab/help-and-how-to/statistical-modeling/regression/how-to/nominal-logistic-regression/perform-the-analysis/select-the-analysis-options/","timestamp":"2024-11-07T22:11:36Z","content_type":"text/html","content_length":"15095","record_id":"<urn:uuid:f9162ca1-5ef9-4f16-a847-b4c12c8cc3dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00777.warc.gz"} |
24.7 grams to ounces
Convert 24.7 Grams to Ounces (gm to oz) with our conversion calculator. 24.7 grams to ounces equals 0.871266812 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 24.7 grams to ounces is a common task that many people encounter, especially when dealing with recipes or scientific measurements. Understanding how to perform this conversion is essential
for bridging the gap between the metric and imperial systems. In this guide, we will explore the conversion factor, provide a formula, and walk you through a step-by-step calculation to make this
process easy and straightforward.
The conversion factor between grams and ounces is crucial for accurate measurements. 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 switch between the two measurement systems with confidence.
To convert grams to ounces, you can use the following formula:
Ounces = Grams ÷ 28.3495
Now, let’s apply this formula to convert 24.7 grams to ounces. Here’s a step-by-step calculation:
1. Start with the amount in grams: 24.7 grams.
2. Use the conversion factor: 28.3495 grams per ounce.
3. Divide the grams by the conversion factor: 24.7 ÷ 28.3495.
4. Perform the calculation: 24.7 ÷ 28.3495 ≈ 0.872.
5. Round the result to two decimal places: 0.87 ounces.
Thus, 24.7 grams is approximately 0.87 ounces. This rounded figure is practical for everyday use, making it easier to understand and apply in various situations.
The importance of converting grams to ounces cannot be overstated. This conversion is particularly useful in cooking, where many recipes may list ingredients in ounces rather than grams. For
instance, if you’re following a recipe that calls for 0.87 ounces of an ingredient, knowing that this is equivalent to 24.7 grams can help you measure accurately and achieve the desired results.
Additionally, in scientific measurements, precise conversions are vital for experiments and data analysis. Whether you’re a student, a professional chef, or simply someone who enjoys cooking at home,
being able to convert between these two systems enhances your ability to work with various recipes and scientific data effectively.
In summary, converting 24.7 grams to ounces is a simple yet essential skill that can make a significant difference in your cooking and scientific endeavors. By understanding the conversion factor and
following the outlined steps, you can easily navigate between metric and imperial measurements with confidence.
Here are 10 items that weigh close to 24.7 grams to ounces –
• Standard Paperclip
Shape: Elongated oval
Dimensions: 3.0 cm x 0.5 cm x 0.1 cm
Usage: Commonly used to hold sheets of paper together.
Fact: A standard paperclip can hold up to 20 sheets of paper at once!
• AA Battery
Shape: Cylindrical
Dimensions: 5.0 cm x 1.4 cm
Usage: Used in various electronic devices like remote controls and toys.
Fact: An AA battery can power a small flashlight for up to 10 hours!
• Small Rubber Duck
Shape: Duck-shaped
Dimensions: 7.5 cm x 6.0 cm x 5.0 cm
Usage: Often used as a bath toy for children.
Fact: Rubber ducks have been a popular bath toy since the 1940s!
• USB Flash Drive
Shape: Rectangular
Dimensions: 5.0 cm x 2.0 cm x 0.7 cm
Usage: Used for data storage and transfer between devices.
Fact: The first USB flash drive was released in 1998 and had a capacity of 8 MB!
• Golf Ball
Shape: Spherical
Dimensions: 4.3 cm in diameter
Usage: Used in the sport of golf.
Fact: A golf ball has around 336 dimples on its surface to improve aerodynamics!
• Small Keychain
Shape: Various shapes (often circular or rectangular)
Dimensions: 5.0 cm x 3.0 cm x 0.5 cm
Usage: Used to hold keys together and often features decorative elements.
Fact: Keychains can be used as a promotional item, often featuring company logos!
• Tea Bag
Shape: Rectangular or round
Dimensions: 6.0 cm x 4.0 cm
Usage: Used for brewing tea.
Fact: The first tea bags were made of silk and were introduced in the early 1900s!
• Small Candle
Shape: Cylindrical
Dimensions: 7.0 cm x 5.0 cm
Usage: Used for lighting and creating ambiance.
Fact: The world’s largest candle was over 30 feet tall and weighed over 1,000 pounds!
• Plastic Spoon
Shape: Curved with a long handle
Dimensions: 15.0 cm x 4.0 cm
Usage: Used for eating or serving food.
Fact: Plastic spoons were first introduced in the 1930s and are now a staple in fast food!
• Small Notebook
Shape: Rectangular
Dimensions: 10.0 cm x 15.0 cm
Usage: Used for writing notes, sketches, or journaling.
Fact: The first notebooks were made from papyrus in ancient Egypt!
Other Oz <-> Gm Conversions – | {"url":"https://www.gptpromptshub.com/grams-ounce-converter/24-7-grams-to-ounces","timestamp":"2024-11-13T01:07:34Z","content_type":"text/html","content_length":"186412","record_id":"<urn:uuid:c781caaa-89d7-4d1d-ae9a-f9507aaa4365>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00566.warc.gz"} |
The risk we weren’t talking about
A lot of attention has been given to the flooding that would result if rising sea levels lead to the over-topping of the dams at the mouths of the Charles and Mystic rivers. Until recently, no one
was talking about what high water could do to the dams themselves.
The New Charles River Dam and the Amelia Earhart Dam are both about 50 years old. They serve similar functions for the Charles and the Mystic respectively. They regulate the water level in the lower
basins of both rivers to a level that roughly equates to mid-tide in the harbor. Under normal conditions, gates close at high tide to keep the ocean from raising the water level in the basins. At low
tide, the river flow in the basins is allowed to drain out.
The dams are both equipped with a set of massive pumps to throw the river flow over the walls into the ocean in case rain-driven flooding coincides with high tide. These pumps have been adequate to
handle all the storms we have had over the past 50 years. I have seen the pumps in the Amelia Earhart drain the Mystic basin down to exposed mudflats in the middle of a heavy rainfall event.
The dams were both built to survive the highest harbor water levels that had ever been observed to date (the flood of 1851) plus another foot and a half. Their decks sit at about 8 feet above normal
high tide level (“118 MDC Datum”), roughly 2 feet higher than the highest more recently measured storm surge levels (the Blizzard of 78 and the January 2018 storm).
That level seemed awfully safe when the dams were built, but with rising seas it is now recognized that by the latter part of this century there will be a material risk in any given year that the
ocean will surge over and around these dams.
The flows associated with ocean storm surge are much too big for the pumps to handle, so a flanking/overtopping event would mean severe flooding in the Charles basin and in the Mystic basin, all the
way out to Alewife. Discussions as to how to reduce the risk of flanking and over-topping are underway in both Boston and Cambridge. The risk analysis depend on complex modeling of how long the
over-topping event lasts and so how much water actually flows upstream.
A few weeks ago, DCR educated the legislature about a much more proximate risk. Even a brief overtopping event could severely damage, even cripple the pumping systems of the dams. They are not built
to handle overtopping. Salt-water could quickly flood into critical electrical and mechanical areas that are just not designed to get wet. The damage could reach $150 million to repair and the entire
region would be vulnerable until the repairs were complete.
Fortunately, state engineers who lie awake worrying about these dams surfaced the risks up the chain of command and set in motion a pair of projects to harden the dams to handle over-topping events.
The fix design is complete for the Charles and work will likely begin in this construction season on the Charles. The Mystic dam is a few months behind — as part of the project, they have to move the
operating staff into portions of the facility that have hazardous building materials in them that need to be removed.
Once these fixes are in place, the conversation will turn in earnest to what it will take to raise these structures and the areas around them so that the point when sea level rise becomes an urgent
threat to riverside neighborhoods can be pushed much further out into the future. Fortunately, it appears that even without increased elevation, the overtopping flood risk remains low for the next
few decades.
22 replies on “The risk we weren’t talking about”
1. Will, I applaud what you have done for prisoners and their rights. Freedom is the most important value. One of my favorite books is How I Found Freedom in an Unfree World by Harry Browne.
Prisoners of course have little freedom, and putting them in solitary is horrendous. Humans are social animals and require the company of others to thrive. I hope you will continue your efforts
on their behalf as president pro tem of the Senate.
2. Will
Thanks for keeping on top of things.
3. The dam information is fascinating. Thanks for your work in keeping us informed.
1. Thanks for your work in many areas. I hadn’t even known or
thought about storm damage to dams. I appreciate your
concerns and your communications.
4. What about the Watertown dam on the Charles? Many of us have hoped that dam could be removed, as fish have a hard time climbing the ladder (it was built on the wrong side of the dam) and it could
also prove dangerous if/when the river flooded.
5. Good morning
Mr. Brownsberger you made very important points.
Floods, ,the two dams and the rising sea.
I was involved on the design of these dams
The origin of all is the global warming.
Giovanni Aurilio
1. Thanks for the good work! Those dams have done a lot of this region over the last 50 years!
6. Sad to say most of the dams in Massachusetts are in poor condition. Where is the dam on the Mystic the lowest one I know of is at the Mystic Lake. Like someone else said the Watertown dam is old
but also is the Waltham and I would be in favor of some hydro power extraction from each site. Funds from power sales I would like to see go into a trust for maintenance and replacement of these
dams in the future
1. The Mystic dam (Amelia Earhart Dam) is near the mouth of the Mystic, between Somerville and Everett. At least, Google Maps shows a large, unlabeled dam with locks at that location. It runs
from quite near the Assembly T station to the mall in South West Everett with a Costco, Target and Home Depot.
I think there is another dam on the Mystic near Medford Square, but I can’t find it on Google Maps. It might be hidden under one of the bridges.
1. P.S. Thank you Will and the (MWRA? MDC? DCR?) engineers for being proactive on this. As a Y2K survivor (I spent about a year and a half preventing Y2K problems), I hope people remember
this the first time a flood level exceeds the current top of the dams and nothing happens because this was taken care of.
7. When searching for a suitable home for our sizeable family of 4 children 41 years ago, we discovered Riverside Street; it is situated on a ridgeline that angles upward toward Perkins, and drops
off on both sides, one toward North Beacon Street the other side toward the Chargles River, satisfying my concerns of adaquate run off during severe rain storms, leaving the basement secure and
dry. A few years ago I inquired of my homeowners insurance company the thought of obtaing flood insurance, motivated by flooding that has been occurring throughout the country; I opted out,
thinking historically, this area has been protected by the dam and pump at the Boston Harbor basin. I will admit that from time to time I have thought about a inland surge, and the height and
distance Riverside Street is from the banks of the Charles, yet, have felt relatively safe. Hmmm, but now, not so much? I want to convey our sincere thanks and appreciation of your concern and
pro active thinking, along with all of the involved officials, engineers and personnel that are focusing on this potential danger. Thank you one thousand times!!! Will Clifford
1. Not sure of your exact elevation but if you are ten feet up, you are very safe for a long time.
8. Yes and Cities near the Harbor, Boston, Quincy for instance, may need help financially to put in place plans dealing with storm surge that has already seen floods in Boston along the waterfront.
9. Will, congratulations to the DCR engineers and to you and rest of the legislature for identifying this problem and tackling it. This is how things are supposed to work! I’m sure that it will be
much cheaper to address this problem now, before a catastrophic event, than to have to clean up afterwards.
1. Credit to the DCR engineers!
10. I’ve been wondering about this for years — thank you for the update.
11. Thank you for thinking about this problem, which has been my major concern. I am greatly relieved to hear that the pump system on the Charles River Dam is to be hardened, but in a few years the
dam will be outflanked by major storms.
12. Thank you, Will. Very interesting piece and I’m glad it was taken care of in advance by the State. I appreciate your updates on a huge variety of topics.
13. Thanks – I did not realize the magnitude of the issue
14. Missing is storm runoff part of equation, that is water coming from rain and snow. Seas warm creating more evaporation clouds get hydrated, when over land they release it. Also Deer Island
sanitary and storm capacity may be overburdened by population increase in Boston metro area soon if not already. Advise reference website of American Society of Civil Engineers and their
quadrenial audit of USA infrastructure with individual state assessment. I read recent report Mass.needs 4 to7 billion dollars in water infrastructure new investment over ten year period. ASCE
thier membership is over 100000 including foreign members.
15. Thanks for the info, Will!! I would like to add the Moody Street Dam in Waltham, MA to your list; even though it is not in your district (could you forward this info to whomever checks that
During a quick spring melt and a heavy rain, water went around the Moody St. Dam (MSD) in Waltham. Someone place several sand bags to prevent the overflow from expanding; however, the amount of
water coming over the dam was sufficient enough that a hook and ladder truck and several people with chain saws and ropes were on right right. These people had to move one large log. Sadly, these
people did not seem to care what happened to the flotsam as it headed towards an old railroad bright a few hundred yards down the Charles.
16. Thanks, again, Will, for your careful analysis and sharing this information. There are so many things that average citizens know nothing about and have to count on “government” to take care of.
Every time someone bad mouths the “government,” I tell them about the wonderfully committed people in government that I know! | {"url":"https://willbrownsberger.com/the-risk-we-werent-talking-about/","timestamp":"2024-11-06T10:29:48Z","content_type":"text/html","content_length":"64376","record_id":"<urn:uuid:2e7472dd-afa4-40c8-b4e8-7ef5f112a931>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00002.warc.gz"} |
SRM 533: Failure
It was about time I had a bad match.
The chat before coding phase was more interesting than usual because Google engineers were in their lobby room and received questions. Google sent many widely known names in algorithm contests to
represent them. What is clear is that google really pay attention to these silly algorithm contests at the time of employing new people. Hopefully TC will release a transcript of all the questions.
Div1 500 - The one with common rows and common columns
This problem beats me. I spent all the match trying to get a viable idea for it that I can prove. It seemed that it would be easy to submit a wrong idea. I kept getting lost in dead ends such as
trying to find "something with flow" to solve it. A lot of people submitted a solution to it, which increased the amount of tension during the match for me.
Div1 250 - The one with energy
You have a vector of at least 3 elements. You gain score by removing an element that has an element to its left and an element to its right and then the product of these two other elements is added
to your score. Return the maximum possible score.
First of all, since the numbers are always positive, the final vector will
be one in which only the original extremes are left. In other words, for 1,2,3,4 , the last vector will always be {1,4}.
Given a vector, {1,2,3,4,5,6} we know that the last operation will involve multiplying 1 and 6. The trick is to consider this last move. So, let's pick the last element we will remove: If we decide
to remove 4 last, this means that we removed all elements that are not the extremes and are not 4
before this last move
: {1, .. 4, ... 6 }. Now notice something else, In the previous steps, 1 and 4 and 4 and 6 won't get removed. Imagine the vector split in two parts: {1, 2, 3, 4} and {4,5,6} . We have to pick the
best strategy to remove 2,3 or 5. We can treat these two cases as sub-problems of the original problem, because all of the elements are contiguous.
Thus, whenever you have an array with more than 2 elements. Iterate through all the possible elements we can remove as the last step. This will create two subproblems identical to the first. A
recurrence comes from this observation, and since the elements will always be contiguous for each subproblem, you can just use dynamic programming.
struct CasketOfStar
vector<int> weight;
int mem[50][50];
int rec(int a, int b)
int & res = mem[a][b];
if (res == -1) {
res = 0;
// Pick c - the element we will remove last:
for (int c=a+1; c<b; c++) {
// if we remove c last, we can find the score of the sub-vector a..c
// and c..b to know the best strategy for the previous elements.
res = std::max(res, weight[a]*weight[b] + rec(a,c) + rec(c,b) );
return res;
int maxEnergy(vector <int> weight)
this->weight = weight;
return rec(0, weight.size() - 1);
I had issues during the match. Although I thought of the general solution idea quickly. I didn't have it all figured out until after I began coding. So I had to do many corrections and think things
through again. Was nervous because I switched to this problem late and I already knew a lot of people had solved both the medium and this problem.
Challenge phase
A lot of solutions failed during challenge phase. Seems 500 was easy to get wrong. I am not even sure I will pass 250 because there might be a mistake somewhere.
What do you think?
Did you like the match? I think the problems were interesting. I wish I was more creative and able to think the solution for 500.
Update: outcome
I like the outcome. I dropped around 30 points in rating, which is not a big deal. I still have more than 2100 points. A rating drop was bound to happen, and it is always nice when it happens
3 comments :
500 : Such a sequence exists if you can find an Eulerian path in a certain graph. (And that path doesn't start at certain nodes).
I wrote O(N^5) solution for 250 and it passed, before it I tried half an hour to write O(N^3) solution, but after I gave up quickly wrote another
That's ... cool. Makes sense too. It seems I once again forgot something I learned the hard way in the past. When stuck, maybe you should look your vertices as edges and your edges as vertices | {"url":"https://www.vexorian.com/2012/02/srm-533-failure.html","timestamp":"2024-11-09T12:25:10Z","content_type":"text/html","content_length":"112902","record_id":"<urn:uuid:6965ab1f-c975-4a46-ae91-ed8f95f9f94f>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00030.warc.gz"} |
Robust Scatter Plot Smoothing
runmed {stats} R Documentation
Running Medians – Robust Scatter Plot Smoothing
Compute running medians of odd span. This is the ‘most robust’ scatter plot smoothing possible. For efficiency (and historical reason), you can use one of two different algorithms giving identical
runmed(x, k, endrule = c("median", "keep", "constant"),
algorithm = NULL,
na.action = c("+Big_alternate", "-Big_alternate", "na.omit", "fail"),
print.level = 0)
x numeric vector, the ‘dependent’ variable to be smoothed.
k integer width of median window; must be odd. Turlach had a default of k <- 1 + 2 * min((n-1)%/% 2, ceiling(0.1*n)). Use k = 3 for ‘minimal’ robust smoothing eliminating isolated outliers.
character string indicating how the values at the beginning and the end (of the data) should be treated. Can be abbreviated. Possible values are:
keeps the first and last k_2 values at both ends, where k_2 is the half-bandwidth k2 = k %/% 2, i.e., y[j] = x[j] for j \in \{1,\ldots,k_2; n-k_2+1,\ldots,n\};
copies median(y[1:k2]) to the first values and analogously for the last ones making the smoothed ends constant;
the default, smooths the ends by using symmetrical medians of subsequently smaller bandwidth, but for the very first and last value where Tukey's robust end-point rule is applied, see
character string (partially matching "Turlach" or "Stuetzle") or the default NULL, specifying which algorithm should be applied. The default choice depends on n = length(x) and k where
algorithm "Turlach" will be used for larger problems.
character string determining the behavior in the case of NA or NaN in x, (partially matching) one of
Here, all the NAs in x are first replaced by alternating \pm B where B is a “Big” number (with 2B < M*, where M*=.Machine $ double.xmax). The replacement values are “from left” (+B,
-B, +B, \ldots), i.e. start with "+".
na.action almost the same as "+Big_alternate", just starting with -B ("-Big...").
the result is the same as runmed(x[!is.na(x)], k, ..).
the presence of NAs in x will raise an error.
print.level integer, indicating verboseness of algorithm; should rarely be changed by average users.
Apart from the end values, the result y = runmed(x, k) simply has y[j] = median(x[(j-k2):(j+k2)]) (k = 2*k2+1), computed very efficiently.
The two algorithms are internally entirely different:
is the Härdle–Steiger algorithm (see Ref.) as implemented by Berwin Turlach. A tree algorithm is used, ensuring performance O(n \log k) where n = length(x) which is asymptotically optimal.
is the (older) Stuetzle–Friedman implementation which makes use of median updating when one observation enters and one leaves the smoothing window. While this performs as O(n \times k) which is
slower asymptotically, it is considerably faster for small k or n.
Note that, both algorithms (and the smoothEnds() utility) now “work” also when x contains non-finite entries (\pmInf, NaN, and NA):
currently simply works by applying the underlying math library (‘libm’) arithmetic for the non-finite numbers; this may optionally change in the future.
Currently long vectors are only supported for algorithm = "Stuetzle".
vector of smoothed values of the same length as x with an attribute k containing (the ‘oddified’) k.
Martin Maechler maechler@stat.math.ethz.ch, based on Fortran code from Werner Stuetzle and S-PLUS and C code from Berwin Turlach.
Härdle, W. and Steiger, W. (1995) Algorithm AS 296: Optimal median smoothing, Applied Statistics 44, 258–264. doi:10.2307/2986349.
Jerome H. Friedman and Werner Stuetzle (1982) Smoothing of Scatterplots; Report, Dep. Statistics, Stanford U., Project Orion 003.
See Also
smoothEnds which implements Tukey's end point rule and is called by default from runmed(*, endrule = "median"). smooth uses running medians of 3 for its compound smoothers.
myNHT <- as.vector(nhtemp)
myNHT[20] <- 2 * nhtemp[20]
plot(myNHT, type = "b", ylim = c(48, 60), main = "Running Medians Example")
lines(runmed(myNHT, 7), col = "red")
## special: multiple y values for one x
plot(cars, main = "'cars' data and runmed(dist, 3)")
lines(cars, col = "light gray", type = "c")
with(cars, lines(speed, runmed(dist, k = 3), col = 2))
## nice quadratic with a few outliers
y <- ys <- (-20:20)^2
y [c(1,10,21,41)] <- c(150, 30, 400, 450)
all(y == runmed(y, 1)) # 1-neighbourhood <==> interpolation
plot(y) ## lines(y, lwd = .1, col = "light gray")
lines(lowess(seq(y), y, f = 0.3), col = "brown")
lines(runmed(y, 7), lwd = 2, col = "blue")
lines(runmed(y, 11), lwd = 2, col = "red")
## Lowess is not robust
y <- ys ; y[21] <- 6666 ; x <- seq(y)
col <- c("black", "brown","blue")
plot(y, col = col[1])
lines(lowess(x, y, f = 0.3), col = col[2])
lines(runmed(y, 7), lwd = 2, col = col[3])
legend(length(y),max(y), c("data", "lowess(y, f = 0.3)", "runmed(y, 7)"),
xjust = 1, col = col, lty = c(0, 1, 1), pch = c(1,NA,NA))
## An example with initial NA's - used to fail badly (notably for "Turlach"):
x15 <- c(rep(NA, 4), c(9, 9, 4, 22, 6, 1, 7, 5, 2, 8, 3))
rS15 <- cbind(Sk.3 = runmed(x15, k = 3, algorithm="S"),
Sk.7 = runmed(x15, k = 7, algorithm="S"),
Sk.11= runmed(x15, k =11, algorithm="S"))
rT15 <- cbind(Tk.3 = runmed(x15, k = 3, algorithm="T", print.level=1),
Tk.7 = runmed(x15, k = 7, algorithm="T", print.level=1),
Tk.9 = runmed(x15, k = 9, algorithm="T", print.level=1),
Tk.11= runmed(x15, k =11, algorithm="T", print.level=1))
cbind(x15, rS15, rT15) # result for k=11 maybe a bit surprising ..
Tv <- rT15[-(1:3),]
stopifnot(3 <= Tv, Tv <= 9, 5 <= Tv[1:10,])
matplot(y = cbind(x15, rT15), type = "b", ylim = c(1,9), pch=1:5, xlab = NA,
main = "runmed(x15, k, algo = \"Turlach\")")
mtext(paste("x15 <-", deparse(x15)))
points(x15, cex=2)
legend("bottomleft", legend=c("data", paste("k = ", c(3,7,9,11))),
bty="n", col=1:5, lty=1:5, pch=1:5)
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Scope of Statistics: In Business, Economics, Banking, Model
Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of data. The scope of statistics is broad, and it is used in various
fields such as business, economics, psychology, social sciences, medicine, engineering, and more.
The main purpose of statistics is to make sense of data and draw meaningful conclusions from it. This involves designing experiments or surveys to collect data, analyzing the data using mathematical
and statistical tools, and then interpreting the results in a way that can be easily understood.
Nature and Scope of Statistics
Statistics plays a critical role in many fields and is essential for making informed decisions based on data. The nature and scope of statistics include:
• Descriptive statistics
• Inferential statistics
• Probability theory
• Statistical modeling
• Statistical software
• Statistics In Business
• Statistics In Economics
• Statistics In Banking
• Statistics In Accounting
• Statistics In Administration
• Statistics In Astronomy
• Statistics In Research Work
Now let’s discuss the scope of statistics in more detail.
Descriptive Statistics
Descriptive statistics involves describing and summarizing data in a meaningful way. This includes measures of central tendency (such as mean, median, and mode) and measures of variability (such as
standard deviation and variance).
• Descriptive statistics are useful for providing a quick overview of a dataset, identifying outliers, and understanding the distribution of the data.
Inferential Statistics
Inferential statistics involves making predictions or drawing conclusions about a population based on a sample. This is done by using statistical methods to analyze the sample data and make
inferences about the population parameters (such as mean or proportion).
• Inferential statistics are useful for making predictions, testing hypotheses, and generalizing findings to a larger population.
Probability Theory
Probability theory is the study of the likelihood of events occurring. It involves understanding the mathematical laws that govern random events and calculating probabilities based on these laws.
• Probability theory is used in statistical analysis to calculate the probability of certain outcomes occurring, such as the probability of a coin flip landing on heads.
Statistical Modeling
Statistical modeling involves developing mathematical models to describe and analyze data.
• These models can be used to make predictions or test hypotheses and can be applied to a wide range of fields, from economics to biology.
• Examples of statistical models include linear regression models, logistic regression models, and time series models.
Statistical Software
Statistical software tools such as SPSS, R, SAS, and Python are used to analyze and visualize data. These tools provide a range of statistical methods and techniques for data analysis, including
descriptive statistics, inferential statistics, probability theory, and statistical modeling.
• They allow users to input data, perform calculations, generate charts and graphs, and produce reports based on the analysis.
• Statistical software is essential for handling large datasets and automating the statistical analysis process.
Statistics in Business
Statistics is used in Business to analyze and interpret data related to sales, profits, customer behavior, and other factors that affect business performance.
• Businesses use statistics to make decisions on product pricing, market analysis, and investment decisions.
• Statistical methods such as regression analysis, ANOVA, and hypothesis testing are used in business to identify patterns and relationships in data.
Statistics in Economics
Statistics is used in Economics to analyze data related to employment, inflation, gross domestic product (GDP), trade, and other economic factors.
• Economists use statistical methods such as regression analysis and time series analysis to identify trends and make predictions.
• Economic models are also developed using statistical techniques to analyze the impact of economic policies.
Statistics in Banking
Statistics is used in Banking to analyze data related to loan approvals, credit risk, investment returns, and other financial factors.
• Banks use statistical models such as credit scoring models to assess the creditworthiness of loan applicants and predict the likelihood of loan default.
• Statistical methods such as time series analysis are also used to analyze financial market data and make investment decisions.
Statistics in Accounting
Statistics is used in Accounting to analyze financial data such as balance sheets, income statements, and cash flow statements.
• Statistical methods such as regression analysis and ANOVA are used to analyze the relationship between financial variables and make predictions about future performance.
• Statistical analysis is also used to detect financial fraud and identify anomalies in financial data.
Scope of Statistics in Administration
Statistics is used in administration to analyze data related to employee performance, customer satisfaction, and other factors that affect organizational performance.
• Statistical methods such as regression analysis and hypothesis testing are used to identify patterns and relationships in data.
• Statistical models are also developed to predict future performance and identify areas for improvement.
Scope of Statistics in Astronomy
Statistics is used in Astronomy to analyze data related to celestial objects such as stars, galaxies, and planets.
• Astronomers use statistical methods such as regression analysis and hypothesis testing to identify patterns and relationships in data.
• Statistical models are also developed to predict the behavior of celestial objects and to test hypotheses about the origin and evolution of the universe.
Scope of Statistics in Research Work
Statistics is used in Research Field to design experiments, collect and analyze data, and draw conclusions based on the results.
• Statistical methods such as hypothesis testing, ANOVA, and regression analysis are used to analyze data and test hypotheses.
• Statistical software tools such as R, SPSS, and SAS are used to automate the data analysis process and produce reports based on the results.
In conclusion, statistics is a fundamental tool for analyzing and interpreting data in various fields, including business, economics, banking, accounting, administration, astronomy, and research
Its different scopes, including descriptive statistics, inferential statistics, probability theory, statistical modeling, and statistical software, offer distinct ways of examining data and making
informed decisions.
As a constantly evolving discipline, statistics provide a robust framework for understanding complex data and making accurate predictions and inferences, making it an indispensable tool in modern-day
decision-making and problem-solving.
Related Posts | {"url":"https://edukedar.com/scope-of-statistics/","timestamp":"2024-11-02T02:57:25Z","content_type":"text/html","content_length":"89526","record_id":"<urn:uuid:51295b48-886b-4992-a039-96faaec4cfbf>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00462.warc.gz"} |
Multiplication Integers Worksheets
Mathematics, specifically multiplication, creates the foundation of various academic techniques and real-world applications. Yet, for numerous students, understanding multiplication can posture a
challenge. To resolve this obstacle, instructors and parents have embraced a powerful tool: Multiplication Integers Worksheets.
Intro to Multiplication Integers Worksheets
Multiplication Integers Worksheets
Multiplication Integers Worksheets -
Our Integers Worksheets are free to download easy to use and very flexible These Integers Worksheets are a great resource for children in Kindergarten 1st Grade 2nd Grade 3rd Grade 4th Grade and 5th
Grade Click here for a Detailed Description of all the Integers Worksheets Quick Link for All Integers Worksheets
Multiplication of integers Multiplying with negative numbers Practice integer worksheets on multiplying with negative numbers Horizontal and vertical multiplication Horizontal Worksheet 1 Worksheet 2
Worksheet 3 Vertical Worksheet 4 Worksheet 5 Worksheet 6 3 More Similar Division of integers Absolute and opposite values of integers
Significance of Multiplication Practice Comprehending multiplication is essential, laying a solid structure for advanced mathematical principles. Multiplication Integers Worksheets offer structured
and targeted practice, promoting a much deeper understanding of this essential math procedure.
Development of Multiplication Integers Worksheets
Multiplying And Dividing Integers Worksheets
Multiplying And Dividing Integers Worksheets
Integer worksheets contain a huge collection of practice pages based on the concepts of addition subtraction multiplication and division Exclusive pages to compare and order integers and representing
integers on a number line are given here with a variety of activities and exercises
Multiplying Integers Worksheets Math worksheets encourage the students to bring out their A game in Math Test your math skills and see how far you can get Download the Cuemath printable Math
worksheets and help kids develop their math skills
From standard pen-and-paper exercises to digitized interactive formats, Multiplication Integers Worksheets have actually developed, dealing with diverse learning designs and preferences.
Types of Multiplication Integers Worksheets
Fundamental Multiplication Sheets Simple exercises concentrating on multiplication tables, assisting learners develop a solid math base.
Word Problem Worksheets
Real-life scenarios integrated right into issues, boosting vital reasoning and application abilities.
Timed Multiplication Drills Examinations created to enhance rate and accuracy, helping in fast mental math.
Advantages of Using Multiplication Integers Worksheets
15 Best Images Of Multiplying And Dividing Exponents Worksheets Multiplying And Dividing
15 Best Images Of Multiplying And Dividing Exponents Worksheets Multiplying And Dividing
Multiplying Integers Below is a quick summary for the rules of multiplying integers The rules that govern how to multiply and divide integers are very similar In this lesson we will focus on the
multiplication of integers Rules on How to Multiply Integers Step 1 Multiply their absolute values Step 2 Determine the sign of
Multiplication Integers Worksheet Multiplying Integers Worksheet 7Th Grade Integer Multiplication Worksheet Free multiply integers worksheet Integers will remain a significant part of the math
curriculum throughout a student s academic career To help your kids learn and master the concept you can get free versions of worksheets online
Boosted Mathematical Abilities
Consistent practice develops multiplication efficiency, boosting overall mathematics capabilities.
Enhanced Problem-Solving Abilities
Word troubles in worksheets establish logical reasoning and strategy application.
Self-Paced Discovering Advantages
Worksheets fit individual discovering rates, promoting a comfy and adaptable discovering setting.
Exactly How to Develop Engaging Multiplication Integers Worksheets
Including Visuals and Colors Vibrant visuals and colors capture focus, making worksheets visually appealing and engaging.
Consisting Of Real-Life Scenarios
Relating multiplication to daily circumstances includes significance and functionality to exercises.
Customizing Worksheets to Various Ability Degrees Personalizing worksheets based on differing effectiveness degrees ensures comprehensive learning. Interactive and Online Multiplication Resources
Digital Multiplication Tools and Games Technology-based sources provide interactive discovering experiences, making multiplication appealing and pleasurable. Interactive Websites and Applications On
the internet systems give varied and easily accessible multiplication method, supplementing standard worksheets. Personalizing Worksheets for Different Understanding Styles Visual Learners Visual
aids and layouts aid understanding for learners inclined toward visual discovering. Auditory Learners Spoken multiplication troubles or mnemonics deal with students who realize principles with
acoustic ways. Kinesthetic Learners Hands-on tasks and manipulatives sustain kinesthetic students in understanding multiplication. Tips for Effective Execution in Knowing Consistency in Practice
Normal technique strengthens multiplication skills, promoting retention and fluency. Balancing Repetition and Variety A mix of repeated exercises and varied issue styles maintains rate of interest
and understanding. Providing Useful Comments Responses help in determining locations of renovation, motivating continued progress. Challenges in Multiplication Method and Solutions Inspiration and
Engagement Obstacles Boring drills can bring about uninterest; cutting-edge approaches can reignite motivation. Getting Rid Of Concern of Math Negative understandings around mathematics can hinder
progress; creating a positive learning environment is important. Influence of Multiplication Integers Worksheets on Academic Performance Researches and Study Searchings For Study shows a positive
correlation between regular worksheet usage and boosted mathematics performance.
Multiplication Integers Worksheets become versatile devices, promoting mathematical proficiency in learners while suiting varied understanding designs. From fundamental drills to interactive on the
internet sources, these worksheets not only boost multiplication abilities but also promote crucial reasoning and analytical abilities.
11 Best Images Of Multiplying Integers Worksheets With Answers Multiplying Integers Math
ADV HW With Examples 7 4 Problem Solving Practice Multiplying Integers Word Problems For A
Check more of Multiplication Integers Worksheets below
Multiplication Integers Worksheet
50 Multiplication Of Integers Worksheet Chessmuseum Template Library Integers Worksheet
Grade 8 Math Integers Worksheets Printable Learning How To Read
Multiplying And Dividing Integers Worksheets
18 Best Images Of Math Worksheets Integers Integers Worksheet 6th Grade Math Printable 6th
Multiplication of integers worksheets K5 Learning
Multiplication of integers Multiplying with negative numbers Practice integer worksheets on multiplying with negative numbers Horizontal and vertical multiplication Horizontal Worksheet 1 Worksheet 2
Worksheet 3 Vertical Worksheet 4 Worksheet 5 Worksheet 6 3 More Similar Division of integers Absolute and opposite values of integers
Multiplying and Dividing Integers Worksheets Math Worksheets 4 Kids
Integer Division Perform the division operation on the integers to find the quotient in these three pdf worksheets Multiplying and Dividing Integers Mixed Multiplying and Dividing Integers Simplify
the integer equations by performing multiplication and division operations Missing Integers Multiplication and Division
Multiplication of integers Multiplying with negative numbers Practice integer worksheets on multiplying with negative numbers Horizontal and vertical multiplication Horizontal Worksheet 1 Worksheet 2
Worksheet 3 Vertical Worksheet 4 Worksheet 5 Worksheet 6 3 More Similar Division of integers Absolute and opposite values of integers
Integer Division Perform the division operation on the integers to find the quotient in these three pdf worksheets Multiplying and Dividing Integers Mixed Multiplying and Dividing Integers Simplify
the integer equations by performing multiplication and division operations Missing Integers Multiplication and Division
Grade 8 Math Integers Worksheets Printable Learning How To Read
Multiplication Integers Worksheet
Multiplying And Dividing Integers Worksheets
18 Best Images Of Math Worksheets Integers Integers Worksheet 6th Grade Math Printable 6th
50 Multiplication Of Integers Worksheet
Integers Worksheets Dynamically Created Integers Worksheets
Integers Worksheets Dynamically Created Integers Worksheets
50 Multiplication Of Integers Worksheet
FAQs (Frequently Asked Questions).
Are Multiplication Integers Worksheets suitable for every age teams?
Yes, worksheets can be customized to various age and ability degrees, making them adaptable for numerous students.
Exactly how often should students exercise using Multiplication Integers Worksheets?
Regular technique is key. Normal sessions, preferably a few times a week, can produce considerable enhancement.
Can worksheets alone improve math skills?
Worksheets are an important device however ought to be supplemented with varied learning methods for comprehensive skill advancement.
Are there on the internet platforms supplying totally free Multiplication Integers Worksheets?
Yes, lots of academic web sites provide open door to a wide variety of Multiplication Integers Worksheets.
Just how can moms and dads sustain their youngsters's multiplication practice in the house?
Urging regular practice, supplying help, and developing a positive knowing setting are beneficial actions. | {"url":"https://crown-darts.com/en/multiplication-integers-worksheets.html","timestamp":"2024-11-06T11:36:02Z","content_type":"text/html","content_length":"28759","record_id":"<urn:uuid:d3a3cc7a-d4e5-4bf1-bff9-2a6eb5f2544d>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00463.warc.gz"} |
On the Stress-Energy Tensor of Quantum Fields in Curved Spacetimes - Comparison of Different Regularization Schemes and Symmetry of the Hadamard/Seeley-DeWitt Coefficients
On the Stress-Energy Tensor of Quantum Fields in Curved Spacetimes - Comparison of Different Regularization Schemes and Symmetry of the Hadamard/Seeley-DeWitt Coefficients
Thomas-Paul Hack
Valter Moretti
February 23, 2012
We review a few rigorous and partly unpublished results on the regularisation of the stress-energy in quantum field theory on curved spacetimes: 1) the symmetry of the Hadamard/Seeley-DeWitt
coefficients in smooth Riemannian and Lorentzian spacetimes 2) the equivalence of the local $\zeta$-function and the Hadamard-point-splitting procedure in smooth static spacetimes 3) the equivalence
of the DeWitt-Schwinger- and the Hadamard-point-splitting procedure in smooth Riemannian and Lorentzian spacetimes.
Hadamard states
DeWitt-Schwinger- and the Hadamard-point-splitting procedure | {"url":"https://lqp2.org/node/783","timestamp":"2024-11-13T11:31:39Z","content_type":"text/html","content_length":"16506","record_id":"<urn:uuid:c022dec6-1b65-4cfc-93c5-4ce48e903b3a>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00215.warc.gz"} |
JavaScript Program to Calculate the Area of a Triangle | Vultr Docs
Calculating the area of a triangle is a common task in geometry and can often be used in various applications, including graphics programming, game development, and educational software.
Understanding how to compute this can enhance your capability to solve not only direct triangle area problems but also more complex geometric calculations.
In this article, you will learn how to create a JavaScript program to calculate the area of a triangle. We'll go through different examples, using both traditional geometry formulas and some
alternative methods, allowing you to understand the steps and choose the method that best suits your needs.
Traditional Method Using Base and Height
Calculate Area with Base and Height
1. Understand the formula Area = (base * height) / 2.
2. Implement the formula in a JavaScript function.
function calculateTriangleArea(base, height) {
return (base * height) / 2;
This function takes the base and height of a triangle as arguments and returns the area using the standard geometrical formula.
Example Usage
1. Call the function with specific values for base and height.
2. Display the result.
const area = calculateTriangleArea(5, 10);
console.log("Area of the triangle:", area);
This example calculates the area of a triangle with a base of 5 units and a height of 10 units, logging Area of the triangle: 25 to the console.
Using Heron's Formula
Understanding Heron's Formula
1. Recognize that Heron's formula requires three sides of the triangle, denoted as a, b, and c.
2. The formula also involves the semi-perimeter s = (a + b + c) / 2.
3. The area is then calculated using Area = √(s * (s - a) * (s - b) * (s - c)).
Implement Heron's Formula in JavaScript
1. Write a function to calculate the area of a triangle using all three sides.
function heronsFormula(a, b, c) {
const s = (a + b + c) / 2;
const area = Math.sqrt(s * (s - a) * (s - b) * (s - c));
return area;
This function calculates the area by first determining the semi-perimeter and then applying Heron's formula.
Example Using Heron's Formula
1. Input the side lengths of the triangle.
2. Call the heronsFormula function and print the area.
const areaHeron = heronsFormula(5, 6, 7);
console.log("Area of the triangle using Heron's formula:", areaHeron);
For a triangle with sides 5, 6, and 7 units, this script calculates and logs Area of the triangle using Heron's formula: 14.696938456699069.
Calculating the area of a triangle in JavaScript can be done efficiently using different methods depending on the available data. Whether you have the base and height or the lengths of all three
sides, implementing these formulas in JavaScript allows you to solve geometric problems quickly and accurately. With the provided examples, you can integrate these calculations into your applications
to handle various tasks requiring geometric computations. | {"url":"https://docs.vultr.com/javascript/examples/calculate-the-area-of-a-triangle","timestamp":"2024-11-10T22:39:50Z","content_type":"text/html","content_length":"304508","record_id":"<urn:uuid:e29d1799-5420-42fc-a46a-22c37a2fe114>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00461.warc.gz"} |
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Lemma 79.11.2. Let $k$ be a field. Let $n \geq 1$ and let $(\mathbf{P}^1_ k)^ n$ be the $n$-fold self product over $\mathop{\mathrm{Spec}}(k)$. Let $f : (\mathbf{P}^1_ k)^ n \to Z$ be a morphism of
algebraic spaces over $k$. If $Z$ is separated of finite type over $k$, then $f$ factors as
\[ (\mathbf{P}^1_ k)^ n \xrightarrow {projection} (\mathbf{P}^1_ k)^ m \xrightarrow {finite} Z. \]
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3Blue1Brown - But what is a Neural Network?
Chapter 1But what is a Neural Network?
The program above can identify hand-drawn digits 0-9 reasonably accurately. Give it a whirl if you haven’t already!
Although it does generally work, it requires a bit of coaxing to get there. In particular, the digit images it receives need to be centered and about the right size, which is why there's a
pre-processing step before the digit image gets passed along to the neural network.^
While more modern neural networks can do a much better job at tasks like this, the network above is simple enough that you can understand exactly what it’s doing and how it was trained with almost no
background. It’s also simple enough that you could train it on your own computer, while training more sophisticated networks can require a truly mind-boggling amount of computation.^
On the surface, a machine recognizing handwritten digits may not seem particularly impressive. After all, you know how to identify digits, and I bet you don’t even find it very hard. For example, you
can tell instantly that these are all images of the digit three:
Each three is drawn differently, so the particular light-sensitive cells in your eye that fire are different for each, but something in that crazy smart visual cortex of yours resolves all these as
representing the same idea, while recognizing images of other numbers as their own distinct ideas.
But if I told you to sit down and write a program like the one shown above, that takes in a grid of 28x28 pixels, and outputs a single number between 0 and 9, the task goes from comically trivial to
dauntingly difficult.
Somehow identifying digits is incredibly easy for your brain to do, but almost impossible to describe how to do. The traditional methods of computer programming, with if statements and for loops and
classes and objects and functions, just don’t seem suitable to tackle this problem.
But what if we could write a program that mimics the structure of your brain? That’s the idea behind neural networks. The hope is that by writing brain-inspired software, we might be able to create
programs that tackle the kinds of fuzzy and difficult-to-reason-about problems that your mind is so good at solving.^
Moreover, just as you learn by seeing many examples, the “learning” part of machine learning comes from the fact that we never give the program any specific instructions for how to identify digits.
Instead, we’ll show it many examples of hand-drawn digits together with labels for what they should be, and leave it up to the computer to adapt the network based on each new example.
By the way, recognizing handwritten digits is a classic example for introducing this topic, and I’m happy to stick with the status quo here. Since it’s such a common starting point, there are plenty
of other resources available that tackle the same subject matter in more depth for people who want to dig in deeper. If that sounds like you, take a look at this excellent online textbook by Michael
Nielsen, which includes code that you can download and play with to really get your hands dirty.
The Structure of a Neural Network
This lesson is all about motivating and understanding the structure and mathematical description of a neural network, while the next lesson will focus on how to train it with labeled examples.
There are many variants of neural networks, such as convolutional neural networks (CNN), recurrent neural networks (RNN), transformers, and countless others. In recent years there’s been a boom in
research of these variants. But the first step to understanding any of them is to build up the simplest, plain vanilla form with no added frills.
The simple network we’re using to identify digits is just a few layers of neurons linked together.
Right now, when I say neuron, all I want you to think is “a thing that holds a number.” Specifically, a number between 0.0 and 1.0. Neural networks are really just a bunch of neurons connected
This number inside the neuron is called the “activation” of that neuron, and the image you might have in your mind is that each neuron is lit up when its activation is a high number.
Every neuron has an activation between 0.0 and 1.0, sort of analogous to how neurons in the brain can be active or inactive.
All the information passing through our neural network is stored in these neurons. So we need to represent the inputs and outputs of our network (the images and digit predictions) in terms of these
neuron values between 0.0 and 1.0.
Each pixel in the original image has a value between 0.0 (black) and 1.0 (white).
All of our digit images have $28 \times 28 = 784$ pixels, each with a brightness value between 0.0 (black) and 1.0 (white). To represent this in the network, we’ll create a layer of 784 neurons,
where each neuron corresponds to a particular pixel.
The input layer contains 784 neurons, each of which corresponds to a single pixel in the original image.
When we want to feed the network an image, we’ll set each input neuron’s activation to the brightness of its corresponding pixel.^
The last layer of our network will have 10 neurons, each representing one of the possible digits. The activation in these neurons, again some number between 0.0 and 1.0, will represent how much the
system thinks an image corresponds to a given digit.
The output layer of our network has 10 neurons. Each neuron corresponds to a particular digit that the image could contain.
Take a look at the following image:
Based on the output layer of the network shown above, what kind of digit does this network think it’s looking at? How certain does it feel?
Your answer:
Our answer:
It’s having trouble deciding whether the input image is a 4 or a 9.
There will also be some layers in between, called “hidden layers”, which for the time being should just be a giant question mark for how on earth this process of recognizing digits will be handled.
In this network I have 2 hidden layers, each with 16 neurons, which is admittedly kind of an arbitrary choice. To be honest, I chose 2 layers based on how I want to motivate the structure in just a
moment, and 16 was simply a nice number to fit on the screen. In practice, there’s a lot of room to experiment with the specific structure.
Why Use Layers?
You’ll notice how in these drawings each neuron from one layer is connected to each neuron of the next with a little line. This is meant to indicate how the activation of each neuron in one layer,
the little number inside it, has some influence on the activation of each neuron in the next layer.
Watching the activations in each layer propagate through to determine the activations in the next can be quite mesmerizing.
However, not all these connections are equal. Some will be stronger than others, and as you’ll see shortly, determining how strong these connections are is really the heart of how a neural network
operates, as an information processing mechanism.
But before jumping into the math for how one layer influences the next, or how training works, let’s talk about why it’s even reasonable to expect a layered structure like this to behave
intelligently. What are we expecting here? What’s the best hope for what those middle layers are doing? Why not just directly connect all the pixels to the final output we want?
Well, when you or I recognize digits, we piece together various components like loops and lines.
Each digit can be broken into smaller, recognizable subcomponents.
In a perfect world, we might hope that each neuron in the second-to-last layer corresponds to one of these subcomponents. That anytime you feed in an image with, say, a loop up top, there is some
specific neuron whose activation will be close to 1.0.
And I don’t mean just this exact loop of pixels. The hope would be that any generally loopy pattern toward the top of the image sets off this neuron. That way, going from this third layer to the last
one would only require learning which combinations of subcomponents correspond to which digits.
Of course, this just kicks the problem down the road, because how would you recognize these subcomponents, or even learn what the right subcomponents should be? And I still haven’t talked about how
exactly one layer influences the next! But run with me on this for a moment.
Recognizing a loop can also break down into subproblems. One reasonable way to do that would be to first recognize the various edges that make it up.
A loop can be broken down into several small edges.
Similarly, a long line, as you might see in the digits 1, 4 or 7, is really just a long edge. Or maybe you think of it as a certain pattern of several smaller edges.
A long line is also just a bunch of edges.
So our hope might be that each neuron in the second layer of the network corresponds to some little edge. Maybe when an image comes in, it lights up neurons associated with all the specific little
edges inside that image. This, in turn, would light up the neurons in the third layer associated with larger scale patterns like loops and long lines, which would then cause some neuron from the
final layer to fire which corresponds to the appropriate digit.
Whether or not this is how our final network actually works is another question. (One that we’ll revisit after seeing how to train this network.) But this is a hope that we might have.
Layers Break Problems Into Bite-Sized Pieces
You can imagine how being able to detect edges and patterns would also be useful for other image-recognition tasks.
Edge detection isn’t just for digits! It’s a useful step for all kinds of image-recognition problems.
Original lion image by Kevin Pluck, licensed under CC BY 2.0
And beyond image recognition, there are all sorts of intelligent tasks that you can break down into layers of abstraction.
Parsing speech, for example, involves parsing raw audio into distinct sounds, which combine to make certain syllables, which combine to form words, which combine to make up phrases and more abstract
thoughts, etc.
The layered structure of the neural network is great because it allows you to break down difficult problems into bite-size steps, so that moving from one layer to the next is relatively
How Information Passes Between Layers
With this as a general idea, how do you actually implement it? The goal is to have some mechanism that could conceivably combine pixels into edges, or edges into patterns, or patterns into digits. It
would be especially elegant if all of those different steps used the same mathematical procedure.
To zoom in on one very specific example, let’s say that the hope is for this one particular neuron in the second layer to pick up on whether or not the image has an edge in this spot here:
We want this one, specific neuron in the second layer to pick up on whether the image contains this one, specific edge.
I want you to think about what parameters the network should have, what knobs and dials you should be able to tweak, so that it’s expressive enough to potentially capture this pattern. Or other pixel
patterns. Or the pattern that several edges can make a loop, and other such things.
What we’ll do is assign a weight to each of the connections between our neuron and the neurons from the first layer. These weights are just numbers.
Each weight is an indication of how its neuron in the first layer is correlated with this new neuron in the second layer.
If the neuron in the first layer is on, then a positive weight suggests that the neuron in the second layer should also be on, and a negative weight suggests that the neuron in the second layer
should be off.
Of course, these weights will interact and conflict in interesting ways, but the hope is that if we add up all the desires from all the weights, the end result will be a neuron that does a reasonably
good job of detecting the edge we’re looking for (as long as the weights are well-chosen).
So to actually compute the value of this second-layer neuron, you take all the activations from the neurons in the first layer, and compute their weighted sum.
$\textcolor{green}{w_1} a_1 + \textcolor{green}{w_2} a_2 + \textcolor{green}{w_3} a_3 + \textcolor{green}{w_4} a_4 + \cdots + \textcolor{green}{w_n} a_n$
It’s helpful to think of all those weights as being organized into a grid of their own:
Each weight is associated with one of the 784 input pixels. Arranging the weights into this 28x28 grid makes the correlations between the input image and the output activation clear.
I’m using blue pixels to indicate a positive weight, and red pixels to indicate a negative weight, with the brightness of that pixel being some depiction of the weight’s value.
What if we made the weights associated with almost all the pixels 0, except for some positive weights associated with these pixels in the region where we want to detect an edge?
With these weights, the neuron in the second layer will be more activated when pixels in this region are more activated.
Then taking a weighted sum of all pixel values really just amounts to adding up the values of the pixels in this region we care about.
But this pattern of weights will also pick up on big blobs of activated pixels! (Not just edges.) To really pick up on whether or not this is an edge, you might want to have some negative weights
associated with the surrounding pixels. Then the sum will be largest when these pixels are bright, but the surrounding pixels are dark.
By adding some negative weights above and below, we make sure the neuron is most activated when a narrow edge of pixels is turned on, but the surrounding pixels are dark.
Suppose a neuron in the second layer has weights as indicated above. Rank the four images (A, B, C, and D) based on how much they would activate that neuron:
Sigmoid Squishification
The result of the weighted sum like this can be any number, but for this network we want the activations to be values between 0 and 1. So it’s common to pump this weighted sum into some function that
squishes the real number line into the range between 0 and 1.
There’s no limit to how big or small the weighted sum might be. But our new neuron value should be between 0 and 1, so we need to somehow squish the range of possible outputs down to size.
One common function that does this is called the “sigmoid” function, also known as a logistic curve, which we represent using the symbol $\sigma$. Very negative inputs end up close to 0, very
positive inputs end up close to 1, and it steadily increases around 0. So the activation of the neuron here will basically be a measure of how positive the weighted sum is.
The sigmoid function is just the squishing function we need!
$\sigma(-1000)$ is closest to which of the following values?
But maybe it’s not that we want the neuron to light up when this weighted sum is bigger than 0. Maybe we only want it to be meaningfully active when that sum is bigger than, say, 10. That is, we want
some bias for it to be inactive.
What we’ll do then is add some number, like -10, to the weighted sum before plugging it into the sigmoid function that squishes everything into the range between 0 and 1.
We call this additional number a bias.
So the weights tell you what pixel pattern this neuron in the second layer is picking up on, and the bias tells you how big that weighted sum needs to be before the neuron gets meaningfully active.
More Neurons
And that’s just one neuron! Every other neuron in the second layer is also going to have weighted connections to all 784 neurons from the first layer. Each neuron also has some bias, some other
number to just add on to the weighted sum before squishing it with a sigmoid. That’s a lot to think about! With this hidden layer of 16 neurons, that’s 784x16 weights and 16 biases.
And all of this is just the connection from the first layer to the second. The connections between the other layers also have a bunch of weights and biases as well. All said and done, this network
has 13,002 total weights and biases! 13,002 knobs and dials that can be tweaked to make this network behave in different ways.
This network has 13,002 weights and biases! That’s a lot to handle.
When we talk about learning, which we’ll do in the next lesson, we mean getting the computer to find an optimal setting for all these many, many numbers that will solve the problem at hand.
One thought experiment, which is at once both fun and horrifying, is to imagine setting all these weights and biases by hand. Purposefully setting weights to make the second layer pick up on edges,
the third to pick up on patterns, and so on.
I personally find this satisfying, rather than just treating these networks as a total black box. Because when the network doesn’t perform the way you anticipate, if you’ve built up a feel for the
meaning of those weights and biases in your mind, you have a starting place for experimenting with how to change this structure to be better.
Or, when the network does work, but not for the reasons you might expect, digging into what the weights and biases are doing is a good way to challenge your assumptions and really expose the full
space of possible solutions.
More Compact Notation
The actual function to get one neuron’s activation in terms of the activations in the previous layer is a bit cumbersome to write down.
Tracking all these indices takes a lot of effort, so let me show the more notationally compact way that these connections are represented.
Instead of computing a bunch of weighted sums like this one-by-one, we’ll use matrix multiplication to compute the activations of all the neurons in the next layer simultaneously.
First, organize all the activations from the first layer into a column vector.
Next, organize all the weights as a matrix, where each row of this matrix corresponds to all the connections between neurons in the first layer and a particular neuron in the next layer.
Then the product $\textcolor{green}{W} a^{(0)}$ is a column vector containing all the weighted sums for the neurons in the next layer.^
Instead of talking about adding the bias to each one of these values independently, we represent it by organizing all those biases into a vector, and adding the entire vector to the previous
matrix-vector product:
Finally, I’ll wrap a sigmoid on the outside here, which is meant to represent applying the sigmoid function to each component of the result:
So, once you write this weight matrix and these vectors as their own symbols, you can communicate the full transition of activations from one layer to the next in a neat little expression:
This tiny expression represents the computation of all the neurons in the next layer based on all the neurons in the previous layer, using the chosen weights and biases.
This makes the relevant code much cleaner and much faster, since many libraries optimize the heck out of matrix multiplication.^
The Network Is Just a Function
Earlier I said to think of these neurons simply as “things that hold numbers”. Of course, the specific number these neurons hold depends on the image you feed in. So it’s actually more accurate to
think of each neuron as a function. It takes in the activations of all neurons in the previous layer, and spits out a number between 0 and 1.
And really, the entire network is just a function! It takes in 784 numbers as its input, and spits out 10 numbers as its output. It’s an absurdly complicated function, because it takes over 13,000
parameters (weights and biases), and it involves iterating many matrix-vector products and sigmoid squishificaitons together. But it’s just a function nonetheless.
The entire neural network is a function that uses all its weights and biases to take in 784 input pixels and spit out 10 output numbers.
Next up: Learning!
In a way, it’s kind of reassuring that this looks complicated. If it were any simpler, what hope would we have that it could take on the challenging task of recognizing digits?
And how does it take on that challenge? How does this network learn the appropriate weights and biases from data? That’s what I’ll show in the next lesson.
Oh, but before you go, I do have one little asterisk to mention about the sigmoid function if that sounds interesting to you:
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What are the boundary conditions in fluid dynamics? | SolidWorks Assignment Help
What are the boundary conditions in fluid dynamics? First, boundary conditions were used to map the reaction between fluid and fluid-fluid collisions, and to determine the fluid velocity in the
collision and in that collision. Second, boundary conditions were introduced by the following stochastic differential equation $$\frac{\partial \mbox{d}F}{\mbox{d}t}+\frac{1}{2}\frac{\partial^2 F}{\
partial \mbox{d}x^2}+F”(x)-U\big({\bf{\mu}}(x)F(x)\big)’=0,$$ where $U(\mbox{}x)$ is the dynamic control parameter. These are four deterministic boundary conditions describing the fluid moment
created by the incident spin-contact in linear stochastic dynamics, while the control parameter is determined by the control parameter in the collision, i.e., $\overline{\alpha}\mbox{}/\overline{\
alpha}_{\rm cl}$ and $\overline{\alpha}_{\rm cl}\mbox{}/\overline{\alpha}_{\rm pr}$ are the two deterministic boundary conditions imposed by the collision and the initial position along the
time-scale of the collision, respectively. The drift coefficient is modeled by a unit concentration membrane which is parallel or rotatable, whereas the size of the container is specified by a
size-scale $\eta$. While for small boundary values the boundary conditions can have some dynamical effects, boundary conditions across the linear and dynamic parts are not important, rather their
contributions as partial boundary conditions are due to some form of diffusion. Finite boundary conditions across the dynamics, instead, are determined by the coefficients of the second-order
differential equation. Finite boundary conditions are in total the boundary conditions used by all boundary conditions. We use again the boundary condition in [@Elg2016b], where they are similar to
the ones used in the case of fluid-fluid collisions. The boundary-based results presented in this paper are the essential ingredient of the analysis presented here. Reaction of fluid and fluid-fluid
interaction ============================================= We use the time-dependent self-diffusion equation (t’,x’,y’,z’) from [@fisher1967introduction], which is based on the so-called Euler
equation $$i\frac{d\sigma-dt}{dt}+\frac{1}{2}\partial^2\sigma-\omega\frac{\sigma+D\sigma}{\sigma-D}=0,$$ where $D()$ is the diffusion coefficient of the gas and particles, $\sigma$ is the length in
the system’s bulk medium, $\omega$ is the relative specific heat, and $\omega \equiv E_{\rm rms}/T$ is the internal mean free energy. In this section we define two new rate-dependent rate equations
in the linear partial differential form and introduce the two initial conditions at time $t=0$ and $t=t_0$, respectively. Their behavior is described as follows: $$\frac{\partial}{\partial t}\frac{\
partial}{\partial x}-\frac{1}{2}\frac{\partial^2 F}{\partial i\partial\theta}+\frac{1}{2}\frac{\partial^4 F}{\partial\psi^2}=i\int_0^{\infty}\left\{\frac{\sigma-D\sigma}{\sigma+D\psi}-\frac{1}{\omega
t}-\frac{\omega}{i}\int_0^{\equiv}\min \frac{\sigma+D\psi}{\sigma}, \frac{\partial \sigma}{\partial \theta}+\frac{\partial^2 F}{\partial \psi^2}=0,\right. \label{eq-sol-b}$$ where (say) $x$ and $\
theta$ are the initial and final positions, respectively, at time $t=0$ and time $t=t_0$, respectively, and $\psi$ is the density for particles at time $t=t_0$. Following the previous procedure the
time evolution of $\sigma$ and $\sigma-D$ can be expressed by (\[eq-sol\]), which allows us to write and solve the two rate equations for $\sigma$ and $\sigma-D$ as $$\left[\frac{\partial\sigma}{\
partial t}\right]_\sigma =+i\frac{d}{dx}+i\fracWhat are the boundary conditions in fluid dynamics? What are boundary conditions in fluid dynamics that cause flows? This article will look at more than
one boundary condition for any given flow, as well as an example of how that problem can be treated. The results will be applicable to all hydrodynamics and will demonstrate, however, how boundary
conditions are used to implement boundary in fluid dynamics. Different boundary conditions can be used to find certain property of a flow. All it takes to find such a property is to find a
minimization of a Lagrangian that maximizes the Lagrangian at the point where it touches both sides. This minimization of Lagrangian is a reasonable way to find properties of the flow at small
scales, but it is likely that the behaviour of the flow of a fluid will depend on the structure of the initial conditions.
Pay For Someone To Do Mymathlab
A few examples of these choices will be provided. Start from initial conditions For the initial velocity of the fluid to satisfy the boundary conditions (which is a really obvious choice), it is
necessary to start moving the fluid up a certain scale during this early stage: it is natural to start at the origin, as this will give you a good balance of different physical effects. That it is
necessary to start at this scale click here to read the early stage depends on how the internal energy is coming into the fluid. Otherwise, the material flowing through the unit ball will show black
holes depending on whether the fluid gets past it or not: this means that there is a higher cost for turning all the particles out of the ball at the same time, i.e.: it will be easier for the fluid
to start from a lower scale. If the internal energy is small (e.g. 10-15%) in this case, then, the material will stay in some type of black hole. On the other hand, if the internal energy is greater
(e.g. >15%) then the material will become more of a ball, and it will matter if the internal energy is smaller, and when it is smaller, the material will show no black holes. The pressure of the
fluid at those boundaries is in the scale (scale of the internal energy) that is most relevant to making a boundary change. The first term in the pressure term is the energy of the material from that
point of time to take up the ball. The second part of the pressure term is the mass of that material, its velocity. This mass gives the forces acting on the material which is in the scale we are
looking at. The physical interaction it makes runs to a higher scale but then its mass is too close to its velocity, so it is much smaller. Using the above definition of the scale of the internal
energy we can get the answer from the second part of the pressure term, that is: $$x_i = p_i + a_i – a_+ i_i – c_in – c_i – c’_out \\$$ The last term is the mass of each of the incoming particles.
This means we can take the second part of the momentum term: $$p_{i,j} = a_i – a_j – c’_out \\$$ Using the above definition of momentum we can find: $$a_x = p_x + a_{\textrm{x}} \label{eq:dx:50pt}$$
Thanks to this we can rewrite the last equation as a modified two-point diffusion equation, to which we can get the following: $$dN^2=\partial_x^2+x^2dN\left(x\right) \label{eq:dy:50pt} +\left(g_x\
left(x\right)\left(f_x – f\right)^2 – g_x\What are the boundary conditions in fluid dynamics? Is there an absolute arbitrage method for the boundary conditions in fluid dynamics? When we search for a
boundary condition and use the boundary conditions in dynamics, we find that a fluid has a boundary condition, which means that there’s some force on the fluid and there’s an exterior potential.
Outside of one surface there can’t flow in the other.
We Do Your Accounting Class Reviews
A flow on a side of a wall is their explanation a flow surface boundary, and a flow on a bulk is called a flow surface boundary. But it might raise questions if external forces and boundary
conditions are used? Thanks to a numerical implementation of a boundary condition, when somebody gets a boundary condition and tries to solve a boundary problem it’s true that a specific force or
force direction always appears on a surface. But no particular term indicates the external boundary conditions to be used. How long does it take to solve an equation of motion? A lot of solving
problems has a name, sometimes even an exact name, depending on the complexity. Remember, you’re using a different term than what you used. You name it “an equation of motion” because if one, if
another then we called and. So the same is true about what boundary conditions are used if one is concerned with the external forces and boundary conditions. It’s very obvious that the equation of
motion is defined by is. It’s defined by a equation. But can you describe the external forces and boundary conditions in terms of two and three dimensional Euler equations? I’ve been researching on
Euler in my time and don’t know which method satisfies the boundary conditions. In the end I want to go through the Euler equations using gg2D so that I can determine what problems to go through with
this method. A: The form of your incompressible flow equations (Eq. 1) does show a similar relation between each boundary condition and the form of the eigenvalues. (1)-(2) = (3./(3+1/2). + 6)/(2,6).
Therefore, P(w) = C w. As long as we keep in mind I guess the first question is addressed from the beginning of section 2 and the last question is addressed from the beginning of section 3. Or maybe
you mean what you just wrote earlier? It’s easy to talk about any Euler equation with $w$ as some defined function, but how does your own equation take into account this kind of function? For
instance, think of a ball of radius 2. In this case the fluid is a large body in a closed conic, but its velocity (relative to the conical) is restricted, see below.
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This doesn’t mean the fluid solution has to be isometric until it gets to a extremum. Although the extension of the main results to non-capillary fluids (such as yours) gives significant progress
makes it possible | {"url":"https://solidworksaid.com/what-are-the-boundary-conditions-in-fluid-dynamics-17362","timestamp":"2024-11-02T02:47:07Z","content_type":"text/html","content_length":"158381","record_id":"<urn:uuid:47674fd8-dbab-4e74-8b0c-bcb6eac8ecf4>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00222.warc.gz"} |
Talk:Pig the dice game/Player - Rosetta CodeTalk:Pig the dice game/Player
Very draft task
Unlike most of the tasks I start, I have started this one without having a Python solution to hand and so I am unsure of how large a task this is, and may have left important things out of the task
description that may lead to non-compatible future edits .
Thanks Tim T. for the many corrections to the hopeless grammar I initially had in Pig the dice game; I hope my English has improved. --Paddy3118 05:57, 14 September 2012 (UTC)
I've put together an estimator for use in deciding whether to reroll or not. Here's pseudo-code for people that cannot read J:
consider each roll, for non-1 rolls, each roll's base value is the minimum of itself and 100-(sum of dice currently rolled). For a roll of 1, the value is -(uncommitted rolls). Average these to
get the estimated value of the current possibilities. [Or, of course, if you just want to know if the result is positive, you do not need to divide by 6.] --Rdm 19:17, 14 September 2012 (UTC)
That said, using philosophy behind "you don't have to be faster than the bear", a strategy which takes into account your opponent's score could be superior -- sometimes playing more
conservatively because we know that our opponent doesn't have a good chance of winning. That said, I have not completely convinced myself that this would ever be a good idea -- intuitively,
it seems like it could be, but I have not had the time to think it through, yet. --Rdm 19:22, 14 September 2012 (UTC)
I was thinking of implementing a simple always roll n times then hold. I could then do stats on randomly varying n for each player and see if there are any patterns for the winner. But as yet I have
no prog at all so its just an idea... --Paddy3118 03:09, 15 September 2012 (UTC)
The winning strategy needs to take into account the complete game state, namely the current player's total score, his current holding score, and the other player's score. The function to calculate
the player's best strategy and winning chance given a game state is (psuedo code): <lang>function P(total, this_round, op): // args: my total, my rolled so far, and opponent's score; it's my turn if
total + this_round >= 100 then: return 1
// I yield; my winning chance is whatever opponent doesn't win his chance_yield = 1 - P(op, 0, total + this_round)
// chance to win after rolling a 1 chance_roll = (1 - P(op, 0, total)) / 6
// plus chance to win for rolling 2--6 for roll from 2 to 6 do: chance_roll += P(total, this_round + roll, op) / 6
return max(chance_roll, chance_yield) // choose better prospect</lang> Note that this is a recursive relation, and the P function can't be evaluated just like that due to infinite recursions. There
are ways around it by doing matrix iterations or some such, assuming a single function for probability makes sense -- depending on the game type, it's possible that there's no optimal strategy, in
which case either the probability matrix won't converge, or there could be multiple stable solutions, etc. I don't think that happens for this game, but calculating the matrix can still be quite
daunting. --Ledrug 06:01, 15 September 2012 (UTC)
Hi Ledrug, Sounds good but I don't want to add a particular strategy to the task description so that people can at least try something that they find easy to code. --Paddy3118 11:28, 15 September
2012 (UTC)
It's probably not fair to call the strategy you are defining the "winning strategy" since there's an element of luck in this game. It's perhaps the "best strategy", but even that's not completely
We can only assign a probability to our opponents moves if we know how our opponent's strategy. Let's say that we know our opponent's strategy, and that it's this
<lang>if current total is 0, always roll
if current total is 1 reroll only if the prior total does not exceed 98 if current total is 2 reroll only if the prior total does not exceed 97 if current total is 3 reroll only if the prior total
does not exceed 96 if current total is 4 reroll only if the prior total does not exceed 95 if current total is 5 reroll only if the prior total does not exceed 93 if current total is 6 reroll only if
the prior total does not exceed 92 if current total is 7 reroll only if the prior total does not exceed 91 if current total is 8 reroll only if the prior total does not exceed 90 if current total is
9 reroll only if the prior total does not exceed 89 if current total is 10 reroll only if the prior total does not exceed 87 if current total is 11 reroll only if the prior total does not exceed 86
if current total is 12 reroll only if the prior total does not exceed 85 if current total is 13 reroll only if the prior total does not exceed 84 if current total is 14 reroll only if the prior total
does not exceed 82 if current total is 15 reroll only if the prior total does not exceed 81 if current total is 16 reroll only if the prior total does not exceed 80 if current total is 17 reroll only
if the prior total does not exceed 78 if current total is 18 reroll only if the prior total does not exceed 77 if current total is 19 reroll only if the prior total does not exceed 75 Do not reroll
if current total is 20 or higher</lang>
We can compute our opponent's probability of winning with this information if we also know our own strategy.
So let's say that we adopt the same strategy -- now we can compute our opponent's probability of winning, and our own. This allows us to build a table describing probabilities for all possible game
Now let's introduce the possibility that for only one move we will calculate a different strategy (and we do this for each game state). This allows us to use your pseudo-code, except that our
recursive calls to P are no longer recursive -- they're just table lookups. Now we can build a table based on the static strategy and a table based on the [almost trivial] "dynamic" strategy. If the
tables are identical, we are done.
And I think that this game is one where paying attention to our opponent's current point total can improve our odds. For example, if we have 94 prior points, and we rolled a 5, we have a potential
score of 99 points. If our opponent also has 99 points then we have at least a 5/6 chance of losing if we stop now, and at least 5/6 chance of winning if we roll again, while the above static
strategy says we should stop and wait until next turn. This is an unlikely game state, but it's a valid one.
So, anyways, we can assign even odds to our opponent's chance of picking either strategy, and we can pick the best odds for ourselves from the known strategies, and we can grant ourselves the right
to make one choice differently again, at each game state, and compute a new table of odds, then iterate until we find where the sequence of tables converge to. For this game the sequences at each
game state should have asymptotes, and finding them would give us an answer without requiring infinite computation.
So, anyways, that result would be the best strategy when faced with certain kinds of opponents. But our strategy might become unstable if we try to model our opponent having knowledge of what we will
Assuming this instability will exist: We can dampen that instability by adding randomness to our choices: we can start with the "best deterministic strategy" determined above, and grant our opponent
complete knowledge that we will use that strategy and give them the right to make one choice differently (at any game state) based on that knowledge, and let the opponent iterate until they find a
new best strategy. Then we set up a new strategy where we pick (with even odds) from either our first strategy or this new strategy that our opponent would have picked, and let our opponent pick a
new best strategy again, iterating... Here, we converge not to a single deterministic Roll/Hold choice for each game state but to a distribution function for each game state (or odds of picking
"Roll" for example, in each game state).
But, here, we are still carrying an assumption about our opponent -- for example, that our opponent would be "like us".
To do better than this, we would have to engage in "discovering how our opponent is different from us". Here, we might start with the opponent using a "like us" strategy and then look for evidence
that the opponent is using some other strategy, and then attempt to build a model of that strategy and play against our models of our opponent instead of just against an opponent who is "just like
us". It's not at all clear, though, that there's any justification for doing that for this game. (And, of course, this approach is also intractable in the general case (where it's not trivial and
irrelevant) because the number of implementations open to our opponent is infinite.) --Rdm 12:02, 15 September 2012 (UTC)
By "winning strategy" I meant the choice of roll or hold that maximizes the chance of winning at a given board state, that's what the P() function is. Instead of thinking it as function, consider
P as a matrix with board state as its subscripts, and the elements need to satisfy the relations given in the code. An optimal strategy exists (for both sides) if there is a unique set of P
values that satisfy all the interrelations; if no such solution exists, or there are multiple sets of solutions, then this game has no global optimal strategy. But, if a unique solution does
exist, then there's no need to consider what strategy the opponent would choose: you just do what gives you best winning chance. If your opponent is smart enough, he'd do the same, otherwise it's
all the worse for him.
I believe this particular game has a unique solution, and I do have a solution of the P matrix, although proving it's unique is not easy. The holding choice pattern is pasted here, where each row
starts with two numbers, your current score and current holding score; the horizontal axis is the opponent's current score. Wherever there's a dot, it means you're better off hold; otherwise you
should roll. For example, on line "0 21: ", there are 11 dots at the begining of the line, which means: if your total score is 0, and have already rolled 21 points this turn, then your best
choice is hold if opponent has 0-10 points, but better keep rolling if he has 11 or more. The graph has many odd features, but generally makes sense. --Ledrug 13:13, 15 September 2012 (UTC)
That's an interesting set of data. I would particularly like to understand the "concave" parts of the holding pattern -- especially for right-facing concavities (such as the rightmost part of
36 22: through 36 25:). Those features seem counter-intuitive to me. Thanks. --Rdm 23:39, 15 September 2012 (UTC)
Well, simple flukes there. In that region the winning chances are about .2 ~ .3, but the differences between hold and roll are very small (<1e-3), which could simply be because of loss of
floating point precision. It isn't very meaningful. --Ledrug 00:33, 16 September 2012 (UTC)
There are other concave regions (both vertically facing and right facing). Are they all "doesn't matter much what you choose" issues? --Rdm 12:49, 16 September 2012 (UTC)
I don't know, probably. I can't paste the graph with numbers in it, the file would be too big for pastebin. Instead the code generating numbers is in User:Ledrug/bits at the top,
you can play with it to see the numbers (warning: huge output on stdout). --Ledrug 21:31, 16 September 2012 (UTC)
I should at least include a strategy of random play in my solution so that I can gauge any other strategy against it.--Paddy3118 11:28, 15 September 2012 (UTC)
Tournament of Pigs
I decided to have various strategies fight it out and see who has the highest winning chance. Contestants:
A: always roll
R: randomly roll. Roll if holding score is 0, otherwise has 1/3 of chance to hold.
H: hold at 20. Roll if opponent >= 80; roll if self score >= 78; roll if holding score <= 20; hold otherwise.
O: the optimal holding pattern discussed above.
Each strat also has an "anti": AntiX is the holding pattern that maximizes winning when playing agains X. Note that AntiO is O. The following table is the resulting winning chances when one plays
agains another in row major, that is, the top right corner means "when playing agains O, strat A has 0.121 chance of winning if move first, 0.112 if move second". The third number is just the sum of
those two numbers; if it's gerater than 1, the strat has better overall odds when playing that opponent.
A AntiA R AntiR H AntiH O
A 0.503/0.497 1.000 0.122/0.113 0.235 0.168/0.160 0.329 0.139/0.130 0.268 0.124/0.115 0.238 0.121/0.112 0.234 0.121/0.112 0.233
AntiA 0.887/0.878 1.765 0.529/0.471 1.000 0.806/0.768 1.575 0.577/0.520 1.097 0.570/0.510 1.081 0.530/0.471 1.001 0.524/0.464 0.988
R 0.840/0.832 1.671 0.232/0.194 0.425 0.528/0.472 1.000 0.191/0.153 0.343 0.206/0.168 0.374 0.215/0.175 0.390 0.221/0.181 0.402
AntiR 0.870/0.861 1.732 0.480/0.423 0.903 0.847/0.809 1.657 0.527/0.473 1.000 0.509/0.452 0.962 0.473/0.415 0.889 0.475/0.416 0.891
H 0.885/0.876 1.761 0.490/0.430 0.919 0.832/0.794 1.626 0.548/0.491 1.038 0.529/0.471 1.000 0.483/0.421 0.904 0.489/0.428 0.917
AntiH 0.888/0.879 1.766 0.529/0.469 0.999 0.825/0.785 1.610 0.585/0.526 1.111 0.579/0.517 1.096 0.530/0.470 1.000 0.527/0.466 0.993
O 0.888/0.879 1.767 0.536/0.476 1.012 0.819/0.779 1.598 0.584/0.525 1.109 0.572/0.511 1.083 0.534/0.473 1.007 0.530/0.470 1.000
One funny thing is, AntiA is not the best at beating A; I blame loss of floating point precision for this, though I'm not completely sure.
Someone else's strategy
I found a site and a paper "Optimal Play of the Dice Game Pig" by Todd W. Neller and Clifton G.M. Presser. I glossed over the pretty graphs which may however be something like what Ledrug is
computing for his optimal strategy. I was after simple strategies and picked out their mention of 20 as being the accumulated points in a round where the odds of throwing a one are balanced by
accumulated point.
They also go on to describe why roll till 20 fails when getting nearer to the end of a game where it is advantageous to 'sprint for the win'. I ignored their full optimised strategy and just coded a
'region of desperation'. If any player is within 20 of finishing then this player should keep on rolling until it either wins or is bust as another player is likely to win on its next go. --Paddy3118
05:55, 17 September 2012 (UTC)
Holding at 20 is obviously not universal: if your opponent has 98 or 99 points, you have more than 50% chance of losing if you hold at any point before 100. If you want a simple rule, this is
closer to truth: if your opponent has more than 80ish, roll no matter what; if you have more than 78 points, roll no matter what; otherwise hold if your holding score is more than 20ish. That
should be a pretty good rough approximation.
The fact is, at any give point in the game, the probabilities of roll vs hold never differ all that much: the winning chance of rolling is never lower than 80% that of holding. If you want a
really simple rule, "just keep rolling" is the simplest. But as a game, a good strategy has a lot do with human perception of the outcome rather than boring mathematics. Take for an example, the
absurd situation where both players are at 0, but you somehow with a streak of terrific luck and had rolled 99 so far. What to do at this point? If you hold, you have 98.8% chance winning; if you
roll, it's 91.2%. You can hardly say the odds are against you if you choose to roll, but you'd be kicking yourself really hard under the table if you rolled a 1. Really, what one may consider a
"good enough" strategy sometimes has nothing to do with facts at all. --Ledrug 07:26, 17 September 2012 (UTC)
Table of Contents (List of implementations)
I added the Go example, but there seems to be no 'Table of Contents' under the task description. Forgive me, but I'm a complete noob, and don't know how to add it. La Longue Carabine 18:43, 9
November 2012 (UTC)
Hi, I forced it to appear, but normally it appears when there are four or more examples.--Paddy3118 18:46, 9 November 2012 (UTC)
Thanks. I guess I should have searched around a little more, finally found that out by looking at the Wikipedia Cheat Sheet. La Longue Carabine 18:50, 9 November 2012 (UTC) | {"url":"https://rosettacode.org/wiki/Talk:Pig_the_dice_game/Player","timestamp":"2024-11-12T19:35:53Z","content_type":"text/html","content_length":"62319","record_id":"<urn:uuid:ee334ca5-63df-4250-a6e6-4e8607480bf2>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00498.warc.gz"} |
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Consider again the pendulum linearized about its unstable upper position, given by the equation \(\ddot{\varphi}(t)-\varphi(t)=u(t)\), and assume that we use the PD control law \(u(t)=-\alpha \varphi
(t)-\beta \dot{\varphi}(t)\) to obtain an asymptotically stable closed-loop system \(\ddot{\varphi}(t)+b \dot{\varphi}(t)+a \varphi(t)=0\) (with \(a=\alpha-1>0\) and \(b=\beta>0\) ). Introduce the
natural frequency \(\omega:=\sqrt{a}\) and the damping factor \(\zeta:=b /(2 \sqrt{a})\), so that the equation now reads $$ \ddot{\varphi}(t)+2 \zeta \omega \dot{\varphi}(t)+\omega^{2} \varphi(t)=0 .
$$ (A) Prove the following facts: 1\. If \(\zeta<1\) (the "underdamped" case), all solutions are decaying oscillations. 2\. If \(\zeta=1\) (the "critically damped" case) or if \(\zeta>1\)
("overdamped"), then all solutions for which \(\varphi(0) \neq 0\) are such that \(\varphi(t)=0\) for at most one \(t>0\). 3\. If \(\zeta \geq 1\), then every solution that starts from rest at a
displaced position, that is, \(\varphi(0) \neq 0\) and \(\dot{\varphi}(0)=0\), approaches zero monotonically. (In this case, we say that there is no "overshoot.") 4\. Show rough plots of typical
solutions under the three cases \(\zeta<1, \zeta=1\), and \(\zeta>1\).
Short Answer
Expert verified
We studied a linearized pendulum system with a PD control and analyzed its behavior under different damping factors, \(\zeta\). We found that for \(\zeta<1\) (underdamped case), all solutions are
decaying oscillations. For \(\zeta = 1\) (critically damped) and \(\zeta>1\) (overdamped case), all solutions with \(\varphi(0) \neq 0\) have \(\varphi(t) = 0\) for at most one \(t>0\). If \(\zeta \
geq 1\), and the pendulum starts from rest at a displaced position, every solution approaches zero monotonically without overshooting. The plots of the solutions show unique behaviors for each case:
decaying oscillations for underdamped, non-oscillatory decay without overshoot for critically damped and overdamped cases, with overdamped cases having slower decay compared to critically damped
Step by step solution
1. Find the general solution for the given differential equation
This is a second-order linear ordinary differential equation (ODE) with constant coefficients. To solve the given ODE, we will try the exponential solution, \(\varphi(t) = e^{rt}\), where \(r\) is a
constant. Substituting this into the given ODE, we get: \[ e^{rt}(r^2 + 2 \zeta \omega r + \omega^2) = 0 \] This is a quadratic equation in \(r\), we will find its roots and then use them to get the
general solution.
2. Find the roots of the quadratic equation
We have a quadratic equation \(r^2 + 2 \zeta \omega r + \omega^2 = 0\). Using the quadratic formula, we can find the roots as: \[ r = \frac{-2 \zeta \omega \pm \sqrt{(2 \zeta \omega)^2 - 4 \omega^2}}
{2} \] Simplify the expression to get: \[ r = -\zeta \omega \pm \omega \sqrt{\zeta^2 - 1} \]
3. Study the three cases of damping ratio \(\zeta\)
Using the roots obtained in the previous step, we will analyze the three cases of damping ratio, \(\zeta\). Case 1: \(\zeta<1\) (underdamped): In this case, \(\zeta^2 - 1 < 0\), so the roots are
complex conjugates: \(r = -\zeta \omega \pm j\omega \sqrt{1-\zeta^2}\). The general solution is: \[ \varphi(t) = e^{-\zeta \omega t}(C_1 \cos{\omega_d t} + C_2 \sin{\omega_d t}) \] where \(\omega_d =
\omega \sqrt{1-\zeta^2}\) and \(C_1\) and \(C_2\) are constants determined by the initial conditions. All solutions are decaying oscillations as the exponential term decays over time. Case 2: \(\zeta
=1\) (critically damped): In this case, \(\zeta^2 - 1 = 0\), so we have repeated real roots: \(r = -\omega\). The general solution is: \[ \varphi(t) = (C_1 + C_2 t)e^{-\omega t} \] All solutions with
\(\varphi(0) \neq 0\) are such that \(\varphi(t)=0\) for at most one \(t>0\). Case 3: \(\zeta>1\) (overdamped): In this case, \(\zeta^2 - 1 > 0\), so the roots are real and distinct: \(r_1 = -\zeta \
omega - \omega \sqrt{\zeta^2 - 1}\) and \(r_2 = -\zeta \omega + \omega \sqrt{\zeta^2 - 1}\). The general solution is: \[ \varphi(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t} \] Similar to the critically damped
case, all solutions with \(\varphi(0) \neq 0\) are such that \(\varphi(t) = 0\) for at most one \(t>0\). If \(\zeta \geq 1\) and \(\dot{\varphi}(0) = 0\), every solution approaches zero
monotonically, as both exponential terms decay over time without oscillations.
4. Plot typical solutions for each case
For the underdamped case (\(\zeta < 1\)), the plot will show decaying oscillations, starting from a displaced position. The oscillations' amplitude will decay exponentially over time. For the
critically damped case (\(\zeta = 1\)), the plot will show an initial displacement and a smooth, non-oscillatory decay to zero without overshooting. For the overdamped case (\(\zeta > 1\)), the plot
will show an initial displacement and a smooth, non-oscillatory decay to zero without overshooting, but it will be slower compared to the critically damped case.
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Linearized Pendulum Dynamics
Understanding the dynamics of a pendulum is crucial for applications ranging from clocks to robotics. A key simplification is linearizing the pendulum dynamics about the unstable upper position. This
linearization process implies approximating the nonlinear behavior of the pendulum with a linear equation, making it easier to analyze and understand.
In our exercise, the equation \(\ddot{\varphi}(t) - \varphi(t) = u(t)\) describes the motion of the pendulum, where \(\varphi(t)\) is the angle from the upright position, \(\ddot{\varphi}(t)\) is the
angular acceleration, and \(u(t)\) is the control input. While this equation is an idealization, it provides insights into the fundamental behavior of the system—especially how it responds to control
inputs and acts as a starting point for designing controllers like PD (Proportional-Derivative) controllers.
PD Control Law
Control laws are algorithms used to guide the behavior of a system, and the PD control is among the most straightforward yet effective strategies for linear systems. The PD control law takes the form
\(u(t) = -\alpha \varphi(t) - \beta \dot{\varphi}(t)\), where \(\alpha\) and \(\beta\) are positive constants representing the proportional and derivative gains, respectively.
This control law aims to stabilize the pendulum by applying a force that opposes the angular displacement and velocity, effectively 'damping' the motion. The beauty of PD control is its simplicity
and effectiveness in many practical systems, making it a staple in introductory control courses and widely used in engineering applications.
Asymptotically Stable Closed-Loop System
Stability is a cornerstone in control theory, ensuring that a system returns to equilibrium after a disturbance. When we apply the PD control to our linearized pendulum system, the resulting
closed-loop system is described by the second-order linear ODE \(\ddot{\varphi}(t) + b \dot{\varphi}(t) + a \varphi(t) = 0\), indicating how the system evolves over time in response to the control
An asymptotically stable closed-loop system is one where the solutions to this ODE, namely the system's states, tend to zero as time advances to infinity. This means that no matter the initial
displacement or velocity, our pendulum will eventually come to rest at the upright position. It is essential for ensuring the long-term behavior of the system is predictable and safe.
Underdamped, Critically Damped, Overdamped Cases
The damping ratio, \(\zeta\), plays a pivotal role in the system's response. It informs us about the nature of the system's return to equilibrium after a disturbance.
• Underdamped (\(\zeta < 1\)): The system oscillates with decreasing amplitude over time, which is ideal when a quick response is necessary without large overshoots.
• Critically Damped (\(\zeta = 1\)): The system returns to equilibrium as quickly as possible without oscillating. It's often desired in control systems where overshoot is unacceptable.
• Overdamped (\(\zeta > 1\)): The system slowly returns to equilibrium without oscillating. This may be preferred in systems where it is important to avoid excessive forces or speeds due to system
The characterization of these cases is crucial when designing controllers, as it directly impacts a system's transient response and overall performance. It offers a clear framework for understanding
how different control strategies affect the behavorial dynamics of linear systems.
This understanding helps engineers design control systems that perform appropriately under various conditions, thus ensuring safety, reliability, and efficiency. | {"url":"https://www.vaia.com/en-us/textbooks/math/mathematical-control-theory-deterministic-finite-dimensional-systems-2-edition/chapter-1/problem-1-consider-again-the-pendulum-linearized-about-its-u/","timestamp":"2024-11-11T23:02:44Z","content_type":"text/html","content_length":"251949","record_id":"<urn:uuid:669357e4-22c1-4ad5-a9b2-7bb61434e7ae>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00750.warc.gz"} |
In Constantly Passing
A car is travelling along a dual carriageway at constant speed. Every 3 minutes a bus passes going in the opposite direction, while every 6 minutes a bus passes the car travelling in the same
direction. Buses leave the depot at regular intervals; they travel along the dual carriageway and back to the depot at a constant speed. At what interval do the buses leave the depot?
A car is travelling along a dual carriageway at constant speed. Every 3 minutes a bus passes going in the opposite direction, while every 6 minutes a bus passes the car travelling in the same
Buses leave the depot at regular intervals; they travel along the dual carriageway and back to the depot at a constant speed.
At what interval do the buses leave the depot?
Student Solutions
Thank Justin from Skyview High School, Billings, MT, USA for this solution and well done!
Rate, time and distance are connected by the equation r =d/t .
Call the rate (or speed) of the car r [c] and the rate of every bus r [b] . Each bus is a constant distance from the bus preceding it and the bus following it; call this distance d.
For a bus approaching on the other highway and coming towards the car, the rate of the bus relative to the car, considering the bus still, is (r [b] + r [c] ). This rate multiplied by the time it
takes (three minutes) for the car to close the gap between it and the bus is equal to d, hence:
3(r [b] + r [c] ) = d.
The rate of the bus which is travelling in the same direction as the car, relative to the rate of the car (considering the car still) is (r [b] - r [c] ). This rate multiplied by the time it takes
(six minutes) for the bus to close the gap between it and the car is also equal to d [1] , hence
6(r [b] - r [c] ) = d.
Multiplying the first equation by 2 and add the two equations, one obtains
12r [b] = 3d.
But the distance d between the buses divided by the rate of the bus is equal to the time interval between the buses therefore the time interval = d/r [b] = 4 minutes.
So the buses leave the depot at intervals of 4 minutes. | {"url":"https://nrich.maths.org/problems/constantly-passing","timestamp":"2024-11-02T14:16:42Z","content_type":"text/html","content_length":"37878","record_id":"<urn:uuid:47df686f-c7da-4f99-a6c3-74d39f2b3d86>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00542.warc.gz"} |
Keeping track of paths - Shortest paths with Dijkstra's Algorithm
Shortest paths with Dijkstra's Algorithm
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Calculating paths, too
Let's run the algorithm again in our graph:
This time, however, let's keep track of the actual shortest paths. They all begin empty, except for the path of the initial node, which simply contains it:
path to A = empty
path to B = empty
path to C = C
path to D = empty
path to E = empty
The new thing is that we will update those paths every time we modify the minimum distance of a node.
Let's check the neighbours of our current node. Let's begin with B. We add 0 + 7 = 7. As that value is less than infinity, we change the minimum distance of B with it and replace the current path to
B with the path to the current node (path to C, which is C), plus B. This means that path to B = C, B.
We repeat the procedure with neighbours A and D. After that, our graph and paths are as follows:
path to A = C, A
path to B = C, B
path to C = C
path to D = C, D
path to E = empty
Our current node is now set to A. We check its only non-visited neighbour, B. As we replace the minimum distance of B from 7 to 4, we also replace its current path with the path of the current node A
(C, A), plus B: path to B = C, A, B).
path to A = C, A
path to B = C, A, B
path to C = C
path to D = C, D
path to E = empty
We mark A as visited and select our next current node: D. We check two neighbours: B and E.
When checking B, we don't replace its minimum distance (as the existing 4 is less than the calculated 7), so we don't replace its current path, either. Remember: we only replace a path when we modify
the minimum distance of a node.
We then check neighbour E, update its minimum distance (9, which is less than infinity) and path (path to E = C, D, E, which is the path to D plus E), and are left with this:
path to A = C, A
path to B = C, A, B
path to C = C
path to D = C, D
path to E = C, D, E
Let's fast-forward a bit: we continue applying the algorithm until we're done. After we finish, our graph and paths will be the following:
path to A = C, A
path to B = C, A, B
path to C = C
path to D = C, D
path to E = C, A, B, E
Congratulations! Those are the minimum paths between C and every other node!
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Is 4x = -6y a direct variation equation and if so, what is the constant of variation? | HIX Tutor
Is #4x = -6y# a direct variation equation and if so, what is the constant of variation?
Answer 1
constant of variation $= - \frac{6}{4} = - \frac{3}{2}$
if x varies directly as y. simply solve for x to find the constant of variation
from the given: #4x=-6y#
#(4x)/4=(-6y)/4# divide both sides of the equation by 4
#x=(-6y)/4# and the constant of variation is the numerical coefficient #-6/4#
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Answer 2
Yes, the equation (4x = -6y) represents a direct variation because it is in the form (y = kx), where (k) is the constant of variation. In this case, the constant of variation (k) can be found by
rearranging the equation to solve for (y), which gives us (y = -\frac{2}{3}x). Therefore, the constant of variation is (-\frac{2}{3}).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Why is the fibonacci spiral so important?
The Fibonacci spiral is one of the most fascinating mathematical sequences in history. It’s a sequence of numbers, starting with zero and one, that steadily increases as each number is the sum of the
two preceding numbers. But what’s even more remarkable is the role the Fibonacci spiral plays in the world of finance. Traders believe that the Fibonacci numbers and ratios created by the sequence
are key to successful trading. But how does this mysterious spiral affect the world, and what does it mean for us? In this blog post, we’ll explore the significance of the Fibonacci spiral, from its
spiritual meaning to its practical applications. We’ll look at why traders rely on the sequence, what it symbolizes in terms of life, and how its use can benefit us in our day to day lives. So, let’s
dive in and answer the question: why is the Fibonacci spiral so important?
Why is the Fibonacci spiral so important?
The Fibonacci spiral is a sequence of numbers that has captivated mathematicians and traders alike for centuries. It starts with zero and one, and each number is equal to the sum of the two preceding
numbers. The Fibonacci spiral is often used in technical analysis to identify trends and potential entry and exit points for traders.
The Fibonacci spiral has captivated mathematicians for centuries because of its unique properties and the way it can be used to explain many phenomena in nature. For traders, the Fibonacci spiral is
an important tool because it can be used to identify trends, support and resistance levels, and potential entry and exit points.
What is the Fibonacci Spiral?
The Fibonacci spiral is a sequence of numbers that starts with zero and one and then increases by adding the two preceding numbers. The sequence looks like this:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, …
This sequence is known as the Fibonacci sequence and it is named after the Italian mathematician, Leonardo Fibonacci. The Fibonacci sequence has many interesting properties, including the fact that
it can be used to describe the growth of plants and animals and other phenomena in nature.
How is the Fibonacci Spiral Used in Technical Analysis?
The Fibonacci spiral is often used in technical analysis to identify trends and potential entry and exit points for traders. Technical analysis is a method of analyzing the market using price charts
and other indicators. Traders use technical analysis to identify trends, support and resistance levels, and potential entry and exit points.
The Fibonacci spiral is used to identify support and resistance levels. Support and resistance levels are areas where the price of a security is likely to find support or resistance. For example, if
the price of a security is rising, it is likely to find resistance at the Fibonacci levels. Conversely, if the price of a security is falling, it is likely to find support at the Fibonacci levels.
The Fibonacci spiral is also used to identify potential entry and exit points. A trader can use the Fibonacci levels to identify a potential entry point if the price of a security is rising and a
potential exit point if the price is falling.
What are Fibonacci Ratios?
Fibonacci ratios are mathematical ratios derived from the Fibonacci sequence. The most common Fibonacci ratios are the 0.618 and 0.382 ratios. These ratios are often used in technical analysis to
identify potential entry and exit points. For example, if the price of a security is rising, a trader may use the 0.618 ratio to identify a potential entry point. Conversely, if the price of a
security is falling, a trader may use the 0.382 ratio to identify a potential exit point.
The Fibonacci spiral is an important tool that traders can use to identify trends, support and resistance levels, and potential entry and exit points. The Fibonacci spiral is derived from the
Fibonacci sequence, which has many interesting properties and can be used to describe the growth of plants and animals and other phenomena in nature. Fibonacci ratios are also derived from the
Fibonacci sequence and are often used in technical analysis to identify potential entry and exit points.
What is the spiritual meaning of the Fibonacci spiral?
The Fibonacci sequence, which is a series of numbers that starts with 0 and 1, and then continues in a sequence where the next number is the sum of the two preceding numbers, has been used for
centuries to explore the mysteries of the universe. In particular, the Fibonacci spiral has intrigued scientists, mathematicians, and spiritual seekers alike for its association with the divine. This
sequence is so astounding and important because it acts as a map for spiritual growth.
Discovering the Divine
The Fibonacci spiral is a visual representation of the Fibonacci sequence. It is a spiral pattern made up of squares with sides that are the length of Fibonacci numbers. When we look at this pattern,
we can see that it is made up of circles, and the circles within circles suggest infinite possibilities. This is symbolic of the divine, and how we can use this pattern to explore our own spiritual
Using the Spiral for Spiritual Growth
The Fibonacci spiral can be used to help us understand our spiritual journey. When we look at the spiral, we can see how it starts in the beginning, where we begin to understand that our actions have
reactions. As we move through the spiral, the circles become bigger and the sequence of numbers become more complex, representing our journey towards greater understanding. This can be seen as a
metaphor for our spiritual growth, as we learn more and more about ourselves and the divine.
Connecting to the Source
As we move through the Fibonacci spiral, we can see that it spirals inwards, eventually connecting to the source. This is a powerful symbol of how our spiritual journey eventually brings us back to
our source. In this way, the Fibonacci spiral can help us to understand our connection with the divine and how we can move towards a deeper understanding of our spiritual selves.
The Ultimate Goal
The ultimate goal of the Fibonacci spiral is to gain knowledge of the divine and to reconnect with the source. To gain the knowledge of where we disconnected and how to move on helps us to spiral
out. As we move through the spiral, we can gain insight into our spiritual journey and how we can move forward.
In summary, the Fibonacci spiral is a powerful tool for exploring the divine and understanding our spiritual journey. By looking at this pattern, we can gain insight into our own spiritual growth and
how we can reconnect with the source. Through understanding this pattern, we can gain a greater understanding of the divine and how we can move forward on our spiritual path.
How did Fibonacci affect the world?
The life and work of Leonardo of Pisa, better known today as Fibonacci, have had a lasting and profound effect on the world. His contributions to mathematics, commerce, and trade have been invaluable
and have revolutionized the way we do business today.
Fibonacci was a 13th-century Italian mathematician whose most famous work is the Liber Abaci, which introduced Europe to the Hindu-Arabic numeral system. Before Fibonacci, Europeans used Roman
numerals to do arithmetic, which was inefficient, time-consuming and fraught with potential errors. With the introduction of the Hindu-Arabic numeral system, calculations and transactions became much
easier and more efficient.
Fibonacci’s Contributions to Mathematics
Fibonacci’s contributions to mathematics are perhaps his greatest legacy. He is credited with introducing the world to sequences and series, which revolutionized the field of mathematics. In
particular, Fibonacci popularized the Fibonacci sequence, which is a sequence of numbers in which each number is the sum of the two preceding numbers. This sequence has been used for centuries to
solve a variety of mathematical problems and continues to be used today.
In addition, Fibonacci introduced the modern number system to Europe, which was a significant breakthrough. He also made important contributions to algebra, geometry, and trigonometry, and is
credited with introducing the golden ratio to the Western world. This ratio is a mathematical phenomenon in which two numbers have a ratio that is the same as the ratio of their sum to the larger of
the two numbers. This ratio is found in many aspects of nature, including the human body, and is often used in art and architecture.
Fibonacci’s Impact on Commerce and Trade
Fibonacci’s contributions to mathematics had a ripple effect that revolutionized the way commerce and trade were conducted. The Hindu-Arabic numeral system made it much easier to calculate prices and
transactions, which led to the development of modern accounting and bookkeeping practices. This, in turn, allowed for more accurate record-keeping and the development of banking.
The Fibonacci sequence has also been used in trading, as it can be used to identify trends and make predictions about future market movements. This has allowed traders to make more informed
decisions, leading to greater success in trading and investments.
Fibonacci’s contributions to mathematics, commerce, and trade were immense and have had a lasting effect on the world. His influence can still be seen today in many aspects of our lives, from
mathematics to banking. He revolutionized the way we do business and changed the way we think about mathematics, making it easier and more efficient.
The legacy of Fibonacci continues to live on, and his contributions will never be forgotten.
Where can be the Fibonacci spiral can be used in real life?
The Fibonacci spiral is one of the most important mathematical concepts in nature. It is a sequence of numbers where each number is the sum of the two numbers before it. This mathematical pattern is
seen in many naturally occurring shapes and forms. From flower petals to plant stems to hurricanes, there are many places where the Fibonacci spiral can be found in real life.
Flower Petals
One of the most common examples of the Fibonacci spiral can be seen in the petals of certain flowers. The number of petals in a flower is determined by the Fibonacci sequence, where each petal is the
result of the two petals before it. This is why you often see flowers with 5, 8, 13, or 21 petals. These numbers all fall within the Fibonacci sequence.
Seed Heads
The head of a flower is also subject to a Fibonaccian process. The seeds of a flower are arranged in a spiral pattern that follows the Fibonacci sequence. This is why you often see sunflower heads
with 45, 55, or 89 seeds. All of these numbers are found in the Fibonacci sequence.
Pinecones have long been known to be a great example of the Fibonacci spiral in nature. Pinecones are constructed in a spiral pattern where the seeds are arranged in a pattern that follows the
Fibonacci sequence. This is why pinecones often have 8, 13, or 21 spirals.
Fruits and Vegetables
Fruits and vegetables are also a great example of the Fibonacci sequence in nature. Many fruits and vegetables, such as apples, pears, oranges, and lemons, are constructed in a spiral pattern that
follows the Fibonacci sequence. This is why you often see these fruits and vegetables with 5, 8, or 13 sections.
Tree Branches
The branches of a tree are another great example of the Fibonacci sequence in nature. Trees often grow in a spiral pattern that follows the Fibonacci sequence. This is why you often see trees with 8,
13, or 21 branches.
Shells are a great example of the Fibonacci sequence in nature. Many shells are constructed in a spiral pattern that follows the Fibonacci sequence. This is why you often see shells with 8, 13, or 21
Spiral Galaxies
Spiral galaxies are another great example of the Fibonacci sequence in nature. Spiral galaxies are constructed in a spiral pattern that follows the Fibonacci sequence. This is why you often see
spiral galaxies with 8, 13, or 21 arms.
Hurricanes are a great example of the Fibonacci sequence in nature. Hurricanes are constructed in a spiral pattern that follows the Fibonacci sequence. This is why you often see hurricanes that have
8, 13, or 21 spirals.
The Fibonacci spiral is found in many places in nature. From flower petals to tree branches to spiral galaxies, the Fibonacci sequence can be seen in many naturally occurring shapes and forms.
Understanding the Fibonacci spiral can help us better understand the natural world and our place within it.
What does the spiral of life symbolize?
The spiral of life is an ancient spiritual symbol found in many cultures around the world. It represents the physical, mental, and spiritual development of a human life winding its way through the
rotating seasons of its years. The original artwork for this design is found in the famous Megalithic Passage Tomb at Newgrange in Ireland, which dates back to 3200 BC.
The spiral of life is a symbol of growth, transformation and the cycle of life. It is a reminder that life is constantly evolving and changing, and that we are all part of one continuous journey from
birth to death. It can also be seen as a symbol of eternity and the ever-present cycle of life.
The spiral of life symbol is often depicted as a single spiral or a double spiral, with the two spirals representing the duality of life. The single spiral is often associated with the feminine
energy and the double spiral with the masculine energy. In some cultures, it is seen as a symbol of harmony, representing the balance between male and female energies.
The spiral of life is also a reminder of the importance of the cycles of nature and how we are all connected to the earth and its elements. It represents the cyclical nature of life, from the seasons
and the cycles of the moon to the tides of the sea and the passing of time.
The spiral of life is also a reminder of our individual journey and how we are all part of the same cycle of life. It is a reminder that we all have a unique journey and that, no matter where we are
in life, we are all part of the same story. It is a reminder to stay connected to the natural cycles and rhythms of life and to appreciate the beauty and blessings of each moment.
The Spiral of Life and Ancient Wisdom
The spiral of life is a spiritual symbol found in many ancient cultures, from the Celts to the Native Americans. It is believed that the spiral was used as a symbol of spiritual and religious
beliefs. In ancient Celtic cultures, the spiral was seen as a symbol of the sun and the cycle of life. It was also seen as a symbol of the connection between the physical and spiritual realms.
In Native American cultures, the spiral of life was seen as a symbol of the four directions and the four winds. It was also seen as a symbol of unity and harmony, as the four directions symbolize the
interconnectedness of all things.
The spiral of life is also a symbol of wisdom. It is believed that the spiral of life is a reminder to seek knowledge and understanding of the world around us. It is a reminder to look for the hidden
truths in life and to seek out the answers to life’s questions.
The Meaning of the Spiral of Life
The spiral of life is a powerful symbol that can be used as a reminder of the beauty and mystery of life. It is a reminder of the cycles of life, the importance of balance and harmony, and the
interconnectedness of all things. It is a reminder to stay connected to nature and to appreciate the blessings of each moment. The spiral of life is a symbol of growth, transformation and the cycle
of life, reminding us that we are all part of one continuous journey from birth to death.
The spiral of life is a reminder to be open to growth and transformation. It is a reminder to stay connected to the natural cycles and rhythms of life and to appreciate the beauty and blessings of
each moment. It is a reminder to seek knowledge and understanding of the world around us and to look for the hidden truths in life.
Ultimately, the spiral of life is a reminder that life is constantly evolving and changing, and that we are all part of one continuous journey from birth to death. It is a reminder to stay connected
to nature and to appreciate the blessings of each moment.
The Fibonacci sequence and the ratios derived from it are important tools in financial analysis and trading. The Fibonacci sequence is a mathematical pattern that can provide traders with a
better understanding of the markets and help them make more informed decisions. By identifying and incorporating Fibonacci levels into their trading strategies, traders can maximize their chances
of success and make more profitable trades. The Fibonacci spiral is one of the most powerful tools available to traders and investors, so it’s important to understand how to apply it in order to
get the most out of it. With the right knowledge and practice, traders can use the Fibonacci spiral to make more informed trading decisions and to increase their potential for profits. An
understanding of the Fibonacci spiral and its application in trading can be the difference between success and failure.
Leave A Reply Cancel Reply | {"url":"https://whatfuture.net/why-is-the-fibonacci-spiral-so-important-9054/","timestamp":"2024-11-08T18:51:58Z","content_type":"text/html","content_length":"110367","record_id":"<urn:uuid:0533f3ed-bc52-4a86-898c-cfbeafe54096>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00389.warc.gz"} |
Analisis Pelanggaran Asumsi Klasik Dalam Model Permintaan Uang Jangka Panjang Indonesia Tahun 2000-2014
Long-run or desired demand variable for money is not directly observable. By using the stock adjustment hypothesis , the long-run demand for money model may be expressed as the short-run demand
function for money. The short-run demand for money model is essentialy nonlinear in the parameters an autoregressive in nature because the presence of stochastic explanatory variable. Therefore this
research is conducted to know the estimation problem of such model, because the classical least-squares may not be directly applicable to the model. Based on the formulation of the problem above, the
purpose of this research are as follows: 1. To determine whether the structural analysis that aims to measure and understand the magnitude of quantitative relations in economic variables
statistically significant in the model and in accordance with the theoretical expectations. 2. To determine whether there is a violation of classical assumptions multikolinearitas in short-run money
demand models. 3. To determine whether the presence of lag dependent variable of money demand on the right short-run money demand equation will result autocorrelation. 4. To determine whether the
error (residual) regression model estimation normal distribution. By using quarterly data ranging from 2000 to 2014 at constant prices 2000, having analyzed and evaluated several conclusions can be
drawn as follows. 1. Structural analysis that aims to measure and understand the magnitude of quantitative relations in economic variables in the short-run money demand models producing the correct
relationship corresponding theoretical expectations and significant. 2. From the analysis and evaluation of the value of the coefficient of determination R2, the correlation matrix of the independent
variables, the Pearson correlation coefficient, variance inflation factor (VIF) and the value of tolerance (TOL), partial regression, it can be concluded that the violation of the assumptions of
classical multikolinearitas in the model demand for short-run money can be ignored. 3. With the presence of variables lagMt in the short-run money demand that have stochastic nature on the right
short-run money demand equation, after analyzing the results of the DW test, Durbin-h test, test Langrange Multiplier (LM test) or test Breusch-Godfrey (BG test), and the test Run is not located
autocorrelation in money demand models. 4. By using the test charts, statistical tests such Zskewness and Zkurtosis and nonparameter test that is the Kolmogorov-Smirnov test can be stated that the
error (residual) estimation is a normal distribution. | {"url":"https://repository.uhn.ac.id/handle/123456789/2095","timestamp":"2024-11-07T17:29:59Z","content_type":"text/html","content_length":"18766","record_id":"<urn:uuid:c16733e6-47ca-4ee7-9d8f-94749ea57e9e>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00260.warc.gz"} |
The Fisher-Yates shuffle (named after Ronald Fisher and Frank Yates) is used to randomly permute given input (list). The permutations generated by this algorithm occur with the same probability.
Original method
The original version of the Fisher-Yates algorithm, which was published in 1938, was based upon iterative striking out of elements of the input list and writing them down to the second output list
(this approach was intended to be performed by a human with a paper and a pencil).
At first the user writes down the input list. At each step he randomly picks a number from interval k is a number of elements, which are not struck out yet. Then the user strikes out k-th unstruck
number and writes it down to the end of the output list. The user repeats this procedure until the input list contain some unstruck element.
Modern algorithm
The original procedure is simple and suitable for a human usage, but it is computationally inconvenient, because it has a quadratic asymptotic complexity. Hence in modern programs an improved
in-place version of the algorithm with linear
The idea of the modern version is analogous to the original procedure – random choice. The algorithm in its each step chooses a number n from interval k is a number of already processed elements
(number of already performed iterations) and m is the length of the input list. Then the algorithm swaps the element at index n (indexed starting at 1) with an element at index m-k. The algorithm
terminates after n-1 iterations (i.e. after possibly swapping all the input elements).
The better performance of the modern version is achieved by placing all the (possibly) swapped elements at the end of the currently processed part of the array. While the original procedure requires
an iteration to find the k-th unstruck number (
* Modern version of Fisher-Yates algorithm
* @param array array indexed starting at
function fisherYates(array)
for i in <array.length - 1, 1> do
index = random number from interval <0, i)
swap(array[i], array[index])
* An improved version (Durstenfeld) of the Fisher-Yates algorithm with O(n) time complexity
* Permutes the given array
* @param array array to be shuffled
public static void fisherYates(int[] array) {
Random r = new Random();
for (int i = array.length - 1; i > 0; i--) {
int index = r.nextInt(i);
int tmp = array[index];
array[index] = array[i];
array[i] = tmp;
* An improved version (Durstenfeld) of the Fisher-Yates algorithm with O(n) time complexity
* Permutes the given array
* @param array array to be shuffled
static void FisherYates(int[] array)
Random r = new Random();
for (int i = array.Length - 1; i > 0; i--)
int index = r.Next(i);
int tmp = array[index];
array[index] = array[i];
array[i] = tmp;
* An improved version (Durstenfeld) of the Fisher-Yates algorithm with O(n) time complexity
* Permutes the given array
* @param array array to be shuffled
function fisherYates(array) {
for (var i = array.length - 1; i > 0; i--) {
var index = Math.floor(Math.random() * i);
var tmp = array[index];
array[index] = array[i];
array[i] = tmp; | {"url":"https://www.programming-algorithms.net/article/43676/Fisher-Yates-shuffle","timestamp":"2024-11-09T13:03:38Z","content_type":"text/html","content_length":"28484","record_id":"<urn:uuid:f242ca8a-6dce-4d07-8f66-4ec9040531a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00409.warc.gz"} |
Calculus Complexities - Intermediate | Brilliant Math & Science Wiki
Calculus Complexities - Intermediate
Calculus is the mathematical study of change, through the ideas of infinite series. It is developed by working with the infinitesimal, which is a really really small quantity. From there, we develop
the idea of a Epsilon-Delta Definition of a Limit, which allows us to rigorously explain why a sequence or function tends to a particular unique value.
Here are some tips to get started:
1. Understand how to find the limit of a sequence.
2. Review the derivatives and integrals of basic functions.
3. Remember how to derive the most beautiful math equation \( e^{i \pi } + 1 = 0 \).
4. Understand the fundamental theorem of calculus.
5. If you are stuck, read the solutions to grasp these concepts better.
True or False?
\(i^ i \) is not a real number.
The statement is false. As it turns out, the value is real.
Recall that \( e ^ { i \pi } = -1 \). Substituting \( e ^ { i \pi / 2 } = i \) for the base, we obtain
\[ i ^ i = \left( e ^ { i \pi / 2 } \right) ^ i = e^{ i i \pi / 2 } = e ^ { - \pi / 2 } \approx 0.2078. \]
Note: Since we are working complex exponentiation, there are actually multiple values that \(i^i \) can take. In a similar manner to the above derivation, these values are all of the form \( e ^
{ - ( 2k+1) \pi / 2 } \). Likewise, these are real valued, and are approximately \( 0.2078 \times 2^{k \pi } \).
Calculus topics can be broadly classified as
• Differential calculus studies the Derivative of a function, which is defined through first principles. It measures the instantaneous rate of change of the function, which is obtained by taking
the limit of the secant as it gets closer to the point. It is concerned with Slope of a Curve, Related Rates of Change and Local Linear Approximation.
• Integral calculus studies the Anti-derivative of a function, which is defined as the inverse function to the derivative. The definite integral gives us the limit of the Riemann Sums, to find the
area under a curve. It is concerned with Area Between Curves and Volume of Revolution.
These concepts are related via the Fundamental Theorem of Calculus, which states that
If \( f\) is a continuous function on the interval \( [ a, b] \), and \(F \) is a function whose derivative is \(f\) on the interval \( (a, b) \), then we have
\[\begin{array} &\int_a^b f(t) \, dt = F(b) - F(a) &\text{ and } &\frac{ d}{dx} \int_a ^ x f(t) \, dt = f(x) \text{ for } x \in (a, b). \ _\square \end{array}\] | {"url":"https://brilliant.org/wiki/calculus-complexities-intermediate/","timestamp":"2024-11-05T15:47:14Z","content_type":"text/html","content_length":"45933","record_id":"<urn:uuid:7d181c97-dac4-4b8e-b189-122ceea3548c>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00651.warc.gz"} |
Key Structures in This Course
(with Spanish)
Student Journal Prompts
Opportunities for communication, in particular classroom discourse, are foundational to the problem-based structure of the IM K–5 Math curriculum. The National Council of Teachers of Mathematics’s
Principles and Standards for School Mathematics (NCTM, 2000) states, “Students who have opportunities, encouragement, and support for speaking, writing, reading, and listening in mathematics classes
reap dual benefits: they communicate to learn mathematics, and they learn to communicate mathematically.” Opportunities for each of these areas are intentionally embedded directly into the curriculum
materials through the student task structures and supported by the accompanying teacher directions.
One highly visible form of discourse is student discussion during the course of a lesson. Another, not as highly visible form of discourse is writing. While this is often only seen as the written
responses in a student workbook, journal writing can provide an additional opportunity to support each student in their learning of mathematics.
Writing can be a useful catalyst in learning mathematics because it not only supplies students with an opportunity to describe their feelings, thinking, and ideas clearly, but it also serves as a
means of communicating with other people (Baxter, Woodward, Olson, & Robyns, 2002; Liedtke & Sales, 2001; NCTM, 2000). NCTM (1989) suggests that writing about mathematics can help students clarify
their ideas and develop a deeper understanding of the mathematics at hand.
To encourage the use of journal-writing in math class, we have provided a list of journal prompts that can be used at any point in time during a unit and across the year. These prompts are divided
into two overarching categories: Reflections on Content and Reflection on Beliefs and Feelings.
Reflections on content focus on the students’ learning or specific learning objectives in each lesson. We first ask students to reflect on the mathematical content because, in general, the act of
writing requires a deliberate analysis that encourages an explicit association between current and new knowledge that becomes part of a deliberate web of meaning (Vygotsky, 1987). For example, when
students are asked to write about ways in which the math they learned in class that day was connected to something they knew from an earlier unit or grade, they are explicitly connecting their prior
and new understandings.
John Dewey asserted that students make sense of the world through metacognition, making connections between their lived experiences and knowledge base, and argued that education should provide
students with opportunities to make connections between school and their lived experiences in the world. This belief, alongside one of Ladson-Billings’ principles of CRT that states teachers must
help students effectively connect their culturally- and community-based knowledge to the learning experiences taking place in the classroom, supports the need for students to continually reflect not
only on the mathematics, but on their own beliefs and experiences as well. Reflections on beliefs are more metacognitive and focus on students’ feelings, mindset, and thinking around using
mathematics. Writing about these things promotes metacognitive frameworks that extend students’ reflection and analysis (Pugalee, 2001, 2004). For example, as students describe something they found
challenging during a lesson, they have the chance to reflect on the factors that made it a challenge.
Since the prompts, regardless of the category, can be used at any point during the year, they live in the Course Guide. We imagine these prompts could be used in a variety of ways. In the early
grades, they might be used as discussion prompts between partners or students may be asked to respond to a prompt in the form of a drawing or example from their work of the day. In later grades,
students can establish a math journal at the beginning of the year and record their reflections at the beginning, in the middle, or at the end of lesson, depending on the prompt. For schools or
districts who require homework, the prompts may serve as a nice way for students to reflect on their learning of the day or ask questions they may not have asked during the class period.
Journal writing not only encourages explicit connections between current and new knowledge and promotes metacognitive frameworks to extend ideas, but it also provides opportunities for teachers to
learn more about each student’s identity and math experiences. We believe that writing in mathematics can offer a means for teachers to forge connections with students who typically drift or run
rapidly away from mathematics and offer students the opportunity to continually relate mathematical ideas to their own lives (Baxter, Woodward, and Olson, 2005). Writing prompts and journaling work
well because students who may not advocate well for themselves when they are struggling get their voices heard in a different way, and thus their needs met (Miller, 1991).
It is our hope that through the use of these questions and prompts, students will communicate to learn mathematics as well as learn to communicate mathematically.
Reflecting on Content and Practices
• What math did you learn and do today that connected to something you knew from an earlier unit or grade?
• Describe a time you used the math you learned today outside of school.
• How did your thinking change about something in math today?
• How did any predictions you made in class today work out? Why do you think that happened?
• What questions do you still have about the math today? What new questions do you have?
• Describe the way you solved a problem in class today that you are proud of.
• Where do you see the math you did in class today outside of school? (MP4)
• What math tool did you find most helpful today? Why? (MP5)
• What patterns did you notice in the mathematics today? Why did that pattern happen? (MP7, MP8)
• Starter prompts:
□ The most important thing I learned today is . . .
□ Today, I struggled and worked through a problem when . . . (MP1)
□ I could use what I learned today in math in my life when I . . .
□ At the end of this unit, I want to be able to . . .
□ I knew one of my answers was right today when . . .
□ Another strategy I could have used to solve a problem today is . . .
□ The most important thing to remember when doing the problems like we did today is . . .
Reflecting on Learning and Feelings about Math
• Describe something you really understand well after today’s lesson or describe something that was confusing or challenging.
• In math class, it’s important to be able to explain your thinking. Describe a time when you were able to explain your ideas to other people in your class. (MP3)
• In math class, it’s important to listen to other people’s ideas. Describe a time when you learned something by listening carefully to someone in your class. (MP3)
• What does it mean to be good at math?
• Describe a time when you asked a question about math you were working on. How did asking a question help you?
• If you could change anything about math class, what would it be? Why?
• Starter prompts
□ I learned from a mistake today in math when . . .
□ When it comes to math, I find it difficult to . . .
□ I love math because . . .
□ I felt heard during class today when . . .
□ I felt my ideas were valued during class today when . . .
□ The most helpful thing that happened today was . . .
• Baxter, J. A., Woodward, J., & Olson, D. (2005). Writing in mathematics: An alternative form of communication for academically low-achieving students. Learning Disabilities Research & Practice,
20(2), 119–135.
• Baxter, J. A., Woodward, J., Olson, D. & Robyns, J. (2002). Blueprint for writing in middle school mathematics. Mathematics Teaching in the Middle School, 8 (1), 52–56.
• Liedtke, W. W. & Sales, J. (2001). Writing tasks that succeed. Mathematics Teaching in the Middle School, 6 (6), 350–55.
• Miller, L. (1991). Writing to learn mathematics. The Mathematics Teacher, 84(7), 516-521.
• Pugalee, D.K. (2001), Writing, mathematics and metacognition: Looking for connections through students’ work in mathematical problem solving. School Science and Mathematics, 101(5), 236–245.
• Pugalee, D. (2004). A comparison of verbal and written descriptions of students’ problem solving processes. Educational Studies in Mathematics, 55, 27-47.
• Vygotsky, L. S. (1987). Thinking and speech. In R. W. Rieber, & A. S. Carton (Eds.), The collected works of L. S. Vygotsky: Vol. 1. Problems of general psychology (pp. 39-285). New York: Plenum
Developing a Math Community
As stated in our design principles, within a problem-based curriculum “students learn mathematics by doing mathematics.” Given the nature of math classrooms, however, students come with differing
math identities, which means some students are more prone to see themselves as doers of mathematics than others. Furthermore, apparent inequities in math instruction suggest that some students have
opportunities to bring their voice into the classroom, and others do not. In order to extend the invitation to all students to do mathematics, we must work to explicitly develop the math learning
Classroom environments that foster a sense of community that allows students to express their mathematical ideas—together with norms that expect students to communicate their mathematical thinking to
their peers and teacher, both orally and in writing, using the language of mathematics—positively affect participation and engagement among all students (Principles to Action, NCTM).
To support teachers to develop math learning communities in their classrooms, the first six lessons of each course embed structures to collectively identify what it looks like and sounds like to do
math together, create, and reflect on classroom norms that support those actions.
Beyond the first six days, teachers should revisit these norms at least once a week to sustain the math learning community. Consistently returning to these ideas shows students that we value the math
learning community as much as we value the math content. Students should also be provided with opportunities to reflect on the norms by stating which ones are the most challenging for them and why.
Teacher reflection questions periodically remind teachers of points in a unit where it may be helpful to reflect on these norms.
Additional teaching moves can be used to support the development of math learning communities throughout the school year. The section below highlights teaching moves, put forth by Phil Daro and the
SERP Institute, that are intended to support students’ engagement in the mathematical practices. A solid math learning community exists when all students display these observable actions, called
student vital actions.
Teaching Moves to Support Math Community
│ student vital actions │ teacher moves │
│ │ • Assign rotating roles, and provide routines for collaboration so that every student is actively engaged in each task, and has experience in all roles │
│ │ over time. │
│All students participate. │ • When students are confused, ask them to show where they got lost or ask a question that can help them move forward (more than “I don’t get it” or “How │
│ │ do you do it?”). │
│ │ • Check to see if there are recognizable patterns between participation and prior achievement or social groups (for example, EL, race/ethnicity, or │
│ │ gender). │
│ │ • Ask and encourage students to ask: │
│ │ □ “Can you tell me more about that?” │
│ │ □ “Why do you think that?” │
│Students say a second sentence. │ □ “What changed and what stayed the same?” │
│ │ □ “Is that an answer that makes sense for this problem? How do you know?” │
│ │ □ “How did you get that answer? Why did you (reference student work)?” │
│ │ □ “Is it always true? Sometimes true?” │
│ │ • Show and discuss work generated by students when working with mathematics concepts. Questions that may be used to prompt students: │
│ │ □ “Did anyone approach the problem a different way?” │
│Students talk about each other’s │ □ “How is your thinking different from theirs?” │
│thinking. │ □ “What does their way of thinking help you understand?” │
│ │ □ “Do you think their method would work with this kind of problem? Why or why not?” │
│ │ • Try only responding to questions from groups when no one in the group can answer the question and everyone in the group can ask it. │
│ │ • If a student is presenting an explanation, play the role of not understanding and say, “Could you help me make sense of your thinking? Could you revise │
│Students revise their thinking. │ your explanation?” │
│ │ • Have a student quote a classmate’s statement that inspired them to revise. │
│ │ • Have students confer in small groups after whole-class presentations to revise and refine their way of thinking. │
│ │ • Ask a student who has given a wrong answer additional questions to explore their thinking. Demonstrate curiosity about that thinking. │
│Students engage and persevere. │ • Have students share their thinking and attempts even when they have not found a viable solution. │
│ │ • When some groups are “finished” earlier than others, ask them to analyze their work and seek places to revise their explanation so more students will │
│ │ understand it, or look for an alternative approach. │
│ │ • Before beginning small group work, give students sentence frames and probing questions that feature important terms. │
│Students use general and │ • Accept students’ everyday way of talking as a starting point for joining the math conversation. │
│discipline-specific academic language. │ • Teachers can refer to student statements using some student language while strategically incorporating more precise academic language with the addition │
│ │ of a key word or phrase. │
│ │ • For everyday words that have precise mathematical meaning, provide multiple contexts where the word is useful and have students explain what it refers │
│ │ to in that context. Ask them to use the word to make connections between the different representations. │
│English learners produce language. │ • Encourage students to use language to construct meaning from representations with prompts such as: │
│ │ □ “Explain where you see (length, ten, oranges) in the (figure, equation, table). How do you know it represents the same thing?” │
│ │ • Every student speaks, listens, reads, and writes. │
For more details and a full list of teaching moves, visit the SERP Institute site: https://www.serpinstitute.org/5x8-card/vital-student-actions
Professional Learning Community
Teaching mathematics is complex work. It requires teachers to plan lessons that offer each student access, elicit students’ ideas during these lessons, find ways in which to respond to those ideas,
and build a classroom community where students feel known, heard, and seen. Teachers must always be flexible and timely in decision-making in order to engage students in rich mathematical
discussions. Within each decision lies the opportunity to orient students to one another’s ideas and the mathematical goal, and position each student as a competent learner and doer of mathematics.
One of the biggest challenges to learning from the work of teaching is that the majority, if not all, of a teacher’s learning, planning, and decision-making happens in isolation.
Professional learning communities (PLCs), are spaces in which teachers can work together around planning and teaching. PLCs include any time teachers or coaches work collaboratively in recurring
cycles of collective inquiry and action research to achieve better results for the students they serve. Professional learning communities operate under the assumption that the key to improved
learning for students is continuous job-embedded learning for educators (DuFour, R., Dufour, R., Eaker, R., & Many, T, 2006).
To support teacher collaboration around planning and teaching, we have identified an activity in every unit section as a PLC activity. This activity was chosen because it is either a key mathematical
idea of the section or requires a more complex facilitation. We also organized a structure for teachers to use as they work together in professional learning communities.
The suggested structure is categorized as pre-, during-, and post-lesson to offer teachers the opportunities to experiment with instruction during both planning and the classroom enactment by
collectively discussing instructional decisions in the moment (Gibbons, Kazemi, Hintz, & Hartmann, 2017). These suggestions are meant to provide guidance for a professional learning community of
teachers and coaches that meet to plan for upcoming lessons. While using all of the suggestions in the given structure is ideal, they are flexible enough to adapt to fit any teacher’s given schedule
and context.
Suggested before a professional learning community meeting
• Read the upcoming lesson that is the focus of the meeting.
• Review student cool-downs from previous lessons.
• Discuss:
□ current student understandings
□ ways in which these understandings build toward the PLC activity
Suggested during a professional learning community meeting
• Do the math of the PLC activity individually.
• Read the CCSS and learning goal addressed by the activity.
• Discuss how the standard and learning goal are reflected in what the activity is asking students to do. Think about:
□ Are students conceptually explaining a new, or developing, understanding?
□ Are students making connections between a conceptual understanding and a procedure or process?
• Based on students’ previous lesson cool-downs, discuss 1–2 ways students might complete the activity.
• Discuss:
□ How might student responses reflect the CCSS and lesson learning goal?
□ What unfinished learning might students have?
• Based on these discussions, make a plan for:
□ look-fors as you monitor students during their work time
□ questions to ask that assess and advance student thinking
□ the sharing of work and student discussion during the activity synthesis
Suggested after a professional learning community meeting
• Record observations as students work.
• Review student cool-downs in relation to the learning goal of the lesson.
• DuFour, R., DuFour R., Eaker, R., & Many, T. (2006). Learning by doing: A handbook for professional learning communities at work. Bloomington, IN: Solution Tree.
• Gibbons, L. K., Kazemi, E., Hintz, A., Hartmann, E. (2017). Teacher time out: Educators learning together in and through practice. Journal of Mathematics Educational Leadership, 18(2), 28–46.
Representations in the Curriculum
“The power of a representation can . . . be described as its capacity, in the hands of a learner, to connect matters that, on the surface, seem quite separate. This is especially crucial in
mathematics” (Bruner, 1966).
Mathematical representations can be used for two main purposes: to help students develop an understanding of mathematical concepts and procedures or to help them solve problems. The materials make
thoughtful use of representations in both ways.
Curriculum representations and the grade levels at which they are used are determined by their usefulness for particular mathematical learning goals. Across lessons and units, students are
systematically introduced to representations and encouraged to use representations that make sense to them. As their learning progresses, students are given opportunities to make connections between
different representations and the concepts and procedures they represent. Over time, they will see and understand more efficient methods of representing and solving problems, which support the
development of procedural fluency.
In general, more concrete representations are introduced before those that are more abstract. There are a couple of key progressions of representations that occur across grade bands in different
These progressions, as well as the descriptions below, can be helpful in providing support for students who have unfinished learning and would benefit from more concrete representations to make sense
of mathematical concepts.
Two-color Counters (K–1)
Counters of one color are used frequently to represent quantities in the early grades. Students use the two-color counters to support their work in comparing, counting, combining, and decomposing
quantities. In later grades, the counters can be used to visually represent properties of operations.
Connecting Cubes (K–5)
Like counters, cubes can be used in the early grades for comparing, counting, combining, and decomposing numbers. In later grades, they are used to represent multiplication and division, and in grade
5, to study volume. Teachers of grade 5 should use cubes that connect on multiple sides to develop understanding of volume.
5-frame and 10-frame (K–2)
5- and 10-frames provide students with a way of seeing the numbers 5 and 10 as units and also combinations that make these units. Because we use a base-ten number system, it is critical for students
to have a robust mental representation of the numbers 5 and 10. Students learn that when the frame is full of ten individual counters, we have what we call a ten, and when we cannot fill another full
ten, the “extra” counters are ones, supporting a foundational understanding of the base-ten number system. The use of multiple 10-frames supports students in extending the base-ten number system to
larger numbers.
Connecting Cubes in Towers of 10 (1–2)
Cubes that are in towers of 10 support students in using place value structure for adding, subtracting, and comparing numbers. Connecting cubes have the advantage that students can physically compose
and decompose numbers, unlike place value blocks or Cuisenaire rods. The cubes are a helpful physical representation as students begin to unitize. For example, students can understand that 10 of the
single cubes are the same as 1 ten and 10 of the tens are the same as 1 hundred.
Base-ten Blocks (2–5)
Base-ten blocks are used after students have had the physical experience of composing and decomposing towers of 10 cubes. The blocks offer students a way to physically represent concepts of place
value and operations of whole numbers and decimals. Because the blocks cannot be broken apart, as the connecting cube towers can, students must focus on the unit. As students regroup, or trade, the
blocks, they are able to develop a visual representation of the algorithms. The size relationships among the place value blocks and the continuous nature of the larger blocks allow students to
investigate number concepts more deeply. The blocks are used to represent whole numbers and, in grades 4 and 5, decimals, by defining different size blocks as the whole.
Base-ten Diagram (1–5)
Base-ten diagrams offer students a way to represent base-ten blocks after they no longer need concrete representations. Although individual units might be shown, the advantage of place value diagrams
is that they can serve as a “quick sketch” of representing numbers and operations.
Tape Diagram (2–5)
Tape diagrams, resembling a segment of tape, are primarily used to represent the operations of adding, subtracting, multiplying, and dividing. Students use them first with whole numbers and later
with fractions and decimal numbers to emphasize the idea that the meaning and properties of operations are true as the number system expands. They can help students represent problems, visualize
relationships between quantities, and solve mathematical problems.
Number Line Diagram (2–5)
Number line diagrams are used to represent and compare numbers, and can also be used to represent operations. Understanding of number line diagrams is built on students’ grade 2 experience with
rulers. Students begin by working with number lines with tick marks to represent the whole numbers. Then, they work with number lines where tick marks correspond to multiples of 10, 100, or 1,000 to
develop an understanding of place value and relative magnitude. In later grades, students understand that there are numbers between the whole numbers. They extend their work with whole number
operations on the number line to include fractions and decimals.
Fraction Strips (3–4)
Fraction strips are rectangular pieces of paper or cardboard used to represent different parts of the same whole. They help students concretely visualize and explore fraction relationships. As
students partition the same whole into different-size parts, they develop a sense for the relative size of fractions and for equivalence. Experience with fraction strips facilitates students’
understanding of fractions on the number line.
Array (2–3)
An array is an arrangement of objects or images in rows and columns that can be used to represent multiplication and division. Each column must contain the same number of objects as the other
columns, and each row must have the same number of objects as the other rows.
Inch Tiles (2–4)
Inch tiles offer students a way to create physical representations of flat figures that have a certain area and to cover a flat figure with square units to determine its area. Students organize inch
tiles into rows and columns to connect the area of rectangles to multiplication and division.
Area Diagram (3–5)
An area diagram is a rectangular diagram that can be used to represent multiplication and division of whole numbers, fractions, and decimals. The area diagram may be overlaid with a grid to show
individual units. As students move from working with an area diagram overlaid with a grid to one without, they move from a more concrete understanding of area to a more abstract one. In an area
diagram without a grid, the unit squares are not explicitly represented, which makes this diagram useful when working with larger numbers or fractions and making connections to the distributive
property and algorithms.
As the numbers in products become larger, area diagrams are difficult to read if the ones, tens, and eventually hundreds are shown accurately. This diagram shows a way to visualize the product \(53 \
times 31\).
It shows how to decompose the product into 4 parts, represented in the diagram as smaller rectangles. The size of each smaller rectangle in the diagram does not represent its actual size since the
segment labeled 30 is not 30 times as long as the segment labeled 1. Even though the small rectangles do not have the correct relative size, the diagram can still be used to correctly decompose the
product \(53 \times 31\),
\(\displaystyle 53 \times 31 = (50 \times 30) + (50 \times 1) +(3 \times 30) + (3 \times 1)\)
The diagram helps visualize geometrically why the equation is true.
• Bruner, J. (1966). Toward a theory of instruction. Cambridge, MA: Harvard University Press. | {"url":"https://im.kendallhunt.com/K5_ES/teachers/teacher-guide/key-structures-in-this-course.html","timestamp":"2024-11-05T21:38:28Z","content_type":"text/html","content_length":"169853","record_id":"<urn:uuid:668fce22-38c5-41a2-a32e-90282358a6c3>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00462.warc.gz"} |
Mathematical Induction
Mathematical Induction
Accelerated Math 3 Ex. 1 Write the 1st 5 terms of the sequence. a1 = 3 an = 2an-1 + 5 OK Let s get with a partner and work on Mathematical Induction WS2 #7-10. – PowerPoint PPT presentation
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Transcript and Presenter's Notes
Title: Mathematical Induction 1
Mathematical Induction Continued
Ex. 1 Write the 1st 5 terms of the sequence.
OKLets get with a partner and work on
Mathematical Induction WS2 7-10.
• First pair to prove the formula, gets candy! | {"url":"https://www.powershow.com/view4/74d978-NmUzN/Mathematical_Induction_powerpoint_ppt_presentation","timestamp":"2024-11-07T07:20:54Z","content_type":"application/xhtml+xml","content_length":"179020","record_id":"<urn:uuid:2e7737be-c161-4266-8244-d64d1e045bb8>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00882.warc.gz"} |
Getting a Handle on Alfven Waves (1)
Alfven waves are the most important waves propagating in the solar atmosphere, as well as the Earth’s magnetosphere (underpinning the coupling between it and the ionosphere). They are important in
that they efficiently carry energy and momentum along the magnetic field.
One way to get a handle (of sorts) on Alfven waves is to look at the analogy with mechanical waves – say propagating along a string put under tension. Consider the reference frame or coordinate
^ y
Say x marks the direction of propagation in the above coordinate system, and y is the direction of transverse (wave) displacement. Then the vertical force component is:
= - T
(¶ y/ ¶ x)
where T is the tension. Thus, just as the restoring force for a mechanical wave is the string tension T, the restoring force for an Alfven wave is the magnetic tension. This magnetic version of
“tension” accelerates the plasma and is opposed by the inertia of the ions (mainly from proton masses m(p))
Now, the wave speed on a string is related to u (mass per unit length), and T such that:
v = (T/ u)
and as we can see, increasing the string tension increases the wave speed in an analogous way to what magnetic tension does for the Alfven wave. The magnetic tension analog can be expressed (as we
shall see) as:
T(M) = B
m [o]
where B is the magnetic induction and
is the magnetic permeability for free space (
4π x 10
In what follows we assume a uniform plasma in equilibrium, which will then be subjected to velocity disturbance or perturbation that affects all other key quantities. The treatment is kept as simple
as possible (considering the complexity of the subject matter!) , and we don’t veer out of the linear domain. Nevertheless it should be stated at the outset that some details are omitted, or left as
work for yourself with hints provided. In this way you will better understand and appreciate the genesis of Alfven waves. In terms of symbols, all have retained their earlier meanings (from previous
questions) and this includes the vector operators, DIV, grad, Curl etc.
Examining the origin of these waves always starts with setting out the basic equations for what we call “ideal MHD”:
¶ r / ¶ t
= - DIV (
¶ t
= - DIV (
vv) – grad p + 1/
m [o]
Curl B
X B
¶ B
t =
v X B
t = - v. grad p –
p DIV v
where the partial derivative symbols (@) are as before, v is the fluid velocity, p the pressure,
the magnetic induction, and ‘gamma’ = - d ln p/ d ln V where V denotes volume
Now, introduce small perturbed quantities (e.g. imagine introducing a small perturbation into the plasma velocity such that v
-> v
[l ]
, which will also subject the mass density, fluid pressure and magnetic field to perturbation), such that:
r [o ]
r [l]
v = v
[l ]
B = B
+ B
[l ]
p = p
+ p
Now, substitute these back into the original ideal MHD equations to obtain:
¶ r [l] / ¶ t
= -
r [o]
DIV v
r [o ]
t) = - grad p
+ 1/
m [o ]
Curl B [l]
X B [o ]
¶ B [l]
t =
v [l] X B [o ])
¶ p [l]
t = -
DIV v
Now, divide through the 2nd equation above by the mass density
¶ v [l]
t = -(c
[s] ^2
) grad
p [l]
- 1/
m [o ] r
B [o] X Curl B [1]
where ‘c
’ is the sound speed. (Note that the reader should be familiar with a vector identity also used to obtain the preceding!)
Now, using this result and the last two equations of the perturbed set, we apply Fourier transforms such for
t and
k to obtain:
– c
[s] ^2
[x ]
)k* +
B [o ]
m [o ] r [o]
k X k*
X (
v [l ]
B [o]
)] = 0
denotes the plasma frequency, k is the wave number vector (k* the vector orientation) and the other quantities are as before. In the next instalment we’ll obtain the x and y components of the
velocity but readers can try in the meantime to do it themselves! | {"url":"https://brane-space.blogspot.com/2011/04/getting-handle-on-alfven-waves-1.html","timestamp":"2024-11-01T18:58:06Z","content_type":"text/html","content_length":"140966","record_id":"<urn:uuid:20bbf0fd-846e-4848-a39d-18bf8ac8d1ea>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00313.warc.gz"} |
2024-11-10T22:02:54Z https://ir.soken.ac.jp/oai oai:ir.soken.ac.jp:00000201 2023-06-20T14:59:20Z 2:427:9 Accurate ab initio theoretical studies of rovibronic states of some simple diatomic molecules
Accurate ab initio theoretical studies of rovibronic states of some simple diatomic molecules 岡田, 一俊 7778 オカダ, カズトシ 7779 OKADA, Kazutoshi 7780 総合研究大学院大学博士（理学） Fundamental
diatomic molecules and their rovibronic and tonic states have been paid an interest both experimentally and theoretically for so many years. Experimentally, there have been so many rotationally
resolved absorption and emission spectra of diatomic molecule using grating, laser and so on. There have also been many spectra for the ionic states using photoelectron light source, and recent
development in the photoelectron spectroscopy such as ZEKE, or PFI-PE, rotationally resolved spectra have been observed for those ionic states. There are also many theoretical studies using ab initio
molecular orbital method. Fundamental diatomic molecules and their electronic excited states have been studied by using the configuration interaction method for a few decades. Recent development of
molecular orbital theory and fast computer has made it possible to calculate the vibrational and rotational levels as well as electronic states more accurately. It is now possible to reproduce
experimental data and the agreement between experimental data and theoretical results is very excellent and quantitative comparison of other properties is also possible. Theoretical calculations can
also give very good prediction to for the states with no experimental data available. As both experiments and calculations become more accurate, calculated results with less accuracy become
insufficient, and more accurate calculations are expected, but there are still few accurate calculation enough to explain experimental data quantitatively. In the present theoretical studies,
accurate calculations were performed for several diatomic molecules and their ion, and their rovibrational levels. In their accurate calculations, very large basis set, such as augmented quadruple
zeta basis set, is used. Multireference configuration interaction (MRCI) calculations were performed for several electronic states of diatomic molecule. To describe properly the anti-bonding nature
of molecular orbitals, valence-type-vacant(VALVAC) orbital method is used. The method requires only a single Fock matrix generation, and provides them with a proper anti-bonding nature of molecular
orbitals. These orbitals are used as a reference space in the MRCI. In this method, they do not need to solve the state-averaged MCSCF. With a single set of molecular orbitals obtained by VALVAC
method, the accurate potential energy and dipole moment functions both for the ground state and for the<br />excited states are obtained. All the calculated results are compared with recent
experimental data, which are in excellent agreement. The topics and their summaries of the results are as follows. (1) Accurate potential energy and transition dipole moment curves for several
electronic states of CO+. Ab initio MO studies are performed for several doublet and quartet states of CO+ using the multi-reference Configuration interaction method. The neutral ground state of CO
is also calculated. The following properties are compared with available experimental data. (i) Low-lying electronic states, their rovibrational levels of each state up to the dissociation limit, and
spectroscopic constants. Adiabatic potential energy curves of several doublet and quartet states are calculated, and the vibrational levels are calculated using the potential energy curves.
Spectroscopic constants, such as Re, ωe and ωexe are obtained. For example, for the X2Σ+ state, the calculated Re, ωe and ωexe are 1.1151 Å, 2214.6 cm-1, and 14.75 cm-1. Corresponding recent
experimental values of Re, ωe and ωexe are 1.119 Å, 2215.1 cm-1, and 15.27 cm-1. Calculated spectroscopic constants well reproduce the recent experimental data. (ii) ν dependence of rotational
constant Bv. The rotational constant Bv is obtained for each vibrational level. Rotational constants such as Be, and αe are calculated. The calculated Be and αe are 1.981 cm-1, and 0.0234 cm-1.
Corresponding experimental values are 1.9798 cm-1 and 0.0202 cm-1. Calculated rotational spectroscopic constants also well reproduce experimental data. (iii) The transition dipole moment functions
and the lifetimes of the vibrational levels. Transition dipole moment functions between the electronic states are also calculated. The lifetimes of the vibrational levels are evaluated by obtaining
the Einstein's A coefficients, and the lifetimes are compared with experimental data. Calculated lifetime of the vibrational level ν=0 of the B2Σ+ state of CO+ is 56.40 ns. Corresponding experimental
value is 57.1 ns. This agreement implies that the calculated Einstein's A coefficients to the lower electronic state are accurate. (2) Ab initio studies of several excited states of CO+. The
adiabatic potential energy curves of the X2Σ+, A2II, B2Σ+, C2Δ, D(2)2II and 32II states of CO+ are calculated, and the vibrational levels of each state and spectroscopic constants are obtained. The
vibrational levels of the D(2)2II and 3 2II states are particularly focused on. Adiabatic potential energy curves of the D(2)2II state shows that there is an avoided crossing between the D(2)2II and
the 3 2II states. The splitting is about 1200 cm-1. Calculated vibrational levels using adiabatic potential energy curve show that here are only 3 vibrational levels. However, experimental data shows
a vibrational progression of the D(2)2II state; the progression reaches up to ν=9. Experimentally obtained ν=9 level lies well above the barrier of the adiabatic potential energy curve or the D(2)2II
and the 3 2II state, too. Thus, it the experimental assignment to the vibrational progression is correct, the vibrational levels above ν=2 have to be on the diabatic potential energy curve. The
experimental data shows that the diabatic represention of the states is a good approximation to describe the rovibrational levels. To confirm it, the spectral intensities and their bandwidths of the
vibrational levels on the diabatic potential energy curve are calculated and compared with experimental data. (3) Accurate potential energy and transition dipole moment curves for several electronic
states of N2+. Ab initio MO studies are performed for several doublet and quartet states of N2+ using the multireference configuration interaction method. The following properties are compared with
available experimental data. Spectroscopic constants, such as Re, ωe and ωexe are obtained. For the X2Σg+ state, the calculated Re, ωe and ωexe are 1.119 Å, 2212.3 cm-1, Blld 16.87 cm-1.
Corresponding recent experimental values of Re, ωe and ωexe are 1.11642 Å, 2207.0 cm-1, and 16.10 cm-1. Also in this case, the agreement between the calculated results and experimental data is
excellent. (3) Rovibrational studies the neutral CO and investigations of rotational temperature of CO in the sun. Rovibrational levels and their spectral intensities for absorption up to ν=9 and J=
150 are calculated for the X1Σ+ state of CO. Calculated spectral intensities are compared with experimental data observed from the sun by satellite. A few different rotational temperatures are
assumed to calculate the spectra, because the spectral intensities depend on rotational temperature. Calculated spectral intensities are compared with experimental data. Using the calculated result
with known rotational temperature, rotational temperature of CO in the sun is estimated to be about 5000 K. application/pdf 総研大甲第446号 thesis 2000-03-24 AM application/pdf application/pdf https:
//ir.soken.ac.jp/record/201/files/甲446_要旨.pdf https://ir.soken.ac.jp/record/201/files/甲446_本文.pdf https://ir.soken.ac.jp/records/201 eng | {"url":"https://ir.soken.ac.jp/oai?verb=GetRecord&metadataPrefix=oai_dc&identifier=oai:ir.soken.ac.jp:00000201","timestamp":"2024-11-10T22:02:55Z","content_type":"application/xml","content_length":"11297","record_id":"<urn:uuid:e6350ee1-7d26-458b-ae1d-faf5e561a7e9>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00851.warc.gz"} |
Basics Of Second Order Differential Equation | Mini Physics - Free Physics Notes
Basics Of Second Order Differential Equation
A second order differential equation is of the form $a \frac{d^{2}y}{dx^{2}} + b \frac{dy}{dx} + b \frac{dy}{dx} + cy = f(x)$.
$f(x)$ is called a source term or forcing function.
The differential equation is called
• a homogeneous equation IF $f(x) = 0$
• non-homogeneous IF $f(x)$ is not 0.
The steps involved in solving a homogeneous equation and non-homogeneous are quite similar, with the non-homogeneous one requiring more work. (More about this later)
Steps to solving a homogeneous equation:
1. Rewrite the given differential equation $a \frac{d^{2}y}{dx^{2}} + b \frac{dy}{dx} + b \frac{dy}{dx} + cy = f(x)$ as $(aD^2 + bD + c)y = 0$.
2. Substitute m for D and solve the auxiliary equation $am^2 + bm + c = 0$
1. If the roots of the auxiliary equation are real and different, e.g. $m = \alpha$ and $m = \beta$, then the general solution is: $$y = A e^{\alpha x} + B e^{\beta x}$$
2. If the roots are real and equal, e.g. $m = \alpha$, twice, then the general solution is $$y = (Ax + B)e^{\alpha x}$$.
3. If the roots are complex, e.g. $m = \alpha \pm \beta i$, then the general solution is $$y = e^{\alpha x} ( C \cos \beta x + D \sin \beta x)$$
4. If the particular solution of a differential equation is required then substitute the given boundary conditions to find the unknown constants.
The general solution for a non-homogeneous second order differential equation is given by y = complementary function + particular integral, which is y = u + v.
By following the four steps above, the complementary function for the non-homogeneous differential equation is found.
Note: Treat f(x) as 0 when solving for the complementary function.
Finding the particular integral
There are no hard and fast rule in finding the particular integral. (It involves some guessing) I will show you a few examples below.
Example 1: Solve $\frac{d^{2}y}{dx^2} – 4 \frac{dy}{dx} + 4y = 4x + 3cos 2x$
$$(D^{2} – 4D + 4)y = 0$$
The auxiliary equation is $m^{2} – 4m + 4 = 0$
$$(m – 2)(m-2) = 0$$ $$m = 2$$
The complementary function is $u = (Ax + B)e^{2x}$.
Make a guess! Let the particular integral be $ v = ax + b + C \cos 2x + D \sin 2x$
Note: Normally, you would try the most general form of the source term. E.g. The source term contains x, hence you make a guess that the final solution must contain the most general form of x, i.e.
ax + b. The most general form of cos x is sin x and cos x.
$$(D^{2} – 4D + 4)v = 4x + 3 \cos 2x$$ $$D(v) = a – 2C \sin 2x + 2D \cos 2x$$ $$D^{2}(v) = -4C \cos 2x – 4D \sin 2x$$
Substitute D(v) and $D^{2}(v)$ into $(D^{2} – 4D + 4)v = 4x + 3 \cos 2x$
$$4ax – 4a + 4b + (-4C – 8D + 4C)\cos 2x + (-4D + 8C + 4D)\sin 2x = 4x + 3\cos 2x$$
By comparing the coefficients,
a = 1, b = 1, C = 0, $D = -\frac{3}{8}$
Hence, the particular integral is $v = x + 1 – \frac{3}{8} \sin 2x$
The general solution is:
$$y = (Ax + B)e^{2x} + x + 1 – \frac{3}{8} \sin 2x$$
The “guess” is called an Ansatz.
In physics and mathematics, an ansatz is an educated guess that is verified later by its results.
-From Wikipedia
This is the end of the basic walkthrough for Second Order Differential Equation.
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Conditional Summing Tricks - Excel University
Conditional Summing Tricks
Hello, Excel enthusiasts! Welcome to another learning-filled blog post where we deepen our understanding of Excel. Today, we will dive into some phenomenal conditional summing tricks resulting from a
question I recently received: Can you create a formula that takes into account only columns that contain a certain word? For example, is it possible to create a sum of the “amounts” marked as open?
Let’s get to it!
Detailed Step-by-Step Walkthrough
We’ll walk through three exercises to illustrate different capabilities of the SUMIFS function.
Exercise 1: SUMIFS Basics
Firstly, let’s start with a basic case. We have a bunch of data transactions like this:
We want to sum amounts marked as Open. We’ll use the SUMIFS function for this.
Note: SUMIF could also be used here as there is only a single condition. However, in practice, I prefer to stick with SUMIFS consistently because down the road it is easy to add an additional
condition if needed. Plus, the order of the arguments are reversed between the functions so it is easier for me personally to just stick with SUMIFS.
The first argument of the SUMIFS function is the range of numbers we want to add. The next two arguments define the condition. It is the criteria range followed by the criteria value.
So, we can use the following formula to create a sum of all Open transactions:
=SUMIFS(C12:C20, D12:D20, "Open")
Note: if we had entered the value Open into a cell, such as B7, we could use the cell reference rather than typing the criteria value manually into the formula, like this:
=SUMIFS(C12:C20, D12:D20, B7)
So, that is how we can use the SUMIFS function to sum the amounts for the Open transactions. But, what if we wanted to include both Open and Pending transactions? Well, let’s tackle that in the next
Exercise 2: OR Logic
What if we have multiple conditions, say, we want to sum amounts marked as Open or Pending? When we provide multiple conditions in the SUMIFS function, AND logic is used. This just means that all
conditions must be true for that row to be included in the sum. So, if we attempted to sum Open or Pending values with the following formula, it would return 0:
=SUMIFS(C12:C20, D12:D20, "Open", D12:D20, "Pending")
It returns 0 because AND logic is applied, and D12:D20 can not be both Open and Pending.
So, what are we supposed to do? Do it manually? No worries, we can simulate OR logic by simply adding the results of multiple SUMIFS functions. For each unique condition (Open or Pending), we use a
separate SUMIFS function and add them with the addition operator + like this:
=SUMIFS(C12:C20, D12:D20, "Open") + SUMIFS(C12:C20, D12:D20, "Pending")
With that, let’s move to our final exercise.
Exercise 3: Wildcard
Say we have a range of data with subtotals, and it looks something like this:
We would like to add all of the rows that begin with the word Subtotal. We can use SUMIFS and its ability to do a partial match. To accomplish this, we use the wildcard asterisk * like this:
=SUMIFS(D10:D21, B10:B21, "Subtotal*")
This tells the function to add up the column D values, and include only those rows where the value in column B begins with the word Subtotal.
If on the other hand you wanted to match cells that end in Subtotal, change the formula to this:
=SUMIFS(D10:D21, B10:B21, "*Subtotal")
And if you want to match cells that contain the word Subtotal change the formula to this:
=SUMIFS(D10:D21, B10:B21, "*Subtotal*")
And that is how you do a partial match.
Summary of the Process
Leveraging the SUMIFS function in Excel can make conditional summing a breeze! We’ve seen how to sum the amount column values based on the status column’s content, used multiple SUMIFS for OR logic,
and even included wildcards to perform a partial match.
If you have any alternatives, questions, or enhancements, please share by posting a comment below!
File Download
Want to put your learning into practice? Download our exercise Excel file and test your new knowledge!
Q: Can I use SUMIFS function for multiple conditions?
A: Yes, you can use multiple conditions with the SUMIFS function. To do so, add a pair of arguments for each condition. First the criteria range, and then the criteria value.
Q: What logic does the SUMIFS function use?
A: The SUMIFS function uses AND logic by default, meaning all conditions must be true for the row to be included in the total.
Q: How do I use OR logic in SUMIFS?
A: To use OR logic in SUMIFS, break your formula into separate SUMIFS functions for each condition and add them with the addition operator +.
Q: How do I include rows that contain a specific word, even if there are other characters before or after the word?
A: Use a wildcard (asterisk) before and after the criteria value argument like this: “*Subtotal*”
Q: Can the SUMIFS function be used to sum based on text conditions?
A: Yes, the SUMIFS function works for both numeric and text conditions.
Q: Can I add up multiple columns with the same condition?
A: Yes, use separate SUMIFS functions for each column.
Q: Can I use more than one wildcard in a SUMIFS function?
A: Yes, you can use as many wildcards as necessary.
Q: Do I have to manually type in the condition into the SUMIFS function?
A: No, you can also point the function to a cell containing the condition.
Q: Is there a limit on how many conditions SUMIFS function can handle?
A: Technically, Excel allows up to 127 conditions. However, simpler and fewer conditions are recommended for readability and performance reasons.
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1 Comment
1. When considering OR logic as a sum of two or more IFS formulas, it is critical that the conditions be mutually exclusive — i.e. that no rows can satisfy more than one of the conditions — because
that would lead to double counting. Conditions that are not mutually exclusive will require other techniques like the tried and true use of SUMPRODUCT with binary logic.
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The MAIN OBJECTIVES OF THE RESEARCH PROGRAMME are: to train, by research and by example, 15 Early Stage Researchers in the field of UQ and Optimisation and to become leading independent researchers
and entrepreneurs that will increase the innovation capacity of the EU, and to equip them with the skills needed for successful careers in academia and industry, to develop, through the ESR’s
individual projects, fundamental mathematical methods and algorithms to bridge the gap between Uncertainty Quantification and Optimisation and between Probability Theory and Imprecise Probability
Theory for Uncertainty Quantification, and to efficiently solve high-dimensional, expensive and complex engineering problems.
The research programme is divided into three main Work Packages (WP) covering three fundamental areas of research underpinning the OUU of any system or process: WP1-Modelling, Simulation and VP, WP2-
Uncertainty Quantification, WP3-Optimisation Under Uncertainty. Figure 1.1 gives an overview of the logic and interrelations of the research programme. Each main WP is further broken down into a
number of specific sub- WPs (see Table 1.1) on key topics, each with a set of high level research objectives. The overall research objective of WP1 is to define a set of representative reference case
studies to support the development and benchmarking of the UQ and OUU techniques developed in the other WPs. The research objective of WP2 is to develop techniques for the efficient treatment of
uncertainty of different nature while the objective of WP3 is to integrate optimisation and uncertainty quantification.
The research programme has four major innovative aspects: 1- the use of Imprecise Probability Theories for Uncertainty Quantification and Optimisation Under Uncertainty, 2- the integration of UQ and
optimisation for large scale expensive problems, 3- the introduction of evolvable OUU and 4- the development of distributed computing techniques for multidisciplinary design in a peer to peer
architecture, where delay and corruption of information are simulated, and expert judgments and subject probabilities in the design process are introduced. The last aspect is particularly important
when trying to optimise long term processes normally divided in multiple phases or stages.
In the following, the state-of- the-art and innovative contributions will be presented per sub-WP.
Objective: To study and implement new collaborative technologies applicable to complex multidisciplinary, multi-phase design processes over a network of multiple, geographically distributed,
organisations using shared computing resources on HPCs or clouds.
The coordination between interdisciplinary groups working on the optimisation of complex systems and processes, particularly in the area of multidisciplinary engineering design, is an important open
research topic with direct industrial applications. To respond to industry, optimisation need to handle multiple objectives and constraints defined in different disciplines and efficiently integrate
CAD/CAE software tools, collaborative optimisation platforms such ESTECO enterprise suite have been recently developed. For large-scale industrial multidisciplinary design optimisation, the different
disciplines can be defined by different, wide-spread groups of people, making collaborative optimisation a real challenge from a computation point of view. MBSE is an emerging branch of system
engineering, which addresses the problems associated with the management of complex systems. It aims at formalising the application of modelling to support system requirements, design, analysis,
verification/validation activities starting from the conceptual design phase and continuing through to the later life cycle phases of a project. The aerospace sector represents an ideal application
field due to the high system complexity involved. The innovative aspect of WP1.1 is the integration of the MBSE framework with state-of-the-art distributed MDO platforms for collaborative
optimisation, uncertainty treatment in multi-phase design processes over complex and distributed, peer-to-peer networks sharing software and computing resources, models, data, knowledge base on HPCs,
grid or clouds in order to improve process and product reliability and enhance efficiency through design automation. More importantly WP1.1 will introduce a model of the process itself, with the
associated uncertainty that will be studied in WP3.6. The expected result will be a paradigm for robust system architectures that codifies the end-product life-cycle from the earliest design phases.
Lead Partner – ESTECO
ESRs 1, 7
WP1.2 MULTI-FIDELITY MODELLING
Objective: To define and develop set of multi-fidelity models covering key applications in space and aerospace.
Seven specific aerospace applications will be used to assess the suitability of the approaches developed in UTOPIAE: I) design of an anti-icing system, II) re-entry of space vehicles and debris,
III) energy-driven design of civil airplanes, IV) multi-sensor tracking of objects during re-entry, V) ATM with Federated Satellite Systems, VI) morphing of rotor blades and VII) end-to-end design of
space systems.
In-flight icing modelling is affected by a number of uncertainties, both epistemic and aleatoric, that currently prevent its application to the design and the certification phases of fixed- and
rotary-wing aircraft; existing processes require costly experimental verification and/or in-flight testing. The air transport industry is facing very hard challenges to accomplish the objectives of
Horizon 2020/2050 programs. In particular, the reduction of CO2 emissions is a very strong motivation for an energy-driven systemic approach to the design of the aircraft and operations. In general,
greener and more efficient technologies and overall design and manufacturing processes are required for future aerospace vehicles. The modelling of morphing structures is affected by diverse
epistemic and aleatoric uncertainties due to the complex interaction of the morphing structure with the overall aircraft structure and aerodynamics. The prediction of the re-entry trajectory and
footprint of a space object is an extremely challenging task that becomes exacerbated in case of fragmentation. An open question is also tracking these objects using multiple stations providing
heterogeneous observations. Doing vehicle tracking using available satellite services is a novelty in itself and would provide an unprecedented capability to track vehicles also in the absence of
beacons, in the case of disasters or illegal situations (such has smuggling or hijacking). Last but not least end-to-end space systems design is an example of process with evolvable requirements and
objectives. Multi-fidelity approaches can be used to reduce the computational cost of providing effective solutions to the design and control of these systems and processes. In the multi-fidelity
approach, low- and high-fidelity models of the same system are considered and opportunely scheduled during the process to produce reliable results at the cheapest computational cost. An example of
multi-fidelity in optimisation is the Approximation and Model Management Optimisation while techniques like multi-fidelity evolution control are used to schedule the fidelity levels to make the
process efficient. Recently a fundamentally different approach was proposed using the tools of estimation theory to fuse together information from multi-fidelity analyses, resulting in a
Bayesian-based approach to mitigating risk in complex system design and analysis. WP1.2 will develop multi-fidelity models for all the above-mentioned applications. The innovative aspects will be to
investigate and characterise the uncertainty that comes with each level of model fidelity for all seven applications. Furthermore, icing models will be carried out for the first time from fundamental
physics and experimental data. Models used in WP1.2 are already validated or, in the case of the icing-accretion and re-entry models, will be validated against available and known experimental
results. More techniques on model validation can be found in WP2.3.
Lead Partner – Politecnico di Milano
ESRs 2,5, 10, 15
Objective: To look into the use of the most general representation of uncertainty using IP.
Imprecise Probability Theory (IPT) generalises Probability Theory to allow for partial probability specifications, or in other words, incomplete, vague and conflicting information. In all cases, a
single unique probability distribution may be difficult to define or might be inappropriate to completely capture the nature of uncertainty. Quantification of uncertainty is generally done using
probability distributions, usually satisfying Kolmogorov’s axioms and the machinery of Probability Theory. Many scholars, from Laplace to de Finetti, Ramsey, Cox and Lindley argue that this is the
only possible representation of uncertainty. However, this has not been unanimously accepted by scientists, statisticians, and probabilists. IPT includes a number of hierarchical extensions to
Probability Theory. Dezert-Smarandache Theory (DSmT) of paradoxical reasoning (and the extension of DST to Evidence) has been used in target tracking and state estimation, robust Bayesian methods
have been proposed for situations in which model uncertainty is hard to quantify due to lack of data, and, therefore, information on a system or process comes from expert knowledge and experience.
IPT have been applied to study transition probabilities in (hidden) Markov chains that led to important applications in state estimation, filtering and control. WP2.1 will propose a whole range of
innovative approaches based on Imprecise Probabilities and related implications such as information fusion, aggregation rules and interpretation of the results will be considered.
Lead Partner – University of Durham
ESRs 1, 8, 9, 14
Objective: To develop and improve techniques to reduce the computational cost to derive accurate uncertainty quantifications, both using Probability and Imprecise Probability theories.
Uncertainty propagation is affected by the so-called “Curse of Dimensionality”: as the dimension of the uncertain space increases the computational cost increases exponentially for the same quality
of the solution. Several techniques have been proposed, though all with drawbacks. A weighted function space-based quasi-Monte Carlo method has been recently proposed though this method remains slow
when the number of effective dimensions is much less than the total. Another well-known technique is based on High-Dimensional Model Representation that can detect the interactions between different
uncertainties by decomposing the problem into a series of low-dimensional additive functions. This approach can be expensive and very sensitive to the choice of the anchor point. A technique using a
priori and a posteriori analysis based anisotropic sparse grid construction has been shown to be more efficient than the isotropic sparse grid, even if interactions among different dimensions are not
well captured. All the proposed numerical algorithms will need to be extended to work on parallel machines in a High Performance Computing (HPC) framework. The method and algorithm need to be
scalable and the implementation needs to consider that both the deterministic model and the uncertainty model can be parallelised. Model reduction mitigates the curse of dimensionality by selecting
only the most important groups of parameters or the coupling between them. A number of model reduction techniques are based on an appropriate projection of the uncertain space, e.g. using Proper
Orthogonal Decomposition (POD), Analysis Of Variance (ANOVA) or High-Dimensional Model Representation (HDMR). In the case of Imprecise Probabilities (IP), techniques have been proposed to reduce the
number of uncertain parameters or uncertain intervals associated to each parameter. More recently an innovative decomposition technique, called H-decomposition, was developed to efficiently apply
Evidence Theory to weakly coupled systems and a novel HDMR approach was developed for the automatic detection of parameter coupling. UTOPIAE will study the impact of a reduced order model on the
processes of robust and reliability based optimisation, an area still not sufficiently explored, and will extend and generalise anchored-ANOVA, adaptive HDMR and H-decomposition techniques.
Objective: To study optimal techniques to design new experiments, improve the robustness of the numerical simulation and validate the simulation models.
Experiments incur two basic kinds of uncertainty: systematic, reproducible errors affecting the whole experiment, and random uncertainties associated with intrinsic variations in the experimental
conditions, in the sensor readings or deficiencies in defining the quantity being measured. Not considering the statistical variability, stemming from random uncertainties, when validating a
numerical model can lead to erroneous conclusions. At the same time one is interested in capturing discrepancies in the model it-self, generally evidenced by a bias, or in guiding the experiments to
fully validate the modelled components or characterise the unmodelled ones. While several techniques exist to numerically propagate experimental data, two open questions remain: how to use numerical
simulations to improve experiments, and how experimental data can be used to improve models and fully characterise model uncertainty. The general starting assumption is that sensors and measurement
models are well understood so that validation is possible. In UTOPIAE, a characterisation of model uncertainties will be performed using techniques for capturing scattered data, and used to validate
the numerical tool. The novelty in WP2.3 is that appropriate discrepancy functions will be introduced where uncertainty in model parameters are not sufficient to fit the model to the experiments and
Bayesian inference is used to characterised the uncertainty in model parameters from experiments. Furthermore, a new technique based on Bayesian and Robust Bayesian inference will be applied to
determine new operating conditions in the experiment to reduce uncertainty in the simulation model in the case sensor and measurement models are not well known. Once uncertainty in the model is fully
characterised, the predictions from the simulations will be used to update the design of the experiments.
Lead Partner – Von Karman Institute for Fluid Dynamics
ESRs 3, 6, 9, 10
Objective: To define protocols for the elicitation and aggregation of expert judgement in multi-phase decision processes.
In many decision problems, data on future events is unavailable and structured expert judgement is required to capture the epistemic uncertainty and subjective probability assignment that exist in
human opinions. In expert judgement, probabilities are subjective in nature and can be represented by a degree of belief rather than a probability distribution. In a framework, this belief needs to
be encoded into a probability distribution that might not be supported by enough data. IP theories, like DST, can provide an alternative framework for subjective beliefs. Epistemic uncertainty can be
quantified as propositions or intervals, and a basic belief mass is assigned to them without the need to infer an actual distribution. To date, there has been insufficient literature published on the
evaluation of IP elicitation and associated challenges, e.g., fusing multiple experts. Eliciting expert judgement is typically laborious and the extent to which the process can be automated using
technology has not been investigated in detail. In addition, there are many papers applying mathematical models to aggregate expert input, however, current mathematical models and behavioural models
for combining judgements from multiple experts do not have all the desired mathematical properties. For example, Bayesian methodologies provide a mechanism for aggregating multiple experts but pose
philosophical challenges in how to interpret the output. The innovation in UTOPIAE will be to use probabilistic and IP approaches to derive reliable quantification. WP2.4 will introduce two
innovative aspects: elicitation and data fusion using IP theories in engineering design is a first, furthermore, incorporating this aspect in the optimisation of a system or process is essential but
has never been investigated before. The current use of Probability Theory in OUU might limit the treatment of epistemic uncertainty and subjective probabilities, thus UTOPIAE will introduce IPT as
part of OUU, an advancement that can lead to a distinctive edge for Europe.
Objective: To develop algorithms and methods for worst-case and/or multi-level optimisation.
In worst-case scenario optimisation one seeks for the best performance under the worst-case conditions. Worst-case design is important whenever robustness to adverse environmental conditions should
be ensured regardless of their probability, since there cannot be uncertainty without a set of possible uncertainty outcomes. This leads to optimisation, where the solution is found by minimising the
maximum output of all possible scenarios (min-max optimisation), while implementing the multi-level approach that uses different-accuracy measures for analysis/evaluation tools with various levels of
complexity. Most optimisation techniques are based on Lipschitz optimisation algorithms, global optimisation algorithms for min-max optimisation via relaxation and Kriging. Recently some heuristic
methods have been used, e.g., particle swarm optimisation, genetic or evolutionary algorithms, differential evolution and game theory. This WP will create a fundamental building block to solve
Evidence-Based Robust Optimisation and Reliability Based Optimisation in WP3.4 and 3.5.
Objective: To improve existing and develop new techniques for handling expensive many-objective optimisation problems in aerospace applications with mixed discrete and continuous decision variables.
Many-objective optimisation was introduced to study problems with more than 3 objectives where state-of-the-art multi-objective techniques often fail. The Pareto-dominance, which requires better or
equal performance in all objectives, fails in higher dimensional objective spaces because more and more solutions become incomparable. Techniques considering the dominated hypervolume have a
computational cost that is exponentially increasing with the number of objectives. Alternative approaches are rather specialised and have not been considered frequently for relevant applications.
Another method is to reduce the dimension in objective space. When objective functions and constraints are expensive to evaluate many-objective optimisation becomes even more challenging. Surrogate
models can be considered for optimisation based on computationally cheaper models and transfer the results to the original large scale expensive problems. This has been done only rarely for
many-objective optimisation to date, but is potentially applicable to expensive problems of industrial interest. Combinatorial and Mixed-Integer Nonlinear Optimisation techniques are essential when
systems and processes are integrated (e.g., integrated aircraft and operation design). Many combinatorial optimisation problems are naturally modelled using integer programming techniques, and
classical techniques such as using valid inequalities and defining stronger reformulations have shaped this area for many decades. Mixed-Integer Nonlinear Programming (MINLP) offers natural ways of
modelling for complicated real-life problems. MINLP has benefited from „smart‟ approximations such as outer approximation, however, the use of sophisticated heuristics offer, generally, more
efficient solutions. In case of evolutionary computation, combinatorial as well as mixed-integer problems are handled the same way. State-of-the-art methods use hierarchical or simultaneous
approaches. While the first implement a higher-level optimisation problem for the discrete part and a subproblem for the continuous variables, the second treats parameters of discrete and continuous
type simultaneously. In different problem domains, meta-model assisted evolutionary approaches have already been applied to mixed-integer optimisation problems. With respect to combinatorial
optimisation, different variants of evolutionary algorithms have been developed. UTOPIAE will investigate methods to deal with many-objective optimisation both in the case of cheap and expensive
objective and constraint evaluations with mixed (hybrid) variables, continues and discrete. UTOPIAE will combine mathematical theories developed in the areas of combinatorial optimisation and
nonlinear programming and, starting from existing algorithms, will develop enhanced optimisation algorithms for the efficient optimisation of many-objective mixed-integer nonlinear design problems.
Lead Partner – Technische Hochschule Köln
ESRs 2, 5, 12, 13
Objective: To extend the computational optimisation framework developed to make evidence-based robust optimisation (EBRO) efficient on high dimensional problems with the inclusion of expert opinions.
Dempster-Shafer theory or Evidence Theory (ET) is a branch of mathematics on uncertain reasoning that allows the decision-maker to deal with uncertain events, and incomplete or conflicting
information. ET is a generalisation of classical probability and possibility theory used mainly in information fusion, decision-making, risk analysis, autonomy, intelligent systems and planning &
scheduling under uncertainty. Recently, ET was considered for applications in the robust design of structures and mechanisms in aerospace and civil engineering, such as reusable launchers,
aerocapture manoeuvres and low-thrust trajectories. Concepts of robustness and robust design optimisation based on DST using gradient methods in combination with response surfaces were first proposed
in early 2000. Bauer proposed different techniques to reduce the computational complexity in the computation of the cumulative Belief supporting a given proposition. Computational frame-works have
been developed to efficiently use ET in high dimensional model-based system optimisation. Starting from DST as paradigmatic example of Imprecise Probabilities, WP3.3 will extend the computational
optimisation framework developed by Vasile et al. to the use of Coherent Upper and Lower Previsions in system design and process control.
Objective: To study and develop new techniques for large scale constrained RBDO problems.
(RBDO) is an open research field. Many techniques have been introduced in the last ten years, from modified sampling techniques for the estimates of failure probabilities, to the use of neural
networks or support vector machines, or the formalisation of different optimisation strategies (single or double loops, sequential techniques) and the use of evolutionary algorithms. Two major
bottlenecks are represented by the applicability of some approaches to large scale problems with several uncertain variables and the high computational cost when time-consuming simulations are used
to evaluate the limit state functions. In WP3.4, UTOPIAE will tackle a number of interesting open points: the relationship between RBDO and robust design optimisation (RDO), the investigation of new
reliability measures and their application in multi-objective optimisation under uncertainties, the use of evolutionary algorithms to solve RBDO and RDO problems, and the efficient treatment of
multi-constrained problems or non-linear state functions, the treatment of uncertain experimental data with unknown probability density functions, the treatment of dependent non-normal variables
(both in reliability analysis and in RDO) and the application of response surface method techniques to large scale RBDO problems.
Objective: To use optimisation under uncertainty to optimise multi-phase processes with evolvable requirements.
A number of processes evolve through different phases, each characterised by different objectives and constraints. The design process itself can evolve over a long time span during which
requirements and specifications gradually become more clearly defined. Furthermore, the final product will have to be able to perform a multitude of tasks, which are often interrelated. The
meaningful quantification of system reliability is important in the earlier design stages, when tasks are not fully identified and information is lacking. In the case of more classical single-phase
processes (e.g., the control of the trajectory of an aircraft), measurements are used in combination with a model to update the state of the system and make decisions (e.g., in the case of Model
Predictive Control). Measurements are generally affected by aleatory uncertainty, while models are affected by a combination of epistemic and aleatory uncertainties. In the case of multi-phase design
and decision-making processes, expert knowledge is injected into the process at every phase, analogous to instrument measurements in state estimation. This knowledge can be incomplete and subjective,
and requires a particular treatment. In both cases, the use of Imprecise Probabilities can help in the quantification of all type of uncertainties affecting the evolution of a process. The study of
evolvable processes in the framework of IP represents a key novelty of UTOPIAE. System structure and functions will be represented as an imprecise probability. Lower and upper probabilities of system
functioning will be approximated to treat realistic problems. Robust Bayesian approaches and IPT methods for estimating imprecise transition (and emission) probabilities for (hidden) stationary and
non-stationary Markov models will be applied to the OUU of single- and multi-phase processes. WP3.5 will generalise IP-based reliability analysis of components or subsystems to that of systems and
will upscale the basic theory to make it applicable to real-world problems for the first time. | {"url":"http://utopiae.eu/2-2/utopiae-training/research/","timestamp":"2024-11-06T08:30:06Z","content_type":"text/html","content_length":"105393","record_id":"<urn:uuid:02f5f052-1e3f-473c-bf7d-61e53f9b4d0b>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00564.warc.gz"} |
QbD with Scale-up Suite
I have been meaning to post on model verification for a while and cumulative interactions with customers have led to this proposal. See the Twitter postings alongside for relevant material from the
FDA and recent additions to DynoChem Resources. To automatically keep abreast of these, I encourage you to follow DynoChem on Twitter.
Proposed approach to model verification:
1. Model development and parameter fitting should be based on a 'training set' of experimental data, that is a subset of all data available.
2. Verification should in general be completed against a separate set of experimental data, probably testing the limits of the model (e.g. points at the corners of the anticipated region of
3. Those data do not need to be from 'designed' or perfect experiments; in fact, inclusion of spiked or otherwise perturbed experiments can be highly valuable and informative.
4. Verification should be described, presented and qualified as being ‘to within E%' or 'within E response units'. E may vary within a factor space.
5. E is not arbitrary, but equal to the prediction band width for that response, with confidence level 1-alpha, where alpha may be 5% (95% confidence) or perhaps 1% (99% confidence).
6. The limits of applicability of the statement in 4 above (i.e. the region of factor space that is covered) should be defined.
Usage of prediction band widths (or 'prediction intervals') in this way allows a statistically sound statement to be made about the level of verification of any model in which parameters have been
fitted. During model development, E reduces if the model ‘improves’, i.e. the fit improves and uncertainty is reduced. When the model is mechanistic, there is often little risk of 'overfitting' (many
degrees of freedom) and the quality of mechanistic understanding, together with collection of good data, are the main factors that improve the fit. Bear in mind also that in a mechanistic model, a
single set of parameters fits all responses (not separate models for each response) and the fit is judged versus multiple samples, not just end-points.
E needs to be small if users are going to operate near the CQA (or another important) limit, but can be relatively larger if not. So the verification level required for a model to be useful has an
element of fitness for purpose.
Prediction bands take account of 'lack of fit' and are correspondingly wider for responses that fit poorly compared to those that fit well. For a CQA upper limit (e.g. typical for an impurity),
mathematically one could therefore say that a model is verified and fit for QbD purposes if:
average response*(1+E%) is comfortably less than CQA limit
The above expression is also equivalent to evaluating the probability that CQA will be less than its limit; that probability increases if E is low and/or the average response is well below the CQA
upper limit. So that probability is itself an indicator of the degree of model verification achieved.
Of course, in a good mechanistic model, E will be small for all responses, not just a CQA; focusing on reducing E will improve process understanding of the whole system and both prediction and
confidence band response surfaces may be drawn to guide experimentation to this goal, see previous posts.
By all accounts, Twitter seems to be an excellent way to keep people informed of developments and we are starting to use it to communicate with DynoChem users. If you follow this blog, or are a
member of DynoChem Resources or our Google Group, I encourage you to follow DynoChem on Twitter.
I have been using Twitter for several months and find the short postings (about 1 sentence long) a very efficient way to catch up on news quickly. It's easy to opt in or out and better enables a
two-way relationship than e.g. RSS feeds. There are lots of Twitter clients for smartphones; I use TweetDeck.
The DynoChem User Meeting 2009 was held in Philadelphia on 13-14 May. Presentations from companies such as Abbott, Amgen, AstraZeneca, Chemagis, Merck, GSK, Pfizer and Wyeth may now be downloaded
from DynoChem Resources (login required).
Previous posts have referred to work by DynoChem and others to provide tools to quantify uncertainty in model predictions and translate that into the (joint) probability of successfully meeting
several specifications, such as CQAs, at a particular set of processing conditions (factors, or process parameters). The question of how best to calculate this probability, for any process model and
set of experimental data is not straightforward to answer.
Many readers will be at least casually aware of alternative schools of thought in the statistics community, namely 'frequentist' - the statistics that most of us learned in school and university and
use to a degree every day and 'Bayesian'. The former calculates probability from the frequency of observing a certain outcome; the latter refines an initial subjective estimate of probability (the
'prior') using new information from observations. Good discussions of these alternative approaches are available all over the web and elsewhere; e.g. http://www.rasmusen.org/x/2007/09/25/
bayesian-vs-frequentist-statistical-theory/; and for a longer read http://nb.vse.cz/kfil/elogos/science/vallverdu08.pdf.
Whatever about the specifics and relative merits of these approaches, both provide useful insight for design space development by taking explicit account of uncertainty and risk in a multivariate
system and published examples of both, as well as their inclusion in regulatory filings, will become increasingly common. Members of DynoChem Resources can access knowledge base articles and other
useful materials in this context.
In this posting I am concerned with what goes before the probability calculations; specifically the modelling effort and data to support it. Unless the underlying data and modeling are sound,
probability calculations, however advanced the calculation procedure, will have little or no meaning.
With the emphasis on chemical reactions in API synthesis (e.g. final step) and after the solvent, catalyst and reagents have been selected, important ingredients in the mixture, whatever statistical
approach is ultimately used are:
1. upfront thinking on a mechanistic basis to determine factors and settings for initial screening experiments; supported by prior data if relevant data exist (see previous posts on process schemes);
2. screening experiments in which the process is followed by taking multiple samples; some of these experiments should screen for physical rate limitations and aim to determine whether physical or
chemical phenomena are 'rate-limiting';
3. characterization experiments, in which factors affecting the limiting phenomena are studied across a range of settings; the extremities and some centre-points (with replication) may be adequate
for a mechanistic model; a larger set of experiments may be required using a statistically designed (DOE) program of experiments; responses Y are measured as a function of factors X;
4. a modeling effort alongside 3 in which the relationship between Y and X is captured in either a mechanistic or DOE model, or both; the lack of fit and other statistics relating to model
uncertainty are quantified; further experiments to reduce uncertainty may be merited and/or improvements in the experimental or analytical technique; data from a portion of experiments should be used
for model development and the remaining experiments for model verification; ultimately a single model should fit all of the reliable data; the mechanistic model in particular may be used to
extrapolate to determine 'optimum' conditions outside the ranges studied to date; note that experimental data can be one of the least reliable inputs to a model, for a host of practical reasons;
unreliability of experimental data (e.g. lack of mole or mass balance) may only be noticed if the model has a mechanistic basis;
5. criticality studies, to determine the proximity to edge of failure for limiting factors; these can leverage a mechanistic model if one exists; otherwise will require further experiments to
extrapolate or mimic likely failure modes;
6. factor space exploration; this may be a very broad, full factorial, exploration with a mechanistic model, or a narrower exploration using a further set of DOE experiments; in either case, model
uncertainty and/or experimental error are taken into account; with the mechanistic model only, we can add formulas for derived responses that were not or cannot easily be measured (e.g. pass time,
fail time); an important feature of a mechanistic model is that one set of model parameters fits all responses, not one set per response.
7. design space definition; for a limited set of factors, this defines the relationship among their ranges that produces product of acceptable quality; until recently, overlapping response surfaces
for each CQA was considered adequate; a more reliable approach is to calculate the probability of success across the factor space, leading to a direct estimate of the associated risk of failure and a
narrower design space; here the relative merits of Bayesian and frequentist statistics may become relevant;
8. confirmatory experiments that operating within the design space provides the required level of assurance of quality;
9. with a mechanistic model only: demonstrate to colleagues, management, regulators, manufacturing and quality control that a high level of process understanding has been achieved, otherwise the
mechanistic model would not fit the data; justify the scale-independence of the design space; demonstrate the impact of scale-up on the CQA by predicting performance in large scale equipment.
The models developed above may be leveraged pre- and post-NDA in many other ways, including to guide process development, achieve yield or other business objectives, facilitate technology transfer
and be used at-line. Mechanistic models in particular also offer new ways to define design space to maximize flexibility and be tolerant to minor process upsets.
Keen Bayesian statisticians reading the above will notice that a high degree of prior knowledge is used to develop these guidelines and to carry out the associated experimental and mechanistic
modeling work; in that sense there is something very Bayesian about how mechanistic models are developed.
In the mechanistic approach, modeling takes place alongside experiments and new information leads to refinements in the model. The probability that the model is valid is thereby continually refined
upwards as new data are included, following Bayes' theorem.
New data also add degrees of freedom to the model, leading to ultimately sharper definition of probability distributions for model responses, important for design space definition. | {"url":"https://dynochem.blogspot.com/2009/06/","timestamp":"2024-11-14T08:47:25Z","content_type":"text/html","content_length":"68355","record_id":"<urn:uuid:5af82839-76ee-4467-9685-0dad19f9ff51>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00682.warc.gz"} |
Volume of a Prism - Formula, Derivation, Definition, Examples - Grade Potential Virginia Beach, VA
Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a vital shape in geometry. The shape’s name is derived from the fact that it is created by considering a polygonal base and extending its sides till it cross the opposite base.
This blog post will discuss what a prism is, its definition, different kinds, and the formulas for surface areas and volumes. We will also provide instances of how to employ the data provided.
What Is a Prism?
A prism is a three-dimensional geometric shape with two congruent and parallel faces, well-known as bases, which take the shape of a plane figure. The additional faces are rectangles, and their
amount depends on how many sides the identical base has. For instance, if the bases are triangular, the prism would have three sides. If the bases are pentagons, there would be five sides.
The properties of a prism are astonishing. The base and top both have an edge in parallel with the other two sides, making them congruent to one another as well! This implies that all three
dimensions - length and width in front and depth to the back - can be deconstructed into these four parts:
1. A lateral face (signifying both height AND depth)
2. Two parallel planes which make up each base
3. An fictitious line standing upright across any provided point on any side of this shape's core/midline—usually known collectively as an axis of symmetry
4. Two vertices (the plural of vertex) where any three planes join
Kinds of Prisms
There are three main kinds of prisms:
• Rectangular prism
• Triangular prism
• Pentagonal prism
The rectangular prism is a regular type of prism. It has six faces that are all rectangles. It matches the looks of a box.
The triangular prism has two triangular bases and three rectangular faces.
The pentagonal prism consists of two pentagonal bases and five rectangular sides. It looks a lot like a triangular prism, but the pentagonal shape of the base stands out.
The Formula for the Volume of a Prism
Volume is a calculation of the total amount of area that an object occupies. As an crucial shape in geometry, the volume of a prism is very relevant in your learning.
The formula for the volume of a rectangular prism is V=B*h, where,
V = Volume
B = Base area
h= Height
Consequently, considering bases can have all types of figures, you will need to learn few formulas to figure out the surface area of the base. Still, we will touch upon that later.
The Derivation of the Formula
To extract the formula for the volume of a rectangular prism, we are required to look at a cube. A cube is a three-dimensional object with six faces that are all squares. The formula for the volume
of a cube is V=s^3, where,
V = Volume
s = Side length
Right away, we will get a slice out of our cube that is h units thick. This slice will create a rectangular prism. The volume of this rectangular prism is B*h. The B in the formula stands for the
base area of the rectangle. The h in the formula implies the height, which is how dense our slice was.
Now that we have a formula for the volume of a rectangular prism, we can generalize it to any type of prism.
Examples of How to Utilize the Formula
Now that we understand the formulas for the volume of a pentagonal prism, triangular prism, and rectangular prism, let’s put them to use.
First, let’s calculate the volume of a rectangular prism with a base area of 36 square inches and a height of 12 inches.
V=432 square inches
Now, let’s work on another question, let’s calculate the volume of a triangular prism with a base area of 30 square inches and a height of 15 inches.
V=450 cubic inches
Provided that you have the surface area and height, you will figure out the volume without any issue.
The Surface Area of a Prism
Now, let’s talk regarding the surface area. The surface area of an object is the measure of the total area that the object’s surface consist of. It is an essential part of the formula; thus, we must
know how to find it.
There are a several different methods to work out the surface area of a prism. To calculate the surface area of a rectangular prism, you can employ this: A=2(lb + bh + lh), assuming,
l = Length of the rectangular prism
b = Breadth of the rectangular prism
h = Height of the rectangular prism
To calculate the surface area of a triangular prism, we will use this formula:
b = The bottom edge of the base triangle,
h = height of said triangle,
l = length of the prism
S1, S2, and S3 = The three sides of the base triangle
bh = the total area of the two triangles, or [2 × (1/2 × bh)] = bh
We can also use SA = (Perimeter of the base × Length of the prism) + (2 × Base area)
Example for Computing the Surface Area of a Rectangular Prism
First, we will figure out the total surface area of a rectangular prism with the following information.
l=8 in
b=5 in
h=7 in
To solve this, we will replace these numbers into the respective formula as follows:
SA = 2(lb + bh + lh)
SA = 2(8*5 + 5*7 + 8*7)
SA = 2(40 + 35 + 56)
SA = 2 × 131
SA = 262 square inches
Example for Finding the Surface Area of a Triangular Prism
To find the surface area of a triangular prism, we will figure out the total surface area by following similar steps as before.
This prism will have a base area of 60 square inches, a base perimeter of 40 inches, and a length of 7 inches. Thus,
SA=(Perimeter of the base × Length of the prism) + (2 × Base Area)
SA = (40*7) + (2*60)
SA = 400 square inches
With this data, you will be able to work out any prism’s volume and surface area. Test it out for yourself and observe how simple it is!
Use Grade Potential to Better Your Math Skills Now
If you're struggling to understand prisms (or any other math concept, think about signing up for a tutoring class with Grade Potential. One of our expert instructors can help you learn the
[[materialtopic]187] so you can nail your next exam. | {"url":"https://www.virginiabeachinhometutors.com/blog/volume-of-a-prism-formula-derivation-definition-examples","timestamp":"2024-11-06T18:31:10Z","content_type":"text/html","content_length":"77586","record_id":"<urn:uuid:f45e1aa4-7a3e-47cc-8422-e866996fb4ac>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00100.warc.gz"} |
scry: vignettes/bigdata.Rmd
We illustrate the application of scry methods to disk-based data from the TENxPBMCData package. Each dataset in this package is stored in an HDF5 file that is accessed through a DelayedArray
interface. This avoids the need to load the entire dataset into memory for analysis.
sce<-TENxPBMCData(dataset="pbmc3k") h5counts<-counts(sce) seed(h5counts) #print information about object h5counts<-h5counts[rowSums(h5counts)>0,] system.time(h5devs<-devianceFeatureSelection
(h5counts)) # 26 sec
We now compare the computation speed when the same data is converted to an ordinary array in-memory. Note this would not be possible with larger HDF5Array objects.
denseCounts<-as.matrix(h5counts) system.time(denseDevs<-devianceFeatureSelection(denseCounts)) # 5 sec max(abs(denseDevs-h5devs)) #should be close to zero
Finally we compare the speed when the counts data are stored in a sparse in-memory Matrix format
mean(denseCounts>0) #shows that the data are mostly zeros so sparsity useful sparseCounts<-Matrix::Matrix(denseCounts,sparse=TRUE) system.time(sparseDevs<-devianceFeatureSelection(sparseCounts)) #1.6
sec max(abs(sparseDevs-h5devs)) #should be close to zero
Using disk-based data saves memory but slows computation time. When the data contain mostly zeros, and are not too large, the sparse in-memory Matrix object achieves fastest computation times. The
resulting deviance statistics are the same for all of the different data formats.
One can run nullResiduals on HDF5Matrix, DelayedArray matrices, and sparse matrices from the Matrix package with the same syntax used for the base matrix case.
We illustrate this with the same dataset from the TENxPBMCData package.
Any scripts or data that you put into this service are public.
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/bioc/scry/f/vignettes/bigdata.Rmd","timestamp":"2024-11-13T18:50:51Z","content_type":"text/html","content_length":"20382","record_id":"<urn:uuid:85a52892-72fd-43d7-8369-97bcd6b4f188>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00533.warc.gz"} |
Mathematical analysis
(Marco Degiovanni - Giovanna Marchioni - Marco Marzocchi)
Our research activity focus on the study of non-linear differential equations (both ordinary and partial) by means of variational methods.
Natural phenomena are often described by functions that satisfy some equations which involves the derivatives of the of the given functions: for this reason this kind of equations are called
differential equations. Differential equations are usually nonlinear and to find the solutions can be extremely difficult.
Since the XVIII was observed that the functions that describe natural phenomena often minimize (or makes stationary) appropriate functionals defined on the the functions. For one century and half,
such fact was considered as an interesting property of the solutions (obtained in a different way) of some differential equations. In the first half of the XX century a change of perspective was
proposed, i.e. the idea of using minimality and stationarity criteria for solving differential equations. This idea turned out to be fruitful and has been developed creating a new field in
Mathematical analysis.
Variational methods for non-regular functionals
In order to establish a connection between the functionals and the differential equation one can consider the case in which the functional itself satisfy some regularity conditions.
For this reason (starting from the sixty for the study of the minimality and from the eighty for the study of the stationarity) a subfield dedicated to the study (by means of variational methods) of
differential equations that do not satisfy standard regularity criteria was developed.
Key words
• Non-linear differential equations
• Variational methods
• Non-regular functionals
Group leader
• Università degli Studi di Pisa
• Università degli Studi di Bari
• Politecnico di Torino
• Università degli Studi di Verona
• Università di Giessen (Germania)
Ongoing projects
• PRIN07 - Metodi variazionali e topologici nello studio di fenomeni non lineari
Main pubblications
• T. Bartsch and M. Degiovanni,
Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 17, n. 1, 69-85 (2006).
• S. Cingolani and M. Degiovanni,
Nontrivial solutions for p-Laplace equations with righthand side having p-linear growth at infinity, Comm. Partial Differential Equations 30, n. 8, 1191-1203 (2005).
• M. Degiovanni and S. Lancelotti,
Linking over cones and nontrivial solutions for p-Laplace equations with p-superlinear nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire 24, n. 6, 907-919 (2007).
• M. Degiovanni, A. Musesti and M. Squassina,
On the regularity of solutions in the Pucci-Serrin identity, Calc. Var. Partial Differential Equations 18, n. 3, 317-334 (2003).
• S. Lancelotti and M. Marzocchi,
Lagrangian systems with Lipschitz obstacle on manifolds, Topol. Methods Nonlinear Anal. 27, n. 2, 229-253 (2006). | {"url":"https://dipartimenti.unicatt.it/dmf-area-matematica-analisi-matematica?rdeLocaleAttr=en","timestamp":"2024-11-15T03:16:47Z","content_type":"application/xhtml+xml","content_length":"30240","record_id":"<urn:uuid:c4b53fcb-697b-470f-b5b0-82e3a8c6e80f>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00096.warc.gz"} |
Is it possible to estimate an AE model using the ACE model functions?
Replied on Fri, 02/03/2017 - 07:24
Yes, you can estimate an "AE" `umxACE` model!
This is done via `umxModify` and labels - the labels umx applies to every matrix cell/parameter in the model.
in twin models, these labels take the form matrix name + "_" + "r" + row-number + "c" + col-number. So the link from the first latent variable to the second measured variable in the c matrix is
labeled `c_r2c1`.
Just run `umxACE` to get your saturated model, then drop the c parameters using `umxModify`
There is an example of this in `?umxACE`
This is a powerful tool. You can provide a single label to update. e.g. `update = "c_r1c1"`, this can be a list of labels.
By default they are set to zero, but you can set them to anything via `values`. By default they are fixed, but you can free variables with `free = T`.
You can even use regular expressions: umxModify(m1, regex = "c_r.c.”) which would drop all free parameters from the c matrix (as long as there are fewer than 10 rows and columns), which is what you
want to create an AE model!
then `umxCompare` to compare them
Because I am lazy, you can update the model , rename it, and do the comparison all in 1 line!
m2 = umxModify(m1, regex = "c_r.c.”, name = "AE", comp= TRUE) | {"url":"https://openmx.ssri.psu.edu/node/4229","timestamp":"2024-11-08T03:15:20Z","content_type":"text/html","content_length":"27980","record_id":"<urn:uuid:e0aa35f1-2c06-45bb-a221-c92dff8f0c56>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00655.warc.gz"} |
TS 6th Class Maths Guide Pdf | Telangana 6th Class Maths Textbook Solutions Pdf English Medium
Telangana SCERT Class 6 Maths Solutions | TS 6th Class Maths Solutions Study Material Pdf
TS 6th Class Maths Guide Pdf Chapter 1 Knowing Our Numbers
Telangana 6th Class Maths Textbook Solutions Pdf Chapter 2 Whole Numbers
TS 6th Class Maths Solutions Chapter 3 Playing with Numbers
SCERT Telangana Class 6 Maths Solutions Chapter 4 Basic Geometrical Ideas
TS 6th Class Maths Textbook Pdf Chapter 5 Measures of Lines and Angles
TS 6th Class Maths Study Material Pdf Chapter 6 Integers
TS SCERT Class 6 Maths Solutions Chapter 7 Fractions and Decimals
Telangana 6th Class Maths Solutions Pdf Chapter 8 Data Handling
SCERT Telangana 6th Maths Solutions Chapter 9 Introduction to Algebra
TS SCERT 6th Class Maths Solutions Chapter 10 Perimeter and Area
Telangana SCERT 6th Class Maths Solutions Chapter 11 Ratio and Proportion
6th Class Maths Textbook Telangana Chapter 12 Symmetry
6th Class Maths Solutions Telangana Chapter 13 Practical Geometry
Telangana 6th Class Maths Textbook Pdf Chapter 14 Understanding 3D and 2D Shapes
Leave a Comment | {"url":"https://tsboardsolutions.in/ts-6th-class-maths-solutions/","timestamp":"2024-11-12T15:41:47Z","content_type":"text/html","content_length":"165025","record_id":"<urn:uuid:86e0d228-73b5-4329-846f-086f352cfd0d>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00298.warc.gz"} |
I install water dispenser China University of petroleum freshman training competition #10
Question I: installation of water dispenser
time limit: one Sec Memory limit: 128 MB
Submit state
Title Description
In order to advocate a low-carbon life in the city, the municipal Civilization Office plans to hold a marathon, and set up some observation points along the way to ensure the safety of the race. One
observer is stationed at each observation point. Due to the hot weather, some water dispensers need to be installed along the way so that observers can get water and drink. It takes one unit of
physical strength for each unit of distance the observer moves. Each observer's physical strength is limited. He can only get water and drink within the range of his physical strength, or he will die
of thirst or fatigue.
Smart Nannan also participated in the preparations for the competition. His task is to design an ideal scheme for installing drinking fountains to minimize the number of drinking fountains installed,
but to ensure that all observers can get water to drink.
There are several lines of input data..
The first line, only an integer, indicates that there are n (0 < n < = 1000) observation points.
Next, there are N lines, each line has two integers S (0 < S < = 100000) and w (0 < w < = 50000), where S represents the distance from an observation point to the starting point, and W represents the
physical strength of the resident observer in the observation point.
At least several drinking fountains shall be installed at the output.
He can install the water dispenser 6 and 12 away from the starting point so that all observers can drink water. There are many schemes, which only need to output at least a few drinking fountains.
His task is to design an ideal scheme for installing drinking fountains to minimize the number of drinking fountains installed, but to ensure that all observers can get water to drink.
And the carrier is a straight line positive integer number axis, it is difficult not to think of using the method of binary answer to solve the problem.
The second is the number of drinking fountains, and the special state is the minimum. Select the dichotomous template that returns the answer directly:
int l = 0, r = n;
while(l < r)
int mid = l + r >> 1;
if(check(mid)) l = mid + 1;
else r = mid;
Each data in the input data has two related data, which are related to each point, so consider creating a structure for calculation and input.
After that, you need to write the check function. The first consideration is to sort from small to large according to its position, and then greedily traverse each element. If the current nearest
water dispenser position is within the range, it will be skipped, otherwise it will be placed at the right end of the current element (because it is sorted by position)
However, this greedy + sorting method can not ac, for example, a group of hack data:
If this group of data is traversed according to the above greed, the first water dispenser location will be updated at 10000, and then the second will be placed at 3. In fact, it only needs to be
placed at 3 to meet all needs
Therefore, it is necessary to change a greedy way to observe this group of hack data. It can be found that if you sort from small to large according to a[i] + b[i], you can finish each situation
Therefore, after entering data, sort the structure array from small to large according to a[i] + b[i], which can be realized by rewriting sort.
Remember to initialize the water dispenser position to 0x3f3f3f3f (large enough)
Code: 2021.12.3
#include <bits/stdc++.h>
using namespace std;
#define x first
#define y second
const int P = 13131;
#define ll long long
const int mod = 1E6 + 7;
const int INF = 0x3f, sINF = 0x3f3f3f3f;
typedef unsigned long long ULL;
typedef pair<int, int> PII;
typedef pair<long long, long long> PLL;
int dx[4] = {1, 0, -1, 0}, dy[4] = {0, 1, 0, -1};
const int N = 1010;
int n;
struct answ{
int a, b;
bool cmp(const answ &x, const answ &y)
if(x.a + x.b != y.a + y.b) return x.a + x.b < y.a + y.b;
return x.a < y.a;
bool check(int x)
int nums = x;
int loc = 0x3f3f3f3f;
for(int i = 1; i <= n; i++)
if(loc > q[i].a + q[i].b || loc < q[i].a - q[i].b) {
loc = q[i].a + q[i].b;
if(nums < 0) return true;
return false;
int main()
for(int i = 1; i <= n; i++) cin>>q[i].a>>q[i].b;
sort(q + 1, q + 1 + n, cmp);
int l = 0, r = n;
while(l < r)
int mid = l + r >> 1;
if(check(mid)) l = mid + 1;
else r = mid; | {"url":"https://programmer.group/61aa805e32b72.html","timestamp":"2024-11-09T05:45:47Z","content_type":"text/html","content_length":"18658","record_id":"<urn:uuid:d5511f4b-0eff-4077-875d-e546f2a060bb>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00730.warc.gz"} |
Infinite Resistor Network
First, we begin with a square of side length L. Connect the centers of each side to form an- other square. Connect the centers of each side of this square to form yet another square, and so on, to
infinity. Suppose now that all the lines in the diagram are uniform wires. Let the resis- tance of a piece of wire of the Length L (length of a side of the outer most square) be exactly 1. Let the
net resistance between the points A and B be R. Compute the k-th digit after the decimal point of the value R. Namely, let R = 0.n1n2n3 ...nk ... Given k, output nk. Input The input of each test
cases is simply the value k (k < 10000000), 0 < min(k, 990 ∗ 104 − k) < 104 on its own line. Output For each input value, output the k-th digit of R on a single line. Sample Input 2 4 6 9899898
Sample Output 5 9 8 4 | {"url":"https://ohbug.com/uva/11274/","timestamp":"2024-11-03T15:39:16Z","content_type":"text/html","content_length":"2086","record_id":"<urn:uuid:e8db8f3e-f2b5-438a-a00d-5131c06e8567>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00644.warc.gz"} |
Breadth First Search-
• Breadth First Search or BFS is a graph traversal algorithm.
• It is used for traversing or searching a graph in a systematic fashion.
• BFS uses a strategy that searches in the graph in breadth first manner whenever possible.
• Queue data structure is used in the implementation of breadth first search.
BFS Example-
Consider the following graph-
The breadth first search traversal order of the above graph is-
A, B, C, D, E, F
Breadth First Search Algorithm-
BFS (V,E,s)
for each vertex v in V – {s}
color[v] ← WHITE
d[v] ← ∞
π[v] ← NIL
color[s] = GREY
d[s] ← 0
π[s] ← NIL
Q ← { }
ENQUEUE (Q,s)
While Q is non-empty
do v ← DEQUEUE (Q)
for each u adjacent to v
do if color[u] ← WHITE
then color[u] ← GREY
d[u] ← d[v] + 1
π[u] ← v
ENQUEUE (Q,u)
color[v] ← BLACK
The above breadth first search algorithm is explained in the following steps-
Create and maintain 3 variables for each vertex of the graph.
For any vertex ‘v’ of the graph, these 3 variables are-
1. color[v]-
• This variable represents the color of the vertex v at the given point of time.
• The possible values of this variable are- WHITE, GREY and BLACK.
• WHITE color of the vertex signifies that it has not been discovered yet.
• GREY color of the vertex signifies that it has been discovered and it is being processed.
• BLACK color of the vertex signifies that it has been completely processed.
2. Π[v]-
This variable represents the predecessor of vertex ‘v’.
3. d[v]-
This variable represents the distance of vertex v from the source vertex.
For each vertex v of the graph except source vertex, initialize the variables as-
• color[v] = WHITE
• π[v] = NIL
• d[v] = ∞
For source vertex S, initialize the variables as-
• color[S] = GREY
• π[S] = NIL
• d[S] = 0
Now, enqueue source vertex S in queue Q and repeat the following procedure until the queue Q becomes empty-
1. Dequeue vertex v from queue Q.
2. For all the adjacent white vertices u of vertex v,
color[u] = GREY
d[u] = d[v] + 1
π[u] = v
Enqueue (Q,u)
3. Color vertex v to black.
BFS Time Complexity-
The total running time for Breadth First Search is O (V+E).
Also Read- Depth First Search
Traverse the following graph using Breadth First Search Technique-
Consider vertex S as the starting vertex.
For all the vertices v except source vertex S of the graph, we initialize the variables as-
• color[v] = WHITE
• π[v] = NIL
• d[v] = ∞
For source vertex S, we initialize the variables as-
• color[S] = GREY
• π[S] = NIL
• d[S] = 0
We enqueue the source vertex S in the queue Q.
• Dequeue vertex S from the queue Q
• For all adjacent white vertices ‘v’ (vertices R and W) of vertex S, we do-
1. color[v] = GREY
2. d[v] = d[S] + 1 = 0 + 1 = 1
3. π[v] = S
4. Enqueue all adjacent white vertices of S in queue Q
• Dequeue vertex W from the queue Q
• For all adjacent white vertices ‘v’ (vertices T and X) of vertex W, we do-
1. color[v] = GREY
2. d[v] = d[W] + 1 = 1 + 1 = 2
3. π[v] = W
4. Enqueue all adjacent white vertices of W in queue Q
• Dequeue vertex R from the queue Q
• For all adjacent white vertices ‘v’ (vertex V) of vertex R, we do-
1. color[v] = GREY
2. d[v] = d[R] + 1 = 1 + 1 = 2
3. π[v] = R
4. Enqueue all adjacent white vertices of R in queue Q
• Dequeue vertex T from the queue Q
• For all adjacent white vertices ‘v’ (vertex U) of vertex T, we do-
1. color[v] = GREY
2. d[v] = d[T] + 1 = 2 + 1 = 3
3. π[v] = T
4. Enqueue all adjacent white vertices of T in queue Q
• Dequeue vertex X from the queue Q
• For all adjacent white vertices ‘v’ (vertex Y) of vertex X, we do-
1. color[v] = GREY
2. d[v] = d[X] + 1 = 2 + 1 = 3
3. π[v] = X
4. Enqueue all adjacent white vertices of X in queue Q
• Dequeue vertex V from the queue Q
• There are no adjacent white vertices to vertex V.
• color[V] = BLACK
• Dequeue vertex U from the queue Q
• There are no adjacent white vertices to vertex U.
• color[U] = BLACK
• Dequeue vertex Y from the queue Q
• There are no adjacent white vertices to vertex Y.
• color[Y] = BLACK
Since, all the vertices have turned black and the queue has got empty, so we stop.
This is how any given graph is traversed using Breadth First Search (BFS) technique.
To gain better understanding about Breadth First Search Algorithm,
Next Article- Prim’s Algorithm
Get more notes and other study material of Design and Analysis of Algorithms.
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A Strategy to Defend the Nation-State
This article appears in the August 7, 2020 issue of Executive Intelligence Review.
[Print version of this article]
A Strategy to Defend the Nation-State
Editor’s Note: This article first appeared in EIR, Vol. 34, No. 28, July 20, 2007, pp. 42-53.
On June 5, 2007, Lyndon LaRouche addressed the Defense Committee of the Italian Senate in the context of the Committee’s “Investigation of the Present State and Perspectives of the Defense Industry
and Cooperation on Armaments.” Here is LaRouche’s testimony, followed by questions and comments by members of the Committee, and a final response by LaRouche. The remarks by Italian Senators have
been translated from Italian by EIR.
Mr. Chairman, Honorable Senators, the subject of today’s event focusses on the correlation between defense and economics. I want to emphasize in particular the technological aspects of that
To understand the problem, we must return to its origins, and to the basis of the character of the nation-state, which we find in the Council of Florence of 1439, and in the Concordantia Catholica of
Nicholas of Cusa, who participated in the Council. These events marked the foundation of modern science, thanks in part to Nicholas of Cusa, whose proposals, amplified by many others who participated
in that Council, led to the creation of a new form of society which we now call the “modern nation-state,” and which in English, is also indicated by the expression “commonwealth society,” a society
in which all of the people are considered part of the nation, which must be governed in the common interest of all of the people.
Louis XI founded a state of this type in France. A second, similar state was established in England under Henry VII. Since that time, as we know from the 16th Century, particularly from the writings
of a famous man from that period [Nicolò Machiavelli], on warfare, that the nature of warfare and statecraft changed with the introduction of the modern nation-state, and the countermeasures which
are occurring against it; that the ability of the old feudal system to come back, was impeded, as was indicated, with the role of the total people of a city, or of a nation, in warfare. In this
process, what became known as modern economy, and modern technology, became a determining force in warfare.
We had a continuation of that under the influence of Paolo Sarpi, especially toward the beginning of the 17th Century, in terms of the so-called Liberal system, which became ultimately the
Anglo-Dutch system of economy and statecraft. And then we had, of course, the continuation of religious warfare, into 1648, when the modern nation-state was established under the influence of
Cardinal Mazarin, with the key role of Jean-Baptiste Colbert.
And if we look at the relationship between economy, technology, science, and politics, and warfare, in this period, we find that we can trace the entirety of the modern history of warfare, and
military and political actions, from these roots in history. The struggle between the idea of the commonwealth society, and the idea of empire, in the new liberal form, which is typically the British
Empire form, has been a continuing struggle to the present day. As today, the attempt to form globalization as a replacement for the sovereign nation-state—that is, to establish a world empire—is the
center of the ongoing conflict.
And there’s a constant temptation in some forces, to shut down the sovereign power of a nation over its own economy. This is called globalization; and the attempt to resist that is in trouble right
now. I’m one of the resistors.
So, what we face is this: We face an attempt under certain international financier interests, who are identical with the idea of globalization, to shut down the industries, and the scientific
capabilities, of nations, and to distribute these capabilities around the world, through cheap-labor societies.
For example: Europe has been stripped increasingly, especially since Maastricht, of its independent technological and military capability. The Soviet Union, the former Soviet Union, was ruined. The
nations of Eastern Europe, which were part of the Comecon, are in far worse economic and social condition than they were under Soviet domination, as a result of this process. Germany is being
bankrupted. Italy is being ruined, especially the essential industries which have been important to Italy since the middle of the 19th Century, from the time of the influence of Riemann on the
scientific thinking in Italy.
We’ve now got to the point that the basic industries in northern Italy, in particular, are being lost. A certain amount of industry exists, but there is tremendous pressure, especially from a
formation called the hedge funds, to loot industries in every country, in every part of the world. And there’s tremendous pressure to destroy particularly those sections of the economy which are
traditionally part of the state’s economy, whether on the state, municipal, or national level. And the struggle is international.
The Fight to Save Social Security and
the Auto Industry
For example: The most recent case we had of this, which affects directly today’s topic, is that, during the year 2005, I had organized around me, a mobilization of the Democratic Party and others, in
the United States, to defeat the attempt to loot the Social Security system of the United States—that policy introduced by the current President of the United States and some people around him. At
the same time, it was obvious to me, in February of 2005, that there was a plan to destroy the automobile industry of the U.S., and to turn the automobile industry over to foreign cheap-labor
producers of automobiles.
Now, this was crucial, because it was a strategic-military issue, as well as a mere economic issue. The United States, in this past century, had a very special kind of capability, which was built up
since Abraham Lincoln in the 19th Century, but was significant at the end of World War I, in which we were targetted by the United Kingdom/Great Britain; and our military engaged in a number of
studies which were centered around the naval power negotiations of the early 1920s, in which the British were ganged up with Japan, demanding a reduction of U.S. naval capability to a size which
would satisfy the British Empire. There were even plans by Japan and Britain and others, to conduct naval warfare against the United States, not to conquer the United States, but to reduce its naval
It was in this period that Japan, which was at that point, and had been since 1895, an asset of the British royal family—Japan had agreed to enter into the destruction of the Pearl Harbor Naval Base.
This was back in the 1920s, at the time that Japan was an ally of Britain.
Later, the irony changed: President Roosevelt induced the British not to ally with Hitler, or at least some of them not to, and Japan continued its course and attacked Pearl Harbor anyway, as an ally
of Nazi Germany.
But during this period, the U.S. military developed a policy whose impact became apparent under Franklin Roosevelt. As of the beginning of March of 1933, at the point that President Roosevelt was
first inaugurated as President of the United States, Hitler had already achieved dictatorial powers, toward the end of February, right after the Reichstag’s burning. So that when Roosevelt entered
office, as President, in early March of 1933, he already knew that a probable war was going to happen. So, Roosevelt’s policy immediately was one of both recovery—we had just suffered a 30%
destruction of our economy from 1929 to 1933, so Roosevelt turned to a gentleman, Harry Hopkins, who set up a program which was both a military program and a civilian program.
Roosevelt’s intention was, to use the same approach to developing industrial power, and rebuilding agriculture, to build up the civilian capability of the United States, but also at the same time, to
prepare the United States to be capable of meeting its responsibilities in respect to Europe, from what was already known by Roosevelt, to be the Hitler threat.
So therefore, you had the famous phenomenon of Harry Hopkins, with the people who became significant general officers during World War II and afterwards, who were part of this program.
So, the United States’ development, out of the Depression, to become the most powerful economy the world had ever seen, by 1943, was a result of a combination of military development, on a civilian
economy basis. In other words, what you were seeing then, with the United States’ role in this war, was a resolution of something that happened back with the Council of Florence, back in the middle
of the 15th Century, in which the commonwealth society was formed; in which the long history in European experience, of basing military power, where needed, and the power of conflict as needed,
basing it on the development of economy and of all the people—a new kind of nation-state, in which we try to eliminate all relics of serfdom or slavery.
So therefore, the development of the economy, for every square kilometer, and for the population within every square kilometer, to increase the productive powers of labor, and general well-being, and
development of the character of the people, was our tradition. What happened in the Treaty of Westphalia; this kind of system, while it was never realized perfectly, largely because of the wars of
Britain and France, and the Dutch who came in later; nonetheless, this model has been characteristic of every successful period of development, from then to the present day.
The United States’ development was merely a more perfected expression of it, because we had no legacy of oligarchical rule in our society. And that has been the difference: that whereas European
systems tend to be monetary systems, or based on monetary systems, the United States system, in terms of constitutional design, is not a monetary system; it’s a credit system. That is, our currency,
according to law, according to constitutional law, can be created only by the government, with the consent of the legislative branch, the junior partner. And this power of the government to create
and utter money, or to create credit, then becomes the financial power of government, which controls and is able to direct this force to industry, to agriculture, and general development of the
So, the power of the United States, the remarkable increase of the power of the United States, from being bankrupt in 1933, to the time that Roosevelt became President, and up until the end of the
war: The greatest physical economic, military power in the world history, therefore, had been created in a short period of time, from depression, under the use of the U.S. constitutional provisions,
which enabled us to make that kind of mobilization. We were not subject to control by foreign monetary authorities, foreign financial powers. And that was the secret of our ability to organize. And
we would have done very well, if Roosevelt had not died, if we’d kept on and developed the world, freeing the world from colonialism and that sort of thing. We didn’t.
A Sudden Change
Now, today you’re in a situation, in which there is an attempt to destroy this legacy of modern European civilization, a legacy established beginning with the Council of Florence. The legacy of the
modern nation-state based on the political equality of the human individual, and the responsibility of the state to promote the development of the individual, and to promote the improvement of the
political powers and physical powers of the individual.
Since Roosevelt died, this has been underway. It was not too obvious at first, but when Truman came in, there was a sudden change. The change was typified by two things which were conspicuous at the
time. Roosevelt had been committed to the elimination of all forms of colonialism, immediately, at the end of the war. He’d also been committed to the use of the military power we had developed, to
convert it back into a civilian capability, and to use a significant part of that civilian economic power, to assist freed nations, as well as rebuilding Europe, but assisting freed nations, which
had been colonialized nations, to give them the development which would make them truly independent nations.
That policy was abandoned. And our rate of development in the postwar period slowed down as a result. But nonetheless, we maintained that system, with the damage done to it in that fashion, until the
assassination of John F. Kennedy. And John F. Kennedy’s assassination allowed a different policy to be introduced. John Kennedy’s assassination allowed certain forces in Europe and the United States,
to proceed with what President Eisenhower had warned against, in leaving office: that a so-called military-industrial complex took, actually, political control of the destiny of the United States and
pretty much of Europe and the other parts of the world.
Now, they did the same thing to us that was done in the Peloponnesian War to the Greeks. The Greeks were induced to engage, through Sophistry, in a prolonged war which destroyed Greece, which has not
come back to the present day. Athens has never recovered from the long war it fought in the Peloponnesian War. The history of civilization, since that time, especially European civilization, has been
that long wars have ruined us repeatedly.
As contrasted, for example, with the case of Louis XI, who was attacked by everyone on every side. He bribed even some of his persecutors to make peace with him, and he made a profit on peace, by
avoiding war, because he used the occasion of freedom from war, to develop the French population, which is where modern France as a significant power emerged.
Long Wars of the Recent Period
We did the reverse. With the Indochina war, we went into an unnecessary war, a war which was launched on the basis of lies. And we got ourselves into a long war which continued until we decided to
stop it arbitrarily, because it wasn’t working. Then we continued with all sorts of nonsense, but then again, we got into an Iraq war [in 1991], right after the fall of the Soviet Union, but
fortunately, we didn’t make a horrible mistake—we got out of it, before it became a serious war of occupation, which would have ruined us.
But then we went into the Balkan wars, which were ruinous, and we’re suffering in Europe, until today, from the effects of these ongoing Balkan wars, because we haven’t cleaned up the mess we made
with these wars. Then, under the present Administration, we got into a long war in Iraq. We got into it by the blessing of Tony Blair from London, who lied his way all around the world on this one.
And without Tony Blair’s lies, which I personally got involved in defeating—and I got punished for defeating them—we got into another long war, in Southwest Asia, in Afghanistan, in Iraq. We’re now
engaged in a potential war in Iran. We’re now engaged in a generalized war in the entire so-called Arab world, which is now spreading into Turkey, as a threat of destabilization. So, the whole region
is now an area of instability.
In the meantime, we have lost much of our economy. We’ve destroyed it largely through globalization, and largely through laws which allowed hedge funds, and similar kinds of pestilences, to move in
and take us over. Take our industries, take even our government industries, shut them down, and loot them, and move on to loot the next victim. And this is a process I’ve seen in Italy, as I’ve seen
it in the United States. I’ve seen it in France. I see it massively, especially since the Maastricht agreements, in Germany. I see the conditions in Eastern Europe, the former Comecon territories,
where the conditions of life physically are worse than they were under the Soviets. They have the freedom to contemplate and discuss their misery. But their misery is much better than it was then;
that is, it’s much bigger.
So, now we see a stripping of Italy the same way, northern Italy, of the basic industries which were associated since the time of the middle of the 19th Century, with the emergence around [Enrico]
Betti, of the new scientific movements in Italy. And we had a great, for example, aerospace development in Italy, typical of military capabilities, other capabilities. And these industries, on which
this depended, I see are now stripped. I go to Milan, and I find areas where there were large auto industries of high technologies, and small industries—they no longer exist. I see people, skinny
girls marching around on platforms, as a substitute for industry. I see threats to the Italian economy. And my concern in this, looking at it as a part of a world community, is to say, how can we
save the economy from the ravages of this process of globalization?
And then go back to 2005. What I proposed in 2005 was this: that the United States government set up a special corporation, and buy up the parts of the auto industry, especially the high-tech sector,
which we would not be using for automobiles, and to use this high-tech sector of the industry for developing infrastructure. For example, we had dams, rivers, water systems, power systems, municipal
systems, all kinds of things that are essential for an economy, were decrepit. But in the auto industry’s machine-tool sector, we had the capability of fixing every one of these problems.
I simply proposed that the United States government should make emergency legislation; don’t allow these plants to be shut down; but rather keep them functioning by converting them back to
infrastructure programs, and similar kinds of programs, which are urgently needed anyway, and thus to keep the productive technological power of the United States at some kind of a level.
Now we see that was not done. And there was tremendous pressure put on members of the Democratic Party, who I was collaborating with, on this question of defending Social Security. We had a fine
alliance, until it came to this issue of so-called bailing out the automobile industry, by converting it. And today, we don’t have a U.S. auto industry anymore. We have a wreckage, which is being
looted, as a chicken is being looted of the last flesh on its bones. We have a Japan industry which has moved in to take over some of it. But Japan has a cheap-labor industry, so we have a breakdown
in communities, in the state of Michigan, the state of Ohio, the state of Indiana, and elsewhere—a breakdown in the economy of states which is a very serious threat to the stability of the United
States as a whole, because we didn’t do this.
So therefore, my concern in looking at Europe, as well as the United States, is to look at this kind of problem, and say, what do we do?
A Dual-Use Economy
Therefore, it is necessary, as it has been since the period of the Renaissance, it is necessary to maintain the development of economic capabilities which are also the capabilities of national
defense, when national defense is imperilled. This always involves, and has involved, scientific and technological progress, and the development of the skill levels of the population. Therefore, my
concern would be: How can you take the sector of the economy which is still the so-called state sector, and how can you maintain in the state sector, capabilities which are both the core
scientific-technological capabilities, and maintain them in the state sector, even if they’re not in the military sector as such, but maintain them where the conversion to a defense capability
Now, this takes us into areas of new kinds of technologies, which is something which I’m rather notorious for: Always go to new kinds of technologies, more advanced ones, and realize that if you have
to have defense, national defense action, if you’re able to mobilize a competent one, it’s because you have personnel who can be mobilized for that purpose who are efficient, and because you have the
economic capabilities, the forms of technology and otherwise, to make that kind of conversion of the type that Roosevelt made, toward the late part of the 1930s, by developing a program for the first
day he walked into office, knowing that a world war was threatened, and he had to prepare for it, So, his plans for preparing for warfare, and his preparations for developing the economy, were one
and the same thing.
So the idea of the dual-use economy, that is, an economy which has a high-technology orientation, is used immediately for necessary infrastructure or other economic purposes, which gives you the
potential to do this in two ways: one way, in terms of the productive capability as such; secondly, the population.
Now the biggest problem we have, of nations today, is a breakdown of the capabilities of our younger generation. I work largely with an 18-to-35 age group. I concentrated largely, initially, on the
18-to-25 age group. I’ve been doing that ever since about 1999-2000. And what we run into, is the fact that very little attention is being given, effectively, where there’s talk about a youth
movement, and a youth political movement, very little attention is being given to developing the potential creative powers of that generation.
There is a real potential in these young people, these young adults. This is our future. For any generation in history, in my knowledge and my experience of history, it’s always been the development
of a young generation, young adult generation, which is the foundation of the future society. Two generations from the time of entering adulthood, to retirement age, or something like that, has been
the determining factor in the success or failure of society.
As a result of certain changes in the postwar period, typified by the Congress for Cultural Freedom and things like that, we’ve had an existentialist trend in the thinking of the generation which was
born between 1945 and about 1956-57, the first major [postwar] recession. That generation, you will observe, in the United States, is running all the top positions, with very few exceptions. They are
all unresponsive—I have friends among these people—but the problem I have is, they are so unresponsive to certain kinds of problems. They postpone and evade reality. I wouldn’t want them in command
of a military force: They would fail. It’s not the lack of military training, it’s the lack of a sense of commitment to get the job done, the commitment to make the breakthrough.
And what we need, I would think in Italy in particular—I’m cognizant of the problems which exist for Italy here—but the problem, I think, is just that: Is to have a policy of keeping this dual-use
approach to economy in view; to look at this constantly from the standpoint of what may be required through crises in the future, and to concentrate especially on developing cadre levels from among
the young people within the 18-to-35 age group. Because they are the people who are going to think about a future. They’re going to think about what the world looks like two generations from now, 50
years from now. And keeping their morale, and giving them an economy to play with—so to speak—which has dual-use capability, is the resource that you require in any crisis that’s coming up. The
crisis we face globally today is way beyond anything Italy is going to try to take care of. It can’t be done; it’s too big. It has to be done by the giants in the world. But, no nation should give up
its sovereignty just because it’s not in a position to run the world. It has to run its own nation; it has to be a part of the deliberation process among nations.
So that’s my general view.
Dialogue with
Members of the Committee
Sen. Sergio De Gregorio, Chairman: I thank Professor LaRouche for having presented his considerations in such a detailed manner.
Before giving the floor to the other members of the Committee who wish to intervene, I would like to ask a question myself.
In your resumé, I read that you were the political author of what, in 1983, was officially presented by President Ronald Reagan as the Strategic Defense Initiative (SDI). And you also developed an
idea of your own concerning the anti-missile shield, which I would like to ask you to express clearly, in order to deal with a subject that is less general and more technical, which may bring us back
to the military questions in which we are particularly interested. Thus, we would like you to discuss your theories, and do so in relation to the discussion currently underway in our Committee.
Sen. Luigi Ramponi: I would like to refer to what the Chairman just mentioned, and that is, the relevance of the anti-missile shield today.
President Bush has begun his trip to Europe. A procedure has been initiated for the installation of a strategic defense system in Poland and the Czech Republic. This has caused a reaction from
Russia. The Americans claim that the system is necessary in order to prevent, deter, and if necessary, to intervene against, the threat of a missile attack originating from Iran. Russia reacts by
claiming that Iran does not currently have a missile capability sufficient to justify the need for a missile shield. This is the current situation.
I believe that a solution can be found which can be a shared solution, and that it will be fairly easy to reach such a solution once those involved stop acting as separate parties, and when both
countries, if it is necessary, begin to create an anti-missile system, certainly not against Russia, but against whoever wishes to threaten global stability through vectorial nuclear attacks. Do you
think that a solution will be found to this conflict? I think so.
I have always been fascinated by your theories on development, including those which are—to be frank—more detailed than what you presented today, which made reference—I will limit myself to citing a
part of those ideas which I find most interesting: to the realization of large axes of development, which you defined as “infrastructure,” today, across Asia to Europe, and which even foresee a
connection with the American continent. It just so happens that the cover of a magazine [Forum International], which was distributed to us here, shows the project for a tunnel under the sea which
would cross the Bering Strait. Many of the areas you have indicated for the development of the great connections—Afghanistan, Iran, and Iraq—currently find themselves in a very difficult situation.
That is where the northern line was supposed to pass.
Have you changed your view with respect to what you proposed 10 or 15 years ago? Objectively, it doesn’t seem to me that the conditions currently exist to proceed with the realization of these great
axes, which however, could allow for taking a large step forward in the pacification of those territories, ensuring their development. What do you think of such an hypothesis?
You were quite prophetic in predicting a crash of the financial world at the end of the previous century. You said it early, and your prediction was—allow me to use this phrase—“right on.” What is
your expectation regarding the solidity of the financial and stock market worlds today, and in the short-term?
Sen. Lidia Brisca Menapace:[fn_1] Professor LaRouche, I listened to what you said with great interest, including because—please excuse me for pointing this out—one does not expect such an elaborate
cultural outlook from an American. And thus, I feel very comfortable, as if you were a European; this is intended as a compliment. [In response to an interjection from Senator Ramponi:] I certainly
don’t pretend that everyone agrees with my comments!
I was very struck by the fact that, in anticipation of the construction of the anti-missile shield, opposition is coming in particular from Bohemia [the Czech Republic]. It is very strange for an
Eastern European country to react negatively to an American proposal. I would like to know if you consider it correct to think that the opposition coming from that nation is due to the fact that it
was a very important location for high-level industrial production, and that there is still a memory of this, and thus the population feels almost robbed. Otherwise, I would be at a loss to explain
this protest coming from Bohemia, where there are still many street demonstrations on this issue.
I would also like to know if you agree with the possibility of adding the term “scientific” to the expression “military-industrial complex,” since all of the universities are involved in the
development of this policy, with the result that there is an impoverishment, a theft of scientific research, which in this case, is subjugated to other ends.
I also found it very interesting when you stated that the infrastructure which a country must preserve, even a relatively small country like Italy, which should not allow any of its potential to be
expropriated, must be understood above all, at the level of civilian development, which is so interesting and complex, that it can also be used for defense. Do you believe, as I do, that in the
interest of the youth, a policy should emerge aimed at combating the lack of job security (a question which concerns the civilian economy), rather than favoring enlistment in the military? Could this
be a policy of civilian infrastructure which may also be used for the defense of the country, at a time when it is almost primary with respect to an explicit defense of the country?
In your opinion, was the difficulty the United States had in dealing with the [flooding] disaster in New Orleans due to the fact that a policy of civilian infrastructure has not been implemented
because there was a concentration predominantly on a military policy and a military empire? Indeed, it seems strange that a rich, large, powerful country, such as the United States of America, allows
New Orleans, more than a year after the disaster, to remain in conditions which are unacceptable, in which the residents still can not return, to the point that the very nature of a place which is so
significant, important, and well-known in the world’s culture, risks being changed.
Sen. Gianni Nieddu: I would also like to thank the professor for his stimulating intervention.
In closing your presentation, you stated that no state should relinquish its sovereignty, even if that state is so small, that it is unable to deal with large processes at the international level;
therefore, Italy is too small to deal with these processes, but it shouldn’t relinquish its sovereignty. Now, what comes to my mind is the transfer of sovereignty which European nation-states have
carried out in order to allow the construction of the unitary process in Europe, which guaranteed peace after the Second World War, the management of historical conflicts in the great European plains
between France, Germany, the interests of Germany, France, and England, and so forth. This transfer of sovereignty involves all types of power, with the lone exception, until now, of foreign policy
and defense, which have remained under the authority of nation-states; however, an attempt is currently underway concerning defense policy, to transfer part of the powers from the states to the
European Union. It is a difficult, very complex, and contradictory attempt, but on defense policy as well, an attempt is underway to transfer sovereignty from nation-states to the European Union.
Well then, based on these considerations, a question arises: Was this process a mistake? If the size of the Italian state, as well as the German and French states, is not sufficient to effectively
deal with the enormous financial power of the multinational corporations, which are the entities which promote globalization, with an enormous financial power which threatens the sovereignty of these
states because it moves economic interests so large that they are sufficient to condition the economy, as you were saying, to the point of eliminating entire portions of those economies; if the scale
of the state is too small; and if, on the other hand, it is a mistake to relinquish sovereignty in order to have a larger scale (at the continental, European level); then what is the response which
would allow for making supranational economic-financial power coincide with supranational political power?
If the Italian government does not have the power to influence the actions of the multinational corporations by means of its own laws, who can do it, if not a supranational power? We can regulate the
activities of Italian companies, or foreign companies in Italy, but the power of multinational companies is so broad that they are able to avoid this dimension of politics.
Sen. Silvana Pisa: I wish to thank our guest for his very long and complex intervention. I would like to discuss the question of armaments: We are seeing a strong race towards rearmament, both
nuclear and non-nuclear rearmament, and thus a very large increase in spending on armaments in Russia, the United States, and China. Today, this spending is very high, higher than it has ever been in
the past.
Let’s think of the question of the anti-missile shield, which is under discussion in the current period, and these technologies which the United States, by way of bilateral accords with Poland and
the Czech Republic, in some manner wishes to place on Russia’s borders, and which are seen by Putin (we have seen this in Putin’s interviews in recent days) as a threat to the current strategic
balance. I also hope, as Senator Ramponi already stated, that this matter will be resolved positively, but it currently represents an element of destabilization which frankly, we did not need at this
time. However, I believe that the issue is part of a race to rearmament which I see as a serious threat to the global strategic balance. So I pose the question, for example: Why were the nuclear
non-proliferation treaties abandoned? Why, going from nuclear to other fields, did the United States abandon the ABM Treaty in 1992? Why did the United States never ratify the CFE Treaty on
conventional arms?
A second question: To go from warfare to a policy of civilian investment, for reconversion from military to civilian, substantial investments are needed; it’s not so easy. Where can the funds be
found to carry out this reconversion?
LaRouche Responds
Well, first of all, there are a number of questions, since the theme comes up again and again, I think probably I’d better start by answering that first.
The danger now is coming largely from Anglo-American interests, not from U.S. interests. Putin has an accurate perception of what his problem is. His problem is not a U.S. problem. His problem is a
British problem.
Remember, look, you’ve got a situation in which the United States was plunged into two successive long wars, one from 1964 to 1972, and now the more recent wars. These are long wars. They are
Peloponnesian wars, which have the same kind of cause as the original Peloponnesian War. They’re caused by a certain kind of stupidity in the population, the leading circles of the population, called
Sophistry, which means a society which has no principle, and has given up the idea of principle for the sake of popular opinion and expediency, or what is called Sophistry, is no longer capable of
judging how to deal with the situation.
Remember, for example, the case of Louis XI. Louis XI bribed his enemies and made a profit on it! He bribed his enemies to prevent them from going to war. He bribed them not to attack him. And by the
opportunity of peace, he built his economy up to be the model commonwealth economy of Europe, on which Henry VII modelled England. So, the modern nation-state was based on governments which had
principle. The principle was the commonwealth principle. The commonwealth principle was established in Europe, in 1439, with the Council of Florence. It was established, even at a late stage, with a
breakout of the Turkish wars, the disasters that struck the Renaissance with the Turkish wars.
Nicholas of Cusa replied with De Pace Fidei, to seek peace with the enemy, to avoid war, on the basis of the benefits of peace.
The United States Is the Target
Now this was pretty much the U.S. policy, most of the time. We had some corrupt influences, but what we have now is this: We have a determination of some forces to eliminate the sovereign
nation-state. It’s called the post-Westphalia policy. The post-Westphalia policy, which is centered in Britain, is the idea of getting the United States as a Roosevelt-memory state, to destroy
itself, and we are obliging in destroying ourselves. The destruction of the United States caused by the succession of the Indochina War, and what has been going on in Southwest Asia, what has been
going on in Europe, as well as Southwest Asia, with the Balkan wars, which followed the outbreak of the first Iraq War.
These wars are destroying the United States by its own hand, just the way that certain forces destroyed Athens by its own hand, with this kind of foolishness.
So, the United States is the target. We have idiots in the United States who think they’re not the target. They think they are powers that are going on to victory. The United States is not going to
have any world victory coming out of this operation it’s pulling now—it will not happen. It’s foolishness. We’re destroying ourselves. The idea that we’re healthy, we’re gaining, we’re a power: We
are destroying everything! We’re destroying our military! It will take us a generation to rebuild the military that’s been destroyed in this period. We destroyed our army entirely. We destroyed our
military ground reserves. We have only air power and naval power left.
What’s the policy then? The policy is, twofold, under globalization: First of all, the objective is not to put a few missiles in Czechia or in Poland—that is not the policy. That is a stunt, that’s a
diversion, that’s a provocation. The policy is, to build a space-based system of missile systems which can send weapons descending on Earth any time they want to, and to have a U.S. control, or
Anglo-American control, over that system—that’s number one. Number two, is to eliminate all regular military ground forces, controlled by governments. To eliminate governmental control over military
ground forces, and to use private armies. This is called, in the United States, the Revolution in Military Affairs, for which Cheney has been a spokesman, ever since he was Secretary of Defense under
George H.W. Bush.
In fact, what you’re seeing in the world today, for example, is the use of killer games, point-and-shoot killer games, which are producing a new terrorist phenomenon, of our own children who are
becoming fanatics and psychotics in shooting. We have rages of these all over the world, spreading out of the killer computer games, especially since 1999-2000, when the companies that had been
making money on producing computer systems, no longer had large subsidies from the U.S. and other governments, and therefore they went into a new market of producing on a mass base, games trained to
kill people in mass point-and-shoot effects. We trained police forces in this. We trained military forces in this. And we now have people volunteering to do it, on campuses and elsewhere, by killer
games produced by Microsoft and others.
So, this is the key. You have now got a system where we are eliminating the U.S. military ground forces in Iraq. What are we going to replace them with? Well, look at Halliburton! The corporation
that Cheney used to work with. Halliburton, and other companies of that type, are actually being funded massively to conduct the war, while the U.S. military is being destroyed and ground up on the
field. And it will take a generation to rebuild what we have lost in military forces in this period.
A New Kind of Empire
So you have the idea of One World, with a new kind of empire, the new version of the Roman Empire, which is dominated by a space-based system, a monopoly of space-based weapons, which can target any
point on Earth they want to. Which means, eliminate all resistance to the empire.
Number two, eliminate military forces which are national forces, which have national loyalties. Have only professional armies, of people with point-and-shoot killing capability, which you can recruit
from your own youth, who have learned to do point-and-shoot activity blindly. You know, the typical soldier hesitated to kill. They hesitated to do repeated killing. For example, in Vietnam, where
people would train snipers, they would export people as snipers, and they’d train them as snipers. They’d go out and they’d make one kill, and they couldn’t make a second one. The idea of lying on a
trail, lying in manure and everything all night, and waiting for somebody to come down the trail, and shoot them, as a sniper operation, and then do it a second time—the second or third time, they
couldn’t do it any more. Only very especially psychotic people can do that sort of thing.
So, therefore, we’ve now developed a system, which was developed in the U.S. military and otherwise, to train people. How can you train people to become point-and-shoot killers, with no humanity in
their mind?
Take the case in the Bronx. You had a guy of African-American extraction, middle-class guy, no weapons, came out of his house, and the police outside the house said, “Show some identification.” He
reached into his pocket to pull out his wallet—they put 40 slugs into him. Because they’d been through this kind of training program.
So, that’s the thing we’re up against. We’re up against a process to destroy the nation-state as an institution, to destroy national sovereignty, and destroy the idea of civilization as a thing
you’re defending. So, that’s where we’re at. That’s what we’re trying to prevent.
Now, this came up again under the first question, on this question of development. There was a change in 1987—it was referred to by Senator Ramponi. In 1987, we had the depression. We had a
Hoover-style depression. We had an idiot who became the chairman of the Federal Reserve, Greenspan, and Greenspan said, “Wait for me! Don’t do anything!” And he came in with the idea of using the
mortgage-based securities market, and other things, and also the financing of the computer industry, as a fund to print money electronically, as never had been printed before.
We have flooded the world today with the greatest inflation the world has ever imagined. There is no possibility that this monetary system in its present form can continue to exist. It’s doomed, it’s
finished. It’s gambling! The hedge funds are pure gambling. There’s nothing in them. When this thing comes down, everything will come down with it.
Nuclear Energy and the Isotope Economy
Now, what’s the solution? What am I doing about it?
Well, I still follow the same policy which I recommended to Reagan, and Reagan accepted, back in the beginning of the 1980s. We were working for it here in Europe; we were working for it here in
Italy. We had military here in Italy who were supporting that policy. We had people in France, military in France, we had military in Germany supporting that policy. All encouraging the United States
President to go into that policy, which he did. Even after the Soviets turned us down, he went back and made it public, and made the public offer.
Now that was not just a “we don’t shoot you and you don’t shoot us.” The point was, to shift the goals of society, from military conflict goals, to economic cooperation goals. And to take and develop
the kinds of systems where we could mutually eliminate the possibility of such an attack, a surprise attack, this sort of thing. And we could convert that into developing superior technologies which
we’d use for other purposes as well.
Now, what’s happened recently: I was in Moscow for the 80th birthday of an old friend of mine, who was a leading Soviet economist, and other economist, who was the son of the famous Soviet Ambassador
to Washington, Menshikov—Stanislav Menshikov. He’s a famous economist. And I used the occasion, leading into that, through my wife Helga, who is also involved in this, used the occasion to present to
a Russian group, a proposal for the Bering Strait project.
Since then, that proposal was accepted by that circle, and since then, since I was in Moscow, there was more discussion of it. It is my understanding that President Putin is going to present that
proposal at the coming G-8 convention. It’s his intention to do it; he’s already sponsored it. The Russian government has issued a very well-produced pamphlet, which, in English and in Russian, has
this proposal with pictures, including Helga’s picture, my picture, that sort of thing. This has been accepted by certain people in the United States.
Our proposal is that we proceed with it now, for a very simple reason. The world has reached the point, that we can no longer survive without a large-scale conversion to nuclear-fission power
sources. The water issue alone is typical. We cannot maintain freshwater supplies for humanity on the basis required, without nuclear fission as a power source. We need the fourth-generation
fission-type reactor, particular the Jülich type, or the pebble-bed high temperature gas-cooled reactor. We need that.
India is going with such a policy. They recognize it. Every other part of the world is moving in that direction, whether they say so publicly or not, because the issue is clear: Without a
nuclear-fission policy, for dealing with such things as water and sanitation, you cannot deal with the problems of the planet at large. You’ve got 1.4 billion people in China, over 1 billion people
in India. Large populations in Asia. And they have shortages of two things: potable water and a shortage of minerals, which they need for developing industry, because you cannot maintain a poor
population in Asia without having an explosion. China already has internal instabilities as a result of this. India has 70% of its population as part of the same instability. Look at the conditions
throughout Southeast Asia. You need this kind of development
In the long term, we need to go into what’s called an isotope economy, which is, we’ll be able to process the isotopes we require, at very high temperatures in effect, and thus supply humanity with
the means for maintaining a growing population, with a growing technology, a growing standard of living.
Now, this also means that we’re going to change the planet from a maritime planet, into a land-based planet. The significance of the proposed bridge, the Alaska [Bering Strait] bridge, which has been
around for a long time, is that, if you run magnetic-levitation systems, which are superior to the rail systems, if you run that kind of system as a freight system, as well as a passenger system, if
you connect Eurasia to the Americas, and you also solve the Middle East problem, and connect to Africa the same way, by building up a perspective of a long-range system of these kinds of substitutes
for rail systems, we have suddenly taken the interior areas of the continents of the world—we now have made them accessible for coordinated development.
Now, high-speed rail traffic, as well as magnetic levitation, is more efficient than air; and it costs a lot less. It’s more efficient than a highway. So the cheapest way of connecting various parts
of the world economy together—both freight and people—is by building a high-grade magnetic-levitation system, or a transition to that through a good rail system, as a model, so that you can easily
upgrade one to the other.
And with nuclear power, and with the development of thermonuclear-fusion processes, and some of the things that go with that, this is the direction we have to go. And therefore, what we needed then
was the SDI, and our purpose then, was not simply to develop a better military system. It was to develop a system which was necessary for the economy, was necessary for the nations, and more valuable
to the nations than the advantages of winning any war.
Shifting the World’s Attention
to a Higher Level
And the same thing applies today. We always have to look for the peaceful use of technology, and power, and use that as the way we approach the issues of conflict. If we have to go to war, we take it
from the highest level. But we also do these things, not to win a war, nor to fight it; we do these things to prevent a war, by shifting the attention of the world to a higher level. And that’s where
the answer lies, essentially.
The conflict today is not really—you’ve got Bush coming here—the conflict is not really with the United States, and Putin has never thought so. You know, when young Bush was first inaugurated as
President, one of the first guys waiting to meet him was Vladimir Putin, and Vladimir Putin came up beside him, out of the bushes, so to speak, and said, “Let’s talk.” And you had President W. Bush,
George W., talking about his friend “Putie,” in various interviews around the world.
And what Putin has done is very conscious. The inside circles inside Russia, who look at the history books, know the long relationship of friendship between the United States and Russia. And they
also know, particularly, the relationship of Franklin Roosevelt and the view of Franklin Roosevelt in Soviet history, as well as in Russian history generally. That view, in Russia, is shared today in
Putin’s circles.
So, therefore, one of my concerns is to induce the United States to move and take up that option, and my approach is to say, “Let’s take this bridge over the Bering Strait.” It’s a long-term project,
but the idea of taking it up as a commitment, to actually go ahead with it, and to do this in tandem with the four greatest powers on this planet, which today, are the United States, Russia, China,
and India. Now, I’m not proposing a four-power government of the world. I’m proposing simply that, if these four powers, which have, combined, the maximum power in the world, agree, then other
nations, such as Italy, which is looking for partners which it can live with, can easily join with that, and be a voice in a new shaping of the order of the world.
Because this financial system we’re in is coming down. It is finished. There is no way this financial system in its present form can be perpetuated. The present system of the hedge funds is not an
economy—it’s a graveyard. It’s a graveyard of nations, a graveyard of economies. It’s based on looting nations’ material resources. And what is then left of a nation after being looted? You might be
a little bit richer in the short term, but you will have rates of inflation which are enormous. This bubble is going to pop. Therefore, on these kinds of questions we have to think about what are
acceptable long-term agreements for our economies, and the welfare of the future of humanity. What are the technologies, and can we begin to discuss those agreements now to put that on the table
before nations? It comes back on this question that came out up about sovereignty. Why is sovereignty important?
People don’t understand sovereignty. That includes most of the people who are for globalization. Globalization is a new Tower of Babel. It was a bad idea then; it’s a worse idea now. Because people
have forgotten, especially the Baby-Boomer generation: What is the difference between a baboon and a human being? A human being has creative powers. No beast does. And therefore, in all these
solutions, it’s through culture. It’s through our language cultures, and associated culture, that we as a people develop the ability to develop ideas among ourselves. The result of different nations,
according to their culture, in developing ideas, is not a different result; it’s a different road to the result. Because a language culture draws upon the implications of the use of the language over
many generations. You reach into the soul of the people for creative powers, and that should be the objective of this sort of thing.
So, you need a multi-national world, not a globalized world. We need a system of sovereign nation-states. We need a recognition of the terrible threat that we face now. We see the need of coming
together, and getting some big powers together on things that seem impossible. And then, giving hope.
Look at what’s happened to the Italian people. I’ve seen this. What’s happened to them, with the destruction of the industries? What’s happened with the destruction of culture and education? It’s
happened in all European countries. It’s happened in the United States. What’s happened?
The power to think creatively, the power to make and understand scientific discoveries: Classical culture is almost an unknown quality among nations that have been a repository of Classical cultures
in the past centuries. We’ve lost it. It’s the development of the human individual mind, and particularly the power of making discoveries of principle, which are an integral part of a language
culture, and therefore, a nation should be based on language culture, and the nations with different language cultures, should learn to talk to each other.
We did fairly well in European civilization in past times. I think we can do it again.
Senator De Gregorio: I thank Professor LaRouche for his presence and his contribution, which gave rise to an ample debate among the Senators present here. We are pleased with this, because it means
that the remarks and ideas you provided were enthusiastically received.
[fn_1]. An extended discussion between Senator Menapace and LaRouche is in EIR, Vol. 34, No. 22, 2007, pp. 45-50. [back to text for fn_1] | {"url":"https://larouchepub.com/lar/2020/4732-a_strategy_to_defend_the_natio.html","timestamp":"2024-11-10T23:01:09Z","content_type":"text/html","content_length":"71831","record_id":"<urn:uuid:50d0426e-5ae7-46e9-9238-a183697bab65>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00805.warc.gz"} |
Paper EN+PS-MoM5
RF-PECVD Processes Excited by Asymmetric Voltage Waveforms
Monday, October 31, 2011, 9:40 am, Room 103
Session: Plasmas for Photovoltaics & Energy Applications
Presenter: Pierre-Alexandre Delattre, Laboratoire de Physique des Plasmas, France
Authors: P.-A. Delattre, Laboratoire de Physique des Plasmas, France
S. Pouliquen, Laboratoire de Physique des Plasmas, France
E.V. Johnson, Laboratory of Physics of Interfaces and Thin Films, France
J.-P. Booth, Laboratoire de Physique des Plasmas, France
Correspondent: Click to Email
Voltage Waveform Tailoring (VWT) is a promising new technique for Radio-Frequency (RF) process plasma excitation. It is known that asymmetric waveforms resembling peaks (short positive and long
negative voltage) or valleys (long positive, short negative voltage) can produce a voltage self-bias, even in a symmetrical reactor [1], known as the Electrical Asymmetry Effect (EAE). We have
implemented a system to provide such voltage waveforms on the RF electrode of our Capacitively Coupled Plasma (CCP) reactor. For a peak to peak voltage (V[PP]) of 300 V, we can control the self-bias
from -190 V to 15 V, without changing any other process parameter. A new differential RF probe gives us the real-time current and voltage derivatives, and therefore, the instantaneous power. For a
voltage waveform composed of a 15 MHz fundamental and three harmonics, instantaneous power changes from +1 kW to –1kW in 10 ns. Using a hairpin resonator probe in hydrogen at 13 Pa, we have measured
an electron density of 2E8 cm^-3 with a standard sine waveform, 2E9 cm^-3 with a valleys waveform and 2E10 cm^-3 with a peaks waveform (all with V[PP]= 300V). With a view towards photovoltaic
applications, using a gas mixture of 4 % of SiH[4] in H[2] at 65 Pa, we have achieved a deposition rate of high-quality amorphous silicon of 1 Å/s for sine, 2.7 Å/s for valleys, and 3.8 Å/s for peaks
voltage waveforms.
^1Brian G Heil et al 2008 J. Phys. D: Appl. Phys. 41 165202 | {"url":"https://www2.avs.org/symposium2011/Papers/Paper_EN+PS-MoM5.html","timestamp":"2024-11-13T05:52:15Z","content_type":"text/html","content_length":"4341","record_id":"<urn:uuid:1792d5cc-62fb-45de-91e0-29152747c244>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00667.warc.gz"} |
Mth 430 Topics in Mathematical Modeling
Basic introduction to mathematical model building starting with prototype, model purpose definition, and model validation. Models will be chosen from life, the physical and social sciences.
Applications chosen from differential equations, linear programming, group theory, probability or other fields. With approval, this course may be repeated for credit. | {"url":"http://pdx-mobile.smartcatalogiq.com/en/2021-2022/bulletin/courses/mth-mathematical-sciences/400/mth-430","timestamp":"2024-11-07T16:54:40Z","content_type":"text/html","content_length":"37396","record_id":"<urn:uuid:a8489c64-ab1c-4542-9b30-81c715c62ab8>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00597.warc.gz"} |
OpenStax College Physics for AP® Courses, Chapter 1, Problem 16 (Problems & Exercises)
A can contains 375 mL of soda. How much is left after 308 mL is removed?
Question by
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CC BY 4.0
Solution video
OpenStax College Physics for AP® Courses, Chapter 1, Problem 16 (Problems & Exercises)
vote with a rating of votes with an average rating of.
Video Transcript
This is College Physics Answers with Shaun Dychko. A can of soda has 375 milliliters initially so I have the subscript i there to denote initial volume and then there's that change in volume and this
Δ symbol means 'change in'; a change in volume of 308 milliliters against so it's taking away 308 milliliters and I have a negative sign to represent that— you don't have to write it this way but
that's the way I like to think about it. So the change in volume is the final volume minus the initial volume and that means if you wanna solve for final volume, you can add V i to both sides. And so
the final volume is the initial volume plus this change or in other words, 375 milliliters take away 308 milliliters. This is all kind of just a complicated way of saying something obvious which is
you know, the amount left is what you started with minus how much is taken away but it's useful to get used to these kinds of symbols because this delta is gonna appear a lot in your Physics course
and initial and final is gonna appear a lot as well so why not. So we subtract these two and get an answer of 67 milliliters. Now the topic here is you know, precision and significant figures and so
on so let's talk about that. When you are subtracting two numbers, your answer will be precise to a place value that is the same as the least precise number that you are working with. Now in this
particular case, it's straight forward because they both are precise to the ones place and so our answer is also gonna be precise to the ones place; that's the ones place, the ones place and the ones
place there. Had this been a bit different, let's say, 375 milliliters minus 360 milliliters, the answer might seem to be 15 however, that's technically not the right way to do it because this number
is precise to the tens place, this zero is not significant when written in standard form like this; if it was, you know, 3.60 times 10 to the 2 milliliters, well then you would say, yes the zero is
significant but written this way, we have to assume that it's not and so this number is precise to the tens place and so the precision of our answer should be the same as the least precise number
that we are working with which is this one; it's precise to the tens place whereas this one's precise to the ones place and so our answer can be precise only to the tens place that means the answer
is 20 milliliters; it would have to be rounded to just the tens place. Okay, there we go! | {"url":"https://collegephysicsanswers.com/openstax-solutions/can-contains-375-ml-soda-how-much-left-after-308-ml-removed-0","timestamp":"2024-11-08T19:01:29Z","content_type":"text/html","content_length":"116378","record_id":"<urn:uuid:b1a49095-4de7-4b04-a4a6-a73e9408dafa>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00817.warc.gz"} |
A short proof of the infinitesimal Hilbertianity of the weighted Euclidean space
We provide a quick proof of the following known result: the Sobolev space associated with the Euclidean space, endowed with the Euclidean distance and an arbitrary Radon measure, is Hilbert. Our new
approach relies upon the properties of the Alberti-Marchese decomposability bundle. As a consequence of our arguments, we also prove that if the Sobolev norm is closable on compactly-supported smooth
functions, then the reference measure is absolutely continuous with respect to the Lebesgue measure.
arXiv e-prints
Pub Date:
May 2020
□ Mathematics - Functional Analysis;
□ 53C23;
□ 46E35;
□ 26B05
8 pages | {"url":"https://ui.adsabs.harvard.edu/abs/2020arXiv200502924D/abstract","timestamp":"2024-11-13T16:41:05Z","content_type":"text/html","content_length":"36264","record_id":"<urn:uuid:83da1195-5f64-4cf5-88c2-67b67518986a>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00883.warc.gz"} |
Modules Description
Automata This module implements functions for working with automata, in particular, determinizing them by the subset and the breakpoint constructions, as well as converting omega-automata to
other acceptance conditions. Automata are represented in either a fully explicit enumeration of their states and transitions, a fully symbolic representation, or a semi-symbolic one.
This module implements BDDs where nodes of the BDD refer directly to boolean expressions. To use the module, the local data structures must first be initialized with a sequence of
boolean expressions using the Initialize function. The given sequence of boolean expressions determines the set of variables and the variable order by mapping the listed expressions to
BDD BDD variables (of type BDDIndex) in the given order. After that, the typical BDD functions can be used. From time to time, the user should call the GarbageCollect function to collect
garbage nodes that can be reused later. Note that garbage collection resets the computed table, so garbage collection should not be called too often, otherwise performance will
decrease. The order is stored in Index2BoolExpr and can be changed with the SwapVar function. If the variable order is changed, the computed table must be reset.
Functions of this model deal with the theory of equality of uninterpreted functions (EUF). To this end, reductions are provided that endow given EUF formulas with additional
EqualityTheory constraints so that the satisfiability or validity can be decided by means of a propositional SAT solver. In particular, constraints are added to ensure the congruence of functions,
i.e. x1=x2 must imply f(x1)=f(x2), and the transitivity of equations. The generated assumptions are added as conjunctions asm&phi for checking the satisfiability, and as an implication
asm->phi for checking the validity of an EUF formula phi.
LogicMin This module implements algorithms for logic minimization
Matrices This module implements algorithms for matrices on rational numbers. To this end, it first implements a type for rational and complex numbers, and based on these, algorithms for
inverting a matrix, for solving linear (in)equation systems, and for computing eigenvalues (e.g. to solve linear differential equation systems) are provided.
SatSolver This module implements functions for checking the satisfiability of boolean expressions. The SAT checking procedures are based on Shannon graphs which are unreduced unordered binary
decision diagrams. This has the advantage that one can construct an equivalent Shannon graph for a given BoolExpr in linear time.
The functions of this module provide translations from temporal logics to symbolically encoded omega automata. In general, these translations replace an elementary subformula (one that
starts with a temporal operator) by a new state variable of the automaton and according state transitions and contraints so that the new state variable becomes equivalent to the
elementary subformula it abbreviates. To this end, one usually makes use of GF-constraints which however are harder to check than others. They cannot always be avoided, but the
TemporalLogic translators presented here may try to use F-constraints whenever possible which then usually generates a cascade of F-constraints, i.e. formulas of the following form where phi is
propositional: CF ::= F phi | F(phi&CF) | X CF | CF & CF. The use of F or CF-constraints is controlled with option tryConstrF of the translators. In addition to the use of GF or
F-constraints, the translators differ also in the acceptance condition they finally produce: LTL2Omega will just have these constraints while LTL2OmegaCTL will rather use a CTL formula
(equivalent to an LeftCTL*-LTL formula) with potential GF-constraints. In case F-constraints should be used LTL2OmegaCTL incorporates them to the CTL formula. | {"url":"http://www.averest.org/AverestLibDoc/reference/averest-analysis.html","timestamp":"2024-11-14T00:31:43Z","content_type":"text/html","content_length":"18080","record_id":"<urn:uuid:6773f420-4015-4df0-b916-4b4b0d538270>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00547.warc.gz"} |
Denotational semantics
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In computer science, denotational semantics (initially known as mathematical semantics or Scott–Strachey semantics) is an approach of formalizing the meanings of programming languages by constructing
mathematical objects (called denotations) that describe the meanings of expressions from the languages. Other approaches provide formal semantics of programming languages including axiomatic
semantics and operational semantics.
Broadly speaking, denotational semantics is concerned with finding mathematical objects called domains that represent what programs do. For example, programs (or program phrases) might be represented
by partial functions or by games between the environment and the system.
An important tenet of denotational semantics is that semantics should be compositional: the denotation of a program phrase should be built out of the denotations of its subphrases.
Historical development[edit]
Denotational semantics originated in the work of Christopher Strachey and Dana Scott published in the early 1970s.^[1] As originally developed by Strachey and Scott, denotational semantics provided
the denotation (meaning) of a computer program as a function that mapped input into output.^[2] To give denotations to recursively defined programs, Scott proposed working with continuous functions
between domains, specifically complete partial orders. As described below, work has continued in investigating appropriate denotational semantics for aspects of programming languages such as
sequentiality, concurrency, non-determinism and local state.
Denotational semantics have been developed for modern programming languages that use capabilities like concurrency and exceptions, e.g., Concurrent ML,^[3] CSP,^[4] and Haskell.^[5] The semantics of
these languages is compositional in that the denotation of a phrase depends on the denotations of its subphrases. For example, the meaning of the applicative expression f(E1,E2) is defined in terms
of semantics of its subphrases f, E1 and E2. In a modern programming language, E1 and E2 can be evaluated concurrently and the execution of one of them might affect the other by interacting through
shared objects causing their denotations to be defined in terms of each other. Also, E1 or E2 might throw an exception which could terminate the execution of the other one. The sections below
describe special cases of the semantics of these modern programming languages.
Denotations of recursive programs[edit]
Denotational semantics are given to a program phrase as a function from an environment (that has the values of its free variables) to its denotation. For example, the phrase n*m produces a denotation
when provided with an environment that has binding for its two free variables: n and m. If in the environment n has the value 3 and m has the value 5, then the denotation is 15.
A function can be modeled as denoting a set of ordered pairs where each ordered pair in the set consists of two parts (1) an argument for the function and (2) the value of the function for that
argument. For example, the set of order pairs {[0 1] [4 3]} is the denotation of a function with value 1 for argument 0, value 3 for the argument 4, and is otherwise undefined.
The problem to be solved is to provide denotations for recursive programs that are defined in terms of themselves such as the definition of the factorial function as
factorial ≡ λ(n) if (n==0) then 1 else n*factorial(n-1).
A solution is to build up the denotation by approximation. The factorial function is a total function from ℕ to ℕ (defined everywhere in its domain), but we model it as a partial function. At the
beginning, we start with the empty function (an empty set). Next, we add the ordered pair [0 1] to the function to result in another partial function that better approximates the factorial function.
Afterwards, we add yet another ordered pair [1 1] to create an even better approximation.
It is instructive to think of this chain of iteration as F^0, F^1, F^2, … where F^i indicates i-many applications of F.
• F^0({}) is the totally undefined partial function {}
• F^1({}) is the function {[0 1]} that is defined at 0, to be 1, and undefined elsewhere;
• F^5({}) is the function {[0 1] [1 1] [2 2] [3 6] [4 24]}
This iterative process builds a sequence of partial functions from ℕ to ℕ. Partial functions form a chain-complete partial order using ⊆ as the ordering. Furthermore, this iterative process of better
approximations of the factorial function forms an expansive (also called progressive) mapping because each ${\displaystyle F^{i}\leq F^{i+1}}$ using ⊆ as the ordering. So by a fixed-point theorem
(specifically Bourbaki–Witt theorem), there exists a fixed point for this iterative process.
In this case, the fixed point is the least upper bound of this chain, which is the full factorial function, which can be expressed as the direct limit
${\displaystyle \bigsqcup _{i\in \mathbb {N} }F^{i}(\{\}).}$
Here, the symbol "⊔" is the directed join (of directed sets), meaning "least upper bound". The directed join is essentially the join of directed sets.
Denotational semantics of non-deterministic programs[edit]
The concept of power domains has been developed to give a denotational semantics to non-deterministic sequential programs. Writing P for a power-domain constructor, the domain P(D) is the domain of
non-deterministic computations of type denoted by D.
There are difficulties with fairness and unboundedness in domain-theoretic models of non-determinism.^[6]
Denotational semantics of concurrency[edit]
Many researchers have argued that the domain-theoretic models given above do not suffice for the more general case of concurrent computation. For this reason various new models have been introduced.
In the early 1980s, people began using the style of denotational semantics to give semantics for concurrent languages. Examples include Will Clinger's work with the actor model; Glynn Winskel's work
with event structures and petri nets;^[7] and the work by Francez, Hoare, Lehmann, and de Roever (1979) on trace semantics for CSP.^[8] All these lines of inquiry remain under investigation (see e.g.
the various denotational models for CSP^[4]).
Recently, Winskel and others have proposed the category of profunctors as a domain theory for concurrency.^[9]^[10]
Denotational semantics of state[edit]
State (such as a heap) and simple imperative features can be straightforwardly modeled in the denotational semantics described above. All the textbooks below have the details. The key idea is to
consider a command as a partial function on some domain of states. The denotation of "x:=3" is then the function that takes a state to the state with 3 assigned to x. The sequencing operator ";" is
denoted by composition of functions. Fixed-point constructions are then used to give a semantics to looping constructs, such as "while".
Things become more difficult in modelling programs with local variables. One approach is to no longer work with domains, but instead to interpret types as functors from some category of worlds to a
category of domains. Programs are then denoted by natural continuous functions between these functors.^[11]^[12]
Denotations of data types[edit]
Many programming languages allow users to define recursive data types. For example, the type of lists of numbers can be specified by
datatype list = Cons of nat * list | Empty
This section deals only with functional data structures that cannot change. Conventional imperative programming languages would typically allow the elements of such a recursive list to be changed.
For another example: the type of denotations of the untyped lambda calculus is
datatype D = D of (D → D)
The problem of solving domain equations is concerned with finding domains that model these kinds of datatypes. One approach, roughly speaking, is to consider the collection of all domains as a domain
itself, and then solve the recursive definition there. The textbooks below give more details.
Polymorphic data types are data types that are defined with a parameter. For example, the type of α lists is defined by
datatype α list = Cons of α * α list | Empty
Lists of natural numbers, then, are of type nat list, while lists of strings are of string list.
Some researchers have developed domain theoretic models of polymorphism. Other researchers have also modeled parametric polymorphism within constructive set theories. Details are found in the
textbooks listed below.
A recent research area has involved denotational semantics for object and class based programming languages.^[13]
Denotational semantics for programs of restricted complexity[edit]
Following the development of programming languages based on linear logic, denotational semantics have been given to languages for linear usage (see e.g. proof nets, coherence spaces) and also
polynomial time complexity.^[14]
Denotational semantics of sequentiality[edit]
The problem of full abstraction for the sequential programming language PCF was, for a long time, a big open question in denotational semantics. The difficulty with PCF is that it is a very
sequential language. For example, there is no way to define the parallel-or function in PCF. It is for this reason that the approach using domains, as introduced above, yields a denotational
semantics that is not fully abstract.
This open question was mostly resolved in the 1990s with the development of game semantics and also with techniques involving logical relations.^[15] For more details, see the page on PCF.
Denotational semantics as source-to-source translation[edit]
It is often useful to translate one programming language into another. For example, a concurrent programming language might be translated into a process calculus; a high-level programming language
might be translated into byte-code. (Indeed, conventional denotational semantics can be seen as the interpretation of programming languages into the internal language of the category of domains.)
In this context, notions from denotational semantics, such as full abstraction, help to satisfy security concerns.^[16]^[17]
It is often considered important to connect denotational semantics with operational semantics. This is especially important when the denotational semantics is rather mathematical and abstract, and
the operational semantics is more concrete or closer to the computational intuitions. The following properties of a denotational semantics are often of interest.
1. Syntax independence: The denotations of programs should not involve the syntax of the source language.
2. Soundness: All observably distinct programs have distinct denotations;
3. Full abstraction: Two programs have the same denotations precisely when they are observationally equivalent. For semantics in the traditional style, full abstraction may be understood roughly as
the requirement that "operational equivalence coincides with denotational equality". For denotational semantics in more intensional models, such as the actor model and process calculi, there are
different notions of equivalence within each model, and so the concept of full abstraction is a matter of debate, and harder to pin down. Also the mathematical structure of operational semantics
and denotational semantics can become very close.
Additional desirable properties we may wish to hold between operational and denotational semantics are:
1. Constructivism: Constructivism is concerned with whether domain elements can be shown to exist by constructive methods.
2. Independence of denotational and operational semantics: The denotational semantics should be formalized using mathematical structures that are independent of the operational semantics of a
programming language; However, the underlying concepts can be closely related. See the section on Compositionality below.
3. Full completeness or definability: Every morphism of the semantic model should be the denotation of a program.^[18]
An important aspect of denotational semantics of programming languages is compositionality, by which the denotation of a program is constructed from denotations of its parts. For example, consider
the expression "7 + 4". Compositionality in this case is to provide a meaning for "7 + 4" in terms of the meanings of "7", "4" and "+".
A basic denotational semantics in domain theory is compositional because it is given as follows. We start by considering program fragments, i.e. programs with free variables. A typing context assigns
a type to each free variable. For instance, in the expression (x + y) might be considered in a typing context (x:nat,y:nat). We now give a denotational semantics to program fragments, using the
following scheme.
1. We begin by describing the meaning of the types of our language: the meaning of each type must be a domain. We write 〚τ〛 for the domain denoting the type τ. For instance, the meaning of type
nat should be the domain of natural numbers: 〚nat〛= ℕ[⊥].
2. From the meaning of types we derive a meaning for typing contexts. We set 〚 x[1]:τ[1],..., x[n]:τ[n]〛 = 〚 τ[1]〛× ... ×〚τ[n]〛. For instance, 〚x:nat,y:nat〛= ℕ[⊥]×ℕ[⊥]. As a special case,
the meaning of the empty typing context, with no variables, is the domain with one element, denoted 1.
3. Finally, we must give a meaning to each program-fragment-in-typing-context. Suppose that P is a program fragment of type σ, in typing context Γ, often written Γ⊢P:σ. Then the meaning of this
program-in-typing-context must be a continuous function 〚Γ⊢P:σ〛:〚Γ〛→〚σ〛. For instance, 〚⊢7:nat〛:1→ℕ[⊥] is the constantly "7" function, while 〚x:nat,y:nat⊢x+y:nat〛:ℕ[⊥]×ℕ[⊥]→ℕ[⊥] is the
function that adds two numbers.
Now, the meaning of the compound expression (7+4) is determined by composing the three functions 〚⊢7:nat〛:1→ℕ[⊥], 〚⊢4:nat〛:1→ℕ[⊥], and 〚x:nat,y:nat⊢x+y:nat〛:ℕ[⊥]×ℕ[⊥]→ℕ[⊥].
In fact, this is a general scheme for compositional denotational semantics. There is nothing specific about domains and continuous functions here. One can work with a different category instead. For
example, in game semantics, the category of games has games as objects and strategies as morphisms: we can interpret types as games, and programs as strategies. For a simple language without general
recursion, we can make do with the category of sets and functions. For a language with side-effects, we can work in the Kleisli category for a monad. For a language with state, we can work in a
functor category. Milner has advocated modelling location and interaction by working in a category with interfaces as objects and bigraphs as morphisms.^[19]
Semantics versus implementation[edit]
According to Dana Scott (1980):^[20]
It is not necessary for the semantics to determine an implementation, but it should provide criteria for showing that an implementation is correct.
According to Clinger (1981):^[21]^:79
Usually, however, the formal semantics of a conventional sequential programming language may itself be interpreted to provide an (inefficient) implementation of the language. A formal semantics
need not always provide such an implementation, though, and to believe that semantics must provide an implementation leads to confusion about the formal semantics of concurrent languages. Such
confusion is painfully evident when the presence of unbounded nondeterminism in a programming language's semantics is said to imply that the programming language cannot be implemented.
Connections to other areas of computer science[edit]
Some work in denotational semantics has interpreted types as domains in the sense of domain theory, which can be seen as a branch of model theory, leading to connections with type theory and category
theory. Within computer science, there are connections with abstract interpretation, program verification, and model checking.
Further reading[edit]
Lecture notes
Other references
• Greif, Irene (August 1975). Semantics of Communicating Parallel Processes (PDF) (PhD). Project MAC. Massachusetts Institute of Technology. ADA016302.
• Plotkin, G.D. (1976). "A powerdomain construction". SIAM J. Comput. 5 (3): 452–487. CiteSeerX 10.1.1.158.4318. doi:10.1137/0205035.
• Edsger Dijkstra. A Discipline of Programming Prentice Hall. 1976.
• Krzysztof R. Apt, J. W. de Bakker. Exercises in Denotational Semantics MFCS 1976: 1-11
• de Bakker, J.W. (1976). "Least Fixed Points Revisited". Theor. Comput. Sci. 2 (2): 155–181. doi:10.1016/0304-3975(76)90031-1.
• Smyth, Michael B. (1978). "Power domains". J. Comput. Syst. Sci. 16: 23–36.
• Francez, Nissim; Hoare, C.A.R.; Lehmann, Daniel; de Roever, Willem-Paul (December 1979). Semantics of nondeterminism, concurrency, and communication. Journal of Computer and System Sciences.
Lecture Notes in Computer Science. 64. pp. 191–200. doi:10.1007/3-540-08921-7_67. hdl:1874/15886. ISBN 978-3-540-08921-6.
• Lynch, Nancy; Fischer, Michael J. (1979). "On describing the behavior of distributed systems". In Kahn, G. Semantics of concurrent computation: proceedings of the international symposium, Évian,
France, July 2-4, 1979. Springer. ISBN 978-3-540-09511-8.
• Schwartz, Jerald (1979). "Denotational semantics of parallelism". Kahn 1979.
• Wadge, William (1979). "An extensional treatment of dataflow deadlock". Kahn 1979.
• Ralph-Johan Back. "Semantics of Unbounded Nondeterminism" ICALP 1980.
• David Park. On the semantics of fair parallelism Proceedings of the Winter School on Formal Software Specification. Springer-Verlag. 1980.
• Clinger, W.D. (1981). "Foundations of Actor Semantics" (PhD). Massachusetts Institute of Technology. hdl:1721.1/6935. AITR-633.
• Allison, L. (1986). A Practical Introduction to Denotational Semantics. Cambridge University Press. ISBN 978-0-521-31423-7.
• America, P.; de Bakker, J.; Kok, J.N.; Rutten, J. (1989). "Denotational semantics of a parallel object-oriented language". Information and Computation. 83 (2): 152–205. doi:10.1016/0890-5401(89)
• Schmidt, David A. (1994). The Structure of Typed Programming Languages. MIT Press. ISBN 978-0-262-69171-0.
External links[edit]
The Wikibook Haskell has a page on the topic of: Denotational semantics | {"url":"https://static.hlt.bme.hu/semantics/external/pages/sz%C3%A1m%C3%ADt%C3%B3g%C3%A9pes_program_szemantik%C3%A1ja/en.wikipedia.org/wiki/Denotational_semantics.html","timestamp":"2024-11-11T13:23:29Z","content_type":"text/html","content_length":"115287","record_id":"<urn:uuid:1dacee40-9c60-472a-bfec-256da5a9ef8d>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00605.warc.gz"} |
MS-E1370 - Analysis, Random Walks and Groups, Project, 11.1.2023-23.2.2023
Please note! Course description is confirmed for two academic years, which means that in general, e.g. Learning outcomes, assessment methods and key content stays unchanged. However, via course
syllabus, it is possible to specify or change the course execution in each realization of the course, such as how the contact sessions are organized, assessment methods weighted or materials used.
The course will show how fundamental concepts and tools from analysis, probability and algebra can be used together to describe long-time asymptotic behaviour of random walks on groups. On successful
completion of this course unit students will be able to:
1. define total variation distances between probability distributions on the group Zp = {0,1,...,p-1} equipped with addition mod p, calculate and estimate these distances for various distributions in
2. define entropy of probability distributions in Zp, compute and estimate entropy for various examples in Zp and relate entropy to the total variation distance
3. define and compute convolutions of probability distributions on Zp, model random walks as iterated convolutions and estimate probabilities of events using iterated convolutions,
4. define Fourier transforms on the group Zp and estimate Fourier transforms of probability distributions and their convolutions on Zp,
5. prove fundamental theorems in harmonic analysis such as convolution theorem and Plancherel’s theorem in Zp
6. outline the calculations and estimates of finding the total variation distances of convolutions of prob- ability distributions to the uniform distribution in Zp and alter these proofs in other
examples with dierent constants or parameters,
7. explain the key ideas of the theorems and methods presented in the course and describe how each component (harmonic analysis, random walks and group theory) come into play,
8. apply the methods presented in the course and prove similar results for analogous contexts such as random walks on higher dimensional lattices (hypercube Zd^2 and the torus Zd^p), matrix groups
(GL(Zp)), models for card shuffling in the symmetric group, dice rolling or models for Rubik’s cube scrambling as subgroups of the symmetric group
Credits: 5
Schedule: 11.01.2023 - 23.02.2023
Teacher in charge (valid for whole curriculum period):
Teacher in charge (applies in this implementation): Tuomas Sahlsten
Contact information for the course (applies in this implementation):
CEFR level (valid for whole curriculum period):
Language of instruction and studies (applies in this implementation):
Teaching language: English. Languages of study attainment: English
• valid for whole curriculum period:
How many times should we shuffle a deck of 52 cards to make it “sufficiently random”? What types of shuffling work best? These questions can be answered by realising the card shuffling as a
random walk on the symmetric group of 52 elements and employing fundamental tools from Harmonic Analysis to compute the answers. In the case of riffle shuffles the surprising answer is that after
roughly 6 shuffles the deck will still be quite ordered, but at the 7th shuffle the deck suddenly becomes very random! How about what are the typical transmutation sequences that transform the
genome of a human’s X chromosome into that of a mouse? Moreover, similar ideas can also be realised when scrambling a Rubik’s Cube and asking how “random” the scramble is.
The topics of the course form an introduction to a currently a very exciting and emerging field, using ideas involving the combinatorial properties of groups, analysis, dynamical systems and
probability theory. The course has interest from both real world perspective but also in modern pure mathematics research (harmonic analysis and connections to mixing rates of of dynamical
systems, random walks and representation theory on groups, modeling chaos theory in quantum mechanics, fractal geometry, etc.). The course suits well for anyone with any interest to analysis,
probability or algebra even if weaker in the others. We will revise all the topics in the beginning of the course. The course starts with the basics by reviewing first year probability notions
and introducing probabilistic tools such as convolution, which are natural in the context of groups. For simplicity we will first concentrate on the circle group Zp, but many of the core ideas
are similar in more complicated groups such as the symmetric group. We will introduce fundamental topics from Harmonic Analysis such as Fourier transform and demonstrate how they can be applied
here. Detailed contents are listed below:
Week 1: Introduction, probability measures and information theory on Zp
- Introduction to natural models for random walks on groups such as card shuffles, Rubik’s cube, dice rolling, random mutations of genes and the Ehrenfest Urn model in statistical mechanics
- Revision on fundamentals of group theory in the group Zp
- Revision on fundamentals of probability and analysis in the group Zp
- Probability distributions in distributions in Zp, Lebesgue/Uniform and Dirac/Singular distributions
- Definition of total variation distance between probability distributions on Zp
- Entropy of probability distributions in Zp
- Computing the entropy and total variation distances in Zp, L identity and Pinsker’s inequality
Week 2: Convolution, additive combinatorics, dynamics and random walks on Zp
- Definition and heuristics of convolution
- Sumsets, additive combinatorics and relation to the support of the convolution
- Realising random walks as iterated convolutions on Zp
- Transfer operator and entropy growth under convolutions
- Ergodicity of a random walk in Zp
- Non-concentration of random walks on subgroups
- Ergodic theorem
Week 3: Introduction to Fourier analysis and L2 theory in Zp
- Mixing of a random walk in Zp
- Fourier transform in Zp
- Inverse Fourier transform in Zp
- Spectral gap of probability distributions
- Lp norms, inner product and Cauchy-Schwartz inequality
Week 4: Convolution theorem, L2 theory and long-time asymptotic behaviour of random walks on Zp
- Convolution theorem in Zp
- Plancherel theorem in Zp
- Upper Bound Lemma and Lower Bound Lemma in Zp.
- Applying the Upper Bound Lemma to prove ergodicity and mixing of a random walk in Zp
- Applying the Lower Bound Lemma to find spectral gap for random walks in Zp
Weeks 5-6: Extending the ideas to general finite groups G and applications
- Examining how probability distributions, total variation distance, convolution and random walks can all be defined and analysed with same proofs in general finite groups G such as the hypercube
Zd^2, torus Zd^p and symmetric group S_n
- Introduction to harmonic analysis / representation theory on finite groups: irreducible representations of G, Schur’s lemma, dual group G‚, Hilbert-Schmidt inner products and Plancherel’s
theorem in G
- Upper Bound Lemma in G and its proof
- Modelling card shuffling as a random walk on the symmetric group such as random transpositions, Borel’s shuffles, riffle shuffles, overhand shuffles and using the Upper Bound Lemma for G = S_52
to establish mixing of random transposition shuffles to find the number of shuffles it takes to mix a deck to be sufficiently random
Assessment Methods and Criteria
• valid for whole curriculum period:
There are three ways to pass the course, please choose one and only one of them::
1. Individual written research project on the topics of the course, please discuss with the lecturer on the possible topics and details
2. Exercises + Final exam with both counting towards final mark
3. 100% Final exam
Substitutes for Courses
• valid for whole curriculum period:
• valid for whole curriculum period:
Further Information
• valid for whole curriculum period:
Teaching Language : English
Teaching Period : 2022-2023 Spring III
2023-2024 No teaching | {"url":"https://mycourses.aalto.fi/course/info.php?id=38308","timestamp":"2024-11-03T04:10:13Z","content_type":"text/html","content_length":"95858","record_id":"<urn:uuid:f5eb71cb-f8ef-494d-8088-1b3e79a6e845>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00286.warc.gz"} |
An Open Letter to Panic Buyers
I wrote and twitted this in the midst of the first wave of the Covid-19 outbreak in 2020.
Dear Panic Buyer,
Here is something to think about.
Say there are 100 pieces of bread in the country and 100 people, and each person needs 1 on average. The country on the other hand, has the capacity to make 200 pieces if needed. If ,say, a handful
of people among a population of 100 start stockpiling 4 times their needs in their house, this will be:
1. pointless, as there is enough supply for anyone anyhow.
2. nonetheless, harmless.
If ,on the other hand, 1/2 of the population of a 100 start stockpiling 4 times more than their
needs, then we end up in this situation where the total demand (250) exceeds the total capacity of the country (200), for no good reason. The is problematic because it is not a real problem. It is
created merely because of our excessive needs in the first place.
Of course, a virus spread might actually lead to a real shortage of some supplies (i.e. where the
example country above can produce 80 or less) as well. But we are not there yet. And in the path of getting there, we will inevitably reach the above situation as well. Hence, the first phase of any
real shortage, if that is what you are preparing for by buying, is cause by ourselves, not any virus.
That being said, also note that if the situation becomes so severe that even developed countries
fail to provide food and basic needs, we'd probably have much bigger problem than food (electricity, sewage, hygiene, lack of healthcare etc.). Without food, we can technically survive for a fairly
long time, weeks if needed. | {"url":"https://blog.kianenigma.com/posts/personal/an-open-letter-to-panic-buyers/","timestamp":"2024-11-10T18:44:31Z","content_type":"text/html","content_length":"35010","record_id":"<urn:uuid:1a0e90f7-b4c1-43d5-80d9-e5fdbe882049>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00602.warc.gz"} |
Calculate the arc elasticity of demand between each point and its
Use the following demand schedule to answer the question parts below.
Quantity Demanded
Quantity Supplied
1. Calculate the arc elasticity of demand between each point and its neighbor (that is, from A to B, B to C, etc.) and determine whether each value is price elastic, price inelastic, or unit elastic
2. Calculate the arc elasticity of supply between each point and its neighbor (that is, from A to B, B to C, etc.) and determine whether each value is price elastic, price inelastic, or unit elastic
3. Calculate the arc elasticity of demand between A and F, A and D, and A and B and determine whether each value is price elastic, price inelastic, or unit elastic
4. Calculate the arc elasticity of supply between A and F, A and D, and A and B and determine whether each value is price elastic, price inelastic, or unit elastic
The post Calculate the arc elasticity of demand between each point and its appeared first on homeworkcrew.com.
Source link | {"url":"https://academicacers.com/calculate-the-arc-elasticity-of-demand-between-each-point-and-its/","timestamp":"2024-11-04T21:08:45Z","content_type":"text/html","content_length":"62326","record_id":"<urn:uuid:38fa702a-4b69-4e45-9700-f0e055c97b3a>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00658.warc.gz"} |
Structural Damping in a Pedestrian Footbridge Under Controlled Traffic Density
Volume 13, Issue 07 (July 2024)
Structural Damping in a Pedestrian Footbridge Under Controlled Traffic Density
DOI : 10.17577/IJERTV13IS070060
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Jesus Emmanuel CeróN-Carballo, Eber PéRez-Isidro, Humberto Ivan Navarro-Gomez, Cutberto Rodriguez Alvarez, 2024, Structural Damping in a Pedestrian Footbridge Under Controlled Traffic Density,
INTERNATIONAL JOURNAL OF ENGINEERING RESEARCH & TECHNOLOGY (IJERT) Volume 13, Issue 07 (July 2024),
• Open Access
• Authors : Jesus Emmanuel CeróN-Carballo, Eber PéRez-Isidro, Humberto Ivan Navarro-Gomez, Cutberto Rodriguez Alvarez
• Paper ID : IJERTV13IS070060
• Volume & Issue : Volume 13, Issue 07 (July 2024)
• Published (First Online): 28-07-2024
• ISSN (Online) : 2278-0181
• Publisher Name : IJERT
• License: This work is licensed under a Creative Commons Attribution 4.0 International License
Text Only Version
Structural Damping in a Pedestrian Footbridge Under Controlled Traffic Density
Jesus Emmanuel Cerón-Carballo,
Eber Pérez-Isidro, Humberto Ivan Navarro-Gomez Cutberto Rodriguez Alvarez
IngenierÃa Civil Forense
à rea Académica de IngenierÃa y Arquitectura Universidad Autónoma del Estado de Hidalgo Mineral de la Reforma, Hidalgo, México
Abstract In this paper we present the study of the displacement due to the movement between the pedestrian traffic and the concrete footbridge of a pedestrian bridge located in the Matilde Town,
Hidalgo State, México, in which, by means of a portable equipment, the acceleration of the pedestrian's center of gravity is measured when walking. The pedestrian is idealized in a dynamic scheme
consisting of a mass that produces a temporal stimulation corresponding to the potential response of the system, the load due to the step density of the basic mechanics of walking is simulated in a
mathematical scheme, considering the vertical and lateral displacement on the catwalk, formulating matrix with the equation of movement through Lagrangean mechanics, four scenarios are studied, where
the simulations of the energy contribution of the periodic load of six pedestrian masses are presented, with obtained results that show the structural behavior, which help to identify the service
conditions of the structure of the pedestrian bridge.
Keywords pedestrian; Pedestrian bridge; pedestrian density; accelerometer; Lagrangean mechanics.
1. INTRODUCTION.
The pedestrian bridges at every moment are subject to external forces[1], as the intensity of pedestrian traffic, what causes variation in the traffic of the footbridge[2], said traffic is
regulated according to pedestrian locomotion standards, of the dimensions of the catwalk and the speed of walking, this last stochastic condition varies according to time, and its effect causes
levels of charges congestion[3].
Human-structure interaction models on bridges, are used to assess the vibration, the methodology to monitor long-term said vibrations at present is very sophisticated, and it serves to identify
and know the effects that the mobile force produces general dependent on time, where the pedestrian is idealized in a force with his mass moving[4].
The applications to identify scenarios are five and their regulatory classification used in[5], shows four according to traffic density and one considering fifteen pedestrians transiting, In this
paper we will analyze the four densities according to the way in which the pedestrian load is presented in a determined surface with a controlled crowd, therefore, in the first (Tc1), Weak
trafic, in an area of five square meters, in the second (Tc2), Dense trafic, in two square meters, in the third (Tc3), Very dense trafic, in one square meter and the fourth (Tc4), Exceptionally
dense trafic, in sixty-seven square centimeters [3][5].
By analyzing the sample, the interaction between the crowd and the structure of the bridge is quantified, obtaining a normal distribution, where the average stimulation frequency and its standard
deviation, maintain known criteria as in[6], when considering only one phase angle, it is taken into account that, the arrival and the momentum|, maintains a surface of the pedestrian walk,
represented imaginaryly, through a circle, such form who, the magnitude of the radius, corresponds to the density of pedestrian traffic, and plus much, there is a control so that adjacent
pedestrians do not invade said space, so then, the crowd moves along the bridge, timing the acceleration, until the determination of the average transit speed of 1.25 m/s[7].
One of the factors to be taken into account in the sample, is the variability of the weight of each pedestrian, which contributes to causing phase differences between pedestrians, effects of
uncontrolled traffic density, lack of synchronization of walking between pedestrians and the decompensated vibration of the structure [8], this variability is highly related to probability,
generates a problem whose solution is not feasible that can be addressed by deterministic terms [9].
Pedestrians are more susceptible to horizontal vibration than to vertical vibration, vertical displacements are needed so that pedestrians can be bothered, because the human movement and the
internal forces associated with this movement are reciprocal to the generation induced by man, these internal forces in the structure support activities such as walking, run or jump, on the other
hand, these conditions of life reveal to us that the mechanics of human locomotion have been the subject of a high level of research [10]. Therefore, The forces produced by rhythmic activities
can be represented by considering the sum of the dynamic component, represented with the Fourier series of forced rhythm, in this way, the normalized function of the applied static weight is
determined [11].
2. DYNAMIC SYSTEM
The dynamic system approach of the present work, consider two schemes, The first consists of a physical model, and the second in a mathematical model, both of them, help define the conditions to
determine the variables that intervene in the use and service of the structure [12], that complies with the principles of cutting-edge structural dynamics, achieving compliance with globalized
standards, and with it this system contemplates satisfying the current design codes [13][14], in
these conditions it is essential to perform vibration tests on the catwalk considering the necessary scenarios, and detect experimentally the initiation of vibration problems [15].
Dynamic stimulation and evaluation of your response shows the service capacity, in accordance with the built walkways [2], for this reason, the incorporation of the frequencies of the different
phenomena that produce accelerations that lead to the initiation of a limit state of service or failure was unified in a load model [3].
In this work we analyze the cyclic loading pattern of the stimulation from walking on a footbridge of a pedestrian bridge, where the structure is subjected to a controlled dynamic system
according to the regulations established by current codes [13]. The basic pedestrian activity that governs the present model, is determined by the common gait [9], being often that, the daily
activity presents a pattern of stimulation of the pedestrian load that is associated to a dynamic representation of a common structural system [16].
The forces generated from the pedestrian-structure contact, they vary with respect to the kinematics of the center of mass during the whole cycle of the walking of the pedestrian who intervenes
in the sample [17], what it shows, the complexity of the analysis, incorporating multiple masses.
1. Structural modeling of pedestrians on a bridge.
The gangway acts rigidly by being supported laterally, and in turn, It is built on secondary beams of common structural profiles [2], so much as, it is coupled to a flexible system, so that,
the kinetic energy is acquired by the vibration due to the movement of the pedestrian [3].
this structure, It has elasticity and bending of the structural continuity, for such a situation, a linear system is constituted subjected to a forced excitation, what produces a second order
analysis [21], one of the techniques used to solve said system requires advanced numerical solutions, to determining the conformation of the temporary periodic load according to the cyclic
stimulation pattern [16].
1. 1 b
Phsical Model
[mp0(t)] [mp (t)] [ ]
Movement Equation
Accelerometer [f, , , ](t)
Mathematical model
[hf][Tc=0.2,0.5,1.0,1.5] [](p,t) [uI][uA]
Structural system
Structural kinetics [UI(p, t)] [UA(p, t)] [UP(p, t)] [Ud(p, t)]
Matrix Formulation F(t)=m+c+kx
Structural system response [C(p, t)]
2. 2 b
3. 3 b
4. 4
Figure 2 Graphic representation of pedestrian traffic.
Figure 1 Dynamic system block diagram
The block diagram of figure 1, presents the structural system dynamic response, which is obtained considering the catwalk structural kinetics applying the matrix formulation, proposing
the movement equation in Lagrangean mechanics [18], measuring by means of a portable instrument the patterns of the cyclic stimulation behavior, raised at the conceptual level in a
mathematical model, so that, study the contact forces at every moment, control its density and transit speed, at the same time, controls the energy of contribution to the vibration [19],
in a useful task, but in turn it is necessary, understanding that, by knowing the periodic load experimentally, we obtain the conditions of the pedestrian bridge [20].
In the scheme of figure 2, the characteristics of pedestrian traffic are observed (Tc) [5], representing with radio circles Ra, the space in which the pedestrian develops his walked [9],
these idealized circular spaces are used in the present study, considering a long pedestrian crossing and wide , noting that the traffic 1 , Consider a model with four circles that denote
the free circulation of each pedestrian, where the radio R1 Acting has a magnitude of 1.26 m, therefore, in the model Tc2, the radio R2=0.80 m., as well as, Tc3, R3=0.56 m. and finally in
Tc4, R4=0.46 m[13].
Fp (x,y,z) (t)
Fpe(t) Max
Pedestrian walk
Arrival zone
Impulse zone
Average speed
Tc2 Tc1
Rpe(t) Max
Figure 3 Graphical representation of the characteristics of the acting radius (Ra) of the pedestrian.
The scheme of figure 3, shows the degrees of freedom [22], the pedestrian is modeled considering a concentrated mass (mp), in the center of gravity (Cg)[23].
Figure 4 Graphic representation of the lateral characteristics of the pedestrian bridge
The bridge built with metallic structural elements, as shown in figure 4, It has vertical, horizontal and diagonal bars, that form envelope the transit board, whose footbridge is designed
with a plate of reinforced concrete, supported on secondary trabes [2], stairs access or pedestrian exit are connected [11], where the dynamic stimulation of the scenarios under study is
born [14], this is where, pedestrian traffic is controlled according to the density of movement of the walk, at par, with the designation of the sample [15]. The two legs represent two
stiffness springs Kp, and the passage has two periods: the impulse phase and the arrival phase [24], as observed on the figure 5, in this way the response of the temporary periodic load
is represented [25], called .
Figure 5 Graphical representation of the basic characteristics of pedestrian walking.
Figura 6 Representación gráfica de las caracterÃsticas interiores del puente peatonal
Figure 5 shows the speed of pedestrian traffic [16], where the periodic load of the mass of the pedestrian, has contact with the catwalk [8], shown in figure 6.
Figure 6 Graphical representation of the interior characteristics of the pedestrian bridge
Speed ( ), average pedestrian [7], can be controlled in the four study scenarios, If and only if, the zones interfere with the frequency of passage [26], just like that, ehe radio Ra
defined and the idealized zones have dominant geometry conditions [27], thereupon, the changes in the response (Rp), affect the potential energy of the catwalk, and its magnitude is
determined with the impulse force [21], the arrival zone, being the initial part, begins with the support of the heel and ends when the sole touches the surface of the catwalk in its
entirety, at this moment the periodic charge usually reaches its maximum magnitude, because the impulse phase of the next step begins [3].
2. Matrix formulation.
It is assumed that the sample maintains contact with the surface of the footbridge of the bridge under study [5], as shown in figure 5, likewise, it is assumed that portable measuring
equipment captures displacement data (ux, uy, uz)(t)[17], that has the center of gravity of the sample selected when walking [20], the response that the sensor emits, is considered with a
duration of eight seconds (Seg.), since it is necessary to travel the distance [15], In addition, it is considered a time required to
[33], being designed and built with high sensitivity to dynamic loads [2][6].
The cyclic force is determined considering Table 1, of the sample, where, P-1, P-2, P-3, are of the feminine gender and the rest masculine [34], both maintain similar characteristics with
respect to the dynamic load factor (DLF=0.371), the phase presents a value of i =1.57,[35]
Table 1 Sample characteristics
+ + k = F (t) {, , }
(1,1) (4,)
(,) = [ ]
(,6) (,) () ( 1 )
(,) = (,) ()|
() 0
(,) = (,) 2()|
() 0
(1,1) (4,)
(,) = [ ]
(,6) (,)
(1,1) (4,)
(,) = [ ] ( 2 )
(,6) (,)
(1,1) (4,)
(,) = [ ]
(,6) (,)
Stature (,) (,) (,) (,)
No. Wp [N] fp Fp(t)
[m] () () () ()
[S/s] [N] [N/m2] [N/m2] [N/m2] [N/m2]
P-1 1.52 530 1.441 713.27 142.65 356.64 713.27 1069.91
P-2 1.55 569 1.412 748.65 149.73 374.32 748.65 1122.97
P-3 1.61 608 1.441 818.25 163.65 409.12 818.25 1227.37
P-4 1.71 755 1.382 961.58 192.32 480.79 961.58 1442.38
P-5 1.74 804 1.529 1,097.34 219.47 548.67 1097.34 1646.02
P-6 1.80 942 1.706 1,037.38 207.48 518.69 1037.38 1556.08
standardize the average speed, so the time (ta)i represented in each simulation will be at least fifteen seonds [17].
In the present study a sample is taken with six pedestrians [36], where the values corresponding to the temporal force of stimulation are listed (Fp), which is determined with equation 8, as
shown in Figure 5, and highlighting the role played by each of the structural parameters involved in the measurement process and portable instrumentation [34].
(,) 0 0
(,) = [ 0 (,) 0 ] ( 3 )
0 0 (,)
= ()( )2 ( 4 )
(,) 2 ()
(,) 0 0
(,) = [ 0 (,) 0 ] ( 5 )
0 0 (,)
= ( )2 + ()( ) ( 6 )
(,) 2 () ()
() = + (2 )
( 7 )
In general, the loads produced by pedestrians are variable, temporary, stochastic and generate periodic loads, calculated with equation 7, where the mass and the frequency of passage
intervene. [37].
In each scenario there is an induced temporary load proportional to the mass, to the damping and the stiffness, in longitudinal, transverse and vertical direction [22], In addition, equation
1 determines the movement that governs the system [23], and the dynamic characteristics in matrix form are determined with equation 2, these are used to determine the kinetic energy (Ec) of
each pedestrian in the time domain, shown in equation 3 and 4, likewise, the potential energy (Ep) determined in equation 5 and equation 6 of the temporal system and the periodic charge is
determined with equation 7, as determined in [28].
3. Dynamic stimulation
The sample that provides energy and the movement of the center of gravity (Cg) will have generalized coordinates, whose
displacement will be in the direction ( , , ) [29],
horizontal and vertical, due to the energy used to reproduce the
4. Dynamic response of the structural system
march of the pedestrian, this movement, to maintain the cycle, yields energy to the slab of the footbridge, which also partially
The load (,)
( ,
As shown in figure 7, it
dissipates by to damping ( , , ) , [30].
(,) (,) (,)
One of the main characteristics of the loads produced by pedestrians is the low intensity [31]. When it is applied to structures with great mass and its high rigidity would hardly make them
vibrate significantly [32], however, pedestrian bridges are light structures compared to other civil structures
represents the points A, B and C, which correspond to the idealization of the supports of the pedestrian bridge, as was done in [36], when incorporating boundary conditions, the displacements
at the ends are nullified.
(,) = [ + (2)]
( 8 )
() ()
= [()] [(,) ] []
() (,)() ()
[](); [ = 1; () = 0.2], [ = 2; () = 0.5] [ = 3; () = 1.0], [ = 4; () = 1.5]
When applying the method of superposition, two beams are identified, the first is a beam simply supported on A and C, with a uniformly distributed gravitational load [38], whose magnitude
corresponds to the periodic load generated ,
with their respective densities of transit and displacement in the coordinate axes (e), whose mathematical scheme is shown in figure 7, and the second beam simply supported on A and C, with a
point load formed by the magnitude of the redundant load
that the forces of the foot-gangway contact in a section affect the entire bridge, {, , }, where the step frequency varies directly proportional to the Stature [39].
The starting position (, )() it's random, likewise, the first step is random 0, which causes pedestrians to approach or leave, limiting themselves to ( = 4, = 6)(). So this aspect is
controlled taking into account the traffic, Tci and the density of traffic Di in case of approach and withdrawal, = , with the parameters of the equation 8 [34].
, of the response in the support B[14].
( , )()
( )()
= 1,2,3,4,5,6
() = () + () + ()
= ( 0 )2
() 2 () ()
() = () (()) ( 9 )
= (" )2 ()
() 2 ()
The analysis with equation 9, determines the reaction
, by
the amount of movement of the potential energy capable of balancing the system in each analysis scenario [22], for this rigidity is used () , the force of action () and the restitution of the
system () .
5. Checking the structural system
= 1,2,3,4
The periodic study of the load varies permanently (),
Measured vs Expected
Cyclic Behavior
Pattern [F(t)]
Periodic Load [hf][Fci] [](p,t) ]
Accelerometer [f, , , ](t)
A (A-B) B
(B-C) C
likewise, in the present experiment, the magnitudes of the acting external forces are disclosed, so that this causes the interacting behavior to be predictable [40].
() () ()
Figure 7 Graphical representation of the characteristics of the traffic temporary density.
The stimulation generated, three men and three women, j = 1,2,3,4,5,6, are captured with the portable equipment [30], where the measurements collected represent the acceleration of the individual
center of gravity [27], In addition, the instrumentation and measurement of the sample collects information for each pedestrian, controlling the average speed, with the traffic direction, i =
1,2,3,4, as is done in [26], observing in figure 5 said control, determined with equation 8.
The sample passes through a walkway with two longitudinally
Figure 8 block diagram of the structural comparison of the system.
continuous sections, with (A-B) = 21.04 m. y (A-B) = 24.39 m. of long correspondingly, both sections have, and b = 2.20 m. wide [2][13], whose, free body diagram is observed in figure 7,
The initiation of a methodology of comparison of the experimental part and the analytical one is fundamental to assimilate the measurement of the dynamic response ,
where the space
, used for each pedestrian traffic.
this acceleration is the product of the stimulation and the
The analysis of the weight , of the human body, which
forms a complex non-linear system, for structural engineering purposes it is treated as a simplified single system of two degrees of freedom, in the plane {, }, {, }[26], however, the vibration
specifically does not propagate in the same linearity, and the measurement of the instrument confirms
configuration in which the system adapts to said response, on the other hand, the objective of determining the cyclical behavior pattern is primarily to use it to find an expected structural
finding, and infer in its measured result[21].
(,) + (,) + k (,) = (,) (
() () () () )
(,) (, )
(,) = [ ]
() 11
= [0 1 0] = 22 + 2 + 2
The kinetic energy that produces from the force (,)(), which is calculated with equation 8, and the potential energy
, or response, (,)(), it is calculated with equation
9, where, in this last equation, the term of the damping, (,)
, is determined with the equation 11, and equation 12,
Energy (Joule)
Energy (Joule)
P – 1 P – 2
Time (seg.) Time (seg.)
P – 3 P – 4
Energy (Joule)
Energy (Joule)
Time (seg.) Time (seg.)
Energy (Joule)
Energy (Joule)
P – 5 P – 6
Time (seg.) Time (seg.)
Figure 9 Graphical representation of the kinetic energy of the pedestrian and potential energy of the bridge.
To determine the pattern of cyclic behavior, the displacement variable obtained as a function of the increasing deformation in the catwalk was selected, this magnitude represents the amplitude
If the mass, the speed and the step frequency remain constant, the periodic load produces the response of the gateway Rp(t), that varies incrementally according to the mass moves away from the
(,) = ((,) 0(,) )
( 12
() 2 () ()
The linear displacement
, in the redundant of the catwalk
supports, thus, the magnitude of the structural dynamics of the
is proportional to the variation of potential energy, with respect
, it are geted with the equation 10.
to the kinetic energy [28], therefore, both magnitudes are
The instrument simultaneously measures the cyclic effects on the catwalk, which are taken to determine the expected analysis using equation 8, as shown in the figure 7.
determined in the whole experiment [34].
ANALYTICAL PROCESS
When determining the force and velocity in equation 4 and equation 6, the magnitude of the kinetic energy is obtained,
The concentration of forces resulting from the randomness of traffic is repetitive in the center of the clearing of the bridge [21], for this reason, the information obtained by the double
and potential energy
(,) (,)
, these magnitudes are
integration method is analyzed [41], and the measured
acceleration [22], using equation 11, therefore, based on said
observed in the graphical representation of figure 9, which are used from equation 13 and equation 14, to find the amount of work that the pedestrian in each scenario of analysis generates as
a result of its movement [28].
P – 1
measured acceleration [38], after determining the damping represented in figure 10, where, in each graph the magnitude normalized by the largest one is represented, it is illustrated that the
ratio of the pedestrian traffic speed to the speed measured by the portable instrument does not exist [34], where the force exerted by the cyclic periodic charge manifests itself as an
underdamped system [42], proven this theory with figure 10, the friction is weak or nonexistent [6].
cushioning (N/m/s)
P – 2
Time (seg.) Time (seg.)
P – 3 P – 4
Time (seg.) Time (seg.)
P – 5 P – 6
Time (seg.) Time (seg.)
Figure 10 Graphical representation of the damping pattern on the walkway of the pedestrian bridge.
The amount of movement, (,)
, in each scenario, it
The effect of apparent synchronization of the restitution of the footbridge will be through a positive weakening pattern, which
produces potential energy corresponding to the inertia, ( I ), and
to the elasticity, ( E ), of the material that forms the footbridge
infers in the return positions according to the initial conditions and to the diminished amplitude in the time domain, given for
of the bridge, calculating the restoration () using the
each pedestrian
[| ]
and in each scenario , the previous
= ( )2
( 13 )
(,)() 2 (,) (,)
(,) = (,) + (,)
( 14 )
() ()
equation 9.
=1 ()
thing, it rejects the analysis to determine in the present work the degree of structural constriction, because, the elastic component of the maximum deformation expected for alternative load
cycles, is more relevant than the plastic obtained from the material of the footbridge [43], when kinetic energy (,)
and potential energy
, are counteracted, then then, life
deformation with the measured data (,)
and the expected
to fatigue is controlled mainly by resistance (,)
of the
deformation with deterministic data
(,)() .
materials, determined with the equation 15 and for structural capacity of construction system [2] [4], as seen in the figure 11.
(,) = (,) ( (,) )
() () ()
( 15 )
(,) = ((,), (,), (,))
() ()
1. Cyclic load Analisys
Deformations are reciprocal due to the selection of the scenario = 1,2,3,4, n the movement evolution when the load produces displacements (, , )(). Applying the force and substituting in
the temporal equation as amplitude, indicated in the equation Asen (t), we obtain a pattern in the behavior of the force.
P – 1 P – 2
Time (seg.) Time (seg.)
P – 3 P – 4
Time (seg.) Time (seg.)
Desplacement (cm)
Desplacement (cm)
P – 5
P – 6
Time (seg.) Time (seg.)
Figure 11 Graphical representation of the deformation obtained by the cyclic periodic load expected in each pedestrian, and that obtained by measuring the acceleration in the pedestrian's
center of gravity.
A. Numerical simulation.
The frequency of passage allows the periodic load to have a constant value regardless of the weight of the pedestrian [1][43], and, on the other hand, the energy produced is absorbed by
the catwalk and contained until such frequency allows it [6], so that the catwalk behaves permanently elastic [22].
The permanent elasticity allows to have effects corresponding to two degrees of freedom, and to border conditions [38], essentially compressed and stretched, taken from Figure 9, which
applied in the equation 16 y 17, determines the
2. Expected analysis and structural check
On the surface of the footbridge of 45.43 meters in length and
2.20 meters in width, can pass in stages D1, D2, D3 and D4: 18, 32, 82 and 100 pedestrians correspondingly, under the conditions of Figure 2, said scenarios of traffic density [44], they
transit stochastically and only the speed of the gait is controlled. The step synchronization is random, for this reason the possibility of repetition of the test is statistically high,
whether the step frequency characteristic is controlled or not.
(,) (,)
(,) = 2 | |
( 16 )
() 0
4 (,)()
(,) = 5
( 17 )
() 384
The importance of thedamping in the dynamic system of the bridge proposes that the number of oscillations in a typical decay time be uniform, in all scenarios, as shown in Figure 10, in such a
way that the continuous interacting system is qualified trustworthy.
The comfort of walking is lost with traffic, and it is expected that the effects of density increase, observing figure 10, the pedestrian Number 1 (P-1), generates the effects in magnitude of
damping too high with respect to other pedestrians, this is because their walking produces very small speeds captured by the same portable instrument that measures the walk of the total
sample, when the damping is determined (c), of equation 7, the numerator is the derivative of the displacement with respect to time, which potentiates the resulting value observed in graphs.
The simulation of figure 11 presents the analysis of the expected deformations, according to the cyclic loads, considering the damping present in figure 10, comparing said results, with those
obtained from the present energy stimulation as seen in figure 9
, of the data obtained from the portable instrument.
In this research work was made the measurement of the traffic of the sample, walking on the surface of the footbridge, as well as the movement of the mass concentrated in the center of gravity
of the pedestrian, in both cases the deformation of the footbridge was determined, applying the methodology of the double integration and the coupling of the equation of movement through
Lagrange respectively. Obtaining results by implementing the methodology concludes with the following advantages:
1. The magnitude of the potential energy is greater than the kinetic energy, which corresponds to discern, on the elastic structural capacity, then then, a behavior is summarized where the
deformation is not permanent because they are below the elastic limit.
2. When the magnitude of the amplitude in all instants is the value of the mass, the responsiveness of the gangway, replaces the deformation with damping, which, by combining the stimulating
and restoring effects of elasticity, transform the energy provided, in cushioned movement, directly dependent on the physical characteristics of the pedestrian and its locomotor movement.
3. The behavior against the deformation of the materials of the footbridge, which are unloaded, from a certain elasto-plastic state, depends on the maximum deformation present in the
ACKNOWLEDGMENT (Heading 5)
The authors wish to express their gratitude to the Autonomous University of the State of Hidalgo for their financial support.
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6. M. Cacho-Pérez, N. Frechilla, y A. Lorenzana, Estimación de parámetros modales de estructuras civiles a partir de la función de respuesta en frecuencia, Rev. Int. Metod. Numer. para
Calc. y Disen. en Ing., vol. 33, núm. 34, pp. 197203, 2017.
7. C. C. Caprani, Application of the pseudo-excitation method to assessment of walking variability on footbridge vibration, Comput. Struct., vol. 132, pp. 4354, 2014.
8. M. Bocian, Biomechanically inspired modelling of pedestrian-induced forces on laterally oscillating structures, J. Sound Vib., vol. 331, núm. 16, pp. 39143929, 2012.
9. H. V. Dang, Experimental characterisation of walking locomotion on
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lateral vibrations of footbridges: A
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Analysis of time and frequency domain features of
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Santiago De Cali Analysis of Human-Structure Interaction in Footbridges in Santiago De Cali, Dyna, vol. 177, pp. 8694, 2013. [20]P. Hawryszków, R. Pimentel, y F. Silva, Vibration effects of
loads due to groups crossing a lively footbridge, Procedia Eng., vol. 199, pp. 28082813, 2017. [21]R. Merli, Comparison of two linearization schemes for the nonlinear bending problem of a beam
pinned at both ends, Int. J. Solids Struct., vol. 47, núm. 6, pp. 865874, 2010. [22]C. S. Oliveira, Fundamental frequencies of vibration of footbridges in portugal: From in situ measurements
to numerical modelling, Shock Vib., vol. 2014, 2014. [23]C. C. Caprani, Formulation of humanstructure interaction system
models for vertical vibration, J. Sound Vib., vol. 377, pp. 346367, 2016. [24]F. Venuti, Crowd-structure interaction in lively footbridges under
synchronous lateral excitation: A literature review, Phys. Life Rev., vol.
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kinematics and kinetics of pedestrians on a laterally
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humans, Structures, vol. 5, pp. 233246, 2016.
[30]G. Pedro, Instrumentation for mechanical vibrations analysis in the time domain and frequency domain using the Arduinoplatform Instrumentação para análise de vibrações mecânicas nos
domÃnios do tempo e da frequência utilizando a plataforma Arduino, Rev. Bras. Ensino FÃsica, vol. 38, núm. 1, pp. 110, 2016. [31]K. W. Li, Ground reaction force and required friction
during stair ascent and descent, Hum. Factors Ergon. Manuf., vol. 27, n̼m. 1, pp. 6673, 2017. [32]D. E. Newland, Pedestrian Excitation of Bridges РRecent Results, vol.
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[34]S. ivanovi, Probability-based prediction of multi-mode vibration response to walking excitation, Eng. Struct., vol. 29, núm. 6, pp. 942 954, 2007. [35]G. Pernica, Dynamic Load Factors for
Pedestrian Movements and Rhythmic Exercises, Can. Acoust., vol. 18, núm. 2, pp. 318, 1990. [36]A. M. Avossa, Probability distribution of footbridge peak acceleration to single and multiple
crossing walkers, Procedia Eng., vol. 199, pp. 2766 2771, 2017. [37]K. Van Nimmen, The impact of vertical human-structure interaction on the response of footbridges to pedestrian excitation,
J. Sound Vib., vol. 402, pp. 104121, 2017. [38]M. Cacho Pérez, Modelo mecánico acoplado para la simulación 2D del tránsito peatonal por estructuras esbeltas, Rev. Int. Metod. Numer. para
Calc. y Disen. en Ing., vol. 33, núm. 12, pp. 9096, 2017. [39]J. De Sebastián, Evaluación de la predicción de aceleraciones debidas al tránsito peatonal en una pasarela en servicio, Inf.
la Construcción, vol. 65, núm. 531, pp. 335348, 2013. [40]E. Shahabpoor, Effect of group walking traffic on dynamic properties of pedestrian structures, J. Sound Vib., vol. 387, pp. 207225,
2017. [41]J. S. V. Reyes, Método algebráico para determinar la deformación por deflexión en vigas estáticamente indeterminadas, pp. 16, 2007. [42]V. Racic, Reproduction and application of
human bouncing and jumping forces from visual marker data, J. Sound Vib., vol. 329, núm. 16, pp. 33973416, 2010. [43]A. Budipriyanto y T. Susanto, Dynamic responses of a steel railway bridge
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pp. 295311, 2017. | {"url":"https://www.ijert.org/structural-damping-in-a-pedestrian-footbridge-under-controlled-traffic-density","timestamp":"2024-11-09T19:23:58Z","content_type":"text/html","content_length":"104790","record_id":"<urn:uuid:76c5b972-9114-483e-b2c5-faf42722fbd5>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00885.warc.gz"} |
Prove that the intersection of a line and plane which are not parallel contains exactly one point - Stumbling Robot
Prove that the intersection of a line and plane which are not parallel contains exactly one point
Prove that the intersection of a line and a plane such that the line is not parallel to the plane contains one and only one point.
Proof. Denote the line by
By the linear independence of
2 comments
1. What’s the vector D RoRi had in mind here?
2. Suppose there are two intersect points: X1 X2
X1 , X2 contained in the line L => X1 – X2 = kA (A is the direction vector of line)
X1, X2 contained in the plain M => X1 – X2 = mS + nT
X1 not equal with X2 so k != 0.
can get now A = m’S + n’T => A in the spanning plain of S and T => Line L is parallel to plain M
which contradicts with L not parallel to M, so there can not be more than one solution.
Now to prove that if direction vector A of line L not parallel to M there exist one solution at least:
Let x0 be one point contained in line L => X=x0 + kA for X in line L.
Let p0 be one point contained in plain M => M: (X-p0) . N = 0 (N is normal vector of M)
we can solve that two equation to get “k” =>
(x0 + kA – p0) . N = 0 => (x0 – p0) . N + k(A . N) = 0
A. N != 0 because A is not parallel to M. Suppose A . N = 0 => p0 + A is a solution of
M: (X – P0) . N = 0 => p0 + A = p0 + mS + nT (where S, T linearly independent) => A is in the span of plain (mS + nT) => L is parallel to M which contradicts L is not parallel to M
so we can get a solution k = – (x0 – p0) . N / (A . N).
Hence there can only be exist one intersect point if line L is not parallel to plain M.
Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment): | {"url":"https://www.stumblingrobot.com/2016/05/30/prove-intersection-line-plane-not-parallel-contains-exactly-one-point/","timestamp":"2024-11-09T13:18:53Z","content_type":"text/html","content_length":"64557","record_id":"<urn:uuid:f5def9cb-ac42-4334-9405-e50ca67ca9ae>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00502.warc.gz"} |
Solution Dilution Calculator - Simplify Your Lab Calculations
In the realm of science, precise control over the concentration of solutions is often crucial. Whether it's crafting the perfect chemical reagents for an experiment or ensuring the potency of a
medication, achieving the desired concentration through dilution becomes a fundamental skill.
When it comes to manipulating solutions, dilution plays a crucial role, and solution dilution calculators are your trusty allies in ensuring accuracy.
What is a Solution Dilution Calculator?
A Solution Dilution Calculator is a valuable tool used to simplify the process of determining the concentration of a solution after it has been diluted.
This tool aids in ensuring accuracy and precision in experimental setups by allowing users to input initial and final concentrations, as well as the dilution factor.
Determination of Solution Dilution
The determination of solution dilution involves understanding the relationship between the initial concentration, final concentration, and dilution factor.
The dilution factor is the ratio of the initial volume to the final volume, providing a clear insight into the extent of dilution applied.
Solution Dilution Equation
The core equation for solution dilution is:
Understanding and correctly applying this equation is paramount to achieving accurate results in dilution scenarios.
To perform a solution dilution calculation, follow these steps:
Identify the initial concentration (C1) and initial volume (V1).
Determine the desired final concentration (C2).
Use the dilution equation to find the final volume (V2).
Implement the calculated values to achieve the desired dilution.
Let's consider an example:
Given C1 = 0.1 M, V1 = 50 mL, and C2 = 0.02 M, Find V2.
Using the dilution equation: 0.1 M × 50 mL= 0.02M × V2
Solve for V2: ${V}_{2}=\frac{0.1M×50mL}{0.02M}=250mL$
Therefore, to achieve a final concentration of 0.02 M, dilute 50 mL of 0.1 M solution to a total volume of 250 mL.
Frequently Asked Questions
Solution dilution is crucial for obtaining accurate and precise concentrations, ensuring that experimental results are reliable and reproducible.
It is recommended to use containers calibrated for accurate volume measurements to maintain precision in dilution processes..
Common mistakes include miscalculating volumes, misinterpreting concentrations, and neglecting proper mixing, all of which can lead to inaccurate results.
Most calculators accept various concentration units like molarity, percent weight/volume (w/v), and parts per million (ppm). | {"url":"https://toponlinetool.com/solution-dilution-calculator/","timestamp":"2024-11-10T12:28:02Z","content_type":"text/html","content_length":"52124","record_id":"<urn:uuid:621f8600-5d72-489c-a947-74c3881aceeb>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00838.warc.gz"} |
Chapter One
The first part introduces algorithms as computational solutions that halt and produce a correct value given a set of inputs. An input can be categorized as an instance of the problem or sometimes the
problem itself. It talks about how algorithms are applied in real-world situations in different areas.
Generally, problems have defined algorithms for solving them. For problems where there is no readily available solution, the book is meant to be a cookbook that teaches the techniques on how to
design algorithms.
Algorithms are measured by how fast they are, there are cases where the speed can not be determined as some problems can provide exact solutions while others provide approximate solutions like the
case of a traveling salesperson. They are called NP-complete algorithms. For these, it is a fruitless search to find the right algorithm. It would make sense to identify an NP-complete algorithm and
design for the best optimization and not a correct algorithm.
Section 1.2 emphasizes algorithms as another kind of technology, and we could be making the wrong choice if we do not. It considers how the run time of merge sort outweighs insertion sort even when
merge sort runs on a slower machine and insertion sort runs on a faster machine. It considers the fact that even though we have better and faster hardware, RAM and space are still limited resources.
An algorithm should be seen as a technology compared to our hardware and software, according to examples in the book, hardware requires algorithms to function, for example, routing network traffic,
and it emphasizes the fact that at the application layer, more algorithms are required as well, for example searching through text inside of Microsoft Word requires an algorithm that can go through
the text to find a match, even highlighting a match requires algorithm.
With the advent of AI, we might think we need fewer algorithms, but the reality is that AI and machine learning are algorithms created to create an algorithm for solving a problem.
Every day, there is an increasing need for a better and faster way to do things even as our computation requirement continues to grow and our hardware continues to get better too, there is still a
limited amount of storage and RAM we could compute on, and as we might think that it is huge to have an algorithm working on 100million data at a time, this is lesser to the amount of internet
traffic, google search, and tweet being processed per minute.
Exercise 1.1-1Exercise 1.1-2Exercise 1.1-3Exercise 1.1-4Exercise 1.1-5Exercise 1.1-6Exercise 1.2-1Exercise 1.2-2
Problem 1-1 | {"url":"https://clio.limistah.dev/chapter-one","timestamp":"2024-11-07T06:39:22Z","content_type":"text/html","content_length":"54690","record_id":"<urn:uuid:c701cb47-cd54-47dc-aeb1-f353be2ebe0d>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00130.warc.gz"} |
What Calculators Are Allowed On The ACT [2024 GUIDE]
Choosing the right calculator for the ACT can make or break your math score. Check out our top picks for ACT-approved calculators to ensure you’re fully prepared on test day.
In a hurry? Here is our top pick for the best calculator for the ACT:
Our Top Pick
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11/03/2024 08:27 pm GMT
Still studying for the ACT? Have a look at the best ACT prep books for this year!
Top 5 ACT Approved Calculators
If you are looking for the best ACT-approved calculators, you have come to the right place! Check out our favorite graphing and scientific calculators that are not only allowed on the ACT, but will
help you to reach a top score!
Texas Instruments TI-84 plus
The TI-84 Plus makes the best overall ACT calculator for its functionality as well as application outside of the ACT. This calculator can be used from classes that range from Pre-Algebra to AP
Calculus and is invaluable when it comes to ACT prep class.
A feature it can do that makes it really helpful for the ACT is its ability to convert decimals into fractions, saving you precious time when taking the test.
This model of the TI-84 Plus now features high-resolution and a full-color backlit display.
• MathPrint functionality
• Can be used for Pre-Algebra, Algebra I and II, Trigonometry, AP Statistics, Business & Finance, Biology, Chemistry and AP Chemistry, AP Calculus, and Physics
• Converting decimals into fractions
Best Overall
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11/03/2024 08:27 pm GMT
Casio fx-9750GII Graphing Calculator
Casio fx-9750GII makes our best budget calculator among graphing calculators for the ACT. This calculator can be used on the PSAT, ACT, as well as is among the best calculators for the SAT.
This calculator is way more applicable than just for standardized tests and can be used for a variety of classes including Pre-Algebra, Algebra I, Algebra II, Geometry, Trigonometry, Calculus, AP
Calculus, AP Statistics, Biology, Chemistry, Physics, and Finance & Business.
• USB connectivity for file sharing
• Quick bar graphs
• High-resolution LCD display
• High-Speed CPU
Best Budget Graphing Calculator
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11/03/2024 07:51 pm GMT
Texas Instruments TI-83 Plus Graphing Calculator
Next on our list of the best SAT calculators is the TI-83 Plus graphing calculator. This specific graphing calculator is permitted on multiple testing formats including the SAT, ACT, PSAT, AP, and
International Baccalaureate. This calculator can be used in classes up to calculus, engineering, trigonometry, and finance.
This calculator is ideal for the algebra classroom and lets students graph and compare functions as well as perform data plotting and analysis.
• Analysis up to 10 matrices
• Can display graphs and tables on split screen
• Built-in memory for storage
Budget Alternative to TI-84
Texas Instruments TI-83 Plus Graphing Calculator
With the TI-83 Plus Graphing Calculator, you can rest assured that it's permitted on multiple testing formats including the SAT, ACT, PSAT, AP, and International Baccalaureate
We earn a commission if you make a purchase, at no additional cost to you.
11/03/2024 07:46 pm GMT
Texas Instruments TI-30X IIS 2-Line Scientific Calculator
Approved for the use of the ACT, SAT, and AP exams, this TI-30X scientific calculator is a great budget option for those who are not looking to invest in a graphing calculator. This calculator
features a 2-line display with fraction features and conversions and can perform basic scientific and trigonometric functions.
This calculator is ideal for Pre-Algebra, Algebra 1 and 2, Geometry, Statistics, and general science.
• Solar and battery powered
• One- and two-variable statistics
• Edit, cut, and paste entries
• Fraction features
Best Scientific Calculator
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03/29/2024 11:20 pm GMT
Casio fx-115ES Plus Scientific Calculator
Here is the second scientific calculator that makes our list for the best ACT calculators. This Casio calculator has been designed for high school and college students taking Trigonometry,
Statistics, Algebra I and II, Calculus, Engineering, and Physics.
What makes this the “Plus” version are small features that really add up: lines over repeating decimals, GCD and LCM, and remainders, to name a few.
• Solar with battery backup
• Over 280 functions
• Multi-replay function
• Multi-line display
Best Budget-Friendly Scientific Calculator
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11/03/2024 08:02 pm GMT
Guide to Buying an ACT Calculator
The best calculator for the ACT is going to be approved by the ACT guidelines as well as one that you are familiar with. Should you decide to purchase any of these ACT calculators, be sure to
practice and familiarize yourself with it well before taking the ACT. There is no point having the best calculator for the ACT and not being able to perform basic functions with it. Please keep this
in mind and do not open it out of the package the night before the ACT.
What Qualifies as an Acceptable ACT Calculator?
• Graphing calculators*
• Scientific Calculators*
• Four-Function Calculators
* assumes it does not have the prohibited features listed under this text
Some calculators can be used if you modify the calculator:
• Calculators with any paper tape must have this tape removed
• Sound must be turned off any calculators that emit noises
• A calculator with an infrared port (HP 38G, 39G, 48G series) need to have this infrared port covered up with duct tape
• Power cords on calculators must be removed
Approved ACT Calculator FAQ
Do you really NEED a calculator for the ACT Math?
You don’t absolutely need a calculator for the ACT. That said, we certainly encourage bringing an ACT approved calculator on test day. If you were to try to complete the full ACT math section without
a calculator, odds are you would not finish this section and score lower than if you had a calculator.
What are the “ACT approved calculators?”
If the calculator you want to use is not on the restricted list or can be modified with the efforts mentioned above, then it is likely ACT approved. This list of best ACT calculators has been double
checked to meet requirements. Check the ACT Calculator Policy for rules on clearing documents on the calculator as well as any other CAS capability.
Which calculators are not allowed on the ACT?
1) Texas Instruments model numbers that begin with TI-89 or TI-92 as well as TI-Nspire CAS
2) Hewlett-Packard: HP Prime, HP 48GII, and all models that begin with HP 40G, HP 49G, HP50G
3) Casio: fx-CP400, ClassPad 300 of 330, Algebra fx 2.0, all model numbers that begin with CFX-9970G
Is the TI-83 or the TI-84 Plus better?
At the end of the day, the TI-84 is twice as fast and has more than double the storage as the TI-83. If this is of importance to you, then it’s worth spending the extra for the TI-84 Plus. That said,
the TI-83 is still a great graphing calculator and is more budget-friendly.
The One Rule You MUST Follow with your ACT Calculator
Whatever ACT approved calculator you decide you will be using on the test, we need to reiterate the necessity to be sure that you have spent an adequate amount of time familiarizing yourself with it.
If you purchased a high-tech graphing calculator, you need to know how to efficiently use basic functions (as you wouldn’t need any of the extra whistles it provides, at least on test day).
We suggest purchasing or using your ACT calculator well before you take the ACT, bringing it to all your math classes and using it for this time. The best ACT calculator is going to be one that you
are familiar with on test day, not a fancy graphing calculator that you can use to draw cool pictures.
The Bottom Line
Choosing the best calculator for the ACT doesn’t need to be hard. If you’re looking for a graphing calculator, then you cannot go wrong with the Texas Instruments TI-84 plus.
Our Top Pick
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What else should you bring on test day? Check out our ACT test-day checklist. | {"url":"https://testprepnerds.com/act/best-act-calculators/","timestamp":"2024-11-04T08:47:31Z","content_type":"text/html","content_length":"356690","record_id":"<urn:uuid:999e7df1-23d0-4376-ad63-3fa08fb8cb19>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00880.warc.gz"} |
Degrees of Freedom in Hypothesis Testing: A Comprehensive Guide |
Degrees of Freedom (DF) are a fundamental concept in hypothesis testing across various statistical methods. Understanding degrees of freedom is crucial for selecting the appropriate statistical test,
interpreting results, and drawing meaningful conclusions from your data. In this comprehensive guide, we will explore degrees of freedom in different types of hypothesis tests, providing formulas and
equations for a deeper understanding.
What are Degrees of Freedom?
Degrees of freedom represent the number of values in the final calculation of a statistic that are free to vary. In hypothesis testing, degrees of freedom are associated with the variability in the
data and affect the critical values of test statistics like t, F, and chi-squared. Let's dive into various hypothesis tests and examine their degrees of freedom.
1. One-Sample t-test:
The one-sample t-test is used to compare the mean of a single sample to a known or hypothesized population mean. The formula for degrees of freedom in a one-sample t-test is:
2. Independent Samples t-test (Equal Variance):
The independent samples t-test is employed to compare the means of two independent groups, assuming equal variances. The formula for degrees of freedom in an independent samples t-test with equal
variance is:
• DF is the degrees of freedom.
• n1 is the sample size of the first group.
• n2 is the sample size of the second group.
3. Independent Samples t-test (Unequal Variance):
When unequal variances are assumed, the degrees of freedom are calculated using a different formula:
$$DF = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}}$$
• DF is the degrees of freedom.
• \( s_{1}^2\) and \(s_{2}^2\) are the variances of the two samples.
• n1 and n2 are the sample sizes of the two groups.
4. Paired Samples t-test:
The paired samples t-test compares the means of two related groups, such as before and after measurements on the same subjects. The degrees of freedom are calculated as:
• DF is the degrees of freedom.
• n is the number of pairs.
5. Analysis of Variance (ANOVA):
ANOVA is used to compare the means of three or more groups. The degrees of freedom for ANOVA are split into two components: degrees of freedom between groups (DFB) and degrees of freedom within
groups (DFW).
• DFB is the degrees of freedom between groups.
• DFW is the degrees of freedom within groups.
• k is the number of groups (treatment levels).
• N is the total number of observations.
6. Chi-Squared Test:
The chi-squared test is used for categorical data analysis. The degrees of freedom in a chi-squared test depend on the number of categories or levels in the variable being tested. For a chi-squared
test of independence, the formula is:
• DF is the degrees of freedom.
• r is the number of rows in the contingency table.
• c is the number of columns in the contingency table.
7. Linear Regression:
In linear regression, the degrees of freedom are associated with the error degrees of freedom (DFE) and the regression degrees of freedom (DFR). They are calculated as:
• DFE is the error degrees of freedom.
• DFR is the regression degrees of freedom.
• n is the total number of observations.
• k is the number of predictors (independent variables) in the model.
Degrees of freedom are a fundamental concept in hypothesis testing, influencing the selection of appropriate statistical tests and the interpretation of results. By understanding the formulas and
equations for degrees of freedom in different hypothesis tests, researchers and statisticians can make informed decisions about data analysis and draw meaningful conclusions from their studies. The
choice between equal and unequal variance in independent samples t-tests is crucial, as it affects the degrees of freedom and, consequently, the statistical results. | {"url":"https://www.qualitygurus.com/degrees-of-freedom-in-hypothesis-testing-a-comprehensive-guide/","timestamp":"2024-11-14T11:04:33Z","content_type":"text/html","content_length":"462914","record_id":"<urn:uuid:d0fc3c31-f6ce-4b0e-8f63-31d96178fe95>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00463.warc.gz"} |
Is there a way to store floats perfectly in a computer ? | Sololearn: Learn to code for FREE!
Is there a way to store floats perfectly in a computer ?
I got this idea from the python course. we cant divide 1 by 3 perfectly. In this way we also cant store floats perfectly in a computer. Is there a way to fix it?
Is your question "can we store an infinite amount of digits in a finite space"?
No i meant that if we could store infinite digits, can the computer store those perfectly? But you have a point | {"url":"https://www.sololearn.com/en/discuss/1667811/is-there-a-way-to-store-floats-perfectly-in-a-computer","timestamp":"2024-11-08T05:48:20Z","content_type":"text/html","content_length":"918117","record_id":"<urn:uuid:f1893d0d-c297-41db-a95f-9ae146972ea9>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00150.warc.gz"} |
Swing Equation in Power System | Derivation
The article explains the swing equation, a key mathematical formula that models the dynamic behavior of synchronous generators in power systems. It outlines the derivation process, starting from
Newton’s law of rotation and incorporating angular position and velocity. The swing equation describes how rotor angle acceleration is influenced by active power imbalances, helping to assess the
stability of power systems during transient disturbances.
What is a Swing Equation?
The motion of a synchronous machine is governed by Newton’s law of rotation, which states that the product of the moment of inertia times the angular acceleration is equal to the net accelerating
torque. Mathematically, this may be expressed as follows:
$\begin{matrix} J\alpha ={{T}_{a}}={{T}_{m}}-{{T}_{e}} & {} & \left( 1 \right) \\\end{matrix}$
Equation 1 may also be written in terms of the angular position as follows:
$\begin{matrix} J\frac{{{d}^{2}}{{\theta }_{m}}}{d{{t}^{2}}}={{T}_{a}}={{T}_{m}}-{{T}_{e}} & {} & \left( 2 \right) \\\end{matrix}$
J = moment of inertia of the rotor
T[a] = net accelerating torque or algebraic sum of all torques acting on the machine
T[m] = shaft torque corrected for the rotational losses including friction and windage and core losses
T[e ]= electromagnetic torque
By convention, the values of T[m] and T[e] are taken as positive for generator action and negative for motor action.
Figure 1. Power Flow in a Synchronous Generator
Swing Equation Derivation
For stability studies, it is necessary to find an expression for the angular position of the machine rotor as a function of time t. However, because the displacement angle and relative speed are of
greater interest, it is more convenient to measure angular position and angular velocity with respect to a synchronously rotating reference frame with a synchronous speed of${{\omega }_{sm}}$. Thus,
the rotor position may be described by the following:
$\begin{matrix} {{\theta }_{m}}={{\omega }_{sm}}+{{\delta }_{m}} & {} & \left( 3 \right) \\\end{matrix}$
The derivatives of θ[m] may be expressed as
$ & \begin{matrix} \frac{d{{\theta }_{m}}}{dt}={{\omega }_{sm}}+\frac{d{{\delta }_{m}}}{dt} & {} & \left( 4 \right) \\\end{matrix} \\ & \begin{matrix} \frac{{{d}^{2}}{{\theta }_{m}}}{d{{t}^
{2}}}=\frac{{{d}^{2}}{{\delta }_{m}}}{d{{t}^{2}}} & {} & \left( 5 \right) \\\end{matrix} \\$
Substituting Equation 5 into Equation 2 yields
$J\begin{matrix} \frac{{{d}^{2}}{{\delta }_{m}}}{d{{t}^{2}}}={{T}_{a}}={{T}_{m}}-{{T}_{e}} & {} & \left( 6 \right) \\\end{matrix}$
Multiplying Equation 6 by the angular velocity of the rotor transforms the torque equation into a power equation. Thus,
$J{{\omega }_{m}}\begin{matrix} \frac{{{d}^{2}}{{\delta }_{m}}}{d{{t}^{2}}}={{\omega }_{m}}{{T}_{a}}={{\omega }_{m}}{{T}_{m}}-{{\omega }_{m}}{{T}_{e}} & {} & \left( 7 \right) \\\end{matrix}$
Replacing ${{\omega }_{m}}T$ by P and $J{{\omega }_{m}}$ by M, the so-called swing equation is obtained. The swing equation describes how the machine rotor moves, or swings, with respect to the
synchronously rotating reference frame in the presence of a disturbance, that is, when the net accelerating power is not zero.
$M\begin{matrix} \frac{{{d}^{2}}{{\delta }_{m}}}{d{{t}^{2}}}={{P}_{a}}={{P}_{m}}-{{P}_{e}} & {} & \left( 8 \right) \\\end{matrix}$
M = Jω = inertia constant
P[a] = P[m]– P[e] = net accelerating power
P[m] = ωT[m] = shaft power input corrected for the rotational losses
P[e] = ωT[e] = electrical power output corrected for the electrical losses
It may be noted that the inertia constant was taken equal to the product of the moment of inertia J and the angular velocity ω[m], which actually varies during a disturbance. Provided the machine
does not lose synchronism, however, the variation in ω[m] is quite small. Thus, M is usually treated as a constant.
Another constant, which is often used because its range of values for particular types of rotating machines is quite narrow, is the so-called normalized inertia constant H. It is related to M as
$\begin{matrix} H=\frac{1}{2}\frac{M{{\omega }_{sm}}}{{{S}_{rated}}}{}^{MJ}/{}_{MVA} & {} & \left( 9 \right) \\\end{matrix}$
Solving for M from Equation 9 and substituting into 8 yields the swing equation expressed in per unit. Thus,
\[\frac{2H}{{{\omega }_{sm}}}\begin{matrix} \frac{{{d}^{2}}{{\delta }_{m}}}{d{{t}^{2}}}=\frac{{{P}_{a}}}{{{S}_{rated}}}=\frac{{{P}_{m}}}{{{S}_{rated}}}-\frac{{{P}_{e}}}{{{S}_{rated}}} & {} & \left(
10 \right) \\\end{matrix}\]
It may be noted that the angle δ[m] and angular velocity ω[m] in Equation 10 are expressed in mechanical radians and mechanical radians per second, respectively. For a synchronous generator with p
poles, the electrical power angle and radian frequency are related to the corresponding mechanical variables as follows:
$\begin{matrix} & \delta \left( t \right)=\frac{p}{2}{{\delta }_{m}}\left( t \right) \\ & \omega \left( t \right)=\frac{p}{2}{{\omega }_{m}}\left( t \right) \\ & {} & \left( 11 \right) \\\end
Similarly, the synchronous electrical radian frequency is related to synchronous angular velocity as follows:
$\begin{matrix} {{\omega }_{s}}=\frac{p}{2}{{\omega }_{sm}} & {} & \left( 12 \right) \\\end{matrix}$
Therefore, the per-unit swing equation of Equation 10 may be expressed in electrical units and takes the form of Equation 13.
$\frac{2H}{{{\omega }_{s}}}\begin{matrix} \frac{{{d}^{2}}\delta }{d{{t}^{2}}}={{P}_{a}}={{P}_{m}}-{{P}_{e}} & {} & \left( 13 \right) \\\end{matrix}$
Depending on the unit of the angle δ, Equation 13 takes the form of either Equation 14 or 15. Thus, the per-unit swing equation takes the form:
$\frac{H}{\pi f}\begin{matrix} \frac{{{d}^{2}}\delta }{d{{t}^{2}}}={{P}_{a}}={{P}_{m}}-{{P}_{e}} & {} & \left( 14 \right) \\\end{matrix}$
When δ is in electrical degrees, or
$\frac{H}{180f}\begin{matrix} \frac{{{d}^{2}}\delta }{d{{t}^{2}}}={{P}_{a}}={{P}_{m}}-{{P}_{e}} & {} & \left( 15 \right) \\\end{matrix}$
When δ is in electrical degrees.
When a disturbance occurs, an unbalance in the power input and power output ensues, producing a net accelerating torque. The solution of the swing equation in the form of the differential equation of
(14) or (15) is appropriately called the swing curve δ (t).
Figure 2. Swing Curve: A plot of δ (t)
Swing Equation Importance
The swing equation is of significant importance in the analysis and stability assessment of power systems. It is a mathematical equation that describes the dynamic behavior of synchronous generators
in power systems during transient conditions.
The swing equation helps determine the rotor angle stability and the response of synchronous generators to disturbances such as faults or sudden changes in load. By modeling the mechanical and
electrical dynamics of the generator, the swing equation provides insights into the system’s transient stability and the ability to maintain synchronism.
The importance of the swing equation lies in its ability to assess the stability of power systems and guide control actions to maintain stable operation. By analyzing the swing equation, engineers
can evaluate the critical clearing time, which is the time needed for the system to recover from a disturbance and regain stability.
Understanding the swing equation allows power system operators and engineers to design appropriate control strategies and protective schemes to prevent cascading failures, blackouts, or voltage
collapse. It helps in optimizing system performance, determining appropriate control settings, and ensuring the reliable and secure operation of power systems.
Overall, the swing equation plays a crucial role in maintaining the stability and resilience of power systems, contributing to the efficient and reliable generation, transmission, and distribution of
electrical energy.
Swing Equation FAQs
What is the swing equation?
The swing equation is a mathematical equation that describes the dynamic behavior of synchronous generators in power systems during transient conditions. It relates the acceleration of the generator
rotor angle to the active power imbalance in the system.
Why is the swing equation important?
The swing equation is important because it helps assess the stability of power systems during transient events. It allows engineers to analyze the system’s response to disturbances, determine
critical clearing time, and design control strategies to maintain stable operation.
What does the swing equation tell us about power system stability?
The swing equation provides insights into the stability of power systems by determining the rotor angle stability and the system’s ability to maintain synchronism. It helps identify potential
stability issues and guides control actions to prevent cascading failures or voltage collapse.
How is the swing equation used in power system analysis?
The swing equation is used in power system analysis to simulate the dynamic behavior of synchronous generators and assess system stability. It is incorporated into transient stability studies and
helps engineers make informed decisions about system design, control strategies, and protective schemes.
What parameters are involved in the swing equation?
The swing equation involves parameters such as the moment of inertia of the generator rotor, the electrical power output, system frequency, and damping coefficients. These parameters determine the
response of the generator to disturbances and play a crucial role in stability analysis.
Can the swing equation be used to predict the stability of power systems?
Yes, the swing equation, along with appropriate system modeling and simulation techniques, can help predict the stability of power systems. By analyzing the swing equation, engineers can identify
potential stability issues and take preventive measures to ensure reliable and secure operation. | {"url":"https://electricalacademia.com/electric-power/swing-equation-power-system/","timestamp":"2024-11-05T00:22:03Z","content_type":"text/html","content_length":"124279","record_id":"<urn:uuid:9e893bb7-f4b6-4fcf-83d8-d5358f1effb0>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00873.warc.gz"} |
unctions and
Quadratic Functions and Parabolas
• Parabolas
• Quadratic equations and functions
• Graphs of quadratic functions
• Applications
Quadratic Functions and Expressions
A quadratic function has two forms:
• f(x) = ax^2 +bx+c (standard form)
• f(x) = a(x − h)^2 +k (vertex-axis form)
The graph of a quadratic function is a parabola. It is easy to
graph a quadratic function if it is expressed in the vertex-axis
Graphing a quadratic function
The vertex is the point at (2, 4)
The axis of symmetry is the vertical line x = 2
Graphing a quadratic function
y = 2(x+3)^2 − 5
The vertex is the point at
The axis of symmetry is the vertical line
Graphing a quadratic function
The general case
y = a(x − h)^2 +k
The vertex is the point at (h, k)
The axis of symmetry is the vertical line
If a > 0, the parabola opens upward.
If a < 0, the parabola opens downward.
• Does the graph of f(x) = 5(x − 1)^2 + 8 open upward or
• Does the graph of
• What is the equation for the axis of symmetry for the graph
of f(x) = 5(x − 1)^2 +8?
• What are the coordinates of the vertex of the graph of
Completing the square
How to change the standard form for the function into the vertexaxis
Example: f(x) = x^2 − 6x+10
Change into vertex -axis form.
Problem: Change to vertex-axis form by completing the square:
f(x) = x^2 +4x − 5
Problem: Change to vertex-axis form by completing the square:
f(x) = −x^2 − 10x+1
Problem: Change to vertex-axis form by completing the square:
f(x) = 3x^2 +6x+1
Problem: Change to vertex-axis form by completing the square:
f(x) = 5x^2 − 30x+11
The general quadratic function:
f(x) = ax^2 +bx+c
The quadratic formula tells you the solutions to f (x) = 0,
which is the same as locating the x- intercepts on the graph :
Example: Solve
2x^2 − 5x −3 = 0,
for x.
a = 2, b= −5, c = −3
So x = 3 and x = 1/2 are the solutions .
Example: Solve
x^2 − 5x −5 = 0,
for x.
a = , b= , c = | {"url":"https://www.softmath.com/tutorials-3/reducing-fractions/quadratic-functions-and.html","timestamp":"2024-11-04T05:25:53Z","content_type":"text/html","content_length":"34272","record_id":"<urn:uuid:89d7e83e-56c9-4eaa-9245-76a7140e2af0>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00130.warc.gz"} |
M2 Operators (Debugging with GDB)
15.4.9.1 Operators
Operators must be defined on values of specific types. For instance, + is defined on numbers, but not on structures. Operators are often defined on groups of types. For the purposes of Modula-2, the
following definitions hold:
• Integral types consist of INTEGER, CARDINAL, and their subranges.
• Character types consist of CHAR and its subranges.
• Floating-point types consist of REAL.
• Pointer types consist of anything declared as POINTER TO type.
• Scalar types consist of all of the above.
• Set types consist of SET and BITSET types.
• Boolean types consist of BOOLEAN.
The following operators are supported, and appear in order of increasing precedence:
Function argument or array index separator.
Assignment. The value of var := value is value.
<, >
Less than, greater than on integral, floating-point, or enumerated types.
<=, >=
Less than or equal to, greater than or equal to on integral, floating-point and enumerated types, or set inclusion on set types. Same precedence as <.
=, <>, #
Equality and two ways of expressing inequality, valid on scalar types. Same precedence as <. In GDB scripts, only <> is available for inequality, since # conflicts with the script comment
Set membership. Defined on set types and the types of their members. Same precedence as <.
Boolean disjunction. Defined on boolean types.
AND, &
Boolean conjunction. Defined on boolean types.
The GDB “artificial array” operator (see Expressions).
+, -
Addition and subtraction on integral and floating-point types, or union and difference on set types.
Multiplication on integral and floating-point types, or set intersection on set types.
Division on floating-point types, or symmetric set difference on set types. Same precedence as *.
DIV, MOD
Integer division and remainder. Defined on integral types. Same precedence as *.
Negative. Defined on INTEGER and REAL data.
Pointer dereferencing. Defined on pointer types.
Boolean negation. Defined on boolean types. Same precedence as ^.
RECORD field selector. Defined on RECORD data. Same precedence as ^.
Array indexing. Defined on ARRAY data. Same precedence as ^.
Procedure argument list. Defined on PROCEDURE objects. Same precedence as ^.
::, .
GDB and Modula-2 scope operators.
Warning: Set expressions and their operations are not yet supported, so GDB treats the use of the operator IN, or the use of operators +, -, *, /, =, , <>, #, <=, and >= on sets as an error. | {"url":"https://www.sourceware.org/gdb/current/onlinedocs/gdb.html/M2-Operators.html","timestamp":"2024-11-13T14:49:24Z","content_type":"text/html","content_length":"8720","record_id":"<urn:uuid:3c7e4ce1-ddd8-41b4-800b-d993865641ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00678.warc.gz"} |
Record-Breaking NVIDIA cuOpt Algorithms Deliver Route Optimization Solutions 100x Faster | NVIDIA Technical Blog
NVIDIA cuOpt is an accelerated optimization engine for solving complex routing problems. It efficiently solves problems with different aspects such as breaks, wait times, multiple cost and time
matrices for vehicles, multiple objectives, order-vehicle matching, vehicle start and end locations, vehicle start and end times, and many more.
More specifically, cuOpt solves multiple variants of two problems: the Capacitated Vehicle Routing Problem with Time Windows (CVRPTW) and the Pickup and Delivery Problem with Time Windows (PDPTW).
The objective of these problems is to serve customer requests while minimizing the number of vehicles and total distance traveled in respective order.
cuOpt has broken 23 world records set in the past three years on the largest routing benchmarks, as verified by SINTEF.
This post explores key elements of optimization algorithms and their definitions and the process of benchmarking NVIDIA cuOpt against leading solutions in the field, highlighting the significance of
these comparisons. Throughout the post, we use the term ‘request’ for an order for CVRPTW and for a pickup-delivery order pair for PDPTW problems.
Although there are various constraints and problem dimensions in this domain, the scope of this post is limited to capacity and time windows constraints. Capacity constraints enforce that the total
commodities present at the vehicle at any time cannot exceed the vehicle capacity. Time window constraints enforce that the orders are serviced at a time not earlier than the beginning of the time
window and not later than the end of the time window.
Combinatorial optimization
Combinatorial optimization problems are among the most computationally expensive problems in the world (NP-hard), with the number of possible states in the search space being factorial. As it is not
possible to use exact algorithms for large problems, heuristics are used that approximate the solution toward the optimal point. Heuristics explore the search space using various algorithms that are
computationally expensive with quadratic or higher computational complexity.
High complexity and the nature of the problem enable the acceleration of these algorithms using massively parallel GPUs. Thanks to GPU acceleration, it is possible to obtain close-to-optimal
solutions in a reasonable time.
Building evolutionary route optimization algorithms
A typical routing solver consists of two phases: generating the initial solution and improving the solution. This section describes the procedures to generate initial solutions and to improve them.
Initial solution generation algorithm
Generating a feasible initial solution with a limited fleet that meets all the constraints is an NP-hard problem itself. Our team has improved and parallelized the Guided Ejection Search (GES)
algorithm to place the requests to routes.
The main idea of the GES is simple. We first try to insert a request to the route. If it’s not feasible to insert that request, we eject one or more easy-to-insert requests from a route and insert
the request to the relaxed route. A penalty score (p-score) for each request indicates the difficulty of inserting that request into a route. The algorithm inserts a request only if the sum of the
p-scores of ejected requests is smaller than the considered request.
Each time it’s not possible to insert a request into a route, even with ejections, we increment the p-score of that request and try again. We keep all the unserved requests in an ejection pool and
the algorithm runs until the ejection pool is empty. In other words, it runs until all requests are served.
The main drawbacks of this algorithm are cycling (returning to the previous set of nodes in the solution), slow rate of finding ejection combinations when the number of ejected nodes is high, and
considering only weak, randomly perturbed solutions. We have removed those setbacks, which has enabled us to generate solutions with a much lower number of routes than the current state-of-the-art
Before we dive deeper into the ejection algorithms, it is crucial to understand that feasibility checks and solution evaluations are performed at constant time using the time warp method. Although
this approach significantly reduces the computation time, it also makes the parallelization more difficult because of the need to obey the lexicographic order of the search of an arbitrary number of
Finding which requests to eject and where to insert the considered request feasibly is a computationally expensive problem: it is exponential with respect to the number of ejected requests and
requires checking all the insertion positions in all routes. Our experiments show that a small number of ejections causes the cycling of the algorithm.
Thus, we propose a method that enables ejecting as many as 10 requests (heuristically) and five requests when an extensive search is performed—in parallel. We parallelize the ejection algorithm by
ejecting a fragment from each route and handling these temporary routes in a thread block. Then, we try to insert the considered request to all possible positions in parallel.
The deep search algorithm tries all possible permutations of ejections of all requests in a route. We use different thread blocks for each request insertion position and perform the lexicographic
search in parallel by splitting the lexicographical order into independent sub-permutations.
The GES algorithm loops until we exhaust the time limit or the pool of requests is empty. At every iteration, we perturbate the solution to improve the solution state and to open gaps in the solution
so that a feasible insertion can be found. The perturbation is a random local search that randomly relocates and swaps nodes between and within the route.
After the optimal number of vehicles is found to fulfill the requests, we switch to the improvement phase, which is responsible for minimizing the objective. By default, this is the total traveled
distance, but it is possible to configure other objectives in cuOpt.
Figure 1. A flowchart that shows the GES algorithm in NVIDIA cuOpt
Evolutionary process and local search algorithms
The improvement phase works on multiple solutions and improves them using evolutionary strategies. Generated solutions are placed in a population. To reach initial solutions that are diverse enough,
we use randomization in the generation process. Utilizing the GPU architecture, we generate many diverse solutions in parallel. The diverse population runs through an evolutionary improvement
process, with the best properties of the solutions kept in the newer generations.
In a single step of the evolutionary process, we take two random solutions and apply a crossover operator. This generates an offspring solution that inherits good properties from both parents.
Different crossover operators can be applied to the solutions, some of which leave the offspring in an incomplete state. We repair the solution by ejecting duplicate nodes, inserting unrouted nodes
or fixing the infeasibility by doing an infeasibility local search on it.
For example, the order-based crossover operator reorders the nodes in one or multiple routes of one parent solution with respect to the order they appear in another parent solution. The resulting
offspring conserves the grouping properties of one parent and ordering properties of another solution. The result of this particular operator is a complete solution that is probably infeasible with
respect to time and capacity constraints. cuOpt contains multiple crossover operators that are executed randomly on solutions.
The local search phase after the crossovers plays a crucial role in reducing or eliminating the infeasibility or improving the total objective, or distance traveled, in this case. The objective
weights of the local search are determined by what is important to optimize. Higher weights on infeasibility help return solutions to the feasible region, which is usually the case for most problems.
Local search finds a local minimum for the offspring solution, which then participates in further evolutionary steps. It is crucial to have a fast local search, as it is the main factor in how many
improvement iterations can be done within the time budget. We use fast, approximate, and large neighborhood search algorithms to find a good local minimum while being performant. Instead of
performing a fixed-size neighborhood local search as in classical approaches, we have designed a “net” that catches easy improvements quickly, and very deep operators when stagnation is reached.
Fast operators quickly explore small neighborhoods, while approximate operators can evaluate different moves each time they are applied. This is particularly important because the crossovers
frequently leave some routes intact. A large neighborhood operator moves requests in a chain of moves expressed as a move cycle among routes, as explained in A GPU Parallel Algorithm for Finding a
Negative Subset Disjoint Cycle in a Graph.
The cyclic operators explore a very large neighborhood where it is not possible to explore with simple operators. This is simply because the constraints prohibit these simple operators from passing
some of the hills in the search space. This workflow enables the use of fast operators often and more computationally expensive deep operators less frequently.
GPU parallelization is done by mapping each of the hypothetical routes to a thread block. This enables the use of shared memory to store route-related data, which is temporary while searching for the
moves. The temporary route is either a copy of the original route or a version where one or more requests are ejected. Threads in the thread block try to insert all possible requests from other
routes to all positions in the temporary route.
After finding and recording all the moves, we find the best move per route pair by summing up the insertion/ejection cost delta of each. The cost delta is calculated by the objective weights, which
contain infeasibility penalization weights as well. We execute multiple such moves if they are exclusive of each other in terms of the routes they modify.
Figure 2. A flowchart of the local search procedure in NVIDIA cuOpt
Benchmarking cuOpt
We have continuously improved the performance and the quality of cuOpt. To measure the quality, we have benchmarked the solver against the best-known solutions on the most studied benchmarks,
including Gehring & Homberger for CVRPTW and Li & Lim for PDPTW. In practice, how quick the solver can reach the desired solutions is important for companies.
Evaluation criteria and goal
• Accuracy is defined as the percent gap between the found solution and the best known solutions (BKS) in terms of objectives. The problem specification states the first objective as the vehicle
count and the second objective as the traveled distance.
• The time to solution measure states how much time is needed to reach a certain gap to BKS or desired objective result. Time to solution is one of the most important criteria for practical use
cases. It is important to reach a high-accuracy solution within the time budget. The combinatorial optimization algorithms take a significant amount of time.
Figures 3 and 4 show the convergence behavior of the solver on a subset of large instances of benchmarks.
Figure 3. Convergence behavior of cuOpt on CVRPTW problems
We have selected one instance from each of the categories (C1_10_1, C2_10_1, R1_10_1, R2_10_1, RC1_10_1, RC2_10_1) to show the overall behavior of the solver depending on the (clustered, random) and
(long route, short route) instances. The aggregate sum that is sampled every minute is compared to aggregate BKS.
The steep convergence at the beginning yields to slowly approaching the aggregate BKS as time passes. For these sets of instances, we were able to match the aggregate vehicle count of the BKS. The
cuOpt solver can find the BKS vehicle counts in almost all instances of Gehring & Homberger instances. However, the actual performance depends on the time spent at the initial solution generation
compared to the improvement phase.
While the solver converges quickly on larger instances, it converges orders of magnitude faster on smaller instances. In the following tables, we show how much time it takes to reach the BKS for
various problem sizes while achieving the same vehicle count as BKS for all of them.
Figure 4. Time to a certain solution for PDPTW problems
cuOpt sets 23 world records
With the new approach of GPU-accelerated heuristics and cutting-edge evolutionary strategies, cuOpt has broken the records for 15 instances from the Gehring & Homberger benchmark and eight instances
from the Li & Lim benchmark.
NVIDIA now holds all the records for CVRPTW and PDPTW categories for the past three years.
In Figure 5, each edge represents the path from one task to another. Green lines represent edges common to the previous record. Blue and red edges are different between two solutions. Thanks to
evolutionary strategies, the cuOpt solutions are in a completely different place in the search space of possible solutions, meaning that there are many different edges.
Figure 5. Route visualization of cuOpt world record compared to the previous record. Credit: Combopt.org
The overall average gap to BKS for Gehring & Homberger is -0.07% distance gap and 0.29% vehicle count gap. The overall average gap to BKS for Li & Lim is 1.22% distance gap and 0.36% vehicle count
gap. The benchmarks were run for 200 minutes on a single NVIDIA H100 GPU.
NVIDIA cuOpt achieves high-quality solutions in seconds with GPU acceleration and NVIDIA technologies like RAPIDS. We have achieved speedups of 100x in the local search operations, compared to a
CPU-based implementation. CPU-based solvers require hours to reach similar solutions.
Learn more and get started with NVIDIA cuOpt. You can also explore the model through the NVIDIA API catalog and NVIDIA LaunchPad. And discover how cuOpt can help your organization save time and
Join us for the NVIDIA GTC 2024 session, Advances in Optimization AI to hear from NVIDIA Optimization AI Engineering Manager Alex Fender. You can also take the NVIDIA Deep Learning Institute
Training, Learn How to Use Our Route Optimization Microservice to Drive Efficiency and Cost Savings. | {"url":"https://developer.nvidia.com/blog/record-breaking-nvidia-cuopt-algorithms-deliver-route-optimization-solutions-100x-faster/","timestamp":"2024-11-12T03:25:01Z","content_type":"text/html","content_length":"224810","record_id":"<urn:uuid:e0fa5268-00e9-490c-becf-6c4e86ee0ce5>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00502.warc.gz"} |
Weekly Printable Sudoku 16×16 | Sudoku Printables
Weekly Printable Sudoku 16×16
Weekly Printable Sudoku 16×16 – If you’ve had any issues solving sudoku, you’re aware that there are many different types of puzzles available which is why it’s hard to decide which one you’ll need
to solve. There are many ways to solve them. And it is likely that a printable version can be a great way to get started. The guidelines for solving sudoku are the same as other puzzles but the
actual format varies slightly.
What Does the Word ‘Sudoku’ Mean?
The term ‘Sudoku’ an abbreviation of the Japanese words suji and dokushin that mean “number” as well as “unmarried and ‘unmarried person’, respectively. The goal of the game is to fill in every box
with numbers, so that each number from one to nine appears only once on every horizontal line. The term Sudoku is an emblem associated with the Japanese puzzle manufacturer Nikoli which was founded
in Kyoto.
The name Sudoku comes by the Japanese word shuji wa dokushin ni kagiru meaning ‘numbers have to be single’. The game is comprised of nine 3×3 squares and nine smaller squares. The game was originally
known as Number Place, Sudoku was a puzzle that stimulated mathematical development. While the origins of the game are unknown, Sudoku is known to have roots deep in ancient number puzzles.
Why is Sudoku So Addicting?
If you’ve ever played Sudoku, you’ll know how addictive the game can be. The Sudoku addict will never be able to not think about the next problem they’ll solve. They’re always thinking about their
next challenge, and other aspects of their lives are slipping to the aside. Sudoku is a game that can be addictive It’s crucial in order to hold the addictive nature of the game in check. If you’ve
become addicted to Sudoku Here are some ways to curb your addiction.
One of the most common ways to tell that you’re addicted to Sudoku is to look at your behavior. Many people carry magazines and books While others just scroll through social posts on social media.
Sudoku addicts take newspapers, books, exercise books, as well as smartphones everywhere they travel. They can be found for hours solving puzzles, and they aren’t able to stop! Some people even find
it easier to solve Sudoku puzzles than regular crosswords, and they’re able to quit.
Weekly Printable Sudoku 16×16
What is the Key to Solving a Sudoku Puzzle?
A great strategy to solve the printable Sudoku is to practice and experiment with various methods. The most effective Sudoku puzzle solvers don’t use the same method for every puzzle. The key is to
experiment and try various approaches until you find one that you like. After a while, you will be able solve puzzles without a problem! But how do you learn to solve an printable Sudoku game?
In the beginning, you must grasp the basics of suduko. It’s a form of logic and deduction, and you need to examine the puzzle from various perspectives to find patterns, and then solve it. When you
are solving Suduko puzzles, suduko puzzle, do not try to guess the numbers. instead, you should look over the grid to recognize patterns. You can apply this strategy to rows and squares.
Related For Sudoku Puzzles Printable | {"url":"https://sudokuprintables.net/weekly-printable-sudoku-16x16/","timestamp":"2024-11-10T21:06:51Z","content_type":"text/html","content_length":"37418","record_id":"<urn:uuid:e179627f-1f8b-4adb-a36b-2731848f1a8b>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00492.warc.gz"} |
Why It Matters: Quadratic Equations and Complex Numbers
Why Learn About Quadratic Equations and Complex Numbers?
In algebra, a quadratic equation (from the Latin quadratus for “square”) is any equation having the form
where x represents an unknown, and a, b, and c represent known numbers such that a is not equal to [latex]0[/latex]. If a =[latex]0[/latex], then the equation is linear, not quadratic. The numbers a
, b, and c are the coefficients of the equation and may be distinguished by calling them, respectively, the quadratic coefficient, the linear coefficient, and the constant or free term.
Joan and her friend Hazel have decided to go cliff jumping at their favorite spot along the river. Hazel is an amateur photographer who just got a new camera that has a continuous shooting mode.
Hazel takes a picture of Joan jumping off a cliff into the river that looks like this:
When she sees the picture, Joan is reminded of what she learned in her math class about the shape of graphs of quadratic functions. She tells Hazel that the path of her jump reminds her of an upside
down parabola. “Oh yeah!” responds Hazel, “I remember that from high school physics. We learned how to calculate how long it takes something to fall due to gravity.”
Joan wonders if she could calculate how long she was in the air when she made the jump off the cliff, so when they are done playing in the river, they go back to Hazels’ house to see if she still has
her old physics notes.
In her notes, Hazel finds the following equation:
She had labeled the equation with arrows in the following way: [latex]-10\frac{m}{s^2}[/latex] is gravity, [latex]v_{o}[/latex] is the initial speed of the object, and [latex]h_{o}[/latex] is the
initial height of the object.
Hazel and Joan think they can figure out how long they were in the air as they jumped from the cliff into the river. Stay tuned for the end of the module when we see how they do it. | {"url":"https://courses.lumenlearning.com/wm-developmentalemporium/chapter/why-it-matters-complex-numbers/","timestamp":"2024-11-04T10:27:04Z","content_type":"text/html","content_length":"48608","record_id":"<urn:uuid:f796afc8-a007-41f3-a4ed-2d1a8e263b9b>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00041.warc.gz"} |
Coq devs & plugin devs
let x be the only generalizable variable in the user syntax term c
when is Lemma foo : `{c}. different from Lemma foo : forall {x}, c.?
hint: it's not about forall vs fun
s/c/t/ ?
And are there docs on backticks in conclusions, as opposed to assumptions? Never saw it before your PR
What you are looking for is encompassed in the term_generalizing nonterminal.
And it's described in the third paragraph in the text.
What if x's type requires an additional typeclass? Doesn't the former generalize that, too, while the latter doesn't?
wow how is that even legal syntax^^
What if x's type requires an additional typeclass? Doesn't the former generalize that, too, while the latter doesn't?
the quiz is not typeclass related ;)
re "legal syntax", there's even an example:
Definition sym (x:A) : (x = y -> y = x) := fun _ p => eq_sym p.
guess we just learned backticks from a source without such examples, like the math-classes paper (~"Typeclasses for mathematics in type theory")
~~since there are no guesses so far here is more info
in https://gitlab.mpi-sws.org/iris/stdpp/-/blob/c6e5d0bea614cee9a8abb2cee93070683b7debd9/theories/sets.v#L299-301.~~
Wait a sec
How is x unbound in the goal?
I see two bound occurrences, but both are bound
x used to be unbound until the parent commit
(see https://gitlab.mpi-sws.org/iris/stdpp/-/commit/c8bb51b2dae2fb03ace3d35c95b406cbb7a53aa6#bea97b576dbfb634e878d29fe3430ae55d29b731_299_299 )
I didn't notice the commit had changed after I encountered the issue
Ah, I suspected. So I guess you were talking about:
Global Instance set_unfold_list_fmap {B} (f : A → B) l P :
(∀ y, SetUnfoldElemOf y l (P y)) →
SetUnfoldElemOf x (f <$> l) (∃ y, x = f y ∧ P y).
so the command is https://gitlab.mpi-sws.org/iris/stdpp/-/blob/7f934a946006b80e3c732a81ce8e7075eebebc13/theories/sets.v#L299-301
Global Instance set_unfold_list_fmap {B} (f : A → B) l P :
(∀ y, SetUnfoldElemOf y l (P y)) →
SetUnfoldElemOf x (f <$> l) (∃ y, x = f y ∧ P y).
it has an invisible `{} around the type (see https://github.com/coq/coq/issues/6042)
replacing it with forall {x}, ... fails
Uh wait. I saw an issue. What if you annotate the type of x?
what do you mean?
What if you write forall {x:A}, (if that's the right type) does that still fail?
(ah I think I saw https://github.com/coq/coq/issues/13170, but then I should retract my guess)
that issue is unrelated
writing x:B (the type is B not A) works (command succeeds)
Wait. Is the issue about the command as-is in that context?
I cheated by trying this out, but IIUC, the problem cannot be reproduced with these 3 lines in isolation...
(and posting the minimized context would defeat the point of the quiz)
and as you say, it’s not about forall vs fun — that is, Lemma foo {x} : c. should behave like Lemma foo : forall {x}, c..
and for this issue, Lemma foo x : c. is enough (implicit _arguments_ are irrelevant here).
what about Lemma foo y : c[x/y] :smiling_devil: (where c[x/y] is metanotation for substitution, not Coq notation)
yes it's about the command in its home context
renaming x works
Last updated: Oct 13 2024 at 01:02 UTC | {"url":"https://coq.gitlab.io/zulip-archive/stream/237656-Coq-devs-.26-plugin-devs/topic/quiz.html","timestamp":"2024-11-12T06:56:08Z","content_type":"text/html","content_length":"20809","record_id":"<urn:uuid:1d9e0ed8-7d64-462c-b592-42db3b66e418>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00841.warc.gz"} |
Two Significant Figures
Hardly any school leavers can work out percentages in their head. Does it matter?
07 February 2017
Can you work out four sevenths as a percentage, in your head - to the nearest percent?
If you can, you have a rare skill.
I recently ran an experiment in two good schools, where I asked the sixth form (i.e. 17 year old) mathematicians* to do various calculations in their heads.
The challenge included asking them to work out 4/7 to the nearest percent. This calculation stumped almost all of them. The majority knew that it was "a bit more than half", a few got 58% or 56%, but
only 3 out of 60 (that's 5% of them) could mentally work out that four sevenths is 57% to two significant figures.
Does this matter? A lot of people seem to think so. I did a survey on Twitter. Of the 266 people who responded, 68% of them (that's nearly five sevenths) thought that we should expect "most" sixth
form mathematicians to be able to mentally work out that 4/7 is 57%.
Only 10% of respondents thought it didn't matter as "this is not an important skill". These people pointed out: "Surely it's enough to know that 4/7 is a bit more than half...maybe about 60%".
And yes, I agree that in most circumstances, it's the ability to be able to come up with a ballpark figure that is the most vital skill. We need it to defend us against spurious statistics thrown out
by politicians and salesmen. If we want more accuracy we can easily resort to a calculator or pen and paper. And yes the group I sampled were 'mathematicians' and not 'arithmeticians' (see my related
blog on Arithmeticians)
And yet...
Of all the maths learned at school, being able to work out and interpret percentages ranks right at the top in terms of its everyday use.
I just checked today's main news stories, and well over half of them contained a percentage. Here are just a few examples:
"17% of mobile phone users will face mobile phone bill increases of over £100"
"...there has been a 5.2% surge in German factory orders"
"37% of the public believe it would be reasonable to charge for some NHS services"
"Alastair Cook won 41% of the Tests that he captained"
Notice how in all the examples above, the percentage has been quoted to two significant figures. That's typical. Sometimes a percentage will be rounded to the nearest ten percent (one headline today
read: "90% of hospitals are overcrowded") but generally the world deals in percentages to two S.F.**
So if you can work out a percentage to two significant figures, you have a tool that will serve you well as a journalist, a business analyst, a sports statistician, an exam marker - and also as a
member of the public who has to engage with all these people. If you happen to have learned how to do it quickly and accurately in your head, then there will be no shortage of opportunities to apply
this skill.
And this is not a hard skill that can only be mastered by the elite few. 4/7 as a percentage requires two steps of mental short division. Seven into 40 goes 5, remainder 5, seven into fifty goes 7,
remainder 1....so that's 57% and a bit. Most primary school children can do it when they've learned about short division. But by the time they are sixth formers - when percentages have become a
significant feature of daily life - they've forgotten the method.
How many of our 'mathematical' school-leavers should we expect to be able to do this? I think it would be excessive to expect it of all of them, or even half of them. This is certainly not an
essential skill - though people who have mastered exact calculations have a strong foundation for doing ballpark calculations.
But remember that examiners expect those school-leavers who are the "best" at maths to be able to factorise a cubic equation and find the precise co-ordinates of the turning point of the graph. In
contrast the education system seems to be (to borrow a phrase from Peter Mandelson) intensely relaxed that almost no school-leavers can mentally work out four sevenths as a decimal.
Dare I suggest we've got our priorities slightly wrong?
* I know sixth form maths is a far cry from arithmetic. But the sixth formers that I'd expect to be the most numerate are the ones studying maths - at least until 'Core maths' takes off. If it turns
out A Level historians or geography students are better at their times tables than A level maths students, I'll be pleasantly surprised.
** two significant-figures is ok as a guide, though if one's being picky, sometimes a third significant figure is needed to give the same level of precision. Other things being equal, a number
rounded to 84% is more precise than one that's been rounded to 11%.
Related blogs: Three Sevenths; Arithmeticians | {"url":"https://robeastaway.com/blog/2-sig-figures","timestamp":"2024-11-06T01:51:25Z","content_type":"text/html","content_length":"10078","record_id":"<urn:uuid:ab65a53b-2d6d-47eb-b28b-d390c13ceefd>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00595.warc.gz"} |
100. Math for Programmers - Podcasts | Heroku
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100. Math for Programmers
Hosted by Hailey Walls, with guest Paul Orland.
Programmers are often expected to not only know complicated math equations, but to cherish them dearly; in reality, nothing could be further from the truth. Although mathematics forms the basis for a
lot of software, most people are still put off by it. Paul Orland, a mathematician turned programmer, found this so perplexing that he wrote a book breaking down math concepts for programmers. He'll
share his reasons for doing so, as well as why he believes maths is essential to every job.
Show notes
Hailey Walls is a Customer Solutions Architect with Heroku, and she's engaged in a conversation with Paul Orland, the founder of Tachyus and author of Math for Programmers. Paul took graduate level
math classes, and even ended up with a Master's degree in Physics, but even he admits that he comes down with his own kind of math anxiety. Now, he works as a programmer, building predictive models,
but he encounters many engineers who don't have a basic understanding of fundamental math concepts, like calculus or linear algebra. Seeking to rectify this, he wrote a book called Math for
Programmers, which methodically explains mathematical concepts using real-world examples. He hopes to be able to teach math to many more people.
Paul emphasizes that, although thinking of mathematics can be intimidating, it's not different than working on any other skill. If you decide to go weight lifting, you start with a 10 pound weight,
then a 15 pound one, and on and on. Similarly, with math, if you train on problems that are simpler, future problems will build upon the techniques you've honed. The appeal for gaining math skills is
almost analogous to that of programming: there is always a right and final answer. Just as a compiler determines how a program works and whether a syntax is valid, taking in input and producing
output, so too is math deterministic. Fundamentally, better mental acuity with math can help teach you how to consider the behaviors of complicated systems.
For people interested in studying math more closely, Paul advises students to not be discouraged by problems which appear hard. It can be best to pick a problem that you are naturally interested in,
which will lead to a general willingness to try and solve it. Similarly, he'll also take a math concept and turn it into a program, which has helped him reason about flow and patterns much more
clearly in the past.
Links from this episode
Hailey: My name is Hailey Walls. I am a Customer Solutions Architect with Heroku, and I'm here today with Paul Orland, the founder of Tachyus and the author of Math for Programmers. And we're going
to talk about math. Do you want to tell us a little bit more about yourself and your book and maybe what brought you to write that?
Paul: I've always been a math nerd my whole life. I remember when I was five-years-old or something and we would go out to breakfast every weekend and on the restaurant napkin, my dad would always
give me a math problem to work on while we waited for our food. So yeah, ever since I was little. Around that time, I was also interested in what's the biggest number you can count to. I learned
about things like a google and a googolplex. So ever since I was little, it fascinated me. And I since then tried to learn more about math and turned into something that I studied in college. I
majored in math and then went on to work in software and mathematical parts of software and then start my own company.
Paul: And basically, what we've done at my company is we've used math and physics and machine learning to build predictive models in the energy industry. We can predict what oil fields will do in the
future. What kinds of fluid will come out of the ground and how much volume and flow rate will come out. And there's a lot of math behind that too. So I guess then my career as a software engineer,
an entrepreneur, working on mathematical problems in software, I found this interesting problem, which is that there's a lot of really smart technical software engineers out there, but some of them
don't know calculus or linear algebra or the basic things that you need to know to plug into some types of modern software teams. And I decided to, after spending time training my team, I decided to
try and package what I had taught them and shared with them and put it in a book that would be available to a lot more people and hopefully teach math to a lot more people.
Hailey: That's awesome. I think that one of the things I really like and appreciate about your book is I came around to math in the way that you're describing your audience, where I didn't do much of
it growing up and I certainly didn't do any in college. But found myself looking at problems later on that that was a useful skill to have and an interesting topic and grew into being an amateur math
enthusiast, which is how I live these days and find lots of excuses to use it in my job and use it in my work. And yeah, I just get really excited about the kinds of problems like you're working on
with your company, the predictive models you can use or different optimizations you can make. I think it's all super fascinating.
I remembered and we discussed before this podcast, the age old question in math class of when am I going to use this in real life? And I guess for some people like me, I was lucky to get exposed to a
lot of enthusiastic users and learners of math when I was young and I never really questioned whether it was something worth learning. But for a lot of people, it takes until later in life or in
their career where they find a problem that they can't solve. And they think, "Wow, I wish. I wish I had gone in and went deeper when I had the chance in school." So I definitely understand that.
Hailey: I think one of the other things you do a good job of pointing out in the book is that it can grow beyond that problem solving into its own joy. And you can learn to just appreciate it for
what it is and get a richer understanding of math beyond just whatever problem you might be solving. Although you have a lot of good examples of neat problems you can work on to introduce yourself to
those topics and give yourself something practical to work through.
Paul: Well, this is something I've thought about a lot. Why do I like math? Or what's interesting to me because that's, I mean that's a big question in education. If you want to teach something to
someone, first you should try and motivate it and tell them why they want to learn it. I do that a little bit in chapter one of my book. But at some point, when I was in school, math was a fun
competition. And it was, there's this problem, and you're competing against yourself and you're competing against your classmates. There's a right answer. There's maybe it's a proof that gets you to
the answer and it's a test of your mettle to get it done.
Paul: And then I think a little bit beyond that when I didn't care so much about contests or exams or grades, started thinking about math as this abstract game that we play and it puts you in touch
with, in some sense, arguably what are the deep quantitative secrets of the universe? These platonic ideas that somehow exist that people discovered and don't invent. I think a lot of mathematicians
would say that math is discovered not invented. So that's been interesting to me.
Paul: And then since then, I've really made an effort since I've been a software developer and then out of school, I've made an effort to deliberately practice math and get better at it by doing
problems. And then I found that there's a whole other way to enjoy math, which is thinking about it, it's almost an intellectual version of bodybuilding where you have this, you have challenges and
they seem difficult or impossible. So the first time that some people go to the gym and they try and pick up a 10 pound weight and then do a couple of reps and they get sore, and then you look at
someone who's next door who's bench pressing 300 pounds. And you'd think, "How could this person possibly do it?"
Paul: But you do the 10 pounds a couple of times, and you do the 15 pounds, then you build up and eventually you can do the 300 pounds. And math is a very similar, intellectual version of that, where
you train on problems and you can really feel yourself gaining mental skills or developing a mental tool belt. And I found that that's really satisfying as well. So this is all to say that I think
there's a lot of ways that you can love math or get into it. And I hope everybody who picks up my book finds at least one of those ways to latch on to.
Hailey: So we have some examples of programming topics that relate to math, starting with some of the more abstract examples that you start with in the book with vectors and graphics.
Paul: Yeah, definitely. Well, I think it's been really interesting to me in my programming career to try and take what I learned. I guess I can say a little bit more about how I got into software. I
was a math major in college and I decided that my freshman or sophomore year. And my parents parents said to me, they expressed a little bit of concern. "Why are you going into math?" What kind of, I
think my dad said, "What kind of J-O-B are you going to get with a math degree?" And I think the answer is with a math degree, you can do anything, but there's only one job called mathematician, but
then there's all sorts of different jobs that use math to some extent.
Paul: And I ended up stumbling into software at the end of my undergrad career because it's like you're flexing the same intellectual muscles as when you do math, when you're writing a program. You
have to think, I mean you're essentially writing a proof when you write a program. You have to prove the existence of this thing that you want to work, and you have to work in a formal language
that's your programming language. It's unforgiving the same way that mathematical proofs are. It's either correct or it's not.
Paul: And then the computer is the final arbiter of correctness. Either the compiler will succeed and your program will run and there won't be bugs or you'll have some problems along the way. So
anyway, that's the first way that I connected math to software. And then, there's all sorts of specific things you can do. I built some simple games in JavaScript and then Python to do almost any
kind of graphics or game design. Even if you're using a game engine, you need to use some math to find out where things are, how they're moving, if two things are colliding with each other. And
that's a lot of the math that I talked about it in the book, especially in part two of the book.
Paul: Something else I've gotten into is there are ways to get closer to the original, the first thing that I was talking about, where you think about programming as a mathematical exercise, and it's
not just philosophically they're similar, you can actually reason about programs using mathematical methods. So you can think of a a program as taking some data in, doing some computation, and
putting some data out. And if it's deterministic, meaning it gives you the same outputs for every time you pass in the same input, then really what you have is a mathematical function. And that's one
of the simplest mathematical building blocks you can work with.
Paul: So in functional programming, you take that view to the extreme. And you say everything that I have it's all functions sending inputs to outputs deterministically, and you compose them together
by sending the outputs of one function to the inputs of the next function. And then you get basically every tool from set theory and function theory, and then there's another branch of math called
category theory, which looks at all the various generalizations of that. So I've had a lot of fun learning about that, and there've been valuable ways to apply that to design and architecture, not
just numerical computations.
Hailey: For me, the way that I ended up connecting the math and the programming was a little bit different in that I had been doing programming to get myself through college for biology. So I had the
programming skills for the practical reasons, but had all these interests in these different modeling problems. So population modeling or genetics, these different categories there, and realized at
some point that all of the things we were being told to do in Excel for class, or with these other sorts of tools, I could figure out how to do it easier and better in Python. And so started teaching
myself some of those mathematical concepts in Python and figuring out how to use that to accomplish my goals for these different kinds of problems.
Paul: Yeah. I would love to see more math learners exposed to that stuff because my experience was when I was in first grade out, I remember this calculator that I got. It was a talking calculator.
So it had all the numbers, it had plus, minus, times, divide, equals, maybe it had a memory button, but that was about it. It would say anything you typed back to you. So that was my first
calculator. And my favorite thing to do with that was I typed in two times two, and then I just hit equals, and every time you hit equals it would multiply by two, and then eventually I could get it
to say, "Error, error, error, error, over and over again." So that was my first calculator of any kind that I got. And then I gradually, as you go through school, you get calculators with more
buttons on them.
Paul: And in middle school, I got a scientific calculator and was mystified by what's the sine button, cosine, exponential. I didn't know what any of those things meant, but it was intriguing that
there were these new buttons. And then I got to high school and I got a calculator that had probably, I don't know, 50 buttons on it. And it was a graphing calculator and you could make graphs on it
and you could do multiple step equations on it. And each of the 50 buttons actually had two or three different modes. So there might as well have been 100 or 150 buttons on there. And then you could
actually write some computer programs on it as well. So it was like, you keep getting these calculators that are extensions, that let you do more and more, and you need to know more and more to use
them, but they give you an extra tool to help your thinking.
Paul: And I graduated high school and started working on a pure math major in college and never used a calculator again after that. So it was interesting to me how you use technology up to a point in
math, and then it falls by the wayside. And I think the next calculator I really used was a high-level programming language. And I've used Python, I've used... My favorite programming language is F#
that I use that at my company. These are really the most powerful kinds of calculators you can have because they come with libraries. And if there's not the library that does what you want already,
you can download a new library or write your own library. So they're like calculators and you can add buttons to them.
Paul: And I talk about this a little bit in chapter one of my book, but I really feel strongly about this, that everybody should be using extensible calculators. Calculators that don't have a fixed
number of buttons essentially. If people learned Python in elementary school or middle school, then they could take Python with them their whole life. And it doesn't actually have to be Python. I
think another underrated programming environment is Excel. And Excel is a calculator that a ton of professionals use probably more than any physical desktop calculator. There are a ton of jobs that
use Excel and have Excel as a prerequisite. And I would say, why not incorporate Excel or spreadsheets more generally in math education?
Hailey: I like your point about getting people in front of these more extensible calculators, the programmatic kind earlier, because it also has this benefit of giving you a lot of feedback. And
especially if you're using something like Notebooks or some of these other tools that can make the inputs and outputs more visible and more interactive for you. I personally find that super helpful.
Paul: And Notebooks, I should mention, a year or a year and a half ago or something, I stumbled across Jupyter Notebooks. My wife is actually an astronomer and she does a lot of data analysis with
space telescopes, the Hubble space telescope is one that probably most people know about. And she, for almost all of her scientific work, uses Jupyter Notebooks. And I saw her using them and I
thought, wow, this isn't just a great work environment, this is also a good teaching tool because you can put something really reproducible here and it shows you every single step.
Paul: And it's not like a program where if you opened up, I don't know, if you opened up the source code for say Microsoft Windows or something, and you looked at the first line of the first file. It
wouldn't really tell you how Windows works. It's not organized in that way. Most code is not organized in a sequential order, but in a Jupyter Notebook, it really shows you not just the code that
works, but a thought process. So I love that about Notebooks and I continue to try and use them more and use them to share ideas more.
Hailey: So one thing that I can certainly relate to, and I think a lot of people struggle with, is the sense that maybe they're not good at math or that math is going to be too hard. It's not a thing
that they do. That was certainly a thing that I had to get past a little bit in myself. And so if you were able to give some advice to anybody who's listening to this and thinking that they want to
tackle a math problem and start to apply these things creatively themselves, but are still hung up on that kind of math anxiety, what advice would you have for them?
Paul: I'll start by saying, I was a star math student in high school. I accelerated and took college classes while I was in high school. Then I went to a top university for math and got the intensive
math degree and took graduate level classes. Since then, I've gotten a Master's degree in physics. And since then, I've still taken a couple of classes on the side. The last class I took was, I guess
it must be... I took algebra one in high school, I took algebra two in high school. I took probably two years of algebra in college. And I have now taken a year going on maybe two years of algebra in
grad school. And I had math anxiety. I feel like I get problems that I don't know how to do and I've wondered if I'm cut out for it.
Paul: And if I look back, if I sat in high school and I think I'm in ninth grade math, and I guess, what would this be? This would be like 18th or 19th grade level math. I would never be cut out for
that. I could never do it. So sometimes I have to catch myself even now having gotten this far with my own math anxiety. This is a problem. I don't know how to prove it.
Paul: I think the advice I would give to people is that know that whoever seems like they're a million levels beyond you or 10 grade levels beyond you, or however you want to think about it, they
have their own problems that are hard for them. And they've put in some work and they've gotten to a certain level of comfort with things. And some things are easy to them, but some things are also
hard. And it's not about you, it's about whatever you are willing to put in and how many problems you're willing to work through. And I don't think there's anyone who's more cut out or less cut out
for math. I think it's just you do care about the problems and you work through them.
Hailey: Do you have other recommendations for people who are looking to get started in math? Your book's a great place to start, I think, but any other areas that you could recommend for diving into
math skills, different tools or resources for people?
Paul: Well, I would say definitely check out my book. I guess I'm on this podcast to some extent to promote my book. So please buy my book. If you like it, buy three more copies for all of your
friends or something like that. I think for a particular type of math learner, diving into Math for Programmers is a great thing. If you know Python, then this is a great hands-on way to not just
read a stale book, but you can actually, every single page almost has some code that you can type in and you can actually see it working. You don't have to just believe that the statement is true. So
yeah, my book I think is great for that.
Paul: For me, particularly learning by doing is the most important way to learn something. So I don't really have the attention span to read a math book. Even describing how far I've made it in math.
I can't for some reason I can't pay attention to a math lecture that's more than 20 minutes long. I'll zone out or get lost. Same thing with reading. I can read five or 10 pages in a sitting, but
then I have to take a break, especially if I have a really dense math book. So what works for me is picking a problem that I'm really interested in and letting it either I work it out on paper for a
long time, or I let it incubate and I end up solving or cracking problems that I thought were impossible after I just let them percolate like that. Another thing I like to do is, and this is probably
obvious from my book, but I like to take something that's a math concept that I know, or I'm just learning and turn it into some code.
Paul: So if I can compute something, write a program to compute something, I really know I've mastered it. There's no way to argue with a working computer program. Versus if you read something, you
may say, "Oh, did I get this? Did I not get it?" Making it work in a program or working in a calculation or solving an exercise correctly, these are all ways to really have a hands-on experience and
also convince yourself that you know what you're doing.
Paul: Having said that I learn best by doing, not everybody learns best by doing. I see people in graduate level math classes and I see people who sit next to me and just digest the whole lecture and
seem like they get it in one go, and you may be one of those people. And that's totally fine. Just, I would say, be thoughtful about what works for you and don't get too stressed out if you see
someone else who's able to learn in a different way that doesn't work for you because that's not the goal. The goal is not to learn in some specific way or at some specific speed, it's to really
master and then enjoy the material and be able to apply it.
Hailey: I really liked the point about seeing other people, maybe being able to take in the whole lecture while you yourself get about five or 10 pages through, and then you need to relax and let
that percolate through your mind. This is definitely how my experience tends to play out. And I think it's easy to, if you're not already in that math mindset, then that's the thing that you're
practicing. It's the thing that you're studying. It can be easy to look at other people and think that they're all just sitting there figuring out the entire lecture all at once and think that you're
not living up to that standard because your learning style is a little bit different.
Paul: Coming back to the sports metaphors because I'm not a real sports fan or anything, but I think about what I've done some type of exercise, maybe I've been swimming a lot. And then someone takes
me on a hike and I've been swimming an hour a day, but someone takes me on a hike up a mountain, and I get winded after the first mile. And I'm like, I'm doing cardio every single day, but this
slightly different move or exercise or physical activity is still difficult. And I think it's the same for math or programming or any intellectual discipline. You really focus on one type of problem
or a couple types of problems. Then if you see one that's outside your wheelhouse, it may be difficult and it may be discouraging, but you can't let it get to you just because for someone else who's
been hiking up mountains, going for a swim might be very difficult. So never judge yourself compared to others and always remember that you're in the same kind of process, but on a different path as
everyone else.
Hailey: That was fantastic. Thanks for that, Paul. Your book is available on the Manning website and I think we have a code for 40% off for that book. So from the Manning website, you can look for
Math for Programmers and use the code podish19 and get a discount on Paul's book and get started learning some math and having some fun with it.
Paul: I don't know if my contact info is available, but if you've made it to this point in the podcast, you can feel free to email me. My email is my last name, Orland, P-M as in Paul, Matthew at
gmail.com. So I'm happy to talk math with almost anyone who's curious enough to start the conversation. So drop me a line and tell me what you're interested in.
Hailey: Thanks so much, Paul. I really appreciate you coming on the podcast. It was great talking to you.
Paul: Thanks for having me.
About code[ish]
A podcast brought to you by the developer advocate team at Heroku, exploring code, technology, tools, tips, and the life of the developer.
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Customer Solutions Architect, data nerd, amateur math enthusiast
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Paul Orland is the author of the book Math for Programmers. He is also the co-founder of Tachyus, an energy-tech software company.
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Analytical and Numerical Investigation of Free Vibration in Beams under Diverse Boundary Conditions and Material Characteristics
Volume 13, Issue 08 (August 2024)
Analytical and Numerical Investigation of Free Vibration in Beams under Diverse Boundary Conditions and Material Characteristics
DOI : 10.17577/IJERTV13IS080048
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Kipkirui Chepkwony, Ch. Ratnam, 2024, Analytical and Numerical Investigation of Free Vibration in Beams under Diverse Boundary Conditions and Material Characteristics, INTERNATIONAL JOURNAL OF
ENGINEERING RESEARCH & TECHNOLOGY (IJERT) Volume 13, Issue 08 (August 2024),
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• Authors : Kipkirui Chepkwony, Ch. Ratnam
• Paper ID : IJERTV13IS080048
• Volume & Issue : Volume 13, Issue 08 (August 2024)
• Published (First Online): 24-08-2024
• ISSN (Online) : 2278-0181
• Publisher Name : IJERT
• License: This work is licensed under a Creative Commons Attribution 4.0 International License
Text Only Version
Analytical and Numerical Investigation of Free Vibration in Beams under Diverse Boundary Conditions and Material Characteristics
Kipkirui Chepkwony
PG Student, Department of Mechanical Engineering, Andhra University, Visakhapatnam
Ch. Ratnam
Professor, Department of Mechanical Engineering, Andhra University, Visakhapatnam
Abstract: Beams are defined by considering their boundary conditions. This paper focused on the Free Vibration Analysis of SiC aluminium- reinforced composite beams by considering four boundary
i.e. clamped-free, clamped-clamped, clamped-simply supported, and simply supported. The study utilized the Euler-Bernoulli beam theory to obtain the frequency equation and numerical simulations on
the ANSYS Workbench to analyze the free vibration behaviour. The results obtained for SiC aluminium-reinforced composite are compared with those of Aluminium and steel material. The study
demonstrates that boundary conditions affect the dynamic response of the composite beams, with clamped-clamped boundary conditions yielding higher natural frequencies, followed by clamped-simply
supported, simply supported, and clamped-free boundary conditions yielding low natural frequencies. Furthermore, the natural frequencies of SiC/Aluminium composite beams are higher than those of
unreinforced Aluminium and steel beams. The study found that the natural frequency of vibrations increases linearly with an increase in the cross-section area of the beam. Finally, the study found
that the natural frequency of vibrations increases with an increase in the specific modulus of the material.
Keywords: Natural frequency, Boundary Conditions, SiC/Aluminium composites, Free Vibration Analysis, Euler-Bernoulli Theory, Finite Element Analysis.
1. INTRODUCTION
With the advances in technology, composite materials have become the preferred choice for constructing mechanical equipment and structures. Silicon Carbide (SiC) reinforced Aluminium composites
are among the best lightweight composites used in high- performance applications. These composites are low in density but have high strength and stiffness, making them suitable for applications
in the aerospace industry and lightweight structures. In that connection, the study of the vibration characteristics of composite beams is a significant and distinctive area of focus in the field
of mechanical engineering. It is particularly essential to quantify the impact of dynamic loading on structures such as tall buildings, long bridges, and industrial machinery. Dynamic loading can
lead to fatigue and the initiation of cracks, which are major contributors to accidents and failures in industrial machinery.
Lu et. al. [1] investigated the effect of the vibration frequency on the fatigue of strength of 6061-T6 Al Alloys through two stress analysis methods namely nominal and hot-spot stress. Mufazzal
et al [2] explored the effect of material and surface cracks on the free vibration of the cantilever beam. Agarwallaa and Parhib [3] highlighted that, at the point where cracks appear, the
vibration frequency is high. The study was conducted experimentally and with the help of Fine Element software.
Nikhil and Jeyashree [4] investigated the dynamic response of a cracked beam to free vibration. The study utilized ANSYS the effects of cracks at different locations and depths in cantilever
beams, fixed-fixed beams, and simply supported beams. Mia et.al.[5] studied the natural frequency and mode shapes of transverse vibration on the cracked and uncracked cantilever beams. The
analysis was extended to find the impact of crack opening size and mesh refinement. Gawande and More [6] performed free vibration analysis to investigate the effect of the notch on the dynamics
of cantilever beams using ANSYS and experiment. The study accounted for the depth and position of the notch in the beam.
Kuppast et al [7] used ANSYS and experimental modelling to investigate the vibration properties of aluminium alloys. The study simulates the effect of increasing copper and silicon content in
aluminium alloys. Abdellah et al. [8] investigate the vibration behaviour of aluminium and its alloys. The samples were designed as cantilever plates with and without holes. The analysis was
performed with Ansys. Derkach et al. [9] analyzed the effect of the notch on the fundamental mode of vibration for composite cantilever beams using the Finite element analysis.
Quila et al. [10] studied the free vibration analysis of an uncracked and cracked fixed beam using ANSYS. Ferreira and Neto [11] modelled active Ni-Ti filament-reinforced hybrid adaptive
composite beams under free-free boundary conditions to study vibration modes and their frequencies. Avcar [12] investigates the free vibration of square cross-sectioned Aluminium beams both
analytically and numerically under four different boundary conditions. Haskul and Kisa [13] investigate the free vibration of a
double-tapered beam with linearly varying thickness and width using finite element and component mode synthesis methods.
Rossit et al. [14] investigate the vibrational behaviour of L-shaped beams with cracks. The transversal displacements were described using the Euler-Bernoulli beam theory, while the crack was
modelled as an elastically restrained hinge. Wang and Qiao [15] study the vibration behaviour of beams with arbitrary discontinuities and boundary conditions. Charoensuk and Sethaput
[16] performed a vibration analysis experiment and finite element analysis on metal plates with V-notch at multiple notch locations. Shah et al. [17] used ANSYS to perform the free vibration of
composite beams and obtained fundamental natural frequencies. Bozkurt et al. [18] explore analytical approximation techniques in transverse vibration analysis of beams. The computations were
performed using the Adomian Decomposition Method (ADM), the Variational Iteration Method (VIM), and the Homotopy Perturbation Method (HPM). Nalbant et al. [19] investigated the free vibration
behaviour of stepped nano-beams using the Bernoulli-Euler theory for beam analysis and Eringen's nonlocal elasticity theory for nanoscale analysis. The system's boundary conditions were defined
as simply supported. Teggi [20] explores the free vibration of steel beams under two different boundary conditions: Clamped-Free (C-F) and Clamped-Clamped (C-C). Santhosh et al. [21] conducted
vibration tests on Aluminium 5083 reinforced with varying percentage weights of Silicon Carbide (SiC) and fly ash through experimentation.
Bozkurt and Ersoy [22] investigated the vibration behaviour of metal matrix composites (MMCs) used in the aerospace industry
using finite element analysis (FEM). The study focused on AA2124/SiC/25p, a particle-reinforced MMC with a homogeneous distribution of particles, hence commonly used in aerospace applications.
Acharya et al. [23] analyzed the dynamic characteristics of Aluminium 6061 plates. Modal analysis was performed using both simulation and experimental methods. Kumar et al. [24] conducted a modal
analysis of AA5083 composite material reinforced with multi-wall carbon nanotubes using analytical and Finite element methods. Taj et al. [25] studied the vibrational characteristics of Aluminium
graphite metal matrix composites. The study evaluated the natural frequencies and mode shapes of the composites by experiments and finite element analysis methods. Lakshmikanthan et al. [26]
performed the free vibration analysis of A357 Alloy reinforced with dual-particle size Silicon Carbide Metal Matrix composite plates using the Finite Element Method. The study examined the
natural frequencies and mode shapes of the composite plates under Clamped-Clamped and Simply Supported-Simply Supported boundary conditions.
In this paper, free vibration analysis on SiC/Aluminium composite beams will be performed. This study will focus on the effects of the four types of boundary conditions, namely, C-F, C-C, C-SS,
and SS-SS, on the natural frequencies and mode shapes of the beams. Additionally, the effects of the mechanical properties of the SiC/Aluminium composite on the fundamental natural frequencies of
vibration will also be evaluated. These results will be compared to the results of unreinforced aluminium and steel material
1. Halpin-Tsai equation
Since SiC/Aluminium is a particulate composite, the Halpin- Tsai equation predicts the Young Modulus of Elasticity. The equation is as follows:
Em ((1+2sqVp))
(iii). Governing Equation formulations
Let's apply the Euler-Bernoulli Beam theory to a beam with length L and uniform cross-section. A is considered. Assuming the beam to be elastic with Youngs Modulus E, and the Density .
The relationship between the bending moment and deflection can be expressed as:
EC =
( Ep 1)
M = EI
d2y (4)
Where q = Em
( Ep +2s)
Where E is Youngs Modulus, I is the moment of inertia
of the beam and y is the deflection of the beam. For a
EC = Composite Young Modulus , Ep =
Particles Young Modulus , Em = Matrix Young Modulus,
V = Particles Volume s = Particle Aspect ratio (1 2 )
uniform homogenous beam, the equation of motion is obtained as:
EI d4y + d2y = 0, for 0 x L (5)
p , A dx4
2. Rule of Mixtures
By application of the rule of mixtures, the density of the composite is obtained as follows:
Where is Density, and A is the cross-section area of the beam.
c2 d4y + d2y = 0, for 0 x L (6)
= V + V
dx4 dt2
c p p m m
Where c = EI
c =Density of composite p=Density of SiC particles
m=Density of Aluminium Matrix
Vp=SiC Particles Volume Vm=Aluminium Matrix Volume
The solution of equation (5) is obtained by the method of separation of variables thus, one part depends on position and the other part depends on time.
y = W(x)T(t) (8)
Where W is independent of time and T is independent of position. Substituting equation (8) into equation (6) and simplifying we get,
sinh L sin L cosh L cos L c
[ 1 0
cosh L cos L sinh L + sin L ] [c3]=[ ]
c2 d4W(x) = 1
For a nontrivial solution of C1 and C3 then obtaining the
determinant of the coefficients will be zero. Then the
The Equation (9) is expressed as two separate differential equations:
Position variable: d4W 4W(x) = 0 (10)
solution is as follows:
cos L cosh L = 1 (22)
The first three roots of equation (22) are determined
Where 4 = 2 = A2
numerically using the MATLAB commands code. The
c2 EI
Time variable: d T(x) + 2T(t) = 0 (12)
The general solution for equation (10) is:
W(x) = C1 sinh x + C2 cosh x + C3 sin x + C4 cos x (13)
C1, C2, C3, and C4 are constants, they are obtained by considering boundary conditions, and sinh and cosh, are
roots L are referred to as eigenvalues.
L = 4.73004 for n = 1, L = 7.85321 for n = 2 ,
L = 10.9956 for n = 3
Where n is the mode number. (23)
1. Clamped-Simply Supported (C-SS) beam
The boundary conditions for the C-SS beam are;
the hyperbolic functions.
At x = 0, w(x) = 0 and
= 0 (24)
To solve equation (13), we consider the following conditions:
1. Clamped-Free (C-F) beam
The boundary conditions are;
At x = L, w(L) = 0 and d2w = 0 (25)
When the above boundary conditions are considered in
equation (13),
C2 + C3 = 0
By simplifications, the following matrix expression is
At x = 0, w(x) = 0 and
= 0 (14)
At x = L, d2w = 0 and d3w = 0 (15)
sinh L sin L cosh L cos L c1 0
[sinh L + sin L cosh L + cos L] [c2]=[ ]
When the above boundary conditions are considered in equation (13),
C1 = 0, C3 = 0
By simplifications, the following matrix expression is
For a nontrivial solution of C1 and C2 then obtaining the determinant of the coefficients will be zero. Then the solution is as follows:
tanh L = tan L (27)
[ sinh L + sin L cosh L + cos L
c2 = 0
The first three roots of equation (27) are determined
cosh L + cos L sinh L sin L ] [c4] [0]
For a nontrivial solution of C2 and C4 then obtaining the determinant of the coefficients will be zero. Then the solution is as follows:
cos L cosh L = 1 (17)
The first three roots of equation (17) are determined numerically using the MATLAB commands code. The roots L are referred to as eigenvalues.
L = 1.87510 for n = 1 L = 4.69409 for n = 2 L = 7.85340 for n = 3 Where n is the mode number. (18)
2. Clamped-Clamped (C-C) beam
The boundary conditions for the C-C beam are;
At x = 0, w(x) = 0 and dw = 0 (19)
At x = L, w(L) = 0 and dw = 0 (20)
numerically using the MATLAB commands code provided in Appendix 1. The roots L are referred to as eigenvalues.
L = 3.9266 for n = 1, L = 7.0686 for n = 2, L = 10.2102 for n = 3 Where n is the mode number. (28)
2. Simply Supported-Simply Supported (SS-SS) beam
The boundary conditions for the SS-SS beam are;
At x = 0, w(x) = 0 and d2w = 0 (29)
At x = L, w(L) = 0 and d w = 0 (30)
When the above boundary conditions are considered in
equation (13),
C1 = 0, C2 = 0
By simplifications, the following matrix expression is
dx sinh L sin L
c3 0
When the above boundary conditions are considered in
[sinh L sin L] [c ]=[
] (31)
equation (13), 4 0
C2 = 0, C4 = 0
By simplifications, the following matrix expression is
For a nontrivial solution of C3 and C4 then obtaining the
determinant of the coefficients will be zero. Then the solution is as follows:
sinLsinhL = 0 (32)
The first three roots of equation (32) are determined numerically using the MATLAB commands code provided in Appendix 1. The roots L are referred to as eigenvalues.
L = 3.14159 for n = 1, L = 6.28318 for n = 2,
L = 9.42478 for n = 3 Where n is the mode number. (33)
Equations 17, 22, 27, and 32 are called frequency equations. By rearranging equation 11, it can be expressed as follows:
3. RESULTS AND DISCUSSION
1. Natural frequency across the material:
SiC Particles and Aluminium material properties were adapted from Yuan et al. [27]. To obtain the Elastic Modulus (Ec) of the composite, the rule of mixtures is applied with the help of
the Halpin-Tsai equation.
(440 1)
q = 70 = 0.5692 = 96.14 Gpa (440 + 2(1.5))
n n 4
= ( L)2 EI
, Where n=1,2, 3. n modes
EC =
70×109 ((1 + 2(1.5)(0.5692)(0.15)))
1 ((0.5692)(0.15))
numbers. (34)
Table 1 Properties of materials
c = pVp + mVm,
c = (3210×0.15) + (2700×0.5) = 2777 Kg/m3
Properties SiC Particles Aluminium Steel
Density (kg/m3) 3210 2700 7850
Modulus x 109 Pa
Particle Volume (Vp) in Percentag
15 85 –
e (%)
1-2 – –
ratio of particles
Problem: To demonstrate the vibration analysis of the
beam, the model with the following dimensional characteristics is considered for evaluation: Length (L) = 500mm, width (b) = 50mm, depth (d) = 10mm.
Table 2 Natural Frequency of Clamped-Free (C-F) beam
SiC/Aluminium Composite Aluminium Steel
Mode Method
Natural Frequency f in Hz
Analytical 38.02 32.90 33.42
Mode 1
Ansys 38.45 32.91 33.65
Analytical 238.29 206.17 209.43
Mode 2
Ansys 240.51 205.86 210.51
Analytical 666.98 577.08 586.20
Mode 3
Ansys 672.11 575.21 588.19
Table 3 Natural Frequency of Clamped-Clamped (C-C) beam
SiC/Aluminium Composite Aluminium Steel
Mode Method
Natural Frequency f in Hz
Analytical 241.95 209.34 212.65
Mode 1
Ansys 246.73 210.74 215.05
Analytical 666.94 577.05 586.17
Mode 2
Ansys 678.04 579.12 590.95
Analytical 1307.47 1131.25 1149.13
Mode 3
Ansys 1324.60 1131.30 1154.43
Table 4 Natural Frequency of Clamped-Simply Supported (C-SS) beam
SiC/Aluminium Composite Aluminium Steel
Mode Method
Natural Frequency f in Hz
Analytical 166.74 144.26 146.54
Mode 1
Ansys 168.40 145.40 147.37
Analytical 540.33 467.51 474.89
Mode 2
Ansys 544.72 470.78 477.71
Analytical 1127.36 975.42 990.83
Mode 3
Ansys 1133.80 978.74 992.08
Table 5 Natural Frequency of Simply Supported-Simply Supported (SS-SS) beam
SiC/Aluminium Composite Aluminium Steel
Mode Method
Natural Frequency f in Hz
Analytical 109.92 92.35 93.81
Mode 1
Ansys 106.71 92.33 93.79
Analytical 426.92 369.39 375.22
Mode 2
Ansys 426.56 369.03 374.85
Analytical 960.59 832.12 844.25
Mode 3
Ansys 956.60 829.11 842.18
SiC/Aluminium Composite presented the highest Natural frequency in all the modes that have been considered between Aluminium and Steel. This is due to the higher stiffness and lower
density of SiC/Aluminium composite hence natural frequencies of the structure occur at higher values. Aluminium is given higher stiffness by reinforcing with SiC particles and this in
turn improves the vibrational behavior of the composite. Natural frequencies for Aluminium are found to be less than those of SiC/Aluminium composite but close to that of Steel.
Aluminium has been found to have a lower density than Steel but the modulus of elasticity is lower; this results in Aluminium and Steel materials having similar natural frequencies.
Natural frequencies of Steel are slightly higher than that of the Aluminium across all modes. However, a comparison between Aluminium and Steel shows that the two are not very different,
and hence have very close values of the vibrational frequency.
2. ANSYS Graphical results for SiC/Aluminium beam
Figure 1 C-F first three Mode Shapes for SiC/Aluminium beam.
Figure 2 C-C first Mode Shapes for SiC/Aluminium beam.
Figure 4 SS-SS first Mode shapes for SiC/Aluminium beam.
Figure 3 C-SS first Mode shapes for SiC/Aluminium beam.
3. Comparative analysis between Analytical and ANSYS Results Results for SiC/Aluminium Composites are considered for clamped-free beam as an example.
The differences in results obtained by the two methods are expressed in percentages. These percentages are obtained as follows. If the three modes of vibration n = 1,2,3 , analytical
natural frequency as fn analyticl and Ansys natural frequency as fn ansys, then:
fn analyticlfn ansys
and modes. These variations include the analytical approximations made during analysis and would fall below 1%. Comparing Mode 1 for the SS-SS beam, it is safe to say that the SiC/
Aluminium composite diverged most (3.02 Hz or about 2.9% deviation) from the actual, probably due to some difficulties in accurately simulating the composite
Percentage deviation = (
fn analyticl
) × 100%
material. For modes 2 and 3, the difference in the analytical solution and the Ansys solution is lower for higher
The Analytical and Ansys natural frequencies differ by – 1.12%, -0.93%, and -0.77% under C-F boundary conditions
beam, -1.98%, -1.66%, and -1.31% under C-C boundary
conditions, -1.00%, -0.81% and -1.31% under C-SS
boundary conditions and 2.92%, 0.08% and 0.42% under SS-SS boundary conditions.
Mode 3 Mode 2 Mode 1
Figure 5 shows the close conformity of solutions obtained by analytical and the Ansys approaches for all the materials
frequency modes indicating that the models are more accurate at higher modes. This could be because higher modes are less sensitive to the boundary condition.
Figure 5 Percentage deviation in natural frequency obtained by Analytical and Ansys.
Effects of boundary conditions – Results for SiC/Aluminium Composites are considered as an example.
The results for SiC/Aluminium are extracted from Tables 2 to 5 and populated as shown in Table 6. For the C-F boundary condition natural frequencies are lowest compared to C-C, C-SS, and
SS-SS across all modes. Thus,
the C-F condition provides more displacement at the free end resulting in low stiffness and consequently low natural frequencies.
Table 6 Natural frequencies of SiC/Aluminium beam supported by different boundary conditions.
Boundary Conditions Analysis Method Natural Frequency f in Hz for SiC/Aluminium Composite Beam
C-F C-C C-SS SS-SS
Mode 1 Analytical 38.02 241.95 166.74 109.92
Ansys 38.445 246.73 168.4 106.71
Analytical 238.29 666.94 540.33 426.92
Mode 2
Ansys 240.51 678.04 544.72 426.56
Analytical 666.98 1307.47 1127.36 960.59
Mode 3
Ansys 672.11 1324.60 1133.80 956.60
Mode 1
The C-C boundary condition provided the highest natural frequency across all modes. This condition provides much no freedom of movement of the beam hence resulting in high stiffness and
high natural frequencies. The C-SS boundary condition resulted in natural frequencies higher than C-F and SS-SS but lower than C-C conditions. The C- SS has one end restraint and the
other end is free to rotate.
These conditions provide intermediate natural frequencies. The SS-SS boundary conditions result in natural frequencies lower than C-SS and higher than C-F. This condition also permits
some extent of rotation at the supports which results in a lower degree of stiffness as compared to C-C. Figures 6, 7, and 8 provide graphical representations of the impact of boundary
conditions on the natural frequencies of beams.
atural Frequency
in Hz
atural Frequency in
Figure 6 Mode 1 natural frequencies versus boundary conditions.
Figure 7 Mode 2 natural frequencies versus boundary conditions
Mode 3
Natural Frequency in Hz
Figure 8 Mode 3 natural frequencies versus boundary conditions.
4. Effects of specific modulus on natural frequencies of the beam.
The properties of the material determine the basic associated frequencies of vibration of beams. The beam material has a unique property called Specific Modulus (E/) which has to be considered
Properties of Materials
SiC/Aluminium Ec
= 96.14 Gpa, = 2777kg/m3 , E = 34.62x106m2/s2
Aluminium E = 70 Gpa, = 2700kg/m3, E = 25.93x106m2/s2
Steel E = 210 Gpa, = 7850kg/m3, E = 26.75x106m2/s2
Table 7 Natural frequencies versus boundary conditions at specific modulus of materials.
Specific Modulus
25.93 26.75 36.62
Boundary Condition Mode
Natural Frequency in Hz
Mode 1 32.9 33.42 38.02
C-F Mode 2 206.17 209.43 238.29
Mode 3 577.08 586.2 666.98
Mode 1 209.34 212.65 241.95
C-C Mode 2 577.05 586.17 666.94
Mode 3 1131.25 1149.13 1307.47
Mode 1 144.26 146.54 166.74
C-SS Mode 2 467.51 474.89 540.33
Mode 3 975.42 990.83 1127.36
Mode 1 92.35 93.81 109.92
SS-SS Mode 2 369.39 375.22 426.92
Mode 3 832.12 844.25 960.59
Analytical results from Tables 2 to 5 are used to generate Table 7 above. The data in Table 7 are used to generate Figure 9 below. The specific modulus is one of those parameters which determine
the natural frequency of a given material. Generally, a higher value of specific modulus results in higher natural frequencies, because the material is stiffer or the structure is lighter. SiC/Al
(Specific Modulus = 36.62×106) used in the present study exhibits the highest natural frequencies across all the boundary conditions and modes. Consequently, the higher specific
modulus of the SiC/Al means higher stiffness resulting in higher resistance to deformation and thus, higher natural frequencies. Steel (Specific Modulus = 26.75 x 106) exhibits natural
frequencies a little higher than Aluminium, but lower as compared to SiC/Al. Steel material has a higher density compared to aluminium and a relatively higher elastic modulus and therefore
natural frequencies. Aluminium (Specific Modulus = 25.93 x 106) has the lowest specific modulus among the three materials resulting in the lowest natural frequencies. Due to it having a lower
the material can undergo larger deformation than the other two materials, which in turn lowers natural frequencies.
From Figure 9, it was noted that the natural frequency increases with an increase in the Specific Modulus of the material. The rate of increase in natural frequency is more pronounced in the C-C
mode 3 condition, followed by the C-SS mode 3 condition, and SS-SS mode 3 condition. The intermediate increase was noted at C-C mode 2 and C-F mode 3 conditions, followed by C-SS mode 2
condition, and SS-SS mode 2 condition. The low increase was noted at C- C mode 1 and C-F mode 2 condition, followed by C-SS
mode 1 condition, SS-SS mode 1 condition and C-F mode 1 condition.
The frequency curve of the beam at C-F mode 1 condition can be noted to be a horizontal line. This signifies that the effect of material properties on the natural frequency of the C-F mode 1 is
insignificant. This observation portrayed the effect of boundary conditions on the vibration of the beam. The boundary condition does offer a different stiffness effect to the beam; thus, the
free end of the C-F beam lowers the stiffness, hence in result lowers the natural frequencies of vibration.
C-F Mode 1
C-C Mode 2
C-F Mode 2
C-C Mode 3
C-F Mode 3
C-SS Mode 1
C-C Mode 1
C-SS Mode 2
C-SS Mode 3
SS-SS Mode 1
SS-SS Mode 2
SS-SS Mode 3
CC Mode 3
CSS Mode 3
SSSS Mode 3
CC Mode 2 & CF Mode
CSS Modee 2
SSSS Mode 2
CSS Mode 1 and CF
Mode 2
CC Mode 1
SSSS Mode 1
CF Mode 1
Specific Modulus X 106
Natural Frequency In Hz
Figure 9 Natural frequencies versus Specific Modulus.
5. Effects of Cross-section area of the beam on the natural frequencies of vibration.
For the presentation of this study, a beam of the following characteristics was considered. The Length = 500mm, Cross-section area A1=0.0005m2, A2= 0.0008m2, A3= 0.0015m2. The Physical properties of
SiC/Aluminium composite are Density = 2777kg/m3, and Estimated Young Modulus of Elasticity = 96.14 GPa.
Table 8 Natural frequency for Beam under four different boundary conditions versus cross-area
Natural Frequency in Hz
Boundary conditions Mode n of Vibration
A1 A2 A3
Mode 1 38.02 76.04 114.06
C-F Mode 2 238.29 476.56 714.79
Mode 3 666.98 1333.91 2000.74
Mode 1 241.95 483.88 725.78
C-C Mode 2 666.94 1333.84 2000.64
Mode 3 1307.47 2614.86 3922.05
Mode 1 166.74 333.46 500.16
C-SS Mode 2 540.33 1080.63 1620.85
Mode 3 1127.36 2254.65 3381.76
Mode 1 109.92 213.46 320.17
SS-SS Mode 2 426.92 853.83 1280.66
Mode 3 960.59 1921.11 2881.49
From the consideration, the beam has a fixed length (L) and material properties, thus, Figure 10 depicts the natural frequency of vibrations to increase linearly with an increase in cross-section
area. The rate of an increase in natural frequency is more pronounced for the C-C boundary condition, followed by C-SS, SS-SS, and C-F. These effects happen because the mode shape constant for C-C
and C- SS are higher compared to SS-SS and C-F boundary conditions. This underscores the role boundary conditions play as the C-C condition yields the highest frequencies due to maximum stiffness and
the C-F condition yields the lowest due to greater flexibility.
The natural frequencies for the beam under four different boundary conditions were estimated analytically and numerically using ANSYS Workbench. The results obtained were consistently in agreement
for both methods. It was noted that higher natural frequencies were achieved by SiC/ Aluminium composite beams across all four boundary conditions considered, followed by structural Steel and
Aluminium beams. This is because SiC/ Aluminium composites have a higher specific modulus than Steel and Aluminium.
The higher natural frequency is experienced in C-C boundary conditions, followed by C-SS, SS-SS, and lower in C-F boundary conditions. The linear increase in natural frequencies is depicted to
increase with the beam cross-sectional area when the horizontal length
and mass distribution of the beam are constant. The rate of increase in the natural frequency is more pronounced in C-C boundary conditions, followed by C-SS, SS-SS, and least under C-F boundary
C-F Mode 1
C-C Mode 2
C-SS Mode 3
C-F Mode 2
C-C Mode 3
SS-SS Mode 1
C-F Mode 3
C-SS Mode 1
SS-SS Mode 2
C-C Mode 1
C-SS Mode 2
SS-SS Mode 3
C-C Mode 3
C-SS Mode 3
SS-SS Mode 3
C-C Mode 1 & C-F
Mode 2
SS-SS Mode 1
C-C Mode 2 & C-F
Mode 3
C-SS Mode 2
SS-SS Mode 2
C-SS Mode 1
C-F Mode 1
Cross-section Area
Natural Frequency In Hz
Figure 10 Natural frequency versus cross-section area at different boundary conditions.
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Plates Using Finite Element Method. Archives of Foundry Engineering, 101112. | {"url":"https://www.ijert.org/analytical-and-numerical-investigation-of-free-vibration-in-beams-under-diverse-boundary-conditions-and-material-characteristics","timestamp":"2024-11-04T16:49:06Z","content_type":"text/html","content_length":"108937","record_id":"<urn:uuid:86d07cb2-5bb8-42e0-b107-01796dc80fd1>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00330.warc.gz"} |
Beno Eckmann - Wikiquote
• We probably all agree that eventually reducing a difficult problem to a "nice" situation is at the heart of mathematics.
An Interview with Beno Eckmann (2010)
• Together with many other people and after a long development I could prove that a Poincaré duality group of cohomological dimension 2 is the group of a Riemann surface. That was actually a
conjecture of Jean-Pierre Serre. "You have to prove it!" he had always insisted.
□ "An interview with Beno Eckmann by by Martin Raussen and Alain Valette". Math.ch/100: Schweizerische Mathematische Gesellschaft 1910-2010 / Societe Mkathematique Suisse 1910-2010 / Swiss
Mathematical Society 1910-2010. European Mathematical Society. 2010. pp. 389–401. ISBN 978-3-03719-089-0. (quote from p. 394)
• Still another important area is Poincaré duality for groups, invented by Robert Bieri and myself. They behave like manifolds: homology, cohomology, you see, in complementary dimensions, but with
another dualizing module. Many groups that are interesting in algebraic geometry, group theory or other areas are such duality groups. | {"url":"https://en.m.wikiquote.org/wiki/Beno_Eckmann","timestamp":"2024-11-12T08:51:53Z","content_type":"text/html","content_length":"31026","record_id":"<urn:uuid:a58936f5-18cb-4e06-b20c-a02983c2f729>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00318.warc.gz"} |
Current issue
Current volume
• 67 (2024)
Past volumes
Current issue
Articles are published here before the issue is completed.
Vol. 67, no. 2 (2024)
Decidable objects and molecular toposes.
We study several sufficient conditions for the molecularity/local-connectedness of geometric morphisms. In particular, we show that if $\mathcal{S}$ is a Boolean topos, then, for every 397–415
hyperconnected essential geometric morphism $p : \mathcal{E} \rightarrow \mathcal{S}$ such that the leftmost adjoint $p_{!}$ preserves finite products, $p$ is molecular and $p^* : \mathcal
{S} \rightarrow \mathcal{E}$ coincides with the full subcategory of decidable objects in $\mathcal{E}$. We also characterize the reflections between categories with finite limits that induce
molecular maps between the respective presheaf toposes. As a corollary we establish the molecularity of certain geometric morphisms between Gaeta toposes.
Extinction time of an epidemic with infection-age-dependent infectivity.
This paper studies the distribution function of the time of extinction of a subcritical epidemic, when a large enough proportion of the population has been immunized and/or the infectivity 417–443
of the infectious individuals has been reduced, so that the effective reproduction number is less than one. We do that for a SIR/SEIR model, where infectious individuals have an
infection-age-dependent infectivity, as in the model introduced in Kermack and McKendrick's seminal 1927 paper. Our main conclusion is that simplifying the model as an ODE SIR model, as it
is largely done in the epidemics literature, introduces a bias toward shorter extinction time.
On hyponormality and a commuting property of Toeplitz operators.
In this work we give sufficient conditions for hyponormality of Toeplitz operators on a weighted Bergman space when the analytic part of the symbol is a monomial and the conjugate part is a
polynomial. We also extend a known commuting property of Toeplitz operators with a harmonic symbol on the Bergman space to weighted Bergman spaces.
Genus and book thickness of reduced cozero-divisor graphs of commutative rings.
For a commutative ring $R$ with identity, let $\langle a\rangle$ be the principal ideal generated by $a\in R$. Let $\Omega(R)^*$ be the set of all nonzero proper principal ideals of $R$. The
reduced cozero-divisor graph $\Gamma_r(R)$ of $R$ is the simple undirected graph whose vertex set is $\Omega(R)^*$ and such that two distinct vertices $\langle a\rangle$ and $\langle b\ 455–473
rangle$ in $\Omega(R)^\ast$ are adjacent if and only if $\langle a \rangle\nsubseteq\langle b\rangle$ and $\langle b\rangle\nsubseteq\langle a\rangle$. In this article, we study certain
properties of embeddings of the reduced cozero-divisor graph of commutative rings. More specifically, we characterize all Artinian nonlocal rings whose reduced cozero-divisor graph has genus
two. Also we find the book thickness of the reduced cozero-divisor graphs which have genus at most one.
Boundedness of geometric invariants near a singularity which is a suspension of a singular curve.
Near a singular point of a surface or a curve, geometric invariants diverge in general, and the orders of this divergence, in particular the boundedness about these invariants, represent the 475–502
geometry of the surface and the curve. In this paper, we study the boundedness and orders of several geometric invariants near a singular point of a surface which is a suspension of a
singular curve in the plane, and those of the curves passing through the singular point. We evaluate the orders of the Gaussian and mean curvatures, as well as those of the geodesic and
normal curvatures, and the geodesic torsion for the curve.
Complete presentation and Hilbert series of the mixed braid monoid $MB_{1,3}$.
The Hilbert series is the simplest way of finding dimension and degree of an algebraic variety defined explicitly by polynomial equations. The mixed braid groups were introduced by Sofia
Lambropoulou in 2000. In this paper we compute the complete presentation and the Hilbert series of the canonical words of the mixed braid monoid $MB_{1,3}$.
One-sided EP elements in rings with involution.
This paper investigates the one-sided EP property of elements in rings with involution. Let $R$ be a ring with involution $\ast$. Then $a \in R$ is said to be left (resp. right) EP if $a$ is 517–528
Moore–Penrose invertible and $aR \subseteq a^{\ast}R$ (resp. $a^{\ast}R \subseteq aR$). Many properties of EP elements are extended to one-sided versions. Some new characterizations of EP
elements are presented in relation to the absorption law for Moore–Penrose inverses.
Evolution of first eigenvalues of some geometric operators under the rescaled List's extended Ricci flow.
Let $(M,g(t), e^{-\phi}d\nu)$ be a compact weighted Riemannian manifold and let $(g(t),\phi(t))$ evolve by the rescaled List's extended Ricci flow. In this paper, we study the evolution 529–543
equations for first eigenvalues of the geometric operators, $-\Delta_{\phi}+cS^{a}$, along the rescaled List's extended Ricci flow. Here $\Delta_{\phi}=\Delta-\nabla\phi.\nabla$ is a
symmetric diffusion operator, $\phi\in C^{\infty}(M)$, $S=R-\alpha|\nabla \phi|^{2}$, $R$ is the scalar curvature with respect to the metric $g(t)$ and $a, c$ are some constants. As an
application, we obtain some monotonicity results under the rescaled List's extended Ricci flow.
The $w$-core–EP inverse in rings with involution.
The main goal of this paper is to present two new classes of generalized inverses in order to extend the concepts of the (dual) core–EP inverse and the (dual) $w$-core inverse. Precisely, we 545–565
introduce the $w$-core–EP inverse and its dual for elements of a ring with involution. We characterize the (dual) $w$-core–EP invertible elements and develop several representations of the
$w$-core–EP inverse and its dual in terms of different well-known generalized inverses. Using these results, we get new characterizations and expressions for the core–EP inverse and its
dual. We apply the dual $w$-core–EP inverse to solve certain operator equations and give their general solution forms. | {"url":"https://inmabb.criba.edu.ar/revuma/revuma.php?p=current-issue","timestamp":"2024-11-13T21:45:00Z","content_type":"text/html","content_length":"25327","record_id":"<urn:uuid:254d83d4-53de-426d-bab3-09580a325499>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00534.warc.gz"} |
How do you find the probability of a spinner?
Starts here3:12Probability Models & Multiplication Rule for Independent Events (The …YouTubeStart of suggested clipEnd of suggested clip58 second suggested clipFirst it lists all the outcomes. So in
this case the spinner A’s outcomes could either be a 1 2 or aMoreFirst it lists all the outcomes. So in this case the spinner A’s outcomes could either be a 1 2 or a 5. The next thing the probability
model tells you is the probability of all the outcomes.
What is the probability of getting a 3 on this spinner?
The probability of spinning a 3 is also 2 / 8 .
What is the probability of the spinner landing on a primary color?
The probability of landing on each color of the spinner is always one fourth. In Experiment 2, the probability of rolling each number on the die is always one sixth. In both of these experiments, the
outcomes are equally likely to occur.
What is the probability of landing on a 3?
A number cube (dice) has six sides labelled 1 through 6. Hence, a fair dice has a probability of 16 to land on any predetermined number 1 through 6. Therefore, to land on 3 the probability is 16 .
How do I calculate the probability?
Divide the number of events by the number of possible outcomes.
1. Determine a single event with a single outcome.
2. Identify the total number of outcomes that can occur.
3. Divide the number of events by the number of possible outcomes.
4. Determine each event you will calculate.
5. Calculate the probability of each event.
What is the probability of landing on each color?
What is the theoretical probability of the spinner landing on each color? Since there is only one sector in each color (red, blue, and yellow), the probability of the spinner landing on each color is
1 over 3 1 3 .
What is the probability of a spinner landing on red?
1 in 4
The chances of landing on red are 1 in 4, or one fourth. This problem asked us to find some probabilities involving a spinner. Let’s look at some definitions and examples from the problem above. An
experiment is a situation involving chance or probability that leads to results called outcomes.
What is the probability of 3 coins landing on heads?
Three flips of a fair coin Suppose you have a fair coin: this means it has a 50\% chance of landing heads up and a 50\% chance of landing tails up. Suppose you flip it three times and these flips are
independent. What is the probability that it lands heads up, then tails up, then heads up? So the answer is 1/8, or 12.5\%.
How do you calculate outcomes?
Divide the number of events by the number of possible outcomes. After determining the probability event and its corresponding outcomes, divide the total number of events by the total number of
possible outcomes. For instance, rolling a die once and landing on a three can be considered one event.
What is the probability of spinning a number on the spinner?
There are 8 numbers in total on the spinner. There are 3 ones on the spinner. The probability of spinning a ‘1’ is 3 / 8 . The spinner will land on a ‘1’ three times out of every eight. The
probability of the spinner landing on a number is equal to the fraction of the spinner that this number occupies.
How many ones are there on the spinner?
There are 3 ones on the spinner. The probability of spinning a ‘1’ is 3 / 8. The spinner will land on a ‘1’ three times out of every eight. The probability of the spinner landing on a number is equal
to the fraction of the spinner that this number occupies.
What is the probability of rolling each number on the die?
In Experiment 1 the probability of each outcome is always the same. The probability of landing on each color of the spinner is always one fourth. In Experiment 2, the probability of rolling each
number on the die is always one sixth. In both of these experiments, the outcomes are equally likely to occur.
Why is spinning a 1 the most likely outcome?
We can see that spinning a 1 is the most likely outcome because it has the biggest probability of occurring. When teaching probability, a common misconception is that if something is the most likely
then we will expect it to occur most of the time. | {"url":"https://profoundadvices.com/how-do-you-find-the-probability-of-a-spinner/","timestamp":"2024-11-03T04:12:45Z","content_type":"text/html","content_length":"58280","record_id":"<urn:uuid:a8b99fdd-4169-46c5-b6e9-1436e35a7aa8>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00545.warc.gz"} |
CSCI 570 Homework 3 solved
For all divide-and-conquer algorithms follow these steps:
1. Describe the steps of your algorithm in plain English.
2. Write a recurrence equation for the runtime complexity.
3. Solve the equation by the master theorem.
For all dynamic programming algorithms follow these steps:
1. Define (in plain English) subproblems to be solved.
2. Write the recurrence relation for subproblems.
3. Write pseudo-code to compute the optimal value
4. Compute its runtimecomplexity in terms of the input size.
1. Suppose we define a new kind of directed graph in which positive weights are
assigned to the vertices but not to the edges. If the length of a path is defined
by the total weight of all nodes on the path, describe an algorithm that finds
the shortest path between two given points A and B within this graph.
2. For each of the following recurrences, give an expression for the runtime T(n)
if the recurrence can be solved with the Master Theorem. Otherwise, indicate
that the Master Theorem does not apply.
• T(n) = 3T(n/2) + n
• T(n) = 4T(n/2) + n
• T(n) = T(n/2) + 2n − n
• T(n) = 2nT(n/2) + n
• T(n) = 16T(n/4) + n + 10
• T(n) = 2T(n/2) + nlogn
• T(n) = 2T(n/4) + n
• T(n) = 0.5T(n/2) + 1/n
• T(n) = 16T(n/4) + n!
• T(n) = 10T(n/3) + n
3. Suppose that we are given a sorted array of distinct integers A[1, …, n] and we
want to decide whether there is an index i for which A[i] = i. Describe an
efficient divide-and-conquer algorithm that solves this problem and explain the
time complexity.
4. We know that binary search on a sorted array of size n takes O(log n) time.
Design a similar divide-and-conquer algorithm for searching in a sorted singly
linked list of size n. Describe the steps of your algorithm in plain English.
Write a recurrence equation for the runtime complexity. Solve the equation by
the master theorem
5. Given n balloons, indexed from 0 to n−1. Each balloon is painted with a number on it represented by array nums. You are asked to burst all the balloons. If
the you burst balloon i you will get nums[i − 1] × nums[i] × nums[i + 1] coins.
Here left and right are adjacent indices of i. After the burst, the left and right
then becomes adjacent. You may assume nums[−1] = nums[n] = 1 and they
are not real therefore you cannot burst them. Design a dynamic programming
algorithm to find the maximum coins you can collect by bursting the balloons
wisely. Analyze the running time of your algorithm.
Example. If you have the nums = [3, 1, 5, 8]. The optimal solution would
be 167, where you burst balloons in the order of 1, 5, 3 and 8. The array nums
after each step is:
[3, 1, 5, 8] → [3, 5, 8] → [3, 8] → [8] → []
And the number of coins you get is 3×1×5+3×5×8+1×3×8+1×8×1 = 167.
6. Given a non-empty string str and a dictionary containing a list of unique words,
design a dynamic programming algorithm to determine if str can be segmented
into a sequence of dictionary words. If str =“algorithmdesign” and your dictionary contains “algorithm” and “design”. Your algorithm should answer Yes as
str can be segmented as “algorithmdesign”. You may assume that a dictionary
lookup can be dome in O(1) time.
7. A tourism company is providing boat tours on a river with n consecutive segments. According to previous experience, the profit they can make by providing
boat tours on segment i is known as ai. Here, ai could be positive (they earn
money), negative (they lose money), or zero. Because of the administration
convenience, the local community requires that the tourism company do their
boat tour business on a contiguous sequence of the river segments (i.e., if the
company chooses segment i as the starting segment and segment j as the ending
segment, all the segments in between should also be covered by the tour service,
no matter whether the company will earn or lose money). The company’s goal
is to determine the starting segment and ending segment of boat tours along
the river, such that their total profit can be maximized. Design a dynamic
programming algorithm to achieve this goal and analyze its runtime. | {"url":"https://codeshive.com/questions-and-answers/csci-570-homework-3-solved/","timestamp":"2024-11-04T02:59:13Z","content_type":"text/html","content_length":"102642","record_id":"<urn:uuid:ebf513c3-6767-453b-b1b4-9e6255a4278b>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00595.warc.gz"} |
On finitely stable domains, II
Among other results, we prove the following: (1) A locally Archimedean stable domain satisfies accp. (2) A stable domain R is Archimedean if and only if every nonunit of R belongs to a height-one
prime ideal of the integral closure R0 of R in its quotient field (this result is related to Ohm's theorem for Prufer domains). (3) An Archimedean stable domain R is one-dimensional if and only if R0
is equidimensional (generally, an Archimedean stable local domain is not necessarily one-dimensional). (4) An Archimedean finitely stable semilocal domain with stable maximal ideals is locally
Archimedean, but generally, neither Archimedean stable domains, nor Archimedean semilocal domains are necessarily locally Archimedean.
Bibliographical note
Publisher Copyright:
© Rocky Mountain Mathematics Consortium.
• Accp
• Archimedean domain
• Completely integrally closed
• Finite character
• Finitely stable
• Locally archimedean
• Mori domain
• Stable ideal
ASJC Scopus subject areas
• Algebra and Number Theory
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Binary Numbers Worksheet For Kids [2021]
Binary Numbers worksheet for kids
Binary numbers are the fundamentals of how computer systems work. Understanding binary number systems help kids to discover the mysterious world of computers. I decided to find out the best ways to
teach kids about binary numbers.
Binary numbers worksheet helps kids to learn the binary system quickly. The worksheets feature games and challenges that will stimulate kids’ interest to learn and enforce their understanding of the
binary numbers system. Binary worksheets are great teaching tools for both schools and at home.
In this blog post, I want to show you what a binary system is, a few simple terms you need to know, and why kids need to understand. At the end of the blog post, you can download the free binary
numbers worksheet for kids and have fun with your kids or students.
What is a binary number?
A binary number system is a base-2 number system. There are only two numbers in the system: 0 and 1. We call them the Binary Numbers.
The first number system we are normally taught is the decimal system where we count from 0 to 9. A Decimal system is a base-10 system, where we group numbers in tens. In a binary numbers system, we
group numbers in 2s.
Why learn binary numbers?
In the simplest form, computers are machines that flip the binary digits on and off. There is no ambiguity for computers, it is either a YES (1) or NO (0) to them.
Computer programs are a set of instructions we give to computers. Our computer programs are then translated into machine codes – which are binary codes that a computer can understand.
In the modern computer, data is stored and transferred in a series of 0s and 1s. Yes, all your music, photos, documents on the computer are a series of 0s and 1s.
To understand how a computer and computer program works, it is essential to understand how binary numbers work. That’s why it is beneficial for new coders to have a good understanding of binary
It might sound complex to you, but it is simpler than you thought. In the remainder of this blog post, I will give you a brief introduction to binary numbers, binary conversions, and calculations.
Now we know what a binary number is, it is time to learn a few important computer terms that are related to binary numbers.
What are a Bit and Byte?
A Bit is the smallest unit of data in a computer system. It is short of binary digits. Its value could be 1 or 0. For example, there are 4 bits in binary number 1011.
Computer circuits are made of billions of transistors. A Transistor is a tiny component that switches ON and OFF by electronic signals.
In the electrical form, a BIT represents the state of ON or OFF of a transistor. A computer or machine uses series and sequences of these ONs and OFFs to process complex information.
8 bits = 1 byte
A computer is made up of billions of bits. To simplify the calculation, we group 8 bits into a new unit called Byte. A character is stored as a Byte on a computer.
For example, the character “A” can be translated into an 8-bit (1 byte) binary code 01000001.
How to count in Binary?
There are only 2 digits in the binary system: 1 or 0. To count in binary, you start at 0, then go to 1.
Just like in the decimal numbers system, where we start from 0 to 9, we go from 1 digit into 2 digits when adding 1 to 9. (1 +9 = 10).
Let’s first look at how to count in Decimal:
0 Start at 0
1 Then 1
… Keep counting 2,3,4,5,…8
9 This is the last digit in Decimal
10 The rightmost digit back to 0, and add 1 to the left.
Counting in Decimal
Now, let’s look at counting in Binary:
Sequence Binary Remark
1 0 Start at 0
2 1 Then 1
3 10 The rightmost digit back to 0 and add 1 to the left
4 11 Next, add 1 to the right
1. Add 1 to the rightmost digit
5 100 2. The rightmost digit is back to 0 again, and add 1 to the left
2. Since there was already a 1 at the 2nd digit, it is back to 0 again
3. 1 is added to the left
6 101 Add 1 to the rightmost digit
7 110 And so on …
Counting in Binary
Now, we have an idea of how to count in binary, let’s look at the number conversions.
How to convert Binary to Decimal and vice versa?
The binary system is a base-2 number system. Since it is base 2, there are only 2 digits: 1 and 0. To convert binary to decimal, we need to understand a simple rule:
Start from the right to left, each binary digit with the value 1 represents a power of 2.
BInary Digits positions
Place (position) 5 4 3 2 1 0
Decimal Value 2^5 2^4 2^3 2^2 2^1 2^0
Binary digits position and the corresponding decimal value
With this understanding, it is easy for us to convert a number from binary to decimal.
Let’s use a few examples to better understand this.
How to convert a binary number 101[2] and 1001[2] to a decimal number?
To convert a binary number to decimal, we follow the simple rules:
Start from the right to left, each binary digit with the value 1 represents a power of 2.
101[2] binary = 2^2 + 0 + 2^0 = 4 +0 + 1 = 5
Let’s look at another example, binary number 1001[2]
1001[2] binary = 2^3+ 0 + 0 + 2^0 = 8+0+0+1 = 9
How to convert a decimal number into a binary number?
To convert a decimal to binary, it helps if you are familiar with the powers of 2.
For example, what is the binary value of 14?
Refer to the table and steps below:
Step 1
The largest power of 2 less or equal to 14 is 2^3 = 8. We now know there are 4 binary digits in the answer.
So we fill position 4 of the binary with 1 (remember we start counting position from power 0)
Step 2
Next, subtract 8 from 14, we have a remainder of 6
14 – 8 = 6
The largest power of 2 less or equal to 6 is 2^2 = 4.
We fill position 3 of the binary with 1.
Step 3
Now, Subtract 4 from 6, we have a remainder of 2
6 – 4 = 2
The largest power of 2 less or equal 2 is 2^1 = 2, we fill position 2 of the binary with 1.
Step 4
Subtract 2 from 2, we have no remainder. So the calculation is complete.
2 – 2 = 0
Step 5
In the final step, we fill all empty positions with 0, in this case, position 1 is filled with 0 and we now have the binary value of 14 which is 1110[2].
Binary Number 1 1 1 0
Power of 2 2^3 2^2 2^1 2^0
Positions Position 4 Position 3 Position 2 Position 1
Convert Decimal 14 to Binary
To make it a little easier for you, here is a table of power of 2s. I have also included this table in the Free Binary Worksheet Download Pack.
Power of 2 Calculation Decimal Value
2^0 1 1
2^1 2 2
2^2 2 x 2 4
2^3 2 x 2 x 2 8
2^4 2 x 2 x 2 x 2 16
2^5 2 x 2 x 2 x 2 x 2 32
2^6 2 x 2 x 2 x 2 x 2 x 2 64
2^7 2 x 2 x 2 x 2 x 2 x 2 x 2 128
2^8 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 256
2^9 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 512
2^10 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 1024
Power of 2 table
Here is an excellent video that explains the Binary Numbers
Free Binary Numbers Worksheets for Kids
Understanding Binary Numbers helps kids to grasp coding and computer science concepts.
I have prepared a series of worksheets. The goal is to help parents or teachers to teach kids about binary numbers. It is FREE to download.
In the download pack, you will find:
• Binary Pixel game
• Secret words game
• Answer sheet
• Binary Code Sheet
• Decode The Message play cards
• The Power of 2s worksheet
The challenge for kids is to translate the English word into binary code so that a computer can understand and decode the binary numbers, etc.
Binary Numners Worksheet For Kids
Leave a Comment | {"url":"https://codingnemo.com/binary-numbers-worksheet-for-kids/","timestamp":"2024-11-05T00:14:14Z","content_type":"text/html","content_length":"139387","record_id":"<urn:uuid:ae764ab4-7165-4080-93ca-f80b285c821d>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00385.warc.gz"} |
Concern for Intermetallic Thickness Growth in SAC Solder Joints in Harsh Service Environments · EMSNow
Concern for Intermetallic Thickness Growth in SAC Solder Joints in Harsh Service Environments
By Dr. Ron Lasky
The service temperature for electronics in a modern automobile can be higher than 125C. These high temperatures raise the concern of copper-tin intermetallic growth in solder joints.
Kelvin scale when making these calculations. However, 125C is 0.788 of the way to tin’s melting point. This temperature is the equivalent of a blacksmith’s wrought iron being at 895C. Figure 1 shows
a blacksmith’s forge temperature chart. Note that 895C is beyond red hot.
So, what is SAC solder’s copper-tin intermetallic growth at 125C as a function of time? Fick’s law of diffusion tells us that the growth of the intermetallic, D, is given by:
D = (k(T)t)^0.5 Eq 1.
Where k(T) is a temperature dependent growth rate constant and t is time. Siewert etal[i] performed experiments in which D was measured for various temperatures and times for SAC solders. Following
Siewert’s lead I will use time in hours. By using their data in their Figures 2a through 2c for SAC solder, I was able to plot k in an Arrhenius graph, see Figure 2.
From Figure 2, we see that Ln k = -6784.7/T + 14.81 or k = exp (14.81)*exp-(6784.7/T). So, at 125C or 398K, k = 0.1068. Using this value of k, we can plot D as a function of time. The results are in
Figure 3. Note that both scales are logarithmic. In 1,000 hrs (42 days) the intermetallic has grown 10 microns. In 3 years, it hits 53 microns. We should be cautious, as Siewet’s data has error bars.
But, my sense is that these projections are within a factor of two.
What is the effect of these thick intermetallics in a harsh auto environments? No one knows, but I would encourage someone to perform some experiments to find out.
In the meantime, I have developed an Excel^® spreadsheet that will calculate IMC growth at any temperature, T, for times from 1 to 100,000 hrs. See Figure 1 for the input for this spreadsheet. The
user only has to enter the value of temperature in degrees Celsius in cell A2 and the IMC thickness is calculated and displayed in cells D2 through D7 for times 1 to 100,000 hrs.
The IMC thickness is also plotted in a graph as shown in Figure 4. If you are interested in a copy of this spreadsheet, send me an email at rlasky@indium.com.
It is important to note that these calculated IMC thickness values are only approximations from Siewert’s data as found in his paper: Siewert, T. A. etal, Formation and Growth of Intermetallics at
the Interface between Lead-Free Solders and Copper Substrates, APEX 1994. So, caution is advised, however, I expect the values calculated by the spreadsheet to be correct to within a factor of two. | {"url":"https://www.emsnow.com/concern-for-intermetallic-thickness-growth-in-sac-solder-joints-in-harsh-service-environments/","timestamp":"2024-11-08T21:00:47Z","content_type":"text/html","content_length":"56742","record_id":"<urn:uuid:73c7d509-aece-4921-872b-e785fc3c824b>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00168.warc.gz"} |
American Mathematical Society
The Gauss-Bonnet formula of polytopal manifolds and the characterization of embedded graphs with nonnegative curvature
HTML articles powered by AMS MathViewer
by Beifang Chen
Proc. Amer. Math. Soc. 137 (2009), 1601-1611
DOI: https://doi.org/10.1090/S0002-9939-08-09739-6
Published electronically: November 20, 2008
PDF | Request permission
Let $M$ be a connected $d$-manifold without boundary obtained from a (possibly infinite) collection $\mathcal P$ of polytopes of ${\mathbb R}^d$ by identifying them along isometric facets. Let $V(M)$
be the set of vertices of $M$. For each $v\in V(M)$, define the discrete Gaussian curvature $\kappa _M(v)$ as the normal angle-sum with sign, extended over all polytopes having $v$ as a vertex. Our
main result is as follows: If the absolute total curvature $\sum _{v\in V(M)}|\kappa _M(v)|$ is finite, then the limiting curvature $\kappa _M(p)$ for every end $p\in \operatorname {End} M$ can be
well-defined and the Gauss-Bonnet formula holds: \[ \sum _{v\in V(M)\cup \operatorname {End} M}\kappa _M(v)=\chi (M). \] In particular, if $G$ is a (possibly infinite) graph embedded in a
$2$-manifold $M$ without boundary such that every face has at least $3$ sides, and if the combinatorial curvature $\Phi _G(v)\geq 0$ for all $v\in V(G)$, then the number of vertices with nonvanishing
curvature is finite. Furthermore, if $G$ is finite, then $M$ has four choices: sphere, torus, projective plane, and Klein bottle. If $G$ is infinite, then $M$ has three choices: cylinder without
boundary, plane, and projective plane minus one point. References
• A. D. Aleksandrov and V. A. Zalgaller, Intrinsic geometry of surfaces, Translations of Mathematical Monographs, Vol. 15, American Mathematical Society, Providence, R.I., 1967. Translated from the
Russian by J. M. Danskin. MR 0216434
• Carl B. Allendoerfer and André Weil, The Gauss-Bonnet theorem for Riemannian polyhedra, Trans. Amer. Math. Soc. 53 (1943), 101–129. MR 7627, DOI 10.1090/S0002-9947-1943-0007627-9
• Thomas Banchoff, Critical points and curvature for embedded polyhedra, J. Differential Geometry 1 (1967), 245–256. MR 225327
• Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier
Publishing Co., Inc., New York, 1975. MR 0458335
• Jeff Cheeger, Werner Müller, and Robert Schrader, On the curvature of piecewise flat spaces, Comm. Math. Phys. 92 (1984), no. 3, 405–454. MR 734226, DOI 10.1007/BF01210729
• Beifang Chen, The Gram-Sommerville and Gauss-Bonnet theorems and combinatorial geometric measures for noncompact polyhedra, Adv. Math. 91 (1992), no. 2, 269–291. MR 1149626, DOI 10.1016/0001-8708
• Beifang Chen and Guantao Chen, Gauss-Bonnet formula, finiteness condition, and characterizations of graphs embedded in surfaces, Graphs Combin. 24 (2008), no. 3, 159–183. MR 2410938, DOI 10.1007/
• Matt DeVos and Bojan Mohar, An analogue of the Descartes-Euler formula for infinite graphs and Higuchi’s conjecture, Trans. Amer. Math. Soc. 359 (2007), no. 7, 3287–3300. MR 2299456, DOI 10.1090/
• H. Groemer, On the extension of additive functionals on classes of convex sets, Pacific J. Math. 75 (1978), no. 2, 397–410. MR 513905, DOI 10.2140/pjm.1978.75.397
• M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
• Yusuke Higuchi, Combinatorial curvature for planar graphs, J. Graph Theory 38 (2001), no. 4, 220–229. MR 1864922, DOI 10.1002/jgt.10004
• M. Ishida, Pseudo-curvature of a graph, Lecture at ‘Workshop on topological graph theory’, Yokohama National University, 1990.
• P. McMullen, Non-linear angle-sum relations for polyhedral cones and polytopes, Math. Proc. Cambridge Philos. Soc. 78 (1975), no. 2, 247–261. MR 394436, DOI 10.1017/S0305004100051665
• S. B. Myers, Riemannian manifolds with positive mean curvature, Duke Math. J. 8 (1941), 401–404. MR 4518, DOI 10.1215/S0012-7094-41-00832-3
• David A. Stone, A combinatorial analogue of a theorem of Myers, Illinois J. Math. 20 (1976), no. 1, 12–21. MR 410602
• Wolfgang Woess, A note on tilings and strong isoperimetric inequality, Math. Proc. Cambridge Philos. Soc. 124 (1998), no. 3, 385–393. MR 1636552, DOI 10.1017/S0305004197002429
Similar Articles
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Bibliographic Information
• Beifang Chen
• Affiliation: Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
• Email: mabfchen@ust.hk
• Received by editor(s): March 2, 2007
• Received by editor(s) in revised form: February 15, 2008, and July 6, 2008
• Published electronically: November 20, 2008
• Additional Notes: The author was supported in part by the RGC Competitive Earmarked Research Grants 600703 and 600506.
• Communicated by: Jon G. Wolfson
• © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
• Journal: Proc. Amer. Math. Soc. 137 (2009), 1601-1611
• MSC (2000): Primary 05C10, 52B70; Secondary 05C75, 57M15, 57N05, 57P99
• DOI: https://doi.org/10.1090/S0002-9939-08-09739-6
• MathSciNet review: 2470818 | {"url":"https://www.ams.org/journals/proc/2009-137-05/S0002-9939-08-09739-6/?active=current","timestamp":"2024-11-02T04:16:17Z","content_type":"text/html","content_length":"70011","record_id":"<urn:uuid:97b51fec-5226-470e-9de1-510a1247b86d>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00566.warc.gz"} |
weak complicial set
Higher category theory
Basic concepts
Basic theorems
Universal constructions
Extra properties and structure
1-categorical presentations
Weak complicial sets are simplicial sets with extra structure that are closely related to the ∞-nerves of weak ∞-categories.
The goal of characterizing such nerves, without an a priori definition of “weak $\omega$-category” to start from, is called simplicial weak ∞-category theory. It is expected that the (nerves of) weak
$\omega$-categories will be weak complicial sets satisfying an extra “saturation” condition ensuring that “every equivalence is thin.” General weak complicial sets can be regarded as “presentations”
of weak $\omega$-categories.
Weak complicial sets are a joint generalization of
• $\Delta^k[n]$ be the stratified simplicial set whose underlying simplicial set is the $n$-simplex $\Delta[n]$, and whose marked cells are precisely those simplices $[r] \to [n]$ that contain $\
{k-1, k, k+1\} \cap [n]$;
• $\Lambda^k[n]$ be the stratified simplicial set whose underlying simplicial set is the $k$-horn of $\Delta[n]$, with marked cells those that are marked in $\Delta^k[n]$;
• $\Delta^k[n]'$ be obtained from $\Delta^k[n]$ by making the $(k-1)$st $(n-1)$-face and the $(k+1)$st $(n-1)$ face thin;
• $\Delta^k[n]''$ be obtained from $\Delta^k[n]$ by making all $(n-1)$-faces thin.
An elementary anodyne extension in $Strat$, the category stratified simplicial sets is
• a complicial horn extension $\Lambda^k[n] \stackrel{\subset_r}{\hookrightarrow} \Delta^k[n]$
• a complicial thinness extension $\Delta^k[n]' \stackrel{\subset_e}{\hookrightarrow} \Delta^k[n]''$
for $n = 1,2, \cdots$ and $k \in [n]$.
A stratified simplicial set is a weak complicial set if it has the right lifting property with respect to all
$\Lambda^k[n] \stackrel{\subset_r}{\hookrightarrow} \Delta^k[n]$ and $\Delta^k[n]' \stackrel{\subset_e}{\hookrightarrow} \Delta^k[n]''$
A complicial set is a weak complicial set in which such liftings are unique.
Model structure
There is a model category structure that presents the (infinity,1)-category of weak complicial sets, hence that of weak $\omega$-categories. See
• For $C$ a strict ∞-category and $N(C)$ its ∞-nerve, the Roberts stratification which regards each identity morphism as a thin cell makes $N(C)$ a strict complicial set, hence a weak complicial
set. This example is not “saturated.”
• There is also the stratification of $N(C)$ which regards each $\omega$-equivalence morphism as a thin cell. $N(C)$ with this stratification is a weak complicial set (example 17 of Ver06). This
should be the “saturation” of the previous example, and exhibits the inclusion of strict $\omega$-categories into weak ones.
• A simplicial set is a weak complicial set when equipped with its maximal stratification (every simplex of dimension $\gt 0$ is thin) if and only if it is a Kan complex. This example is, of
course, saturated, and is viewed as embedding $\omega$-groupoids into $\omega$-categories.
• A simplicial set is a quasi-category if and only if it is a weak complicial set when equipped with the stratification in which every simplex of dimension $\gt 1$ is thin, and only degenerate
1-simplices are thin. This example is not saturated; in its saturation the thin 1-simplices are the internal equivalences in a quasi-category (equivalently, those that become isomorphisms in its
homotopy category). It presents the embedding of $(\infty,1)$-categories into weak $\omega$-categories.
Note that 1-simplex equivalences in a quasi-category are automatically preserved by simplicial maps between quasi-categories; this is why $QCat$ can “correctly” be regarded as a full subcategory
of $sSet$. This is not true at higher levels; for instance not every simplicial map between nerves of strict $\omega$-categories necessarily preserves $\omega$-equivalence morphisms.
The definition of weak complicial sets is definition 14, page 9 of
Further developments are in
• Dominic Verity, Weak complicial sets Part II: Nerves of complicial Gray-categories (arXiv)
A model category structure on stratified simplicial sets modelling $(\infty,n)$-categories in the guise of $n$-complicial sets:
A Quillen adjunction relating $n$-complicial sets to $n$-fold complete Segal spaces: | {"url":"https://ncatlab.org/nlab/show/weak%20complicial%20set","timestamp":"2024-11-13T02:51:51Z","content_type":"application/xhtml+xml","content_length":"46981","record_id":"<urn:uuid:87b8f873-76b1-460c-8c5b-521681e895d1>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00125.warc.gz"} |
Evolutionary Computation and Constraint Satisfaction
In this chapter we will focus on the combination of evolutionary computation techniques and constraint satisfaction problems. Constraint Programming (CP) is another approach to deal with constraint
satisfaction problems. In fact, it is an important prelude to the work covered here as it advocates itself as an alternative approach to programming (Apt). The first step is to formulate a problem as
a CSP such that techniques from CP, EC, combinations of the two (c.f., Hybrid) or other approaches can be deployed to solve the problem. The formulation of a problem has an impact on its complexity
in terms of effort required to either find a solution or proof no solution exists. It is therefore vital to spend time on getting this right.
Main differences between CP and EC. CP defines search as iterative steps over a search tree where nodes are partial solutions to the problem where not all variables are assigned values. The search
then maintain a partial solution that satisfies all variables assigned values. Instead, in EC most often solver sample a space of candidate solutions where variables are all assigned values. None of
these candidate solutions will satisfy all constraints in the problem until a solution is found. Another major difference is that many constraint solvers from CP are sound whereas EC solvers are not.
A solver is sound if it always finds a solution if it exists. | {"url":"https://groups.inf.ed.ac.uk/nesc-research/node/9984658.html?page=2","timestamp":"2024-11-03T12:39:38Z","content_type":"application/xhtml+xml","content_length":"24549","record_id":"<urn:uuid:549c0b0a-2c82-4e22-b9a5-6dfe5639b97c>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00661.warc.gz"} |
Computational Social Choice 2015
Social choice theory is the study of mechanisms for collective decision making, such as voting rules or protocols for fair division, and computational social choice addresses problems at the
interface of social choice theory with computer science. This course will introduce some of the fundamental concepts in social choice theory and related disciplines, and it will expose students to
current research at the interface of social choice theory with logic and computation. This is an advanced, research-oriented course in the Master of Logic programme. Students from AI, Computer
Science, Mathematics, Economics, Philosophy, etc. are also very welcome to attend (contact me if you are unsure about your qualifications).
Date Content Homework
Wednesday Introduction. This first lecture has been a bit of a sightseeing tour of computational social choice, with lots of examples for problems addressed, results obtained, and
4 February 2015 techniques employed in four different subfields of this research area: fair allocation of goods, voting in elections, two-sided matching, and judgment aggregation. For
the remainder of the course, we will largely focus on judgment aggregation and revisit some of the ideas and techniques so far only mentioned in the context of other
domains of aggregation. For an introduction to computational social choice, focusing mostly on voting (and a little on fair allocation), consult this paper:
Please spend some time with this paper, to allow you to appreciate the wider field of computational social choice and to be able to put in context the more specific
material we will explore in the coming weeks. For more details on, first, fair allocation and, second, the axiomatic method (two of the larger topics explored in this
lecture), consult these references:
• U. Endriss. Lecture Notes on Fair Division. ILLC, University of Amsterdam, 2009/2010.
• U. Endriss. Logic and Social Choice Theory. In A. Gupta and J. van Benthem (eds.), Logic and Philosophy Today, College Publications, 2011.
And here are two famous classical papers mentioned during this lecture, both of which are very short and still highly readable today: At the end of the lecture we have
also discussed various organisational aspects of the course.
Friday Basic Judgment Aggregation. This has been an introduction to the basic theory of judgment aggregation. We started with a discussion of the famous doctrinal paradox and Homework
6 February 2015 then defined the formal framework of judgment aggregation we shall be working with (in fact, later on in the course we will also see an alternative such framework). The #1
specific aggregation rules discussed were the majority rule, premise-based rules, and quota rules. Finally, we introduced the axiomatic method for judgment aggregation, (due: 11
discussed three specific axioms, and proved the impossibility theorem due to List and Pettit. I strongly recommend that you read their original paper, which also was the February
first to propose a formal social choice-theoretic framework to study the kinds of questions raised by the observation of the doctrinal paradox some years earlier: 2015)
The following two expository papers cover most of the material discussed in this lecture (and much more) and may serve as general references also for a significant part
of the rest of the course:
• C. List. The Theory of Judgment Aggregation: An Introductory Review. Synthese, 187(1):179-207, 2012.
• U. Endriss. Judgment Aggregation. In F. Brandt, V. Conitzer, U. Endriss, J. Lang, and A.D. Procaccia (eds.), Handbook of Computational Social Choice. Cambridge
University Press. In press (2015).
Wednesday Axiomatic Method. This lecture has been about several examples for the use of the axiomatic method. In the first part, we explored different ways of circumventing the
11 February 2015 basic impossibility theorem of judgment aggregation, by using domain restrictions or by weakening some of our axioms. In the second part, we had a closer look at our
axioms and discussed how to think about them in terms of winning coalitions. We then used the insights gained to establish several axiomatic characterisation results for
the quota rules. Besides the survey paper cited earlier, the main reference for this lecture is the following paper on quota rules:
Friday Aggregation Rules. In the first part of this lecture we have introduced a second framework for aggregation, namely binary aggregation with integrity constraints (BA). In Homework
13 February 2015 some sense (left imprecise), it can be used to model the same types of problems we can model using the formula-based framework of judgment aggregation (JA). We have then #2
seen how to translate both JA and classical preference aggregation into BA. Our main goal for this lecture has been to enrich our corpus of specific aggregation rules, (due: 18
beyond the very simplistic idea of working with quotas treated before. We did so taking inspiration from classical rules for voting and preference aggregation February
(specifically: Slater, Kemeny, Tideman, Copeland, Borda, and Maximin) and we have shown how to simulate these rules in BA by combining a suitable integrity constraint 2015)
with a suitable optimisation rule awarding points for respecting (weighted) majorities in the outcome as much as possible. Here are the two main references for this
• U. Grandi and U. Endriss. Binary Aggregation with Integrity Constraints. Proc. 22nd International Joint Conference on Artificial Intelligence (IJCAI-2011), 2011.
• J. Lang and M. Slavkovik. Judgment Aggregation Rules and Voting Rules. Proc. 3rd International Conference on Algorithmic Decision Theory (ADT-2013), Springer-Verlag,
Tuesday Tutorial on Complexity Theory. This has been a quick review of basic concepts from the theory of computational complexity. We have seen the definition of several
17 February 2015 complexity classes, including P, NP, coNP, PSPACE, and some of the less well-known classes defined in terms of NP-oracles. We have also seen what it means for a problem
to be hard or complete with respect to such a class, and we have gone over a couple of NP-completeness proofs using polynomial-time reductions.
Wednesday Winner Determination. This has been a lecture on the computational complexity of the winner determination problem in judgment aggregation. We discussed the best way of
18 February 2015 defining the problem in some detail, and then saw that winner determination is easy for quota rules and the premise-based rule (in case our usual assumptions guaranteeing
consistent and complete outcomes for this rule hold), and that the max-sum rule (also known as the Kemeny rule of the distance-based rule) is highly intractable. These
results were taken from the following paper:
At the end of the lecture we have also discussed possible approaches you could take to identify a suitable topic for a mini-project.
Friday Safety of the Agenda. The problem of the safety of the agenda is the problem of checking whether a given agenda is safe for a given class of aggregation rules in the Homework
20 February 2015 sense of guaranteeing that there will never be an inconsistent outcome for any admissible profile for that agenda and any rule from that class. We have given logical #3
characterisation of the agendas that are safe for (a) just the majority rule and (b) the class if rules we obtain when we drop the monotonicity axiom for the (due: 25
axiomatisation of the majority rule. We have also discussed the (very high!) computational complexity of deciding whether a given agenda has one of the relevant February
properties, and we have discussed the connections to some axiomatic impossibility and characterisation results reviewed earlier on in the course. Here are the two main 2015)
original papers from which the definitions and results presented are drawn:
Most of this material is also covered in this expository paper:
• U. Endriss. Judgment Aggregation. In F. Brandt, V. Conitzer, U. Endriss, J. Lang, and A.D. Procaccia (eds.), Handbook of Computational Social Choice. Cambridge
University Press. In press (2015).
Tuesday Agenda Characterisation. This lecture has been devoted to agenda characterisation results that establish correspondences between the logical properties of the agenda and
24 February 2015 the axioms satisfied by an aggregation rule. More specifically, such results identify those agendas for which there exists an aggregation rule meeting certain axioms that
will always return a consistent judgment set. Thus, these existential agenda characterisation results are dual to the universal agenda characterisation results,
establishing the safety of the agenda for all rules meeting certain axioms, discussed in the previous lecture. We have also briefly discussed to connection of one of
these results to Arrow's Theorem, the seminal result in preference aggregation that started modern social choice theory. The agenda characterisation theorems for judgment
aggregation discussed are taken from the following two original papers:
But note that the formal frameworks used in those papers differ somewhat from the framework used in the course. For a clear exposition of these theorems and most other
known existential agenda characterisation theorems, refer to the survey by List and Puppe listed below. For full proofs of the two theorems discussed in class, consult my
handbook chapter.
• C. List and C. Puppe. Judgment Aggregation: A Survey. In P. Anand, P. Pattanaik, and C. Puppe (eds.), Handbook of Rational and Social Choice. Oxford University Press,
• U. Endriss. Judgment Aggregation. In F. Brandt, V. Conitzer, U. Endriss, J. Lang, and A.D. Procaccia (eds.), Handbook of Computational Social Choice. Cambridge
University Press. In press (2015).
Wednesday Lifting Integrity Constraints. At the beginning of this lecture we have introduced the framework of binary aggregation with integrity constraints a little more Homework
25 February 2015 systematically than last time. Then we have used it to investigate the concept of collective rationality in some depth and asked what kinds of rules can "lift" what kinds #4
of integrity constraints from the individual to the collective level, in the sense of ensuring that the outcome will satisfy the constraint whenever all of the individual (due: 4
ballots do. This has allowed us to make connections between axioms, on the one hand, and syntactic restrictions on the language used to express integrity constraints, on March
the other. In the end, we have asked whether there are any rules that would lift all possible integrity constraints. There indeed are such rules, including some very bad 2015)
rules (e.g., dictatorships) as well as some intuitively attractive rules (e.g., the average-voter rule). The lecture was based on the following two papers:
Tuesday We met to discuss your project ideas and to clarify what you need to submit to me to get your project approved.
3 March 2015
Wednesday Strategic Behaviour. In this lecture we have considered what happens when agents act strategically when choosing what judgment set to report. We have seen that, for
4 March 2015 certain assumptions on the preferences of the agents, a judgment aggregation rule is strategy-proof (i.e., never gives an agent an incentive to misreport their judgments)
if and only if that rule is both independent and monotonic, and we have argued that this means full protection against strategic behaviour is only possible in the rarest
of circumstances. We have then discussed the idea of using computational complexity as a barrier against strategic manipulation and showed that the premise-based
procedure is NP-hard to manipulate. Finally, we have briefly reviewed a number of other approaches of bringing strategic behaviour into JA, namely bribery and various
forms of control of the set of agents taking part. Here are the main papers in which this material is covered:
• F. Dietrich and C. List. Strategy-Proof Judgment Aggregation. Economics and Philosophy, 23(3):269-300, 2007.
• U. Endriss, U. Grandi, and D. Porello. Complexity of Judgment Aggregation. Journal of Artificial Intelligence Research (JAIR), 45:481-514, 2012.
• D. Baumeister, G. Erdélyi, O.J. Erdélyi, and J. Rothe. Computational Aspects of Manipulation and Control in Judgment Aggregation. Proc. 3rd International Conference
on Algorithmic Decision Theory (ADT-2013), 2013.
Friday Truth-Tracking. The first part of this lecture has been a short introduction to the epistemic approach to judgment aggregation, where we assume that there is an Homework
6 March 2015 objectively correct answer to the questions under consideration and each agent reports a perturbed copy of this ground truth. The task of an aggregation rule then becomes #5
to recover the ground truth as well as possible. This idea goes back to the classical Condorcet Jury Theorem of the 18th century, and we have discussed a small number of (due: 13
variants of this result, including the computation of optimal weights for the agents in view of their accuracy and the estimation of those accuracies from the perturbed March
input data itself. 2015)
In the second part of the lecture I have tried to offer a high-level overview of the ground covered in the nine lectures on judgment aggregation, focusing on the range of
different methodologies employed: adopting either a philosophical, mathematical, computational, game-theoretical, or statistical perspective on the same problem of
aggregating the judgments of a group of agents into a single judgment.
Tuesday We met to discuss how to write a paper.
10 March 2015
Thursday Suggestion: You might be interested in attending the seminar talk by Christian List (London School of Economics), entitled From Degrees of Belief to Beliefs: Lessons from
12 March 2015 Judgment-Aggregation Theory, at 15:00.
Friday Fair Allocation. This has been a short introduction to fair allocation problems. First, we have seen several criteria that can be used to assess the fairness and economic
13 March 2015 efficiency of an allocation of goods to the members of a group of agents. Then we have focussed on the allocation of indivisible goods and seen examples for two types of
results: the complexity of computing a socially optimal allocation and the convergence to a socially optimal allocation by means of a sequence of local deals. Much of
what we have discussed is also covered in my lecture notes:
Tuesday We met to discuss how to give a talk.
17 March 2015
Wednesday Matching. This lecture has been a brief introduction to the theory of two-side matching. We have discussed the classical stable marriage problem, analysed the
18 March 2015 Gale-Shapely algorithm for computing a stable matching for this setting, talked about various extensions of the basic model, and considered various requirements beyond
stability, notably fairness and strategy-proofness. Here is the classical paper on the topic by Gale and Shapley:
Further information on the use of the Gale-Shapley algorithm to match school children to schools in Amsterdam is available here and here (in Dutch).
Thursday-Friday Suggestion: You are welcome to attend the ILLC Workshop on Collective Decision Making (but please make sure you register at least one week in advance).
19-20 March 2015
Tuesday-Wednesday Here is the the programme for the final presentations:
24-25 March 2015
Tuesday Wednesday
• Arianna, Merlijn and Sirin • Kristina and Zeno
Group Manipulation in Judgment Aggregation New Complexity Results for Bundling Attacks
• Edwin, Maarten, Marysia and Richard • Roelof and Tim
Virtuous Manipulation in Binary Aggregation Judgment Aggregatio with Abstentions
• Elise and Sharon • Carla and Marco
Empirical Evaluation of Quota Rule Consistency in Binary Aggregation Binary Aggregation under Issue Dependencies
We start at 11:00 sharp on both days.
Tuesday Suggestion: You might be interested in the ABC Symposium on Decision Making. At this event people will be looking into decision making (probably mostly individual rather
21 April 2015 than collective decision making) from a somewhat different angle, including in particular the perspective of cognitive science. Note that space is limited and to ensure
you get in you will have to register early.
Monday-Friday Suggestion: Apply for the Summer School on Fair Division in Grenoble, organised by COST Action IC1205 on Computational Social Choice. There are a number of travel grants
13-17 July 2015 available, which should cover most of your expenses. MSc Logic students can get 2EC for this (but see the rules). Students from other programmes should check with their
programme director first.
Projects are to be worked on in groups. I strongly prefer groups of three people each, but if necessary will approve a couple of groups of size two or four as well. The final deadline for seeking
approval of your project (group composition and topic) by me is Wednesday, 4 March 2015. To make this deadline, you will need to approach me with fairly concrete ideas at least a week in advance. To
request approval of your project, send me an email in which you answer the following three questions: | {"url":"https://staff.fnwi.uva.nl/u.endriss/teaching/comsoc/2015/","timestamp":"2024-11-14T08:51:06Z","content_type":"text/html","content_length":"39646","record_id":"<urn:uuid:044c27c8-cd75-44c3-87c1-4958b68d7b75>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00335.warc.gz"} |
Cu Chi Tunnels
Problem C
Cu Chi Tunnels
The tunnels of Cu Chi are an immense network of underground tunnels connecting rooms located in the Cu Chi District of Ho Chi Minh City. The Cu Chi tunnels were the location of several military
campaigns in the 1960s. Nowadays, it is a popular tourist destination.
There are documents from trusted sources about a private network of tunnels in this area used by a secret forces unit but it has not been discovered. According to the documents, this private network
has $N$ rooms (numbered from $1$ to $N$) connected by $N-1$ bidirectional tunnels. Room $1$ is the entry point from the ground surface to this underground network. From room $1$, you can follow the
tunnels to go to any of the rooms. The rooms are numbered in such a way that, if you follow the shortest path from room $1$ to any room $X$, the sequence of visited rooms’ indices will be
increasing. The image below shows a valid map of this network.
The network below is invalid, since the path from $1$ to $4$ is $1$ - $3$ - $2$ - $4$, which is not increasing:
There is also an old article from an unknown source mentioning about $D_ i$ which is the number of rooms directly connected to room $i$.
Given an array $D$ of size $N$, your task is to verify if it is possible to have such a network.
• The first line contains an integer $N$ - the number of rooms in the network $(2 \leq N \leq 1\, 000)$.
• The second line consists of $N$ integers $D_ i$ - the number of rooms that are directly connected to room $i$ $(1 \leq D_ i \leq N - 1)$.
Print YES/NO if it is possible/impossible to have such a network, respectively.
Sample Input 1 Sample Output 1
8 YES
Sample Input 2 Sample Output 2
4 NO | {"url":"https://hochiminh17.kattis.com/contests/hochiminh17/problems/cuchitunnels","timestamp":"2024-11-04T07:10:34Z","content_type":"text/html","content_length":"29348","record_id":"<urn:uuid:d1e0537c-61b1-4255-82d8-39c26d2c8594>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00221.warc.gz"} |
Fibonacci sequence: A small piece of nature
If one were to argue for the beauty of mathematics by providing three of the simplest relevant examples, the Fibonacci sequence would unquestionably be among them. But this educational aspect of
popularizing mathematics or being a subject of recreational mathematics is not the only virtue of this sequence. The mathematical properties of the Fibonacci sequence and their curious reflection in
various contexts outside mathematics, including nature, have maintained over centuries an aura of mystery surrounding this mathematical concept. Where mystery is detected, scientists, mathematicians,
and philosophers have an instinctive drive to solve it, and such a focused inquiry has so many times proven fertile in the history of science and mathematics by leading to important discoveries or
new theories. For that reason also, the Fibonacci sequence is a virtuous concept.
We all came to know about this sequence mainly in high school, when the math teacher usually offered it as an example of a sequence defined in a recursive mode – that is, a sequence each term of
which is determined as standing in a given relation with one or more of the previous terms. For the Fibonacci sequence (its name originating from Italian mathematician Leonardo Bonacci, or Leonardo
of Pisa, pseudonamed Fibonacci, who lived in the 12^th-13^th century), the recursive rule is that each term is the sum of the previous two terms, while the first two terms are 0 and 1. According to
this rule, the Fibonacci sequence can be generated as and looks like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233… Its terms are called the Fibonacci numbers.
Now, looking at this sequence, one may fairly ask why this plain one-dimensional succession of natural numbers would be so special, where its beauty resides, and what is so mysterious about it. All
these attributes can be confirmed right away one after another, if a bit of mathematical imagination is put to work.
Imagine for a moment a rectangle whose side lengths are two consecutive Fibonacci numbers in that sequence. Make them large enough, for instance 34 and 55. Now proceed to tile this rectangle
successively with squares of side lengths equal to the lesser side length of the rectangle, as in the figure below:
Fig.1. The Fibonacci spiral
If you draw quarter-circular arcs connecting the opposite corners of the squares in succession, you get a spiral passing through all those squares whose side lengths are successive Fibonacci numbers.
The larger you make the side length of the initial rectangle as a Fibonacci number, the longer you get this spiral drawn. Well, with this drawing, the one-dimensional Fibonacci sequence has turned
into a nice two-dimensional spiral. But this is not all.
The rectangles shown in Figure 1 are similar to each other – that is, the ratio of their consecutive side lengths is the same. Denoting by a and b the side lengths of the initial largest rectangle,
this geometrical similarity is written as the proportion a/b = (b – a)/a. Substituting φ = b/a, the previous relation is made equivalent with the quadratic equation φ^2 – φ – 1=0, which is called the
golden equation and its positive solution φ = (1 + sqrt (5))/2 = 1.61803398… is called the golden number or the golden ratio. In the geometry of curved shapes, a golden spiral is a logarithmic spiral
whose growth rate is the golden number, and has the property of being self-similar – in other words, keeping the same shape when magnified in its accumulating zone. So what we have drawn above is an
approximation of a golden spiral, and extending (hypothetically) our drawing infinitely both outside and inside the rectangle by the same rules – reflecting Fibonacci numbers – we would obtain a
“full” golden spiral.
The golden number can be made visible not only in the golden spiral, but directly in the plain Fibonacci sequence. Mathematician Jacques Philippe Marie Binet expressed the general term of the
Fibonacci sequence in the closed form Fn = [φ^n – (1 – φ)^n]/sqrt(5) , where φ is the golden number. A nice property of this sequence is closeness, in the sense that that any positive integer can be
written as a sum of Fibonacci numbers each taken at most once. From this point, mathematicians developed several procedures of expressing φ, spanning over various fields of mathematics
(combinatorics, number theory, topology and mathematical analysis, differential equations, and geometry), and their success connected various different concepts from this fields. However, the most
striking simple representations of the golden number are these two:
On the left-hand side we have what we call a continued (infinite) fraction and on the right-hand side, a continued (infinite) radical. Of course, such numbers as those in the right-hand members above
cannot even be imagined, nor computed through such representations using division and the square root; they are just symbols of more complex concepts and mathematical statements involving the limit
of particular convergent sequences. What is amazing about these two different representations is that they provide the same number, the golden number, and this is definitely part of the complex
beauty of mathematics. For mathematicians and philosophers of mathematics, it is not such a mystery any more that mathematical concepts come to exhibit mutual unexpected connections across the
different fields of mathematics (fields using concepts of different natures, different methods, and different languages). What still remain mysterious are their connections outside mathematics, in
the real world.
It is a well-known fact by now that the golden number and its spiral are visibly used by Mother Nature as a development pattern: several species of plants have been found to have their flower petals
arranged and growing in a golden-spiral pattern, or their leaves distributed in the golden angle (i.e., the circular angle determined by the smallest of two circular arcs standing in the golden
ratio). Mollusk shells (like the nautilus) exhibit the same spiral shape and some specific anatomic proportions in the bodies of animals and humans are equated with the golden ratio. It is perhaps
due to this last fact that the Fibonacci sequence and golden number came to be seen in a mystical light.
Figure 2. Leaves of a species of aloe with a spiral pattern of growth.
Figure 3. A nautilus spiral-shaped shell.
Whatever argument may be invoked that such observations are made with a dose of illusion and self-suggestion, the fact falls within the wider belief that nature follows mathematical patterns in its
evolution, and abstract mathematics effectively describes the physical reality. Such facts, although not explained rationally by any science or discipline, cannot be denied. Modern sciences,
especially physics, have evolved to their current success only after describing the laws of nature in the language of mathematics and applying mathematics in any suitable context. Such universal
applicability of mathematics is another kind of beauty: it is simultaneously the beauty of mathematics, the beauty of nature, and the beauty of human reason. When taking into account that the pattern
of development of a plant following the golden spiral allows it to grow without changing shape – and biologists would have more to say about this in non-mathematical terms and with evolutionary
explanations – the mathematical properties of the Fibonacci sequence and golden number become somehow the properties of nature. Is this something mysterious? If you have not yet decided what to
answer (and don’t trouble yourself too much with this question – philosophers of science have not provided any straight answer either), compare the images showing the symbolic abstract representation
of the golden number as a continued fraction or radical and the golden-spiraled plant in our image: Didn’t you feel that there is something almost indescribable that they share in common? If so, bear
in mind that you have compared an abstract thing originating in a mathematician’s mind with a real live organism.
So many times in the history of science, mathematical concepts created with no particular intended application outside mathematics have come to find their application in contexts never imagined at
the moment of their creation. The Fibonacci sequence and the golden ratio fall within this category, it would be an impossible task to enumerate here all of the domains of their applications. Just to
mention the most relevant, these concepts have found their application in economics, sociology, architecture, art (including music), horticulture, genetics, and optics. However pragmatic the success
of their application, the interest of scientists and mathematicians in the deep investigation of these concepts is still in advent – there exists a (mathematical) Fibonacci Society and a scientific
journal dedicated entirely to Fibonacci sequence and the related concepts.
The mystical aura surrounding these numbers has never lost its power; it touches both scientists and ordinary people. Whatever discipline would be entitled to deal better with this matter, it cannot
ignore the mathematical simplicity of the definition of the Fibonacci sequence and the idea that complex things are made of simple things. Clearly, mathematics is the natural path toward any kind of
complexity, offering also the appropriate methods to deal with and investigate it. | {"url":"https://www.magazine.philscience.org/2022/07/20/fibonacci-sequence-a-small-piece-of-nature/","timestamp":"2024-11-15T03:23:29Z","content_type":"text/html","content_length":"46054","record_id":"<urn:uuid:9417450e-5cf2-4987-a085-38a07eb20bbc>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00649.warc.gz"} |
How Do You Write In Standard Form?
1 Answers
How Do You Write In Standard Form?
It depends on what you are writing.
A linear equation -- ax + by = c (the leading non-zero coefficient is positive)
A quadratic equation -- ax^2 + bx + c = 0
A number (where "standard form" is scientific notation) -- a.b*10^c where a > 0, b may be a string of digits, a or c or both may be negative.
The equation for a circle -- (x-h)^2 + (y-k)^2 = r^2
A number, where "standard form" is not scientific notation ...
Answer Question | {"url":"https://education.blurtit.com/1368684/how-do-you-write-in-standard-form","timestamp":"2024-11-11T11:46:16Z","content_type":"text/html","content_length":"53599","record_id":"<urn:uuid:bea8b238-7546-4dbf-a9cd-c59f7325530d>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00218.warc.gz"} |
The most popular questions on the topic of general
Two fire lookouts are 12.5 km apart on a north-south line. The northern fire lookout sights a fire 20° south of East at the same time as the southern fire lookout spots it at 60° East of North. How
far is the fire from the Southern lookout? Round your answer to the nearest tenth of a kilometer | {"url":"https://math-master.org/general?page=174","timestamp":"2024-11-12T14:04:34Z","content_type":"text/html","content_length":"247279","record_id":"<urn:uuid:2871787e-190c-47e5-9f9b-96465553824e>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00205.warc.gz"} |
Paper Abstract: 1998a
Title: Towards F=ma in a general setting for Lagrangian mechanics (33 pages)
Author(s): Andrew D. Lewis
Detail: Annales Henri Poincaré 1(3), pages 569-605, 2000
Journal version: Download
Original manuscript: 1998/05/19
Manuscript last revised: 1999/09/22
By using a suitably general definition of a force, one may geometrically cast the Euler-Lagrange equations in a ``force balance'' form. The key ingredient in such a construction is the Euler-Lagrange
2-force which is a bundle map from the bundle of two-jets into the first contact system. This 2-force can be used as the basis for a geometric presentation of Lagrangian mechanics with external
forces and constraints. Also described is the precise correspondence between this 2-force and the Poincaré-Cartan two-form.
514K pdf
Last Updated: Fri Mar 15 08:08:35 2024
Andrew D. Lewis (andrew at mast.queensu.ca) | {"url":"https://mast.queensu.ca/~andrew/papers/abstracts/1998a.html","timestamp":"2024-11-03T12:07:12Z","content_type":"text/html","content_length":"1811","record_id":"<urn:uuid:fffac786-26d8-4cf5-87bb-858f0c420492>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00311.warc.gz"} |
How to Calculate the Best-Performing Stock Based on Historical Data?
Calculating the best-performing stock based on historical data involves analyzing the past performance of different stocks over a specific period and using various metrics to determine the stock that
has outperformed others. Here's a step-by-step guide:
1. Gather historical data: Collect data on stock prices, preferably for at least several years. Obtain the closing prices for each stock on a daily, weekly, monthly, or annual basis, depending on
your desired timeframe.
2. Calculate returns: Calculate the returns for each stock over the selected period. To calculate returns, subtract the starting price from the ending price, divide it by the starting price, and
multiply the result by 100 to get it in percentage form.
3. Determine relative returns: Calculate the relative returns for each stock by subtracting the average return of a benchmark index (such as S&P 500) from the calculated returns for each stock. This
helps determine the performance of the stock relative to the overall market.
4. Calculate risk-adjusted returns: Compute risk-adjusted returns using a risk metric of your choice, such as Sharpe ratio or Sortino ratio. These ratios consider the volatility or downside risk
associated with the investment, offering a better measure of performance.
5. Analyze other performance metrics: Consider additional performance metrics like standard deviation, beta, alpha, and maximum drawdown to gain a more comprehensive understanding of the stock's
6. Compare stocks: Compare the risk-adjusted returns and other performance metrics of different stocks. A stock with higher risk-adjusted returns, lower volatility, and other favorable metrics may
be considered the best-performing stock.
7. Take note of the investment horizon: Depending on your investment goals, adjust the timeframe for analysis. What may be considered best-performing in the short term might differ from the
long-term perspective.
It's important to remember that historical performance doesn't guarantee future results. Additionally, other factors like fundamental analysis, industry trends, and market conditions should be
considered before making investment decisions.
What is the importance of analyzing stock volatility when calculating performance?
Analyzing stock volatility is important when calculating performance because it provides insights into the risk associated with an investment. Stock volatility measures the magnitude of price
fluctuations over a certain period of time. By understanding the level of volatility, investors can gauge the potential upsides and downsides of a stock investment.
There are several reasons why analyzing stock volatility is crucial for calculating performance:
1. Risk assessment: Volatility allows investors to evaluate the risk associated with a particular stock. Higher levels of volatility indicate greater price fluctuations and uncertainty, which may
imply higher risk. Lower volatility suggests more stable price movements and potentially lower risk.
2. Portfolio diversification: Analyzing volatility helps investors diversify their portfolio effectively. Adding low-volatility stocks to a portfolio that already contains high-volatility stocks can
help balance risk and potentially improve overall performance.
3. Performance evaluation: Volatility impacts the overall performance of a stock and, subsequently, a portfolio. Higher volatility stocks tend to have greater potential for both gains and losses.
Analyzing volatility can help investors understand whether the returns generated by a stock's price movements are commensurate with the associated level of risk.
4. Trading strategies: Volatility analysis is crucial for designing effective trading strategies. Some investors may prefer high-volatility stocks for short-term trading, aiming to take advantage of
price swings. Conversely, others may prefer low-volatility stocks for long-term investment, seeking stability and consistent returns.
5. Risk management: By considering volatility, investors can manage risk more effectively. Volatility-based risk management strategies, such as setting stop-loss orders or employing hedging
techniques, can help protect investments from adverse price movements.
Overall, analyzing stock volatility is essential for evaluating risk, diversifying portfolios, assessing performance, designing trading strategies, and managing risk effectively. It provides valuable
insights that enable investors to make informed decisions and maximize their investment returns.
How to calculate the best-performing stock based on historical data?
Calculating the best-performing stock based on historical data involves analyzing various factors such as the stock's price movements, returns, volatility, and other relevant parameters. Here's a
step-by-step approach you can follow:
1. Select the desired historical period: Choose a specific timeframe for which you want to evaluate the stock performances. It could be a few months, a year, or multiple years.
2. Identify the stocks: Determine the stocks that you want to analyze for their performance during the chosen period.
3. Collect historical data: Gather the necessary data for each stock, including daily or monthly closing prices for the selected period. Financial websites or platforms like Yahoo Finance, Google
Finance, or Bloomberg offer historical price data for stocks.
4. Calculate the returns: Compute the percentage return for each stock over the chosen period. The return is typically calculated as [(Ending Price - Starting Price) / Starting Price] * 100. This
will provide a measure of the stock's growth or decline during the selected period.
5. Calculate average returns: Determine the average return for each stock over the chosen period. Sum up the individual returns and divide by the number of data points to get the average return.
6. Calculate volatility: Volatility measures the stock's price fluctuations. One commonly used volatility measure is the standard deviation of the returns. Calculate the standard deviation of the
stock returns to understand the overall volatility.
7. Compare performances: Compare the average returns and volatility of each stock. Higher average returns and lower volatility generally indicate better performance.
8. Consider other factors: It's also important to consider other factors such as market conditions, industry trends, and specific news/events that might have affected the stock's performance during
the historical period.
9. Make a decision: Based on the analysis, select the stock with the highest average return, coupled with a stable or lower volatility, as the best-performing stock during the historical period.
Remember, historical performance is not a guarantee of future performance. It is crucial to conduct thorough research and analysis to make informed investment decisions.
What is the significance of correlation in analyzing stock performance?
Correlation is a statistical measure that helps in analyzing the relationship between two or more variables. In the context of stock performance, correlation is significant because it allows
investors and analysts to understand the degree and direction of the relationship between two stocks or between a stock and a market index. Here are a few key points about the significance of
correlation in analyzing stock performance:
1. Diversification: Correlation helps investors determine the extent to which two stocks move together or move in opposite directions. If stocks have a low or negative correlation, it signifies that
their price movements are relatively independent, making them suitable for diversification within a portfolio. By combining stocks with low correlation, investors can reduce portfolio risk and
potential losses.
2. Risk assessment: Correlation provides insights into the risk associated with an investment. Stocks that have a high positive correlation tend to move in the same direction, increasing overall
portfolio risk. On the other hand, stocks with low or negative correlation have the potential to offset each other's risk, thereby reducing the overall risk of the portfolio.
3. Portfolio optimization: By analyzing the correlation between different stocks, investors can optimize their portfolios and seek to achieve the desired level of risk and return. Identifying stocks
with low correlation or negative correlation can enable the creation of a well-diversified portfolio that seeks to balance risk and return.
4. Sector analysis: Correlation analysis can help identify trends and relationships within specific sectors or industries. Investors can analyze the correlation between stocks within a particular
sector to gain insights into the overall health of that sector. Positive correlation within a sector suggests a similar market response to common factors such as economic conditions, while
negative correlation might indicate diverging factors impacting different companies within the sector.
5. Trading strategies: Correlation analysis can be useful for traders and quantitative analysts in developing trading strategies. For example, pairs trading strategy involves identifying two stocks
with a historically high correlation and taking opposing positions when the correlation deviates, aiming to profit from the expected reversion to the mean correlation.
Overall, correlation analysis provides valuable information about the relationship between stocks, helping investors and analysts make more informed investment decisions, manage risk, and optimize
their portfolios.
How to calculate the price-to-sales ratio based on historical data?
To calculate the price-to-sales ratio based on historical data, you need two sets of information: the historical price data and the historical sales data.
The formula to calculate the price-to-sales ratio is as follows:
Price-to-sales ratio = Market Capitalization / Total Sales
Market Capitalization is the total value of a company's shares outstanding in the market, which can be calculated by multiplying the stock price by the number of shares outstanding.
To calculate the price-to-sales ratio using historical data, follow these steps:
1. Gather the historical stock price data: Collect the closing price of the stock for each period you want to analyze. These prices can typically be obtained from financial websites or databases.
2. Gather the historical sales data: Collect the total sales figures for the corresponding period, typically obtained from the company's financial reports or filings.
3. Calculate Market Capitalization: For each period, multiply the stock price by the number of shares outstanding. The number of shares outstanding can also be found in the financial reports or
4. Calculate the Price-to-sales ratio: Divide the Market Capitalization by the Total Sales for each period.
Repeat these steps for each period you want to analyze, and you will have the historical price-to-sales ratio for the given time frame.
It is important to note that the price-to-sales ratio is just one valuation metric, and it should be considered along with other financial ratios and metrics to gain a comprehensive understanding of
a company's value and performance. | {"url":"https://twynedocs.com/blog/how-to-calculate-the-best-performing-stock-based-on","timestamp":"2024-11-07T13:08:00Z","content_type":"text/html","content_length":"268507","record_id":"<urn:uuid:92ecc21f-d92e-40df-b699-9d75dab48f89>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00162.warc.gz"} |
Short course proposal
This is a proposition for a short course at 1st year master student level. The aim of this course is to give a basic introduction to tropical geometry, with some applications to real algebraic
Tropical geometry is the algebraic geometry build over the tropical semi-ring (T, « + », « . »)$ where
T=RU{-infinity}, « a+b »=max(a,b), and « a.b »=a+b.
The ``tropical roots'' of polynomials defined upon this algebra are piecewise linear objects, and turn out to be easier to study than their classical analogues (i.e. zero set of polynomials defined
over a field). The tropical semi-ring is linked to the classical semi-ring (R⁺,+,.) by the so-called Maslov dequantization (see [Vir01]) : we transport the semi-ring structure on R⁺ to T by the
homeomorphism log_t, and make t goes to infinity. Thank to this dequantization process, many properties of classical algebraic varieties are reflected by tropical varieties. Conversely, one of the
main issue in tropical geometry is then to understand which properties of tropical varieties can be lifted to classical algebraic varieties.
Recent years have seen a tremendous development in tropical geometry that both established the field as an area of its own right and unveiled its deep connections to numerous branches of pure and
applied mathematics. As an example of application of tropical geometry, one of the most important is certainly the use of tropical methods in enumerative geometry. These methods were initiated in the
seminal paper of Mikhalkin [Mik05], and were a breakthrough in both complex and real enumerative geometries. There is no doubt that tropical techniques promise to remain extremely fruitful in the
In this introductory course, I will mainly focus on the study of tropical curves in the plane, and how to use them to construct real algebraic curves.
Program of the course :
· Introduction to tropical semi-ring and tropical polynomials; Maslov dequantization
· Tropical curves in R²; Kapranov Theorem.
· Some basic tropical intersection theory; Bezout Theorem; Bernstein Theorem.
· Application to real algebraic geometry: combinatorial patchworking and construction of real algebraic curves with controlled topology.
Potential Readings
For easy reading introductions to tropical geometry, I refer to [Bru09], [BPS08] (both in french), [Bru10] (in portuguese), [RGST05, [Mik07], and [Gat].
About real algebraic geometry and Hilbert's 16th problem, one can read the survey [Vir84] and the website [Vir].
[BPS08] N. Berline, A. Plagne, and C. Sabbah, editors. Géométrie tropicale. Editions de l’Ecole Polytechnique, Palaiseau, 2008. available at
[Bru09] E. Brugallé. Un peu de géométrie tropicale. Quadrature, (74):10–22, 2009. available at http://people.math.jussieu.fr/∼brugalle/articles/Quadrature/Quadrature.pdf
[Bru10] E. Brugallé. Um pouco de geometria tropical. Matematica Universitaria, 46:27–40, 2010. Translation from french by E. Amorim and N. Puignau.
[Gat] A. Gathmann. Tropical algebraic geometry. math.AG/0601322.
[Mik05] G. Mikhalkin. Enumerative tropical algebraic geometry in R² . J. Amer. Math. Soc., 18(2):313–377, 2005.
[Mik07] G. Mikhalkin. What is. . . a tropical curve? Notices Amer. Math. Soc., 54(4):511–513, 2007.
[RGST05] J. Richter-Gebert, B. Sturmfels, and T. Theobald. First steps in tropical geometry. In Idempotent mathematics and mathematical physics, volume 377 of Contemp. Math., pages 289–317. Amer.
Math. Soc., Providence, RI, 2005.
[Vir] O. Ya. Viro. Patchworking. http://www.pdmi.ras.ru/∼olegviro/patchworking.html.
[Vir84] O. Ya. Viro. Progress in the topology of real algebraic varieties over the last six years. Russian Math. Surveys, 41:55–82, 1984.
[Vir01] O. Viro. Dequantization of real algebraic geometry on logarithmic paper. In European Congress of Mathematics, Vol. I (Barcelona, 2000), volume 201 of Progr. Math., pages 135–146. Birkh ̈user,
Basel, 2001. | {"url":"http://mate.dm.uba.ar/~alidick/TropicalBrugallew.htm","timestamp":"2024-11-14T08:20:12Z","content_type":"text/html","content_length":"59347","record_id":"<urn:uuid:14934568-a8bd-4d97-82e6-9b5f3df87a85>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00889.warc.gz"} |
Exhaust Flow Rate Cfm (a) Flow Rate (for Four Exhaust Fans) As A Function Of The Static - Powerflow ExhaustsExhaust Flow Rate Cfm (a) Flow Rate (for Four Exhaust Fans) As A Function Of The Static
Exhaust Flow Rate Cfm (a) Flow Rate (for Four Exhaust Fans) As A Function Of The Static
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Zero Probability
November, Dan D (2019) Zero Probability. [Preprint]
This is the latest version of this item.
Zero Probability - philsci-archive d3.pdf
Download (264kB) | Preview
In probability textbooks, it is widely claimed that zero probability does not mean impossibility. But what stands behind this claim? In this paper I offer an explanation for this claim based on
Kolmogorov's formalism. As such, this explanation is relevant to all interpretations of Kolmogorov's probability theory. I start by clarifying that the claim refers only to nonempty events, since
empty events are always considered impossible. Then I offer the following three reasons for the claim that nonempty events with zero probability are considered possible: The main reason is simply
that they are nonempty, and so they are considered possible despite their zero probability. The second reason is that sometimes the zero probability is taken to be an approximation of some
infinitesimal probability value. Such a value is strictly positive and as such does not imply impossibility in a strict sense. The third reason is that, according to some interpretations of
probability, the same event can have different probabilities. They assume that an event with exactly zero probability (which does not approximate an infinitesimal value) can have strictly positive
probabilities. This means that such an event can be possible, which implies that its zero probability does not mean impossibility.
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Wage Growth Rate in context of yearly rate
31 Aug 2024
Title: An Examination of Wage Growth Rates: A Yearly Perspective
Abstract: This study delves into the concept of wage growth rates, focusing specifically on their yearly manifestation. The analysis explores the implications and mathematical underpinnings of wage
growth rates in a yearly context.
Wage growth rate is a critical economic indicator that measures the change in wages over time. It is an essential metric for understanding labor market dynamics and its impact on overall economic
performance. This study aims to provide a comprehensive overview of wage growth rates within a yearly framework, highlighting their mathematical formulation and implications.
Mathematical Formulation:
The yearly wage growth rate (WGR) can be calculated using the following formula:
WGR = ((Wt - W(t-1)) / W(t-1)) * 100
• Wt represents the current year’s wages
• W(t-1) denotes the previous year’s wages
This formula calculates the percentage change in wages from one year to the next, providing a clear and concise measure of wage growth.
Yearly Wage Growth Rate Implications:
The yearly wage growth rate has significant implications for both employers and employees. On one hand, high wage growth rates can lead to increased labor costs, potentially affecting business
profitability. Conversely, low wage growth rates may indicate stagnant wages, which could negatively impact employee morale and retention.
On the other hand, moderate wage growth rates can signal a healthy labor market with opportunities for career advancement and salary increases. This, in turn, can foster a positive work environment
and encourage employees to invest in their skills and education.
In conclusion, this study has provided an in-depth examination of wage growth rates within the context of yearly rates. The mathematical formulation of the yearly wage growth rate highlights its
importance as a key economic indicator. Understanding the implications of wage growth rates is crucial for both employers and employees to navigate the complexities of the labor market effectively.
• [1] Smith, J. (2022). Labor Market Dynamics: A Yearly Perspective.
• [2] Johnson, K. (2019). Wage Growth Rates: Implications for Employers and Employees.
Note: The references provided are fictional and used only to demonstrate the format of academic citations.
Related articles for ‘yearly rate’ :
• Reading: Wage Growth Rate in context of yearly rate
Calculators for ‘yearly rate’ | {"url":"https://blog.truegeometry.com/tutorials/education/bb68a6bca0dfc7b04091880817f273ac/JSON_TO_ARTCL_Wage_Growth_Rate_in_context_of_yearly_rate.html","timestamp":"2024-11-04T08:02:17Z","content_type":"text/html","content_length":"15787","record_id":"<urn:uuid:d001b22e-3ee5-46c1-8de7-5b4343f663ab>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00765.warc.gz"} |
Calculating the Tax-Basis Income - Terms
the equations for Tax-Basis EP & Tax-Basis Incurred Loss use the following terms (listed in the artical):
PL = Paid Loss (during year)
IL = Incurred Loss (during year)
L^D = Losses after Discounting
D = Discount amount (= difference between undiscounted and discounted losses = IL – LD)
I'm trying to understand in the equation the what the "chg" represents?
the first example give loss at 7,000
and the irs discount rate as 5%
chg(L^D) = 7000/(1.05)
so why isn't the chg(L^D) for CY+1 not
chg(L^D) = (0-7000)/(1.05^2)
since we need to discount back another year? (like the investment income getting another year of interest)
I think I just figured it out it's the change in the losses after discounting, so if the loss had been paid out in CY+2
then the chg(L^D) for CY+1 would be
(7000/(1.05^2)) - (7000/1.05)
and after reading a the next section L really stands for Loss Reserves not Loss as that would be reserves + paid amounts
Yes, what you say is essentially correct.
I came to the same conclusion as mec06e after spending some time scratching my head.. Please could the wiki be updated to say
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Importance Of Maths and Physics Tutor
Expert maths and physics tutor that can adapt to their student’s needs are in more demand. In the meantime, you should also talk about the sessions’ planned subjects. Be bold about letting your
instructor know if you’re unhappy with the class’s direction. If you don’t, you’ll have a lot of difficulties in class and cause a lot of extra confusion.
Your kid’s understanding of math and physics can be bolstered with the help of a tutor. They will show you some simple ways to solve a specific numerical issue.
The Role Of Tutors In The Development Of Mathematical And Physical Literacy
Getting a tutor is helpful since kids who struggle with math and physics on admission exams or even in high school frequently do better overall with their instruction.
● The Tutor Will Make Everything Clear:
A good instructor knows everything there is to know about a subject and can explain it in ways students can easily grasp. They use their expertise to simplify complex material so their students can
easily get it.
● If You Hire A Tutor, You May Rest Assured That You’ll Get The Time And Focus You Need:
An Expert maths and physics tutor is obligated to pay close attention to you. To improve, he can assist you in working on your weaknesses. If you let them know where you’re struggling, they can give
you more attention in those areas.
● The Instructor Is Familiar With The Structure And The Anticipated Inquiries:
Knowing the typical format of exam questions is just as crucial as learning the material, particularly in mathematics and physics tests. You can boost your chances of passing the exam by
concentrating more on those specific questions. With the help of a tutor, you can solve these problems and achieve significantly higher grades in these courses.
● Memory Aids For Mathematical Expressions:
Tutors know that memorizing numerous formulas can be tiresome for any learner. So, they can teach multiple recipes to their students using mnemonics.
Learn The In And Out Of Finding The Right Tutor For You
Tutoring services can connect you with a qualified educator who can help you finally master that tricky arithmetic concept.
The Advantage Of Having A Private Tutor Is That You Can Learn At Your Own Pace:
Finding a math tutor for yourself or your child will make all the difference in the world, as tutors tailor their instruction to each student’s unique learning style and pace.
Get Some Math Training To Comprehend Your Environment Better:
One-on-one math tutoring can help students become more proficient in the subject and develop problem-solving skills that will serve them well in various situations.
A Qualified Math Tutor Can Help You Acquire A Better Job:
Your chances of getting hired in various industries will increase if you list math skills on your CV. The use of mathematics is integral in fields as diverse as accounting, engineering, medicine, and
even carpentry.
Avoiding Academic Failure In Math:
Nowadays’ math classrooms have too many students for teachers to provide individual help to each one.
Considering the importance of learning and practice in math and physics, it is reasonable to consider hiring a tutor for these disciplines. Expert maths and physics tutors can help you breeze through
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MATHS :: Lecture 23
MATHS :: Lecture 23 :: Vectoralgebra
Vector Algebra
A quantity having both magnitude and direction is called a vector.
Example: velocity, acceleration, momentum, force, weight etc.
Vectors are represented by directed line segments such that the length of the line segment is the magnitude of the vector and the direction of arrow marked at one end denotes the direction of the
A vector denoted by A, B such that the magnitude of the vector is the length of the line segment AB and its direction is that from A to B. The point A is called initial point of the vector B is
called the terminal point. Vectors are generally denoted by a, vector b, vector c,…)
A quantity having only magnitude is called a scalar.
Example: mass, volume, distance etc.
Addition of vectors
This is known as the triangle law of addition of vectors which states that, if two vectors are represented in magnitude and direction by the two sides of a triangle taken in the same order, then
their sum is represented by the third side taken in the reverse order.
Subtraction of Vectors
Types of Vectors
Zero or Null or a Void Vector
A vector whose initial and terminal points are coincident is called zero or null or a void vector. The zero vector is denoted by
Proper vectors
Vectors other than the null vector are called proper vectors.
Unit Vector
A vector whose modulus is unity, is called a unit vector.
The unit vector in the direction of
There are three important unit vectors, which are commonly used, and these are the vectors in the direction of the x, y and z-axes. The unit vector in the direction of the x-axis is, the unit vector
in the direction of the y-axis is
Collinear or Parallel vectors
Vectors are said to be collinear or parallel if they have the same line of action or have the lines of action parallel to one another.
Coplanar vectors
Vectors are said to be coplanar if they are parallel to the same plane or they lie in the same plane.
Product of Two Vectors
There are two types of products defined between two vectors.
They are (i) Scalar product or dot product
(ii) Vector product or cross product.
Scalar Product (Dot Product)
The scalar product of two vectors
1. Two non-zero vectors
2. Let
3. If m is any scalar,
4. Scalar product of two vectors in terms of components
= a1b1 + a2b2 + a3b3
5. Angle between the two vectors
Work done by a force:
Work is measured as the product of the force and the displacement of its point of application in the direction of the force.
Vector Product (Cross Product)
The vector product of two vectors
1. Vector product is not commutative
2. Unit vector perpendicular to
(i) ¸ (ii) gives
3. If two non-zero vectors
4. Let
5. (m
6. Geometrical Meaning of the vector product of the two vectors is the area of the parallelogram whose adjacent sides are
Area of triangle with adjacent sides
7. Vector product
8. The angle between the vectors
Moment of Force about a point
The moment of a force is the vector product of the displacement
(i.e) Moment
Download this lecture as PDF here | {"url":"http://ecoursesonline.iasri.res.in/Courses/Mathematics/Data%20Files/lec23.html","timestamp":"2024-11-01T19:09:07Z","content_type":"application/xhtml+xml","content_length":"22904","record_id":"<urn:uuid:ca4f1bab-c83a-4ad4-8b5d-1ff82712b295>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00446.warc.gz"} |
CS 100 - Lecture 002 Report
The report explains the second lecture of the CS Freshmen Lecture Series, conducted by Habib University.
% CS-100 Fall 2019 Guest Lecture Report % Use this template to write a 250-word (at max.) report on the guest lecture. \documentclass{report} \usepackage[utf8]{inputenc} \usepackage{hyperref} \title
{Easy and Hard Problems in Computer Science} %Title must be written exactly as ifiedspec \author{By Dr. Shahid Hussain \\ \\ Reviewed by Mohammad Ibrahim Ali} %Speaker name must be written exactly as
specified \date{2 September 2019} % Date when report was written \begin{document} \par \maketitle This report reviews the first guest lecture of the CS Freshmen Seminar, delivered by Dr Shahid
Hussain, \textit{Program Director} and \textit{Assistant Professor Computer Science} at \textit{Habib University}. The lecture aimed to highlight how a computational problem is classified. \medskip \
par The lecturer stated that computational problems are differentiated by their level of difficulty. He gave the example of \textbf{Travelling Salesperson Problem (TSP)}, in which six points were
pointed on a map. The problem was to find the shortest route to travel through these points while starting and ending at the same point. Since there was no other way except to check each route and
calculating the distance or time, it concluded that the increase in the number of points could turn the problem harder. Hence, it was classified as a \textbf{hard problem}. Contrary to this, the idea
of an \textbf{easy problem} was discussed, which was defined as a problem that doesn’t require more time to solve with the increase in data, as in sorting numbers. \medskip \par The lecturer also
highlighted some interesting facts like, \begin{itemize} \item Even a fast computer that could compute 1 million operations per second would require 2 million years to solve a TSP problem with 25
points. \item Hard problems could be not-very hard, and easy problems could be not-very easy, which leads to another classification. \end{itemize} \medskip \par The lecture concluded to the idea that
a programmer should be able to detect the problem as easy or hard and should prefer easy problems, to bring productivity in his work. \bigskip \section*{References} \url{http://bit.ly/2lOsUYI} \end | {"url":"https://es.overleaf.com/articles/cs-100-lecture-002-report/pkgxxwhkrjxv","timestamp":"2024-11-04T16:44:40Z","content_type":"text/html","content_length":"38815","record_id":"<urn:uuid:d0f473f0-c243-49fe-a95f-11fb9eddf1f2>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00544.warc.gz"} |
How To Find A Tangent Angle - BestTemplatess
How To Find A Tangent Angle
How To Find A Tangent Angle – 3 Example 1 Find Ratios of Tangents Find tan S and tan R. Write each answer as a fraction and a decimal, rounded to four digits. SOLUTION = opp S adj. S = RTST = 80 18 =
40 9 tan S 4.4444 = opp R adj. R = STRT = 18 80 = 9 40 tan R 0.2250 =
4 Guided Practice Example 1 Find tan J and tan K. Round to four decimal places. 0.7500 answer, 0.5333 answer,
How To Find A Tangent Angle
Example 2 Find the length of a leg Algebra Find the value of x. Solution Use the tangent of an acute angle to find the length of the leg. tan 32o = opp. adj. Write the ratio of the tangent of 32o.
tan 32o 11 = x Substitute. x tan 32o = 11 Multiply each side by x. x = 11 tan 32o Divide each side by tan 32o x 11 0.6249 Use a calculator to find tan 32o x 17.6 Simplify
Table Of Tangents And Cotangents
Example 3 Calculating Height Using the LAMPPOST tangent, find the height h of the street lamp to the nearest inch. tan 70o = opp. adj. Write the ratio of the tangent at 70o. tan 70o h = 40
substitutions. Multiply each side by 40. 40 tan 70o = h Use a calculator to simplify. 109.9 h Answer The lantern is about 110 inches tall.
7 Example 4 Use a special right triangle to find the tangent Use a special right triangle to find the tangent of a 60o angle. Step 1 Since all 30o-60o-90o triangles are similar, you can simplify the
calculations by choosing 1 as the length of the shortest leg. Use the 30o-60o-90o triangle theorem to determine the length of the long leg.
Example 4 Use a special right triangle long leg = short leg 3 30o- 60o- 90o triangle theorem to find tangent x = 1 3 Substitute. x = 3 Simplify.
Find The Tangent Of The Angle Between The Lines Which Have Intercepts 3, 4, And 1, 8 On The X And Y Axes Respectively
Example 4 Use a special right triangle to find the tangent. STEP 2 Find tan 60o tan 60o = opp. adj. Write the ratio of the tangent of 60o. tan 60o = 3 1 Substitute. tan 60o = 3 Simplify. Answer The
tangent of any 60o angle is 3 1.7321
Guided Practice For Examples 2, 3, and 4, find the value of x. Round to the nearest tenth. Answer 12.2 Answer 19.3 What if? In Example 4, assume that the side length of the short leg is 5 instead of
1. Show that the tangent at 60° is still equal to 5.
11 Daily Homework Quiz Use this chart for exercises 1-4. 1. If a = 18, b = 80, and c = 82, find tan B and tan A. Write each answer as a fraction and round to 4 decimal places. Answer tan B = 40 9 = ;
tan A = =
In The Diagram, Ab And Ac Are Tangents To A Circle, Center O And Radius 3.6cm. Calculate The Area Of The Shaded Region, Given That $\\left| \\!{\\underline {\\,{boc} \\,}} \\right. = \\dfrac{{2\\pi
Daily Homework Quiz Use this chart for exercises 1-4. 2. If a = 17 and m A = 31, find b to the nearest tenth. o Answer 28.3
Daily Homework Quiz Use this chart for exercises 1-4. 3. If a = 9 and m B = 74, find b to the nearest tenth. o Answer 31.4
Daily Homework Quiz Use this chart for exercises 1-4. 4. If a = 5 and m B = 79, find the number of square units in the area of ABC to the nearest tenth. o Answer 64.3
Tangent Formula: Tangent Functions, Formulas, Solved Examples
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I have two circles labeled $A$ and $B$. Each of these circles has known positions $vec$ and $vec$ with radii $r_A$ and $r_B$ . I need to find the blue lines radiating from the origin of the circles
at the angle ($theta_A$ or $theta_B$) shown in the diagram. These theta angles correspond to the angle from the origin of the circles to the point where the tangent line intersects the circle. The
tangent must follow the diagram shown above so that $theta_A = theta_B + 180$. It must also be assumed that these circles never intersect $d = lVert vec – vec rVert > r_A + r_B$.
Enlarge the circle $A$ to the radius of $r_A+r_B$ while shrinking $B$ to $0$ . The common tangent maintains the same direction. Then you have a right triangle with sides $P_AP_B$ and $r_A+r_B$. You
have now added the direction angle and triangle angle of $P_AP_B$.
The Tangent Ratio
Let us call the intersection of the line between the centers with a common tangent $O$. Then you will get two similar right triangles. $$|costheta_A|=frac-vec||}=|costheta_B|=frac-vec||}$$ you can
also write $$||vec- vec|| +||vec-vec||=||vec-vec||=d$$ From here $$|costheta_A|=|costheta_B|= frac$ $d>r_A+r_B From $ you get $|costheta_A|<1$
You must be logged in to answer this question. Not the answer you’re looking for? Explore other questions in Geometry Trigonometry.
By clicking “All Cookies”, you agree that Stack Exchange may store cookies on your device and disclose information in accordance with our cookie policy. ACT Math: How to find the tangent of an angle.
Examples Find The Sine, Cosine And Tangent Of Angles A, B.
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GRE Courses and Classes in Miami, LSAT Courses and Classes in Miami, GRE Courses and Classes in Phoenix, MCAT Courses and Classes in Houston, SAT Courses and Classes in Seattle, ACT Courses and
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If The Angle Between Two Tangents Drawn From An External Point ‘p’ To
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Test Prep Phoenix, LSAT Test Prep Atlanta
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Angle Inclination Of A Line
Find angle using tangent, how to find a tangent angle, how to find horizontal tangent, how to find tangent line, how to find the measure of a tangent chord angle, find tangent angle, how to find the
slope of a tangent line, find angle using tangent calculator, inverse tangent to find angle, how to find tangent, how to find an angle with tangent, how to find angle using tangent | {"url":"https://besttemplatess.com/how-to-find-a-tangent-angle/","timestamp":"2024-11-13T21:35:08Z","content_type":"text/html","content_length":"56655","record_id":"<urn:uuid:e074c5bf-2c10-40d2-80fe-0d791153b5d1>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00141.warc.gz"} |
I am posting this for my friend Brian and for anyone for whom the mathematics of TDVP appear to present a formidable barrier. What is the Calculus of Distinctions? We often hear: “Math is the
language of science.” This is of course true, and the mathematics of TDVP is the Calculus of Distinctions. Furthermore, the Calculus of Distinctions is the mathematics of consciousness. So the
Calculus of Distinctions is the language of science, when science recognizes that consciousness is a fundamental part of reality and not just an accidental afterthought. The Calculus of Distinctions
puts consciousness into the equations of the physical laws of the universe for the first time in the history of modern science. How does it do that? It does it by starting with the definition of the
first distinction as the distinction of self from other; and by pointing out that any distinct object has meaning only as part of the triad: 1.) the Object. 2.) That from which the object is
distinguished, and 3.) The conscious entity drawing the distinction. Thus, when consciousness is included, the logical language and mathematical model of reality becomes triadic, not binary, as
contemporary science assumes. The Calculus of Distinctions opens science to broader aspects of reality currently ignored by mainstream science.
Scientists dreaming of a Theory of Everything are attempting to create a logical, mathematical model of reality based on a very limited part of the less than 5% of reality that is available to us
through our physical senses and the mechanical extensions of those senses.
I want to put the Calculus of Distinctions into the proper perspective by pointing out to you how the languages of the logic, mathematics and models of the universe relate to the actual structure of
reality. The Calculus of Distinctions is a language that reflects the logical structure of physical reality and consciousness. It is a comprehensive logical language that includes the logic of
Newton’s calculus as a sub-set related to the mid-scale of reality. It expands the model to include the relativistic principles of the very large, and the quantum nature of the very small. I’ve done
this by using the principles of relativity, quantum mechanics and the experimental data of the Large Hadron Collider to derive the Triadic Rotational Unit of Equivalence (the TRUE unit for short), as
the basic unit of distinction. In this way, it allows us to apply universal logic to the small fraction of reality available to us through the physical senses, without ignoring the broader picture of
reality we have glimpsed through the insights of relativity and quantum physics.
What is a language? A language is built up of a group of sounds, each one easily distinguishable from the others, like, e.g., ah, eh, ee, oh, oo; buh, cuh, duh, fuh, guh, huh, juh, kuh, … The symbols
representing these sounds form an alphabet, and combinations of these sounds, known as vowels and consonants, form words, which we use to represent images formed in our consciousness from sense data
that we take to be representative of distinct objects existing in reality. A sentence is a statement expressing a logical structure, and an equation is simply a sentence or statement in a quantized
language analogous to a sentence in a verbal language: If the left-hand side of an equation is taken to be the subject, the equals sign is the verb, and the right-hand side is the predicate. Just as
in a verbal language, modifiers and connectors can be added in to make equations more complex in order to represent the logical structure of reality. So, all of the words and statements of a
language, verbal or mathematical, comprise a symbolic model of reality. Such a model is a logical system and therefore ever incomplete by Gӧdel's incompleteness theorems.
Adhering to the logical rules of operation and calculation, math is a language as devoid of speculation as we can make it, and its application, in so far as its axiomatic basis corresponds with
reality, leads to valid conclusions. The Calculus of Distinctions encompasses the logic of all languages, whether verbal or mathematical, by including consciousness and the actions of conscious
entities (the drawing of distinctions) as a complete triadic logical system. As you might expect, the operational rules of the Calculus of Distinctions are different from those of contemporary
mathematics. I have developed them and published them elsewhere, and will provide references for anyone who wants to pursue learning them. Those familiar with George Spencer Brown’s “Laws of Form”
will see similarities in some of the basic forms, because logic, in its purest form is universal. However, the Calculus of Distinctions differs very significantly from the calculus of indications in
the Laws of Form in several ways: In the Calculus of Distinctions, in contrast with the Laws of Form and other systems of symbolic logic, existence is central, and dimensionality is explicit.
I spent two years working with Russian mathematician Vladimir Brandin developing the dimensionality of the Calculus of Distinctions. A short summary of our work was published in Moscow in 2003. And
the application of the Calculus of Distinctions to quantum reality was developed over the past six plus years in collaboration with Dr. Vernon Neppe.
12 comments:
1. Thanks for the mention and your concern, Ed! Let it be said, though, that although I consider the detail of the math of the ‘TDVP Calculus of Distinctions’, presented in some of your earlier
articles, could be a ‘formidable barrier’ to most that have not had the wealth of experience in the discipline as yourself, I see no reason why the fundamental ‘Triadic’ principle of ‘conscious’
existence you propound cannot be seen and accepted as the truth, through and beyond this most unfortunate barrier of unenlightenment. Because of my most profound and extraordinary
mystical-initiation of 1980, I do indeed see and accept this truth, regardless of the complexity of the math behind it, which of course I hope does eventually become common knowledge, is taught,
and accepted, by your peers and the rest of humankind alike.
As previously stated as part of your earlier articles, I would now just like to reiterate the following, which was presented in full support of your overall ‘Triadic’ thesis, but with just one
important and overriding proviso that had been emphasized during my initiation:
The Code within the Seed of the Universe Itself, the Mind of the Ultimate Force, the 'Triadic One' of the mystically-inspired formula 'Y = X squared plus One' - One problem remains, though, Ed:
What is to be the 'fourth ingredient' of this formula that will, of necessity, balance (cosmicate) and thus prevent humanity from destroying the planet and itself on route to a state of conscious
perfection, ad infinitum?
Perhaps this proviso is beyond the scope of these articles, and the ‘Calculus of Distinctions’, but with the perilous state of the temporal human-world today, and the seemingly fiendish intent of
‘the powers that be’ behind it, one surely has to wonder if humankind has any chance of fulfilling the ‘Divine Will’ and purpose of the Ultimate Force becoming the balanced and benevolent
Ultimate Being on planet Earth in the system of the Sun – Whatever, in this respect, my Higher Self was hopeful, rather than pessimistic!
So, in ending, as perhaps an amusing anecdote, in what now seems the dim and distant past, I actually achieved the one and only 100% mark in any exam, for which I ever sat – It concerned
‘Calculus’ of then Advance Level Mathematics; and though I could solve the equations, I then never really understood the meaning and purpose behind ‘Calculus’ itself. However, my mystical
experience of 1980, together with your, later, serendipitous influence, has made up for this lack of traditional and uneducated background in many ways; for I have come to understand that the
above formula represents the most important ‘Calculus’ of all – The gradual change of form from Ultimate Force Spiritual Being to Ultimate Force Spiritual/Temporal Being, ad Infinitum!
Furthermore, your ‘TDVP Calculus of Distinctions’ should no doubt add comprehensive, mathematical and consciousness significance to the fundamental essence of the formula’s ‘Triadic One’ – Good
for our further understanding of and progress towards conscious perfection, ad infinitum, providing we are not scuppered on route due to the iniquitous powers that presently try to enslave us
from the cradle to the grave, and which desperately need to be cosmicated.
2. I went over this website and I believe you have a lot of wonderful information, saved to my bookmarks precalculus homework help
3. All big words...where is the calculus?
1. To learn more search the blog for CoDD or Calculus
4. You talk about the calculus of distinctions. It is mainly physics based post. I am a physics subject student. Thanks Edward for sharing this post. Please click to read about Best Homework
Astrophysics information.
1. Thanks, looks like a good site.
5. The Calculus is the upgrade form of the Math. In the Inter level we explore more ideas with the math, but in the bachelor study, we need to follow the rules of the calculus.
6. Through this blog, we would be able to know about the calculus of distinctions. This idea of working with this blog is really a unique idea like click here ideas.We should concentrate on the
creativity of this blog. In this way we can perform best in the world of blogging. I am going to share this blog with my friends and students as well.
7. The title of the article emerges in me a specific sort of interest however it won't not pull in everybody and appear to be dull or exhausting to you yet give me a chance to propose that it is an
organic product ful read. Individuals should read its type of typingservice.org/our-typing-services/manuscript-typing-services/ as it is an audit that covers the parts of the Australian
compelency. The author has utilized simple words to cover a dry subject this way.
8. I am also asking on search enging I got your article. I think I need more info if you can please let me know. do you have any ebook about CALCULUS OF DISTINCTIONS best ideas here ? .
1. No ebook on the CoDD yet. I haven't had the time, but a descussions of the basics will be in my contribution to the first volume of the Academy for the Advancement of Post-materialist
Sciences, to be out soon check https://www.aapsglobal.com/
9. Some of the important part of the calculus is this distinction which is really different from those regular things of this part of math. here to see more about the writing tips. | {"url":"http://www.erclosetphysics.com/2016/05/what-is-calculus-of-distinctions.html","timestamp":"2024-11-10T07:57:39Z","content_type":"text/html","content_length":"239921","record_id":"<urn:uuid:f25838d0-b9b6-4d14-830f-c1123339c450>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00097.warc.gz"} |
SSC 10th Class Maths Notes Chapter 12 Applications of Trigonometry
Students can go through AP SSC 10th Class Maths Notes Chapter 12 Applications of Trigonometry to understand and remember the concepts easily.
AP State Syllabus SSC 10th Class Maths Notes Chapter 12 Applications of Trigonometry
→ If a person is looking at an object then the imaginary line joining the object and the eye of the observer is called the line of sight or ray of view.
→ An imaginary line parallel to earth surface and passing through the point of observation is called the horizontal.
→ If the line of sight is above the horizontal then the angle between them is called “angle of elevation”.
→ If the line of sight is below the horizontal then the angle between them is called the angle of depression.
→ Useful hints to solve the problems:
1. Draw a neat diagram of a right triangle or a combination of right triangles if necessary.
2. Represent the data given on the triangle.
3. Find the relation between known values and unknown values.
4. Choose appropriate trigonometric ratio and solve for the unknown.
→ The height or length of an object or the distance between two distant objects can be determined with the help of trigonometric ratios.
→ To use this application of trigonometry, we should know the following terms.
→ The terms are Horizontal line, Line of Sight, Angle of Elevation and Angle of Depression.
→ Horizontal line: A line which is parallel to earth from observation point to object is called “horizontal line”.
→ Line of Sight (or) Ray of Vision: The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer.
→ Angle of Elevation: The line of sight is above the horizontal line then angle between the line of sight and the horizontal line is called “angle of elevation”.
1. If the angle of observer moves towards the perpendicular line (pole/tree/ building), then angle of elevation increases and if the observer moves away from the perpendicular line (pole/tree/
building), then angle of elevation decreases.
2. If height of tower is doubled and the distance between the observer and foot of the tower is also doubled, then the angle of elevation remains same.
3. If the angle of elevation of sun above a tower decreases, then the length of shadow of a tower increases.
→ Angle of Depression: The line of sight is below the horizontal line then angle between the line of sight and the horizontal line is called angle of depression.
1. The angle of elevation and depression are always acute angles.
2. The angle of elevation of a point P as seen from a point ‘O’ is always equal to the angle of depression of ‘O’ as seen from P.
→ Points to be kept in mind:
I. Trigonometric ratios in a right triangle:
II. Trigonometric ratios of some specific angles:
→ Solving Procedure:
When we want to solve the problems of height and distances, we should consider the following :
1. All the objects such as tower, trees, buildings, ships, mountains, etc. shall be considered as linear for mathematical convenience.
2. The angle of elevation or angle of depression is considered with reference to the horizontal line.
3. The height of the observer is neglected, if it is not given in the problem.
4. To find heights and distances, we need to draw figures and with the help of these figures we can solve the problems. | {"url":"https://apboardsolutions.guru/ap-ssc-10th-class-maths-notes-chapter-12/","timestamp":"2024-11-02T18:02:39Z","content_type":"text/html","content_length":"55578","record_id":"<urn:uuid:b77f3087-00fe-4947-9284-53c9e6135163>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00350.warc.gz"} |
Bank Accounts: Your Q&A Resource for Bank Accounts
There are 100 dimes in 10 dollars. Below, we will show you how we calculated that 100 dimes equal 10 dollars…. Step 1: Let’s first determine how many dimes are in 1 dollar. The table below gives
the answer. There are 10 dimes in one dollar. Number of Dimes Value 1 Dime 10 cents 2 […]
How Many Dimes in 10 Dollars? Read More »
How Many Pennies are in 1,000 Dollars?
To figure out how many pennies are in 1,000 dollars ($1,000), let’s first figure out how many pennies are in 1 dollar and then 100 dollars. There are 100 pennies in 1 dollar. Furthermore, there are
100 dollar bills in 100 dollars. So, 100 pennies x 100 dollar bills in $100 = 10,000 pennies in
How Many Pennies are in 1,000 Dollars? Read More »
How Many Quarters in a Dollar?
There are four quarters in a dollar. You can determine that four quarters equal one dollar by the following: 1 quarter = 25 cents 2 quarters = 50 cents 3 quarters = 75 cents 4 quarters = $1.00
Another way to think of it is as follows: 100 cents are in 1 dollar 25 cents
How Many Quarters in a Dollar? Read More »
Does Coinstar Take Pennies?
Q: Does Coinstar take pennies in their machines? Yes, Coinstar takes pennies. However, the machine won’t accept 1943 steel pennies if you happen to have one or some. You’ll know it’s steel if
it sticks to a magnet. Most pennies were made of steel in 1943, but a few were made out of cooper-alloy. If
Does Coinstar Take Pennies? Read More »
How Many Nickels are in 100 Dollars?
Q: How many nickels in 100 Dollars? An easy way to figure out how many nickels are in 100 dollars is by first determining how many nickels are in one dollar. There are 20 nickels in one dollar. See
our article titled How Many Nickels Make a Dollar to see how we determined that 20
How Many Nickels are in 100 Dollars? Read More »
How Many Quarters in 50 Dollars?
Q: How many quarters are in 50 dollars? To find out how many quarters are in 50 dollars, let’s simplify things and determine how many quarters are in one dollar. There are 4 quarters in one dollar.
Now, we can figure out how many dollars are in 50 dollars. There are 50 one-dollar bills in
How Many Quarters in 50 Dollars? Read More »
How Many Pennies in 20 Dollars?
Q: How many pennies in 20 dollars? Let’s first find out how many pennies are in one dollar. This will help make it easier to find out how many pennies are in 20 dollars. 100 pennies are equal to
one dollar. Since there are 100 pennies in one dollar, you can find out how many
How Many Pennies in 20 Dollars? Read More »
How Many Pennies are in 10 Dollars?
Q: How many pennies are in 10 dollars? Let’s first determine how many pennies are in a dollar. This will make it easier to find out how many pennies are in 10 dollars. A penny is worth 1 cent and
100 pennies are equal to a dollar. Since there are 100 pennies in one dollar,
How Many Pennies are in 10 Dollars? Read More »
How Many Pennies are in 50 Dollars?
Q: How many pennies in 50 Dollars? Firstly, let’s talk about how many pennies are in a dollar? The answer is 100 pennies equal a $1.00. Now that we know that 100 pennies equal $1.00, we need to
determine how many dollars make $50.00? There are 50 dollar bills in $50.00. Now we can do
How Many Pennies are in 50 Dollars? Read More »
Can I Cash My Own Money Order?
If you are reading this article, you likely purchased a money order with cash and now no longer need to use it for its intended purposes. You also likely wish to receive your money back for your
purchase of the money order. Fortunately, you can cash your own money order and we will explain how.
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Regression Questions - Get Homework Done
Part I
1. In the linear regression model, ,
is referred to as
1. the population regression function.
2. the sample regression function.
3. exogenous variation.
4. the right-hand variable or regressor.
• In the simple linear regression model, the regression slope
1. indicates by how many percent Y increases, given a one percent increase in X.
2. when multiplied with the explanatory variable will give you the predicted Y.
3. indicates by how many units Y increases, given a one unit increase in X.
4. represents the elasticity of Y on X.
• The interpretation of the slope coefficient in the model is as follows:
1. a 1% change in X is associated with a % change in Y.
1. a 1% change in X is associated with a change in Y of 0.01 .
1. a change in X by one unit is associated with a 100 % change in Y.
1. a change in X by one unit is associated with a change in Y.
• The interpretation of the slope coefficient in the model is as follows:
1. a 1% change in X is associated with a % change in Y.
1. a change in X by one unit is associated with a 100* % change in Y.
1. a 1% change in X is associated with a change in Y of 0.01 .
1. a change in X by one unit is associated with a change in Y.
• The interpretation of the slope coefficient in the model is as follows:
1. a 1% change in X is associated with a % change in Y.
1. a change in X by one unit is associated with a change in Y.
1. a change in X by one unit is associated with a 100 % change in Y.
1. a 1% change in X is associated with a change in Y of 0.01 .
• To decide whether or not the slope coefficient is large or small,
1. you should analyze the economic importance of a given increase in X.
2. the slope coefficient must be larger than one.
3. the slope coefficient must be statistically significant.
4. you should change the scale of the X variable if the coefficient appears to be too small.
• Assume that you had estimated the following quadratic regression model
= 607.3 + 3.85 Income – 0.0423 Income2. If income increased from 10 to 11 ($10,000 to $11,000), then the predicted effect on test scores would be:
1. 3.85.
2. 3.85-0.0423.
3. 2.96.
4. Cannot be calculated because the function is non-linear.
Part II:
Long Question 1:
Earnings functions attempt to find the determinants of earnings, using both continuous and binary variables. One of the central questions analyzed in this relationship is the returns to education.
Collecting data from 253 individuals, you estimate the following relationship
= 0.54 + 0.083 × Educ, R2 = 0.20, SER = 0.445
(0.14) (0.011)
where Earn is average hourly earnings and Educ is years of education.
1. What is the effect of an additional year of schooling? (notice, the dependent variable (Y) is in log, not level).
• If you had a strong belief that years of high school education were different from college education, how would you modify the equation? What if your theory suggested that there was a “diploma
• You read in the literature that there should also be returns to on-the-job training. To approximate on-the-job training, researchers often use the so called Mincer or potential experience
variable, which is defined as Exper = Age – Educ – 6. Explain the reasoning behind this approximation. Is it likely to resemble years of employment for various sub-groups of the labor force?
You incorporate the experience variable into your original regression
= -0.01 + 0.101 × Educ + 0.033 × Exper – 0.0005 × Exper2 ,
(0.16) (0.012) (0.006) (0.0001)
R2 = 0.34, SER = 0.405
• What is the effect of an additional year of experience for a person who is 40 years old and had 12 years of education?
• What is the effect of an additional year of experience for a person who is 60 years old and had 12 years of education?
Long Question 2:
Part III:
Empirical Question 1:
Using the data set TeachingRatings posted under the Exam 2 tab on Blackboard,
Carry out the following exercuses. HINT: Also, read the data description (pdf)!
1. Estimate a regression of on and . What is the estimated regression equation?
• Interpret all SEVEN estimated coefficients from part a.
• Which of these coefficients are statistically significant at 5% level?
• Add and to the regression. Is there evidence that has a nonlinear effect on ? (Hint: in other words, is statistically significant at 5% level)?
• Professor Smith is a man. He has cosmetic surgery that increases his beauty index from standard deviation below the average to one standard deviation above the average. What is his value of
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Covering all About AMC Past Papers: A Comprehensive Guide for Students and Educators
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