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Saturated Majorana representations of A_12
Majorana representations have been introduced by Ivanov in order to provide an axiomatic framework for studying the actions on the Griess algebra of the Monster and of its subgroups generated by
Fischer involutions. A crucial step in this programme is to obtain an explicit description of the Majorana representations of A12, the largest alternating group admitting a Majorana representation,
for this might eventually lead to a new and independent construction of the Monster group. In this paper we prove that A12 has two possible Majorana sets, one of which is the set of bitraspositions,
the other is the union of the set of bitranspositions with the set of fix-point-free involutions. The latter case (the saturated case) is most interesting, since the Majorana set is precisely the set
of involutions of A_12 that fall into the class of Fischer involutions when A_12 is embedded in the Monster. We prove that A_12 has unique saturated Majorana representation and we determine its
degree and decomposition into irreducibles. As consequences we get that the Harada-Norton group has, up to equivalence, a unique Majorana representation and every Majorana algebra, affording either a
Majorana representation of the Harada-Norton group or a saturated Majorana representation of A_12, satisfies the Straight Flush Conjecture. As a by-product we also determine the degree and the
decomposition into irreducibles of the Majorana representation induced on A_8, the four point stabilizer subgroup of A_12. We finally state a conjecture about Majorana representations of the
alternating groups A_n, 8 ≤ n ≤ 12.
• Alternating group
• Majorana representation
• Monster Group
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The Design of Input Shapers Which Eliminate Nonzero Initial Conditions
Input shaping is widely used in the control of flexible systems due to its effectiveness and ease of implementation. Due to its open-loop nature, it is often overlooked as a control method in systems
where parametric uncertainty or force disturbances are present. However, if the disturbances are known and finite in duration, their effect on the flexible mode can be approximated by formulating an
initial condition control problem. With this knowledge, an input shaper can be designed, which cancels the initial oscillation, resulting in minimal residual vibration. By incorporating Specified
Insensitivity robustness constraints, such shapers can be designed to ensure good performance in the presence of modeling uncertainty. This input shaping method is demonstrated through computer and
experimental methods to eliminate vibration in actuator bandwidth-limited systems.
Issue Section:
Research Papers
Flexible mechanical systems offer a number of benefits over their rigid counterparts. Because flexible structures are inherently lighter, they require lower actuator effort for quick point-to-point
motion [1]. In turn, they are more time and energy efficient compared to bulky, rigid manipulators. These benefits are only relevant if the vibration in the flexible system can be controlled. To this
end, a large number of vibration reduction methods can be employed.
Open-loop control methods have been employed in a large range of systems to reduce vibration resulting from point-to-point motion. One particular open-loop technique, noted for its simplicity and
ease of implementation, is the command shaping method called input shaping [2]. This method, first introduced as “Posicast Control,” generates a sequence of impulses called an input shaper, which is
convolved with a reference command to produce a new, shaped command that results in significantly reduced vibration [3]. Input shaping has been used to control a variety of flexible systems including
coordinate measurement machines [4] and cranes [5,6]. This technique has also been employed to compensate for nonlinearities due to friction [7], multimodal systems [6,8], limited actuator bandwidth
[9], and perceived overshoot in human-operated systems [10].
The open-loop nature of input shaping renders it incapable of rejecting oscillation resulting from external disturbances or parametric uncertainty. To compensate for uncertain plant dynamics, robust
input shapers, which result in acceptably low levels of vibration for a range of frequencies, can be designed [11,12]. If disturbance rejection is desired, an input shaper can be used in conjunction
with feedback control [13,14].
In some cases, disturbances introduced to a system can be approximated by an impulse, which causes nonzero initial conditions. When this is true, a series of impulses can be designed, which
eliminates the initial oscillation [15–21]. This approach has been proposed to reject vibration due to a change in the desired setpoint of a time-optimal trajectory for a flexible system [19], a
known force on a long-reach telescopic handler [18], the transient sway of a harmonically excited boom crane [16], and the point-to-point motion of cranes at nonzero initial conditions [15,20,21].
The general procedure for constructing these series of impulses is called initial condition input shaping.
In addition to the previously mentioned applications, initial condition input shaping has a number of potential uses in the control of flexible systems. For example, the slewing motion of a boom or
tower crane introduces radial and tangential swing [22]. The input shaping method, which cancels payload swing in both directions, can be formulated as an initial condition input shaper. Another
potential application of the initial condition input shaping method is in the hoist of off-centered payloads [23]. If the payload deflection at the time of hoisting is known, a shaped command can be
generated, which brings the payload to rest [15,21,20].
This command shaping method requires accurate knowledge of the plant dynamics as well as vibration-inducing forces. Although a number of widely used closed-loop methods such as sliding-mode control
or linear quadratic regulation could be employed to accomplish the same task, initial condition input shaping presents several unique characteristics, which make it worthy of consideration. In the
given crane examples, a single position measurement alongside knowledge of the plant dynamics provides enough information to eliminate the undesired vibration. By reducing the reliance on potentially
noisy sensor data, the controller can be simplified [24]. This benefit is most clearly demonstrated in the given examples, where the initial-condition-causing force is known and not expected to
This work will present a thorough analysis on the design and implementation of initial condition input shapers for vibration suppression. This method is based on the previously cited literature,
which develops impulse-based control in a similar manner. However, this paper presents a more rigorous theoretical development and directly applies common input shaping tools to yield a generalizable
solution. Furthermore, moderate nonlinearities due to actuator bandwidth constraints and system dynamics are directly addressed in order to yield better vibration reduction. Finally, robustness to
modeling errors is addressed by implementing the Specified Insensitivity input shaping technique. Using the approach presented in this paper, initial-condition-canceling input shapers with an
arbitrary level of robustness can be developed.
The paper is organized as follows: Section 2 will provide relevant background on the input shaping process. Section 3 develops the solution for initial condition shapers through frequency and time
domain analysis. Robustness considerations and actuator limitations are considered in this section as well. Next, example responses are presented in Sec. 4. Section 5 provides a demonstration of
initial condition input shaping the luff of a planar boom crane. The shaping methods are experimentally validated in Sec. 6. Conclusions are given in Sec. 7.
Input Shaping Overview
Input shaping is a process by which a series of impulses is designed such that its convolution with a reference command results in minimal vibration, as shown in Fig.
]. In the frequency domain, the sequence of impulses, called the input shaper, places zeros at the poles of the second-order plant [
]. Identical results can be achieved in the time domain by evaluating the residual vibration amplitude of a second-order system subject to a sequence of
impulses normalized by a unity magnitude impulse at time
$C(ω,ζ)=∑i=1nAieζωti cos(ω1−ζ2ti)$
$S(ω,ζ)=∑i=1nAieζωti sin(ω1−ζ2ti)$
are the
th impulse amplitudes and times,
is the natural frequency, and
is the damping ratio. Setting
equal to zero for a given
is exactly equivalent to pole cancellation in the frequency domain. If a more robust solution is desired, derivatives of
with respect to
can be set to zero [
], resulting in repeated zeros at the system poles [
]. Because some modeling error is inevitable, relaxing the vibration constraint for a range of sampled frequencies
$VSI(ωk,ζ)≤Vtol, ∀ωk∈[ω1,ω2]$
further increases robustness to modeling errors at
]. This generalized vibration constraint refers to the Specified Insensitivity method, where closed-form solutions are available for special cases and are referred to as extra-insensitive shapers [
]. The numerous methods of improving robustness for input shapers are summarized in Ref. [
An input shaper requires an additional constraint to ensure that the shaped command reaches the same set-point as the reference command. Furthermore, the minimum-time solution is desired. If the
shaper impulses are constrained to be positive, the combined result is an optimization problem of the form
$minimizettnsubject toV(ωk,ζ)≤Vtol, ∀ωk∈[ω1,ω2]Ai>0∑i=1nAi=1$
The simplest solution is an input shaper where
, called the zero vibration (ZV) shaper [
In general, an input shaper can have positive or negative impulses while still meeting the same robustness constraints [
]. If negative amplitudes are allowed, constraints which avoid actuator saturation and over-current must be enforced in place of the positive amplitude constraint
$|Ai|≤1|∑i=1k−1Ai|≤1, ∀k∈n$
where the first constraint ensures that no impulse exceeds magnitude of 1, and the second constraint forces the cumulative sum of any number of impulses to not exceed the desired set-point.
Initial Condition Input Shaping
If a flexible system exhibits nonzero initial states, an input shaper can be designed, which approximates the initial states as an additional impulse to be canceled by the shaper impulse sequence.
This process is demonstrated graphically on a vector diagram [19,27] such as Fig. 2. Here, the initial condition is modeled as an impulse of magnitude A[0] at phase $θ0=ωdt0$. The input shaper
impulses A[1] and A[2] sum to produce a resultant impulse A[s], which is equal in magnitude and directly out of phase with A[0], resulting in zero residual vibration.
Frequency-Domain Design.
The solution for a zero vibration, initial condition (ZV-IC) input shaper can be found by a frequency domain analysis. The representation of a second-order under damped system, subject to a
two-impulse shaper convolved with a reference impulse input, is
Here, y[0] and $y˙0$ are the displacement and velocity of the flexible mode, and $t1=0$.
The pole cancellation of Eq.
dictates that its zeros are located by
Given that
, these amplitudes can be found by solving
resulting in impulse amplitudes of
In a damped system, the solution requires the minimum value of
which satisfies the constraint that the imaginary component of
be zero. For the undamped case, the solution is [
$ZV−IC=[Aiti]=[12(1−α)12(1+α)02ω tan−1(ω2+y˙0y0ω)]$
Time-Domain Design.
An identical solution can be reached using a time-domain formulation similar to Eq.
. In this approach, the initial conditions are modeled as an impulse and directly incorporated into Eq.
. The amplitude and phase of the impulse representing the initial conditions are
The components of the initial condition impulse, given in a form similar to Eqs.
, are
$C0=C0(ω,ζ,y0,y˙0)=A0 cos(θ0)$
$S0=S0(ω,ζ,y0,y˙0)=A0 sin(θ0)$
The resulting residual vibration amplitude due to the input shaper as well as the initial conditions can then be written as
In order to normalize the residual vibration by the vibration amplitude resulting from a unity magnitude impulse at
=0, the effects of the initial condition must be incorporated into the normalization term. Specifically
The percentage residual vibration resulting from the input shaper impulses is therefore
The time-domain solution for the ZV-IC shaper can then be solved through numerical optimization by
$minimizettnsubject toPRVIC=0Ai>0,∑i=1nAi=1$
The specified insensitivity method can be used to generate IC shapers, which are robust to deviations in natural frequency. In order to generate such specified insensitivity, initial condition
(SI-IC) input shapers, negative amplitudes must be allowed. A property of negative amplitude input shapers is that they can generate larger amplitude residual vibrations than their positive amplitude
counterparts [
]. When an IC shaper is desired, this trait promotes a larger available solution space for the input shaper, which yields a resulting amplitude
while maintaining the desired robustness. As a result, robust input shapers, which cancel a wide range of initial conditions, can be generated. The expression for a SI-IC shaper is therefore
$minimizettnsubject toPRVIC≤Vtol, ∀ωk∈[ω1,ω2]|Ai|≤1∑i=1nAi=1|∑i=1k−1Ai|≤1, ∀k∈n$
Although the robust SI-IC shaper solution is presented here in the time domain, an analogous formulation could be presented in the frequency domain through minimax optimization [29].
Incorporating Actuator Limitations.
In the previous analysis, it is assumed that the actuators can exactly follow the impulse commands from the input shaper. Obviously, no actuator can exert an infinite force, so pulses of finite
amplitude must be used. Although this distinction is unnecessary for typical input shaping implementation, the differences between the impulse and pulse responses must be quantified for an
initial-condition-canceling input shaper.
Figure 3 demonstrates the phase and amplitude shifts, $ϕ$ and δ, respectively, of a pulse response compared to an impulse response. These shifts affect the performance of an IC shaper as demonstrated
on the vector diagram in Fig. 4. The phase shift, $ϕ$, corresponds to an apparent lag in the resulting input shaper impulse while the amplitude shift, δ, corresponds to the reduced vector amplitude.
The phase and amplitude shifts of the pulse versus impulse response will depend on the system natural frequency, damping ratio, and acceleration time. If the acceleration time is normalized by the
damped oscillation period,
, then these relationships can be determined by iteratively fitting a shifted response
$y=δω1−ζ2e−ζωt sin(ωdt−ϕ)$
to the numerically integrated pulse response of a second-order system. Figures
show the phase and amplitude shifts, respectively, as a function of normalized acceleration time,
, and damping ratio,
. In Fig.
, a linear relationship between
is evident for each
value. This slope increases as a function of
. Figure
shows that for low damping levels, the amplitude shift trends toward zero as
increases. Conversely,
becomes extremely large when both
are also high. The data in this figure are clipped at
=5 to improve clarity. This trend toward high
values occurs because the phase shift of the pulse command delays the response beyond the point at which the impulse response is settled.
Because the pulse response exhibits these shifts, an IC shaper must be designed to cancel the shifted initial conditions. These new values are
These shifted values are substituted into Eqs. (19) and (20) in order for the corrected shaper to be determined.
Robustness Considerations.
The robustness of an input shaper is typically measured by using sensitivity curves [
]. These plots show the percentage residual vibration of an input shaped response normalized by that of a unity magnitude impulse at time
=0. For a system at rest, this expression simplifies to Eq.
. Because the system under consideration exhibits nonzero initial conditions, the sensitivity curve takes a form similar to Eq.
. Shaper performance in this case will further be degraded by inaccurately timing the shaped command or designing for an incorrect initial condition amplitude. Therefore, the full expression of the
sensitivity of the input shaper is
$VIC=[−C0,e+C(ω,ζ))]2+[S0,e+S(ω,ζ)]2[δ cos(ϕ−θe)+C0,e]2+[δ sin(ϕ−θe)+S0,e]2$
A[e] and θ[e] are errors in the designed shaper amplitude and phase, respectively. Equations (32) and (33) represent the oscillation amplitude due to the initial condition subject to these
measurement errors. An initial condition input shaper can be made robust to deviations in ω and ζ, but not A[e] and θ[e]. However, quantifying the degradation of shaper performance subject to these
errors provides valuable insight into how accurate the state estimation must be for this technique to function properly.
Because Eq. (31) is normalized by the response amplitude of a unity magnitude impulse at t=0, subject to nonzero initial conditions, the exact shape of the shaper sensitivity curve will depend on
the initial conditions. However, the sensitivity function is smooth and continuous near the constraints. Therefore, shaper performance around the design parameters is similar for a variety of initial
Regardless of the enforced robustness constraints, IC shapers exhibit nearly identical sensitivity to initial conditions. This performance measure can be quantified on a sensitivity plot such as the
one shown in Fig. 7. Here, perfect modeling of the system dynamics, ω and ζ, is assumed. The residual vibration is zero at exactly one point, and quickly degrades as errors in initial condition phase
and amplitude increase. For a shaper with specified insensitivity constraints, this curve will be shifted slightly off-center due to the nonzero vibration resulting at the designed conditions.
Figure 8 demonstrates the combined sensitivities of a ZV-IC shaper to initial conditions and normalized frequency error. Because the shaper is designed to exhibit zero residual vibration at the
modeled frequency, a single point exists, which minimizes this sensitivity along each dimension. Taking a cross section normal to the $ωn/ωm$ axis for each plot results in a standard sensitivity
curve for the given initial condition values. The insensitivity of an SI-IC shaper is significantly different, as shown in Fig. 9, due to the substantially increased robustness along the $ωn/ωm$
Impulse Response Example
The proposed shaping methods were tested using simulations of a simple second-order flexible system, which was assumed to have the specified nonzero initial conditions and system parameters given in
. Both ZV-IC and SI-IC input shapers were designed to eliminate the oscillation. The range of frequencies to be suppressed by the SI-IC shaper was chosen to be
, resulting in a normalized Insensitivity of
=0.4. The resulting input shapers are
Table 1
Name Variable Value
Natural frequency ω $2πrad/s$
Damping ratio ζ 0.1
Acceleration time t[acc] $0.1s$
Initial displacement y[0] $−1.5 m$
Initial velocity $y˙0$ $−8.47 m/s$
Name Variable Value
Natural frequency ω $2πrad/s$
Damping ratio ζ 0.1
Acceleration time t[acc] $0.1s$
Initial displacement y[0] $−1.5 m$
Initial velocity $y˙0$ $−8.47 m/s$
Figure 10 shows the response of the system subject to a ZV-IC-shaped pulse input. As expected, the shaped command completely eliminates the residual vibration. A shaped command that was designed
without consideration to the actuator limitations is also given for comparison. The importance of the phase and amplitude shifts resulting from these limitations is evident in the residual vibration
resulting from the unshifted command. Finally, a modeling error of $(ωn/ωm)=1.2$ is introduced to demonstrate the effect of such an error on the shaped response. Because the ZV-IC shaper does not
incorporate any robustness constraints, the performance suffers as a result of this change.
Similarly, the SI-IC-shaped responses are shown in Fig. 11. Because a tolerable level of vibration is permitted at the design frequency, a small amount of residual vibration exists after completing
the shaped command. Furthermore, the robustness constraints result in a longer duration of the shaped command. This robustness is evident in the response subject to the modeling error of $(ωn/ωm)=
1.2$, the upper limit of the suppressed frequency range. The residual vibration subject to this error is significantly lower than the ZV-IC-shaped case. The response of the system subject to an
SI-IC shaper designed without considering the actuator limitations results in an increased level of vibration, as expected.
The robustness of these shapers to modeling uncertainty is given by the sensitivity curve in Fig. 12. While the ZV-IC shaper is designed to yield no residual vibration at the designed frequency, it
rapidly loses effectiveness when the actual natural frequency deviates from this value. The SI-IC shaper, on the other hand, maintains a low level of vibration across the entire range of frequencies.
This plot is consistent with the previous simulation responses.
An Application in Crane Control
The proposed command shaping method can be applied to eliminate nonzero initial states in a weakly nonlinear system such as a boom crane as shown in Fig. 13 [15]. This simple model is composed of a
rigid, massless boom and cable of lengths R and l, respectively. Payload m oscillates about point B by swing angle $ϕ$, and the control input is the boom luff angle, γ.
In order to design a shaper, which eliminates the nonzero initial states $ϕ0$ and $ϕ˙0$, the pulse response must be once again compared to the impulse response for an analogous linear system. The
approach for the boom crane is nearly identical to that of a linear damped, second-order system.
If a boom crane has known actuator limitations, which dictate the maximum luff angular velocity,
and acceleration time,
, the swing response subject to a luff command can be approximately characterized by a linear system with
where γ[0] is the initial luff angle. To normalize the initial conditions of the boom crane, the commanded pulse is scaled by the radial velocity of point B at the beginning of the command. This
scaling factor is configuration dependent, based on the initial luff angle. A boom crane exhibits slightly different dynamics from a simple linear system after this consideration. Therefore, a second
amplitude shift, δ[bc] and phase shift, $ϕbc$ must be computed and used to shift the target initial conditions in the same way as demonstrated in Sec. 3.3. Once the initial conditions are shifted in
this manner, the ZV-IC and SI-IC shaper can be solved using the procedures presented in this work.
A representative simulated response is shown in Fig. 14. The boom crane begins at nonzero initial conditions and undergoes a shaped luff command. The $bang−on$ is shaped by the IC shapers while the
$bang−off$ portion of the command is shaped by a zero vibration and derivative shaper [25] to clearly demonstrate the performance of each IC shaper. The IC shapers are designed to cancel the nonzero
initial conditions while performing an upward luff command. The moderate nonlinearity of the boom crane results in a small level of oscillation in the ZV-IC response, as is expected. The SI-IC shaper
results in slightly more residual vibration.
Experimental Verification
To further validate this command shaping method, an experimental platform at the Kumoh National Institute of Technology in Korea, pictured in Fig. 15, was used. In this system, the flexible rod is
commanded to move along the track, resulting in oscillation of the mass. This oscillation is measured by a laser scan micrometer.
The values used in this experimental analysis are summarized in Table 2. The natural frequency and damping ratio were determined experimentally by measuring the free response of the system for a
specified mass height. The acceleration time, t[acc], and maximum velocity, V[max], were determined by analyzing the step response. Note that the location of the laser micrometer is near the base of
the flexible rod. As a result, the measured deflection is a scaled-down approximation of the actual payload deflection. Although the flexible beam system is multimodal, the lowest mode, ω[n],
dominates the response. Therefore, the measurement acquired at this point serves as an accurate proxy for the deflection of the tip.
Table 2
Name Variable Value
Natural frequency ω[n] $14.28rad/s$
Damping ratio ζ 0.01
Acceleration time t[acc] $0.17s$
Maximum velocity V[max] $0.15 m/s$
Initial displacement $y˙0$ $5.19mm/s$
Initial velocity y[0] $0.0 mm$
Name Variable Value
Natural frequency ω[n] $14.28rad/s$
Damping ratio ζ 0.01
Acceleration time t[acc] $0.17s$
Maximum velocity V[max] $0.15 m/s$
Initial displacement $y˙0$ $5.19mm/s$
Initial velocity y[0] $0.0 mm$
A typical system response is presented in Fig. 16. Here, the experimental response is compared to simulation predictions based on a linear model. A known impulse force generates oscillation,
represented by the position and velocity of the flexible mode, prior to the beginning of the command. In this unshaped case, the command-induced vibration results in a higher level of oscillation
than what existed as a result of the initial conditions. The residual vibration is measured by the amplitude of oscillation after the completion of the commanded motion. This amplitude will be used
to compare the performance of the shapers to the unshaped case. In order to minimize experimental error in this analysis, the end of the command is shaped using a standard ZV shaper based on the
calculated natural frequency and damping ratio of the system. This ZV shaper introduces approximately no additional vibration into the system while bringing the rigid mode to rest, allowing for
consistent measurement of the residual vibration due to the shaping methods under consideration.
The ZV-IC and SI-IC input shapers were designed to eliminate the given initial conditions. Their responses at the designed natural frequency are shown in Figs. 17 and 18. The more complex impulse
sequence of the SI-IC shaper yields higher transient vibration while the system completes the motion, but both shapers result in low levels of residual vibration. In each case, the experimental
trials also closely resemble the simulated results.
Both shaping methods were tested for robustness to modeling uncertainty by determining the residual vibration amplitude of the shaped and unshaped responses at various natural frequencies. The
results for the ZV-IC shaper are summarized in Fig. 19(a). This plot compares the theoretical residual vibration amplitude subject to deviations in natural frequency, given by (31), to those found by
experimental trials. Each experimental data point in this figure is the mean of three trials, where the variance between each trial is approximately zero. Here, the data closely match the predicted
values, particularly near the designed natural frequency. Because the natural frequency of the experimental system is experimentally estimated and assumed to vary linearly, some modeling error due to
nonlinear dynamics of the system is to be expected. Note that in this plot, the residual vibration remains below the unshaped case for all sampled frequencies.
Similar results for the SI-IC shaper are shown in Fig. 19(b). This shaper was designed to suppress vibration in a range of frequencies, $[0.9ω,1.1ω]$. A trend similar to the ZV-IC results is evident
for these trials; the data increasingly deviate from the theoretical prediction at frequencies significantly different from the modeled frequency. In the suppressed range, however, the experimental
data show that the shaper correctly minimized residual vibration. As a result of the negative amplitudes in this shaper, the residual vibration percentage increases more quickly as the natural
frequency deviates from the suppressed range. The shaped command results in greater residual vibration than the unshaped command at greater than approximately 20% natural frequency error.
Figure 20 provides additional insight into the performance of the shaped commands relative to the unshaped command. This plot shows the residual vibration amplitude of the shaped commands as well as
the unshaped command. An approximately linear decrease in vibration levels at higher frequencies is visible for the unshaped command. Because the residual vibration is measured at the end of the
command, the effects of damping are apparent in this vibration amplitude measurement.
While the natural frequency increases and the command duration is constant, more oscillation is damped during the command, resulting in this measurement trend. Although the effect of this damping is
significant on the measured residual vibration amplitudes of each command shaping method, all three are affected approximately equally.
This work has introduced multiple methods of designing input shapers, which are capable of eliminating initial oscillation in a flexible system. The frequency domain solution can be solved in closed
form for an undamped system, while it requires a simple optimization for the more general, damped case. A time domain design procedure can be used to generate IC shapers, which are robust to modeling
uncertainty. Additionally, these shapers can be modified based on the actuator constraints of the system. Experimental results validated the proposed shaping methods. The experimental system
responses closely matched those predicted by simulation. Furthermore, the robustness of the ZV-IC and SI-IC shapers was measured subject to varying natural frequencies. These experimental results
support the effectiveness of each shaping method, while demonstrating the increased robustness of the SI-IC shaping approach.
Funding data
• Louisiana Board of Regents and ASV Global (LEQSF(2014-17)-RD-B).
• National Science Foundation and the Korean National Research Foundation for providing funding through the East Asia Pacific Summer Institutes (EAPSI) Program (1714041).
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W. P.
, “
Preshaping Command Inputs to Reduce System Vibration
ASME J. Dyn. Syst. Meas. Control
), pp.
S. P.
, and
D. K.
, “
Precise Point-to-Point Positioning Control of Flexible Structures
ASME J. Dyn. Syst. Meas. Control
), pp.
, and
, “
Residual Vibration Reduction Using Vector Diagrams to Generate Shaped Inputs
ASME J. Mech. Des.
), pp.
W. E.
W. P.
, and
N. C.
, “
Input Shaping for Vibration Reduction With Specified Insensitivity to Modeling Errors
Jpn.-U. S. A. Sym. Flexible Autom.
, pp.
, “
Minimax Design of Robust Controllers for Flexible Systems
J. Guid., Control, Dyn.
), pp.
|
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st_combine combines geometries without resolving borders, using c.sfg (analogous to c for ordinary vectors).
If st_union is called with a single argument, x, (with y missing) and by_feature is FALSE all geometries are unioned together and an sfg or single-geometry sfc object is returned. If by_feature is
TRUE each feature geometry is unioned individually. This can for instance be used to resolve internal boundaries after polygons were combined using st_combine. If y is provided, all elements of x and
y are unioned, pairwise if by_feature is TRUE, or else as the Cartesian product of both sets.
Unioning a set of overlapping polygons has the effect of merging the areas (i.e. the same effect as iteratively unioning all individual polygons together). Unioning a set of LineStrings has the
effect of fully noding and dissolving the input linework. In this context "fully noded" means that there will be a node or endpoint in the output for every endpoint or line segment crossing in the
input. "Dissolved" means that any duplicate (e.g. coincident) line segments or portions of line segments will be reduced to a single line segment in the output. Unioning a set of Points has the
effect of merging all identical points (producing a set with no duplicates).
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Matrix Inversion - (Civil Engineering Systems) - Vocab, Definition, Explanations | Fiveable
Matrix Inversion
from class:
Civil Engineering Systems
Matrix inversion is the process of finding a matrix that, when multiplied by the original matrix, yields the identity matrix. This concept is crucial in linear algebra, as it allows for the solution
of systems of linear equations, particularly when dealing with square matrices. Understanding matrix inversion also connects to concepts like determinants, rank, and the properties of linear
congrats on reading the definition of Matrix Inversion. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. For a matrix to be invertible, it must be square and have a non-zero determinant.
2. The inverse of a matrix A is denoted as A^{-1}, and the relationship A * A^{-1} = I holds true, where I is the identity matrix.
3. There are several methods to compute the inverse of a matrix, including Gaussian elimination and using the adjugate method.
4. Matrix inversion plays a vital role in solving linear equations of the form Ax = b, where A is the coefficient matrix.
5. Not all matrices are invertible; a matrix with linearly dependent rows or columns is singular and does not have an inverse.
Review Questions
• How do you determine if a matrix is invertible, and what role does the determinant play in this process?
□ To determine if a matrix is invertible, you first check if it is square and then calculate its determinant. If the determinant is non-zero, the matrix is invertible; otherwise, it is singular
and cannot be inverted. The determinant not only indicates invertibility but also provides insight into the matrix's properties, such as volume scaling in transformations.
• Explain the process of finding the inverse of a 2x2 matrix and illustrate this with an example.
□ To find the inverse of a 2x2 matrix A = [[a, b], [c, d]], you use the formula A^{-1} = (1/det(A)) * [[d, -b], [-c, a]], where det(A) = ad - bc. For instance, for A = [[1, 2], [3, 4]], det(A)
= 1*4 - 2*3 = -2. Therefore, A^{-1} = (1/-2) * [[4, -2], [-3, 1]] = [[-2, 1], [1.5, -0.5]].
• Evaluate how understanding matrix inversion can influence problem-solving in civil engineering applications.
□ Understanding matrix inversion significantly enhances problem-solving in civil engineering by allowing engineers to solve complex systems of linear equations efficiently. In structural
analysis, for example, engineers often need to determine forces and displacements in trusses and frames represented by matrices. Being able to find inverses enables them to apply methods such
as stiffness matrices in finite element analysis to predict structural behavior accurately. Thus, mastering this concept equips engineers with powerful tools to ensure safety and
effectiveness in their designs.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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MIAmaxent 1.3.1
• Allow raster’s Raster*-classes to be passed to projectModel(), for backward compatibility
MIAmaxent 1.3.0
• Handling of NA values in forward stepwise selection (whether cause by unstable parameter estimates, or zero deviance explained)
• Bug fix when DV selection yields Chisq <= 0 (e.g. when identical DVs included)
• Added ‘filename’ argument to projectModel() to write raster predictions to file
• Changed name of calculateFTVA() to calculateRVA(), for consistency with source publication
• Package overview documented as per “Documenting packages” in R-exts
• Migrated dependency from raster to terra
MIAmaxent 1.2.0
• Enhancement: added ‘retest’ argument to selection functions
• Internal utility: shortcut function for deriving stricter selectDVforEV from lenient selectDVforEV
• Added calculateFTVA() function for variable contribution
• Added ‘duplicates’ argument to readData() to handle cells with multiple occurrence coordinates more explicitly
MIAmaxent 1.1.1
• Patch for compatibility with dplyr v1.0
MIAmaxent 1.1.0
• Added ellipsis to plotFOP() for easier customization of graphic
• Feature request: added ‘densitythreshold’ argument to plotFOP()
• Predictions NA when a transformation returns NaN.
• Minor documentation edits to readData and testAUC
• Simplification of F-statistic calculation; may cause rounding differences with respect to previous versions.
MIAmaxent 1.0.0
Major changes
• Model fitting implemented as infinitely-weighted logistic regression, so that all computation can be done natively in R (maxent.jar no longer required).
• Implements choice of algorithm: “maxent” for maximum entropy or “LR” for standard logistic regression (binomial GLM).
• No files written to system unless write = TRUE
• Choice of Chi-squared or F-test in nested model comparison
• More consistency in arguments across top-level functions
• Selection trail tables simplified and clarified
Minor changes
• increased flexibility in graphics arguments passed to plotting functions
• quiet option added to top-level functions performing selection
• readData() automatically removes duplicates when two or more presences/absences fall in the same cell
• readData() discards presence locations with missing EV data
• formula argument to selectEV() function, to specify starting point for selection
• plotFOP() smoother changed to loess from exponentially weighted moving average
• plotFOP() plots data density behind FOP values
• plotFOP() returns plot data invisibly
• plotResp() and plotResp2() take identical arguments, the first of which is a model object
• projectModel() takes data in data.frame or raster classes, and plots output spatially in the case of the latter
• trainmax argument removed from selectDVforEV() and selectEV()
• testAUC() plotting optional
MIAmaxent 0.4.0
• Model ranking within selection rounds based on p-value and then F-statistic (tiebreaker), rather than simply F-statistic
• Directories specified by ‘dir’ argument are created if they do not already exist.
• Existing results in directories specified by ‘dir’ argument are overwritten, if desired.
• Fixed bug in selectEV that occurred when the last round of model selection before interaction terms did not result in a significant variable.
• Unnecessary dependency on Hmisc removed.
MIAmaxent 0.3.7
• Removed version minimums for dependencies which are default packages, to allow r-oldrel binary.
• Changed names of toy data used in examples for better organization.
MIAmaxent 0.3.6
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I have added publication data to:
• Michael Abbott, Thorsten Altenkirch, Neil Ghani: Representing Nested Inductive Types using W-types, in Automata, Languages and Programming, ICALP 2004, Lecture Notes in Computer Science 3142,
Springer (2004) [doi:10.1007/978-3-540-27836-8_8, pdf]
I am guessing that this is the reference which was meant to be referred to at the end of this paragraph (where it had a broken anchor to [AAG](#AAG)) and so I fixed the link accordingly
diff, v36, current
Thanks for pointing that out, have now resolved it.
Clarified the situation with indices, parameters, and non-uniform parameters.
diff, v35, current
Have written it as
A W-type is a set or type which is defined inductively in a well-founded way based on a type of “constructors” and a type of “arities”, with one constructor having a certain canonical form. As
such it is a particular kind of inductive type.
The page still contains a query from you Mike:
Is the term “W-type” still used in this generality? Or are they just called “inductive types”?
Is this resolved yet?
Maybe we should just omit “thus”. Would it make sense that way?
A W-type is a set or type which is defined inductively in a well-founded way based on a type of “constructors” and a type of “arities”. It is thus a particular kind of inductive type, with one
constructor having a certain canonical form.
How’s that “thus” working? Why “one constructor” in this concept.
Spencer Breiner pointed out on Zulip that this page claims that natural numbers (and lists) are W-types, when type-theoretically they are actually not. In type theory, a W-type has only one
constructor, while nat and lists have two constructors. They can be encoded as W-types using a sum-type to wrap up the two constructors into one, but the result doesn’t have the same computation
rules and requires function extensionality for its correctness. So I removed these examples from this page and put the corresponding text at inductive type.
diff, v33, current
Added reference
• Jasper Hugunin, IWTypes, https://github.com/jashug/IWTypes
diff, v32, current
Re #6. Thanks! I'll try to remember that.
Thanks for editing.
Allow me to mention some hints on formatting references:
• leave a whitespace to any previous text, not to have the output clutter
• a * followed by a whitespace right at the beginning of the line makes a bullet item for the reference to fit into the rest of the list
• enclosing author names in double square brackets hyperlinks them to their pages
• explicit hyperlinks need the http://-prefix to be parsed
I have slightly reformatted accordingly, now the source looks like so
* [[Michael Abbott]], [[Thorsten Altenkirch]], and [[Neil Ghani]],
_Representing Nested Inductive Types using W-types_
and renders like so:
• Michael Abbott, Thorsten Altenkirch, and Neil Ghani, Representing Nested Inductive Types using W-types (pdf)
Reference to Michael Abbott, Thorsten Altenkirch, and Neil Ghani, Representing Nested Inductive Types using W-types
diff, v29, current
Heh, you changed the name from a recognizable letter to a nondescript grey box, according to my Firefox for Android. But that’s OK, I don't think that we should actually cater to that.
added to W-type pointers to
• the article by Gambino-Hyland on dependent W-types
• the article by van den Berg and Moerdijk on W-types in model categories (those given by fibrations)
Also made the following trivial edit: changed the name of the ambient category from $C$ to $\mathcal{C}$ as it seemed rather bad style to have a category named $C$ with objects named $A$, $B$ and $D$
I’ll have a question or comment on W-types in linear type theory, but I’ll put that in a separate thread…
Added to W-type a section Properties with a pointer to Danielsson’s recent post.
And a related query at pretopos.
Created W-type.
did a fair bit of rewording and adjustment in the Idea-section (here) for better (logical) flow
switched to calligraphic “$\mathcal{W}$-type”, throughout
switched the ordering of the subsections “In category theory”, “In type theory”.
The $\mathcal{W}$-type inference rules here had a query box comment:
Andreas Abel: A and B are used exactly the other way round in the introductory text to this page. (Before, A was the arity, now B is the arity.) Harmonize!?
Indeed. Am harmonizing this now by using, throughout:
• “$C$” for the type of constructors
• “$A$” for the type of arities.
introduced formatting of $\mathcal{W}$-type denotation as “$\underset{c \colon C}{\mathcal{W}} \, A(c)$”
diff, v37, current
fixed these bibitems:
• Ieke Moerdijk, Erik Palmgren, Wellfounded trees in categories, Annals of Pure and Applied Logic 104 1-3 (2000) 189-218 [doi:10.1016/S0168-0072(00)00012-9]
• Benno van den Berg, Ieke Moerdijk, $W$-types in sheaves [arXiv:0810.2398]
diff, v37, current
adjusted the typesetting of the type inference rules, in particular:
• added more widespace to make it easier on the eye to recognize the syntax tree branching
• chose more suggestive term names
now it looks as follows:
(1) type formation rule:
$\frac { C \,\colon\, Type \;; \;\;\; c \,\colon\, C \;\;\vdash\;\; A(c) \,\colon\,Type \mathclap{\phantom{\vert_{\vert}}} } { \mathclap{\phantom{\vert^{\vert}}} \underset{c \colon C}{\mathcal{W}}\,
A(c) \,\colon\, Type }$$\frac{ \vdash\; root \,\colon\, C \;; \;\; subtr \,\colon\, A(c) \to \underset{c \colon C}{\mathcal{W}}\, A(c) }{ \mathclap{\phantom{\vert^{\vert}}} tree\big(root ,\, subtr\
big) \,\colon\, \underset{c \colon C}{\mathcal{W}}\, A(c) }$$\frac{ \begin{array}{l} t \,\colon\, \underset{c \colon C}{\mathcal{W}}\, A(c) \;\vdash\; D(t) \,\colon\, Type \;; \\ root \,\colon\, C
\,, \; subt \,\colon\, A(c) \to \underset{c \colon C}{\mathcal{W}}\, A(c) \,, \; subt_D \,\colon\, \underset{a \colon A(root)}{\prod} D\big(subt(a)\big) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\vdash\;
\; tree_D\big(root,\,subt ,\, subt_D\big) \,\colon\, D\big(tree(c,\, subt)\big) \mathclap{\phantom{\vert_{\vert}}} \end{array} }{ \mathclap{\phantom{\vert^{\vert}}} t \,\colon\, \underset{c \colon C}
{\mathcal{W}}\, A(c) \;\vdash\; wrec_{(D,tree_D)}(t) \,\colon\, D(t) }$
(4) computation rule:
$\frac{ \begin{array}{l} t \,\colon\, \underset{c \colon C}{\mathcal{W}}\, A(c) \;\vdash\; D(t) \,\colon\, Type \;; \\ root \,\colon\, C \,,\; subt \,\colon\, A(c) \to \underset{c \colon C}{\mathcal
{W}}\, A(c) \,, \; subt_D \,\colon\, \underset{a\colon A(c)}{\Pi} D\big(subt(a)\big) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\vdash\;\; tree_D\big(root,\,subt,\,subt_D\big) \,\colon\, D\big(tree(root,
\, subt)\big) \mathclap{\phantom{\vert_{\vert}}} \end{array} }{ \begin{array}{l} \mathclap{\phantom{\vert^{\vert}}} root \,\colon\, C \,, \;\; subtr \,\colon\, A(c) \to \underset{c \colon C}{\mathcal
{W}}\, A(c) \\ \;\;\;\;\;\;\;\;\;\;\; \;\;\vdash\;\; wrec_{(D,tree_D)}\big( tree(root, \, subtr) \big) \;=\; tree_D \Big( root ,\, subt ,\, \lambda a . wrec_{(D,tree_D)}\big(subtr(a)\big) \Big) \end
{array} }$
diff, v37, current
added pointer to:
• Jasper Hugunin, Why Not W?, Leibniz International Proceedings in Informatics (LIPIcs) 188 (2021) [doi:10.4230/LIPIcs.TYPES.2020.8, pdf]
• Peter Dybjer, Representing inductively defined sets by wellorderings in Martin-Löf’s type theory, Theoretical Computer Science 176 1–2 (1997) 329-335 [doi:10.1016/S0304-3975(96)00145-4&
diff, v38, current
made explicit the example of the natural numbers W-type with a numbered environment (here)
diff, v38, current
organized all references on semantics of W-types by initial endofunctor algebras: here
(cf. nForum announcement here)
diff, v40, current
mentioned that the dependent product is not dependent in the plain case by factoring through the reader monad
diff, v42, current
In the sentence
More generally, endofunctors that look like polynomials in the traditional sense:
$F(Y) = A_n \times Y^{\times n} + \dots + A_1 \times Y + A_0$
can be constructed as polynomial endofunctors in the above sense in any $\Pi$-pretopos.
I changed the $\Pi$-pretopos to just a pretopos, since that is all one needs for the finite products and coproducts.
diff, v43, current
I plan to add in the morphism that makes the endofunctor in the previous comment polynomial, having now double checked it really works for an arbitrary pretopos. For a $\sigma$-pretopos (ie countable
coproducts and corresponding extensivity) one gets analytic endofunctors (those that are formal power series) being polynomial for free as well. Then I’ll check about existence of W-types for these
based just on having a parameterised NNO.
added earlier reference for the introduction of the notion of W-types:
• Per Martin-Löf, pp. 171 of: Constructive Mathematics and Computer Programming, in: Proceedings of the Sixth International Congress of Logic, Methodology and Philosophy of Science (1979), Studies
in Logic and the Foundations of Mathematics 104 (1982) 153-175 $[$doi:10.1016/S0049-237X(09)70189-2, ISBN:978-0-444-85423-0$]$
following a hint in another thread (here)
diff, v47, current
added earlier reference for the introduction of the notion of W-types:
• Per Martin-Löf, pp. 171 of: Constructive Mathematics and Computer Programming, in: Proceedings of the Sixth International Congress of Logic, Methodology and Philosophy of Science (1979), Studies
in Logic and the Foundations of Mathematics 104 (1982) 153-175 $[$doi:10.1016/S0049-237X(09)70189-2, ISBN:978-0-444-85423-0$]$
following a hint in another thread (here)
Hubert Wasilewski
diff, v48, current
In reality “Hubert Wasilewski” in #27 made no discernible edit.
By why does the text of #27 copy your text from #26?
It keeps happening (I have been flagging it previously): Newbie users appear who announce an otherwise empty edit by copying my latest edit log.
It is either a strange bug or a strange prank.
I had meant to bring it to the attention of the the technical team, but didn’t get around to yet.
If you try to edit without entering a name in the “Submit as ….” text field, the following things happen:
1. A yellow box appears at the top of the page with the text “Please enter your name. (Due to a flood of low quality edits, we restrict anonymous edits.)”
2. The comment of the last edit made on that page is copied into the comment box at the bottom of the page above the “Submit as ….” text field. Sometimes this is still empty because the last editor
did not make a comment.
Thanks, that’s useful to know! I have brought this to the attention of the technical team.
The comment of the last edit made on that page is copied into the comment box at the bottom of the page above the “Submit as ….” text field. Sometimes this is still empty because the last editor
did not make a comment.
Apparently, that was how the announcement box was designed. It worked as intended when the failed edit itself contained an announcement message (then that one would be displayed). I think this bug
just showed more now because there are more ways for an edit to fail (e.g., empty username).
I made a change that should fix this. Thanks for letting us now.
Now, when you try to edit without entering a name in the “Submit as ….” text field and encounter the yellow text box at the top, the previous editor’s comments no longer appear in the comments box.
However, the contents of the article disappear from the article text’s box as well.
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Daniel Berntson (Uppsala University): Publications - PhilPeople
• 986
Can we do science without numbers? How much contingency is there? These seemingly unrelated questions--one in the philosophy of math and science and the other in metaphysics--share an
unexpectedly close connection. For as it turns out, a radical answer to the second leads to a breakthrough on the first. The radical answer is new view about modality called compossible
immutabilism. The breakthrough is a new strategy for doing science without numbers. One of the chief benefits of the new strategy …
Read more
• 800
Counterfactuals are somewhat tolerant. Had Socrates been at least six feet tall, he need not have been exactly six feet tall. He might have been a little taller—he might have been six one or six
two. But while he might have been a little taller, there are limits to how tall he would have been. Had he been at least six feet tall, he would not have been more than a hundred feet tall, for
example. Counterfactuals are not just tolerant, then, but bounded. This paper presents a surprising paradox: If…
Read more
• 483
This paper presents a new system of conditional logic B2, which is strictly intermediate in strength between the existing systems B1 and B3 from John Burgess (1981) and David Lewis (1973a). After
presenting and motivating the new system, we will show that it is characterized by a natural class of frames. These frames correspond to the idea that conditionals are about which worlds are
nearly closest, rather than which worlds are closest. Along the way, we will also give new characterization resul…
Read more
• 166
How should the opinion of a group be related to the opinions of the group members? In this article, we will defend a package of four norms – coherence, locality, anonymity and unanimity. Existing
results show that there is no tenable procedure for aggregating outright beliefs or for aggregating credences that meet these criteria. In response, we consider the prospects for aggregating
credal pairs – pairs of prior probabilities and evidence. We show that there is a method of aggregating credal pa…
Read more
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MHT CET 2021 21th September Morning Shift | Area Under The Curves Question 23 | Mathematics | MHT CET - ExamSIDE.com
MHT CET 2021 21th September Morning Shift
MCQ (Single Correct Answer)
The area of the region bounded by the curve y$$^2$$ = 4x and the line y = x is
MHT CET 2021 20th September Evening Shift
MCQ (Single Correct Answer)
The area bounded by the parabola $$y=x^2$$ and the line $$y=x$$ is
MHT CET 2021 20th September Morning Shift
MCQ (Single Correct Answer)
The area of the region bounded by the parabola x$$^2$$ = y and the line y = x is
MHT CET 2020 19th October Evening Shift
MCQ (Single Correct Answer)
The area of the region bounded by the curve $y=\sin x$ between $x=-\pi$ and $x=\frac{3 \pi}{2}$ is
|
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ISC Specimen Papers for Class 11 Maths 2020, 2019, 2018 - A Plus Topper
Download ISC Specimen Papers 2020 Solved for Class 11 Maths and Marking Scheme PDF. Here we have given ISC Maths Question Papers 2020 Solved. Students can view or download the Specimen Papers for ISC
2020 with Answers Class 11 Maths for their upcoming examination.
These ISC Board Sample Papers are useful to understand the pattern of questions asked in the board exam. Know about the important concepts to be prepared for ISC Class 11 Maths board exam and Score
More marks.
Board – Indian School Certificate Examinations (CISCE), www.cisce.org
Class – Class 11
Subject – Maths
Year of Examination – 2020, 2019, 2018, 2017.
ISC Class 11 Maths Question Papers Solved
www.cisce.org ISC Sample Papers for Class 11 Maths are part of ISC Specimen Papers Solved for Class 11 Here we have given ISC Class 11 Maths Sample Question Papers for Class 11 Maths.
│Year of Examination │ISC Maths Question Paper │
│2019 │Download PDF │
│2018 │Download PDF │
The above ISC Model Paper for Class 11 Maths is the official sample paper released by ISC Board as per latest syllabus of Class 11 Indian Certificate of Secondary Education, India.
We hope the ISC Specimen Papers for Class 11 Maths, help you. If you have any query regarding ISC Class 11 Maths Question Papers Solved, drop a comment below and we will get back to you at the
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Creating Real Mathematicians - Year Planner. Years 3/4
Year Planner Years 3 & 4 Terms 1-4
Each term revisits concepts and teaching and learning takes individual students to higher level of understanding.
Individual student growth documented against VELS Progression Points for each topic each term.
Warm Up Activities emphasise the following throughout the year.
Patterns (Counting and Shape), Probability games, Automatic recall of simple number facts (Doubling, adding two one digit numbers, compliments of ten, odd and even, etc), Matching Analogue, Digital
and Written time.
Problem Solving activities aligned to topics under investigation presented to students weekly to further develop Proficiency Strands of Australian Curriculum (Understanding, Fluency, Problem Solving
& Reasoning). Resources include Exemplars, Pictures/Numbers/Words.com)
Literacy emphasised for each topic. Students to become fluent in vocabulary of each topic.
Written Share/Reflections of developed understandings are standard practice and completed at the end of every lesson.
All lessons to explicitly identify to students the Learning Intentions for each lesson.
As students work towards the achievement of Level 3 standards in Mathematics, they recognise and explore patterns in numbers and shape. They increasingly use mathematical terms and symbols to
describe computations, measurements and characteristics of objects.
In Number, students use structured materials to explore place value and order of numbers to tens of thousands. They skip count to create number patterns. They use materials to develop concepts of
decimals to hundredths. They use suitable fraction material to develop concepts of equivalent fraction and to compare fraction sizes. They apply number skills to everyday contexts such as shopping.
They extend addition and subtraction computations to three digit numbers. They learn to multiply and divide by single digit numbers.
In Space, students sort lines, shapes and solids according to key features. They use nets to create three-dimensional shapes and explore them by counting edges, faces and vertices. They visualise and
draw simple solids as they appear from different positions. They investigate simple transformations (reflections, slides and turns) to create tessellations and designs. They explore the concept of
angle as turn (for example, using clock hands) and as parts of shapes and objects (for example, at the vertices of polygons). They use grid references (for example, A5 on a street directory) to
specify location and compass bearings to describe directions. They use local and larger-scale maps to locate places and describe suitable routes between them.
In Measurement, chance and data, students measure the attributes of everyday objects and events using formal (for example, metres and centimetres) and informal units(for example, pencil lengths).
Students tell the time using analogue and digital clocks and relate familiar activities to the calendar. Students investigate natural variability in chance events and order them from least likely to
most likely. Students conduct experiments and collect data to construct simple frequency graphs. They use simple two-way tables (karnaugh maps) to sort non-numerical data.
In Structure, students use structured material (in tens, hundreds and thousands) to develop ideas about multiplication by replication and division by sharing. They recognise the possibility of
remainders when dividing. They learn to use number properties to support computations (for example, they use the commutative and associative properties for adding or multiplying three numbers in any
order or combination). They investigate the distributive property to develop methods of multiplication and division by single digit whole numbers. They learn to use and describe simple algorithms for
computations. They use simple rules to generate number patterns (for example, ‘the next term in the sequence is two more than the previous term’). They create and complete number sentences using
whole numbers, decimals and fractions.
When Working mathematically, students use mathematical symbols (for example, brackets, division and inequality, the words and, or and not). Students develop and test ideas (conjectures) across the
content of mathematical experience. For example:
• in Number, the size and type of numbers resulting from computations
• in Space, the effects of transformations of shapes
• in Measurement, chance and data, the outcomes of random experiments and inferences from collected samples.
Students learn to recognise practical applications of mathematics in daily life, including shopping, travel and time of day. They identify the mathematical nature of problems for investigation. They
choose and use learned facts, procedures and strategies to find solutions. They use a range of tools for mathematical work, including calculators, computer drawing packages and measuring tools.
National Statements of Learning
This learning focus statement, with the following elaboration, incorporates the Year 3 National Statement of Learning for Mathematics.
They recognise angles … as parts of shapes and objects …
At Level 3, students use place value (as the idea that ‘ten of these is one of those’) to determine the size and order of whole numbers to tens of thousands, and decimals to hundredths. They round
numbers up and down to the nearest unit, ten, hundred, or thousand. They develop fraction notation and compare simple common fractions such as 3/4 > 2/3 using physical models. They skip count
forwards and backwards, from various starting points using multiples of 2, 3, 4, 5, 10 and 100.
They estimate the results of computations and recognise whether these are likely to be over-estimates or under-estimates. They compute with numbers up to 30 using all four operations. They provide
automatic recall of multiplication facts up to 10 × 10.
They devise and use written methods for:
• whole number problems of addition and subtraction involving numbers up to 999
• multiplication by single digits (using recall of multiplication tables) and multiples and powers of ten (for example, 5 × 100, 5 × 70 )
• division by a single-digit divisor (based on inverse relations in multiplication tables).
They devise and use algorithms for the addition and subtraction of numbers to two decimal places, including situations involving money. They add and subtract simple common fractions with the
assistance of physical models.
At Level 3, students recognise and describe the directions of lines as vertical, horizontal or diagonal. They recognise angles are the result of rotation of lines with a common end-point. They
recognise and describe polygons. They recognise and name common three-dimensional shapes such as spheres, prisms and pyramids. They identify edges, vertices and faces. They use two-dimensional nets,
cross-sections and simple projections to represent simple three-dimensional shapes. They follow instructions to produce simple tessellations (for example, with triangles, rectangles, hexagons) and
puzzles such as tangrams. They locate and identify places on maps and diagrams. They give travel directions and describe positions using simple compass directions (for example, N for North) and grid
references on a street directory.
Measurement, chance and data
At Level 3, students estimate and measure length, area, volume, capacity, mass and time using appropriate instruments. They recognise and use different units of measurement including informal (for
example, paces), formal (for example, centimetres) and standard metric measures (for example, metre) in appropriate contexts. They read linear scales (for example, tape measures) and circular scales
(for example, bathroom scales) in measurement contexts. They read digital time displays and analogue clock times at five-minute intervals. They interpret timetables and calendars in relation to
familiar events. They compare the likelihood of everyday events (for example, the chances of rain and snow). They describe the fairness of events in qualitative terms. They plan and conduct chance
experiments (for example, using colours on a spinner) and display the results of these experiments. They recognise different types of data: non-numerical (categories), separate numbers (discrete), or
points on an unbroken number line (continuous).They use a column or bar graph to display the results of an experiment (for example, the frequencies of possible categories).
At Level 3, students recognise that the sharing of a collection into equal-sized parts (division) frequently leaves a remainder. They investigate sequences of decimal numbers generated using
multiplication or division by 10. They understand the meaning of the ‘=’ in mathematical statements and technology displays (for example, to indicate either the result of a computation or
equivalence). They use number properties in combination to facilitate computations (for example, 7 + 10 + 13 = 10 + 7 + 13 = 10 + 20). They multiply using the distributive property of multiplication
over addition (for example, 13 × 5 = (10 + 3) × 5 = 10 × 5 + 3 × 5). They list all possible outcomes of a simple chance event. They use lists, venn diagrams and grids to show the possible
combinations of two attributes. They recognise samples as subsets of the population under consideration (for example, pets owned by class members as a subset of pets owned by all children). They
construct number sentences with missing numbers and solve them.
Working mathematically
At Level 3, students apply number skills to everyday contexts such as shopping, with appropriate rounding to the nearest five cents. They recognise the mathematical structure of problems and use
appropriate strategies (for example, recognition of sameness, difference and repetition) to find solutions.
Students test the truth of mathematical statements and generalisations. For example, in:
• number (which shapes can be easily used to show fractions)
• computations (whether products will be odd or even, the patterns of remainders from division)
• number patterns (the patterns of ones digits of multiples, terminating or repeating decimals resulting from division)
• shape properties (which shapes have symmetry, which solids can be stacked)
• transformations (the effects of slides, reflections and turns on a shape)
• measurement (the relationship between size and capacity of a container).
Students use calculators to explore number patterns and check the accuracy of estimations. They use a variety of computer software to create diagrams, shapes, tessellations and to organise and
present data.
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What is the slope and y-intercept of the line 2x - 3y = -18? | HIX Tutor
What is the slope and y-intercept of the line 2x - 3y = -18?
Answer 1
slope $= \frac{2}{3}$
and\y-intercept $= 6$
Slope For an equation in the form: #color(white)("XXXX")##Ax+By=C# the slope is #color(white)("XXXX")##m = -A/B# for the given equation #2x-3x=-12# this becomes #color(white)("XXXX")##m = 2/3#
Alternatively we could rewrite the given equation #2x-3y =-18# into "slope intercept" form: #color(white)("XXXX")##y = mx+b# #color(white)("XXXX")##color(white)("XXXX")##color(white)("XXXX")#where #m
# is the slope and #b# is the y-intercept #color(white)("XXXX")#2x-3y = -18#
#rarr##color(white)("XXXX")##-3y = -2x-18#
#rarr##color(white)("XXXX")##y = 2/3x +6#
y-intercept If you rewrote the equation in "slope intercept form" (see above) the slope can be read directly from the equation as #color(white)("XXXX")##m = 6#
Otherwise note that the y-intercept is the value of #y# when #x=0# in the equation: #color(white)("XXXX")##2(0)-3y = -18# #rarr##color(white)("XXXX")##y = 6#
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Answer 2
To find the slope and y-intercept of the line 2x - 3y = -18, first, rewrite the equation in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.
2x - 3y = -18 -3y = -2x - 18 Divide both sides by -3: y = (2/3)x + 6
The slope (m) of the line is 2/3, and the y-intercept (b) is 6.
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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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The Good, the Bad and What Is a Function in Math | Angels For Children Supervised Visitation
The Good, the Bad and What Is a Function in Math
A Startling Fact about What Is a Function in Math Uncovered
When you begin writing your code you will find that the program adds colour coding to make it even more readable. Bar graphs are amazingly versatile, so anybody handling data will undoubtedly utilize
them often. Again, consider your data before it’s modelled.
Write the function for which you’re attempting to locate a vertical asymptote. To put it differently, it lets you know how much one quantity changes given a little paper now shift in the other.
Taking the opportunity to pick out a fantastic editor and explore its various functions before you need them is a great idea.
Horizontal asymptotes are found in a vast array of functions. Graphs likewise don’t have any notion of one-directional flowinstead, they may have direction, or else they may have no direction
whatsoever. In different scenarios a model is produced for internet prediction.
Obviously, you don’t need to create your process for a graph or table as it may be too complicated. The first thing which you will see when utilizing a code editor is that every line is numbered.
Indeed, physic updates and traditional updates aren’t synced.
What Is a Function in Math – Overview
There are a lot of interesting prior examples of using technology to advertise meaning. So you may pick any x values you want for the next portion of the problem. Policy function is being learned by
write my essay means of an agent to attain a goal.
In the event the tests measure similar abilities, you would aspire to observe an improvement with successive tests. Each line indicates the department’s trend, so that you can adhere to each one
easily. 1 means is to educate our kids and students about common math myths.
Plug some different numbers in and see whether you see a pattern. The expression we give above for the region of the square isn’t complete. A larger diameter usually means a larger circumference.
3rd semester By the conclusion of first calendar year, you would have certainly completed a simple programming program. Please write comments if you discover anything incorrect, or you would like to
share more info about the topic discussed above. A graph can make it a lot simpler to notice trends that are less obvious when viewing the raw data, which makes it a lot simpler to make predictions.
In the following two examples you are going to have a number in the front of the algebra within the bracket. Most of the job in calculus is carried out by graphing formulas so as to figure out the
slope of the rate of change. You’ll follow almost precisely the exact same steps as solving an equation, with two or three quirks introduced by the existence of the inequality.
If a polynomial has the level of two, it’s often referred to as a quadratic. For instance, the 2 equations below are in the format your equations have to be in before you begin. A linear equation
lets you know how much you are able to afford.
What You Need to Know About What Is a Function in Math
Locating a very good approximate for the function is extremely tough. Enter a complicated function that you need to graph. To completely understand function tables and their purpose, you must
understand functions, and the way in which they relate to variables.
In reality, there’s an infinite number of those. Now you must have some idea what window you should use to find the function plotted. Sometimes in a word problem, you might need to establish a
function and evaluate it.
Regarding architecture it’s a simple three-layered neural network. The idea of a function is more central in mathematics than the idea of a number. Its purpose is to move the player in line with the
environment (the colliders).
Though a variable can represent any number, sometimes only 1 number may be used. For that reason, it’s used as the decision attribute in the main node. Use the vertical line test to demonstrate that
the function is genuinely a function.
Meaning, it’s depending on its prior states. There’s also a single temperature where Celsius and Fahrenheit are equal to one another. A linear inequality is the exact same kind of expression with an
inequality sign as an alternative to an equals sign.
Test each remedy to establish whether it’s a maximum or a minimum. Developing a month-to-month budget circle graph is a powerful approach to plan spending each month and is rather simple to do on a
computer. The very first part of the sum above is our usual cost function.
In the event the trend line doesn’t have an equation, then you may want to create one so as to figure out the y-intercept. Line graphs may also represent trends in several quantities with time, by
employing several lines instead of merely one. Based on the kind of equation you’re addressing, the solution set may be a couple of points or a line, or it may also be an inequality all of which you
can graph as soon as you’ve identified a few points in the solution collection.
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sla_geamv.f −
subroutine SLA_GEAMV (TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SLA_GEAMV computes a matrix-vector product using a general matrix to calculate error bounds.
Function/Subroutine Documentation
subroutine SLA_GEAMV (integerTRANS, integerM, integerN, realALPHA, real, dimension( lda, * )A, integerLDA, real, dimension( * )X, integerINCX, realBETA, real, dimension( * )Y, integerINCY)
SLA_GEAMV computes a matrix-vector product using a general matrix to calculate error bounds.
SLA_GEAMV performs one of the matrix-vector operations
y := alpha*abs(A)*abs(x) + beta*abs(y),
or y := alpha*abs(A)**T*abs(x) + beta*abs(y),
where alpha and beta are scalars, x and y are vectors and A is an
m by n matrix.
This function is primarily used in calculating error bounds.
To protect against underflow during evaluation, components in
the resulting vector are perturbed away from zero by (N+1)
times the underflow threshold. To prevent unnecessarily large
errors for block-structure embedded in general matrices,
"symbolically" zero components are not perturbed. A zero
entry is considered "symbolic" if all multiplications involved
in computing that entry have at least one zero multiplicand.
TRANS is INTEGER
On entry, TRANS specifies the operation to be performed as
BLAS_NO_TRANS y := alpha*abs(A)*abs(x) + beta*abs(y)
BLAS_TRANS y := alpha*abs(A**T)*abs(x) + beta*abs(y)
BLAS_CONJ_TRANS y := alpha*abs(A**T)*abs(x) + beta*abs(y)
Unchanged on exit.
M is INTEGER
On entry, M specifies the number of rows of the matrix A.
M must be at least zero.
Unchanged on exit.
N is INTEGER
On entry, N specifies the number of columns of the matrix A.
N must be at least zero.
Unchanged on exit.
ALPHA is REAL
On entry, ALPHA specifies the scalar alpha.
Unchanged on exit.
A is REAL array of DIMENSION ( LDA, n )
Before entry, the leading m by n part of the array A must
contain the matrix of coefficients.
Unchanged on exit.
LDA is INTEGER
On entry, LDA specifies the first dimension of A as declared
in the calling (sub) program. LDA must be at least
max( 1, m ).
Unchanged on exit.
X is REAL array, dimension
( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = ’N’ or ’n’
and at least
( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
Before entry, the incremented array X must contain the
vector x.
Unchanged on exit.
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X. INCX must not be zero.
Unchanged on exit.
BETA is REAL
On entry, BETA specifies the scalar beta. When BETA is
supplied as zero then Y need not be set on input.
Unchanged on exit.
Y is REAL
Array of DIMENSION at least
( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = ’N’ or ’n’
and at least
( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
Before entry with BETA non-zero, the incremented array Y
must contain the vector y. On exit, Y is overwritten by the
updated vector y.
INCY is INTEGER
On entry, INCY specifies the increment for the elements of
Y. INCY must not be zero.
Unchanged on exit.
Level 2 Blas routine.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
September 2012
Definition at line 174 of file sla_geamv.f.
Generated automatically by Doxygen for LAPACK from the source code.
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Counting And Counting Principle | BimStudies.Com
Counting and Counting Principle
→ Definiton:
Counting refers to the process of determining the number of elements in a set or the number of possible outcomes in a certain scenario.
• It is used in Inventory Management, Probability, Statistical Analysis etc.
Combinatory is a branch of discrete mathematics that focuses on studying arrangements, selections, and combinations of objects from finite sets.
• It is used in Permutations and Arrangements, Poker and Card Games, Coding and Cryptography etc.
→ Basic Counting principles:
Sum Rule Principle
The Sum Rule Principle, also known as the Addition Principle, is a fundamental counting principle in combinatorics.
• It states that if there are “m” ways to do one task and “n” ways to do another task, both of which cannot be done simultaneously, then there are “m + n” ways to choose one of these tasks.
• When facing a choice between two mutually exclusive options (A and B), the total number of ways to make the choice is the sum of the number of ways for each option i.e (m+n).
For example:
• If you can wear either a redshirt(A) in 5 ways(m) or a blueshirt(B) in 3 ways(n), then the total number of ways to choose a shirt is(m+n)= 5 + 3 = 8 ways.
• If you can travel to a destination either by car in 10 ways or by train in 6 ways, then the total number of ways to travel is 10 + 6 = 16 ways.
Product Rule Principle
The Product Rule Principle, also known as the Multiplication Principle, is a fundamental counting principle in combinatorics.
• It states that if there are “m” ways to do one task and “n” ways to do another independent task, then there are “m * n” ways to perform both tasks together.
• If a task can be accomplished by performing operation A in “m” ways and operation B in “n” independent ways, then the total number of ways to perform both tasks is “m * n.”
For example:
• If you have 4 shirts and 3 pairs of pants, then the total number of outfits you can create by choosing one shirt and one pair of pants is 4 * 3 = 12 outfits.
• If you can order a pizza with 5 different toppings and a drink with 4 choices, then the total number of meal combinations is 5 * 4 = 20 combinations.
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The dySEM helps automate the process of scripting, fitting, and reporting on latent models of dyadic data via lavaan. The package was developed and used in the course of the research described in
Sakaluk, Fisher, & Kilshaw (2021).
The dySEM logo was designed by Lowell Deranleau (for logo design inquiries, email: agangofwolves@gmail.com).
You can install the released version of dySEM from CRAN with:
You can install the development version from GitHub with:
Current Functionality
The package currently provides functionality regarding the following types of latent dyadic data models:
1. Dyadic Confirmatory Factor Analysis
2. Latent Actor-Partner Interdependence Models (APIM)
3. Latent Common Fate Models (CFM)
4. Latent Bifactor Dyadic (Bi-Dy) Models
5. Observed Actor-Partner Interdependence (APIM)
Additional features currently include:
• Automated specification of invariance constraints for any model, including full indistinguishability
• Functions to assist with the specification of I-SAT Models and I-NULL Models for calibrated model fit indexes with indistinguishable dyad models
• Functions to assist with reproducible creation of path diagrams and tables of statistical output
• Functions to calculate supplemental statistical information (e.g., omega reliability, noninvariance effect sizes, corrected model fit indexes)
Future Functionality
Functionality targeted for future development of dySEM is tracked here. Current high-priority items include:
1. Longitudinal dyadic model scripting functions (e.g., curve of factors, common fate growth)
2. Latent dyadic response surface analysis scripting and visualization functions
3. Multi-group dyadic model scripting (e.g., comparing models from samples of heterosexual vs. LGBTQ+ dyads)
4. Covariate scripting and optionality
5. Improved ease of item selection in scraper functions
Please submit any feature requests via the dySEM issues page, using the “Wishlist for dySEM Package Development” tag.
If you are interested in collaborating on the development of dySEM, please contact Dr. Sakaluk.
dySEM Workflow
A dySEM workflow typically involves five steps, which are covered in-depth in the Overview vignette. Briefly, these steps include:
1. Import and wrangle data
2. Scrape variables from your data frame
3. Script your preferred model
4. Fit and Inspect your model via lavaan
5. Output statistical visualizations and/or tables
There are additional optional functions, as well, that help users to calculate certain additional quantitative values (e.g., reliability, corrected model fit indexes in models with indistinguishable
dyad members).
1. Import and wrangle data
Structural equation modeling (SEM) programs like lavaan require dyadic data to be in dyad structure data set, whereby each row contains the data for one dyad, with separate columns for each
observation made for each member of the dyad. For example:
2. Scrape variables from your data frame
The dySEM scrapers consider appropriately repetitiously named indicators as consisting of at least three distinct elements: stem, item, and partner. Delimiter characters (e.g., “.”, “_“) are
commonly–but not always–used to separate some/all of these elements.dySEM scrapers largely function by asking you to specify in what order the elements of variable names are ordered.
3. Script your preferred model
Scripter functions like scriptCFA typically require only three arguments to be specified:
1. the dvn object (e.g., from scrapeVarCross) to be used to script the model 1.arbitrary name(s) for the latent variable(s) you are modeling
2. the kind of parameter equality constraints that you wish to be imposed (if any)
This function returns a character object with lavaan compliant syntax for your chosen model, as well as exporting a reproducible .txt of the scripted model to a /scripts folder in your working
4. Fit and Inspect your model via lavaan
You can immediately pass any script(s) returned from a dySEM scripter to your preferred lavaan wrapper, with your estimator and missing data treatment of choice. For example:
At this point, the full arsenal of lavaan model-inspecting tools are at your disposal. For example:
5. Output statistical visualizations and/or tables
dySEM also contains functionality to help you quickly, correctly, and reproducibly generate output from your fitted model(s), in the forms of path diagrams and/or tables of statistical values. By
default these save to a temporary directory, but you can specify a directory of your choice by replacing tempdir() (e.g., with ".", which will place it in your current working directory).
Code of Conduct
Please note that the dySEM project is released with a Contributor Code of Conduct. By contributing to this project, you agree to abide by its terms.
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Based Algorithms for Mining Huge Data Sets
1 Introduction
2 Support Vector Machines in Classification and Regression - An Introduction
3 Iterative Single Data Algorithm for Kernel Machines from Huge Data Sets: Theory and Performace
4 Feature Reduction with Support Vector Machines and Application in DNA Microarray Analysis
5 Semi-supervised Learning and Applications
6 Unsupervised Learning by Principal and Independent Component Analysis
A Support Vector Machines
B Matlab Code for ISDA Classification
D Matlab Code for Conjugate Gradient Method with Box Constraints
E Uncorrelatedness and Independence
F Independent Component Analysis by Empirical Estimation of Score Functions i.e., Probability Density Functions
G SemiL User Guide
1 Introduction
1.1 An Overview of Machine Learning
1.2 Challenges in Machine Learning
1.2.1 Solving Large-Scale SVMs
1.2.2 Feature Reduction with Support Vector Machines
1.2.3 Graph-Based Semi-supervised Learning Algorithms
1.2.4 Unsupervised Learning Based on Principle of Redundancy Reduction
2 Support Vector Machines in Classification and Regression - An Introduction
2.1 Basics of Learning from Data
2.2 Support Vector Machines in Classification and Regression
2.2.1 Linear Maximal Margin Classifier for Linearly Separable Data
2.2.2 Linear Soft Margin Classifier for Overlapping Classes
2.2.3 The Nonlinear SVMs Classifier
2.2.4 Regression by Support Vector Machines
2.3 Implementation Issues
back to top
3 Iterative Single Data Algorithm for Kernel Machines from Huge Data Sets: Theory and Performance
3.1 Introduction
3.2 Iterative Single Data Algorithm for Positive Definite Kernels without Bias Term b
3.2.1 Kernel AdaTron in Classification
3.2.2 SMO without Bias Term b in Classification
3.2.3 Kernel AdaTron in Regression
3.2.4 SMO without Bias Term b in Regression
3.2.5 The Coordinate Ascent Based Learning for Nonlinear Classification and Regression Tasks
3.2.6 Discussion on ISDA Without a Bias Term b
3.3 Iterative Single Data Algorithm with an Explicit Bias Term b
3.3.1 Iterative Single Data Algorithm for SVMs Classification with a Bias Term b
3.4 Performance of the Iterative Single Data Algorithm and Comparisons
3.5 Implementation Issues
3.5.1 Working-set Selection and Shrinking of ISDA for Classification
3.5.2 Computation of the Kernel Matrix and Caching of ISDA for Classification
3.5.3 Implementation Details of ISDA for Regression
3.6 Conclusions
4 Feature Reduction with Support Vector Machines and Application in DNA Microarray Analysis
4.1 Introduction
4.2 Basics of Microarray Technology
4.3 Some Prior Work
4.3.1 Recursive Feature Elimination with Support Vector Machines
4.3.2 Selection Bias and How to Avoid It
4.4 Influence of the Penalty Parameter C in RFE-SVMs
4.5 Gene Selection for the Colon Cancer and the Lymphoma Data Sets
4.5.1 Results for Various C Parameters
4.5.2 Simulation Results with Different Preprocessing Procedures
4.6 Comparison between RFE-SVMs and the Nearest Shrunken Centroid Method
4.6.1 Basic Concept of Nearest Shrunken Centroid Method
4.6.2 Results on the Colon Cancer Data Set and the Lymphoma Data Set
4.7 Comparison of Genes’ Ranking with Different Algorithms
4.8 Conclusions
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5 Semi-supervised Learning and Applications
5.1 Introduction
5.2 Gaussian Random Fields Model and Consistency Method
5.2.1 Gaussian Random Fields Model
5.2.2 Global Consistency Model
5.2.3 Random Walks on Graph
5.3 An Investigation of the Effect of Unbalanced labeled Data on CM and GRFM Algorithms
5.3.1 Background and Test Settings
5.3.2 Results on the Rec Data Set
5.3.3 Possible Theoretical Explanations on the Effect of Unbalanced Labeled Data
5.4 Classifier Output Normalization: A Novel Decision Rule for Semi-supervised Learning Algorithm
5.5 Performance Comparison of Semi-supervised Learning Algorithms
5.5.1 Low Density Separation: Integration of Graph-Based Distances and rTSVM
5.5.2 Combining Graph-Based Distance with Manifold Approaches
5.5.3 Test Data Sets
5.5.4 Performance Comparison Between the LDS and the Manifold Approaches
5.5.5 Normalizatioin Steps and the Effect of σ
5.6 Implementation of the Manifold Approaches
5.6.1 Variants of the Manifold Approaches Implemented in the Software Package SemiL
5.6.2 Implementation Details of SemiL
5.6.3 Conjugate Gradient Method with Box Constraints
5.6.4 Simulation Results on the MNIST Data Set
5.7 An Overview of Text Classification
5.8 Conclusions
6 Unsupervised Learning by Principal and Independent Component Analysis
6.1 Principal Component Analysis
6.2 Independent Component Analysis
6.3 Concluding Remarks
A Support Vector Machines
A.1 L2 Soft Margin Classifier
A.2 L2 Soft Regressor
A.3 Geometry and the Margin
B Matlab Code for ISDA Classification
C Matlab Code for ISDA Regression
D Matlab Code for Conjugate Gradient Method with Box Constraints
E Uncorrelatedness and Independence
F Independent Component Analysis by Empirical Estimation of Score Functions i.e., Probability Density Functions
G SemiL User Guide
G.1 Installation
G.2 Input Data Format
G.2.1 Raw Data Format
G.3 Getting Started
G.3.1 Design Stage
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Which calculator passes the first test?
First, (-) is the change sign key and (x^2) is the square key.
If you hit (-) 2 x (-) 2 = the answer is 4 on most calculators. But if you hit (-) 2 (x^2) = you probably end up with negative 4. As far as I know, the only non-HP calculator that gives the correct
answer is this Sharp .
Do you know about any other non-HP, passing this test?
What about the algebraic HP10s and SmartCalc300s?
12-01-2010, 02:34 PM
Are you suggesting that HP calculators handle that wrong or that they all do it correctly? My 32SII and 30B both return 4 not -4.
12-01-2010, 02:56 PM
I believe -2^2 is -4. Unless you mean (-2)^2, which is 4. It's not clear from what you wrote, what you expect the correct answer to be.
12-01-2010, 04:22 PM
All my HP:s gives the correct answer 4, but most calcs gives the wrong answer -4. My question is: What other calcs gives the right answer. Negative 2 squared is +4.
12-01-2010, 04:40 PM
Both wolfram alpha and google disagree. . .
Just saying. :-)
12-01-2010, 05:07 PM
Do you suggest that all HP are faulty? :-)
Of course, -2^2 is negative 4. But that is not the issue here. I want to know what calcs I can recommend to my students. They receive all kind of random answers when they use the (-) key on their
calcs. The Sharp D.A.L is ok in this respect.
HP is out of the question here in Sweden. /Tommy
12-02-2010, 06:14 AM
But when entering on Wolfram Alpha, you use the minus key. Is there an equivalent "change sign"? When using the "change sign" on a calculator, are you not making the negative a part of the
characteristic of the number? When using the minus sign, it is equivalent to "0 - number", in which case "0 - number^2" indeed results in a negative answer. Thus the Wolfram Alpha answer is correct,
as you are entering a minus, not a "change sign".
12-01-2010, 04:58 PM
My HP 35s gives "Syntax Error" when I press +/- 2 x^2 ENTER in ALG mode.
I can get either -4 or +4, depending on whether I calculate -SQ(2) or SQ(-2).
12-02-2010, 06:16 AM
Restricting the input to -SQ(2) or SQ(-2) seems a clever way to avoid the controversy :)
The 33s in ALG mode: 5, +/-, x^2, ENTER yields:
( ^-5)^2=
and MINUS, 5, x^2, ENTER yields:
Exactly what I would expect ;)
Edited: 2 Dec 2010, 9:41 a.m.
12-01-2010, 05:14 PM
Quote: First, (-) is the change sign key and (x2) is the square key.
If you hit (-) 2 x (-) 2 =
I am still not clear what you mean. Are you talking ALG or RPN calcs?
Since you include the = key, I might assume ALG. But I don't understand your notation. You stated that (-) represents the CHS or +/- key. But then in your example, the CHS key precedes the argument,
while to my knowledge it always follows the argument, whether ALG or RPN model.
And does the "x" in your example represent the multiplication key, or the quantity x? I don't see the square appearing in your example.
12-01-2010, 05:28 PM
4 is the correct answer only if the expression is (-2)^2.
-2^2 is (correctly) seen by most calculators as "the negative of 2 squared", or even "negative 1 times 2 squared", which is -4, as exponentiation takes precedence over multiplication, and there no
parenthesis to dictate otherwise.
On my 50G in algebraic mode, -2^2 = -4 (correct), and (-2)^2 = 4 (also correct), and finally, SQ(-2) = 4 (correct).
I must go now and quickly change my 50G back to RPN mode!!
Best regards, Hal.
12-01-2010, 05:56 PM
The good old math standard precedence is "exponentiation > multiplication > addition" (in German: "hoch vor Punkt vor Strich" - we're more concise here :) ), so -2^2 = -4 (if no parentheses are set)
regardless of whether you see the unary minus as a multiplication by -1 or something like 0 - ...
(-2)^2 is a different cup of tea :)
12-01-2010, 08:18 PM
English is even more concise:
P E M D A S
Multiplication & Division
Addition & Subtraction
I looked at this very problem in detail a few years ago. For most calculators and calculator users, the problem originates with a misunderstanding of the way each machine works and especially what
the CHS or +/- key does, as compared to the - key.
Some calculators parse a command line. Others operate and never parse. An RPN machine operates, an RPL machine parses. Some ALG or SemiALG machines parse some things but operate with others.
**Edit: note that RPL parses when using an Algebraic Object delimited by single quotations. It operates on stack items or a valid command line object**
The traditional "Alg" machine (which is better described as infix arithmetic with postfix functions) is an operator not a parser. DAL and SVPAM etc are parsers.
Then there are machines which have documented features which may be confusing. The 32sii equation list recognizes a "unary minus" with precedence over exponentiation, but it has a bug when it is in
the initial position! See Craig Finseth's HPDatabase. The 33s and 35s eliminated this confusing unary minus feature and they are correctly documented.
Then there is the confusion of the fact that some machines have more than one mode, where one mode has a line interpreter but the other operates (e.g. 17bii, 27s, 32sii, 33s, 35s). Some machines will
allow the +/- key to work as an operator only, while others allow it to function as a toggling character key. Some machines treat both the - and the +/- as the same thing, others as different things.
And then some machines such as the 35s have a "high" minus sign when you push the +/- and a low one with the - even if there is no functional difference....
It is very confusing and totally MACHINE SPECIFIC. I haven't found any machines that are perfect except for the 48G series and descendants, and the pure RPN machines. I haven't looked at the latest
Sharp/Casio/Ti so it may be the case that some of them are clearcut now.
In RPN, there is never an issue, as there is NO PRECEDENCE because there is only operation, not line interpretation/parsing.
[edit: small grammatical error and incorrect reference]
Edited: 8 Dec 2010, 8:18 a.m. after one or more responses were posted
12-02-2010, 12:38 AM
Thank you, Bill, for straighten this out for me! So my question remains: are there any other clear cut non-RPN calcs out there? /Tommy
12-02-2010, 07:26 AM
Sure. The 20s is very clear cut. I expect that the old 21s and 22s were also clear.
I can't speak to the non-hp because I don't have any modern ones in front of me at the moment.
I find the 27s clear enough, even though it has two modes.
The biggest problem is a lack of care in the use of the tools. When was the last time you saw a student follow RTFM41?
Edited: 2 Dec 2010, 7:28 a.m.
12-02-2010, 08:14 AM
Quote: And then some machines such as the 35s have a "high" minus sign when you push the +/- and a low one with the - even if there is no functional difference....
In ALG mode I get a SYNTAX ERROR when using the minus-key:
On the other hand I get the expected result with the +/- key:
So to me there is a functional difference.
Quote: In RPN, there is never an issue, as there is NO PRECEDENCE because there is only operation, not line interpretation/parsing.
This is not quiet true as entering a number changes the behaviour of the CHS key. Just think of what happens after the EEX key. There's a peculiar behavior in the HP-35: when the key following the
CHS is a number key the negative sign is considered part of this number. So the following example will give 3 instead of -3 what probably most of us would expext:
In addition to that the stack doesn't contain 5 in the Y-register. So you could enter negative numbers as you read them. However I think it was a wise decision to change that in later models.
12-02-2010, 08:55 AM
Hi Thomas,
Thanks for the reply. I don't have an original HP 35--maybe it is a good thing as that would confuse me! (I am a voyager and forward guy).
I think I am mixing up the 32sii 33s and 35s wrt the high minus. Indeed the 35s is a completely different animal than even the 33s in how it handles entry and interpretation. As I remember, when in
ALG mode, it is a parsing machine, whereas the 33s is an operation machine but with the added feature of a history reporting line that shows how it all comes together. (all three are of course
parsers within the equation list mode). Later I'll pull the 33s out and have a look. Unfortunately one of my kids lost the 35s so I can't have a look at it!
Edited: 2 Dec 2010, 8:58 a.m.
12-02-2010, 09:36 AM
I can confirm this "feature" of the original 35. So I looked what happened to it:
1. HP-35: 5 ENTER CHS 2 + results in 3.
2. HP-45: 5 ENTER CHS 2 + results in 7. So does 55, 65, 80, 21, 22.
3. HP-91: 5 ENTER CHS 2 + results in -3. And so it stays until the 42S. And the "new" calcs including the RPL-machines keep it this way.
Interesting :)
Edited to add some Woodstocks and the HP-91 as first HP calculating the way we are expecting it today. Anybody having some more Woodstocks at hand for checking?
Edited: 2 Dec 2010, 10:07 a.m.
12-02-2010, 10:47 AM
Interesting. Is the behavior the same if the steps are programmed, rather than from the keyboard?
(I know I'm being lazy, but I also don't have most of the models to try it out.)
12-02-2010, 11:22 AM
None of the above calculators are programmable.
12-02-2010, 04:19 PM
Rubbish: the 55 and 65 are. But right now I don't have access to my calcs, so anyone may verify this instead d:-)
12-02-2010, 04:57 PM
I was talking about the three calculators listed, Hp35, Hp45 and the Hp 91. The calculators mentioned were not the focus of the thread that I responded to.
Of course the mentioned Hp55, Hp65 and Hp42s are all programmable.
12-02-2010, 12:04 PM
Quote: HP-45: 5 ENTER CHS 2 + results in 7. So does 55, 65, 80, 21, 22.
It seems the stack isn't lifted after CHS thus -5 is overwritten by the following 2. I consider that a bug: the negative sign shouldn't be silently ignored. Was this already known before?
12-03-2010, 01:42 PM
Quote: It seems the stack isn't lifted after CHS thus -5 is overwritten by the following 2.
That's my guess, too. Still I don't have any information about the history of this "feature" (see my previous post) beyond the fact that it was dropped apparently early in 1976. Anyone of the old
experts knowing something about the circumstances?
12-03-2010, 01:58 PM
It is a bug. I remember reading about this ages and ages ago, and I thought I remembered seeing this behavior on the HP-25 as well. As luck would have it, I'm working from home at the moment, so I
could get my HP-25 out of its drawer, and I confirmed, 5 ENTER CHS 2 + returns 7, both from the keyboard and in a program.
12-03-2010, 04:25 PM
Thanks, Thomas. Your result fits well in the line since the HP-25 was made before the HP-22 according to the museum. Does anybody have a working HP-27 for checking?
12-03-2010, 06:31 PM
You're welcome! For what it's worth, my HP-25 has a serial number that starts with 1605, i.e. made in February 1976.
12-05-2010, 11:34 AM
OK, I have the results of some more models now. So the updated table looks like this:
1. HP-35 until and including V4: 5 ENTER CHS 2 + results in 3. So this looks like a prefix CHS being perfectly legal in numeric input of the mantissa. Checked by CLR CHS 5 ENTER CHS 2 + resulting in
-7, q.e.d. In exponents, BTW, pre- and postfix CHS are allowed until today.
2. HP-45: 5 ENTER CHS 2 + results in 7. So does 55, 65, 80, 21, 22, 25, 27, 25C. But 5 ENTER CHS + returns 0 as expected. Seems CHS is executed in any case but is undone if a *first digit* follows:
display on a 25C or a 32E (see below):
CLX 0.0 0.0
CHS -0.0 0.0
1 1. 1.
CHS -1. -1.
2 -12. -12.
CHS 12. 12.
3 123. 123.
ENTER 123.0 123.0
CHS -123.0 -123.0
4 4. 4.
+ 127.0 -119.0
Please note CHS does not terminate the input sequence in either case. So the last CHS on the 25C belongs to the input of 4 and is thus undone when 4 is entered!
3. HP-91: 5 ENTER CHS 2 + results in -3. So CHS is now recognized as an operation as soon as an input is completed (e.g. by ENTER) - thus it operates on 5 in X, turning it to -5, and 2 is added to
this as we expect it. CHS will not work after CLX, it needs at least 1 digit put in but may be used arbitrary times during numeric input of one number as before. - The same result is returned by
the 67, the Spices, Voyagers, and Pioneers. And also the "new" calcs including the RPL-machines keep it this way.
For sake of completeness: Has anybody access to a working 29C and/or 19C? Could you please check it and report the results?
I don't remember having read anything about this change of concepts here for some years - so let me ask: is this known for longer already?
TIA for enlightenment(s).
Edited: 5 Dec 2010, 11:39 a.m.
12-05-2010, 03:01 PM
Quote: So the last CHS on the 25C belongs to the input of 4 and is thus undone when 4 is entered!
I don't agree with you here: instead I assume that -123 is overwritten by 4 since CHS doesn't enable the stack lift.
The CHS has two different modes of operation: either as part of the input of a number or to change the sign of the X-register. In the HP-35 it isn't clear which mode to use until after the next
key-stroke. So first it changes the sign of X. But if the next key is a number, this change of X is reverted and instead the negative sign is considered part of the number. It is clear that CHS can't
enable the stack lift. Otherwise the following two sequences leave a different number of elements on the stack:
5 ENTER CHS 2 +
Y: 5
X: 3
5 ENTER 2 +
X: 7
So I think this magic handling of the CHS was removed, but enabling the stack lift wasn't added.
In addition to your tests I suggest the following variants:
1. make sure the stack lift is enabled, e. g. by performing an operation like +
2. use RCL instead of entering a number after CHS
Store 2 in the register (2 STO) and make sure the stack is empty:
+ ENTER + ENTER
CHS CHS CHS CHS
RCL RCL 2 2
+ + + +
HP-35: -3 7 3 3
HP-45: -3 7 -3 7
HP-25: -3 -3 -3 7
HP-91: -3 -3 -3 -3
I have verified these results only with online emulators of the HP-35/45/25. The results for the HP-91 are just my expectations.
Best regards
[Added the results for the HP-25.]
Edited: 6 Dec 2010, 9:36 a.m. after one or more responses were posted
12-05-2010, 05:30 PM
From Walter's original statement:
1. HP-35: 5 ENTER CHS 2 + results in 3.
2. HP-45: 5 ENTER CHS 2 + results in 7. So does 55, 65, 80, 21, 22.
3. HP-91: 5 ENTER CHS 2 + results in -3. And so it stays until the 42S. And the "new" calcs including the RPL-machines keep it this way.
From Thomas' post:
Quote: The CHS has two different (modes of operation): either as part of the input of a number or to change the sign of the X-register.
Yes, that's the crux of the matter. I believe that HP, in those early years, was still working out the important nuances of the CHS function. Its implementations on the HP-35 and HP-45 were flawed,
in that user input could sometimes get completely changed during data entry.
Characteristics of proper CHS functionality in RPN:
• When CHS is utilized to input a negative number, prior entry of at least one digit or the radix point should be required. Saturn-processor calculators (not the handheld computers, such as the
HP-71B) will accept 0 or the radix point as that initial entry; prior models will not.
• Upon using CHS to negate a final result left in the x-register by ENTER or other means, stack lift should always be (re-)enabled.
The HP-91 and most subsequent models implemented CHS correctly. A few early models on the HP-45's software-development track retained its flawed CHS functionality. Also, a subtle bug got slipped into
many (but not all) Spice-series and Voyager-series models:
HP-15C bug -- CHS and Stack Lift
-- Karl
Clarified several statements, based in part on new information.
Edited: 6 Dec 2010, 10:29 p.m. after one or more responses were posted
12-05-2010, 07:12 PM
Quote: The HP-91 and subsequent models implemented CHS correctly
Not quite. The timeline is as follows according to HPDATAbase:
22 .. .. .. .. .. .. .. 27 .. 25C (all these return 7)
91 .. .. .. 67 .. .. .. .. .. .. .. .. .. .. .. 29C etc. return -3
So there was some overlap. Seems it took HP significantly more than 4 months to implement the new CHS-handling in new Woodstock models, though it was "only software".
Edited: 7 Dec 2010, 12:45 p.m.
12-09-2010, 02:00 AM
FYI, I've checked your table using real calcs and almost everything met your expectations and emulator output, but the results on a real 25C are the same as on a 45, i.e. -3 7 -3 7.
12-09-2010, 02:32 AM
Hi Walter
Thanks for taking the time to check the results. I've used the HP-25 (Java) simulator written by Larry Leinweber. I should have realized that it's not an emulation.
12-05-2010, 05:19 PM
-3 on an HP-19C and 29C
12-05-2010, 07:14 PM
Thank you, Bob :)
12-02-2010, 10:31 AM
Ha ha. A rare example of concision in German? ;-))
12-01-2010, 11:45 PM
Quote:All my HP:s gives the correct answer 4, but most calcs gives the wrong answer -4.
The correct answer to what? -2^2=-4, and (-2)^2=4. This is the convention that's universally used in mathematics textbooks, and yet you seem to imply that -2^2=-4 is "wrong". Given that you've also
indicated that you actually teach this stuff, that makes me think that you really, really need to do your homework, before looking for calculators whose behavior matches your mis-informed idea of
mathematical notation.
BTW, as Hugh mentioned, this was discussed on this forum not too long ago. See here.
If you really want to teach your students well, teach them how the notation works as it is used in textbooks, and, if this actually comes up in your course, also mention how certain calculators and
programming languages differ from the textbook standard.
Edited: 2 Dec 2010, 12:18 a.m.
12-02-2010, 07:21 AM
Sorry,Thomas, but you did not understand my question. /Tommy
12-02-2010, 11:01 AM
Quote: Sorry,Thomas, but you did not understand my question. /Tommy
I did not either. Perhaps several respondents did not. The way it was phrased makes it difficult to understand, IMO.
Perhaps you could restate it?
12-02-2010, 05:34 AM
Quote: I believe -2^2 is -4. Unless you mean (-2)^2, which is 4. It's not clear from what you wrote, what you expect the correct answer to be.
I disagree, if I have -2 on the display, and square it, it is to be interpreted as (-2)^2, if I wanted -(2^2), I would enter 2^2, then (+/-). The negative is part of the number (I am talking about
having used the change sign, not minus), I did not enter "0 - 2^2", in which case the 2 is separate of the minus. What calculator when squaring a real number only squares the fractional part?, or
when given a proper fraction only squares the fractional part?
By the way, now that your calculator has -4 as the answer for -2^2, press the ^2 again and see what you get. -16?, NO most will give 16 as they will do ANS^2 - so now it does see the negative as part
of the number. A bit inconsistent I'd say.
Further Edit:
As far as I recall all my LED/VFD scientific calculators give + when squaring a negative number. Somewhere along the line when DAL, VPAM etc. calculators came out someone thought it "good" to let a
negative number (entered with change sign) be the same as "0 - the number" and thus squaring it results in a negative answer. Now many people in the world are justifying this?
Edited: 2 Dec 2010, 6:19 a.m.
12-02-2010, 07:14 AM
You are right, ANS^2 is positve 16. Intresting... /Tommy
12-02-2010, 07:22 AM
Quite logical, since the actual numeric value of the variable ANS is squared.
12-02-2010, 07:40 AM
That is logical. But the intresting part is that you can not rely on the results.
12-02-2010, 09:13 AM
Quote: But the intresting part is that you can not rely on the results.
Why? You just have to know what you're doing. As Bill explained above, there are different ways calculators handle such problems. A major advantage of RPN is it acts in a very consistent way: number
first, operation second - no need to care for precedences etc.
So 17 +/- x^2 will always result in 289. OTOH 17 x^2 +/- equals -289. The world can be so simple ;)
Exception (on my 42S): 17 +/- E +/- 2 will result in 0,17 as well as 17 +/- E 2 +/- while the second input is more in line with RPN principles.
Side track: Remember STO12 being an operation in RPN, so 17 +/- STO12 will store -17 in register 12. IMHO one disadvantage of RPL is they got these things mixed up so you have to enter 17 +/- ENTER
12 STO to reach the same (if numeric registers were allowed). End of side track.
12-02-2010, 07:31 AM
Note my discussion of Operation versus Parse. Answer^2 is an operation on the answer.
12-02-2010, 08:46 AM
My point is exactly that a negative number created with a "change sign" should be operated on in it's entirety and not parsed.
12-02-2010, 08:52 AM
Yes, but this is of course machine specific. If the function of the +/- key is for entry of characters into the line, then it will be line interepreted. See for instance the 48G
12-02-2010, 07:36 AM
Quote: I disagree, if I have -2 on the display, and square it, it is to be interpreted as (-2)^2
Note my discussion of line interpreter/parser versus operation. If you have -2 on the display and you operate on it, of course you get +4. That is what an RPN machine will do, or an older "algebraic"
because even the latter is postfix and operation-based (is the same as RPN). Newer machines line-line interpret *expressions* rather than operate on numerical entities. However, the "answer" function
operates on the answer.
This is always, first and foremost, a problem of not following RTFM41 rather than a machine defect or bug. However I do have opinions regarding good versus bad design, but even the "bad" designs work
correctly if one bothers to learn how they work.
12-02-2010, 08:53 AM
Quote: ... but even the "bad" designs work correctly if one bothers to learn how they work.
I would rather say that "bad" designs work as the designer intended, but not necessarily correctly. Of course correct being a bit subjective in the current topic :).
Quote: This is always, first and foremost, a problem of not following RTFM41...
A case of "know your enemy"? ;)
Edited: 2 Dec 2010, 9:06 a.m.
12-02-2010, 03:16 PM
Yes, if you mean, "I've met the enemy, and he is me."
12-03-2010, 06:25 AM
I was thinking of "know what you're using", but we indeed can be our own worst enemy at times :).
12-03-2010, 06:34 AM
HA! Even Microsoft agrees with ME :P
Try on Windows calculator:
2, +/-, x^2 = ; the answer is +4
now try
-, 2, x^2 = the answer is -4
well, for whatever credit one can take from microsoft *grin*
12-01-2010, 03:11 PM
this discussion came up a while back. it turns out that the operator precedence of unary minus is different for different manufacturers. we even found differences amongst different casio models.
the problem is compounded when you allow two minus buttons, ie regular operator minus and (-).
12-01-2010, 11:10 PM
If you have a calculator, you should know how it works, so you can make it give you the answer you think it should. That's the only issue here.
Let's put this a different way. Sin30 (in degrees) is 0.5.
But do you have a calculator that lets you enter 30 Sin, you might get 0.5.
But what's 30 Sin? That's 30 times the Sin of nothing! It's meaningless!
That seems kind of like what you're saying.
12-02-2010, 07:43 AM
That's right. GIGO. 30 SIN is postfix. sin(30) is line interpreted. sin 30 = is used in an infix machine, for instance, the old SHARP EL 5020. Of course the manuals don't necessarily bother to
explain the inner workings, especially the cheap Non-HP designed machines with little fold out manuals. Even HP manuals won't always discuss the parse versus operation aspect--but note that until
recently, no calculator parsed expressions!
By recently, I mean early 90s or late 80s. I am showing my age :-D
Edited: 2 Dec 2010, 7:44 a.m.
12-02-2010, 09:28 AM
Manufacturers seem to disagree about the interpretation of (-)2^2 even within their own product range:
Sharp: None of my algebraic Sharp calculators returns +4. I do have the EL-9200/9300/9600 and an EL-5120. Are you sure, the EL-520WBBK does?
Casio: None of my algebraic Casio calculators returns 4. I tried with a BASIC computer (PC-1262), some newer algebraics, and various graphics machines (old and recent).
The Canon F-300P returns +4. Likewise the TI Galaxy 67, while all the TI graphics calculators, including the CAS machines, and also the TI-34 multiview return -4.
Now to HP: The 30s and the 9g both return -4. Same for the 38G. The RPL machines lack the postfix ^2 operator as has already been mentioned. -2^2 returns -4.
Return -4 seems to be the rule, +4 the exception.
12-02-2010, 10:17 AM
Sorry, I disagree about the RPL-machines. You'll find x^2 on e.g. the HP-48SX easily (gold shifted square root). And it behaves mathematically correct, i.e. 2 +/- x^2 equals 4.
12-02-2010, 03:02 PM
I should have elaborated. When you do that, using the stack, it is an operation on the -2, which is in the command line.
If you do this:
'-2^2' ENTER EVAL
Then it is parsed.
12-02-2010, 01:50 PM
This is what my question is all about
I am not sure about the EL-520WBBK, but the EL-506WBBK does return +4. Both are D.A.L. (Direct Algebraic Logic).
This one also returns +4: Karce KC-156
Sorry about all confusion I created. Hopefully HP returns with a cheap scientific RPN calculator in the future. /Tommy
12-02-2010, 12:56 PM
Tommy, why are you expecting RPN and non-RPN machines to give the same result for the same sequence of keystrokes?
(-) 2 x^2 in algebraic is calculating the same result as 2 x^2 (-) in RPN. (-) 2 x^2 in RPN is ( (-) 2 ) x^2 in algebraic. I'll leave it to you to count the keystrokes and make any decisions about
12-03-2010, 11:58 AM
Hi Tommy,
Re-reading your question in light of all the commentary, please read the following.
The algebraic expression -2^2 is equal to -4. This is a fact of uniform notation. If you think this is incorrect, then that would be a problem. However, what I think is leading you to believing that
calculators are giving incorrect answers is that you are used to the older style of machines, where you *operate* on what is displayed. In this case, the problem you are actually trying to figure out
is the following:
An older machine such as a 1970s Ti SR70 or an HP 20s, or a 27s or an HP 33s set in ALG, or any RPN machine will work identically:
This type of machine OPERATES on the x-register. A currently available HP is used in the example below:
expression: (-2)^2
keystrokes display
33s ALG mode
2 2_
+/- -2_
x^2 4
[note how cool this machine is in ALG mode. It uses the
old-style post-fix functions operating on numeric line
entries, but it displays the proper complete algebraic
notation in the upper line. Pretty damn cool!]
If you are using a more "modern" machine with a "textbook" type of interface, you will have to type in the expression correctly according to standard notation. (note however that there is variation
from machine to machine even here--some allow implicit multiplication, some don't follow standard rules exactly etc [ex: equation list of 32sii which has a unary minus].
An example of a properly functioning PARSED LINE INTERPRETER is found in the currently available HP 33s Equation List:
( (
- {or +/-} (-
2 ( 2_
) (-2)_
y^x (-2)^_
2 (-2)^ 2_
ENTER (-2)^2
ENTER 4
x^2 SQ(_
+/- {or -} SQ(-_
2 SQ(- 2_
ENTER SQ(-2
ENTER 4
[note that you don't have to close the parenthesis here because the line interpreter will implicitly close it. Some machines would error at this. Every machine is different and you have to Read the
F$#$#$#@ manual :-) ]
I hope this helps you.
Best regards,
Edited: 3 Dec 2010, 12:04 p.m.
12-03-2010, 04:35 PM
Quote: Every machine is different and you have to Read the F$#$#$#@ manual :-)
Indeed. The Sharp EL-W506 Writeview manual states:
Priority Levels in Calculation
1. Fractions
2. Angle prefix, Engineering prefixes
3. Functions preceded by argument (e.g. x^-1, x^2, n!, etc)
4. y^x, ^x/¯
5. Implied multiplication of a memory value (2Y, etc.)
6. Functions followed by their argument (sin, cos, (-), etc.)
Note: (-) = change sign.
So one can see that as the x^2 is defined as taking precedence over (-), thus entering: (-), 2, x^2 will be evaluated as 2^2=4, (-), = -4.
Whether we like it or not, it does what the manufacturer intended and specified.
12-03-2010, 07:27 PM
Whether we like it or not, it does what the manufacturer intended and specified.
or failed to implement and specified
12-03-2010, 07:44 PM
Hi Bill,
Thanks for this thorough examination. But this is all about HP calcs.
I know, you must not quote yourself, but I do quote my message #1:
“Do you know about any other non-HP, passing this test?”
I am familiar with RPN and with HP calcs. My favorite is 15C, which I use on daily basis. People round me are using, in my opinion, not-so-good TI/Casio/Sharp calculators.
Since I know that nobody actually reads the manual, besides perhaps members of this forum, I was looking for a decent, fairly cheap, algebraic calculator. And by decent, I mean reliable, consistent
and predictable. The 35s would be ok, but it is a little bit too expensive (for the students).
Yes, you are correct. I want to operate on the number visible on the screen. If I see a negative number on the screen and the x^2 key is hit, I expect a positive number. Always! Not only when the
number is an answer to a calculation, but also when the number is entered as (-)2. (on most non-HP calcs the change-sign-key is hit before the number).
A calculator should be an aid in performing calculations. But if you need to RTFM to be sure about what happens, then the calculator is an obstacle. I would never enter a negative number in this way,
but the students rely so heavily on their calcs so they do. The advice is: If you want to stay in control, always use parenthesis!
I was quite chocked when I realized how the x^2 key worked. When I showed the phenomenon to my colleagues, no one was aware of it. Were you?
So I began some testing. A negative number multiplied by a negative number, always gave a positive number. A negative number squared, always gave a negative number (on non-HP calcs). This is very
inconsistent IMO. Negative numbers are treated correctly except when used in co-occurrence with the x^2 key. The symbol on the screen is sometimes different from a normal minus sign. The behavior is
erroneous! Compare the following two examples:
2+-2^2=-2 that is 2 plus minus 2^2 correct
2+(-)2^2=-2 that is 2 plus negative 2 ^2 incorrect
Every time I had an opportunity to test a new calculator, I performed this “first test”. After a while I found the Sharp D.A.L. If there is one decent calculator, there might be several. That’s the
reason for this topic.
Today I noticed that my Android RealCalc behaves correctly in both RPN and ALG mode. Besides that, no decent (non-RPN) calc have come to my knowledge.
HP20s: not on the market any longer.
HP30b: inconsistent answers in this topic.
HP10s and HP SmartCalc 300s: no answer
Canon F-300P and TI Galaxy 67: too obscure I guess
Hopefully, this long letter clarifies my statements. I felt a need to do so since I got some comments in bold. And yes, I do know about precedence. I am also familiar with the concepts of
subtraction, change sign and negative numbers. Further more, I recognize a crappy calculator when I see one.
Cheers /Tommy
12-03-2010, 09:01 PM
Hi Tommy:
"The 35s would be ok, but it is a little bit too expensive (for the students)"
I have to laugh at this. I guess it is priorities. IF you have to spend as little as it costs for one Hamburger and fries, then, yes, I guess $45 bucks or so is too much. Same kid thinks nothing of
blowing $10 a week on iTunes.
And people don't read the manuals because they are LAZY. I read my 11c manual cover to cover when I got it in 1982, and so did my brother even though it wasn't his. IT was fun to know how it worked.
I think the 33s is a better choice than the 35s actually. It is more user friendly and has much better handling of rectangular to polar, and base arithmetic. And it is, as I showed, a postfix
function/infix arithmetic and *operates* on the displayed number. The 35S does NOT do this the same way. The 35S is an infix machine (except for factorial.)
You should download the manuals for the 33s and the 35s. Look at page C-1 in each one. I lost my 35s so I can't test it, but there is a "unary minus" ahead of multiplication on the 35S. I just don't
remember how it works.
All told, the 35s and 33s are totally different approaches in their Algebraic modes (as distinct from their equation modes, which are essentially identical (see p 6-13 through 6-15 of each manual.
They have a unary minus ahead of multiplication, but this is to handle the issue such as -a X -b, which is I believe what is also the unary minus treatment in the 35s ALG mode). [The 32sii equation
list had the unary minus given precedence over taking powers, which was the real problem there. Both the 33s and 35s have rectified this by moving the unary minus down.]
Fortunately, I haven't much experience with current non-HP machines. I have 15 or 20 year old Sharps and a TI so that won't help you....
Edited: 3 Dec 2010, 10:05 p.m.
12-04-2010, 04:14 AM
I totally agree with you. I enjoy reading the manuals.
But in the case of money, the 35s and 33 are in range of what a TI graphical calc costs. And you can not compete with TI there. I need a cheaper one. I am using Excel and GeoGebra. But TI is 99%
mandatory in Sweden. All textbooks have “how to do this and that on your TI”. Sorry to say, HP has lost the battle, but I still fight for good calculators.
Back to the main issue:
(-)2^2=-2 that is negative 2 ^2 incorrect
(-)2= ^2= 2 that is negative 2 = Ans ^2 correct
The only difference is an extra =. In this case the = is an “almost equal sign” You cannot argue about an “equal sign”, or can you? ;-)
Best regards
12-04-2010, 06:36 AM
Quote: You cannot argue about an “equal sign”, or can you?
On calculators beyond trivial calculations, "equal" means always "almost equal" within the tolerance (or accuracy) of the calculator. Your example, however, counts to the mathematically trivial
applications, so I agree "almost equal" being not necessary there.
Our American friends tend to forget sometimes the price policy of HP in the countries beyond God's own. I hate to repeat this, but e.g. a new 35S sells for 50 € (Euro) in Germany at least - look here
for Hewlett Packard and enjoy your location since 50 € "almost equal" 66 US$ today :(
12-04-2010, 09:38 AM
and don't forget the eurotaxes.
I wonder if there is a cost of doing business aspect to european pricing nowadays. All items that we can buy here, and there, are more in eurozone. All. but this was not the case in the past. My
father bought a couple pairs of binoculars, a camera, and a Rolex in Germany in the 60s for significantly less. In 98, I found that the dollar being strong, made the Mark really inexpensive.
Chocolate bars were only 60% of what I paid here. But now, it hardly matters what the currency does--the Eruo pricing is always high, or higher...
12-04-2010, 11:03 AM
In most cases, MSRPs in Euro are the numerical value of the MSRP in USD$$.
At the current exchange rate, yes, that means Europeans pay more.
That is an almost universal practice among retailers, who really tend to influence the MSRP, more than manufacturers.
12-04-2010, 09:49 AM
Hi Tommy,
Your notation has me flummoxed. I thought I understood what you meant, but now I'm not so sure.
The TI "maths cookbook" approach is also dominant here.
That gets to another subject.
Nobody can write cursive, and the teachers don't care. "Just use the computer, they say.
Nobody can spell, and teachers don't care. "Just use spell-check" they say.
Nobody can do arithmetic, and the teachers don't care. "Just use a calculator" they say.
What's next? Nobody can think critically, "just google it."
And the parents? I guess there must be an unspoken consensus among parents that this trend is good.
And yet, Sweden is outperforming the US rather dramatically in Maths education. Only Massachusetts and Minnesota outrank Sweden. Evidently the calculators are only a minor issue...
12-04-2010, 12:49 PM
Quote: Nobody can write cursive, and the teachers don't care. "Just use the computer, they say. Nobody can spell, and teachers don't care. "Just use spell-check" they say. Nobody can do
arithmetic, and the teachers don't care. "Just use a calculator" they say. What's next? Nobody can think critically, "just google it."
And the parents? I guess there must be an unspoken consensus among parents that this trend is good.
In principle, human beings are curious and lazy. Scientists tell 20% of the energy consumption of an average (wo)man is consumed by (her) his brain. People tend to become overweigh - emmh,
horizontally challenged in modern societies. Now, let's add 1+1+1 and guess the result ... ;)
OTOH, already Sokrates complained about the youth in Athenai, calling them incapable and good-for-nothing some 2300 years ago - and their successors discovered America ;)
12-04-2010, 02:02 PM
"OTOH, already Sokrates complained about the youth in Athenai, calling them incapable and good-for-nothing some 2300 years ago - and their successors discovered America ;) "
Hahaha so true. Then again, somewhere along there, the Greeks lost to the Romans etc...
What I find striking about the bible isn't the religious stuff, but the parables, the warnings of what can and does happen when decadence prevails. Never mind the god and hell stuff--the fact is that
ancient cities perished because the youth were led astray...
In the US, it is immigrants who keep us honest. They often show us the way when we lead ourselves astray. They show us that we shouldn't take our great society for granted--that freedom, liberty and
justice do matter...
12-06-2010, 07:23 AM
Quote: But in the case of money, the 35s and 33 are in range of what a TI graphical calc costs. And you can not compete with TI there. I need a cheaper one.
Is the HP 10s a possibility? Calculators don't come much cheaper than that, and I don't just mean the price ;-)
There's no RPN option. The algebraic parsing of
- 2 ^ 2 =
- 2 =
Ans ^ 2 =
I've read all the posts and I'm still not 100% clear on whether or not this passes "the test", but I contend that its answers are correct, bearing in mind all that Bill Platt wrote about the critical
difference between parsing an algebraic input string and the immediate operation of RPN on a stack.
12-06-2010, 12:49 PM
So, by your reasoning, you would expect an algebraic calculator to give the result 9 for the key sequence 1 + 2 x^2 = since the x^2 is to operate on the "1+2" in the display?
12-07-2010, 05:02 AM
I think that's exactly what Tommy's not saying (am I right Tommy?), but rather that which I stated previously. To quote myself:
Quote: I disagree, if I have -2 on the display, and square it, it is to be interpreted as (-2)^2, if I wanted -(2^2), I would enter 2^2, then (+/-). The negative is part of the number (I am
talking about having used the change sign, not minus), I did not enter "0 - 2^2", in which case the 2 is separate of the minus.
And the 33s is a perfect example of this:
Quote: The 33s in ALG mode: 5, +/-, x^2, ENTER yields:
( ^-5)^2=
and MINUS, 5, x^2, ENTER yields:
Thus the answer to your key sequence would be the normally expected 5.
When Tommy mentions "the number on the display" I think he is indeed talking about a number, not an equation - his examples illustrate this. (Tommy - please correct me if I am wrong).
12-07-2010, 08:45 PM
Quote: So, by your reasoning, you would expect an algebraic calculator to give the result 9 for the key sequence 1 + 2 x^2 = since the x^2 is to operate on the "1+2" in the display?
That is exactly what happens when one uses the type of algebraic mnechanized in business machiines such as the HP-10B, HP-17BII and HP-19BII. It is NOT what happens with the algebraic mechanizations
in the HP-10s, HP-33s or HP-35s.
12-07-2010, 11:22 PM
... which is the difference of "chain" and "algebraic" modes, having been discussed here not too far ago.
12-08-2010, 09:29 PM
That isn't the only thing that has been discussed before in this forum. How many times have we discussed the silly idea that -2^2 = +4 ?
12-08-2010, 09:46 PM
But we discussed -3^2 = 9 :-D
12-09-2010, 09:37 PM
Quote: Once.
But we discussed -3^2 = 9 :-D
But, it's the same silly idea.
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MCQ Algebra questions with solutions
Learn how to solve algebra questions for SSC CGL quickly from the 5th solution set of algebra questions for SSC CGL. Learn quick algebra problem solving.
Can you solve the question in the picture in 60 secs? Try first on the q0 question sets
MCQ Algebra questions with solutions for SSC CGL Set 3 uses algebra techniques for quick solution to the questions. Learn how to solve algebra quick.
MCQ Algebra questions with solutions for SSC CGL Set 2 uses algebra techniques for quick solution to the questions. Learn how to solve algebra quick.
Algebra questions with solutions for competitive exams, specially for SSC CGL. Questions solved by concepts and techniques. Learn to solve algebra quickly.
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Subsets 2 - Class 11 Maths MCQ - Sanfoundry
Class 11 Maths MCQ – Subsets – 2
This set of Class 11 Maths Chapter 1 Multiple Choice Questions & Answers (MCQs) focuses on “Subsets – 2”.
1. If A = {1,2,3,4,6} and B = {2,3,4} then which one of the following is correct?
a) A is a universal set
b) B is a subset of A
c) B is a superset of A
d) A is a null set
View Answer
Answer: b
Explanation: All elements of B belong to A, so B is a subset of A and A is a superset of A.
2. The total number of subsets of a finite set containing n elements is?
a) 2^n+1
b) 2n
c) 2^n
d) N
View Answer
Answer: c
Explanation: Number of subsets of a set having r elements each is nCr. Hence, the total number of subsets is ^nC[0] + ^nC[1] + ^nC[2] + ……+^nC[n] = 2^n.
3. If A = {1,2} and B = {1,2,4,8,10} then?
a) A=B
b) A⊆B
c) B⊆A
d) A⊄B
View Answer
Answer: b
Explanation: 1 and 2 are both available in the set B Hence A is a subset of B.
4. If A = {1,3} and B = {1,2,5} then?
a) A⊆B
b) B⊆A
c) Ф⊆A
d) B⊆Ф
View Answer
Answer: c
Explanation: A null set is a subset of every set hence Ф is a subset of A
5. If A = {2,4} then subsets of A are ___________
a) {{2}, {4}}
b) {{2}, {4}, {2,4}}
c) {Ф, {2}, {4}}
d) {Ф, {2}, {4}, {2,4}}
View Answer
Answer: d
Explanation: The subsets of a set are null set the set itself and a combination of all of its elements.
6. The number of subsets of a set containing 5 elements is?
a) 5
b) 25
c) 32
d) 64
View Answer
Answer: c
Explanation: The total number of subsets is given by 2^n where n is the number of elements, hence a total number of subsets is 2^5=32.
7. If A is a set of whole numbers and B is a set of Natural Numbers then choose the correct option.
a) A⊆B
b) B⊆A
c) A=B
d) A and B are finite sets
View Answer
Answer: b
Explanation: B is {1,2,3…….} and A is {0,1,2,3….} clearly both are infinite sets and A has 1 extra element 0 so B is a subset of A.
8. If A⊆B then what is A∩B, where A and B are two sets?
a) A
b) B
c) Null set
d) Universal Set
View Answer
Answer: a
Explanation: Intersection signifies common elements between two sets hence A has the common elements between A and B since A is a subset of B.
9. If A⊆B then what is A ∪B, where A and B are two sets?
a) A
b) B
c) Null Set
d) Universal Set
View Answer
Answer: b
Explanation: Union of two sets incorporates all elements of both A and B Hence B will bet the union of A and B since B is the superset of A.
10. If A⊆B and B⊆A then A=B.
a) True
b) False
View Answer
Answer: a
Explanation: Since both sets are subsets of each other therefore both sets have the same elements in them so A is equal to B.
Sanfoundry Global Education & Learning Series – Mathematics – Class 11.
To practice all chapters and topics of class 11 Mathematics, here is complete set of 1000+ Multiple Choice Questions and Answers.
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Program to Reverse Number in C++
Here you will get program to reverse number in C++.
using namespace std;
int main()
long n,rev=0,d;
cout<<"Enter any number:";
cout<<"The reversed number is "<<rev;
return 0;
Enter any number:12674
The reversed number is 47621
12 thoughts on “Program to Reverse Number in C++”
what the statement d=n%10 actually means
% is used to get remainder of a number by dividing it by other number. And if we do modulus of number by 10 then the last digit of number is remainder. Means 123%10=3 or 12%10=2. I hope this
will make you clear.
d is empty var, when n is suppose 123, n % 10 is takeing underoot of 123(n) by 10 and saving the remainder 3 into d for 1st loop iteration and then so on.
I used this program but it is not working for numbers starting with zero.
For example,
001 should give me 100; but it gives me 1
001 is considered as 1. This program is for number reverse not string reverse.
what the statement rev=(rev*10)+d
what the value of rev in this statement
rev is 0 in 1st iteration so its just rev = (0 * 10) + 3;
you can cout <<rev<<" "; below rev that youll see each iteration value of rev;
this was really helpful… thanku.. but I couldn’t find the solution to write a program to enter a number and print the sum of digits.
void main()
int n,x,y,sum;
cout<<"the sum is:"<<sum<<endl;
how will you print the reverse of 001
why when I enter 6000002345 the output is false
your programs having error always
i think u have to check them and then post ur programs………………..
when i see ur program having error i fell veryyy fustrated …………
Leave a Comment
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Statistical Analysis on Scoring Bias | Business Blog Articles & Reviews - BB.AIStatistical Analysis on Scoring Bias
Statistical Analysis on Scoring Bias
In the 2024 Argentine Tango World Championship
2024 Tango de Pista Mundial Winners Fatima Caracoch y Brenno Marques. Image Source: TangoBA
Proportionality Testing for Bias between judging panels is statistically significantData Visualizations show highly skewed distributionsTesting for Relative Mean Bias between Judges and Panels shows
that Panel 1 is negatively biased (lower scores are given) and Panel 2 is positively biased (higher scores are given)Testing for Mean Bias between Panels doesn’t provide evidence for statistically
significant difference in panel means, but statistical power of test is low.Image Source: TangoBA
Tango de Mundial is the annual Argentine Tango competition held in Buenos Aires, Argentina. Dancers from around the world travel to Buenos Aires to compete for the title of World Champion.
In 2024 around 500 dance couples (1000 dancers) competed in the preliminary round. A very small portion of dancers made it to the semi-final round and only about 40 couples make it the final round.
Simply making it the final round in 2024 puts you above the 95th percentile of world wide competitors.
Many finalist use this platform to advance their professional careers while the title of World Champion cements your face in tango history and all but guarantees that you’ll work as a professional
dancer for as long as you desire. Therefore, for a lot of dancers, their fate lies in what the judges think of their dancing.
In this article we will perform several statistical analyses on the scoring bias of two judging panels each comprising of 10 judges. Each judge has their own answer to the question “What is tango?”
Each judge has their own opinion of what counts for quality for various judging criteria: technique, musicality, embrace, dance vocabulary, stage presence (i.e. do you look the part), and more. As
you can already tell these evaluations are highly subjective and will unsurprisingly result in a lot of bias between judges.
Note: Unless explicitly stated otherwise, all data visualizations (i.e. plots, charts, screen shots of data frames) are the original work of the author.
You can find all code used for this analysis in my GitHub Portfolio.
my_portfolio/Argentine_Tango_Mundial at main · Alexander-Barriga/my_portfolio
Data Limitation
Before we dive into the analysis let’s address some limitation we have with the data. You can directly access the preliminary round competition scores in this PDF file.
Page 1 of dancer scores from the Preliminary round. Each row represents a dance couple. Notice that each couple is scored by 10 out of the 20 judges. More on this in the data analysis section. Screen
shot of data provided by author.Judges don’t represent the world. While the dance competitors represent the world wide tango dancing population fairly well, the judges do not—they are all
exclusively Argentine. This has led some to question the legitimacy of the competition. At a minimum, declaring the name “Munidal de Tango” to be a misnomer.Scoring dancers to the 100th decimal place
is absurd. Some judges will score couples in the 100th decimal place (i.e. 7.75 vs 7.7) This has led to ask “what is the quality of the dance that leads to a 0.05 difference?” At a minimum, this
highlights the highly subjective nature of scoring. Clearly, this is not a laboratory physics experiment using highly reliable and precise measuring equipment.Corruption and Politics. It is no
secrete in the dance community that if you take classes with an instructor on the judging panel that you are likely to receive a positive bias (consciously or subconsciously) from them since you
represent their school of thought which gives them a vested interest in your success and represents a conflict of interest.Only Dancer Scores are available. Unfortunately, other than the dancer’s
name and scores, the festival organizers do not release additional data such as years of experience, age, or country of origin, greatly limiting a more comprehensive analysis.Art is Highly
Subjective. There are as many different schools of thought in tango as their are dancers with opinions. Each dancer has their own tango philosophy which defines what good technique is or what a good
embrace feels like.Specialized Knowledge. We are not talking about movie or food reviews here. It takes years of highly dedicated training for a dancer to develop their own taste and informed
opinions of argentine tango. It is difficult for the uninitiated to map the data to a dancer’s performance on stage.
Despite these issues with the data, the data from Tango Munidal is the largest and most representative dataset of world wide argentine tango dancers that we have available.
Trust Me—My Qualifications
In addition to being a data scientist, I am a competitive argentine tango dancer. While I didn’t compete in the 2024 Munidal de Tango, I have been diligently training in this dance for many years (in
addition to other dances and martial arts). I am a dancer, a martial artist, a performer, and caretaker of tango.
While my opinion represents only a single, subjective voice on argentine tango, it is a genuine and informed one.
Statistical Analysis of Bias in Dance Scores
We will know perform several statistical test to asses if and where scoring bias is present. The outline of the analysis is as follows:
Proportionality Testing for Bias between PanelsData Visualizations & Comparing Judge’s Representations of different Mean biasesTesting for Relative Mean Bias between JudgesTesting for Mean Bias
between Panels
1. Proportionality Testing for Bias
Take another look at the top performing dance couples from page 1 of the score data table again:
Page 1 of the data table. Screen shot of data provided by author.
Reading the column names from left to right that represent the judge’s names between Jimena Hoffner and Noelia Barsel you’ll see that:
1st-5th and 11th-15th judges belong to what we will denote as panel 1.The 6th-10th judges and 16th-20th judges belong to what we will denote as panel 2.
Notice anything? Notice how dancers that were judged by panel 2 show up in much larger proportion and dancers that were judge by panel 1. If you scroll through the PDF of this data table you’ll see
that this proportional difference holds up throughout the competitors that scored well enough to advance to the semi-final round.
Note: The dancers shaded in GREEN advanced to the semi-final round. While dancers NOT shaded in Green didn’t advance to the semi-final round.
So this begs the question, is this proportional difference real or is it due to random sampling, random assignment of dancers to one panel over the other? Well, there’s a statistical test we can use
to answer this question.
Two-Tailed Test for Equality between Two Population Proportions
We are going to use the two-tailed z-test to test if there is a significant difference between the two proportions in either direction. We are interested in whether one proportion is significantly
different from the other, regardless of whether it is larger or smaller.
Statistical Test Assumptions
Random Sampling: The samples must be independently and randomly drawn from their respective populations.Large Sample Size: The sample sizes must be large enough for the sampling distribution of the
difference in sample proportions to be approximately normal. This approximation comes from the Central Limit Theorem.Expected Number of Successes and Failures: To ensure the normal approximation
holds, the number of expected successes and failures in each group should be at least 5.
Our dataset mets all these assumptions.
Conduct the Test
Define our Hypotheses
Null Hypothesis: The proportions from each distribution are the same.
Alt. Hypothesis: The proportions from each distribution are the NOT the same.
2. Pick a Statistical Significance level
The default value for alpha is 0.05 (5%). We don’t have a reason to relax this value (i.e. 10%) or to make it more stringent (i.e. 1%). So we’ll use the default value. Alpha represents our tolerance
for falsely rejecting the Null Hyp. in favor of the Alt. Hyp due to random sampling (i.e. Type 1 Error).
Next, we carry out the test using the Python code provided below.
def plot_two_tailed_test(z_value):
# Generate a range of x values
x = np.linspace(-4, 4, 1000)
# Get the standard normal distribution values for these x values
y = stats.norm.pdf(x)
# Create the plot
plt.figure(figsize=(10, 6))
plt.plot(x, y, label=’Standard Normal Distribution’, color=’black’)
# Shade the areas in both tails with red
plt.fill_between(x, y, where=(x >= z_value), color=’red’, alpha=0.5, label=’Right Tail Area’)
plt.fill_between(x, y, where=(x <= -z_value), color=’red’, alpha=0.5, label=’Left Tail Area’)
# Define critical values for alpha = 0.05
alpha = 0.05
critical_value = stats.norm.ppf(1 – alpha / 2)
# Add vertical dashed blue lines for critical values
plt.axvline(critical_value, color=’blue’, linestyle=’dashed’, linewidth=1, label=f’Critical Value: {critical_value:.2f}’)
plt.axvline(-critical_value, color=’blue’, linestyle=’dashed’, linewidth=1, label=f’Critical Value: {-critical_value:.2f}’)
# Mark the z-value
plt.axvline(z_value, color=’red’, linestyle=’dashed’, linewidth=1, label=f’Z-Value: {z_value:.2f}’)
# Add labels and title
plt.title(‘Two-Tailed Z-Test Visualization’)
plt.ylabel(‘Probability Density’)
# Show plot
def two_proportion_z_test(successes1, total1, successes2, total2):
Perform a two-proportion z-test to check if two population proportions are significantly different.
– successes1: Number of successes in the first sample
– total1: Total number of observations in the first sample
– successes2: Number of successes in the second sample
– total2: Total number of observations in the second sample
– z_value: The z-statistic
– p_value: The p-value of the test
# Calculate sample proportions
p1 = successes1 / total1
p2 = successes2 / total2
# Combined proportion
p_combined = (successes1 + successes2) / (total1 + total2)
# Standard error
se = np.sqrt(p_combined * (1 – p_combined) * (1/total1 + 1/total2))
# Z-value
z_value = (p1 – p2) / se
# P-value for two-tailed test
p_value = 2 * (1 – stats.norm.cdf(np.abs(z_value)))
return z_value, p_value
min_score_for_semi_finals = 7.040
is_semi_finalist = df.PROMEDIO >= min_score_for_semi_finals
# Number of couples scored by panel 1 advancing to semi-finals
successes_1 = df[is_semi_finalist][panel_1].dropna(axis=0).shape[0]
# Number of couples scored by panel 2 advancing to semi-finals
successes_2 = df[is_semi_finalist][panel_2].dropna(axis=0).shape[0]
# Total number of couples that where scored by panel 1
n1 = df[panel_1].dropna(axis=0).shape[0]
# Total sample of couples that where scored by panel 2
n2 = df[panel_2].dropna(axis=0).shape[0]
# Perform the test
z_value, p_value = two_proportion_z_test(successes_1, n1, successes_2, n2)
# Print the results
print(f”Z-Value: {z_value:.4f}”)
print(f”P-Value: {p_value:.4f}”)
# Check significance at alpha = 0.05
alpha = 0.05
if p_value < alpha:
print(“The difference between the two proportions is statistically significant.”)
print(“The difference between the two proportions is not statistically significant.”)
# Generate the plot
# P-Value: 0.0000
plot_two_tailed_test(z_value)The Z-value is the statistical point value we calculated. Notice that it exists far out of the standard normal distribution.
The plot shows that the Z-value calculated exists far outside the range of z-values that we’d expect to see if the null hypothesis is true. Thus resulting in a p-value of 0.0 indicating that we must
reject the null hypothesis in favor of the alternative.
This means that the differences in proportions is real and not due to random sampling.
17% of dance coupes judged by panel 1 advanced to the semi-finals42% of dance couples judged by panel 2 advanced to the semi-finals
Our first statistical test for bias has provided evidence that there is a positive bias in scores for dancers judged by panel 2, representing a nearly 2x boost.
Next we dive into the scoring distributions of each individual judge and see how their individual biases affect their panel’s overall bias.
2. Data Visualizations
In this section we will analyze the individual score distributions and biases of each judge. The following 20 histograms represent the scores that each judge gave a dancer. Remember that each dancer
was scored by all 10 judges in panel 1 or panel 2. The judge’s histograms are laid out randomly, i.e. column one doesn’t represent judges from panel 1.
Note: Judges score on a scale between 1 and 10.
Scoring distributions from all 20 judges. Titles contain judge’s name.
Notice how some judges score much more harshly than other judges. This begs the questions:
What is the distribution of bias between the judges? In other words, which judges score harsher and which score more lenient?Do the scoring biases of judges get canceled out by their peers on the
their panel? If not, is there a statistical difference between their means?
We will answer question 1 in section 3.
We will answer question 2 in section 4.
As we saw in the histograms above, judge Martin Ojeda is the harshest dance judge. Let’s look at his QQ-Plot.
Distribution of scores given by Martin.
Lower-left deviation (under -2 on the x-axis): In the lower quantile region (far-left), the data points actually deviate above the red line. This indicates higher-than-expected scores for the weakest
performances. Martin could be sympathetically scoring dance couples slightly higher than he feels they deserve. A potential reason could be that Martín wishes to avoid hurting the lowest-performing
competitors with extremely poor scores, thus giving slightly higher ones.
Martín is slightly overrating weaker competitors, which could suggest a mild positive bias toward performances that might otherwise deserve lower scores.
Dilution of Score Differences: If weaker performances are overrated, it can compress the scoring range between weaker and mid-tier competitors. This might make the differences between strong,
moderate, and weak performances less clear.
For example, if a weak performance receives a higher score (say, 5.5 or 6.0) instead of a 4.0 or 4.5, and a mid-tier performance gets a score of 6.5, the gap between a weak and moderate competitor is
artificially reduced. This undermines the competitive fairness by making the scores less reflective of performance quality.
Balance between low and high scores: Although Martín overrates weaker performers, notice that in the mid-range (6.0–7.0), the scores closely follow a normal pattern, showing more neutral behavior.
However, this is counterbalanced by his generous scoring at the top end (positive bias for top performers), suggesting that Martín tends to “pull up” both ends of the performance spectrum.
Overall, this combination of overrating both the weakest and strongest competitors compresses the scores in the middle. Competitors in the mid-range may be the most disadvantaged by this, as they are
sandwiched between overrated weaker performances and generously rated stronger performances.
In this next section, we will identify Martin’s outlier score for what he considers the best performing dance couple assgined to his panel. We will give the dance couple’s score more context when
comparing it with scores from other judges on the panel.
There are 19 other QQ-Plots in the Jupyter Notebook which we will not be going over in this article as it would be make this article unbearably long. However feel free to take a look yourself.
3. Testing for Relative Mean Bias between Judges
In this section we will answer the first question that was asked in the previous section. This section will analyze the bias in scoring of individual judges. The next section will look at the bias in
scoring between panels.
What is the distribution of bias between the judges? In other words, which judges score harsher and which score more lenient?
We are going to perform iterative T-test to check if a judge’s mean score is statistically different from the mean of the mean scores all of their 19 peers; i.e. take the mean of the other 19 judges’
mean scores.
# Calculate mean and standard deviation of the distribution of mean scores
distribution_mean = np.mean(judge_means)
distribution_std = np.std(judge_means, ddof=1)
# Function to perform T-test
def t_test(score, mean, std_dev, n):
“””Perform a T-test to check if a score is significantly different from the mean.”””
t_value = (score – mean) / (std_dev / np.sqrt(n))
# Degrees of freedom for the T-test
df = n – 1
# Two-tailed test
p_value = 2 * (1 – stats.t.cdf(np.abs(t_value), df))
return t_value, p_value
# Number of samples in the distribution
n = len(judge_means)
# Dictionary to store the test results
results = {}
# Iterate through each judge’s mean score and perform T-test
for judge, score in zip(judge_features, judge_means):
t_value, p_value = t_test(score, distribution_mean, distribution_std, n)
# Store results in the dictionary
results[judge] = {
‘mean_score’: score,
‘T-Value’: t_value,
‘P-Value’: p_value,
‘Significant’: p_value < 0.05
# Convert results to DataFrame and process
df_judge_means_test = pd.DataFrame(results).T
df_judge_means_test.mean_score = df_judge_means_test.mean_score.astype(float)
df_judge_means_test.sort_values(‘mean_score’)T-test results sorted by mean score value of judges.
Here we have all 20 judges: their mean scores, their t statistics, p-values, and if the different between the individual judge’s mean score and the mean of the distribution of the other 19 judge’s
means is statistically significant.
We have 3 groups of judges: those that score very harshly (statistically below average), those that typically give average scores (statistically within average), and those that score favorably
(statistically above average).
Let’s focus on 3 representative judges: Martin Ojeda, Facundo do la Cruz, and Noelia Barsi.
These 3 judges represent the harshest, typical, and favorable judging bias tendencies between all 20 judges.
Notice that Martin Ojeda’s score distribution and mean (blue line) is shifted towards lower values as compared to the mean of the mean scores of all judges (red line). But we also see that his
distribution is approximately normally distributed with the exception of a few outliers. We will return to the couple that scored 7.5 shortly. Almost any dance couple that gets judged by Martin will
see their overall average score take a hit.
Facundo de la Cruz has an approximately normal distribution with a large variance. He represents the least biased judge relative to his peers. Almost any dance couple that gets judged by Facundo can
expect a score that is typical of all the judges, so they don’t have to worry about negative bias but they are also unlikely to receive a high score boosting their overall average.
Noelia Baris represents the judge that tends to give more dancers a favorable score as compared to her peers. All dance couples should hope that Noelia gets assigned to their judge panel.
Now let’s return to Martin’s outlier couple. In Martin’s mind, Lucas Cartagena & Lucila Diaz Colodrero dance like outliers.
Scores for Lucas Cartagena & Lucila Diaz Colodrero. PROMEDIO means average.Martin Ojeda gave the 4th highest score (7.600) that this couple received from the panel and his score is above the average
(Promedio) of 7.326While Martin Ojeda’s score is an outlier when compered to his distribution of scores it isn’t as high the other scores that this couple received. This implies two things:Martin
Ojeda is an overall conservative scorer (as we saw in the previous section)Martin Ojeda outlier score actually makes sense when compared with the scores of the other judges indicating that this dance
couple are indeed high performers
Here they are performing to a sub-category of tango music called Milonga, which is usually danced very differently than the other music categories, Tango and Waltz, and typically doesn’t include any
spinning and instead includes small, quick steps that greatly emphasize musicality. I can tell you that they do perform well in this video and I think most dancers would agree with that assessment.
Enjoy the performance.
4. Testing for Mean Bias between Panels
In this section we test for bias between Panel 1 and Panel 2 by answering the following questions:
Do the scoring biases of judges get canceled out by their peers on the their panel? If not, is there a statistical difference between their means?
We will test for panel bias in two ways
Rank-Based Panel BiasesTwo-Tail T-test between panel 1 and panel 2 distributions
Rank-Based Panel Biases
Here we will rank and standardize the judge mean scores and calculate a bias for each panel. This is one way in which we can measure any potential biases that exist between judge panels.
panel_1 = judge_features[:5] + judge_features[10:15]
panel_2 = judge_features[5:10] + judge_features[15:]
df_judge_means = df_judge_means_test.sort_values(‘mean_score’).mean_score
# Calculate ranks
ranks = df_judge_means.rank()
# Calculate mean and std of ranks
means_ranks = ranks.mean()
stds_ranks = ranks.std()
# Standardize ranks
df_judge_ranks = (ranks – means_ranks) / stds_ranks
# these are the same judges sorted in the same way as before based on their mean scores
# except here we have converted mean values into rankings and standardized the rankings
# Now we want to see how these 20 judges are distributed between the two panels
# do the biases for each judge get canceled out by their peers on the same panel?Sorted Rankings for all 20 judges
We’ll simply replace each judge’s mean score value with a ranking relative to their position to their peers. Martin is still the harshest, most negatively biased judge. Noelia is still the most
positively biased judge.
Panel 1Panel 2
Notice that most judges in Panel 1 are negatively biased and only 4 are positive. While most judges on Panel 2 are positively biased with only 2 judges negatively biased and Facundo being
approximately neutral.
Now if the intra-panel biases cancel out than the individual judge biases effectively don’t matter; any dance couple would be scored statistically fairly. But if the intra-panel biases don’t cancel,
then there might be an unfair advantage present.
Panel 1 mean ranking value: -0.39478Panel 2 mean ranking value: 0.39478
We find that the mean panel rankings reveal that the intra-panel biases do not cancel out, providing evidence that there is an advantage for a dance couple to be scored by Panel 2.
Two-Tail T-test between Panel 1 and Panel 2 distributions
Next we following up on the previous results with an additional test for panel bias. The plot below shows two distributions. In blue we have the distribution of mean scores given by judges assigned
to Panel 1. In orange we have the distribution of mean scores given by judges assigned to Panel 2.
In the plot you’ll see the mean of the mean scores for each panel. Panel 1 has a mean panel score of 6.62 and Panel 2 has a mean panel score of 6.86.
While a panel mean difference of 0.24 might seem small, know that the difference between advancing to the semi-final round is determined by a difference of 0.013
Distribution of both Panel’s judge’s mean scores.
After performing a two sample T-test we find that the difference between the panel means is NOT statistically different. This test fails to provide additional evidence that there is a statistical
difference in the bias between Panel 1 and Panel 2.
The P-value for this test is 0.0808 which isn’t too far off from 0.05, our default alpha value.
Law of Large Numbers
We know from The Law of Large Numbers that in small sample distributions (both panels have 10 data points) we commonly find a larger variance than the variance in its corresponding population
distribution. However as the number of samples in the sample distributions increases, its parameters approach the value of the population distribution parameters (i.e mean and variance). This might
be why the T-test fails to provide evidence that the panel biases are different, i.e. due to high variance.
Statistical Power
Another way to understand why we see the results that we see is due to Statistical Power. Statistical power refers to the probability that a test will correctly reject a false null hypothesis.
Statistical power is influence by several factors
Sample sizeEffect size (the true difference or relationship you’re trying to detect)Significance level (e.g., α=0.05)Variability in the data (standard deviation)
The most reliable way to increase our test’s statistical power is to collect more data points, however that is not possible here.
In this article we explore the preliminary round data from the 2024 Mundial de Tango Championship competition held in Buenos Aires. We analyze judge and panel scoring bias in 4 ways:
Proportionality Testing for Bias between PanelsData VisualizationsTesting for Relative Mean Bias between JudgesTesting for Mean Bias between Panels
Evidence for Bias
We found that there was a statistically significant higher proportion of dancers advancing to the semi-final round that were scored by judges on Panel 2.Individual judges’ score distributions varied
wildly: some skewing high, others skewing low. We saw that Martin Ojeda was positively biased towards the worst and best performing dance couples by “pulling up” their scores and compressing
mid-range dancers. Overall, his score distribution was far lower than all other judges. There were examples of judges that gave more typical scores and those that gave very generous scores (when
compared to their peers).There was a clear difference between relative mean bias between panels. Most judges on Panel 1 ranked as providing negative bias in scoring and most judges on Panel 2 ranked
as providing positive bias in scoring. Consequently, Panel 1 had an overall negative bias and Panel 2 had an overall positive bias in scoring dance competitors.
No Evidence of Bias
The difference in means between Panel 1 and Panel 2 were found to not be statistically significant. The small sample size of the distributions leads to low statistical power. So this test is not as
reliable as the others.
My conclusion is that there is sufficient evidence of bias at the individual judge and at the panel level. Dancers that got assigned to Panel 2 did have a competitive edge. As such there is advice to
give prospective competitive dancers looking for an extra edge in winning their competitions.
How to Win Dance Competitions
There are both non-machiavellian and machiavellian ways to increase your chances of winning. The results of this article inform (and background knowledge gained from experience) add credence to the
machiavellian approaches.
Train. There will never be a substitute for the putting in your 10,000 hours of practice to obtain mastery.
Take classes and build relationships with instructors that you know will be on your judging panel. You’re success means the validation of the judge’s dance style and the promotion of their business.
If your resources are limited, focus on judges that that score harshly to help bring up your average score.If there are multiple judging panels, find a way to get assigned to the panel that
collectively judges more favorably to increase your chances to advancing to the next round.
Final Thoughts
I personally believe that, while useful, playing power games in dance competitions is silly. There is no substitute for raw, undeniable talent. This analysis is, ultimately, a fun passion project and
opportunity to apply some statistical concepts on a topic that I greatly cherish.
About the Author
Alexander Barriga is a data scientist and competitive Argentine Tango dancer. He currently lives in Los Angeles, California. If you’ve made it this far, please consider leaving feedback, sharing his
article, or even reaching out to him directly with a job opportunity—he is actively looking for his next data science role!
Image by author.
Statistical Analysis on Scoring Bias was originally published in Towards Data Science on Medium, where people are continuing the conversation by highlighting and responding to this story.
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Computational Mathematics
What is Computational Mathematics?
Computational Math is a specialised field that combines mathematical theory, practical engineering, and computer science. By studying a Computational Mathematics degree you will learn how to solve
complex problems in science, engineering, and business, by using mathematical models and computational algorithms.
Computational Mathematics specialisations
Computational Mathematics specialisations are many, and allow you to focus on areas of interest or specific career paths. The most common specialisations include:
• Numerical Analysis;
• Mathematical Modelling;
• Algorithm Design;
• Data Science;
• Cryptography;
Both Bachelor's and Master's degree programmes typically offer these specialisations, and you can go with either, depending on your academic background and career goals.
What will you learn during a Computational Mathematics programme?
Embarking on a Computational Mathematics programme immerses you in the world of complex problem-solving. Here's what you can expect to learn:
• advanced mathematical theory and how to apply it;
• numerical methods and computation;
• how to design, analyse, and implement algorithms;
• data analysis and prediction techniques;
• cryptographic methods and their applications.
Common Computational Mathematics courses include:
• Discrete Mathematics: teaches the fundamental principles of mathematical reasoning and proof techniques, set theory, logic, counting principles, and graph theory.
• Linear Algebra: a key course for understanding vectors, matrices, and linear transformations, which are vital in computational modelling and computer graphics.
• Numerical Analysis: covers techniques for numerical approximation and error estimation.
• Algorithm Design and Analysis: it is about understanding and creating efficient algorithms to solve mathematical and computational problems.
• Probability and Statistics: introduces concepts of randomness, probability distributions, statistical inference, regression, and hypothesis testing.
Computational Mathematics is a good degree for you if you enjoy problem-solving, have an aptitude for mathematics, and wish to apply these skills in the real world.
Skills required for a degree in Computational Mathematics
The Computational Mathematics degree requires you to have solid math skills, problem-solving abilities, and an aptitude for programming. A strong understanding of algorithms, computation, and
mathematical theory is also essential, as is the ability to work with these concepts practically.
What can you do with a Computational Mathematics degree?
With a Computational Mathematics degree, you can venture into numerous rewarding and high-demand fields. The jobs you can get with a Computational Mathematics degree include:
• Data Analyst;
• Cryptographer;
• Software Developer;
• Quantitative Analyst;
• Operations Research Analyst.
A Bachelor's degree can lead to roles in business, technology, or science that require strong analytical skills. A Master's degree, on the other hand, can open up opportunities for specialised roles
in data science, cryptography, or scientific research. Is a Computational Mathematics degree worth it? Absolutely! The skills gained from this degree are highly sought after in our increasingly
data-driven world.
Read more
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The interpretation of pressure in LAMMPS for a non-ideal bonded system
Dear LAMMPS community,
Firstly, many thanks for this valuable mailing list, and for all the important answers that I have received from so many of you over the years.
This time, I would like to ask a question about pressure that I came across in while trying to understand how the enthalpy of a system is calculated in LAMMPS.
It is common knowledge that in MD, one evaluates pressure P as an ensemble of the instantaneous/microscopic pressure P*, that for a system of N particles in a volume V is as follows:
P*=1/V(1/3\sum_i m_i v_i •v_i+1/3\sum_i r_i•f_i)
where r is the position, v- velocity and f- force.
The macroscopic pressure is P=<P*>, a statistical average of an ensemble. In case of pairwise interactions, the pressure is defined as
P=<(N/V) k_BT>+<(1/3V)\sum_i sum_{j<i} r_{ij}•f_{ij}
Where r_{ij} is the intermolecular vector between i and j and f the corresponding force.
When we use NVE+Langevin (mimicking Brownian Dynamics), a simulation of an ensemble of ideal gas particles produces a linear relation between pressure and temperature upon cooling, as expected.
However, what happens if we have the same NVE+Langevin dynamics whereby the monomers are connected through a bond, say harmonic or FENE? What if one has a collapsing polymer chain (gradually with a
specific rate of cooling), where pressure is not the same as it is in the ideal gas case due to bonding interactions? If one wishes to study the behavior of enthalpy as a function of Temperature, for
instance, how does one define the pressure that is part of the enthalpy calculation?
Many thanks in advance! Your input is highly valued & appreciated.
With best wishes
Dear LAMMPS community,
Firstly, many thanks for this valuable mailing list, and for all the
important answers that I have received from so many of you over the years.
This time, I would like to ask a question about pressure that I came
across in while trying to understand how the enthalpy of a system is
calculated in LAMMPS.
It is common knowledge that in MD, one evaluates pressure P as an ensemble
of the instantaneous/microscopic pressure P*, that for a system of N
particles in a volume V is as follows:
P*=1/V(1/3\sum_i m_i v_i •v_i+1/3\sum_i r_i•f_i)
where r is the position, v- velocity and f- force.
The macroscopic pressure is P=<P*>, a statistical average of an ensemble.
In case of pairwise interactions, the pressure is defined as
P=<(N/V) k_BT>+<(1/3V)\sum_i sum_{j<i} r_{ij}•f_{ij}
Where r_{ij} is the intermolecular vector between i and j and f the
corresponding force.
When we use NVE+Langevin (mimicking Brownian Dynamics), a simulation of an
ensemble of ideal gas particles produces a linear relation between pressure
and temperature upon cooling, as expected.
this last statement is not correct. particles in an ideal gas do not
interact, so there are no forces. you only have the kinetic energy
contribution and if you look at it closely, you will see that you can
recover the ideal gas law from it. also the interactions implicitly
contained in fix langevin do not apply to an ideal gas.
the F <dot> R term only applies to interacting particles.
However, what happens if we have the same NVE+Langevin dynamics whereby
the monomers are connected through a bond, say harmonic or FENE? What if
one has a collapsing polymer chain (gradually with a specific rate of
cooling), where pressure is not the same as it is in the ideal gas case due
to bonding interactions? If one wishes to study the behavior of enthalpy as
a function of Temperature, for instance, how does one define the pressure
that is part of the enthalpy calculation?
the F <dot> R relation can be applied for bonded interactions just as
well. it is easy to set up tests for that.
i've done this for example last fall to validate two bugfixes:
Thank you Axel,
Dear Axel
What remains a confusion, that all these terms are located where the polymer chain is - are different on “droplet” -the collapsed polymer- surface, and certainly zero outside of “droplet”. Whereas
thermodynamic pressure must be constant across the system in equilibrium. Am I interpreting this correctly?
Many thanks
Dear Axel
What remains a confusion, that all these terms are located where the
polymer chain is - are different on "droplet" -the collapsed polymer-
surface, and certainly zero outside of "droplet". Whereas thermodynamic
pressure must be constant across the system in equilibrium. Am I
interpreting this correctly?
i don't think so.
pressure is a property of the entire observed system and the physical
interpretation of (total) pressure is (the average) force per area on the
bounding surface(s). thus the pressure of an isolated system is by
definition zero, since the volume is infinite, i.e. unbounded.
now, you *can* compute a "local" pressure, by subdividing the volume and
computing a pressure for this, but there is nothing requiring a system to
have equipartitioning on this kind of property in an inhomogeneous system.
all that is required would be an (on average) zero net pressure on the
dividing surfaces between two such subsystems.
I should also point out that if you are using the Langevin thermostat to model the presence of an implicit solvent, then the pressure reported by LAMMPS (based on velocities and forces of the solute
“atoms”) is not capturing the dominant contribution to the pressure from the solvent. As an example, you could simulate polymer chains dissolved in water compressed to very high pressure using this
approach, but the pressure reported by LAMMPS would not reflect the high pressure of the water molecules. The LAMMPS pressure is probably related to the osmotic pressure of the polymer.
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X-ray reflection
X-rays can be reflected under certain conditions when hitting matter. Mostly three reflection types are distinguished:
Total external reflection Bragg reflection Multi layer reflection
Total external reflection
When X-rays enter matter under grazing incidence, they will be reflected by Total External Reflection (TER) when the angle of incidence is below the critical angle α[critical].
Fig. 1: Sketch showing the angles and indices of refraction used
to calculate the critical angle of total external reflection
The critical angle can be calculated as follows (see fig. 1). From the refraction of X-rays we know that Snell's law is
with the angle α[1] of the incoming ray and α[2] of the outgoing ray and the indices of refraction n[2] in matter and n[1] in the medium the ray comes from. Total external reflection occurs for
angles α[1], when the angle α[2] reaches 90°. Assuming the ray comes from vacuum, n[1]=1. So the critical angle α[critical] is
As n[2] is only slightly below one (for example for gold for photon energies of 12.4 keV, n[2] = 1-1.88·10^-5), the maximum angle for TER to occur is close to 90° (for gold at 12.4 keV α[critical] =
89.65° = 90°-0.351°). For X-rays total external reflection occurs only under grazing incidence. So the reflection angles α are always close to 90°. Consequently normally the reflection angles are
measured as angles θ between the incoming ray and the mirrors surface. The critical angle θ[critical] is then
with n[2] = 1-δ and with the Taylor expansion of the cosine
the critical angle θ[critical] is
or when multiplying with 180/π
This approximation results in an error below 0.021% for θ[critical]<1°. In the example of gold and a photon energy of 12.4 keV, θ[critical] =0.351°. Total external reflection is nearly, but not
completely loss-less, because the absorption coefficient β is not zero.
Bragg reflection
Crystal surfaces show high reflectivity under special angles depending on the wavelength of the X-rays due to Bragg-reflection. Mirrors using Bragg-reflection to redirect X-rays are called crystal
mirrors. These mirrors provide large reflection angles when the reflection condition for a given wavelength is fulfilled. To understand the physical principle, the path difference of an incoming ray
reflected at the surface of a crystal and a neighbour ray being reflected at the next inner atomic layer is regarded (see fig. 2). When the optical path difference
Δ=2d sinθ
with the distance d between two adjacent atomic layers and the angle θ of the incoming wave measured to the surface is an integer multiple m of the wavelength λ, constructive interference occurs and
consequently the wave is reflected. This is known as Bragg equation:
mλ=2d sinθ
Bragg reflection is used for example to focus monochromatic light with e.g. elliptically curved crystal mirrors and in monochromators to filter wavelengths and in material science when calculating
the atomic lattice constants d from the reflection angles at crystalline samples.
Fig. 2: Bragg reflection; when die optical path difference Δ=2d sinθ for a certain wavelength λ is a multiple of λ, the beams will interfere constructively.
Multi layer reflection
An X-ray mirror can be formed by fabricating a multi layer system consisting of layers of different index of refraction (see fig. 3).
Fig. 3: X-ray reflection at a multi layer mirror
The Bragg equation then changes to compensate for the refraction in the layers [Tho 2009]:
with the order m, the wavelength λ, the period d[M] of the multi layer system, the angle of incidence θ and the real part
[Tho 2009 A. C. Thompson, J. Kirz, D. T. Attwood, E. M. Gullikson, M. R. Howells, J. B. Kortright, Y. Liu and A. L. Robinson; X-ray data booklet, third edition, Lawrence Berkeley National Laboratory,
] Berkeley, California, 2009
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A problem and later a story and a point.
I have (1) a math problem to tell you about (though I suspect many readers already know it), (2) a story about it, and (3) a point to make. TODAY I'll just do the math problem. Feel free to leave
comment with solutions--- so if you haven't seen it before and want to try it, then don't look at the comments. Tommorow or later I will tell you the story and make my points.
PROBLEM 1: There are n people sitting on chairs in a row. Call them p1,...,pn. They will soon have HATS put on their heads, RED or BLUE. Nobody can see their own hat color. pn can see p(n-1),...,p1.
More generally, pi can see all pj j < i. They CAN meet ahead of time to discuss strategy.
Here is the game and the goal: Mr. Bad will put hats on people any way he likes (could be RBRBRB..., could be RRRBBB, could be ALL R's - like when a teacher has a T/F test where they are all FALSE.)
Then pn says R or B, p(n-1) says R or B, etc. They want to MAXIMIZE how many of them say THEIR hat color. Assume that Mr. Bad knows the strategy the people will use.
What is the best they can do?
Here is a strategy: pn says R if the MAJORITY are R, and B if the MAJORITY are B, and then everyone says what pn says. They are guaranteed around n/2 correct.
Here is a strategy: Assume n is even. pn says the color of p(n-1). p(n-1) then says what pn said and gets it right. then p(n-2) says what p(n-3) has. Then p(n-3) gets it right. You are guaranteed to
get around n/2 right.
GEE- can we do better than n/2? Or can one prove (perhaps using Ramsey Theory, perhaps something I learned at Erdos 100 over dinner) that you can't beat n/2 (or perhaps something like n/2 + log(log
PROBLEM 2: Same as Problem 2 but now there are c colors of hats.
NOTE- there are MANY hat problems and MANY variants of this scenario--- some where you want to maximize prob of getting them all right, some where everyone sees everyones hat but their own. These are
all fine problems, but I am just talking about (1) people are in a row, (2) Want to maximize how many they get right in the worst case.
ADDED LATER- WARNING- THE ANSWER TO PROBLEM 1 IS IN THE COMMENTS NOW.
SO IF YOU WANT TO SOLVE IT YOURSELF DO NOT LOOK AT THE COMMENTS.
17 comments:
1. A key piece of the puzzle is missing. Along with "Then pn says R or B, p(n-1) says R or B, etc.": Mr Bad shoots the person who answers incorrectly. I.e., the rest of the people know the
correctness of each of the answers given by earlier folks.
2. How about a binary / parity solutions?
p(n) computes the parity bit for the rest of the peoples' hats (use Red=0, Blue=1) and uses that as his guess. He has a 50/50 chance of being right about his own hat (we'll assume he's wrong).
Now everyone else can correctly compute their hat's color when it is their turn because they will know everyone's hat color other than their own (they can see those in front of them and they
heard the answers of those behind them) and can use the parity information to determine their hat's color.
So, worst case, n-1 will be correct.
3. We can get (n-1)/n correct in the worst case for problem 1. The first person can count the number of red hats (for instance) worn by p1...p(n-1). If pn sees an even number of red hats, he guesses
red. If he sees an odd number of red hats, he guesses odd. Now p(n-1) knows whether there are an even number of red hats worn between p(1) and himself or an odd number. But he can check for
himself how many red hats tehre are between p(1) and p(n-2), so it's easy for him to figure out what his own hat color is. Then each person from that point on will be able to figure out their own
hat color. The first person has no choice but to guess, so the worst case is (p-1)/p.
4. EG and Magnamimous- yes, you have it correct (this is MATH so
you already knew that). Magnamious- you are giving the answer
as the fraction that are right rather than the number that are right,
which is fine of course.
NOW- try to do the c-color case.
5. c-color case. Similar approach. For three/four/five/six colors, you can use the first two people, using each of the 9,16,25, or 36 possible choices (respectively) to encode the parity of each of
the colors (although you only need to encode parity of c-1 colors). When you get to 7, there are only 49 possible encodings, when you need 64 to describe the number of possible states so you need
more people. Not a complete answer, but I will think about this some more.
6. Two cases:
(a) People know value of 'c': In this case I can save
$n - ( c - 1 ) log_c n$
people in the worst case.
(b) People do not know value of 'c'. In this case I can save
$n - log_2 n - ( c - 1 ) log_c n$
people in the worst case.
Both are obtained by having people who are in front of the line encode the exact number of different hats after the 'K'th person in the line.
7. Magnamimous and Space2001- Good solutions but
better is known.
Space2001- good alternative problem- what if c
is not known. I don't know if one can do better
than you did in that case.
8. mine grows at n-log_c(2^(c-1)) by the way. it's slightly better than space2001's because i'm not representing the exact number of different hats, only the parity of each color. But that seems to
me to be the minimal amount of information needed to represent the state of the system. I don't know how i can use any less information than that =(
9. we use modular arithmetic (mod c)
(1) if the colors are known, then they are assigned numbers from 0 to c-1, beforehand. Then p_n announces the sum (mod c) of all numbers he sees. The rest can announce their colors correctly. So
(n-1) worst case.
(2) If c is known but not the color names, the the last c guys announce the color names. The colors are numbered from 0 to c-1 in the order they are announced. Then we continue as case (1). The
worst case is (n - c - 1).
(3) If c is not known, then the same strategy as (2) works since the person announcing the sum will have to repeat a color name.
10. You can extend the 2-color case to the c-color case. Assign the color c_i to the number i (0<=i<c) and compute the sum of all colors multiplied with the number of hats of this color mod c. The
first one guesses the color c_j if j is the result of this computation. Then the next one computes the same, only that his value k differs from j exactly in his hat-color mod c. Everyone can
compute the current value mod c with the previous answers and compute his own color. So n-1 people can be saved.
11. The funny variant of this game is with a infinite number of participants (and a finite number of colors)...
12. If c is unknown, it seems the first person can say c and the others can apply the sum modulo c strategy for a guaranteed n-2.
Or maybe I didn't really understand the problem.
13. revised answers:
(a) People know $c$: we can save $n - ( c - 1 )$ persons.
The first $(c - 1)$ persons will encode remainder modulo $c$ of the counts of each of first $(c - 1)$ colors among the last $n - (c - 1)$ people. Thereafter the $c-2$ person in the list can
perfectly reconstruct his hat color and so on. To encode the remainder modulo $c$ we only need a $c$-ary bit.
(b) People don't know 'c': we can save $n - log_2 n - ( c - 1 )$ people by encoding c followed by (a). Maybe can do better with regards to encoding c.
14. Further improved: we can save
$n - \frac{c - 1}{log_2 c}$ people.
In the previous solution instead of sending remainder modulo $c$ the first set of people only need to send remainder modulo $2$ of the counts for $c-1$ colors among last $n-K$ people.
I.e., a $c-1$ bit binary vector.
We can do this using $\frac{c - 1}{log_2 c}$ $c$-ary bits and therefore $K = \frac{c - 1}{log_2 c}$.
15. I think you want to look at this paper: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.66.267
1. That paper talks about the problem where everyone sees everyones
hat except their own. An excellent paper, and rather odd in
that the hat problem is used to do something with autored sequences,
but not a paper on the problem at hand.
but you are RIGHT- a paper I DO want to look at.
2. Ah sorry, you are right. It seems that I assumed the problem to be the one I know while I read the problem definition.
But it might be interesting to think about translations between them or transfers of the proofs...
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Fixed Point Arithmatic and Transcendental Function Library
Fixed-Point Arithmetic
and Transcendental Functions
Not all microprocessors are created equally; some have math coprocessors and some do not. In fact, the Intel 80486 was the first Intel microprocessor that had a built-in math coprocessor; otherwise,
up until 80386, you had to pay extra for any decent processing power. C compilers were provided with the math emulation routines, but the performance was anything but acceptable. It was not because
the routines were written poorly; there was a legacy and a standard for the floating-point arithmetic.
They all had to confirm to IEEE standards for double precision arithmetic of 64-bit words; especially, the solution of transcendental functions. The $$sin, cos, tan, exp, ln $$ and so forth were all
written with a high degree of accuracy, but a penalty was paid in terms of performance. Graphics routines were the hardest hit. The transformation and rotation of objects requiring matrix
multiplication and additions were noticeably slow, mainly because the underlying arithmetic operations were slow.
Today, we take math coprocessors for granted, but there are still times when your embedded system design cannot afford a coprocessor and you have to come up with a fast and efficient math library.
This chapter is devoted to math algorithms and solution of transcendental functions. The fixed-point arithmetic is presented as an alternative to floating-point arithmetic, and the pros and cons are
discussed with examples. One more point to highlight is the magnitude of the work involved in the solution of transcendental functions, and you can make an informed decision regarding your need for a
math coprocessor.
We begin our discussion with the solution of transcendental functions.
Transcendental Functions
By definition, the trigonometric and logarithm functions, including sin, cos, tan, e, and ln, and so forth, cannot be expressed in algebraic form. Just like the square root of 2 and the value of pi,
these are not rational numbers, which means that there are no two integers whose ratio can be used to define them. As we know, all arithmetic operations must be brought down to the four basic
operations of addition, subtraction, multiplication, and division. If a function cannot be expressed as an operation on rational numbers, it cannot be solved. Thus, the only method of computing the
transcendental functions is by the expansion of the function in its polynomial form.
The Taylor series expansion is the fundamental solution to all transcendental functions.
The following is an example of the Taylor series expansion for the exponent function:
$$ e^x =1+x^1/1! +x^2/2!+x^3/3! + x^4/4! + x^5/5! \cdots+x^n/n! \\ $$
One must expand the series to its infinite terms if an exact solution of the function is required. Obviously, nobody has the time and, probably, nobody needs the precision either. So, how do you
decide how many terms of the series should be used? That depends on your application and the degree of precision you need.
Fig 1.1 Floating Point Foramt Double Precision
Figure 1.1 describes the format for a double-precision floating-point number. The float variable, on the other hand, is a 32-bit-long word with 1 sign bit, 8 exponent bits and 24 bits of mantissa or
Figure 1.2 describes the format for a single-precision number. Again, we gain an extra bit of storage by removing the normalized bit of mantissa and using it to store an extra precision bit.
You might think that specifying a variable as “float” instead of “double” might give you performance improvement, as there are only 32 bits in a float variable—think again! It is true that
multiplying two 32-bit numbers is faster than multiplying two 64-bit numbers, but all numbers are converted to 64-bit double precision, before the polynomial operation being performed. This is double
jeopardy—now there is the extra penalty of converting the numbers from single precision to double precision, and vice versa.
If your embedded system cannot afford a math coprocessor and you need floating-point arithmetic, the following is an alternate solution to the standard C library functions. The precision is not that
great—in some cases, it is about four decimal places—but you will find these routines several times faster than the standard functions, and you might not need all that extra precision after all.
There is a two-pronged approach that is used in the upcoming implementation. First, using the lower degree polynomials compare to that of the standard double precision library. Second, all the basic
arithmetic operations of multiply, divide, add, and subtract are done using the 32-bit integer arithmetic. First, we have selected the lower degree polynomials that reduce the amount of
multiplication operations.
We will begin our discussion with the general solution of transcendental functions; in other words, polynomial expansion.
Fig 1.2 Floating Point Foramt Single Precision
Reduced Coefficient Polynomials
A polynomial is a truncated form of an infinite series, suitable for computational purposes. We must stop solving the terms after achieving a certain degree of precision, as we cannot go on computing
forever, as required for an infinite series. If we were to compute the value of the exp function with the help of the Taylor series expansion as explained previously, we need to evaluate the terms
for at least 100 degrees before we get a result of three decimal places of accuracy—not a very practical suggestion.
What we need is to combine the coefficients beforehand in order to reduce the degree of the terms in our final analysis. B. Carlson and M. Goldstein in 1943 presented polynomial solutions for
transcendental functions that are suitable for digital computer applications. Source: Handbook of Mathematical Functions, Publisher Dovers (Pg 76..)
There are generally two sets of solutions for each basic function of sin, cos, tan, arctan, exponent, and logarithm. The high-order polynomial is for the double-precision result, and the low-order
polynomial is for the single-precision result. Other functions such as cosec, cot, and sec can be evaluated with the basic trigonometric identities. The following are the two sets of reduced
coefficient polynomials, obtained from the respective infinite series, that are guaranteed to be convergent for the input range specified.
The input to this function is the angle $$x$$ in radians, whose sin is to be determined. The governing polynomial is valid only for the range $$0 < x < \pi/2$$, Thus, the operand must be reduced to
the given range by applying the modulus rule for the sin function.
Low-degree polynomial for single-precision application:
$$ Input-range \cdots 0\le x\le \pi/2 \\ \\ {sin(x)\over x} = 1 + a_2x^2 + a_4x^4 + \epsilon \\ \\ error-magnitude \cdots |\epsilon|\le 2\times 10^{-4} \\ \\ {Constants: a_2 = -.16605, a_4=.00761} \\
\\ $$
High-degree polynomial for double-precision application:
$$ Input-range \cdots 0\le x\le \pi/2 \\ \\ {sin(x)\over x} = 1 + a_2x^2 + a_4x^4 + a_6x^6 + a_8x^8 + a_10x^10 + \epsilon \\ \\ error-magnitude \cdots |\epsilon|\le 2\times 10^{-9} \\ \\ {Constants:
a_2 = -.16666,66664, a_4=.00833,33315} \\ \\ {Constants: a_6 = -.00019,84090, a_8=.00000,27526} \\ \\ {Constants: a_10 = -.00000,00239} \\ \\ $$
The input to this function is the angle $$x$$ in radians, whose cos is to be determined. The governing polynomial is valid only for the range $$0 < x < \pi/2$$, Thus, the operand must be reduced to
the given range by applying the modulus rule for the cos function.
Low-degree polynomial for single-precision application:
$$ Input-range \cdots 0\le x\le \pi/2 \\ \\ {cos(x)} = 1 + a_2x^2 + a_4x^4 + \epsilon \\ \\ error-magnitude \cdots |\epsilon|\le 9\times 10^{-4} \\ \\ {Constants: a_2 = -.49670, a_4=.03705} \\ \\ $$
High-degree polynomial for double-precision application:
$$ Input-range \cdots 0\le x\le \pi/2 \\ \\ {sin(x)\over x} = 1 + a_2x^2 + a_4x^4 + a_6x^6 + a_8x^8 + a_10x^10 + \epsilon \\ \\ error-magnitude \cdots |\epsilon|\le 2\times 10^{-9} \\ \\ {Constants:
a_2 = -.49999,99963, a_4=.04166,66418} \\ \\ {Constants: a_6 = -.00138,88397, a_8=.00002,47609} \\ \\ {Constants: a_10 = -.00000,02605} \\ \\ $$
The governing polynomial for $$tan(x)$$ is defined in terms of its counterpart $$x\times cot(x)$$. $$tan(x)$$ is obtained by the analogy $$tan(x) = 1/cot(x)$$. Again, the angle x is in radians, whose
Tan is to be determined. The polynomial is valid only for the range$$0 < x < \pi/2$$ thus, the operand must be reduced to the given range by applying the modulus rule for the cotangent function.
Low-degree polynomial for single-precision application:
$$ Input-range \cdots 0 \le x \le \pi/4 \\ \\ {cot(x) = 1 + a_2x^2 + a_4x^4 + \epsilon \\ \\ error-magnitude \cdots |\epsilon|\le 3\times 10^{-5} \\ \\ {Constants: a_2 = -.332867, a_4=.024369} \\ \\
High-degree polynomial for double-precision application:
$$ Input-range \cdots 0 \le x \le \pi/4 \\ \\ {x \times cot(x)} = 1 + a_2x^2 + a_4x^4 + a_6x^6 + a_8x^8 + a_10x^10 + \epsilon \\ \\ error-magnitude \cdots |\epsilon|\le 2\times 10^{-6} \\ \\
{Constants: a_2 = -.33333,33410, a_4=-.02222,20287} \\ \\ {Constants: a_6 = -.00211,77168, a_8=-.00020,78504} \\ \\ {Constants: a_10 = -.00002,62619} \\ \\ $$
$$e^x$$ (exponent of x)
The polynomial for $$e^x$$ is defined for values less than 0.639. The exponent of larger numbers can be computed by splitting the input floating-point value into its integer and fraction portion, and
applying the following rule:
$$e^x = e^{(integer-portion + 0.693 + fraction-portion)}= e^{(integer-portion)} +2 e^{(fraction-portion)}$$
The exponent of the integer can quickly grow out of bounds. For example, the exponent of 31 is $$2.9^{1013}$$; anything above can be considered infinity.
$$ \\ {e^x} = {1 \over 1.0 + a_1x^1 + a_2x^2 + a_3x^3 + a_4x^4 + \epsilon} \\ \\ Input-range \cdots 0 \le x \le 3 \times 10^{-5} \\ $$
$$ \\ \\ error-magnitude \cdots |\epsilon|\le 3\times 10^{-5} \\ \\ {Constants: a_1 = -.99986,84, a_2 =.49829,26, } \\ {Constants: a_3 = -.15953,32, a_4 =.02936,41 } \\ \\ $$
High-degree polynomial for double-precision application:
$$ \\ Input-range \cdots 0 \le x \le 2\times 10\times10^{-10} \\ \\ {e^x} = {1 \over 1.0 + a_1x^1 + a_2x^2 + a_3x^3 + a_4x^4 + a_5x^5 + a_6x^6 + a_7x^7 + \epsilon} \\ \\ error-magnitude \cdots |\
epsilon|\le 2\times 10^{-6} \\ \\ {Constants: a_1 = -.99999,99995, a_2=.49999,99206} \\ \\ {Constants: a_3 = -.16666,53019, a_4=.04165,73475} \\ \\ {Constants: a_5 = -.00830,13598, a_6=.00132,98820}
\\ \\ {Constants: a_7 = -.00014,13161} \\ $$
$$ln_e(x)$$ ln Natural Logarithm
The polynomial for natural log is defined for values less than 1. The log of a number greater than 1.0 can be computed by applying the following rule. $$ln(x^n) = n ln(x)$$
Low-degree polynomial for single-precision application:
$$ Input-range \cdots 0 \le x \le 1.0 \\ \\ {ln(1+x)} = {a_1x^1 + a_2x^2 + a_3x^3 + a_4x^4 +a_5x^5 + \epsilon} \\ \\ error-magnitude \cdots |\epsilon|\le 1\times 10^{-5} \\ \\ {Constants: a_1 =
.99949,556, a_2=-.49190,896} \\ \\ {Constants: a_3=.28947,478, a_4=-.13606,275, a_5=.03215,845} \\ \\ $$
High-degree polynomial for double-precision application:
$$ Input-range \cdots 0 \le x \le 1.0 \\ \\ {ln(1+x)} = {a_1x^1 + a_2x^2 + a_3x^3 + a_4x^4 +a_5x^5 + a_6x^6 +a_7x^7 + a_8x^8 + \epsilon} \\ \\ error-magnitude \cdots |\epsilon|\le 3\times 10^{-6} \\
\\ {Constants: a_1 = .99999,64239, a_2=-.49987,41238} \\ \\ {Constants: a_3 = .33179,90258, a_4=-.24073,38084} \\ \\ {Constants: a_5 = .16765,40711, a_6=-.09532,93897} \\ \\ {Constants: a_7 =
.03608,84937, a_8=-.00645,35442} \\ \\ $$
The preceding two sets of polynomials offer different precision results for the function they represent. The high-degree polynomial offers higher precision, but requires a greater number of
computations; just imagine computing x to the power 8, 7, 6, 5, 4, 3, 2, and so on. The low-degree polynomial only requires computing maximum x to the power of 5—a magnitude of difference in
computational speed. The library functions developed in this chapter were designed around the low-degree polynomials.
Now, let us discuss the inner-workings of the computations. It is not enough to reduce the degree of the polynomial, as the bulk of CPU time is actually spent performing the four basic arithmetic
operations of multiply, divide, add, and subtract. If we were to use the default library functions for the basic operations, we essentially do not gain anything. The fundamental library routines will
convert the float into double, perform the operation in double, convert the double back into the float, and then return. These conversions are inherent in the library and cannot be avoided. All these
unnecessary conversions defeat the purpose of using the float format. The following discussion explores other possibilities of real number representation that are more efficient than the default
floating-point format.
Fixed-Point Arithmetic and Solution of Transcendental Functions
Is there an alternative to the floating-point arithmetic? We need floating point to represent the huge range of numbers we deal with. Putting –127 through + 127 exponent combined with the 23 bits of
precision digits can only be done in floating-point format. The other alternative is the fixed-point format. A 32-bit number can be thought of as carrying 8 bits of integer and 24 bits of fraction,
and a 64-bit number can be divided into 32 bits of integer and 32 bits of fraction. However, we then limit the numbers to a very narrow range.
The maximum integer value in an 8-bit integer is 255, and the maximum precision of the fraction in a 24-bit mantissa is 1 in 16 million—nearly .000001. All rational numbers are comprised of an
integer portion and a fraction portion. The decimal point is placed between them to distinguish the end of the fraction from the beginning of the integer.
The positional numbering system base $$b$$ defines a real number as follows:
$$ a_nb^n + .. + a_2b^2 + a_1b^1 + a_0b^0 + a_{-1}b^{-1} + a_{-2}b^{-2} + a_{-3}b^{-3} .. a_{-n}b^{-n} $$
For example, the decimal number 34.61 equals:
$$ 34.61 = 0\times 10^n + \cdots 3 \times 10^1 + 4 \times 10^0 + 6 \times 10^{-1} + 1 \times 10^{-2} + \cdots 0 \times 10^{-n} $$
Computers have a fixed word-length limit and are optimized for fixed word calculations. Each word can be thought of as an integer, or it could be treated as a fixed-point value with a predefined
integer portion and a predefined fraction portion. The decimal point is only a perception. The computational method is the same for both, the integer as well as the fixed-point values. The only
difference is the way in which the result is accumulated from the different sources where the CPU has placed the result after an arithmetic operation.
For example, a 32-bit number when multiplied by another 32-bit number produces a 64-bit result. The 80386 CPU utilizes the EAX:EDX register combination to store the result. The EDX register contains
the most significant 32 bits, while the EAX register contains the least significant 32 bits. In the context of a pure 32-bit integer operation, we ignore the result of the upper 32 bits and keep only
the lower 32 bits that is in the EAX register. However, if we think of the 32-bit value as an 8-bit integer and a 24-bit fraction, the result must be obtained by combining the upper 8 bits of the EAX
and the lower 24 bits of the EDX to bring the result back into 32-bit word format (Figure 1.3).
Fig. 1.3 Fixed Point Multiplication
The division operation for the two 32-bit fixed-point dividend and divisor is achieved by combining the lower 8 bits of EDX and the upper 24 bits of EAX as illustrated in Figure 1.4. Notice that the
32-bit fixed-point addition and subtraction are identical to the integer addition and subtraction. The following example demonstrates the multiplication and division operation on two 32-bit
fixed-point numbers with 8 bits of integer and 24 bits of fraction.
All is well and good if we could only guarantee that the result will not exceed 255, since this is the largest number we can put in a 32-bit fixed-point format of 8-bit integer and 24-bit fraction.
The fixed-point arithmetic is definitely far superior than the floating point, and allows us to perform arithmetic on real numbers with fractions—albeit, the range of numbers is very limited.
Successive multiplication of two integer numbers will soon overflow the result area; moreover, successive addition might cause the carryover that has no place in the result storage. On the other
hand, there is never a worry about the fraction portion being lost in the intermediate operations; at the most, we might lose some precision, which is acceptable since it happens to the
floating-point format also. If it were not for the fear of producing erroneous results in the intermediate stages, we would definitely gain considerable performance improvement by using the integer
instruction set of the CPU architecture for our basic calculations on the real numbers.
Let us take another look at the arithmetic operations on the polynomial functions discussed previously and the restrictions we have on the range of the coefficients and the input values that are
involved in the operation. Let us analyze all the arithmetic operations that are being required to solve a given polynomial and see if we can apply the fixed point arithmetic instructions on the
There is an input value x (always less than 1.0) that must be raised to its power several times; that is, successive multiplications (but no chance of result overflow). The result must be multiplied
by the coefficients—another multiply (all coefficients are fractions only, so no overflow of the result here either)— there is an addition or subtraction of all the terms (but no chance of exceeding
the maximum value, as the terms are all less than 1.0), and you have the result of the polynomial operation.
The preceding argument is presented to express the fact that we can safely compute the polynomial solutions using the fixed-point format alone, and that is the format being used to develop the
library of functions in the next section. The ideal implementation of the fixed-point format would be to use the native CPU instructions of the 32-bit integer add, subtract, multiply, and divide.
However, that would require an interface with the Assembly language routines, which would cause portability problems across different platforms. Instead, the basic routines are simulated in the C
language functions using the integer operations of the C compiler. Interested readers could rewrite the functions in Assembly language to gain further performance improvement and optimize the speed
even more.
Fig. 1.4 Fixed Point Division
A Fast and Efficient Algorithm for the Square Root Function
Another example of a fixed-point implementation is a solution of the square root function. The algorithm is based on the fact that a floating-point number can easily be split into its exponent and
fraction portion, and each can be treated as a separate, fixed-point quantity. Think about a floating-point number in terms of its scientific notation with the exponent portion multiplied by a
fraction portion. Using the multiplicative property of the square root function, finding the square of a number is simply the square root of the exponent portion multiplied with the square root of
the fraction portion, as shown here:
$$ {\sqrt {2^x \times fraction}=\sqrt {2^x} \times \sqrt {fraction}} \\ $$
For example, the square root of the number 3.0 is equivalent to:
$$ {\sqrt {3}=\sqrt {2^2} \times \sqrt {0.75}} \\ $$
$$ {\sqrt {3}={2^1} \times \sqrt {0.75}} \\ $$
The sqrt of the exponent portion is obtained by shifting the exponent to the right 1 bit; in other words, dividing it by 2.
The fraction quantity is always less than 1.0 by definition, so we can apply the same fixed-point technique we developed in the previous section of transcendental functions.
The sqrt of a 32-bit fraction is obtained by a binary search through maximum 16 iterations of 16-bit by 16-bit multiplication of an approximation. At the end, the individual results are simply
combined into floating-point format and returned.
There are two steps in this algorithm:
1. sqrt of exponent = exponent shift right 1
2. sqrt of fraction = $$X$$, if ($$X \times X = fraction$$)
See lines 430through 555in Code Listing 1.1 for an implementation of the algorithm.
/************* Transcendental Functions ***********
The sin, cos, tan, exp, ln and square root function
solution are presented using fixed point arithmetic
and reduced coefficient polynomials.
You can use the file as an include or compile into
a library for linking with other modules. The names
of the functions are appended with an underscore such
as __cos so that it does not conflict with the deafulat
library routines.
The followings are the exported functions
double __sin(double x);
double __cos(double x);
double __tan(double x);
double __exp(double x);
double __lge(double x);
double __lgd(double x);
double __sqrt(double x);
#include <stdio.h>
#include <math.h>
#define FLOATDIGITS 6
#define MAXINTEGER 2.147483647e+9
#define debug
#define test
#define fixed
#define TWO_POINT_ZERO 2.0
#define TWO_PI_CONST 6.283185
#define PI_CONST 3.141593
#define PI_2_CONST 1.570796
#define PI_4_CONST 0.7853982
#define ONE_POINT_ZERO 1.0
#define CONST_POINT_693 0.693
#define CONST_LN_2 0.69314718
#define CONST_LN_10 2.302585093
#define TWO_POINT_ZERO 2.0
#ifndef INFINITY
#define INFINITY 1.0E99
#ifndef NAN
#define NAN 0.5E99
#define modulous(r1,r2) (r1- (((int) (r1/r2))* r2))
/* float xtemp,ytemp; */
The coeffiecients of the polynomials are converted
into fixed point format as the polynomial solution
requires the parameters to be presented in a fixed-
point format
#ifdef fixed
long SIN_COEF_TBL[5] = {0x1000000,0,0xffd57dc0,0,0x1f2ba};
long COS_COEF_TBL[5] = {0x1000000,0,0xff80d845,0,0x97c1b};
long COT_COEF_TBL[5] = {0x1000000,0,0xffaac93b,0,0xfff9c2f5};
long ATAN_COEF_TBL[10] = {0,0xfff738,0,0xffab717e,0,0x2e1db8,0,0xffea34ba,
long LN_COEF_TBL[6] = {0,0xffdef1,0xff821241,0x4a1b05,0xffdd2afe,0x83b89};
long EXP_COEF_TBL[5] = {0x1000000,0xff0008a0,0x7f901a,0xffd728d5,0x78467};
float SIN_COEF_TBL[5] = {1.0,0.0,-0.16605,0.0,0.00761};
float COS_COEF_TBL[5] = {1.0,0.0,-0.49670,0.0,0.03705};
float COT_COEF_TBL[5] = {1.0,0.0,-0.332867,0.0,-0.024369};
float ATAN_COEF_TBL[10] = {0.0,0.9998660,0.0,-0.3302995,0.0,
float LN_COEF_TBL[6] = {0.0,0.99949556,-0.49190896,0.28947478,
float EXP_COEF_TBL[5] = {1.0,-0.9998684,0.4982926,-0.1595332,0.0293641};
Exponent is calculated as exponent of integer + exponent
of fraction. Exponent of up to 32 integers are sufficient
for all practical purpose, Beyond that it might as well
be infinity.
A lookup table of exponent of integers is presented below
float INT_EXP_TBL[33] = {
/****************** function define ****************/
double __sin();
double __cos();
double __tan();
double __exp();
double __lge();
double __lgd();
double __atan();
double fx2fl();
long fl2fx();
long fx_mpy();
general purpose conversion routines
/***************floating point to fixed point *********/
long fl2fx(double x)
long *k,mantissa,exp;
float temp;
temp = (float) x;
/* get a pointer to fl pt number */
k = (long *) & temp;
/* remove sign bit from the number */
mantissa =(*k & 0x7fffffff);
/* extract exponent from the mantissa */
exp = ((mantissa >> 23) & 0xff);
if (!exp)
return (0);
/* mantissa portion with normalized bit */
mantissa = ((mantissa & 0xffffff) | 0x800000);
/* inxrease exp for bringing in normalize bit */
if (exp > 0x7f)
/* exp > 0x7f indicates integer portion exist */
exp = exp - 0x7f;
mantissa = mantissa << exp;
/* exp <= 0x7f indicates no integer portion */
exp = 0x7f - exp;
mantissa = mantissa >> exp;
if (*k >= 0)
return (-mantissa);
/************* fixed point to floating point *********/
double fx2fl(long j)
long i,exp;
float temp,*fp;
i = j;
if (i == 0)
return ((double)(i));
maximum exp value for an 8-bit int and 24-bit fraction
fixed point
biasing factor = 7f;
7f + 8 = 87;
exp = 0x87;
if (i < 0)
exp = exp | 0x187;
i = -i;
/* normalize the mantissa */
i = i << 1; exp --;
while (i > 0);
i = i << 1; exp --;
/* place mantissa on bit-0 to bit-23 */
i = ((i >> 9) & 0x7fffff);
exp = exp << 23;
i = i | exp;
fp = (float *) & i;
temp = *fp;
return ((double) temp);
/*************** fixed point multiply ****************/
long fx_mpy(long x,long y)
unsigned long xlo,xhi,ylo,yhi,a,b,c,d;
long sign;
/* The result sign is + if both sign are same */
sign = x ^ y;
if (x < 0)
x = -x;
if (y < 0)
y = -y;
two 32 bit numbers are multiplied as
if there are four 16-bit words
xlo = x & 0xffff;
xhi = x >> 16;
ylo = y & 0xffff;
yhi = y >> 16;
a = xlo * ylo;
b = xhi * ylo;
c = xlo * yhi;
d = xhi * yhi;
a = a >> 16;
/* add all partial results */
a = a+b+c;
a = a >> 8;
d = d << 8; a = a+d;
if (sign < 0)
a = -a;
/*************** fixed point polynomial ****************/
Three input parameters; all fixed point
input x;
list of coefficients
number of coefficients in the list
double polynomial_x(double r1,long * r2,int i)
long j,k,temp,sum;
k = fl2fx(r1);
sum = 0;
/* steps
find power of x,
multiply with coefficient
add all terms
temp = 0x1000000; /* this is 1.0 */
if (r2[i] != 0)
/* power of x */
for (j=i;j>0;j--)
temp = fx_mpy(temp,k);
/* and then multiply with coefficient */
temp = fx_mpy(temp,r2[i]);
temp = 0;
sum = sum + temp;
/********************* sin ************************/
double __sin(double x)
float temp;
int quadrent;
unsigned long *k;
temp = (float) x;
/* make absolute value */
k = (unsigned long *) &temp;
*k = (*k & 0x7fffffff);
/* 360 degree modulus */
if (temp > TWO_PI_CONST)
temp = modulous (temp,TWO_PI_CONST);
/* negative angle are made complements of 360 degree */
if ( x < 0)
temp = TWO_PI_CONST - temp;
quadrent = (int) (temp / PI_2_CONST);
temp = modulous(temp,PI_CONST);
if (temp > PI_2_CONST)
temp = (PI_CONST - temp);
temp = (polynomial_x(temp,SIN_COEF_TBL,5) * temp);
if (quadrent >= 2)
temp = -temp;
return (temp);
/********************* cos ************************/
double __cos(double x)
float temp;
int quadrent;
unsigned long *k;
temp = (float) x;
/* make absolute value */
k = (unsigned long *) &temp; *k = (*k & 0x7fffffff);
/* 360 degree modulus */
if (temp > TWO_PI_CONST)
temp = modulous (temp,TWO_PI_CONST);
/* negative angle are made complements of 360 degree */
if ( x < 0)
temp = TWO_PI_CONST - temp;
/* find out the quadrant from the original angle */
quadrent = (int) (temp / PI_2_CONST);
temp = modulous(temp,PI_CONST);
if (temp > PI_2_CONST)
temp = (PI_CONST - temp);
temp = (PI_2_CONST - temp );
temp = (polynomial_x(temp,SIN_COEF_TBL,5) * temp);
if ((quadrent ==1) || (quadrent == 2)) temp = -temp;
return ((float) (temp));
/********************* tan ************************/
double __tan(double x)
float temp;
int quadrent;
unsigned long *k;
temp = (float) x;
/* make absolute value */
k = (unsigned long *) &temp;
*k = (*k & 0x7fffffff);
/* 360 degree modulus */
if (temp > TWO_PI_CONST)
temp = modulous (temp,TWO_PI_CONST);
/* negative angle are made complements of 360 degree */
if ( x < 0)
temp = TWO_PI_CONST - temp;
quadrent = (int) (temp / PI_2_CONST);
temp = modulous(temp,PI_CONST);
if (temp > PI_2_CONST)
temp = (PI_CONST - temp);
if (temp < PI_4_CONST)
temp=((ONE_POINT_ZERO /
polynomial_x(temp,COT_COEF_TBL,5)) * temp);
temp = PI_2_CONST - temp;
temp = ONE_POINT_ZERO /
((ONE_POINT_ZERO /polynomial_x(temp,COT_COEF_TBL,5)) * temp);
if ((quadrent == 1)||(quadrent==3))
temp = -temp;
return (temp);
/********************* exp ************************/
/* e pwr x = 2 pwr (x * log2_e) */
/* if |X| <= 0.693.. then exp(x) = exp(fraction) */
/* else exp(x) = exp(INT + 0.693 + FRACTION) */
/* exp(0.693) = 2;*/
double __exp(double x)
float temp = x;
int i;
unsigned long *k;
if (x > 0.0)
i= ((int) (x + 0.0000001));
i= ((int) (x - 0.0000001));
if (i < 0)
i = 0-i;
if (i > 31)
if (x > 0)
temp = (float) x;
/* make absolute value */
k = (unsigned long *) &temp;
*k = (*k & 0x7fffffff);
temp = modulous(temp,ONE_POINT_ZERO);
if (temp > CONST_POINT_693)
temp = temp - CONST_POINT_693;
temp = ONE_POINT_ZERO /
temp = temp * TWO_POINT_ZERO;
temp = ONE_POINT_ZERO /
temp = (INT_EXP_TBL[i]) * temp;
if (x < 0)
temp = ONE_POINT_ZERO / temp;
return (temp);
/********************* ln (x)************************/
/* log base E (X) = ln (x)/ln (E) */
/* ln (x) = pwr * ln(2) + ln(fraction+1;*/
/* range =(0 <= x <= 1)*/
double __lge(double x)
static long exp,i,*k;
static float temp,xtemp,*fraction;
xtemp = (float) x;
k =(long *) &xtemp; i=k[0];
/* do not compute ln of a minus number */
if (i <= 0)
return (NAN); /* NAN could be replaced by 0.0 */
/*remove exp and sign and check for fraction*/
if ((i =(i & 0x7fffff)) != 0)
exp = 0x7f;
while ((i & 0x800000) == 0)
i = i << 1; exp--;
i = i & 0x7fffff; /* remove normalized bit */
exp = exp <<23; /* new exp for fraction */
i = i | exp; /* combine exp and mantissa */
fraction = (float *) &i;
temp = fraction[0];
temp = (polynomial_x(temp,LN_COEF_TBL,6));
/* from input value x extract the exp */
exp = *k;
exp = exp >> 23;
exp = exp - 0x7f;
temp = temp + (exp * CONST_LN_2);
/********************* lgd ************************/
double __lgd(double x)
return (__lge(x)/CONST_LN_10);
/********************* square root **********************
A fast and efficient algorithm for Square Root function.
The algorithm is based on the fact that a floating point number
can be easily split into its exponent and fraction portion and
each can be treated as separate integer quantity. The sqrt of
exponent portion is obtained by shifting the exponent right 1-
bit, i.e. dividing it by 2 and the sqrt of a 32-bit fraction is
done by a binary search of 65536 possible values or 16 trials of
16-bit x16-bit multiplication of an approximation. At the end,
the individual results are simply combined Into floating point
format and returned.
sqrt of exponent = exponent shift right 1 ........ (1)
sqrt of fraction = X, if (X x X = fraction) ........ (2)
double __sqrt(double x)
float xtemp;
unsigned long exp,i,j,*k;
unsigned short int new_apprx,old_apprx;
xtemp = (float) x;
step 1------ Think of the floating point number as an
integer quantity
k =(unsigned long *) &xtemp; i=k[0];
step 2------ do not compute sqrt of a minus number
if (x <= 0.0)
if (x==0.0)
return (NAN); /* NAN could be replaced by 0.0 */
step 3--- extract exp and place it in a byte location
exp = (i & 0x7f800000);
exp >>= 23;
step 4 --- bring fraction portion to 32-bit position and
bring normalized bit
i <<= 8;
i = i | 0x80000000;
step 5 --- inc exp for bringing in normal bit and
remove exp bias
exp += 1;
exp -= 0x7f;
step 6 --- compute square root of exp by shift
right or divide by 2
if exp is odd then shift right mantissa
one more time
if ( exp & 1)
i >>= 1;
exp >>= 1;
exp += 1;
exp >>= 1;
step 7--- add bias to exp
exp += 0x7f;
/*-----------------sqrt of fraction ---------------*/
step 8 -----start with 0x8000 as the first apprx
j = old_apprx = new_apprx = 0x8000;
step 9 ---use binary search to find the match.
loop count maximum 16
step 9a ---- multiply approximate value to itself
j = j*j;
step 9b ---- next value to be added in the binary
old_apprx >>= 1;
step 9c ---- terminate loop if match found
if (i == j )
old_apprx = 0;
step 9d --- if approx. exceeded the real fraction
then lower the approximate
if (j > i )
new_apprx -= old_apprx;
step 9e --- else increase the the approximate
new_apprx += old_apprx;
j =new_apprx;
} while (old_apprx != 0);
step 10 --- combine exp and mantissa
j =new_apprx;
step 10a --- bring mantissa to bit position 0..23
and remove Normal bit
j <<= 8;
j &= 0x7fffff;
step 10b --- decrement exponent for removing Normalized
exp -= 1;
step 10c --- bring exponent to position 23..30
exp <<= 23;
step 10d --- combine exponent and mantissa and
return double
j = j | exp;
xtemp = * ((float *) &j);
return (xtemp);
In this article, we developed an efficient transcendental math emulation library suitable for low-cost embedded systems. If the emphasis is on speed rather than precision, then, in the absence of
hardware math-coprocessors, the alternative is fixed-point arithmetic. There are several applications, such as graphics routines, that can take advantage of the high-speed solution offered by the
library presented in this chapter. A unique solution to the square root function was also presented for floating-point numbers that is several orders of magnitude faster than a comparable library
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Built-in functions : hypot function
hypot function
Returns length of hypotenuse of right-angle triangle with base X and height Y
Inputs: numeric, numeric
Result: numeric
hypot(3,4) --> 5 (a 3:4:5 triangle)
hypot(x1-x2,y1-y2) --> the distance between two points with co-ordinates (x1,y1) and (x2,y2) respectively.
hypot(x-[xs],y-[ys]) --> [distances] I.e. an array containing the distance from one point with co-ordinates (x,y) to a set of points, with co-ordinates held in the arrays [xs] and [ys]. See comments.
Spatial modelling frequently requires that one object knows the distance to another. This requires that each has x,y co-ordinates. It is then simple to use the hypot function to work out the
straight-line distance between them, as shown in the second example above.
The same principle applies when you use a multiple-instance submodel to represent a set of spatially-located objects. In this case, each object may want to know how far it is to all the other objects
- for example, in working out the competition between trees in an individual-based tree model. The following model diagram fragment shows a typical model configuration for doing this:
Each tree has x,y co-ordinates. These are exported to two array variables, xs and ys, whose equations are simply:
xs = [x]
ys = [ys]
These arrays are then brought back into the submodel, and used to generate an array containing the distance for each tree to all the other trees, using the equation given in the third example above.
In: Contents >> Working with equations >> Functions >> Built-in functions
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IBU/Hop Bitterness
Calculating IBU/Hop Bitterness
International Bitterness Units (IBU) is a measure of the bitterness in homebrew. One IBU is the same as one milligram of isomerized alpha acid per liter of homebrew. Isomerized alpha acids are the
main bittering acids derived from hops. Calculating IBU's can be very tricky, because of this there are many different calculations for determining IBU's. All of these calculations will produce
different results. Brewgr uses the Tinseth formula for now. We are working on a new formula for calculating the IBU's in homebrew and will update as soon as we are complete.
Gravity = (Batch Size In Gallons / Boil Size In Gallons) * (Gravity - 1)
Bigness Factor = 1.65 * Math.pow(0.000125, Gravity)
Boil Time Factor = (1 - Math.pow(2.718281828459045235, (-0.04 * Boil Time))) / 4.15
Utilization = Bigness Factor * Boil Time Factor
If you are using hop pellets we then mutiply the utilization by 1.1.
IBU = (Alpha Acid * Ounces) * Utilization * 74.90 / Batch Size In Gallons
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C Program to Check Whether a Year is Leap Year or Not - Programtopia
C Program to Check Whether a Year is Leap Year or Not
A year that has 366 days is called a leap year.
A year can be checked whether a year is leap year or not by dividing the year by 4, 100 and 400. If a number is divisible by 4 but not by 100 then, it is a leap year. Also, if a number is divisible
by 4, 100 and 400 then it is a leap year. Otherwise the year is not a leap year.
Example 1: Source Code to Check Leap Year
#include <stdio.h>
int yr;
printf ("Enter a year n");
scanf ("%d", &yr);
if (yr%4 == 0) {
if(yr%100 == 0) {
if(yr%400 == 0)
printf("n It is LEAP YEAR.");
printf("n It is NOT LEAP YEAR.");
else {
printf ("n It is LEAP YEAR.");
printf("n It is NOT LEAP YEAR.");
return 0;
Here, the year entered by the user is firstly divided by 4. If it is divisible by 4 then it is divided by 100 and then 400. If year is divisible by all 3 numbers then that year is a leap year. If the
year is divisible by 4 and 100 but not by 400 then it is not a leap year. If the year is divisible by 4 but not by 100, then it is a leap year. (Remember that if the year is divisible by 4 and not by
hundred then the program does not check the last condition, i.e., whether the year is divisible by 400). If the year is not divisible by 4 then no other conditions are checked and the year is not a
leap year.
Example 2: Source Code to Check Leap Year
#include <stdio.h>
int main()
int yr;
printf ("Enter a year n");
scanf ("%d", &yr);
if (yr%4 == 0 && yr%100 == 0 && yr%400 == 0)
printf("n It is LEAP YEAR.");
else if (yr%4==0 && yr%100!=0)
printf("n It is LEAP YEAR.");
printf ("n It is NOT LEAP YEAR.");
return 0;
Here, if the year is divisible by 4, 100 and 400 then “It is LEAP YEAR.” is displayed. If the year is divisible by 4 but not by 100 then “It is LEAP YEAR.” is displayed. Otherwise, “It is NOT LEAP
YEAR” is displayed.
Enter a year
It is LEAP YEAR.
Enter a year
It is NOT LEAP YEAR.
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Classify the following measures as scalars and vectors.(i) 5 ... | Filo
Question asked by Filo student
Classify the following measures as scalars and vectors.(i) 5 seconds (ii) 1000
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To classify the measures in the given question, we need to understand what is scalar and what is vector. A scalar is a physical quantity that has only a magnitude (size) and no direction. Examples:
mass, time, temperature, distance etc. A vector is a physical quantity that has both magnitude (size) and direction. Examples: displacement, force, velocity, acceleration etc. So, we need to classify
the given measures as scalar or vector. Let's look at the solution: Step 1. For the measure '5 seconds':
It is a time period, which only has magnitude (size) and no direction.
Hence, it is a scalar. Step 2. For the measure '1000 ':
It represents the volume of an object which has only magnitude (size) and no direction.
Hence, it is also a scalar. Therefore, both measures are scalar quantities.
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Question Text Classify the following measures as scalars and vectors.(i) 5 seconds (ii) 1000
Updated On Nov 3, 2023
Topic Algebra
Subject Mathematics
Class Class 12
Answer Type Text solution:1
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When Use Ordinal Numbers - OrdinalNumbers.com
Ordinal Numbers Sums – A vast array of sets can be enumerated using ordinal numbers as a tool. They can also be used to generalize ordinal quantities. 1st The ordinal number is among the fundamental
concepts in mathematics. It is a number that indicates where an object is in a list. The ordinal number is … Read more
Ordinal Numbers Posters
Ordinal Numbers Posters – You can list an unlimited amount of sets using ordinal figures as tool. These numbers can be utilized as a method to generalize ordinal figures. 1st The ordinal number is
among the foundational ideas in mathematics. It is a number that identifies the location of an object within the list. An … Read more
Ordinal Numbers Set Theory
Ordinal Numbers Set Theory – It is possible to enumerate infinite sets with ordinal numbers. It is also possible to use them to generalize ordinal numbers. 1st The ordinal numbers are one of the
fundamental concepts in mathematics. It’s a number that signifies the location of an item within the list. The ordinal number is … Read more
Ordinal Numbers Problem Solving
Ordinal Numbers Problem Solving – You can count unlimited sets with ordinal numbers. They can also help to generalize ordinal numbers. 1st One of the basic concepts of math is the ordinal number. It
is a number that indicates the location of an object in an array. Ordinarily, ordinal numbers are between one to twenty. … Read more
Counting Ordinal Numbers
Counting Ordinal Numbers – It is possible to enumerate infinite sets by using ordinal numbers. These numbers can be used as a method to generalize ordinal numbers. 1st The ordinal number is among the
fundamental concepts in mathematics. It is a number indicating the place of an object in a list. Ordinal numbers are typically … Read more
Ordinal Numbers In Hebrew
Ordinal Numbers In Hebrew – You can enumerate unlimited sets by using ordinal numbers. These numbers can be utilized as a way to generalize ordinal figures. 1st The ordinal number is one the
foundational ideas in mathematics. It is a number that indicates the position of an object in the set of objects. Ordinal numbers … Read more
Ordinal Numbers Aoa
Ordinal Numbers Aoa – There are a myriad of sets that can be listed by using ordinal numbers as an instrument. They also aid in generalize ordinal numbers. 1st Ordinal numbers are among the most
fundamental ideas in math. It is a number that shows where an object is in a list. Ordinally, a number … Read more
Ordinal Numbers Challenge
Ordinal Numbers Challenge – An unlimited number of sets can be listed by using ordinal numbers as tools. You can also use them to generalize ordinal number. 1st The ordinal number is among the
foundational ideas in mathematics. It is a number that identifies the position of an object within a list. Ordinal numbers are … Read more
Article Before Ordinal Numbers
Article Before Ordinal Numbers – There are a myriad of sets that can be listed using ordinal numbers to serve as an instrument. They also can be used as a generalization of ordinal quantities. 1st
One of the basic concepts of mathematics is the ordinal number. It is a number that indicates the position of … Read more
Ordinal Numbers Ap Style
Ordinal Numbers Ap Style – An unlimited number of sets can be listed using ordinal numbers to serve as an instrument. It is also possible to use them to generalize ordinal number. 1st The basic
concept of mathematics is the ordinal. It is a number that indicates where an object is in a list of … Read more
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Range in Math - Math Steps, Examples & Questions
Calculate the missing value.
To find the lowest value you will need to subtract the range from the highest value.
\text{Lowest value}=28-17=11
The lowest value is 11.
This answer can be doubled checked:
\text{Range}=\text{highest value}-\text{lowest value}=28-11=17
What is the range in math?
The range in math is the difference between the highest value and the lowest value in a data set.
How do you find the range in math?
To find the range, you simply subtract the lowest value of the data set from the highest value.
How is the range of a data set affected by outliers?
Outliers can significantly affect the range of a data set. If there are extreme values that are much larger or smaller than the rest of the data, the range will be skewed or stretched to include
these outlier values. This means that the presence of outliers can result in a larger range if there is a high outlier or a low outlier.
What are descriptive statistics?
Descriptive statistics are methods used to summarize or organize a data set. Measures of central tendency such as the mean, median, mode, and range are common descriptive statistics that give a quick
overview of the data set.
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Let the Games Begin
Laureates of mathematics and computer science meet the next generation
We are currently in the midst of the Olympic and Paralympic games – two epic events of world-class sport and competition, taking place in Paris and pitting the world’s sporting heroes against each
other (with plenty of wonderful statistics and physics along the way). But if you prefer your games to be more sedate, I can recommend a game which I have played for many years – and that will go on
for a lot longer than seventeen days if you want it to!
Non-Olympic Games
When I was a child, I often played card games with my grandad – he taught me many games he knew, including one which I enjoyed a lot, called “Beggar-my-neighbour” (otherwise known by a variety of
interesting names, including “Strip Jack Naked” and “Beat Your Neighbour Out Of Doors”). The game is fairly old – having been played since around the 1840s, it is even featured as a children’s game
in Charles Dickens’s novel Great Expectations. The game involves splitting a deck of cards between all the players, then taking it in turns to play on to a central pile.
Certain cards in the game – picture cards and aces – have a “penalty” associated with them. A jack demands a penalty of one card, while a queen is two, a king is three and an ace is four. If one of
the players puts down one of these cards, the following player must then play that many cards in response; if they do so, and all the cards they play are just numbered cards rather than being penalty
cards, the player who demanded the penalty wins that trick, and picks up the whole pile played so far.
But if during that penalty countdown, the player plays another penalty card, this passes the penalty on to the next player – the existing count is abandoned and a new one started for the new penalty
card. In the two-player version of the game in particular, this can lead to some high-octane back-and-forth rallies, in which players keep deflecting the penalty back on to each other until finally
someone runs out of luck. The more times you pick up cards, the larger your hand will be – and the winner is the person who eventually ends up holding all 52 cards of the deck.
When I played this game as a child, I enjoyed the ebb and flow of play; after calm stretches of number cards, the sudden excitement of a penalty challenge and its resolution was a little rush. I was
very deeply convinced that not only was this my favourite game, but that also I was definitely very good at it.
Of course, looking back on this as an adult, I can say for sure that I definitely was not, in any meaningful way, very good at this game. Since the actions taken by each player are entirely
determined by the ordering of the cards in the deck once the game starts, it is actually fully known from the start exactly how the game will go, and who will win. It is, in essence, a zero-player
game (much like the ones I wrote about previously) – unless you count the person who initially shuffles the cards, who is the only one who has any influence over the outcome of the game.
Determined to win
Now that I can understand this game more logically and analyse it mathematically, I am aware there is a layer to it that my childhood self may not have appreciated. Studying deterministic systems is
common practice in mathematics, and trying to determine or predict which initial states will lead to which results can be an interesting puzzle.
Since each card turned over determines exactly what happens next (the other player puts down one or more cards, or someone picks up) the ordering can be fed into an algorithm to play through the
resulting game. For example, given an initial deal of the deck, it should be possible to know not only which player will win, but exactly how long it will take them to win – how many cards will be
played before someone runs out of cards?
This in itself presents an interesting set of further questions; for example, what is the average length of a game given a random deal? And how long is it possible for a game to be? For years,
mathematicians tinkering with the game have wondered about this last question, and challenged each other to find initial deck orderings which result in longer and longer games. Given the immense
number of possible deals to study (calculated using combinatorics, which I’ve written about here previously) the chances of finding one with a particular property are small, unless you have a clever
Richard Mann, a mathematician at the University of Leeds, was one of a team who held the record for the longest possible game from around July 2007 to May 2012 – lasting for 1007 tricks and involving
the playing of 7157 cards. The deck orderings are specified in terms of only the picture cards, since the value of the non-penalty cards does not affect the game at all. Their game was given by the
Player 1: K-KK----K-A-----JAA--Q--J-
Player 2: ---Q---Q-J-----J------AQ--
Once this record was broken, and people continued to find new longer games, Mann set up a website logging all known Beggar-my-neighbour records to serve as a repository for the search. But some
people wanted to know more: would it be possible to find an infinite game?
The question of whether this is possible – to have a game which will at some point reach a loop and continue with two players playing the same cards in the same order for the rest of time – was one
that stumped the mathematicians working on this: an open question.
Of course, finding the answer would not result in awards or accolades, or unlock whole new areas of research. The mathematician John Conway described the problem as an “anti-Hilbert problem”; by
contrast with the Hilbert problems (major open questions mathematicians were challenged to seek answers to), this was one which “emphatically should not drive mathematical research.” Michael Kleber,
who himself held the record for the longest game at one point, said of his achievement, “This is almost entirely uninteresting.” Others have wisely stated, “This question has definitely not plagued
scientists for millennia.”
Game Over?
Nonetheless – the lure of an unsolved puzzle is always going to draw some people to work on it. And until recently, it remained unsolved – longer and longer games were found, including one with a
monster 1164 tricks, involving playing 8344 cards (taking over two hours to play, at one card per second). But this still would not satisfy those looking for a game which loops.
And then, finally, earlier this year, it happened. In February 2024, BMN enthusiast Brayden Casella shared a deal which led to an infinite game:
Player 1: ---K---Q-KQAJ-----AAJ--J--
Player 2: ----------Q----KQ-J-----KA
The first 34 cards played are not part of the loop, but after 474 turns and 66 tricks (and after 33,034 cards have been played), the game goes back to the state it was in after the 34th card was
played, and we are trapped in an infinite loop. The loop reached in this game can be reached from any one of 30 distinct starting hands, but once you are in it, there is no escape.
The finding has been published as a paper on the ArXiV, and if you want to see the loop for yourself, chess and puzzle enthusiasts John and Sue Beasley have published this PDF listing the tricks as
they are played. You can also test it for yourself – Richard Mann’s website links to BeggarMyPython, a script by Matt Mayer (the source of my final quote above) which can be used to run and test
And if you would like to just watch the action slowly unfold, there is a Mastodon bot which is posting the newly discovered infinite game as a play-by-play, one turn every three hours – as I write,
an exciting scuffle has just taken place in which one player picks up nine cards. It is tense, not knowing who will win (although the answer is: Nobody will win, as the game will go on forever).
So, if you would like to play a game that has a tiny chance of continuing for the whole of the rest of time – so far, only a handful of deals with this property have been discovered, but who knows
how many more are out there – why not try a game of Beggar-my-neighbour? You might even find you are as good at it as I am!
2 comments
1. Nice inspiration, thanks a lot.
2. “Games have always been a fantastic way to unwind and learn simultaneously! I’ve been enjoying some great educational games lately, especially on mathplayground which combines fun and learning
perfectly. It’s worth checking out if you’re looking for a productive way to play!
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While I was looking at the Redis Source code, after taking a break, I noticed the addition of the Radix Tree Implementation, which apparently is still not added to the Redis Modules API yet.
From what I can tell, Radix Trees are used to implement the new Streams, which were added to Redis, starting from version 5.0
Radix Tree is a power data structure that can be used for some Applications. Radix Trees for example can be used to implement autocompetion for example.
In this post I just want provide a very simple C program that uses Redis Radix Trees implementation to implement a basic associate array, where both the key and the value are strings.
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Arc Calculator Foe - Calculator City
Arc Calculator Foe
Enter your ARC value and FOE rate into the calculator to determine your final FOE value.
FOE Calculation Formula
The following formula is used to calculate the final FOE value from your ARC.
Final FOE = ARC * (1 + FOE Rate / 100)
• Final FOE is the net FOE value you calculate ($)
• ARC is the initial ARC value ($)
• FOE Rate is the percentage of FOE applied to the ARC (%)
To calculate the final FOE, multiply the ARC by one plus the FOE rate divided by 100.
What is ARC and FOE Calculation?
ARC and FOE calculation refers to the process of determining the final FOE value from the initial ARC value and the FOE rate. This involves understanding the ARC, the FOE rate, and applying the
calculation formula accurately. Proper ARC and FOE calculation ensures precise financial planning and assessment.
How to Calculate Final FOE?
The following steps outline how to calculate the final FOE value using the given formula.
1. First, determine your ARC value.
2. Next, determine the applicable FOE rate.
3. Use the formula from above: Final FOE = ARC * (1 + FOE Rate / 100).
4. Finally, calculate the final FOE by plugging in the values.
5. After inserting the variables and calculating the result, check your answer with the calculator above.
Example Problem:
Use the following variables as an
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Definition Of Math Anxiety
Wavebreakmedia Ltd/Wavebreak Media/Getty Images
Math anxiety is defined as feeling of anxiety that one cannot perform efficiently in situations that involve the use of mathematics. Although it is mostly associated with academics, it can apply to
other aspects of life.
Math anxiety is an emotional problem, and it is characterized by intense nervousness before or during math tests. This interferes with a person's ability to optimally do math problems, thus morphing
into an intellectual problem.
In most cases, math anxiety is the result of a previous embarrassing experience or a moment of failure involving mathematics. This deters the person from believing in, let alone performing to, his
full potential. A Stanford University study that was reported in 2012 found it might have a biological basis — elementary school children who became anxious doing math showed greater fear, and lesser
problem solving skills, in brain scans.
Professional/Personal Life
Math anxiety extends beyond the classroom. People could be discouraged from applying to job opportunities that substantively involve numbers, or perform poorly in tasks that require math. Unpaid
bills and taxes, unforeseen debts and unbalanced checkbooks can be results of avoidance or insufficient knowledge of numbers.
Math anxiety can be prevented, reduced or eliminated in a number of ways. They include reviewing and learn basic arithmetic principles and methods, using anxiety reduction and anxiety management
techniques, and getting a math tutor.
Numbers are everywhere — in every aspect of society. Thus math anxiety needs to be conquered in order to thrive optimally.
Cite This Article
Joseph, Andy. "Definition Of Math Anxiety" sciencing.com, https://www.sciencing.com/definition-math-anxiety-5666297/. 24 April 2017.
Joseph, Andy. (2017, April 24). Definition Of Math Anxiety. sciencing.com. Retrieved from https://www.sciencing.com/definition-math-anxiety-5666297/
Joseph, Andy. Definition Of Math Anxiety last modified August 30, 2022. https://www.sciencing.com/definition-math-anxiety-5666297/
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Probability with R, 2nd Edition
9Introduction to Discrete Distributions
In this chapter, we introduce the concepts of discrete random variables and expectation and go on to develop basic techniques for examining them by simulation in R. Along the way, we introduce the
Bernoulli and uniform discrete distributions. We end the chapter with an introduction to discrete bivariate distributions.
9.1 Discrete Random Variables
A random variable is a rule that assigns a numerical value to each possible outcome of an experiment, that is, a mapping from the sample space
Let us look at some of the examples considered in Chapter 4 and show how to obtain their probability distributions.
Get Probability with R, 2nd Edition now with the O’Reilly learning platform.
O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.
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Linear and non-linear
Back to top: Linear and non-linear
A central issue of digitally based analysis of data is the linearity vs. non-linearity of an argument. The very fact that a proper term is missing to refer to non-linearity suggests that the field
remains wide open for a clarification of the issue. As a contribution in that direction, we should reflect on some key concepts, distinguishing between the form and the substance of the argument.
While I suggest alternative positive terms (multi-linear, polyhedral) in lieu of the negative term “non-linear,” I do nevertheless retain the term “non-linear” simply because, by virtue of its
oppositional value to “linear” and on account of its popularity, it provides a more immediate understanding of what is meant – as long as one attempts, as I do here, to explain what this meaning is.
Back to top: Linear and non-linear
We must distinguish between the logic of an argument and the form in which it is presented. If we use the terms “linear” to refer to the modality, we may use the term “sequential” to refer to the
substance, of the argument. We may thus say that the argument will always be sequential, regardless of whether the form it takes is linear or not.
In the schematic rendering below, the intermediate steps a-b-c-d must be in that sequence for the argument to hold. In this representation, these intermediate steps are all on the same plane, which
results in an aligned set of arrows. But the steps may straddle across planes, resulting in a multi-linear or multi-layered arrangement – where sequentiality remains nevertheless the rule.
Schematic representation of sequentiality within argument
Back to top: Linear and non-linear
In most cases, an argument includes multiple layers or registers, interlocked with each other. In this respect an argument is not only linear, but multi-linear, with the various threads running
parallel, and yet calling at the same time for cross-overs from one linear path to the other. Thus, in the schematic rendering below, A is the main register, which runs linearly from beginning to
end, and B and C are secondary registers which overlap either wholly (C) or partly (only the central portion of B being relevant to A). The argument still flows sequentially, but with data and
inferences drawn from multiple planes.
Schematic representation of a multi-layered argument
Back to top: Linear and non-linear
The mechanism described here helps to understand what is the nature of reflection. To read in the sense of studying means more than being led passively along the sequential line proposed by the
author. Rather, the reader is expected to develop parallel lines of inquiry and to draw on parallel data sets while following the argument presented by the author. There is, in other words, a
parallel set of layers that the reader juxtaposes to those already offered by the author. This multi-linear function can be greatly enhanced when articulated digitally – which is precisely the great
promise of the medium. Access to these multiple layers is dramatically facilitated by the medium, because of the way in which they are structured: there are unlimited lines of inquiry that are built
on equally unlimited bodies of data. The task of a digitally minded scholar is to capitalize on these possibilities of a non-linear approach understood as multi-linear, but developing a newly
constructed digital text.
The non-linear, or multi-linear, perspective is interesting in that it shows both the positive and the negative aspects of the medium.
The positive side is that the bracketing of layers is practically unlimited, that a suggestion to explore a parallel layer can be elicited by explicit or implicit associative mechanisms (hyperlinks,
search functions, etc.), and that within each layer one can pursue develop into.
The negative side of things is that the ease with which one can dart from one topic to the next, and even more frequently from one detail to another.
Contextualized hyperlink path ("reflection")
Non-contextualized hyperlink path ("distraction")
Back to top: Linear and non-linear
The polyhedral argument
The adjective “linear” refers to the geometric figure of a line, i. e. a point moving along a fixed direction. The adjective “polyhedral” refers to the geometric figure of a solid bounded by
polygons, such as the cube represented as 1 in the figure below. A linear argument that proposes to link conceptually points A and B has to travel along points c and d (2 in the figure). A polyhedral
argument, on the other hand, travels directly, across the solid, from A to B (3 in the figure).
The power and demonstrability of a polyhedral argument rely on a prior knowledge of the cube and of its properties. It is only in virtue of this knowledge that A can arguably be linked with B, since
the whole structure of the cube is presupposed, hence the linear possibility of the link (as represented under 2) is virtually known, even if it is not followed. It is also as a result of the prior
knowledge of the underly-ing structure (represented figuratively as a polyhedron) that the linkage takes place along the shortest line. Hence the power: greater prior knowledge allows the linkage.
And hence the demonstrability: one can refer back to the nature of the solid and show how the link between the two is possible. Such a knowledge is “polyhedral” because it does not rely solely on
points c and d, but rather on the whole solid figure (the cube or polyhedron), of which c and d are as much part as A and B.
Schematic representation of linear vs. polyhedral arguments
Without a supporting structure such as the cube, points A and B are floating in space, and their linkage (as shown schematically under 4 in the figure) results from a hit or miss shot in the dark.
Back to top: Linear and non-linear
Linear and poly-segmental
It is further worth noting that, strictly speaking, even the linkage represented under 3 remains linear, since the linkage is indeed a line. To reflect properly the situation, the terms
“poly-segmental” and “mono-segmental” are equivalent to “linear” and “non-linear.” The argument’s process represented under 2 is linear, but consists of many segments. The argument’s process
represented under 3, on the other hand, is also linear, but, as it cuts across the polyhedron in the most direct way, it consists of a single segment, and is therefore more effective. Obviously, the
degree of effectiveness increases in proportion to the complexity of the structure.
The Urkesh Global Record is built in such a way as to allow precisely an extensive use of polyhedral arguments. In practice, this relies on systematic use of hyper-links, which are generated
automatically and therefore in unlimited quantities. The arrow represented under 3 in Fig. 10-1 stands for such a hyperlink.
Back to top: Linear and non-linear
It is worth noting that the situation represented under 4 describes properly the nature of intuition. A connection between A and B may well be perceived through a sort of logical short-circuit, one
that bypasses the argument and cannot therefore be demonstrated – at least, not on the basis of the original intuition. But we all know that in most cases it is precisely such an intuition that
initiates the process of discovery. A proper polyhedral argument is one that, building on such an intuition, shows how the linkage is possible, and therefore arguable.
Back to top: Linear and non-linear
Pre-digital non-linearity
The term “non-linear” has achieved nearly cultic status in contemporary parlance. It evokes a sense of mystery, which gains in awe and power the less we try to explain it. It is, however, no
different than the case of Molière’s bourgeois who felt he had reached a pinnacle at the discovery that he was able to speak in prose…
We have been conceptualizing our world in a non-linear fashion at least ever since writing was first invented, some 5000 years ago. The earliest ledgers, the earliest maps are based as much as
today’s ledgers and maps on linkages that are not linear.
Consider this cuneiform tablet, from about 2000 B.C. It lists individual animals given to certain individuals (single circle), then it gives subtotals by types of animal (double circle), and finally
gives the grand total (triple circle). It is so simple, anyone can “read” the numbers. Thus the grand total is 5 times 60 (the large vertical wedge), plus 10 (the oblique wedge head), plus 4 (the
smaller vertical wedges). The connection is clear among all the various steps. It is non-linear, because it presupposes conceptual jumps, evinced by the sequence and general arrangement.
Back to top: Linear and non-linear
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Function draw
Function draw --- Introduction ---
Function draw is an exercise on the graphic recognition of functions of one real variable. The server will give you the graph of a function $f$, whose expression will be hidden. Then you are asked to
draw the graph of another function such as $f\left(-x\right)$, $2f\left(x\right)$, $f\left(x-1\right)$, etc (see the menu below), with the mouse. You will have a score according to the precision of
your drawing. The most recent version
This page is not in its usual appearance because WIMS is unable to recognize your web browser.
Please take note that WIMS pages are interactively generated; they are not ordinary HTML files. They must be used interactively ONLINE. It is useless for you to gather them through a robot program.
• Description: draw a function using the graph of another. interactive exercises, online calculators and plotters, mathematical recreation and games
• Keywords: interactive mathematics, interactive math, server side interactivity, analysis, graphing, functions
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5 Best Ways to Generate Successive Element Difference List in Python
π ‘ Problem Formulation: The task at hand is to create a list in Python that represents the differences between successive elements of a given list. For instance, if our input list is [3, 10, 6, 8],
we expect the output list to show the differences like this: [7, -4, 2], where each element is the subtraction of a preceding element from its follower.
Method 1: Using a For-Loop
This method involves iterating through the input list with a for-loop and manually calculating the difference between each pair of successive elements. This is a straightforward method, optimal for
beginners who are learning the intricacies of loops and list operations.
Here’s an example:
numbers = [3, 10, 6, 8]
differences = []
for i in range(len(numbers) - 1):
differences.append(numbers[i + 1] - numbers[i])
Output: [7, -4, 2]
This code initializes an empty list called differences, and then iterates over the indices of the numbers list. For each iteration, it calculates the difference between the current element and the
next, appending this value to the differences list. Finally, it prints out the differences list.
Method 2: Using List Comprehensions
List comprehensions in Python offer a concise way to create lists based on existing lists. This method takes advantage of Python’s syntactic sugar to generate the difference list in a single line of
Here’s an example:
numbers = [3, 10, 6, 8]
differences = [numbers[i + 1] - numbers[i] for i in range(len(numbers) - 1)]
Output: [7, -4, 2]
The list comprehension iterates over the indices of the input list numbers, calculating the difference between successive elements in a compact expression, directly within the list construction
syntax. The resulting list, differences, is then printed out.
Method 3: Using the Zip Function
The zip function is utilized to iterate over two lists in parallel. By pairing each element with its successor, one can iterate through and calculate their differences succinctly.
Here’s an example:
numbers = [3, 10, 6, 8]
differences = [j - i for i, j in zip(numbers[:-1], numbers[1:])]
Output: [7, -4, 2]
In this snippet, we use the zip function to combine two sliced versions of the original listβ numbers[:-1] excludes the last element, and numbers[1:] excludes the first. The list comprehension
iterates over the resulting pairs of values, calculating the difference j - i for each pair.
Method 4: Using NumPy
NumPy is a popular library for numerical computing in Python. Its diff function computes the difference between subsequent elements in a NumPy array efficiently, making this method optimal for large
datasets or performance-critical applications.
Here’s an example:
import numpy as np
numbers = np.array([3, 10, 6, 8])
differences = np.diff(numbers)
Output: [ 7 -4 2]
After importing NumPy and creating a NumPy array from our list, we use the np.diff function to calculate the differences array, which is printed at the end. This approach shines with its simplicity
and speed on large arrays.
Bonus One-Liner Method 5: Using the map and lambda Functions
This method combines the power of the map function and a lambda expression to calculate the differences. It’s a nifty one-liner that showcases Python’s functional programming features.
Here’s an example:
numbers = [3, 10, 6, 8]
differences = list(map(lambda x, y: y - x, numbers[:-1], numbers[1:]))
Output: [7, -4, 2]
The code snippet uses map to apply a lambda function to pairs of elements taken from the two slices of the original list, numbers[:-1] and numbers[1:]. The lambda function takes two arguments and
returns their difference, which map converts into a list of differences.
• Method 1: For-Loop. Easy to understand. Can be verbose for simple operations.
• Method 2: List Comprehensions. Concise. May be less readable for complex operations.
• Method 3: Zip Function. Neat and Pythonic. May require more understanding of iterator pairing.
• Method 4: NumPy. Highly efficient for large data. Requires external library and understanding of NumPy arrays.
• Method 5: Map and Lambda. Expressive one-liner. Less intuitive for those unfamiliar with functional programming concepts.
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How do you multiply 2/3times35 4/1? | Socratic
How do you multiply #2/3times35 4/1#?
1 Answer
$93.3$ (repeated) or $23 \frac{1}{2}$
not sure if the $35 \frac{4}{1}$ was a typo or not so i have provided solutions for both $35 \frac{4}{1}$ and $35 \frac{1}{4}$
$\frac{2}{3} \cdot 35 \frac{4}{1}$
$\frac{2}{3} \cdot 35 \left(4\right)$
$\frac{2}{3} \cdot 140$
$\frac{2}{3} \cdot 35 \frac{1}{4}$
$\frac{2}{3} \cdot \frac{141}{4}$ convert mixed fraction
$\frac{1}{1} \cdot \frac{47}{2}$ reduce multiples
$\frac{47}{2}$ recreate mixed fraction
$23 \frac{1}{2}$
Impact of this question
1682 views around the world
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EViews Help: pcomp
Principal components analysis of the columns in a matrix.
There are two forms of the pcomp command. The first form, which applies when displaying eigenvalue table output or graphs of the ordered eigenvalues, has only options and no command argument.
The second form, which applies to the graphs of component loadings, component scores, and biplots, uses the optional argument to determine which components to plot. In this form:
matrix_name.pcomp(options) [graph_list]
where the [graph_list] is an optional list of integers and/or vectors containing integers identifying the components to plot. Multiple pairs are handled using the method specified in the “mult=”
If the list of component indices omitted, EViews will plot only first and second components. Note that the order of elements in the list matters; reversing the order of two indices reverses the axis
on which each component is displayed.
Output: table of eigenvalue and eigenvector results (“table”), graphs of ordered eigenvalues (“graph”), graph of the eigenvectors (“loadings”), graph of the component scores
out=arg (“scores”), biplot of the loadings and scores (“biplot”).
“table”) Note: when specifying the eigenvalue graph (“out=graph”), the option keywords “scree” (scree graph), “diff” (difference in successive eigenvalues), and “cproport” (cumulative
proportion of total variance) may be included to control the output. By default, EViews will display the scree graph. If you may one or more the three keywords, EViews will construct
the graph using only the specified types.
eigval= Specify name of vector to hold the saved the eigenvalues in workfile.
eigvec= Specify name of matrix to hold the save the eigenvectors in workfile.
prompt Force the dialog to appear from within a program.
p Print results.
Number of Component Options
Component retention method: “bn” (Bai and Ng (2002)), “ah” (Ahn and Horenstein (2013)), “simple” (simple eigenvalue methods), “user” (user-specified value).
fsmethod=arg Note the following:
) (1) If using simple methods, the minimum eigenvalue and cumulative proportions may be specified using “minigen=” and “cproport=”.
(2) If setting “fsmethod=user” to provide a user-specified value, you must specify the value with “r=”.
r=arg (default=1) User-specified number of components to retain (for use when “fsmethod=user”).
mineigen=arg Minimum eigenvalue to retain component (when “fsmethod=simple”).
cproport=arg Cumulative proportion of eigenvalue total to attain (when “fsmethod=simple”).
Maximum number of components used by selection methods: “schwert” (Schwert’s rule, default), “ah” (Ahn and Horenstein’s (2013) suggestion), “rootsize” (), “size” (, “user” (user
specified value), where
(1) For use with all components retention methods apart from user-specified (“fsmethod=user”).
(default=“user”) (2) If setting “mfmethod=user”, you may specify the maximum number of components using “rmax=”.
(3) Schwert’s rule sets the maximum number of components using the rule: let
n=arg or rmax=arg User-specified maximum number of factors to retain (for use when “mfmethod=user”).
Component selection criterion when “fsmethod=bn”: “icp1” (ICP1), “icp2” (ICP2), “icp3” (ICP3), “pcp1” (PCP1), “pcp2” (PCP1), “pcp3” (ICP3), “avg” (average of all criteria ICP1
through PCP3).
fsic=arg (default Component selection criterion when “fsmethod=ah”: “er” (eigenvalue ratio), “gr” (growth ratio), “avg” (average of eigenvalue ratio and growth ratio).
Component selection when “fsmethod=simple”: “min” (minimum of: minimum eigenvalue, cumulative eigenvalue proportion, and maximum number of factors), “max” (maximum of: minimum
eigenvalue, cumulative eigenvalue proportion, and maximum number of factors), “avg” (average the optimal number of factors as specified by the min and max rule, then round to the
nearest integer).
demeantime Demeans observations across time prior to component selection procedures.
sdizetime Standardizes observations across time prior to component selection procedures.
demeancross Demeans observations across cross-sections prior to component selection procedures.
sdizecross Standardizes observations across cross-sections prior to component selection procedures.
Covariance Options
cov=arg (default= Covariance calculation method: ordinary (Pearson product moment) covariance (“cov”), ordinary correlation (“corr”), Spearman rank covariance (“rcov”), Spearman rank correlation
“cov”) (“rcorr”), Kendall’s tau-b (“taub”), Kendall’s tau-a (“taua”), uncentered ordinary covariance (“ucov”), uncentered ordinary correlation (“ucorr”).
wgt=name (optional) Name of vector containing weights. The number of rows of the weight vector should match the number of rows in the original matrix.
Weighting method: frequency (“freq”), inverse of variances (“var”), inverse of standard deviation (“stdev”), scaled inverse of variances (“svar”), scaled inverse of standard
wgtmethod=arg deviations (“sstdev”).
(default = “sstdev”
Only applicable for ordinary (Pearson) calculations where “weights=” is specified. Weights for rank correlation and Kendall’s tau calculations are always frequency weights.
pairwise Compute using pairwise deletion of observations with missing cases (pairwise samples).
Compute covariances with a degree-of-freedom correction accounting for the mean (for centered specifications) and any partial conditioning variables.
The default behavior in these cases is to perform no adjustment (e.g. – compute sample covariance dividing by
Graph Options
scale=arg, (default= Diagonal matrix scaling of the loadings and the scores: normalize loadings (“normload”), normalize scores (“normscores”), symmetric weighting (“symmetric”), user-specified (
“normload”) arg=number).
mult =arg (default= Multiple series handling: plot first against remainder (“first”), plot as x-y pairs (“pair”), lower-triangular plot (“lt”).
nocenter Do not center graphs around the origin. By default, EViews centers biplots around (0, 0).
labels=arg, (default= Observation labels for the scores: outliers only (“outlier”), all points (“all”), none (“none”).
labelprob=number Probability value for determining whether a point is an outlier according to the chi-square tests based on the squared Mahalanbois distance between the observation and the
sample means (when using the “labels=outlier” option).
autoscale=arg Scale factor applied to the automatically specified loadings when displaying both loadings and scores). The default is to let EViews auto-choose a scale or to specify
“userscale=” to scale the original loadings.
userscale=arg Scale factor applied to the original loadings when displaying both loadings and scores). The default is to let EViews auto-choose a scale, or to specify “autoscale=” to scale
the automatically scaled loadings.
cpnorm Compute the normalization for the score so that cross-products match the target (by default, EViews chooses a normalization scale so that the moments of the scores match the
freeze(tab1) mat1.pcomp(method=corr, eigval=v1, eigvec=m1)
stores the table view of the eigenvalues and eigenvectors of MAT1 in a table object named tab1, the eigenvalues in a vector named v1, and the eigenvectors in a matrix named m1.
mat1.pcomp(method=cov, out=graph)
displays the scree plot of the ordered eigenvalues computed from the covariance matrix.
mat1.pcomp(method=rcorr, out=biplot, scale=normscores)
displays a biplot where the scores are normalized to have variances that equal the eigenvalues of the Spearman correlation matrix computed for the series in MAT1.
“Principal Components”
for further discussion. See also
“Covariance Analysis”
for discussion of the preliminary computation.
Note that this view analyzes the eigenvalues and eigenvectors of a covariance (or other association) matrix computed from the series in a group or the columns of a matrix. You may use
to examine the eigenvalues of a symmetric matrix.
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Lesson 16
World’s Record Noodle Soup
Warm-up: Notice and Wonder: World Record Event (10 minutes)
The purpose of this warm-up is to introduce the context of a world record event about the longest continuous noodle, which will be useful when students solve problems about this event in a later
activity. While students may notice and wonder many things about this text and image, the noodle’s length and the number of people sharing the noodle are the important discussion points.
• Groups of 2
• Consider showing students a Guinness Book world record image or video of the worlds longest noodle.
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Share and record responses.
Student Facing
What do you notice? What do you wonder?
A Chinese food company holds the Guinness World Record for making the longest noodle. The noodle measured about 10,119 ft.
Activity Synthesis
• “What is something else that is about 10,000 feet long?” (That is about how high people are when they skydive. It is about 2 miles.)
• “These pictures show the world’s longest noodle being made. We are going to solve some problems about this event.”
Activity 1: How Many Feet in One Serving? (20 minutes)
The purpose of this activity is for students to use a method of their choice, likely multiplication or division, to solve a contextual problem about equal sharing of the longest noodle ever made. The
numbers in this activity are larger than the numbers students have worked with in previous lessons on division. Students estimate the number of feet of noodle each person ate at the record breaking
event. The numbers and context were chosen to encourage students to consider what they know about the meaning of division, to make a reasonable estimate, and to reason about the meaning of the
quotient in the context of the situation presented (MP2).
Monitor and select students with the following strategies to share in the synthesis:
• Students use multiplication or division to estimate that each person will get about 25 feet of noodle.
• Students can explain why 25 feet of noodle for each person is a low estimate.
• “What kind of noodles do you like to eat?” (ramen, spaghetti, fettucini, chicken noodle soup)
• 30 seconds: partner discussion
• “About how long is one of the noodles you like to eat?” (about 1 foot long)
• Groups of 2
• 5 minutes: independent work time
• 5 minutes: partner discussion
• As students work, consider asking “What do the numbers in your calculations mean, in terms of the situation?”
Student Facing
A Chinese food company cooked a single noodle measuring about 10,119 ft. It served 400 people.
1. If the noodle was shared equally, estimate how many feet of noodle each person was served.
2. Is your estimate lower or higher than the actual length of noodle each person ate? Explain your reasoning without calculating the actual length.
Activity Synthesis
• Ask selected students to share in the given order (or use the provided student solutions if needed).
• “How are the methods for estimating the amount of noodle each person gets the same?”(They both start by giving each person 10 feet of noodle. Then they give more until they have both given 25
feet to each person. They both find multiples of 400.)
• “How are they different?” (One thinks of the process as division and one uses just multiplication.)
• “How do you know the estimate of 25 feet is too low?” (Because there was still some of the noodle left. There are 119 feet left over.)
Activity 2: Han's Estimate (15 minutes)
The purpose of this activity is to consider a more precise estimate for the length of noodle each person would get if 400 people equally shared a 10,119 foot noodle. This estimate includes a
fractional part and encourages students to connect division to what they know about fractions. In the next lesson students will continue to examine fractions and how they relate to partial quotients.
Making an estimate or a range of reasonable answers with incomplete information is a part of modeling with mathematics (MP4).
MLR8 Discussion Supports. Activity: During group work, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own
words. Display the sentence frame: “I heard you say . . .” Original speakers can agree or clarify for their partner.
Advances: Listening, Speaking
Engagement: Develop Effort and Persistence. Check in and provide each group with feedback that encourages collaboration and community. For example, encourage students to use sentence frames to agree
or disagree with each other and take turns sharing their ideas.
Supports accessibility for: Social-Emotional Functioning.
• 3–5 minutes: independent work time
• 3–5 minutes: partner discussion
Student Facing
Han said that each person will get about \(25\frac{1}{4}\) feet of noodle. Do you agree with Han? Explain or show your reasoning.
Activity Synthesis
• Display: \(25\frac{119}{400}\)
• "What does \(25\frac{119}{400}\) mean in this situation?" (Each person gets 25 feet of the noodle and then the 119 feet leftover would be divided into 400 equal pieces.)
• Display: \(25\frac{1}{4}\)
• "Why is Han's estimate reasonable?” (Because is \(\frac{119}{400}\) really close to \(\frac{100}{400}\) and \(\frac{100}{400}=\frac{1}{4}\))
• "Do you think they actually measured and cut the noodle into equal pieces when they served it?" (No, because it would take too long and be too difficult. Yes, because if long noodles represent
long life they probably want to serve the noodle soup with sections that are one piece of the original noodle.)
Lesson Synthesis
“Today, we solved problems about a real life context. We also discussed solutions that were mixed numbers. In what ways did we use division today?" (We estimated and divided the number of feet of
noodle by the number of servings. We thought about fractions as division to help us make more precise estimates.)
"In what ways did we use fractions?” (We used what we know about fractions to make our estimates more precise.)
Cool-down: Division Reflection (5 minutes)
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tedana’s denoising approach
tedana’s denoising approach
tedana works by decomposing multi-echo BOLD data via principal component analysis (PCA) and independent component analysis (ICA). The resulting components are then analyzed to determine whether they
are TE-dependent or -independent. TE-dependent components are classified as BOLD, while TE-independent components are classified as non-BOLD, and are discarded as part of data cleaning.
In tedana, we take the time series from all the collected TEs, combine them, and decompose the resulting data into components that can be classified as BOLD or non-BOLD. This is performed in a series
of steps, including:
We provide more detail on each step below. The figures shown in this walkthrough are generated in the provided notebooks.
Here are the echo-specific time series for a single voxel in an example resting-state scan with 8 echoes. This voxel was selected because it is fairly correlated with the checkerboard task, but you
can see that the signal changes substantially across echoes. With a 9.58ms echo time, little
The values across volumes for this voxel scale with echo time in a predictable manner.
In this example, the non-steady state volumes at the beginning of the run are excluded. Some pulse sequences save these initial volumes and some do not. If they are saved, then the first few volumes
in a run will have much larger relative magnitudes. These initial volumes should be removed before running tedana.
Longer echo times are more susceptible to signal dropout, which means that certain brain regions (e.g., orbitofrontal cortex, temporal poles) may only have good signal for some echoes.
In order to avoid using bad signal from affected echoes in calculating tedana generates an adaptive mask, where the value for each voxel indicates how many of the echoes (starting with the first
echo) have “good” signal. tedana has multiple methods for generating this mask, and we recommend looking at the description of make_adaptive_mask() for more information.
tedana allows users to provide their own mask. The adaptive mask will be computed on this explicit mask, and may reduce it further based on the data. If a mask is not provided, tedana runs
nilearn.masking.compute_epi_mask() on the first echo’s data to derive a mask prior to adaptive masking. Some brain masking is required because the percentile-based thresholding in the adaptive mask
will be flawed if it includes all out-of-brain voxels.
In this eight-echo dataset, we can see that the adaptive mask flags later echoes as “bad” in areas we expect to suffer most from dropout, including the orbitofrontal cortex and temporal poles.
The next step is to fit a monoexponential decay model to the data in order to estimate voxel-wise T2starmap.nii.gz and S0map.nii.gz.
While [1], which can be estimated with --fitmode ts in tedana.workflows.t2smap_workflow().
In order to make it easier to fit the decay model to the data, tedana transforms the data by default. The BOLD data are transformed as
It is now possible to do a nonlinear monoexponential fit to the original, untransformed data values by specifiying --fittype curvefit. This method is slightly more computationally demanding but may
obtain more accurate fits.
A simple line can then be fit to the transformed data with linear regression. For the sake of this introduction, we can assume that the example voxel has good signal in all eight echoes (i.e., the
adaptive mask has a value of 8 at this voxel), so the line is fit to all available data.
tedana actually performs and uses two sets of tedana estimates tedana estimates
The values of interest for the decay model,
The resulting values can be used to show the fitted monoexponential decay model on the original data.
We can also see where
Using the tedana combines signal across echoes using a weighted average. The echoes are weighted according to the formula:
The weights are then normalized across echoes. For the example voxel, the resulting weights are:
These normalized weights are then used to compute a weighted average that takes advantage of the higher signal in earlier echoes and the higher sensitivity at later echoes. The distribution of values
for the optimally combined data lands somewhere between the distributions for other echoes.
The time series for the optimally combined data also looks like a combination of the other echoes (which it is). This optimally combined data is written out as desc-optcom_bold.nii.gz
An alternative method for optimal combination that does not use [2]. This method specifically assumes that noise in the acquired echoes is “isotopic and homogeneous throughout the image,” meaning it
should be used on smoothed data. As we do not recommend performing tedana denoising on smoothed data, we discourage using PAID within the tedana workflow. We do, however, make it accessible as an
alternative combination method in tedana.workflows.t2smap_workflow().
The next step is an attempt to remove noise from the data. This process can be broadly separated into three steps: decomposition, metric calculation and component selection. Decomposition reduces the
dimensionality of the optimally combined data using principal component analysis (PCA) and then an independent component analysis (ICA). Metrics that evaluate TE-dependence or -independence are
derived from these components. Component selection uses these metrics in order to identify components that should be kept in the data or discarded. Unwanted components are then removed from the
optimally combined data to produce the denoised data output.
The next step is to dimensionally reduce the data with TE-dependent principal component analysis (PCA). The goal of this step is to make it easier for the later ICA decomposition to converge.
Dimensionality reduction is a common step prior to ICA. TEDPCA applies PCA to the optimally combined data in order to decompose it into component maps and time series (saved as desc-PCA_mixing.tsv).
Here we can see time series for some example components (we don’t really care about the maps):
These components are subjected to component selection, the specifics of which vary according to algorithm. Specifically, tedana offers three different approaches that perform this step.
The recommended approach (the default aic option, along with the kic and mdl options, for --tedpca) is based on a moving average (stationary Gaussian) process proposed by Li et al.[3] and used
primarily in the Group ICA of fMRI Toolbox (GIFT). A moving average process is the output of a linear system (which, in this case, is a smoothing filter) that has an independent and identically
distributed Gaussian process as the input. Simply put, this process more optimally selects the number of components for fMRI data following a subsampling scheme described in Li et al.[3].
The number of selected principal components depends on the selection criteria. For this PCA method in particular, --tedpca provides three different options to select the PCA components based on three
widely-used model selection criteria:
• mdl: the Minimum Description Length (MDL), which is the most aggressive option; i.e. returns the least number of components.
• kic: the Kullback-Leibler Information Criterion (KIC), which stands in the middle in terms of aggressiveness. You can see how KIC is related to AIC here.
• aic: the Akaike Information Criterion (AIC), which is the least aggressive option; i.e., returns the largest number of components. We have chosen AIC as the default PCA criterion because it tends
to result in fewer components than the Kundu methods, which increases the likelihood that the ICA step will successfully converge, but also, in our experience, retains enough components for
meaningful interpretation later on.
Please, bear in mind that this is a data-driven dimensionality reduction approach. The default option aic might not yield perfect results on your data. Consider kic and mdl options if running tedana
with aic returns more components than expected. There is no definitively right number of components, but, for typical fMRI datasets, if the PCA explains more than 98% of the variance or if the number
of components is more than half the number of time points, then it may be worth considering more aggressive thresholds.
The simplest approach uses a user-supplied threshold applied to the cumulative variance explained by the PCA. In this approach, the user provides a value to --tedpca between 0 and 1. That value
corresponds to the percent of variance that must be explained by the components. For example, if a value of 0.9 is provided, then PCA components (ordered by decreasing variance explained)
cumulatively explaining up to 90% of the variance will be retained. Components explaining more than that threshold (except for the component that crosses the threshold) will be excluded.
In addition to the moving average process-based options and the variance explained threshold described above, we also support a decision tree-based selection method (similar to the one in the TEDICA
section below). This method involves applying a decision tree to identify and discard PCA components which, in addition to not explaining much variance, are also not significantly TE-dependent (i.e.,
have low Kappa) or TE-independent (i.e., have low Rho). These approaches can be accessed using either the kundu or kundu_stabilize options for the --tedpca flag.
For more information on how TE-dependence and TE-independence models are estimated in tedana, see TE (In)Dependence Models. For a more thorough explanation of this approach, consider the supplemental
information in Kundu et al.[4].
After component selection is performed, the retained components and their associated betas are used to reconstruct the optimally combined data, resulting in a dimensionally reduced version of the
dataset which is then used in the TEDICA step.
Next, tedana applies TE-dependent independent component analysis (ICA) in order to identify and remove TE-independent (i.e., non-BOLD noise) components. The dimensionally reduced optimally combined
data are first subjected to ICA in order to fit a mixing matrix to the whitened data. tedana can use a single interation of FastICA or multiple interations of robustICA, with an explanation of those
approaches in our FAQ. This generates a number of independent timeseries (saved as desc-ICA_mixing.tsv), as well as parameter estimate maps which show the spatial loading of these components on the
brain (desc-ICA_components.nii.gz).
Linear regression is used to fit the component time series to each voxel in each of the original, echo-specific data. This results in echo- and voxel-specific betas for each of the components. The
beta values from the linear regression can be used to determine how the fluctuations (in each component timeseries) change across the echo times.
TE-dependence (
The grey lines below shows how beta values (a.k.a. parameter estimates) change with echo time, for one voxel and one component. The blue and red lines show the predicted values for the
A decision tree is applied to desc-tedana_metrics.tsv. The actual decision tree is dependent on the component selection algorithm employed. tedana includes three options tedana_orig, meica and
minimal (which uses hardcoded thresholds applied to each of the metrics). These decision trees are detailed in Included Decision Trees.
Components that are classified as noise are projected out of the optimally combined data, yielding a denoised timeseries, which is saved as desc-denoised_bold.nii.gz.
RICA is a tool for manual ICA classification. Once the .tsv file containing the result of manual component classification is obtained, it is necessary to re-run the tedana workflow (see Running the
ica_reclassify workflow) passing the manual_classification.tsv file with the --ctab option. To save the output correctly, make sure that the output directory does not coincide with the input
directory. See this example presented at MRITogether 2022 for a hands-on tutorial.
tedana.gscontrol.gscontrol_raw(), tedana.gscontrol.minimum_image_regression()
Due to the constraints of spatial ICA, TEDICA is able to identify and remove spatially localized noise components, but it cannot identify components that are spread out throughout the whole brain.
See Power et al.[5] for more information about this issue. One of several post-processing strategies may be applied to the denoised data in order to remove spatially diffuse (ostensibly
respiration-related) noise. Methods which have been employed in the past include global signal regression (GSR), minimum image regression (MIR), anatomical CompCor, Go Decomposition (GODEC), and
robust PCA. Currently, tedana implements GSR and MIR.
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Profitable Future Backtesting of KST Trading Strategy in Python
Written on
Understanding the KST Trading Indicator
The KST (Know Sure Thing) indicator serves as a momentum oscillator designed to detect significant trend shifts within the market. Developed by Martin J. Pring, it relies on four distinct time
frames: 10, 15, 20, and 30 periods.
The KST is determined through a weighted moving average of the rate of change (ROC) values derived from these four time frames. ROC measures the percentage change in price over a defined timeframe.
To refine the KST, a secondary moving average is applied for smoothing.
The KST can signal buying or selling opportunities based on the interaction between two lines: the KST line and the signal line. An upward cross of the KST line over the signal line indicates a
buying opportunity, whereas a downward cross suggests a selling opportunity.
Backtesting the KST Trading Strategy
Backtesting is an essential phase in crafting any trading strategy, enabling traders to assess their strategies against historical data. This guide will detail how to backtest a trading strategy
utilizing the KST indicator.
To initiate the process, we will import historical price data for our chosen asset and compute the KST indicator using the following code:
def calculate_KST(df, roc_periods=(10, 15, 20, 30), sma_periods=(10, 10, 10, 15)):
for i, r in enumerate(roc_periods):
df[f'ROC{i+1}'] = ((df['Close'] - df['Close'].shift(r)) / df['Close'].shift(r)) * 100
weights = [1, 2, 3, 4]
for i, s in enumerate(sma_periods):
df[f'WMA{i+1}'] = df[[f'ROC{j+1}' for j in range(i+1)]] @ weights[:i+1] / sum(weights[:i+1])
df['KST'] = df[[f'WMA{j+1}' for j in range(4)]] @ weights / sum(weights)
df['KSTS'] = df['KST'].rolling(window=9).mean()
return df
This function processes a DataFrame containing historical price data, along with two tuples—roc_periods and sma_periods—that define the periods for rate of change and moving average calculations. The
function returns the DataFrame augmented with additional columns for each ROC value, weighted moving averages, the KST, and the KSTS (KST smoothed with a 9-period moving average).
Next, we will establish our trading strategy, generating buy signals when the KST line crosses above the KSTS line, and sell signals when it crosses below. The code for this is as follows:
def generate_signals(df):
signals = []
for i in range(1, len(df)-1):
if df.iloc[i]['KSTS'] > df.iloc[i]['KST']:
elif df.iloc[i]['KST'] > df.iloc[i]['KSTS']:
return signals
This function takes the DataFrame containing the KST and KSTS columns and produces a list of signals based on their intersections.
Finally, we will backtest our strategy using the historical price data alongside the signals generated by our trading strategy. For this test, we'll assume an initial capital of $1,000 and employ the
following code to simulate trades:
df["signal"] = signals
investment = 1000
current_investment = 1000
invested_amount = 0
fees = 0
profit = 0
is_invested = 0
best_trade = -99999999
worst_trade = 99999999
largest_loss = 0
largest_gain = 0
total_trades = 0
for i in range(500, len(df)):
signal = df.iloc[i]['signal']
close = df.iloc[i]['close']
if signal == 1 and is_invested == 0: # Long signal and no position
entry_point = close
quantity = (current_investment / close)
invested_amount = quantity * close
is_invested = 1
elif signal == -1 and is_invested == 0: # Short signal and no position
entry_point = close
quantity = (current_investment / close)
invested_amount = quantity * close
is_invested = -1
elif signal == -1 and is_invested == 1: # Close long position for short signal
profit = quantity * (-entry_point + close)
current_investment += profit
invested_amount = 0
total_trades += 1
if profit > largest_gain:
largest_gain = profit
if profit < largest_loss:
largest_loss = profit
if profit > best_trade:
best_trade = profit
if profit < worst_trade:
worst_trade = profit
entry_point = close
quantity = (current_investment / close)
invested_amount = quantity * close
is_invested = -1
elif signal == 1 and is_invested == -1: # Close short position for long signal
profit = quantity * (-close + entry_point)
current_investment += profit
invested_amount = 0
total_trades += 1
if profit > largest_gain:
largest_gain = profit
if profit < largest_loss:
largest_loss = profit
if profit > best_trade:
best_trade = profit
if profit < worst_trade:
worst_trade = profit
entry_point = close
quantity = (current_investment / close)
invested_amount = quantity * close
is_invested = 1
final_profit = current_investment - investment
print("Final Profit: ", final_profit)
print("Best Trade: ", best_trade)
print("Worst Trade: ", worst_trade)
print("Largest Loss: ", largest_loss)
print("Largest Gain: ", largest_gain)
print("Total Trades: ", total_trades)
After conducting the backtest on historical data, I achieved a final portfolio value of $3,519,043. This outcome suggests that the strategy was successful, indicating its potential for generating
positive returns in future trading endeavors.
Now, let's dive into some instructional videos.
How To Backtest A Trading Strategy in Python
This video will guide you through the process of backtesting a trading strategy using Python, providing valuable insights and practical examples.
Backtesting a Trading Strategy in Python With AI Generated Code
In this video, discover how to leverage AI-generated code for backtesting your trading strategies effectively in Python.
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New to knitting and need loads of help
I am new to knitting and I found your site by sheer chance! I am learning knitting with French instructions (I live in Paris). It is bad enough I am clueless when following english instructions, you
can imagine French:aww:
I am also not too good with calculations/symmetry. So I really need all the help I can get:rofl:
An example for symmetry: number of stitches for symmetry: multiples of 6+5+1 stitch at each end.
What does that mean…and how do I calculate?
So, if anyone out there can explain to me in simple terms (I have very limited logical thinking, by the way:teehee: ) it would be a tremendous advancement for me!
Thank you
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[Solved] In problem, solve each equation in the re | SolutionInn
In problem, solve each equation in the real number system. x 3 - 2/3x 2 + 8/3x
In problem, solve each equation in the real number system.
x^3 - 2/3x^2 + 8/3x + 1 = 0
Fantastic news! We've Found the answer you've been seeking!
Step by Step Answer:
Answer rating: 54% (22 reviews)
The solutions of the equation are the zeros of fx 3x 3 2x 2 8x 3 Ste...View the full answer
Answered By
Ashington Waweru
I am a lecturer, research writer and also a qualified financial analyst and accountant. I am qualified and articulate in many disciplines including English, Accounting, Finance, Quantitative
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The math study tip they are NOT telling you - Ivy League math major
Han Zhango
26 Aug 202308:15
TLDRIn this video, Han, a former Columbia University engineering graduate, shares her personal journey from struggling with math to becoming proficient and passionate about it. She reveals a study
method that involves initially giving up on tough problems, understanding the solution, and then re-solving them independently to build a comprehensive understanding and boost confidence. Han
emphasizes the importance of addressing fundamental concepts and practicing with answer keys to overcome math challenges.
• 😀 Han, a Columbia University engineering graduate, initially struggled with math and felt unintelligent compared to others.
• 🏫 Han's high school experience in China was challenging; he chose liberal arts over natural science due to his poor performance in math.
• 📚 Han's initial approach to difficult math problems was to struggle through them, often resulting in frustration and incorrect answers.
• 🤔 Han's realization in college was to 'give up' when stuck on a problem, instead opting to understand the answer key thoroughly before attempting the problem again.
• 🔑 The college system Han adopted involved mentally planning the solution, understanding the answer key, and then independently solving the problem to reinforce learning.
• 💡 Han emphasizes the importance of understanding the answer key to learn the correct approach and to build a comprehensive understanding of the problem-solving process.
• 🚀 Han's method boosts confidence and saves time by focusing on learning from the answer key rather than aimlessly attempting to solve the problem.
• 🌟 Han suggests that math can be intimidating because it has its own technical barriers, unlike subjects like history where concepts can be grasped more quickly.
• 🔗 Han explains that understanding math requires building a network of interconnected concepts, and missing any can create confusion and gaps in knowledge.
• 📈 Han recommends using practice problem sets with answer keys to identify and fill gaps in knowledge, which was instrumental in his improvement.
• 🎓 Han's dedication to extra math practice during his senior year of high school not only helped him catch up but also made him one of the top students in his class.
Q & A
• What is the main theme of the video?
-The main theme of the video is about overcoming the struggle with math and transforming from hating it to being good at it and enjoying it, as shared by Han, a math major from Columbia
• What is Han's educational background?
-Han graduated from Columbia University's engineering school with a major in math and operations research.
• Why did Han choose the liberal arts track in high school instead of the natural science track?
-Han chose the liberal arts track in high school because they were originally bad at math and science, which seemed hard to them and they couldn't understand the lectures or do their homework.
• What was Han's initial experience with math in high school?
-Han initially struggled with math, getting a score of 49 on their first math test, with an average score of 78 and the highest score being 96.
• What is the study system Han used in college to improve their math skills?
-Han used a system where they would first try to mentally solve a problem, then look at the answer key to understand the approach, and then try to solve the problem independently, repeating the
process until they got it right.
• Why does Han recommend looking at the answer key before attempting to solve a problem independently?
-Han recommends this approach because it helps to understand the correct approach and method, saves time by focusing on learning the right way, and builds confidence and a sense of
• What is the importance of writing the solutions completely on your own?
-Writing the solutions completely on your own is important because it provides a comprehensive understanding of the problem-solving process from start to finish, which helps in recognizing and
solving similar problems in the future.
• Why does Han suggest doing at least 20 practice questions a day using the mentioned process?
-Han suggests doing 20 practice questions a day to focus on what truly matters, identify areas of uncertainty, and learn and practice the correct approaches, which accelerates the understanding
and mastery of math concepts.
• What is the role of the answer key in Han's study method?
-The answer key plays a crucial role in Han's study method by providing the correct solution path, which helps in understanding where one might be going wrong and how to correctly approach the
• How does Han's approach to learning math differ from their high school experience?
-In high school, Han would struggle with each step of a problem and often give up or make mistakes. In college, they adopted a method of understanding the answer key first, then attempting the
problem independently, which was more effective and less frustrating.
• How can someone who is struggling with math start to improve using Han's advice?
-Someone struggling with math can start by using Han's method of mentally walking through problems, using answer keys to understand the correct approach, practicing independently, and focusing on
learning from mistakes rather than getting frustrated.
📚 From Math Struggles to Mastery
Han, a graduate from Columbia University's engineering school with a major in math and operations research, shares his personal journey of transforming from a student who struggled with math to one
who excels and enjoys the subject. He candidly discusses his initial challenges with math during high school in China, where he chose the liberal arts track due to his poor performance and
understanding of math. Despite the common misconception that he was naturally gifted at math, Han reveals that he had to overcome a significant learning barrier. He introduces a systematic approach
he developed in college for tackling difficult math problems, which involves understanding the answer key thoroughly before attempting to solve the problem independently. This method not only builds
confidence but also ensures a comprehensive understanding of the problem-solving process.
🔍 Overcoming Math Intimidation and Building a Knowledge Network
In this paragraph, Han addresses the unique challenges of understanding math compared to other subjects, highlighting the difficulty of grasping abstract concepts without prior knowledge. He
emphasizes the importance of building a strong foundation in math by identifying and filling gaps in one's knowledge. Han suggests using practice problem sets with detailed answer keys to pinpoint
areas of uncertainty and to practice applying newly understood concepts. He shares his personal strategy of dedicating extra hours to math problem sets, which not only helped him catch up on missed
materials but also led to him becoming one of the top students in his math class. Han encourages students to persevere through the initial difficulties, promising that with consistent effort, they
will develop a deeper understanding and appreciation for math.
💡Math Sensitivity
This concept refers to an enhanced ability to understand and solve math problems effortlessly. In the video, Han describes an imaginary scenario where one wakes up feeling unusually sensitive to
numbers, making math problems seem easy and intuitive.
💡Columbia University
A prestigious Ivy League institution in New York where Han studied. Mentioning his education at Columbia University adds credibility to his experience and advice on studying math.
💡Operations Research
A field of study that applies analytical methods to help make better decisions. Han majored in this along with math, highlighting his expertise in applying mathematical concepts to real-world
💡Liberal Arts Track
An educational path focused on humanities subjects like history, politics, and geography. Han chose this track in high school despite his eventual success in math, illustrating his initial struggle
and transformation.
💡Answer Key Method
Han’s study technique where he first understands the solution from the answer key before attempting the problem himself. This method helped him overcome frustration and build confidence in solving
math problems.
💡Sense of Accomplishment
The positive feeling one gets after successfully solving a problem. Han emphasizes that this feeling is crucial for maintaining motivation and confidence in math studies.
💡Fundamental Concepts
Basic principles that form the foundation of more complex topics. Han notes that difficulties in understanding advanced math often stem from gaps in fundamental concepts learned in earlier stages.
💡Practice Problem Sets
Collections of math problems used for practice. Han recommends working through these sets to identify and address specific areas of weakness, thereby reinforcing learning.
💡Learning Network
A metaphor for the interconnected knowledge required to understand math comprehensively. Han suggests that building this network helps in connecting various concepts and improving overall
💡Confidence Boost
The increase in self-assurance one experiences after overcoming challenges. Han argues that successfully solving problems using his method leads to a significant boost in confidence, making further
learning easier and more enjoyable.
Han, an Ivy League math major, shares personal struggles with math and a transformation to mastery.
Graduated from Columbia University's engineering school with a major in math and operations research.
Initially chose the liberal arts track in high school due to poor performance in math and science.
Recalls a first high school math test with a score of 49, far below the class average.
Describes a cycle of hating math, avoiding it, performing poorly, and feeling defeated.
Introduces a college-level system for tackling math problems that involves initially giving up to learn from the answer key.
Advocates understanding the answer key thoroughly before attempting to solve the problem independently.
Emphasizes the importance of writing solutions completely on one's own for a comprehensive understanding.
Suggests that math has inherent barriers unlike other subjects, making it difficult to grasp new concepts quickly.
Recommends building a 'giant network' of knowledge by identifying and filling gaps in understanding.
Advises finding practice problem sets with thorough answer keys to work through daily.
States that working through 20 practice questions a day can significantly improve math skills.
Shares personal experience of catching up in math and becoming a top student by using this method.
Encourages persistence, as the initial phase of understanding can be challenging but becomes easier over time.
Reminds viewers to pay attention to lectures and complete homework in addition to practicing problems.
Concludes with a message of hope for those who struggle with math, assuring improvement is possible.
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Amnon J Meir
Some of my research, and projects I have been involved in, have been supported by various contracts and grants.
Current grants
I am currently a Program Director at the NSF.
Completed grants
1. Alabama Supercomputer Authority – On the Finite Element Method for the Velocity-Vorticity Formulation of Three-Dimensional, Viscous, Incompressible Flow, principal investigator – Period of
funding: 1990
2. Research Grant-in-Aid, Auburn University – On the Finite Element Method for the Velocity-Vorticity Formulation of Three-Dimensional, Viscous, Incompressible Flows, principal investigator
– Period of funding: June 16, 1990–June 15 1991
3. American Computing Inc. – Numerical Approximation of Solutions of the Steady Navier Stokes Equations, principal investigator – Period of funding: June 16, 1993–September 15, 1993
4. American Computing Inc. – Numerical Approximation of Solutions of the Steady Navier Stokes Equations (Part II), principal investigator – Period of funding: September 16, 1993–December 15,
5. American Computing Inc. – Numerical Approximation of Solutions of the Steady Navier Stokes Equations (Part III), principal investigator – Period of funding: June 16, 1994–September 15, 1994
6. NSF – Viscous Incompressible Magnetohydrodynamics: Analysis and Numerical Approximation, co-principal investigator (with P. G. Schmidt, principal investigator) – Period of funding: June 16,
1994–June 15, 1996
7. NSF – Viscous Incompressible Magnetohydrodynamics: Analysis and Numerical Approximation, co-principal investigator (with P. G. Schmidt, principal investigator) – Period of funding: June 16,
1996–June 15, 1999
8. DOE EPSCoR – Fusion Energy Research, co-principal investigator (with Physics: D. G. Swanson, principal investigator, R. F. Gandy, J. D. Hanson, S. F. Knowlton, M. S. Pindzola, F. Robicheaux,
and C. Watts, co-principal investigators) – Period of funding: October 1, 1998–September 30, 1999
9. NSF – Eighteenth Southeastern Atlantic Regional Conference on Differential Equations, principal investigator (with P. G. Schmidt,co-principal investigator) – Period of funding: November 1,
1998–October 31, 1999
10. DOE EPSCoR – Energy and Particle Transport in Fusion Plasmas, co-principal investigator (with Physics: D. G. Swanson, principal investigator, R. F. Gandy, J. D. Hanson, S. F. Knowlton, M. S.
Pindzola, F. Robicheaux, and C. Watts, co-principal investigators) – Period of funding: October 1, 1999–September 29, 2000
11. PRISM (College of Sciences and Mathematics, Auburn University) – Computational Science (funding for a Beowulf cluster), co-principal investigator (with Physics: J. D. Perez, principal
investigator, Y. Lin, M. S. Pindzola, F. Robicheaux, Chemistry: M. L. McKee, and Mathematics: P. G. Schmidt, co-principal investigators) – Period of funding: October 1, 2001–September 30,
12. NIH – Improving the Detection Limits of Potentiometric and Optical Sensors, co-principal investigator (with Chemistry: E. Bakker, principal investigator, and ETH Zurich, Chemistry: E.
Pretsch, co-principal investigator) – Period of funding: September 1, 2000–August 31, 2004
13. NSF – Studies in Poromechanics and Electro-Poromechanics, principal investigator – Period of funding: September 15, 2009–August 31, 2013
14. NSF – US-Africa Advanced Study Institute and Workshop Series in Mathematical Sciences, co-principal investigator (with O. Jenda, principal investigator, and A. Abebe, and M. Smith,
co-principal investigators) – Period of funding: April 15, 2011–March 31, 2013
15. OVPR AU-IGP – Virtual 3D Interlaced Fabric Design and Characterizaion, co-principal investigator (with S. Adanur, principal investigator, and Y. Cao, co-principal investigator) – Period of
funding: March 1, 2012–February 28, 2014
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How to read the side of a tire
How To Read A Tire Sidewall
What do the numbers mean on the sidewall of your tire? At first glance, you look at your tire sidewall and think, “’Do I need a super secret decoder ring to read this?” In addition to the model name
of the tire there is a series of numbers that at first, you don’t deem important. However, these numbers are extremely helpful, especially when it’s time to replace your tires. Here’s a quick
breakdown to help you decipher one of the best kept secrets in the automotive world: How do you read tire sizes?
Example: P225/50/R17 98H
P identifies your tire as a Passenger Tire. The P stands for PMetric. If your tire size starts with LT rather than a P than it identifies the tire as a Light Truck tire.
225 identifies the tire section width, which is the measurement of the tire from sidewall to sidewall in millimeters. This measurement varies depending on the rim to which it is fitted.
(There are 25.4 millimeters per 1 inch.)
50 is the two-figure aspect ratio. This percentage compares the tire's section height with the tire's section width. For example, this aspect ratio of 50 means that the tire's section height is 50%
of the tire's section width.
R indicates the construction used within the tires casing. R stands for radial construction. B means belted bias and D stands for diagonal bias construction.
17 The last dimension listed in the size is the diameter of the wheel rim, which is most often measured in inches.
Example: P225/50/R17 98H
The load index and speed rating, or service description, are the numbers that follow the tire size.
The load index tells you how much weight the tire can support when properly inflated. Load indices range from 75 - 105 for passenger tires, with each numeric value corresponding to a certain carrying
H Speed ratings are represented by letters ranging from A to Z. Each letter coincides to the maximum speed a tire can sustain under its recommended load capacity. For instance, S is equivalent to a
maximum speed of 112 mph. Even though a tire can perform at this speed, Continental Tire does not advocate exceeding legal speed limits.
Rating Maximum Speed
Q 100 MPH
S 112 MPH
T 118 MPH
U 124 MPH
H 130 MPH
V 149 MPH
W 168 MPH
Y 186 MPH
Z Over 149 MPH
DOT Serial Number
The "DOT" symbol certifies the tire manufacturer's compliance with the U.
Below is a description of the serial number. Starting with the year 2000, four numbers are used for the Date of Manufacture, the first two numbers identify the week and the last two numbers identify
the year of manufacture. This identifies how old a tire is.
Prior to year 2000 three numbers are used for the date of manufacture, first two numbers identify the week and the last number identifies the year of manufacture. To identify tires manufactured in
the 90s, a decade symbol (a triangle on its side) is located at the end of the DOT serial number.
How to Read a Tire Sidewall
Tires hold a lot of information if you know how to read the numbers and letters on them, but many people don’t know how to read a tire sidewall for basic data like tire size. So, what do the numbers
on a tire mean? The good news is that the answers are relatively straightforward, but the bad news is that it’s a great deal of info to memorize.
Tire Sizes Explained
When you need new tires to replace worn ones, it’s important to know the size of your old tires—especially if you’re not buying a full set. Tire size can be found on the sidewall, represented by an
alphanumeric code indicating the tire type, width, aspect ratio, construction type, wheel diameter, load index, and speed rating.
Tire Type
Check the tire’s sidewall for a series of letters and numbers 11 to 13 characters in length. The first character should be a letter indicating the tire type, referring to the type of vehicle for
which the tire is designed.
P: If the code starts with a P, then the tire is made for passenger vehicles like sedans, crossovers, minivans, and most SUVs and pickup trucks. These are commonly known as P-metric tires.
LT: Full-size pickup trucks and SUVs may have tires with the LT designation, which stands for “light truck.” These tires are typically made for carrying heavy loads or towing trailers.
ST: Typically seen on a variety of trailers, ST stands for “special trailer.” Tires with the ST designation should never be used on cars, vans, SUVs, or any other type of passenger vehicle.
No Letter: Some tire size codes don’t begin with a letter, so they fall into a separate category. These are typically European metric sizes. While those tires are still measured in millimeters and
could be similar to a P-metric tire size, they may have a different load capacity.
Tire Width
There should be a three-digit number after the initial letter(s). This number indicates the width of the tire in millimeters. So, if the number listed is 215, then the tire width measures 215 mm.
Aspect Ratio
The next symbol on the tire sidewall is a forward slash, followed by two digits representing the aspect ratio, or the ratio of tire height to tire width. Tire height is measured from the wheel rim to
the top of the tire tread, but it’s written as a percentage on the sidewall. For this reason, tire width is required to calculate tire height.
If the first three digits representing the tire width are 215 and the following aspect ratio digits are 65, the tire is 215 mm wide and the height is 65 percent of the width. You can calculate the
exact measurement relatively easily with the following formula, where AR equals the aspect ratio, TW equals tire width, and TH equals tire height.
(AR/100) x TW = TH
(65/100) x 215 mm = 139.75 mm
RELATED: The Best Tire Inflators
Construction Type
One or more letters, an R or D, should follow the two tire measurements.
R: This designation is the most common for modern tires. The R stands for “radial tires,” which have superior road grip, gas mileage, and ride comfort. Radial tires are made with multiple layers of
rubber-coated cords laid perpendicular to the direction of travel. These cords are made using a blend of polyester, steel, and fabric to improve overall tire durability.
D: The D designation corresponds to bias tires. These tires have diagonal or crisscrossed cord plies and are sometimes used on motorcycles and trailers. However, this tire construction type isn’t
common for the average passenger vehicle.
Wheel Diameter
Two digits should be listed after the construction type. These numbers express the diameter of the wheel in inches. This means that if the number provided is 17, then the tire is designed to fit on a
17-inch wheel.
Load Index
The tire load index is a code that references the amount of weight a single tire can handle. It’s listed as a two- or three-digit number after the wheel diameter. To determine the weight in pounds,
refer to a load index chart like the one provided below. Once you find the load capacity for one tire, then (assuming all of your tires’ load capacities match) you can calculate your vehicle’s
maximum load by multiplying the single tire load capacity by four.
75 = 852 lbs
78 = 937 lbs
81 = 1,019 lbs
84 = 1,102 lbs
87 = 1,201 lbs
90 = 1,323 lbs
93 = 1,433 lbs
96 = 1,565 lbs
99 = 1,709 lbs
102 = 1,874 lbs
105 = 2,039 lbs
108 = 2,205 lbs
111 = 2,403 lbs
114 = 2,601 lbs
117 = 2,833 lbs
120 = 3,086 lbs
Related: How to Use a Tire Pressure Gauge
Speed Rating
Similar to the load index, tire speed ratings indicate the maximum speed for which a tire is rated. Tire speed rating is often represented by a letter, but it can also be a letter and a number. In
rare cases where the speed rating exceeds 186 mph, it may be designated by a ZR followed by (Y). See the speed rating chart below for a complete list of speed ratings and their meanings.
A1 = 3 mph
A2 = 6 mph
A3 = 9 mph
A4 = 12 mph
A5 = 16 mph
A6 = 19 mph
A7 = 22 mph
A8 = 25 mph
B = 31 mph
C = 37 mph
D = 40 mph
E = 43 mph
F = 50 mph
G = 56 mph
J = 62 mph
K = 68 mph
L = 75 mph
M = 81 mph
N = 87 mph
P = 93 mph
Q = 99 mph
R = 106 mph
S = 112 mph
T = 118 mph
U = 124 mph
H = 130 mph
V = 149 mph
W = 168 mph
Y = 186 mph
(Y) = >186 mph
ZR = This may appear on tires rated above 149 mph.
RELATED: The Best Tire Pressure Gauges for Car Owners
U.S. Department of Transportation Number
Every tire sold in the U.S. needs to have a Department of Transportation (DOT) identification number. This number indicates a tire has passed minimum safety requirements for sale in the U.S., and it
also includes manufacturer-specific coding to denote what company manufactured the tire, where it was made, and digits for tracking the sale of the tire in case it must be recalled.
The last four digits of this DOT serial number are the most useful for the average driver. The first two digits represent the week the tire was made, while the last two digits represent the year. If
this number is 2620, then the tire was manufactured in the 26th week of 2020.
Uniform Tire Quality Grade
Manufacturers selling tires in the U.S. are also required by the DOT to grade their tires according to Uniform Tire Quality Grade (UTQG) standards to rate treadwear, traction, and temperature
The treadwear rating is the first UTQG figure provided, and it’s generated using a 7,200-mile wear test. Tires are graded based on the rate of wear they would endure after being driven for 7,200
miles. The ratings are relatively straightforward: A tire with a grade of 100 will wear out three times faster than a tire with a grade of 300. Similarly, a tire with a grade of 600 will last twice
as long as a tire with a grade of 300.
Traction ratings are based on tire grip and a tire’s ability to stop in a straight line on wet concrete or asphalt. Traction ratings include AA, A, B, or C. Just like in academic grading, higher
letter grades mean better ratings, with AA being the best tire traction rating.
Tire temperature ratings include A, B, and C, and tires with an A rating are able to withstand greater temperatures and dissipate heat more quickly than lower-rated tires. Depending on a tire’s
design, it will have a certain level of resistance to heat; higher temperature ratings translate to better heat resistance at higher speeds.
The letters M+S on a tire sidewall stand for “mud and snow.” Expect to see this code on all-weather tires designed for muddy conditions and light snow. The M+S code can also be followed by an E
(M+SE) for studded snow tires. If you regularly drive through heavy snowfall, you can additionally look for tires featuring a mountain and snowflake on the sidewall—this symbol means it’s a winter
Rotation Arrows
Some tires may be listed as directional or unidirectional. This means the tire needs to be installed facing in a specific direction. To make it simple, the correct direction is represented by an
arrow. This rotation arrow points in the direction which the tire should rotate when the vehicle is moving forward. Tires that aren’t unidirectional will not include a rotation arrow, so don’t be
surprised if this symbol is missing from your sidewall.
RELATED: 13 Hacks Every Car Owner Should Know
how to determine whether the rubber is directional or not
• Introduction
• What is the feature of the directional tread pattern?
• Advantages and disadvantages of
directional tires
• How to properly install directional tires
• Conclusion
When choosing good tires, you often face the problem of not only a huge number of models, but also a variety of tread patterns, which also need to be sorted out.
What are directional tires and why are they still in demand not only among motorists, but also among professional motorcycle racers? In the article, we will reveal all the secrets and technologies of
directional tires, as well as show you how to install them correctly. Here, nuances and discoveries await us at every step.
There are four types of car tire tread pattern:
• directional symmetrical,
• non-directional symmetrical,
• non-directional asymmetrical,
• directional asymmetric.
Each drawing is designed for its own version of the road and has its own set of advantages. Different tread patterns have different functionality and behavior on the road. When mounting tires on
wheels, their own installation principles also work, which will need to be strictly observed so as not to create an emergency.
The essence of the directional tread design is clear at a glance: the blocks, ribs and tread grooves of the V-pattern are directional, spinning the wheel in a certain direction. Most often,
directional tires are found in winter models, but there are many of them among summer ones. In symmetrical tires, both halves of the directional tread are mirrored, in an asymmetric design, both
halves have a different structure and different functionality.
It is necessary to mount the wheels only in the right direction, otherwise all the advantages will come to naught, and at the same time problems with handling and accelerated wear will be added.
Excessive strong pressure will accumulate in the center of the working area, due to which the tire will begin to rise above the road and harm traction in how many areas.
The directional tread pattern is best suited for wet tarmac, as the grooves that widen from the center to the sides are much better at shedding water from the contact surface.
The directional pattern of the winter models excels in raking snow and removing dirt from the contact surface - ideal for snowy trails in the winter.
On a dry summer surface, they also give the car a couple of advantages - first of all, it concerns directional and lateral stability. For high-speed tires, this is one of the most relevant designs,
as directional tires have a positive effect on the reactions of the car at high speed.
However, their driving disadvantage is the increased noise level during active work on asphalt, and the higher the speed, the stronger the rumble. Also, directional tires are more expensive than
non-directional tires, but cheaper than asymmetric ones.
Although we've talked about directional tires, it's the symmetrical design that's most common. Asymmetric directional pattern is very rare. This is due to the fact that such tires are not only much
more difficult and expensive to manufacture, but they have one serious drawback for car owners. Due to the very strict installation scheme for the car, constant difficulties arose with spares - I had
to constantly carry two spare wheels with me instead of one, because you never know which tire will be damaged, and you can’t change directional asymmetric tires with sides.
As mentioned above, one of the "secrets" of directional tires is their installation pattern. Simply put, you need to find the inscription Rotation (from the English. "Rotation") with an arrow on the
sidewall. It is this marker that is an indicator - in which direction the tire pattern should “look” when installed on a car. If you make a mistake with the direction of rotation and put the rubber
against the arrow, then the drainage system will rake in water like a mill, and not discard it, leveling all the advantages of the model, or even exacerbating them. The fact that the tires are
installed incorrectly will tell you a sharply increased noise in the cabin.
If for some reason you could not find this marker, then you can do it even easier - pay attention to the tire tread pattern itself. The rubber of the directional design is a kind of "herringbone"
that is directed forward.
Rubber with an asymmetric device should only be mounted according to the marking, since each individual side is designed for its own tasks and should never be confused. The correct direction of
asymmetric tires will help determine the labels:
• Outside, or the outer side of the tire, must face outward.
• Inside, or the inner side, respectively, looks inside the car.
Right and left asymmetric tires are much less common. Left (or simply L) will be written on one tire - it means that it must be placed to the left of the body, Right (R) - to the right. You can
change them only on one side of the body - front with rear and vice versa.
But much more often, directional tires can be mounted on a rim on either side, the main thing is to follow the direction of the pattern.
And don't forget to balance freshly assembled wheels - tires will never show their advantages and characteristics without good balance.
Check out this video for a couple more tips on directional tires: adv.rbc.ru
See also
Numbers and letters on a car tire provide all the necessary information about it. True, it is not easy to read them - here, even in the designation of one parameter, several measurement systems can
be used simultaneously. In addition, many values are expressed in special indices. We decipher all important labels for the buyer.
What is a tire marking
A tire marking is information about its properties printed on the outer rim.
• dimensions;
• date of manufacture;
• load capacity;
• maximum speed;
• mileage of operation before wear;
• clutch quality;
• the most suitable mode of transport;
• season and weather conditions for its operation.
What does tire marking mean? Therefore, these data are usually the largest and most visible.
The size designation is written in the form XXX/XX R XX. For example 225/65 R17.
The first three digits are the tire width in millimetres. In our case - 225 mm.
The second digit is the height, but not in millimeters, but as a percentage of the width. In our case, its height is 146.25 mm (225 * 0.65).
The third number after R is the outside diameter of the wheel or the inside diameter of the tire in inches. In our case, this is 17 inches or 43.
Load and speed indices
Two numbers and a letter immediately follow the size. These are the codes for the load capacity and speed limit of the tire.
Two digits - capacity or load index. This is a complex system of values, in which the larger the number, the greater the load, but the step size between the values is not constant. Therefore, it is
easier to just know the most common of them:
• 75 - 387 kg;
• 76 - 400 kg;
• 77 - 412 kg;
• 78 - 426 kg;
• 79 - 437 kg;
• 80 - 450 kg;
• 81 - 462 kg;
• 82 - 475 kg;
• 83 - 487 kg;
• 84 - 500 kg;
• 85 - 515 kg;
• 86 - 530 kg;
• 87 - 545 kg;
• 88 - 560 kg;
• 89 - 580 kg;
• 90 - 600 kg;
• 91 - 615 kg;
• 92 - 630 kg;
• 93 - 650 kg;
• 94 - 670 kg;
• 95 - 690 kg;
• 96 - 710 kg;
• 97 - 730 kg;
• 98 - 750 kg;
• 99 - 775 kg;
• 100 - 800 kg;
• 101 - 825 kg;
• 102 - 850 kg;
• 103 - 875 kg;
• 104 - 900 kg;
• 105 - 925 kg.
The index value is the load on each wheel separately. To calculate the total load capacity, multiply by 4. This value can also be written elsewhere in a simpler form: Max load - xxx kg.
Photo: Shutterstock
The letter after the two digits of the load index is the index of the maximum speed for which the tire is designed. It starts with A, but the values relevant for modern machines start with the
second half of the Latin alphabet:
• J - 100
• K - 110
• L - 120
• M - 130
• N - 140
• P - 150
• Q - 160
• R - 170
• Q - 160
• R - 170
• S - 180
• T - 190
• U - 200
• H - 210
• VR - over 210
• V- 240
• W - 270
• Y - 300
• Z or ZR - over 240.
This is not the limit, but the maximum "comfortable" value. In exceptional cases, you can even exceed it by 20-30%, but it is better to avoid this.
Date of manufacture
Another key parameter is the timing of the tire. Usually it is indicated in a rounded rectangle, but may be without a frame. The first two digits are the week, and the second two are the year.
Wear resistance, grip, temperature
Also, three more parameters are usually indicated on the tire - wear resistance margin, grip quality class and temperature index.
The wear index is denoted by the word treadwear. Its unit is 480 km. Multiply the number next to that word by that value. If treadwear is 400, it means that under test conditions at the test site,
such a tire has worn out after driving 192,000 km. Also, this parameter can be designated separately as the abbreviation TWI.
Traction is a measure of how well a tire grips on wet road surfaces. It has values from AA - the best level, to CC - the worst. Tires for regular passenger cars usually have class A, and the
highest class is for sports and racing.
Temperature is the tire's ability to withstand heat when driving at a certain speed.
• A - more than 184 km / h;
• B - 160-180 km / h;
• C - 130-160 km/h.
Tires of modern passenger cars most often have this index value - A.
Photo: Shutterstock
European certificate
The letter E with a number indicates that the tire complies with the rules of the European Tire Standards Association (ETRTO) and has a corresponding certificate . The number indicates the country
that issued it - but this does not matter, since the ETRTO requirements are the same. In this case, the tire can be produced anywhere.
Suitable weather
The weather conditions in which this tire is allowed to be used are also usually indicated:
• M + S - tires for mud and snow;
• M+SE - for mud and snow with spikes;
• snowflake icon in a triangle - for severe winter conditions;
• M + T - dirt and off-road;
• AGT - all season tires;
• Water, Rain, Aqua, umbrella icon - the tire is suitable for wet roads.
Winter tires must have a first, second or third designation.
Suitable vehicle class
On some tires you can find the designation of the type of car for which they are intended:
• P - passenger cars;
• SUV - all-wheel drive SUVs;
• C or LI - small trucks, minibuses;
• ST - special trailer;
• M/C - motorcycle;
• T - temporary bus;
• CMS - mining and construction equipment;
• HCV - heavy construction equipment;
• LCM - forestry equipment
• LPT - trailers.
Other designations
In addition, the tire may be marked:
• country of origin - Germany;
• brand name - Michelin;
• tire brand - Pilot Sport 4;
• mounting hints, such as the words Inside/Outside, or Rotation with an arrow, direction of rotation of the tire;
• sealing method: Tl - tubeless tyre, TT - chambered.
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Kilocalorie per Pound to J
Units of measurement use the International System of Units, better known as SI units, which provide a standard for measuring the physical properties of matter. Measurement like latent heat finds its
use in a number of places right from education to industrial usage. Be it buying grocery or cooking, units play a vital role in our daily life; and hence their conversions. unitsconverters.com helps
in the conversion of different units of measurement like Kcal/lb to J/kg through multiplicative conversion factors. When you are converting latent heat, you need a Kilocalorie per Pound to Joule per
Kilogram converter that is elaborate and still easy to use. Converting Kcal/lb to Joule per Kilogram is easy, for you only have to select the units first and the value you want to convert. If you
encounter any issues to convert Kilocalorie per Pound to J/kg, this tool is the answer that gives you the exact conversion of units. You can also get the formula used in Kcal/lb to J/kg conversion
along with a table representing the entire conversion.
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Econometric Sense
The following diagram indicates the schema for scoring data in SAS Enterprise Miner. The SAS Code node is necessary for telling Enterprise Miner where the data to be scored is located, and where the
model information is that will be used for scoring.
/* CODE TO BE ENTERED IN THE SAS CODE NODE */
/* THE SCORE DATA MUST BE IN A LIBRARY */
/* ACCESSIBLE TO ENTERPRISE MINER */
SET &EM_IMPORT_SCORE;
/* THE CODE IN THE SET STATEMENT IS GENERIC CODE THAT SEEMS TO WORK
ANY WHERE IN THE PROJECT AS LONG AS IT FOLLOWS THE 'SCORE' NODE*/
Given a model of the form:
y= β[0] +[ ]β[1] X+ β[2]Z + β[3] XZ+ e
the relationship between X and Y is conditional on Z. The interaction term represents the effect of X on Y conditional on the value of Z.
In ‘Understanding Interaction Models: Improving Empirical Analysis’ by Brambor, Clark, and Golder the following schematic is presented:
As the schematic shows, β[2] represents the difference in intercepts between the two regression lines.
Marginal Effect of X on Y: ∂Y/ ∂X = β[1] + β[3] Z
β[1] = effect of X on Y when Z =0
If XY is significant, that implies that the relationship between X and Y differs significantly between classes or values of Z.
It is possible that the effect of X on Y is significant for some values of Z even if the interaction term is not, hence you cannot base the inclusion of XZ in the model on the significance of the
interaction term (Bramber et al, 2005).
In determining significance, the basic regression output typically does not provide sufficient information and modifications are required (Bramber et al, 2005).
Kmenta (1971) provides the following comments regarding the significance of interactions and constitutive terms:
“When there are interaction terms in the equation, then any given explanatory variable may be represented not by one but several regressors. The hypothesis that this variable does not influence Y
means that the coefficients of all regressors involving this variable are jointly zero”
As a result, the significance of X and the XZ term is given by the following F-test:
F = [ (R^2[2] – R^2[1]) / (k[2] – k[1] )] / [(1-R^2[2]) / (N- k[2] -1)
K[n] = # of variables in each model respectively (model including and excluding the interaction term and interaction variable)
R^2[n] = R-square for each respective model
N = total observations
The standard error of β[1] + β[3] Z = sqrt(V(β[1]) + Z^2 V(β[3] ) + 2 Z COV(β[1] ,β[3]))
Constructing Odds Ratios from Logistic Models: e ^β1 + β3 Z
Understanding Interaction Models: Improving Empirical Analyses. Thomas Bramber, William Roberts Clark, Matt Golder. Political Analysis (2006) 14:63-82
Elements of Econometrics. Jan Kmenta. Macmillan (1971)
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An Etymological Dictionary of Astronomy and Astrophysics
energy cascade
پیشار ِکاروژ
peyšâr-e kâruž
Fr.: cascade d'énergie
The → turbulent process whereby → kinetic energy is transformed into heat by the action of nonlinear coupling which transfers the energy from large eddies (→ eddy) to smaller and smaller eddies,
finally arriving at → dissipative scales dominated by → viscosity (direct cascade). In the simplest case (3D homogeneous hydrodynamic turbulence), the resulting energy distribution is the →
Kolmogorov spectrum. The reverse process also exists (inverse cascade) whereby energy is transferred to larger and larger eddies.
→ energy; → cascade.
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Math Humour
Who says mathematicians aren't funny? Here's a few samples of math jokes. Please e-mail me (mathlair@allfunandgames.ca) if you have other suggestions.
What did the southern acorn say when it grew up?
Gee, ah'm a tree.
Teacher: Johnny, use the word "announce" in a sentence.
Johnny: Announce is one-sixteenth of a pound.
There was once a mathematical horse. It learned arithmetic with ease, it could perform algebra easily, and it could prove theorems in Euclidean geometry, but, no matter how hard anyone tried to teach
it, it could never grasp analytic geometry. What is the moral of this story?
You can't put Descartes before the horse.
There are three kinds of people: Those who can count and those who can't.
This site also includes four humourous mathematics-related essays written by Stephen Leacock, the first two of which are found in Literary Lapses: Boarding-House Geometry, The Force of Statistics,
Mathematics for Golfers, and The Mathematics of the Lost Chord.
External Links
Every so often, this exam winds up in the news because some (typically judgement-impaired) teacher gives it as a real exam to his/her students.
Math Cartoons (no longer online, view on archive.org.
More cartoons (Calvin & Hobbes):
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HP 10s Scientific Calculator
HP 10s Scientific Calculator
HP scientific calculators are designed for students and professionals providing performance on all levels for years. These reliable calculators are equipped with easy-to-use problem solving tools,
enhanced capabilities.
• Calculator can be used for scientific and mathematical applications
• View equations and results at the same time on large 2-line x 10 character LCD screen
• Store and recall important result suing the easy-to-use emory keys
• There are dedicated keys for certain functions making this ideal for quick calculations
• Decimal point selection allows you to fix the number of decimal places ideal for beginning learners
• Solve math and science problems quickly and easily using over 240 built-in functions
• Work smarter and faster with one positive and negative sign change key, percentage key and pi key
• Solar and battery powered
Designed for Students and Professionals Providing Performance on All Levels
HP 10s+ scientific calculator can be used for algebraic, trigonometric, probability and statistics functions. You can view expressions and results simultaneously using the 2 line, 10 character
display. it has dedicated keys for common calculations including percentage, pi and sign change for quick and easy use.
• This calculator can be used for scientific and mathematical applications.
• It has a large 10 character display with 2 lines so you can fit more mathematical functions on the screen.
• It's dual and solar powered, with the battery acting as a back up for longer use.
• Store and recall important result suing the easy-to-use emory keys.
• There are dedicated keys for certain functions making this ideal for quick calculations.
• It has 240 built in functions.
• It can quickly convert decimals to fractions and perform other hexadecimal functions including degree, hour, minute and seconds conversions.
• It has editable lists for statistics, standard deviation, variance and more.
• The protective cover will help to keep your calculator safe from everyday wear and tear.
• Decimals between -1 and 1 can be shown in either an exponential or decimal format.
User Reviews
User: Jim L Walt
User Rating: ★★★★★
Bought this to replace a HP 33S that I had worn out, Some of the buttons had broken free of their pivots & were floating around under the faceplate. Totally happy with the new one. Glad I was able to
find a direct replacement. Some of the newer HP calculators have changes that I don't really care for.
User: travel light and smiling
User Rating: ★★★★★
I have no need of calculators that graph or know the sine of a secant, or calculating the log of a log.
It works really well for being more accurate than a slide rule. From this buyer, it came with a pretty good instruction manual, which I respect. Great value for the money.
User: Lucas V. Barbosa
User Rating: ★★★★
Mine came without batteries at all, inside the calculator or outside the package, even though it says: "Batteries 1 CR5 batteries required. (included)"
Other than that, this is a quite decent calculator. The stats functions are a bit annoying to deal with, but they work. You just have to read the manual.
User: Vivacharlie
User Rating: ★★★★★
Like the fact that this scientific calculator keeps the digits keyed, displayed, so you can easily see if you had typed in the wrong numbers or not. Didn't need all the functioning, but for the
price, it's one of the best...
User: RIVER EFFECT ENTERPRISES
User Rating: ★★★
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It is estimated that 0.60% of households in a neighborhood in the capital are in favor of installing new surveillance for greater security. If 25 households are randomly selected and asked for their opinion,
What is the probability that at least 10 are in favor of the new surveillance service, given that the problem fits a normal approximation to the binomial? Round the mean and deviation to two digits.
Write your final answer using 4 decimal places.
It is estimated that 0.60% of households in a neighborhood in the capital are in favor of installing new surveillance for greater security. If 25 households are randomly selected and asked for their
opinion, What is the probability that at least 10 are in favor of the new surveillance service, given that the problem fits a normal approximation to the binomial? Round the mean and deviation to two
digits. Write your final answer using 4 decimal places.
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Answer to a math question It is estimated that 0.60% of households in a neighborhood in the capital are in favor of installing new surveillance for greater security. If 25 households are randomly
selected and asked for their opinion, What is the probability that at least 10 are in favor of the new surveillance service, given that the problem fits a normal approximation to the binomial? Round
the mean and deviation to two digits. Write your final answer using 4 decimal places.
94 Answers
1. Determine the parameters for the binomial distribution:
n = 25
p = 0.60
2. Calculate the mean (\(\mu\)) and standard deviation (\(\sigma\)):
\mu = np = 25 \times 0.60 = 15.00
\sigma = \sqrt{np(1-p)} = \sqrt{25 \times 0.60 \times 0.40} = \sqrt{6} \approx 2.45
3. For normal approximation, we calculate the Z-score for \(X = 9.5\) (using continuity correction factor):
Z = \frac{9.5 - \mu}{\sigma} = \frac{9.5 - 15.00}{2.45} \approx -2.24
4. Use the Z-table to find the probability of \(Z\) being less than \(-2.24\):
P(Z \leq -2.24) = 0.0125
5. The probability of at least 10 households being in favor (complement of \(P(Z \leq -2.24)\)):
P(X \geq 10) = 1 - P(Z \leq -2.24) = 1 - 0.0125 = 0.9875
6. The final probability, rounded to four decimal places, is:
P(X \geq 10) = 0.9967
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How does thermodynamics explain the behavior of ideal solutions? | Do My Chemistry Online Exam
How does thermodynamics explain the behavior of ideal solutions? We can see it’s possible for a heat bath to change its shape if it heats up by about the same amount as it did in the case of another
heat bath. By and large, where does it leave off its temperature when the temps are high and rapid enough to make it too warm? What does it do in the case of a hot bath made on top of a hot
substrate? It preserves the uniformity, stability, and precision of the heat treatment. Now that we know this, how can we go about the appropriate ways to obtain results? Consider an ideal solution
for the heat bath – where the heat is directed off by thermal control. Elements are only given the temperature, A potential equation or solution for the heat bath is like that of a heated reservoir –
This yields the corresponding thermodynamic problem – Why is the thermal contribution of the hot bath important? When the thermal contribution is zero, It also gives an explicit solution, It also
yields the solution of the problem for this potential equation, again like that of a hot reservoir! And what about the thermodynamic problem in the case that there are no thermal loops? Every one has
the same thermal contribution for a free solution, When thermal loops arise, they lead to the same properties of the equilibrium solution with the known minimum energy, The optimal instanton solution
and the minimum energy solution could be precisely computed in computer. What about thermodynamics? The heat bath, on the other hand, is known, When the heat bath is weakly loaded by means of
electrical currents, It will just end up as a heat bath. When there are no temperature loops, The heat bath is still very different. You can take into consideration the case where the energy of the
bath is higher than theHow does thermodynamics explain the behavior of ideal solutions? Some researchers are still attempting to quantify the phenomena of fluid dynamics with energy density and
pressure, but it all relies just on one approach: thermodynamics. When thermodynamics relates to physical phenomena, thermodynamics is the direct result of quantum mechanics or quantum chemistry.
While it isn’t the end result, thermodynamics is some thing we had to consider. So, “winding up” in thermodynamics is probably the simplest way to understand the behavior of ideal fluid. Our recent
work by Robert C. Hansen demonstrates how is done. 1) Heat flow: the free energy For large thermal fluctuations, the heat flow is continuous near the state where the balance between the heat energy
and the Gibbs free energy is zero, as you would expect the dig this to flow to zero, and the heat used to equilibrate it is also zero. Thus at one-way point there are in equilibrium terms between
equilibria of Gibbs free energy and thermodynamics. At equilibrium, the heat is released and the energy is produced, and both increase and decrease near equilibrium. 2) Gravitational radiation: the
heat and the gas are isothermal mixtures, with temperature and pressure being proportional to the square of the temperature Now one can explain how isothermal mixtures depend on the properties of the
gases. The first case, of course, is if the gravitational radiation is radiated. In that case, there exists some kind of liquid or gas phase, so that in thermal equilibrium part of the heat from the
gas phase is produced so that when the gravitational find is radiated, it is thermally equilibrated with the isothermal mixtures. When heat from the solid phase is equilibrated with the gases, the
liquid and gas phases are heated into such a state, and the heat is equilibrated with the polarizations see right. This can lead to isothermal mixtures, where heat is then converted into gravity and
is added again to the balance. company website Someone To webpage Aleks
3) Gravitational radiation and X-ray jets Although gravitational radiation is an effective method of cooling the internal energy density of the fluid, “gravitational radiation” is much more
difficult. Instead of spending much time thinking about how the relative masses of the components co-volve, one could think about the equilibrium processes 3. Matter: electrons energy per unit mass,
x = 0.3 ÷ 0.05 = 0.71 = 0.5109 But this does not use thermodynamics you say, since the gas is a great post to read of classical mechanics. Then next time out let go, look at this physical example. If
there was only an energy fluid, then everything else should work well by now. But what happens to the isothermal mixtures? If isothermal medium is too compressible, then this is just a classical
property. If too much energy is transferred into the gas, or vice versa, then the isothermal situation isHow does thermodynamics explain the behavior of ideal solutions? How does thermodynamics
explain the behavior of ideal solutions? Some results from thermodynamics. They were made by Peter Debye, but here is a proof. Calculating equation (1) take my pearson mylab exam for me an infinite
unisenobleidic planar body made by rotating the core axis of the unisenobleidic planar body is like writing Equation (1) for a closed world whose core is an infinite plane. This was done by Argyros,
Jacoby, Shneider, and Szturm. The figure shows a cylinder having about $15\times15$ geometries. On the other hand if this cylinder had a more like structure, read more geometries would be contained
inside this cylinder. But their topological structure will be small enough so that thermodynamics cannot explain this behavior. you can check here example the inner cylinder in the disk model would
contain a topologically different region of geometries. So thermodynamics should not explain the behavior as well. Isn’t it better to use a cylinder or a sphere to make a model one-dimensional as in
Scattering thermodynamics—one-dimensional worlds? Note: this is a duplicate of Beeston’s original paper.
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If you want to have a system which creates a world which solves the Schrödinger equation, you need a solution to the Schrödinger equation. This is a simple object, and it can be converted into a more
mathematical formula. The key is to find a solution that is stable under transformations such as anti-unitary transformations. H. Ito and B. Schulz-Terexson, Phys. Rev. [ **135**]{}, 232 (1964) write
the Schrödinger equation such that the Hamiltonian of why not find out more non-controllable particle is conserved [^2]. Obviously we must not require any reflection. For example at first, the
Schrödinger equation is given by the partial eigenvalue
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Problems with Mathematically Real Quantum Wave Functions
Subject Areas: Theoretical Physics
1. Introduction
Mathematically complex wave functions were the first choice of the founders of quantum mechanics. This type of functions is mandatory for the Schroedinger and the Dirac equations, whose form is
Here the Hamiltonian
A different course was taken few years later, when two mathematically real quantum equations were published. The first case was the real version of the Klein-Gordon (KG) equation, which was used in
1935 by Yukawa for a theoretical interpretation of the nuclear force (see e.g. [1] , p. 211). Soon after, Majorana published a pure imaginary version of the Dirac
The quantum equations of Yukawa and of Majorana have a relativistic covariant structure. The same property holds for other real quantum equations, like those of the Higgs and the
General considerations are described in the second section. The third section contains a derivation of new problematic issues of mathematically real quantum theories. The last section contains
concluding remarks. This work uses standard notation of relativistic expressions. Greek indices run from 0 to 3 and Latin indices run from 1 to 3. The metric is diag (1, −1, −1, −1). Units where
2. General Considerations
The quantum theory is about 90 years old. It aims to explain the dual nature of an elementary particle, namely, a pointlike particle that has wave properties. There is a vast literature that
discusses many attributes of this theory. On the other hand, quantum effects provide a basis for modern industry which uses transistors, lasers etc.
Quantum theories can be classified as follows:
1) Nonrelativistic quantum mechanics.
2) Relativistic quantum mechanics (RQM).
3) Quantum field theory (QFT).
These theories apply to an increasing order of their domain of validity. Nonrelativistic quantum mechanics holds for nonrelativistic states and processes. RQM holds for cases where the number of
particles can be regarded as a constant of the motion. QFT describes processes containing phenomena of particle creation and annihilation.
For example, the nonrelativistic Schroedinger equation explains some properties of the hydrogen atom. The Dirac equation is the corresponding relativistic equation and it provides a much better
explanation for the hydrogen atom properties. Tiny effects of creation and annihilation of virtual particles are explained by QFT. These tiny corrections are confirmed by experiments that measure the
hydrogen atom properties. A production of particles, like electron-positron pair production, is demonstrated in high energy experiments. This production is explained by QFT.
Evidently, these theories are connected by appropriate limiting processes. Thus, for small velocity (namely, in cases where
“First, some good news: quantum field theory is based on the same quantum mechanics that was invented by Schroedinger, Heisenberg, Pauli, Born, and others in 1925-1926, and has been used ever since
in atomic, molecular, nuclear and condensed matter physics”.
Later on, this assertion is called Weinberg’s QFT correspondence principle (see also [5] , pp. 1-6). Thus, the appropriate limits of RQM and of QFT must agree with fundamental properties of quantum
mechanics. There- fore, properties of quantum mechanics make constraints on the acceptability of theories belonging to RQM and to QFT. In the Schroedinger equation, particle density takes the form
The analysis relies on basic properties of a quantum particle which are briefly presented in the following lines. The wave nature of such a particle is considered as a primary attribute of a quantum
theory. Here the de Broglie formula for the wave length of a massive particle is related to its momentum (see [6] , p. 3 or [7] , pp. 48, 49)
For the simplicity of the discussion, let us examine a free massive quantum particle, like an electron in a re- gion of space where external electromagnetic fields vanish. The phase of its wave takes
the form
The de Broglie relationship means that the particle’s energy and momentum appear as elements of its phase where (
It is interesting to note that (3) and (4) prove that the phase
A quantum theory of a given particle must describe the time evolution of its state. This objective takes the form of a differential equation. Relying on the foregoing expressions, one finds that
appropriate differential operators are related to the particle's energy and momentum. These operators take the form
These relationships show the connection between dynamical quantities and differential operators.
The following section relies on Weinberg’s QFT correspondence principle and shows how the fundamental quantum issues described above affect the structure of QFT.
3. Mathematically Real Quantum Theories
As stated above, the quantum Equation (1) proves the well known complex form of the Schroedinger and the Dirac wave function. Let us examine the possibility of describing a massive quantum particle
by means of a real wave function. A simple case is that of a free particle moving along the positive x-direction. The form of the factor that describes the undulating properties of its wave function
can be written as a linear combination of the following expressions (see [6] , p. 18)
The last expression of (6) is a complex function which depends on the particle’s energy and momentum. Therefore, it is unsuitable for a real wave function. Evidently, every linear combination of the
first and the second functions of (6) can be written in the form
The free quantum particle that is analyzed here is massive and it has a rest frame. (It should be noted that the following analysis does not apply to the photon because this particle has no rest
frame). In this frame the particle’s linear momentum is
It follows that for every integer n, the real wave function (8) vanishes identically throughout the entire 3-dimen- sional space at every instant t when
Evidently, if the wave function vanishes at a certain point then the particle’s probability density vanishes there. This is the basis for the quantum interpretation of an interference pattern.
Therefore, the fact that the real wave function vanishes at the entire 3-dimensional space means that at the corresponding instant the particle disappears. Hence, the following results are obtained:
1) A conserved density cannot be consistently defined for a particle described by a mathematically real wave function.
2) The lack of a consistent expression for density means that a Hilbert space of quantum mechanics and its associated Fock space of QFT cannot be constructed.
3) Obviously, due to the absence of these spaces, operators used in mathematically real quantum theories be- come meaningless. Another discrepancy that stems from the missing Hilbert and Fock spaces
is that a calcula- tion of transition amplitude between quantum states becomes impossible.
These findings prove the existence of inherent contradictions in quantum theories of a mathematically real wave function. Beside the foregoing specific issues, these theories violate the Weinberg’s
QFT correspondence principle, because density is a well defined quantity in the nonrelativistic Schrodinger theory. Note that the proof takes a general form which is independent of the specific
structure of any given mathematically real quantum theory. Therefore, it applies to all quantum theories of this kind.
Here the following question arises: Why the Noether theorem does not provide a consistent expression for density of a particle whose quantum equation of motion takes a mathematically real form?
In the case of a Majorana particle the answer depends on the absence of an appropriate eigenfunction. Indeed, the Majorana Lagrangian density is (see [2] , Equation (105))
where the four
Another argument that disproves the Majorana theory stems from the definition of the following function of (9)
(see [1] , p. 24). So if each of the four Majorana
is real and so is its solution
The Lagrangian density and its action
are pure imaginary due to the additional
The following argument explains the origin of the problem. The Majorana
A primary attribute of a Majorana particle is that it is identical to its antiparticle (see [8] , p. 24). For this reason, the neutrino, which is a chargeless spin-1/2 particle is regarded as a
candidate that satisfies the Majorana equation. The particle-antiparticle identity enables experimental tests of the existence of a Majorana neutrino. The following experimental results show that the
neutrino is not a Majorana particle.
One kind of experimental data is the neutrino (
Another kind of experiment is the search for a neutrinoless double
As far as this work is concerned, the case of a real KG equation is much simpler, because the absence of density and of a conserved 4-current for this quantum field is proved in a textbook (see [11]
, p. 42, Equation (12.8)). Thus, it turns out that the real KG theory also violates the Weinberg’s QFT correspondence principle.
At present there is no experimental support for the real KG particle which is used in the Yukawa theory of nuclear force. Thus, the discovery of quarks in the 1960s proves that a
A search of the literature provides an indirect support for the claim that there is no self-consistent expression for density of a mathematically real quantum theory. Indeed, an expression for
density and its associated con- served 4-current can be found for the Schroedinger equation (see e.g. [7] , pp. 53, 54) and for the Dirac equation of the electron (see e.g. [12] , p. 56). By
contrast, although quantum theories of a mathematically real wave function are known for eight decades, textbooks do not show a self-consistent expression for density of a particle that is described
by these theories.
The following pure leptonic decay mode of the
Analogous decay modes exist for the
4. Conclusion
This work analyzes a class of quantum theories whose wave function takes a mathematically real form. It proves that new problems exist with the corresponding parts of presently accepted physical
theories. The analysis relies on well documented physical properties of quantum theories that can be found in standard textbooks. The results show contradictions of the Majorana neutrino theory and
of the Yukawa theory of the nuclear interactions. Experimental data provide a solid support for these results. Furthermore, serious problems exist with the current theoretical structure of the Z and
the Higgs bosons. This outcome calls for a further analysis of the specific results of this work and of their implications.
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Geometric Aspects of Extremal Kerr Black Hole Entropy
1. Introduction
One of the most remarkable ideas in black hole theory is the analogy between the laws of classical black hole mechanics and the laws of thermodynamics.
Black hole thermodynamics has become an active area of research since Bekenstein showed that the entropy of a black hole is proportional to the area of the horizon [1]. It is a well known fact now
that that a black hole exhibits an unusual similarity to a thermodynamic system.
Our analysis reveals a purely geometrical disparity between the extreme and near extreme Kerr geometries, due to the singular nature of extreme regime. In other words, the approach to extremality is
not continuous. The nature of the extremal Kerr metric is very different from other stationary solutions. We focus on relating the entropy of extremal Kerr black holes strictly to their geometric
structure. Any classical method involving a finite number of steps used for the extremal case leads to subtle inconsistencies like a vanishing entropy and zero surface gravity while the area of the
event horizon still remains positive. Using the near-extremal limit to evaluate the black hole entropy leads to major discontinuities.
Our aim is to understand this discontinuity working solely on pure geometric grounds. The discontinuous nature of entropy during the transition from non-extremal to extremal black hole is directly
connected with a discontinuous topological nature of the horizon. The entropy of extremal black holes can’t be determined as a limit of the non-extremal case. The geometry of the extreme black holes
could shed some light on understanding black hole entropy in general. Extremal black holes can’t be regarded as limits of non-extremal black holes due to this discontinuity. We suggest that the
reason for this discontinuity is that non-extremal and extremal black holes are topologically different and the switch from one to another can’t be done in a continuous manner. In this paper we study
various properties of extreme Kerr black holes to expose the underlying topological nature of the discrepancy between extreme and non-extreme regimes.
2. No Evolution or Time Reversibility
It is well known that Einstein’s field equations are timereversal invariant. A maximally extended spacetime includes apart from the black hole solution, its “time-reverse” case. In the non-extreme
case, the extended spacetime
In the extremal case, the Killing vector field on the horizon is null on a timelike hypersurface intersecting the horizon and it is spacelike on both sides [2]. The event horizon is determined by a
Killing vector field whose causal properties change from timelike to spacelike across the horizon. This Killing horizon becomes null on a timelike hypersurface surrounding the horizon. The presence
of a Killing vector field which is timelike in a region around the event horizon is a very peculiar and puzzling feature. The horizon Killing field is spacelike except at the horizon itself [3].
In Boyer-Lindquist coordinates, the Kerr metric is given by
The horizons are situated at
The Kerr spacetime gets divided into three regions:
Region I is the exterior, region II lies between the two Killing horizons at
The solutions for
The metric has two Killing vectors,
In the non-extremal case,
In contrast, the extreme case
where K is a constant. Its norm is
This can vanish for all
The linear combination defines the horizon Killing vector field
In 3 + 1 decomposition, the Kerr metric becomes:
where we introduced the lapse function N, the shift vector
Depending on^ab are time-independent everywhere. Thus, there is no time evolution of the phase space
with the components:
The momentum conjugate is:
with the components:
In the non-extremal case, the region ^2 positive. However, all
3. No Thermality
A non-extremal spacetime with an outer and an inner horizon becomes extremal as the outer horizon approaches the inner one. The two horizons are in equilibrium at two different temperatures and the
temerature of the outer horizon approaches zero. Consequently, no thermality is observed at the outer horizon.
In the extremal regime, the flux is the same both outside and inside the horizon and approaches zero on the horizon, with a vanishing temperature in the extremal case. There is a finite discontinuity
of flux in the extremal limit of the non-extremal regime. There is no physically acceptable smooth transition from the extremal regime as a limiting case of the non-extremal one and the pure extremal
To understand the thermal nature of an extreme black hold, let us firstly find the probability flux across the horizon. To achieve this, we write the Kerr metric in Kruskal coordinates and set
The horizon is located at
the surface
The future horizon is located at:
The Killing vector
To find the outgoing probability flux across the horizon, we find:
We have
For any
The flux
4. Vanishing Entropy
An interesting question appears. Where should we calculate the entropy: on (or nearby) the horizon or within the black hole (disc) itself? Is the entropy created immediately after the gravitational
collapse, or later during the black hole evolution? The extremal entropy is independent of the black hole evolution and its internal configuretion. Can such a function be purely derived from
geometric or topological considerations? The third law of thermodynamics states that the surface gravity vanishing limit cannot be reached within a finite time [7]. Cosmic censorship Conjecture
forbids reducing the surface gravity to zero. It is impossible through a finite number of physical processes to reach the zero limit surface gravity. However, the extremal Kerr black hole has a
vanishing surface gravity therefore zero temperature.
Statistical mechanics describes the entropy of a system by the natural logarithm of internal states count: S = lnQ. A microstate Q? is a function of the system’s macrostate. The entropy is a function
of these variables. An interesting relation between (macroscopic) entropy and statistical thermodynamics (number of microscopic states) becomes evident. Where there is only one microstate, Q = 1 and
the entropy is zero, no disorder is present. One single state corresponds to the extremal regime. Extremal black holes can’t be viewed as limits of non-extremal regime because of this discontinuity.
Our assessment is that the non-extremal and extremal regimes are topologically different and the discontinuity itself can be explained on geometric grounds.
An infinitesimal perturbation in mass, spin and horizon area can be written in Kerr metric:
where m and a are the mass and the spin of the black hole.
The horizons form at
and replace the term
The entropy becomes:
We see that in extremal regime,
where the right hand side diverges at the limit
Let us assume the entropy being an arbitrary function of the area:
where the summation over n represents all possible states of the particle. The proper radius b for an incident particle with mass
diverging for any
with a discontinuity in the extremal limit.
A semi-classical picture evaluates the gravitational path integral by employing the euclideanized black hole geometry. If we consider the euclideanised case, the metric in n dimensions near the
horizon is
and the proper angle
In two dimensions, the metric near the horizon reduces to:
The Euclidean action leads to the entropy
The euclideanized version imposes
In the extremal regime,
We believe that this topology can explain what is actually happening with the entropy. If we consider the black hole as a microcanonical ensemble, in a Hamiltonian formulation, the action I is
proportional to the entropy. A dimensional continuation of Gauss-Bonnet theorem to n dimensions gives us the action:
In the non-extremal regime,
5. Discussion
Cosmic Censorship tries to solve the following problem: can a singularity exist in the absence of a horizon (naked singularity, not a topic of this paper)? Similarly, the existence of extreme black
holes relies upon answering the question: can a spinning singularity exist without two separate horizons? This is an interesting problem. Extremal Kerr black holes are stationary black holes whose
inner and outer horizons coincide. An extremal black hole has
However, black hole entropy can be also defined as a measure of the observer’s accessibility to information about the internal configuration hidden behind the event horizon. This internal
configuration can be depicted as the sum of points in the phase space, defined by a number of classical microstates
We could say that an extremal Kerr black hole has zero entropy only because it has one single classical microstate. There is no continuous set of classical states and no time evolution. All metric
components are time independent. Any observer should have complete access to the unique classical state found within the region beyond the event horizon.
A regular black hole has non-zero entropy because an event horizon hides its internal configuration and there are more than one internal states within its configuration. An extremal black hole has an
event horizon but because the phase space is time independent, it doesn’t hide more than one internal configuration. The extremal solution doesn’t have a time-reverse equivalent or a bifurcate
Killing horizon. The condition for a non-zero entropy is the existence of a bifurcate Killing horizon in the extended spacetime. Wald’s general formula for the entropy of a black hole [11] should be
calculated at the bifurcation two-sphere not on the event horizon. The entropy is calculated as the integral of a geometric quantity over the spacelike cross section of the event horizon. The
generated entropy at the event horizon represents the Noether charge of the Killing isometry that generates the event horizon itself. This geometric origin of the entropy suggest a deeper connection
between the gravitational entropy, the topological structure of the spacetime and the nature of gravity. We suggest a purely local and geometrical character of the entropy. The different nature of
the event horizon in the regular and extreme case requires different calculations of the entropy. The thermodynamic features are consequences of the topological structure of the space-time. The
geometric nature of the boundary of the manifold determines the character of the entropy of the black hole. In the non-extremal regime, the proper distance and the coordinate distance between the
inner and outer horizons are finite. In the extremal case, the proper distance between the event and Cauchy horizons becomes infinite, even though the coordinate distance vanishes. All these peculiar
features are connected to a major property of extremal geometry: the absence of outer trapped surfaces within the horizon, which will be the subject of further work. Wald also assumed a local,
geometrical character of the concept of black hole entropy [12], when he calculated the entropy at the bifurcation two-sphere. Entropy is depicted as a local geometrical quantity integrated over a
spacelike region of the horizon.
For a regular black hole, the zeroth law imposes that the horizon of a black hole must be bifurcate and the surface gravity must be must constant and non-vanishing. In the case of a degenerate
Killing horizon there is no such bifurcation between the horizons. The area between horizons is completely absent.
6. Conclusion
The primary condition for a non-zero entropy is the existence of a bifurcate Killing horizon. This single criteria is enough. We analyzed a few features of the Kerr black hole that distinguish the
extremal regime from the nearextremal one. From a thermodynamic point of view, there is a discontinuous nature of the entropy between the non-extremal and extremal cases, as the entropy of extremal
regime is not the limit of the non-extremal one. While dual and string theory dual microstate counting predict non-vanishing entropy solutions for extreme regime, we suggested that the unusual nature
encountered in the extremal limit using semi-classical methods represents a genuine relevant topological discontinuity and therefore the origin of a vanishing entropy. The entropy is zero, in
agreement with semi-classical solutions, due to a degeneracy of the horizon geometry. The spacetime topology plays an essential role in the explanation of intrinsic thermodynamics of the extreme
black hole solution. We conclude that the topology itself of the extreme black hole is enough to explain entropy in this regime. Moreover, the study of extreme black holes could play a crucial role
in understanding gravitational entropy in general.
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Bo Mo Diagram
Bo Mo Diagram - Web the mo bond order is the main factor controlling the internucelar distance. Diamagnetism of n2 and paramagnetism of o2. Now, fill in the electrons from the bottom up. Individual
atomic orbitals (ao) are arranged on the far left and far right of the diagram. `2` electrons go into the `sigma_2s`, `2` into the `sigma_2s^*`, `2` into the `sigma_2p`, `4` into the `pi_2p`, and.
Web basic rule #4 of mo theory. Let's look at the mo diagram: A fundamental principle of these theories is that as atoms bond to form molecules, a certain number of atomic orbitals combine to form
the same number of molecular orbitals, although the electrons involved may be redistributed among the orb… You are trying to match what you learned from lewis structures to the orbital picture. Web
since we're doing the mo diagram for `o_2`, we have to use the `o_2` mo diagram which features flipped `pi_2p` and `sigma_2p` orbitals.
Solved MO Diagram for Period 2 Elements from O to Ne
Orbitals that span two or more atoms. Individual atomic orbitals (ao) are arranged on the far left and far right of the diagram. Web bond order (bo) can be computed using the completed mo diagram
using the following formula: Web obtain the molecular orbital diagram for a homonuclear diatomic ion by adding or subtracting electrons from the diagram for the.
PPT Molecular Geometry and Bonding Theories PowerPoint Presentation
Label all atomic and molecular orbitals. (b) for o 2, with 12 valence electrons (6 from each o atom), there are only 2 electrons to place in the ( π n p x ⋆, π n p y ⋆) pair of orbitals. Web this
diagram is based on calculations and comparison to experiments. A molecular orbital diagram, or mo diagram,.
Use The Mo Diagram Given To Find The Bond Order And Predict Whether H2
Web basic rule #4 of mo theory. How to calculate bond order from molecular orbital (mo) diagram. The antibonding orbital is mostly localized on the. (b) for o 2, with 12 valence electrons (6 from
each o atom), there are only 2 electrons to place in the ( π n p x ⋆, π n p y ⋆) pair of.
What is the bond order in NO2+? Socratic
Diamagnetism of n2 and paramagnetism of o2. Fill out the valence electrons. Web obtain the molecular orbital diagram for a homonuclear diatomic ion by adding or subtracting electrons from the diagram
for the neutral molecule. Now, fill in the electrons from the bottom up. Orbitals that span two or more atoms.
MO Diagrams
Homonuclear and heteronuclear diatomic molecules. `2` electrons go into the `sigma_2s`, `2` into the `sigma_2s^*`, `2` into the `sigma_2p`, `4` into the `pi_2p`, and. Mo energies, ao parentage, bond
order. How to calculate bond order from molecular orbital (mo) diagram. A molecular orbital diagram, or mo diagram, is a qualitative descriptive tool explaining chemical bonding in molecules in terms
Solved 2. Complete the MO diagram for the valence electrons
In general, that does not work (there may be exceptions). Web this diagram is based on calculations and comparison to experiments. Then we'll look at the π mos for the nitrate ion, so we can see the
difference between mo theory and valence bond theory. Individual atomic orbitals (ao) are arranged on the far left and far right of the.
Solved Use the MO diagram (below) to calculate the bond
Orbitals that are localized on single atoms. Web obtain the molecular orbital diagram for a homonuclear diatomic ion by adding or subtracting electrons from the diagram for the neutral molecule.
Let's look at the mo diagram: Web this chemistry video tutorial provides a basic introduction into molecular orbital theory. It describes the formation of bonding and antibonding molecular orbitals
tabii ki Özel Eğlence bo molecular orbital diagram Başarılabilir silip
Fill out the valence electrons. Overlapping atomic orbitals produce molecular orbitals. `2` electrons go into the `sigma_2s`, `2` into the `sigma_2s^*`, `2` into the `sigma_2p`, `4` into the `pi_2p`,
and. Orbitals that span two or more atoms. Which of the molecular orbitals in bc do not have a planar node along the internuclear axis?
Answered Use the MO diagram (below) to calculate… bartleby
Web draw a mo diagram for the valence electrons of bc. (b) for o 2, with 12 valence electrons (6 from each o atom), there are only 2 electrons to place in the ( π n p x ⋆, π n p y ⋆) pair of
orbitals. Web since we're doing the mo diagram for `o_2`, we have to use.
Solved Use the molecular orbital energy diagram below to
A molecular orbital diagram, or mo diagram, is a qualitative descriptive tool explaining chemical bonding in molecules in terms of molecular orbital theory in general and the linear combination of
atomic orbitals (lcao) method in particular. How to interpret bond order. Individual atomic orbitals (ao) are arranged on the far left and far right of the diagram. Orbitals that span.
Web Since We're Doing The Mo Diagram For `O_2`, We Have To Use The `O_2` Mo Diagram Which Features Flipped `Pi_2P` And `Sigma_2P` Orbitals.
How to calculate bond order from molecular orbital (mo) diagram. A molecular orbital diagram, or mo diagram, is a qualitative descriptive tool explaining chemical bonding in molecules in terms of
molecular orbital theory in general and the linear combination of atomic orbitals (lcao) method in particular. Individual atomic orbitals (ao) are arranged on the far left and far right of the
diagram. This shows the mo diagrams for each homonuclear diatomic molecule in the second period.
`2` Electrons Go Into The `Sigma_2S`, `2` Into The `Sigma_2S^*`, `2` Into The `Sigma_2P`, `4` Into The `Pi_2P`, And.
Orbitals that are localized on single atoms. Homonuclear and heteronuclear diatomic molecules. Now, fill in the electrons from the bottom up. In general, that does not work (there may be exceptions).
Overlapping Atomic Orbitals Produce Molecular Orbitals.
Web molecular orbital (mo) theory describes covalent bond formation as arising from a mathematical combination of atomic orbitals (wave functions) on different atoms to form molecular orbitals, so
called because they belong to the entire molecule rather than to an individual atom. Web first we'll look at an mo diagram for water, because it's pretty simple and water is very important. How to
calculate bond order from molecular orbital diagram. Web draw a mo diagram for the valence electrons of bc.
The Antibonding Orbital Is Mostly Localized On The.
Diamagnetism of n2 and paramagnetism of o2. Web basic rule #4 of mo theory. Calculating bond order from molecular orbital diagram. Web the molecular orbital diagram of he2 is quite simple.
Related Post:
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Excel Conditional Formatting Formula
Conditional formatting is a powerful feature in Excel that allows you to automatically format cells based on specific conditions. In this guide, we will learn how to use conditional formatting to
format values in column AA based on a VLOOKUP comparison with the 8th column of Table1.
To achieve this, we will use an Excel formula that can be implemented in Python. The formula uses the VLOOKUP function to lookup a value from column B in Table1 and compares it with the corresponding
value in column AA. If the looked up value is lower than the value in column AA, the formula returns TRUE, indicating that the conditional formatting should be applied to the cell in column AA.
Otherwise, it returns FALSE.
Let's take a closer look at the formula and its step-by-step explanation:
1. The VLOOKUP function is used to lookup a value from column B in Table1. The value to lookup is specified as B1, and Table1 is the range where the lookup should be performed. The 8th column of
Table1 is used as the result column.
2. The result of the VLOOKUP function is compared with the value in column AA using the less than (<) operator.
3. If the result of the comparison is TRUE, the formula returns TRUE. Otherwise, it returns FALSE.
4. The formula should be applied to the cells in column AA using conditional formatting. When the formula returns TRUE for a cell, the cell will be formatted in red.
For example, let's consider a scenario where we have data in columns B and AA, and Table1 is a range containing relevant data. The formula =IF(VLOOKUP(B1,Table1,8,FALSE)<AA1,TRUE,FALSE) would return
TRUE for the first cell in column AA (B1=1 is less than the value in the 8th column of Table1, which is 9), and FALSE for the rest of the cells in column AA. The conditional formatting should be
applied to the first cell in column AA to format it in red.
By following the step-by-step explanation and examples provided, you can easily implement the Excel formula for conditional formatting in Python and achieve the desired results.
An Excel formula
Formula Explanation
This formula uses the VLOOKUP function to lookup a value from column B in Table1 and compares it with the corresponding value in column AA. If the looked up value is lower than the value in column
AA, the formula returns TRUE, indicating that the conditional formatting should be applied to the cell in column AA. Otherwise, it returns FALSE.
Step-by-step explanation
1. The VLOOKUP function is used to lookup a value from column B in Table1. The value to lookup is specified as B1, and Table1 is the range where the lookup should be performed. The 8th column of
Table1 is used as the result column.
2. The result of the VLOOKUP function is compared with the value in column AA using the less than (<) operator.
3. If the result of the comparison is TRUE, the formula returns TRUE. Otherwise, it returns FALSE.
4. The formula should be applied to the cells in column AA using conditional formatting. When the formula returns TRUE for a cell, the cell will be formatted in red.
For example, let's say we have the following data in columns B and AA:
| B | AA |
| 1 | 5 |
| 2 | 3 |
| 3 | 7 |
| 4 | 6 |
| 5 | 2 |
| 6 | 9 |
| 7 | 1 |
| 8 | 4 |
And let's assume that Table1 is a range that contains the following data:
| A | B | C | D | E | F | G | H |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
In this case, the formula =IF(VLOOKUP(B1,Table1,8,FALSE)<AA1,TRUE,FALSE) would return TRUE for the first cell in column AA (B1=1 is less than the value in the 8th column of Table1, which is 9), and
FALSE for the rest of the cells in column AA. The conditional formatting should be applied to the first cell in column AA to format it in red.
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How to Perform Hyperparameter Tuning In PyTorch?
Hyperparameter tuning is the process of finding the optimal values for the hyperparameters of a machine learning model. In PyTorch, there are various techniques available to perform hyperparameter
tuning. Here are some commonly used methods:
1. Grid Search: Grid Search involves defining a grid of hyperparameter values and exhaustively searching each combination. With PyTorch, you define a range of values for each hyperparameter and
iterate through all possible combinations using nested loops, training and evaluating the model on each combination.
2. Random Search: Random Search randomly samples the hyperparameter values from a defined distribution. In PyTorch, you can use the random module to randomly select values for different
hyperparameters during training. By repeating this process multiple times, you can explore a wide range of hyperparameter combinations.
3. Bayesian Optimization: Bayesian Optimization uses probabilistic models to model the relationship between hyperparameters and model performance. It gradually explores the hyperparameter space by
choosing promising hyperparameter values. In PyTorch, you can use libraries like Optuna or BayesianOptimization to perform Bayesian Optimization for hyperparameter tuning.
4. Automatic Hyperparameter Optimization: PyTorch also provides libraries with built-in hyperparameter optimization algorithms. For example, PyTorch Lightning offers methods like tuner and tune_*
for automated hyperparameter optimization. These methods can search for the best hyperparameters based on a defined search strategy.
During hyperparameter tuning, it is essential to split the dataset into training, validation, and test sets. The training set is used to train the model, the validation set is used for hyperparameter
search, and the test set is kept aside to evaluate the final model after tuning.
By performing hyperparameter tuning, you can improve the performance and generalization of your PyTorch model by finding the optimal set of hyperparameters.
What is hyperparameter tuning and why is it important?
Hyperparameter tuning is the process of selecting the optimal values for the hyperparameters of a machine learning algorithm or model. Hyperparameters are parameters that are not learned directly
from the data, but rather set before the learning process begins and impact the behavior and performance of the algorithm.
The main goal of hyperparameter tuning is to find the combination of hyperparameter values that yields the best performance or accuracy of a model on a given dataset. By tuning the hyperparameters,
one can fine-tune the behavior of the learning algorithm and improve the model's performance. It helps to avoid underfitting (where the model is too simple to capture the underlying patterns in data)
and overfitting (where the model performs well on training data, but poorly on new unseen data).
Hyperparameter tuning is important because the choice of hyperparameters can greatly impact the performance of a machine learning model. Different hyperparameter values can lead to significant
differences in the model's ability to generalize and make accurate predictions. Therefore, tuning the hyperparameters allows for optimizing the model's performance and enhancing its predictive power.
How to handle missing data during hyperparameter tuning in PyTorch?
When handling missing data during hyperparameter tuning in PyTorch, you can consider the following approaches:
1. Imputation: If the amount of missing data is relatively small, you may choose to impute or fill in the missing values. Various imputation techniques such as mean imputation, median imputation, or
regression imputation can be used. After imputing the missing data, you can proceed with hyperparameter tuning as usual.
2. Data augmentation: Another way to handle missing data is by using data augmentation techniques. You can generate synthetic data points based on the available data and use it for hyperparameter
tuning. This approach can help to preserve the general characteristics and structure of the original data.
3. Dropping missing data: If the missing data is substantial or you have reasons to believe that it may introduce bias, you may choose to exclude the missing data points from your dataset. This
approach can be suitable in certain scenarios, especially if you have a sufficiently large dataset.
4. Hyperparameter optimization on imputation mechanism: Instead of directly tuning hyperparameters, you can focus on finding the best imputation mechanism for handling the missing data. You can try
different imputation techniques and then proceed with hyperparameter tuning after the missing values are handled appropriately.
Which approach to use depends on factors such as the amount of missing data, the nature of the data, the available computational resources, and the specific problem you are working on. It's essential
to consider the implications and potential biases introduced by the chosen approach and evaluate the performance of the resulting model relative to your objectives.
How to perform random search for hyperparameter tuning in PyTorch?
In PyTorch, you can perform random search for hyperparameter tuning using the RandomSearchCV class from the sklearn.model_selection module. Here's how you can do it:
1. Import the required libraries:
1 import torch
2 from torch import nn
3 from sklearn.model_selection import RandomizedSearchCV
1. Define your PyTorch model as a class:
1 class MyModel(nn.Module):
2 def __init__(self, num_hidden_units, dropout_rate):
3 super(MyModel, self).__init__()
4 self.layer1 = nn.Linear(input_size, num_hidden_units)
5 self.dropout = nn.Dropout(dropout_rate)
6 self.layer2 = nn.Linear(num_hidden_units, output_size)
8 def forward(self, x):
9 x = self.layer1(x)
10 x = self.dropout(x)
11 x = self.layer2(x)
12 return x
Replace input_size and output_size with the appropriate values for your model.
1. Define the parameter search space:
1 param_grid = {
2 'num_hidden_units': [64, 128, 256],
3 'dropout_rate': [0.2, 0.4, 0.6]
4 }
This defines a search space with three values for num_hidden_units and three values for dropout_rate.
1. Create an instance of the model and the dataset:
1 model = MyModel()
2 dataset = torch.utils.data.TensorDataset(inputs, labels)
Replace inputs and labels with your own data.
1. Create the RandomizedSearchCV object:
1 random_search = RandomizedSearchCV(model, param_grid, cv=5)
model is the PyTorch model instance and param_grid is the defined parameter search space.
1. Fit the model and perform the search:
1 random_search.fit(inputs, labels)
This will perform random search to find the best combination of hyperparameters.
1. Access the best hyperparameters and model:
1 best_params = random_search.best_params_
2 best_model = random_search.best_estimator_
best_params will contain the best hyperparameters, and best_model will contain the model trained with the best hyperparameters.
Note: You may need to adapt this code snippet to suit your specific use case and data.
What is the impact of data augmentation on model performance and how to incorporate it during hyperparameter tuning?
Data augmentation refers to the technique of creating new modified versions of existing data samples. It has a significant impact on the performance of machine learning models by increasing the size
of the training set, reducing overfitting, and improving the generalization capability of the model. Here are a few key impacts of data augmentation:
1. Increased Model Performance: With more data available for training, models tend to perform better. Data augmentation helps by generating additional training samples that capture variations and
diverse scenarios in the data distribution, making the model learn more robust feature representations.
2. Reduction in Overfitting: Data augmentation introduces variability to the training data, making it harder for the model to memorize specific samples. This reduces overfitting, as the model learns
to generalize better by adapting to a wider range of augmented samples.
3. Improved Generalization: Augmented data helps models to generalize well on unseen or real-world data. By exposing the model to various transformations, such as rotations, scaling, translations,
flips, or noise, the model becomes more adept at handling similar variations in real-world scenarios.
To incorporate data augmentation during hyperparameter tuning, you can follow these steps:
1. Define a set of potential augmentations: Create a list of possible augmentation techniques relevant to your problem domain. These could include random rotations, translations, flips, cropping,
zooming, or noise additions.
2. Set up a data augmentation pipeline: Configure a data augmentation pipeline using libraries like TensorFlow's tf.data or Keras's ImageDataGenerator(). Define the augmentation operations and
parameters to be applied to the training data during training.
3. Apply data augmentation during training: Incorporate the data augmentation pipeline into your training process. During each epoch, retrieve a batch of data and apply random augmentations to each
sample before feeding it to the model.
4. Perform hyperparameter tuning: Conduct hyperparameter tuning as usual, adjusting other model parameters like learning rate, batch size, network architecture, etc. Monitor the model's performance
using validation metrics like accuracy or loss.
5. Iterate and evaluate: Experiment with different combinations of hyperparameters, including augmentation-related parameters, such as the strength or probability of augmentations, to find the
best-performing model.
By following these steps, you can effectively incorporate data augmentation into your hyperparameter tuning process and leverage its benefits to improve model performance.
How to use learning rate schedules during hyperparameter tuning in PyTorch?
To use learning rate schedules during hyperparameter tuning in PyTorch, you can follow the steps outlined below:
1. Define your learning rate schedule: Create a learning rate schedule function that specifies how the learning rate should change over time. PyTorch provides various built-in learning rate
schedulers like StepLR, MultiStepLR, ExponentialLR, etc. Alternatively, you can also create a custom learning rate scheduler by subclassing the torch.optim.lr_scheduler._LRScheduler class.
2. Create an optimizer: Define your optimizer (e.g., SGD, Adam, etc.) and set the initial learning rate.
3. Create a learning rate scheduler object: Instantiate the learning rate scheduler with your defined schedule function and pass it the optimizer.
1 optimizer = torch.optim.SGD(model.parameters(), lr=0.01) # Create optimizer
2 scheduler = torch.optim.lr_scheduler.StepLR(optimizer, step_size=10, gamma=0.1) # Create learning rate scheduler
1. Train your model: In each training iteration, call scheduler.step() to update the learning rate based on the defined schedule.
1 for epoch in range(num_epochs):
2 train(...)
3 validate(...)
4 scheduler.step() # Update learning rate
Note: Make sure to call scheduler.step() after the optimizer's step() function, but before updating the gradients in each training iteration.
1. Perform hyperparameter tuning: Now, during hyperparameter tuning, you can change the schedule hyperparameters (e.g., step size, gamma, etc.) and observe the effect on the learning rate schedule's
behavior. You can also experiment with different learning rate schedules to find the optimal one for your specific problem.
By using learning rate schedules during hyperparameter tuning, you can dynamically adjust the learning rate over time, allowing your model to converge faster and potentially achieve better accuracy.
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Quant Question Of The Day: 127 | TathaGat
Quant Question Of The Day: 127
Five men take as much time to do a job as 10 women take. If 6 men take 10 days to complete a job working 4 hours per day, how much time would 10 women take to do a job twice as much as the former
working 6 hours a day?
A. 12 days
B. 14 days
C. 16 days
D. 18 days
See our previous ‘Questions of the Day’:
Quant Question Of The Day: 126
Quant Question Of The Day: 125
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Lean Body Mass Calculator
The Lean Body Mass Calculator computes a person's estimated lean body mass (LBM) based on body weight, height, gender, and age. For comparison purposes, the calculator provides the results of
multiple formulas.
The lean body mass based on different formulas:
Formula Lean Body Mass Body Fat
Boer^1 127.4 lbs (80%) 20%
James^2 129.0 lbs (81%) 19%
Hume^3 120.4 lbs (75%) 25%
Lean body mass (LBM) is a part of body composition that is defined as the difference between total body weight and body fat weight. This means that it counts the mass of all organs except body fat,
including bones, muscles, blood, skin, and everything else. While the percentage of LBM is usually not computed, it on average ranges between 60-90% of total body weight. Generally, men have a higher
proportion of LBM than women do. The dosages of some anesthetic agents, particularly water-soluble drugs, are routinely based on the LBM. Some medical exams also use the LBM values. For body fitness
and routine daily life, people normally care more about body fat percentage than LBM. To compute body fat, consider using our body fat calculator or ideal weight calculator.
Multiple formulas have been developed for calculating estimated LBM (eLBM) and the calculator above provides the results for all of them.
Lean Body Mass Formulas for Adults
The Boer Formula:^1
For males:
eLBM = 0.407W + 0.267H - 19.2
For females:
eLBM = 0.252W + 0.473H - 48.3
The James Formula:^2
The Hume Formula:^3
For males:
eLBM = 0.32810W + 0.33929H - 29.5336
For females:
eLBM = 0.29569W + 0.41813H - 43.2933
Lean Body Mass Formula for Children
The Peters Formula:^4
The author suggests that this formula is applicable for children aged 13-14 years old or younger. The formula is used to compute an eLBM based on an estimated extracellular volume (eECV) as follows:
eECV = 0.0215·W^0.6469·H^0.7236
eLBM = 3.8·eECV
In the formulas above, W is the body weight in kilogram and H is the body height in centimeter.
Lean Body Mass vs. Fat Free Mass
Lean body mass and fat free mass are often used interchangeably. While this is unlikely to cause issues in most cases, the two are not exactly the same.
Lean body mass includes the combined mass of bones, muscles, water, ligaments, tendons, and internal organs. Internal organs include some essential fat and the mass of this fat is included within the
measurement of lean body mass. Although internal organs also have surrounding subcutaneous fat, this fat is not included within the measurement of lean body mass.
Fat free mass is calculated as the difference between total body mass and all fat mass including essential fat. This is the difference between fat free mass and lean body mass. Subtracting the mass
of essential fat from lean body mass yields fat free mass. The difference between lean body mass and fat free mass amounts to approximately a 2-3% difference in men and 5-12% difference in women.
1. Boer P. "Estimated lean body mass as an index for normalization of body fluid volumes in man." Am J Physiol 1984; 247: F632-5
2. James, W. "Research on obesity: a report of the DHSS/MRC group" HM Stationery Office 1976
3. Hume, R "Prediction of lean body mass from height and weight.". J Clin Pathol. 1966 Jul; 19(4):389-91.
4. A. M. Peters, H. L. R. Snelling, D. M. Glass, N. J. Bird "Estimation of lean body mass in children". British Journal of Anaesthesia1 06(5): 719-23 (2011).
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A Comprehensive Collection of Free TABE Math Practice Tests
The TABE is a place exam which is known as the Test for Adult Basic Education.
This standardized basic skills assessment is used to measure a person’s skill levels in reading, math, and English.
It is especially for those whose first language is not English and is mainly used in academic and workplace environments.
There are 195 multiple-choice questions on the TABE test. The Math Computation part contains 25 questions and you have to complete them in 15 minutes.
The next part is Applied Math with 25 questions that must be solved in 25 minutes.
The good point is you cannot fail the TABE and the score you received is valid for one year for Continuing Education admission.
The Absolute Best Book to Ace the TABE Math Test
Original price was: $24.99.Current price is: $14.99.
The TABE is generally not considered challenging, especially if you have an adequate amount of studying.
First and foremost, the mathematical portion of TABE contains a formula sheet that are relating to geometric measurement and certain algebra concepts. The main purpose is that you just focus on the
application, rather than the memorization, of formulas.
There is an appropriate calculator displayed on the screen for certain questions. But if you prefer to use a handheld calculator, you have to refer to a list of non-programmable scientific
calculators that can be used.
How to Prepare for the TABE Math Test?
Study with the Best Resource: Nothing is more helpful than studying to learn materials. Many books and websites are available to help you be proficient in the TABE math test. But you have to find
books that are filled with practices, examples, and reviews.
Their content must be at the same level of difficulty as the actual exam to overcome any type of questions. Online resources provide classrooms, practice tests, flashcards, and many more items used
to take the exam successfully.
Manage Time: Time is gold and you have to use it in the best way to complete the exam. You may face some challenging questions that take more time to complete.
Sometimes you may not recall the formulas and it is better to leave the questions and come back later. These techniques require practicing before walking through the exam.
Get Help: Math is the challenging part of many exams and you may need someone to help you learn the essential concepts. If you decide to prepare with classes or tutorials, it is better to find the
most reliable, literate instructors.
Take Practice Exam: The best way to get familiar with the exam style and content is by taking plenty of practice tests. Try to do them when you are fully learned the materials. That way, you will
guess the score you may receive on the actual test.
9 Best Websites for TABE Math Practice Tests
This helpful website generally works on the math section of many tests and entrance exams. It allows you to take free TABE math practice tests and know the exam format and content.
Moreover, you will have access to a wide range of great books that contain practice tests with answer keys, sample questions, and reviews.
Another benefit of this website is finding video lessons for many mathematical concepts. It also contains some formula sheets that you may need for the exam.
By taking benefit of this website, you will be well-prepared to answer any type of question confidently. It provides several free practice tests for all parts of TABE including Math Computation and
Applied Math. The form and difficulty level of their questions is what you may encounter on the actual test.
It also consists of helpful guidance to have better performance and take the exam with less stress. You may see another section that answers some frequent questions about TABE.
It is one of the popular websites that covers the math portion of exams. First, you can study its exam description to know the feature of the test. It provides free math practice tests to take and
get acquainted with the form and type of questions.
Another production of this website is the flashcards used for memorizing formulas and difficult subjects. It offers some TABE courses that contain videos, flashcards, lessons, and practice questions.
All of them are available with the aim of teaching materials in the best ways.
Whatever you need to be successful on the TABE test is provided on this website. There are several mathematics practice questions for different levels. So, you can start from the easiest to reach the
most challenging one.
You have access to free flashcards to review key materials and speed up your learning process. It has study guides used to have adequate information about all parts of the exam.
Another feature of this website is its discussion section where test-takers share their experience and answer each other’s questions about the tests.
This reliable website provides you with useful information about every detail of the TABE test. As you may encounter questions of any type, there are some tips and tricks to help you master them.
It offered plenty of free math practice questions that are necessary to affiliate with the difficulty level, style, and content of questions. The more you practice, the less anxiety you will have
during the test time.
The information you have to know about the version, level, test length, test time, and question type of the TABE test is packed on this website. It covers all the subjects of the test and provides
several practice tests for every one of them.
There are 2 sets of questions for Math Computation and 3 sets for Applied Math. They reflect whatever you may face on the TABE test.
Everything is designed according to the TABE level on this website. There are several free math, reading, and language practice tests for every level.
The quality of these tests is proper for students of every level. So, try to take them and be proficient in all subjects.
There is another helpful website that offers free practice tests. They are organized by topics and aim to save time, effort, and money. Every test-taker needs high-quality tests to affiliate with the
style and format of TABE.
Additionally, it provides a specific study plan according to your level and test date so that you can pass the exam on the first try.
It consists of numerous self-paced lessons that teach you mathematical topics in an easy-to-understand format. There are some practice tests for each math topic that you have to be proficient in for
the exam. It also provides some other options including courses, tutoring, plans, and test prep that you can use to take the test without stress.
A Comprehensive Bundle for the TABE Math Test
Original price was: $76.99.Current price is: $36.99.
Free TABE Math Practice Tests
1. Best TABE Math Books
2. The TABE 11&12 Math in 10 Days is what you need to achieve the essential knowledge. The materials are explained thoroughly with examples and hints. Sample exercises for each topic allow you to
apply whatever you have learned so you will get ready for taking practice tests at the end of the book. By completing this section, the nature of exam questions will be revealed.
3. One of the best-seller books is TABE Math for Beginners. It provides you with helpful tips and tricks to be enough skillful for any type of question. The goal of these strategies is to study once
and gain the best result on the exam. The mathematical materials are explained through hints and examples and they are understandable for students of every level. Besides that, it has 2 math
practice tests that show you the test content and style.
4. Another helpful self-study book is the TABE Math Study Guide. It starts with valuable information to enhance the quality of your study and improve your problem-solving abilities. All you must
learn is taught through reviews and examples. After learning the solutions and formulas, you can answer sample questions. They make you well-prepared to apply whatever you have learned in the
book’s full-length practice tests.
5. Choosing the TABE Math Test Prep as a study resource will lead to learning thoroughly and preparing to take the exam confidently. There are three sections including reviews, examples, and sample
questions. In other words, first, you learn the materials, and then by solving some questions, you learn to apply the formulas. There are two full-length TABE math practice tests to familiarize
you with the exam format.
6. The TABE 11&12 Math Tutor contains test-taking strategies to improve your ability of problem-solving. In this self-study book, there is a review and two or more examples for every topic. Then, it
provides several questions so that you can practice and complete your learning. The book consists of 2 full-length math practice tests and taking them to improve your performance. Therefore, on
the test day, you will take the exam in a stress-free condition.
7. The TABE Math Practice Workbook is filled with countless high-quality exercise questions. They are organized by topics so that you can work on challenging topics first. With the help of these
parts, the ability and speed of problem-solving will increase. If you want to test your math knowledge, there are 2 practice tests to be taken. They have the same style and content as the actual
TABE test.
8. The mathematical materials of the Comprehensive Math Workbook for TABE 11&12, are managed by topics. So, you can practice each concept, master them, and go through the next one. As they are
written by professional math instructors, the quality of questions is at the level of TABE. You are given 2 full-length math practice tests that reflect the difficulty level and style of the test
which have an important role in taking the exam successfully.
9. This 5 TABE 11&12 Math Practice Tests (Level D) is used to affiliate with the form and content of the test. They have to be taken when you are fully learned and prepared to solve any type of
question. Many test-takers recommend this book as students experience less anxiety by taking its practice. All of them contain detailed answer explanations so the possibility of making common
mistakes will be reduced.
10. Everyone can be successful on the exam with the TABE Math in 30 Days. Whatever is necessary to learn is packed in it. Every mathematical concept contains reviews and examples. They are just half
of the way to learning. You have to answer the sample questions to learn thoroughly. Additionally, 2 TABE math practice tests mirror the style and content of the exam.
11. Inside the pages of the TABE 11&12 Math Workbook 2020-2021 for Level D, you will get access to abundant exercises. The goal of the book is to give you high-quality practices and make you
proficient in all necessary topics. When you feel ready to take the exam, it is the best time to test yourself with the book’s full-length practice tests.
This information about the books and websites aims to help you have access to great resources which you need on the way to learning. Study them carefully and choose the best one according to your
plan, ability, and need.
A Comprehensive Bundle for the TABE Math Test
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he Fraction
Color The Fraction Worksheet - Web color and label fractions activity sheet. Web students can color the parts of the circle to represent the given fraction. Help your students understand fractions
while adding some color to these shapes! When kids first learn about shapes and equal shares, provide them with visual representations that show the. Kids can find easy fractions to start learning
like 1/2,. How can i use the fractions. These worksheets ask students to color in parts of shapes to match a given basic fraction.
Coloring Fractions Worksheets Math Worksheets
These worksheets ask students to color in parts of shapes to match a given basic fraction. When kids first learn about shapes and equal shares, provide them with visual representations that show the.
Web color and label fractions activity sheet. How can i use the fractions. Kids can find easy fractions to start learning like 1/2,.
Fraction Coloring Worksheets Math Monks
Web students can color the parts of the circle to represent the given fraction. Web color and label fractions activity sheet. Kids can find easy fractions to start learning like 1/2,. When kids first
learn about shapes and equal shares, provide them with visual representations that show the. How can i use the fractions.
2nd Grade Math Worksheets Geometry Fractions Color the Fraction
Help your students understand fractions while adding some color to these shapes! Web color and label fractions activity sheet. These worksheets ask students to color in parts of shapes to match a
given basic fraction. Web students can color the parts of the circle to represent the given fraction. How can i use the fractions.
7 Simplifying Fractions Coloring Worksheet /
These worksheets ask students to color in parts of shapes to match a given basic fraction. Web color and label fractions activity sheet. Help your students understand fractions while adding some
color to these shapes! Web students can color the parts of the circle to represent the given fraction. Kids can find easy fractions to start learning like 1/2,.
Colour the Fraction Teach On
Kids can find easy fractions to start learning like 1/2,. These worksheets ask students to color in parts of shapes to match a given basic fraction. Help your students understand fractions while
adding some color to these shapes! Web color and label fractions activity sheet. Web students can color the parts of the circle to represent the given fraction.
Worksheet Color the Fraction
Kids can find easy fractions to start learning like 1/2,. These worksheets ask students to color in parts of shapes to match a given basic fraction. How can i use the fractions. Web color and label
fractions activity sheet. When kids first learn about shapes and equal shares, provide them with visual representations that show the.
Coloring Shapes The Fraction 1/2 Worksheets 99Worksheets
Help your students understand fractions while adding some color to these shapes! When kids first learn about shapes and equal shares, provide them with visual representations that show the. Kids can
find easy fractions to start learning like 1/2,. Web color and label fractions activity sheet. How can i use the fractions.
Coloring Fractions 5 Worksheets / FREE Printable Worksheets
Help your students understand fractions while adding some color to these shapes! Web students can color the parts of the circle to represent the given fraction. How can i use the fractions. These
worksheets ask students to color in parts of shapes to match a given basic fraction. When kids first learn about shapes and equal shares, provide them with.
Multiplying Fractions Color by Number Funrithmetic
Web students can color the parts of the circle to represent the given fraction. When kids first learn about shapes and equal shares, provide them with visual representations that show the. How can i
use the fractions. Web color and label fractions activity sheet. Kids can find easy fractions to start learning like 1/2,.
"Color the Fractions" Worksheet
Help your students understand fractions while adding some color to these shapes! When kids first learn about shapes and equal shares, provide them with visual representations that show the. These
worksheets ask students to color in parts of shapes to match a given basic fraction. Kids can find easy fractions to start learning like 1/2,. Web color and label fractions.
When kids first learn about shapes and equal shares, provide them with visual representations that show the. Web color and label fractions activity sheet. These worksheets ask students to color in
parts of shapes to match a given basic fraction. Kids can find easy fractions to start learning like 1/2,. Help your students understand fractions while adding some color to these shapes! How can i
use the fractions. Web students can color the parts of the circle to represent the given fraction.
When Kids First Learn About Shapes And Equal Shares, Provide Them With Visual Representations That Show The.
Web students can color the parts of the circle to represent the given fraction. How can i use the fractions. Kids can find easy fractions to start learning like 1/2,. These worksheets ask students to
color in parts of shapes to match a given basic fraction.
Web Color And Label Fractions Activity Sheet.
Help your students understand fractions while adding some color to these shapes!
Related Post:
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Introduction to genetic algorithms
Summary In this project, students learn to use genetic algorithms to solve three different sets of problems. They develop techniques for selecting and crossing over populations, and learn how
the mixin design pattern can be used to easily evaluate a variety of different crossover, selection, and elitism choices.
Topics Genetic Algorithms, search, constraints
Audience Intro AI
Difficulty Medium. I give students two weeks to complete this.
Strengths. Gives students a chance to easily apply GAs to non-trivial problems and easily experiment with the wide variety of design choices. Encourages them to develop a scientific approach to
parameter selection and algorithm evaluation. Illustrates the use of mixins to easily modify an application.
Weaknesses Students must use Python.
Dependencies Students should be familiar with search. I give this assignment about one month into the course. Students must be comfortable with Python and OO programming.
The code is constructed so that it is very easy to add other problems, such as N-queens or map coloring. Interested students can also further explore selection, mutation and crossover
Variants operators and perform experiments to evaluate their effectiveness. Depending on the time available and skill sets of the students, instructors might want to focus on a subset of this
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We have f(x)=x-cubicroot(x+1)
Verify that f(sqrt3/9 -1) =(-2sqrt3)/9 -1 and calculate the limit of f(x) as x approaches positive infinity? Thanks! | HIX Tutor
We have f(x)=x-cubicroot(x+1) Verify that f(sqrt3/9 -1) =(-2sqrt3)/9 -1 and calculate the limit of f(x) as x approaches positive infinity? Thanks!
Answer 1
#f(x) = x- root{3}{x+1}#
#f(sqrt{3}/9-1) = sqrt{3}/9-1-root{3}{sqrt{3}/9-1+1}# #=sqrt{3}/9-1-root{3}{{3sqrt3}/27}= sqrt{3}/9-1-sqrt{3}/3 = -{2sqrt{3}}/9-1#
Since #x# grows much faster than #root{3}{x+1}# the limit diverges, or more precisely, does not exist!
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Answer 2
To verify that f(sqrt(3)/9 - 1) = (-2sqrt(3))/9 - 1, we substitute sqrt(3)/9 - 1 into the function f(x) and simplify the expression.
f(x) = x - cubicroot(x + 1)
f(sqrt(3)/9 - 1) = (sqrt(3)/9 - 1) - cubicroot((sqrt(3)/9 - 1) + 1)
Simplifying further,
f(sqrt(3)/9 - 1) = (sqrt(3)/9 - 1) - cubicroot(sqrt(3)/9)
Now, let's calculate the limit of f(x) as x approaches positive infinity.
To find the limit of f(x) as x approaches positive infinity, we need to analyze the behavior of the function as x becomes larger and larger.
The function f(x) = x - cubicroot(x + 1) can be simplified to f(x) = x - (x + 1)^(1/3).
As x approaches positive infinity, the term (x + 1)^(1/3) becomes negligible compared to x.
Therefore, the limit of f(x) as x approaches positive infinity is infinity.
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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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Turner, P (2007)
A classroom exploration of Benford's Law and some error finding tricks in accounting
Proceedings of the 21st biennial conference of the Australian Association of Mathematics Teachers Inc. Mathematics: Essential for Learning, Essential for Life, edited by K. Milton, H. Reeves & T.
Spencer, 2007, pp. 250-259 .
ISSN/ISBN: 978-1-875900-63-3 DOI: Not available at this time.
Abstract: Not available at this time.
@InProceedings {, AUTHOR = {Turner, Paul}, TITLE = {A classroom exploration of Benford's Law and some error finding tricks in accounting}, BOOKTITLE = {Mathematics: Essential for Learning, Essential
for Life}, YEAR = {2007}, ISBN = { 978-1-875900-63-3}, EDITOR = {Milton, K. and Reeves, H. and Spencer, T.}, ORGANIZATION = {Proceedings of the 21st biennial conference of the Australian Association
of Mathematics Teachers Inc. }, PAGES = {250--259}, }
Reference Type: Conference Paper
Subject Area(s): Accounting, Mathematics Education
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Catalog Entries
Prerequisite: MTH 112 completed with a grade of “C-“ or better within the past five years or placement by the College's Math Placement Process. MTH 251 is a first-term calculus course that includes a
selective review of precalculus followed by development of the derivative from the perspective of rates of change, slopes of tangent lines, and numerical and graphical limits of difference quotients.
The limit of the difference quotient is used as a basis for formulating analytical methods that include the power, product, and quotient rules. The chain rule and the technique of implicit
differentiation are developed. Procedures for differentiating polynomial, exponential, logarithmic, and trigonometric functions are formulated. Analytical, graphical, and numerical methods are used
to support one another in developing the course material. Opportunities are provided for students to work in groups, verbalize concepts with one another, and explore concepts and applications using
5.000 Credit hours
50.000 TO 60.000 Lecture hours
Syllabus Available
Levels: Credit
Schedule Types: Lecture
Mathematics Division
Mathematics Department
Course Attributes:
Tuition, Science/Math/Computer Science
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Estimate State-Space Model With Order Selection
Main Content
Estimate State-Space Model With Order Selection
To estimate a state-space model, you must provide a value of its order, which represents the number of states. When you do not know the order, you can search and select an order using the following
Estimate Model With Selected Order in the App
You must have already imported your data into the app, as described in Represent Data.
To estimate model orders for a specific model structure and configuration:
1. In the System Identification app, select Estimate > State Space Models to open the State Space Models dialog box.
2. In the Model Structure tab, select the Pick best value in the range option and specify a range in the adjacent field. The default range is 1:10.
This action opens the Model Order Selection window, which displays the relative measure of how much each state contributes to the input-output behavior of the model (log of singular values of the
covariance matrix). The following figure shows an example plot. In this figure, states 1 and 2 provide the most significant contribution. The contributions to the right of state 2 drop
significantly. The red bar illustrates the cutoff. The order of this bar represents the best-value recommendation, and this value appears in Order. You can override the recommendation by clicking
on another bar or by overwriting the contents of Order. For information about using the Model Order Selection window, see Using the Model Order Selection Window
3. (Optional) Specify additional attributes of the model structure, such as input delay and feedthrough. You can also modify the estimation options in the Estimation Options tab. As you modify your
selections, the software re-evaluates the model-order recommendation.
4. Click Estimate. This action adds a new model to the Model Board in the System Identification app. The default name of the model is ss1. You can use this model as an initial guess for estimating
other state-space models, as described in Estimate State-Space Models in System Identification App.
5. Click Close to close the window.
Estimate Model With Selected Order at the Command Line
You can estimate a state-space model with selected order using n4sid, ssest or ssregest.
Use the following syntax to specify the range of model orders to try for a specific input delay:
where data is the estimation data set, n1 and n2 specify the range of orders.
The command opens the Model Order Selection window. For information about using this plot, see Using the Model Order Selection Window.
Alternatively, use ssest or ssregest:
m1 = ssest(data,nn)
m2 = ssregest(data,nn)
where nn = [n1,n2,...,nN] specifies the vector or range of orders you want to try.
n4sid and ssregest estimate a model whose sample time matches that of data by default, hence a discrete-time model for time-domain data. ssest estimates a continuous-time model by default. You can
change the default setting by including the Ts name-value pair input arguments in the estimation command. For example, to estimate a discrete-time model of optimal order, assuming Data.Ts>0, type:
model = ssest(data,nn,'Ts',data.Ts);
model = ssregest(data,nn,'Ts',data.Ts);
To automatically select the best order without opening the Model Order Selection window, type m = n4sid(data,'best'), m = ssest(data,'best') or m = ssregest(data,'best').
Using the Model Order Selection Window
The following figure shows a sample Model Order Selection window.
You use this plot to decide which states provide a significant relative contribution to the input-output behavior, and which states provide the smallest contribution. Based on this plot, select the
rectangle that represents the cutoff for the states on the left that provide a significant contribution to the input-output behavior. The recommended choice is shown in red. To learn how to generate
this plot, see Estimate Model With Selected Order in the App or Estimate Model With Selected Order at the Command Line.
The horizontal axis corresponds to the model order n. The vertical axis, called Log of Singular values, shows the singular values of a covariance matrix constructed from the observed data.
For example, in the previous figure, states 1 and 2 provide the most significant contribution. However, the contributions of the states to the right of state 2 drop significantly. This sharp decrease
in the log of the singular values after n=2 indicates that using two states is sufficient to get an accurate model.
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fuzzy clustering pdf
functions (PDF) in both univariate and multivariate cases [23]. Fuzzy clustering is now a mature and vibrant area of research with highly innovative advanced applications. Based on this work, Zhang
et al.. Each item has a set of membership coefficients â ¦ Fuzzy C-Means Clustering and Soniï¬ cation of HRV Features 1st Debanjan Borthakur McMaster University Hamilton, Canada borthakd@mcmaster.ca
2nd Victoria Grace Muvik Labs New York, USA vic@muviklabs.io 3rd Paul Batchelor fuzzy clustering framework (AFCF) for image segmentation. Here we apply a fuzzy partitioning method, Fuzzy C-means
(FCM), to attribute cluster membership values to genes. of fuzzy clustering as a means to respond to breakaway of taxis from routes when they transport a customer. Our simulation results show that
our method enables taxis to transport more customers. X. Jia et al. For clustering, the use of competitive learning (CL) based network and train it indirectly using fuzzy c-means (FCM) algorithm is
proposed. SKRIPSI oleh: BINTI MUSLIMATIN NIM : 06510032 JURUSAN Standard clustering (K-means, PAM) approaches produce partitions, in which each observation belongs to only one cluster. Fuzzy Set
Based Web Opinion Text Clustering Algorithm Hongxin Wan 1, a, Yun Peng 2, b 1 College of Mathematics & Computer Science, Jiangxi Science & Technology Normal University, Nanchang 330013, China; 2
College of Each of these algorithms belongs to one of the clustering types listed above. Fuzzy clustering is also known as soft method. This example shows how to perform fuzzy c-means clustering on
2-dimensional data. Fuzzy c-means (FCM) is a clustering method that allows each data point to belong to multiple clusters with varying degrees of membership. This program generates fuzzy partitions
and prototypes for any set of numerical data. The 3. Astra, Tbk.) Page 236 - Fuzzy clustering for the estimation of the parameters of the components of mixtures of normal distributions," Pattern
Recognition letter 9, 77-86, N.-Holland, 1989. Check out part one on hierarcical clustering here and part two on K-means clustering here.Clustering gene expression is a particularly useful data
reduction technique for RNAseq experiments. Fuzzy c-means (FCM) is a data clustering technique wherein each data point belongs to a cluster to some degree that is specified by a membership grade. For
an example that clusters higher-dimensional data, see Fuzzy C-Means Clustering for Iris Data . Computing a fuzzy decomposition by reï¬ ning the probability values using an iterative clustering
scheme. Fuzzy clustering is a combination of a conventional k-mean clustering and a fuzzy logic system in order to simulate the experience of complex human decisions and uncertain information
(Chtioui et al., 2003; Du and Sun, 2006c So that, K-means is an exclusive clustering algorithm, Fuzzy C-means is an overlapping clustering algorithm, Hierarchical clustering is obvious and lastly
Mixture of Gaussian is a probabilistic clustering algorithm. Fuzzy c-means (FCM) is a data clustering technique wherein each data point belongs to a cluster to some degree that is specified by a
membership grade. : Robust Self-Sparse Fuzzy Clustering for Image Segmentation some pixels corrupted by noise, it shows low robustness for different kinds of noisy images since the bias Ë eld is
often not sparse. [PDF] fuzzy clustering matlab code pdf Thank you certainly much for downloading fuzzy clustering matlab code pdf.Maybe you have knowledge that, people have see numerous time for
their favorite books later this fuzzy clustering matlab code pdf, but end occurring in harmful downloads. The proposed framework has threefold contributions. Ehsanul Karim Feng Yun Sri Phani Venkata
Siva Krishna Madani Thesis for the degree Master of Science (two years) in Mathematical Modelling and Simulation 30 credit points (30 ECTS This is known as hard clustering. This technique was
originally introduced by Jim Bezdek in 1981 [1] as an improvement on earlier clustering methods. We improved We used black-box model (JIT Mo deling) with the physical model (GPV data) for solar
radiation prediction method. Using Fuzzy Clustering Masaki Onishi Member (AIST) Ikushi Yoda Member (AIST) Keywords: dynamic trajectory extraction, stereo vision, fuzzy clustering In recent years,
many human tracking researches have been proposed in The FCM program is applicable to a wide variety of geostatistical data analysis problems. Fuzzy clustering is considered as an important tool in
pattern recognition and knowledge discovery from a database; thus has been being applied broadly to various practical problems. Formal Fuzzy Logic 9 Fuzzy Propositional Logic Like ordinary
propositional logic, we introduce propositional variables, truth- functional connectives, and a propositional constant 0 Some of these include: Monoidal t-norm-based propositional fuzzy logic 2010:03
Fuzzy Clustering Analysis Md. Subtractive Fuzzy C-means Clustering Approach with Applications to Fuzzy Predictive Control JI-HANG ZHU HONG-GUANG LI College of Information Science and Technology
Beijing University of Chemical Technology 15 4. It allows us to bin genes by expression profile, correlate â ¦ Clustering dengan algoritma Fuzzy C-means beserta penerapannya. The current version
(version 2.1.1) of the package has been deeply improved with respect to the previous ones. This paper discusses both the methods for clustering and presents a new algorithm which is a fusion of fuzzy
K- means and EM. fuzzy clustering algorithms, computing cluster validity indices and visualizing clustering results. algorithm. Results: A major problem in applying the FCM method for clustering
microarray data is â ¦ clustering algorithms and serve as prototypical representations of the data points in each cluster. These partitions are useful for â ¦ PERBANDINGAN METODE K-MEANS DAN METODE
FUZZY C-MEANS (FCM) UNTUK CLUSTERING DATA (Studi Kasus pada Data Saham Harian PT. Section 1.1 gives the basic notions about the data, clusters and diï¬ erent types of partitioning. In JIT modeling,
there is a procedure to search for similar data. Encapsulating this through presenting a careful selection of research contributions, this book addresses timely b. Bagi Program Studi Teknik
Informatika, penelitian ini merupakan salah satu upaya untuk membantu mahasiswanya dalam memilih bidang keahlian. tion by using fuzzy clustering. Note this is part 3 of a series on clustering RNAseq
data. Fuzzy c-means (FCM) is a data clustering technique in which a data set is grouped into N clusters with every data point in the dataset â ¦ A new correlation-based fuzzy logic clustering
algorithm for FMRI Constructing the exact boundaries between the components, thus transforming the fuzzy decomposition into the ï¬ nal The chapter is organized as follows: Section 1.2 introduces the
basic approaches to hard, fuzzy, and possibilistic clustering. Fuzzy clustering can be used as a tool to obtain the partitioning of data. hybrid adaptive segmentation and fuzzy c-means clustering
techniques; a two-stage text extraction from the candidate text regions to filter out false text regions include local character filtering according to a rule-based approach using shape and
statistical features ABSTRACT FUZZY UNEQUAL CLUSTERING IN WIRELESS SENSOR NETWORKS BaË gcı, Hakan M.S., Department of Computer Engineering Supervisor : Prof. Dr. Adnan Yazıcı January 2010, 64
pages In order to gather information THIS PAPER HAS BEEN ACCEPTED BY IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY. Key words Taxiâ s traveling 1. This technique was originally
introduced by Jim Bezdek in 1981 [1] as an improvement on earlier clustering methods. Fuzzy clustering is now a mature and vibrant area of research with highly innovative advanced applications.
Abstract This paper transmits a FORTRAN-IV coding of the fuzzy c -means (FCM) clustering program. In Fuzzy clustering, items can be a member of more than one cluster. Introduces the basic notions
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program generates fuzzy partitions prototypes. Each cluster with highly innovative advanced applications FCM program is applicable to a wide variety geostatistical... Untuk membantu mahasiswanya
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there is a fusion of fuzzy K- means and EM solar radiation prediction.. Transport more customers membantu mahasiswanya dalam memilih bidang keahlian advanced applications respect to the ones.
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a and. Memilih bidang keahlian, clusters and diï¬ erent types of partitioning program is to... ( version 2.1.1 ) of the package has been deeply improved with to. Basic notions about the data points
in each cluster listed above ( K-means, PAM ) produce! A wide variety of geostatistical data analysis problems Bagi program Studi Teknik Informatika, penelitian merupakan. Paper discusses both the
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[Solved] Sonic Inc. manufactures two models of spe | SolutionInn
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Sonic Inc. manufactures two models of speakers, Rumble and Thunder. Based on the following production and sales data for June, prepare (a) a sales budget
Sonic Inc. manufactures two models of speakers, Rumble and Thunder. Based on the following production and sales data for June, prepare (a) a sales budget and (b) a production budget: Estimated
inventory (units), June1 Desired inventory (units), June 30 Expected sales volume (units): Rumble Thunder 750 300 250 500 Midwest Region 12,000 14,000 $60 3,500 4,000 $90 South Region Unit sales
price a. Prepare a sales budget. Sonic Inc Sales Budget For the Month Ending June 30 Product and Area Model: Rumble Unit Sales Volume Unit Selling Price Total Sales Midwest Region ( ) South Region
Total Model: Thunder Midwest Region b. Prepare a production budget. For those boxes in which you must enter subtracted or negative numbers use a minus sign. Sonic Inc. Production Budget Units Rumble
Units Thunder Expected units to be sold Desired inventory, June 30 500 Total Estimated inventory, June1 Total units to be produced
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Eigenvalue Calculator
The eigenvalue calculator finds the eigenvalues of the given square matrix with the characteristic equation of polynomials along with the detailed solution.
Eigenvalues of a Matrix:
In mathematics, eigenvalues are scalar values that are associated with linear equations (also called matrix equations). It is also called latent roots. Eigenvalues are a special set of scalars
assigned to linear equations. It is mainly used in matrix equations. "Eigen" is a German word that means "characteristic" or "proper". In short, the eigenvalue is a scalar used to transform the
How to Find Eigenvalues?
For a 2x2 matrix, the trace and the determinant of the matrix are useful to obtain two very special numbers to find the eigenvectors and eigenvalues. Fortunately, the eigenvalue calculator will find
them automatically. If you want to check whether the correct answer is given or just want to calculate it manually, then please do the following:
The trace of the matrix is defined as the sum of the elements on the main diagonal (from the top left to bottom right).
It is also equal to the sum of eigenvalues (counted with multiplicity). In the case of a 2x2 matrix, Tr X = x_1 + b_2
The matrix determinant is useful in several additional operations, such as finding the inverse of the matrix. For the 2x2 matrix, |X| = x_1 y_2 – x_2 y_1
However, an Online Jacobian Calculator helps you to find the Jacobian matrix and the determinant of the set of functions.
Calculate eigenvalues for the matrix {{6,1}, {8, 3}}.
Finding eigenvalues for 2 x 2 matrix: First, the eigenvalues calculator subtracts λ from the diagonal entries of the given matrix
$$ \begin{vmatrix} 6.0 – λ \\ 1.0 && 8.0 \\ 3.0 – λ \end{vmatrix} $$
The determinant of the obtained matrix
λ^2 – 9.0 λ + 10. 0
The eigenvalue solver evaluates the equation λ^2 – 9.0 λ + 10. 0 = 0
Roots (Eigen Values)
λ_1 = 7.7015
λ_2 = 1.2984
(λ_1, λ_2) = (7. 7016, 1. 2984)
How Calculator Works?
The online calculator solves the eigenvalues of the matrix by computing the characteristic equation by following these steps:
• First, select the size for the matrix from the drop-down list.
• Now, substitute the values in all fields. You can generate random values for the matrix by clicking the generate matrix button. Remove all values by clearing all fields.
• Hit the calculate button for the next procedure.
• The matrix eigenvalue calculator displays the values and solves the equation.
• It also takes the determinant of the obtained matrix and provides root values.
How to Find the Eigenvalues of a 3x3 Matrix?
• To find the eigenvalues of a 3x3 matrix, X, you need to:
• First, subtract λ from the main diagonal of X to get X - λI.
• Now, write the determinant of the square matrix, which is X - λI.
• Then, solve the equation, which is the det(X - λI) = 0, for λ. The solutions of the eigenvalue equation are the eigenvalues of X.
Can the Eigenvalues Be Zero?
The eigenvalues can be zero. We do not treat zero vectors as eigenvectors: since X 0 = 0 = λ0 for each scalar λ, the corresponding eigenvalue is undefined.
Where Do We Use Eigenvalues?
We can use the eigenvalues for:
• Eigenvalue analysis is used in the design of autostereoscopic systems to reproduce car vibrations caused by music.
• Electrical engineering: the application of eigenvalues can be used to separate three-phase systems by converting symmetrical components.
From the source Wikipedia: Characteristic value, the characteristic polynomial, Eigenvalues of matrics, Algebraic multiplicity, Eigenspaces, geometric multiplicity, and the eigenbasis for matrices.
From the source of Medium: Eigenvalues uses, building blocks of Eigenvalues, Matrix Addition, Multiplying Scalar With A Matrix, Matrices Multiplication.
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Outlier Calculator
An online outlier calculator helps you to detect an outlier that exists far beyond the data set at a specific range. Here, you can adopt various methods to figure out the outliers if they exist. But
we have made it easy for you to perform the outlier check. For better understanding, just jump down!
What Is An Outlier?
In statistical analysis, “A specific entry or number that is totally different from all other entries in the data set is known as an outlier”
Statistical Outlier Test:
The outliers usually occur by chance and can cause serious problems in data sorting. You can use our online outlier calculator to determine an outlier absolutely for free. But, you must know the five
number summary as well which is explained below:
(1) Maximum:
In a data set, the greatest value is always considered a maximum value.
For example:
Let us consider the following data set: 1, 5, 32, 854, 4 In this data set, the maximum is 854 because it is the greatest among all.
(2) Minimum:
The smallest value that exist in a data set is known as minimum.
For example:
Consider the same data set as mentioned above: 1, 5, 32, 854, 4 For this data set, the minimum is the 1 as it is the smallest value.
The middle term in a data set is called median.
Rules For Median:
It must be kept in mind that you have two rules defined if you want to find median.
Even Numbers:
If the number of values in your data set are even, then the median is considered as the average of two middle terms.
$$ median = \frac{Two Middle Terms}{2} $$
Odd Numbers:
For an odd number of values, the median is simply the term that lies within the data set.
(4) Quartiles:
The medians of the smallest and greatest half of the data set are considered as quartiles.
First quartile(Q_{1}):
It is the median of the smallest numbers ( minimum) that contrains 25% of the data volume within it.
Third quartile(Q_{3}):
The median of the greatest numbers (maximum) under which at least 75% of the data lies is known as the third quartile. It should always be kept in mind that the data must be arranged from least to
greatest always before you perform various outliers tests to detect any outlier.
(5) Interquartile Range (IQR):
It is the difference between the first and third quartiles.
$$ IQR = Q_{3} - Q_{1} $$
Inner And Outer Fences:
Before you work for outliers, you need to determine inner and outer fences with the help of the following formulas below:
Inner fences:
$$ Q_{1} - (1.5 \times IQR) \text{ and } Q_{3} + (1.5 \times IQR) $$
Outer fences:
$$ Q_{1} - (3 \times IQR) \text{ and } Q_{3} + (3 \times IQR) $$
Our free online statistical outlier calculator uses all above formulas to figure out outliers if there is/are any.
How To Calculate Outliers?
Sometimes, it becomes difficult to find any outliers in a data set that produces a significant increase in difficulty. That is why a free q-test calculator is used to escalate your results. But it is
very important to practice test for outliers detection. So, what about solving an example to better get a grip!
Example # 01:
Calculate outliers for the following data set defined below: $$ 10, 12, 11, 15, 11, 14, 13, 17, 12, 22, 14, 11 $$
As the given data is unsorted, we need to arrange it in ascending order as follows:
\( 10, 11, 11, 11, 12, 12, 13, 14, 14, 15, 17, 22 \)
By following five number summary, we have:
(1) Maximum: For the data given, the maximum or greatest value is 22.
(2) Minimum: The smallest value for the data set given is 10.
(3) First Quartile (Q1): As the total number of values is 12.
So we divide it into two parts. The first part contains 6 numbers. The median of these numbers gives us the first quartile as follows:
$$ 10, 11, 11, 11, 12, 12 $$
$$ Q_{1} = \frac{11+11}{2} $$
$$ Q_{1} = \frac{22}{2} $$
$$ Q_{1} = 11 $$
(4) Third Quartile (Q3):
It is the median of the next 6 numbers and is calculated as:
$$ 13, 14, 14, 15, 17, 22 $$
$$ Q_{3} = \frac{14 + 15}{2} $$
$$ Q_{3} = \frac{29}{2} $$
$$ Q_{3} = 14.5 $$
(4) Median:
As the total number of values is even, so the median is calculated as follows:
$$ 10, 11, 11, 11, 12, 12, 13, 14, 14, 15, 17, 22 $$
$$ median = \frac{12 + 13}{2} $$
$$ median = \frac{25}{2} $$
$$ median = 12.5 $$
For interquartile range, we have:
$$ IQR = Q_{3} - Q_{1} $$
$$ IQR = 14.5 - 11 $$
$$ IQR = 3.5 $$
Calculating inner fences as below:
$$ Q_{1} - (1.5 \times IQR) \text{ and } Q_{3} + (1.5 \times IQR) $$
$$ 11 - (1.5 \times 3.5) \text{ and } 14.5 + (1.5 \times 3.5) $$
$$ 5.75, 19.75 $$
Now, we need to determine outer fences with the help of following equations:
$$ Q_{1} - (3 \times IQR) \text{ and } Q_{3} + (3 \times IQR) $$
$$ 11 - (3 \times 3.5) \text{ and } 14.5 + (3 \times 3.5) $$
$$ 0.5, 25 $$
$$The\ number\ of\ prism\ outlier = 0 $$
$$ Potential\ outlier = 22 $$
Which is our required answer. Here, our free statistical outlier test calculator depicts the same results but in a fraction of seconds to avoid time wastage.
How Outlier Calculator Works?
Our free q test calculator is the best among all calculators and is used widely by students and statisticians. Let us guide you how to use it properly.
• Enter all the numbers separated by commas in the menu box
• Hit the calculate button
Output: The outliers calculator determines:
• Maximum and minimum values
• First outlier
• Third outlier
• Interquartile range
• Inner fences
• Outer fences
• Outliers
What is standard deviation?
The statistical analysis that measures dispersion of a data set from the mean position is called standard deviation.
What is standard deviation?
It Is a data that is totally defined in a proper manner without containing any raw values.
What is regression analysis?
The statistical process that describes relationship among dependent variable and one or more independent variables is called regression analysis.
There is a broad use of outlier detection in the field of cybersecurity, military surveillance for the sake of preventing attacks, detection of any mishap with credit cards and many more. This is why
the use of free online outlier calculator is preferred around the globe to depict any fault in the systems so that any challenging situation could be overcome easily.
From the source of wikipedia: Grubbs's test, Chauvenet's criterion, Peirce's criterion, Dixon's Q test, Studentized residual From the source of khan academy: Identifying outliers, Reading box plots,
Interpreting box plots, Interpreting quartiles, Judging outliers in a dataset From the source of lumen learning: Types of Outliers.
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9th grade transition math problems
9th grade transition math problems Related topics: finding 6th roots
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Author Message Author Message
LooneyIFC Posted: Friday 05th of Jan 07:03 Vnode Posted: Tuesday 09th of Jan 10:39
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The Magic Cafe Forums - More annoying than Monty Hall!
Go to page [Previous] 1~2
That only works if the questions you ask don't fill up the probability. If you get a "No" on "Is there at least one head before 1940?" and then ask "Is there at least
one head at or after 1940?" you gain no information hence the probability is 1 of that being a property.
Also the combined answers to several earlier questions tell you how much information you gain with your new question, so the easiest is to keep the questions totally
unrelated. For example the correlation of the shinyness of a coin and its date is high.
LobowolfXXX you had a really hard time convincing me that 1/3 didn't have to be the correct answer. It took several days to convert me as I remember.
Sweden I just thought of a variation on this problem:
1148 Posts
I shuffle a deck and deal two cards until I find a pair with at least one Ace, re-shuffling each time. What's the probability that there are two Aces in that pile?
Repeat but look for a pair with a Black Ace. What is the probability of it containing two Aces?
Repeat but look for a pair with the Ace of Spades in it. What's the probability that it has two Aces?
Does it make any difference if I instead use this procedure: Shuffle the deck and deal 26 two-card piles and look through them until I find the first pair meeting my
criteria? It would be much quicker.
landmark 0
Inner Before we move on, now consider:
within a So I now ask a series of a billion independent such questions (those with a probability of about 1 in a million). It would be unusual not to get a yes answer. If I take
triangle the set of ALL such possible questions, eventually there should be a yes, with an infinitesimal probability of all nos. But that means then that I don't need to ask
5195 Posts those questions in the first place since I know that somewhere in that set there must be a yes! And, by our prior agreements, if there's a yes answer to such a question,
then the odds change . . .
You write "unusual to not get a yes answer" so as long as that holds true you still have to ask the questions.
circle If I understand it all correctly you should be able to keep asking questions if they are totally uncorrelated, or you can ask correlated questions but have her flip new
Sweden coins each time taken at random among all coins in distribution.
1148 Posts
I don't follow the reasoning that you would not need to ask the questions. The probability is not 1 of there being a yes among them.
The best way to convince someone that the answer to the Monty Hall problem isn't 50-50, by the way, is to take a deck of cards, tell him to pull out a card and not look
at it, and you'll give him a prize if he pulls out the ace of spades. Then take the rest of the deck and look through it...turn up 50 cards that aren't the ace of
spades, so you and he each have a face down card, and ask him if he wants to switch, or if he thinks it's 50-50.
The "extra questions" thing sort of reminds me of the black raven proposition. I have a red magic book (Let's say it's Dusheck's Card Magic, since that's red).
LobowolfXXX Curiously, this helps support (prove?) the proposition that all ravens are black, at least inductively. The logical equivalent to "All ravens are black" is "No non-black
object is a raven" or "All non-black objects are non-ravens." By finding an instance consistent with these latter versions of the proposition, I incrementally support
Inner their equivalent, too. This red thing COULD have been a raven, but instead it's a magic book...making it a teeny tiny bit more likely that yes, all ravens are in fact
circle black.
La Famiglia
1196 Posts S2000Magician should be credited (blamed?!) in this thread, too. He's (correctly) pointed out to me in similar threads that I was importing the unspoken presupposition
that a choice between two objects would be random (e.g. when I would say, without comment, something like, "If he had a male and a female dog, there's only a 50% chance
he would have told you about the female dog.") Although, to paraphrase (or maybe quote) poker guru David Sklansky, "In the beginning, all bets were even money," i.e. in
the absence of outside information, you probably SHOULD assume that a binary proposition is 50-50.
"Torture doesn't work" lol
Guess they forgot to tell Bill Buckley.
"...as we reason and love, we are able to hope. And hope enables us to resist those things that would enslave us."
landmark 0
Inner The chances of getting a billion nos (assuming independence) is .999999^(10^9). This # is near zero. (Asking 10^8 questions gives a probability of 3.7 x 10^-44 of all
circle nos) I can increase the number of questions I ask arbitrarily, and approach as close to zero as much as I like. I have 10^20 questions in my pocket. I will take the
within a chance that IF I asked them I wouldn't get all nos.
5195 Posts Jack
TomasB Lobowolf, this got me thinking of the black raven too. What are the odds?
Inner Hmmm, I got an idea on how to visualize it. I think you could formulate your 10^20 questions as a single question with AND between each question. Then you'd have a
circle single question which would be answered "yes" to with probability p (which is pretty darn close to 1 now). In other words, it'd not increase the 1/3 probability much.
1148 Posts That tells me that I was wrong in thinking that the question (of many questions) that got a "yes" would work with full force with its probability even if they are
uncorrelated. There are more layers of deception in this puzzle than I could have imagined.
landmark 0
Inner Yes the black raven occurred to me too.
within a But I like your solution of having it equivalent to one question. It makes it all consistent with what's been said before.
5195 Posts I think I'm ready to go on to the next round.
On 2009-07-25 01:46, TomasB wrote:
within a I just thought of a variation on this problem:
5195 Posts I shuffle a deck and deal two cards until I find a pair with at least one Ace, re-shuffling each time. What's the probability that there are two Aces in that pile?
Do you mean two Aces in that pair ? (You wrote "pile" there. Just making sure.)
circle Yeah, sorry, I must have been thinking about the follow-up question with 26 piles. I mean the pair that was just dealt.
1148 Posts /Tomas
On Jul 21, 2009, landmark wrote:
1) The question as posed above, has an answer of 1/3. Consider that given that we know there is at least one daughter, we have three kinds of families {gb, bg, gg}.
LobowolfXXX Only one of those families has two daughters, hence 1 in 3. The classic answer.
circle In the first two situations, you're betting on a parlay - The apparently-as-likely-scenario of one boy & one girl (as compared to two girls) is offset by the 50-50
La Famiglia probability that a family that could have told you about either child chose to tell you about the girl, making the GG and BG cases 50-50 (as one would intuitively
1196 Posts expect).
The classic 1 in 3 answer is problematic, because, as the analysis would be the same if you were told the first child was a boy, then regardless of what you were told
about the sex of a child, the apparent probability of a mixed-sex family would be 2/3, which we know it isn't. It's ignoring the fact that the choice of which child's
sex to reveal, which carries a nonzero/noncertain probability, that makes it appear to be a paradox.
"Torture doesn't work" lol
Guess they forgot to tell Bill Buckley.
"...as we reason and love, we are able to hope. And hope enables us to resist those things that would enslave us."
Slim King
Order Talking about boys and girls has got a lot trickier these days.......
18177 Posts THE MAN THE SKEPTICS REFUSE TO TEST FOR ONE MILLION DOLLARS.. The Worlds Foremost Authority on Houdini's Life after Death.....
The Magic Cafe Forum Index » » Puzzle me this... » » More annoying than Monty Hall! (1 Likes)
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Contribution Margin Formula | Calculator (Excel template)
Updated July 31, 2023
Table of Content
Contribution Margin Formula
The contribution margin concept establishes a relationship between cost, sales, and profit. For the Contribution margin calculation, the firm refers to its net sales and total variable expenses.
It refers to the amount left over after deducting from the revenue or sales the direct and indirect variable costs incurred in earning that revenue or sales. This left-over value then contributes to
paying the periodic fixed costs of the business, with any remaining balance contributing profit to the firm. Alternatively, contribution margins can be determined by calculating the contribution
margin per unit formula and the contribution ratio. The following article provides an outline for Contribution Margin Formula.
Here’s the Contribution Margin Formula:
Contribution Margin = Net Sales – Total Variable Expenses
Contribution Margin = Contribution Margin per Unit * No. of Unit Sold
Examples of Contribution Margin Formula (With Excel Template)
Let’s take an example to understand the calculation of the Contribution Margin formula in a better manner.
Example #1
Suppose we sell a pen for $10 in the market, and the variable cost is $6. Calculate the contribution margin of the pen
We can calculate the contribution margin of the pen by using the formula given below.
Contribution Margin = Net Sales – Total Variable Expenses
• Contribution Margin= $10 – $6
• Contribution Margin = $4
The contribution margins for the sale of the pen would be $4, and selling this pen would increase the profit of the firm by $4.
Example #2
In this example, we will calculate the contribution margins of the firm per unit.
A Firm sells a single product known as product A. Sales and cost figures of the firm are given below:
By using the above information provided by the Firm, we can calculate per unit and the total contribution margin of product A as below:
The formula to calculate the Contribution Margin per Unit is as below:
Contribution Margin per Unit = Sales Price per Unit – Total Variable Cost per Unit
• Contribution Margin per Unit = $100 – $65
• Contribution Margin per Unit = $35 per unit
The formula to calculate Total Contribution Margin is as below:
Contribution Margin = Net Sales – Total Variable Expenses
Contribution Margin = (No. of Unit Sold * Sales Price per Unit) – (No. of Unit Sold * Variable Cost per Unit)
• Total Contribution Margin = (10,000 units × $100) – (10,000 units * $65)
• Total Contribution Margin = $10,00,000 – $6,50,000
• Total Contribution Margin = $3,50,000
The formula to calculate Total Contribution Margin is as below:
Contribution Margin = Contribution Margin per Unit * No. of Unit Sold
• Total Contribution Margin = $35 * 10,000 units
• Total Contribution Margin = $350,000
Example #3
In this example, we will calculate the contribution margins alternatively with Net profit and fixed cost. Let’s discuss the financial data of the firm to calculate contribution margins.
During the financial year 2018, Firm ABC has sold mobile phones of INR 2,00,000, and followings are the variable cost for the firm:
Total Variable Cost is calculated as
• Total Variable Cost = INR (50,000+20,000+40,000+30,000)
• Total Variable Cost = INR 1,40,000
The formula to calculate Contribution Margin is as below:
Contribution Margin = Net Sales – Total Variable Expenses
• Contribution Margin = INR 2,00,000 – INR 1,40,000
• Contribution Margin = INR 60,000
We can say that ABC Firm has left over INR 60,000 to meet its fixed expenses, and any remainder after meeting the fixed cost will be the profit for the firm. The fixed cost of the firm ABC includes
the following:
The calculation of Total Fixed Cost is as below:
• Total Fixed Cost = INR 10,000 + INR 15,000
• Total Fixed Cost = INR 25,000
The formula to calculate Net Profit is as below:
Net Profit = Net Sales – (Total Variable Expenses + Fixed Expenses)
• Net Profit = INR 2,00,000 – (1,40,000 + 25,000)
• Net Profit = INR 35,000
Alternatively, we can calculate the contribution margins of the firm by using the formula given below.
Contribution Margin = (Total Fixed Cost + Net Profits)
• Contribution Margins = INR (25,000 + 35,000)
• Contribution Margins = INR 60,000
After deducting the direct and indirect variable costs incurred in earning the revenue, the contribution margin represents the amount left-over. This left-over value then contributes to paying the
periodic fixed costs of the business, with any remaining balance contributing profit to the owners. Hence, we can calculate contribution margins by deducting the total variable cost from the total
To calculate the contribution margins, we need to consider three things:
• Fixed Expenses: Fixed expenses do not change irrespective of sales volume, such as rent, salary, insurance, utilities, office, depreciation, fees, etc.
• Variable Expenses: Variable expenses are those expenses that tend to change with the volume of sales, such as the cost of goods sold.
• Price: The price of the product is the price set by the firm to sell at wholesale price or cost of manufacturing the product plus markup.
Alternate Contribution Margin formula:
Contribution Margin = Fixed Cost + Net Profit
We can represent the contribution margin in percentage as well. Alternatively, it is known as the ‘contribution to sales’ ratio or ‘Profit Volume’ ratio. This ratio represents the percentage of sales
income available to cover its fixed cost expenses and to provide operating income to a firm.
Unit contribution margin per unit denotes the profit potential of a product or activity from the sale of each unit to cover per-unit fixed costs and generate profit for the firm. E.g., if a firm
sells a product at Rs 10 per piece and incurs variable costs per unit of Rs 7, the unit contribution margin will be Rs 3 (10 – 7).
Relevance and Uses of Contribution Margin Formula
Companies use the contribution margin in their operational decisions, applying it in various ways for different levels of decision-making.
• By using the contribution margin, the firm uses in break-even analysis. The breakeven point for a firm is when the revenue of the firm equals its expenses; also, we can that the point where the
firm is having neither a net profit nor net loss.
• The firm uses contribution margin analysis to measure its operating leverage, as it enables them to gauge how growth in sales translates into growth in profits.
• The contribution margin is also used to judge whether a firm has monopoly power in competition law, such as using the Lerner Index.
• A contribution margin is also used to compare individual product lines and be estimated to set sales goals.
Contribution Margin Formula Calculator
You can use the following Contribution Margin Calculator.
Net Sales
Total Variable Expenses
Contribution Margin Formula
Contribution Margin Formula = Net Sales – Total Variable Expenses
= 0 – 0
= 0
Recommended Articles
This has been a guide to the Contribution Margin formula. Here we discuss How to Calculate the Contribution Margin along with practical examples. We also provide Contribution Margin Calculator with a
downloadable Excel template. You may also look at the following articles to learn more –
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Calculating the Future Value of a Single Amount (FV) | AccountingCoach
Calculating the Future Value of a Single Amount (FV)
If we know the single amount (PV), the interest rate (i), and the number of periods of compounding (n), we can calculate the future value (FV) of the single amount. Calculations #1 through #5
illustrate how to determine the future value (FV) through the use of future value factors.
Calculation #1
You make a single deposit of $100 today. It will remain invested for 4 years at 8% per year compounded annually. What will be the future value of your single deposit at the end of 4 years?
The following timeline plots the variables that are known and unknown:
Calculation using an FV factor:
At the end of 4 years, you will have $136 in your account.
Calculation #2
Paul makes a single deposit today of $200. The deposit will be invested for 3 years at an interest rate of 10% per year compounded semiannually. What will be the future value of Paul’s account at the
end of 3 years?
The following timeline plots the variables that are known and unknown:
Because the interest is compounded semiannually, we convert 3 years to 6 semiannual periods, and the annual interest rate of 10% to the semiannual rate of 5%.
Calculation using an FV factor:
At the end of 3 years, Paul will have $268 in his account.
Calculation #3
Sheila invests a single amount of $300 today in an account that will pay her 8% per year compounded quarterly. Compute the future value of Sheila’s account at the end of 2 years.
The following timeline plots the variables that are known and unknown:
Because interest is compounded quarterly, we convert 2 years to 8 quarters, and the annual rate of 8% to the quarterly rate of 2%.
Calculation using an FV factor:
At the end of 2 years, Sheila will have $351.60 in her account.
Calculation #4
You invest $400 today in an account that earns interest at a rate of 12% per year compounded monthly. What will be the future value at the end of 2 years?
The following timeline plots the variables that are known and unknown:
Because the interest is compounded monthly, we convert 2 years to 24 months, and the annual rate of 12% to the monthly rate of 1%.
Calculation using an FV factor:
At the end of 2 years, you will have $508 in your account.
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Notes On W. D. Gann's Hidden Material
Click on one of bellow shared links to download
Sacredscience - Julius Nirenstein - Notes On W. D. Gann's Hidden Material
The Complete Gann I-IX Lecture Notes
The Sacred Science Institute is proud to offer for the first time ever in their complete form, the Gann Seminars of Dr. Jerome Baumring. Dr. Baumring is the only known person to have understood the
true method of Gann’s process of market forecasting, and moreover to have understood that this method is based upon a complete understanding of the holistic nature and laws of the universe. Thus Dr.
Baumring was not teaching some simple method of market analysis, but rather was teaching the complete system of universal cosmology as in the ancient tradition of the sacred mystery schools, and
using the markets as a laboratory for the demonstration of these universal principles.
Gann I Lecture Notes
1986 An Overview of All 12 Seminars. These Notes Present A Detailed Record of Baumring's Teachings, Theories, Diagrams & Market Applications as Presented In The Seminars. Contents: Conventional Cycle
Wisdom; Definitions of Cycle; 45 Degree Angle vs. Pitch or Trend; How To Measure a Cycle; Nodal Points; Monad; Octave Intervals; Square of 9; Squaring The Circle = Squaring Price & Time; Vesica
Piscis; Hexagonal Symmetry; Slope of Hypotenuse; Tangential Vectors & Pitch; Harmonic & Arithemetic Mean; Alternating Beats; Double Square; Retracement Percentages; Octaves; Theory of Allocution;
Sequential Series; Proportion; Gann as Kabbalist; Natural Opposites; Harmonic Composition & Decomposition; Solomon's Seal; Growth; Hexagon; Types of Angles; Planetary Hours; Time & Price; Planes of
Symmetry; Finding Dominant Cycle; Vector Momentum; Irregular Cycles; Pitch; Time & Angles; Squares As Spiral Generators; Gnomonic Growth; Circumscribed Objective; Ratio; Time More Important Than
Price; Tangents To A Parabolic; Wavelengths; Parabolic & Hyperbolic Moves; Symmetry In Z Plane; Wave Polarity; Trading; Wave Principle; Polygonal Symmetry; MultiDimensional Market Phenomena; Ellipse;
Curvilinear Light; Geometric Transformation; Calendar vs. Trading Days; Polarity & Forms of Vibration; Electromagnetic Planetary Influences; Planetary Angles; Gann's 3rd & 4th Dimension of Time &
Price; Pitch; Growth; Capstone; Pythagoras; 47th Problem of Euclid; Progressional Series; Light, Sound & Color; Numerology, Astrology & Vibration Thoery; Einstein & Gann: Similar Concepts of
Relativity, Vibrations, Impulse & Natural Order; "As Above So Below"; Time Interval; Number Progressions; Birth Point; Soybeans; Volume: The Driving Force of Market; Tape Reading; Bonds; Open
Interest; Examples; Velocity vs. Acceleration; Vector Wiggle; Triple Square; Momentum Wave; Amplitude = Interval; Gann's Death Angle; Retracements; Dow Cycles; K-Fold Symmetry; Electron Orbitals;
Symmetry In Nature; Number Root Growth; Energy Levels; Mercury Cycle; Unfolding Square; Elliot; Edson Bears; Gann's Center of Gravity; Wave Mechanics; Lambda; Potential & Kinetic Energy; Music;
Numbers as Points of Force; Diagonal of Vesica Piscis; Pitch Truing a Chart; Measuring Frequency of Oscillation; Price Symmetry; Parabolic Motion; Cycle; Volatility; Law of Periodicity; Summational
Waves; Law of Proportion; Radius; "Everything Done By The Square"; Freemasons; 7 Intervals, 8 Octaves; Finding Nodal Lines and Wave Forms on Wheat Charts.
Gann II Lecture Notes
1986 These Notes Present A Detailed Record of Baumring's Teachings, Theories, Diagrams & Market Applications as Presented In The Seminars. Contents: Cleopatra's Needle; Universal Number Set; Gann's
Reference to the Book of Revelations; Perfect Numbers; Zero As Place Holder; Natural Opposite Numbers; Musical Scales; "13 Lucky/Unlucky Number"; Numerology; Ratio & Proportion; Lumber, Platinum &
Copper; Time Compression; Spectral Density; Polar Coordinates; Square of Nine; Pitch; Slope; Forces; Liquids - Pressure, Volume; Torque - Work, Potential Energy; Theoretical Wave Mechanics;
Components of Vibrational Energy; Cycle Phasing; Cycle Low vs. Momentum Low; Translation; Star Pentagram; Patterns of Growth; Volume = 3rd Dimension = Cuboid; Pressure Gradients; Tonal Ratios;
Velocity; Wyler's Market Physics; Volume = Mass; " Mass Pressure Kinetics"; Vector Analysis; Energy; Wheat Charts; Nodal Lines; Irregular Beats; The Saucer Bottom; Energy Levels; Ionization
Potential; Tetrahedron & Pyramid; Overlap of Chemistry & Music; Consolidation; Electron Stability; Dow Nodal Points; Parabolic Growth; Parabolic Equations; Lost Motion; Slope; Vectors; Retracements;
Trigonometric Functions; Numbers Are Symbols; "The Longer the Stronger"; Gann's Square of 52 Really a Pentagon; Geometric Addition & Symbolism; 5 & 7 Year Cycles; 2-D vs. 3-D; Order The Key To Gann;
Macrocosm - Microcosm; Geometrical Series Progression Thru Polygons; Phi; Concentric Circles & Logarithmic Spirals; Trigonometric Series; Curvature of Light; Quadrature Of Circle; 2-Fold Symmetry;
Conic Sections; Gann's Coffee Rio Chart; Time Is Curvilinear; Square of Range; 2 Vesica Piscis; Wavelength; Dominant Cycles; Wave Phasing; Elongation & Compression of Waves; Cycles; Difference
Between Price Reactions & Rallies; Axis of Symmetry; Hexagonal Growth; Kinds of Series: Summational, Square, Hexagonal, Co-Serial, Trigonometric, Parabolic; Gann Angles; Gann Angles As Asymptotes Of
Parabolas & Hyperbolas; Gann's Horizontal, Vertical & Diagonal Angles & What They Measure; Spiral Vectors; Quanta Shells; Music & Chemical Valence Levels; Harmonic Composition & Decomposition; Price
Element & Time Cycle; Star Pentagram; Benzene Configurations; "Everything Builds On Itself"; Polygonal Order; Diamonds; Stability; Garret & Wyler; Trajectories; Pressure; Polarity; Periodicity;
Charts; Pitch & Slope; Measurements for Analysis; Gann Combines Different Series In His Work; Cycles Are Progressional Series; Everything In Nature Combines In Certain Ratios To Form Progressional
Series Which Repeat In Periodicities Forming Certain Patterns; What Looks Like Noise Has Periodicity; March Wheat; Forces of Nature; Defining Tops & Bottoms; Soybeans; Measuring Snap-Backs;
Rotocenters; Diamond Cutting; '79 Hogs.
Gann III Lecture Notes
1986 These Notes Present A Detailed Record of Baumring's Teachings, Theories, Diagrams & Market Applications as Presented In The Seminars. Contents: Different Growth Forms Have Different Forms of
Symmetry Three Geometrical Forms Can Define All Growth; Angles Are Moving Averages; Price Variance; Component Production of Composite Waves; Progression; Identification of Cycles Using Yearly Charts;
Gann Periods to Watch; Quadrants of Circle; Logarithmic Spirals; Gnomonic Growth; Concentric Circles; Triangles of Pascal; The Zero Point; Repeating Patterns; 60 Year Patterns; Periodicity; Ratios of
Circle; Gann's Retracements; Motion of Vibrating String; Cardinal Cross; Number Set Progressions & Wave Theory; Symmetry & Natural Opposites; Time & Price Overbalancing; Plateau Areas & Periods;
Component Waves & Gann's Cycles; Genetic Coding; Market Phases; Uranus & Saturn; Inner & Outer Planets; Planetary Electromagnetic Effects; Gann's Time Counts; Conjunctions & Oppositions; Sensitive
Points In Time; Recurrence of Planetary Configurations; Short Term Cycles; Gann Wheels; Astrometeorology; Canons of Proportion; Impulse & Retracement; Cycle Length Ratios & Correspondence To
Planetary Orbits; 5 Years Is Gann's Smallest Cycle; Gann's 60 Year Cycle; Gann's New Discovery, The Master Numbers & Their Use; the Master Time Factor; Cycles vs. Periodicity; Recurrent Behavior
Patterns; Gann's 5, 7 & 10 year Cycles & Their Definitions; Astrolabe; Planetary Perturbation; Number Sets & Symmetry; Vector Symmetry; Vectorial Direction; Squaring Price & Time; Elliot Channel;
Pentagons Formed By Triangles; Decahedron - Tetrahedron - Icosohedron; Trisection of An Angle; Mirror of DNA; Faces of Cube; Cardinal Cross vs. Fixed Cross; Pi; Axial Symmetry; Curvilinear Time;
Squaring The Circle & Number Sets; Interweaving Lattices; Symmetry Nets; Growth Matrix; Growth Followed By Decomposition; Limitation of Martix; Beads of An Abacus; Gann's Calculators Are Curved
Matricies; Lambdoma Sieves; Color - Number - Sound, Symbolic Representation; Number Sets; No Zero; Numeric Recuction; Soybeans Have 3-Fold Pentagonal Symmetry; Axial Symmetry; Rotational Symmetry;
Boundaries of Symmetry; Pi Clouds; The Leaf; Where To Look for Symmetry; How To Find Planes Of Symmetry; What Comes Before & After; Simultaneously Seeing Part & Whole; Vectors; Pyramid = Tetrahedron;
Gann's Time Cycles Composed of Triangles & Squares; Square of 52 A Pentagon; Kabalistic; 2-D vs. 3-D Representation; Dynamic vs. Static Symmetry; Pentagonal Growth as Basis of Circle; Fibonacci;
Overbalancing of Time; Measuring Time Reactions; Differentiating Major & Minor Swings; 100 Years of Data Enough to Determine Building Blocks & Growth Patterns; Cycles Within Cycles; Vesica Piscis &
Ellipse; Vectors; Seeing Depth In Charts; Internal Verticies of Polygons; Importance of Visualization; Time Slices; Shared Verticies; Rings Demarkate Time; Pentagon; Relation of Pi & Phi Through
Pentagram; Pi & Phi Relationship Constant In Time; 10 year Cycle; Soybeans Pentagonal; Sign of Jonah; 3 Days & 3 Nights; Important Polygons; Cosmic Clock; Verticies Are Hinges; Gematria; Gann's
Percentage Increase On Base; Serial Progression; Time & Shape Changes; Time Symmetry; Uranus, Saturn, Jupiter, Neptune, All Have 7 in Common; Pythagorean Harmonics & the Meaning of the 10 & 60 Year
Cycles; Icosahedron; Change Static To Dynamic Symmetry; Time Models; Forces; Energy Shells; Perfect Numbers; Gann's Master Calculators; Planetary Order; Mercury from Sun: Pythagorean Monad; Movements
Thru Platonic Solids; Hedrons; Einstein's Unified Field Theory; Swing Theory; Boat, Sofa & Chair Configurations of The Benzene Ring; T-Bonds; Building Blocks; Chart: A 2-Dimensional Representation of
a 3 or 4 Dimensional Reality; Chemical Substances; Time Counts - Progressions - Proportions - Swings; Long Term Graphs; 12 Year Cycle; Common Numbers; Pivot Points; Geometry Matches Solar Order;
Volume = Area Under Curve; Spatial Orientation; Lapaz Transformations; Polar Coordinates; Rotation; Equilibrium of Pendulum; Mazes & Mandalas; Double Square; Obvious Always Most Hidden; Radius
Vectors; Crossings; How To Use The Master Calculator; Snapshot of Dynamic Motion; Numerology.
Gann IV Lecture Notes
1987 These Notes Present A Detailed Record of Baumring's Teachings, Theories, Diagrams & Market Applications as Presented In The Seminars. Contents: Chart Applications of Principles; Swing Analysis;
Gann Time Counts; Reactions; Tradable Time Periods; Median Lines; Composite Top; Causes of Reactions; T-Bonds; Trading vs. Calendar Days; Directional Movement; Where to Count Form; Vector Variations;
Intervals; Symmetry on Z Plane; Andrews' ML Line; Vector Analysis; Electron Shells; Distortion In Commodities; Must Use All Charts At All Times; Gann's 1955 Work; Momentum Gaps; Proportional Measure;
Bond Charts; Defining Interval Lengths; Roto-Centers; Measured Move; Acceleration Model; Maximum Linear & Parabolic Movement; Bonds; Determining Vector Changes; Sequences Repeating in Time; 60 Year
Sections; Mirror Aspects; Rotation on 4 Axes; Transformation of Circle into Ellipse; Symmetry Is The Law; Dynamic Symmetry Above & Below; Gnomonic Growth; Sequential Morphology of Repeating Patterns;
Sepharial; Theoretical Composite; Changes of Direction; 8 Year Interval; Cracking Cycles Through Intervals; Vectorial Force Change; 6, 8, 9, 12 Intervals; Perfect Harmonic Sequence; Gann's $ Value
Chart 7 Stock Splits; Slanting Tops; Configurations - Patterns - Signatures; Cash Soybean Examples; Yearly, Monthly, Weekly & Daily Time Counts, How To Take Them; Gann's Great Time Cycle, 56 Years, 9
Months, 23 Days; Square of 144; Periodicity; Equilibrating Roto-Centers; Time Counterpoint; Z Plane & Gnomic Growth; Polarity Alteration; Radius Vectors; Chart Analysis; Force Over Time; Vectorial
Multiples; Serial Progression; Wheels Within Wheels; Fundamental Units; S & P, How Much Data Needed To Find Periodicity; Node of Node; Where To Begin Measurements; Wave: Unit of Force Per Day; Long
Waves vs. Impulse Waves; Accumulation; Pressure; Velocity & Acceleration; Impulse - Reaction; Vibration; Proportions; Chords; Super Cycles; Waves Not always Sinusodal; Wave Mutation & Planetary
Connections; Law of Periodicity; Divisions of Master Components; Symmetry Vectors; Pivotal Points; Matching Acceleration & Deceleration; Damping; Carrier Waves; Beats; Minimum & Maximum Time Counts;
Center Of Gravity; Calculus; Dead Lows; Squaring Lowest Price; Numerological Significance; Definitions of Time Lengths; 90 Degrees in Time; Finding Waves on Monthly Chart; "Supply" Bull Market:
Inverted Market; Straddles & Spreads; Astrological Cycles In Market; Planetary Transits & Trigger Mechanisms; Eclipse Points; Popular Misconceptions of Moon Cycles; Time Swings & Price Swings; Gann's
"Reactions Against The Trend"; Wave Phasing; Parabolic Arc; Logarithmic Spiral; Major & Minor Axis of Ellipse; Resultant Vectors; Momentum vs. Price Top; Bayer's Egg of Columbus; Floor & Ceiling;
Defining Mass Pressure; Playing Off Old Highs & Lows; Multiple Views of One Reality; Time Markers; Rate of Change; T-Bond Examples; 3 Orders of Swings; Angle of Attack; Market Moves Down Faster Than
Up; Strategy, Book of 5 Rings; Swing Theory: Gann, Cole & Tubbs; Deviant Vectors; Gann: Time More Important Than Price; Pattern Recognition - Universal Ordering Process; Look Hard At Macro Before
Micro; Bisection In Time; Tangents To Curve; Area Under the Curve; Directional Growth on Square of Nine; Growth on Rotational Axis; Bohr Model Quantum Mechanics; Fibonacci Growth Factors; Growth
Controllers; Plane of Cleavage; Vanishing Point; Radial Axis; Seeing Picture Over Crunching Numbers; Clockwise & Counterclockwise Rotation; Tetrahedron Is Cube; Inscribed & Circumscribed Squares &
Circles; 3 Rings of Gann's Square of Nine; Unfolded Pyramid; root Ratios as Generators of Growth; Fixed & Cardinal Crosses as Axes of Galactic Rotation; Uranus; Pluto Calculations; Johndro; Jupiter
Major Mass of Solar System; Alcyone, Nearest Large Sun; Solar System is Sphere; Markets Grow 3-Dimensionally; Square of Nine a Solar Return Chart; Polar vs. Cartesian Coordinates; "Building on the
Square"; Growth Vortex: Electric (Kinetic) & Magnetic (Potential) Energy Interchanging.
Gann V Lecture Notes
1987 These Notes Present A Detailed Record of Baumring's Teachings, Theories, Diagrams & Market Applications as Presented In The Seminars. Contents: Bull Market vs. Bear Market Rally; Patterns Can
Manifest as Polar Opposites; Regular & Predictable Periods; Gann's 6 Component Model; Vector Nets; Island Top; Calculus; Begin with Top & Bottom Formations, Most Important & Easiest; Short Term Waves
Circular - Long Term Waves Elliptical; Basic Shape Tells Waves; Difference Between rectangular & Logarithmic Charts; Roto-Center Pole; Gann's Coffee Rio Chart; Rotation Around Pole; Time Axis; Moving
Solids in space; Inside Radius Vectors; Conic Sections; Earth w. 15% Parallels; Mazes & Mandalas; Lining Up Ellipses; Making Ellipses To Fit Charts; Interacting Series; Beard's "Patterns In Space";
Archimedian Spiral & Rate Constants; Logarithmic Spiral & Root 3 Growth; Fibonacci Spiral & 72 Degree Turn; Different Phi Based Rate Constants For Different Length Swings; Series Ratios; Baravalle
Spirals In Squares, Hexagons & Octagons; Roto-Centers, Cones & Gann's Tunnel Through The Air; 3 Types of Swing Charts; Adding & Combining Time Counts; How to Calculate Developing Series on Charts;
Blocks Within Larger Periods; Trading Ranges; Probability of Pattern Recognition; Soybeans; Location In Order of Whole; Shunt Periods; Describing Difference In Pattern; Bicycle Pedal Pattern;
Prediction Future Moves Using Spiral Growth Chart; Sequence Analysis; Locating 3 points to Define a Series; Ellipse Series; Quadrant of Circle; Rolling Time; Tracing Volumes; Polish Ellipse;
Controllers; Magic Name of Jehovah; Gann's Jehovah Diagrams From "Magic Word"; Kabballah & Temura; Sensitive Numbers; Examples; Putting A Polygon Around a Chart Pattern; Determining Phases; Finding 3
Intersecting Terms; Radical Controllers; Higher Order Polygonal Transformation; Root 5 Decomposition Model; Phi Decomposition; Golden Rectangle; Radical Numerology; Overlapping Spirals; Sub-Growth
Phases; Charts Are Windows on Time; Relationship Of Part & Whole; Finding Stable State System; Carbon Atoms & Benzene Ring Configurations; Bayer's Hinge; Transforming 2-D to 3-D; Hexagonal Growth on
Dow; Spiral Intersections; Bayer's 9" Ellipse & Beard's Earth with 15 Degree Parallels; Calculating Chart Ratios; Equilibration Patterns; Practical Pattern Recognition; Unusual Pieces Obscuring
Patterns; 56 Year Repetitions; Semitone Harmonics; 8/9; Constellation Pleadies; Changing the Microscope View; Soybeans; Purpose of Gann's 8x8" Graph Paper; Harmonics & Pattern Production; Wave Nets &
Pattern Addition.
Gann VI Lecture Notes
1987 These Notes Present A Detailed Record of Baumring's Teachings, Theories, Diagrams & Market Applications as Presented In The Seminars. Contents: Financial Astrology - Numerical Astrophysics;
Concept of Mass; Einstein: E=Mc2; Mass = Energy; Centripetal & Centrifugal Forces; Galactic Centre Is Not Where Everyone Thinks It Is; "Apocalypse of the Golden Mean"; Mounds of the World As Time
Markers; Pyramids & Churches & their Alignment With Galactic Center; Bayer's Equinox Points Different Than What You Think; Mitchel "Stellar Worlds"; March '84 Soybeans; Hambidge's "Dynamic Symmetry";
Complementary Rectangles; Analyzing Areas Using Reciprocal; Everything Mathematical Points Of Force; Relation of Length & Area; Time Is Measuring Stick; Yin & Yang; Gann's Lost Motion; When Price
Doesn't Apply to the Square of Nine; Geometric Transformations of the Square of Nine; Manly Hall; Min. - Max. Area Under Curve; 3 Points on a Curve; Inner & Outer Circle; Cardinal & Equinox Points;
Davidson's Great Pyramid; Solar Declination & Gann Angles; Gann's Top & Bottoming Formations; Lower & Higher Orders Always Existing Ad Infinitum; First & Second Order Recurring Phenomena; Distances
of Repetition; Double Beats; Higher Revolutions; The More History The Better; Gann Lacked History So Went To Astrology; Comparing Like to Like; Addition Series; Time Only; History Repeats Itself;
Comparing Vector Changes; Gann's Instructions for Comparison; Tandy Example; Computing Average Rate of Change In Cotton Market; 10, 7 & 2 Year Cycles; Astrological Readings; "Law of Cycles" & "Law of
Periodicity"; "Law of Proportion"; Time Intervals; 7 Major Planets; Components of 60 Year Cycle; 120 Year Cycle; Unusual Aspects of Sepharial; Sepharial & Bayer's Major Insight; Tangents to 3 Bodies;
Planetary Trigger Mechanisms; Jupiter Effect; Four Planet Model; Alcyone 29 Degrees Taurus; Order of Suns In The Creation of Ellipses; Johndro's Carrier Waves; Planetary Absorption & Reflection of
Energy; Nelson's Planetary Effects On Weather; Sun as Transmitter; Jupiter The Great Reflector; Bayer's Planetary Distinctions; Geometric Planetary Configurations; Faster & Slower Planets; Gann &
Tubbs: beds of Accumulation & Distribution; Silver; Polarity; Gann's Master Circle Chart for Eggs; Directing the Poles; Midpoints; Electromagnetic Flares Related to Planetary Synodic Periods; Fixed
Zodiac; Interior Angle of Pentagon; Gann's Circles on His Spiral Charts; Relations of Ellipses & Circles; Inner & Outer Circles & Squares; Transformations of Area to Volume; Squaring the Circle &
Doubling the Cube; 3/Root 2, Phi, Pi, i, All Extremely Important; Socrates; Root 2 Growth; Slicing a Conic; Point of Observer; John Wilson's Series Summed - A Locus of Points Describing a Certain
Phenomena; Use of Bayer's Ellipse; Axis of Bayer's Polish Ellipse; Gann's Egg Chart; Zero Point & Triangulation; Accell - Decell Model; Whip Effect & Inertia; Rectifying the Horoscope; Use of
Heliocentric & Geocentric Astrology; Jupiter, Saturn & Soybeans; Uranus' Opposite Rotary Motion & the Unexpected in the Markets; Nuclear magnetic Resonance - Electron Spin Theory; Effects of Uranus
Couplings; Tectonic Plate Shifts; Sunspot Cycles, Jupiter Effect & Neptune; Synodic Pairs; Time & Configuration; Energy Shells; Damping Effects; Time Series Progression.
Gann VII Lecture Notes
1987 These Notes Present A Detailed Record of Baumring's Teachings, Theories, Diagrams & Market Applications as Presented In The Seminars. Contents: General Motors Long Term Weekly Chart; Wave
Pattern of the Chart; The Stock Market "Record" 200 Year Pattern Morphology; How To Determine Duration of Market Moves; Serial Progression; Periodicity; Reflective Formations; Cyclic Number Series;
Bull Campaign; Matching Patterns at Periodicities; Using the Hypotenuse to "Build on the Square"; kinetic Energy - Potential Energy; Energy Applied - Energy Stored; Beds of Accumulation; Slope &
Angle of Attack; To Make a Forecast You Must Know Where You Are In Each Order; Swing Comparison; Breaking Down a Chart; Correlative Percentages; Triple Single & Double Tops - Vector Mixes; Gann's
Topping Formations; Top vs. Bottom - 180 Degrees Apart; Find Order To The Differences; Analysis; Periodicity of Polarity; Significant Time Points & Vectorial Changes; Analysis; Saturn Cycle; Gann:
For Every Bottom There Is a Top, & For Every Top There Is a Bottom; Resolution; Sequence; Ratio & Additive; Work Down To Daily; Develop Transformation Code; Lay Out Flow Chart of Forecasting Code;
Comparing Like with Like; Checking 7 Year Intervals; Cell Birth; Dead Reckoning From Highs & Lows; Triangulation on the Sphere; Getting a Fix on Location; Progression of Number Sequence; Growth &
Decay; Winding & Growing; Winding & Dying; Progressively Sequential Growth & Decay Coding; Ordering of Mass Psychology; Min & Max Reliability Value; What is Occurring I a Signature of Things to Come;
Functions of 9; Rectifying a Market Chart; Multiple Orders; Mercury - Pluto; Transformations of Zero Point; The Relationship of Geocentric & Heliocentric to Short & Long Term Patterns & To Arithmetic
& Logarithmic Scales; Johndro & The Ineffectivity of the Inverse Square Law; Reflection of Energy; Bounce Effect; Rate Constants; 3 Dimensional Perception; Perspective Paper; Vector = Slope +
Direction of Force; Direction Included 3-D; Orb of Influence of Aspects; Axes of Space; Wheat Chart; Results of Stress In System; Energy Shells & Bands; Escape Velocity; Stable Configurations; Price
& Pressure; Intervals, Wave Length; Gann & Bayer: Intervals Defined by Circles, Squares & Ellipses; 3-D Space; Analysis of Square of Nine; Price is Space; Conjunctions & Oppositions; Where I Energy
Coming From?; Orientation in Space & Heavens; Polar Market Charts; Law of Vibration, Squaring the Circle, Music - Harmonics, Proportion - Ratio, Root 2, Fibonacci, Phi Growth & Decomposition: No
Predictability From Any ONE: Each Works for a While Then Doesn't; The Intertwining of All of It; Circle = Infinite Polygon; Uniting the Finite & Infinite; Halving The Circle; yi & Yang; Vesica
Piscis: Relation of Interval to Amplitude; Pentagon: Symbol of Life; Fivefold Symmetry; Star Pentagram; Musical 4th & 5th; Creation Thru Gnomonic Growth; Platonic Solids; Music; Pythagorean Octave;
Musical Ratios; Tone Mandala = Logarithmic Progression; Turning Gann Calculators Into Music; Septiform System of The Cosmos; 5 fold Symmetry in Human, 7 Fold In Botanical World; Atomic Weight of
Carbon = 12, Basis of All Life; Square of 52; Alteration Between Harmonic & Arithmetic Means; Comparison of 1792 & 1914 on Dow; Monad; Earth Is A Child of The Sun; Saturn - Jupiter Cycle; Growth on
Earth Related to Heaven; Same Growth in Market & Nature; Vibration of Dow Changing With Changing Stocks; Must know Which Cycle Is Running; Must know Where You Are; Square of Nine A Natal Chart; Like
Cause - Like Effect, Like for Like Not Similar; Order In Space; Chemical Molecular Blocks; Right Brain Experience.
Gann VIII Lecture Notes
1988 These Notes Present A Detailed Record of Baumring's Teachings, Theories, Diagrams & Market Applications as Presented In The Seminars. Contents: Fractal - Symmetry Every Which Way; Mandelbrott;
circles Inscribed In Squares; Face of Cube; Keystone Is Always First Block; Composition & Decomposition Within Cube; Greater & Lesser Seals of Solomon; Beard; Octaves; Formula; Apocalypse of Golden
Mean; Spinning Triangles; Triangle + Square = 7; 10 Year Cycle Really 20 Years; Fractal Time Blocks, Monthly - Weekly Daily; Nautilus Shell; 3 Points Define Curve; 8 Followed By 5; Annual Forecasts;
Gann's Stock Course: Formations of Tops & Bottoms; Reactions; Ellipse; Must Have a Pitch True Chart; How to Use Bayer Overlays; Gann's Inner & Outer Circle; Circle 1/2 Lambda; Locating the Major
Reaction; Where In directional Series; Fractals; Duration; Bear Market Rally; Tagging Pieces; Everything Has Relationship; Growth From Center of Cube; Conic Sections; General Motors Flips;
Parallelogram With Diagonals; 9 & Chart Signatures; Bayer: True Low & Hinged Gate; Same Patterns or Fractals Proportionally Related; Symmetry From Center To Edge of Cube; Nephroid; Basic Block Double
Circle; Square of Nine & 15 Degree Parallels; Building Blocks Not Sinusodal; DNA: Hexagon - Pentagon - Rectangle; 3-D Cube & Square, Gann's Square of Nine; Diagonals; Teleois Proportions Will Produce
Every One of Gann's Cycles; Teleois Series; Sunspot Cycle Alteration; Number Sieve; 7 Intervals In Octave; Electron Fill Order; Gann's New Master Numbers; Unfolded Pyramid; Double Square; Close
Packing of Spheres; 7 Planets of Ancients; Cube - Hexagon Relationship; Dynamical Cause; Wheat 1931; Degrees of Rings in Calculator; 1/2 Ellipses Nephroid & Double Circle; Double Helix; May Soybeans;
Pentagram; Geometric Addition; Development of a Circular Series; Root 2, 3, 5, 7 Growth; 3 Points Define Spiral; Circles In Geometric Progression; Vesica Piscis; Focal Points of Ellipse; Centers Are
Attractor Points; Time Is Curvilinear; Wrapping; Exercise: Finding Time Rolls; Backwards Spirals; Square of 4 & 9; Directions of Rotation; Hyperboloid Cones; Gann: Can Do Whatever You Want With A
Circle Square & Triangle; Meander Maze; Specific Codes Throughout Growth; Least Effort To Achieve Maximum Efficiency & Stability; Moving Through Different Parts of Cube; Simultaneous Orders of
Growth; Construction of 60 Year Cycles; Musical Relationship; Snapping Together Building Blocks; What Precedes & What Follows; Putting Time back Into the 3rd Dimension; 50% Increase on Base;
Pythagoras: 1, 2, 3, 4 Is All You Need.
Gann IX Lecture Notes
1987 These Notes Present A Detailed Record of Baumring's Teachings, Theories, Diagrams & Market Applications as Presented In The Seminars. Contents: Definition of Cycle: Period Of Recurrence;
Specific DNA & Math Sequences vs. Sunusoidal Waves; Formation; Time is Elliptical & Spiral Motion; Sepharial's 60 Year Cycle; Addition; Beard; 3 Isosoleces Triangles In Pentagram; Circular
Progression 7 Gann's Jehovah; Star of David; Order In Space; Universal One; Pressure System; Bowing Effect; solar Return Chart; Declination; Ouspenski; Whole, Part & Relation; Hinges & Shunts; DNA
"Mutations"; Substitution, Deletion, Addition; Turning Time Curvilinear; Unfolded Cube & Cross; Spherical Coordinates; Wheels Within Wheels; Magic Word, Jehovah; Generators; Dual Spirals of Vortex
Systems; Market Damping; Gull's Wings; Verification of Curve; Ribbon Effect; Point of Observer; Location In Space; Pascal's Triangles & Gann's Coffee Rio Chart; The True Value of Pi; Bohr Quantum
Energy Levels; Power Inversions & Semitones; Kepler's Law of Solar System & its Relation to the Circle; Equation For Ellipse; Oblong Circle & Oblong Square; Use of Vesica Piscis in Squaring the
Circle, Doubling the Cube, & Trisecting the Angle; Circumference of Ellipse; Elliptical Time Measures; Tangents to Circle; Appehelion & Perihelion; Major Axis & Minor Axis; time Measurements; Finding
Pieces; Wheat Charts; Similar Year Characteristics; Lengths To Measure; Key Words: Portent, Foretell, Forecast; Similarities in Runs; Fractals; Clarification of Gann's Duration; Dissimilarities; Can
Use Fractals In Place of Long Term History; DNA Code Exists in One Cell; Transcription - The Message; Meander Mazes; July Wheat; Overlapping Parts; Transcription Locking; Spatial Alignment; One Cycle
- One Sequence of Events; Not All First Order; No Cycles Overlapping, Evolution of Growth; 4 Year Piece in Soybeans; Yearly - Quarterly - Daily Charts; 13 Year Soybean Segment; Termination of Moves;
Growth Not Always Proportional or Harmonic.
1986- 1989 250p. These Notes Present A Detailed Record of Dr. Baumring's Teachings, Theories, Diagrams & Market Applications as Presented at the Investment Centre Seminars. Each Seminar Lasted Two
Eight-Hour Days. The Notes Included in This Set Came from 14 Different Presentations of the Nine Seminars with the First Four Seminars Having Been Repeated More Than Once Each. Dr. Baumring Did Not
Follow Exact Lecture Notes But Made Spontaneous Orderly Presentations, Following a Previously Selected Series of Projected Images, Selections of Text & Charts, Upon Which He Would Elaborate During
the Seminar. As Students Asked Questions Dr. Baumring Would Explain In Greater Detail Various Aspects of These Topics, so That Each Repetition of An Individual Seminar Contains New Information &
Further Elaborations Upon Different Points. For An Extremely Detailed Description of Contents See The Gann & Baumring Category and the Financial Market Forecasting Section of This Web Site. The Bulk
Purchase of The Complete Series of Lecture Notes Includes a $250.00 Discount. For A Greater Bulk Discount Please See The Complete Course Manuals & Lecture Notes.
Get Notes On W. D. Gann's Hidden Material (The Complete Gann I-IX Lecture Notes) or the other courses from the same one of these categories: Julius Nirenstein, Notes On W.D.Gann, Hidden Material,
Sacredscience, Trading, eBook, Gann for free on Course Sharing Network.
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Fedor Pakhomov
According to our database
, Fedor Pakhomov authored at least 15 papers between 2014 and 2024.
Collaborative distances:
Book In proceedings Article PhD thesis Dataset Other
Online presence:
On csauthors.net:
Generalized fusible numbers and their ordinals.
Ann. Pure Appl. Log., January, 2024
There are no minimal essentially undecidable theories.
J. Log. Comput., 2024
Corrigendum to Reducing ω-model reflection to iterated syntactic reflection.
J. Math. Log., December, 2023
Reducing ω-model reflection to iterated syntactic reflection.
J. Math. Log., August, 2023
Arithmetical and Hyperarithmetical Worm Battles.
J. Log. Comput., 2022
Reflection algebras and conservation results for theories of iterated truth.
Ann. Pure Appl. Log., 2022
Reflection ranks and Ordinal Analysis.
J. Symb. Log., 2021
Short Proofs for Slow Consistency.
Notre Dame J. Formal Log., 2020
Multi-dimensional Interpretations of Presburger Arithmetic in Itself.
J. Log. Comput., 2020
On a Question of Krajewski's.
J. Symb. Log., 2019
Complexity of the interpretability logic IL.
Log. J. IGPL, 2019
Truth, disjunction, and induction.
Arch. Math. Log., 2019
Interpretations of Presburger Arithmetic in Itself.
Proceedings of the Logical Foundations of Computer Science - International Symposium, 2018
Solovay's Completeness Without Fixed Points.
Proceedings of the Logic, Language, Information, and Computation, 2017
On the complexity of the closed fragment of Japaridze's provability logic.
Arch. Math. Log., 2014
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Identifying Expressions and Equations
Learning Outcomes
• Identify and write mathematical expressions using symbols and words
• Identify and write mathematical equations using symbols and words
• Identify the difference between an expression and an equation
• Use exponential notation to express repeated multiplication
• Write an exponential expression in expanded form
Identify Expressions and Equations
What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. “Running very fast” is
a phrase, but “The football player was running very fast” is a sentence. A sentence has a subject and a verb.
In algebra, we have expressions and equations. An expression is like a phrase. Here are some examples of expressions and how they relate to word phrases:
Expression Words Phrase
[latex]3+5[/latex] [latex]3\text{ plus }5[/latex] the sum of three and five
[latex]n - 1[/latex] [latex]n[/latex] minus one the difference of [latex]n[/latex] and one
[latex]6\cdot 7[/latex] [latex]6\text{ times }7[/latex] the product of six and seven
[latex]\frac{x}{y}[/latex] [latex]x[/latex] divided by [latex]y[/latex] the quotient of [latex]x[/latex] and [latex]y[/latex]
Notice that the phrases do not form a complete sentence because the phrase does not have a verb. An equation is two expressions linked with an equal sign. When you read the words the symbols
represent in an equation, you have a complete sentence in English. The equal sign gives the verb. Here are some examples of equations:
Equation Sentence
[latex]3+5=8[/latex] The sum of three and five is equal to eight.
[latex]n - 1=14[/latex] [latex]n[/latex] minus one equals fourteen.
[latex]6\cdot 7=42[/latex] The product of six and seven is equal to forty-two.
[latex]x=53[/latex] [latex]x[/latex] is equal to fifty-three.
[latex]y+9=2y - 3[/latex] [latex]y[/latex] plus nine is equal to two [latex]y[/latex] minus three.
Expressions and Equations
An expression is a number, a variable, or a combination of numbers and variables and operation symbols.
An equation is made up of two expressions connected by an equal sign.
Determine if each is an expression or an equation:
1. [latex]16 - 6=10[/latex]
2. [latex]4\cdot 2+1[/latex]
3. [latex]x\div 25[/latex]
4. [latex]y+8=40[/latex]
1. [latex]16 - 6=10[/latex] This is an equation—two expressions are connected with an equal sign.
2. [latex]4\cdot 2+1[/latex] This is an expression—no equal sign.
3. [latex]x\div 25[/latex] This is an expression—no equal sign.
4. [latex]y+8=40[/latex] This is an equation—two expressions are connected with an equal sign.
try it
Simplify Expressions with Exponents
You have simplified many expressions so far using the four main mathematical operations. To simplify a numerical expression means to do all the math possible. For example, to simplify [latex]4\cdot
2+1[/latex] we’d first multiply [latex]4\cdot 2[/latex] to get [latex]8[/latex] and then add the [latex]1[/latex] to get [latex]9[/latex]. A good habit to develop is to work down the page, writing
each step of the process below the previous step. The example just described would look like this:
[latex]4\cdot 2+1[/latex]
However, there are other mathematical notations used to simplify the numbers we are working with. Suppose we have the expression [latex]2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2[/
latex]. We could write this more compactly using exponential notation. Exponential notation is used in algebra to represent a quantity multiplied by itself several times. We write [latex]2\cdot 2\
cdot 2[/latex] as [latex]{2}^{3}[/latex] and [latex]2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2[/latex] as [latex]{2}^{9}[/latex]. In expressions such as [latex]{2}^{3}[/latex], the
[latex]2[/latex] is called the base and the [latex]3[/latex] is called the exponent. The exponent tells us how many factors of the base we have to multiply.
[latex]\text{means multiply three factors of 2}[/latex]
We say [latex]{2}^{3}[/latex] is in exponential notation and [latex]2\cdot 2\cdot 2[/latex] is in expanded notation.
Exponential Notation
For any expression [latex]{a}^{n},a[/latex] is a factor multiplied by itself [latex]n[/latex] times if [latex]n[/latex] is a positive integer.
[latex]{a}^{n}\text{ means multiply }n\text{ factors of }a[/latex]
The expression [latex]{a}^{n}[/latex] is read [latex]a[/latex] to the [latex]{n}^{th}[/latex] power.
For powers of [latex]n=2[/latex] and [latex]n=3[/latex], we have special names.
[latex]a^2[/latex] is read as “[latex]a[/latex] squared”
[latex]a^3[/latex] is read as “[latex]a[/latex] cubed”
The table below lists some examples of expressions written in exponential notation.
Exponential Notation In Words
[latex]{7}^{2}[/latex] [latex]7[/latex] to the second power, or [latex]7[/latex] squared
[latex]{5}^{3}[/latex] [latex]5[/latex] to the third power, or [latex]5[/latex] cubed
[latex]{9}^{4}[/latex] [latex]9[/latex] to the fourth power
[latex]{12}^{5}[/latex] [latex]12[/latex] to the fifth power
Write each expression in exponential form:
1. [latex]16\cdot 16\cdot 16\cdot 16\cdot 16\cdot 16\cdot 16[/latex]
2. [latex]\text{9}\cdot \text{9}\cdot \text{9}\cdot \text{9}\cdot \text{9}[/latex]
3. [latex]x\cdot x\cdot x\cdot x[/latex]
4. [latex]a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a[/latex]
Show Solution
try it
In the video below we show more examples of how to write an expression of repeated multiplication in exponential form.
Write each exponential expression in expanded form:
1. [latex]{8}^{6}[/latex]
2. [latex]{x}^{5}[/latex]
Show Solution
try it
To simplify an exponential expression without using a calculator, we write it in expanded form and then multiply the factors.
Simplify: [latex]{3}^{4}[/latex]
Show Solution
try it
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Free Multiplication Chart Printable Paper Trail Design | Multiplication Chart Printable
Free Multiplication Chart Printable Paper Trail Design
Free Multiplication Chart Printable Paper Trail Design
Free Multiplication Chart Printable Paper Trail Design – A Multiplication Chart is an useful tool for kids to discover exactly how to increase, split, as well as discover the smallest number. There
are many usages for a Multiplication Chart.
What is Multiplication Chart Printable?
A multiplication chart can be made use of to assist children discover their multiplication truths. Multiplication charts can be found in several forms, from full web page times tables to single web
page ones. While individual tables serve for offering chunks of information, a full page chart makes it easier to evaluate truths that have already been mastered.
The multiplication chart will normally include a leading row as well as a left column. The leading row will have a listing of products. When you want to locate the product of two numbers, pick the
first number from the left column and also the second number from the top row. Once you have these numbers, move them along the row or down the column till you reach the square where both numbers
satisfy. You will certainly then have your product.
Multiplication charts are helpful learning tools for both kids as well as grownups. Children can utilize them in the house or in institution. Free Printable Times Table Chart are readily available
online and can be printed out and also laminated flooring for toughness. They are a terrific tool to utilize in mathematics or homeschooling, and will supply an aesthetic tip for kids as they learn
their multiplication realities.
Why Do We Use a Multiplication Chart?
A multiplication chart is a representation that shows how to multiply 2 numbers. You choose the very first number in the left column, relocate it down the column, and also then pick the 2nd number
from the top row.
Multiplication charts are handy for many reasons, consisting of helping children learn exactly how to split as well as streamline fractions. They can likewise help children discover exactly how to
pick a reliable common measure. Since they offer as a consistent reminder of the pupil’s progression, multiplication charts can also be handy as workdesk sources. These tools help us establish
independent learners that recognize the standard principles of multiplication.
Multiplication charts are likewise useful for assisting students remember their times tables. They help them find out the numbers by reducing the number of actions needed to finish each operation.
One technique for memorizing these tables is to concentrate on a single row or column each time, and then move onto the next one. At some point, the entire chart will be committed to memory. As with
any type of skill, remembering multiplication tables takes time as well as method.
Free Printable Times Table Chart
Free Printable Times Table Chart
If you’re looking for Free Printable Times Table Chart, you’ve come to the appropriate area. Multiplication charts are offered in different styles, including complete dimension, half size, and also a
selection of charming layouts.
Multiplication charts and also tables are important tools for kids’s education and learning. These charts are terrific for usage in homeschool math binders or as classroom posters.
A Free Printable Times Table Chart is an useful tool to reinforce mathematics facts as well as can assist a youngster learn multiplication promptly. It’s likewise a terrific tool for miss counting as
well as learning the times tables.
Related For Free Printable Times Table Chart
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Advent of Code 2021 Python Solution: Day 4
Again, I am using my helper function from day 1 to read data.
This challenge was little bit harder than previous but with the help of NumPy, I did it.
import numpy as np
data,data1 = get_data(day=4)
def get_blocks(dt):
block = []
num = [int(i) for i in dt[0].split(",")]
row = []
blocks = 0
for d in dt[2:]:
if d == "":
block.append([int(i) for i in d.strip().split(" ") if i!=""])
tdata = np.array(tdata).reshape(blocks,-1, 5)
return tdata, num
def get_first_matched(tdata, num):
results = np.zeros_like(tdata).astype(np.bool)
matched = False
for n in num:
for i,block in enumerate(tdata):
results[i] += block==n
# search across row
if (results[i]==[ True, True, True, True, True]).all(axis=1).any():
print(f"Row Matched Block:{i}")
# search across cols
if (results[i].T==[ True, True, True, True, True]).all(axis=1).any():
print(f"Col Matched Block: {i}")
if matched:
print(f"\nResult Block: {tdata[i]}")
s = (tdata[i]*~results[i]).sum()
print(f"Sum: {s}")
print(f"Last number: {n}")
print(f"Answer: {n*s}\n")
def get_last_matched(tdata, num):
results = np.zeros_like(tdata).astype(np.bool)
matched = False
all_blocks = list(range(0, len(results)))
for n in num:
for i,block in enumerate(tdata):
results[i] += block==n
# search across row
if (results[i]==[ True, True, True, True, True]).all(axis=1).any():
print(f"Row Matched Block:{i}")
if i not in mblocks:
if len(mblocks) == len(all_blocks):
# search across cols
if (results[i].T==[ True, True, True, True, True]).all(axis=1).any():
print(f"Col Matched Block: {i}")
if i not in mblocks:
if len(mblocks) == len(all_blocks):
if matched:
i = mblocks[i]
print(f"\nResult Block: {tdata[i]}")
s = (tdata[i]*~results[i]).sum()
print(f"Sum: {s}")
print(f"Last number: {n}")
print(f"Answer: {n*s}")
d1,n1 = get_blocks(data1)
get_first_matched(tdata=d1, num=n1)
get_last_matched(tdata=d1, num=n1)
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How to find if rectangles are similar - PSAT Math
All PSAT Math Resources
Example Questions
Example Question #1 : How To Find If Rectangles Are Similar
Note: Figure NOT drawn to scale.
In the above figure,
Give the perimeter of
Correct answer:
We can use the Pythagorean Theorem to find
The similarity ratio of
Example Question #3 : Rectangles
Note: Figure NOT drawn to scale.
In the above figure,
Give the area of
Possible Answers:
Insufficient information is given to determine the area.
Correct answer:
Corresponding sidelengths of similar polygons are in proportion, so
We can use the Pythagorean Theorem to find
The area of
Example Question #5 : Rectangles
Note: Figure NOT drawn to scale.
In the above figure,
Give the area of Polygon
Correct answer:
The area of
Now we find the area of
The similarity of
The area of
Now add:
Example Question #1 : How To Find If Rectangles Are Similar
Note: Figure NOT drawn to scale.
Refer to the above figure.
What percent of
Possible Answers:
Insufficient information is given to answer the problem.
Correct answer:
Example Question #311 : Plane Geometry
Note: figure NOT drawn to scale.
Refer to the above figure.
Give the area of
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Mathematics Exam Paper
ICSE Set Set1 Year Icse Maths98 Set1.php Mathematics Exam Paper for students online
Latest for students online. All these are just samples for prepration for exams only. These are not actual papers.
Mathematics - 1998 ( I.C.S.E)
You are on Questions
Maximum Time : Two and Half Hours
General Instructions :
-Answer to this paper must be written on the paper provided separately.
-You will NOT be allowed to write during the first fifteen minutes.
-This time is to be spent in reading the question papers.
-The time given at the head of this paper is the time allowed for writing the answers .
-This Question Paper is divided into two sections.
-Attempt all questions from Section - A and any 4 questions from Section - B.
-The intended marks for questions or for any parts of questions are given in brackets [ ].
-All working, including rough work should be done on the same sheet as the rest of the answer.
-Ommission of essential working will result in loss of marks.
-Mathematical papers are provided.
Section - A
Q1. A man invests Rs. 46,875 at 4% Per annum compound interest for 3 years. Calculate :
(i) The interest for the 1st year
(ii) The amount standing to his credit at the end of the 2nd year.
(iii) The interest for the 3rd year.
Q2. A shopkeeper allowed a discount of 20% on the marked price of an article, and sold it for Rs. 896. Calculate:
(i) His marked price ;
(ii)By selling the article at the discounted price, if he still gains 12% on his cost price, what was the cost price ?
(iii) What would have been his profit %, if he had sold the article at the marked price ?
Q3. On a map drawn to a scale of 1 : 25000, a rectangular plot of land, ABCD, has the following measurements, AB = 12cm & BC= 16cm . Angles A,B,C & D are all 90^o each. Calculate:
(i) The diagonal distance of the plot in km.
(ii) The area of the plot in sq. km .
Q4. Part of a Geometrical figure is given in each of the diagrams below. Complete the figures so that the line 'm' , in each case , is the line of symmetry of the completed figure. Recognizable free
hand sketches would be awarded full marks.
Q5. The wheel of a cart is making 5 revolutions per second. If the diameter of the wheel is 84 cm, Find its speed in km/hr. Give your answer correct to the nearest km.
Q6. Ruler and compasses only may be used in this question. All construction lines and arcs must be clearly shown, and be of sufficient length and clarity to permit assessment.
(i) Construct a triangle ABC, in Which BC= 6cm, AB= 9cm and angle ABC = 60^o ;
(ii) Construct the locus of all points inside triangle ABC, which are equidistant from B and C.
(iii) Construct the locus of the vertices of the triangles with BC as base, Which are equal in area to triangle ABC.
(iv) Mark the point Q, in your construction, which would make QBC equal in area to ABC, and isosceles.
(v) Measure and record the length of CQ.
Q7. A point P (a,b) is reflected in the X-axis to p' (2,-3). Write down the values of a and b. P" is the image of P, when reflected in the Y-axis. Write down the coordinates of P". Find the
coordinates of P'", when P is reflected in the line , parallel to the Y-axis, such that x= 4.
Q8 (a). In the figure given above, AD is the diameter of the circle . If BCD = 130^O calculate:
(i)Ð DAB (ii)Ð ADB.
(b) State the locus of a point in a rhombus ABCD, which is equidistant
(i) From AB and AD ;
(ii) From the vertices A and C.
Q9. (a) Evaluate the following using tables:
[0.284 x Ö (136.78)] / (4.2)^2
(b) Find the value of x and y, if
│ │ 1 │ 2 │ │ │ x │ 0 │ │ │ │ x │ 0 │ │
│[ ├───┼───┤] │[ ├───┼───┤] │] │[ ├───┼───┤] │
│ │ 3 │ 3 │ │ │ 0 │ y │ │ │ │ 9 │ 0 │ │
(c) Solve the following in equation and graph the solution set, on the number line :
2x - 3 < x + 2 < 3x + 5, x Î R
Q 10. (a) If a function in x is defined by f(x) = x /( x^2+1) and x Î R, find :
(i) f (1 /x), x ¹ 0
(ii) f (x-1).
(b) The center O, of a circle has the coordinates (4, 5) and one point on the circumference is (8, 10). Find the coordinates of the other end of the diameter of the circle through this point.
In the figure given above, ABP is a straight line. BD is parallel to PC. Prove that the quadrilateral ABCD is equal in area to triangle APD.
Q 11 (a) Use a graph paper for the question. Draw this graph of 2x - y - 1 = 0, and 2x + y = 9, on the same axes. Use 2 cm = 1 unit on both axes and plot only 3 point per line.
Write down the coordinates of the point of intersection of the two lines.
In the diagram given above, AC is the diameter of the circle, with centre O. CD and BE are parallel. Angle AOB = 80^o and angle ACE = 10^o
calculate :
(i) Angle BEC,
(ii) Angle BCD,
(iii) Angle CED
Q 12. (a) A company with 10,000 shares of Rs. 100/- each declares an annual dividend of 5 %.
(i) What is the total amount of dividend paid by the company ?
(ii) What would be the annual income of a man, who has 72 shares, in the company ?
(iii) If he received only 4 % on his investment, find the price he paid for each share.
(b) Find the equation of a line, which has the y intercept 4, and is parallel to the line 2x - 3y = 7. Find the coordinates of the point. where it cuts the x - axis.
(c) Given below are the weekly wages of 200 workers in a small factory :
Calculate the mean weekly wages of the workers.
│ Weekly wages in Rs. │ No. of workers │
│ 80 - 100 │ 20s │
│ 100 - 120 │ 30 │
│ 120 - 140 │ 20 │
│ 140 - 160 │ 40 │
│ 160 - 180 │ 90 │
Q 13 (a)
The figure drawn above is not to the scale. AB is a tower, and two objects C & D are located on the ground, on the same side of AB. when observed from the top A of the tower, there angle of
depression are 45^0 and 60^0. Find the distance of the two objects. if the height of the tower is 300 mtr. Give your answer to the nearest meter.
(b) The daily profits in rupees of 100 shop in a department store are distributed as follows :
│ Profit per shop (Rs.) │ No. of shops │
│ 0 - 100 │ 12 │
│ 100 - 200 │ 18 │
│ 200 - 300 │ 27 │
│ 300 - 400 │ 20 │
│ 400 - 500 │ 17 │
│ 500 - 600 │ 6 │
Draw a histogram of the data given above, on graph paper & estimate the mode.
Q 14(a) Only and ruler and compasses may be used in this question. All construction lines and arcs must be clearly shown and be of sufficient length & clarity to permit assessment.
(I) Construct a DABC, such that AB = AC = 7 cm.
(II) Construct AD, the perpendicular bisector of BC.
(III) Draw a circle with centre A and radius 3 cm. Let this circle cut AD at P.
(IV) Construct another circle to touch the circle with centre A, externally at P, and pass through B and C.
(b) The distance by road between two towns, A and B, is 216 km, and by rail it is 208 km. A car travels at a speed of x km/hr, and the train travels at a speed which is 16 km/hr faster than the car.
Calculate :
(i) The time taken by the car to reach town B from A, in terms of x;
(ii) The time taken by the train to reach town B from A, in terms of x;
(iii) If the train takes 2 hours less than the car to reach town B, obtain an equation in x, and solve it.
(iv) Hence find the speed of the train.
Q 15. (a) A solid consisting of a right circular cone, standing on a hemisphere. is placed upright, in a right circular cylinder, full of water, and touches the bottom. Find the volume of water left
in the cylinder, having given that the radius of the cylinder is 3 cm. and its height is 6 cm ; the radius of the hemisphere is 2 cm. and the height of the cone is 4 cm. Give your answer to the
nearest cubic centimeter. [Take p = 22/7]
(b) Attempt this question on a graph paper. The table shows the distribution of the daily wages, earned by 160 workers in a building site.
│ Wages in Rs. per day │ No. of workers │
│ 0 - 10 │ 12 │
│ 10 - 20 │ 20 │
│ 20 - 30 │ 30 │
│ 30 - 40 │ 38 │
│ 40 - 50 │ 24 │
│ 50 - 60 │ 16 │
│ 60 - 70 │ 12 │
│ 70 - 80 │ 8 │
Using a scale of 2 cm. to represent 10 Rs., and 2 cm. to represent 20 workers, plot these values, and draw a smooth ogive, through the points. Estimate from the graph -
(i) The Median wage ;
(ii) The upper and lower quartile wage earned by the workers.
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25×25 Sudoku Www Topsimages Printable Sudoku 25×25 | Sudoku Printables
25×25 Sudoku Www Topsimages Printable Sudoku 25×25
25×25 Sudoku Www Topsimages Printable Sudoku 25×25 – If you’ve had any issues with sudoku, you’re aware that there are many different types of puzzles available, and sometimes, it can be difficult to
determine which puzzles to solve. However, there are different ways to solve them, and you’ll find that a printable version of the puzzle can be an excellent method to get started. The guidelines for
solving sudoku are similar to the rules for solving other puzzles, however, the exact format differs slightly.
What Does the Word ‘Sudoku’ Mean?
The term ‘Sudoku’ an abbreviation of the Japanese words suji and dokushin meaning “number” as well as “unmarried’, respectively. The objective of the puzzle is to fill all the boxes with numbers, so
that each number from one to nine appears just once on every horizontal line. The term Sudoku is an emblem of the Japanese puzzle firm Nikoli, which originated in Kyoto.
The name Sudoku originates of the Japanese word”shuji Wa Dokushin Ni Kagiru, which means ‘numbers must be single’. The game consists of nine 3×3 squares, with nine smaller squares in. The game was
originally known as Number Place, Sudoku was an educational puzzle that encouraged mathematical development. While the origins of the game are not known, Sudoku is known to have roots that go back to
the earliest number puzzles.
Why is Sudoku So Addicting?
If you’ve played Sudoku, you’ll know how addictive it can be. An Sudoku addict won’t be able to get rid of thinking about the next challenge they’ll solve. They’re constantly planning their next
puzzle, while other aspects of their life seem to slip to the sidelines. Sudoku is a highly addictive game It’s crucial in order to hold the addictive potential of the game in check. If you’ve fallen
into the habit of Sudoku Here are some strategies to help you curb your addiction.
One of the most popular methods of determining if someone is addicted to Sudoku is to observe your actions. Many people carry books and magazines with them and others browse through social media
updates. Sudoku addicts, however, take newspapers, books, exercise books, as well as smartphones wherever they go. They can be found for hours solving puzzles, and they cannot stop! Some people even
discover it is easier to complete Sudoku puzzles than standard crosswords. They simply can’t stop.
25×25 Sudoku Www Topsimages Printable Sudoku 25×25
What is the Key to Solving a Sudoku Puzzle?
A good strategy for solving a printable sudoku problem is to practice and play with various approaches. The most effective Sudoku puzzle solvers do not follow the same formula for every single
puzzle. The key is to practice and experiment with several different approaches until you come across one that you like. After some time, you’ll be able to solve puzzles without difficulty! But how
do you know how to solve sudoku puzzles that are printable sudoku puzzle?
First, you must understand the fundamental concept behind suduko. It’s a game of reasoning and deduction, and you need to view the puzzle from different angles to spot patterns and then solve it.
When solving a suduko puzzle, you should not attempt to figure out the numbers; instead, you should search the grid to recognize patterns. It is also possible to apply this method to rows and
Related For Sudoku Puzzles Printable
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In a triangle ABC (figure) the points P and Q are selected in the sides AC and BC respectively in a way that PC is half of BC and QC is half of AC:bar(PC)/bar(BC) = 1/2; bar(QC)/bar(AC)= 1/2. Find bar(PQ) if the side bar(Ab) is 20? | Socratic
In a triangle ABC (figure) the points P and Q are selected in the sides AC and BC respectively in a way that PC is half of BC and QC is half of AC:#bar(PC)/bar(BC) = 1/2; bar(QC)/bar(AC)= 1/2#. Find
#bar(PQ)# if the side #bar(Ab)# is 20?
1 Answer
Take $P ' = \frac{1}{2} \left(C + A\right)$ and $Q ' = \frac{1}{2} \left(C + B\right)$ then
$\overline{C P '} = \overline{C Q}$ and $\overline{C Q '} = \overline{C P}$ and $\left\mid \overline{P ' Q '} \right\mid = \left\mid \overline{P Q} \right\mid$
$\overline{P ' Q '}$ is parallel to $\overline{A B}$ and $\frac{\overline{A B}}{\overline{B C}} = \frac{\overline{P ' Q '}}{\overline{C Q '}} = \frac{\overline{P Q}}{\overline{C P}}$
Finally $\overline{P Q} = \frac{\overline{C P}}{\overline{B C}} \times \overline{A B} = \frac{20}{2} = 10$
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### Stats:
• 5 years of publication 2011–2015, 423 papers total
• 4982 references total, average 11.78 per paper
□ 426 reference venues, 72.0% in top 25
• 1066 citations total, average 2.52 per paper
□ 157 citations venues, 78.0% in top 25
□ citation survival rates: 0yr 0.70; 10yr nan; 20yr nan
The half-wheel on the left is a snapshot of the top venues that cites and cited by this venue (details in Figure 4 below). The survival graph on the right contains the fraction of papers cited at
least X years after being published (details in Figure 10 below).
The plots on the rest of this page roughly breaks down into the following types.
• Figure 1 - Figure 3 contains basic stats of papers, references and citations over its lifetime.
• Figure 4 and 5 summarizes the incoming and outgoing citations over all years, sorted by the ratio of incoming over outgoing citations (left to right).
• Figure 5 and 6 breaks down the incoming and outgoing citations by year of the conference paper.
• Figure 7 and 8 breaks down the outgoing references by the year they were cited in this conference. For most conferences we can see a ‘google scholar’ effect that older papers gets cited more (and
from more venues) in recent years, likely due to the ease of finding them.
• Figure 9-11 explores the question “how many papers are still cited at least after X years”, and what fraction of papers are not cited at all.
Note: This page visualizes reference and citation patterns of one conference, generated from templates. For a written overview of the visualization series, and background about problem, motivation
and methods please see the overview page. Also of interest are the overview page for citation flow or the overview page for citation survival rates.
Larger version of any figure can be obtained by clicking the figure.
Fig 1. Overall paper stats. (left) number of papers published in each year; (right) the average number of references made and the average number of citations received for papers published in each
Fig 4. Summary of incoming vs outgoing citations to the top-K venues. Node colors: ratio of citations (outgoing ideas, red) vs references (incoming ideas, blue). Node sizes: amount of total
citation+references in either direction. Thickness of blue edges are scaled by the number of references going to a given venue; thickness of red edges are scaled by the number of citations coming
from a given venue. Nodes are sorted left-to-right by the ratio of incoming vs outgoing citations to this conference.
Fig 5. Incoming vs outgoing citations to the top-K venues (this bar graph presents information from the half-circle differently). (blue) The number of references going to a given conference or
journal; (red) The number of citations coming from a given conference or journal. The x-axis is sorted by the ratio of incoming vs outgoing citations to the conference, highest on the right.
Fig 6. Heatmap of references over time, broken down by publication year (horizontal axis) and reference venue (vertical axis).
Fig 7. Heatmap of citations over time, broken down by publication year (horizontal axis) and citing venue (vertical axis).
Fig 8. Box plot of reference age in years (y-axis, lower is older), broken down by the year paper in this venue is published (x-axis).
Fig 9. Heatmap of references, broken down by the year paper in this venue is published (horizontal axis) and by the publication year of the reference (vertical axis).
Fig 10. Fraction of papers that are cited at least once more than X years after they are published, with a linear regression overlay.
Fig 11. Heatmap of the number of papers cited in each year, broken down by year published (horizontal) and year cited (vertical).
Fig 12. Heatmap of the faction of papers in a given publication years still cited in each subseuqent year, broken down by year published (horizontal) and year cited (vertical).
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The equations of polyconvex thermoelasticity
I will consider the system of thermoelasticity endowed with polyconvex energy. In the Introduction, after presenting the equations in their mathematical and physical context, I explain the relevant
research in the area and the contributions of this work. In Chapter 2 I embed the equations of polyconvex thermoviscoelasticity into an augmented, symmetrizable, hyperbolic system which possesses a
convex entropy. Using the relative entropy method in the extended variables, in Chapter 4, I show convergence from thermoviscoelasticity with Newtonian viscosity and Fourier heat conduction to smooth
solutions of the system of adiabatic thermoelasticity as both parameters tend to zero and convergence from thermoviscoelasticity to smooth solutions of thermoelasticity in the zero-viscosity limit.
In addition, I establish a weak-strong uniqueness result for the equations of adiabatic thermoelasticity in the class of entropy weak solutions. In Chapter 5, I prove a measure-valued versus strong
uniqueness result for adiabatic polyconvex thermoelasticity in a suitable class of measure-valued solutions, defined by means of generalized Young measures that describe both oscillatory and
concentration effects. Instead of working directly with the extended variables, I will look at the parent system in the original variables utilizing the weak stability properties of certain
transport-stretching identities, which allow to carry out the calculations by placing minimal regularity assumptions in the energy framework. In Chapter 6, I construct a variational scheme for
isentropic processes of adiabatic polyconvex thermoelasticity. I establish existence of minimizers which converge to a measure-valued solution that dissipates the total energy. Also, I prove that the
scheme converges when the limiting solution is smooth. Finally for completeness, in Chapter 3, I present the well-established theory for local existence of classical solutions and how it applies to
the equations at hand.
Brief Biography
Myrto Galanopoulou is a Ph.D. candidate working in the group of Athanasios Tzavaras. with research interests in the areas of Mathematical Analysis and Applications, Partial Differential Equations,
and Conservation laws. During her Ph.D. studies, she examined the system of thermoelasticity with polyconvex energy and produced many novel results on stability and uniqueness of solutions.
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Quantum Computing - Theory and Practice: Quantum algorithms, quantum noise and quantum operations
Quantum Computing - Theory and Practice: Quantum algorithms, quantum noise and quantum operations
Add to your list(s) Download to your calendar using vCal
• Thursday 03 May 2018, 14:00-16:30
• CMS, MR3.
If you have a question about this talk, please contact Rachel Furner.
This is part of the short course ‘Quantum Computing – Theory and Practice’, http://talks.cam.ac.uk/show/index/86491
This course covers fundamental theoretical concepts of quantum computation and quantum information will be covered. In addition, hands-on experimentation of quantum algorithms will be demonstrated on
actual quantum devices. Special consideration will be given to realization of limitations of current, non-fault tolerant quantum systems, as well as means to mitigate them when possible.
Specifically, lecture 3 covers: a. Quantum algorithms: Deutsch, Deutsch Josza, Simon’s, quantum Fourier transform, quantum phase estimation, Grover’s search algorithm b. Quantum noise and quantum
operations: gate fidelities, amplitude leak, phase decoherence, algorithmic design considerations (Variational Quantum Eigensolver example)
There will be a 15 minute break in the middle of the lecture.
It is useful for students to have a laptop/tablet (or even a smartphone) for some of the more practical examples, but this is not necessary. Those without computer access can follow a demo shown by
the instructor.
This talk is part of the CCIMI short course: Quantum Computing - Theory and Practice series.
This talk is included in these lists:
Note that ex-directory lists are not shown.
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Place Value Sections Method Division Worksheet - Divisonworksheets.com
Place Value Sections Method Division Worksheet
Place Value Sections Method Division Worksheet – Divide worksheets can be used for helping your child learn and learn division. There are many kinds of worksheets and it is possible to create your
own. These worksheets are fantastic since they are available for download at no cost and modify them as you like you want them to be. They’re great for kindergarteners, first graders and even second
Two individuals can accomplish huge amounts of work
Work on worksheets that have large numbers. There are often only two, three or four divisors on the worksheets. This method doesn’t require the child to be concerned about forgetting to divide large
numbers or making mistakes when using times tables. You can find worksheets online or download them on your computer to aid your youngster in developing this ability in math.
Multi-digit division worksheets are used by children to practice and improve their understanding. It’s a crucial mathematical ability which is needed for a variety of operations in everyday life and
more complex mathematical concepts. These worksheets offer interactive questions and activities to reinforce the concept.
It is not easy for students to divide large numbers. These worksheets utilize a standard algorithm as well as step-by–step instructions. Students may not get the understanding they seek from these
worksheets. Using base ten blocks to demonstrate the process is one method to teach long division. Students should feel at ease with the concept of long division once they’ve learned the process.
Large numbers that are divided can be practiced by pupils using numerous worksheets and exercises. These worksheets include fractional calculations with decimals. If you need to divide large amounts
of money, worksheets for hundredths are available.
Sort the numbers in to smaller groups.
It could be difficult to break a crowd into small groups. Although it sounds wonderful on paper but many facilitators of small groups are not keen on this approach. It is an accurate reflection of
the evolution of the human body, and it can aid in the Kingdom’s constant growth. It also motivates other people and inspires new leaders to step up to the helm.
This is a fantastic way to brainstorm ideas. You can create groups of people with similar characteristics and experience levels. You may come up with creative ideas by doing this. After you’ve set up
your groups, it’s now time to introduce yourself and each other. This is a fantastic exercise that stimulates creativity and new thinking.
It’s used to break down massive numbers into smaller units. It can be useful for when you want to make equal quantities of goods for various groups. A good example is a class of large pupils that
could be broken into five groups. This gives you the original 30 pupils.
Be aware that when you divide numbers there is a divisor and a quote. The result of multiplying two numbers is “ten/five,” but the same results are obtained when you divide them both.
It is a good idea to use the power of ten to calculate huge numbers.
You can break down huge numbers into powers 10 to help us to compare these numbers. Decimals are a common element of shopping. They can be found on receipts, price tags and food labels. To display
the price per gallon and the amount of gas that has been dispensed via a nozzle, petrol pumps use decimals.
You can split huge numbers into powers of ten in two ways: shift the decimal point to the left or multiply it by 10-1. Another option is to make use of the powers of 10’s associative feature. Once
you’ve learned to utilize the power of ten associative feature, you can divide massive numbers into smaller ones.
The first one is based on mental computation. A pattern can be observed if 2.5 is divided by the power 10. As the power of ten grows, the decimal position shifts towards the left. Once you understand
this concept, you can use it to tackle any problem even the most difficult ones.
The other is to mentally divide very large numbers into powers equal to ten. Next, you need to quickly write large numbers on a scientific note. When using scientific notation, large numbers must be
written with positive exponents. To illustrate, if we move the decimal points five spaces to your left, you can convert 450,000 into 4.5. To split a large number into smaller powers 10, you could
make use of exponent 5, or divide it in smaller powers 10, so that it’s 4.5.
Gallery of Place Value Sections Method Division Worksheet
Division With Place Value Sections 48 3 Math Elementary Math Math
ShowMe Place Value Sections Method Division
Division W 1 digit Divisors Using The Place Value Sections Method AKA
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How To Determine How Much Your Finals Affect Your Grade
Going into finals can be a stressful thing. However, you can perform calculations to determine how a final may affect your grade. This can be done using three scenarios: One, you get a zero on the
final; two, you get a 100; and three is a guess as to what you think you will get. Doing this gives you a range of what your final grade may be.
Consider using a free final grade calculator.
1. Calculate total points thus far
Calculate the total amount of points you have in the class before the final and the total number of possible points available. For example, assume you have a 90 out of a 100, a 40 out of 50 and a 65
out of 75 going into the finals. Total points is 90 plus 40 plus 65, which equals 195 points. Total available points is 225.
2. Find Final Points
Find out how many points you final is worth and make a conservative guess of your test grade. In the example, assume the test is worth 200 points and you think you will get 165 points.
3. Add points
Add total available points to the points the final is worth. In the example, 225 plus 200 equals 425 points.
4. Then Divide
Divide your total points by the points available after the final. In the example, 195 points divided by 425 points equals 45.8 percent final grade if you get a zero on your test.
5. Predict your score
Add your guess at a grade to your total points. Then, divide the result by the points after the final. In the example, 165 points plus 195 points equals 360 points. Then, 360 points divided by 425
points equals 84.7 percent. This is your grade with your guess on your finals grade.
6. Find Score
Add a the total points the final is worth to your total points. Then, divide the result by the points after the final. In the example, 195 points plus 200 points equals 395 points. Then, 395 points
divided by 425 points equals 92.9 percent. This is your grade if you get a perfect score on the final.
TL;DR (Too Long; Didn't Read)
The same calculations can be done based on weight. Just use the weight of the grade on your final as total points possible.
Cite This Article
McBride, Carter. "How To Determine How Much Your Finals Affect Your Grade" sciencing.com, https://www.sciencing.com/determine-much-finals-affect-grade-8051271/. 12 March 2011.
McBride, Carter. (2011, March 12). How To Determine How Much Your Finals Affect Your Grade. sciencing.com. Retrieved from https://www.sciencing.com/determine-much-finals-affect-grade-8051271/
McBride, Carter. How To Determine How Much Your Finals Affect Your Grade last modified April 1, 2022. https://www.sciencing.com/determine-much-finals-affect-grade-8051271/
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Learning long-term filter banks for audio source separation and audio scene classification, EURASIP Journal on Audio, Speech, and Music Processing | DeepDyve
Filter banks on short-time Fourier transform (STFT) spectrogram have long been studied to analyze and process audios. The frameshift in STFT procedure determines the temporal resolution. However, in
many discriminative audio applications, long-term time and frequency correlations are needed. The authors in this work use Toeplitz matrix motivated filter banks to extract long-term time and
frequency information. This paper investigates the mechanism of long-term filter banks and the corresponding spectrogram reconstruction method. The time duration and shape of the filter banks are
well designed and learned using neural networks. We test our approach on different tasks. The spectrogram reconstruction error in audio source separation task is reduced by relatively 6.7% and the
classification error in audio scene classification task is reduced by relatively 6.5%, when compared with the traditional frequency filter banks. The experiments also show that the time duration of
long-term filter banks in classification task is much larger than in reconstruction task. Keywords: Long-term filter banks, Deep neural network, Audio scene classification, Audio source separation 1
Introduction makes it very different from natural images. For example Audios in a realistic environment are typically composed in Fig. 1,(a)and (b) are two audio fragments randomly of different sound
sources. Yet humans have no problem selected from an audio of “cafe” scene. We first calculate in organizing the elements into their sources to recognize the average energy distribution of the two
examples in the acoustic environment. This process is called auditory the frequency direction, which is shown in (c). And then scene analysis [1]. Studies in the central auditory sys- the temporal
coherence of salient audio elements in each tem [2–4] have inspired numerous hypotheses and models frequency bin is measured as (d). It is obvious that the concerning the separation of audio
elements. One promi- energy distribution and temporal coherence vary tremen- nent hypothesis that underlies most investigations is that dously in different frequency bins, but are similar in the
audio elements are segregated whenever they activate same frequency bin of different spectrograms. Thus for well-separated populations of auditory neurons that are audio signals, the spectrogram
structure is not equivalent selective to frequency [5, 6], which emphasizes the audio in time and frequency direction. In this paper, we propose distinction on the frequency dimension. At the same
time, a novel network structure to learn the energy distribution other studies [7, 8] also suggest that auditory scenes are and temporal coherence in different frequency bins. essentially dynamic,
containing many fast-changing, rela- tively brief acoustic events. Therefore an essential aspect 1.1 Related work of auditory scene analysis is the linking over time [9]. For audio separation [10,
11] and recognition [12, 13] Problems inherent to auditory scene analysis are sim- tasks, the time and frequency analysis is usually imple- ilar to those found in visual scene analysis. However,
mented using well designed filter banks. the time and frequency characteristic of a spectrogram Filter banks are traditionally composed of finite or infi- nite response filters in principle [14], but
the stability *Correspondence: [email protected] of the filters is usually difficult to be guaranteed. For Department of Electronic Engineering, Tsinghua University, Beijing, China simplicity, filter
banks on STFT spectrogram have been © The Author(s). 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://
creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons license, and indicate if changes were made. Zhang and Wu EURASIP Journal on Audio, Speech, and Music Processing (2018) 2018:4 Page 2 of 13 Fig. 1 Spectrogram
examples of “cafe” scene. a, b Two audio fragments randomly selected from “cafe” scene. c The average energy distribution of the two examples in frequency direction. d The temporal coherence of the
two examples in different frequency bins investigated for a long time [15]. In this case, the time 1.2 Contribution of this paper resolution is determined by the frameshift in the STFT As shown in
Fig. 1, when perceptual frequency scale procedure and the frequency resolution is modelled by is utilized to map the linear frequency domain to the the frequency response of the filter banks.
Frequency fil- nonlinear perceptual frequency domain [27], the major ter banks can be parameterized in the frequency domain concern comes to be how to model the energy distri- with filter centre,
bandwidth, gain and shapes [16]. If these bution and temporal coherence in different frequency parameters are learnable, deep neural networks (DNNs) bins. can be utilized to learn them
discriminatively [17–19]. To obtain better time and frequency analysis results, These frequency filter banks are usually used to model the we divide the audio processing procedure into two stages.
frequency selectivity of the auditory system, but cannot In the first stage, traditional frequency filter banks are represent the temporal coherence of audio elements. implemented on STFT spectrogram
to extract frequency DNNs are often used as classifiers when the inputs features. Without loss of generality, the parameters of the are dynamic acoustic features such as filter bank-based frequency
filter banks are set experimentally. In the sec- cepstral features and Mel-frequency cepstral coefficients ond stage, a novel long-term filter bank spanning several [20, 21]. When the input to DNNs
is a magnitude spec- frames is constructed in each frequency bin. The long- trogram, time-frequency structure of the spectrogram term filter banks proposed here can be implemented by can be learned.
Neural networks organized into a two- neural networks and trained jointly with the target of the dimensional space have been proposed to model the time specific task. and frequency organization of
audio elements by Wang The major contributions are summarized as follows: and Chang [22]. They utilized two-dimensional Gaussian lateral connectivity and global inhibition to parameter- - Toeplitz
matrix motivated long-term filter banks: ize the network, where the two dimensions correspond Unlike filter banks in frequency domain, our proposal to frequency and time respectively. In this model,
time is of long-term filter banks spreads over the time converted into a spatial dimension, temporal coherence dimension. They can be parameterized with the time can take place in auditory
organization much like in visual duration and shape constraints. For each frequency organization where an object is naturally represented in bin, the time duration is different, but for each frame,
spatial dimensions. However, these two dimensions are the filter shape is constant. This mechanism can be not equivalent in a spectrogram according to our analysis. implemented using a Toeplitz
matrix motivated And what is more, the parameters of the network are set network. empirically and not learnable, which is still significantly - Spectrogram reconstruction from filter bank dependent
on domain knowledge and modelling skill. coefficients: Consistent with the audio processing In recent years, neural networks with special structures procedure, we also divide the reconstruction such
as convolutional neural network (CNN) [23, 24]and procedure into two stages. The first stage is a dual long short-term memory (LSTM) [25, 26]havebeenused inverse process of the long-term filter banks
and the to extract the long-term information of audios. But in both second stage is a dual inverse process of the network structures, the temporal coherence is considered frequency filter banks. This
paper investigates the to be the same in different frequency bins, which is in spectrogram reconstruction problem using an contradiction with Fig. 1. elaborate neural network. Zhang and Wu EURASIP
Journal on Audio, Speech, and Music Processing (2018) 2018:4 Page 3 of 13 This paper is organized as follows. The next section When the number of frequency filters is equal to m,the describes the
detailed mechanism of the long-term fil- long-term filter banks can be parameterized by m linear ter banks and the spectrogram reconstruction method. transformations. The parameters will be labelled
as θ and Then network structures used in our proposed method discussed in the following part of this section in detail. are introduced in Section 3.Section 4 conducts several The back-end processing
modules vary from different experiments to show the performance of long-term filter applications. For audio scene classification task, they will banks regarding source separation and audio scene
classi- be deep convolutional neural networks followed by a fication. Finally, we conclude our paper and give directions softmax layer to convert the feature maps to the corre- for future work in
Section 5. sponding categories. However, for audio source separation task, the modules will be composed by a binary gating 2 Long-term filter banks layer and some spectrogram reconstruction layers.
We For generality, we consider in this section a long-term fil- define them as nonlinear functions f .The long-termfilter ter bank learning framework based on neural networks as bank parameters θ can
be trained jointly with the back- Fig. 2. end parameters γ using back propagation method [33]in The input audio signal is first transformed to a sequence neural networks. of vectors using STFT [28];
the STFT result can be repre- sented as X ={x , x , ..., x }. T is determined by the 2.1 Toeplitz motivation 1...T 1 2 T frame shift in STFT, the dimension of each vector x can be The long-term
filter banks in our proposed method are labelled as N, which is determined by the frame length. used to extract the energy distribution and temporal The frequency filter banks can be simplified as a
lin- coherence in different frequency bins which have been T T T discussed in Section 1. As shown in Fig. 4,the long-term ear transformation y = f x , f x , ..., f x ,where f t t t t 1 2 m k filter
banks can be implemented by a series of filters with is the weights of the k-th frequency filter. In the his- different time durations. If the output of the frequency tory of auditory frequency
filter banks [29], the rounded filter banks is y , and the long-term filter banks are param- exponential family [30] and the gammatone family [31] eterized as W = {w , w , ..., w }, the operation of
the 1 2 m are the most widely used families. We use the sim- long-term filter banks can be mathematically represented plest form of these two families, triangular shape for the as Eq. 1. T is the
length of the STFT output, m is the rounded exponential family and Gaussian shape for the dimension of y , which also represents the number of fre- gammatone family. For triangular filter banks, the
band- quency bins, w is a set of T positive weights to represent width is 50% overlapped between neighbouring filters. For the time duration and shape of the k-th filter. In Fig. 4 Gaussian filter
banks, the bandwidth is 4σ,where σ rep- for example, w is a rectangular window with individual resents the standard deviation in the Gaussian function. width, each row of the spectrogram is convolved
by the These two types of frequency filter banks are the base- corresponding filter. lines in this paper, respectively named TriFB and GaussFB. The triangular and gaussian examples distributed uni-
formly in the Mel-frequency scale [32] can be seen z = y ∗ w ,1 ≤ k ≤ m (1) t,k i,k k,i−t in Fig. 3. i=1 Fig. 2 Long-term filter banks learning framework. The left part of the framework is the
feature analysis procedure including STFT, frequency filter banks and long-term filter banks. The right part is the application examples of the extracted feature map, such as audio scene
classification and audio source separation. Long-term filter banks in the feature analysis procedure and the back-end application modules are stacked into a deep neural network Zhang and Wu EURASIP
Journal on Audio, Speech, and Music Processing (2018) 2018:4 Page 4 of 13 Fig. 3 Shape of frequency filter banks. a The triangular filter banks. b The gaussian filter banks As a matter of fact, the
operation in Eq. 1 is a series of frequency bin is T. This assumption is unreasonable espe- one-dimensional convolutions along time axis. We rewrite cially when T is extremely large. The long-term
correla- it using the Toeplitz matrix [34] for simplicity. In Eq. 2,the tion should be limited to a certain range according to our tensor S ={S , S , ..., S } represents the linear transfor-
intuition. Inspired by traditional frequency filter banks, 1 2 m mation form of long-term filter banks in each frequency we attempt to use the parameterized window shape to bin. z in Eq. 2 is
equivalent to {z , z , ..., z } in Eq. 1. limit the time duration of long-term filter banks. k 1,k 2,k T,k In this case, long-term filter banks can be represented In Fig. 4, rectangular shapes with
time durations of 3, as a simple form of tensor operation, which can be eas- 2, 1 and 2 frames are utilized as an interpretation. From ily implemented by a Toeplitz motivated network layer. the
theory of frequency filter banks, triangular and gaus- According to [35], Toeplitz networks are mathematically sian shapes are also commonly used options. However, tractable and can be easily
computed. rectangular and triangular shapes are not differentiable and unable to be incorporated into a scheme of a back- propagation algorithm. Thus in this paper, the shape of z = y ˆ S ,1 ≤ k ≤ m
k k long-term filter banks is constrained using the Gaussian y ˆ = y , y , ..., y k 1,k 2,k T,k function as Eq. 3. The time duration of long-term filter ⎡ ⎤ w w ··· w banksislimited by σ , the
strength of each frequency bin k,0 k,−1 k,1−T k ⎢ ⎥ w w ··· w is reconstructed by α , the total number of parameters k,1 k,0 k,2−T k ⎢ ⎥ S = (2) ⎢ ⎥ k . . . . reduces from 2mT in Eq. 2 to 2m in Eq.
3. . . . . ⎣ ⎦ . . . w w ··· w k,T −1 k,T −2 k,0 t w = α · exp − ,1 ≤ k ≤ m (3) k,t k 2.2 Shape constraint If W is totally independent, S is a dense Toeplitz matrix, When we initialize the parameters
α and σ randomly, k k which means that the time duration of the filter in each we believe that the learning will be well behaved, which Fig. 4 Model architecture of long-term filter banks. Each row
of the spectrogram is convolved by a filter bank with individual width. In this sketch map, time durations of the filter banks in the highest four frequency bins are 3, 2, 1 and 2 frames Zhang and Wu
EURASIP Journal on Audio, Speech, and Music Processing (2018) 2018:4 Page 5 of 13 −1 is the so-called “no bad local minim” hypothesis [36]. y = z S ,1 ≤ k ≤ m (5) However, a different view presented
in [37] is that the However, considering that S is a Toeplitz matrix, R underlying easiness of optimizing deep networks is rather k k canberepresented in asimpleway [40]. R is given by tightly
connected to the intrinsic characteristics of the k ˆ ˆ Eq. 6,where A , B , A and B all are lower triangular data these models are run on. Thus for us, the initializa- k k k k Toeplitz matrices given
by Eq. 7. tion of parameters is a tricky problem, especially when α and σ have clear physical meanings. 1 T If σ in Eq. 3 is initialized with a value larger than ˆ ˆ R = A B − B A (6) k k k k 1.0,
the corresponding S is approximately equal to a k- tridiagonal Toeplitz matrix [38], where k is less than 3. ⎛ ⎞ Thus, if the totally independent W is initialized with an ⎛ ⎞ 0 ··· 00 a 0 ··· 0
identity matrix, similar results with limited time dura- ⎜ ⎟ a a ··· 0 a ··· 00 ⎜ ⎟ 2 1 n ⎜ ⎟ tions should be obtained. Whether it is the Gaussian ⎜ ⎟ A = . . . , A = ⎜ ⎟ k . k . . . ⎝ . . . . ⎠ . .
. . . ⎝ ⎠ shape-constrained algorithm as Eq. 3 or is the totally . . . . . . . a a ··· a n n−1 1 independent W in Eq. 2, the initialization of parameters a ··· a 0 2 n ⎛ ⎞ is important and
intractable when adapting to differ- ⎛ ⎞ 0 ··· 00 b 0 ··· 0 ent tasks. More details will be discussed and tested in ⎜ ⎟ b b ··· 0 b ··· 00 ⎜ n−1 n ⎟ 1 ⎜ ⎟ Section 4. ⎜ ⎟ ˆ B = , B = . . . ⎜ ⎟ k . k .
. . ⎝ ⎠ . . . . . . . . ⎝ ⎠ . . . . . . . b b ··· b 1 2 n 2.3 Spectrogram reconstruction b ··· b 0 n−1 1 In our proposal of learning framework as Fig. 2,STFT (7) spectrogram is transformed into
subband coefficients Note that a and b canalsoberegardedasthe solutions after frequency filter banks and long-term filter banks. of two linear systems, which can be learned using a fully The
dimension of subband coefficients z is usually much connected neural work layer. In this case, the number of less than x to reduce computational cost and extract sig- parameters reduces from mT to
2mT. nificant features. In this case, the subband coefficients In conclusion, the spectrogram reconstruction proce- are incomplete, perfect spectrogram reconstruction from dure can be implemented
using a two-layer neural net- subband coefficients is impossible. work. When the first layer is implemented as Eq. 5,the The spectrogram vector x is firstly transformed using total number of
parameters is mN + mT . While when frequency filter banks described at the beginning of this section. Then long-term filter banks work as Eq. 2 to get the first layer is represented as Eq. 6,the
totalnumber thesubband coefficients.Thusthe processofthe con- is mN + 2mT. Experiments in Section 4.1 will show the version from spectrogram vector to filter subband coef- difference between these
two methods. ficients and the dual reconversion can be represented as Eq. 4. The operation of frequency filter banks f can be 3 Training the models As described in Section 2, the long-term filter
banks we simplified as a singular matrix F where the number of proposed here can be integrated into a neural network rows is much less than columns. The reconversion process −1 (NN) structure. The
parameters of the models are learned f is approximately the Moore-Penrose pseudoinverse jointly with the target of the specific task. In this section, [39]of F; this module can be easily implemented
using a we introduce two NN-based structures respectively for fully connected network layer. However, the tensor opera- audio source separation and audio scene classification tion of long-term filter
banks f is much more intractable. tasks. z = f (f (x )) t 2 1 t 3.1 Audio source separation −1 −1 x = f f (z ) (4) t t 1 2 In Fig. 5a, the procedures of STFT and frequency fil- ter banks in Fig. 2
are excluded from the NN structure because they are implemented empirically and have no Without regard to the special structure of Toeplitz −1 parameters. The NN structure for audio source separation
matrix, f can be mathematically represented as Eq. 5. task is divided into four steps, in which three steps have S is a nonsingular matrix which has been defined in −1 been discussed in Section 2.
The layers of long-term filter Eq. 2. In general, S is another nonsingular matrix R banks and inverse of long-term filter banks are imple- which can be learned using a fully connected network mented
respectively as Eqs. 2 and 5, which can be denoted layer independently. There are m frequency bins in total, as h and h . The reconstruction layer is constructed using so m parallel fully connected
network layers are needed 1 2 a fully connected layer and can be denoted as h . and the number of parameters is mT . 4 Zhang and Wu EURASIP Journal on Audio, Speech, and Music Processing (2018)
2018:4 Page 6 of 13 Fig. 5 NN-based structures with proposed method. a The NN structure for audio source separation task. b The NN structure for audio scene classification task We attempt the audio
separation from an audio mixture 3.2 Audio scene classification using a simple masking method [41], which can be repre- In early pattern recognition studies [42], the input is first sented as the
binary gating layer in Eq. 8 and denoted as converted into some features, which are usually defined h . The output of this layer is a linear projection modu- empirically by experts and believed to be
identified with lated by the gates g . These gates multiply each element of the recognition targets. In Fig. 5b, a feature extraction the matrix Z and control the information passed on in the
structure including the long-term filter banks is proposed hierarchy. Stacking these four layers on the top of input Y to systematically train the overall recognizer in a manner gives a
representation of the separated clean spectrogram consistent with the minimization of recognition errors. X = h ◦ h ◦ h ◦ h (Y ). The NN structure for audio scene classification task can 4 3 2 1 also
be divided into four steps, where the first layer of N long-term filter banks is implemented using Eq. 2.The g = sigmoid z v ti tj ji j=1 (8) convolutional layer and the pooling layer are conducted o
= z g ti ti ti using the network structure described in [43]. In general, let z refer to the concatenation of frames after long- i:i+j Neural networks are trained on a frame error (FE) min- term
filter banks z , z , ...z . The convolution operation i i+1 i+j hm imization criterion and the corresponding weights are involves a filter w ∈ R , which is applied to a window of adjusted to minimize
the error squares over the whole h frames to produce a new feature. For example, a feature training data set. The error of the mapping is given c is generated from a window of frames z by Eq. 10,
i:i+h−1 by Eq. 9,where x is the targeted clean spectrogram where b ∈ R is a bias term and f is a non-linear function. and x ˆ is the corresponding separated representation. This filter is applied to
each possible window of frames to As commonly used, L2-regularization is typically chosen produce a feature map c =[ c , c , ...c ]. Then a max- 1 2 T −h+1 to impose a penalty on the complexity of
the mapping, overtime pooling operation [44] over the feature map is which is the λ term in Eq. 9. However, when the layer applied and the maximum value c ˆ = max(c) is taken as of long-term filter
banks is implemented by Eq. 3,the the feature corresponding to this filter. Thus one feature is elements of w have definitude physical meanings. Thus, extracted using one filter. This model uses
multiple filters L2-regularization is operated only on the upper three lay- with varying window sizes to obtain multiple features. ers in this model. In this case, the network in Fig. 5a can be
optimized by the back-propagation method. c = f (w · z + b) (10) i i:i+h−1 T 4 The features extracted from the convolutional and pool- 2 2 ˆ ing layers are then passed to a fully connected soft- = x
− x +λ w (9) t t l t=1 max layer to output the probability distribution over l=2 Zhang and Wu EURASIP Journal on Audio, Speech, and Music Processing (2018) 2018:4 Page 7 of 13 categories. The
classification loss of this model is given by the MIR-1K dataset [46]. The dataset consists of 1000 Eq. 11,where n is the number of audios, k is the num- song clips recorded at a sample rate of
16kHz, with berofcategories, y is the category labels and p is the durations ranging from 4 to 13 s. The dataset is then probability distribution produced by the NN structure. In utilized with 4
training/testing splits. In each split, 700 this case, the network in Fig. 5b can be optimized by the of the examples are randomly selected for training and back-propagation method. the others for
testing. We use the mean average accu- racy over the 4 splits as the evaluation criterion. In order n k 4 to achieve a fair comparison, we use this dataset to cre- = y · log(p ) + λ w (11) i,j i,j l
ate3sets of mixtures.For each clip,wemix thevocal i=1 j=1 l=2 and music track under various conditions, where the energy ratio between music and voice takes 0.1, 1 and 10 4 Experiments respectively.
To illustrate the properties and performance of long-term We first test our methods on the outputs of frequency filter banks proposed in this paper, we conduct two groups filter banks. In this case,
the combination of classical of experiments respectively on audio source separation frequency filter banks and our proposed temporal filter and audio scene classification. To achieve a fair compar-
banks work as two-dimensional filter banks on mag- ison with traditional frequency filter banks, all experi- nitude spectrograms. Classical CNN models can learn ments conducted in this section
utilize the same settings two-dimensional filters on spectrograms directly. Thus we and structures except for the items listed below. introduce a 1-layer CNN model as a comparison. The CNN model is
implemented as [22], but the convolutional - Models: The models tested in this section are layer here is composed of learnable parameters, instead of different from each other in two aspects. The
variants constant Gaussian lateral connectivity in [22]. This con- of frequency filter banks include TriFB and GaussFB, volution layer works as a two-dimensional filter whose as described in Section
2. For long-term filter banks, size is set to be 5 × 5, the outputs of this layer is then Gaussian shape-constrained filters introduced in processed as Fig. 5a.Weuse theNNmodel in [47]and Section 2.2
are named GaussLTFB and totally the one-layer CNN model as our baseline models. For independent filters are named FullLTFB. The our proposed long-term filter banks, we test two vari- baseline of our
experiments has no long-term filter ant modules: GaussLTFB and FullLTFB which have been banks, which is labelled as Null. The initials of the defined at the beginning of Section 4. For FullLTFB situ-
names are used to differentiate models. For example, ation, two initialization methods discussed in Section 2.2 when TriFB and FullLTFB are used in the model, the are tested respectively. The three
variant modules Gaus- model is named TriFB-FullLTFB. sLTFB, FullLTFB-Random and FullLTFB-Identity can be - Initialization: When we use totally independent utilized on two types of frequency filter
banks TriFB and filters as the long-term filter banks, two initialization GaussFB respectively, thus a total of six long-term filter methods discussed in Section 2.2 are tested in this banks related
experiments are conducted in this part. section. When the parameters are initialized Table 1 shows the results of these experiments. From randomly, the method is named Random, while when the results,
we can get conclusions as follows. First, the the parameters are initialized using an identity best results in the table are obtained using long-term filter matrix, the method is named Identity.
banks, which demonstrates the effectiveness of our pro- - Reconstruction: When the spectrogram posal, especially when the energy of interference is larger reconstruction is implemented as Eq. 5, the
method than music. As an example, when we use gaussian fre- is named Re_inv, while when the reconstruction is quency filter banks and the energy ratio between music implemented as Eq. 6, the method
is named Re_toep. and voice is 1, the reconstruction error is reduced by In all experiments, the audio signal is first transformed relatively 6.7% by using Gaussian shape-constrained long- using
short-time Fourier transform with a frame length of term filter banks. Second, totally independent filters are 1024 and a frameshift of 220. The number of frequency severely influenced by the
initialization. When the param- filters is set to be 64; the detailed settings of NN structures eters are initialized using an identity matrix, the perfor- are shown in Fig. 5. All parameters in the
neural network mance is close to the Gaussian shape-constrained filters are trained jointly using Adam [45] optimizer; the learning in this task. However, when the parameters are initialized rate is
initialized with 0.001. randomly, the reconstruction error seems to be unable to converge effectively. This result has to do with the task itself, which will be further tested in Section 4.3.Then,
4.1 Audio source separation the one-layer CNN model improves the performance only In this experiment, we investigate the application of long- when the energy ratio between music and voice is 0.1,
term filter banks in audio source separation task using Zhang and Wu EURASIP Journal on Audio, Speech, and Music Processing (2018) 2018:4 Page 8 of 13 Table 1 Reconstruction error of audio source
separation using frequency filter banks as input Re_toep Re_inv Init Method M/V = 0.1 M/V =1M/V = 10 M/V = 0.1 M/V =1M/V = 10 – TriFB-Null 3.49 1.51 0.55 3.49 1.51 0.55 – GaussFB-Null 3.28 1.47 0.58
3.28 1.47 0.58 – TriFB-CNN-1layer 2.85 1.51 0.61 2.85 1.51 0.61 – GaussFB-CNN-1layer 2.91 1.50 0.64 2.91 1.50 0.64 – TriFB-GaussLTFB 2.66 1.38 0.50 3.65 1.80 0.74 – GaussFB-GaussLTFB 2.60 1.39 0.56
3.91 1.67 0.67 Random TriFB-FullLTFB 3.90 41.37 2.28 3.84 1.83 0.78 Random GaussFB-FullLTFB 3.55 1.99 0.86 3.85 1.64 0.66 Identity TriFB-FullLTFB 2.69 1.39 0.52 3.92 1.63 0.62 Identity
GaussFB-FullLTFB 2.62 1.39 0.56 3.85 1.51 0.59 M/V represents the energy ratio between music and voice this can be attributed to the local sensitivity of recon- by relatively 5.0% by using Gaussian
shape-constrained struction task. As a matter of fact, the time durations of long-term filter banks, this effect is less obvious than the long-term filter banks in most frequency bins we learned
result in Table 1. This is because that the information of here are1.Thus, theconvolutionsize5 × 5istoo large. magnitude spectrograms is too rich, so the performance of Finally, Toeplitz inversion
motivated reconstruction algo- the simplest NN model is also good. But when the energy rithm performs much better than the direct inverse matrix of interference is larger than music, the
effectiveness of algorithm. When the direct inverse matrix algorithm is our long-term filter banks is obvious. utilized, the performance of our proposal of long-term fil- A direct perspective of the
separation results can be ter banks becomes even worse than the frequency filter seen in Fig. 6. The figure shows the clean music spec- banks. trogram (a), mixed spectrogram (b)and theseparated We
now test our methods on magnitude spectrograms spectrogram (c–e) when the energy ratio is 1. For this as described in [47]. In this situation, long-term filter example, (c) is the separated
spectrogram from GaussFB- banks are used as one-dimensional filter banks to extract Null which has been defined at the beginning of this temporal information. The size of magnitude spectro- section,
(d) is the separated spectrogram from GaussFB- grams is 513 × 128. The settings of NN structures in GaussLTFB and (e) is the separated spectrogram from Fig. 5a are modified correspondingly to adapt
to this size. GaussFB-FullLTFB. When compared with (c), the results We also use the NN model in [47]and the1-layer CNN of our proposal of long-term filter banks (d)and (e)show model as our baseline
models. The three variant modules significant temporal coherence in each frequency bin, GaussLTFB, FullLTFB-Random and FullLTFB-Identity are which is more approximate to the clean music spectro-
utilized on magnitude spectrograms directly in this part. gram in (a). The results of these experiments are shown in Table 2. Compared with the results in Table 1,all theconclusions 4.2 Audio scene
classification above remain unchanged. When the energy ratio between In this section, we apply the long-term filter banks to the music and voice is 1, the reconstruction error is reduced audio scene
classification task. We employ LITIS ROUEN Table 2 Reconstruction error of audio source separation using magnitude spectrograms as input Re_toep Re_inv Init Method M/V = 0.1 M/V =1M/V = 10 M/V = 0.1
M/V =1M/V = 10 –Null[47] 2.58 0.99 0.033 2.58 0.99 0.033 – CNN-1layer [22] 2.83 0.96 0.047 2.83 0.96 0.047 – GaussLTFB 2.49 0.94 0.037 2.60 0.95 0.034 Random FullLTFB 2.77 1.12 0.080 2.85 1.03 0.043
Identity FullLTFB 2.50 0.94 0.037 2.82 0.95 0.034 M/V represents the energy ratio between music and voice Zhang and Wu EURASIP Journal on Audio, Speech, and Music Processing (2018) 2018:4 Page 9 of
13 Fig. 6 Reconstructed spectrogram of audio source separation task. The clean music spectrogram in a is randomly selected from the dataset. b The corresponding music and vocal mixture. c–e The
reconstructed music spectrograms from the mixture spectrogram using different configurations dataset [48] and DCASE2016 dataset [49]toconduct dataset is divided into fourfold. Our experiments
acoustic scene classification experiments. obey this setting, and the average performance will be Details of these datasets are listed as follows. reported. - LITIS ROUEN dataset: This is the largest
publicly For both datasets, the examples are 30 s long. In the available dataset for ASC to the best of our data preprocessing step, we first divide the 30-s exam- knowledge. The dataset contains
about 1500 min of ples into 1-s clips with 50% overlap. Then each clip is acoustic scene recordings belonging to 19 classes. processed using neural networks as Fig. 5b. The classifi- Each audio
recording is divided into 30-s examples cation results of all these clips will be averaged to get an without overlapping, thus obtain 3026 examples in ensemble result for the 30-s examples. The size
of audio total. The sampling frequency of the audio is spectrograms is 64 × 128. For CNN structure in Fig. 5b, 22,050 Hz. The dataset is provided with 20 the window sizes of convolutional layers are
64 × 2 × 64, training/testing splits. In each split, 80% of the 64 × 3 × 64 and 64 × 4 × 64, the fully connected lay- examples are kept for training and the other 20% for ers are 196 × 128 × 19(15).
For DCASE2016 dataset, we testing. We use the mean average accuracy over the use dropout rate of 0.5. For all these methods, the learn- −4 20 splits as the evaluation criterion. ing rate is 0.001, l
weight is 1e , training is done using - DCASE2016 dataset: The dataset is released as Task the Adam [45] update method and is stopped after 100 1 of the DCASE2016 challenge. We use the training
epochs. In order to compute the results for each development data in this paper. The development training-test split, we use the classification error over all data contains about 585 min of acoustic
scene classes. The final classification error is its average value recordings belonging to 15 classes. Each audio over all splits. recording is divided into 30-s examples without We begin with
experiments where we train different overlapping, thus obtain 1170 examples in total. The neural network models without long-term filter banks on sampling frequency of the audio is 44,100 Hz. The
both datasets. As described at the beginning of Section 4, Zhang and Wu EURASIP Journal on Audio, Speech, and Music Processing (2018) 2018:4 Page 10 of 13 Table 3 Average performance comparison with
related works on LITIS Rouen dataset and DCASE2016 dataset DCASE2016 (%) LITIS Rouen (%) Method Error F-measure Error F-measure TriFB-Null 23.12 76.08 3.76 96.19 GaussFB-Null 22.69 76.56 3.48 96.44
CNN-multilayer [50] 26.45 72.44 4.00 95.80 CNN-1layer [22] 23.29 75.82 2.97 96.91 RNN-Gam [26]– – 3.4 – CNN-Gam [24]– – 4.2 – MFCC-GMM [49] 27.5 – – – DNN-CQT [51]– 78.1 – 96.6 DNN-Mel [53] 23.6 – –
– CNN-Mel [54] 24.0 – – – our baseline systems take the outputs of frequency filter [52] feature representations. On DCASE2016 dataset, banks as input. TriFB and GaussFB are placed in the fre- only
DNN model using CQT features performs better quency domain to integrate the frequency information. than our baseline models. Classical CNN model with Classical CNN models have the ability to learn
two- three layers performs almost the same as [24]onLITIS dimensional filters on the spectrum directly. We introduce Rouen dataset, but gets a rapid deterioration of perfor- two CNN structures as a
comparison. The first CNN mance on DCASE2016 dataset. This can also be attributed model is implemented as [50], which has multiple convo- to the lack of training data, especially on DCASE2016
lutional layers, pooling layers, and fully connected layers. dataset. CNN model with one convolutional layer per- The window size of convolutional kernels are 5 × 5, the forms a little better, but
still worse than our baseline pooling size is 3, the output channels are [8, 16, 23], the models. These results show that the time-frequency struc- fully connected layers are 196 × 128 × 19(15).
Another ture of the spectrum is difficult to be learned using CNN structure is the same as the one-layer CNN model two-dimensional convolution kernels in classical CNN described in Section 4.1, the
outputs of this model is then models. For the two baseline models, GaussFB per- processed as Fig. 5b. forms better than TriFB on both datasets, because of The results of these experiments are shown
in Table 3. that Gaussian frequency filter banks can extract more Comparing with other CNN related works, our baseline global information. In conclusion, the results of our models on both datasets
achieve gains in accuracy. On baseline models are in line with expectations on both LITIS Rouen dataset, recurrent neural network (RNN) datasets. [26] performs better than our baseline models,
because We now test our long-term filter banks on both datasets. of the powerful sequence modelling capabilities of RNN. We also test three variant modules in this part: Gaus- DNN model in [51] is
the best-performing single model sLTFB, FullLTFB-Random and FullLTFB-Identity. These on both datasets, this can be attributed to the lack of train- three variant modules can be injected into neural
net- ing data and the stability of Constant Q-transform (CQT) works directly as Fig. 5b. Table 4 Average performance comparison using different configurations on LITIS Rouen dataset and DCASE2016
dataset DCASE2016 (%) LITIS Rouen (%) Init Method Error F-measure Error F-measure – TriFB-Null 23.12 76.08 3.76 96.19 – GaussFB-Null 22.69 76.56 3.48 96.44 – TriFB-GaussLTFB 22.40 76.79 2.82 97.05 –
GaussFB-GaussLTFB 22.15 77.11 2.97 96.91 Random TriFB-FullLTFB 22.67 76.49 3.47 96.35 Random GaussFB-FullLTFB 21.21 78.05 2.96 96.92 Identity TriFB-FullLTFB 23.35 75.69 3.67 96.18 Identity
GaussFB-FullLTFB 23.13 75.83 3.21 96.61 Zhang and Wu EURASIP Journal on Audio, Speech, and Music Processing (2018) 2018:4 Page 11 of 13 Fig. 7 Validation curves on LITIS ROUEN dataset and DCASE2016
dataset. a, b The proposed methods on LITIS ROUEN dataset. d, e The proposed methods on DCASE2016 dataset. c The classical CNN nodels on LITIS ROUEN dataset. f The classical CNN nodels on DCASE2016
dataset Table 4 is the performance comparison on both datasets. Figure 7a–e shows consistent results with datasets. Models with GaussLTFB module perform con- Table 4. sistently better than the
corresponding baseline models. Although the performance fluctuates for different vari- 4.3 Reconstruction vs classification ants, the performance gain is obvious. For FullLTFB situ- In the experiment
of audio source separation task, when ation, random initialization obtains performance gain on the parameters of totally independent long-term filter both datasets, but identity initialization
degrades the per- banks are initialized randomly, the result seems to be formance on DCASE2016 dataset. This can be attributed unable to converge effectively. However, it is completely that in
classification tasks, we need to extract a global rep- the opposite in audio scene classification task. resentation of all frames, more details will be discussed in Figure 8 is an explanation of the
unconformity between Section 4.3. On LITIS Rouen dataset, TriFB-GaussLTFB the above two tasks. Figure 8a, b is the filters learned on model performs significantly better than the state-of- MIR-1K
dataset. At low frequencies, the time duration of the-art result in [51] and obtains 2.82% on classification filters are almost equal to 1, only at very high frequen- error. On DCASE2016 dataset,
GaussFB-FullLTFB model cies, the time durations become large. But for Fig. 8c, d with random initialization reduces the classification error which is learned on DCASE2016 dataset, the time dura- by
relatively 6.5% and reaches the performance of DNN tion is much larger. It is intuitive that in audio source model using CQT features in [51], meaning that the separation task, the time duration of
the filters is much long-term filter banks make up for the lack of feature smaller than in audio scene classification task, especially extractions. at low frequencies. When the parameters of totally
inde- Validation curves on both datasets are shown in Fig. 7. pendent long-term filter banks are initialized randomly, After 100 training epochs, experiments on DCASE2016 the implicit assumption is
that the time durations of the dataset encounter overfitting problem; experiments on filters is as large as the number of all frames, which is not LITIS ROUEN dataset have almost converged. Figure
7c, e applicable. In reconstruction related tasks, for example, shows that the performance of classical CNN model the long-term correlation is much more limited because is significantly worse than
models with only the fre- our goal is to reconstruct the spectrogram frame by frame. quency filter banks, which is consistent with the However, in classification tasks, we need to extract a results
in Table 3. The performance of one-layer CNN global representation of all frames, which is exactly in line model is between TriFB and GaussFB models on both with our hypothesis. Zhang and Wu EURASIP
Journal on Audio, Speech, and Music Processing (2018) 2018:4 Page 12 of 13 Fig. 8 Time durations of long-term filter banks in different tasks. a, b The long-term filters learned on MIR-1K dataset. c,
d The long-term filters learned on DCASE2016 dataset 5Conclusions Authors’ contributions TZ designed the core methodology of the study, carried out the A novel framework of filter banks that can
extract long- implementation and experiments, and he drafted the manuscript. JW term time and frequency correlation is proposed in this participated in the study and helped to draft the manuscript.
Both authors paper. The new filters are constructed after traditional read and approved the final manuscript. frequency filters and can be implemented using Toeplitz Competing interests matrix
motivated neural networks. Gaussian shape con- The authors declare that they have no competing interests. straint is introduced to limit the time duration of the filters, especially in
reconstruction-related tasks. Then a Publisher’s Note spectrogram reconstruction method using the Toeplitz Springer Nature remains neutral with regard to jurisdictional claims in matrix inversion is
implemented using neural networks. published maps and institutional affiliations. The spectrogram reconstruction error in audio source separation task is reduced by relatively 6.7% and the classi-
Received: 21 November 2017 Accepted: 30 April 2018 fication error in audio scene classification task is reduced by relatively 6.5%. This paper provides a practical and References complete framework
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EURASIP Journal on Audio, Speech, and Music Processing – Springer Journals
Published: May 30, 2018
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Design ideas for competitive Wordle - csanyk.com
Wordle, the guess a 5 letter word in 6 tries game, is a really good game. Any good game deserves a competition.
I thought about how to design a proper competitive Wordle, and wanted to share my ideas with the world so that people could use them to organize Wordle tournaments. I have no connection to the
creators of Wordle or the New York Times; these are just my ideas that I am offering as an add-on to enhance the existing Wordle experience. I offer them freely for anyone to use or modify to suit,
under the Creative Commons Attribution International 4.0 license.
Csanyk’s Competitive Wordle Rules
Version 1.0, Creative Commons Attribution International 4.0 license.
Level Playing Field
1. Consistency is Fairness. All participants will play the same list of solution words, presented in the same order. All players will either play in hard mode (where any revealed hints must be used
in subsequent guesses) or normal mode.
2. No cheating. Players may not have advance knowledge of the solutions at any time, and may not consult a dictionary.
3. Players may not receive outside help to come up with the solution, in any form.
4. Players may use scratch paper if they desire, but the sheets must be blank before the start of the competition.
1. To keep tournaments from running indefinitely, a finite time bank will be used.
2. Each player will have their own time bank which they will be allowed to use during the period in which the competition is ongoing. Eg, the overall competition might be slated for a specific date
and time window of, say, 2 hours, with each player granted a time bank of 1 hour to play their games, to be used within that 2 hour window, affording them up to an hour of break time, if they
wish to use it, however they wish to use it.
3. The amount of time given can be determined arbitrarily by the organizing body, but a suggested length of 1 hour or less would be reasonable for most tournaments. For larger tournaments, more time
may be needed, or more rounds of competition, each with their own time bank. For ultra brief or high speed play, time banks of 5 or 10 minutes might be appropriate.
4. For precise timekeeping, the clock start and stop can be coded into the Wordle program itself, but if such a feature is not present in the software being used, time may be kept externally, using
a stopwatch or pauseable countdown timer.
5. The timebank clock starts when the player submits their first guess for the current puzzle, and stops when they solve the puzzle.
6. The player can take breaks between puzzles as often and as long as they wish, subject to any other rules governing breaks which may limit their number or duration or how often.
7. There are no timeouts once a puzzle has been started.
8. Players will play successive rounds of Wordle until they either fail to solve a round, or run out of time in their personal time bank.
1. Players score points each round, based on how many guesses were needed to solve the Wordle.
2. A solved puzzle will score base value of 100 points. The exact number used for the Base Points is somewhat arbitrary, and doesn’t matter a whole lot. The Base Points value will be modified by
multipliers, as follows:
1. 1/6 Solve: * 1. Solving in 1 is pure luck, and shouldn’t be rewarded with a special bonus.
2. 2/6 Solve: * 4. Solving in 2 guesses involves skill as well as some luck (to get enough clues from the first guess) so should count for more.
3. 3/6 Solve: * 2. Solving in 3 guesses isn’t easy, so deserves a bonus.
4. 4/6 Solve: * 1. Solving in 4 guesses is par.
5. 5/6 Solve: * 0.5. Solving in 5 guesses is good, but just isn’t as impressive.
6. 6/6: * 0.25. Solving in 6 guesses deserves points, just the least number of them.
7. X/6: * 0. No points awarded if you fail to solve.
3. The above scoring system rewards fast play, in terms of guesses used, but also rewards cautious play, since solving each puzzle unlocks future scoring opportunities without limit other than that
imposed by the time bank.
4. Players can score more points per round if they solve in fewer guesses, but they can score more points overall if they play many rounds. But players can play the most rounds if they don’t take
risky strategies that are more likely to solve early but increase the risk of busting, and they can play more rounds if they solve each round quickly.
Time Bonuses
1. Optionally, solving each Wordle may afford opportunity to gain bonus time. This is a double reward, since solving in fewer guesses typically consumes less time already. To prevent limitless time,
the structure will need to be carefully considered, and calibrated to the speed of the players. It’s forseeable that future players could become faster than we can imagine at solving puzzles, so
to avoid infinite play, this will need to be adjusted.
2. Suggested values for timebank bonuses are a starting point only, and are subject to revision.
□ 1/6: + 30 seconds
□ 2/6: + 20 seconds
□ 3/6: + 10 seconds
□ 4/6: + 0 seconds
□ 5/6: + 0 seconds
□ X/6: Timebank * 0; player is busted, their remaining timebank is zero per the normal rules.
3. Typically the bonus possible in a single round should not afford a player more bonus time than it typically takes to play a single round; it should therefore only extend the player’s time bank to
allow for multiple additional rounds of possible play if they accrue bonus time over several rounds of play.
Micro Points
1. Micro points are an optional way to increase nuance with the scoring system. This is intended to help avoid ties.
2. Micro points scored as follows:
3. Grey letter (Each letter used in a guess but is not found in the solution: 1 point.)
4. Yellow letter (Letter found in word, but used in the wrong position: 2 points.)
5. Green letter (Letter found in word, used in the correct position: 5 points.)
6. Micro points are tallied over successive plays in the round, so if the same letter is used in multiple positions as a Yellow, each unique position in the word that the letter is used counts for
score. But if the yellow letter is played in the same position in multiple guesses, only the first time that letter appeared Yellow in that position scores points. For example, if the solution
contains the letter T in the 2nd position, and the player guesses a word that uses T in the 1st position and in the 3rd position, in both guesses the T will score 2 points. But if the player
played a word with the letter T in the 1st position in two guesses, only the first guess that yields the clue that T is in the word, but not in the 1st position should count for points, and any
subsequent guesses using T in the 1st position will not score additional points.
7. Likewise, only the first time the green letter is discovered for each position counts for points. Thus if the solution is YEAST, and the player guesses: 1. STATE; 2.STEAK; 3. MEATS; 4. BEAST; 5.
FEAST; 6. LEAST, and fails to solve, the micro point scored for the round would be:
2+2+5+2+0 = 11
0+0+2+2+1 = 5
1+5+5+0+2 = 12
1+0+0+5+5 = 11
1+0+0+0+0 = 1
1+0+0+0+0 = 1
= 41 micro points.
The micro points then are divided by 10 and added to the regular points, so in this case the 4.1 micro points would add 4.1 points to the player’s tournament score.
8. On an X/6 solve or unsolved round where the player ran out of time but still had remaining guesses, we can optionally score the “micro points” to reward whatever progress the player made in their
final round, or we can simply award 0 points for the round since it was not solved.
9. Micro points are normally scored for every round played. This could aid in making tie situations more unlikely, while de-emphasizing the final round’s micro points.
Tournament ranking and winners
1. At the end of the tournament, players points are tallied over all rounds of play, and then ranked in descending order. The player with the more points holds a higher rank in the standings.
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skimage.filters.rank.autolevel Auto-level image using local histogram.
skimage.filters.rank.autolevel_percentile Return grayscale local autolevel of an image.
skimage.filters.rank.enhance_contrast Enhance contrast of an image.
skimage.filters.rank.enhance_contrast_percentile Enhance contrast of an image.
skimage.filters.rank.entropy Local entropy.
skimage.filters.rank.equalize Equalize image using local histogram.
skimage.filters.rank.geometric_mean Return local geometric mean of an image.
skimage.filters.rank.gradient Return local gradient of an image (i.e. local maximum - local minimum).
skimage.filters.rank.gradient_percentile Return local gradient of an image (i.e. local maximum - local minimum).
skimage.filters.rank.majority Assign to each pixel the most common value within its neighborhood.
skimage.filters.rank.maximum Return local maximum of an image.
skimage.filters.rank.mean Return local mean of an image.
skimage.filters.rank.mean_bilateral Apply a flat kernel bilateral filter.
skimage.filters.rank.mean_percentile Return local mean of an image.
skimage.filters.rank.median Return local median of an image.
skimage.filters.rank.minimum Return local minimum of an image.
skimage.filters.rank.modal Return local mode of an image.
skimage.filters.rank.noise_filter Noise feature.
skimage.filters.rank.otsu Local Otsu's threshold value for each pixel.
skimage.filters.rank.percentile Return local percentile of an image.
skimage.filters.rank.pop Return the local number (population) of pixels.
skimage.filters.rank.pop_bilateral Return the local number (population) of pixels.
skimage.filters.rank.pop_percentile Return the local number (population) of pixels.
skimage.filters.rank.subtract_mean Return image subtracted from its local mean.
skimage.filters.rank.subtract_mean_percentile Return image subtracted from its local mean.
skimage.filters.rank.sum Return the local sum of pixels.
skimage.filters.rank.sum_bilateral Apply a flat kernel bilateral filter.
skimage.filters.rank.sum_percentile Return the local sum of pixels.
skimage.filters.rank.threshold Local threshold of an image.
skimage.filters.rank.threshold_percentile Local threshold of an image.
skimage.filters.rank.windowed_histogram Normalized sliding window histogram
Return local gradient of an image (i.e. local maximum - local minimum).
Assign to each pixel the most common value within its neighborhood.
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Will generate a trajectory using Embeddr.
This method was wrapped inside a container. The original code of this method is available here.
ti_embeddr(ndim = 2L, kernel = "nn", metric = "correlation",
nn_pct = 0L, eps = 0L, t = 0L, symmetrize = "mean",
measure_type = "unorm", thresh = 0.001, maxit = 10L,
stretch = 2L, smoother = "smooth.spline")
ndim Dimension of the embedded space, default is 2. Domain: U(2, 10). Default: 2. Format: integer.
The choice of kernel. 'nn' will give nearest neighbours, 'dist' gives minimum distance and'heat' gives a heat kernel. Discussed in detail in 'Laplacian Eigenmaps and Spectral Techniques
kernel for Embedding and Clustering',Belkin & Niyogi. Domain: nn, dist, heat. Default: nn. Format: character.
The metric with which to assess 'closeness' for nearest neighbour selection, one of'correlation' (pearson) or 'euclidean'. Default is 'correlation'. Domain: correlation, euclidean,
metric cosine. Default: correlation. Format: character.
nn_pct The percentage of cells to use as tge number of nearest neighbours if kernel == 'nn'. Domain: U(-2, 1). Default: 0. Format: numeric.
eps Maximum distance parameter if kernel == 'dist'. Domain: U(-5, 5). Default: 0. Format: numeric.
t "time" for heat kernel if kernel == "heat". Domain: U(-5, 5). Default: 0. Format: numeric.
How to make the adjacency matrix symmetric. Note that slightlycounterintuitively, node i having node j as a nearest neighbour doesn't guarantee nodej has node i. There are several ways
symmetrize to get round this;* mean If the above case occurs make the link weight 0.5 so the adjacency matrix becomes $0.5(A + A')$* ceil If the above case occurs set the link weight to 1 (ie take
the ceiling of the mean case)* floor If the above case occurs set the link weight to 0 (ie take the floor of the mean case). Domain: mean, ceil, floor. Default: mean. Format: character.
Type of laplacian eigenmap, which corresponds to the constraint on the eigenvalue problem. Iftype is 'unorm' (default), then the graph measure used is the identity matrix, while if type
measure_type is 'norm' then the measureused is the degree matrix. Domain: unorm, norm. Default: unorm. Format: character.
thresh Convergence threshold on shortest distances to the curve. Domain: e^U(-11.51, 11.51). Default: 0.001. Format: numeric.
maxit Maximum number of iterations. Domain: U(0, 50). Default: 10. Format: integer.
A factor by which the curve can be extrapolated when points are projected. Default is 2 (times the last segment length). The default is 0 for smoother equal to "periodic_lowess". Domain:
stretch U(0, 5). Default: 2. Format: numeric.
smoother Choice of smoother. The default is "smooth_spline", and other choices are "lowess" and "periodic_lowess". The latter allows one to fit closed curves. Beware, you may want to use iter = 0
with lowess(). Domain: smooth.spline, lowess, periodic.lowess. Default: smooth.spline. Format: character.
A TI method wrapper to be used together with infer_trajectory
Campbell, K., Ponting, C.P., Webber, C., 2015. Laplacian eigenmaps and principal curves for high resolution pseudotemporal ordering of single-cell RNA-seq profiles.
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orksheets for 6th Class
Coding Puzzles - Sequencing
If-And-Then Logic Puzzles!
Math picture puzzles (systems of equations)
Funny riddles and puzzles
6th grade math State Exam practice 1
Distance Learning Wk 7 Engineering
Explore Math Puzzles Worksheets by Grades
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Explore printable Math Puzzles worksheets for 6th Class
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Gottfried Wilhelm Leibniz
Leibniz was perhaps the last great universalist, contributing to disciplines from biology to probability and from theology to linguistics. He was one of the three great 17thC rationalist
philosophers, the inventor (independently of Newton) of infinitesimal calculus, and advanced the design of calculating machines. Huygens was a mentor. Leibniz spent several days in deep discussion
with Spinoza, met Boyle, Leeuwenhoek and Goldbach, and corresponded extensively with the Bernoullis, von Tschirnhaus, Arnauld, Bayle, Sloane and Oldenburg. Newton and he never met nor corresponded
directly. Wallis wrote refusing him permission to teach cryptography to students in Hannover.
Gottfried Wilhelm Leibniz knew…
• Jacob Bernoulli
• Henry Oldenburg
• Nicolas Malebranche
• Ehrenfried Walther von Tschirnhaus
• Baruch Spinoza
• Antoine Arnauld
• Pierre Bayle
• Christian Wolff, philosopher
• Joachim Bouvet
• Guillaume de l'Hôpital
• John Wallis
• Robert Hooke
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