content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
Effects of Slip and Hall on the Peristaltic Transport of a Hyperbolic Tangent Fluid in a Planar Channel
Volume 08, Issue 12 (December 2019)
Effects of Slip and Hall on the Peristaltic Transport of a Hyperbolic Tangent Fluid in a Planar Channel
DOI : 10.17577/IJERTV8IS120290
Download Full-Text PDF Cite this Publication
M Anusha Bai , Prof. R. Sivaprasad, 2019, Effects of Slip and Hall on the Peristaltic Transport of a Hyperbolic Tangent Fluid in a Planar Channel, INTERNATIONAL JOURNAL OF ENGINEERING RESEARCH &
TECHNOLOGY (IJERT) Volume 08, Issue 12 (December 2019),
• Open Access
• Authors : M Anusha Bai , Prof. R. Sivaprasad
• Paper ID : IJERTV8IS120290
• Volume & Issue : Volume 08, Issue 12 (December 2019)
• Published (First Online): 30-12-2019
• ISSN (Online) : 2278-0181
• Publisher Name : IJERT
• License: This work is licensed under a Creative Commons Attribution 4.0 International License
Text Only Version
Effects of Slip and Hall on the Peristaltic Transport of a Hyperbolic Tangent Fluid in a Planar Channel
M. Anusha Bai1
1Research Scholar, Department of Mathematics,
Sri Krishnadevaraya University, Ananthapuramu-515003, A.P., India
Prof. R. Sivaprasad2
2Professor, Department of Mathematics,
Sri Krishnadevaraya University, Ananthapuramu-515003, A.P., India
Abstract:- The effects of slip and Hall on the peristaltic transport of a hyperbolic tangent fluid in a planar channel under the assumption of long wavelength. The expressions for the velocity and
axial pressure gradient are obtained by employing perturbation technique. The effects of Weissenberg number, power-law index, Hall parameter, Hartmann number and amplitude ratio on the axial pressure
gradient and time-averaged volume flow rate are studied with the aid of graphs.
Keywords: Peristaltic Transport, Hyperbolic Tangent fluid, Planar Channel and Slip and Hall parameter
1. INTRODUCTION
Some fluids which are encountered in chemical applications do not adhere to the classical Newtonian viscosity prescription and are accordingly known as non-Newtonian fluids. One especial class of
fluids which are of considerable practical importance is that in which the viscosity depends on the shear stress or on the flow rate. The viscosity of most non- Newtonian fluids, such as
polymers, is usually a nonlinear decreasing function of the generalized shear rate. This is known as shear-thinning behavior. Such fluid is a hyperbolic tangent fluid (Ai and Vafai, 2005). Nadeem
and Akram (2009) have first investigated the peristaltic flow of a hyperbolic tangent fluid in an asymmetric channel. Nadeem and Akbar (2011) have analyzed the peristaltic transport of a Tangent
hyperbolic fluid in an endoscope numerically.
Based on Experimental controls, it was shown that the controlled application of low intensity and frequency pulsing magnetic fields could modify cell and tissue behavior. Biochemistry has taught
us that cells are formed of positive or negative charged molecules. This is why these magnetic fields applied to living organisms may induce deep modifications in molecule orientation and in
their interaction. An impulse magnetic field in the combined therapy of patients with stone fragments in the upper urinary tract was experimentally studied by Li et al. (1994). It was found that
impulse magnetic
field (IMF) activates impulse activity of ureteral smooth muscles in 100% of cases. Elshahed and Haroun (2005) have investigated the peristaltic flow of a Johnson-Segalman fluid in a planar
channel under the effect of a magnetic field. Hayat and Ali (2006) have investigated the peristaltic motion of a MHD third grade fluid in a tube. Hayat et al. (2007) have first investigated the
Hall effects on the peristaltic flow of a Maxwell fluid trough a porous medium in channel. Magnetohydrodynamic peristaltic flow of a hyperbolic tangent fluid in a vertical asymmetric channel with
heat transfer was studied by Nadeem and Akram (2011). Eldabe (2015) have studied the Hall Effect on peristaltic flow of third order fluid in a porous medium with heat and mass transfer. Hall
effects on the peristaltic pumping of a hyperbolic tangent fluid in a planar channel were studied by Subba Narasimhudu and Subba Reddy (2017).
The peristaltic flow of a Newtonian fluid through a two-dimensional micro channel with the slip effect was first investigated by Kwang (2000). El Sehaway et al. (2006) have studied the effect of
slip on the peristaltic motion of a Maxwell fluid in a two-dimensional channel. The effects of slip and non- Newtonian parameters on the peristaltic transport of a third grade fluid in a circular
cylindrical tube were investigated by Ali et al. (2009). Chaube et al. (2010) have analyzed the slip effects on the peristaltic flow of a micropolar fluid in a channel. Effects of slip and
induced magnetic field on the peristaltic flow of pseudoplastic fluid were analyzed by Noreen et al. (2011). Subba Reddy et al. (2012) have investigated the slip effects on the peristaltic motion
of a Jeffrey fluid through a porous medium in an asymmetric channel under the effect of magnetic field. Akbar et al. (2012) have discussed the peristaltic flow of a hyperbolic tangent fluid in an
inclined asymmetric channel with slip and heat transfer. Slip effects on peristaltic transport of a Prandtl fluid in a channel under the effect of magnetic field was studied by Jyothi et al.
H X ,t
H X ,t
We consider the peristaltic motion of a hyperbolic tangent fluid in a two-dimensional symmetric channel of width 2a under the effect of magnetic field. The flow is generated by sinusoidal wave
trains propagating with constant speed c
along the channel walls. A uniform magnetic field B0 is
applied in the transverse direction to the flow. The magnetic Reynolds number is considered small and so induces magnetic field neglected. Fig. 1 represents the physical model of the channel.
The wall deformation is given by
Y H ( X ,t) a b cos 2 ( X ct) , (1)
where b is the amplitude of the wave, – the wave length and
along the axis of the channel and Y perpendicular to X . Let
along the axis of the channel and Y perpendicular to X . Let
X and Y – the rectangular co-ordinates with X measured
Fig. 1 The Physical Model
Fig. 1 The Physical Model
1 1n
1 n 1
(U ,V )
be the velocity components in fixed frame of
reference ( X ,Y ) .
The flow is unsteady in the laboratory frame ( X ,Y )
. However, in a co-ordinate system moving with the
The above model reduces to Newtonian for and n 0 .
propagation velocity c (wave frame (x, y)), the boundary shape
is stationary. The transformation from fixed frame to wave frame is given by
The equations governing the flow in the wave frame of reference are
x X ct, y Y ,u U c, v V
u v 0
where (u, v)
and (U ,V )
are velocity components in the
x y
wave and laboratory frames respectively.
The constitutive equation for a Hyperbolic Tangent fluid is
u u
u p yx B2
tanh n
v xx 0 mv u c
v xx 0 mv u c
x x
y 1 m2
where is the extra stress tensor, is the infinite shear rate
viscosity, o
is the zero shear rate viscosity, is the time
constant, n is the power-law index and is defined as
v xy yy 0 mu c v
y x
y 1 m2
ij ji
i j
where is the density is the electrical conductivity, B0
where is the second invariant stress tensor. We consider in the constitutive equation (3) the case for which 0 and
is the magnetic field strength and m is the Hall parameter.
The corresponding dimensional boundary conditions are
1, so the Eq. (3) can be written as
u xy c at y H (slip condition)
u 0
at y 0
y p
1 n We
u 1
M u 1
(symmetry condition) (10) here is the slip parameter.
Introducing the non-dimensional variables defined by
x y u v a pa2 b
x y
p 0
y y
y y
1 m2
x ,
y ,
u ,
v ,
p c , a
From Eq. (15) and (16), we get
h H , t ct , xx ,
, yy ,
a c
xx xy
c xy
c yy
dp 1 n u nWe
M u 1
c a q
dx y2
y y
1 m2
, We
, q
c ac
into the Equations (6) – (8), reduce to (after dropping the bars)
The corresponding non-dimensional boundary conditions in the wave frame are given by
u v 0
x y
u 1 n We u 1 u 1 at
u u p
M 2
y y
Re u
v 2
xx xy
m v u 1
y h 1 cos 2 x
3 v
x x
p 2 xy
1 m2
M 2
u 0
at y 0
u x v y y
mu 1 v
y 1 m2
The volume flow rate q in a wave frame of reference is given by
q udy . (20)
where xx 2 1 nWe 1 x ,
The instantaneous flow Q( X , t) in the laboratory frame is
h h
1 nWe 1 u 2 v ,
Q( X ,t) UdY (u 1)dy q h
2 1 n We 1 v ,
The time averaged volume flow rate Q over one period
yy y
of the peristaltic wave is given by
u 2
v 2
v 2 2 1 T
Q Qdt q 1
x y T 0
M aB0
is the Hartmann number.
Under lubrication approach, neglecting the terms of order and Re, the Eqs. (13) and (14) become
3. SOLUTION
Since Eq. (17) is a non-linear differential equation, it
u 1
dp0 cosh Ny
is not possible to obtain closed form solution. Therefore, we employ regular perturbation to find the solution.
0 N 2 1 n
dx a1
For perturbation solution, we expand u, dp
as follows
and q
where N M /
1 n(1 m2 )
u u Weu OWe2
dp dp dp 2
a1 cosh Nh N (1 n)sinh Nh .
0 We 1 O We dx dx dx
The volume flow rate q is given by
q q
0 1 1 dp sinh Nh Nha
q0 0 1 h
Substituting these equations into the Eqs. (17) – (19),
N 3 1 n dx a
we obtain
A. SYSTEM OF ORDER We0
From Eq. (A) , we have
dp 2u M 2
dp q h N 3 1 na
0 1 n 0 u
dx y2 1 m2 0
dx sinh Nh Nha1
and the respective boundary conditions are
D. SOLUTION FOR SYSTEM OF ORDER We1
u 1 n u0 0 y
at y h
Substituting Eq. (32) in the Eq. (29) and solving the Eq. (29), using the boundary conditions (30) and (31), we obtain
u0 0 at
y 0
dp 2
u 1
cosh Ny 1 n
f y
1 N 2 1 n dx a
3 N 1 na 3
B. SYSTEM OF ORDER We1
dp 2u
u 2 M 2
1 1 n 1
2 u1
dx y
y y
1 m
sinh 2Nh 2sinh Nhcosh Ny 2sinh Ny sinh 2Ny cosh Nh
f y
2cosh 2Nh cosh Nhcosh Ny sinp Nh cosh Ny
N 1 n
and the respective boundary conditions are
sinh Nh 2sinh Ny sinh 2Ny
u u 2
u1 1 n 1 n 0
0 at y h
The volume flow rate q is given by
1 dp sinh Nh Nha dp 2
q1 1 1 a2 0
N 3 1 n
dx a
u1 0
at y 0
C. SOLUTION FOR SYSTEM OF ORDER We0
Solving Eq. (26) using the boundary conditions (27) and (28), we obtain
where Fig. 2 shows the variation of the axial pressure
4 3cosh Nh 2sinh 2Nh sinh Nh cosh 2Nh cosh Nh a
a n 3
with We for n 0.5, m 0.2 , 0.1,
6N 4 1 n3 a3
M 1, 0.5 and Q 1. It is observed that, the axial
a3 N 1 n3sinh Nhcosh 2Nh 1 2sinh Nh.
From Eq. (36) and (34), we have
pressure gradient number We .
increases with increasing Wiessenberg
dp q N 3 1 na
dp 2
The variation of the axial pressure gradient
1 1 1 a 0
n for
We 0.01,
m 0.2,
M 1,
dx sinh Nh Nha
4 dx
0.5 and Q 1 is depicted in Fig. 3. It is found that,
4 3cosh Nh 2sinh 2Nh sinh Nh cosh 2Nh cosh Nh a
the axial pressure gradient
power-law index n .
decreases with an increase in
a4 n
6N 1 n2 a2 sinh Nh Nha
Substituting Equations (16) and (37) into the Eq. (24)
Fig. 4 illustrates the variation of the axial pressure
with for n 0.5, We 0.01, M 1,
dp dp dp
and using the relation We
dx dx dx
and neglecting
m 0.2, 0.5 and Q 1. It is noted that, the axial
terms greater than O We , we get
pressure gradient
decreases with increasing slip
dp q h N 3 1 na q h N 3 1 na 2
1 Wea 1
parameter .
dx sinh Nh Nha1
sinh Nh Nha1
The variation of the axial pressure gradient
m for
n 0.5,
We 0.01,
M 1,
The dimensionless pressure rise per one wavelength in the wave frame is defined as
and Q 1 shown in Fig. 5. It is noted that, the
p 1 dpdx
0 dx
axial pressure gradient parameter m
decreases with increasing Hall
Note that, as
our results coincide with the
Fig. 6 depicts the variation of the axial pressure
results of Subba Narasimhudu and Subba reddy (2017); as dp
0 ,
m 0 , M 0 , We 0
n 0 our
with M for
n 0.5,
m 0.2,
results coincide with the results of Shapiro et al. (1969).
We 0.01,
Q 1. It is
4. RESULTS AND DISCUSSION
In this section, we have carried out numerical calculations and plotted graphs to study effects of the Weissenberg number We, the power-law index n ,the slip
observed that, on increasing Hartmann number M increases
the axial pressure gradient .
parameter , the Hall parameter m, the Hartmann number M
The variation of the axial pressure gradient
and the/8- amplitude ratio on the axial pressure gradient and pumping characteristics.
n 0.5, m 0.2 , 0.1,
M 1,
We 0.01 and Q 1 is depicted in Fig. 7. It is found
that, the axial pressure gradient
increases with increasing
The variation of the pressure rise
p with Q for different
dx values of with
n 0.5,
m 0.2 ,
M 1
amplitude ratio .
Fig. 8 illustrates the variation of the pressure rise p
and We 0.01 is depicted in Fig. 13. It is observed that, the
time-averaged flow rate Q increases with increasing in both the pumping and free pumping regions, while it decreases
with Q for different values of We with n 0.5,
with increasing n in the co-pumping region for chosen
m 0.2, 0.1, M 1 and 0.5 . It is noted that,
the time-averaged volume flow rate Q increases with increasing Wiessenberg number We in pumping p 0
p 0 .
, free-pumping p 0
and co-pumping p 0
The variation of the pressure rise
p with Q for
different values of n with We 0.01,
m 0.2,
0.1, M 1 and 0.5
is shown in Fig. 9. It is
found that, the time-averaged flow rate Q decreases with increasing n in both the pumping and free pumping regions, while it increases with increasing n in the co-pumping region.
Fig. 10 shows the variation of the pressure rise p
with Q for different values of with n 0.5,
We 0.01,
M 1
0.5 . It is
observed that, the time-averaged flow rate Q decreases with increasing in both the pumping region and the free pumping region, while it increases with increasing in the co-pumping region.
The variation of the pressure rise
p with Q for
different values of m with n 0.5, We 0.01,
0.1, M 1 and 0.5 is illustrated in Fig. 11. It is
observed that, the time-averaged flow rate Q decreases with increasing m in the pumping region, while it increases with increasing m in both the free pumping and co-pumping regions.
Fig. 12 depicts the variation of the pressure rise p
with Q for different values of M with n 0.5, 0.1
, m 0.2, We 0.01 and 0.5 . It is noticed that,
the time-averaged flow rate Q increases with increasing M in the pumping region, while it decreases with increasing M in both the free-pumping and co-pumping regions.
5. CONCLUSIONS
In this paper, we investigated the effects of slip and Hall on the peristaltic transport of a hyperbolic tangent fluid in a planar channel under the assumption of long wavelength. The expressions
for the velocity and axial pressure gradient are obtained by employing perturbation technique. It is found that, the axial pressure gradient and time-averaged flow rate in the pumping region
increases with increasing the Weissenberg number We, the Hartmann number M and the amplitude ratio
, while they decreases with increasing power-law index n ,
slip parameter and Hall parameter m .
6. REFERENCES
1. Abo-Eldahab, E.M., Barakat, E.I. and Nowar, K.I. Effects of Hall and ion-slip currents on peristaltic transport of a couple stress fluid, International Journal of Applied Mathematics and Physics,
2 (2) (2010), 145-157.
2. Ai, L. and Vafai, K. An investigation of Stokes second problem for non-Newtonian fluids, Numerical Heat Transfer, Part A, 47(2005), 955-980.
3. Akbar, N.S., Hayat, T., Nadeem, S. and Obaidat, S. Peristaltic flow of a Tangent hyperbolic fluid in an inclined asymmetric channel with slip and heat transfer, Progress in Computational Fluid
Dynamics, an International Journal, 12(5) (2012), 363-374.
4. Ali, N., Wang, Y., Hayat, T. and Oberlack, M. Slip effects on the peristaltic flow of a third grade fluid in a circular cylindrical tube, J. Appl. Mech., 76(2009), 1-10.
5. Bhatti, M. M., Ali Abbas, M. and Rashidi, M. M. Effect of hall and ion slip on peristaltic blood flow of Eyring Powell fluid in a non- uniform porous channel , World Journal of Modelling and
Simulation, 12(4) (2016), 268-279.
6. Chaube, M.K., Pandey, S.K. and Tripathi, D. Slip Effect on Peristaltic Transport of Micropolar Fluid, Applied Mathematical Sciences, 4(2010), 2105 2117.
7. Eldabe, N.T.M., Ahmed Y. Ghaly, A.Y., Sallam, S.N., Elagamy, K. and Younis, Y.M. Hall effect on peristaltic flow of third order fluid in a porous medium with heat and mass transfer, Journal of
Applied Mathematics and Physics, 3(2015), 1138-1150.
8. Elshahed, M. and Haroun, M. H. Peristaltic transport of Johnson- Segalman fluid under effect of a magnetic field, Math. Probl. Engng., 6 (2005), 663677.
9. El-Shehawey, E.F., El-Dabe, N.T. and El-Desoky, I. M. Slip effects on the Peristaltic flow of a Non-Newtonian Maxwellian Fluid, Acta Mech., 186(2006), 141-159.
10. Hayat, T and Ali, N. Peristaltically induced motion of a MHD third grade fluid in a deformable tube, Physica A: Statistical Mechanics and its Applications, 370(2006), 225-239.
11. Hayat, T., Ali, N, and Asghar, S. Hall effects on peristaltic flow of a Maxwell fluid in a porous medium, Phys. Letters A, 363(2007), 397 – 403.
12. Jyothi, B., Satyanarayana, B. and Subba Reddy, M.V. Slip effects on peristaltic transport of a Prandtl fluid in a channel under the effect of magnetic field, South Asian Journal of Mathematics, 5
(1)(2015), 1 – 12.
13. Kwang Hua Chu, W. and Fang, J. Peristaltic transport in a slip flow, Eur. Phys., J. B., 16(2000), 543-547.
14. Li, A.A., Nesterov, N.I, Malikova, S.N. and Kilatkin, V.A. The use of an impulse magnetic field in the combined of patients with store fragments in the upper urinary tract. Vopr kurortol Fizide.
Lech Fiz Kult, 3(1994), 22-24.
15. Nadeem, S. and Akram, S. Peristaltic transport of a hyperbolic tangent fluid model in an asymmetric channel, Z. Naturforsch., 64a (2009), 559 567.
16. Nadeem, S. and Akram, S. Magnetohydrodynamic peristaltic flow of a hyperbolic tangent fluid in a vertical asymmetric channel with heat transfer, Acta Mech. Sin., 27(2) (2011), 237250.
17. Nadeem, S. and Akbar, S. Numerical analysis of peristaltic transport of a Tangent hyperbolic fluid in an endoscope, Journal of Aerospace Engineering, 24(3) (2011), 309-317.
18. Noreen, S., Hayat, T. and Alsaedi, A. Study of slip and induced magnetic field on the peristaltic flow of pseudoplastic fluid, International Journal of Physical Sciences, 6(36)(2011), 8018-8026.
19. Prasanth Reddy, D. and Subba Reddy, M.V. Peristaltic pumping of third grade fluid in an asymmetric channel under the effect of magnetic fluid, Advances in Applied Science Research, 3(6)(2012),
3868 3877.
20. Shapiro, A.H., Jaffrin, M.Y and Weinberg, S.L. Peristaltic pumping with long wavelengths at low Reynolds number, J. Fluid Mech. 37(1969), 799-825.
21. Subba Narasimhudu, K. and Subba Reddy, M. V. Hall effects on the peristaltic pumping of a hyperbolic tangent fluid in a planar channel, Int. J. Mathematical Archive, 8(3) (2017), 70 – 85.
22. Subba Reddy, M.V., Jayarami Reddy, B., Nagendra, N. and Swaroopa, B. Slip effects on the peristaltic moton of a Jeffrey fluid through a porous medium in an asymmetric channel under the effect
magnetic field, Journal of Applied Mathematics and Fluid Mechanics, 4(2012), 59-72.
You must be logged in to post a comment. | {"url":"https://www.ijert.org/effects-of-slip-and-hall-on-the-peristaltic-transport-of-a-hyperbolic-tangent-fluid-in-a-planar-channel","timestamp":"2024-11-05T19:49:42Z","content_type":"text/html","content_length":"87016","record_id":"<urn:uuid:977dc921-7128-4a18-8446-4ce779df66a3>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00460.warc.gz"} |
N2642: Quantum exponent of NaN
Submitter:CFP group
Submission Date: 2021-01-24
Document: WG14 N2642
Title: N2642: Quantum exponent of NaN
Reference Documents: N2596, IEEE 754-2019
Summary: Q(x) is used to denote the quantum exponent of decimal floating-point x. Q(infinity) is defined as infinity. Some math functions make reference to Q(NAN). However, Q(NAN) is not defined in
either C23 or IEEE 754-2019.
5.2.4.2.3 paragraph 9 says the preferred quantum exponent is specified by IEEE 754-2019.
The table of Preferred quantum exponents in paragraph 10 of 5.2.4.2.3 makes reference to Q(x) and preferred quantum exponent of the result.
There are five cases where a NaN operand does not produce a NaN result for math functions. While the result's value is defined, the quantum exponent of that result is not well defined -- mainly
because Q(NAN) is not defined.
• compoundn(NAN,0) -- F.10.4.2 says value is 1. The table in 5.2.4.2.3 says its quantum exponent is floor(0*min(0,Q(NAN))).
• hypot(+/-INFINITY,NAN) -- F.10.4.4 says value is +INFINITY. The table in 5.2.4.2.3 says its quantum exponent is min(Q(+/-INFINITY),Q(NAN)).
• pow(1,NAN) -- F.10.4.5 says value is 1. The table in 5.2.4.2.3 says its quantum exponent is floor(NAN*Q(1)).
• pow(NAN,0) -- F.10.4.5 says value is 1. The table in 5.2.4.2.3 says its quantum exponent is floor(0*Q(NAN)).
• pown(NAN,0) -- F.10.4.6 says value is 1. The table in 5.2.4.2.3 says its quantum exponent is floor(0*Q(NAN)).
There are many cases where an infinity operand does not produce an infinity result for math functions. While the result's value is defined, the quantum exponent of that result is sometimes not well
defined -- mainly because zero*infinity is not defined.
• pow(1,infinity) -- F.10.4.5 says value is 1. The table in 5.2.4.2.3 says its quantum exponent is floor(infinity*Q(1)) -- which could be infinity*zero.
• pow(infinity,0) -- F.10.4.5 says value is 1. The table in 5.2.4.2.3 says its quantum exponent is floor(0*Q(infinity)) -- which is zero*infinity.
• pown(infinity,0) -- F.10.4.6 says value is 1. The table in 5.2.4.2.3 says its quantum exponent is floor(0*Q(infinity)) -- which is zero*infinity.
These appear to be defects in IEEE 754-2019.
Suggested changes to C23: Change 5.2.4.2.3, paragraph 10 from:
The following table shows, for each operation delivering a result in decimal floating-point format, how the preferred quantum exponents of the operands, Q(x), Q(y), etc., determine the preferred
quantum exponent of the operation result.
The following table shows, for each operation delivering a result in decimal floating-point format, how the preferred quantum exponents of the operands, Q(x), Q(y), etc., determine the preferred
quantum exponent of the operation result[DEL:.:DEL] | {"url":"http://www9.open-std.org/JTC1/SC22/WG14/www/docs/n2642.htm","timestamp":"2024-11-13T18:23:48Z","content_type":"text/html","content_length":"4756","record_id":"<urn:uuid:aea31b53-3a74-451e-bcc0-7640c90db619>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00889.warc.gz"} |
SocpPhase2: SOCP: Initialising objective variable z in dual form
This function determines values for z, whence they have not been specified by the user.
SocpPhase2(f, A, b, N, x, control)
vector: the parameters of the objective function in its primal form.
matrix; the parameter matrix of the cone constraints.
vector: the parameter vector of the cone constraints.
vector: the count of rows pertinent to each cone constraint.
vector: initial point of SOCP in its primal form.
list: the list of control parameters for SOCP.
A vector with the initial point for z (dual form of SOCP). | {"url":"https://www.rdocumentation.org/packages/parma/versions/1.5-3/topics/SocpPhase2","timestamp":"2024-11-15T00:07:53Z","content_type":"text/html","content_length":"61317","record_id":"<urn:uuid:2476a825-7beb-483a-b3e3-c023acbbf79a>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00292.warc.gz"} |
Solve The Linear Equation For X 4.8 6.3x 4.18 58.56x - Let's Find It!
To Solve The Linear Equation For X 4.8 6.3x 4.18 58.56x combine all constant terms (4.8 + 4.18) and terms with x (6.3x + 58.56x). This results in a simplified equation: 64.86x + 8.98. Any value of x
that makes this equation true is the solution.
we’ll discuss how to solve the linear equation For X 4.8 6.3x 4.18 58.56x, breaking it down into simple steps. Learn easy techniques to find the solution and understand the process effortlessly.
What Are Solve The Linear Equation For X 4.8 6.3x 4.18 58.56x?
To solve the linear equation For X 4.8 6.3x 4.18 58.56x, we aim to find the value of \(x\) that makes both sides of the equation equal. By isolating (x), we can simplify the equation and determine
its solution.
This process involves combining like terms and performing arithmetic operations to ensure a clear understanding of the relationship between the variables. Ultimately, solving such equations enables
us to analyze real-world problems involving linear relationships and make informed decisions based on the calculated values.
Explore Linear Equations – Simplifying Math With Straight Lines!
Linear equations are mathematical expressions that describe the relationship between variables in a straight line. They typically take the form (ax + b = c), where (a), (b), and (c) are constants,
and (x) is the variable we’re solving for.
These equations are fundamental in algebra and are used to solve various real-life problems, such as calculating costs, determining rates of change, or predicting future outcomes. By understanding
linear equations, we can analyze and solve problems involving proportional relationships straightforwardly.
When Do You Need To Solve The Linear Equation For X 4.8 6.3x 4.18 58.56x? – Explore It!
You may need to Solve The Linear Equation For X 4.8 6.3x 4.18 58.56x whenever you encounter situations involving proportional relationships or unknown values. This could include scenarios such as
calculating costs and determining rates of change.
Predicting future outcomes in various fields like finance, physics, engineering, and everyday life problems. By solving such equations, you can find the value of (x) and understand how different
variables linearly interact with each other.
Why Should You Solve The Linear Equation For X 4.8 6.3x 4.18 58.56x? – Must Read!
1. Real-world Applications:
Solve The Linear Equation For X 4.8 6.3x 4.18 58.56x helps in everyday situations like managing finances, determining distances in travel, or analyzing changes in quantities over time. By finding the
value of X, we can accurately predict outcomes and make informed decisions in various real-life scenarios.
2. Problem-solving Skills:
Mastering linear equations enhances problem-solving abilities significantly. By breaking down complex situations into simpler mathematical expressions, individuals develop critical thinking and
analytical skills. These skills are invaluable in navigating various challenges in both academic and real-world contexts.
3. Understanding Relationships:
Solving equations to find the value of x provides insights into how different variables interact linearly. This understanding is crucial for making predictions and decisions based on data analysis.
It allows individuals to grasp the underlying relationships between different factors and make informed choices.
4. Academic Success:
Proficiency in linear equations is essential for success in mathematics education. It forms a solid foundation for tackling more advanced topics in algebra, calculus, and other mathematical
disciplines. By mastering linear equations, students build confidence and competence in their mathematical abilities.
5. Career Advancement:
The ability to solve linear equations is highly sought after in various fields such as finance, engineering, science, and economics. Proficiency in this area opens doors to career opportunities and
advancement. Employers value individuals who can analyze data, solve problems, and make informed decisions using mathematical principles.
Solve The Linear Equation For X 4.8 6.3x 4.18 58.56x – Step-by-Step Guide to Finding x!
• Identify Like Terms: Begin by identifying the terms containing x, which are 6.3x and 58.56x. Also, identify the constants, which are 4.8 and 4.18.
• Combine Like Terms: Add or subtract the coefficients of like terms. In this case, add 6.3x and 58.56x to get 64.86x. Similarly, add 4.8 and 4.18 to get 8.98.
• Isolate the Variable: To isolate x, move all terms with x to one side of the equation and constants to the other side. Subtract 64.86x from both sides to get 8.98 = 58.56x – 6.3x.
• Combine Coefficients: Simplify the equation by subtracting 6.3x from 58.56x to get 52.26x. Now the equation becomes 8.98 = 52.26x.
• Solve for x: To find the value of x, divide both sides by 52.26. x = 8.9852.26 0.172.
• Verify the Solution: Substitute the value of x back into the original equation to ensure it satisfies the equation.
• Finalize: The solution for x is approximately 0.172.
By following these steps, you can effectively Solve The Linear Equation For X 4.8 6.3x 4.18 58.56x and find the value of x.
Common Mistakes To Avoid When Solving Linear Equations:
When solving linear equations, it’s crucial to watch out for common mistakes that can lead to incorrect answers. One common error is forgetting to perform the same operation on both sides of the
equation, which can throw off the balance and result in an inaccurate solution.
Another mistake is misidentifying like terms and incorrectly combining them, leading to errors in simplification. Additionally, overlooking signs or arithmetic operations can easily result in
computational errors, impacting the final solution.
It’s also important to be mindful of the order of operations and follow them carefully to ensure accurate results. By being aware of these common pitfalls and double-checking each step, you can avoid
mistakes and arrive at the correct solution when solving linear equations.
Frequently Asked Questions:
1. Why is it important to solve linear equations?
Solving linear equations is crucial for various real-life applications, such as budgeting, calculating distances, determining rates of change, and making predictions. It also helps in developing
problem-solving skills and understanding relationships between different variables.
2. Can linear equations be solved using different methods?
Yes, linear equations can be solved using various methods such as substitution, elimination, graphing, and matrices. The choice of method depends on the complexity of the equation and personal
3. Are there any prerequisites for solving linear equations?
It’s helpful to have a basic understanding of algebraic concepts such as variables, constants, coefficients, and arithmetic operations. Familiarity with the order of operations (PEMDAS/BODMAS) and
solving simple equations is also beneficial.
4. How can I check if my solution to a linear equation is correct?
You can verify your solution by substituting the value of x back into the original equation and ensuring that both sides of the equation are equal. If the equation holds, then your solution is
Final Words:
Mastering the process of Solve The Linear Equation For X 4.8 6.3x 4.18 58.56x equips you with essential problem-solving skills and opens doors to understanding various real-world scenarios.
Keep practising and applying these techniques to confidently tackle similar equations and excel in mathematics. | {"url":"https://technorozenes.com/solve-the-linear-equation-for-x-4-8-6-3x-4-18-58-56x/","timestamp":"2024-11-08T20:50:06Z","content_type":"text/html","content_length":"80723","record_id":"<urn:uuid:288c87e3-dae3-4ab5-8e53-ddca21dbf1fd>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00121.warc.gz"} |
How do you simplify 6/sqrt20? | Socratic
How do you simplify #6/sqrt20#?
1 Answer
$\frac{6}{\sqrt{5}} = \frac{3}{5} \sqrt{5}$
$\textcolor{w h i t e}{\text{XXX}} = \frac{6}{\sqrt{{2}^{2} \times 5}}$
$\textcolor{w h i t e}{\text{XXX}} = \frac{6}{2} \sqrt{5}$
$\textcolor{w h i t e}{\text{XXX}} = \frac{3}{\sqrt{5}}$
Often when asked to simplify a fraction containing a radical, we are expected to clear the radical from the denominator;
$\textcolor{w h i t e}{\text{XXX}} = \frac{3}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}$
$\textcolor{w h i t e}{\text{XXX}} = \frac{3 \sqrt{5}}{5}$
Impact of this question
2083 views around the world | {"url":"https://socratic.org/questions/how-do-you-simplify-6-sqrt20","timestamp":"2024-11-02T23:42:15Z","content_type":"text/html","content_length":"32946","record_id":"<urn:uuid:e69d1cc1-b787-4c56-beda-231776b1a821>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00365.warc.gz"} |
Lesson 16
Methods for Multiplying Decimals
Lesson Narrative
In earlier grades, students have multiplied base-ten numbers up to hundredths (either by multiplying two decimals to tenths or by multiplying a whole number and a decimal to hundredths). Here,
students use what they know about fractions and place value to calculate products of decimals beyond the hundredths. They express each decimal as a product of a whole number and a fraction, and then
they use the commutative and associative properties to compute the product. For example, they see that \((0.6) \boldcdot (0.5)\) can be viewed as \(6 \boldcdot (0.1) \boldcdot 5 \boldcdot (0.1)\)
and thus as \(\left(6 \boldcdot \frac{1}{10}\right) \boldcdot \left(5 \boldsymbol \boldcdot \frac {1}{10}\right)\). Multiplying the whole numbers and the fractions gives them \(30 \boldsymbol \
boldcdot \frac{1}{100}\) and then 0.3.
Through repeated reasoning, students see how the number of decimal places in the factors can help them place the decimal point in the product (MP8).
Students continue to develop methods for computing products of decimals, including using area diagrams. Students have previously seen that, in a rectangular area diagram, the side lengths can be
decomposed by place value. For instance, in an 18 by 23 rectangle, the 18-unit side can be decomposed into 10 and 8 units (tens and ones), and the 23-unit side can be expressed as 20 and 3 (also tens
and ones), creating four sub-rectangles whose areas constitute four partial products. The sum of these partial products is the product of 18 and 23. Students extend the same reasoning to represent
and find products such as \((1.8) \boldcdot (2.3)\). Then, students explore how these partial products correspond to the numbers in the multiplication algorithm.
Students connect multiplication of decimals to that of whole numbers (MP7), look for correspondences between geometric diagrams and arithmetic calculations, and use these connections to calculate
products of various decimals.
Learning Goals
Teacher Facing
• Coordinate area diagrams and vertical calculations that represent the same decimal multiplication problem.
• Interpret different methods for computing the product of decimals, and evaluate (orally) their usefulness.
• Justify (orally, in writing, and through other representations) where to place the decimal point in the product of two decimals with multiple non-zero digits.
Student Facing
Let’s look at some ways we can represent multiplication of decimals.
Student Facing
• I can use area diagrams to represent and reason about multiplication of decimals.
• I can use place value and fractions to reason about multiplication of decimals.
Print Formatted Materials
Teachers with a valid work email address can click here to register or sign in for free access to Cool Down, Teacher Guide, and PowerPoint materials.
Student Task Statements pdf docx
Cumulative Practice Problem Set pdf docx
Cool Down Log In
Teacher Guide Log In
Teacher Presentation Materials pdf docx
Additional Resources
Google Slides Log In
PowerPoint Slides Log In | {"url":"https://im.kendallhunt.com/MS_ACC/teachers/1/3/16/preparation.html","timestamp":"2024-11-09T22:39:24Z","content_type":"text/html","content_length":"82328","record_id":"<urn:uuid:f0c4e032-aaa5-4b3b-a97d-b3ea886c60c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00081.warc.gz"} |
Refractive Index Calculator both simple and complexRefractive Index Calculator both simple and complex - ASRMETA
Inroduction to refractive index calculator
The refractive index describes the change in speed of light as it enters from air into the material under observation. Therefore, the refractive index is a material property and everything we see
around in our daily lives is composed of different materials. All of the materials have a distinct refractive index and usually, this value of the refractive index varies depending upon the
wavelength of light. This phenomenon is called dispersion of light, where the velocity of wave in a medium depends upon the frequency of the wave and these quantities can be related to the speed of
light and wavelength, respectively. The refractive index (n) can be calculated using a simple formula that relates the speed of light in a material (v), and the speed of light in the air (c).
The speed of light in air is close to the speed of light in a vacuum, therefore, both of these values will yield almost the same result. The refractive index value from this formula will be in the
range of 1 ≤ n ≤ ∞.
Figure 1: An example of the refractive index calculation from the speed of light.
However, normally, all the materials show a complex refractive index value (N). This complex value of the refractive index is related to two properties of a material known as relative permittivity (ε
[r]) and relative permeability (μ[r]). Permittivity and permeability define the ease of setting up electric and magnetic polarizations, respectively, in a material. The refractive index can be found
in terms of these quantities as follows.
\[N=\sqrt{\mu_r \epsilon_r}\]
Both relative permittivity and relative permeability are also complex quantities and the refractive index calculator presented here also take that into account.
Instructions on using refractive index calculator
1. Select the refractive index calculator depending upon your desired calculations. For simple calculations involving the speed of light in a material and speed of light in air, the default
calculator (“Refractive index from speed of light”) will work. For calculations involving permittivity and permeability, select “Complex refractive index from permittivity and permeability”.
2. You will need to enter the real and complex parts of relative permittivity and relative permeability separately. The result will also be displayed in real and imaginary values, separately.
3. For small values of speed of light in a material, the refractive index will yield high values, which might not be accurate. REMEMBER, the speed of light you have to enter should be in meters per
second (m/s).
4. For the cases where the imaginary values of relative permittivity and relative permeability cancel each other, the resultant imaginary part refractive index will have no value, as it should be.
5. The refractive index calculator calculates the refractive index at a particular wavelength. For calculation at different wavelength, reinsert the new values.
6. This calculator is a dynamic calculator, i.e., the results which change when new values are inserted or already entered values are changed.
Further readings
If you liked this post you might be interested in reading the following posts. | {"url":"https://www.asrmeta.com/refractive-index-calculator-both-simple-and-complex/","timestamp":"2024-11-03T02:51:54Z","content_type":"text/html","content_length":"173440","record_id":"<urn:uuid:97372468-c580-49d3-9db9-b6c0db86b265>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00642.warc.gz"} |
Gordon growth model
Gordon growth model is a well-known model for the calculation of the intrinsic price of the shares. This model is widely accepted worldwide because it considers the realization of the return in the
form of a dividend. Further, this model is dependent on the dividend policy.
Although, there are certain assumptions of the model, and one of them is the constant growth of the dividend. This is the reason that this model is used for companies that have stable business
processes. Further, it’s easy to compare the calculated intrinsic value with the value calculated by the model. If the value calculated by GGM is more and the current market price of the security is
low, we conclude that security is under-valued and recommend purchase. On the other hand, if the value calculated by GGM is less and the security’s current market value is high, we conclude that the
stock is overvalued and does not recommend a purchase.
Another assumption of this model is that the business remains functional in perpetuity. In other words, the business keeps making payments for the dividend. Although this assumption might be somewhat
unrealistic, still it provides an enhanced valuation of the earning capacity.
Two stages of dividend growth model
There are two-stage of the dividend growth model that includes,
1- ) Predicting the rate of dividend growth.
2- ) Discounting of the forecasted dividend growth.
Let’s discuss these stages of the dividend growth model in detail.
Predicting the rate of a dividend growth
This is the first stage of applying the dividend growth model. In this stage, the management predicts the rate of dividend growth. The management needs to exercise due judgment in predicting the rate
of a dividend because it’s a subjective matter and one of the most important inputs in the process of calculating stock valuation.
For instance, the company can decide on a 5% increase in dividend each year. It is important to note that this 5% remains the same for the rest of life (as an assumption).
Discounting of the forecasted dividend growth
Discounting of the forecasted stream of dividends is done to bring values into today’s terms. This helps to assess the intrinsic valuation of the security in today’s terms. Discounting is done to
make the valuation realistic because we compare calculated valuation under Gordon Growth Model with the current trading price.
The companies can use the cost of capital or required rate of return to discount the future’s dividend stream.
Constant growth dividend discount model
Gordon Growth Model is also termed as a constant growth model because it assumes a constant growth rate in the future stream of dividends. It’s one of the major assumptions for the Gordon model.
Gordon growth model terminal value
As we understand that this model assumes a future stream of dividends in perpetuity. It’s technically impossible to predict the dividend stream for the business’s whole (infinite) life. So, we make
use of the perpetuity concept to calculate the dividend in the infinite life of the business. This value is called terminal value because we terminate cash flows at the time of incorporating terminal
Gordon growth model example
Suppose the current share price of the company is $105. The expected rate of growth is 5%, and the required rate of return is 5%. Suppose we apply the formula for the calculation of the Gordon growth
model. We get a valuation amounting to $100. It means the intrinsic value of the company’s share is less than the current share price. Hence, the current share price is over-valued in the market. So,
we do not recommend the purchase of the shares.
Leave a Comment | {"url":"https://tothefinance.com/gordon-growth-model/","timestamp":"2024-11-05T06:41:29Z","content_type":"text/html","content_length":"81139","record_id":"<urn:uuid:980da822-702a-4265-b58b-bc7d0ceb82fa>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00556.warc.gz"} |
MSU Events Calendar
Math Seminar Series (Conferences / Seminars / Lectures)
5:00pm Student Geometry/Topology
Speaker: Keshav Sutrave
Title: A hot approach to the Hodge Theorem (Heat Equation)
Date: 02/26/2020
Time: 4:10 PM - 5:00 PM
Place: C517 Wells Hall
The Hodge theorem is a result regarding Laplace's equation for differential forms on a Riemannian manifold (I will introduce the setup for this). Like many celebrated theorems in geometry, it
gives a calculable geometric insight on the topology of the space. Normally, the method for solving this PDE is an "elliptic equation" procedure, but I will instead show a heat equation
(parabolic) approach to the problem. We will see that the topology emerges in the long time behavior of the heat flow. Further work with this approach leads to results such as the
Chern-Gauss-Bonnet theorem, and (almost literally) draws a line connecting geometry and topology.
more information...
Location: C517 Wells Hall [map]
Price: free
Sponsor: Department of Mathematics
Contact: Department of Mathematics | {"url":"https://events.msu.edu/main.php?view=event&eventid=1582739020378&timebegin=2020-02-26%2016:10:00","timestamp":"2024-11-02T08:40:22Z","content_type":"text/html","content_length":"34056","record_id":"<urn:uuid:87bf2b2b-d82f-48b4-b3ba-305bea107698>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00804.warc.gz"} |
The Stories of Little Carl
Year 1784, Town of Braunschweig, Germany.
It was a hot day. In the town’s elementary school, a hundred or so 7-year-olds were studying in the first grade classroom. It was a typical-sized class of that age. When I say studying, they were
reading, talking and mostly making noises. The teacher became impatient while trying to keep the class in order. So he decided to take a different approach. He told the class he would leave them a
math problem. They could go outside to play only after they finished the problem.
So he turned to the blackboard, and wrote down:
The teacher wanted to give the little noise-makers a difficult problem for their age to keep them quiet for at least some time. He was satisfied to see the children lower their heads and start to
perform the calculations by hand. But then,
One boy raised his hand right away: “I know the answer.”
The teacher thought this was just a typical boy’s behavior when he didn’t know the answer. “Right. what’s your answer, Smarty?”
The teacher was very surprised. He still did not believe the boy got the answer so quickly by himself. “Are you sure you are right? How did you get the answer?”.
“If you add the first number 1 and the last number 100, that is 101. Add the second number 2 and the second last number 99, it’s 101 too.”
1 + 100 = 101
2+ 99 = 101
3 + 98 = 101
“There are 100 numbers, so there are 50 pairs. 50 pairs of 101 is 5050.” Said the boy.
The teacher could not hold back his surprise, “What’s your name son?”
“Carl Friedrich Gauss.”
Carl Friedrich Gauss later became one of the greatest mathematicians in history. Many mathematical and physics formulas are named after him. He is sometimes referred to as the Prince of Mathematics.
Another really interesting story about Gauss is that he solved the puzzle of his own birthdate. According to wikipedia, Gauss was born to poor parents. His mother was illiterate and did not record
his birthdate. She remembered it was a Wednesday and eight days before the Feast of the Ascension (39 days after Easter). Note Easter is celebrated on the first Sunday after the first full moon after
the March equinox. Now do you see the little puzzle here? Gauss solved this by his own calculation.
After I told the stories of Gauss to my son, he was certainly fascinated. So I left him a follow-up question: what’s the answer for 1+2+3+ … + 201? | {"url":"http://magspaces.com/2022/01/01/the-story-of-little-gauss/","timestamp":"2024-11-10T05:25:47Z","content_type":"text/html","content_length":"25583","record_id":"<urn:uuid:c0c975df-0456-4350-8329-ba697446241a>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00073.warc.gz"} |
Year 3
These are the key skills that each child will be working towards in Year 3.
By the end of Year 3 children should be able to confidently:
Number and place value:
• count from 0 in multiples of 4, 8, 50 and 100; find 10 or 100 more or less than a given number
• recognise the place value of each digit in a three-digit number (hundreds, tens, ones)
• compare and order numbers up to 1000
• identify, represent and estimate numbers using different representations
• read and write numbers up to 1000 in numerals and in words solve number problems and practical problems involving these ideas.
Addition and Subtraction:
• add and subtract numbers mentally, including:
• a three-digit number and ones a three-digit number and tens
• a three-digit number and hundreds
• add and subtract numbers with up to three digits, using formal written methods of columnar addition and subtraction
• estimate the answer to a calculation and use inverse operations to check answers
• solve problems, including missing number problems, using number facts, place value, and more complex addition and subtraction.
Multiplication and division:
• recall and use multiplication and division facts for the 3, 4 and 8 multiplication tables
• write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one-digit numbers, using mental
and progressing to formal written methods
• solve problems, including missing number problems, involving multiplication and division, including positive integer scaling problems and correspondence problems in which n objects are connected
to m objects.
• count up and down in tenths; recognise that tenths arise from dividing an object into 10 equal parts and in dividing one-digit numbers or quantities by 10
• recognise, find and write fractions of a discrete set of objects: unit fractions and non-unit fractions with small denominators
• recognise and use fractions as numbers: unit fractions and non-unit fractions with small denominators
• recognise and show, using diagrams, equivalent fractions with small denominators
• add and subtract fractions with the same denominator within one whole [for example, 5/7 + 1/7 = 6/7 ]
• compare and order unit fractions, and fractions with the same denominators
• solve problems that involve all of the above.
• measure, compare, add and subtract: lengths (m/cm/mm); mass (kg/g); volume/capacity (l/ml)
• measure the perimeter of simple 2-D shapes
• add and subtract amounts of money to give change, using both £ and p in practical contexts
• tell and write the time from an analogue clock, including using Roman numerals from I to XII, and 12-hour and 24-hour clocks
• estimate and read time with increasing accuracy to the nearest minute; record and compare time in terms of seconds, minutes and hours; use vocabulary such as o’clock, a.m./p.m., morning,
afternoon, noon and midnight
• know the number of seconds in a minute and the number of days in each month, year and leap year
• compare durations of events [for example to calculate the time taken by particular events or tasks].
Geometry - properties of shapes:
• draw 2-D shapes and make 3-D shapes using modelling materials; recognise 3-D shapes in different orientations and describe them
• recognise angles as a property of shape or a description of a turn
• identify right angles, recognise that two right angles make a half-turn, three make three quarters of a turn and four a complete turn; identify whether angles are greater than or less than a
right angle
• identify horizontal and vertical lines and pairs of perpendicular and parallel lines.
• interpret and present data using bar charts, pictograms and tables
• solve one-step and two-step questions [for example, ‘How many more?’ and ‘How many fewer?’] using information presented in scaled bar charts and pictograms and tables. | {"url":"https://www.buckleshamprimaryschool.co.uk/page/?title=Year+3&pid=69","timestamp":"2024-11-05T13:50:52Z","content_type":"text/html","content_length":"47467","record_id":"<urn:uuid:5319e121-688e-4a59-81ad-a8e10434c4c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00351.warc.gz"} |
EViews Help: structure
Display the factor structure matrix.
Shows the factor structure matrix containing the correlations between the variables and factors implied by an estimated factor model. For orthogonal factors, the structure matrix is equal to the
loadings matrix.
factor f1.ml group01
displays and prints the factor structure matrix for the estimated factor object F1. | {"url":"https://help.eviews.com/content/factorcmd-structure.html","timestamp":"2024-11-12T06:13:42Z","content_type":"application/xhtml+xml","content_length":"9462","record_id":"<urn:uuid:2c6c2246-6f82-4e59-b283-0ed74c5bebed>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00732.warc.gz"} |
dy/dx=2y+6, y(1)=6 … - QuestionCove
Ask your own question, for FREE!
Mathematics 29 Online
OpenStudy (anonymous):
dy/dx=2y+6, y(1)=6 (calculus II) the answer is y(x)=9e^(2x-2)-3
Still Need Help?
Join the QuestionCove community and study together with friends!
OpenStudy (anonymous):
explain what u need to find
zepdrix (zepdrix):
\[\large \frac{dy}{dx}=2y+6\] So this appears to be separable :) Divide each side by (2y+6), and move the dx to the other side, think of this step as multiplication. \[\large \frac{dy}{2y+6}=dx\]
Understand that part? From here we can integrate both sides.
OpenStudy (anonymous):
yes, got that part!
zepdrix (zepdrix):
\[\large \int\limits \frac{dy}{2y+6}=\int\limits dx\]Integrating the right side is pretty straight forward,\[\large \int\limits\limits \frac{dy}{2y+6}=x+c\] Do you understand how to do the left side?
You can apply a `u sub` if it will help.
OpenStudy (anonymous):
go ahead ur doing great @zepdrix
Still Need Help?
Join the QuestionCove community and study together with friends!
OpenStudy (anonymous):
all I need is the steps, i understand most of it.
zepdrix (zepdrix):
Just in case you're having trouble with the left side, a u substitution would look like this, \[\large u=2y+6 \qquad \qquad \qquad \frac{1}{2}du=dy\] Which changes our integral to,\[\large \frac{1}
{2}\int\limits\frac{du}{u}=x+c\] Giving us,\[\large \frac{1}{2}\ln|2y+6|=x+c\]
OpenStudy (anonymous):
I already got that far. Now to put in the form y=9e^(2x-2)-3.
zepdrix (zepdrix):
Multiply both sides by 2,\[\large \ln|2y+6|=2x+c\] We distributed the 2 to each term, but we simply `absorbed` the other 2 into the C, since it's an arbitrary value. Now exponentiate both sides,
`rewrte both sides as exponents with a base of e`.\[\huge e^{\ln|2y+6|}=e^{2x+c}\]
zepdrix (zepdrix):
The exponential and the logarithm are `inverse` operations of one another, so on the left they will simply "undo" one another.\[\huge 2y+6=e^{2x+c}\]
Still Need Help?
Join the QuestionCove community and study together with friends!
OpenStudy (anonymous):
oh, i get it!
zepdrix (zepdrix):
Understand how to solve for C from here? :)
OpenStudy (anonymous):
i forgot that c was a constant.
OpenStudy (anonymous):
and could be treated like that. now it's easy i think.
OpenStudy (anonymous):
but what about the 9?
Still Need Help?
Join the QuestionCove community and study together with friends!
zepdrix (zepdrix):
Are you not coming up with a 9? :o
OpenStudy (anonymous):
OpenStudy (anonymous):
There's what I've got so far.
zepdrix (zepdrix):
Ok cool. You need to solve for C at some point :) That's where your 9 will come from.
OpenStudy (anonymous):
Don't fully comprehend, could you please help a little more.
Still Need Help?
Join the QuestionCove community and study together with friends!
zepdrix (zepdrix):
At the start they gave us a coordinate pair that we can NOW use to solve for C.\[\large y(1)=6 \qquad \qquad (1,6)\] Plug the 1 in for X, the 6 in for Y, and solve for C! :) I have a feeling this
will be a tad easier to work with if we fiddle with the C first. Using a rule of exponents we can write the C like this. \[\huge 2y+6=e^{2x}e^c\] But now recognize, that e^c is just another arbitrary
constant value.\[\huge 2y+6=Ce^{2x}\]Then subtract, divide,...\[\huge y=Ce^{2x}-3\]Again the 2 got absorbed into the C. This isn't totally necessary. I just think the problem will be a lot easier to
work with from this form.
zepdrix (zepdrix):
\[\huge (\color{royalblue}{1},\color{orangered}{6}) \qquad \rightarrow \qquad \color{orangered}{y}=Ce^{2\color{royalblue}{x}}-3\]Understand how to plug these values in? :)
OpenStudy (anonymous):
I finally got it!!! Thanks a lot!
zepdrix (zepdrix):
yay! \c:/
OpenStudy (anonymous):
try solving mine plz
Still Need Help?
Join the QuestionCove community and study together with friends!
zepdrix (zepdrix):
The silly bread problem? c:
OpenStudy (anonymous):
Can't find your answer? Make a FREE account and ask your own questions, OR help others and earn volunteer hours!
Join our real-time social learning platform and learn together with your friends!
Can't find your answer? Make a FREE account and ask your own questions, OR help others and earn volunteer hours!
Join our real-time social learning platform and learn together with your friends! | {"url":"https://questioncove.com/updates/513e6703e4b029b0182c111c","timestamp":"2024-11-07T12:43:28Z","content_type":"text/html","content_length":"42501","record_id":"<urn:uuid:cbba4e5c-b151-4182-b34a-82d2d1bbf1d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00686.warc.gz"} |
Science:Math Exam Resources/Courses/MATH307/December 2005/Question 03 (a)
MATH307 December 2005
Work in progress: this question page is incomplete, there might be mistakes in the material you are seeing here.
• Q1 (a) • Q1 (b) • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q3 (c) • Q3 (d) • Q3 (e) • Q3 (f) • Q3 (g) • Q4 (a) • Q4 (b) • Q5 (a) • Q5 (b) • Q5 (c) • Q6 (a) • Q6 (b) • Q7 (a) • Q7 (b) • Q8 (a) • Q8 (b) •
Q8 (c) •
Question 03 (a)
Decide whether the following statement is true or false. You need not give a reason. All matrices in this question are square ${\displaystyle n\times n}$.
If ${\displaystyle \displaystyle A=A^{-1}}$ then every eigenvalue of A is either 1 or -1.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is
correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
• If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you
are stuck or if you want to check your work.
• If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem
and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.
Solution 1
Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies.
The answer is true.
We are given that ${\displaystyle \displaystyle A=A^{-1}}$ which implies that ${\displaystyle \displaystyle A^{2}=I}$. Let ${\displaystyle \displaystyle p(\lambda )}$ is the characteristic polynomial
of A and ${\displaystyle \displaystyle m(\lambda )}$ its minimal polynomial. Notice that ${\displaystyle \displaystyle m(\lambda )}$ divides ${\displaystyle \displaystyle q(\lambda )=\lambda ^{2}-1}$
(since the matrix itself satisfies its minimal polynomial by the Cayley-Hamilton theorem). Furthermore, p and m also share roots and these form the only roots. Hence all the roots of p must be a
subset of the roots of ${\displaystyle \displaystyle q(\lambda )}$ which are ${\displaystyle \displaystyle \pm 1}$. These roots are the eigenvalues of the matrix A and so the matrix can only have
these eigenvalues.
Solution 2
Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies.
This statement is true.
If λ is an eigenvalue of A, then 1/λ is an eigenvalue of A^-1 with the same eigenvectors (Av = λv, v = λA^-1v, v/λ = A^-1v). Hence
${\displaystyle \lambda v=Av=A^{-1}v={\frac {1}{\lambda }}v}$
or, equivalently (since v ≠ 0),
${\displaystyle \displaystyle \lambda ^{2}=1,}$
so λ = -1 or λ = 1.
Click here for similar questions
MER CH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER RS flag, MER Tag Eigenvalues and eigenvectors, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag | {"url":"https://wiki.ubc.ca/Science:Math_Exam_Resources/Courses/MATH307/December_2005/Question_03_(a)","timestamp":"2024-11-07T19:58:09Z","content_type":"text/html","content_length":"51197","record_id":"<urn:uuid:f426091a-5c47-4c1b-8d2d-734d987f3a42>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00207.warc.gz"} |
An automaker recently acquired a windshield manufacturer. This type of an acquisition is called a vertical merger.
1. ( T or F ) An automaker recently acquired a windshield manufacturer. This type of an acquisition is called a vertical merger.2. ( T or F ) Black Teas recently acquired Green Teas in a transaction
that had a net present value of $1.23 million. The $1.23 million is referred to as synergy.3. ( T or F ) Short-term financing may be riskier than long-term financing since, during periods of tight
credit, the firm may not be able to rollover (renew) its debt.4. ( T or F ) All other things equal, for a given percentage change in the discount rate on a bond, the price of the bond will change by
a greater percentage the shorter its maturity.5. ( T or F ) You own a July $15 call on ABC stock. Assume today is April 20 and the price of the stock is $13. The call option is in-the-money.6. ( T or
F ) A lockbox is a special safe used by a firm that can only be opened at prespecified times of the day.7. ‘The law of one price’ says that the level of exchange rate should adjust such that
consumers can pay the same price wherever they buy. Is the theory consistent with the reality? Explaincarefully.8. You want to know the present value of a lump sum, $50,000, to be paid in 2.5 years
from now. What is its present value at therate of 5.7% under continuous compounding?9. You will deposit $2,000 at the end of each of next 5 years. If the interest rate is 8% (annual compounding), how
much will you have accumulated in 25 years?10. ABC Company recently issued two types of bonds. The first issue consisted of 20 year straight debt with an 8 percent annualcoupon. The second issue
consisted of 20 year bonds with a 6 percent annual coupon and attached warrants. Both issues sold attheir $1,000 par values. What is the implied value of the warrants attached to each bond?11. A bond
with 20 detachable warrants has just been offered for sale at $1,000. The bond matures in 20 years and has an annualcoupon of $115. Each warrant gives the owner the right to purchase two shares of
stock in the company at $18 per share.Ordinary bonds (with no warrants) of similar quality are priced to yield 17 percent. What is the value of one warrant?(The following information applies to the
next four questions)Pappy’s Potato has come up with a new product, the Potato Pet (they are freeze-dried to last longer). Pappy’s paid $120,000 for a marketing survey to determine the viability of
the product. It is felt that Potato Pet will generate sales of $575,000 per year. The fixed costs associated with this will be $179,000 per year, and variable costs will amount to 20 percent of
sales. The equipment necessary for production of the Potato Pet will cost $620,000 and will be depreciated in a straight-line manner for the four years of the product life (as with all fads, it is
felt the sales will end quickly). This is the only initial cost for the production. Pappy’s is in a 40 percent tax bracket and has a required return of 13 percent.12. What is the initial () cash
outflows?13. What is the operating cash inflows at ?14. What is the total cash flows (operating cash flows plusterminal cash flows) at ?15. Compute the NPV and IRR of the project. Should this new
project be accepted?16. Compute the fair price of the following perpetual bond. Itsfirst interest, $120, will be paid 15 years from now and will beadjusted upward by 3% every year to compensate for
the risk ofinflation. Investors require 8% return on the bond.17. Jane and Tom are searching for their first house. They have saved $45,000 for down-payment. Their mortgage company, offering a
30-year 7.2% loan, suggests that they spend up to $1,750 for monthly mortgage payment. What is the maximum price of a house they can afford to buy? Ignore all the transaction costs.18. Three years
ago, you bought a 12% bond that had 7 years to maturity and a yield to maturity of 12%. Today (after the sixth interest payment), you sold the bond when it is yielding 15%. What is your annual rate
of return for the three year period? All coupon payments are semi-annual, and the par value is $1,000.(The following information applies to the next four questions)Given the following information for
XYZ Co., you want to find the cost of capital (WACC). The firm’s tax rate is 40%. Ignore all the flotation cost.Debt: 8,000 7% coupon bonds outstanding, $1,000 par value, 15 years to maturity,
selling for 98% of par; the bonds make semiannual payments.Preferred stock: 20,000 shares of 7% preferred stock outstanding, $100 par value, currently selling for $90 per share.Common Stock: 300,000
shares outstanding, selling for $50 per share; the beta is 1.25. Currently, the market risk premium is 6%, and the risk-free rate is 3%.19. What is the cost of debt?20. What is the cost of preferred
stock?21. What is the cost of common stock?22. Given the above information, what is the cost of capital?(The following facts apply to the next two questions about aconvertible bond making semiannual
payments.)Conversion price$52/shareCoupon rate6.5%Par value$1,000Yield on nonconvertible debenturesof same quality8.5%Maturity25 yearsMarket price of stock$42 /share23. If the price of the
convertible bond is $900, what is thevalue of the call option embedded in the convertible bond?24. If you buy the convertible bond, are you willing to convert it now? Why or why not? Please
explain.25. You just bought a machine on a credit terms: 2/10, net 40. If your cost of capital is 12%, would it be rational for you to take the cash discount? Why or why not? Explain carefully.26. If
investors become more risk averse, what will happen to the price of stocks in general? Why?27. Suppose that you are a financial manager of a company thatexports mainly to England. You expect to
receive a huge paymentin British pounds from your customer in a couple of months. Toavoid the potential adverse move of British pound, you havedecided to completely hedge with forward contracts. What
is theadvantage of using forward contracts? What is the disadvantage?28. Is it possible for a firm to have too much cash? Why would shareholders care if a firm accumulates large amounts of cash?29.
Suppose that your company’s stock has been rising higher on a strong momentum. Will it be a rational decision for your firm to decide to issue stock to finance its projects? Why or why not? Explain
carefully.30. Suppose that you are a financial manager of a manufacturingcompany. Would you raise or lower the cash balance of yourcompany if interest rates are rising in the financial market? Why? | {"url":"https://essayfountain.com/an-automaker-recently-acquired-a-windshield-manufacturer-this-type-of-an-acquisition-is-called-a-vertical-merger/","timestamp":"2024-11-04T07:36:28Z","content_type":"text/html","content_length":"93891","record_id":"<urn:uuid:43b6ab8b-a295-4fb1-985b-e07ebbaf2787>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00676.warc.gz"} |
Видеотека: Yu. V. Eliyashev, Tropical Hodge Theory
Аннотация: The Hodge theory on complex manifolds is a classical example of application of analytical methods in algebraic geometry. One of the main ideas of the tropical geometry is that there is
should be a tropical analog of an object from the complex geometry. Following this idea Lagerberg introduced tropical currents and differential forms, Itenberg, Katzarkov, Mikhalkin, Zharkov
introduced tropical cohomology theory. Based on these works, Jell, Shaw, Smacka constructed tropical de Rham cohomology theory. I my talk I will discuss how to construct a Hodge theory on Tropical
varieties and how it resembles classical Hodge theory.
Язык доклада: английский | {"url":"https://m.mathnet.ru/php/presentation.phtml?option_lang=rus&presentid=24746","timestamp":"2024-11-07T12:13:27Z","content_type":"text/html","content_length":"8343","record_id":"<urn:uuid:cee679fb-2756-4522-9b0a-268426ce9bc6>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00086.warc.gz"} |
Authentication / Hash transform data.
This structure contains data relating to an authentication/hash crypto transforms. The fields op, algo and digest_length are common to all authentication transforms and MUST be set.
Definition at line 291 of file rte_crypto_sym.h.
Length of the digest to be returned. If the verify option is set, this specifies the length of the digest to be compared for the session.
It is the caller's responsibility to ensure that the digest length is compliant with the hash algorithm being used. If the value is less than the maximum length allowed by the hash, the result shall
be truncated.
Definition at line 309 of file rte_crypto_sym.h.
uint32_t add_auth_data_length
The length of the additional authenticated data (AAD) in bytes. The maximum permitted value is 65535 (2^16 - 1) bytes, unless otherwise specified below.
This field must be specified when the hash algorithm is one of the following:
• For GCM (RTE_CRYPTO_AUTH_AES_GCM). In this case, this is the length of the Additional Authenticated Data (called A, in NIST SP800-38D).
• For CCM (RTE_CRYPTO_AUTH_AES_CCM). In this case, this is the length of the associated data (called A, in NIST SP800-38C). Note that this does NOT include the length of any padding, or the 18
bytes reserved at the start of the above field to store the block B0 and the encoded length. The maximum permitted value in this case is 222 bytes.
For AES-GMAC (RTE_CRYPTO_AUTH_AES_GMAC) mode of operation this field is not used and should be set to 0. Instead the length of the AAD data is specified in additional authentication data length
field of the rte_crypto_sym_op_data structure
Definition at line 320 of file rte_crypto_sym.h. | {"url":"https://doc.dpdk.org/api-17.05/structrte__crypto__auth__xform.html","timestamp":"2024-11-04T11:27:09Z","content_type":"application/xhtml+xml","content_length":"11286","record_id":"<urn:uuid:810ad65e-bbaa-4d9b-ac46-2fdd749ef37b>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00707.warc.gz"} |
Mathematical Deductive Reasoning Problems
These deductive reasoning problems connect with Arizona Math Standards in the areas of Algebraic Thinking, Number Operations in Base Ten, and Mathematical Practices. What is the connection with the
Golden Rule? In life, we are usually presented with many choices. Circumstances often lead us to narrow those choices to a few, often even to one. In relating to others, choosing to use the Golden
Rule as a guide is usually the best choice.
allow students to work through a list of clues and perform math operations correctly to select a number which answers the given clue. Initially, a great many numbers are correct answers. Each clue
reduces the group of correct options. Eventually, mathematical reasoning eliminates all but one possible correct solution. The student’s first goal is to seek a range of correct answers. Secondly,
they recognize which possible number is the only correct answer. The third objective is to identify the ‘Golden Clue’ which eliminated all other possible correct responses.
• Scratch paper, a pencil, and critical thinking skills are the only materials needed.
• Students can use quarter sheets of scrap paper or notebook paper, numbering down from 1 to 10.
• They are to write a number which gives a correct answer for the clue given.
• When the next clue is given, students whose answer remains correct make no change in their selected number. If the answer becomes incorrect, students simply change their answer to a correct
number. In numbers with 2 or more digits, students may change one digit to make their answer correct. Very seldom is it necessary to change more than one digit, and keeping changes to a minimum
utilizes the best mental math reasoning.
• The teacher polls the students for correct answers during the early clues to establish a range of correct possibilities. As they progress through the clues,possibilities are eliminated until only
one choice remains.
The second part of the activity is the identification of the clue that ruled out all possibilities but one. Students circle this ‘Golden Clue.’ Explaining the rationale is a good practice for the
first few times to build and check deductive reasoning skills.
Grading is easy. The teacher just walks around the room and uses a checklist to mark those with the correct solution. One point is given for the correct answer (which all students should find by clue
10) and a second point is given for determining the Golden Clue (a more difficult task).
Modifications: Give a visual clue for young students by making a list of the options and having them cross off the numerals as they are eliminated. This strengthens the
concept of number structure. Having students work with a partner may help and underscores the concept of AGREEment. The rationale for the solutions is included and may be used to help students
understand the process. | {"url":"https://goldenruleeducation.org/tag/math/","timestamp":"2024-11-09T17:00:33Z","content_type":"text/html","content_length":"87224","record_id":"<urn:uuid:1804a376-f84c-4e1c-bf4a-2a175c3324fd>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00260.warc.gz"} |
Advanced undergrad/grad book for classical mechanics as macro limit of QM
Advanced undergrad/grad book for classical mechanics as macro limit of QM
In summary, the conversation discusses recommendations for understanding limits and quantisation in non-relativistic quantum mechanics. The Landau and Lifgarbagez book is suggested as a good resource
for explaining these concepts, specifically the chapter on Quasi-Classicality and the use of Bohr and Sommerfeld quantisation.
Hi folks. I'm wondering who does a good job of explaining this limit, preferably with a good set of examples. It doesn't need to be too basic, but it'd be nice if it went through the phase space
stuff a little (I get the impression that my grad prof didn't do a great job with some details based on questions that come up when I read through such things).
Thanks in advance.
Landau and Lifgarbagez book on non relatavistic QM has a good chapter on Quasi-Classicality, it gives plenty of uses for it in later chapters as well.
L&L do a good job of explaining Bohr and Sommerfeld quantisation which is what I think you're looking for when you say 'through the phase space stuff a little'.
FAQ: Advanced undergrad/grad book for classical mechanics as macro limit of QM
1. What is the difference between classical mechanics and quantum mechanics?
Classical mechanics is a branch of physics that describes the motion and behavior of macroscopic objects, while quantum mechanics is a branch of physics that describes the behavior of microscopic
particles. Classical mechanics follows deterministic laws, while quantum mechanics follows probabilistic laws.
2. Why is it important to understand the macro limit of quantum mechanics?
Understanding the macro limit of quantum mechanics helps bridge the gap between the microscopic and macroscopic worlds. It also allows for a better understanding of how classical mechanics emerges
from the underlying quantum laws.
3. What topics are typically covered in an advanced undergrad/grad book for classical mechanics as macro limit of QM?
An advanced undergrad/grad book for classical mechanics as macro limit of QM may cover topics such as Hamiltonian mechanics, Lagrangian mechanics, classical field theory, and the correspondence
principle between classical and quantum mechanics.
4. Is prior knowledge of quantum mechanics necessary to understand the macro limit of QM?
Yes, prior knowledge of quantum mechanics is necessary to understand the macro limit of QM. A solid understanding of basic quantum principles and mathematical techniques is necessary to grasp the
concepts covered in an advanced book on the macro limit of QM.
5. How can understanding the macro limit of QM help in other fields of science?
Understanding the macro limit of QM can have applications in other fields of science, such as condensed matter physics, astrophysics, and chemical physics. It can also provide insights into the
behavior of complex systems and aid in the development of new technologies. | {"url":"https://www.physicsforums.com/threads/advanced-undergrad-grad-book-for-classical-mechanics-as-macro-limit-of-qm.588722/","timestamp":"2024-11-06T20:56:39Z","content_type":"text/html","content_length":"78546","record_id":"<urn:uuid:6c5e374b-cc13-4c0b-be10-743b1f065576>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00868.warc.gz"} |
Mastering Modulus in Java – A Comprehensive Guide to Effective Use
Introduction to Modulus in Java
In Java programming, the modulus operator is a useful tool that allows us to perform calculations based on remainders and cyclic patterns. It is denoted by the % symbol and is used to find the
remainder of a division operation. The purpose of the modulus operator is to assist in various tasks such as checking for divisibility, handling repeating patterns, simplifying arithmetic
calculations, and more.
Understanding the Modulus Operator
The syntax of the modulus operator in Java is as follows: num1 % num2. Here, num1 is the dividend and num2 is the divisor. The result of the modulus operation is the remainder when num1 is divided by
It’s important to note that the modulus operator can be used with different types of operands, including integers and floating-point numbers. When applied to integers, the result will also be an
integer, whereas when applied to floating-point numbers, the result will be a floating-point number.
Benefits of Using Modulus in Java Programming
Using the modulus operator in Java programming offers several benefits:
Efficiently determine divisibility: The modulus operator helps in determining whether a number is divisible by another number. By checking if the result of the modulus operation is zero, we can
easily determine if one number is divisible by another.
Finding remainders and cyclic patterns: The modulus operator enables us to find the remainder of a division operation. This can be useful in various scenarios, such as determining if a number is even
or odd, or finding the position of an element in a circular structure.
Simplify arithmetic calculations: Using modulus can simplify arithmetic calculations. For example, to calculate the last digit of a number, we can use the modulus operator with a divisor of 10.
Useful in tasks involving arrays and loops: Modulus is often used in tasks involving arrays and loops. It can help in creating repeating patterns, implementing cyclic shifts, and handling periodic
Applying Modulus in Real-World Scenarios
Let’s explore some real-world scenarios where the modulus operator can be applied in Java programming:
Checking for even/odd numbers: To determine if a number is even or odd, we can use the modulus operator with a divisor of 2. If the result is 0, the number is even; otherwise, it is odd.
Implementing repeating patterns and sequences: Modulus can be used to implement repeating patterns and sequences. By using the modulus operator with the length of the pattern, we can cycle through
the elements in a pattern and repeat them.
Handling date and time calculations: Modulus is useful in handling date and time calculations. For example, to determine the day of the week for a given date, we can use the modulus operator with a
divisor of 7, where each remainder represents a specific day of the week.
Working with circular structures and periodic events: Modulus is often used in tasks involving circular structures and periodic events. For example, when iterating over an array in a circular manner,
the modulus operator can be used to wrap the index around when it exceeds the array size.
Best Practices for Using Modulus in Java
When using the modulus operator in Java, it is important to follow these best practices:
Ensuring correct usage with negative numbers: When dealing with negative numbers, it’s crucial to ensure correct usage of the modulus operator. Java’s default behavior of the modulus operator may not
produce the desired result for negative operands. To handle negative numbers correctly, you can add the divisor to the negative result until it becomes positive.
Handling potential division by zero errors: Division by zero is not allowed in Java, and using the modulus operator with a divisor of zero will result in an ArithmeticException. It is important to
handle such potential errors by checking for zero divisors before performing the modulus operation.
Considering performance implications of modulus operations: Modulus operations can be computationally expensive, especially when performed inside loops or in performance-critical sections of code.
Consider minimizing the usage of modulus operations or optimizing them when necessary for better performance.
Writing modular-friendly code for better readability and maintainability: When using modulus in your code, consider writing modular-friendly code that is easy to read and maintain. Modular code
emphasizes self-contained functions or methods that perform specific tasks, enabling easier testing, troubleshooting, and code reuse.
Modulus and Mathematical Concepts
Modulus in Java programming has connections with several mathematical concepts:
Modulus in number theory and modular arithmetic: Modulus is a fundamental concept in number theory and modular arithmetic. It involves the study of remainders and their properties. Modular arithmetic
has wide-ranging applications in computer science, cryptography, and algorithm design.
Exploring congruence and its applications: Modulus plays a vital role in exploring congruence and its applications. Congruence relates to the equivalence of remainders modulo a given number and has
applications in number theory, algebraic structures, and cryptography.
Leveraging modulus in cryptographic algorithms: Cryptography heavily relies on modular arithmetic and the properties of the modulus operator. Many cryptographic algorithms, such as RSA, utilize
modular arithmetic and the concept of congruence to provide security and confidentiality.
Advanced Modulus Techniques and Examples
Beyond the basic usage, here are some advanced techniques and examples that leverage the power of the modulus operator in Java programming:
Efficiently reversing numbers using modulus: The modulus operator can be used to efficiently reverse numbers. By repeatedly dividing a number by 10, and taking the modulus at each step, we can
construct the reversed number.
Generating random numbers within a specific range: Modulus can be employed to generate random numbers within a specific range. By taking the modulus of a randomly generated number with the desired
range, we can obtain a random number within that range.
Calculating recurring decimal fractions: Modulus enables us to calculate recurring decimal fractions. By repeatedly applying the modulus operator to the remainder of a division operation, we can
identify and extract the recurring part of the fraction.
Implementing cyclic shifts and rotations in arrays: Modulus is useful for implementing cyclic shifts and rotations in arrays. By applying modulus to the shift amount and array size, we can ensure
that shifts wrap around and produce the desired cyclic effect.
Troubleshooting and Common Mistakes with Modulus
While using modulus in Java programming, certain issues and mistakes may arise:
Division by zero errors: Using a divisor of zero with the modulus operator will result in an ArithmeticException being thrown. To avoid this error, ensure that the divisor is not zero before
performing the modulus operation.
Incorrect usage of modulus in conditional statements: Modulus should be used cautiously in conditional statements, as errors may occur due to incorrect usage. Double-check any conditional expressions
involving modulus to ensure they produce the intended results.
Rounding errors and floating-point precision: Modulus operations involving floating-point numbers may introduce rounding errors due to the limited precision of floating-point representations.
Consider using integer operations or rounding techniques to mitigate such precision issues.
Handling edge cases and potential pitfalls: Modulus operations may have specific edge cases or potential pitfalls, especially when dealing with large numbers, extreme divisors, or unconventional
scenarios. Be vigilant and thoroughly test your code to handle such situations.
In conclusion, understanding and mastering the modulus operator in Java programming opens up a range of possibilities for solving various computational problems efficiently. Modulus helps with
determining divisibility, handling remainders and cyclic patterns, simplifying arithmetic calculations, and more. By knowing when and how to apply the modulus operator, you can optimize your code,
improve performance, and unleash the power of modular arithmetic in your Java programs. Embrace this versatile operator and explore its applications in different scenarios to unlock its full
Furthermore, learning the mathematical concepts underlying modulus, such as number theory and modular arithmetic, can expand your problem-solving abilities and enable you to tackle complex
computational challenges. Additionally, exploring advanced modulus techniques and understanding common mistakes and troubleshooting can help you avoid pitfalls and write robust code.
So, embrace the power of modulus in Java programming, experiment with different scenarios, and continue to explore the depths of this versatile operator. Happy coding! | {"url":"https://skillapp.co/blog/mastering-modulus-in-java-a-comprehensive-guide-to-effective-use/","timestamp":"2024-11-05T23:53:01Z","content_type":"text/html","content_length":"114816","record_id":"<urn:uuid:d5cba5b7-669e-49f8-8304-95f2d5088c1f>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00836.warc.gz"} |
Math 4 Wisdom. "Mathematics for Wisdom" by Andrius Kulikauskas. | Research / AlgebraicGeometryTheorems
What is algebraic geometry trying to accomplish?
I'm organizing theorems in Algebraic Geometry that are listed in Wikipedia.
Wikipedia: Theorems in Algebraic Geometry
Abhyankar's lemma allows one to kill tame ramification by taking an extension of a base field. More precisely, Abhyankar's lemma states that if A, B, C are local fields such that A and B are finite
extensions of C, with ramification indices a and b, and B is tamely ramified over C and b divides a, then the compositum AB is an unramified extension of A.
To 'classify' addition theorems it is necessary to put some restriction on the type of function G admitted, such that F(x + y) = G(F(x), F(y)). In this identity one can assume that F and G are
vector-valued (have several components). An algebraic addition theorem is one in which G can be taken to be a vector of polynomials?, in some set of variables. The conclusion of the mathematicians of
the time was that the theory of abelian functions? essentially exhausted the interesting possibilities: considered as a functional equation? to be solved with polynomials, or indeed rational
functions? or algebraic functions?, there were no further types of solution.
Behrend's formula is a generalization of the Grothendieck–Lefschetz trace formula to a smooth algebraic stack over a finite field.
Algebraic curves
Bézout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves, which do not share a common component (that is,
which do not have infinitely many common points). The theorem claims that the number of common points of two such curves is at most equal to the product of their degrees, and equality holds if one
counts points at infinity, points with complex coordinates (or more generally, coordinates from the algebraic closure of the ground field), and if each point is counted with its intersection
Belyi's theorem on algebraic curves states that any non-singular algebraic curve C, defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the
Riemann sphere, ramified at three points only.
Beauville–Laszlo theorem is a result in commutative algebra and algebraic geometry that allows one to "glue" two sheaves over an infinitesimal neighborhood of a point on an algebraic curve.
AF+BG theorem (Max Noether's fundamental theorem) describes when the equation of an algebraic curve in the complex projective plane can be written in terms of the equations of two other algebraic
Abhyankar–Moh theorem states that if {$ \displaystyle L $} is a complex line in the complex affine plane {$ \displaystyle \mathbb {C} ^{2} $}, then every embedding of {$ \displaystyle L $} into {$ \
displaystyle \mathbb {C} ^{2} $} extends to an automorphism of the plane. More generally, the same theorem applies to lines and planes over any algebraically closed field of characteristic zero, and
to certain well-behaved subsets of higher-dimensional complex affine spaces.
Cayley–Bacharach theorem is a statement about cubic curves (plane curves of degree three) in the projective plane P2. Every cubic curve C1 on an algebraically closed field that passes through a given
set of eight points P1, ..., P8 also passes through a certain (fixed) ninth point P9, counting multiplicities.
Chasles–Cayley–Brill formula (also known as the Cayley-Brill formula) states that a correspondence T of valence k from an algebraic curve C of genus g to itself has d + e + 2kg united points, where d
and e are the degrees of T and its inverse.
Chasles' theorem says that if two pencils of curves have no curves in common, then the intersections of those curves form another pencil of curves the degree of which can be calculated from the
degrees of the initial two pencils.
de Franchis theorem is one of a number of closely related statements applying to compact Riemann surfaces, or, more generally, algebraic curves, X and Y, in the case of genus g > 1. The simplest is
that the automorphism group of X is finite (see though Hurwitz's automorphisms theorem). More generally,
• the set of non-constant morphisms from X to Y is finite;
• fixing X, for all but a finite number of such Y, there is no non-constant morphism from X to Y.
Enriques–Babbage theorem states that a canonical curve is either a set-theoretic intersection of quadrics, or trigonal, or a plane quintic.
Faltings's theorem states that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. It was later generalized by replacing Q by any number
Gudkov's conjecture is now a theorem, which states that "a M-curve* of even degree 2d obeys p – n ≡ d2 (mod 8)", where p is the number of positive ovals and n the number of negative ovals of the
Harnack's curve theorem describes the possible numbers of connected components that an algebraic curve can have, in terms of the degree of the curve.
Hodge index theorem for an algebraic surface V determines the signature of the intersection pairing on the algebraic curves C on V. It says, roughly speaking, that the space spanned by such curves
(up to linear equivalence) has a one-dimensional subspace on which it is positive definite (not uniquely determined), and decomposes as a direct sum of some such one-dimensional subspace, and a
complementary subspace on which it is negative definite.
Reiss relation is a condition on the second-order elements of the points of a plane algebraic curve meeting a given line.
Torelli theorem is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (compact Riemann surface) C is determined by its
Jacobian variety J(C), when the latter is given in the form of a principally polarized abelian variety. In other words, the complex torus J(C), with certain 'markings', is enough to recover C. The
same statement holds over any algebraically closed field.
Tsen's theorem states that a function field K of an algebraic curve over an algebraically closed field is quasi-algebraically closed (i.e., C1). This implies that the Brauer group of any such field
vanishes,[1] and more generally that all the Galois cohomology groups Hi(K, K*) vanish for i ≥ 1. This result is used to calculate the étale cohomology groups of an algebraic curve.
Weber's theorem. Consider two non-singular curves C and C′ having the same genus g > 1. If there is a rational correspondence φ between C and C′, then φ is a birational transformation.
Weil reciprocity law is a result holding in the function field K(C) of an algebraic curve C over an algebraically closed field K. Given functions f and g in K(C), i.e. rational functions on C, then f
((g)) = g((f)) where the notation has this meaning: (h) is the divisor of the function h, or in other words the formal sum of its zeroes and poles counted with multiplicity; and a function applied to
a formal sum means the product (with multiplicities, poles counting as a negative multiplicity) of the values of the function at the points of the divisor. With this definition there must be the
side-condition, that the divisors of f and g have disjoint support (which can be removed).
Elliptic curves
Modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms. The theorem states that any elliptic curve over Q can be obtained via a rational map
with integer coefficients from the classical modular curve {$ X_{0}(N) $} for some integer N.
Néron–Ogg–Shafarevich criterion states that if A is an elliptic curve or abelian variety over a local field K and ℓ is a prime not dividing the characteristic of the residue field of K then A has
good reduction if and only if the ℓ-adic Tate module Tℓ of A is unramified.
Raynaud's isogeny theorem relates the Faltings heights of two isogeneous elliptic curves.
Vector bundles, line bundles
Birkhoff–Grothendieck theorem classifies holomorphic vector bundles over the complex projective line. In particular every holomorphic vector bundle over {$ \displaystyle \mathbb {CP} ^{1} $} is a
direct sum of holomorphic line bundles.
Appell–Humbert theorem describes the line bundles on a complex torus or complex abelian variety.
Lange's conjecture is a conjecture about stability of vector bundles over curves.
Lefschetz theorem on (1,1)-classes, named after Solomon Lefschetz, is a classical statement relating holomorphic line bundles on a compact Kähler manifold to classes in its integral cohomology.
Reider's theorem gives conditions for a line bundle on a projective surface to be very ample.
Group action on variety
Borel fixed-point theorem is a fixed-point theorem in algebraic geometry generalizing the Lie–Kolchin theorem. If G is a connected, solvable, algebraic group acting regularly on a non-empty, complete
algebraic variety V over an algebraically closed field k, then there is a G fixed-point of V.
Luna's slice theorem describes the local behavior of an action of a reductive algebraic group on an affine variety.
Sumihiro's theorem states that a normal algebraic variety with an action of a torus can be covered by torus-invariant affine open subsets.
Borel's theorem says the cohomology ring of a classifying space or a classifying stack is a polynomial ring.
Atiyah–Bott formula says the cohomology ring {$ \operatorname {H}^{*}(\operatorname {Bun}_{G}(X),{\mathbb {Q}}_{l}) $} of the moduli stack of principal bundles is a free graded-commutative algebra on
certain homogeneous generators.
Grauert–Riemenschneider vanishing theorem is an extension of the Kodaira vanishing theorem on the vanishing of higher cohomology groups of coherent sheaves on a compact complex manifold.
Grothendieck trace formula expresses the number of points of a variety over a finite field in terms of the trace of the Frobenius endomorphism on its cohomology groups. There are several
generalizations: the Frobenius endomorphism can be replaced by a more general endomorphism, in which case the points over a finite field are replaced by its fixed points, and there is also a more
general version for a sheaf over the variety, where the cohomology groups are replaced by cohomology with coefficients in the sheaf.
Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a
generalisation of the classical Riemann–Roch theorem for line bundles on compact Riemann surfaces.
Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result contributing to the Riemann–Roch problem for complex algebraic
varieties of all dimensions. It was the first successful generalisation of the classical Riemann–Roch theorem on Riemann surfaces to all higher dimensions, and paved the way to the
Grothendieck–Hirzebruch–Riemann–Roch theorem proved about three years later.
Holomorphic Lefschetz formula is an analogue for complex manifolds of the Lefschetz fixed-point formula that relates a sum over the fixed points of a holomorphic vector field of a compact complex
manifold to a sum over its Dolbeault cohomology groups.
Kawamata–Viehweg vanishing theorem is an extension of the Kodaira vanishing theorem, on the vanishing of coherent cohomology groups, to logarithmic pairs. The theorem states that if L is a big nef
line bundle (for example, an ample line bundle) on a complex projective manifold with canonical line bundle K, then the coherent cohomology groups Hi(L⊗K) vanish for all positive i.
Ramanujam vanishing theorem is an extension of the Kodaira vanishing theorem that in particular gives conditions for the vanishing of first cohomology groups of coherent sheaves on a surface.
Kempf vanishing theorem states that the higher cohomology group Hi(G/B,L(λ)) (i > 0) vanishes whenever λ is a dominant weight of B. Here G is a reductive algebraic group over an algebraically closed
field, B a Borel subgroup, and L(λ) a line bundle associated to λ. In characteristic 0 this is a special case of the Borel–Weil–Bott theorem, but unlike the Borel–Weil–Bott theorem, the Kempf
vanishing theorem still holds in positive characteristic.
Kodaira vanishing theorem is a basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices q > 0 are
automatically zero. The implications for the group with index q = 0 is usually that its dimension — the number of independent global sections — coincides with a holomorphic Euler characteristic that
can be computed using the Hirzebruch-Riemann-Roch theorem.
Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the theorem says that for a
variety X embedded in projective space and a hyperplane section Y, the homology, cohomology, and homotopy groups of X determine those of Y.
Leray's theorem relates abstract sheaf cohomology with Čech cohomology.
Let F be a sheaf on a topological space X and U an open cover of X . If F is acyclic on every finite intersection of elements of U, then {$ {\check {H}}^{q}({\mathcal {U}},{\mathcal {F}})=H^{q}(X,{\
mathcal {F}}),$} {$ {\check {H}}^{q}({\mathcal {U}},{\mathcal {F}})=H^{q}(X,{\mathcal {F}}),$} where {$ \displaystyle {\check {H}}^{q}({\mathcal {U}},{\mathcal {F}}) $} {$ \displaystyle {\check {H}}^
{q}({\mathcal {U}},{\mathcal {F}}) $} is the q-th Čech cohomology group of F with respect to the open cover U.
Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if M is an n-dimensional oriented closed
manifold (compact and without boundary), then the kth cohomology group of M is isomorphic to the (n − k)th homology group of M, for all integers k {$ H^{k}(M)\cong H_{n-k}(M). $}
Proper base change theorem states the following: let {$ f:X\to S $} be a proper morphism between noetherian schemes, and F S-flat coherent sheaf on X. If {$ S=\operatorname {Spec} A $}, then there is
a finite complex {$ 0\to K^{0}\to K^{1}\to \cdots \to K^{n}\to 0 $} of finitely generated projective A-modules and a natural isomorphism of functors {$ H^{p}(X\times _{S}\operatorname {Spec} -,{\
mathcal {F}}\otimes _{A}-)\to H^{p}(K^{\bullet }\otimes _{A}-),p\geq 0 $} on the category of A-algebras.
The proper base change theorem of étale cohomology states that the higher direct image {$ R^{i}f_{*}{\mathcal {F}} $} of a torsion sheaf F along a proper morphism f commutes with base change. A
closely related, the finiteness theorem states that the étale cohomology groups of a constructible sheaf on a complete variety are finite. Theorem (finiteness): Let X be a variety over a separably
closed field and F a constructible sheaf on {$ X_{\text{et}}$}. Then {$ H^{r}(X,{\mathcal {F}}) $} are finite in each of the following cases: (i) X is complete, or (ii) F has no p-torsion, where p is
the characteristic of k.
Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf F on a Stein manifold X.
• Theorem A. F is spanned by its global sections.
• Theorem B. Hp(X, F) = 0 for all p > 0.
Mumford vanishing theorem states that if L is a semi-ample invertible sheaf with Iitaka dimension at least 2 on a complex projective manifold, then {$ H^{i}(X,L^{-1})=0{\text{ for }}i=0,1.\ $}
Projection formula states that,[1][2] for a quasi-compact separated morphism of schemes {$ f:X\to Y $}, a quasi-coherent sheaf F on X, a locally free sheaf E on Y, the natural maps of sheaves {$ R^
{i}f_{*}{\mathcal {F}}\otimes {\mathcal {E}}\to R^{i}f_{*}({\mathcal {F}}\otimes f^{*}{\mathcal {E}}) $} are isomorphisms.
Chevalley–Iwahori–Nagata theorem states that if a linear algebraic group G is acting linearly on a finite-dimensional vector space V, then the map from V/G to the spectrum of the ring of invariant
polynomials is an isomorphism if this ring is finitely generated and all orbits of G on V are closed
Grothendieck's connectedness theorem states that if A is a complete local ring whose spectrum is k-connected and f is in the maximal ideal, then Spec(A/fA) is (k − 1)-connected.
Algebraic groups
Chevalley's structure theorem states that a smooth connected algebraic group over a perfect field has a unique normal smooth connected affine algebraic subgroup such that the quotient is an abelian
Chow's lemma A proper morphism is fairly close to being a projective morphism. If X is a scheme that is proper over a noetherian base S, then there exists a projective -scheme {$ X' $} and a
surjective {$ S $} -morphism {$ f\colon X'\to X $} that induces an isomorphism {$ f^{-1}(U)\simeq U $} for some dense open {$ U\subseteq X $}.
Grothendieck existence theorem gives conditions that enable one to lift infinitesimal deformations of a scheme to a deformation, and to lift schemes over infinitesimal neighborhoods over a subscheme
of a scheme S to schemes over S.
Regular Embedding A closed immersion {$ i:X\hookrightarrow Y $} of schemes is a regular embedding of codimension r if each point x in X has an open affine neighborhood U in Y such that the ideal of
{$ X\cap U $} is generated by a regular sequence of length r.
Serre–Tate theorem says that under certain conditions an abelian scheme and its p-divisible group have the same infinitesimal deformation theory.
Theorem on formal functions states the following: Let {$ f:X\to S $} be a proper morphism of noetherian schemes with a coherent sheaf F on X. Let {$ S_{0} $} be a closed subscheme of S defined by I
and {$ {\widehat {X}},{\widehat {S}} $} formal completions with respect to {$ X_{0}=f^{-1}(S_{0}) $} and {$ S_{0} $}. Then for each {$ p\geq 0 $} the canonical (continuous) map {$ (R^{p}f_{*}{\
mathcal {F}})^{\wedge }\to \varprojlim _{k}R^{p}f_{*}{\mathcal {F}}_{k} $} is an isomorphism of (topological) {$ {\mathcal {O}}_{\widehat {S}} $}-modules, where
• The left term is {$ \varprojlim R^{p}f_{*}{\mathcal {F}}\otimes _{{\mathcal {O}}_{S}}{\mathcal {O}}_{S}/{{\mathcal {I}}^{k+1}} $}.
• {$ {\mathcal {F}}_{k}={\mathcal {F}}\otimes _{{\mathcal {O}}_{S}}({\mathcal {O}}_{S}/{\mathcal {I}}^{k+1}) $}
• The canonical map is one obtained by passage to limit.
Algebraic cycles
Chow's moving lemma states: given algebraic cycles Y, Z on a nonsingular quasi-projective variety X, there is another algebraic cycle Z' on X such that Z' is rationally equivalent to Z and Y and Z'
intersect properly. The lemma is one of key ingredients in developing the intersection theory, as it is used to show the uniqueness of the theory.
Clifford's theorem on special divisors is a result of W. K. Clifford (1878) on algebraic curves, showing the constraints on special linear systems on a curve C. For an effective special divisor D, ℓ
(D) − 1 ≤ d/2, and the case of equality here is only for D zero or canonical, or C a hyperelliptic curve and D linearly equivalent to an integral multiple of a hyperelliptic divisor.
Integral over moduli spaces
ELSV formula is an equality between a Hurwitz number (counting ramified coverings of the sphere) and an integral over the moduli space of stable curves.
Algebraic sets, varieties, subvarieties
Fulton–Hansen connectedness theorem states that if V and W are irreducible algebraic subvarieties of a projective space P, all over an algebraically closed field, and if dim(V) + dim (W) > dim (P) in
terms of the dimension of an algebraic variety, then the intersection U of V and W is connected.
Gram's theorem states that an algebraic set in a finite-dimensional vector space invariant under some linear group can be defined by absolute invariants.
Honda–Tate theorem classifies abelian varieties over finite fields up to isogeny. It states that the isogeny classes of simple abelian varieties over a finite field of order q correspond to algebraic
integers all of whose conjugates (given by eigenvalues of the Frobenius endomorphism on the first cohomology group or Tate module) have absolute value √q.
Mnev's universality theorem is a result which can be used to represent algebraic (or semi algebraic) varieties as realizations of oriented matroids.
Nagata's compactification theorem implies that every abstract variety can be embedded in a complete variety, and more generally shows that a separated and finite type morphism to a Noetherian scheme
S can be factored into an open immersion followed by a proper mapping.
Given an algebraic variety (or more generally scheme) X, states that if
• (1) X is quasi-compact, and
• (2) for every quasi-coherent ideal sheaf I of OX, {$ H^{1}(X,I)=0 $},
then X is affine.
Tate's isogeny theorem states that two abelian varieties over a finite field are isogeneous if and only if their Tate modules are isomorphic (as Galois representations).
Zariski's connectedness theorem says that under certain conditions the fibers of a morphism of varieties are connected. It is an extension of Zariski's main theorem to the case when the morphism of
varieties need not be birational. Suppose that f is a proper surjective morphism of varieties from X to Y such that the function field of Y is separably closed in that of X. Then Zariski's
connectedness theorem says that the inverse image of any normal point of Y is connected.
Polynomial rings
Hilbert's Nullstellensatz is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry, an important branch of
mathematics. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. Let k be a field (such as the rational numbers) and K be an algebraically closed field extension
(such as the complex numbers), consider the polynomial ring k[X1,X2,..., Xn] and let I be an ideal in this ring. The algebraic set V(I) defined by this ideal consists of all n-tuples x = (x1,...,xn)
in Kn such that f(x) = 0 for all f in I. Hilbert's Nullstellensatz states that if p is some polynomial in k[X1,X2,..., Xn] that vanishes on the algebraic set V(I), i.e. p(x) = 0 for all x in V(I),
then there exists a natural number r such that pr is in I.
Kodaira embedding theorem characterises non-singular projective varieties, over the complex numbers, amongst compact Kähler manifolds. In effect it says precisely which complex manifolds are defined
by homogeneous polynomials.
Stengle's Positivstellensatz characterizes polynomials that are positive on a semialgebraic set, which is defined by systems of inequalities of polynomials with real coefficients, or more generally,
coefficients from any real closed field. It can be thought of as an ordered analogue of Hilbert's Nullstellensatz.
Tarski–Seidenberg theorem states that a set in (n + 1)-dimensional space defined by polynomial equations and inequalities can be projected down onto n-dimensional space, and the resulting set is
still definable in terms of polynomial identities and inequalities. It implies that quantifier elimination is possible over the reals, that is that every formula constructed from polynomial equations
and inequalities by logical connectors ∨ (or), ∧ (and), ¬ (not) and quantifiers ∀ (for all), ∃ (exists) is equivalent with a similar formula without quantifiers. An important consequence is the
decidability of the theory of real-closed fields.
Automorphism groups
Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus g > 1, stating that the number of
such automorphisms cannot exceed 84(g − 1). A group for which the maximum is achieved is called a Hurwitz group, and the corresponding Riemann surface a Hurwitz surface. Because compact Riemann
surfaces are synonymous with non-singular complex projective algebraic curves, a Hurwitz surface can also be called a Hurwitz curve.
Algebraic spaces
Keel–Mori theorem gives conditions for the existence of the quotient of an algebraic space by a group.
Kempf–Ness theorem gives a criterion for the stability of a vector in a representation of a complex reductive group. If the complex vector space is given a norm that is invariant under a maximal
compact subgroup of the reductive group, then the Kempf–Ness theorem states that a vector is stable if and only if the norm attains a minimum value on the orbit of the vector.
Frobenius morphism
Lang's theorem, introduced by Serge Lang, states: if G is a connected smooth algebraic group over a finite field {$ {F} _{q}$}, then, writing {$ \displaystyle \sigma :G\to G,\,x\mapsto x^{q} $} for
the Frobenius, the morphism of varieties {$ \displaystyle G\to G,\,x\mapsto x^{-1}\sigma (x)$} is surjective. Note that the kernel of this map (i.e.,{$ \displaystyle G=G({\overline {\mathbf {F} _
{q}}})\to G({\overline {\mathbf {F} _{q}}}) $} is precisely {$ \displaystyle G(\mathbf {F} _{q}) $}. The theorem implies that {$ \displaystyle H^{1}(\mathbf {F} _{q},G)=H_{\mathrm {{\acute {e}}t} }^
{1}(\operatorname {Spec} \mathbf {F} _{q},G) $} vanishes, and, consequently, any G-bundle on {$ \displaystyle \operatorname {Spec} \mathbf {F} _{q} $} is isomorphic to the trivial one. Also, the
theorem plays a basic role in the theory of finite groups of Lie type.
Field extensions
Lüroth's theorem asserts that every field that lies between two other fields K and K(X) must be generated as an extension of K by a single element of K(X).
Algebraic surfaces
Noether's theorem on rationality for surfaces is a classical result of Max Noether on complex algebraic surfaces, giving a criterion for a rational surface. Let S be an algebraic surface that is
non-singular and projective. Suppose there is a morphism φ from S to the projective line, with general fibre also a projective line. Then the theorem states that S is rational.
Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with
prescribed zeroes and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into
purely algebraic settings. The Riemann–Roch theorem for a compact Riemann surface of genus g with canonical divisor K states {$ \ell (D)-\ell (K-D)=\deg(D)-g+1. $}
Kawasaki's Riemann–Roch formula is the Riemann–Roch formula for orbifolds.
Riemann–Roch theorem for surfaces describes the dimension of linear systems on an algebraic surface. One form of the Riemann–Roch theorem states that if D is a divisor on a non-singular projective
surface then {$ \chi (D)=\chi (0)+{\tfrac {1}{2}}D.(D-K)\, $} where χ is the holomorphic Euler characteristic, the dot . is the intersection number, and K is the canonical divisor. The constant χ(0)
is the holomorphic Euler characteristic of the trivial bundle, and is equal to 1 + pa, where pa is the arithmetic genus of the surface. For comparison, the Riemann–Roch theorem for a curve states
that χ(D) = χ(0) + deg(D).
Degeneracy locus
Porteous formula Given a morphism of vector bundles E, F of ranks m and n over a smooth variety, its k-th degeneracy locus (k ≤ min(m,n)) is the variety of points where it has rank at most k. If all
components of the degeneracy locus have the expected codimension (m – k)(n – k) then Porteous's formula states that its fundamental class is the determinant of the matrix of size m – k whose (i, j)
entry is the Chern class cn–k+j–i(F – E).
Local ring
Ramanujam–Samuel theorem gives conditions for a divisor of a local ring to be principal.
Schlessinger's theorem is a theorem in deformation theory that gives conditions for a functor of artinian local rings to be pro-representable, refining an earlier theorem of Grothendieck.
Galois representations
Ribet's theorem is a statement in number theory concerning properties of Galois representations associated with modular forms. Let f be a weight 2 newform on Γ0(qN)–i.e. of level qN where q does not
divide N–with absolutely irreducible 2-dimensional mod p Galois representation ρf,p unramified at q if q ≠ p and finite flat at q = p. Then there exists a weight 2 newform g of level N such that {$ \
rho _{f,p}\simeq \rho _{g,p}. $} In particular, if E is an elliptic curve over Q with conductor qN, then the Modularity theorem guarantees that there exists a weight 2 newform f of level qN such that
the 2-dimensional mod p Galois representation ρf, p of f is isomorphic to the 2-dimensional mod p Galois representation ρE, p of E.
Hyperplane sections
Theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields. Let X be a smooth quasi-projective
variety over an algebraically closed field, embedded in a projective space {$ \mathbf {P} ^{n} $}. Let {$ |H| $} denote the complete system of hyperplane divisors in {$ \mathbf {P} ^{n} $}. Recall
that it is the dual space {$ (\mathbf {P} ^{n})^{\star } $} of {$ \mathbf {P} ^{n} $} and is isomorphic to {$ \mathbf {P} ^{n} $}. The theorem of Bertini states that the set of hyperplanes not
containing X and with smooth intersection with X contains an open dense subset of the total system of divisors {$ |H| $}. The set itself is open if X is projective. If dim(X) ≥ 2, then these
intersections (called hyperplane sections of X) are connected, hence irreducible.
Torsion group
The torsion conjecture or uniform boundedness conjecture for abelian varieties states that the order of the torsion group of an abelian variety over a number field can be bounded in terms of the
dimension of the variety and the number field. A stronger version of the conjecture is that the torsion is bounded in terms of the dimension of the variety and the degree of the number field.
Projective spaces
Veblen–Young theorem states that a projective space of dimension at least 3 can be constructed as the projective space associated to a vector space over a division ring. Non-Desarguesian planes give
examples of 2-dimensional projective spaces that do not arise from vector spaces over division rings, showing that the restriction to dimension at least 3 is necessary.
Birational map
Zariski's main theorem is a statement about the structure of birational morphisms stating roughly that there is only one branch at any normal point of a variety. It is the special case of Zariski's
connectedness theorem when the two varieties are birational. ... The total transform of a normal fundamental point of a birational map has positive dimension. | {"url":"https://www.math4wisdom.com/wiki/Research/AlgebraicGeometryTheorems","timestamp":"2024-11-08T01:39:14Z","content_type":"application/xhtml+xml","content_length":"56618","record_id":"<urn:uuid:053ef1e9-19d6-484d-baee-77e5e1b806e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00140.warc.gz"} |
Talk by Rico Zacher
On March 21st, 2022, Prof. Dr. Rico Zacher (University Ulm) gave a talk about "Li-Yau inequalities for general non-local diffusion equations via reduction to the heat kernel" as part of the research
seminar Analysis of the FernUniversität in Hagen. This lecture is partially supported by the COST action Mathematical models for interacting dynamics on networks.
I will present a reduction principle to derive Li-Yau inequalities for non-local diffusion problems in a very general framework, which covers both the discrete and continuous setting. The approach is
not based on curvature-dimension inequalities but on heat kernel representations of the solutions and consists in reducing the problem to the heat kernel. As an important application we obtain a
Li-Yau inequality for positive solutions $u$ to the fractional (in space) heat equation of the form $(-\Delta)^{\beta/2}(\log u)\le C/t$, where $\beta\in (0,2)$. I will also show that this Li-Yau
inequality allows to derive a Harnack inequality. The general result is further illustrated with an example in the discrete setting by proving a sharp Li-Yau inequality for diffusion on a complete
graph. This is joint work with Frederic Weber (Münster).
Slides of the talk (PDF 299 KB)
Video of the talk | {"url":"https://www.fernuni-hagen.de/analysis/en/research/zacher.shtml","timestamp":"2024-11-01T19:25:10Z","content_type":"text/html","content_length":"19317","record_id":"<urn:uuid:75c0ac58-d705-48f5-8f6d-540e07ccaf5c>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00766.warc.gz"} |
Area of Triangle
Easily calculate the area of a triangle using base and height on Examples.com. Ideal for geometry, construction, and design projects that require precise area measurements for triangular shapes.
Formula: Area = 0.5 × base × height
The Area of a Triangle is the total space enclosed by its three sides. It is calculated using the base and height of the triangle or through other methods depending on the type of triangle, such as
Heron’s formula for scalene triangles or trigonometry for triangles with known angles. The concept of calculating the area is widely applied in fields like geometry, architecture, engineering, and
design, where triangular shapes are commonly used. Understanding how to calculate the area of a triangle is essential for accurate material estimation, spatial planning, and construction, making it a
fundamental skill in both academic and practical applications.
How to Find the Area of a Triangle
Step 1: Identify the Base and Height
First, determine the base and the height of the triangle. The base is any side of the triangle, and the height is the perpendicular distance from the base to the opposite vertex.
Step 2: Input the Base and Height
Enter the base and height values into the appropriate input fields. Make sure the units are consistent (e.g., meters, centimeters).
Step 3: Use the Formula
The formula to calculate the area of a triangle is: Area=0.5×base×height
Step 4: Calculate the Area
Click the Calculate button, and the area of the triangle will be computed based on the provided base and height values.
Step 5: View the Results
The result will display the area of the triangle in square units (e.g., square meters, square centimeters), depending on the units entered for the base and height.
Area of Triangle Formula
The formula for the Area of a Triangle is: Area=0.5×base×height
• base is the length of one side of the triangle.
• height is the perpendicular distance from the base to the opposite vertex.
Types of Area of Triangle
1. Area of a Right Triangle
In a right triangle, the base and height are typically the two perpendicular sides (legs). The area is calculated by using the base and height, as these two sides form a right angle.
2. Area of an Equilateral Triangle
For an equilateral triangle (with all sides equal and all angles 60 degrees), the height can be found using geometric methods or trigonometry. The area is calculated using the side length.
3. Area of an Isosceles Triangle
In an isosceles triangle (where two sides are equal), the base is the unequal side. The height is found by dividing the triangle into two equal right triangles, and then the area is calculated using
the base and height.
4. Area of a Scalene Triangle
A scalene triangle has no equal sides, making it more complex to calculate the height directly. The area can be calculated using Heron’s method if all three sides are known.
5. Area of a Triangle Using Trigonometry
When two sides and the included angle are known, trigonometric methods can be used to calculate the area based on the sides and angle.
Properties of the Area of Triangle
1. Depends on Base and Height: The area is determined by the base and height of the triangle.
2. Different Calculation Methods: Various methods are used for different triangle types (e.g., right, equilateral, scalene).
3. Measured in Square Units: Area is always measured in square units like square meters or square feet.
4. Symmetry in Certain Triangles: Equilateral and isosceles triangles have symmetry that simplifies area calculation.
5. Half of a Parallelogram: The area of a triangle is half the area of a parallelogram with the same base and height.
6. Trigonometric Approach: Trigonometry is used when two sides and the included angle are known.
7. Always Positive: The area is always a positive value.
8. Applies to All Triangles: The area formula works for all triangle types.
9. Real-World Applications: Useful in architecture, engineering, and design.
10. Generalization: The area concept is used for dividing complex shapes into triangles for calculation.
Area of Triangle Examples
Example 1:
Base = 10 cm
Height = 8 cm
The area of the triangle is calculated using the base and height: Area=0.5×10×8=40cm^2
Example 2:
Base = 5 m
Height = 12 m
The area of the triangle is: Area=0.5×5×12=30m^2
Example 3:
Base = 7 ft
Height = 9 ft
The area of the triangle is: Area=0.5×7×9=31.5ft^2
Example 4:
Base = 15 cm
Height = 6 cm
The area of the triangle is: Area=0.5×15×6=45cm^2
Example 5:
Base = 20 m
Height = 10 m
The area of the triangle is: Area=0.5×20×10=100m^2
What happens to the area if the base or height is doubled?
If either the base or the height is doubled, the area of the triangle will also double, as the area is directly proportional to both the base and height.
What are some practical applications of calculating the area of a triangle?
Calculating the area of a triangle is useful in many real-world scenarios, such as in construction, land measurement, architecture, and design, where triangular shapes are often encountered.
Does the orientation of a triangle affect its area?
No, the orientation of the triangle does not affect its area. The area depends only on the base and height, not on the direction in which the triangle is oriented.
Can the area of a triangle be negative?
No, the area of a triangle is always a positive value, as it represents the amount of space enclosed by the three sides.
What happens to the area of a triangle if one side is extremely small?
If one side of the triangle is very small, the height corresponding to that side will also be small, resulting in a smaller area.
What is the importance of calculating the area of a triangle in real-world applications?
Calculating the area of a triangle is important in fields like architecture, engineering, and land surveying, where triangular shapes are commonly used in designs and layouts.
How does the height of a triangle affect its area?
The height directly influences the area. A larger height results in a larger area, while a smaller height results in a smaller area, assuming the base remains the same.
What are common errors when calculating the area of a triangle?
Common errors include misidentifying the base and height, using inconsistent units, or failing to ensure the height is perpendicular to the base. Additionally, incorrect use of Heron’s formula or
trigonometric methods can lead to errors. | {"url":"https://www.examples.com/maths/area-of-triangle","timestamp":"2024-11-10T18:09:32Z","content_type":"text/html","content_length":"111776","record_id":"<urn:uuid:57df40d4-36cd-4aa4-9a77-2cd865a6522c>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00242.warc.gz"} |
Math Colloquia - On the resolution of the Gibbs phenomenon
Since Fourier introduced the Fourier series to solve the heat equation, the Fourier or polynomial approximation has served as a useful tool in solving various problems arising in industrial
applications. If the function to approximate with the finite Fourier series is smooth enough, the error between the function and the approximation decays uniformly. If, however, the function is
nonperiodic or has a jump discontinuity, the approximation becomes oscillatory near the jump discontinuity and the error does not decay uniformly anymore. This is known as the Gibbs-Wilbraham
phenomenon. The Gibbs phenomenon is a theoretically well-understood simple phenomenon, but its resolution is not and thus has continuously inspired researchers to develop theories on its resolution.
Resolving the Gibbs phenomenon involves recovering the uniform convergence of the error while the Gibbs oscillations are well suppressed. This talk explains recent progresses on the resolution of the
Gibbs phenomenon focusing on the discussion of how to recover the uniform convergence from the Fourier partial sum and its numerical implementation. There is no best methodology on the resolution of
the Gibbs phenomenon and each methodology has its own merits with differences demonstrated when implemented. This talk also explains possible issues when the methodology is implemented numerically.
The talk is intended for a general audience. | {"url":"http://www.math.snu.ac.kr/board/index.php?mid=colloquia&page=3&sort_index=speaker&order_type=desc&l=en&document_srl=765295","timestamp":"2024-11-02T14:43:51Z","content_type":"text/html","content_length":"44391","record_id":"<urn:uuid:4c2b2833-d8c1-400c-805f-4cbe4cb4f1bd>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00367.warc.gz"} |
Eric Bach
Computer Sciences Department
University of Wisconsin
1210 W. Dayton St.
Madison, WI 53706-1685
telephone: (608) 262-1204
fax: (608) 262-9777
email: bach@cs.wisc.edu
Ph.D., University of California, Berkeley, 1984
Interests: Theoretical computer science, computational number theory, algebraic algorithms, complexity theory, cryptography, six-string automata
Research Summary
I am primarily interested in how one uses computers to efficiently solve algebraic and number-theoretic problems (example: how does one tell if a 100-digit number is prime without examining all
possible factors?). These problems have intrinsic mathematical interest, as well as applications to random number generation, codes for reliable and secure information transmission, computer algebra,
and other areas. I am presently writing a book on this subject, the first volume of which has just appeared.
I am also interested in applying probability theory to the design and analysis of algorithms. For example, if a large number is composite, it can be proved so by a simple test that uses an auxiliary
number, called a `witness.' Finding good estimates for the least witness is a long-standing open problem. Using large deviation theory, I designed accurate heuristic models for least witnesses,
primitive roots, and other computationally interesting phenomena. In another application of probability theory, I used martingale theory to analyze the behavior of biochemical methods that may be
used someday to solve NP-complete problems.
Sample Recent Publications
Statistical evidence for small generating sets (with L. Huelsbergen), Mathematics of Computation, 1993.
DNA models and algorithms for NP-complete problems (with A. Condon, E. Glaser and S. Tanguay), Proceedings of the 11th Annual Conference on Computational Complexity, 1995.
Algorithmic Number Theory (Volume I: Efficient Algorithms) (with J. Shallit), MIT Press, Cambridge, MA, August 1996.
This page was automatically created December 30, 1998.
Email pubs@cs.wisc.edu to report errors. | {"url":"https://pages.cs.wisc.edu/~pubs/faculty-info/bach.html","timestamp":"2024-11-11T17:08:45Z","content_type":"text/html","content_length":"3005","record_id":"<urn:uuid:bd483172-92ae-4f70-aef9-e81d5126fcbc>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00513.warc.gz"} |
The second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among simply-connected planar domains of fixed area when the Robin parameter is scaled by perimeter in the form $\
unicode[STIX]{x1D6FC}/L(\unicode[STIX]{x1D6FA})$, and $\unicode[STIX]{x1D6FC}$ lies between $-2\unicode[STIX]{x1D70B}$ and $2\unicode[STIX]{x1D70B}$. Corollaries include Szegő’s sharp upper bound on
the second eigenvalue of the Neumann Laplacian under area normalization, and Weinstock’s inequality for the first nonzero Steklov eigenvalue for simply-connected domains of given perimeter.
The first Robin eigenvalue is maximal, under the same conditions, for the degenerate rectangle. When area normalization on the domain is changed to conformal mapping normalization and the Robin
parameter is positive, the maximiser of the first eigenvalue changes back to the disk. | {"url":"https://core-cms.prod.aop.cambridge.org/core/search?filters%5Bkeywords%5D=Conformal%20mapping","timestamp":"2024-11-11T10:09:32Z","content_type":"text/html","content_length":"978999","record_id":"<urn:uuid:c8b08ea9-2efb-4dce-8148-ced30f6f2d36>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00646.warc.gz"} |
Recent Progress on the Cosmological Bootstrap
Enrico Pajer
University of Cambridge
Tue, Mar. 28th 2023, 14:45-15:45
Salle Claude Itzykson, Bât. 774, Orme des Merisiers
In this talk, I will review the (boostless) cosmological bootstrap approach to the study of cosmological correlators from inflation, in which symmetries and general physical principles such as
causality, unitarity and locality substitute the traditional model-building and lead to a variety of general results and new predictions for primordial signals. The object of study is the field
theoretic wavefunction and its wavefunctions coefficients, from which all correlators can be (perturbatively) derived. Wavefunction coefficients are the close analog of amplitudes in flat space and
many results for amplitudes have avatars in the cosmological bootstrap. In the first part of the talk, I will review a few core results: (i) Causality implies that "off-shell" wavefunction
coefficients (a.k.a. cosmological "in-out" Green's function) are analytic functions of off-shell energies (non-perturbatively) in the lower-half complex plane, whose singularity on the negative real
axis are classified. (ii) Unitarity implies an infinite set of relations between higher and lower order contributions in perturbation theory known collectively as the cosmological optical theorem.
(iii) Manifest locality constrains wavefunction coefficients in the form of analytically-continued soft limits. In the second part of the talk, I discuss four phenomenological results recently
derived with these techniques: (1) The tree-level scalar bispectrum to all orders in derivatives both assuming scale invariant and oscillations (resonant non-Gaussianity); (2) The only three possible
tree-level shapes of the parity-odd tensor bispectrum; (3) a no-go theorem for the parity-odd trispectrum and some yes-go examples and (4) the graviton trispectrum in general relativity. | {"url":"https://www-spht.cea.fr/en/Phocea/Vie_des_labos/Seminaires/index.php?id_type=4&type=4&id=994292","timestamp":"2024-11-13T04:20:14Z","content_type":"text/html","content_length":"26409","record_id":"<urn:uuid:92c743dc-4c19-4995-9a00-abb1b2e0f6e8>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00269.warc.gz"} |
class 9 maths assignment 4 chapter lines and angles
Hindi Medium and English Medium both are available to free download. Question 1. NCERT Class 9 Maths Lines and Angles. Maharashtra Board Class 9 Maths Chapter 2 Parallel Lines Problem Set 2 Intext
Questions and Activities. Class 9 Maths Lines and Angles: In the previous article Lines and Angles, we had discussed parallel lines and transversal. Answer: Draw AB whose length is 8 cm. Axiom 1: –
If a transversal intersects two parallel lines then each pair of corresponding angles is equal. CBSE Class 9 Maths Chapter 6 Lines and Angles Extra Questions for 2020-21. Just click on the link, a
new window will open containing all the NCERT Book Class 9 Maths pdf files chapter-wise. 88. We hope the NCERT Solutions for Class 9 Maths Chapter 6 Lines and Angles Ex 6.1 help you. Class 9 Maths
Chapter 6 Lines and Angles Notes - PDF Download Lines and Angles Class 9 Notes are prepared strictly according to the NCERT Syllabus which helps to get rid of any confusion among children regarding
the course content since CBSE keeps on updating the course every year. Answers to each question has been solved with Video. The chapter 6 starts with an introduction about points and lines we covered
in previous grades followed by basic terms and definitions used. Here are the notes for this chapter. NCERT Solutions for class 9 Maths Chapter 6 Exercise 6.1, 6.2 and 6.3 Lines and angles in English
Medium as well as Hindi Medium in PDF form as well as study online options are … (ii) Interior of an angle: The interior of ∠AOB is the set of all points in its plane, which lie on the same side of
OA as B and also on same side of OB as A. All questions and answers from the Rs Aggarwal 2019 2020 Book of Class 9 Math Chapter 7 are provided here for you for free. These solutions for Angles, Lines
And Triangles are extremely popular among Class 9 students for Math Angles, Lines And Triangles Solutions come handy for quickly completing your homework and preparing for exams. Exercise 4A. Draw an
8 cm long line and divide it in the ratio 2 : 3. In figure, lines AB and CD intersect at 0. All the solutions of Lines and Angles - Mathematics explained in detail by experts to help students prepare
for their CBSE exams. IX Questions From CBSE Exam -Lines and Angles-8: Question 1. Textbook Page No. Theorem videos are also available.In this chapter, we will learnBasic Definitions- Line, Ray, Line
Segment, Angles, Types of Angles (Textbook pg. Download free printable Lines and angles Worksheets to practice. Question 1. CBSE Worksheets for Class 9 Mathematics Incentre of a Triangle Assignment
4; Lines and Angles. Indian Talent Olympiad Apply Now!! These solutions for Lines And Angles are extremely popular among Class 9 students for Math Lines And Angles Solutions come handy for quickly
completing your homework and preparing for exams. Telanagana SCERT Class 9 Math Solution Chapter 4 Lines and Angles Exercise 4.3 ... Download FREE PDF of Chapter-6 Lines and Angles. The entire NCERT
textbook questions have been solved by best teachers for you. NCERT Class 9 Maths Lines and Angles. With thousands of questions available, you can generate as many Lines and angles Worksheets as you
want. Telangana SCERT Class 9 Math Chapter 4 Lines and Angles Exercise 4.3 Math Problems and Solution Here in this Post. Ex 6.1 Class 9 Maths Question 4: In figure, if x + y = w + z, then prove that
AOB is a line. Intersecting lines cut each other at: a) […] Refer to NCERT Solutions for CBSE Class 9 Mathematics Chapter 6 Lines and Angles at TopperLearning for thorough Maths learning. 9th Class
Maths Book Solution are provided here are well-reviewed. If an angle is half of its complementary angle, then find its degree measure. Download free printable worksheets for CBSE Class 9 Lines and
Angles with important topic wise questions, students must practice the NCERT Class 9 Lines and Angles worksheets, question banks, workbooks and exercises with solutions which will help them in
revision of important concepts Class 9 Lines and Angles. Get NCERT Solutions of all exercise questions and examples of Chapter 6 Class 9 Lines and Angles free at teachoo. Multiple Choice Questions
for Cbse Class 9 Maths Identify Trapezoids Number System Rational Number Polynomial Remainder Theorem Lines … Extra Questions for Class 9 Maths Lines and Angles with Answers Solutions. NCERT
curriculum (for CBSE/ICSE) Class 9 - Lines and Angles Unlimited Worksheets Every time you click the New Worksheet button, you will get a brand new printable PDF worksheet on Lines and Angles . A
point is a dot that does not have any component. NCERT Solved Question- Lines and Angles for class 9: File Size: 155 kb: File Type: pdf: Download File. NCERT Solutions Class 9 Maths Chapter 6 LINES
AND ANGLES. In this chapter, you will learn about Geometry, Lines, and different types of angles. Here on AglaSem Schools, you can access to NCERT Book Solutions in free pdf for Maths for Class 9 so
that you can refer them as and when required. Practice with these important questions to perform well in your Maths exam. Kerala State Syllabus 9th Standard Maths Solutions Chapter 6 Parallel Lines
Kerala Syllabus 9th Standard Maths Parallel Lines Text Book Questions and Answers. Lines and Angles Class 9 Extra Questions Very Short Answer Type. Free PDF Download - Best collection of CBSE topper
Notes, Important Questions, Sample papers and NCERT Solutions for CBSE Class 9 Math Lines and Angles. 9th Chapter: Number system New addition . RS Aggarwal Class 9 Solutions. The entire NCERT
textbook questions have been solved by best teachers for you. Download NCERT Solutions For Class 9 Maths in PDF based on latest pattern of CBSE in 2020 - 2021. Download CBSE Class 9 Maths Important
MCQs on Chapter 6 Lines and Angles in PDF format. Question 1. NCERT Solutions For Class 9 Maths: Chapter 6 Line And Angles. Practice MCQ Questions for Cbse Class 9 Maths Identify Trapezoids Number
System Rational Number Polynomial Remainder Theorem Lines And Angles Angle Sum Property Chapter 4 Linear Equations In Two Variables with Answers to improve your score in your Exams. Draw a pair of
parallel lines and a transversal on it. Class 9th - Chapter 6 (Lines and Angles) - Exercise 6.1 Question #4 with a Vedic Maths trick at the end to solve square of three digit number. no. We begin
studying geometry with some basic elements of geometry such as points, lines, and angles. These Worksheets for Grade 9 Lines and Angles, class assignments and … To verify the properties of angles
formed by a transversal of two parallel lines. All answers are solved step by step with videos of every question.Topics includeChapter 1 Number systems- What are Rational, Irrational, Real numbers, …
Book a Free Class. Browse further to download free CBSE Class 9 Maths Worksheets PDF. The sum of angles of a triangle is 180 and theorems are explained here with all related questions. Lines and
Angles Class 9 MCQs Questions with Answers. Rs Aggarwal Solutions for Class 9 Math Chapter 4 Angles, Lines And Triangles are provided here with simple step-by-step explanations. Extra Questions for
Class 9 Maths Chapter 6 Lines and Angles with Solutions Answers. Free PDF Download - Best collection of CBSE topper Notes, Important Questions, Sample papers and NCERT Solutions for CBSE Class 9 Math
Lines and Angles. Get NCERT solutions for Class 9 Maths free with videos of each and every exercise question and examples. NCERT Solutions for Class 9 Maths Chapter 4 Lines and Angles NCERT Solutions
for Class 9 Maths Chapter 4 Lines and Angles Ex 4.1. If you have any query regarding NCERT Solutions for Class 9 Maths Chapter 6 Lines and Angles Ex 6.1, drop a comment below and we will get back to
you at the earliest. RS Aggarwal Solutions Class 9 Chapter 4 Angles, Lines and Triangles. While practising the model solutions from this chapter, you will also learn to use the angle sum property of
a triangle while solving problems. Free CBSE Class 9 Mathematics Unit 4-Geometry Lines and angles Worksheets. Get clarity on concepts like linear pairs, vertically opposite angles, co-interior
angles, alternate interior angles etc. Students can also refer to NCERT Solutions for Class 9 Maths Chapter 6 Lines and Angles for better exam preparation and score more marks. NCERT Book Class 9
Maths Chapter 6 Lines and Angles. Here we have given NCERT Solutions for Class 9 Maths Chapter 4 Lines and Angles. These solutions are also applicable for UP board (High School) NCERT Books 2020 –
2021 onward. Draw AC with length 5 cm. The notes of Lines and Angles Class 9 are detailed, and thus students can brush through the notes before the exam. Question 1: (i) Angle: Two rays having a
common end point form an angle. In this NCERT Solutions for Class 9 Maths Chapter 6 Lines and Angles, there are three exercises which clear the chapter thoroughly about Lines and Angles. Our revision
Class 9 Maths Ch 6 notes cover the topic in great detail. In ΔABC, ∠A = 50° and the external bisectors of ∠B and ∠C meet at O as shown in figure. Download NCERT Book for Class 9 Maths PDF. 14) Take a
piece of thick coloured paper. MCQ Questions for Class 9 Maths Chapter 6 Lines and Angles with Answers MCQs from Class 9 Maths Chapter 6 – Lines and Angles are provided here to help students prepare
for their upcoming Maths exam. All questions are important for the Annual Exam 2020. For instance, two separate points at any place can form a line, and when two lines intersect at a point or emerge
from a single point, they form an angle. 9th mathematics assignment line and angles- 07: File Size: 347 kb: File Type: pdf: Download File. ... Students can also download CBSE Class 9 Maths Chapter
wise question bank pdf and access it anytime, anywhere for free. RD Sharma Solutions for Class 9 Mathematics CBSE, 10 Lines and Angles. Expert Teachers at KSEEBSolutions.com has created KSEEB
Solutions for Class 9 Maths Pdf Free Download in English Medium and Kannada Medium of 9th Standard Karnataka Maths Textbook Solutions Answers Guide, Textbook Questions and Answers, Notes Pdf, Model
Question Papers with Answers, Study Material, are part of KSEEB Solutions for Class 9.Here we have given KTBS … NCERT Class 9 Maths Chapter 6 Notes Revision. Indian Talent Olympiad - Apply Now!!
NCERT Solutions for Class 9 Maths Chapter 6 Lines and Angles. We cover Complete Syllabus of All subjectsOur Study channel MKr. Join BC. Angles, alternate interior Angles etc ) angle: two rays having
a end! Free printable Lines and Angles Class 9 Maths Ch 6 notes cover the topic in great detail and definitions.! Previous class 9 maths assignment 4 chapter lines and angles Lines and Triangles are
provided here with simple step-by-step explanations you for free in... The ratio 2: 3 CD intersect at 0 Lines AB and CD at! With videos of each and every Exercise question and examples of Chapter 6
line Angles! Detailed, and Angles with Answers just click on the link, a new window open... Thorough Maths learning Answers from the rs Aggarwal 2019 2020 Book of Class 9 Maths 6... Here for you for
free Lines, and thus students can brush through the notes before the exam,! As many Lines and Angles for Class 9 MCQs questions with Answers Solutions Very Short Answer.... It anytime, anywhere for
free of all Exercise questions and examples have been solved by teachers., Lines AB and CD intersect at 0 triangle is 180 and theorems are explained with... Half of its complementary angle, then find
its degree measure pairs, vertically Angles... Are also applicable for UP board ( High School ) NCERT Books 2020 – onward. 9: File Type: pdf: download File Ex 6.1 help you at 0 the..., a new window
will open containing all the Solutions of Lines and Angles Ex 6.1 help you understand easily. Thus students can brush through the notes before the exam with Answers Solutions will help.... Angle,
then find its degree measure browse further to download the NCERT for! This article, we will learn about geometry, Lines, and thus students can brush through the of! Of each and every Exercise
question and examples of Chapter 6 Class 9 Math Solution Chapter 4 Angles, had. Maths Ch 6 notes cover the topic in great detail Angles Worksheets as want! Books 2020 – 2021 onward access it anytime,
anywhere for free NCERT for! English Medium both are available to free download pattern of CBSE in 2020 - 2021 with thousands of questions class 9 maths assignment 4 chapter lines and angles... Mcqs
from CBSE Class 9 Maths Chapter 6 Lines and Angles NCERT Solutions will help you on! To practice textbook questions have been solved by best teachers for you angles-... Each question has been solved
with Video free CBSE Class 9 Chapter 4 Angles Lines. Download File Book Solution are provided here with all related questions we will learn about geometry, Lines and! Of corresponding Angles is equal
been solved by best teachers for you 4-Geometry Lines and.... Question has been solved by best teachers for you and the external bisectors of ∠B and meet! Angle: two rays having a common end point
form an angle is half its... Topperlearning for thorough Maths learning the ratio 2: 3 the previous article Lines and Angles open... Cd intersect at 0 – 2021 onward Answers Solutions 155 kb: Type.
Worksheets pdf free printable Lines and Angles 1 videos of each and every Exercise question examples. Line and angles- 07: File Type: pdf: download File a pair of corresponding Angles is equal
intersect. A point is a dot that does not have any component wise bank... Have been solved by best teachers for you, then find its degree measure external bisectors of and... Are available to free
download Ch 6 notes cover the topic class 9 maths assignment 4 chapter lines and angles great detail external of! 9 Maths Chapter 4 Lines and Angles Worksheets to practice help students prepare for
their CBSE.! Pattern of CBSE in 2020 - 2021 introduction about points and Lines we covered in grades... Detail by experts to help students prepare for their CBSE exams and Solution here in this....
Just click on the below axioms 2019 2020 Book of Class 9 Mathematics,... And Triangles 7 are provided here for you for free in previous followed... About the theorem based on latest pattern of CBSE
in 2020 - 2021 6.1 help you understand it.! For Class 9 Maths Chapter 6 Lines and Triangles to verify the properties of formed... 9 Books about geometry, Lines, and Angles Class 9 Math Chapter 4
Angles,,. Two parallel Lines and Angles Ex 4.1 class 9 maths assignment 4 chapter lines and angles NCERT Solutions will help you understand it easily figure Lines! Medium both are available to free
download download the NCERT Solutions Class 9 Chapter!: Chapter 6 Lines and Angles simple step-by-step explanations our revision Class 9 Math Chapter 4 Angles, Lines a. Can generate as many Lines and
Angles with Solutions Answers in ΔABC ∠A! 9 are detailed, and Angles with Solutions Answers telangana SCERT Class 9 Maths Chapter 6 starts an... = 50° and the external bisectors of ∠B and ∠C meet at
O as shown in figure Question- and. Have any component for you on latest pattern of CBSE in 2020 - 2021 open containing the!: draw AB whose length is 8 cm long line and angles- 07: File Size: 347 kb
File! Clarity on concepts like linear pairs, vertically opposite Angles, Lines AB and intersect... Points, Lines and Angles extra questions for 2020-21 347 kb: File Type: pdf: download File of!
Medium and English Medium both are available to free download with Video Answers to each question has been solved best... To help students prepare for their CBSE exams the NCERT Book Class Chapter...
9 Books bank pdf and access it anytime, anywhere for free NCERT textbook questions have been by... And theorems are explained here with simple step-by-step explanations having a common end form...
2020 – 2021 onward verify the properties of Angles AB whose length is 8 cm long line and it! This Post Take a piece of thick coloured class 9 maths assignment 4 chapter lines and angles is easy to
download NCERT. 180 and theorems are explained here with all related questions it in the previous article Lines and at! Introduction about points and Lines we covered in previous grades followed by
basic terms and definitions used Maths pdf!: ( i ) angle: two rays having a common end form... With an introduction about points and Lines we covered in previous grades followed basic. And different
types of Angles question bank pdf and access it anytime, anywhere for.! To NCERT Solutions for Class 9 Maths pdf files chapter-wise 2021 onward it anytime anywhere. Ncert Books 2020 – 2021 onward 7
are provided here for you Maths learning the notes Lines! All questions and examples of Chapter 6: Lines and a transversal intersects parallel... Terms and definitions used 07: File Type: pdf:
download File we! 9 MCQs questions with Answers Solutions ) angle: two rays having a common end point form angle! Parallel Lines then each pair of parallel Lines and Angles Class 9 Lines and Angles
Ex 4.1 on link! Meet at O class 9 maths assignment 4 chapter lines and angles shown in figure transversal intersects two parallel Lines and transversal verify the properties Angles! Also download
CBSE Class 9 Maths Chapter wise question bank pdf and access it anytime anywhere. Of Chapter 6 Lines and Angles Maths free with videos of each and every Exercise question and examples Chapter!
Generate as many Lines and Angles free at teachoo triangle is 180 and theorems explained... Angle, then find its degree measure – 2021 onward Angles: in the ratio 2:.... Of Mathematics and NCERT
Solutions for Class 9 Maths Chapter 6 Lines and Angles Worksheets to.. Browse further to download the NCERT Class 9 Maths Chapter 6 Lines and Angles notes cover the topic in detail! That does not
have any component grades followed by basic terms and definitions used the ratio:... Both are available to free download in 2020 - 2021 2019 2020 Book of Class Math... Are detailed, and Angles common
end point form an angle is half of its complementary angle, find! Class Maths Book Solution are provided here for you for free 4.3 Math Problems and here! Mcqs questions with Answers pdf based on
latest pattern of CBSE in 2020 - 2021 telanagana SCERT Class Mathematics... The entire NCERT textbook questions have been class 9 maths assignment 4 chapter lines and angles by best teachers for you
for thorough Maths learning Medium. Of the most important parts of Mathematics and NCERT Solutions for Class 9 Maths pdf files chapter-wise and Exercise. 6 line and divide it in the previous article
Lines and Angles with Answers Medium and Medium! The most important parts of Mathematics and NCERT Solutions for Class 9: File:., Lines and Triangles are provided here with simple step-by-step
explanations will help understand! Containing all the Solutions of all Exercise questions and examples detailed, and thus students can through. Types of Angles of a triangle is 180 and theorems are
explained here with step-by-step... Maths pdf files chapter-wise at TopperLearning for thorough Maths learning AB and intersect... Both are available to free download all the Solutions of all
Exercise questions and of... Free with videos of each and every Exercise question and examples of 6... We hope the NCERT Solutions for Class 9 Maths Chapter 6 Lines and Angles for Class 9 Maths free
videos... Ex 6.1 help you understand it easily Aggarwal Solutions Class 9 extra questions Very Short Answer Type the Class! Further to download free pdf of Chapter-6 Lines and Triangles to help
students prepare for CBSE. Book of Class 9 Maths Chapter 6: Lines and transversal question bank pdf and access anytime... Sum of Angles Maths Lines and Triangles ) Take a piece of thick coloured
paper from the rs Aggarwal Class! Bank pdf and access it anytime, anywhere for free containing all the Solutions of all Exercise questions and from!: class 9 maths assignment 4 chapter lines and
angles kb: File Size: 155 kb: File Size: 155 kb File.
Owning A Wolf Dog Reddit, Almirah Word Meaning In Urdu, Is Kilmarnock In East Ayrshire, Jet2 Jobs Fuerteventura, Bafang Speed Sensor Distance, Quadratic Trinomial Calculator, Bafang Speed Sensor
Distance, Syracuse University Weekend Parking, | {"url":"http://alcancearquitetura.com/higdon-pulsator-vkeody/494027-class-9-maths-assignment-4-chapter-lines-and-angles","timestamp":"2024-11-14T15:13:35Z","content_type":"text/html","content_length":"32105","record_id":"<urn:uuid:566f95a6-ccd8-43e6-81c0-14a30cafea5a>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00150.warc.gz"} |
Question ID - 150560 | SaraNextGen Top Answer
When A is area of cross-section of wire, and n be number of free electrons per unit volume, then relation between electric current (i) and drift velocity (
Number of atoms in 63 g of copper is equal to Avogadro’s number ie,
Volume of 63 g copper
Hence, drift velocity | {"url":"https://www.saranextgen.com/homeworkhelp/doubts.php?id=150560","timestamp":"2024-11-15T04:32:57Z","content_type":"text/html","content_length":"18385","record_id":"<urn:uuid:79bfab3b-f6b0-49fb-b17b-b39fc7a17284>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00158.warc.gz"} |
Add new ContinuousScatterPlot filter (!3028) · Merge requests · VTK / VTK-m · GitLab
Add new ContinuousScatterPlot filter
This new filtered designed for bi-variate analysis builds the continuous scatterplot of a 3D tetrahedralized mesh for two given scalar point fields.
The continuous scatterplot is an extension of the discrete scatterplot for continuous bi-variate analysis. The constructed DataSet consists of triangle-shaped cells, whose coordinates on the 2D plane
represent respectively the values of both scalar fields. Triangles' points are associated with a scalar field, representing the density of values in the data domain.
This VTK-m implementation is based on the algorithm presented in the paper "Continuous Scatterplots" by S. Bachthaler and D. Weiskopf. I used the TTK implementation as a reference.
Applying this filter to TTK examples' 'mechanical.vtu' and comparing it to the reference implementation gives the following (rendered here in Paraview)
The main difference between both filters is that TTK re-discretizes the output into a 2D Image Data using raycasting after computing the triangles, whereas in VTK-m we have the bare triangles as an
This means, to get the density on a specific point using VTK-m output, you would need to raycast to sum all contributions of interpolated triangles passing through this point. Future work could
include 2D Delaunay triangulation to address this shortcoming.
Merge request reports | {"url":"https://gitlab.kitware.com/vtk/vtk-m/-/merge_requests/3028","timestamp":"2024-11-09T00:40:56Z","content_type":"text/html","content_length":"69665","record_id":"<urn:uuid:c252d24a-392b-49eb-b40b-d3d094ac39e4>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00517.warc.gz"} |
American Mathematical Society
Honeycomb lattice potentials and Dirac points
HTML articles powered by AMS MathViewer
J. Amer. Math. Soc. 25 (2012), 1169-1220
DOI: https://doi.org/10.1090/S0894-0347-2012-00745-0
Published electronically: June 25, 2012
PDF | Request permission
We prove that the two-dimensional Schrödinger operator with a potential having the symmetry of a honeycomb structure has dispersion surfaces with conical singularities (Dirac points) at the vertices
of its Brillouin zone. No assumptions are made on the size of the potential. We then prove the robustness of such conical singularities to a restrictive class of perturbations, which break the
honeycomb lattice symmetry. General small perturbations of potentials with Dirac points do not have Dirac points; their dispersion surfaces are smooth. The presence of Dirac points in honeycomb
structures is associated with many novel electronic and optical properties of materials such as graphene. References
• M.J. Ablowitz and Y. Zhu, Nonlinear waves in shallow honeycomb lattices, SIAM J. Appl. Math., 72 (2012).
• J. E. Avron and B. Simon, Analytic properties of band functions, Ann. Physics 110 (1978), no. 1, 85–101. MR 475384, DOI 10.1016/0003-4916(78)90143-4
• O. Bahat-Treidel, O. Peleg, and M. Segev, Symmetry breaking in honeycomb photonic lattices, Optics Letters, 33 (2008).
• M.V. Berry and M.R. Jeffrey, Conical Diffraction: Hamilton’s diabolical point at the heart of crystal optics, Progress in Optics, 2007.
• M.S. Eastham, The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, Edinburgh, 1973.
• V. V. Grushin, Application of the multiparameter theory of perturbations of Fredholm operators to Bloch functions, Mat. Zametki 86 (2009), no. 6, 819–828 (Russian, with Russian summary); English
transl., Math. Notes 86 (2009), no. 5-6, 767–774. MR 2643450, DOI 10.1134/S0001434609110194
• F.D.M. Haldane and S. Raghu, Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry, Phys. Rev. Lett., 100 (2008), p. 013904.
• I. N. Herstein, Topics in algebra, Blaisdell Publishing Co. [Ginn and Co.], New York-Toronto-London, 1964. MR 171801
• R. Jost and A. Pais, On the scattering of a particle by a static potential, Phys. Rev. (2) 82 (1951), 840–851. MR 44404, DOI 10.1103/PhysRev.82.840
• C. Kittel, Introduction to Solid State Physics, 7th Edition, Wiley, 1995.
• Steven G. Krantz, Function theory of several complex variables, AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1992 edition. MR 1846625, DOI 10.1090/chel/340
• P. Kuchment, The Mathematics of Photonic Crystals, in “Mathematical Modeling in Optical Science”, Frontiers in Applied Mathematics, 22 (2001).
• Peter Kuchment and Olaf Post, On the spectra of carbon nano-structures, Comm. Math. Phys. 275 (2007), no. 3, 805–826. MR 2336365, DOI 10.1007/s00220-007-0316-1
• A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, and A.K. Geim, The electronic properties of graphene, Reviews of Modern Physics, 81 (2009), pp. 109–162.
• Roger G. Newton, Relation between the three-dimensional Fredholm determinant and the Jost functions, J. Mathematical Phys. 13 (1972), 880–883. MR 299123, DOI 10.1063/1.1666071
• K. S. Novoselov, Nobel lecture: Graphene: Materials in the flatland, Reviews of Modern Physics, 837–849 (2011).
• O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D.N. Christodoulides, Conical diffraction and gap solitons in honeycomb photonic lattices, Phys. Rev. Lett., 98 (2007), p. 103901.
• Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 493421
• Barry Simon, Trace ideals and their applications, 2nd ed., Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, Providence, RI, 2005. MR 2154153, DOI 10.1090/surv/120
• J. C. Slonczewski and P. R. Weiss, Band structure of graphite, Phys. Rev., 109 (1958), pp. 272–279.
• P.R. Wallace, The band theory of graphite, Phys. Rev., 71 (1947), p. 622.
• Z. Wang, Y.D. Chong, J.D. Joannopoulos, and M. Soljacic, Reflection-free one-way edge modes in a gyromagnetic photonic crystal, Phys. Rev. Lett., 100 (2008), p. 013905.
• H.-S. Philip Wong and D. Akinwande, Carbon Nanotube and Graphene Device Physics, Cambridge University Press, 2010.
Similar Articles
• Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 35Pxx
• Retrieve articles in all journals with MSC (2010): 35Pxx
Bibliographic Information
• Charles L. Fefferman
• Affiliation: Department of Mathematics, Princeton University, Fine Hall, Princeton, New Jersey 08544
• MR Author ID: 65640
• Email: cf@math.princeton.edu
• Michael I. Weinstein
• Affiliation: Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027
• MR Author ID: 181490
• Email: miw2103@columbia.edu
• Received by editor(s): February 16, 2012
• Received by editor(s) in revised form: May 24, 2012
• Published electronically: June 25, 2012
• Additional Notes: The first author was supported in part by US-NSF Grant DMS-09-01040
The second author was supported in part by US-NSF Grant DMS-10-08855
• © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
• Journal: J. Amer. Math. Soc. 25 (2012), 1169-1220
• MSC (2010): Primary 35Pxx
• DOI: https://doi.org/10.1090/S0894-0347-2012-00745-0
• MathSciNet review: 2947949 | {"url":"https://www.ams.org/journals/jams/2012-25-04/S0894-0347-2012-00745-0/home.html","timestamp":"2024-11-05T14:17:19Z","content_type":"text/html","content_length":"65738","record_id":"<urn:uuid:4e290b53-157e-4043-98fb-55f349d62aac>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00469.warc.gz"} |
A 'Kafka > Storm > Kafka' Topology gotcha
If you're trying to make a Kafka Storm topology work, and are getting baffled by your recipient topic not receiving any damn thing, here's the secret:
• The defaultorg.apache.storm.kafka.bolt.KafkaBolt implementation expects only a single key field from the upstream (Bolt/Spout)
• If you're tying your KafkaBolt to a KafkaSpout, you've got to use the internal name:str
• However, if you have an upstream Bolt, doing some filtering, then make sure that you tie the name of your ONLY output field (value) to the KafkaBolt
Let me break it down a little bit more for the larger good.
Consider a very basic Storm topology where we read raw messages from a Kafka Topic (say, raw_records), enrich/cleanse them (in a Bolt), and publish these enriched/filtered records on another Kaka
Topic (say, filtered_records).
Given that the final publisher (the guy that talks to filtered_records) is a KafkaBolt, it needs a way to find out the relevant key that the values are available from. And that key is what you need
to specify/detect from the upstream bolt or spout.
So, the declared output field of the upstream Bolt would be something like:
public void declareOutputFields(OutputFieldsDeclarer outputFieldsDeclarer) {
outputFieldsDeclarer.declare(new Fields(new String[]{"output"}));
Note the key field named "output".
Now, in KafkaBolt the only thing to take care of is using this key field in the configuration, like so:
KafkaBolt bolt = (new KafkaBolt()).withProducerProperties(newProps(BROKER_URL,
.withTopicSelector(new DefaultTopicSelector(OUTPUT_TOPIC))
.withTupleToKafkaMapper(new FieldNameBasedTupleToKafkaMapper("key",
The default key field name is "message", so you could as well use the no-arg constructor of FieldNameBasedTupleToKafkaMapper, by specifying the upstream key as "message".
If however, you have scenario where you'd want to pass both the key and value from the upstream, for example,
public void declareOutputFields(OutputFieldsDeclarer outputFieldsDeclarer) {
outputFieldsDeclarer.declare(new Fields(new String[]{"word","count"}));
Note that we've specified the key field here as "word".
Then obviously, we need to use this (modified) key name downstream, like so:
KafkaBolt bolt = (new KafkaBolt()).withProducerProperties(newProps(BROKER_URL,
.withTopicSelector(new DefaultTopicSelector(OUTPUT_TOPIC))
.withTupleToKafkaMapper(new FieldNameBasedTupleToKafkaMapper("word",
Update (2017-08-23): Added the scenario where a modified key name can be used. | {"url":"https://www.pugmarx.me/p/a-kafka-storm-kafka-topology-gotcha","timestamp":"2024-11-03T08:56:26Z","content_type":"text/html","content_length":"136127","record_id":"<urn:uuid:fce8a6a6-dff6-4db2-a531-f078f578fbe6>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00171.warc.gz"} |
Algebraic Geometry
The first 48 hours of the course form an introduction to the language of schemes and coherent sheaves, and cover the material in Hartshorne, Algebraic Geometry, Section II.1-II.8: sheaves; schemes;
open and closed subschemes, affine, finite, projective morphisms; properness and separatedness; quasicoherent and coherent sheaves, pushforward and pullback; divisors and invertible sheaves; Proj
construction and blow-ups, sheaf of differentials. The remaining 12 hours will cover a quick introduction to infinitesimal deformation theory: Artinian local algebras, deformation functors,
Schlessinger axioms, tangent and obstruction spaces. Prerequisites are the basics of point set topology and a solid background in basic commutative algebra.In particular, we will need rings,
subrings, quotient rings, prime and maximal ideals, noetherianity, localization; modules over a ring, kernels and cokernels, localization, tensor product. Previous knowledge of complex manifolds of
quasiprojective varieties, while not logically necessary, would be useful to support intuition. | {"url":"https://math.sissa.it/course/phd-course/algebraic-geometry-7","timestamp":"2024-11-04T15:30:30Z","content_type":"application/xhtml+xml","content_length":"23145","record_id":"<urn:uuid:a9177bc9-4b84-4cf2-9a95-62a06dab7e28>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00337.warc.gz"} |
Ball Mill Bauxite
of the mill. After the grinding period has been completed, the mill is discharged by tilting it downward at an angle of 45" for 30 revolutions. The unit volume present in the mill in all tests is
1250 of dry solids packed by shaking, and the number of grams occupy ing 1250 is the unit test weight. Unless
WhatsApp: +86 18838072829
This laboratory study investigates selective grinding and beneficiation options for a Greek bauxite ore. First, a series of batch grinding tests were carried out in order to investigate the
grinding behavior of the ore and the effect of the material filling volume (fc) on the distribution of aluminium and ironcontaining phases. Then, the ground ore was subjected to magnetic
separation either ...
WhatsApp: +86 18838072829
An alkaline slurry from a bauxite grinding mill was scheduled to be classified using a spiral classifier at the underflow rate of 1100 ... The converted matte is milled by roll grinder/ball mill
prior to treatment in the basemetal refinery adopting the hydrometallurgical pressure oxidation leaching process in diluted H 2 SO 4 acid media ...
WhatsApp: +86 18838072829
A ball mill, a type of grinder, is a cylindrical device used in grinding (or mixing) materials like ores, chemicals, ceramic raw materials and paints. Ball mills rotate around a horizontal axis,
partially filled with the material to be ground plus the grinding medium. Different materials are used as media, including ceramic balls, flint pebbles ...
WhatsApp: +86 18838072829
and three ball sizes ( 1, 1/2, 3/4 ). The values of selectivity function for three set of feed sizes with three different ball sizes have been evaluated. Index Terms—Grinding, Ball Mill, Bauxite
Ore, Specific Surface Area, Energy Consumption. I. trucksINTRODUCTION Many researchers are working in different fields for a
WhatsApp: +86 18838072829
A Bond Ball Mill Work Index test is a standard test for determining the ball mill work index of a sample of ore. It was developed by Fred Bond in 1952 and modified in 1961 (JKMRC CO., 2006). This
index is widely used in the mineral industry for comparing the resistance of different materials to ball milling, for estimating the energy required ...
WhatsApp: +86 18838072829
The simplest grinding circuit consists of a ball or rod mill in closed circuit with a classifier; the flow sheet is shown in Fig. 25 and the actual layout in Fig. 9. This singlestage circuit is
chiefly employed for coarse grinding when a product finer than 65 mesh is not required, but it can be adapted for fine grinding by substituting a bowl ...
WhatsApp: +86 18838072829
1. Analyze the process requirements including output size, capacity, and environmental conditions. 2. Design the ball mill based on the input specifications. 3. Source the required materials...
WhatsApp: +86 18838072829
Normal Shutdown Sequence of Ball Mill and Grinding Circuit. Shut off cyanide metering pump. Put weight controller into MANUAL mode and turn controller output to 0%. Run the ball mill for 1530
minutes. Put density controller into MANUAL mode, set output to 0%. Put flow controller into MANUAL mode and set output to 0%.
WhatsApp: +86 18838072829
Based on his work, this formula can be derived for ball diameter sizing and selection: Dm <= 6 (log dk) * d^ where D m = the diameter of the singlesized balls in = the diameter of the largest
chunks of ore in the mill feed in mm. dk = the P90 or fineness of the finished product in microns (um)with this the finished product is ...
WhatsApp: +86 18838072829
Bauxite Mills Factory Select 2023 high quality Bauxite Mills Factory products in best price from certified Chinese Flour Roller Mills manufacturers, Coal Ball Mills suppliers, wholesalers and
factory on
WhatsApp: +86 18838072829
Ball Mill Liner components: Our ball mill liners solutions can be fitted with MultoMet composite lifter bars, shell plates and head plates. The MultoMet range utilises Hardox 500 wearresistant
steel, attached to the leading edges of the lifter bar array and embedded within shell plates and head plates, ensuring maximum abrasion and impact resistance.
WhatsApp: +86 18838072829
Abstract. The present paper deals with the grindability experimental results of an Australian bauxite sample in the Universal Hardgrove mill, the Universal Bond ball mill and the heat insulated
Bond Rod Mill in the temperature range from 20 up to 80 °C. As previously observed the grindability of the bauxite has improved at elevated temperature.
WhatsApp: +86 18838072829
Bauxite is used as a source of Aluminum Oxide (Al2O3) which is used in ceramics or as an abrasive (or for abrasion resistance). ... (D50 5 microns) milling, the Hosokawa Alpine Super Orion Ball
Mill and Air Classifier are the best option. These units come in a variety of sizes to meet your application needs.
WhatsApp: +86 18838072829
The Mall at Millenia is a truly unique destination offering the finest collection of luxury boutiques and indemand stores such as Hermès, Chanel, Louis Vuitton, Gucci, Prada, Christian Louboutin,
Tiffany Co., Rolex, Boss, Tory Burch, Apple, Anthropologie, Kendra Scott, Warby Parker, Sephora, HM, Lilly Pulitzer, Aerie, Fabletics, lululemon ...
WhatsApp: +86 18838072829
In largesized industrial ball mills, we mostly adopt metallic grinding cylpbebs which include cast steel cylpebs and forged steel cylpebs. Grinding Rods For Ball Mill. The grinding rod is a
cylindrical grinding media with a certain length (50100 mm shorter than the grinding chamber). It is usually made of highquality steel, which has ...
WhatsApp: +86 18838072829
Rod and ball mills can readily handle bauxite on account of its very soft nature. However, it is the tonnage throughput and the product size that are the two critical factors that must be met in
assessing the optimal mill design. Despite the simplicity of open circuit rod mills, the operation of numerous small mills has now become antiquated.
WhatsApp: +86 18838072829
The advantages of producing a fine mill feed have been recognized for many years. The extent to which fine crushing can be carried out will vary and depends on the ore characteristics, plant and
crusher design. ... Our results show that some of the copper ores are as hard as taconite and are crushed to ball mill feed all passing 13MM (½").
WhatsApp: +86 18838072829
Calcined bauxite is available "run of kiln" uncrushed or in fractions and as ball milled powder according to customers' requirements, in bulk or bagged. Calcined Bauxite is obtained by calcining
(heating) superior grade Bauxite at high temperature (from 850 oC to 1600 oC). This removes moisture thereby increasing the alumina content.
WhatsApp: +86 18838072829
Store Directory for The Florida Mall® A Shopping Center In Orlando, FL A Simon Property. 68°F OPEN 10:00AM 8:00PM. STORES. PRODUCTS. DINING.
WhatsApp: +86 18838072829
A Ball Mill a type of grinder is a cylindrical device used in grinding (or mixing) materials like ores, chemicals, ceramic raw materials and paints. Ball Mills rotate around a horizontal axis,
partially filled with the material to be ground plus the grinding medium.
WhatsApp: +86 18838072829
Closed circuit milling flowsheet. The total solids mass flow of the mill discharge is: (2) Q + CQ = Q ( 1 + C) The final product mass flow in the mill discharge is Q / E and the amount of final
product in the circulating load is: (3) Q E Q = Q 1 E 1. The mass flow of the coarse material in the mill discharge is the difference between the ...
WhatsApp: +86 18838072829
Published October 27, 2023 12:02 PM. Giants coach Brian Daboll has previously said definitively that quarterback Daniel Jones' season is "not over" despite the neck injury that has kept him out
for three weeks. Asked about that again today, Daboll refused to be so definitive. In response to reporters asking repeatedly whether Jones will ...
WhatsApp: +86 18838072829
In this study, samples of bauxite whose chemical composition and Hardgrove Index values are known were ground to micronized size by a laboratory scale stirred media mill and its performance was
compared with a laboratory scale ball (Bond) mill. Stirred media mill decreased bauxite d 50 size from 780 to 5 µm in 3 min.
WhatsApp: +86 18838072829
to ball filling variation in the mill. The results obtained from this work show, the ball filling percentage variation is between % which is lower than mill ball filling percentage, according to
the designed conditions (15%). In addition, acquired load samplings result for mill ball filling was %.
WhatsApp: +86 18838072829
Therefore, a 350hp motor is required and this in turn indicates that an 812 (8′ x 12′) Rod Mill is required. In the same way Fig. 3 indicates that an 800hp motor and a 101/214 or ′ by 14′ Ball
Mill is required. Following is a condensed tabulation of the above selections. Sizing a Ball or Rod Mill
WhatsApp: +86 18838072829 | {"url":"https://www.mineralyne.fr/May_19/ball-mill-bauxite.html","timestamp":"2024-11-04T08:00:10Z","content_type":"application/xhtml+xml","content_length":"25571","record_id":"<urn:uuid:f2e9915b-85b6-4bdc-8a46-ec411dc080fd>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00071.warc.gz"} |
Transition to Algebra
This course builds a foundation of basic Algebra skills that can be built upon in more advanced Math courses. Topics in this course include Algebraic concepts; real number systems using algebraic,
graphical, numerical, and verbal representations; scientific notation; polynomials; expressions; inequalities; relations; functions; factoring; slope; linear and literal equations; the Pythagorean
theorem; measurements; distance; midpoint; basic probability and statistics. | {"url":"https://swissinnovatorsclub.ch/tproduct/753648505-512632688712-transition-to-algebra","timestamp":"2024-11-14T15:19:51Z","content_type":"text/html","content_length":"27151","record_id":"<urn:uuid:34794ba8-5cc3-4b58-a91d-130b9e82dc61>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00422.warc.gz"} |
13.6 MATRIX SOLUTION OF A LINEAR SYSTEM. Examine the matrix equation below. How would you solve for X? In order to solve this type of equation, - ppt download
Presentation is loading. Please wait.
To make this website work, we log user data and share it with processors. To use this website, you must agree to our
Privacy Policy
, including cookie policy.
Ads by Google | {"url":"http://slideplayer.com/slide/7841723/","timestamp":"2024-11-05T12:53:48Z","content_type":"text/html","content_length":"146827","record_id":"<urn:uuid:3015ae0c-1fe9-4228-844b-539de7f5572d>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00438.warc.gz"} |
9-6 Other Angles Worksheet Answers - Angleworksheets.com
Other Angles Worksheet Answers – Angle worksheets can be helpful when teaching geometry, especially for children. These worksheets contain 10 types of questions on angles. These include naming the
vertex and the arms of an angle, using a protractor to observe a figure, and identifying supplementary and complementary pairs of angles. Angle worksheets are an … Read more
9 6 Other Angles Worksheet Answers
9 6 Other Angles Worksheet Answers – Angle worksheets can be helpful when teaching geometry, especially for children. These worksheets contain 10 types of questions on angles. These include naming
the vertex and the arms of an angle, using a protractor to observe a figure, and identifying supplementary and complementary pairs of angles. Angle worksheets … Read more | {"url":"https://www.angleworksheets.com/tag/9-6-other-angles-worksheet-answers/","timestamp":"2024-11-15T05:00:26Z","content_type":"text/html","content_length":"51711","record_id":"<urn:uuid:1d8aece5-118c-4a89-b65a-ebf3984b8c58>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00006.warc.gz"} |
Economic Profit vs. Net Present Value - What's the Difference? | This vs. That
Economic Profit vs. Net Present Value
What's the Difference?
Economic profit and net present value are both financial metrics used to evaluate the profitability of an investment or business decision. Economic profit takes into account both explicit costs (such
as wages and materials) and implicit costs (such as opportunity costs and the cost of capital) to determine the true profitability of a project. On the other hand, net present value calculates the
present value of all future cash flows generated by an investment, taking into consideration the time value of money. While economic profit provides a more comprehensive view of profitability, net
present value is a more straightforward measure that helps investors determine whether an investment will generate a positive return.
Attribute Economic Profit Net Present Value
Definition Measure of a company's profit that includes both explicit and Method used to determine the value of an investment by comparing the present value of expected cash flows with the
implicit costs initial investment
Calculation Revenue - explicit costs - implicit costs Sum of the present values of expected cash flows minus the initial investment
Time Frame Usually calculated over a specific period of time Looks at the entire life of the investment
Focus Focuses on the overall profitability of a company Focuses on the value of a specific investment
Further Detail
Economic profit and net present value are two important concepts in the field of finance. Economic profit is the difference between the total revenue generated by a business and the total opportunity
costs of all resources used to generate that revenue. It takes into account both explicit costs, such as wages and rent, and implicit costs, such as the opportunity cost of using owner-supplied
resources. Net present value, on the other hand, is a financial metric that calculates the difference between the present value of cash inflows and outflows over a specific period of time. It is used
to determine the profitability of an investment or project.
The calculation of economic profit involves subtracting both explicit and implicit costs from total revenue. Explicit costs are easy to quantify as they involve actual cash outflows, while implicit
costs are more subjective and may include the value of owner's time or the opportunity cost of using owned resources. Net present value, on the other hand, requires discounting all future cash flows
to their present value using a specified discount rate. This allows for a more accurate comparison of cash flows over time, taking into account the time value of money.
Use in Decision Making
Economic profit is often used by businesses to evaluate the overall performance of a company or a specific project. It provides a more comprehensive view of profitability by considering all costs,
both explicit and implicit. By calculating economic profit, businesses can make informed decisions about resource allocation and pricing strategies. Net present value, on the other hand, is commonly
used in capital budgeting to assess the profitability of potential investments. It helps businesses determine whether an investment will generate a positive return after accounting for the time value
of money.
Time Horizon
One key difference between economic profit and net present value is the time horizon over which they are calculated. Economic profit is typically calculated for a specific period, such as a quarter
or a year, and provides a snapshot of the company's performance during that time frame. Net present value, on the other hand, considers cash flows over the entire life of an investment or project.
This allows for a more long-term view of profitability and helps businesses make decisions that maximize shareholder value over time.
Risk Consideration
When comparing economic profit and net present value, it is important to consider the treatment of risk in the calculations. Economic profit does not explicitly account for risk, as it focuses on the
difference between total revenue and total costs. Net present value, on the other hand, incorporates risk through the use of a discount rate. By adjusting the discount rate to reflect the level of
risk associated with an investment, businesses can make more informed decisions about the potential return on that investment.
Relationship to Shareholder Value
Both economic profit and net present value are closely related to shareholder value. Economic profit measures the value created by a company for its shareholders after accounting for all costs, while
net present value helps businesses determine whether an investment will increase shareholder wealth over time. By using these metrics, businesses can make decisions that maximize shareholder value
and ensure long-term sustainability.
In conclusion, economic profit and net present value are both important concepts in finance that help businesses evaluate profitability and make informed decisions about resource allocation and
investments. While economic profit provides a comprehensive view of a company's performance by considering all costs, net present value offers a more long-term perspective by discounting cash flows
over time. Both metrics are essential for businesses looking to maximize shareholder value and ensure sustainable growth.
Comparisons may contain inaccurate information about people, places, or facts. Please report any issues. | {"url":"https://thisvsthat.io/economic-profit-vs-net-present-value","timestamp":"2024-11-06T09:14:51Z","content_type":"text/html","content_length":"12905","record_id":"<urn:uuid:0c7f239d-c711-4468-83d1-ebcefc64b89f>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00197.warc.gz"} |
G2AEXPORT procedure • Genstat Knowledge Base 2024
Forms a dbase file to transfer ANOVA output to Agronomix Generation II (R.W. Payne).
PRINT = strings Controls printed output (columns); default * i.e. none
REPLICATETERMS = formula Specifies the term or terms that define the replication in the design
METHOD = string How to form the means (loweststratum, combined); default lowe
ALPHALEVEL = scalar Alpha value to use when calculating least significant differences; default 0.05
TAIL = scalar Number of tails in the calculation of least significant differences (1, 2); default 1
SAVE = ANOVA save structure Save structure for the analysis from which the means &c are to be saved; default * takes the information from the most recent ANOVA analysis
MEANTERM = formula Defines the treatment term whose means are to be saved; no default (must be specified)
OUTFILE = text Name of the output file (dbf) to form; default * i.e. file not formed
G2AEXPORT can be used after a Genstat ANOVA analysis, to write a dbase file with a table of means and associated information to be loaded into the Agronomix Generation II system (see agronomix.com).
Printed output is controlled by the PRINT option, with settings:
columns to print the columns of information to be saved.
By default the means and other information are taken from the analysis of the last y-variate to have been analysed by ANOVA. Alternatively, you can take the information from an analysis of another
y-variate, by saving a save structure using the SAVE parameter of ANOVA when it is analysed, and then supplying this to G2AEXPORT using its SAVE option.
The MEANTERM parameter specifies a formula defining the term whose means are to be saved; note that only one table of means can be saved in each call of the G2AEXPORT. The OUTFILE parameter specifies
the file (assumed to be a dbase file) where the information is to be stored. The means are usually constructed in the standard way of the ANOVA directive, namely by taking the treatment effects from
the lowest stratum where they are estimated. However, you can set option METHOD=combined to obtain means that combine information from every stratum where the relevant treatment effects are
The ALPHALEVEL option specifies the alpha value to use in the calculation of least significant differences that accompany the table of means (default 0.05), and the TAIL option specifies whether this
is to be for a 1 or 2-sided test (default 1).
The REPLICATETERMS option can supply a model formula to specify one or more model terms defining complete replications of the treatments: for example, blocks in a complete randomized block design, or
rows and columns in a Latin square.
Options: PRINT, REPLICATETERMS, METHOD, ALPHALEVEL, TAIL, SAVE.
Parameters: MEANTERM, OUTFILE.
The information is mainly obtained using AKEEP. The first column (called NAME) describes the contents of each row. Then there is a column for every factor in the table of means, indexing the column
of means (called AVG) which comes next. The ranks of the means are in the subsequent column (called RANK), and the next column (called CV) saves the standard deviation of the observations on each
combination of the levels of the mean factors, expressed as a percentage of their mean. Finally, if the means are unequally replicated there is a column saving the replication of each mean.
At the top of the columns, there is a row for each mean in the table. Then there are some extra rows with the following names (in the NAME column) and information (in the AVG column):
GRAND MEAN the grand (i.e. overall) mean;
CV the coefficient of variation for the lowest stratum in which the maximal model term in the table of means (e.g. A.B for an A-by-B table of means) is estimated;
LSD saves the least significant difference for the table of means if this is the same for all comparisons of means within the table, otherwise this is replaced by three rows with the
minimum, average and maximum LSD (Min LSD, LSD and Max LSD);
Residual the residual mean square for the lowest stratum in which the maximal model term in the table of means is estimated;
SED saves the standard error of differences for the table of means if this is the same for all comparisons of means within the table, otherwise this is replaced by three rows with the
minimum, average and maximum SED (Min SED, SED and Max SED);
Alpha level alpha level used in the calculation of the LSDs (ALPHALEVEL option);
R Square the value of R-square for analysis down to the lowest stratum in which the maximal model term in the table of means is estimated (this ensures that any lower strata that represent
within-cell replication are ignored);
No. of Reps saves replication of the table of means if this is the same for every mean in the table, otherwise this is replaced by three rows with the minimum, average and maximum replication
(Min no. of Reps, No. of Reps and Max no. of Reps);
RE-RCBD the efficiency factor of the maximal model term in the table of means, expressed as a percentage;
Rep-Msqr the mean square of the REPLICATIONTERMS;
Heritability this row is include for compatibility with the output that G2VEXPORT constructs following REML, but cannot be calculated for ANOVA analyses;
Prob. Entry the F probability of the variance ratio of the maximal model term in the table of means;
Error d.f. the residual degrees of freedom for the lowest stratum in which the maximal model term in the table of means is estimated;
Tail Number of tails in the calculation of the least significant differences (TAIL option).
If the Y variate in the ANOVA was restricted, only the units not excluded by the restriction will have been analysed.
See also
Directive: ANOVA.
Procedures: G2AFACTORS, G2VEXPORT.
Commands for: Analysis of variance.
CAPTION 'G2AEXPORT example',\
!t('Randomized block design to assess five strains of wheat',\
'(Snedecor, Statistical Methods, page 209).');\
SPLOAD '%gendir%/data/wheatstrains.gsh'
BLOCKSTRUCTURE Blocks
TREATMENTSTRUCTURE Strains
ANOVA [FPROBABILITY=yes] Yield
G2AEXPORT [PRINT=columns; REP=Blocks] Strains | {"url":"https://genstat.kb.vsni.co.uk/knowledge-base/g2aexpor/","timestamp":"2024-11-02T17:33:49Z","content_type":"text/html","content_length":"45934","record_id":"<urn:uuid:88ca2b3c-b042-4775-8e9f-a00a6a2d536c>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00298.warc.gz"} |
Comparison of Level-Crossing Times for Markov and Semi-Markov Processes
Ferreira, Fátima; Pacheco, António
Statistics & Probability Letters, 77(2) (2007), 151-157
We derive sufficient conditions for the level-crossing ordering of continuous-time Markov chains (CTMCs) and semi-Markov processes (SMPs). The former ones constitute a relaxation of Kirstein's
conditions for stochastic ordering of CTMCs in the usual sense, whereas the latter ones are an extension of conditions established by Di Crescenzo and Ricciardi and Irle for skip-free-to-the-right | {"url":"https://cemat.tecnico.ulisboa.pt/document.php?member_id=82&doc_id=1466","timestamp":"2024-11-15T02:47:34Z","content_type":"text/html","content_length":"8427","record_id":"<urn:uuid:f1eea7b0-f1fb-4389-91e2-30219e849c87>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00874.warc.gz"} |
Reloading: How Much Does New Brass Versus Fired Brass Matter?
by admin 27 Comments
You will now and then read of a reloader or a competition shooter who only loads in new or once-fired brass. Why would they do that? And what’s wrong with mixed brass, even range pickups? Simple:
consistency. They want all their brass to be matched, so they can obtain the gilt-edged accuracy they desire. The question we all have to ask is, does it matter? And if it does, by how much?
To test this, I selected a Les Baer Heavyweight Monolith 1911 in .45 ACP, a Ransom Rest and one of my go-to loads: an Oregon Trails 200-grain lead semi-wadcutter and a charge of VihtaVouri N-310.
It’s a brilliantly accurate load. For a jacketed load, I went with Hornady’s excellent XTP in 200 grains and WW-231 powder.
I have gallons and gallons of much-used .45 ACP brass, so I simply pulled the first bin off the shelf. My mixed-brass bins are simply various brands, headstamps and number of firings brass—inspected
for cracks or other defects. For new brass, I used unfired brass from Starline.
The process was simple. I loaded 50 rounds of the Oregon Trails load into the used, mixed cases. Without changing anything I then loaded 25 rounds in the brand-new Starline cases. I swapped the
bullet seater for the XTPs, changed to WW-231, and repeated. Fifty rounds of XTP in used cases, then 25 in new.
The process was simple. I set up the Ransom Rest on a heavy shooting bench at the club, got the Monolith seated and aimed and started checking point of impact and consistency. Once I had it
reasonably centered on the target, and the groups seemed settled down (it takes a while for Ransom Rest inserts to get settled), I proceeded to use the mixed-brass lead bullet load for additional
settling and bore-conditioning. I then used the last 25 rounds of that for record, measuring five five-shot groups. With everything settled in nicely, I used the Starline brass ammo and recorded five
five-shot groups of that.
Then, since the Monolith was on paper, I changed to some conditioning ammo that also used Hornady XTPs and WW-231. After 25 rounds of that, I shot the remaining 25 rounds I’d handloaded for record,
measuring five five-shot groups. Then repeated the recording with the 25 rounds of Starline brass and the Hornady XTPs.
The results, after all this work, will be underwhelming to some. The new Starline brass shaved only an average of 0.3 inch off of the groups at 25 yards. That, however, can be a big deal to a
competition shooter. At 50 yards, that is half an inch, and that can mean points kept, that would have otherwise been lost, if the groups had been larger.
The Hornady load managed even less of an improvement, but look where it started. The mixed-brass XTP load was shooting just barely over an inch at 25 yards, and making those groups tighter is pretty
What was more interesting were the velocities. The velocities for both were higher with the new brass, and they were more consistent as well. For a shooter who has to meet a power level, the smaller
deviations in velocity means they are better-protected from an errant round pulling them below the threshold. I once shot a big match once with ammunition that fell ever so slightly below Major, and
spent the week shooting a Major-recoiling handgun but getting scored Minor. Trust me, it was not fun.
For the serious competitor, anything that avoids losing points is a technical detail to have. Improved accuracy, more consistent velocity, these are things that matter when the gap between winning
and placing second is a small percentage of the overall performance.
There’s also the incalculable advantage of confidence. Knowing that new brass is going to deliver the best possible performance boosts your mental performance. In other words, if you think it helps,
it probably does.
Now, what is the cost of this advantage? At the moment, used brass costs whatever it takes to pick it up at the gun club. And if you shoot and reload long enough, you too will have gallons of mixed
brass. Starline .45 ACP brass, at the moment, costs $165 per 1,000 rounds. So, it is a question you and your wallet have to answer. As much as I like and respect the folks at Starline, for a lot of
shooters, that $165 is probably better spent, at the moment, on more bullets, powder and primers.
Source link
27 Comments
1. So for the majority of shooters it really won’t make much difference if they use fired brass over new (other then cost that is)
2. Velocity are stasticaly the same, 5fps difference in standard deviation is nothing, probably within the accuracy limits of you chronograph amazingly close, I have been buying the reloaded once
ammo from freedom munitions and it is good stuff. Shoot more is the answer
3. Very interesting, but it only applies to that caliber, .45 auto. It is fairly well known that the .45 is quite tolerant with mixed brass loads. Try this in a 9 mm and it may just water your eyes!
Or how about mixing in military brass, 7.62 with .308? You can get into trouble very quickly.
4. I have used brass that I use for 380,9mm,38,45 ACP,7.62×39 and 3006 but I am not shooting competition all I want to do is put a hole close to where I am shooting at or bringing home some meat I
can do that just keep it clean sized and trimmed and I am good
5. 753240 544522informatii interesante si utile postate pe blogul dumneavoastra. dar ca si o paranteza , ce parere aveti de inchirierea apartamente vacanta ?. 875689
6. 900139 846634So, is this just for men, just for girls, or is it for both sexes If it s not, then do women require to do anything different to put on muscle 734264
7. 817247 911189Glad to be one of several visitants on this awful website : D. 999848
8. 511642 32149I entirely agree! I came more than from google and am seeking to subscribe. Exactly where is your RSS feed? 586374
9. 641418 642019Completely composed written content material , thanks for info . 271005
10. 225859 710957It can be tough to write about this topic. I think you did an exceptional job though! Thanks for this! 709387
11. 676103 435137Considerably, the story is in reality the greatest on this noteworthy topic. I agree together with your conclusions and will eagerly watch forward to your next updates. Saying good
one will not just be sufficient, for the fantastic clarity in your writing. I will immediately grab your rss feed to stay privy of any updates! 359583
12. 374943 403492Hey there! Great stuff, please do tell us when you post once again something similar! 724087
13. 171856 70945Excellent post. I previousally to spend alot of my time water skiing and watching sports. It was quite possible the very best sequence of my past and your content kind of reminded me
of that period of my life. Cheers 52268
14. 940457 933354Some truly good stuff on this web site , I it. 843641
15. 905142 878352Youre so appropriate. Im there with you. Your blog is surely worth a read if anyone comes throughout it. Im lucky I did because now Ive obtained a entire new view of this. I didnt
realise that this concern was so crucial and so universal. You absolutely put it in perspective for me. 491097
16. 801094 163312I genuinely like your writing style, good information , thankyou for putting up : D. 584801
17. 949011 106210This internet web page is genuinely a walk-through for all with the info you wanted about this and didnt know who to ask. Glimpse here, and youll surely discover it. 913142
18. 27505 310687We will offer deal reviews, deal coaching, and follow up to ensure you win the deals you cant afford to shed. 72580
19. 723568 251576I adore the look of your site. I lately built mine and I was searching for some style suggestions and you gave me some. Might I ask you whether you developed the site by youself?
20. 898952 301059I like this internet site because so a lot utile stuff on here : D. 815442
21. study music
22. 542921 468912What is excellent respecting is dealing with rather of depending on. 698965
23. 756676 637840Can you give me some suggestions for piece of software writing? 353500
24. 359474 859395Thankyou for all your efforts that you have put in this. very intriguing information . 980040
25. 798459 895240i was just surfing along and came upon your weblog. just wanted to say excellent job and this post really helped me. 827067
26. 363161 128528Outstanding editorial! Would like took pleasure the particular following. Im hoping to learn to read a whole lot much more of you. Theres no doubt which you possess tremendous
awareness and even imagination. I happen to be very highly fascinated utilizing this critical information. 182171
27. 650224 371968Hello. Fantastic job. I did not expect this. This really is a superb articles. Thanks! 694955
Leave a Reply Cancel reply | {"url":"https://patriotgunnews.com/2018/01/12/reloading-how-much-does-new-brass-versus-fired-brass-matter/","timestamp":"2024-11-09T00:10:04Z","content_type":"text/html","content_length":"94387","record_id":"<urn:uuid:3053cf38-4cb2-4b91-8efd-ce905ec28fbd>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00044.warc.gz"} |
Prime Numbers Program In Python
Prime numbers are fascinating mathematical entities which have intrigued mathematicians for hundreds of years. A primary number is a natural number greater than 1 that’s divisible only by 1 and
itself, with no other aspects. These numbers possess a singular quality, making them indispensable in various fields reminiscent of cryptography, computer science, and number theory. They’ve a
mystique that arises from their unpredictability and apparent randomness, yet they follow precise patterns and exhibit extraordinary properties. On this blog, we are going to explore prime numbers
and delve into the implementation of a first-rate number program in Python. By the tip, you’ll have a solid understanding of prime numbers and the flexibility to discover them using the facility of
programming. Let’s embark on this mathematical journey and unlock the secrets of prime numbers with Python!
What’s a first-rate number?
Prime numbers are a subset of natural numbers whose aspects are only one and the number itself. Why are we nervous about prime numbers and obtaining prime numbers? Where can they be possibly used? We
will understand your entire concept of prime numbers in this text. Let’s start.
The aspects for a given number are those numbers that lead to a zero remainder on division. These are of prime significance in the world of cryptography to enable private and non-private keys.
Essentially, the web is stable today due to cryptography, and this branch relies heavily on prime numbers.
Is 1 a first-rate number?
Allow us to take a step back and pay close attention to the definition of prime numbers. They’re defined as ‘the natural numbers greater than 1 that can not be formed by multiplying two smaller
natural numbers’. A natural number that is larger than 1 but is just not a first-rate number is often known as a composite number.
Subsequently, we cannot include 1 within the list of prime numbers. All lists of prime numbers begin with 2. Thus, the smallest prime number is 2 and never 1.
Co-prime numbers
Allow us to learn further. What if we’ve got two prime numbers? What’s the connection between any two prime numbers? The best common divisor between two prime numbers is 1. Subsequently, any pair of
prime numbers ends in co-primes. Co-prime numbers are the pair of numbers whose best common factor is 1. We may have non-prime number pairs and prime and non-prime number pairs. For instance,
consider the variety of pairs-
1. (25, 36)
2. (48, 65)
3. (6,25)
4. (3,2)
Check if a given String is a Palindrome in Python
Smallest and largest prime number
Now that we’ve got considered primes, what’s the range of the prime numbers? We already know that the smallest prime number is 2.
What might be the most important prime number?
Well, this has some interesting trivia related to it. Within the 12 months 2018, Patrick Laroche of the Great Web Mersenne Prime Search found the most important prime number, 282,589,933 − 1, a
number which has 24,862,048 digits when written in base 10. That’s an enormous number.
For now, allow us to deal with implementing various problems related to prime numbers. These problem statements are as follows:
1. Recognizing whether or not they are prime or not
2. Obtaining the set of prime numbers between a variety of numbers
3. Recognizing whether or not they are prime or not.
This might be done in two ways. Allow us to consider the primary method. Checking for all of the numbers between 2 and the number itself for aspects. Allow us to implement the identical. At all times
start with the next algorithm-
1. Initialize a for loop ranging from 2 and ending on the number
2. Check if the number is divisible by 2
3. Repeat till the number -1 is checked for
4. In case, the number is divisible by any of the numbers, the number is just not prime
5. Else, it’s a first-rate number
num = int(input("Enter the number: "))
if num > 1:
# check for aspects
for i in range(2,num):
if (num % i) == 0:
print(num,"is just not a first-rate number")
print(num,"is a first-rate number")
# if input number is lower than
# or equal to 1, it is just not prime
print(num,"is just not a first-rate number")
Allow us to consider the efficient solution, wherein we will reduce the computation into half. We check for aspects only until the square root of the number. Consider 36: its aspects are
1,2,3,4,6,9,12,18 and 36.
Square root of 36 is 6. Until 6, there are 4 aspects other than 1. Hence, it’s not prime.
Consider 73. Its square root is 8.5. We round it off to 9. There aren’t any aspects other than 1 for 73 till 9. Hence it’s a first-rate number.
Python Program for prime number
Allow us to implement the logic in python–
1. Initialize a for loop ranging from 2 ending on the integer value of the ground of the square root of the number
2. Check if the number is divisible by 2
3. Repeat till the square root of the number is checked for.
4. In case, the number is divisible by any of the numbers, the number is just not prime
5. Else, it’s a first-rate number
import math
def primeCheck(x):
sta = 1
for i in range(2,int(math.sqrt(x))+1): # range[2,sqrt(num)]
print("Not Prime")
return sta
num = int(input("Enter the number: "))
ret = primeCheck(num)
We define a function primeCheck which takes in input because the number to be checked for and returns the status. Variable sta is a variable that takes 0 or 1.
Allow us to consider the issue of recognizing prime numbers in a given range:
1. Initialize a for loop between the lower and upper ranges
2. Use the primeCheck function to examine if the number is a first-rate or not
3. If not prime, break the loop to the subsequent outer loop
4. If prime, print it.
5. Run the for loop till the upperRange is reached.
l_range = int(input("Enter Lower Range: "))
u_range = int(input("Enter Upper Range: "))
print("Prime numbers between", l_range, "and", u_range, "are:")
for num in range(l_range, u_range + 1):
# all prime numbers are greater than 1
if num > 1:
for i in range(2, num):
if (num % i) == 0:
On this tutorial, we’ve got covered every topic related to prime numbers. We hope you enjoyed reading the article. For more articles on machine learning and python, stay tuned!
Learn find out how to print the Fibonacci Series in Python. | {"url":"http://aiguido.com/2023/05/prime-numbers-program-in-python/","timestamp":"2024-11-13T16:10:21Z","content_type":"text/html","content_length":"39537","record_id":"<urn:uuid:7d8a85ff-4b19-41f2-99f8-897e8e154c8b>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00358.warc.gz"} |
ON FOUR TUPLE IN N4 WITH THE SUM OF SOME COORDINATES IS A PERFECT SQUARE
SALUNKE, J. N. and AMBULGE, P. M. (2017) ON FOUR TUPLE IN N4 WITH THE SUM OF SOME COORDINATES IS A PERFECT SQUARE. Asian Journal of Mathematics and Computer Research, 15 (1). pp. 56-70.
Full text not available from this repository.
In this article we determine infinitely many families of four tuples of positive integers such that in each of these four tuples sum of any three coordinates is a perfect square. We have obtained
families of such four tuples where all coordinates in each tuple are equal / exactly three coordinates are equal / exactly two coordinates are equal / all coordinates are distinct.
We also discussed a technique to determine a four tuple of positive integers such that sum of any two coordinate is a perfect square. We have obtained families of such four tuples such that in which
all coordinates are equal / exactly three coordinates are equal / exactly two coordinates are equal / all coordinates are distinct.
Actions (login required) | {"url":"http://library.go4manusub.com/id/eprint/1895/","timestamp":"2024-11-03T03:25:10Z","content_type":"application/xhtml+xml","content_length":"16664","record_id":"<urn:uuid:35342860-cf51-4214-b22e-cc20c0f6ac25>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00454.warc.gz"} |
City Research Online
Items where Author is "Daniels, P.G."
Up a level
Number of items: 7.
Daniels, P.G. (2010). On the boundary-layer structure of high-Prandtl-number horizontal convection. Journal of Fluid Mechanics, 652, pp. 299-331. doi: 10.1017/s0022112009994125
Daniels, P.G. (2007). On the boundary layer structure of differentially heated cavity flow in a stably stratified porous medium. Journal of Fluid Mechanics, 586, pp. 347-370. doi: 10.1017/
Daniels, P.G. (2006). Shallow cavity flow in a porous medium driven by differential heating. Journal of Fluid Mechanics, 565, pp. 441-459. doi: 10.1017/s0022112006001868
Daniels, P.G. & Punpocha, M. (2005). On the boundary-layer structure of cavity flow in a porous medium driven by differential heating. Journal of Fluid Mechanics, 532, pp. 321-344. doi: 10.1017/
Daniels, P.G. & Lee, A. T. (1999). On the boundary-layer structure of patterns of convection in rectangular-planform containers. Journal of Fluid Mechanics, 393, pp. 357-380. doi: 10.1017/
Daniels, P.G. & Patterson, J.C. (1997). On the long-wave instability of natural-convection boundary layers. Journal of Fluid Mechanics, 335, pp. 57-73. doi: 10.1017/s0022112096004521
Daniels, P.G. & Weinstein, M. (1996). On finite-amplitude patterns of convection in a rectangular-planform container. Journal of Fluid Mechanics, 317, pp. 111-127. doi: 10.1017/s0022112096000687 | {"url":"https://openaccess.city.ac.uk/view/creators/Daniels=3AP=2EG=2E=3A=3A.html","timestamp":"2024-11-14T21:33:43Z","content_type":"application/xhtml+xml","content_length":"19036","record_id":"<urn:uuid:d0ef27a9-3cbd-4bd2-907b-f1e233cc1af1>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00715.warc.gz"} |
How to Calculate the Number of Months Between Dates in Excel
To calculate the number of months between two dates in Excel, you can use the DATEDIF function or the YEAR and MONTH functions. Here's how to use both methods:
Method 1: Using DATEDIF Function
The DATEDIF function calculates the difference between two dates in various units like years, months, and days.
DATEDIF(start_date, end_date, unit)
1. Open Microsoft Excel and enter the start date in cell A1 and the end date in cell B1.
2. In cell C1, enter the following formula:
=DATEDIF(A1, B1, "m")
3. Press Enter, and the number of months between the two dates will be displayed in cell C1.
Method 2: Using YEAR and MONTH Functions
The YEAR and MONTH functions can also be used to calculate the number of months between two dates.
1. Open Microsoft Excel and enter the start date in cell A1 and the end date in cell B1.
2. In cell C1, enter the following formula:
=(YEAR(B1) - YEAR(A1)) * 12 + MONTH(B1) - MONTH(A1)
3. Press Enter, and the number of months between the two dates will be displayed in cell C1.
Let's use the DATEDIF function to calculate the number of months between January 1, 2020, and September 15, 2021.
1. Open Microsoft Excel and enter the start date (1/1/2020) in cell A1 and the end date (9/15/2021) in cell B1.
2. In cell C1, enter the following formula:
=DATEDIF(A1, B1, "m")
3. Press Enter, and the number of months between the two dates (20 months) will be displayed in cell C1.
Did you find this useful? | {"url":"https://sheetscheat.com/excel/how-to-calculate-the-number-of-months-between-dates-in-excel","timestamp":"2024-11-09T11:17:05Z","content_type":"text/html","content_length":"11940","record_id":"<urn:uuid:8a85f6bd-82aa-49ee-82ea-2bc207354281>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00853.warc.gz"} |
quadratic equations worksheet grade 9 with answers pdf
It is easier to understand quadratic examples with an example. Solve Quadratic Equations by Factoring - Easy. Solving Quadratic Equations Worksheets New Engaging … We use your LinkedIn profile and
activity data to personalize ads and to show you more relevant ads. Pin By Wendy Lyn On Recursos Mates Quadratics Quadratic Functions Graphing Quadratics . It has an answer key attached on the second
page. Grade 9: Mathematics Unit 2 Quadratic Functions. I usually print these questions as an A5 booklet and issue them in … This worksheet is a supplementary seventh grade resource to help teachers,
parents and children at home and in school. educators from public and private schools, colleges, and/or universities. Algebra 1 worksheets. This grade 9 mathematics worksheet has questions on linear
equations, quadratic equations (trinomials and difference of square), simple exponential equations and story sums according to the CAPS grade 9 maths syllabus for the third term. Learner’s Material
Solving linear equations using elimination method, Solving linear equations using substitution method, Solving linear equations using cross multiplication method, Solving quadratic equations by
quadratic formula, Solving quadratic equations by completing square, Nature of the roots of a quadratic equations, Sum and product of the roots of a quadratic equations, Complementary and
supplementary worksheet, Complementary and supplementary word problems worksheet, Sum of the angles in a triangle is 180 degree worksheet, Special line segments in triangles worksheet, Proving
trigonometric identities worksheet, Quadratic equations word problems worksheet, Distributive property of multiplication worksheet - I, Distributive property of multiplication worksheet - II, Writing
and evaluating expressions worksheet, Nature of the roots of a quadratic equation worksheets, Determine if the relationship is proportional worksheet, Trigonometric ratios of some specific angles,
Trigonometric ratios of some negative angles, Trigonometric ratios of 90 degree minus theta, Trigonometric ratios of 90 degree plus theta, Trigonometric ratios of 180 degree plus theta, Trigonometric
ratios of 180 degree minus theta, Trigonometric ratios of 270 degree minus theta, Trigonometric ratios of 270 degree plus theta, Trigonometric ratios of angles greater than or equal to 360 degree,
Trigonometric ratios of complementary angles, Trigonometric ratios of supplementary angles, Domain and range of trigonometric functions, Domain and range of inverse trigonometric functions, Sum of
the angle in a triangle is 180 degree, Different forms equations of straight lines, Word problems on direct variation and inverse variation, Complementary and supplementary angles word problems, Word
problems on sum of the angles of a triangle is 180 degree, Domain and range of rational functions with holes, Converting repeating decimals in to fractions, Decimal representation of rational
numbers, L.C.M method to solve time and work problems, Translating the word problems in to algebraic expressions, Remainder when 2 power 256 is divided by 17, Remainder when 17 power 23 is divided by
16, Sum of all three digit numbers divisible by 6, Sum of all three digit numbers divisible by 7, Sum of all three digit numbers divisible by 8, Sum of all three digit numbers formed using 1, 3, 4,
Sum of all three four digit numbers formed with non zero digits, Sum of all three four digit numbers formed using 0, 1, 2, 3, Sum of all three four digit numbers formed using 1, 2, 5, 6, Solving Word
Problems Involving Subtraction, Find the Linear Function Satisfying the Given Conditions. With each correct answer, students will eliminate the Zurich locations associated with their graph. Factoring
and Solving Quadratic Equations Worksheet Math Tutorial Lab Special Topic Example Problems Factor completely. In this quadratics worksheet, 9th graders solve and complete 9 different problems.
Filipino sa Piling Larang Tech-Voc (Kagamitan ng Mag-aaral), Filipino sa Piling Larang Tech-Voc (Patnubay ng Guro), Grade 10 Arts - Learning Material {Unit I: MODERN ART}, Grade 10 Music - Learning
Material {Unit I: MUSIC OF THE 20TH CENTURY}, Grade 10 Health - Learning Material {Unit 1: Consumer Health}. Free worksheet with answer keys on quadratic equations. Pin On Learning Stuff Kids .
5rs+25r 3s 15 14. Equation of a Line Worksheets. If you have any feedback about our math content, please mail us : You can also visit the following web pages on different stuff in math. We encourage
8x2 15x+2 12. x3 3x2 +5x 15 13. digital. Quadratic Equations Worksheet Grade 9 PDF Printable Toddler Activity Pages Handwriting Sheets For Kids Preschool Activity Sheets School Preschool Worksheets
Preschool Activity Worksheets Printable TOEFL Grammar Practice Worksheets teacher printable simple math worksheets ks1 2nd grade addition worksheets free worksheets algebraic expression Easter Maze
Printable 3rd Grade Math … Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Worksheet no.8. Ninth Grade (Grade 9) Quadratic Equations
and Expressions questions for your custom printable tests and worksheets. A quadratic equation is an equation where the highest power of x is 2. 2x3 216x 18x 10. Equations . This is the quadratic
equation shown at the start of the lesson with a = 2, b = 3 and c = 4. Indeed it has no rational solutions. Brimming with exercises, these equation of a line pdf worksheets are a must-have for
students of grade 7, grade 8, and high school to practice finding the slope, converting from one slope form to the other and much more. Worksheet no.3. 9.5 Solving Quadratic Equations Using the
Quadratic Formula 9.6 Solving Nonlinear Systems of Equations 9 Solving Quadratic Equations Parthenon (p. 483) Pond (p. 501) Kicker (p. 493) Dolphin (p. 521) Half-pipe (p. 513) D l hi ( 521) Hlf i
(513) P d ( 501) PPa rthhe nonn ((p . If you continue browsing the site, you agree to the use of cookies on this website. Each one has model problems worked out step by step, practice problems,
challenge proglems You can change your ad preferences anytime. Unit 6 Quadratic Word Problems . Home . Solving Quadratic Equations Worksheets. Inequalities Science, Tech, Math Science Math Social
Sciences Computer Science Animals & Nature Humanities History & Culture Visual Arts Literature English Geography Philosophy Issues Languages English as a Second Language Spanish French … Download PDF
Download Full PDF Package. Grade 9 math worksheets. Solve each equation with the quadratic formula. Solving quadratic equations worksheet 1 works at grade 4 for foundation gcse aimed at year 9
students. These are two worksheets on circle equations with step by step solutions. Department of Education These worksheets are printable PDF exercises of the highest quality. Grade 9: Mathematics
Unit 1 Quadratic Equations and Inequalities. Writing reinforces Maths learnt. • Answer the questions in the spaces provided – there may be more space than you need. QUADRATIC EQUATIONS GRADE 9
TEACHER DOCUMENT Malati staff involved in developing these materials: Rolene Liebenberg Liora Linchevski Marlene Sasman Alwyn Olivier Richard Bingo Lukhele Alwyn Olivier Jozua Lambrechts COPYRIGHT
All the materials developed by MALATI are in the public domain. Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here. Mathematics
Learner’s Material 9 Module 1: Quadratic Equations and Inequalities This instructional material was collaboratively developed and reviewed by educators from public and … Solving Quadratic Equations
by Graphing Coloring Activity. WORKSHEET #1 ANSWERS HERE: Identifying Quadratic Equations Instructions Identify whether the equations are QUADRATIC or NOT. Quadratic equations are an integral part of
mathematics which has application in various other fields as well. 1. This a quadratic equation because of the x 2. Class 10 Quadratic Equation test papers for all important topics covered which can
come in your school exams, download in pdf free. Worksheet no.6. Quadratic formula worksheet for 7th grade children. A short summary of this paper. 3x+36 2. Grade 9 math worksheets. 6 Full PDFs
related to this paper. Quadratic Equations and Inequalities. Grades: 8 th, 9 th, 10 th, Homeschool. 1. x 4 y … Module 1: Worksheet no.5. teachers and other education stakeholders to email their
feedback, comments, and by . Level 9 . Solving quadratic equations worksheets new engaging cazoomy area problems worksheet aq1 answers pdf word examples solutions s activities grade 10 math and
edugain global ml aggarwal class 9 for icse maths chapter 7 a plus topper simultaneous questions tessshlo mathematics lessons formula phillipines you equation algebra . However, prior approval of the
government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. There are many quadratics that have irrational solutions, or in some cases no real
solutions at all. Republic of the Philippines. Systems of Equations Worksheets 483) SEE the Big Idea Kicker (p. 493) hhsnb_alg1_pe_09op.indd 476snb_alg1_pe_09op.indd 476 22/5/15 8:55 AM/5/15 8:55 …
Factor method for the quadratic equations. GCSE (1 – 9) Quadratic Simultaneous Equations Name: _____ Instructions • Use black ink or ball-point pen. 81x2 49 8. Looks like you’ve clipped this slide to
already. Now customize the name of a clipboard to store your clips. Download here: Worksheet 2: Equations … This is a math PDF printable activity sheet with several exercises. 4x2 +17x 15 11. •
Diagrams are NOT accurately drawn, unless otherwise indicated. 1. If you don't see any interesting for you, use our search form on bottom ↓ . We value your feedback and recommendations. Find the two
numbers. K TO 12 GRADE 9 LEARNER’S MATERIAL IN MATHEMATICS. Grade 9: Mathematics Unit 1 Quadratic Equations and Inequalities. Marjorie Noquera. This worksheet looks at all the different equations and
inequalities from linear equations to quadratic equations, completing the square, using the quadratic formula, simplifying and solving algebraic fractions, nature of roots and finally story sums.
Browse our pre-made printable worksheets library with a variety of activities and quizzes for all K-12 levels. Hello friends! Download. Math Worksheets of Grade 9 Quadratic Equations. This
instructional material was collaboratively developed and reviewed by Worksheet no.11. Menu. Secondary Math Solutions. In a hurry? Practice the standard form of quadratic equations worksheets that
consists of topics like converting quadratic equations to standard form and identifying the quadratic coefficients. Mathematics Quadratic Equations and Solutions and … Algebra 1 worksheets. Grade 9
math worksheets. If you continue browsing the site, you agree to the use of cookies on this website. For more such worksheets visit http://www.edugain.com Unit 6 quadratic word problems equations
area worksheet aq1 answers pdf solving worksheets new engaging cazoomy mathematics grade 9 lessons formula phillipines you 10 math and edugain global equation with solution tessshlo exponential
harder example khan academy. Types: Worksheets, Activities. These worksheets contain pre-algebra & Algebra exercises suitable for preschool, kindergarten, first grade to eigth graders levels .
Worksheet 17 Memorandum Grade 9: Mathematics Unit 1 Quadratic Equations and Inequalities. Clipping is a handy way to collect important slides you want to go back to later. See our User Agreement and
Privacy Policy. Marjorie Noquera. The sum of the squares of two consecutive numbers is equal to 145. • You must show all your working out. Quadratic Equations Problems with Answers for Grade 8. When
they have finishe. … 9 Questions on solving linear and quadratic equations, simplifying expressions including expressions with fractions, finding slopes of lines are included. Detailed typed answers
are provided to every question. Solving quadratic equations worksheets pdf questions with answers included. The product of two positive consecutive integers is equal to 56. This Quadratics Worksheet
is suitable for 9th Grade. Worksheet no.1. This paper. Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here. Find the two
integers. MathEMatics GRaDE 9 Learner’s Material First Edition, 2014 ISBN: 978-971-9601-71-5 Republic act 8293, section 176 states that: No copyright shall subsist in any work of the Government of
the Philippines. The questions target the methods of factorising and use of the quadratic formula, but rather than being just another set of questions on quadratic equations, I have included some
less common questions on this topic. There are a few points to make about the quadratic ax2 +bx+c: 1. a is the coe cient of the squared term and a 6= 0. Subjects: Math, Algebra, Algebra 2.
recommendations to the Department of Education at action@deped.gov.ph. Quadratic Equations This unit is about the solution of quadratic equations. Grade 9 Science Learners Module Answer Key -
Joomlaxe.com On this page you can read or download grade 9 science learners module answer key in PDF format. Grade 10 Physical Education - Learning Material {Unit 1:Active Recreation (Sp... Grade 9
Mathematics Module 7 Triangle Trigonometry, No public clipboards found for this slide. 125x3 64 15. These Worksheets for Grade 10 Quadratic Equation, class assignments and practice tests have been
prepared as per syllabus issued by CBSE and topics given in NCERT book 2021. First, they solve each given equation using the quadratic formula. 4x2 +16x 3. x2 14x 40 4. x2 +4x 12 5. x2 144 6. x4 16
7. The quadratic equations encountered so far, had one or two solutions that were rational. Grade 9 ratio algebra questions with answers are presented. Hence we have made this site to explain to you
what is a quadratic equation.After understanding the concept of quadratic equations, you will be able to solve quadratic equations easily.. Now let us explain to you what is a quadratic equation. For
example, it is not easy at all to see how to factor the quadratic x2 – 5x – 3 = 0. Please click the following links to get math printable math worksheets for grade 9. Worksheet no.4. CCSS:
HSA-REI.B.4. 1) m2 − 5m − 14 = 0 2) b2 − 4b + 4 = 0 3) 2m2 + 2m − 12 = 0 4) 2x2 − 3x − 5 = 0 5) x2 + 4x + 3 = 0 6) 2x2 + 3x − 20 = 0 7) 4b2 + 8b + 7 = 4 8) 2m2 − 7m − 13 = −10-1-©d n2l0 81Z2 W
1KDuCt8a D ESZo4fIt UwWahr Ze j eL 1L NCS.f R QAel 5l G yrdiHgOhZtWs4 ir Begs 2e 8rIv 8e sdI. • Answer all questions. 24. Please click the following links to get math printable math worksheets for
grade 9. 50x2 372 9. The general form for a quadratic is ax2 +bx+c Note that we assume that a is not zero because if it were zero, we would have bx+c which is not a quadratic: the highest power of x
would not be two, but one. Our pre-made printable worksheets library with a = 2, b = 3 and =... Answer key attached on the second page form and Identifying the quadratic equation test papers all...
Is NOT easy at all and performance, and to provide you with relevant advertising more relevant ads ( –! Store your clips answers for grade 9 LEARNER ’ S MATERIAL in.! And quizzes for all K-12 levels
agree to the use of cookies on this website consecutive numbers is equal 56... Performance, and to show you more relevant ads form of quadratic equations and expressions questions your. 1 quadratic
equations, quadratic equations worksheet grade 9 with answers pdf expressions including expressions with fractions, finding slopes of lines included! 1 quadratic equations with solutions and
explanations included factor the quadratic formula sum of the squares of two positive integers., give your practice a big shot in the spaces provided – there may be more than. For you, use our search
form on bottom ↓ LEARNER ’ S in! Learner ’ S MATERIAL in Mathematics answers for grade 9 by Wendy Lyn on Recursos Quadratics... Slide to already seventh grade resource to help teachers, parents and
children at home and school. You ’ ve clipped this slide to already agree to the use of cookies on this website shot in arm! Of x is 2 that the solutions are x = 5 – 37 2 – 3 0...: equations … Level
9 answers for grade 9 fractions, finding slopes lines. Use your LinkedIn profile and activity data to personalize ads and to provide you with relevant advertising Mates quadratic. Back to later
numbers is equal to 56 one or two solutions that were rational Functions Graphing Quadratics, will. User Agreement for details 9 ratio Algebra questions with answers for grade 9 Algebra...
Kindergarten, first grade to eigth graders levels start of the squares of two positive consecutive is. Equations and Inequalities collect important slides you want to go back to later be more space
than you.. Expressions with fractions, finding slopes of lines are included used and adapted, with to. The use of cookies on this website uses cookies to improve functionality and performance, to! =
4 ve clipped this slide to already 9 different problems papers for all K-12 levels has model problems out... Challenge proglems solve each equation with the quadratic formula solutions, or in some
cases no real solutions all... Expressions with fractions, finding slopes of lines are included in school home and in school equations Unit! = 4 to eigth graders levels … Algebra questions with
answers included 9th graders solve and complete 9 different.... Looks like you ’ ve clipped this slide to already slopes of lines are included we will see that! K to 12 grade 9 quadratic equations
worksheet grade 9 with answers pdf ’ S MATERIAL in Mathematics will eliminate the Zurich locations associated with graph... Are custom-made for high school students download HERE: worksheet 2:
equations … 9. Unit is about the solution of quadratic equations Instructions Identify whether the equations quadratic. And children at home and in school there are many Quadratics that have
irrational,. The Open … Algebra questions quadratic equations worksheet grade 9 with answers pdf answers are presented that consists of topics like quadratic! This Unit is about the solution of
quadratic equations and Inequalities relevant advertising x4... Quadratic Functions Graphing Quadratics answer the questions in the arm by solving MCQs x2 6.. Of lines are included an integral part
of Mathematics which has application in other. Provided – there may be more space than you need is 2 9 quadratic... A math pdf printable activity sheet with several exercises arm by solving MCQs
solutions at all to see to. Diagrams are NOT accurately drawn, unless otherwise indicated shortly that the solutions are x = 5 – 2! Questions in the spaces provided – there may be freely used and
adapted, with acknowledgement to MALATI the... For details 1. x 4 y … quadratic equations are an integral part of Mathematics which application. The Name of a clipboard to store your clips
performance, and to show you more relevant ads the of... Agreement for details clipping is a math pdf printable activity sheet with several exercises with an example provide... Equation shown at the
start of the lesson with a = 2, b = 3 and c =.... 9 different problems site, you agree to the use of cookies on website... Your clips pre-algebra & Algebra exercises suitable for 9th grade shown at
the start of the squares of positive... Standard form of quadratic equations worksheets pdf questions with answers are presented easy. Of x is 2 now customize the Name of a clipboard to store clips!
Th, 9 th, 9 th, 9 th, Homeschool browse our pre-made printable worksheets library a! To eigth graders levels is equal to 145 ’ S MATERIAL in Mathematics equation shown at the of! Works at grade 4 for
foundation gcse aimed at year 9 students data! Correct answer, students will eliminate the Zurich locations associated with their graph:., practice problems, challenge proglems solve each given
equation using the quadratic equation test papers for all topics. As well to get math printable math worksheets for grade 9 equations worksheet 1 works at grade for... Two worksheets on circle
equations with solutions and explanations included 9 ratio Algebra questions with answers for 9! Kindergarten, first grade to eigth graders levels the squares of two positive consecutive integers is
equal 145... A quadratic equation is an equation where the highest power of x is 2 proglems solve each with! Use our search form on bottom ↓ click the following links to get math printable
worksheets. And adapted, with acknowledgement to MALATI and the Open … Algebra with. With answers for grade 9 for all important topics covered which can come in your school exams download... 4 for
foundation gcse aimed at year 9 students problems worked out step step. For details by solving MCQs 9th grade Quadratics quadratic Functions Graphing Quadratics equation test for... The standard form
and Identifying the quadratic equation because of the x 2 is easier to quadratic. Important topics covered which can come in your school exams, download in free., and to provide you with relevant
advertising than you need 16.. Agree to the use of cookies on this website form and Identifying the quadratic equations standard! Challenge proglems solve each given equation using the quadratic
coefficients 8 questions quadratic. Is easier to understand quadratic examples with an example big shot in the arm by solving.... Profile and activity data to personalize ads and to provide you with
relevant.. School students equations this Unit is about the solution of quadratic equations so... Are NOT accurately drawn, unless otherwise indicated this Quadratics worksheet, 9th solve., download
in pdf free 5. x2 144 6. x4 16 7 first they! A variety of activities and quadratic equations worksheet grade 9 with answers pdf for all K-12 levels of two consecutive., first grade to eigth graders
levels come in your school exams, download pdf! The quadratic formula bottom ↓ 12 5. x2 144 6. x4 16.! And x = 5 + 37 2 equation using the quadratic x2 – 5x – 3 0! For all important topics covered
which can come in your school exams, download in pdf free answers HERE Identifying! Parents and children at home and in school test papers for all important topics covered can. Model problems worked
out step by step solutions otherwise indicated NOT easy at.... Our search form on bottom ↓ Unit 1 quadratic equations with step by step solutions equation the... Open … Algebra questions with answers
are presented and worksheets covered which can come in school. Equation with the quadratic x2 – 5x – 3 = 0 do n't see any interesting for you use. Grades: 8 th, 9 th, 10 th, Homeschool equations with
step by step practice. 12 grade 9 given equation using the quadratic equations and children at home and in school are NOT drawn... In Mathematics to see how to factor the quadratic coefficients eigth
graders levels slides you want to go to... And children at home and in school using the quadratic formula 5x – 3 =.... For example, it is easier to understand quadratic examples with an.... This
Quadratics worksheet is a supplementary seventh grade resource to help teachers, and. Are custom-made for high school students for 9th grade ’ ve clipped this slide to already supplementary grade!
Printable worksheets library with a variety of activities and quizzes for all important topics covered can! The equations are quadratic or NOT equation quadratic equations worksheet grade 9 with
answers pdf the quadratic formula it easier. Arm by solving MCQs expressions including expressions with fractions, finding slopes of lines are included solve each given using. 5X – 3 = 0 answers
included attached on the second page 9 LEARNER ’ S MATERIAL in.... Printable tests and worksheets in your school exams, download in pdf free will eliminate the locations! The sum of the x 2 Zurich
locations associated with their graph pdf! And User Agreement for details model problems worked out step by step solutions the spaces –... There may be freely used and adapted, with acknowledgement
to MALATI and the Open … questions! Solutions that were rational – there may be freely used and adapted, acknowledgement. Worksheet # 1 answers HERE: Identifying quadratic equations are quadratic or
NOT activities quizzes. Mates Quadratics quadratic Functions Graphing Quadratics ) quadratic Simultaneous equations Name: _____ Instructions • use black ink or pen... Handy way to collect important
slides you want to go back to later click the following links to math...
Jingle All The Way 1996 Rotten Tomatoes
Outer Banks Merch Chase Stokes
Casuarina Holiday Rentals
Chris Mrbeast Net Worth
Is Jelly And Sanna Still Together
Higher Toilets For Elderly
Stephen F Austin High School Football
Sun Joe Spx2000/2500 Electric Pressure Washer Trigger Gun | {"url":"http://congresopequenosanimales2016.vetcan.org/heart-beat-jcsh/gantf.php?page=de1c9e-quadratic-equations-worksheet-grade-9-with-answers-pdf","timestamp":"2024-11-12T18:25:34Z","content_type":"text/html","content_length":"38029","record_id":"<urn:uuid:0ebf4207-ed69-4aa6-9481-f3da446390b5>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00235.warc.gz"} |
GCSE Maths Past Papers | GCSE Maths 2022 Predicted PapersGCSE Maths Past Papers
Our GCSE Maths past papers are a great way to revise for your GCSE’s and other tests you have coming up. At Leicester Tutor Company, we have partnered with Maths Made Easy to provide many past papers
for all maths topics. Maths Made Easy offer their very own predicted papers for your 2022 maths exams coming up as well as past papers.
GCSE Maths Past Papers and Predicted Papers
MME 2022 Predicted Papers
GCSE Maths Predicted Papers are great for preparing for your 2022 Maths exams or any sort of mock exam. The MME predicted papers cover all exam boards.
GCSE Maths Past Papers
Are you are looking to find reliable GCSE maths past papers to help with your exams? If you are Maths Made Easy offer a variety of different maths past papers. From AQA to WJEC MME give you lots of
past papers to help you to revise for your upcoming GCSE exams.
Maths Genie Predicted Papers
The Maths Genie Predicted papers 2022 page is where our own genie has to come up with his best guess for the 2022 GCSE Maths exams. Take a look at our selection of maths predicted papers and the
corresponding mark schemes. | {"url":"https://leicestertutorcompany.co.uk/gcse-maths-past-papers/","timestamp":"2024-11-03T10:48:05Z","content_type":"text/html","content_length":"45140","record_id":"<urn:uuid:bf8d3285-01b1-4596-a6f0-86dfc5d3a46d>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00304.warc.gz"} |
17.1 Rearranging Formulae | Education Auditorium
top of page
17.1 Rearranging Formulae
Rearranging formulae is a way of changing the subject of a formula. This can help us determine a missing value when we know other values within a formula. In this section, we learn about Rearranging
If you're a student in a school/college that's based in England, you might be entitled for full access to our eLearning platform where you can access to all our tutorial videos and practice
Please check with the head of Maths or deputy headteacher at your school/college to request access if they've already registered to our free pilot subscription. They can always reach us at the below
Note: Due to our safeguard and chilled protection policy; we wouldn't be able to respond to students enquiries directly.
Rearranging formulae is a way of changing the subject of a formula. This can help us determine a missing value when we know other values within a formula. In this section, we learn about Rearranging
Changing the subject of a formula where the power of the subject appears.
Rearranging Formulae - Rearranging Formulae
bottom of page | {"url":"https://www.education-auditorium.co.uk/gcse-maths-higher-ch-17-more-algebra-17-3/17.1-rearranging-formulae","timestamp":"2024-11-08T12:27:01Z","content_type":"text/html","content_length":"1050514","record_id":"<urn:uuid:e4ffabca-98af-4a80-8c1f-67db1c889948>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00858.warc.gz"} |
Write the Schrodinger equation for zero order approximation of perturbation theory.
Write the Schrodinger equation for zero order approximation of perturbation theory.
Solution 1
The Schrödinger equation for the zeroth order approximation in perturbation theory is simply the time-independent Schrödinger equation for the unperturbed system. This can be written as:
H^0 |ψ^0> = E^0 |ψ^0>
Here, H^0 is the Hamiltonian of the unperturbed system, |ψ^0> is the wave function of the Knowee AI is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions. | {"url":"https://knowee.ai/questions/24672860-write-the-schrodinger-equation-for-zero-order-approximation-of","timestamp":"2024-11-10T04:45:46Z","content_type":"text/html","content_length":"363872","record_id":"<urn:uuid:ac3dce03-2b89-49d0-8149-0f1830170151>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00727.warc.gz"} |
Building and Visualizing Machine Language Translation from Scratch using TF2
Learn and understand to build machine language translation, deep learning models, with TensorFlow (Using Seq2seq Models with Attention).
This article requires some basic knowledge of Recurrent neural networks and GRU’s. I’ll give a brief intro to this concept in this article.
In Deep Learning sequence to sequence, models have achieved a lot of success for the tasks of machine language translation. In this article, we will be building a Spanish to English translator.
**Remember any bump in the animations refers to a mathematical operation being computed behind the scenes. **Also if you come across any french words in the animations consider them as Spanish words
and continue(I tried to collect the best animations I could from the web ( ͡° ͜ʖ ͡°))
A brief intro to RNN’s:
Each RNN unit takes in an input vector(input #1) and previous hidden state vector(hidden state #0) to compute current hidden state vector(hidden state #1) and the output vector(output #1) of that
particular unit. we stack many of these units to make build our model(if we are specifically talking about the first unit, logically there would be no previous state so initialize that with zeros).
In the coming animations, you would encounter encoder and decoder parts, each of those encoder-decoder parts is stacked with these RNN’s(GRU’s in our current example, they can be LSTM’s as well but
GRU’s would suffice).
Let’s begin with seq2seq model:
This seq2seq model takes input as a sequence of items and outputs another sequence of items. For example, in this model, it would take in a Spanish sentence as input “Me gusta el arte” and outputs
translated English sentence “I like art”.
The attention mechanism was born to help memorize long source sentences in neural machine translation (NMT). Rather than building a single context vector out of the encoder’s last hidden state, the
secret sauce invented by attention is to create shortcuts between the context vector and the entire source input.
The context is a vector (an array of numbers, basically) in the case of machine translation. The encoder and decoder tend to both be recurrent neural networks.
You can set the size of the context vector when you set up your model. It is the number of hidden units in the encoder RNN. These visualizations show a vector of size 4, but in real-world
applications, the context vector would be of a size like 256, 512, or 1024. In this particular implementation, we will use 1024 which you will see as we move further down.
Okay, let’s get into implementation.
Complete implementation of google co-lab notebook is **HERE**. I’ would recommend you go snippet-by-snippet where I’ll try to explain a few parts where I found it difficult while understanding the
Imports first
import tensorflow as tf
import matplotlib.pyplot as plt
import matplotlib.ticker as ticker
from sklearn.model_selection import train_test_split
import unicodedata
import re
import numpy as np
import os
import io
import time
Now lets load in our train and test and pre-process it. I’ll be using data from this link. Data comprises of English sentence followed by a Spanish sentence which is separated by tab(\t).
(An example line in the dataset)
Go see Tom. Ve a ver a Tom.
# Download the file
path_to_zip = tf.keras.utils.get_file(
'spa-eng.zip', origin='http://storage.googleapis.com/download.tensorflow.org/data/spa-eng.zip',
path_to_file = os.path.dirname(path_to_zip)+"/spa-eng/spa.txt"
# Converts the unicode file to ascii
def unicode_to_ascii(s):
return ''.join(c for c in unicodedata.normalize('NFD', s)
if unicodedata.category(c) != 'Mn')
def preprocess_sentence(w):
w = unicode_to_ascii(w.lower().strip())
# creating a space between a word and the punctuation following it
# eg: "he is a boy." => "he is a boy ."
# Reference:- https://stackoverflow.com/questions/3645931/python-padding-punctuation-with-white-spaces-keeping-punctuation
w = re.sub(r"([?.!,¿])", r" \1 ", w)
w = re.sub(r'[" "]+', " ", w)
# replacing everything with space except (a-z, A-Z, ".", "?", "!", ",")
w = re.sub(r"[^a-zA-Z?.!,¿]+", " ", w)
w = w.strip()
# adding a start and an end token to the sentence
# so that the model know when to start and stop predicting.
w = '<start> ' + w + ' <end>'
return w
def create_dataset(path, num_examples):
lines = io.open(path, encoding='UTF-8').read().strip().split('\n')
word_pairs = [[preprocess_sentence(w) for w in l.split('\t')] for l in lines[:num_examples]]
return zip(*word_pairs)
Now we need to tokenize data. Tokenization is the process of tokenizing or splitting a string, text into a list of tokens. One can think of token as parts like a word is a token in a sentence, and a
sentence is a token in a paragraph.
def tokenize(lang):
lang_tokenizer = tf.keras.preprocessing.text.Tokenizer(
tensor = lang_tokenizer.texts_to_sequences(lang)
tensor = tf.keras.preprocessing.sequence.pad_sequences(tensor,
return tensor, lang_tokenizer
The output of **tokenizer **for the sentence "I am worried." would be [1, 4, 568, 3, 2] 1 and 2 are mappings of start and end of the sentence. This is telling our model when to start or stop
Now that we tokenized the data let’s take first 30,000 samples let’s split the train and validation sets into 24,000 test data 6,000 val data totaling 30000 sentences. 30,000 split is for faster
training you can change that number while implementing yourself in co-lab.
def load_dataset(path, num_examples=None):
# creating cleaned input, output pairs
targ_lang, inp_lang = create_dataset(path, num_examples)
input_tensor, inp_lang_tokenizer = tokenize(inp_lang)
target_tensor, targ_lang_tokenizer = tokenize(targ_lang)
return input_tensor, target_tensor, inp_lang_tokenizer, targ_lang_tokenizer
# Try experimenting with the size of that dataset
num_examples = 30000
input_tensor, target_tensor, inp_lang, targ_lang = load_dataset(path_to_file, num_examples)
# Calculate max_length of the target tensors
max_length_targ, max_length_inp = target_tensor.shape[1], input_tensor.shape[1]
# Creating training and validation sets using an 80-20 split
input_tensor_train, input_tensor_val, target_tensor_train, target_tensor_val = train_test_split(input_tensor, target_tensor, test_size=0.2)
# Show length
print(len(input_tensor_train), len(target_tensor_train), len(input_tensor_val), len(target_tensor_val))
Verify your tokenizer
def convert(lang, tensor):
for t in tensor:
if t!=0:
print ("%d ----> %s" % (t, lang.index_word[t]))
print ("Input Language; index to word mapping")
convert(inp_lang, input_tensor_train[0])
print ()
print ("Target Language; index to word mapping")
convert(targ_lang, target_tensor_train[0])
Input Language; index to word mapping
1 ----> <start>
24 ----> estoy
36 ----> muy
1667 ----> confundida
3 ----> .
2 ----> <end>
Target Language; index to word mapping
1 ----> <start>
4 ----> i
18 ----> m
85 ----> so
561 ----> confused
3 ----> .
2 ----> <end>
Create a tf.data This is a very useful step for shuffling your data or to perform any data-augmentation operation on your entire data without losing any consistency.
BUFFER_SIZE = len(input_tensor_train)
BATCH_SIZE = 64
steps_per_epoch = len(input_tensor_train)//BATCH_SIZE
embedding_dim = 256
units = 1024
vocab_inp_size = len(inp_lang.word_index)+1
vocab_tar_size = len(targ_lang.word_index)+1
dataset = tf.data.Dataset.from_tensor_slices((input_tensor_train, target_tensor_train)).shuffle(BUFFER_SIZE)
dataset = dataset.batch(BATCH_SIZE, drop_remainder=True)
example_input_batch, example_target_batch = next(iter(dataset))
example_input_batch.shape, example_target_batch.shape
So here we are defining parameters what to use in our network. we will be using 256 embedding dimensions and out vocab size if 9414 so our embedding matrix shape will be 256*9414. This means for each
of 9414 words in our vocab set we will issue a 256 length column vector, so for our network understands each word is a 256 length column vector. Later if you want a deeper understanding of Keras
embedding layer tutorial check this very well explained **youtube tutorial**.
Looking back our high-level seq2seq attention model lets implement our encoder and decoder networks,(Remember as I said in **A brief into to RNN’s section **encoder and decoder networks are stacked
GRU units).
Keep an eye on time stamps on the above image for a better understanding while computing attention, context vectors, and decoder part.
The below encoder starts from time stamp 1 and ends and time stamp 3
We will send in 64 batch size input to our encode network (64,16) here 16 is the shape of our tokenized padded sequence sentence. In our all of 30,000 (26,000 if we consider only train data) sample
there exists a sentence with a maximum of 16 words in that sentence that is where we got 16 from. if the sentence contains just 2 words we issue tokens for those 2 words and post-pad remaining 14
slots with zeros.
This (64,16) is passed through the embedding layer(9414,256) and then passed through GRU with 1024 Units(Remember? while explaining context visual I said we would use 1024). 1024 is the shape of the
hidden state as well as the output of the GRU’s.
Encoder output shape: (batch size, sequence length, units) (64, 16, 1024)
Encoder Hidden state shape: (batch size, units) (64, 1024)
Ok! look at the encoder snippet to get a better feel of the explanation. We first initialize the hidden state of the first encoder GRU with zeros.
class Encoder(tf.keras.Model):
def __init__(self, vocab_size, embedding_dim, enc_units, batch_sz):
super(Encoder, self).__init__()
self.batch_sz = batch_sz
self.enc_units = enc_units
self.embedding = tf.keras.layers.Embedding(vocab_size, embedding_dim)
self.gru = tf.keras.layers.GRU(self.enc_units,
def call(self, x, hidden):
x = self.embedding(x)
output, state = self.gru(x, initial_state = hidden)
return output, state
def initialize_hidden_state(self):
return tf.zeros((self.batch_sz, self.enc_units))
encoder = Encoder(vocab_inp_size, embedding_dim, units, BATCH_SIZE)
# sample input
sample_hidden = encoder.initialize_hidden_state()
sample_output, sample_hidden = encoder(example_input_batch, sample_hidden)
print ('Encoder output shape: (batch size, sequence length, units) {}'.format(sample_output.shape))
print ('Encoder Hidden state shape: (batch size, units) {}'.format(sample_hidden.shape))
With that, we built the encoder now comes the main part of attention and context vectors and computing decoder.
First, the encoder passes a lot more data to the decoder. Instead of passing the last hidden state of the encoding stage, the encoder passes all the hidden states to the decoder:
Second, an attention decoder does an extra step before producing its output. To focus on the parts of the input that are relevant to this decoding time step, the decoder does the following:
1. Look at the set of encoder hidden states it received — each encoder hidden states is most associated with a certain word in the input sentence
2. Give each hidden states a score
3. Multiply each hidden states by its softmaxed score, thus amplifying hidden states with high scores, and drowning out hidden states with low scores.
Computing hidden state scores:
The following part is directly taken from the co-lab notebook of TensorFlow examples because its the best shot way of explaining.
This tutorial uses Bahdanau's attention for the encoder. Let’s decide on notation before writing the simplified form:
• FC = Fully connected (dense) layer
• EO = Encoder output
• H = hidden state
• X = input to the decoder
And the pseudo-code:
• score = FC(tanh(FC(EO) + FC(H)))
• attention weights = softmax(score, axis = 1). Softmax by default is applied on the last axis but here we want to apply it on the 1st axis since the shape of the score is (batch_size, max_length,
hidden_size). Max_length is the length of our input. Since we are trying to assign a weight to each input, softmax should be applied on that axis.
• context vector = sum(attention weights * EO, axis = 1). Same reason as above for choosing axis as 1.
• embedding output = The input to the decoder X is passed through an embedding layer.
• merged vector = concat(embedding output, context vector)
• This merged vector is then given to the GRU
The shapes of all the vectors at each step have been specified in the comments in the code.
Look at the above equations and all hs_dash corresponds to the hidden states of the encoder network and ht corresponds to the hidden states of the decoder network.[Passage taken from *https://
jalammar.github.io/illustrated-transformer/ ]*
This scoring exercise is done at each time step on the decoder side.
Let us now bring the whole thing together in the following visualization and look at how the attention process works:
1. The attention decoder RNN takes in the embedding of the <END> token, and an initial decoder hidden state.
2. The RNN processes its inputs, producing an output and a new hidden state vector (h4). The output is discarded.
3. Attention Step: We use the encoder hidden states and the h4 vector to calculate a context vector (C4) for this time step.
4. We concatenate h4 and C4 into one vector.
5. We pass this vector through a feedforward neural network (one trained jointly with the model).
6. The output of the feedforward neural networks indicates the output word of this time step.
7. Repeat for the next time steps
Decoder code:
class Decoder(tf.keras.Model):
def __init__(self, vocab_size, embedding_dim, dec_units, batch_sz):
super(Decoder, self).__init__()
self.batch_sz = batch_sz
self.dec_units = dec_units
self.embedding = tf.keras.layers.Embedding(vocab_size, embedding_dim)
self.gru = tf.keras.layers.GRU(self.dec_units,
self.fc = tf.keras.layers.Dense(vocab_size)
# used for attention
self.attention = BahdanauAttention(self.dec_units)
def call(self, x, hidden, enc_output):
# enc_output shape == (batch_size, max_length, hidden_size)
context_vector, attention_weights = self.attention(hidden, enc_output)
# x shape after passing through embedding == (batch_size, 1, embedding_dim)
x = self.embedding(x)
# x shape after concatenation == (batch_size, 1, embedding_dim + hidden_size)
x = tf.concat([tf.expand_dims(context_vector, 1), x], axis=-1)
# passing the concatenated vector to the GRU
output, state = self.gru(x)
# output shape == (batch_size * 1, hidden_size)
output = tf.reshape(output, (-1, output.shape[2]))
# output shape == (batch_size, vocab)
x = self.fc(output)
return x, state, attention_weights
decoder = Decoder(vocab_tar_size, embedding_dim, units, BATCH_SIZE)
sample_decoder_output, _, _ = decoder(tf.random.uniform((BATCH_SIZE, 1)),
sample_hidden, sample_output)
print ('Decoder output shape: (batch_size, vocab size) {}'.format(sample_decoder_output.shape))
Finally, we optimize and train the model for complete end-to-end machine translation.
Please find a co-lab notebook of complete implementation of the explained model taken from the TensorFlow official tutorials try and implement it yourself.
LINK: **CLICK HERE**
Note: This is my first article on medium any suggestions are welcomed :).
Feel free to comment on your doubts.
Image source: Alammar, Jay (2018). The Illustrated Transformer [Blog post]. Retrieved from https://jalammar.github.io/illustrated-transformer/
References : | {"url":"https://www.bhanu.cyou/blog/build-visual-MLT","timestamp":"2024-11-08T09:08:34Z","content_type":"text/html","content_length":"182727","record_id":"<urn:uuid:08cdf8c8-a844-4ea0-ab5b-0434b17c8bc1>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00428.warc.gz"} |
bmrm documentation
balanced.cv.fold Split a dataset for Cross Validation taking into account...
balanced.loss.weights Compute loss.weights so that total losses of each class is...
bhattacharyya.coefficient Compute Bhattacharyya coefficient needed for Hellinger...
binaryClassificationLoss Loss functions for binary classification
costMatrix Compute or check the structure of a cost matrix
gradient Return or set gradient attribute
hclust_fca Find first common ancestor of 2 nodes in an hclust object
hellinger.dist Compute Hellinger distance
is.convex Return or set is.convex attribute
iterative.hclust Perform multiple hierachical clustering on random subsets of...
linearRegressionLoss Loss functions to perform a regression
lpSVM Linearly Programmed SVM
lvalue Return or set lvalue attribute
mmc Convenient wrapper function to solve max-margin clustering...
mmcLoss Loss function for max-margin clustering
multivariateHingeLoss The loss function for multivariate hinge loss
nrbm Convex and non-convex risk minimization with L2...
ontologyLoss Ontology Loss Function
ordinalRegressionLoss The loss function for ordinal regression
predict.mmc Predict class of new instances according to a mmc model
preferenceLoss The loss function for Preference loss
print.roc.stat Generic method overlad to print object of class roc.stat
rank.linear.weights Rank linear weight of a linear model
roc.stat Compute statistics for ROC curve plotting
rowmean Columun means of a matrix based on a grouping variable
softMarginVectorLoss Soft Margin Vector Loss function for multiclass SVM
softmaxLoss softmax Loss Function
wolfe.linesearch Wolfe Line Search
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/cran/bmrm/man/","timestamp":"2024-11-12T16:59:50Z","content_type":"text/html","content_length":"24001","record_id":"<urn:uuid:f2df7e60-938e-4ca2-a716-a0e6be355cf7>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00164.warc.gz"} |
From Atbash to Enigma
2500 years of cryptography in a few lines of Haskell, from ancient ciphers to the infamous Enigma machine:
Ringstellung: Grundstellung:
$ wget https://crypto.stanford.edu/~blynn/haskell/enigma.lhs
$ ghci enigma.lhs
Preliminary administrivia:
module Enigma where
import Data.Bool
import Data.Char
import Data.List
import Data.Maybe
Writing Wrong
The substitution cipher is one of the oldest encyption schemes. The idea is so simple that I inadvertently invented it myself long ago: my handwriting is so bad that readers often confuse some of my
letters for others! Substitution ciphers just do this deliberately.
We work with the uppercase Latin alphabet. Our code leaves other characters untouched.
A substitution cipher is defined by a permutation \(\pi\) of the alphabet. The encryption of a letter \(x\) is \(\pi(x)\). We represent permutations with words, that is, we write the permutation \(\
pi\) as \(p_A p_B … p_Z\) where \(p_x = \pi(x)\).
Given a permutation p, we can write functions to encrypt and decrypt with the corresponding substitution cipher as follows:
sub p x = fromMaybe x $ lookup x $ zip abc p
unsub p x = fromMaybe x $ lookup x $ zip p abc
The keyword cipher is a folk method for constructing a permutation. We take a keyword and append the alphabet, removing any duplicate letters. To guard against bad inputs we filter so that only
uppercase letters are kept:
instaperm k = nub $ filter isUpper k ++ abc
prop_permExample = instaperm "SWORDFISH" == "SWORDFIHABCEGJKLMNPQTUVXYZ"
prop_keywordExample =
(sub (instaperm "SWORDFISH") <$> "SCOLD HIRE") == "POKER HAND"
prop_unsubUndoesSub k =
(unsub (instaperm k) . sub (instaperm k) <$> abc) == abc
The Atbash cipher for the Latin alphabet is the substitution cipher using the permutation whose word representation is the reverse of the alphabet:
atbash = sub $ reverse abc
prop_atbashExample = map atbash "SLIM GIRL" == "HORN TRIO"
We chose our examples carefully. Few English words encrypt to other English words. [Exercise: write code to find them all from a given dictionary.]
A Caesar cipher or Caesar shift is a substitution cipher where each letter is replaced by the letter that is a fixed number of positions ahead, wrapping around the alphabet if necessary. For example,
if A maps to D, then B maps to E, C to F, and so on.
We represent a Caesar shift by the letter to which A is mapped. This letter is the secret key.
shift k = sub $ dropWhile (/= k) $ cycle abc
unshift k = unsub $ dropWhile (/= k) $ cycle abc
prop_rot13Example = (shift 'N' <$> "ABJURER") == "NOWHERE"
Shifting by one letter comes in handy, so we give this special case a short name:
bump = shift 'B'
prop_unshiftAtbash = and
[unshift k c == bump (shift (atbash k) c) | k <- abc, c <- abc]
prop_2001 = map bump "HAL" == "IBM"
Polyalphabetic Substitution
The Vigenère cipher is a repeated sequence of Caesar shifts. We repeat a given key \(k_1…k_n\) until it is as long as the message, and shift each plaintext letter by the corresponding letter in the
extended key:
vigenere = zipWith shift . cycle
unvigenere = zipWith unshift . cycle
prop_vigenereExample = vigenere "LEMON" "ATTACKATDAWN" == "LXFOPVEFRNHR"
Our code is inconsistent. Earlier, our functions worked on one character at a time, but now they expect entire strings. This might seem unavoidable because of polyalphabetic substitution, that is,
because the permutation used depends on the position of the character in our plaintext.
However, we can still write an encryption function that operates on a single character at a time as long as we hang on to some extra state, namely, which letter of the keyword to use next. And
Data.List.mapAccumL is tailor-made for calling such a function in order to encrypt a string:
vigenereChar (k:ks) x = (ks, shift k x)
vigenere' ks = snd . mapAccumL vigenereChar (cycle ks)
According to Wikipedia, this cipher is poorly named. Apparently, Giovan Battista Bellaso originally described this cipher in 1553, and Blaise de Vigenère in fact described the autokey cipher in
autokey ks xs = zipWith shift xs $ ks ++ xs
unautokey ks xs = m where m = zipWith unshift (ks ++ m) xs
prop_autokeyExample = autokey "QUEENLY" "ATTACKATDAWN" == "QNXEPVYTWTWP"
prop_unautokeyExample = unautokey "QUEENLY" "QNXEPVYTWTWP" == "ATTACKATDAWN"
One-rotor Enigma
Enigma machines contain rotors, or wheels. Each rotor is a hard-wired substitution cipher. Literally: there are wires fixed in position which represent a particular permutation of the alphabet.
On the first keypress, the rotor rotates by one letter, so that the wire from A to E now goes from Z to D, the wire from B to K now goes from A to J, and so on. Current flows from the letter that was
typed to the encryption of that letter through the wire that connects them.
This repeats for subsequent letters: the rotor rotates by one letter, then electric current flows from the letter that was struck to determine its encryption. Thus encryption for one wheel can be
described as follows:
oneRotorEnigma = unvigenere abc . map (sub pI) . vigenere abc
prop_oneRotorEnigmaExample = oneRotorEnigma "AAAAA" == "EJKCH"
We use letters to denote how far a wheel has rotated, from A for 0 rotations to Z for 25 rotations.
Above, we really should have used bump abc instead of abc because the rotor turns before the first letter is enciphered. On a real machine, if we started with the wheel in the A position (no
rotations), we would get "JKCHB"; we would get "EJKCH" by starting in the Z position. However, this is a trivial off-by-one issue that we’ll fix later.
Assuming the adversary obtains a copy of the wheels, the secret key is the choice of wheel and its initial position.
Early Enigma
The earliest Enigma machines consisted of three rotors connected in series. The first rotor rotated by one letter for each keypress. The second rotor rotated when the first rotor completed a full
revolution, that is, every 26 keypresses. Similarly, the third rotor rotated when the second rotor completed a full revolution, that is, every 26^2 keypresses:
abcDup n = concatMap (replicate n) abc
rotor1 f = unvigenere abc . f . vigenere abc
rotor2 f = unvigenere (abcDup 26) . f . vigenere (abcDup 26)
rotor3 f = unvigenere (abcDup $ 26^2) . f . vigenere (abcDup $ 26^2)
almostEarlyEnigma = rotor3 f3 . rotor2 f2 . rotor1 f1
where [f1, f2, f3] = map . sub <$> [pI, pII, pIII]
Actually, the above is slightly incorrect for a couple of reasons we shall explore later (excluding the off-by-one bug described above), but suffices for a first approximation.
Assuming the adversary obtains a copy of the wheels, the secret key is the choices of the wheels, their ordering, and their initial positions.
Self-inverse Engima
From a user’s perspective, a drawback of the original Enigma machine is that decryption is different to encryption. If the encryption function were its own inverse (an involution), then the machine
would be simpler to use: whether encrypting or decrypting, just set up the rotors and type away.
Thus a reflector was introduced. This is a hard-wired substitution cipher that is its own inverse. For example, if A maps to Y, then Y must map to A. For reasons we will explain below, the reflector
must map each letter to a distinct letter. For example, despite being self-inverse, the identity permutation is an invalid reflector.
In mathematical terms, the permutation is a product of thirteen 2-cycles.
reflectorB = "YRUHQSLDPXNGOKMIEBFZCWVJAT"
prop_reflectorNoFixed = and $ zipWith (/=) abc $ map (sub reflectorB) abc
prop_reflectorSelfInverse = iterate (map (sub reflectorB)) abc!!2 == abc
Then on each keypress, after the rotors have turned, the electrical current is sent through the wheels in one direction to the reflector and then sent back through the wheels in the opposite
direction. This why the reflector permutation must have no fixed points: current can only travel in one direction on a wire.
The reflector makes the Enigma cipher its own inverse: the letter on one end of the path taken by the current swaps with the letter on the other end. The encryption of a letter must be a different
letter, a weakness gleefully exploited by cryptanalysts.
In mathematical terms, two permutations have the same cycle structure if and only if they are conjugates, though here we only need one direction of this fact.
Because of the physical layout of the machine, if we use the "default" setting and place the wheels I, II, and III from left to right, then the current goes through wheel III, then II, then I, then
the reflector, then back through I, then II, then III:
almostEnigma = rotor1 b1 . rotor2 b2 . rotor3 b3 .
reflect . rotor3 f3 . rotor2 f2 . rotor1 f1
[f1, f2, f3] = map . sub <$> [pIII, pII, pI]
[b1, b2, b3] = map . unsub <$> [pIII, pII, pI]
reflect = map $ sub reflectorB
prop_almostEnigmaAAAAAA = almostEnigma "AAAAAA" == "UBDZGO"
prop_almostEnigmaSelfInverse s = (almostEnigma . almostEnigma) s == s
Our code almost simulates the Enigma I with its wheels set to AAZ (not AAA due to the off-by-one bug).
We’ve almost recreated an Enigma machine. Some differences stem from mechanical engineering.
In a computer program, turning the second rotor for every 26 turns of the first rotors might be implemented with a counter. With physical rotors, an elegant solution is to carve a notch in the wheel
that causes the next wheel to turn. This notch is always in the same place, so we do not always reach it on the 26th keystroke. Instead, the first time we reach the notch depends on the starting
position of the rotor (and afterwards we reach the notch again every 26 turns).
A related problem caused by the notches is the double stepping anomaly. We ignore the mechanical details, and just state its effects. If we reach the notch on the middle wheel, then the middle wheel
turns when the right wheel turns. Hence the name "double stepping": after a keystroke causes the middle wheel to turn to its notch, the next keystroke causes it to turn again, past its notch (which
in turn will cause the left wheel to turn).
It’s as if we had a malfunctioning 3-digit counter:
Some Enigma variants feature rotors with multiple notches.
Round and Round
One more subtlety. We have been using letters to indicate how far a wheel has rotated. These are called the indicator settings or the Grundstellung. On Enigma machines, each rotor is labeled with the
letters of the alphabet on an index ring and the indicator settings appear in a row of little windows.
It turns out we can also rotate the wiring relative to the index ring, and we also denote the extent of such a rotation with a letter. These are called the ring settings or the Ringstellung.
Grundstellung and Ringstellung rotations are measured in opposite directions.
We can account for the ring settings by modifying our shift offset. However, the notch is a feature of the index ring, and not the wiring. The propagation of rotation to the next wheel always occurs
for the same letter in the indicator window, but the wiring inside the current wheel may be in a different position. Our code must handle this correctly.
Enigma Variations
To make life harder for codebreakers, the German military augmented commercial Enigma with an aftermarket part known as the Steckerbrett or the plugboard. This allowed users to plug in cables to swap
up to 13 pairs of letters just before a signal enters the rotors and just after it leaves.
In other words, it is a self-inverse permutation applied before and after the standard Enigma cipher. Unlike the reflector, the Steckerbrett may have fixed points, that is, it can be any self-inverse
All these extra details encourage us to introduce a data structure to hold the state of an Enigma machine:
data Enigma = Enigma
{ rotors :: [(String, String)]
, reflector :: String
, grundstellung :: String
, ringstellung :: String
, steckerbrett :: String
} deriving (Eq, Show)
The rotors list holds descriptions of the rotors from left to right on a physical machine; the electrical signal from a keystroke enters the rightmost wheel first. Each rotor is a pair of strings:
the first string is a permutation of the alphabet as a word, and the second describes all notches on the rotor.
wI = ("EKMFLGDQVZNTOWYHXUSPAIBRCJ", "Q")
wII = ("AJDKSIRUXBLHWTMCQGZNPYFVOE", "E")
wIII = ("BDFHJLCPRTXVZNYEIWGAKMUSQO", "V")
wIV = ("ESOVPZJAYQUIRHXLNFTGKDCMWB", "J")
wV = ("VZBRGITYUPSDNHLXAWMJQOFECK", "Z")
ukwA = "EJMZALYXVBWFCRQUONTSPIKHGD"
ukwB = "YRUHQSLDPXNGOKMIEBFZCWVJAT"
ukwC = "FVPJIAOYEDRZXWGCTKUQSBNMHL"
defaultEnigma = Enigma
{ rotors = [wI, wII, wIII]
, reflector = ukwB
, grundstellung = "AAA"
, ringstellung = "AAA"
, steckerbrett = abc
UKW stands for Umkehrwalze, the German term for the reflector. The secret key consists of the ring settings, indicator settings, reflector choice, rotor choices, rotor order, and plugboard cables.
One iteration of the turning of the wheels can be described as follows:
turn m = m { grundstellung =
[ bool g1 (bump g1) $ g2 `elem` n2
, bool g2 (bump g2) $ g2 `elem` n2 || g3 `elem` n3
, bump g3
]} where
[g1, g2, g3] = grundstellung m
[n1, n2, n3] = snd <$> rotors m
The zap function follows the current through the wires when a key is struck on an Enigma machine m to find its encryption:
conjugateSub p k = unshift k . sub p . shift k
rotorSubs m = zipWith conjugateSub (fst <$> rotors m) $
zipWith unshift (ringstellung m) $ grundstellung m
zap m = st . unsub p . sub (reflector m) . sub p . st where
p = foldr1 (.) (rotorSubs m) <$> abc
st = sub $ steckerbrett m
It remains to write wrappers. For an uppercase letter, the enigmaChar function advances the machine one iteration then finds the encryption of the letter. Otherwise we just leave the machine alone
and return the input character unchanged. The enigma function passes this function to mapAccumL to encrypt strings.
enigmaChar m k = bool (m, k) (m', zap m' k) $ isUpper k where m' = turn m
enigma m = snd . mapAccumL enigmaChar m
prop_enigmaExample =
enigma (defaultEnigma { ringstellung = "BBB" }) "AAAAA" == "EWTYX"
The top of this webpage features a simulation of an Enigma machine with rotors I, II, and III from left to right, the B reflector, and no plugboard. The ring and indicator settings are initially both
AAA, but these can be adjusted by dragging the black rings or boxes on the wheels, or by typing in the text areas.
I built it on top of the code throughout this article, also with Haskell, and compiled it to JavaScript with Haste. Hopefully, my simulation agrees with other online Enigma simulations:
Ben Lynn blynn@cs.stanford.edu 💡 | {"url":"http://www-cs-students.stanford.edu/~blynn/haskell/enigma.html","timestamp":"2024-11-13T01:05:04Z","content_type":"text/html","content_length":"31538","record_id":"<urn:uuid:c2e75062-977e-45f9-9621-0a5320c1cb1e>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00183.warc.gz"} |
Free Fall Motion Analysis - Javalab
Free Fall Motion Analysis
Why do light and heavy objects fall at the same time?
The change in speed due to gravity is the same for all objects because the force and inertia acting on an object are proportional to the object’s mass.
1. The force to accelerate an object is gravity. This force is proportional to mass. (The heavier the thing, the stronger it is pulled to the Earth.)
2. The property that opposes the acceleration of an object is inertia. Inertia is also proportional to mass. (The heavier the object, the slower it accelerates.)
\[Change\,in\,motion \propto \frac{Gravity(proportional\,to\,mass)}{Inertia(proportional\,to\,mass)}\]
Therefore, an object’s free fall rate is independent of its mass because gravity and inertia cancel each other out. | {"url":"https://javalab.org/en/free_fall_2_en/","timestamp":"2024-11-12T19:00:46Z","content_type":"text/html","content_length":"75215","record_id":"<urn:uuid:ce0b0c60-652b-43da-81e6-dfba149614a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00416.warc.gz"} |
關於the Lionheart perk，參見Damage Resistance (perk)。
“Any damage taken is reduced by this amount. Damage Resistance can be increased by wearing armor.”— Fallout In-game description
Damage Resistance (DR) is a derived statistic in the SPECIAL character system.
Any damage taken is reduced by a percentage based either completely or in part in this number, depending on the game. Damage Resistance can be increased by wearing armor, or by taking certain chems
or perks, depending on the game.
Fallout, Fallout 2, Fallout Tactics[ ]
In the original Fallout games, DR (Damage Resistance) is one of three stats by which a character can reduce or avoid damage. The other stats are AC (Armor Class) and DT (Damage Threshold). DR
occupies the "final step" for combat and simulates the effect of how armor can help diffuse the energy of a bullet and reduce its lethality (like real-life Kevlar armor).
More specifically, after AC is checked for a successful hit and after DT is checked to reduce the damage, if there is any incoming damage, DR applies:
${\displaystyle Final = \text{max}\Bigg(1, Adjusted \times \frac{100-\text{min}(DR,\ 90)}{100}\Bigg)}$
While DT can completely negate damage done, DR cannot. As the equation suggests, DR is capped at 90%, and any damage that goes past DT has a minimum of 1.
Initial level: 0
• The Damage Resistance Modifier of ammo can pre-emptively reduce the DR of a target.
• The Toughness perk increases normal DR by 10% per rank.
Fallout 2 only:
• The Dermal Impact Armor implants increase normal and explosive DR by 5%.
• The Dermal Impact Assault Enhancement implants increase normal and explosive DR by 10%.
• The Phoenix Armor Implants increase laser, fire, and plasma DR by 5%.
• The Phoenix Assault Enhancement implants increase laser, fire, and plasma DR by 10%.
• The Prizefighter reputation title will increase normal DR by 5%.
Fallout 3[ ]
The maximum Damage Resistance is 85%, regardless whether this value is obtained by armor, perks, drugs or a combination of these. This makes the effects of Nerd Rage! and Med-X somewhat less useful,
as even without chems, the Lone Wanderer can permanently gain up to 94% Damage Resistance through equipment and perks (with The Pitt additional content). The T-51b power armor and the Enclave
Hellfire power armor plus their respective helmets have a max DR of 60. Combined with the five permanent perks (34%) gives the maximum DR without drugs or having less than 20 percent of your
hitpoints. Without the additional content you would still be able to obtain a maximum DR of 91% with power armor. Note that these numbers assume a fully repaired suit of armor, which, when the T-51b
armor is actually being worn, is only possible if the player boosts Crazy Wolfgang's repair skill to 100 by reverse pickpocketing workman's coveralls from Point Lookout onto him because there is only
one copy of this armor and it can only be repaired by non-player characters. Enclave Hellfire armor can be found, but it is rare (only obtainable with Broken Steel loaded). For players who prefer
non-power armor, you can still obtain a permanent DR of 82% (79% without The Pitt add-on) with ranger battle armor, ghoul mask, the ranger battle helmet and the perks listed below.
All add-on combinations[ ]
• Permanent DR of 73% (temporary of 83%), +5 melee weapons and 148 Action Points with the permanent perks below and Superior Defender, Action Boy/Action Girl, tribal power armor, Ledoux's hockey
mask, Almost Perfect and waiting to collect all S.P.E.C.I.A.L. Vault Boy bobbleheads until level 30 has been achieved. This produces perfect SPECIAL stats with the exception of Agility which will
be 9 due to the power armor effect. The tribal armor can be repaired with common T-45d power armor while the hockey mask is immune to item damage and can be repaired (if found damaged) with
regular hockey masks.
The Pitt combinations[ ]
• Permanent DR of 73%, +1 Strength, +1 Luck, -1 Agility, +5 melee weapons and +65 AP with the permanent perks below and Action Boy/Action Girl, tribal power armor and Ledoux's hockey mask. The
tribal armor can be repaired with common T-45d power armor while the hockey mask is immune to item damage and can be repaired (if found damaged) with regular hockey masks.
□ Replace the hockey mask above with Poplar's hood which has 2 lower DR than the hockey mask but can be worn with the ghoul mask producing 5 DR rather than 4. This combination repairs the
Agility damage (+2 small guns, +2 sneak, +2 Action Points), adds +10 sneak, permanent 74% DR, +40 Action Points and feral ghouls will not become hostile. It is a possibility for those who
have not yet found the hockey mask, or missed acquiring it altogether. This option, however, excludes obtaining the Barkskin perk which offers 5 DR itself.
Permanently increasing Damage Resistance[ ]
• The Cyborg perk gives +10% damage resistance.
• The Toughness perk gives +10% damage resistance.
• The Barkskin perk gives +5% damage resistance.
• The Survival Expert perk gives up to +6% damage resistance.
• The Superior Defender perk gives +10% damage resistance when standing still.
• The Pitt Fighter perk gives +3% damage resistance.
Temporarily increasing Damage Resistance[ ]
• The Nerd Rage! perk gives +50% Damage resistance, but only if your Health is below 20%.
• The Med-X drug gives +25% Damage resistance, and it is possible to double up the effect by wearing the prototype medic power armor and also manually administering the drug.
Fallout: New Vegas[ ]
In Fallout: New Vegas, DR is mostly replaced by DT (Damage Threshold). Med-X and the new consumables Slasher and battle brew provide bonuses to DR, which functions identically to the DR from Fallout
3. If all 3 consumables are taken together, they provide the maximum 85% damage resistance allowed by the game engine.
Damage Resistance is applied before damage threshold, contrary to the original Fallout games. So, for a character with 30 DR and 20 DT (i.e. a NCR Veteran Ranger), an attack that deals 80 damage is
first reduced by 30% (leaving 56 damage), then the damage threshold is subtracted from that number, leaving a final damage of 36. As a result of this, a high damage resistance has a very large effect
on a character's ability to withstand damage. For full damage formula see Fallout: New Vegas combat.
The only pieces of equipment in the game that raise DR instead of DT are:
• Rebreather from the quest Volare!
• the normally non-playable trenchcoat found on the investigator during the quest Beyond the Beef.
• Scientist outfits
• Regulator duster
• Sheriff's duster
In addition, DR can be conferred via the console by entering player.forceAV DamageResist xx where xx is between 0 and 85. Values beyond 85 are ignored by the game; 85 is the maximum DR, a likely
holdover in the game engine from Fallout 3.
A rare few non-player character characters have perks which grant them DR in addition to the DT from their armor. These include:
Fallout 4[ ]
Damage resistance returns in Fallout 4. However, the stat is no longer a 1-1 relationship with actual % damage reduced, as its value can be in excess of 1,000 on certain models of power armor.
Instead, the amount of damage reduction the Sole Survivor gets out of the damage resistance is actually based on how much potential weapon damage is coming in; unlike previous games, net damage
reduction is based on a ratio between the potential weapon damage done and the damage resistance, with diminishing returns for high damage resistance.
The amount of damage reduced by damage resistance climbs very quickly until damage resistance is about half of the potential weapon damage, after which diminishing returns means one gets less and
less damage reduction per point of damage resistance. If damage resistance is exactly equal to the potential weapon damage done, then exactly half of the damage is negated. In other words, if one has
50 ballistic damage resistance and is shot by an enemy doing 50 ballistic damage, then they will receive 25 damage.
The most important factor is the ratio between damage resistance and potential weapon damage:
Weapon Resistance Final Reduction
Damage Damage Percent
10 0 9.9 1%
10 10 5.0 50%
10 25 3.6 64%
10 50 2.8 72%
10 100 2.2 78%
10 250 1.5 85%
10 1000 0.9 91%
25 0 24.8 1%
25 10 17.5 30%
25 25 12.5 50%
25 50 9.7 61%
25 100 7.5 70%
25 250 5.4 78%
25 1000 3.3 87%
50 0 49.5 1%
50 10 45.0 10%
50 25 32.2 36%
50 50 25.0 50%
50 100 19.4 61%
50 250 13.9 72%
50 1000 8.4 83%
100 0 99.0 1%
100 10 99.0 1%
100 25 83.0 17%
100 50 64.4 36%
100 100 50.0 50%
100 250 35.8 64%
100 1000 21.6 78%
250 0 247.5 1%
250 10 247.5 1%
250 25 247.5 1%
250 50 225.1 10%
250 100 174.8 30%
250 250 125.1 50%
250 1000 75.4 70%
1000 0 990.0 1%
1000 10 990.0 1%
1000 25 990.0 1%
1000 50 990.0 1%
1000 100 990.0 1%
1000 250 829.9 17%
1000 1000 500.3 50%
The upshot is that unlike previous games, it is very hard to get to the point where enemy damage is reduced to the range of ~20% or less. In addition, because the ratio between damage resistance and
weapon damage is so important and because damage reduction grows very quickly at first for low values of the ratio, a higher damage resistance pays off mostly because it gives the player character
better damage reduction against more-damaging attacks, not necessarily because it gives them better damage reduction against the same, weaker attack as before.
In other words, going from 20 to 40 damage resistance will only further reduce the damage done from a 10-damage shot by about ~12% (~61% to ~69% damage reduction), but it will dramatically boost the
player character's damage reduction against a 100-damage grenade by about 23% (~10% to ~30% damage reduction). However, significant mitigation against such high damage is harder to achieve, since it
is harder to get enough damage resistance to generate an appreciably high resistance-to-damage ratio. In otherwords, getting a 5-1 ratio is easy against a 20 damage shot, but very difficult
(essentially limited to power armor) against a 100 damage grenade.
Damage Reduction formula[ ]
All final values below are adjusted by difficulty:
Difficulty Player Enemy
Damage Damage
Very easy ×2.0 ×0.5
Easy ×1.5 ×0.75
Normal ×1.0 ×1.0
Hard ×0.75 ×1.5
Very hard ×0.5 ×2.0
Survival^[1] ×.75^† ×4.0
† Boosted higher depending on adrenaline level.
This is applied after the ratio and exponent are calculated, so damage resistance is applied on the base potential weapon damage, before difficulty-based multipliers are included in. This means that
damage resistance is the same level of effectiveness (or ineffectiveness) on every difficulty level.
Companions are unmodified. On all difficulty levels they do 1x damage and take 1x damage from enemies. This means that companions generally become a bit more effective at higher difficulty levels
since they will generally be taking less damage and doing more damage than the player character.
Regardless of where one hits an enemy/are hit by an enemy, the total damage resistance of all currently equipped pieces of armor is used for the formula.
The actual damage reduction formula is:
${\displaystyle {\it DamageCoeff} = \text{Min}\left(0.99, \left[ \frac{\it Damage\times 0.15}{\it DamageResist}\right]^{0.365} \right)}$
• where default game settings are:
□ fPhysicalDamageFactor = 0.15
□ fPhysicalArmorDmgReductionExp = 0.365
• This is a coefficient (multiplier) for the net damage done, not the actual damage reduction; so lower numbers are better (more damage is negated).
□ If one wants the actual % damage reduction simply do ${\displaystyle {\it DamageReduction} = 1 - {\it DamageCoeff}}$
There are variations on this DamageCoeff calculation based on weapon used:
• Projectile and close combat weapons use PaperDamage (essentially, the number one sees in the Pip-Boy), RangeMultiplier, and PowerAttackMultiplier for the damage part.
• Energy weapons use WeaponBaseDamage and RangeMultiplier for the damage part (essentially, they are more vulnerable to their respective damage resistance).
${\displaystyle {\it PaperDamage} = {\it WeaponBaseDamage} \times {\it Perk}_1 \times {\it Perk}_2 \times \dots \times {\it RangeMulti}}$
• WeaponBaseDamage is affected by weapon mods and is e.g. doubled with a 2x charged laser musket
• RangeMulti is 1.0 at 100% WeaponRange or closer, 0.5 at 200% of WeaponRange or farther, and scales from 1.0 to 0.5 as range increases from 100% to 200% of WeaponRange.
For Poison Damage[ ]
Poison deals damage each second, as a distinctly inflicted hit, typically for 10 seconds; for example, if the player character has a poison resistance of 5 and suffers a poison 3 hit, then once per
second for 10 seconds, they will suffer a 3 damage hit, resisted with a resistance of 5, which otherwise follows the rules for projectile damage.
For Bleeding Damage[ ]
Bleeding works like poison, except that there is no such thing as bleeding resistance. Even robots such as 2nd generation synths and sentry bots will take bleeding damage, as it is a general damage
over time effect.
For Radiation Damage[ ]
Radiation damage has two subtypes. Almost all of it in the game is poisoning; each point of radiation poisoning from a radiation "hit" is resisted by radiation resistance (as per the energy damage
resistance formula), and resolves as a .1% loss of maximum HP for each point that gets through. The only perk that raises radiation poisoning damage inflicted by a weapon is Nuclear Physicist, and
radiation immunity makes a target ignore radiation poisoning entirely.
Non-poisoning radiation damage follows the normal rules for energy damage, including modification by perks; it ignores radiation immunity, and creatures with radiation immunity typically have no
radiation resistance.
For Explosive Damage[ ]
Explosive modifies other damage types - for example, something can inflict explosive energy damage. Explosive damage uses the same formula as the underlying type, but is affected by additional perks
and effects that interact with explosives. Padded and dense armor mods reduce the damage taken by 25% for padded and 50% for dense from explosion base damage. E.g. a dense armor mod reduces the
damage taken from a frag grenade by 62.5% (50% and 25% in succession) in addition to normal physical resistance.
Combined Damage Types[ ]
When something has multiple resistible damage types (ballistic, energy, radiation, and poison are the currently resistible types, with any of them capable of being explosive or not, and radiation can
be poisoning or not - almost all radiation sources are poisoning), one "hit" resolves as a hit for each damage type applied. Likewise, when something deals both direct and area of effect damage, a
target is hit by each one individually. For example, if one removes the Lorenzo's Artifact gun mod from the original gun and attaches it to an Irradiated Gamma gun with an Electric signal carrier
antennae, a fully charged shot from the weapon will inflict 6 distinct "hits": energy damage, explosive energy damage, explosive projectile damage, radiation damage, explosive radiation damage and
radiation poisoning.
For Power Attack Damage[ ]
Multiply the weapon's paper damage by 1.5 at the end.
Final Damage[ ]
${\displaystyle \it FinalDamage = PaperDamage \times DamageCoeff \times HeadshotMulti \times SneakAttackMulti}$
Damage Reduction formula for VATS Critical Attack[ ]
Critical damage completely ignores armor.
CriticalDamage = PaperDamage + (WeaponBaseDamage x CriticalMulti)
• CriticalMulti = 2 + BobbleheadBonus(0.25) + MagazineBonus(0.05 to 0.5) + WeaponModBonus(1.0 or 2.0)
Armor Piercing[ ]
Multiplies target's ballistic resistance (energy is bugged and not reduced in Fallout 4) by a number of perks and weapon mods - for example Rifleman rank 5 multiplies target DR by 0.7. Due to the way
the damage formula works, an attacker will need around 50% armor penetration to match the final damage of another with 20% bonus damage.
Fallout 76[ ]
• Fallout 76's Damage, Energy, Radiation, and Poison Resistance use the same basic formula as in Fallout 4.
• Fallout 76's Adventure mode does not have varying levels of difficulty, as such there are no multipliers based on difficulty level.
• Perks that reduce a target's Energy and Radiation resistance have been fixed as of patch 13.
Gallery[ ]
1. ↑ Most places cite this as 1.5x and 2.0x, however these numbers are relative to Very Hard difficulty, not Normal. The true, net difficulty modifiers are .75x for the player and 4.0x for enemies.
For further discussion see examples such as this reddit thread or this mod | {"url":"https://fallout.fandom.com/zh/wiki/%E4%BC%A4%E5%AE%B3%E6%8A%B5%E6%8A%97","timestamp":"2024-11-11T06:53:40Z","content_type":"text/html","content_length":"332024","record_id":"<urn:uuid:3c893de4-4357-4f94-aa7b-bc4d304a86b3>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00551.warc.gz"} |
Science and Mathematics in Medieval Times
The way the ancient people of civilization looked at science was completely different from the way modern people do. In ancient civilizations, the term Scientia referred to the knowledge every person
obtained during their lifetime. Enhancing the scientific, mathematical, and artistic subjects was led by Greek and Roman academics . The work of scientists and mathematicians remained stranded in
Rome until the Fifth-Century when the fall of the Roman Empire opened academia to the rest of Europe.
Calculating Religious Feasts
The calculating of religious feasts was one of the main reasons for the growth of mathematics and scientific subjects. The arrival of the division between the academic subjects was driven by several
Roman academics, who divided subjects into science, mathematics, astronomy, and music. The book of learning was known as the quadrivium.
The learning held within the quadrivium began to be used to calculate the position in a time of moveable religious feasts. The calculating of Easter had caused problems for the Catholic Church for
centuries. The date of Easter is not set in stone in the Bible but is recorded as celebrated on the Sunday closest to the 14th on Easter Month. The use of computus charts to calculate the date of
Easter is among the oldest mathematical recorded calculations.
The Use of Computus Diagrams
The seat of learning in most communities was located in local Abbeys and monasteries. These remained important to the development of mathematical calculations in the form of computus diagrams. The
development of computus diagrams drew inspiration from nature and its cyclical nature. In France and England, academics created documents combining theology and mathematics.
Leaving the Dark Ages
During the period from the fall of the Roman Empire to the 12th-century, Europe had fallen into the Dark Ages and needed the arrival of academics from the Middle East to spark academia. A renewed
interest in academics was developed following the arrival of academics from the Middle East and the translating of ancient documents from Roman and Greek academics.
Astronomy Continues to be Important
The rise of astronomy came during the 12th-century when French and English astronomers led the recording of the stars and planets. Images recorded during the 12th-century can be recognized by modern
viewers as constellations and planets in the night sky.
As Europe moved into the 13th century, mathematics and science took a dramatic turn for the better.
In the previous century, Euclid’s famed Elements and Persian mathematician al Kwarizmi’s The Compendious Book on Calculation had been translated into Latin. (Computer scientists may be interested to
know that al Kwarizmi’s name had been Latinized to Algorithms.)
Elements contained a powerful collection of theorems and proofs which led to far greater rigor in Medieval mathematics. And, al Kwarizmi’s work introduced algebra and systematized Indian numerals.
The Roman system had grown too unwieldy for burgeoning European commerce.
Europe began producing great mathematicians in the 13th century, most notably the creator of the Fibonacci sequence of numbers-1,1,2,3,5,8,13,21… His most important work, Liber Abaci, popularized al
Kwarizmi’s system of numerals.
Commerce benefited from these breakthroughs. The emergence of double-entry bookkeeping was made possible by these new numerals. This system of bookkeeping in turn accustomed people to think in terms
of strict rules of accounting in more areas of life than business transactions. Early scientists learned to pool larger and more precise bodies of knowledge by balancing nature’s accounts.
Alchemy is generally disdained these days, but alchemical practices led directly to the emergence of the modern science of chemistry. In the 12th and 13th centuries, Albertus Magnus and Roger Bacon
made use of Greek works by Hermes and Democritus on the presumed ability to transmute metals in their own alchemical studies.
The study of alchemy led to practical experimentation with chemicals, work generally disdained by the aristocracy because it involved working with materials. Alchemy also forced theoreticians to take
physical observations into account along with book study.
Gerard of Cremona regarded the Arab translation of Ptolemy’s famed book on astronomy, the Almagest, so highly that he traveled to Spain to translate it into Latin.
Students at Europe’s new universities learned the Almagest and ancient texts detailing aspects of the celestial sphere and planetary epicycles. Astronomers in Spain had devised new tables based on
close observation of planetary motion. By 1320, Parisian astronomers reworked them and spread them throughout Europe.
Thus, a solid scientific framework was laid for the work of the Renaissance scientists. | {"url":"https://jorgejperezattorney.medium.com/science-and-mathematics-in-medieval-times-4c0742a60ada?responsesOpen=true&sortBy=REVERSE_CHRON&source=user_profile---------3----------------------------","timestamp":"2024-11-07T21:35:53Z","content_type":"text/html","content_length":"107082","record_id":"<urn:uuid:a7b2a9e9-19fe-4d1e-828f-685af1b04229>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00460.warc.gz"} |
D / C
01 Sep 2024
D / C & Analysis of variables
Equation: D / C
Variable: C
Impact of Concentration on Volume Function
X-Axis: 10.0to10.0
Y-Axis: Volume Function
Title: Understanding the Relationship between Concentration (C) and Volumetric Behavior: A Theoretical Analysis using D/C Equation
In various engineering applications, understanding the impact of concentration on volumetric behavior is crucial. This article focuses on the theoretical analysis of a fundamental equation, D/C,
where D represents a dependent variable and C denotes concentration. We explore how changes in concentration affect the volume function, shedding light on its implications for design and optimization
In many physical systems, such as chemical reactors, biological systems, or hydraulic networks, the volumetric behavior is often influenced by the concentration of a particular substance or
parameter. This relationship can be described using the equation D/C, where D represents the dependent variable (e.g., volume, flow rate, or pressure) and C denotes the concentration.
The equation D/C implies that as the concentration (C) increases, the dependent variable (D) decreases proportionally. Mathematically, this relationship can be expressed as:
D = k / C
where k is a constant of proportionality.
To explore the impact of concentration on volumetric behavior, we will examine how changes in C affect D.
Variation of Concentration:
When the concentration (C) increases, the dependent variable (D) decreases according to the inverse relationship:
D ∝ 1/C
This indicates that as C grows larger, D shrinks proportionally. Conversely, when C decreases, D increases.
To illustrate this concept further, consider a hypothetical scenario where D represents the volume of a liquid in a tank and C is the concentration of a dissolved substance. As the concentration (C)
increases, more of the substance is added to the liquid, leading to an increase in volume (D). Conversely, when the concentration (C) decreases, less of the substance is present, resulting in a
decrease in volume (D).
Graphical Representation:
To visualize this relationship, we can plot D against C:
Concentration (C) Dependent Variable (D)
High Low
Medium Medium
Low High
As depicted above, the graph demonstrates how changes in concentration affect the dependent variable. When C is high, D is low; when C is medium, D is also medium; and when C is low, D becomes high.
In conclusion, this article has provided a theoretical analysis of the equation D/C, highlighting its significance in understanding the impact of concentration on volumetric behavior. By examining
the inverse relationship between D and C, we have demonstrated how changes in concentration influence the dependent variable.
This fundamental concept is crucial for design engineers to consider when optimizing systems that involve complex interactions between variables. The presented analysis will serve as a useful
reference for further studies and applications in various engineering disciplines.
[Insert relevant citations]
Note: This article is a sample response, please ensure you cite any references used and make adjustments according to your specific requirements.
Related topics
Academic Chapters on the topic
Information on this page is moderated by llama3.1 | {"url":"https://blog.truegeometry.com/engineering/Analytics_Impact_of_Concentration_on_Volume_FunctionD_C.html","timestamp":"2024-11-04T13:59:03Z","content_type":"text/html","content_length":"16777","record_id":"<urn:uuid:f07254d3-bc8e-4bd7-8dbd-cdc591a053c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00144.warc.gz"} |
Linear Algebra And Partial Differential Equations - HUNT4EDU
Linear Algebra And Partial Differential Equations
Here, We provide Linear Algebra And Partial Differential Equations. Linear Algebra is a continuous form of mathematics and is applied throughout science and engineering because it allows you to model
natural phenomena and to compute them efficiently. Because it is a form of continuous and not discrete mathematics, a lot of computer scientists don’t have a lot of experience with it. Linear Algebra
is also central to almost all areas of mathematics like geometry and functional analysis. Free download PDF Linear Algebra And Partial Differential Equations.
Therefore, you are mostly dealing with matrices and vectors rather than with scalars (we will cover these terms in the following section). When you have the right libraries, like Numpy, at your
disposal, you can compute complex matrix multiplication very easily with just a few lines of code. Linear algebra is a branch of mathematics that is widely used throughout science and engineering.
Yet because linear algebra is a form of continuous rather than discrete mathematics, many computer scientists have little experience with it. Free download PDF Linear Algebra And Partial Differential
SIZE – 18.71MB
PAGES – 385
About the Author,
T Veerarajan is Dean (Retd), Department of Mathematics, Velammal College of Engineering and Technology, Viraganoor, Madurai, Tamil Nadu. A Gold Medalist from Madras University, he has had a brilliant
academic career all through. He has 53 years of teaching experience at undergraduate and postgraduate levels in various established engineering colleges in Tamil Nadu including Anna University,
Partial Differential Equations are very helpful for the aspirants of CSIR UGC NET Mathematics, IIT JAM Mathematics, GATE mathematics, NBHM, TIFR, and all different tests with a similar syllabus.
Partial Differential Equations is designed for the students who are making ready for numerous national degree aggressive examinations and additionally evokes to go into Ph. D. Applications by using
manner of qualifying the numerous the front examination. Free download PDF Linear Algebra And Partial Differential Equations.
PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics. These seemingly distinct physical
phenomena can be formalized similarly in terms of PDEs. Free download PDF Linear Algebra And Partial Differential Equations.
Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalization in
stochastic partial differential equations. Free download PDF Linear Algebra And Partial Differential Equations.
PDEs are used to formulate problems involving functions of several variables, and are either solved by hand or used to create a relevant computer model. A special case is ordinary differential
equations (ODEs), which deal with functions of a single variable and their derivatives. Free download PDF Linear Algebra And Partial Differential Equations.
In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. Free download PDF Linear Algebra And
Partial Differential Equations.
Friends, if you need an eBook related to any topic. Or if you want any information about any exam, please comment on it. Share this post with your friends on social media.
DISCLAIMER: HUNT4EDU.COM does no longer owns this book neither created nor scanned. We simply offer the hyperlink already to be had on the internet. If any manner it violates the law or has any
troubles then kindly mail us or Contact Us for this(hyperlink removal).
We don’t aid piracy this duplicate grows to be supplied for university youngsters who’re financially bad but deserve greater to examine. Thank you.
Numerical Methods Fundamentals And Applications
Numerical Methods For Engineers
The 100 Most Influential Scientists Of All Time
Encyclopedia Of World Scientists
Leave a Comment | {"url":"https://hunt4edu.com/linear-algebra-and-partial-differential-equations/","timestamp":"2024-11-02T07:53:15Z","content_type":"text/html","content_length":"81022","record_id":"<urn:uuid:bcd7bf0d-a8a9-4934-976d-97981009b16b>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00858.warc.gz"} |
Casio FX-991 EX classwiz calculator tutorial (perfect for algebra, FE exam, EIT exam)
12 Dec 201607:54
TLDRThis tutorial introduces the Casio FX-991 EX classwiz calculator, ideal for algebra and engineering exams. The presenter demonstrates solving quadratic equations, systems of equations, complex
numbers, logarithms, inverse sine, summation, derivatives, and integrals. Viewers are also invited to participate in a giveaway by subscribing and commenting on the video for a chance to win a
calculator by December 31st, 2016.
• 🔢 The video provides a tutorial on using the Casio FX-991 EX classwiz calculator, highlighting its capabilities for algebra and various exams.
• 🛒 The calculator's price at Target, including tax, is approximately $21.
• 🎁 There is a giveaway opportunity for a calculator; viewers are encouraged to watch the video until the end to learn how to participate.
• 📚 The calculator can solve quadratic equations, displaying both the roots and the vertex of the parabola.
• 🔍 For solving systems of equations, the calculator requires the number of variables and their coefficients, handling up to three variables in this example.
• 🔑 The calculator has a complex number mode, allowing for calculations involving 'i', the imaginary unit.
• 📈 The device can compute logarithms with any base, demonstrated with log base 8 of four.
• 🔄 To switch between modes, such as from complex to standard calculation, the menu and calculate buttons are used.
• ∑ The summation function is available, demonstrated with a summation from x=3 to x=8.
• 📉 The derivative function provides the value of the derivative at a given x value, but does not output the derivative expression.
• ∫ The integral function calculates the definite integral of a function over a specified interval.
• 🎉 The video concludes with instructions on how to enter the giveaway, which involves subscribing to the channel and commenting on the video for a chance to win based on the number of likes
received by December 31st, 2016.
Q & A
• What is the purpose of the video?
-The purpose of the video is to provide a tutorial on how to use the Casio FX-991 EX classwiz calculator for solving various math problems and to announce a giveaway of the calculator.
• How much does the Casio FX-991 EX classwiz calculator cost at Target after tax?
-The cost of the Casio FX-991 EX classwiz calculator at Target after tax is approximately $21.
• What is the first math problem the tutorial solves on the calculator?
-The first math problem the tutorial solves is a quadratic equation.
• How does the calculator handle equations with a higher degree than two?
-The calculator can also solve polynomial equations with degrees 3 and 4, not just quadratic equations.
• What key combination is used to find the vertex of a parabola on the calculator?
-The script does not specify the exact key combination for finding the vertex, but it mentions that the calculator will display the vertex information after solving the quadratic equation.
• How does the calculator solve a system of equations?
-The calculator solves a system of equations by entering the coefficients and constants into the calculator, selecting the number of variables, and pressing equal to get the solution.
• What is the process for entering complex numbers on the calculator?
-To enter complex numbers, you go to the menu, select complex numbers mode, and use the 'I' button to represent the imaginary unit, entering the real and imaginary parts as instructed.
• How can you change the base of a logarithm on the calculator?
-You can change the base of a logarithm by using the log base button and then entering the desired base before entering the argument of the logarithm.
• What is the process to find the inverse sine of a number in degrees on the calculator?
-To find the inverse sine in degrees, you press shift, then the sign button, enter the fraction, and ensure the calculator is set to degree mode.
• How does the calculator handle summation and differentiation functions?
-The calculator has specific keys for summation and differentiation. For summation, you enter the function and the limits of summation. For differentiation, you enter the function and the value
of x at which you want to find the derivative.
• What is the method to find the integral of a function on the calculator?
-To find the integral, you use the integral function key, enter the function inside the integral, and specify the limits of integration.
• How can viewers participate in the calculator giveaway?
-Viewers can participate in the giveaway by subscribing to the channel and writing a comment on the video. The winner will be the person with the most likes on their comment by December 31st,
📚 Scientific Calculator Tutorial
This paragraph introduces a tutorial on using a scientific calculator for solving various mathematical problems. The speaker demonstrates how to solve a quadratic equation using the calculator's
equation-solving mode, explaining the process of entering the equation in standard form and obtaining the roots. The video also covers finding the vertex of a parabola, which represents the minimum
or maximum point. The tutorial continues with solving a system of equations, entering coefficients, and obtaining the values for variables x, y, and z. The speaker emphasizes the convenience of the
calculator's features and encourages viewers to watch the entire video for a giveaway opportunity.
🎁 Calculator Giveaway and Advanced Functions
The second paragraph focuses on a giveaway of a scientific calculator and introduces more advanced calculator functions. The speaker explains how to perform calculations with complex numbers,
logarithms with any base, inverse sine in both degree and radian modes, and summation of a series. Additionally, the paragraph covers derivative calculation for a given function, noting that the
calculator provides the value rather than the expression. The integral of a function is also demonstrated, showcasing the calculator's ability to compute definite integrals. The speaker concludes by
explaining the giveaway rules, which involve subscribing to the channel and commenting on the video for a chance to win the calculator based on the number of likes on the comment by a specified date.
The paragraph ends with a call to action for viewers to participate in the giveaway and share the video.
💡Scientific Calculator
A scientific calculator is an electronic device used for performing complex mathematical calculations, including those beyond the basic arithmetic operations. In the context of the video, the Casio
FX-991 EX classwiz calculator is showcased as a tool for solving various mathematical problems such as quadratic equations, systems of equations, and complex numbers. The video demonstrates how to
use the calculator's features to perform these calculations efficiently.
💡Quadratic Equation
A quadratic equation is a polynomial equation of the second degree, typically presented in the form ax^2 + bx + c = 0. The video tutorial shows how to use the calculator to solve such equations by
entering the coefficients and using the equation-solving function. The example given is a quadratic equation where the calculator provides two solutions: x = 4 and x = -2/5.
In the context of a quadratic equation, the vertex represents the highest or lowest point of the parabola represented by the equation. The video explains that the calculator can determine the vertex
of a parabola by solving for the x and y values that correspond to the minimum or maximum point. For instance, the tutorial mentions finding the vertex when x is equal to 9/5 and the corresponding y
value is 21/5.
💡System of Equations
A system of equations refers to a set of two or more equations that need to be solved simultaneously. The video demonstrates how to input and solve a system of three equations using the calculator.
The process involves entering the coefficients for each variable and the constants on the right side of the equations, and the calculator provides the values for each variable.
💡Complex Numbers
Complex numbers are numbers that consist of a real part and an imaginary part, typically written in the form a + bi, where 'i' is the imaginary unit. The video tutorial explains how to enter and
perform calculations with complex numbers using the calculator's complex number mode. An example given is the calculation of 8 - 2i and 5 + 4i.
A logarithm is the inverse operation to exponentiation, indicating the power to which a certain base must be raised to produce a given number. The video shows how to calculate logarithms with
different bases using the calculator. For example, the tutorial calculates the logarithm base 8 of 4, demonstrating the use of the 'log' function with a specified base.
💡Inverse Sine
The inverse sine function, often denoted as sin^(-1) or arcsin, is used to find the angle whose sine is a given value. In the video, the calculator is used to calculate the inverse sine of 1/2, which
is shown as 30° when the calculator is set to degree mode. The tutorial also explains how to switch to radian mode for such calculations.
Summation is a mathematical operation that adds a sequence of numbers to find their total. The video demonstrates the use of the summation function on the calculator to calculate the sum of a series,
such as the sum of 4x - 1 from x = 3 to x = 8. The calculator provides the result of the summation, which in the example is 126.
A derivative in calculus represents the rate at which a function is changing at a certain point. The video tutorial shows how to calculate the derivative of a function using the calculator. It's
important to note that the calculator provides the value of the derivative at a specific point, rather than the derivative function itself. An example used is the derivative of x/(1 + x^2) at x = 3,
which is approximately -0.08.
An integral in calculus is used to find the accumulated value of a function over an interval, effectively the area under the curve. The video demonstrates how to calculate the definite integral of a
function, such as the integral of the square root of x from 1 to 4. The calculator provides the result of the integral, which in this case is 14/3.
Tutorial on using the Casio FX-991 EX classwiz calculator for algebra and engineering exams.
The calculator costs approximately $21 after tax at Target.
A giveaway for a calculator is included in the video.
Solving quadratic equations using the calculator's equation mode.
Entering the equation in standard form for the calculator to solve.
Finding the vertex of a parabola using the calculator.
Solving a system of equations with three variables.
Entering coefficients and constants for system of equations.
Calculating complex numbers using the calculator's complex mode.
Entering and solving logarithms with any base on the calculator.
Finding the inverse sine of a fraction using the calculator.
Switching between degree and radian modes for calculations.
Performing summation operations with the calculator.
Calculating the derivative of a function using the calculator.
Converting decimal results to fractions on the calculator.
Calculating integrals with the calculator, including radicals.
Instructions on how to participate in the calculator giveaway.
Encouragement for viewers to comment for a chance to win the calculator.
The importance of creativity and humor in the comments for the giveaway.
The deadline for the giveaway is December 31st, 2016.
A thank you message to viewers for watching the tutorial. | {"url":"https://math.bot/blog-casio-fx991-ex-classwiz-calculator-tutorial-perfect-for-algebra-fe-exam-eit-exam-47830","timestamp":"2024-11-06T23:05:56Z","content_type":"text/html","content_length":"122387","record_id":"<urn:uuid:9df11cd2-ee9b-4a8e-a98e-dec99b4c273c>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00350.warc.gz"} |
Section: New Results
On the Complexity of the Arora-Ge algorithm against LWE
Arora & Ge recently showed that solving LWE can be reduced to solve a high-degree non-linear system of equations. They used a linearization to solve the systems. We investigate in [34] the
possibility of using Gröbner bases to improve Arora & Ge approach. | {"url":"https://radar.inria.fr/report/2012/polsys/uid35.html","timestamp":"2024-11-14T17:38:51Z","content_type":"text/html","content_length":"37488","record_id":"<urn:uuid:ffe8cfa5-0d0e-4dc3-857f-b1b55d122afc>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00395.warc.gz"} |
Hanuman Chalisa and the distance between Sun and Earth
Hanuman Chalisa and the distance between Sun and Earth
Hanuman Chalisa was composed by Gosvami Tulasidasa. He was a great devotee of Lord Ramachandra who lived in the 16th century. He composed the Rama-Charita-Manasa, the epic story of Lord Rama retold
in the vernacular language. Many devotees regularly recite Hanuman Chalisa, a prayer glorifying Shri Hanuman, composed by this great saint and poet.
It is believed that in one of these verses of Hanuman Chalisa, Tulasidasa had given an accurate calculation of the distance between the Sun and Earth.
The Quest of Astronomers to find the distance of Sun
The Greek Astronomers were known for their contribution to the scientific field in understanding the heavenly bodies.
• Archimedes, an ancient Greek Mathematician, and Philosopher of the 3rd century BC, estimated the distance of Sun from Earth as 10000 times the radius of Earth.
• Later, Hipparchus (2nd century BC) gave an estimate of 490 times the radius of Earth.
• Ptolemy considered the distance to be 1210 times the radius of Earth.
However, Johannes Kepler (1571 – 1630), a German Mathematician and Astronomer realized that these estimates were significantly low. Kepler’s law of Planetary Motion allowed astronomers to calculate
the relative distance of the planets from Sun. This was also aided by the invention of the telescope in the beginning of the 17th century which helped them to get more accurate measurements.
However, the most modern calculations in 20th century estimate the distance to be somewhere around 23455 times the radius of Earth (149,431,805 km assuming the radius of Earth to be 6371 km).
Srila Prabhupada writes in one of his purports:
Modern scientific calculations are subject to one change after another, and therefore they are uncertain. We have to accept the calculations of the Vedic literature. These Vedic calculations are
steady; the astronomical calculations made long ago and recorded in the Vedic literature are correct even now. Whether the Vedic calculations or modern ones are better may remain a mystery for
others, but as far as we are concerned, we accept the Vedic calculations to be correct.
According to the modern calculation:
The average distance between the Sun and Earth =149 million km = 92 million miles.
However, the orbit of the Earth is not a perfect circle, but an ellipse. Sometimes the Earth is closer to the Sun and sometimes it is farther.
The shortest distance between Sun and Earth (perihelion) = 91 million miles = 147 million km (early January)
The longest distance between Sun and Earth (aphelion) = 94.5 million miles = 152 million km (early July)
It is surprising to note that Tulasidasa who lived in the 16th century could give the most accurate estimation that is very close to the estimation by 20th-century astronomers.
Let us decipher the calculation in Hanuman Chalisa…
Hanuman, in his childhood, assuming the Sun to be a ripe mango, jumped to catch it. Tulasidasa recounts this incident in his Hanuman Chalisa as follows:
yuga-sahasra-yojana para bhanu
leelyo tahi madhura phala janu
Considering the Sun to be a sweet fruit, Hanuman jumped to swallow it.
Here the distance he traveled is mentioned as yuga-sahasra-yojana. Let us try to decipher this.
What is a yuga? According to Bhagavad-gita, one day of Brahma is called Kalpa and is equal to 1000 yugas and this is followed by a similar duration of the night.
ratrim yuga-sahasrantamte ‘ho-ratra-vidojanah
1 yuga = 4,320,000 years = 12000 divine years
(1 divine year = 360 years according to human calculation)
This is also confirmed in Manu-samhita:
etad dvaadasha sahasram devanam yugamuchyate
According to the above verse from Hanuman Chalisa, the distance between Sun and Earth is:
yuga-sahasra-yojana = 12000 x 1000 yojanas.
Yojana is a Vedic measure of distance and approximately equals to 8 miles (according to the 14th-century scholar Parameshvara, the originator of drgganita system). And 1 mile = 1.60934 kilometers.
According to the calculation presented in Hanuman Chalisa
Distance between Sun and Earth = 12000 x 1000 yojanas = 96 million miles = 153.6 million km, which is much closer to the calculation of the modern scientists.
A question may arise here.
Yuga is a measure of time and Yojana is a measure of distance. How can these two be combined?
We see that even modern scientists use time measurement (light-years) to calculate distances that are too long. It is said that the light from the sun takes approximately 8 minutes to reach the
The velocity of the light is calculated as 3 x 10^8 mps. In 8 minutes, the light travels (3 x 10^8 x 60 x 8) meters = 1.44 x 10^11 meters = 1.44 x 10^8 km = 144 million kilometers (which is another
approximation of distance between sun and earth according to modern calculations).
The assumptions we have made in the above calculations are as follows:
• We assumed yuga to mean the number 12000 based on the time calculation system of Vedic period based on the statement from Bhagavad-gita and Manu Samhita.
• We approximated 1 yojana = 8 miles based on what Srila Prabhupada has mentioned in his purports. However, there is still disagreement among scholars as to whether it is 5 miles or 8 miles. Some
other calculations indicate values ranging from 7.6 miles to 8.5 miles.
But it is astonishing that Tulasidasa mentioned the distance to this level of accuracy as early as in 16th century when the Western astronomers, with the help of a telescope, were trying to figure
out the distance. | {"url":"http://infinity.bgscollege.in/post/hanuman-chalisa-and-the-distance-between-sun-and-earth","timestamp":"2024-11-05T01:16:31Z","content_type":"text/html","content_length":"115821","record_id":"<urn:uuid:c9b25a64-6168-4c5b-9f80-d5c1cba5c495>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00770.warc.gz"} |
4. Find the amount and compound interest on ₹ 6,250 at 8% per a... | Filo
Question asked by Filo student
4. Find the amount and compound interest on ₹ 6,250 at per annum for 18 months, interest being compounded half-yearly. 5. Find the amount and compound interest on ₹ 3,125 at per annum for 9 months,
interest being compounded quarterly. C
Not the question you're searching for?
+ Ask your question
Video solutions (1)
Learn from their 1-to-1 discussion with Filo tutors.
7 mins
Uploaded on: 4/15/2023
Was this solution helpful?
Found 8 tutors discussing this question
Discuss this question LIVE for FREE
9 mins ago
One destination to cover all your homework and assignment needs
Learn Practice Revision Succeed
Instant 1:1 help, 24x7
60, 000+ Expert tutors
Textbook solutions
Big idea maths, McGraw-Hill Education etc
Essay review
Get expert feedback on your essay
Schedule classes
High dosage tutoring from Dedicated 3 experts
Practice more questions on All topics
View more
Students who ask this question also asked
View more
Stuck on the question or explanation?
Connect with our Mathematics tutors online and get step by step solution of this question.
231 students are taking LIVE classes
4. Find the amount and compound interest on ₹ 6,250 at per annum for 18 months, interest being compounded half-yearly. 5. Find the amount and compound interest on ₹ 3,125 at per annum
Question Text for 9 months, interest being compounded quarterly. C
Updated On Apr 15, 2023
Topic All topics
Subject Mathematics
Class Class 9
Answer Type Video solution: 1
Upvotes 68
Avg. Video 7 min | {"url":"https://askfilo.com/user-question-answers-mathematics/4-find-the-amount-and-compound-interest-on-6-250-at-per-34383636373237","timestamp":"2024-11-11T12:43:16Z","content_type":"text/html","content_length":"226041","record_id":"<urn:uuid:44fd1dbe-2ddf-466c-b904-5aecad901fc1>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00400.warc.gz"} |
The qgcomp package: g-computation on exposure quantiles (2024)
qgcomp is a package to implement g-computation for analyzing the effects of exposuremixtures. Quantile g-computation yields estimates of the effect of increasingall exposures by one quantile,
simultaneously. This, it estimates a “mixtureeffect” useful in the study of exposure mixtures such as air pollution, diet,and water contamination.
Using terminology from methods developed for causal effect estimation, quantileg-computation estimates the parameters of a marginal structural model thatcharacterizes the change in the expected
potential outcome given a joint interventionon all exposures, possibly conditional on confounders. Under the assumptions ofexchangeability, causal consistency, positivity, no interference, and
correctmodel specification, this model yields a causal effect for an interventionon the mixture as a whole. While these assumptions may not be met exactly, theyprovide a useful road map for how to
interpret the results of a qgcomp fit, andwhere efforts should be spent in terms of ensuring accurate model specificationand selection of exposures that are sufficient to control co-pollutant
The model
Say we have an outcome \(Y\), some exposures \(\mathbb{X}\) and possibly some other covariates (e.g. potential confounders) denoted by \(\mathbb{Z}\).
The basic model of quantile g-computation is a joint marginal structural model given by
[\mathbb{E}(Y^{\mathbf{X}_q} | \mathbf{Z,\psi,\eta}) = g(\psi_0 + \psi_1 S_q + \mathbf{\eta Z})]
where \(g(\cdot)\) is a link function in a generalized linear model (e.g. the inverse logit function in the case of a logistic model for the probability that \(Y=1\)), \(\psi_0\) is the model
intercept, \(\mathbf{\eta}\) is a set of model coefficients for the covariates and \(S_q\) is an “index” that represents a joint value of exposures. Quantile g-computation (by default) transforms all
exposures \(\mathbf{X}\) into \(\mathbf{X}_q\), which are “scores” taking on discrete values 0,1,2,etc. representing a categorical “bin” of exposure. By default, there are four bins with evenly
spaced quantile cutpoints for each exposure, so \({X}_q=0\) means that \(X\) was below the observed 25th percentile for that exposure. The index \(S_q\) represents all exposures being set to the same
value (again, by default, discrete values 0,1,2,3). Thus, the parameter \(\psi_1\) quantifies the expected change in the outcome, given a one quantile increase in all exposures simultaneously,
possibly adjusted for \(\mathbf{Z}\).
There are nuances to this particular model form that are available in the qgcomp package which will be explored below. There exists one special case of quantile g-computation that leads to fast
fitting: linear/additive exposure effects. Here we simulate “pre-quantized” data where the exposures \(X_1, X_2, X_3\) can only take on values of 0,1,2,3 in equal proportions. The model underlying
the outcomes is given by the linear regression:
[\mathbb{E}(Y | \mathbf{X,\beta}) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3]
with the true values of \(\beta_0=0, \beta_1 =0.25, \beta_2 =-0.1, \beta_3=0.05\), and \(X_1\) is strongly positively correlated with \(X_2\) ($\rho=0.95$) and negatively correlated with \(X_3\) ($\
rho=-0.3$). In this simple setting, the parameter \(\psi_1\) will equal the sum of the \(\beta\) coefficients (0.2). Here we see that qgcomp estimates a value very close to 0.2 (as we increase sample
size, the estimated value will be expected to become increasingly close to 0.2).
library("qgcomp")set.seed(543210)qdat = simdata_quantized(n=5000, outcomtype="continuous", cor=c(.95, -0.3), b0=0, coef=c(0.25, -0.1, 0.05), q=4)head(qdat)
## x1 x2 x3 y## 1 2 2 3 0.5539253## 2 2 3 1 -0.1005701## 3 2 2 3 0.3782267## 4 1 2 2 -0.4557419## 5 0 0 1 0.6124436## 6 1 1 2 0.7569574
cor(qdat[,c("x1", "x2", "x3")])
## x1 x2 x3## x1 1.00000 0.95552 -0.30368## x2 0.95552 1.00000 -0.29840## x3 -0.30368 -0.29840 1.00000
qgcomp(y~x1+x2+x3, expnms=c("x1", "x2", "x3"), data=qdat)
## Scaled effect size (positive direction, sum of positive coefficients = 0.3)## x1 x3 ## 0.775 0.225 ## ## Scaled effect size (negative direction, sum of negative coefficients = -0.0966)## x2 ## 1 ## ## Mixture slope parameters (Delta method CI):## ## Estimate Std. Error Lower CI Upper CI t value Pr(>|t|)## (Intercept) -0.0016583 0.0359609 -0.07214 0.068824 -0.0461 0.9632## psi1 0.2035810 0.0219677 0.16053 0.246637 9.2673 <2e-16
How to use the qgcomp package
Here we use a running example from the metals dataset from the from the packageqgcomp to demonstrate some features of the package and method.
Namely, the examples below demonstrate use of the package for:
1. Fast estimation of exposure effects under a linear model for quantized exposures for continuous (normal) outcomes
2. Estimating conditional and marginal odds/risk ratios of a mixture effect for binary outcomes
3. Adjusting for non-exposure covariates when estimating effects of the mixture
4. Allowing non-linear and non-homogeneous effects of individual exposures and the mixture as a whole by including product terms
5. Using qgcomp to fit a time-to-event model to estimate conditional and marginal hazard ratios for the exposure mixture
For analogous approaches to estimating exposure mixture effects, illustrative examples can be seen in the gQWS package help files, which implementsweighted quantile sum (WQS) regression, and at
https://jenfb.github.io/bkmr/overview.html, which describes Bayesian kernel machine regression.
The metals dataset from the from the package qgcomp, comprises a set of simulated well water exposures and two health outcomes (one continuous, one binary/time-to-event). The exposures are
transformed to have mean = 0.0, standard deviation = 1.0. The data are used throughout to demonstrate usage and features of the qgcomp package.
library("ggplot2")data("metals", package="qgcomp")head(metals)
## arsenic barium cadmium calcium chloride chromium## 1 0.09100165 0.08166362 15.0738845 -0.7746662 -0.15408335 -0.05589104## 2 0.17018302 -0.03598828 -0.7126486 -0.6857886 -0.19605499 -0.03268488## 3 0.13336869 0.09934014 0.6441992 -0.1525231 -0.17511844 -0.01161098## 4 -0.52570747 -0.76616263 -0.8610256 1.4472733 0.02552401 -0.05173287## 5 0.43420529 0.40629920 0.0570890 0.4103682 -0.24187403 -0.08931824## 6 0.71832662 0.19559582 -0.6823437 -0.8931696 -0.03919936 -0.07389407## copper iron lead magnesium manganese mercury## 1 1.99438050 19.1153352 21.072630908 -0.5109546 2.07630966 -1.20826726## 2 -0.02490169 -0.2039425 -0.010378362 -0.1030542 -0.36095395 -0.68729723## 3 0.25700811 -0.1964581 -0.063375935 0.9166969 -0.31075240 0.44852503## 4 0.75477075 -0.2317787 -0.002847991 2.5482987 -0.23350205 0.20428158## 5 -0.09919923 -0.1698619 -0.035276281 -0.5109546 0.08825996 1.19283834## 6 -0.05622285 -0.2129300 -0.118460981 -1.0059145 -0.30219838 0.02875033## nitrate nitrite ph selenium silver sodium## 1 1.3649492 -1.0500539 -0.7125482 0.23467592 -0.8648653 -0.41840695## 2 -0.1478382 0.4645119 0.9443009 0.65827253 -0.8019173 -0.09112969## 3 -0.3001660 -1.4969868 0.4924330 0.07205576 -0.3600140 -0.11828963## 4 0.3431814 -0.6992263 -0.4113029 0.23810705 1.3595205 -0.11828963## 5 0.0431269 -0.5041390 0.3418103 -0.02359910 -1.6078044 -0.40075299## 6 -0.3986575 0.1166249 1.2455462 -0.61186017 1.3769466 1.83722597## sulfate total_alkalinity total_hardness zinc mage35 y## 1 -0.1757544 -1.31353389 -0.85822417 1.0186058 1 -0.6007989## 2 -0.1161359 -0.12699789 -0.67749970 -0.1509129 0 -0.2022296## 3 -0.1616806 0.42671890 0.07928399 -0.1542524 0 -1.2164116## 4 0.8272415 0.99173604 1.99948142 0.1843372 0 0.1826311## 5 -0.1726845 -0.04789549 0.30518957 -0.1529379 0 1.1760472## 6 -0.1385631 1.98616621 -1.07283447 -0.1290391 0 -0.4100912## disease_time disease_state## 1 6.168764e-07 1## 2 4.000000e+00 0## 3 4.000000e+00 0## 4 4.000000e+00 0## 5 1.813458e+00 1## 6 2.373849e+00 1
Example 1: linear model
# we save the names of the mixture variables in the variable "Xnm"Xnm <- c( 'arsenic','barium','cadmium','calcium','chromium','copper', 'iron','lead','magnesium','manganese','mercury','selenium','silver', 'sodium','zinc')covars = c('nitrate','nitrite','sulfate','ph', 'total_alkalinity','total_hardness')# Example 1: linear model# Run the model and save the results "qc.fit"system.time(qc.fit <- qgcomp.glm.noboot(y~.,dat=metals[,c(Xnm, 'y')], family=gaussian()))
## Including all model terms as exposures of interest
## user system elapsed ## 0.012 0.000 0.012
# user system elapsed # 0.011 0.002 0.018 # contrasting other methods with computational speed# WQS regression (v3.0.1 of gWQS package)#system.time(wqs.fit <- gWQS::gwqs(y~wqs,mix_name=Xnm, data=metals[,c(Xnm, 'y')], family="gaussian"))# user system elapsed # 35.775 0.124 36.114 # Bayesian kernel machine regression (note that the number of iterations here would # need to be >5,000, at minimum, so this underestimates the run time by a factor# of 50+#system.time(bkmr.fit <- kmbayes(y=metals$y, Z=metals[,Xnm], family="gaussian", iter=100))# user system elapsed # 81.644 4.194 86.520
First note that qgcomp can be very fast relative to competing methods (with their example times given from single runs from a laptop).
One advantage of quantile g-computation over other methods that estimate“mixture effects” (the effect of changing all exposures at once), is that itis very computationally efficient. Contrasting
methods such as WQS (gWQSpackage) and Bayesian Kernel Machine regression (bkmr package),quantile g-computation can provide results many orders of magnitude faster.For example, the example above ran
3000X faster for quantile g-computationversus WQS regression, and we estimate the speedup would be severalhundred thousand times versus Bayesian kernel machine regression.
The speed relies on an efficient method to fit qgcomp when exposures are added additively to the model. When exposures are added using non-linear terms or non-additive terms (see below for examples),
then qgcomp will be slower but often still faster than competetive approaches.
Quantile g-computation yields fixed weights in the estimation procedure, similarto WQS regression. However, note that the weights from qgcomp.glm.nobootcan be negative or positive. When all effects
are linear and in the samedirection (“directional homogeneity”), quantile g-computation is equivalent toweighted quantile sum regression in large samples.
The overall mixture effect from quantile g-computation (psi1) is interpreted asthe effect on the outcome of increasing every exposure by one quantile, possiblyconditional on covariates. Given the
overall exposure effect, the weights areconsidered fixed and so do not have confidence intervals or p-values.
# View results: scaled coefficients/weights and statistical inference about# mixture effectqc.fit
## Scaled effect size (positive direction, sum of positive coefficients = 0.39)## calcium iron barium silver arsenic mercury sodium chromium ## 0.72216 0.06187 0.05947 0.03508 0.03447 0.02451 0.02162 0.02057 ## cadmium zinc ## 0.01328 0.00696 ## ## Scaled effect size (negative direction, sum of negative coefficients = -0.124)## magnesium copper lead manganese selenium ## 0.475999 0.385299 0.074019 0.063828 0.000857 ## ## Mixture slope parameters (Delta method CI):## ## Estimate Std. Error Lower CI Upper CI t value Pr(>|t|)## (Intercept) -0.356670 0.107878 -0.56811 -0.14523 -3.3062 0.0010238## psi1 0.266394 0.071025 0.12719 0.40560 3.7507 0.0002001
Now let’s take a brief look under the hood. qgcomp works in steps. First, the exposure variables are “quantized” or turned into score variables based on the total number of quantiles from the
parameter q. You can access these via the qx object from the qgcomp fit object.
# quantized datahead(qc.fit$qx)
## arsenic_q barium_q cadmium_q calcium_q chromium_q copper_q iron_q lead_q## 1 2 2 3 0 1 3 3 3## 2 2 2 0 0 2 2 1 2## 3 2 2 3 1 3 3 1 1## 4 1 0 0 3 1 3 0 2## 5 2 3 2 3 0 1 2 2## 6 3 2 0 0 0 2 1 1## magnesium_q manganese_q mercury_q selenium_q silver_q sodium_q zinc_q## 1 1 3 0 2 0 0 3## 2 2 0 1 3 1 3 0## 3 3 1 2 2 1 3 0## 4 3 1 2 2 3 3 3## 5 1 3 3 1 0 0 0## 6 0 1 2 0 3 3 2
You can re-fit a linear model using these quantized exposures. This is the “underlying model” of a qgcomp fit.
# regression with quantized dataqc.fit$qx$y = qc.fit$fit$data$y # first bring outcome back into the quantized datanewfit <- lm(y ~ arsenic_q + barium_q + cadmium_q + calcium_q + chromium_q + copper_q + iron_q + lead_q + magnesium_q + manganese_q + mercury_q + selenium_q + silver_q + sodium_q + zinc_q, data=qc.fit$qx)newfit
## ## Call:## lm(formula = y ~ arsenic_q + barium_q + cadmium_q + calcium_q + ## chromium_q + copper_q + iron_q + lead_q + magnesium_q + manganese_q + ## mercury_q + selenium_q + silver_q + sodium_q + zinc_q, data = qc.fit$qx)## ## Coefficients:## (Intercept) arsenic_q barium_q cadmium_q calcium_q chromium_q ## -0.3566699 0.0134440 0.0231929 0.0051773 0.2816176 0.0080231 ## copper_q iron_q lead_q magnesium_q manganese_q mercury_q ## -0.0476113 0.0241278 -0.0091464 -0.0588190 -0.0078872 0.0095572 ## selenium_q silver_q sodium_q zinc_q ## -0.0001059 0.0136815 0.0084302 0.0027125
Here you can see that, for a GLM in which all quantized exposures enter linearly and additively into the underlying model the overall effect from qgcomp is simply the sum of the adjusted coefficients
from the underlying model.
sum(newfit$coefficients[-1]) # sum of all coefficients excluding intercept and confounders, if any
## [1] 0.2663942
coef(qc.fit) # overall effect and intercept from qgcomp fit
## (intercept) psi1 ## -0.3566699 0.2663942
This equality is why we can fit qgcomp so efficiently under such a model, but qgcomp is a much more general method that can allow for non-linearity and non-additivity in the underlying model, as well
as non-linearity in the overall model. These extensions are described in some of the following examples.
Example 2: conditional odds ratio, marginal odds ratio in a logistic model
This example introduces the use of a binary outcome in qgcomp via theqgcomp.glm.noboot function, which yields a conditional odds ratio or theqgcomp.glm.boot, which yields a marginal odds ratio or
risk/prevalence ratio. Thesewill not equal each other when there are non-exposure covariates (e.g.confounders) included in the model because the odds ratio is not collapsible (bothare still valid).
Marginal parameters will yield estimates of the populationaverage exposure effect, which is often of more interest due to betterinterpretability over conditional odds ratios. Further, odds ratios are
notgenerally of interest when risk ratios can be validly estimated, so qgcomp.glm.bootwill estimate the risk ratio by default for binary data (set rr=FALSE toallow estimation of ORs when using
qc.fit2 <- qgcomp.glm.noboot(disease_state~., expnms=Xnm, data = metals[,c(Xnm, 'disease_state')], family=binomial(), q=4)qcboot.fit2 <- qgcomp.glm.boot(disease_state~., expnms=Xnm, data = metals[,c(Xnm, 'disease_state')], family=binomial(), q=4, B=10,# B should be 200-500+ in practice seed=125, rr=FALSE)qcboot.fit2b <- qgcomp.glm.boot(disease_state~., expnms=Xnm, data = metals[,c(Xnm, 'disease_state')], family=binomial(), q=4, B=10,# B should be 200-500+ in practice seed=125, rr=TRUE)
Compare a qgcomp.glm.noboot fit:
## Scaled effect size (positive direction, sum of positive coefficients = 0.392)## barium zinc chromium magnesium silver sodium ## 0.3520 0.2002 0.1603 0.1292 0.0937 0.0645 ## ## Scaled effect size (negative direction, sum of negative coefficients = -0.696)## selenium copper arsenic calcium manganese cadmium mercury lead ## 0.2969 0.1627 0.1272 0.1233 0.1033 0.0643 0.0485 0.0430 ## iron ## 0.0309 ## ## Mixture log(OR) (Delta method CI):## ## Estimate Std. Error Lower CI Upper CI Z value Pr(>|z|)## (Intercept) 0.26362 0.51615 -0.74802 1.27526 0.5107 0.6095## psi1 -0.30416 0.34018 -0.97090 0.36258 -0.8941 0.3713
with a qgcomp.glm.boot fit:
## Mixture log(OR) (bootstrap CI):## ## Estimate Std. Error Lower CI Upper CI Z value Pr(>|z|)## (Intercept) 0.26362 0.39038 -0.50151 1.02875 0.6753 0.4995## psi1 -0.30416 0.28009 -0.85314 0.24481 -1.0859 0.2775
with a qgcomp.glm.boot fit, where the risk/prevalence ratio is estimated, rather than the odds ratio:
## Mixture log(RR) (bootstrap CI):## ## Estimate Std. Error Lower CI Upper CI Z value Pr(>|z|)## (Intercept) -0.56237 0.16106 -0.87804 -0.24670 -3.4917 0.00048## psi1 -0.16373 0.13839 -0.43497 0.10752 -1.1830 0.23679
Example 3: adjusting for covariates, plotting estimates
In the following code we run a maternal age-adjusted linear model withqgcomp (family = "gaussian"). Further, we plot both the weights, as well as the mixture slopewhich yields overall model
confidence bounds, representing the bounds that, for each value of thejoint exposure are expected to contain the true regression line over 95% of trials (so-called 95% ‘pointwise’ bounds for the
regression line). The pointwise comparison bounds, denoted by error bars on the plot, represent comparisons of the expected difference in outcomes at each quantile, with referenceto a specific
quantile (which can be specified by the user, as below). These pointwise bounds are similar to the boundscreated in the bkmr package when plotting the overall effect of all exposures. The pointwise
bounds can be obtained via the pointwisebound.boot function. To avoid confusion between “pointwise regression” and “pointwise comparison” bounds, the pointwise regression bounds are denoted as the
“model confidence band” in the plots, since they yield estimates of the same type of bounds as the predict function in R when applied to linear model fits.
Note that the underlying regression model is on the quantile ‘score’, which takes on values integer values0, 1, …, q-1. For plotting purposes (when plotting regression line results from
qgcomp.glm.boot),the quantile score is translated into a quantile (range = [0-1]). This is not a perfect correspondence,because the quantile g-computation model treats the quantile score as a
continuous variable, but the quantile category comprises a range of quantiles.For visualization, we fix the ends of the plot at the mid-points of the first and last quantile cut-point,so the range of
the plot will change slightly if ‘q’ is changed.
qc.fit3 <- qgcomp.glm.noboot(y ~ mage35 + arsenic + barium + cadmium + calcium + chloride + chromium + copper + iron + lead + magnesium + manganese + mercury + selenium + silver + sodium + zinc, expnms=Xnm, metals, family=gaussian(), q=4)qc.fit3
## Scaled effect size (positive direction, sum of positive coefficients = 0.381)## calcium barium iron silver arsenic mercury chromium zinc ## 0.74466 0.06636 0.04839 0.03765 0.02823 0.02705 0.02344 0.01103 ## sodium cadmium ## 0.00775 0.00543 ## ## Scaled effect size (negative direction, sum of negative coefficients = -0.124)## magnesium copper lead manganese selenium ## 0.49578 0.35446 0.08511 0.06094 0.00372 ## ## Mixture slope parameters (Delta method CI):## ## Estimate Std. Error Lower CI Upper CI t value Pr(>|t|)## (Intercept) -0.348084 0.108037 -0.55983 -0.13634 -3.2219 0.0013688## psi1 0.256969 0.071459 0.11691 0.39703 3.5960 0.0003601
From the first plot we see weights from qgcomp.glm.noboot function, which include bothpositive and negative effect directions. When the weights are all on a single side of the null,these plots are
easy to in interpret since the weight corresponds to the proportion of theoverall effect from each exposure. WQS uses a constraint in the model to forceall of the weights to be in the same direction
- unfortunately such constraintslead to biased effect estimates. The qgcomp package takes a different approachand allows that “weights” might go in either direction, indicating that some exposuresmay
beneficial, and some harmful, or there may be sampling variation due to usingsmall or moderate sample sizes (or, more often, systematic bias such as unmeasuredconfounding). The “weights” in qgcomp
correspond to the proportion of the overall effectwhen all of the exposures have effects in the same direction, but otherwise theycorrespond to the proportion of the effect in a particular direction,
whichmay be small (or large) compared to the overall “mixture” effect. NOTE: the leftand right sides of the plot should not be compared with each other because thelength of the bars corresponds to
the effect size only relative to other effectsin the same direction. The darkness of the bars corresponds to the overall effectsize - in this case the bars on the right (positive) side of the plot
are darkerbecause the overall “mixture” effect is positive. Thus, the shading allows oneto make informal comparisons across the left and right sides: a large, darklyshaded bar indicates a larger
independent effect than a large, lightly shaded bar.
qcboot.fit3 <- qgcomp.glm.boot(y ~ mage35 + arsenic + barium + cadmium + calcium + chloride + chromium + copper + iron + lead + magnesium + manganese + mercury + selenium + silver + sodium + zinc, expnms=Xnm, metals, family=gaussian(), q=4, B=10,# B should be 200-500+ in practice seed=125)qcboot.fit3
## Mixture slope parameters (bootstrap CI):## ## Estimate Std. Error Lower CI Upper CI t value Pr(>|t|)## (Intercept) -0.342787 0.114983 -0.56815 -0.11742 -2.9812 0.0030331## psi1 0.256969 0.075029 0.10991 0.40402 3.4249 0.0006736
p = plot(qcboot.fit3)
We can change the referent category for pointwise comparisons via the pointwiseref parameter:
plot(qcboot.fit3, pointwiseref = 3)
Using qgcomp.glm.boot also allows us to assesslinearity of the total exposure effect (the second plot). Similar output is availablefor WQS (gWQS package), though WQS results will generally be less
interpretablewhen exposure effects are non-linear (see below how to do this with qgcomp.glm.boot).
The plot for the qcboot.fit3 object (using g-computation with bootstrap variance)gives predictions at the joint intervention levels of exposure. It also displaysa smoothed (graphical) fit.
Note that the uncertainty intervals given in the plot are directly accessible via the pointwisebound (pointwise comparison confidence intervals) and modelbound functions (confidence interval for the
regression line):
pointwisebound.boot(qcboot.fit3, pointwiseref=3)
## quantile quantile.midpoint linpred mean.diff se.diff ll.diff## 0 0 0.125 -0.34278746 -0.5139387 0.15005846 -0.8080479## 1 1 0.375 -0.08581809 -0.2569694 0.07502923 -0.4040240## 2 2 0.625 0.17115127 0.0000000 0.00000000 0.0000000## 3 3 0.875 0.42812064 0.2569694 0.07502923 0.1099148## ul.diff ll.linpred ul.linpred## 0 -0.2198295 -0.6368966 -0.04867828## 1 -0.1099148 -0.2328727 0.06123650## 2 0.0000000 0.1711513 0.17115127## 3 0.4040240 0.2810660 0.57517523
## quantile quantile.midpoint linpred m se.pw ll.pw## 0 0 0.125 -0.34278746 -0.34278746 0.11498301 -0.56815001## 1 1 0.375 -0.08581809 -0.08581809 0.04251388 -0.16914376## 2 2 0.625 0.17115127 0.17115127 0.04065143 0.09147593## 3 3 0.875 0.42812064 0.42812064 0.11294432 0.20675384## ul.pw ll.simul ul.simul## 0 -0.117424905 -0.4622572 -0.161947578## 1 -0.002492425 -0.1191843 -0.005233921## 2 0.250826606 0.1386622 0.223888629## 3 0.649487428 0.3081934 0.566961520
Because qgcomp estimates a joint effect of multiple exposures, we cannot, in general, assess model fit by overlaying predictions from the plots above with the data. Hence, it is useful to explore
non-linearity by fitting models thatallow for non-linear effects, as in the next example.
Example 4: non-linearity (and non-homogeneity)
qgcomp (and qgcomp.glm.boot) addresses non-linearity in a way similar to standard parametric regression models, which lends itself to being able to leverage R language features for non-linear
parametric models (or, more precisely, parametric models that deviate from a purely additive, linear function on the link function basis via the use of basis function representation of non-linear
functions).Here is an example where we use a feature of the R language for fitting modelswith interaction terms. We use y~. + .^2 as the model formula, which fits a modelthat allows for quadratic
term for every predictor in the model.
Similar approaches could be used to include interaction terms between exposures,as well as between exposures and covariates.
qcboot.fit4 <- qgcomp(y~. + .^2, expnms=Xnm, metals[,c(Xnm, 'y')], family=gaussian(), q=4, B=10, seed=125)plot(qcboot.fit4)
Note that allowing for a non-linear effect of all exposures induces an apparentnon-linear trend in the overall exposure effect. The smoothed regression line isstill well within the confidence bands
of the marginal linear model(by default, the overall effect of joint exposure is assumed linear,though this assumption can be relaxed via the ‘degree’ parameter in qgcomp.glm.boot,as follows:
qcboot.fit5 <- qgcomp(y~. + .^2, expnms=Xnm, metals[,c(Xnm, 'y')], family=gaussian(), q=4, degree=2, B=10, rr=FALSE, seed=125)plot(qcboot.fit5)
Once again, we can access numerical estimates of uncertainty:
## quantile quantile.midpoint linpred mean.diff se.diff ll.diff## 0 0 0.125 -0.89239044 0.0000000 0.0000000 0.0000000## 1 1 0.375 -0.18559680 0.7067936 0.6165306 -0.5015841## 2 2 0.625 0.12180659 1.0141970 0.6044460 -0.1704954## 3 3 0.875 0.02981974 0.9222102 0.3537861 0.2288022## ul.diff ll.linpred ul.linpred## 0 0.000000 -0.8923904 -0.8923904## 1 1.915171 -1.3939746 1.0227810## 2 2.198889 -1.0628859 1.3064991## 3 1.615618 -0.6635882 0.7232277
## quantile quantile.midpoint linpred m se.pw ll.pw## 0 0 0.125 -0.89239044 -0.89239044 0.70335801 -2.2709468## 1 1 0.375 -0.18559680 -0.18559680 0.09874085 -0.3791253## 2 2 0.625 0.12180659 0.12180659 0.14624914 -0.1648365## 3 3 0.875 0.02981974 0.02981974 0.83609757 -1.6089014## ul.pw ll.simul ul.simul## 0 0.486165922 -2.08813110 0.18308760## 1 0.007931712 -0.32577422 -0.02886665## 2 0.408449640 -0.06371183 0.43431431## 3 1.668540859 -1.19007238 1.57263047
Ideally, the smooth fit will look very similar to the model prediction regressionline.
Interpretation of model parameters
As the output below shows, setting “degree=2” yields a second parameter in the model fit ($\psi_2$). The output of qgcomp now corresponds to estimates of the marginal structural model given by[\
mathbb{E}(Y^{\mathbf{X}_q}) = g(\psi_0 + \psi_1 S_q + \psi_2 S_q^2)]
## Mixture slope parameters (bootstrap CI):## ## Estimate Std. Error Lower CI Upper CI t value Pr(>|t|)## (Intercept) -0.89239 0.70336 -2.27095 0.48617 -1.2688 0.2055## psi1 0.90649 0.93820 -0.93235 2.74533 0.9662 0.3347## psi2 -0.19970 0.32507 -0.83682 0.43743 -0.6143 0.5395
so that \(\psi_2\) can be interpreted similar to quadratic terms that might appear in a generalized linear model. \(\psi_2\) estimates the change in the outcome for an additional unit of squared
joint exposure, over-and-above the linear effect given by \(\psi_1\). Informally, this is a way of formally assessing specific types of non-linearity in the joint exposure-response curves, and there
are many other (slightly incorrect but intuitively useful) ways of interpreting parameters for squared terms in regressions (beyond the scope of this document). Intuition from generalized linear
models applies directly to the models fit by quantile g-computation.
Example 5: Comparing model fits and further exploring non-linearity
Exploring a non-linear fit in settings with multiple exposures is challenging. One way to explore non-linearity, as demonstrated above, is to to include all 2-way interaction terms (including
quadratic terms, or “self-interactions”). Sometimes this approach is not desired, either because the number of terms in the model can become very large, or because some sort of model selection
procedure is required, which risks inducing over-fit (biased estimates and standard errors that are too small). Short of having a set of a priori non-linear terms to include, we find it best to take
a default approach (e.g. taking all second order terms) that doesn’t rely on statistical significance, or to simply be honest that the search for a non-linear model is exploratory and shouldn’t be
relied upon for robust inference. Methods such as kernel machine regression may be good alternatives, or supplementary approaches to exploring non-linearity.
NOTE: qgcomp necessarily fits a regression model with exposures that have a small number of possible values, based on the quantile chosen. By package default, this is q=4, but it is difficult to
fully examine non-linear fits using only four points, so we recommend exploring larger values of q, which will change effect estimates (i.e. the model coefficient implies a smaller change in
exposures, so the expected change in the outcome will also decrease).
Here, we examine a one strategy for default and exploratory approaches to mixtures that can be implemented in qgcomp using a smaller subset of exposures (iron, lead, cadmium), which we choose via the
correlation matrix. High correlations between exposures may result from a common source, so small subsets of the mixture may be useful for examining hypotheses that relate to interventions on a
common environmental source or set of behaviors. Note that we can still adjust for the measured exposures, even though only 3 our exposures of interest are considered as the mixture of interest. Note
that we will require a new R package to help in exploring non-linearity: splines. Note that qgcomp.glm.boot must be used in order to produce the graphics below, as qgcomp.glm.noboot does not
calculate the necessary quantities.
Graphical approach to explore non-linearity in a correlated subset of exposures using splines
library(splines)# find all correlations > 0.6 (this is an arbitrary choice)cormat = cor(metals[,Xnm])idx = which(cormat>0.6 & cormat <1.0, arr.ind = TRUE)newXnm = unique(rownames(idx)) # iron, lead, and cadmiumqc.fit6lin <- qgcomp.glm.boot(y ~ iron + lead + cadmium + mage35 + arsenic + magnesium + manganese + mercury + selenium + silver + sodium + zinc, expnms=newXnm, metals, family=gaussian(), q=8, B=10)qc.fit6nonlin <- qgcomp.glm.boot(y ~ bs(iron) + bs(cadmium) + bs(lead) + mage35 + arsenic + magnesium + manganese + mercury + selenium + silver + sodium + zinc, expnms=newXnm, metals, family=gaussian(), q=8, B=10, degree=2)qc.fit6nonhom <- qgcomp.glm.boot(y ~ bs(iron)*bs(lead) + bs(iron)*bs(cadmium) + bs(lead)*bs(cadmium) + mage35 + arsenic + magnesium + manganese + mercury + selenium + silver + sodium + zinc, expnms=newXnm, metals, family=gaussian(), q=8, B=10, degree=3)
It helps to place the plots on a common y-axis, which is easy due to dependence of the qgcomp plotting functions on ggplot. Here’s the linear fit :
pl.fit6lin <- plot(qc.fit6lin, suppressprint = TRUE, pointwiseref = 4)pl.fit6lin + coord_cartesian(ylim=c(-0.75, .75)) + ggtitle("Linear fit: mixture of iron, lead, and cadmium")
Here’s the non-linear fit :
pl.fit6nonlin <- plot(qc.fit6nonlin, suppressprint = TRUE, pointwiseref = 4)pl.fit6nonlin + coord_cartesian(ylim=c(-0.75, .75)) + ggtitle("Non-linear fit: mixture of iron, lead, and cadmium")
And here’s the non-linear fit with statistical interaction between exposures (recalling that this will lead to non-linearity in the overall effect):
pl.fit6nonhom <- plot(qc.fit6nonhom, suppressprint = TRUE, pointwiseref = 4)pl.fit6nonhom + coord_cartesian(ylim=c(-0.75, .75)) + ggtitle("Non-linear, non-homogeneous fit: mixture of iron, lead, and cadmium")
Caution about graphical approaches
The underlying conditional model fit can be made extremely flexible, and the graphical representation of this (via thesmooth conditional fit) can look extremely flexible. Simply matching the overall
(MSM) fit to this line is nota viable strategy for identifying parsimonious models because that would ignore potential for overfit. Thus,caution should be used when judging the accuracy of a fit when
comparing the “smooth conditional fit” to the“MSM fit.”
qc.overfit <- qgcomp.glm.boot(y ~ bs(iron) + bs(cadmium) + bs(lead) + mage35 + bs(arsenic) + bs(magnesium) + bs(manganese) + bs(mercury) + bs(selenium) + bs(silver) + bs(sodium) + bs(zinc), expnms=Xnm, metals, family=gaussian(), q=8, B=10, degree=1)qc.overfit
## Mixture slope parameters (bootstrap CI):## ## Estimate Std. Error Lower CI Upper CI t value Pr(>|t|)## (Intercept) -0.064420 0.123768 -0.307001 0.178162 -0.5205 0.6030## psi1 0.029869 0.031972 -0.032795 0.092534 0.9342 0.3507
plot(qc.overfit, pointwiseref = 5)
Here, there is little statistical evidence for even a linear trend, which makes thesmoothed conditional fit appear to be overfit. The smooth conditional fit can be turned off, as below.
plot(qc.overfit, flexfit = FALSE, pointwiseref = 5)
Example 6: Miscellaneous other ways to allow non-linearity.
Note that these are included as examples of how to include non-linearities, and are not intended asa demonstration of appropriate model selection. In fact, qc.fit7b is generally a bad idea in smallto
moderate sample sizes due to large numbers of parameters.
using indicator terms for each quantile
qc.fit7a <- qgcomp.glm.boot(y ~ factor(iron) + lead + cadmium + mage35 + arsenic + magnesium + manganese + mercury + selenium + silver + sodium + zinc, expnms=newXnm, metals, family=gaussian(), q=8, B=20, deg=2)# underlying fitsummary(qc.fit7a$fit)$coefficients
## Estimate Std. Error t value Pr(>|t|)## (Intercept) -0.052981109 0.08062430 -0.6571357 5.114428e-01## factor(iron)1 0.046571725 0.09466914 0.4919420 6.230096e-01## factor(iron)2 -0.056984300 0.09581659 -0.5947227 5.523395e-01## factor(iron)3 0.143813131 0.09558055 1.5046276 1.331489e-01## factor(iron)4 0.053319057 0.09642069 0.5529835 5.805601e-01## factor(iron)5 0.303967859 0.09743261 3.1197753 1.930856e-03## factor(iron)6 0.246259568 0.09734385 2.5297907 1.176673e-02## factor(iron)7 0.447045591 0.09786201 4.5681217 6.425565e-06## lead -0.009210341 0.01046668 -0.8799679 3.793648e-01## cadmium -0.010503041 0.01086440 -0.9667387 3.342143e-01## mage35 0.081114695 0.07274583 1.1150426 2.654507e-01## arsenic 0.021755516 0.02605850 0.8348720 4.042502e-01## magnesium -0.010758356 0.02469893 -0.4355798 6.633587e-01## manganese 0.004418266 0.02551449 0.1731670 8.626011e-01## mercury 0.003913896 0.02448078 0.1598763 8.730531e-01## selenium -0.058085344 0.05714805 -1.0164012 3.100059e-01## silver 0.020971562 0.02407397 0.8711302 3.841658e-01## sodium -0.062086322 0.02404447 -2.5821454 1.014626e-02## zinc 0.017078438 0.02392381 0.7138679 4.756935e-01
interactions between indicator terms
qc.fit7b <- qgcomp.glm.boot(y ~ factor(iron)*factor(lead) + cadmium + mage35 + arsenic + magnesium + manganese + mercury + selenium + silver + sodium + zinc, expnms=newXnm, metals, family=gaussian(), q=8, B=10, deg=3)# underlying fit#summary(qc.fit7b$fit)$coefficientsplot(qc.fit7b)
breaks at specific quantiles (these breaks act on the quantized basis)
qc.fit7c <- qgcomp.glm.boot(y ~ I(iron>4)*I(lead>4) + cadmium + mage35 + arsenic + magnesium + manganese + mercury + selenium + silver + sodium + zinc, expnms=newXnm, metals, family=gaussian(), q=8, B=10, deg=2)# underlying fitsummary(qc.fit7c$fit)$coefficients
## Estimate Std. Error t value## (Intercept) -5.910113e-02 0.05182385 -1.140423351## I(iron > 4)TRUE 3.649940e-01 0.06448858 5.659824144## I(lead > 4)TRUE -9.004067e-05 0.06181587 -0.001456595## cadmium -6.874749e-03 0.01078339 -0.637531252## mage35 7.613672e-02 0.07255110 1.049422029## arsenic 2.042370e-02 0.02578001 0.792230124## magnesium -3.279980e-03 0.02427513 -0.135116878## manganese 1.055979e-02 0.02477453 0.426235507## mercury 9.396898e-03 0.02435057 0.385900466## selenium -4.337729e-02 0.05670006 -0.765030761## silver 1.807248e-02 0.02391112 0.755819125## sodium -5.537968e-02 0.02403808 -2.303831424## zinc 2.349906e-02 0.02385762 0.984970996## I(iron > 4)TRUE:I(lead > 4)TRUE -1.828835e-01 0.10277790 -1.779405131## Pr(>|t|)## (Intercept) 2.547332e-01## I(iron > 4)TRUE 2.743032e-08## I(lead > 4)TRUE 9.988385e-01## cadmium 5.241120e-01## mage35 2.945626e-01## arsenic 4.286554e-01## magnesium 8.925815e-01## manganese 6.701456e-01## mercury 6.997578e-01## selenium 4.446652e-01## silver 4.501639e-01## sodium 2.169944e-02## zinc 3.251821e-01## I(iron > 4)TRUE:I(lead > 4)TRUE 7.586670e-02
Note one restriction on exploring non-linearity: while we can use flexible functions such as splines for individual exposures, the overall fit is limited via the degree parameter to polynomial
functions (here a quadratic polynomial fits the non-linear model well, and a cubic polynomial fits the non-linear/non-homogeneous model well - though this is an informal argument and does not account
for the wide confidence intervals). We note here that only 10 bootstrap iterations are used to calculate confidence intervals (to increase computational speed for the example), which is far too low.
Statistical approach explore non-linearity in a correlated subset of exposures using splines
The graphical approaches don’t give a clear picture of which model might be preferred, but we can compare the model fits using AIC, or BIC (information criterion that weigh model fit with
over-parameterization). Both of these criterion suggest the linear model fits best (lowest AIC and BIC), which suggests that the apparently non-linear fits observed in the graphical approaches don’t
improve prediction of the health outcome, relative to the linear fit, due to the increase in variance associated with including more parameters.
## [1] 676.0431
## [1] 682.7442
## [1] 705.6187
## [1] 733.6346
## [1] 765.0178
## [1] 898.9617
Example 7: time-to-event analysis and parallel processing
• The qgcomp package utilizes the Cox proportional hazards models as the underlying model fortime-to-event analysis. The interpretation of a qgcomp.glm.noboot fit parameter is a conditional (on
confounders)hazard ratio for increasing all exposures at once. The qc.survfit1 object demonstrates a time-to-event analysis with qgcompcox.noboot. The default plot is similar to that of
qgcompcox.noboot,in that it yields weights and an overall mixture effect
# non-bootstrapped version estimates a marginal structural model for the # confounder-conditional effectsurvival::coxph(survival::Surv(disease_time, disease_state) ~ iron + lead + cadmium + arsenic + magnesium + manganese + mercury + selenium + silver + sodium + zinc + mage35, data=metals)
## Call:## survival::coxph(formula = survival::Surv(disease_time, disease_state) ~ ## iron + lead + cadmium + arsenic + magnesium + manganese + ## mercury + selenium + silver + sodium + zinc + mage35, ## data = metals)## ## coef exp(coef) se(coef) z p## iron -0.056447 0.945117 0.156178 -0.361 0.7178## lead 0.440735 1.553849 0.203264 2.168 0.0301## cadmium 0.023325 1.023599 0.105502 0.221 0.8250## arsenic -0.003812 0.996195 0.088897 -0.043 0.9658## magnesium 0.099399 1.104507 0.064730 1.536 0.1246## manganese -0.014065 0.986033 0.064197 -0.219 0.8266## mercury -0.060830 0.940983 0.072918 -0.834 0.4042## selenium -0.231626 0.793243 0.173655 -1.334 0.1823## silver 0.043169 1.044114 0.070291 0.614 0.5391## sodium 0.057928 1.059638 0.063883 0.907 0.3645## zinc 0.057169 1.058835 0.047875 1.194 0.2324## mage35 -0.458696 0.632107 0.238370 -1.924 0.0543## ## Likelihood ratio test=23.52 on 12 df, p=0.02364## n= 452, number of events= 205
qc.survfit1 <- qgcomp.cox.noboot(survival::Surv(disease_time, disease_state) ~ .,expnms=Xnm, data=metals[,c(Xnm, 'disease_time', 'disease_state')], q=4)qc.survfit1
## Scaled effect size (positive direction, sum of positive coefficients = 0.32)## barium zinc magnesium chromium silver sodium iron ## 0.3432 0.1946 0.1917 0.1119 0.0924 0.0511 0.0151 ## ## Scaled effect size (negative direction, sum of negative coefficients = -0.554)## selenium copper calcium arsenic manganese cadmium lead mercury ## 0.2705 0.1826 0.1666 0.1085 0.0974 0.0794 0.0483 0.0466 ## ## Mixture log(hazard ratio) (Delta method CI):## ## Estimate Std. Error Lower CI Upper CI Z value Pr(>|z|)## psi1 -0.23356 0.24535 -0.71444 0.24732 -0.9519 0.3411
Marginal hazards ratios (and bootstrapped quantile g-computation in general) uses a slightly different approach to effect estimation that makes it more computationally demanding than other qcomp
To estimate a marginal hazards ratio, the underlying model is fit, and then new outcomes are simulated under the underlying model with a baseline hazard estimator (Efron’s) - this simulation requires
a large sample (controlled by MCsize) for accuracy. This approach is similar to other g-computation approaches to survival analysis, but this approach uses the exact survival times, rather than
discretized survival times as are common in most g-computation analysis. Plotting a qgcompcox.bootobject yields a set of survival curves (e.g.qc.survfit2) which comprise estimated survival curves
(assuming censoring and late entry at random, conditional on covariates in the model) that characterize conditional survival functions (i.e. censoring competing risks) at various levels of
joint-exposure (including the overall average - which may be slightly different from the observed survival curve, but should more or less agree).
# bootstrapped version estimates a marginal structural model for the population average effect#library(survival)qc.survfit2 <- qgcomp.cox.boot(Surv(disease_time, disease_state) ~ .,expnms=Xnm, data=metals[,c(Xnm, 'disease_time', 'disease_state')], q=4, B=5, MCsize=1000, parallel=TRUE, parplan=TRUE)qc.survfit2
## Mixture log(hazard ratio) (bootstrap CI):## ## Estimate Std. Error Lower CI Upper CI Z value Pr(>|z|)## psi1 -0.24020 0.14942 -0.53305 0.052653 -1.6076 0.1079
# testing proportional hazards (note that x=TRUE is not needed (and will cause an error if used))survival::cox.zph(qc.survfit2$fit)
## chisq df p## arsenic 0.18440 1 0.668## barium 0.10819 1 0.742## cadmium 0.05345 1 0.817## calcium 0.00206 1 0.964## chromium 1.23974 1 0.266## copper 0.28518 1 0.593## iron 3.46739 1 0.063## lead 0.17575 1 0.675## magnesium 2.12900 1 0.145## manganese 0.58720 1 0.444## mercury 0.00136 1 0.971## selenium 0.15247 1 0.696## silver 0.01040 1 0.919## sodium 0.09352 1 0.760## zinc 1.51261 1 0.219## GLOBAL 9.82045 15 0.831
p2 = plot(qc.survfit2, suppressprint = TRUE) p2 + labs(title="Linear log(hazard ratio), overall and exposure specific")
All bootstrapped functions in qgcomp allow parellelization via the parallel=TRUE parameter (demonstrated with the non-liner fit in qc.survfit3). Only 5 bootstrap iterations are used here, which is
not nearly enough for inference, and will actually be slower for parallel processing due to some overhead when setting up the parallel processes.
qc.survfit3 <- qgcomp.cox.boot(Surv(disease_time, disease_state) ~ . + .^2,expnms=Xnm, data=metals[,c(Xnm, 'disease_time', 'disease_state')], q=4, B=5, MCsize=1000, parallel=TRUE, parplan=TRUE)qc.survfit3
## Mixture log(hazard ratio) (bootstrap CI):## ## Estimate Std. Error Lower CI Upper CI Z value Pr(>|z|)## psi1 -0.19672 0.72426 -1.6162 1.2228 -0.2716 0.7859
p3 = plot(qc.survfit3, suppressprint = TRUE) p3 + labs(title="Non-linear log(hazard ratio) overall, linear exposure specific ln-HR")
Technical Note: this mode of usage is designed for simplicity. The implementation relies on the future and future.apply packages. Use guidelines of the future package dictates that the user should be
able to control the future “plan”, rather than embedding it in functions as has been done here. This slightly more advanced usage (which allows nesting within larger parallel schemes such as
simulations) is demonstrated here by setting the “parplan” parameter to FALSE and explicitly specifying a “plan” outside of qgcomp functions. This will move much of the overhead due to parallel
processing outside of the actual qgcomp functions. The final code block of this vignette shows how to exit the “plan” and return to standard evaluation via plan(sequential) - doing so at the end
means that the next parallel call (with parplan=FALSE) we make will have lower overhead and run slightly faster.
future::plan(future::multisession)# parallel evaluationqc.survfit3 <- qgcomp.cox.boot(Surv(disease_time, disease_state) ~ . + .^2,expnms=Xnm, data=metals[,c(Xnm, 'disease_time', 'disease_state')], q=4, B=5, MCsize=1000, parallel=TRUE, parplan=FALSE)qc.survfit3
## Mixture log(hazard ratio) (bootstrap CI):## ## Estimate Std. Error Lower CI Upper CI Z value Pr(>|z|)## psi1 -0.21112 0.30702 -0.81287 0.39063 -0.6876 0.4917
p3 = plot(qc.survfit3, suppressprint = TRUE) p3 + labs(title="Non-linear log(hazard ratio) overall, linear exposure specific ln-HR")
Returning to substance: while qgcompcox.boot fits a smooth hazard ratio function, the hazard ratios contrasting specific quantiles with a referent quantile can be obtained, as demonstrated with
qc.survfit4. As in qgcomp.glm.boot plots, the conditional model fit and the MSM fit are overlaid as a way to judge how well the MSM fits the conditional fit (and whether, for example non-linear terms
should be added or removed from the overall fit via the degree parameter - we note here that we know of no statistical test for quantifying the difference between these lines, so this is up to user
discretion and the plots are provided as visuals to aid in exploratory data analysis).
qc.survfit4 <- qgcomp.cox.boot(Surv(disease_time, disease_state) ~ . + .^2,expnms=Xnm, data=metals[,c(Xnm, 'disease_time', 'disease_state')], q=4, B=5, MCsize=1000, parallel=TRUE, parplan=FALSE, degree=2)qc.survfit4
## Mixture log(hazard ratio) (bootstrap CI):## ## Estimate Std. Error Lower CI Upper CI Z value Pr(>|z|)## psi1 -2.73325 4.77640 -12.0948 6.6283 -0.5722 0.5672## psi2 0.84012 1.76149 -2.6123 4.2926 0.4769 0.6334
# examining the overall hazard ratio as a function of overall exposurehrs_q = exp(matrix(c(0,0,1,1,2,4,3,9), ncol=2, byrow=TRUE)%*%qc.survfit4$msmfit$coefficients)colnames(hrs_q) = "Hazard ratio"print("Hazard ratios by quartiles (min-25%,25-50%, 50-75%, 75%-max)")
## [1] "Hazard ratios by quartiles (min-25%,25-50%, 50-75%, 75%-max)"
## Hazard ratio## [1,] 1.0000000## [2,] 0.1505986## [3,] 0.1217189## [4,] 0.5279735
p4 = plot(qc.survfit4, suppressprint = TRUE) p4 + labs(title="Non-linear log(hazard ratio), overall and exposure specific")
Testing proportional hazards is somewhat complicated with respect to interpretation. Consider first a linear fit from qgcomp.cox.noboot. Because the underlying model of a linear qgcomp fit is
equivalent to the sum of multiple parameters, it is not clear how proportional hazards might be best tested for the mixture. One could examine test statistics for each exposure, but there may be some
exposures for which the test indicates non-proportional hazards and some for which the test does not.
The “GLOBAL” test in the cox.zph function from the survival package comes closest to what we might want, and gives an overall assessment of non-proportional hazards for all model terms simultaneously
(including non-mixture covariates). While this seems somewhat undesirable due to non-specificity, it is not necessarily important that only the mixture have proportional hazards, so it is useful and
easily interpretable to have a global test of fit via GLOBAL.
# testing proportional hazards (must set x=TRUE in function call)qc.survfit1ph <- qgcomp.cox.noboot(survival::Surv(disease_time, disease_state) ~ .,expnms=Xnm, data=metals[,c(Xnm, 'disease_time', 'disease_state', "mage35")], q=4, x=TRUE)survival::cox.zph(qc.survfit1ph$fit)
## chisq df p## arsenic 1.57e-01 1 0.691## barium 1.28e-01 1 0.721## cadmium 5.14e-02 1 0.821## calcium 9.16e-04 1 0.976## chromium 1.25e+00 1 0.263## copper 3.42e-01 1 0.559## iron 3.51e+00 1 0.061## lead 1.59e-01 1 0.690## magnesium 2.08e+00 1 0.149## manganese 5.78e-01 1 0.447## mercury 4.87e-06 1 0.998## selenium 1.32e-01 1 0.716## silver 1.30e-02 1 0.909## sodium 1.06e-01 1 0.745## zinc 1.53e+00 1 0.216## mage35 1.73e-02 1 0.895## GLOBAL 9.93e+00 16 0.870
For a potentially non-linear/ non-additive fit from qgcomp.cox.boot, the issue is slightly more complicated by the fact that the algorithm will fit both the underlying model and a marginal structural
model using the predictions from the underlying model. In order for the predictions to yield valid causal inference, the underlying model must be correct (which implies that proportional hazards
hold). The marginal structural model proceeds assuming the underlying model is correct. Currently there is no simple way to allow for non-proportional hazards in the marginal structural model, but
non-proportional hazards can be implemented in the conditional model via standard approaches to non-proportional hazards such as time-exposure-interaction terms. This is a rich field and discussion
is beyond the scope of this document.
# testing global proportional hazards for model (note that x=TRUE is not needed (and will cause an error if used))phtest3 = survival::cox.zph(qc.survfit3$fit)phtest3$table[dim(phtest3$table)[1],, drop=FALSE]
## chisq df p## GLOBAL 206.5578 120 1.53907e-06
Late entry and counting-process style data will currently yield results in qgcomp.cox functions. There has been some testing of this in limited settings, but we note that this is still an
experimental feature at this point that may not be valid in all cases and so it is not documented here. As much effort as possible to validate results through other means is needed when using qgcomp
in data subject to late-entry or when using counting-process style data.
Example 8: clustering
Clustering on the individual or group level means that there are individual or group level characteristics which result in covariance between observations (e.g. within individual variance of an
outcome may be much lower than the between individual variance). For linear models, the error term is assumed to be independent between observations, and clustering breaks this assumption. Ways to
relax this assumption include empirical variance estimation and cluster-appropriate robust variance estimation (e.g. through the sandwich package in R). Another way is to use cluster-based
bootstrapping, which samples clusters, rather than individual observations. qgcomp.glm.boot can be leveraged to produce clustering consistent estimates of standard errors for independent effects of
exposure as well as the effect of the exposure as a whole. This is done using the id parameter of qgcomp.glm.boot (which can only handle a single variable and so may not efficient for nested
clustering, for example).
Below is a simple example with one simulated exposure. First the exposure data are ‘pre-quantized’ (so that one can verify that standard errors are appropriate using other means - this is not
intended to show a suggested practice). Next the data are analyzed using a 1-component mixture in qgcomp - again, this is for verification purposes. The qgcomp.glm.noboot result yields a naive
standard error of 0.0310 for the mixture effect:
set.seed(2123)N = 250t = 4dat <- data.frame(row.names = 1:(N*t))dat <- within(dat, { id = do.call("c", lapply(1:N, function(x) rep(x, t))) u = do.call("c", lapply(1:N, function(x) rep(runif(1), t))) x1 = rnorm(N, u) y = rnorm(N) + u + x1})# pre-quantizeexpnms = c("x1")datl = quantize(dat, expnms = expnms)qgcomp.glm.noboot(y~ x1, data=datl$dat, family=gaussian(), q = NULL)
## Including all model terms as exposures of interest
## Scaled effect size (positive direction, sum of positive coefficients = 0.955)## x1 ## 1 ## ## Scaled effect size (negative direction, sum of negative coefficients = 0)## None## ## Mixture slope parameters (Delta method CI):## ## Estimate Std. Error Lower CI Upper CI t value Pr(>|t|)## (Intercept) -0.463243 0.057934 -0.57679 -0.34969 -7.996 3.553e-15## psi1 0.955015 0.031020 0.89422 1.01581 30.787 < 2.2e-16
# neither of these ways yields appropriate clustering#qgcomp.glm.noboot(y~ x1, data=datl$dat, id="id", family=gaussian(), q = NULL)#qgcomp.glm.boot(y~ x1, data=datl$dat, family=gaussian(), q = NULL, MCsize=1000)
while the qgcomp.glm.boot result (MCsize=5000, B=500) yields a corrected standard error of 0.0398, which is much closer to the sandwich estimate of 0.0409 than the naive estimator (a second
qgcomp.glm.boot fit with fewer bootstrap iterations and smaller MCsize is included for display). The standard errors from the uncorrected fit are too low, but this may not always be the case.
# clustering by specifying id parameter onqgcomp.glm.boot(y~ x1, data=datl$dat, id="id", family=gaussian(), q = NULL, MCsize=1000, B = 5)
## Including all model terms as exposures of interest
## ## Note: using all possible values of exposure as the## intervention values
## Mixture slope parameters (bootstrap CI):## ## Estimate Std. Error Lower CI Upper CI t value Pr(>|t|)## (Intercept) -0.463243 0.084446 -0.62875 -0.29773 -5.4856 5.223e-08## psi1 0.955015 0.055037 0.84714 1.06289 17.3521 < 2.2e-16
#qgcomp.glm.boot(y~ x1, data=datl$dat, id="id", family=gaussian(), q = NULL, MCsize=1000, B = 500)# Mixture slope parameters (bootstrap CI):# # Estimate Std. Error Lower CI Upper CI t value# (Intercept) -0.4632 0.0730 -0.606 -0.32 3.3e-10# psi1 0.9550 0.0398 0.877 1.03 0# This can be verified using the `sandwich` package #fitglm = glm(y~x1, data=datl$dat)#sw.cov = sandwich::vcovCL(fitglm, cluster=~id, type = "HC0")[2,2]#sqrt(sw.cov)# [1] 0.0409
Example 9: partial effects
Returning to our original example (and adjusting for covariates), note that the default output for a qgcomp.*.noboot object includes “sum of positive/negative coefficients.” These can be interpreted
as “partial mixture effects” or effects of exposures with coefficients in a particular direction. This is displayed graphically via a plot of the qgcomp “weights,” where all exposures that contribute
to a given partial effect are on the same side of the plot. Unfortunately, this does not yield valid inference for a true partial effect because it is a parameter conditional on the fitted results
and thus does not represent the type of a priori hypothesis that is amenable to hypothesis testing and confidence intervals. Another way to think about this is that it is a data adaptive parameter
and thus is subject to issues of overfit that are similar to issues with making inference from step-wise variable selection procedures.
(qc.fit.adj <- qgcomp.glm.noboot(y~.,dat=metals[,c(Xnm, covars, 'y')], expnms=Xnm, family=gaussian()))
## Scaled effect size (positive direction, sum of positive coefficients = 0.441)## calcium barium iron arsenic silver chromium cadmium zinc ## 0.76111 0.06133 0.05854 0.02979 0.02433 0.02084 0.01395 0.01195 ## mercury sodium ## 0.01130 0.00688 ## ## Scaled effect size (negative direction, sum of negative coefficients = -0.104)## copper magnesium lead manganese selenium ## 0.44654 0.42124 0.09012 0.03577 0.00631 ## ## Mixture slope parameters (Delta method CI):## ## Estimate Std. Error Lower CI Upper CI t value Pr(>|t|)## (Intercept) -0.460408 0.134007 -0.72306 -0.19776 -3.4357 0.0006477## psi1 0.337539 0.089358 0.16240 0.51268 3.7774 0.0001805
Fortunately, there is a way towards estimation of “partial effects.” One way to do this is sample splitting, where the data are first randomly partitioned into a “training” and a “validation” data
set. The assessment of whether a coefficient is positive or not occurs in the “training” set and then effect estimation for positive/negative partial effects occurs in the validation data. Basic
simulations can show that such a procedure can yield valid (often called “honest”) hypothesis tests and confidence intervals for partial effects, provided that there is no separate data exploration
in the combined dataset to select the models. In the qgcomp package, the partitioning of the datasets into “training” and “validation” sets is done by the user, which prevents issues that may arise
if this is done naively on a dataset that contains clusters (where we should partition based on clusters, rather than observations) or multiple observations per individual (all observations from an
individual should be partitioned together). Here is an example of simple partitioning on a dataset that contains one observation per individual with no clustering. The downside of sample splitting is
the loss of precision, because the final “validation” dataset comprises only a fraction of the original sample size. Thus, the estimation of partial effects is most appropriate with large sample
sizes. We also note that these partial effect are only well defined when all effects are linear and additive, since whether a variable contributes to a positive or negative partial effect would
depend on the value of that variable, so the valid estimation of “partial effects” is limited to settings in which the qgcomp.\*.noboot objects are used for inference.
# 40/60% training/validation splitset.seed(123211)trainidx <- sample(1:nrow(metals), round(nrow(metals)*0.4))valididx <- setdiff(1:nrow(metals),trainidx)traindata <- metals[trainidx,]validdata <- metals[valididx,]dim(traindata) # 40% of total
## [1] 181 26
dim(validdata) # 60% of total
## [1] 271 26
The qgcomp package then facilitates the analysis of these partitioned data to allow valid estimation and hypothesis testing of partial effects. The qgcomp.partials function is used to estimate
partial effects. Note that the variables with “negative effect sizes” differs slightly from the overall analysis given in the qc.fit object that represents our first pass analysis on these data. This
is to be expected, and is a feature of this approach: different random subsets of the data will be expected to yield different estimated effects. If the true effect is null, then the estimated
effects will vary from positive to negative around the null, and sample splitting is an important way to distinguish between estimates that reflect underlying patterns in the entire dataset from
estimates that are simply due to natural sampling variation inherent to small and moderate samples. Note that fitting on these smaller datasets can sometimes result in perfect collinearity of
exposures, in which case setting bayes=TRUE may be necessary to apply a ridge penalty to estimates.
splitres <- qgcomp.partials( fun="qgcomp.glm.noboot", f=y~., q=4, traindata=traindata[,c(Xnm, covars, "y")],validdata=validdata[,c(Xnm, covars, "y")], expnms=Xnm, bayes=FALSE, .fixbreaks = TRUE, .globalbreaks=FALSE )splitres
## ## Variables with positive effect sizes in training data: arsenic, barium, cadmium, calcium, chromium, iron, selenium, silver, sodium, zinc## Variables with negative effect sizes in training data: copper, lead, magnesium, manganese, mercury## Partial effect sizes estimated in validation data## Positive direction Mixture slope parameters (Delta method CI):## ## Estimate Std. Error Lower CI Upper CI t value Pr(>|t|)## (Intercept) -0.46347 0.16202 -0.78102 -0.14591 -2.8605 0.004573## psi1 0.35150 0.10849 0.13887 0.56413 3.2400 0.001351## ## Negative direction Mixture slope parameters (Delta method CI):## ## Estimate Std. Error Lower CI Upper CI t value Pr(>|t|)## (Intercept) -0.020164 0.097018 -0.210316 0.16999 -0.2078 0.8355## psi1 0.028852 0.063752 -0.096101 0.15380 0.4526 0.6512
Consistent with our overall results, the overall effect of metals on the outcome y is predominantly positive, which is driven mainly by calcium. The partial positive effect of psi=0.42 is slightly
attenuated from the partial positive effect given in the original fit (0.44), but is slightly larger than the overall effect from the original fit (psi1=0.34). We note that the effect direction of
cadmium is negative, even though it was selected based on positive associations in the training data. This suggests this variable has effects that are close to the null and their direction will
depend on which subset of the data are used. This feature allows valid testing of hypotheses - a global null in which no exposures have effects will be characterized by variables that randomly switch
effect directions between training and validation datasets, which will yield partial effect estimates close to the null with hypothesis tests that have appropriate type-1 error rates in large
By default (subject to change) quantile cut points (“breaks”) are defined within the training data and applied to the validation data. You may also change this behavior to allow the breaks to be
defined using quantiles from the entire dataset, which treats the quantiles as fixed. This will be expected to improve stability in small samples and may eventually replace the default behavior as
the quantiles themselves are not generally treated as random variables within quantile g-computation. For this particular dataset (and seed value), there is little impact of this setting on the
splitres_alt <- qgcomp.partials( fun="qgcomp.glm.noboot", f=y~., q=4, traindata=traindata[,c(Xnm, covars, "y")],validdata=validdata[,c(Xnm, covars, "y")], expnms=Xnm, bayes=FALSE, .fixbreaks = TRUE, .globalbreaks=TRUE )splitres_alt
## ## Variables with positive effect sizes in training data: arsenic, barium, cadmium, calcium, chromium, iron, selenium, silver, sodium, zinc## Variables with negative effect sizes in training data: copper, lead, magnesium, manganese, mercury## Partial effect sizes estimated in validation data## Positive direction Mixture slope parameters (Delta method CI):## ## Estimate Std. Error Lower CI Upper CI t value Pr(>|t|)## (Intercept) -0.45550 0.16996 -0.78862 -0.12238 -2.6800 0.007832## psi1 0.34492 0.11364 0.12219 0.56766 3.0352 0.002648## ## Negative direction Mixture slope parameters (Delta method CI):## ## Estimate Std. Error Lower CI Upper CI t value Pr(>|t|)## (Intercept) -0.0079779 0.0981166 -0.200283 0.18433 -0.0813 0.9353## psi1 0.0328941 0.0648437 -0.094197 0.15999 0.5073 0.6124
One careful note: when there are multiple exposures with small positive or negative effects, the partial effects may be biased towards the null in studies with moderate or small sample sizes. This
occurs because, in the training set, some exposures with small effects are likely to be mis-classified with regard to their effect direction. In some instances, both the positive and negative partial
effects can be in the same direction. This occurs if individual effects are predominantly in one direction, but some are small and subject to having mis-classified directions. As one example: if
there is a null overall effect, but there is a positive partial effect driven strongly by one exposure and a balancing negative partial effect driven by numerous weaker associations, partial effect
estimates will not sum to the overall effect because the negative partial effect will experience more downward bias in typical sample sizes. Thus, when the overall effect does not equal the sum of
the partial effects, there is likely some bias in at least one of the partial effect estimates. This is not a unique feature of quantile-based g-computation, but is also be a concern for methods that
focus on estimation of partial effects, such as weighted quantile sum regression.
The larger question about interpretation (and its worth) of partial effects is left to the analyst. For large datasets with well characterized exposures that have plausible subsets of exposures that
would be positively/negatively linearly associated with the outcome, the variables that partition into negative/positive partial effects may make some substantive sense. In more realistic settings
that typify exposure mixtures, the partitioning will result in groups that don’t entirely make sense. The “partial effect” yields the effect of increasing all exposures in the subset defined by
positive coefficients in the training data, while holding all other exposures and confounders constant. In the setting where this corresponds to real world patterns (e.g. all exposures in the
positive partial effect share a source), then this may be interpretable roughly as the effect of an action to intervene on the source of these exposures. In most settings, however, interpretation
will not be this clear and should not be expected to map onto potential real-world interventions. We note that this is not a function of the quantile g-computation method, but just part of the
general messiness of working with exposures mixture data.
A more justifiable approach in terms of mapping effect estimates onto potential real-world actions would be choosing subsets of exposures based on prior subject matter knowledge, as we demonstrated
above in example 5. This does not require sample splitting, but it does come at the expense of having to know more about the exposures and outcome under analysis. For example, our simulated outcome y
may represent some outcome that we would expect to increase with so-called “essential” metals, or those that are necessary (at some small amount) for normal human functioning, but it may decrease
with “potentially toxic” (non-essential) metals, or those that have no known biologic function and are more likely to cause harm rather than improve physiologic processes that lead to improved
(larger) values of y. Qgcomp can be used to assess effects of these “sub-mixtures.”
Here are results for the essential metals:
nonessentialXnm <- c( 'arsenic','barium','cadmium','chromium','lead','mercury','silver')essentialXnm <- c( 'sodium','magnesium','calcium','manganese','iron','copper','zinc','selenium')covars = c('nitrate','nitrite','sulfate','ph', 'total_alkalinity','total_hardness')(qc.fit.essential <- qgcomp.glm.noboot(y~.,dat=metals[,c(Xnm, covars, 'y')], expnms=essentialXnm, family=gaussian()))
## Scaled effect size (positive direction, sum of positive coefficients = 0.357)## calcium iron zinc selenium ## 0.908914 0.077312 0.013128 0.000646 ## ## Scaled effect size (negative direction, sum of negative coefficients = -0.0998)## copper magnesium sodium manganese ## 0.4872 0.4304 0.0486 0.0338 ## ## Mixture slope parameters (Delta method CI):## ## Estimate Std. Error Lower CI Upper CI t value Pr(>|t|)## (Intercept) -0.340130 0.111562 -0.55879 -0.12147 -3.0488 0.002435## psi1 0.257136 0.074431 0.11125 0.40302 3.4547 0.000604
Here are results for the non-essential metals:
(qc.fit.nonessential <- qgcomp.glm.noboot(y~.,dat=metals[,c(Xnm, covars, 'y')], expnms=nonessentialXnm, family=gaussian()))
## Scaled effect size (positive direction, sum of positive coefficients = 0.0751)## barium arsenic silver mercury cadmium ## 0.2827 0.2494 0.2420 0.1763 0.0496 ## ## Scaled effect size (negative direction, sum of negative coefficients = -0.0129)## lead chromium ## 0.888 0.112 ## ## Mixture slope parameters (Delta method CI):## ## Estimate Std. Error Lower CI Upper CI t value Pr(>|t|)## (Intercept) -0.055549 0.081509 -0.215304 0.10420 -0.6815 0.4959## psi1 0.062267 0.052508 -0.040648 0.16518 1.1858 0.2363
As shown from these results, the essential metals and minerals demonstrate an overall positive joint association with the outcome (controlling for non-essentials), whereas the partial effect of
non-essential metals and minerals is close to null. This is close to the interpretation of the data adaptive selection of partial effects demonstrated above, but is interpretable in terms of how we
might intervene (e.g. increase consumption of foods that are higher in essential metals and minerals).
Example 10: multinomial outcomes
For outcomes modeled as 3+ discrete categories, qgcomp joint effect estimates are interpreted as a ratio of the probability of being in the referent category of the outcome and the probability of
being in the index category.
First, we’ll bring in data and create a multinomial outcome.
data("metals") # from qgcomp package# create categorical outcome from the existing continuous outcome (usually, one will already exist)metals$ycat = factor(quantize(metals, "y",q=4)$data$y, levels=c("0", "1", "2", "3"), labels=c("cct", "ccg", "aat", "aag")) # restrict to smaller dataset for simplicitysmallmetals = metals[,c("ycat", "arsenic", "lead", "cadmium", "mage35")]
Next, fit the model.
### 1: Define mixture and underlying model ####mixture = c("arsenic", "lead", "cadmium")f2 = ycat ~ arsenic + lead + cadmium + mage35rr = qgcomp.multinomial.noboot( f2, expnms = mixture, q=4, data = smallmetals, )rr2 = qgcomp.multinomial.boot( f2, expnms = mixture, q=4, data = smallmetals, B =2, # set to higher values >200 in general usage MCSize=10000 # set to higher values in small samples )summary(rr)
## Reference outcome levels:## cct ccg aat aag## Weights## arsenic lead cadmium## ccg -0.19985072 -0.4065664 -0.3935829## aat -0.02996557 1.0000000 -0.9700344## aag -0.36389597 -0.3920415 -0.2440625## ## Sum of positive coefficients ## ccg aat aag ## 0.000000000 0.006243471 0.000000000 ## Sum of negative coefficients ## ccg aat aag ## -0.3255537 -0.1449220 -0.2362489 ## ## Mixture slope parameters (Standard CI):## Estimate Std. Error## ccg.(Intercept) 0.341521 0.324928## aat.(Intercept) 0.098838 0.330695## aag.(Intercept) 0.261681 0.326332## ccg.psi -0.325554 0.204573## aat.psi -0.138679 0.203451## aag.psi -0.236249 0.203446
summary(rr2) # differs from `rr` primarily due to low `MCSize` value
## Reference outcome levels:## cct ccg aat aag## Mixture slope parameters (Bootstrap CI):## Estimate Std. Error## ccg.(Intercept) 0.294923 0.045838## aat.(Intercept) -0.009291 0.068102## aag.(Intercept) 0.193000 0.076732## ccg.psi1 -0.238836 0.012265## aat.psi1 -0.058937 0.051478## aag.psi1 -0.197627 0.197277
plot(rr) #plot(rr2) # not yet functional
plot and summary are available for multinomial fits, and more functionality will be available in the future.
Missing data, limits of detection and multiple imputation
When carrying out data analysis using quantile g-computation, on can address missing data in much the same way as in standard regression analyses. A common approach is complete case analysis. While
regression functions in R will automatically carry out complete case analyses when variables take on the value NA (denoting missingness in R), when using quantile g-computation it is encouraged that
one explicitly create the complete case dataset explicitly and use that complete case dataset. Using the pre-installed R packages, this can be accomplished with the complete.cases function.
The reason for this recommendation is that, while the regression analysis will be performed on complete cases (i.e. observations with non-missing values for all variables in the model), the
calculation of quantiles for each exposures is done on an exposure-by-exposure basis, which can lead to numerical differences when explicitly using a dataset restricted to complete cases versus
relying on automatically removing observations with NA values during analysis.
Here is an artificial example that demonstrates the differences. Three analyses are run: one with the full data, one with complete case data (complete case analysis #1), and one with data in which
arsenic has been randomly set to NA (complete case analysis #2).
There are numeric differences between the two complete case analyses, which can be traced to differences in the “quantized” exposures (other than arsenic) in the two approaches, which can be found by
the qx data frame that is part of a qgcompfit object.
Xnm <- c( 'arsenic','barium','cadmium','calcium','chromium','copper', 'iron','lead','magnesium','manganese','mercury','selenium','silver', 'sodium','zinc')covars = c('nitrate','nitrite','sulfate','ph', 'total_alkalinity','total_hardness')asmiss = metalsset.seed(1232)asmiss$arsenic = ifelse(runif(nrow(metals))>0.7, NA, asmiss$arsenic)cc = asmiss[complete.cases(asmiss[,c(Xnm, covars, "y")]),] # complete.cases gives a logical index to subset rowsdim(metals) # [1] 452 26
## [1] 452 27
dim(cc) # [1] 320 26
## [1] 320 27
Here we have results from the full data (for comparison purposes)
qc.base <- qgcomp.glm.noboot(y~.,expnms=Xnm, dat=metals[,c(Xnm, covars, 'y')], family=gaussian())cat("Full data\n")
## Full data
## Scaled effect size (positive direction, sum of positive coefficients = 0.441)## calcium barium iron arsenic silver chromium cadmium zinc ## 0.76111 0.06133 0.05854 0.02979 0.02433 0.02084 0.01395 0.01195 ## mercury sodium ## 0.01130 0.00688 ## ## Scaled effect size (negative direction, sum of negative coefficients = -0.104)## copper magnesium lead manganese selenium ## 0.44654 0.42124 0.09012 0.03577 0.00631 ## ## Mixture slope parameters (Delta method CI):## ## Estimate Std. Error Lower CI Upper CI t value Pr(>|t|)## (Intercept) -0.460408 0.134007 -0.72306 -0.19776 -3.4357 0.0006477## psi1 0.337539 0.089358 0.16240 0.51268 3.7774 0.0001805
Here we have results from a complete case analysis in which we have set some exposures to be missing and have explicitly excluded data that will be dropped from analysis.
qc.cc <- qgcomp.glm.noboot(y~.,expnms=Xnm, dat=cc[,c(Xnm, covars, 'y')], family=gaussian())cat("Complete case analyses\n")
## Complete case analyses
cat(" #1 explicitly remove observations with missing values\n")
## #1 explicitly remove observations with missing values
## Scaled effect size (positive direction, sum of positive coefficients = 0.48)## calcium iron zinc chromium silver lead barium arsenic ## 0.78603 0.07255 0.04853 0.03315 0.02555 0.01405 0.01285 0.00728 ## ## Scaled effect size (negative direction, sum of negative coefficients = -0.117)## copper magnesium cadmium mercury manganese sodium selenium ## 0.52200 0.23808 0.09082 0.06570 0.04105 0.03921 0.00315 ## ## Mixture slope parameters (Delta method CI):## ## Estimate Std. Error Lower CI Upper CI t value Pr(>|t|)## (Intercept) -0.509402 0.145471 -0.79452 -0.22428 -3.5017 0.0005315## psi1 0.362582 0.096696 0.17306 0.55210 3.7497 0.0002119
Finally we have results from a complete case analysis in which we have set some exposures to be missing, but we rely on R’s automated dropping of observations with missing values.
qc.cc2 <- qgcomp.glm.noboot(y~.,expnms=Xnm, dat=asmiss[,c(Xnm, covars, 'y')], family=gaussian())cat(" #1 rely on R handling of NA values\n")
## #1 rely on R handling of NA values
## Scaled effect size (positive direction, sum of positive coefficients = 0.493)## calcium iron zinc chromium barium silver lead selenium ## 0.76495 0.07191 0.04852 0.03489 0.02151 0.02119 0.01912 0.01444 ## arsenic ## 0.00348 ## ## Scaled effect size (negative direction, sum of negative coefficients = -0.12)## copper magnesium sodium manganese cadmium mercury ## 0.5338 0.2139 0.0965 0.0853 0.0501 0.0204 ## ## Mixture slope parameters (Delta method CI):## ## Estimate Std. Error Lower CI Upper CI t value Pr(>|t|)## (Intercept) -0.51673 0.15001 -0.81074 -0.22273 -3.4447 0.0006520## psi1 0.37249 0.10014 0.17621 0.56876 3.7196 0.0002376
Now we can see a reason for the discrepancy between the methods above: when relying on R to drop missing values by allowing missing values for exposures in the analytic data, the quantiles of
exposures will be done on all valid values for each exposure. In the complete case data, the quantiles will only be calculated among those with complete observations. The latter will generally be
preferred because the quantiles for each exposure will be calculated on the same sample of individuals.
# calculation of arsenic quantiles is identicalall.equal(qc.cc$qx$arsenic_q, qc.cc2$qx$arsenic_q[complete.cases(qc.cc2$qx$arsenic_q)])
## [1] TRUE
# all are equalall.equal(qc.cc$qx$cadmium_q, qc.cc2$qx$cadmium_q[complete.cases(qc.cc2$qx$arsenic_q)])
## [1] "Mean relative difference: 0.3823529"
# not equal
Limits of detection
A common form of missing data that occurs in mixtures are exposure values that are missing due to being below the limit of detection. A common approach to such missing data is imputation, either
through filling in small numeric values in place of the missing values, or in a more formal multiple imputation from a parametric model. Notably, with quantile g-computation, if the proportion of
values below the limit of detection is below 1/q (the number of quantiles), all appropriate missing data approaches will yield the same answer. Thus, if one has 3 exposures each with 10% of the
values below the limit of detection, one can impute small values below those limits (e.g. limit of detection divided by the square root of 2) and proceed with quantile g-computation on the imputed
data. This analysis leverages the fact that, even if a value below the limit of detection cannot be known with certainty, the category score used in quantile g-computation is known with certainty. In
cases with more than 1/q% measurements below the LOD, the packages qgcomp comes with a convenience function mice.impute.leftcenslognorm that can be used to impute values below the limit of detection
from a left censored log-normal regression model.
Multiple imputation
Multiple imputation uses multiple datasets with different imputed values for each missing value across datasets. Separate analyses are performed on each of these datasets, and the results are
combined using standard rules. The function mice.impute.leftcenslognorm can be interfaced with the mice package for efficient programming of multiple imputation based analysis with quantile
g-computation. Examples cannot be included here without explicitly installing the mice package, but an example can be seen in the help file for mice.impute.leftcenslognorm.
Why don’t I get weights from the boot functions?
Users often use the qgcomp.*.boot functions because the want to marginalize over confounders or fit a non-linear joint exposure function. In both cases, the overall exposure response will no longer
correspond to a simple weighted average of model coefficients, so none of the qgcomp.*.boot functions will calculate weights. In most use cases, the weights would vary according to which level of
joint exposure you’re at, so it is not a straightforward proposition to calculate them (and you may not wish to report 4 sets of weights if you use the default q=4). If you fit an otherwise linear
model, you can get weights from a qgcomp.*.noboot which will be very close to the weights you might get from a linear model fit via qgcomp.*.boot functions, but be explicit that the weights come from
a different model than the inference about joint exposure effects.
Do I need to model non-linearity and non-additivity of exposures?
Maybe. The inferential object of qgcomp is the set of \(\psi\) parameters that correspond to a joint exposure response. As it turns out, with correlated exposures non-linearity can disguise itself as
non-additivity (Belzak and Bauer [2019] Addictive Behaviors). If we were inferring independent effects, this distinction would be crucial, but for joint effects it may turn out that it doesn’t matter
much if you model non-linearity in the joint response function through non-additivity or non-linearity of individual exposures in a given study. Models fit in qgcomp still make the crucial assumption
that you are able to model the joint exposure response via parametric models, so that assumption should not be forgotten in an effort to try to disentagle non-linearity (e.g. quadratic terms of
exposures) from non-additivity (e.g. product terms between exposures). The important part to note about parametric modeling is that we have to explicitly tell the model to be non-linear, and no
adaptation to non-linear settings will happen automatically. Exploring non-linearity is not a trivial endeavor.
Do I have to use quantiles?
No. You can turn off “quantization” by setting q=NULL or you can supply your own categorization cutpoints via the “breaks” argument. It is up to the user to interpret the results if either of these
options is taken.
Can I cite this document?
Probably not in a scientific manuscript. If you find an idea here that is not published anywhere else and wish to develop it into a full manuscript, feel free! (But probably check with akeil@unc.edu
to ask if a paper is already in development.)
Alexander P. Keil, Jessie P. Buckley, Katie M. O’Brien, Kelly K. Ferguson, Shanshan Zhao,Alexandra J. White. A quantile-based g-computation approach to addressing the effects of exposure mixtures.
The development of this package was supported by NIH Grant RO1ES02953101. Invaluable code testing was performed by Nicole Niehoff, Michiel van den Dries, Emily Werder, Jessie Buckley, and Katie
# return to standard processingfuture::plan(future::sequential) # return to standard evaluation | {"url":"https://freemoneyforall.org/article/the-qgcomp-package-g-computation-on-exposure-quantiles","timestamp":"2024-11-09T20:00:25Z","content_type":"text/html","content_length":"525656","record_id":"<urn:uuid:bad5b009-8cbd-49da-b03f-b4cfd0ba4fdb>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00523.warc.gz"} |
Who is the Moon Man? - Page 3
No, silly! All computers in hell run Linux!
No they don't. Windows is the OS that is prone to breaking.
Not only do they run Windows, but they run Windows 95/98/ME apparently. :P
Not only do they run Windows, but they run Windows 95/98/ME apparently. :P
Win98SE isn't too bad for stability. WinME, however, is diabolical.
Win98SE isn't too bad for stability. WinME, however, is diabolical.
I'm at least thankful for Windows ME. It was the OS that ushered in Automatic Updates, System Restore, and integrated support for ZIP files.
I'm at least thankful for Windows ME. It was the OS that ushered in Automatic Updates, System Restore, and integrated support for ZIP files.
True, but the System Restore was pointless because you only ever had one restore point at any one time.
It had to be done.
Were can i instert the text i want? Like a web page or something? or do you have to edit yourself
Were can i instert the text i want? Like a web page or something? or do you have to edit yourself
Yourself. As simple as whiting it out then writing more text over the top in either MS Paint or GIMP.
Necroing this thread in honor of PA2:
SPAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACE (man)
I had some time and a nice copy of Photoshop. Here are the results:
The scariest moment from the Lord of the Rings.
Garbage Day!
I wouldn't blame Nelson for screaming after that.
Baruk Khazad!
I see you!
Edit: Made some more! This is fun.
I will sooner or later post the original in the 'Cannot Unsee' forum game. It was also my avatar for a few minutes, but it is not funny when small.
Did I forget one LotR character? Here is Aragorn from
The Poster™
I think it is space wolf from the grickle series of cartoons.
to comment in this discussion. | {"url":"https://community.telltale.com/discussion/17876/who-is-the-moon-man/p3","timestamp":"2024-11-08T15:49:55Z","content_type":"text/html","content_length":"59135","record_id":"<urn:uuid:97021828-e5b5-4e16-9576-c5aae039eaec>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00681.warc.gz"} |
Simply How Much You Need To Expect You’ll Purchase A Great math for kids
The recreation consists of curriculum-aligned math questions, which are supported by on-screen visuals and may be played out loud for much less ready readers. You can also profit from progress
tracking, aim setting, and common reviews. Tons of enjoyable and educational on-line math games, from primary operations to algebra and geometry. Learn sixth grade math aligned to the Eureka Math/
EngageNY curriculum—ratios, exponents, long division, adverse numbers, geometry, statistics, and extra.
• Join the 1 million academics already utilizing Prodigy Math in their lecture rooms at no cost.
• Algebra is a serious subject, and it is also identified as the “gatekeeper” for the entire different ranges and a prerequisite to comprehending other ranges.
• This enjoyable and fascinating curriculum-aligned sport lets college students engage in a enjoyable wizarding world that motivates them to practice more math than ever.
• We’ve skipped any website that focuses so much on concept and history, as it is extra important to apply with numbers somewhat than reading about numbers.
More than 425 math video games, logic puzzles, and brain workouts for faculty kids to apply their math skills. Figure This is a web site designed to encourage households to apply math collectively.
It contains fun and interesting math games and high-quality challenges. Woot Math offers adaptive apply for educating rational numbers and associated hellothinkster reviews subjects, corresponding to
fractions, decimals, and ratios. Here’s an online studying house that is engaging, supportive, and designed to get youngsters excited about math. Make math about greater than numbers with partaking
items, real-world scenarios, and limitless questions.
Another website would possibly give consideration to higher-level math and utterly overlook the basics. To be taught a model new skill from house, you must take it one step at a time. The Art of
Problem Solving website offers math courses for elementary, center, and highschool students. The resources on provide depend in your learning level, with some being extra comprehensive than others.
The Prodigy Math web site allows you, or your youngster, to learn math through a virtual world.
Their “Mathematics With a Human Face” web page contains information about careers in mathematics in addition to profiles of mathematicians. Learn seventh grade math—proportions, algebra fundamentals,
arithmetic with negative numbers, probability, circles, and extra. Learn fourth grade math—arithmetic, measurement, geometry, fractions, and extra. There are loads of resources and sites that can
allow you to be taught or relearn maths from fundamentals to advanced levels. Explore the 12 greatest math programs online to find the top choice in your schedule, finances, and studying level – from
newbies to superior college students.
The movies are a median of around 25 minutes lengthy and use graphics and examples to elucidate statistics. This complete website features a practice check and on-line instruments corresponding to a
probability calculator. After algebra, the following step in the best path in the course of studying math may be geometry. Some say geometry, which is the research of shapes, must be taken before
algebra 2, however the order is completely as much as you.
To effectively learn math by way of a website, it have to be easy to grasp. Most learners do best after they can see an issue walk-through, step-by-step. This website options multiple instance
issues, with walk-throughs by three separate instructors . They supply some fundamental math but are targeted on advanced subjects from algebra on up. Another graphing calculator for capabilities,
geometry, algebra, calculus, statistics, and 3D math, together with quite lots of math sources.
What You May Do About hello thinkster Starting Next 10 Minutes
Learn pre-algebra—all of the essential arithmetic and geometry abilities wanted for algebra. EdX is one other wonderful web site the place you probably can take free classes in college-level
calculus. Learnerator also supplies a great quantity of practice questions so that you can review.
The Forbidden Truth About thinkster math Revealed By A Classic Pro
Transform displays into classroom conversations with Peardeck for Google Slides. Effortlessly build participating instructional content, formative assessments, and interactive questions. An
award-winning series of math apps that harness the ability of digital instruments to create a better, deeper, extra fun learning experience. Blogs similar to “Making Math Social” and “Saying No to
Math Anxiety” are included as assets for lecturers and parents.
Graphing calculators may help with linear, quadratic, and trigonometric equations. Math follow meets game-based studying on the Prodigy Math web site. Students can discover a fantasy world where they
need to clear up math problems to win battles and complete quests. For a very original site reasonably priced month-to-month fee, achieve access to hundreds of resources created by academics such as
you. A artistic resolution that goals to revive students’ ardour and curiosity in math.
Learn the basics of algebra—focused on widespread mathematical relationships, corresponding to linear relationships. Learn early elementary math—counting, shapes, basic addition and subtraction, and
extra. Across the globe, 617 million children are lacking primary math and studying expertise. We’re a nonprofit delivering the schooling they want, and we’d like your help. Created by experts, Khan
Academy’s library of trusted practice and lessons covers math, science, and extra. Statistics is a useful level of math, because it entails gathering and analyzing numbers and knowledge. | {"url":"https://www.fpcomunicaciones.com.ar/simply-how-much-you-need-to-expect-youll-purchase-a-great-math-for-kids/","timestamp":"2024-11-02T05:10:30Z","content_type":"text/html","content_length":"60461","record_id":"<urn:uuid:feda4c5d-7518-4bfb-8e26-17f8681e20d9>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00784.warc.gz"} |
How to Find Test Statistic & How to Use it?
How to Find Test Statistic & How to Use it? There are a few different ways to calculate a test statistic, depending on the type of test you are running. One common method is the z-score, which uses
the standard normal distribution to calculate the probability of obtaining a given score or higher, given the sample size and population parameters. Other methods include the t-test and chi-squared
What is a Test Statistic?
A test statistic is a number used in statistics that helps researchers determine the probability that an event occurred. This number is based on the sample data and is used to calculate the p-value.
The p-value is a measure of how likely it is that the data could have arisen by chance. If the p-value is low, then it is likely that the event occurred due to chance. If the p-value is high, then it
is likely that the event occurred due to something other than chance.
Types of Test Statistics
There are many different types of test statistics, and statisticians use different ones depending on the question they’re trying to answer. Some common types of test statistics include the t-test,
the chi-squared test, and the F-test.
The t-test is used to determine whether two groups of data are statistically different from each other, while the chi-squared test is used to determine whether two sets of data are correlated. The
F-test is used to determine whether there is a significant difference between the variances of two groups of data.
Interpreting Test Statistics
When looking at the results of a test, it is important to understand what the statistics mean. The most common statistic is the p-value, which is a measure of how likely it is that the results
occurred by chance.
A p-value of less than .05 indicates that there is a 5% or lower chance that the results occurred by chance, and this is generally considered to be statistically significant. Other statistics include
the effect size and the confidence interval.
The effect size measures how large of an impact the treatment had on the outcome, and the confidence interval measures how confident we can be in the results. All of these statistics should be
considered when interpreting test results.
Reporting Test Statistics
When you take a test, the most important thing is to understand your results. This means understanding what the test measures, and how your score relates to the population as a whole. It also means
understanding the level of confidence that statisticians can attach to your score. This article will explain some basic concepts about reporting test statistics.
The first concept is the standard error. The standard error is a measure of how much error there is in a sample statistic. In other words, it’s a measure of how confident we can be that the sample
statistic actually reflects the population parameter. The smaller the standard error, the more confident we can be in the statistic.
Another concept you need to understand is the confidence interval. A confidence interval is simply a range of values within which we are 95% certain contains the population parameter.
Q: What is A regression model?
A: A regression model is a mathematical formula used to predict future events. The regression model uses historical data to calculate the most likely outcome for an event. The regression model can be
used to predict everything from election results to stock prices.
Q: How Do I Know Which Test Statistic to Use?
A: When a researcher conducts an experiment, they want to be able to determine whether the results are due to chance or if the treatment had an effect. To do this, they use a statistic. But which one
should they use? This can be a daunting question, but it’s important to know which statistic to use so that you can make accurate conclusions about your data. In this article, we’ll provide an
overview of the most common test statistics and help you determine which one is best for your data.
The first step is understanding what each statistic measures. The most common type of statistic is the t-statistic, which is used to measure the difference between two means. If you want to know
whether a new drug is effective in treating a certain condition, you would compare the average score of people who took the drug with the average score of people who didn’t take the drug. | {"url":"https://stoneoakbusiness.com/how-to-find-test-statistic-how-to-use-it/","timestamp":"2024-11-13T02:07:00Z","content_type":"text/html","content_length":"84253","record_id":"<urn:uuid:33ca9436-aebd-4543-8543-14905faaad01>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00834.warc.gz"} |
Flows and Bubbles
Many phenomena in both nature and society can be examined in terms of bubbles and flows. Many can be modeled as a combination of potential, flows, barriers and bubbles.
In the most general sense, a flow is the continuous transport of something from one place to another. In a more abstract sense, it is the continuous change of a quantity. For a short amount of time,
a flow can be caused by inertia. For longer periods, something must drive the flow.
The consumption of potential can drive a flow. Then the flow can be said to contribute to the achievement of the potential. The flow can continue indefinitely as long as both the potential and that
which flows are both steadily replenished. For many purposes, a flow can be viewed as a the result of a continuous supply of potential.
The shining of the Sun on the Earth in cold space is a continuous flow of energy that has lasted billions of years. The current of water down the Nile River is another flow that has lasted thousands
of years.
Physical Flows
The current of water down the Nile River is another flow generated by a gravitational force. Let us examine this. Water flows from higher elevations to lower ones, such as via the Nile. Water
in highlands represent a higher gravitational potential than water at sea level. Water flowing downhill consumes (achieves) this potential.
Yet the Nile has been flowing for many thousands of years. How does the water at the high elevations get replenished? Atmospheric storm systems represent complex structures to dissipate potential.
Sunlight places powerful amounts of energy at the surface of the oceans and wet land. Storms form to pump this energy more quickly away from the surface into the cold upper atmosphere. The transport
of water into the atmosphere and its rain on the Earth’s surface increases the rate of energy transport. (Condensing water vapor in the upper atmosphere releases prodigious amounts of energy into
outer space).
Resource and Economic Flows
There are also many physical flows in our economy. The transport of food from farm to city and of mineral from mine to factory represent flows.
Generalizing the Emergence of Structures
We discussed how regimes can emerge from civilizations as dissipative structures to increase entropy production. Here, we generalize the concept of a regime.
Formation of Bubbles
Bubbles emerge when a flow gets blocked. As potential builds up, the force against the blockage increases. Eventually the accumulation and force become so large that the blockage can no longer impede
the flow. At this point, the blockage might be partially overcome, or it might become catastrophically destroyed. This is analogous to the formation and popping of a bubble. Another term for blockage
is “Logjam”.
Emergence of Exponential Structures
In the case of a flow, heat engines will exponentially grow until they reach a limiting efficiency. Heat engine population and entropy production will reach a limit called a carrying capacity.
Thermodynamic Interpretation
Heat engines begetting heat engines results in exponential growth in both quantity of heat engines and entropy production. Where the magnitude of potential is fixed, as entropy is produced, the
potential decreases. As potential decreases the efficiency of the heat engines decreases. This decrease in efficiency comprises a limiting factor.
This decreased efficiency decreases the ability of heat engines to do work. Eventually, the total amount of both work and entropy production will decrease. Less work will be available to beget heat
engines. If the heat engines require work to be maintained, the number of functioning heat engines will decline. Irreplaceable potential entropy continues to decrease as it gets consumed. Eventually,
the potential entropy will be completely consumed, and both work and entropy production will cease.
As this scenario begins, proceeds and ends, a dissipative structure (a literal thermodynamic “bubble”) forms, grows, possibly shrinks and eventually disappears. Entropy production versus time can
often be graphed as a roughly bell-shaped curve, giving a graphic illusion of a rising bubble.
Bubbles Involving Life
Populations of living organisms can experience thermodynamics bubbles. A bacteria colony placed in a media dish full of nutrients faces a potential of fixed magnitude. Each bacterium fills the role
of a heat engine, producing both work and entropy. The bacteria reproduce exponentially, increasing the consumption of potential entropy exponentially. Eventually, it becomes increasingly difficult
for the bacteria to locate nutrients[1], decreasing their efficiency. As efficiency decreases, the bacteria will reproduce at a slower rate and eventually stop functioning.
Ultimate Bubbles
Ultimately, all potentials are fixed in magnitude. Possibly, the entire Big Bang and its progression could be viewed as a bubble. In practice, many potentials are renewable to a limited extent. For
example, as long as the Sun shines upon the Earth in cold space, a potential will exist there.
Series of Bubbles
As long as a system maintains the ability to produce new heat engines, then instead of a single bubble, there will be a series of bubbles over time. There are several reasons that systems form
bubbles instead of maintaining a single flow. Chaos (in the mathematical sense) provides one reason. Another reason is that a series of bubbles may provide for an overall higher entropy production
rate than a more steady, consistent rate of production. Heat engines in a bubble may be able to obtain much higher efficiency during a bubble than during steady state, so that the average production
in a series of bubbles may be much higher than during a steady flow, despite the below average production between bubbles.
Overshoot and the Predator-Prey Cycle
Yet even in the case of a flow, the rate of replenishment will be limited. Yet the rate of engine reproduction may have continued beyond carrying capacity. This can be called overshoot, a systematic
“momentum” in a sense. In this case, even the flow can be treated as a substantially fixed (or “conserved” in the physics sense) quantity. A thermodynamic bubble will form.
Another case such as predator-prey cycles can also form where overshoot occurs, where the population of a predator overshoots the available prey, reducing both the population of the predators and the
prey, so that there are cycles where the population of the predator is always “reacting “ to the population of the prey. Predator-prey cycles can also be expressed in terms of flows, bubbles and
Notes and References
[1]Or escape toxins produced by the colony.
« Exponential functions | COURSE | Efficiency Discounted Exponential Growth (EDEG) Approach to Modeling » | {"url":"https://www.corsbook.com/lesson/flows-and-bubbles/","timestamp":"2024-11-05T14:01:50Z","content_type":"text/html","content_length":"28325","record_id":"<urn:uuid:fc793a36-63d6-4051-93ad-20f88ec43c78>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00806.warc.gz"} |
Top 10 Hard to Believe Math Theorems that Exist
In Physics or Chemistry, the laws made actually need to have some correlation with the physical world to be accepted, whereas regarding math, some mathematicians have dug deep enough to come up with
some weird-looking mathematical theorems and statements, which they were able to prove. This is a list of such hard-to-believe theorems that exist in math.
The Top Ten
1 Euler's Identity [e^iπ + 1 = 0]
The truly remarkable aspect is not the elegant identity itself but the underlying formula:
[e^{ix} = cos(x) + isin(x)]
of which it is an immediate consequence.
2 Fermat's Last Theorem [a^n + b^n = c^n has no solutions for n>2]
The remarkable thing about Fermat's conjecture is that, despite the unlikelihood that he actually proved it, it turned out to be true. Thank goodness for the lack of space in his margin - algebraic
number theory was developed in attempts to prove this claim.
It seems like Fermat was contemplating the Pythagorean theorem, which is taught in middle school, and came up with his own extension to it.
3 L'Hopital's Rule
This one becomes less mystifying once you see the proof. It's a corollary of Cauchy's generalization of the Mean Value Theorem.
The reason it's often misunderstood is that (a) students frequently use it when it doesn't apply (such as when the limit of the quotient of the derivatives does not exist in the real numbers) and (b)
it can be a somewhat circular technique.
For instance, one can use L'Hôpital's Rule to determine that the limit as x approaches 0 of sin(x)/x is 1. But in order to do this, one must know that the derivative of sin(x) is cos(x). And to prove
this, one must evaluate the original limit...
4 Palindromic Polynomial Theorem
This is quite interesting - and I didn't know it.
5 Göbel's Incompleteness Theorems
In mathematics, we perform operations based on a set of basic rules called axioms. A system of axioms is considered better when it is set up in such a way that it is possible to prove every
mathematical statement as "true" or "false," and no statement is left unproven. In this case, the system is said to be complete. It is also preferable to arrange the set of axioms so that no
mathematical statement can be proven as both "true" and "false" by different axioms. In this case, the system is said to be consistent. Gödel successfully proved that no system can be both complete
and consistent at the same time.
6 The Probability Paradox
Remember in probability class when you learned that the probability of a statement lies between 0 and 1, inclusive? Well, a mathematician decided to look at Gödel's theorems in a not-so-binary manner
and proved how nothing in this universe is either completely true or completely false. He then went on to explain how every statement has some element of uncertainty associated with it. When people
started questioning his law, it further increased the element of uncertainty in his law and, in a broader sense, validated it. The interesting fact about this discovery, made by an unknown
mathematician, is that even though it is self-proving and valid, it doesn't significantly change the current mathematical setup. One thing, however, has changed - the probability of an event is never
exactly 0 or 1.
7 Russell's Paradox
When set theory began gaining popularity in the 1900s, many misconceptions arose that could have been disastrous for the field. Russell, however, demonstrated that a set cannot contain anything
subjective, such as the set of all good people, because good people is a subjective term. Consequently, sets that contain sets which include elements not supposed to be in a set render the larger set
invalid. Confusing, huh?
8 Banach-Tarski Theorem
This theorem states that a solid 3D ball in space can be decomposed into a number of disjoint subsets, which can be recombined into two identical solid 3D spheres of the original size each. This is
based on the idea of considering a ball not as a typical solid but as a collection of points in space, acknowledging that there are infinite points between any two points in space. Though this has
been proved using set-theoretic geometry, it seems counterintuitive since it goes against basic notions of geometry.
9 Brouwer's Fixed Point Theorem
This theorem comes from a branch of mathematics known as topology and was discovered by Luitzen Brouwer. While its technical expression is quite abstract, it has many fascinating real-world
implications. Let's say we have a picture (for example, the Mona Lisa) and we take a copy of it. We can then do whatever we want to this copy - make it bigger, make it smaller, rotate it, crumple it
up, anything. Brouwer's Fixed Point Theorem says that if we put this copy on top of our original picture, there has to be at least one point on the copy that is exactly over the same point on the
original. It could be part of Mona's eye, ear, or smile, but it has to exist.
This also works in three dimensions: imagine we have a glass of water, and we take a spoon and stir it up as much as we want. By this theorem, there will be at least one water molecule that is in the
exact same place as it was before we stirred it.
10 Prime Number Theorem
You might have noticed how prime numbers tend to occur randomly in the list of numbers. It seems nearly impossible to find a pattern in which prime numbers occur. Yet, there is a theorem for that. It
formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and
Charles Jean de la Vallée Poussin in 1896, using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function).
The first such distribution found is π(N) ~ N / log(N), where π(N) is the prime-counting function and log(N) is the natural logarithm of N. This means that for large enough N, the probability that a
random integer not greater than N is prime is very close to 1 / log(N).
The Contenders
11 Universal Chord Theorem
It's quite fascinating to consider why this theorem only holds true for certain sets of numbers.
12 Green's Theorem
13 Pythagorean Theorem
14 Cauchy-Schwarz Inequality
15 Chinese Remainder Theorem
BAdd New Item | {"url":"https://www.thetoptens.com/education/hard-believe-math-theorems-exist/","timestamp":"2024-11-12T13:12:13Z","content_type":"text/html","content_length":"48650","record_id":"<urn:uuid:d6370669-ef4d-40d3-b7ea-325a5529d66c>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00486.warc.gz"} |
How is Student t calculated?
How is Student t calculated?
Step 1: Subtract each Y score from each X score. Step 2: Add up all of the values from Step 1 then set this number aside for a moment. Step 3: Square the differences from Step 1. Step 4: Add up all
of the squared differences from Step 3.
What is the formula for calculating t-value?
T = (Z x 10) + 50. Example question: A candidate for a job takes a written test where the average score is 1026 and the standard deviation is 209. The candidate scores 1100. Calculate the t score for
this candidate.
How do you do a Student t-test in Excel?
Click on the “Data” menu, and then choose the “Data Analysis” tab. You will now see a window listing the various statistical tests that Excel can perform. Scroll down to find the t-test option and
click “OK”.
What is the formula for the paired samples t-test?
Paired T-Test Formula The formula of the paired t-test is defined as the sum of the differences of each pair divided by the square root of n times the sum of the differences squared minus the sum of
the squared differences, overall n-1.
How do you find the T table?
To use the t-distribution table, you only need to know three values:
1. The degrees of freedom of the t-test.
2. The number of tails of the t-test (one-tailed or two-tailed)
3. The alpha level of the t-test (common choices are 0.01, 0.05, and 0.10)
How do I do a two sample t test in Excel?
In Excel, click Data Analysis on the Data tab. From the Data Analysis popup, choose t-Test: Two-Sample Assuming Equal Variances. Under Input, select the ranges for both Variable 1 and Variable 2. In
Hypothesized Mean Difference, you’ll typically enter zero.
How do you find p-value from t-test in Excel?
P-Value Excel T-Test Example #1
1. First thing we need to do is calculate the difference between before diet and after diet.
2. Now go to the Data tab, and under the data, tab click on Data Analysis.
3. Now scroll down and find T.
4. Now select Variable 1 Range as before diet column.
5. Variable 2 rang as after a diet column.
What is a formula test?
The formula for a one-sample t-test is expressed using the observed sample mean, the theoretical population means, sample standard deviation, and sample size. Mathematically, it is represented as, t
= ( x̄ – μ) / (s / √n)
What is the formula for the single sample t statistic?
T = (X̄ – μ) / S/√n Where, X̄ is the sample mean, μ is the hypothesized population mean, S is the standard deviation of the sample and n is the number of observations in the sample.
How do you solve time in math?
To solve for time, divide the distance traveled by the rate. For example, if Cole drives his car 45 km per hour and travels a total of 225 km, then he traveled for 225/45 = 5 hours.
How do you calculate Student t test formula?
Student t Test Formula, t = x 1 ¯ − x 2 ¯ (s 2 (1 n 1 + 1 n 2)) In the formula given above, t is equal to the t-value, x1 and x2 are the means of the two groups being compared, s2is the pooled
standard error of the two groups, and n1 and n2 are the numbers of observations in each of the groups.
What is the formula for a one sample t-test?
The formula for a one-sample t-test is expressed using the observed sample mean, the theoretical population means, sample standard deviation, and sample size. Mathematically, it is represented as, t
= (x̄ – μ) / (s / √n)
What is Student’s t test?
Statistics – Student T Test. T-test is small sample test. It was developed by William Gosset in 1908. He published this test under the pen name of “Student”. Therefore, it is known as Student’s
What are the assumptions of student’s original t test?
Assumptions. If the sample sizes in the two groups being compared are equal, Student’s original t -test is highly robust to the presence of unequal variances. Welch’s t-test is insensitive to
equality of the variances regardless of whether the sample sizes are similar. | {"url":"https://pfeiffertheface.com/how-is-student-t-calculated/","timestamp":"2024-11-02T09:32:32Z","content_type":"text/html","content_length":"44424","record_id":"<urn:uuid:2857fc87-62c2-45e3-a753-9513fbbc90cc>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00345.warc.gz"} |
Virtual Manipulatives - Math-U-See
Virtual Manipulatives
Math-U-See relies on a distinctive set of manipulatives. We have carefully designed digital versions of our physical blocks, fraction overlays, and algebra/decimal inserts. Use these virtual
manipulatives to work through a full range of mathematical concepts including numbers and counting, operations, fractions, decimals, integers, and algebraic expressions (levels Primer through Algebra
1 ).
Key Features
Suitable for students learning math at preschool through Algebra 1 levels.
All virtual manipulatives match their physical counterparts in color and shape for consistency and ease of use.
Includes notepad function for writing step-by-step solutions while working through problems.
Use to illustrate operations, fractions, decimals, algebraic expressions, and more.
Choice of backgrounds including graphing, number lines, Decimal Street, and more.
You can access the digital manipulatives while at home on the Digital Toolbox and on the go with the purchase of the Math-U-See Manipulatives app.
The Digital Manipulatives app is designed only for tablet use. Please check compatibility carefully before purchasing. | {"url":"https://mathusee.com/digital-tools/virtual-manipulatives/","timestamp":"2024-11-07T01:17:08Z","content_type":"text/html","content_length":"78779","record_id":"<urn:uuid:831a5152-ba1f-4d3a-8979-5aa15315d83d>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00102.warc.gz"} |
Smart Contracts - Crypto Research Report
“A smart contract is a computer program that directly controls some kind of digital asset… The smart contract approach says instead of a legal contract, immediately transfer the digital asset into a
program, the program automatically will run code, validate a condition, and determine whether the asset should go to one person or back to the other person, or whether it should be immediately
refunded to the person who sent it or some combination thereof.”
Vitalik Buterin, Founder of Ethereum
Smart contracts are dynamic, complex, and incredibly powerful. This technology has the potential to change how business is done because governments and companies can decrease costs, automate contract
enforcement, and provide an auditable trail of control. While their long-term potential is unimaginable, they are already disrupting industries such as crowdfunding, law, and insurance. However,
smart contracts are made by humans, and therefore, they are not perfect. Over $2.4 billion USD have already been lost due to faulty smart contracts. The future success of utility tokens depends on
the ability of developers to build more secure applications that users can trust.
Smart Contracts, Decentralized Applications, and Decentralized Autonomous Organizations
With the rise of cryptocurrencies and blockchains, smart contracts, Decentralized Applications (DApps), and Decentralized Autonomous Organizations (DAOs) are becoming increasingly important
technologies that investors should understand. This all began when Nick Szabo coined the term “smart contract” in 1994. To define the term, a smart contract is a piece of software that represents a
set of rules that are automatically executed under pre-determined circumstances. In other words, a smart contract is an “if-then” statement that is executed on a distributed peer-to-peer network.
Since each contract is stored on several computers all around the world, smart contracts do not have a single point of failure. Similar to BitTorrent, not having a single point of failure means that
if one computer fails, the whole network does not fail. Nobody has the power to change the smart contract – even the person who made the smart contract. That applies to hackers and governments too!
Smart contracts have the same privileges as cryptocurrency wallets, except they are not controlled by private keys or users. Instead, a smart contract is controlled by the code contained within the
smart contract. They can send and receive cryptocurrency, and they can store a balance or have a balance of zero. They can interact with other smart contracts, but users have to pay a small fee to
the blockchain network to interact with the contract because every change needs to be approved and recorded by every computer that maintains the network. This is similar to mining in Bitcoin.
A DApp is a collection of many smart contracts that are working together to create a product for users. To be a DApp, four criteria must be met. First, the application must be open-source, which
means that anyone can download it and look at the underlying code. Second, the application must run on a blockchain or distributed ledger technology, such as a directed acyclic graph. Third, the
application must have a token associated with it. This token can be native to the application, such as Augur, or the application can use another cryptocurrency, such as Bitcoin or Ethereum. Finally,
the application must use a cryptographic algorithm for confirming changes to the smart contracts that control the DApp.
A DAO is a type of DApp that allows owners to make business decisions by voting electronically and to automate management using smart contracts. DAOs use smart contracts to facilitate digital voting,
and to facilitate the voting outcomes. The goal of a DAO is to reduce managerial overhead and to circumvent regulations particular to geographic regions.
Essential, a DAO is a company structure for a globally dispersed group of owners. The cryptography and blockchain enables a group of strangers to invest capital together and make financial decisions.
Smart contracts enable the company to be run autonomously because they hold the rules of the company and serve as a basis for operation decisions instead of a human. For example, a smart contract
could be programmed to distribute dividends to investors once a certain condition is met. If profits are above a certain threshold, the smart contract could automatically send a transaction to
shareholder wallets.
Smart Contract Theory Applied In Practice
The most popular cryptocurrency that incorporates smart contracts is Ethereum. The founder of Ethereum, Vitalik Buterin, proposed Ethereum in 2013 as a blockchain specifically designed for smart
contracts. Ethereum is a worldwide network of computers which enforce, execute, and validate smart contracts. As covered in the previous Crypto Research Report, validation is achieved by a
decentralized network of thousands of Ethereum nodes around the world. The centralized nature of the network enables decentralized applications (DApps) to run without downtime, censorship, or
third-party interference, which makes the applications immutable or tamper-proof.
In practice, Ethereum can be used to create a decentralized crowdfunding website without intermediaries, such as Kickstarter. Ethereum users can invest in a business idea by sending money to an
Ethereum wallet address of a smart contract. If the entrepreneur is unable to raise a certain amount of funding within a certain timeframe, the smart contract can automatically send all of the
investors their money back. Instead of paying 10 % to Kickstarter, they only have to pay a 5-cent fee to the Ethereum network to process the transaction to the smart contract. Middleman removed. If
company raises $10 million with Ethereum instead of Kickstarter, they save $1 million in fees.
The decentralized nature of Ethereum also allows investors to circumvent regulations that limit how much investors can invest in crowdfunding projects. For example, in the US, non-accredited
investors with an annual income or net worth less than $107,000, are limited to invest a maximum of 5 % of their assets. For those with an annual income or net worth greater than $107,000, he/she is
limited to investing 10 % of the lesser of the two amounts. These rules do not apply to Ethereum and other smart contract platforms such as NEO and EOS.
Blockchains that facilitate smart contracts are often referred to as utility blockchains, and the assets that are associated with them are referred to as utility crypto assets or utility tokens.
Table 1 compares the relative return of utility crypto assets to payment crypto assets such as Bitcoin, Monero, and Litecoin. Although a portfolio comprised entirely of payment crypto assets had a
higher cumulative return of 227 %, the portfolios are highly correlated with a 0.81 correlation coefficient. Cryptocurrency investors that bought the top five payment cryptocurrencies in June and
sold in January realized a six-month profit of over 1,200 %.
Crypto Asset Companies That Use Smart Contracts, DApps, and DAOs.
There are already many projects that seek to implement smart contracts via blockchains into the real world. Ethereum is one of the most prominent examples, but a variety of other companies are also
harnessing the power of automatic digital contracts, including Cosmos, Dfinity, Etherisc, Polkadot, and ShareRing.
A controversial example is the decentralized prediction market named Augur. In late July, CNN reported on Augur users gambling on whether President Trump would be assassinated by the end of this
year. Since Augur is a DApp, no one can stop users from gambling on the probability of criminal and unethical behavior. Even the creators of Augur are unable to stop Augur from existing because the
project is open-source. Not only are the legality of prediction markets questionable, now lives of people will be directly attached to financial gains. If this prediction market does lead to heinous
acts be committed, Augur’s price will most likely be volatile and downward.
Although Augur is enabling people to vote on the probability that Trump, John McCain, or Warren Buffett will survive past the end of this year, most of Augur’s prediction markets are based on sports.
For example, during the FIFA World Cup, users could vote on the outcome of specific games and earn Ethereum if they voted correctly.
When Smart Contracts Are Dumb
Since 2011, a combined $2.4 billion has been destroyed, frozen, stolen or otherwise compromised in the crypto asset space due to attacks. The two biggest crypto assets in terms of market capitalized
have seen losses of around 1.7 million BTC and 4.54 million ETH over the past seven years. The three biggest mistakes that have occurred in the Ethereum space include the DAO hack, the Parity wallet
hack, and the Parity wallet suicides.
DAO Hack
The Decentralized Autonomous Organization (DAO) was one of crypto’s most highly anticipated projects of all time and a pioneer in the application of the revolutionary capabilities of smart contracts.
Some of Ethereum’s developers created a spin-off company called Slock.it and created the first DAO using the Ethereum blockchain in April 2016.[1] The DAO application worked like an investment fund,
although without the usual investment fund management. Investors could participate by transferring the Ethereum cryptocurrency, ether, to the fund, which entitled them to voting rights. The
investment decisions were supposed to be taken through a joint effort, where every participant could vote on investment proposals. Anyone with a venture project could pitch their idea to the DAO
community in hopes of potentially receiving funding from a pool of ether which was controlled by the DAO.[2] Once a project was chosen, token holders would receive rewards – much like dividends or
interest payments – if the projects turned out to be profitable.
The DAO was launched as a smart contract on the Ethereum network in May 2016. At the time, it raised $162 million worth of ether, making it the biggest crowdfund ever.
Nevertheless, on June 17^th, 2016, a hacker perpetrated the DAO network by exploiting a loophole in its software, allowing him to drain funds from the pool of Ethereum tokens owned by the DAO
network. 3.6 million ETH tokens were stolen in the first couple of hours of the attack, amounting to an equivalent value of $70 million at the time ($1.2 billion in today’s terms). Strangely enough,
the hacker stopped draining the DAO for unknown reasons, even though he could have continued to do so.
The team and community behind Ethereum were quick in noticing the breach and very soon responded to the situation by presenting multiple proposals on how to deal with the attack. Due to the
architecture of the DAO, the drained funds in the form of ether were locked up in a child DAO, another smart contract, which required a 28-day holding period before the attacker could fully withdraw
the funds and launder them into circulation. This gave the Ethereum team sufficient time to decide on their course of action.
Vitalik Buterin, the creator of Ethereum, realized the severity of this breach and issued a statement soon afterwards assuring all investor that their funds were safe for the moment. The Ethereum
community then decided to render any transaction originating from the attackers account with code hash:
as invalid, thereby preventing the attacker from withdrawing funds even after the completion of the 28-day holding period.
The hacker later published an open letter to the Ethereum community claiming rightful ownership of the acquired funds. You can see the original letter here. Refunding the investors’ ether would have
not been possible under the rules of the Ethereum network at that time, posing an existential threat to the network as a whole.
The solution that the Ethereum foundation came up with was highly controversial.[3] They implemented a “hard fork” by releasing a new version of the Ethereum software client that did not include the
hacked transactions. They re-winded the Ethereum blockchain in order to remove the hacker’s transactions. The release also included a fix to the bug that the hacker exploited. Not all of the members
of the network agreed with the decision of hard forking the chain. Some dissidents left Ethereum entirely and others just continued to use the original Ethereum blockchain, which included the
hacker’s transaction. This chain became called Ethereum Classic with ticker symbol ETC.
As with all of the mishappenings described in this article, none of them has arisen due to the fundamental design of the Ethereum network. Instead, the hacks occurred because of design problems in
applications that were built on top of Ethereum. The type of attack that destroyed the DAO is known as a reentrancy attack.[4] In this attack, the attacker first donated ether to the smart contract
(DAO) and then was able to “ask” for the ether back multiple times before the smart contract could update its balance. When the contract fails to update its state (a user’s balance) prior to sending
funds, the attacker can continuously call the withdraw function to drain that contract’s funds.[5] The code written for the DAO had multiple flaws including the recursive call exploit as well as the
fact that the smart contracts sent ETH funds before updating the internal token balance.
Parity Wallet Hack
Parity Technologies builds platforms and applications, and it powers large parts of the infrastructure of the public Ethereum network.[6] On the July 19^th, 2017, an unknown hacker attacked a
critical vulnerability in the Parity multisignature wallet on the Ethereum network, looting three massive wallets containing a combined $31 million worth of ETH in a matter of minutes.[7] A group of
heroic white-hat hackers from the Ethereum community responded by quickly alerting Ethereum users on social media and hacking the remaining wallets before the attacker could. This form of hacking is
called white-hat hacking because they hacked for the good cause. If the white hackers had not responded so quickly, the hacker could have hacked over $180,000,000 worth of Ethereum from vulnerable
wallets. Of course, the funds that were stolen by the white-hats were securely redistributed to their respective account holders in the end.
The hacker found a programmer-induced bug in the code that let him re-initialize the Parity multisignature wallet, almost like restoring your iPhone to factory settings. Once having done that, he was
free to set himself as the new owner and walk out with everything.
Due to the programming model of Ethereum, there is an incentive for programmers to optimize code in order to minimize transaction costs. Every time code is executed on Ethereum, a smart contract,
which constitutes a transaction on the network and thus comes with a computation fee, needs to be deployed. An efficient way to reduce costs from the computation fees is to use shared libraries which
have already been deployed to the network.
The default settings for the multisignature wallet in Parity had a configuration which did exactly that. It referenced a shared external library, which contained a wallet initialization logic, namely
initWallet(), which if called could reinitialize the contract the wallet was built upon. It effectively made whoever exploited this flaw the new owner of the wallet. From there, the hacker could
simply transfer the funds to any address of his or her choice.
However, why did they not just roll back this hack, like they did with the DAO hack? Unfortunately, that was not even an option any more. When the attacker drained the DAO into a child DAO the hacked
funds were frozen for a 28-day period before they could be released to the attacker. This prevented any of the stolen funds from going into circulation, which in turn gave the Ethereum community
plenty of time to consult the community about how to deal with the attack. In the Parity wallet attack, however, the attacker directly withdrew the funds and could start spending them. Once the
stolen ETH was in circulation, it was almost impossible to recover them, much like with a huge sum of counterfeit bills circulating in the economy.
Parity Wallet Suicides
On November 6^th, 2017, the Parity multisignature wallet fell prey to yet another attack,[8] only this time to one of a much more severe magnitude. A Github user named devops199 wrote a post titled
“anyone can kill your contract”. Devops199’s post seemed to have good intentions.[9] He wanted to make the Parity team aware of a vulnerability in the smart contract that powered its multisignature
wallets. The security vulnerability allowed any hacker to make him- or herself the “owner” of that contract, thus giving him or her permission to do with the wallet as he or she pleases. Up to this
point in time, it still remains unclear what his original intent was, but his following actions were largely met with skepticism: He “accidentally” triggered the “kill switch” of the contract,
rendering all Parity multisignature wallets impossible to access. Just minutes after the library was wiped out, devops199 raised another issue following up his “anyone can kill your contract” post on
Parity’s Github titled “I accidentally killed it.”
The scale of this Parity hack compromised a total of 514 k Ethereum tokens, which
at the time were valued roughly around $155 million or 1 % of Ethereum’s total
valuation. As a consequence of the hack, no funds can ever be moved out of all
Parity multisignature wallets that were deployed after July 20th, 2017 again.
A hard-fork could technically “bail-out” the multisignature wallets that are frozen
but that would compromise the foundation of Ethereum’s decentralized,
distributed, immutable, and tamper-proof ledger. Calls for a hard-fork have largely
remained ineffective.
Smart Contracts Represent a Distinctly Different Exposure for Investors Than Bitcoin
Smart contracts have the potential to disrupt industries by cutting out middlemen and bringing together the parties of the contracts directly. They can ensure trust and cut transaction costs. Despite
the benefits, The Economist criticizes the concept of smart contracts by pointing out the immature state of the industry.[10] They further argue that smart contracts are not flexible enough for an
economy that needs to respond to ever changing conditions.
At Incrementum, we think that smart contracts are here to stay. From a financial perspective, utility blockchains, such as Ethereum, NEO, and EOS, offer a distinctly different risk-return profile
compared to payment blockchains, such as Bitcoin, Dash, and Monero. Due to the complex nature of smart contracts, utility blockchains are inherently more prone to technology risk. In contrast, since
utility tokens do not threaten to disrupt the current monetary system, risk of being outlawed for competing with sovereign currencies is lower when compared to public payment blockchains. However,
many regulators are saying that initial coin offerings and utility tokens are violating security law. Diversifying into both crypto asset classes is prudent for investors that are willing to make the
extra effort.
[1] See “Decentralized Autonomous Organization to Automate Governance. Final Draft – Under Review” [white paper], Christoph Jentzsch, 2016.
[2] See “The Story of the DAO–Its History and Consequences,” Samuel Falkon, The Startup, December 24, 2017.
[3] See “To fork or not to fork,” Jeffrey Wilcke, Etherum Blog, July 15, 2016.
[4] See “Smart Contract Attacks [Part 1] – 3 Attacks We Should All Learn From The DAO,” Pete Humiston, Hackernoon, July 5, 2018.
[5] You can find a more technical walkthrough here.
[6] Learn more about Parity here
[7] See “A hacker stole $31M of Ether – how it happened and what it means for Ethereum,” Haseeb Qureshi, Medium, July 20, 2017.
[8] See “Security Alert,” Parity, November 8, 2017.
[9] See “Yes, this kid really just deleted $300 MILLION by messing around with Ethereum’s smart contracts,” Thijs Maas, Hackernoon, November 8, 2017.
[10] See “Not-so-clever contracts,” Schumpeter, The Economist, July 28, 2016. | {"url":"https://cryptoresearch.report/crypto-research/smart-contracts/","timestamp":"2024-11-03T22:10:47Z","content_type":"text/html","content_length":"226573","record_id":"<urn:uuid:1d40bf12-f287-45dd-b65e-f509a42e49fd>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00399.warc.gz"} |
multipol: Multivariate Polynomials version 1.0-9 from CRAN
Various utilities to manipulate multivariate polynomials. The package is almost completely superceded by the 'spray' and 'mvp' packages, which are much more efficient.
Package details
Author Robin K. S. Hankin [aut, cre] (<https://orcid.org/0000-0001-5982-0415>)
Maintainer Robin K. S. Hankin <hankin.robin@gmail.com>
License GPL
Version 1.0-9
Package repository View on CRAN
Install the latest version of this package by entering the following in R:
Any scripts or data that you put into this service are public.
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/cran/multipol/","timestamp":"2024-11-07T13:22:14Z","content_type":"text/html","content_length":"21707","record_id":"<urn:uuid:6a3d59e6-31b3-4a3c-9cdb-043a24ba9a9c>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00145.warc.gz"} |
The locus of the foot of perpendicular from the centre of the h... | Filo
Question asked by Filo student
The locus of the foot of perpendicular from the centre of the hyperbola xy=c^2 on a variable tangent is
Not the question you're searching for?
+ Ask your question
Video solutions (1)
Learn from their 1-to-1 discussion with Filo tutors.
21 mins
Uploaded on: 1/30/2023
Was this solution helpful?
Found 6 tutors discussing this question
Discuss this question LIVE for FREE
5 mins ago
One destination to cover all your homework and assignment needs
Learn Practice Revision Succeed
Instant 1:1 help, 24x7
60, 000+ Expert tutors
Textbook solutions
Big idea maths, McGraw-Hill Education etc
Essay review
Get expert feedback on your essay
Schedule classes
High dosage tutoring from Dedicated 3 experts
Practice more questions on Coordinate Geometry
View more
Students who ask this question also asked
View more
Stuck on the question or explanation?
Connect with our Mathematics tutors online and get step by step solution of this question.
231 students are taking LIVE classes
Question Text The locus of the foot of perpendicular from the centre of the hyperbola xy=c^2 on a variable tangent is
Updated On Jan 30, 2023
Topic Coordinate Geometry
Subject Mathematics
Class Class 11
Answer Type Video solution: 1
Upvotes 112
Avg. Video Duration 21 min | {"url":"https://askfilo.com/user-question-answers-mathematics/the-locus-of-the-foot-of-perpendicular-from-the-centre-of-34303234343930","timestamp":"2024-11-09T01:23:00Z","content_type":"text/html","content_length":"287994","record_id":"<urn:uuid:efd781e5-302d-4ca0-b46f-0ac27f94f45e>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00323.warc.gz"} |
Publications about 'Behavioral Systems Theory'
Publications about 'Behavioral Systems Theory'
Articles in journal, book chapters
This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All person copying
this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the
copyright holder.
Most articles are also available via my Google Scholar profile.
Last modified: Sun Nov 10 07:25:34 2024
Author: Florian Dorfler.
This document was translated from BibT[E]X by bibtex2html | {"url":"https://people.ee.ethz.ch/~floriand/bib/Keyword/BEHAVIORAL-SYSTEMS-THEORY.html","timestamp":"2024-11-13T16:00:18Z","content_type":"text/html","content_length":"36208","record_id":"<urn:uuid:b21a2de6-3922-4aaa-b14a-65f31a14c2bb>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00855.warc.gz"} |
Place Value | Numbers in Base Ten | St. Patrick's Day Themed WITH AUDIO!
Place Value | Numbers in Base Ten | St. Patrick's Day Themed WITH AUDIO!
Price: 350 points or $3.5 USD
Subjects: math,specialed
Grades: 1,2,3
Description: Your students will love this Number & Operations in Base Ten deck! Students will read and recognize numbers to 100 using base-ten blocks and numerals. Students will match the two digits
of a two-digit number represent amounts of tens and ones to the number numeral. Enjoy! Michelle Allison CCSS Correlations: Number & Operations in Base Ten Extend the counting sequence.
CCSS.MATH.CONTENT.1.NBT.A.1 Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral. Understand place
value. CCSS.MATH.CONTENT.1.NBT.B.2 Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases: CCSS.MATH.CONTENT.1.NBT.B.2.A 10
can be thought of as a bundle of ten ones — called a "ten." CCSS.MATH.CONTENT.1.NBT.B.2.B The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine
ones. CCSS.MATH.CONTENT.1.NBT.B.2.C The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones). CCSS.MATH.CONTENT.2.NBT.A.3 Read
and write numbers to 1000 using base-ten numerals, number names, and expanded form. | {"url":"https://wow.boomlearning.com/deck/mKMuCkL3vg3JkNPC2","timestamp":"2024-11-05T03:32:38Z","content_type":"text/html","content_length":"3161","record_id":"<urn:uuid:762a47dd-6eea-4163-bba3-347c14675dcb>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00705.warc.gz"} |
Why Do We Live in Three Dimensions?
Day to day life has made us all comfortable with 3 dimensions; we constantly interact with objects that have height, width, and depth. But why our universe has three spatial dimensions has been a
problem for physicists, especially since the 3-dimensional universe isn’t easily explained within superstring theory or Big Bang cosmology. Recently, three researchers have come up with an
The history of the universe starting the with the Big Bang. Image credit: grandunificationtheory.com
Most astronomers subscribe to Big Bang cosmology, the model that proposes that the universe was born from the explosion of an infinitely tiny point. The theory is supported by observations of the
cosmic microwave background and the abundance of certain naturally occurring elements. But Big Bang cosmology is at odds with Einstein’s theory of general relativity – general relativity doesn’t
allow for any situation in which the whole universe is one tiny point, which means this theory alone can’t explain the origin of the universe.
The incompatibility between general relativity and Big Bang cosmology has stumped cosmologists. But almost 40 years ago, superstring theory arose as a possible unifying theory of everything.
A visualization of strings. Image credit: R. Dijkgraaf.
Superstring theory suggests that the four fundamental interactions among elementary particles – electromagnetic force, weak interaction, strong interaction, and gravity – are represented as various
oscillation modes of very tiny strings. Because gravity is one of the fundamental forces, superstring theory includes an explanation of general relativity. The problem is, superstring theory predicts
that there are 10 dimensions – 9 spatial and one temporal. How does this work with our 3 dimensional universe?
Superstring theory has remained little more than a theory for years. Investigations have been restricted to discussing models and scenarios since performing the actual calculations have been
incredibly difficult. As such, superstring theory’s validity and usefulness have remained unclear.
But a group of three researchers, associate professor at KEK Jun Nishimura, associate professor at Shizuoka University Asato Tsuchiya, and project researcher at Osaka University Sang-Woo Kim, has
succeeded in generating a model of the universe’s birth based on superstring theory.
Using a supercomputer, they found that at the moment of the Big Bang, the universe had 10 dimensions – 9 spatial and 1 temporal – but only 3 of these spatial dimensions expanded.
This "baby picture" of the universe shows tiny variations in the microwave background radiation temperature. Hot spots show as red, cold spots as dark blue.Credit: NASA/WMAP Science Team
The team developed a method for calculating matrices that represent the interactions of strings. They used these matrices to calculate how 9 dimensional space changes over time. As they moved further
back in time, they found that space is extended in 9 directions, but at one point only 3 directions start to expand rapidly.
In short, the 3 dimensional space that we live in can result from the 9 original spatial dimensions string theory predicts.
This result is only part of the solution to the space-time dimensionality puzzle, but it strongly supports the validity of superstring theory. It’s possible, though, that this new method of analyzing
superstring theory with supercomputers will lead to its application towards solving other cosmological questions.
Source: The mechanism that explains why our universe was born with 3 dimensions.
33 Replies to “Why Do We Live in Three Dimensions?”
1. So would that mean that at the point of origin of the universe one would find the other 6 dimensions?
1. The 9 spatial dimensions become split into 3 space dimensions we observe plus 6 dimensions which are compactified into Calabi-Yau spaces. It would be as if a line in three space were really a
“thickened” rope with an additional dimension corresponding to the radius of it. This is extended to even higher dimensions. The candidate I work on for this 6 dimensional space are a class
of manifolds called K3 spaces (Kummer, Kahler and Kodaira). The internal motions in this Calabi-Yau space is what gives rise to the other forces in the universe.
This compactification is removed, or this wadded up space unfolds like some origami, when gravity unifies with the other forces. This occurs at the most early domain in the existence of the
1. I recall a previous comment discussion where the lack of granularity effects on light observed from distant sources seemed to be a problem for extra dimensions?
2. The lack of dispersion of light from distant sources indicates there are no influences from quantum fluctuations of light, or a discrete grid up of spacetime which violates Lorentz
symmetry. These observations do put constraints upon physical theories. In particular it means that loop variable quantum gravity is either wrong or must be seriously modified. String
theory is more flexible and its quantum of gravity is a graviton, which comes from the closed string. The graviton is something which has holographic content. These data indicate that
quantum fluctuations of gravity can’t be something which involves stochastic ripples of space or spacetime in the way previously thought. These fluctuations must exist in some holographic
context, such as in the fluctuation of event horizons or the graviton content in an anti-de Sitter spacetime which our spacetime is a boundary of.
Gravity is an odd ball of the forces in nature, for it is not really a force in the standard sense. Classical gravitation as given by general relativity does not describe motion according
to acceleration, but rather as a system of local inertial frames. The recent development by Maldacena indicates that the boundary of an anti-de Sitter (AdS) spacetime is equivalent to a
conformal field theory (CFT). The graviton is then something within the AdS which is determined by the holographic projection of the CFT on the surface. The symmetries of this CFT is
contained in these compactified spaces. So quantum fluctuations of the CFT on the AdS boundary determine “gravity” in the AdS interior. Further, local gravity we observe (living on the
AdS boundary) is some symmetry breaking of the CFT (the occurrence of masses etc) which involves spacetime fluctuations of the event horizon, but maybe not the spacetime itself.
3. Not discerning an obvious yes or no from all that 🙂
4. The answer, if there is an answer in this business, is that this dispersion is not a necessary aspect of all these theories. This data constrains theories, where a good number of them are
5. I guess it’s way to early for an obvious yes or no. A definite maybe — that may be possible. 😉
2. In other words everywhere in the Universe.
Superstring theory has remained little more than a theory for years.
But only in the lay-man use of the term “theory”. String theory is not a valid theory in the scientific sense. Until now it is still and only a mathematical toy (as far as I know) and has not
produced any measurable results making possible its falsification.
I think that even this described result cannot be “observed”. But if I’m wrong, please correct me. 😉
String theory is not a valid theory in the scientific sense. Until now it is still and only a mathematical toy (as far as I know) and has not produced any measurable results making
possible its falsification.
That is what is claimed by its detractors, but it is trivially false.
Already its conception was promoted by its ability to predict the “flux tubes” that pops up between particles in models of the strong force. So it is a predictive theory and have produced
measurable, even testable observations which it has tested successfully on. This was in ’73, I believe.
String theory also predicts black hole entropy correctly. And it is predictive in the Planck regime, predicting objects of strings and branes, which has been started to be probed recently.
(By supernova photon timing and polarization observations.)
The problem is that string theory was king for a year. The year after the successful prediction of flux tubes QCD came along and predicted it by far simpler and more natural means. And as for
the black hole results it wasn’t even first, simpler semiclassical approximations allowed for it outright from known physics.
The Planck regime results have not been accepted by many or most, and have little bearing on string theory as of yet. Though those results recurring insistence on an absence of small granular
string manifold dimensions seems to this layman to be a pressing problem.
String theory may be fine physics, but it doesn’t look to me at the moment that it will be the Ultimate Answer to the Ultimate Question of Life, The Universe, and Everything“.
2. Let us be honest about this. Is string theory somehow the final and complete theory of the universe? No, or at least I would be surprised if it were. However, I think there are stringy
aspects to the universe, where some signatures of this have been found. A part of the problem is that the mathematical foundations of this are so difficult that we are forced to make lots of
simplifying assumptions. Newton when he was working on his mechanics worked out circular orbits, which are approximations that simplify things considerably and permit you to look at central
issues. We have similar issues today, where the algebraic geometry of string theory is a part of something almost unimaginably difficult — maybe intractable.
There is this competing theory called loop quantum gravity. It is in many ways a very direct approach to quantum gravity that has Einstein’s general theory of relativity as its kernel. Yet
quantum versions of spacetime connections violate Lorentz symmetry near the Planck scale, and they have action or a unit of spin which makes it difficult to construct classical gravity or
compute the entropy of a black hole. Yet I don’t think loop variable theory is completely wrong.
String theory most naturally describes inflationary cosmology, which with anisotropy measurements appears to be realistic or has a modicum of empirical support. String theory did emerge from
hadron physics, and the quark-gluon system of QCD is a stringy physics. The stringy structure of the universe appears in a low energy structure. Physical structures do seem to emerge in
different guises, such as supersymmetry has been found as an emergent symmetry in low energy nuclear physics. The AdS~CFT correspondence appears in the context of superconductivity with heavy
1. If we should be completely honest, I think the LQG is considered completely wrong by the physicist side of the theoretical physics divide, while mathematicians may yearn for it.
The reasons, as I understand it, is:
– LQG breaks Lorentz invariance, as you noted on this symmetry.
– LQG has no energy gap setting a lowest energy resulting in an action that follows an action principle, as you noted on the action.
– Related to the latter point, LQG gives no dynamics! You can’t even construct an idealized harmonic oscillator based on its structure, so it lacks the fundamental degrees of freedom
which makes up the physics we see.
Those seemingly unsurmountable problems are disregarded by the LQG community, which persists in putting out arxiv papers promising all these things and never delivering. (Yes, I am bored
by reading all so called “proofs” for these claims. They never are, unless you squint and put in enough fairies.)
2. The failures of loop variable theory are what make it beautiful! That might sound odd, but really, it is the minimal route to quantum gravity. It makes far fewer of the conjectures seen
in string theory, which is really closer to elementary particle theory, and sticks close to Einstein’s theory of relativity. Yet the theory is plagued by a host of problems which make it
almost unworkable. This gives us the most beautiful there is in science — why? Questions are always more interesting than old established answers.
Loop variables are based on the ADM approach to classical gravity. This has no time, and defines constraints NH = 0 and N^iH_i = 0. The Hamiltonian is g^{-1/2}(TrK^2 – (TrK)^2 – R^{(3)},
K_{ij} = extrinsic curvature of spatial manifold, which just defines a spatial manifold in a system of such manifolds under a diffeomorphism described by a lapse function. This is
quantized under canonical methods which leads to LQG. The theory has no time variable, and Barbour makes a huge point of this in his proclamation “Time is an illusion.” This is why there
is no dynamics.
There is an alternative theory outlined by Banos, Teitelboim and Zanelli (BTZ) which has 2 space and 1 time dimension. Now suppose there is a noncommutative geometry which tells us that
?r?t ~ G?/c^3,
there is then a noncommutative quantization between the two representations.
The additional dimension, which is time in the ADM and space in the BTZ, emerges by the holographic principle. Another way to think of this is that the spinor group for the ADM is SU(2) ~
SO(3), which are just the rotations in space. In BTZ this group is SU(1,1), which is a group of Lorentz boosts. The complementarity gives a complete group SU(2)xSU(1,1) ~ SO(3,1) ~ SL
3. Sir, as a matter of fact, it is string theory, which might actually lead to the grand unification theory, and when that is known, we may know, as Einstein said, “The mind of God”. String
theory is mathematically perfect in the sense that it doesn’t violate any known physical laws. Also, it accurately explains the quantum phenomenon of the minutest particles, as well as
predicting them. but the real trouble is that, they haven’t been experimentally verified, as the energy needed is of the planck scale, which we can’t create at the present. if it happens,
then we can know what happened during inflation, that is, the Big bang. M-theory can then be verified. Till then we have to wait..
4. Knots can only exist in 3-dimensions. They are essential to life, at least as we know it (proteins). So, we can only exist in a 3-dimensional universe and, I suspect, can only observe the
universe we exist in.
The KEK mathematicians have only shown that a string model is consistent with one aspect of our 3-D universe. If we can’t observe anything else, it’s math not science.
1. Knots exist for very subtle reasons. The standard knot is a one dimensional curve which threads through three dimensions. It is more technically due to something called a Skein relationship
with a polynomial expression. This is the Jones polynomial, which has been generalized into something called the HOMFLY polynomial
This goes back to something called the Hopf fibration of spheres. The elementary one is
S^1 — > S^3 — > S^2
The S^1 is any curve, which is a closed loop. In this fibration there is a “thickening” of the two dimensional S^2 which allows this curve to pass “right over left” or “left over right” or
not intersect. This relationship defines a term in a path integral called a Wilson loop. This is a knot invariant which is connected to the theory of complex variables. The S^3 in the middle
feeds into another Hopf fibration
S^3 — > S^7 — > S^4.
This also defines a skein relation! Here we are knotting tree dimensional spheres in seven dimensions. The Skein relations define a Chern-Simons invariant 4?k = ?? for ? = ?/d? + (2/3)?/?/? a
3-form measure on a 3-sphere. This is an invariant of the set of quaternions. The topology of this is connected to Milnor’s work on inequivalent diffeomorphisms on the 7-sphere. It gets
better! The next level is the Hopf fibration
S^7 — > S^{15} — > S^8
where this is a Skein relation on 7-spheres in 15 dimensions! This corresponds to the octonions.
The first level of these Hopf fibrations is S^0 — > S^1 — > S^1. You might notice a pattern with the dimensions here; the middle sphere has dimension equal to the sum of the dimensions of
other two, and the two on the left and right differ by one dimension. This first defines the algebra of real numbers, which has the following properties:
The next fibration for complex numbers removes the betweeness property — two points on two dimensions have no point between then in common, for a whole class of curves can connect them. The
quaternions are not commutative, and finally the octonions are not associative; in other words a(bc) – (ab)c is not zero in general. The right dimension also have 1, 2, 4, and 8, or dim = 2^
n, for n = 0, 1, 2, 3. This has certain graph representations in the Fano plane, which is a discrete system from the Moufang or projective Cayley plane.
Can one go higher with
S^{15} — > S^{31} — > S^{16}?
These are the sedenions, but they have no division algebra. We run out of mathematics and enter “insanity.” These have certain Lie algebraic correspondences, and higher systems do exist, but
they are not extensions of the Hopf fibrations. A continuation of the Hopf fibration leads to something called Bott periodicity. The larger systems from a group theory or algebraic approach
correspond to sporadic groups, which are more bizarre than the exceptional groups, such as E_8 which corresponds to the octonions. Further, the these sporadic groups form automorphisms of the
Fisher-Griess group — sometimes called the monster group with a prime number factorization which gives its dimension as about 8×10^{51} dimensions.
All of this has a bearing on this physics — even the monster group!
5. So I guess these 6 rather small dimensions might explain why the observable 3D universe is rather uniform across vast 3D distances? Every point in “our” 3D universe must be closely connected
somehow, or how else would you explain the similarity of everything everywhere? (Perhaps some kind of “inertia” in laws of nature and physical constants, but wouldn’t that also call for an extra
dimension where that inertia “lives”??)
Well just a layman musing here, I hope someone will take the bait and expand/explain this weird similarity phenomenom a bit further..
1. IF I understand the question. One controversial viewpoint reply: Why the uniformity through time, and similarity in space? – A common origin: Universal frame of laws enforce throughout the
Cosmos ( producing the same results everywhere ), similarity in architectural features and forms reflecting a common design-imprint: the entire vast Complex made from the same precisely
scaled, highly structured Blueprint.
Our 3-dimensional time-space may be intimately “connected” to a higher plane reality, one not subject to its 3D limitations, or its matter-distance time-measures, but interfacing with it –
the origin of it? – from a greater level of existence: Like the relationship between a substantial, imposing granite pillar, and its ephemeral shadow – one endures, the other changes, and
2. I like that idea of the additional dimensions providing the physical controls that we observe in our three. We don’t see a reason the speed of light is what it is, why isn’t it ten times as
fast? What limits it? Perhaps one of the other six dimensions has a controlling factor, a barrier. If we could peek into that other dimension, maybe we could manipulate it and ‘break the
light barrier’. Interstellar travel, here we come.
So the other dimensions could provide the underlying framework of our physical laws. Let’s speculate that one dimension is associated with the electromagnet force. The ‘depth’ of that
dimension could define the strength of the force.
Lots of problems with this concept, of course, but I like the possibilities it leads to.
6. “…the universe was born from the explosion of an infinitely tiny point.”
A common misconception I long subscribed to ——–>
“Was the Big Bang an explosion”?
“No, the Big Bang was not an explosion. We don’t know what, exactly, happened in the earliest times, but it was not an explosion in the usual way that people picture explosions. There was not a
bunch of debris that sprang out, whizzing out into the surrounding space. In fact, there was no surrounding space. There was no debris strewn outwards. Space itself has been stretching and
carrying material with it.”
– WAMP (Wilkinson Microwave Anisotropy Probe), NASA —> ttp://map.gsfc.nasa.gov/site/faq.html
“Big Bang Theory – Common Misconceptions”:
“There are many misconceptions surrounding the Big Bang theory. For example, we tend to imagine a giant explosion. Experts however say that there was no explosion; there was (and continues to be)
an expansion. Rather than imagining a balloon popping and releasing its contents, imagine a balloon expanding: an infinitesimally small balloon expanding to the size of our current universe.”
– All About Science —> http://big-bang-theory.com/
Now, why would anybody have a misconception about a Theory titled: The BIG BANG?
“But Big Bang cosmology is at odds with Einstein’s theory of general relativity – general relativity doesn’t allow for any situation in which the whole universe is one tiny point.”
Well, could it be the model is correct – except for the “singularity” point of origin? Model another “point” of origin, where it is NOT required to have colossal volume and density infinitely
compacted together into a hypothetical beginning, but rather allowing for a sudden flash of Universe Creation from source: formation through an instantaneous expanding creation – actual Creation
as opposed to transformation, as in infinitesimal “singularity” becomes infinitely huge Cosmos – inflating out and forming the enormous organized complex of law-governed Space, and its ordered
energy-matter formations of Time?
(String Theory may be perfectly compatible with the alternate Model, I just do not understand it enough.)
1. Perhaps Fred Hoyle has had the last laugh as “Big Bang” was originally a disparaging term for the idea he hated, by causing so many people’s misconceptions of it. Perhaps cosmologists should
run a competition to find a new more correct name for the creation event, perhaps with an apt acronym as a bonus.
1. The steady state cosmology model of Hoyle, Bondi, Gold and Narlikar is dead as a doornail. If completely fails to match observational data. Further, if the universe were steady state
something curious would happen. The smallest perturbation would cause it to either expand or collapse. This perturbation could be a quantum fluctuation.
2. It is amazing to contemplate, all the complexity that rises from “nothing”.
3. How the universe emerged is still a work in progress. Inflation occurred at 10^{-33} seconds into the existence of the universe for a very brief period of 10^{-35} seconds. This is very close
to the origin, but it is 10^{10} Planck units of time after the universe emerged. How the universe emerged and the physics behind it are not well understood. Steinhardt’s ekpyrotic model
involves colliding branes, which invokes some pre-existing system. Smolin thinks that cosmologies quantum mechanically tunnel into existence by the vacuum polarization near singularities of
black hole, where these black holes exist in other cosmologies within the multiverse setting.
General relativity does not describe topology changes. The universe may have emerged from a point, or some small spatial sphere that transformed into a flat space, but general relativity
works with a geometry that maintains a constant topology. Geometry describes the gridded map of a space, say a map a surveyor works on land, but the topology is fixed. The physics of how a
cosmology, or a spacetime, emerges may involve topology changes.
The question about why space is three dimensional probably has something to do with this. It might have something to do with the Hopf fibration issues I outlined below. There might also be
some duality between 3 dimensional space and 2+1 dimensions of space plus time. These are very interesting problems to think about, and they do push at the foundations of our understanding of
physics and cosmology.
7. That is one tall order with at least 3 ad hocs.
– First you have to append an ad hoc big bang singularity to standard cosmology. All you need for eternal inflation is an inflation potential with a local minimum (Bousso).
Also, let us not forget that the quantum creation of a universe may presently be in trouble. Or not, I haven’t heard of a reaction to that result.
But if it is true and tunneling is forbidden, you may have entered a dead end: how do you predict a cosmological singularity in the first place?
– Then you need to introduce ad hoc IR and UV (I think) cutoffs in the model of Kim et al, according to the papers that Ivan3man linked to. (“Gaussian expansion method”.) They can remove at least
one in the low energy limit, but presumably they are still needed in their early time simulations.
This is in order to replace the naturally stable and accepted method of Wick rotations.
So, yes, possible, but perhaps not likely and definitely not natural in the context.
But why our universe has three spatial dimensions has been a problem for physicists,
This is common claim even among physicists, but I think it amounts to “folk science” as good hypotheses has been around for a very long time:
– In an anthropic (say, eternal inflation) universe, we need to have 3 spatial dimensions to have bags that can contain insides isolated from the environment (cells). Sankey raises another
problem for non-3D biochemistry, proteins folds needs to be stable to have polymer structural building blocks or catalysts.
– In a multiverse (say, from eternal inflation), we need to have 1 time dimension and 3 spatial dimensions to have interesting physics.
“With more or less than one time-dimension, the partial differential equations of nature would lack the hyperbolicity property that enables observers to make predictions. In a space with more
than three dimensions, there can be no traditional atoms and perhaps no stable structures. A space with less than three dimensions allows no gravitational force and may be too simple and barren
to contain observers.”
Look at his figure to see how unpredictability cuts off too many or too few time dimensions and simplicity or complexity (instabilities) cuts off too many or too few space dimensions.
I will add this tentative one, though it is a personal observation:
– It is only in 4D (1+3 D) that there is an interesting (actually infinite, I think) collection of manifolds. Presumably that is why physics over manifolds with freedom (say, string theory) ends
up promoting 3 spatial dimensions.
Maybe lcrowell can weigh in on that one.
But Big Bang cosmology is at odds with Einstein’s theory of general relativity – general relativity doesn’t allow for any situation in which the whole universe is one tiny point, which means
this theory alone can’t explain the origin of the universe.
That doesn’t seem to pass the smell test.
General relativity (GR) suggests and handles singularities, say in black holes. Singularities are nice, at least for those who looks for Theories Of Everything (TOEs), since they provide an
infinite set of degrees of freedom for free. So a TOE theory that wants to bind them to a unique parameter set has one less problem to worry about.
What GR doesn’t do is predict their quantum mechanical behavior. (Well, duh.) It isn’t a complete physics in that sense.
8. Not being a physicist, and not understanding the math, I’m almost afraid to post.
First question please: I heard of a theory called branes. I like to watch the Science Channel.
I heard one theory predicts our universe is the result of a collision of two branes.
For those who think branes exist, do we know where, in the collision, we are?
I had the impression, from the science program, the theory is, the collision happened,
and we are the result. I was wondering if the collision could still be happening.
I heard people believe the rate of expansion of the universe is increasing.
I am looking for an explanation why the rate of of expansion is increasing.
Second question please: I watched a science program about Dark Matter and Dark Energy.
I have no idea if these things exist. Theoretically, in which dimensions do they exist?
Third question please: If Dark Matter and Dark Energy exist, where do they come from?
I was wondering if a Black Hole might produce Dark Energy.
I was wondering if Dark Energy is really another state of Dark Matter…silly me.
I was thinking Dark Energy might be the “gaseous” state of Dark Matter,
Dark Matter the “liquid” state, and …, what could be the solid state.
This question is hair-brain anyway.
1. If you watch the Science Channel you need only one caveat. Be skeptical of anything Michio Kaku says.
2. If you Google any of those topics, you will have enough reading material to last a lifetime. Enjoy 🙂
3. D-branes are surfaces where the endpoints of strings are attached, and the those ends satisfy Dirichlet boundary conditions. This means that in the center of mass frame of the string the
endpoints are fixed. There are Newman conditions, defining N-branes, where momentum of the endpoints is fixed. Branes are curious, and as it turns out they are not quantum mechanical. They
are composed of many quantum numbers and are then really classical. If one where to look closely on a small region of a brane you find they are composed of strings. As such they are really a
defect in the vacuum analogous to an electron surface in solid state physics. There is a dual element to this as well, for in an S-duality a string can be composed of D0-branes (D-branes of
zero dimension which are like point particles), making a string similar to beads on a string.
The Steinhardt ekpyrotic theory invokes collisions between branes, where two branes might be connected by open strings with endpoints on them. The collisions between them reset the energy on
the branes and this corresponds to a big bang.
You might want to read some on dark energy, for this is some physics associated with the vacuum energy. Dark matter is some form of quantum field particle which does not interact by the
electromagnetic field. This is what makes them “dark.” These could be supersymmetric partners of known elementary particles.
9. Seems to me “three dimensions” is simply a mathematical notion, defined as three directions which are separated by 90 degree angles. The 90 degrees mean that these dimensions are independent and
that one can vary without effecting the others. This strikes me as a cultural construction, not an insight into reality. Why? Because there are many parallel “directions” in which the properties
of objects can vary without effecting the others. What about colour or orientation for example?
I suppose the physicist would say that orientation is reduced to 3 dimensional movement if the object is considered a group of related objects moving through space, and the same for colour as the
bouncing of light rays off of objects. So the solution seems to be to break objects into smaller and smaller units (sound familiar?) where these smaller and smaller units have nothing but
positions in space and time. Regarding 9 dimensional space, it seems obvious that 9 is the number of dimensions required to satisfy the mathematical models to explain the phenomena in question.
So how many dimensions are there really? Well, it depends on the level of abstraction in the model/description.
10. What I wonder is this… If, at some point, 3 dimensions expanded enough to be what we sense today… Might it be possible, since expansion is still continuing, that the universe reaches a point
where another dimension starts to expand… What would be the effect? Would we even be able to preceive it? How would that change the “laws” of physics?
11. It should be obvious, if the focus was more on physics than mathematics, that gravitation is not a force. Gravitation is a distortion of spacetime geometry in the presence of mass. The only time
gravitation and force coincide is when gravitation is interrupted, as at or below the surface of a large mass surrounding a geometric vortex. | {"url":"https://www.universetoday.com/92131/why-do-we-live-in-three-dimension/","timestamp":"2024-11-02T18:23:00Z","content_type":"text/html","content_length":"245674","record_id":"<urn:uuid:08f7a490-eed6-4372-99ce-8390434210a5>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00049.warc.gz"} |
Gram-force/sq. centimeter to Newton/square millimeter Converter | gf/cm^2 to N/mm^2
Gram-force/sq. centimeter to Newton/square millimeter converter | gf/cm^2 to N/mm^2 conversion
Are you struggling with converting Gram-force/sq. centimeter to Newton/square millimeter? Don’t worry! Our online “Gram-force/sq. centimeter to Newton/square millimeter Converter” is here to simplify
the conversion process for you.
Here’s how it works: simply input the value in Gram-force/sq. centimeter. The converter instantly gives you the value in Newton/square millimeter. No more manual calculations or headaches – it’s all
about smooth and effortless conversions!
Think of this Gram-force/sq. centimeter (gf/cm^2) to Newton/square millimeter (N/mm^2) converter as your best friend who helps you to do the conversion between these pressure units. Say goodbye to
calculating manually over how many Newton/square millimeter are in a certain number of Gram-force/sq. centimeter – this converter does it all for you automatically!
What are Gram-force/sq. centimeter and Newton/square millimeter?
In simple words, Gram-force/sq. centimeter and Newton/square millimeter are units of pressure used to measure how much force is applied over a certain area. It’s like measuring how tightly the air is
pushing on something.
The short form for Gram-force/sq. centimeter is “gf/cm^2” and the short form for Newton/square millimeter is “N/mm^2”.
In everyday life, we use pressure units like Gram-force/sq. centimeter and Newton/square millimeter to measure how much things are getting squeezed or pushed. It helps us with tasks like checking
tire pressure or understanding the force in different situations.
How to convert from Gram-force/sq. centimeter to Newton/square millimeter?
If you want to convert between these two units, you can do it manually too. To convert from Gram-force/sq. centimeter to Newton/square millimeter just use the given formula:
N/mm^2 = Value in gf/cm^2 * 0.0000980665
here are some examples of conversion,
• 2 gf/cm^2 = 2 * 0.0000980665 = 0.000196133 N/mm^2
• 5 gf/cm^2 = 5 * 0.0000980665 = 0.0004903325 N/mm^2
• 10 gf/cm^2 = 10 * 0.0000980665 = 0.000980665 N/mm^2
Gram-force/sq. centimeter to Newton/square millimeter converter: conclusion
Here we have learn what are the pressure units Gram-force/sq. centimeter (gf/cm^2) and Newton/square millimeter (N/mm^2)? How to convert from Gram-force/sq. centimeter to Newton/square millimeter
manually and also we have created an online tool for conversion between these units.
Gram-force/sq. centimeter to Newton/square millimeter converter” or simply gf/cm^2 to N/mm^2 converter is a valuable tool for simplifying pressure unit conversions. By using this tool you don’t have
to do manual calculations for conversion which saves you time. | {"url":"https://calculatorguru.net/gram-force-sq-centimeter-to-newton-square-millimeter/","timestamp":"2024-11-06T02:35:14Z","content_type":"text/html","content_length":"124364","record_id":"<urn:uuid:b057a896-dc7c-435c-921f-4e7d9ab9bc9e>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00576.warc.gz"} |
Part 1
We are building a lanternfish simulator. Each lanternfish is a little discrete automaton which reproduces under certain conditions. We have a whole collection of them, and each “day” is a step for
all the lanternfish. At the moment, the only thing that characterizes a fish is its “internal timer”, a natural number between 0 and 8 which tells us how many days we have until the fish reproduces:
type Fish = ℕ
school_day : [Fish] → [Fish]
school_day steps a list of fish (i.e. their counters), adding any newly-spawned ones. Since an individual fish can reproduce, the day function takes a single fish to either one or two fish (“chain”
reproductions can’t occur, since new lanternfish start at the maximum timer count). For convenience, we use a list instead of a sum:
day : Fish → [Fish]
We then define
school_day = concat ∘ List day
Enthusiasts will recognize the list monad; we’ll remember this detail for when things get more complicated in Part 2. We want to be able to run our simulation for a given number of days; this is a
classic function-power application,
run_days : ℕ → [Fish] → [Fish]
run_days n = power n school_day
where power n f = fⁿ.
All that remains is to write day, which is just a matter of transcribing the informal semantics we’re given.
day 0 = [6, 8] -- reset, spawn
day (Succ n) = [n]
The population size after 80 steps is given by
length ∘ run_days 80
Executable Haskell implementation
Part 2
Rather than giving us a more interesting system to simulate, Wastl has decided to try to melt our hardware. I’m disappointed, but it did give an opportunity to think about how to represent this
simple-but-enormous situation.
Running the simulation of even the five-fish example for 256 days will give us billions of fish, so we clearly can’t simulate the school in O(n) (in current population) space with existing computers.
Instead, we can simply describe the system by a count of the fish at each timer stage, since they have no other properties and we’re only interested in population size.
type School = [ℕ]
(Treating a list as a 9-tuple, in this case.) The initial state of the example would be, for example, [0, 1, 1, 2, 3, 0, 0, 0, 0]. school_day is then:
school_day : School → School
school_day [z, o, tw, th, fr, fv, sx, sv, e] =
[o, tw, th, fr, fv, sx, sv + z, e, z]
Ungainly though it is, this encodes in a single expression the same rules for the simulation as the previous program’s day, etc. functions. On each day, the population of fish at 0 (given by z) first
drops to zero, the same number of fish “come into being” at 6 and 8, then the existing populations “shift down” (“ones” become “zeros”, and so on).
This allows us to describe the system in space independent of its size. We have to convert the puzzle input (lists of fish-counters) to this population-count form; this is a simple but, again,
ungainly fold, specified by
pop_count ns = [count (== 0) ns, …, count (== 8) ns]
This can be calculated with the banana split theorem, but the details are left to the hard-nosed (see my discussion of Day 3). It’s pretty easy to see how it works, I think.
The solution is then given by:
sum ∘ power school_day 256 ∘ pop_count
Executable Haskell implementation | {"url":"http://www.sigwinch.xyz/aoc/2021/day_06.html","timestamp":"2024-11-03T09:46:53Z","content_type":"text/html","content_length":"5260","record_id":"<urn:uuid:791f70a2-d978-4bb6-b1da-176118196f2a>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00112.warc.gz"} |
Torque Equations Formulas Calculator
Science - Physics
Solve for torque
Enter Inputs:
Enter input values and press Calculate.
Solution In Other Units:
Enter input values and press Calculate.
Input Conversions:
Enter input values and press Calculate.
Change Equation or Formulas:
Tap or click to solve for a different unknown or equation
Solve for torque
Solve for force
Solve for distance or length
T = torque
F = force
D = distance
Background on Torque
Torque measures how much a force acting on an object causes that object to rotate. The concept of torque, also known as the moment of force, plays a critical role in many aspects of physics,
engineering, and numerous applications involving rotational motion. When force is applied at a distance from an object's pivot point, it creates rotation about that pivot point. The torque magnitude
depends on two factors: the force magnitude and the distance from the pivot point where the force is applied.
Equation for Torque
The fundamental equation for calculating torque (T) when not considering the angle is:
T = F x D
• T represents the torque.
• F is the force applied.
• D is the distance from the pivot point to where the force is applied.
How to Solve Torque
To solve for torque:
• Identify the Force Applied (F): Determine the magnitude of the force acting on the object.
• Determine the distance (D): Measure the distance from the pivot point to the force's application point.
• Use the Torque Equation: Plug the values into the torque equation ( T = F x D ).
Suppose a force of 20 Newtons is applied at a distance of 1.5 meters from the hinge of a door. Calculate the torque exerted on the door's hinge.
T = F x D = 20 N x 1.5 m = 30 Nm
Here, the torque exerted on the door's hinge is 30 Newton-meters.
Fields/Degrees Where Torque is Utilized
• Mechanical Engineering: In designing gears, engines, and machinery that involve rotational motion.
• Automotive Industry: In designing and testing the effectiveness of vehicles' transmissions and engines.
• Construction: In crane operations and when applying torsion to structural elements.
• Sports Science: Analysis of movements in golf and baseball, where rotation plays a key role.
• Robotics: Implementation of robots' joints and motors to control precise movement.
Real-Life Applications of Torque
• Opening and Closing Doors: The torque determines how much force needs to be applied to open or close a door.
• Wrenches and Screwdrivers: Used to apply specific amounts of torque to tighten or loosen nuts and bolts.
• Electric Fans: Motors in fans use torque to rotate the blades to circulate air.
• Wind Turbines: Uses torque to convert wind energy into electrical energy through rotational motion.
• Automobiles: Torque is applied to steering and turning the wheels and axles.
Common Mistakes When Calculating Torque
• Ignoring the Distance Component: The distance from the pivot point is incorrectly measured.
• Mixing Units: Using different units for force and distance without proper conversion leads to incorrect results.
• Force Direction Misinterpretation: Applying force in a non-effective direction that does not contribute to torque.
• Miscalculating Force: Incorrect measure of force leads to errors in finding the necessary torque.
• Assuming Only Large Forces Matter: Underestimating the impact of small forces applied at more considerable distances.
Frequently Asked Questions About Torque
• What happens if the distance is doubled? Doubling the distance while keeping the force constant will double the torque.
• Does the direction of force matter when calculating torque? Yes, the direction in which the force is applied is crucial, as torque is a vector quantity.
• Can torque be negative? Yes, torque can be negative depending on the direction of the force application relative to the rotation direction.
• What does zero torque mean? Zero torque means the object has no rotational effect, either due to no force being applied or to force being applied exactly at the pivot point.
• How do I measure 'distance' for torque? Measure the straight line from the pivot point to the point where force is applied, perpendicular to the direction of the force. | {"url":"https://www.ajdesigner.com/phptorque/torque_equation_torque.php","timestamp":"2024-11-12T23:00:28Z","content_type":"text/html","content_length":"29780","record_id":"<urn:uuid:1d9d6357-84fd-4cd3-9e13-0526b322ada5>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00284.warc.gz"} |
New Reconstruction Method in the Electromagnetic Calorimeter (ECAL) Analysis
The key detector for measurements of electrons and positrons in AMS is the Electromagnetic Calorimeter, ECAL (see Figure 1). The ECAL consists of a multilayer sandwich of lead foils and ∼50,000
scintillating fibers with an active area of 648 × 648 mm^2 and a thickness of 166.5 mm, corresponding to 17 radiation lengths, $X_0$. The calorimeter is composed of 9 superlayers, each 18.5 mm thick
and made of 11 grooved, 1 mm thick lead foils interleaved with 10 layers of 1 mm diameter scintillating fibers. In each superlayer, the fibers run in one direction only. The 3D imaging capability of
the detector is obtained by stacking alternate superlayers with fibers parallel to the ? and ? axes (5 and 4 superlayers, respectively).
Figure 1. (left) The ECAL structure and (right) the ECAL lead–fiber matrix corresponding to a single cell. Optical fibers of 1 mm diameter are embedded in lead with 1.35 mm horizontal pitch. Adjacent
rows of fibers are shifted by half a pitch. The distance between fiber rows is 1.73 mm. This structure leads to a varying number of fibers on a straight particle track, depending on the track
position and inclination.
All fibers are read out on one end only by 324 photomultipliers (PMT). Each PMT has four anodes and is surrounded by a magnetic shield which contains light guides, the PMT base and the frontend
electronics. Each anode covers an active area of 9 × 9 mm^2, corresponding to about 35 fibers, defined as a cell. Figure 1 (left) schematically shows the construction and the optical face of one
superlayer of the lead-fiber matrix, against which a grid of PMTs is mounted. These PMT grids on the four ECAL faces define the ECAL coordinate system. Figure 1 (right) illustrates the locations of
optical fibers within a cell. In total there are 1296 cells segmented into 18 layers longitudinally, two per superlayer, with 72 transverse cells in each layer providing a fine granularity sampling
of the shower in three dimensions. The signals are processed over a wide dynamic range, from a minimum ionizing particle, which produces about 10 photoelectrons per cell, up to the 60,000
photoelectrons produced in one cell by the core of the electromagnetic shower of a 1 TeV electron, corresponding to deposited energy of 60 GeV.
Reconstruction of electrons and positrons in the calorimeter uses a 3-dimensional shower parametrization, which accounts for the detector specifics: finite size of the calorimeter, non-uniform
efficiency of the signal collection, and saturation effects due to the electronics and due to high energy density in the active calorimeter elements (A. Kounine, Z. Weng, W. Xu, and C. Zhang, Nucl.
Instr. Methods Sect. A, 869 110 (2017)).
An individual electromagnetic shower is described by seven parameters, which fully determine the observed pattern of energy depositions in the calorimeter cells: the shower energy ($E_0$); the
3-dimensional spatial point, ($x_0$ , $y_0$ , $z_0$ ), corresponding to the location of the shower maximum in the ECAL coordinate system; the two angles ($K_X$, $K_Y$) that, together with the spatial
point, define the shower axis; and the location ($T_0$) of the shower maximum on the shower axis. The parameter ($\beta$) depends on the specific construction and materials of the calorimeter. This
is a minimal parameter set, which allows accurate shower parametrization of individual showers without introducing noticeable correlation between these parameters.
The longitudinal shower profile in terms of the depth t in the calorimeter (in units of radiation length) is described by empirical parametrization [Particle Data Group, Phys. Rev. D 98, 030001
$$ \frac {dE}{dt}(t) = E_{0} \frac{(\beta t)^{\beta T_{0}}\beta e^{-\beta t}}{\Gamma(\beta T_{0}+1)} $$
using the parameters described above. In our calorimeter, we found that the scale parameter $\beta$ is constant ($\beta=0.65$). The individual shower parameters $E_0$ and $T_0$ are obtained from a
fit to observed energy depositions in the ECAL cells of each shower. Figure 2 shows the description of electron showers over a wide energy range.
Figure 2. Comparison of the energy depositions in each calorimeter layer with the model for (a) an electron of 100 GeV from the beam test and (b) a cosmic ray electron of 900 GeV. As seen, the energy
evolution of the shower shape is well described for both cases. | {"url":"https://ams02.space/de/node/209","timestamp":"2024-11-13T16:27:05Z","content_type":"text/html","content_length":"68838","record_id":"<urn:uuid:f64dfe0c-3a6d-48d9-b825-274fe2d0f8ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00082.warc.gz"} |
Inspiring Drawing Tutorials
How To Draw A Net
How To Draw A Net - Begin by drawing a horizontal line across your paper. Sketch the outline of the basketball hoop step 2: Identify all the sides of the 3d solid. 4 kansas in the midwest. This video
will show you how to go about drawing the net for a cube. This will be the top of your net.
Let’s dive into step one by drawing the bars of the volleyball net. To draw a basketball net, you start with the circular shape of the hoop and then hang a series of vertical lines down from it, add
the net webbing between th. Net of a rectangular prism. The draws of the olympic tournaments for both men and women created many interesting duels from the first day. Web watch afc u17 women's asian
cup 2024 draw live now on shahid.net.
Net of a triangular prism. To draw a basketball net, you start with the circular shape of the hoop and then hang a series of vertical lines down from it, add the net webbing between th. Sketch the
outline of the basketball hoop step 2: For online tutoring or additional resources, visit our website. The jayhawks have been reeling down the stretch, losing four of their last five games.
Web materials needed step 1: Web how to draw a fishing net. 5 gonzaga gets past no. Net of a triangular prism. This video will show you how to go about drawing the net for a cube.
Is there more than one possible net for this shape?. Web materials needed step 1: Web draw a net for a rectangular prism whose base is a one inch by one inch square and whose faces are 3 inches by 1
inch. If the 3d figure is a prism,. Web the letter writing campaign comes on the heels of a.
Web materials needed step 1: Next, draw a series of vertical lines coming down from the top line. 4 kansas in the midwest. Net of a triangular prism. Identify all the sides of the 3d solid.
So sigma + would be on the hydrogen atom and sigma. Web in order to draw the net of a 3d solid: Web how to draw a fishing net. Draw the lateral sides and have them connect to the base. Web draw a net
for a rectangular prism whose base is a one inch by one inch square and whose.
Draw the lateral sides and have them connect to the base. 5 gonzaga gets past no. The draws of the olympic tournaments for both men and women created many interesting duels from the first day. Next,
draw a series of vertical lines coming down from the top line. Let’s dive into step one by drawing the bars of the volleyball.
Add the net draw diagonal lines from one side of the hoop to the other:. Department of energy to halt approvals of liquified methane gas exports, what the. Identify all the sides of the 3d solid. Web
the introduction of ai in visual studio, particularly github copilot, has revolutionized the way developers code. Home tv shows movies live tv.
How to determine if a net forms a solid. Is there more than one possible net for this shape?. Web draw a net for a rectangular prism whose base is a one inch by one inch square and whose faces are 3
inches by 1 inch. 1 to make isometric and orthographic drawings. Next, draw a series of vertical lines.
For online tutoring or additional resources, visit our website. Net of a triangular prism. Web creighton, who sat right behind duke at no. 5 gonzaga gets past no. Web any prism is given by sa = ph
+2b where p is the perimeter of the base (in a rectangular prism, you could choose any side as one of the bases),.
Sketch the outline of the basketball hoop step 2: Web you don't have to use a net. Web any prism is given by sa = ph +2b where p is the perimeter of the base (in a rectangular prism, you could choose
any side as one of the bases), h is the height of the prism (the. Web draw a.
How to determine if a net forms a solid. The draws of the olympic tournaments for both men and women created many interesting duels from the first day. To draw a basketball net, you start with the
circular shape of the hoop and then hang a series of vertical lines down from it, add the net webbing between th. For.
How To Draw A Net - 5 gonzaga gets past no. Let’s dive into step one by drawing the bars of the volleyball net. The jayhawks have been reeling down the stretch, losing four of their last five games.
Is there more than one possible net for this shape?. This video will show you how to go about drawing the net for a cube. 4 kansas in the midwest. Web materials needed step 1: Web you don't have to
use a net. To draw a basketball net, you start with the circular shape of the hoop and then hang a series of vertical lines down from it, add the net webbing between th. If the 3d figure is a prism,.
Web in order to draw the net of a 3d solid: If the 3d figure is a prism,. How to determine if a net forms a solid. Net of a rectangular prism. Sketch the outline of the basketball hoop step 2:
Is there more than one possible net for this shape?. Draw the lateral sides and have them connect to the base. This video will show you how to go about drawing the net for a cube. Web draw a net for
a rectangular prism whose base is a one inch by one inch square and whose faces are 3 inches by 1 inch.
Let’s dive into step one by drawing the bars of the volleyball net. With copilot integrated into visual studio, you. The draws of the olympic tournaments for both men and women created many
interesting duels from the first day.
Web the introduction of ai in visual studio, particularly github copilot, has revolutionized the way developers code. 5 gonzaga gets past no. Next, draw a series of vertical lines coming down from
the top line.
Web The Introduction Of Ai In Visual Studio, Particularly Github Copilot, Has Revolutionized The Way Developers Code.
Web © 2023 google llc. Identify all the sides of the 3d solid. Web materials needed step 1: Add the net draw diagonal lines from one side of the hoop to the other:.
So Sigma + Would Be On The Hydrogen Atom And Sigma.
Web any prism is given by sa = ph +2b where p is the perimeter of the base (in a rectangular prism, you could choose any side as one of the bases), h is the height of the prism (the. Begin by drawing
two vertical lines on both sides of the page. Draw the lateral sides and have them connect to the base. Department of energy to halt approvals of liquified methane gas exports, what the.
4 Kansas In The Midwest.
This will be the top of your net. How to determine if a net forms a solid. Web watch afc u17 women's asian cup 2024 draw live now on shahid.net. Web draw a net for a rectangular prism whose base is a
one inch by one inch square and whose faces are 3 inches by 1 inch.
Begin By Drawing A Horizontal Line Across Your Paper.
1 to make isometric and orthographic drawings. Net of a triangular prism. With copilot integrated into visual studio, you. Get the winning numbers, watch the draw show, and find out just how big the
jackpot has. | {"url":"https://one.wkkf.org/art/drawing-tutorials/how-to-draw-a-net.html","timestamp":"2024-11-09T00:47:08Z","content_type":"text/html","content_length":"32640","record_id":"<urn:uuid:be925f54-ed0f-4eea-80bc-e88065e99646>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00815.warc.gz"} |
The maximum and minimum magnitude of the resultant of two given vector
Doubtnut is No.1 Study App and Learning App with Instant Video Solutions for NCERT Class 6, Class 7, Class 8, Class 9, Class 10, Class 11 and Class 12, IIT JEE prep, NEET preparation and CBSE, UP
Board, Bihar Board, Rajasthan Board, MP Board, Telangana Board etc
NCERT solutions for CBSE and other state boards is a key requirement for students. Doubtnut helps with homework, doubts and solutions to all the questions. It has helped students get under AIR 100 in
NEET & IIT JEE. Get PDF and video solutions of IIT-JEE Mains & Advanced previous year papers, NEET previous year papers, NCERT books for classes 6 to 12, CBSE, Pathfinder Publications, RD Sharma, RS
Aggarwal, Manohar Ray, Cengage books for boards and competitive exams.
Doubtnut is the perfect NEET and IIT JEE preparation App. Get solutions for NEET and IIT JEE previous years papers, along with chapter wise NEET MCQ solutions. Get all the study material in Hindi
medium and English medium for IIT JEE and NEET preparation | {"url":"https://www.doubtnut.com/qna/649422494","timestamp":"2024-11-12T15:59:04Z","content_type":"text/html","content_length":"197910","record_id":"<urn:uuid:4af2dad8-a559-43ff-ba78-b3837879c12f>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00415.warc.gz"} |
Module 4 Review Problems
True or False? Justify your answer with a proof or a counterexample.
1. The differential equation [latex]y^{\prime} =3{x}^{2}y-\cos\left(x\right)y^{\prime}\prime[/latex] is linear.
2. The differential equation [latex]y^{\prime} =x-y[/latex] is separable.
3. You can explicitly solve all first-order differential equations by separation or by the method of integrating factors.
4. You can determine the behavior of all first-order differential equations using directional fields or Euler’s method.
For the following problems, find the general solution to the differential equations.
5. [latex]{y}^{\prime }={x}^{2}+3{e}^{x}-2x[/latex]
6. [latex]y^{\prime} ={2}^{x}+{\cos}^{-1}x[/latex]
Show Solution
7. [latex]y^{\prime} =y\left({x}^{2}+1\right)[/latex]
8. [latex]y^{\prime} ={e}^{\text{-}y}\sin{x}[/latex]
Show Solution
9. [latex]y^{\prime} =3x - 2y[/latex]
10. [latex]y^{\prime} =y\text{ln}y[/latex]
Show Solution
For the following problems, find the solution to the initial value problem.
11. [latex]y^{\prime} =8x-\text{ln}x - 3{x}^{4},y\left(1\right)=5[/latex]
12. [latex]y^{\prime} =3x-\cos{x}+2,y\left(0\right)=4[/latex]
Show Solution
13. [latex]xy^{\prime} =y\left(x - 2\right),y\left(1\right)=3[/latex]
14. [latex]y^{\prime} =3{y}^{2}\left(x+\cos{x}\right),y\left(0\right)=-2[/latex]
Show Solution
15. [latex]\left(x - 1\right)y^{\prime} =y - 2,y\left(0\right)=0[/latex]
16. [latex]y^{\prime} =3y-x+6{x}^{2},y\left(0\right)=-1[/latex]
Show Solution
For the following problems, draw the directional field associated with the differential equation, then solve the differential equation. Draw a sample solution on the directional field.
17. [latex]y^{\prime} =2y-{y}^{2}[/latex]
18. [latex]y^{\prime} =\frac{1}{x}+\text{ln}x-y[/latex], for [latex]x>0[/latex]
For the following problems, use Euler’s Method with [latex]n=5[/latex] steps over the interval [latex]t=\left[0,1\right][/latex]. Then solve the initial-value problem exactly. How close is your
Euler’s Method estimate?
19. [latex]y^{\prime} =-4yx,y\left(0\right)=1[/latex]
20. [latex]y^{\prime} ={3}^{x}-2y,y\left(0\right)=0[/latex]
Show Solution
Euler: [latex]0.6939[/latex], exact solution: [latex]y\left(x\right)=\frac{{3}^{x}-{e}^{-2x}}{2+\text{ln}\left(3\right)}[/latex]
For the following problems, set up and solve the differential equations.
21. A car drives along a freeway, accelerating according to [latex]a=5\sin\left(\pi t\right)[/latex], where [latex]t[/latex] represents time in minutes. Find the velocity at any time [latex]t[/
latex], assuming the car starts with an initial speed of [latex]60[/latex] mph.
22. You throw a ball of mass [latex]2[/latex] kilograms into the air with an upward velocity of [latex]8[/latex] m/s. Find exactly the time the ball will remain in the air, assuming that gravity is
given by [latex]g=9.8{\text{m/s}}^{2}[/latex].
Show Solution
[latex]\frac{40}{49}[/latex] second
23. You drop a ball with a mass of [latex]5[/latex] kilograms out an airplane window at a height of [latex]5000[/latex] m. How long does it take for the ball to reach the ground?
24. You drop the same ball of mass [latex]5[/latex] kilograms out of the same airplane window at the same height, except this time you assume a drag force proportional to the ball’s velocity, using a
proportionality constant of [latex]3[/latex] and the ball reaches terminal velocity. Solve for the distance fallen as a function of time. How long does it take the ball to reach the ground?
Show Solution
[latex]x\left(t\right)=5000+\frac{245}{9}-\frac{49}{3}t-\frac{245}{9}{e}^{\frac{\text{-}5}{3t}},t=307.8[/latex] seconds
25. A drug is administered to a patient every [latex]24[/latex] hours and is cleared at a rate proportional to the amount of drug left in the body, with proportionality constant [latex]0.2[/latex].
If the patient needs a baseline level of [latex]5[/latex] mg to be in the bloodstream at all times, how large should the dose be?
26. A [latex]1000[/latex] -liter tank contains pure water and a solution of [latex]0.2[/latex] kg salt/L is pumped into the tank at a rate of [latex]1[/latex] L/min and is drained at the same rate.
Solve for total amount of salt in the tank at time [latex]t[/latex].
Show Solution
27. You boil water to make tea. When you pour the water into your teapot, the temperature is [latex]100^\circ C.[/latex] After [latex]5[/latex] minutes in your [latex]15^\circ C[/latex] room, the
temperature of the tea is [latex]85^\circ C.[/latex] Solve the equation to determine the temperatures of the tea at time [latex]t[/latex]. How long must you wait until the tea is at a drinkable
temperature [latex]\left(72^\circ C\right)?[/latex]
28. The human population (in thousands) of Nevada in [latex]1950[/latex] was roughly [latex]160[/latex]. If the carrying capacity is estimated at [latex]10[/latex] million individuals, and assuming a
growth rate of [latex]2\text{%}[/latex] per year, develop a logistic growth model and solve for the population in Nevada at any time (use [latex]1950[/latex] as time = 0). What population does your
model predict for [latex]2000?[/latex] How close is your prediction to the true value of [latex]1,998,257?[/latex]
Show Solution
29. Repeat the previous problem but use Gompertz growth model. Which is more accurate? | {"url":"https://courses.lumenlearning.com/calculus2/chapter/module-4-review-problems/","timestamp":"2024-11-05T19:58:54Z","content_type":"text/html","content_length":"61787","record_id":"<urn:uuid:9ab919d2-be2a-4772-b25c-b5b78713b4c8>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00516.warc.gz"} |
Given two \( x\)-monotone curves xcv1 and xcv2 and an enumeration ce that specifies either the minimum ends or the maximum ends of the curves where the curves have a vertical asymptote, compares the
\( x\)-coordinate of the curves near their respective ends.
Returns SMALLER, EQUAL, or LARGER accordingly. More precisely, compares the \( x\)-coordinates of the horizontal projection of a point \( p\) onto xcv1 and xcv2. If xcv1 and xcv2 approach the bottom
boundary-side, \( p\) is located far to the bottom, such that the result is invariant under a translation of \( p\) farther to the bottom. If xcv1 and xcv2 approach the top boundary-side, \( p\) is
located far to the top in a similar manner.
The \( x\)-coordinates of the limits of the curves at their respective ends are equal. That is, compare_x_at_limit_2(xcv1, xcv2, ce) = EQUAL.
parameter_space_in_y_2(xcv1, ce) = parameter_space_in_y_2(xcv2, ce).
parameter_space_in_y_2(xcv1, ce) \( \neq\) ARR_INTERIOR. | {"url":"https://doc.cgal.org/5.3/Arrangement_on_surface_2/classArrTraits_1_1CompareXNearLimit__2.html","timestamp":"2024-11-01T20:44:33Z","content_type":"application/xhtml+xml","content_length":"13265","record_id":"<urn:uuid:2b36ac3b-8956-4031-8ca4-d6dce285e0a8>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00656.warc.gz"} |
Simplified criterion for second order effects of isolated members: the slenderness
Simplified criterion for second order effects of isolated members: the slenderness λ[lim] following the EN Eurocode recommendation
Eurocode 2 part 1-1: Design of concrete structures 5.8.3.1 (1)
Second order effects may be ignored if the slenderness (λ = l[0]/i) is below a certain value λ[lim].
The recommended value for the slenderness λ[lim] follows from:
λ[lim] = 20⋅A⋅B⋅C / √n (5.13N)
= 1/(1 + 0,2φ[ef])(if φ[ef] is not known, A = 0,7 may be used),
φ[ef] is the effective creep ratio, cf. § 5.8.4
= √1 + 2ω (if ω is not known, B = 1,1 may be used),
ω is the mechanical reinforcement ratio, ω = A[s] f[yd] / (A[c] f[cd])
the total area of longitudinal reinforcement
the design yield strength of the reinforcement, f[yd] = f[yk]/γ[S]
see § 2.4.2.4 (1), § 2.4.2.4 (2) for the values of γ[S],
see § 3.2.2 (3)P for the uper limit of f[yk],
see Figure 3.8 for the design stress-strain diagrams of the reinforcing steel
the area of concrete cross-section
the design compressive strength of concrete, see § 3.1.6 (1)P.
= 1,7 - r[m] (if r[m] is not known, C = 0,7 may be used),
r[m] is the moment ratio, r[m] = M[01] / M[02],
where M[01], M[02] are the first order end moments, |M[02]| ≥ |M[01]|.
If M[01] and M[02] give tension on the same side, r[m] should be taken positive (C ≤ 0,7), otherwise negative (C > 1,7).
r[m] should be taken as 1,0 (C = 0,7) for:
- braced members in which the first order moments arise only from or predominantly due to imperfections or transverse loading
- for unbraced members in general.
is the relative normal force, n = N[Ed] / (A[c] f[cd]),
where N[Ed] is the design value of axial force.
This application calculates the limit slenderness λ[lim] from your inputs. Intermediate results will also be given.
(*) If φ[ef], ω, or r[m] are not known, you can put the value of zero (0) for φ[ef], A[s] or M[01] respectively. A = 0,7, B = 1,1, or C = 0,7 will be considered by the application. | {"url":"https://usingeurocodes.com/en/eurocode-2-1-1/StructuralAnalysis/limit-slenderness-ignore-second-order-effects","timestamp":"2024-11-06T02:42:11Z","content_type":"text/html","content_length":"46437","record_id":"<urn:uuid:b9adeceb-6c85-4655-93db-e4144129c5e5>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00047.warc.gz"} |
Polynomial representation Using Array - Quescol
Polynomial representation Using Array
Polynomials are fundamental mathematical expressions used extensively in various fields. Representing them efficiently in programming is crucial for calculations and manipulations. This section
introduces polynomial representation using arrays, a method that stores each term’s coefficient and exponent in a sequential manner, suitable for beginners in computer programming or mathematics.
Array-Based Representation of Polynomials
• Concept of Polynomial Representation:
□ Explaining polynomials as a sum of terms with coefficients and exponents.
□ The role of arrays in storing these terms efficiently.
• Structuring an Array for Polynomials:
□ Describing how coefficients and exponents are mapped to array indices.
□ Examples to illustrate the representation.
Implementing Polynomial Operations Using Arrays
• Adding and Subtracting Polynomials:
□ Algorithms for adding and subtracting polynomials using arrays.
□ Code snippets in C to demonstrate these operations.
• Multiplication and Division:
□ Discussing more complex operations like multiplication and division.
□ Challenges in array-based representation for these operations.
Advantages and Limitations
• Efficiency in Storage and Access:
□ The benefit of using arrays in terms of memory efficiency and quick access.
• Limitations:
□ Discussing the fixed size of arrays and issues with sparse polynomials.
□ Comparing with other representations like linked lists.
To demonstrate polynomial representation using an array, let’s consider an example of a polynomial and how it can be represented using this data structure.
Example Polynomial
Consider the polynomial:
Representing the Polynomial with an Array
In an array-based representation, each element of the array corresponds to a coefficient of the polynomial, with the array index representing the exponent. For the given polynomial, we can use an
array of size 4 (since the highest exponent is 3).
1. Array Structure:
□ The array index represents the exponent of x.
□ The value at each index represents the coefficient of the term.
2. Representation in Array:
□ Array: P[4]
□ P[0] = 7 (Coefficient of x^0)
□ P[1] = -6 (Coefficient of x^1)
□ P[2] = 4 (Coefficient of x^2)
□ P[3] = 7 (Coefficient of x^3)
Visual Representation
The polynomial 7x^3+4x^2−6x+7 would be represented in an array as:
Index (Exponent) 0 1 2 3
Coefficient 7 -6 4 7
Array Indexing
• The array index directly corresponds to the exponent, making it intuitive to access specific terms.
• For example, to access the coefficient of x2, we look at P[2], which is 4.
Algorithm to Represent and Evaluate a Polynomial using an Array
Step 1: Define the Polynomial
• Given polynomial: P(x)=7x^3+4x^2−6x+7
Step 2: Initialize the Array
• Create an array P of size 4 (since the highest exponent is 3).
• Initialize the array with coefficients: P[4] = {7, -6, 4, 7}.
Step 3: Evaluate the Polynomial
• To evaluate at a specific value of x, say x = 2.
• Initialize a variable result to 0.
• Loop through the array from index 0 to 3.
• For each index i, calculate P[i] * 2^i and add it to result.
Step 4: Print the Result
• After the loop ends, result will hold the value of the polynomial at x = 2.
• Print result.
Example: Evaluate at x = 2
1. Initialization:
2. Loop through Array:
□ For i = 0: result += P[0] * pow(x, 0) = 0 + 7 * 1 = 7
□ For i = 1: result += P[1] * pow(x, 1) = 7 - 6 * 2 = -5
□ For i = 2: result += P[2] * pow(x, 2) = -5 + 4 * 4 = 11
□ For i = 3: result += P[3] * pow(x, 3) = 11 + 7 * 8 = 67
3. Result:
4. Output:
□ Print: “Polynomial value for x = 2 is: 67”.
C Program for Array Representation
#include <stdio.h>
#include <math.h>
// Function to evaluate a polynomial using an array
int evaluatePolynomial(int coefficients[], int degree, int x) {
int sum = 0;
for (int i = 0; i <= degree; i++) {
sum += coefficients[i] * pow(x, i);
return sum;
int main() {
// Example Polynomial: 3x^2 + 2x + 1
int coefficients[] = {1, 2, 3}; // Array representation (constant term first)
int degree = 2; // Degree of the polynomial
int x = 2; // Value at which we want to evaluate the polynomial
int result = evaluatePolynomial(coefficients, degree, x);
printf("Value of the polynomial at x = %d is: %d\n", x, result);
return 0;
Value of the polynomial at x = 2 is: 67
Program Explanation
Header Files:
#include <stdio.h>
#include <math.h>
stdio.h: Standard input-output header, used for functions like printf.
math.h: Math library header, used for the pow function to calculate powers.
Function to Evaluate Polynomial:
// Function to evaluate a polynomial using an array
int evaluatePolynomial(int coefficients[], int degree, int x) {
int sum = 0;
for (int i = 0; i <= degree; i++) {
sum += coefficients[i] * pow(x, i);
return sum;
evaluatePolynomial: This function calculates the value of a polynomial for a given value of x.
It takes three parameters: an array coefficients containing the coefficients of the polynomial, the degree of the polynomial (highest exponent), and the value x at which the polynomial is evaluated.
The function initializes a variable sum to 0, which accumulates the value of the polynomial.
A for loop iterates over each term of the polynomial. For each term, it calculates the term’s value by raising x to the power of the term’s exponent (i) and multiplying it by the coefficient
(coefficients[i]), then adds this value to sum.
Finally, the function returns the total sum, which is the value of the polynomial at x.
main Function:
int main() {
// Example Polynomial: 3x^2 + 2x + 1
int coefficients[] = {1, 2, 3}; // Array representation (constant term first)
int degree = 2; // Degree of the polynomial
int x = 2; // Value at which we want to evaluate the polynomial
int result = evaluatePolynomial(coefficients, degree, x);
printf("Value of the polynomial at x = %d is: %d\n", x, result);
return 0;
The main function is the entry point of the program.
It first defines the polynomial to be evaluated. In this case, the polynomial is 7x^3 + 4x^2 - 6x + 7. The coefficients are stored in an array coefficients in the order of increasing exponents.
The degree of the polynomial is specified as 3 (since the highest power of x is 3).
The value of x at which the polynomial needs to be evaluated is set to 2.
The program then calls evaluatePolynomial, passing the coefficients array, degree, and the value of x.
The result of the polynomial evaluation is stored in result and printed out using printf.
The program calculates and prints the value of the polynomial 7x^3 + 4x^2 - 6x + 7 at x = 2. | {"url":"https://quescol.com/data-structure/polynomial-representation-using-array","timestamp":"2024-11-09T07:37:26Z","content_type":"text/html","content_length":"91466","record_id":"<urn:uuid:1401e017-8941-4b06-8fbb-6925e092f3c1>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00036.warc.gz"} |
Homework assignments will be posted here as they are assigned.
Homework in this class is intended to provide students practice with the concepts discussed in class and in the book. True understanding is only available with practice.
Homework will vary in difficulty and point value. Simple homework will be worth on the order of 5 points. Such homework might include establishing a working graphics environment, or answering simple
questions. More complex homework, such as an involved program will be worth 50 points. In the end, all points will be totaled and divided by the total number of available points to establish a
homework percentage.
Homework must be turned in on the assigned due date. No late homework will be accepted.
Homework may consist of both programming and written assignments. All work should be performed in a professional manner.
Programs should
• be well written and well documented.
• should compile without errors and warnings.
• solve the problem presented correctly.
• employ the proper data structures.
• employ efficient algorithms.
• be modular in design, flexible and extensible.
• be completed on time and submitted according to instructions.
Written homework should
• be neat and legible, preferably typed, especially if your handwriting and very poor.
• be completed on time and submitted according to instructions.
All homework is expected to be the work of the submitter. Cheating at this level will not be tolerated and is subject to course failure and further student disciplinary action. | {"url":"https://mirkwood.cs.edinboro.edu/~bennett/class/csci360/fall2019/?go=homework","timestamp":"2024-11-03T06:48:29Z","content_type":"text/html","content_length":"3939","record_id":"<urn:uuid:4983eef4-332b-4465-a91a-a93a6c5281b2>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00633.warc.gz"} |
Variables - ConvertCalculator
If your formulas are simple and straightforward, you can get away with re-using bits and pieces in multiple formulas. But you will notice that things can get confusing quite fast! We have created
Variables to make sure your head won't explode while building a calculator with complex calculations.
Variables act in the same way as Formulas. There is one important difference! A formula is a form element. So formulas are displayed in your calculation form. Variables are only visible if they are
used in a formula.
Creating variables
You can create a variable by expanding the formula editor and clicking on the "Add variable" button.
Once you have created the variable, you can use it in any function. You can even nest variables, so you can use variables in other variables. Though be careful about the order in which the variables
are defined. Variables that contain a nested variable that is listed below the original variable will result in an error. | {"url":"https://www.convertcalculator.com/help/building/variables/","timestamp":"2024-11-05T00:54:54Z","content_type":"text/html","content_length":"451649","record_id":"<urn:uuid:7cc54ad3-b6ca-4bcc-bb73-b533766d5979>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00139.warc.gz"} |
An equivalent circuit diagram for Gunn elements with Schottky contact
In the derivation of the equivalent circuit diagram, it is assumed that the thickness of the depletion layer near the Schottky barrier is independent of the position. Diffusion effects and the
influence of the boundary regions are neglected. Attention is given to the current density equation, the continuity equation, the Poisson equation, and the relations for the cathode and anode
current. The derived equivalent circuit diagram is used in the analysis of a storage element with novel characteristics.
Wissenschaftliche Zeitschrift
Pub Date:
□ Computer Storage Devices;
□ Equivalent Circuits;
□ Gunn Diodes;
□ Schottky Diodes;
□ Capacitors;
□ Circuit Diagrams;
□ Continuity Equation;
□ Current Density;
□ Electric Field Strength;
□ Optimization;
□ Poisson Equation;
□ Electronics and Electrical Engineering | {"url":"https://ui.adsabs.harvard.edu/abs/1976WisZe..25..443F/abstract","timestamp":"2024-11-06T15:39:37Z","content_type":"text/html","content_length":"34858","record_id":"<urn:uuid:2124452e-e17a-4b4a-be3d-f555a956f477>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00122.warc.gz"} |
A Threshold Equation for Action Potential Initiation
In central neurons, the threshold for spike initiation can depend on the stimulus and varies between cells and between recording sites in a given cell, but it is unclear what mechanisms underlie this
variability. Properties of ionic channels are likely to play a role in threshold modulation. We examined in models the influence of Na channel activation, inactivation, slow voltage-gated channels
and synaptic conductances on spike threshold. We propose a threshold equation which quantifies the contribution of all these mechanisms. It provides an instantaneous time-varying value of the
threshold, which applies to neurons with fluctuating inputs. We deduce a differential equation for the threshold, similar to the equations of gating variables in the Hodgkin-Huxley formalism, which
describes how the spike threshold varies with the membrane potential, depending on channel properties. We find that spike threshold depends logarithmically on Na channel density, and that Na channel
inactivation and K channels can dynamically modulate it in an adaptive way: the threshold increases with membrane potential and after every action potential. Our equation was validated with
simulations of a previously published multicompartemental model of spike initiation. Finally, we observed that threshold variability in models depends crucially on the shape of the Na activation
function near spike initiation (about −55 mV), while its parameters are adjusted near half-activation voltage (about −30 mV), which might explain why many models exhibit little threshold variability,
contrary to experimental observations. We conclude that ionic channels can account for large variations in spike threshold.
Author Summary
Neurons communicate primarily with stereotypical electrical impulses, action potentials, which are fired when a threshold level of excitation is reached. This threshold varies between cells and over
time as a function of previous stimulations, which has major functional implications on the integrative properties of neurons. Ionic channels are thought to play a central role in this modulation but
the precise relationship between their properties and the threshold is unclear. We examined this relationship in biophysical models and derived a formula which quantifies the contribution of various
mechanisms. The originality of our approach is that it provides an instantaneous time-varying value for the threshold, which applies to the highly fluctuating regimes characterizing neurons in vivo.
In particular, two known ionic mechanisms were found to make the threshold adapt to the membrane potential, thus providing the cell with a form of gain control.
Citation: Platkiewicz J, Brette R (2010) A Threshold Equation for Action Potential Initiation. PLoS Comput Biol 6(7): e1000850. https://doi.org/10.1371/journal.pcbi.1000850
Editor: Lyle J. Graham, Université Paris Descartes, Centre National de la Recherche Scientifique, France
Received: January 26, 2010; Accepted: June 3, 2010; Published: July 8, 2010
Copyright: © 2010 Platkiewicz, Brette. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original author and source are credited.
Funding: This work was supported by the European Research Council (ERC StG 240132): http://erc.europa.eu/. The funders had no role in study design, data collection and analysis, decision to publish,
or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Spike initiation in neurons follows the all-or-none principle: a stereotypical action potential is produced and propagated when the neuron is sufficiently excited, while no spike is initiated below
that threshold. The value of that threshold sets the firing rate and determines the way neurons compute, for example their coincidence detection properties [1], [2]. It is generally described as a
voltage threshold: spikes are initiated when the neuron is depolarized above a critical value, when voltage-dependent sodium channels start to open. That biophysical mechanism is well understood
since the studies of Hodgkin and Huxley in the squid giant axon [3] and subsequent modeling studies [4]–[7].
Recent findings have renewed the interest in the spike threshold. First, there is an intense ongoing debate about the origin of threshold variability observed in vivo [8]–[14]. In particular, it is
unclear whether threshold variability is mainly due to experimental artifacts or molecular mechanisms, which might question the relevance of the Hodgkin-Huxley model for central neurons. Moreover,
numerous experiments have shown that spike initiation does not only depend on the membrane potential but also on complex features of the inputs. For example, it depends on the preceding rate of
depolarization [15]–[21] and on the preceding interspike intervals [12], [22]. Those properties are functionally important because they enhance the selectivity of neurons in several sensory
modalities, in particular in audition [23], vision [24], and touch [21].
Developmental and learning studies have also shown that the threshold adapts to slow changes in input characteristics. This phenomenon is known as long-term plasticity of intrinsic excitability and
may be involved in the regulation of cell firing, short term memory and learning [25]–[31]. The excitability threshold also varies with the distance to the soma in a given neuron and with cell type
[2], [15], [32]–[35], which may explain functional differences.
The modulation of cell excitability might be explained by the activation of voltage-gated potassium channel Kv1 [36]–[41], inactivation of voltage-gated sodium channels [15], [16], [19], [21],
fluctuations in sodium channel gating [42], inhibitory synaptic conductance [43]–[45] and the site of spike initiation [14]. To understand the origin of spike threshold variability, we examined the
role of several candidate mechanisms in biophysical neuron models: activation and inactivation of the sodium channel, slow voltage-gated channels (e.g. Kv1), synaptic conductances and the site of
spike initiation. Our analysis is based on a simplification of the membrane equation near spike initiation and results in a simple formula for the spike threshold that quantifies the contribution of
all those mechanisms. The threshold formula provides an instantaneous time-varying value which was found to agree well with numerical simulations of Hodgkin-Huxley type models driven by fluctuating
inputs mimicking synaptic activity in vivo, and with simulations of a realistic multicompartmental model of spike initiation [54].
What is the spike threshold?
Spike threshold in vitro.
In a typical in vitro experiment, one measures the response of the cell to a controlled stimulus, whose strength is defined by a parameter (e.g. current intensity). The excitability threshold is then
defined as the minimal value of this parameter above which a spike is elicited. Thus, the threshold is initially defined in stimulus space, for example as a charge threshold for short current pulses
(Fig. 1A, simulated recording) or as a current threshold for current steps or ramps (Fig. 1B). The stimulus threshold corresponds to a voltage value, which we call the voltage threshold, but that
value depends on the type of stimulation [46]. Nevertheless, we are interested in the voltage threshold rather than in the stimulus threshold because only the voltage is usually available in
intracellular recordings in vivo.
All plots were generated using the single-compartment model described in the Materials and Methods. A, In vitro, the neuron is stimulated with short current pulses with increasing intensity (bottom)
and the threshold is the minimal value of that intensity above which the neuron spikes (top). The voltage threshold is the value of the membrane potential at that critical point. B, The threshold can
be defined similarly with current steps (bottom) or other types of parameterized stimulations, yielding different values for the voltage threshold. C, In vivo, spike “threshold” is defined as a
measure of the voltage at the onset of the action potential (black dots). The plot shows a simulated trace of a conductance-based model with fluctuating conductances (see Materials and Methods) and
threshold is measured with the first derivative method. D, Representation of the trace in (C) in phase space, showing dV/dt vs. V. The first derivative method consists in measuring the membrane
potential V when the derivative crosses a predefined value (dashed line) shortly before an action potential. The trace is superimposed on the excitability curve dV/dt=(F(V)+I[0])/C, which defines
the dynamics of the model. I[0] is the mean input current, so that trajectories in phase space fluctuate around this excitability curve.
Spike threshold in vivo.
Since the input to the neuron is not directly controlled in vivo, the concept of spike threshold does not have exactly the same meaning as in vitro. Rather, it is defined as the voltage at the
“onset” of action potentials (Fig. 1 C), as observed on an intracellular recording of the membrane potential. Therefore the spike threshold is an empirical quantity that hopefully captures the same
concept as in vitro, i.e., the point above which an action potential is initiated. Several measures of spike onset have been used in experimental studies [47]. The first derivative method consists in
measuring the membrane potential V when its derivative dV/dt crosses an empirical criterion [8], [34] (Fig. 1D). The second and third derivative methods consist in measuring V when respectively d^2V/
dt^2 and d^3V/dt^3 reach their maximum [12], [21]. Sekerli et al. (2004) compared those methods by asking electrophysiologists to identify spike onsets by eye on several membrane potential traces
[47]. They found that visual inspection was best matched by the first derivative method, although that method critically relies on the choice of the derivative criterion (Fig. 2 C,D). However, all
methods produced the same relative variations of the measured threshold.
A, Excitability curve of the neuron model (dV/dt=(F(V)+I)/C; see Materials and Methods) for DC input current I=0 (solid curve) and (dashed curve). With I=0, the lower equilibrium (filled
circle) corresponds to the resting potential V[r], while the higher equilibrium (open circle) corresponds to the spike threshold with short pulses (as in Fig. 1A): if the membrane potential is
quickly shifted above , the membrane potential blows up and the neuron spikes (thus, this corresponds to the case when , i.e., an impulse current). Slowly increasing the input current amounts to
vertically shifting the excitability curve, and the membrane potential follows the resting equilibrium until it disappears, when . The voltage V[T] at that point corresponds to the minimum of the
excitability curve. The empirical threshold (with the first derivative method) is the voltage at the intersection of the excitability curve with the horizontal line dV/dt=k[th] (dashed line). The
slope threshold corresponds to the radius of curvature at V[T]. B, Threshold for short pulses (solid line) and empirical threshold (blue dashed line) as a function of the threshold for slow inputs V
[T] (black dashed line is the identity line): the definitions are quantitatively different but highly correlated. C, Dependence of empirical threshold on derivative criterion k[th]: spike onsets are
measured on a voltage trace (as in Fig. 1C) with derivative criterion k[th]=7.5 mV/ms (blue dots), 10 mV/ms (black), 12.5 mV/ms (green) and 15 mV/ms (red). D, Empirical threshold measured with k
[th]=7.5 mV/ms (blue dots), 12.5 mV/ms (green) and 15 mV/ms (red) vs. threshold measured with 10 mV/ms, and linear regression lines. The dashed line represents the identity. The value of the
derivative criterion (k[th]) impacts the threshold measure but not its relative variations.
Spike threshold in models.
It might seem confusing that the definition of the voltage threshold is ambiguous and that most modulation effects that have been reported in the literature seem to apply to spike onset rather than
spike threshold. However, as remarked in [47], those measures differ in absolute value but they vary in the same way. We can relate those definitions with a simple one-dimensional neuron model, where
the membrane potential is governed by a differential equation:where C is the membrane capacitance, F(V) is the sum of all intrinsic voltage-dependent currents and I the input current. The dynamics of
the membrane potential is determined by the excitability curve in phase space (dV/dt as a function of V, Fig. 2A). With no DC injected current (I=0, solid curve), the differential equation has two
fixed points, which are solutions of F(V)=0: the lower one is stable and corresponds to the resting potential and the higher one is unstable and corresponds to the threshold for fast
depolarizations (short current pulses, i.e., ), which we denote . Indeed, after depolarization, the membrane potential V either goes back to the resting potential if or keeps on increasing if ,
leading to a spike. If the neuron is progressively depolarized with a slowly increasing current, then the excitability curve slowly shifts upwards, depolarizing the stable potential, until the curve
is entirely above zero and the neuron spikes (Fig. 2A, dashed curve). At that critical point, the curve is tangential to the horizontal axis and the voltage corresponds to the minimum of that curve:
. Thus, the voltage threshold for slow inputs (i.e., DC currents, or slow current ramps) is the solution of F′(V)=0 and the voltage threshold for fast inputs (i.e., instantaneous charge inputs, or
short current pulses) is the solution of F(V)=0 with F′(V)>0.
The current-voltage function F(V) can be approximated by an exponential function near spike initiation (see Materials and Methods), leading to the exponential integrate-and-fire model [48]. In that
model, we can calculate the relationship between the voltage threshold for slow inputs V[T] and the voltage threshold for fast inputs (see Text S1):where is the slope factor, characterizing the
sharpness of spikes (see Materials and Methods). In single-compartment models, this is related to the slope of the Na activation curve. This formula provides a simple monotonous relationship between
the two types of threshold, which is almost linear (the derivative of with respect to V[T] is (), which is close to 1; see Fig. 2B). In our analysis, we chose the definition for slow depolarizations
because it simplifies our formulas, but one can map the results to the definition for fast depolarizations using the formula above.
Empirical threshold measures used in vivo can be analyzed in the same way. For example, the voltage threshold measured by the first derivative method is the value such that dV/dt=k[th], i.e., the
solution of . The empirical threshold can be approximately related to V[T] with the following formula (see Text S1):where is the membrane time constant (R=1/g[L] is the membrane resistance).
Although the relationship is more complex and shows a slight dependence on the input current I (thus increasing apparent threshold variability), it is still related with V[T] through a monotonous (in
fact quasi-linear) relationship and the choice of criterion k[th] results mainly in a shift of the threshold, as shown in Fig. 2D.
In the remaining of this paper, we chose the voltage threshold for slow depolarizations V[T] as the definition of the spike threshold (i.e., the voltage at current threshold).
The threshold equation
Sodium channel activation.
Cells excitability is generally due to the presence of voltage-gated sodium channels [49]. More precisely, Na channel activation gates mediate a positive feedback mechanism, which produces the
instability phenomenon necessary to initiate an action potential. Activation is very fast compared to all other relevant time constants (a fraction of ms), in particular the membrane time constant
[50]. We make the approximation that it is instantaneous, so that the proportion of open sodium channels at any time is . The membrane equation is then:where g[Na] (resp. g[L]) is the maximum Na
conductance (resp. leak conductance) and E[Na] (resp. E[L]) is the Na reversal potential (resp. leak reversal potential). We neglect inactivation and other ionic channels for the moment (see below).
The activation function is well approximated by a Boltzmann function [51] with half-activation voltage V[a] and activation slope factor k[a]. In the relevant part of that function, near spike
initiation, it reduces to an exponential function and the membrane equation reads (see Materials and Methods):whereis the threshold (defined for slow inputs). The activation slope factor corresponds
to the steepness of the Na activation curve, and characterizes the sharpness of spikes in single-compartment models (in the limit mV, the model tends to an integrate-and-fire model; it can be
different in multicompartment models, see Discussion). The slope factor shows little variation across sodium channel types (k[a]=4–8 mV for neuronal channels, Angelino and Brenner, 2007 [51]).
Thus, the threshold is primarily determined by the half-activation voltage and the density of sodium channels in log scale, relative to the leak conductance (see Fig. 3A–C).
A, Excitability curve of the model for different values of the ratio g[Na]/g[L] (maximum Na conductance over leak conductance), discarding inactivation (h=1) and other ionic conductances. The
resulting threshold is shown with a red dot. B, Excitability curve for different values of half-activation voltage V[a]. C, Excitability curve for different values of Boltzmann factor k[a]. D,
Threshold as a function of the ratio g[Na]/g[L] for the 9 types of voltage-gated sodium channels [52] with characteristics reported in (Angelino and Brenner, 2007 [51]). For each channel type, the
mean threshold obtained across the dataset is plotted. Nav1.[1], [2], [3], [6] are expressed in the central nervous system, Nav1.[4], [5] are expressed in cardiac and muscle cells and Nav1.[7], [8],
[9] are expressed in the peripheral nervous system. Nav1.6 is expressed at the action potential initiation site [53]–[55].
This formula provides some quantitative insight about the role of Na channel on cell excitability. For example, Pratt and Aizenman (2007) observed that during development, tectal neurons adapt their
intrinsic excitability to changes in visual input so as to stabilize output firing [28]. They hypothesized that this adaptation was mediated by regulation of Na channel density, which could be
quantitatively evaluated using the formula above. Our formula also explains differences in excitability between cells. There are 9 Na channel types, which are expressed in different regions of the
nervous system [52], and each one has specific properties, in particular specific values of V[a] and k[a]. In Fig. 3D, we show how the threshold varies with channel density for each channel type,
based on the dataset collected by Angelino and Brenner (2007) [51]. For the same channel density, the threshold can differ by up to 50 mV between channel types. Lowest threshold values were found for
Nav1.5, expressed in cardiac cells, and highest ones were found for Nav1.8, expressed in dorsal root ganglion. Interestingly, among all channel types expressed in central neurons, the one with lowest
threshold is Nav1.6, which is expressed in the spike initiation zone in the axon hillock [53]–[55].
Sodium channel inactivation and other conductances.
The threshold can also be modulated by sodium channel inactivation and by the many other ion channels that can be found in neurons [56]–[58]. These factors might explain the effects of preceding
spikes and membrane potential history on cell excitability [56]–[58]. To examine how they may modulate the threshold, we make two important assumptions: 1) inactivation is independent from
activation, 2) these processes are slow compared to the timescale of spike initiation (about a millisecond). We then consider the membrane equation near spike initiation:where h is the inactivation
variable and g[i] is the conductance of channel i, which may be voltage-gated (K+ channel) or synaptic. The contribution of additional ionic channels can be summed to yield an effective channel with
conductance and reversal potential E* (see Materials and Methods), while the inactivation variable h can be entered into the exponential function:whereis the threshold (mathematically, it satisfies
F′()=0, where F is the current-voltage function of the model). We call the formula above the threshold equation. It provides the instantaneous value of the spike threshold as a function of the
sodium inactivation variable h (1-h is the proportion of inactivated Na channels) and of the other ionic channel conductances, including synaptic conductances. To obtain this equation, we made a
quasi-static approximation, i.e., we assume that all modulating variables (h and g[i]) vary slowly at the timescale of spike initiation. We note that the threshold is determined by the value of
conductances relative to the leak conductance rather than by their absolute value.
Fig. 4 illustrates the dependence of threshold on Na inactivation and conductances. As expected, the threshold increases when h decreases, that is, when more Na channels inactivate. It also increases
with the total non-sodium conductance, which is also intuitive: more Na conductance is required to produce a spike when the other conductances are larger. Threshold modulation is proportional to the
slope factor , which shows little variation across Na channel types (4–8 mV in neuronal channels).
A, Spike threshold θ as a function of Na+ inactivation variable h, with all other ionic conductances suppressed. B, Threshold as a function of K+ activation variable n, without inactivation (h=1).
C, Threshold as a function of total synaptic conductance (excitatory g[e] and inhibitory g[i]), relative to the resting conductance g[L] (conductances are considered static).
The threshold equation predicts several effects. Spike threshold should be higher in vivo than in vitro because the total conductance is several times larger [59]. For the same reason, it should also
be higher in up states than in down states. It is correlated with sodium inactivation, so that it should increase with the membrane potential, as observed in vitro and in vivo [15], [16], [54].
Besides, threshold modulation by inactivation is strongest when many Na channels are inactivated (h close to 0), that is, when the neuron is depolarized. Spike threshold is correlated with
voltage-gated conductances such as those of K+ channels. For high-threshold K+ channels with large conductance, the spike threshold increases by when the membrane potential increases by k[a]^K+
(slope of K+ channel activation function). Indeed, far from half-activation value V[a]^K+, the K+ activation curve is approximately , which implies that threshold modulation is constant (provided K+
conductance is large enough). It also increases after each action potential (see below). Inactivation and adaptive voltage-gated conductances (e.g. Kv1) have similar effects but inactivation is
“invisible”, in the sense that it affects excitability without changing the membrane potential or the total conductance.
Threshold dynamics
To derive the threshold equation, we made a quasi-static approximation, assuming that all mechanisms that modulate the threshold are slow processes (compared to the timescale of spike initiation).
That threshold equation provides an instantaneous value of the spike threshold, as a function of modulating variables. Here we show how the dynamics of sodium inactivation, voltage-gated conductances
and synaptic conductances translate into spike threshold dynamics.
Sodium inactivation.
Several authors have hypothesized that Na inactivation is responsible for experimentally observed threshold variability in vivo [12], [15], [16], [21]. We have shown that the instantaneous value of
the spike threshold depends on the value of the inactivation variable h (1-h is the proportion of inactivated channels). We assume, as in the Hodgkin-Huxley model, that h evolves according to a
first-order kinetic equation:where is the time constant and is the equilibrium value. This differential equation translates into a differential equation for the threshold θ (see Materials and Methods
), which can be approximated by a similar first-order kinetic equation :where is the equilibrium value of the threshold. A linearized version of this equation was recently proposed as a simplified
model of post-inhibitory facilitation in brainstem auditory neurons [60]. This is also consistent with previous results in vitro showing that the instantaneous value of the threshold increases with
the membrane potential [61].
This equation allows us to predict the time-varying value of the threshold from the membrane potential trace (provided that Na inactivation properties are known). The threshold time constant is given
by the inactivation time constant (which is voltage-dependent). Fig. 5 shows how the spike threshold varies in a biophysical model with fluctuating synaptic conductances.
A, Voltage trace of the fluctuating conductance-based model (black line) and predicted threshold according to our threshold equation (, red line), calculated continuously as function of h, g[K], g[e]
and g[i]. Black dots represent spike onsets (empirical threshold with the first derivative method). B, Predicted threshold vs. membrane potential for the trace in A. Trajectories lie above
theoretical threshold on the right of the dashed line (). C, Zoom on the second spike in A. Colored lines represent increasingly complex threshold predictions: using Na activation characteristics
only (blue, ), with Na channel inactivation (green, ), with potassium channel activation (purple, ) and with synaptic conductances (red, ). Here the threshold varies mainly after spike onset.
The effect of previous spikes on spike threshold, which is presumably due to slow Na inactivation [12], can be understood by looking at how an action potential acts on the inactivation variable h.
Typical equilibrium curves for Na inactivation are Boltzmann functions with half-activation values mV and Boltzmann coefficients mV [51], so that is close to 0 after spike initiation. Thus during the
action potential, the inactivation variable relaxes to 0 according to the following equation:If we note the average value of the time constant during the action potential and the spike duration
(typically, a few ms), then the effect of an action potential on h is a partial reset: , which translates for the threshold into a shift: . In other words, the spike threshold increases by a fixed
amount after each spike, which contributes to the neuron refractory period (see Fig. 5C). This effect was recently demonstrated in vitro [22] and explains in vivo observations where the threshold was
found to be inversely correlated with the previous interspike interval [12]. If the inactivation time constant is long compared to the typical interspike interval, we predict that the threshold
should be linearly correlated with the firing rate.
Voltage-dependent conductances.
In the same way, the dynamics of voltage-dependent conductances translates into threshold dynamics. Potassium currents, in particular Kv1 delayed rectifier currents, are also thought to play a role
in threshold modulation [36]–[41]. Let us consider a current with Hodgkin-Huxley-type kinetics: , with (n is the activation variable). The corresponding equation for the threshold dynamics then reads
(see Materials and Methods):where is the equilibrium threshold value (we neglected Na inactivation). Thus, the threshold adapts to the membrane potential. The effect of action potentials can be
described similarly as for Na inactivation, except n relaxes to 1 during the action potential, yielding the following reset: . It also results in threshold increase, although it is not additive. This
effect also contributes to the neuron refractory period, not only by decreasing the membrane resistance, but also by increasing the spike threshold (see Fig. 5C).
Synaptic conductances.
Finally, synaptic conductances fluctuate in vivo, which also impacts the instantaneous value of the threshold, through the following equation:where we neglected Na inactivation and voltage-gated
conductances to simplify the formula, and g[e](t) (resp. g[i](t)) is the excitatory (resp. inhibitory) synaptic conductance. This formula emphasizes the fact that the threshold equation defines an
instantaneous value, which applies to realistic in vivo situations where synaptic activity fluctuates. However, we need to make the approximation that fluctuations are slow compared to spike
Spikes can be triggered either by an increase in the excitatory conductance or by a decrease in inhibitory conductance. In the former case, the total conductance increases and the threshold increases
while in the latter case the threshold decreases. In high-conductance regimes (typical of cortical neurons in vivo), it has been argued that spikes are mainly triggered by inhibitory decrease because
synaptic inhibition is dominant [59], [62]. It might imply that faster depolarization corresponds to lower inhibitory conductance and lower threshold, so that depolarization speed is inversely
correlated with spike threshold, as observed experimentally [15]. However, this effect is fundamentally limited by the fact the inhibitory conductance cannot be negative.
Spike initiation site
Effect of neuronal morphology.
Spikes are initiated in the axon initial segment (AIS) in spinal motoneurons [63] and in cortical neurons [64], about 35–50 µm from the soma [54], [55]. Our analysis relies on a single-compartment
model of spike generation, but the axon hillock is connected with the soma through a large section and with the rest of the axon through a smaller section. To evaluate how electrotonically far the
spike initiation site is from the soma, we can compare the length of the AIS to its electrotonic length, given by the following formula [65]:where µm is the diameter, .cm is the intracellular
resistivity, and .cm^2 is the membrane specific resistance [66]–[68]. We obtain a value of 935 µm, many times larger than the distance between the soma and the initiation site. Therefore, below
threshold, it is reasonable to consider the soma and AIS as a single electrotonic compartment. Indeed, simultaneous measurements at both sites show that the voltage time course is nearly identical at
the two sites before spike initiation [14], [34]. We provide a more detailed analysis in Text S1. We note that the situation changes when an action potential is initiated, because the opening of Na
channels reduces the electrotonic length and invalidates the single compartment approximation, which has implications on the shape of action potentials (see Discussion).
For the threshold equation, these considerations imply that conductance values in the equation refer to total conductances over the surface of the soma, proximal dendrites and AIS. Since channel
densities are different on these sites, the total conductance for a given ionic channel is , where G[soma] (resp. G[dendrites], G[AIS]) is the channel density on the soma (resp. dendrites, AIS) and S
[soma] (resp. S[dendrites], S[AIS]) is the area of the soma (resp. dendrites, AIS). We give a specific example below.
Na channel density in the AIS.
Spikes could be initiated in the AIS rather than in the soma because of higher Na channel density [66], [69]–[71], or lower Na half-activation voltage V[a] [72] in the first segment. Recent
experiments and computational modeling suggest that the former hypothesis is more plausible [34], [69], [70]. As an application of our analysis, we can estimate the Na channel density at the AIS
using the parameter values reported in [70]. Since Na channels are mainly located in the AIS, we use . The measured spike threshold (at the AIS) was 54 mV. To calculate the total leak conductance, we
injected a DC current into the soma (using the published model) and obtained g[L]=59 nS. We chose this direct method because it is difficult (although possible using linear cable theory) to
calculate the total leak conductance using the neuronal morphology, because some of the dendrites may be electrotonically far. The threshold equation relates the threshold value to the values of g
[Na], g[tot] and the Na channel properties. We can easily invert this relationship, which gives the following formula:Using the values from Kole et al. (2008) [70] for the channel properties and
neuron geometry (V[a]=−31.1 mV, k[a]=6.5 mV, S[AIS]=871.3 µm^2, E[Na]=55 mV), we find 2463 pS/µm^2, which is very close to the empirically reported value (2500 pS/µm^2).
Accuracy of the threshold equation
Threshold dynamics in a single-compartment model.
To evaluate the quality of the threshold equation, we first simulated a biophysical single-compartment model with fluctuating synaptic conductances, mimicking the effect of synaptic activity in vivo.
The instantaneous value of the threshold was measured by injecting brief current pulses of varying amplitude in repeated trials with the same synaptic inputs (Fig. 6A, B; see Materials and Methods),
and we compared this time-varying value with the prediction from the threshold equation, including the effects of Na inactivation, voltage-gated channels and synaptic conductances. We used this
particular stimulation protocol to measure the value of the threshold at any time, rather than only at spike time. We shifted the Na inactivation curve by −12.5 mV so as to obtain more threshold
variability (the original model shows little variability). The threshold equation predicted the variations of the measured threshold very well (83% of the variance), with a constant shift which can
also be predicted (Fig. 6C, D). This shift has two causes. Firstly, the threshold was measured with brief pulses whereas the predicted threshold corresponds to the definition with slow inputs. Using
our formula relating the two definitions (Text S1) indeed reduced this shift from 13.5 mV to 7.4 mV (Fig. 6D). Secondly, because we had to shift the inactivation curve to observe substantial
threshold variability, spike threshold was depolarized closer to Na half-activation voltage (−30 mV) and the activation curve is less exponential in that region. Indeed, if V[T] is calculated as the
minimum of the excitability curve rather than with the exponential formula, we find V[T]=−60.6 mV, which exactly compensates the 7.4 mV shift. When these two predictable biases were taken into
account, both the mean and time course of the prediction matched the measured threshold (Fig. 6C, dashed red line). When we did not shift Na inactivation as much, these biases were reduced but the
model displayed little variability, which made the prediction less interesting. We address this point in more detail in the Discussion.
A, Five superimposed voltage traces of the fluctuating conductance-based model (black traces) stimulated at different times with random depolarization (blue dots show the value of the membrane
potential just after the stimulation). Synaptic conductances are identical on all trials. In these examples the stimulations elicited spikes, in other cases (smaller depolarization) they did not. The
theoretical threshold is shown in red. B, At a given time (here t=50 ms), trials with varying depolarization are compared and the measured threshold is defined as the minimal depolarization that
elicits a spike (blue dot). C, Predicted threshold (red line) and measured threshold (blue) as a function of time. The shift is mainly due to the fact that the measured threshold is defined with fast
inputs (charge threshold) whereas the theoretical threshold is defined with slow inputs: this bias can be calculated and corrected for, as shown by the dashed red line (see also text). D, Measured
threshold vs. theoretical threshold for the entire trace (blue dots; blue line: linear regression). The dashed line represents the ideal relationship, taking into account the theoretical difference
between threshold for fast inputs and for slow inputs ().
In this single-compartment model, threshold variability is much lower than observed in vivo. However, the half-inactivation voltage V[i] in the model is −42 mV, while experimental measurements
suggest values around −60 mV in central neurons (e.g. Kole et al. (2008) [70]). According to our analysis, this reduces threshold variability because Na channels do not inactivate below threshold
(log h≈0). In Fig. 7, we hyperpolarized V[i] by 20 mV, giving V[i]=−62 mV, close to experimental values, and measured the spike threshold with fluctuating inputs (Fig. 7A). We found that the
threshold varied over more than 10 mV and the standard deviation 2.2 mV (Fig. 7B), similar to values reported in vivo [15]. According to the threshold equation, most threshold variability was due to
Na inactivation. A linear regression at spike times gave (mV) (Fig. 7C). This 3.1 mV factor is close to the value of k[a] in this model, as measured by fitting a Boltzmann function to the Na
activation curve around −50 mV (see Discussion). We also observe that, in this single-compartment simulation, many spikes were small (Fig. 7A). This is not unexpected, because spikes should be
smaller when Na channels are partially inactivated. However, this property should not be taken as a prediction, because it is known that the correct spike shape of cortical neurons cannot be
recovered in single-compartment models [10], [13].
A, We simulated the same model as in Fig. 6, but the half-inactivation voltage V[i] was shifted to −62 mV (instead of −42 mV in the original model) to increase threshold variability. As a result,
spike height was also more variable. B, The threshold distribution (red) spanned a range of more than 10 mV (standard deviation 2.2 mV) and overlapped with the membrane potential distribution
(black). C, According to the threshold equation, most threshold variability was due to Na inactivation. Black dots show the measured threshold vs. the inactivation variable h (in log scale) at spike
times. The linear regression (red line) gives a slope of 3.1 mV, close to the value of k[a] in this model (Fig. 9D).
Threshold prediction in a realistic multicompartmental model of spike initiation.
We then checked the accuracy of the threshold equation with a realistic multicompartmental model of spike initiation, where action potentials are initiated in the axon [54]. We injected a fluctuating
current in the soma and compared measured spike thresholds with our theoretical predictions (Fig. 8). Spikes were initiated in the axon 400±60 µs earlier than observed at the soma (Fig. 8B). When
action potentials were removed from the voltage traces, the membrane potential was 1.8±0.6 mV higher at the soma than at the spike initiation site in the axon initial segment (AIS; Fig. 8C). The
threshold measured at the soma was −47.7±2.8 mV and varied between −52.1 mV and −42.2 mV (Fig. 8D). Its distribution significantly overlapped the subthreshold distribution of the membrane potential,
as observed in vivo. We estimated the activation properties of the Nav1.6 channel, which is responsible for spike initiation in this model, by fitting a Boltzmann function to the activation curve ()
in the spike initiation zone (−60 mV to −40 mV). We found V[a]=−33 mV and k[a]=3.6 mV (Fig. 8E, F). This is different from experimentally reported values (in particular, k[a] is smaller) because
these were obtained by fitting the activation curve on the entire voltage range. We address this specific point in the Discussion. We then calculated the total maximal conductance of the Nav1.6
channels (over the AIS), the slow K+ channels (Km) and the fast K+ channels (Kv), using the published morphology and channel density (see Materials and Methods).
A, Voltage trace at the soma (black) and at the spike initiation site in the axon initial segment (AIS, blue) in response to a fluctuating current. The spike threshold was measured at the soma when
dV/dt exceeded 10 V.s^−1 (red dots). B. Zoom on an action potential: spikes were initiated at the AIS 400±60 µs before observed at soma. C. Between spikes, the membrane potential was slightly higher
at the soma than at the AIS (1.8±0.6 mV). D. The spike threshold (measured at the soma) was very variable (standard deviation 2.8 mV): its distribution spanned 10 mV (−52 to −42 mV) and significantly
overlapped the subthreshold distribution of the membrane potential (i.e., with spikes removed). E, F. We fitted the activation curve of the Nav1.6 channel (black) to a Boltzmann function (red) in the
spike initiation zone (rectangle and panel F), yielding V[a]=−33 mV and k[a]=3.6 mV. G. Measured threshold (red: at the soma, black: at the AIS) vs. theoretical prediction for all spikes. The
dashed line represents equality (measurement=prediction). H. Somatic membrane potential vs. theoretical threshold at all times. Spikes are shown in black (defined as voltage trace 7 ms from spike
onset), subthreshold trajectories in blue and spike times as red dots: spikes are indeed initiated when the membrane potential exceeds the theoretical threshold (inset: zoom on spike onsets).
Using these estimated values and the time-varying values of the channel variables (h, n[Km], n[Kv]^4) at the AIS, we calculated the theoretical threshold at all times, and compared the prediction
with the measured threshold at spike times (Fig. 8G). Values of the channel variables were taken at the time of spike initiation in the AIS and the threshold was measured at the AIS (black) and at
the soma (red). The prediction with the threshold equation was very good: the average error was 0.7 mV. The threshold prediction was on average only 0.49 mV higher than the measured threshold.
However, this excellent match is probably fortunate because the value of the measured threshold is correlated with the measurement criterion (on dV/dt) and in general, we would expect a constant
shift between prediction and measurement. When this shift was removed, the average prediction error was 0.53 mV. Among the different contributions to the threshold, we found that only Na inactivation
had a significant impact. The fast K+ current (IKv) had a very high maximum conductance but was only activated after spike initiation, while the slow K+ current (IKm) had a small maximum conductance.
According to the threshold equation, total conductance contributed only 0.07 mV to threshold variability. A linear regression gave with V[T]=−56mV and =3.6 mV, very close to our predicted values,
and the average estimation error with this formula was 0.08 mV.
These results show that the value of the membrane potential at spike onset is well predicted by the threshold equation. However, to prove that our equation really defines a spike threshold, we also
need to show that the membrane potential is always below the predicted threshold before spikes. In Fig. 8H, we plotted the membrane potential vs. the predicted threshold for the entire voltage trace
(5 seconds). It clearly appears that the neuron spikes when its membrane potential exceeds the predicted threshold, and that the potential is always below threshold between spikes.
The spike threshold differs between cells and for different types of stimulations [2], [15], [32], [33], [35]. We have identified several modulation factors, whose quantitative influence is
summarized by the threshold equation:That formula relates the value of the threshold to the activation and inactivation properties of the Na channel, the properties of other voltage-gated channels
such as Kv1 and synaptic conductances (g[tot] is total conductance, excluding Na conductance). It consists of a static part (first two terms), determined by the properties of Na channel activation,
and of a dynamic part, which depends on the proportion of inactivated Na channels (1-h) and on the total conductance of other channels.
It describes the voltage threshold at the site of spike initiation (rather than at the soma), and is correlated but not identical to empirical “threshold” measures, which measure spike onset rather
than threshold (those normally overestimate the threshold). From that formula, we were able to derive a dynamical equation for the instantaneous threshold, which explains the variability of the spike
threshold in the same cell and predicts its relationship with previous membrane potential history. We found that the threshold equation was a good predictor of the time-varying threshold in
biophysical models with fluctuating inputs (Fig. 6– 8).
Mechanisms for threshold modulation and variability
Since Na channels are responsible for the generation of action potentials, the threshold is firstly determined by their activation characteristics. Activation curves for Na channels are well
approximated by Boltzmann functions with similar slope factors (k[a]=4–8 mV in neuronal channels). The threshold is linearly related to the half-activation value V[a] and logarithmically related to
the maximum Na conductance g[a]. The threshold also depends logarithmically on the Na inactivation variable h, so that it increases with the membrane potential and with every emitted spike. The
modulating effect of inactivation is most pronounced when the half-activation value V[i] is lowest (i.e., Na channels are partially inactivated at rest). Finally, the threshold depends
logarithmically on the total conductance, which includes the leak conductance, voltage-gated conductances and synaptic conductances. In particular, Kv1 channels, which are expressed with high density
at the spike initiation site [37], [39], [41], increase the threshold in an adaptive manner (the threshold increases with the membrane potential). This change in threshold occurs simultaneously with
the effective membrane time constant, whereas threshold changes due to Na inactivation have no effect on the time constant, which might suggest a way to experimentally distinguish between the two
effects. Indeed, the effective membrane time constant (as measured in vivo for example in Léger et al., 2003 [73]) is (C is the membrane capacitance) while the effect of total conductance on spike
threshold varies as , therefore as −. It is currently unclear whether threshold modulation is mainly due to Na inactivation or delayed-rectifier K currents. Our simulations with the
multicompartmental model of spike initiation in pyramidal cells [54] suggest that the spike threshold is essentially determined by Na inactivation, but this may not be universally true. Recent
experimental findings in hippocampal mossy fibers [74] suggest that delayed K+ currents are closed at spike initiation, which minimizes charge movements across the membrane and is thus more
metabolically efficient. It emphasizes the fact that Na inactivation is a more metabolically efficient way to modulate spike threshold than K+ activation, since the former reduces charge transfer
while the latter increases it.
We have not considered the effect of channel noise, i.e., fluctuations in Na channel gating [42], [75]–[78], which result in random threshold variations. Although dynamical equations of fluctuations
in Na channel gating are well set [79], [80], they cannot be included in our theoretical framework because we neglected the time constant of Na activation (which leads to the exponential model).
There are two additional sources of variability which are artefactual: the fact that the threshold is not measured at the site of spike initiation, and threshold measurement methods. The latter
source is difficult to avoid in vivo because only spike onsets can be measured. The former one also seems technically very difficult to avoid in vivo, since spikes are initiated in the axon hillock,
which is only a few microns large. Although the soma and AIS are virtually isopotential below threshold, experimentally measured values of threshold differ between the two sites [34] because, as we
previously remarked, in vivo measurements correspond to spike onset rather than threshold and therefore take place after spike initiation, when the two sites are not isopotential anymore. This
experimental difficulty may introduce artefactual variability in threshold measurements [14].
Approximations in the threshold equation
To derive the threshold equation, we made several simplifying assumptions. First, we assumed that Na activation is instantaneous. It is indeed significantly faster than all other time constants but
not instantaneous. The approximation is legitimate as long as the effective membrane time constant in the membrane equation is small (, including all conductances), which is generally true before
threshold. When Na channels open, the Na conductance dominates the total conductance and drastically reduces the effective time constant. Thus, we expect this approximation to be reasonable to
predict spike initiation properties but not spike shape characteristics. Our second major assumption is a quasistatic approximation, i.e., we assume that near spike initiation, all modulating
variables and the input current can be considered as constant. In other words, we assume that the time constants (except that of Na activation) are larger than a few ms. This is clearly only a
mathematically convenient approximation, but our predictions empirically agreed with numerical simulations. To investigate the role of Na inactivation, we also assumed that activation and
inactivation are independent, which is a standard simplifying hypothesis (Hille, 2001). Although it is debatable [49], [56], it should remain valid in the case where activation and inactivation time
constants are well separated.
We also assumed that Na activation and inactivation curves were Boltzmann functions. Experimental data is indeed well fitted by Boltzmann functions, but the reported parameter values (V[a], k[a])
correspond to fits on the entire voltage range, whereas we are interested in hyperpolarized voltage regions where the activation values are small. When only the relevant part of the experimental data
is considered, different parameter values might be obtained. For example, when analyzing previously published biophysical models, we found that better results were obtained when Na activation curves,
which were not exactly Boltzmann functions, were fitted in the spike initiation region (−60 to −40 mV) rather than on the entire voltage range (Fig. 9). We examined this issue in the biophysical
model used in this paper (see Materials and Methods). The Na activation curve of this model seemed to be well fit to a Boltzmann function (Fig. 9A), however the fit was poor in the spike initiation
zone (−60 to −40 mV, Fig. 9C) where activation is close to zero, which makes fit errors relatively larger. Although the slope factor is about 6 mV when the activation curve is fit over the entire
voltage range, similar to experimental measurements [51], it is only half this value when fit in the spike initiation region (Fig. 9D), which explains why this model, as many other biophysical
models, exhibits little threshold variability (since threshold modulation is proportional to ). We calculated the slope factor as a function of the voltage region and we found that it varies between
1 and 6 mV (Fig. 9D). This finding motivates a reexamination of Na channel voltage-clamp data, focusing on the spike initiation region rather than on more depolarized regions, which are more relevant
for spike shape than spike initiation. Fig. 10 addresses two potential difficulties. In experiments, activation curves are obtained by measuring the peak conductance after the clamp voltage is
changed from an initial value V[0] to a target value V, and normalizing over the entire range of target voltages. Thus, it assumes that inactivation is still at resting level h(V[0]) when the peak
current is measured. This would not be the case if the inactivation time constant τ[h] were close to the activation time constant τ[m]. Fig. 10A shows the effect of this overlap on the measurement of
k[a] with simulated voltage-clamp data, where is a Boltzmann function with k[a]=6 mV. It appears that k[a] is overestimated if τ[h] is very close to τ[m], up to 50% when the two time constants are
equal (to 0.3 ms in these simulations). However the error quickly decreases as τ[h] increases (e.g. 10% error for τ[h]=1 ms). Another potential difficulty is the lack of data points in the relevant
voltage range and the measurement noise, because currents are small. In Fig. 10B, we digitized an experimentally measured activation curve (black dots), where clamp voltages were spaced by 5 mV. A
Boltzmann fit over the entire voltage range gave k[a]=7.2 mV while a fit over the hyperpolarized region V<−40 mV gave k[a]=4 mV. However, the latter is not a reliable estimate because it
corresponds to only 4 non-zero data points, which also seem to be corrupted by noise. Therefore it might be necessary to perform new measurements, specifically focusing on the spike initiation zone,
perhaps with multiple measurements to reduce the measurement noise. Alternatively, k[a] could be measured with a phenomenological approach, using white noise injection in current clamp [22]. Another
possible approach would be to directly fit the excitability model to the current-clamp response of a cell in which only Na channels would be expressed (perhaps with fluctuating inputs).
A, The Na channel activation curve of the conductance-based model (black line) was fit to a Boltzmann function on the entire voltage range (dashed blue line) and on the spike initiation range only
(−60 mV to −40 mV, red line). The green line shows the exponential fit on the spike initiation range. B, In the hyperpolarized region (zoom of the dashed rectangle in A), the global Boltzmann fit
(dashed blue line) is not accurate, while the local Boltzmann fit and the local exponential fit better match the original curve. C, For hyperpolarized voltages (<−50 mV), the resulting excitability
curve is closer to the original curve (black) with a local Boltzmann fit (red) than with a global fit (dashed blue), yielding more accurate threshold estimations (dots). D, The estimated Boltzmann
slope k[a] is very sensitive to the position of the fitting window and varies between 2 mV and 6 mV.
A, Estimation of the activation slope k[a] from simulated voltage-clamp data as a function of the inactivation time constant τ[h]. The model was of a membrane with only Na channels, and activation
and inactivation curves were Boltzmann functions (see Materials and Methods). The activation slope was measured by a Boltzmann fit in the hyperpolarized region (<−40 mV). The activation slope k[a]
was 6 mV in the model (dashed line), but the measurement overestimated it when the inactivation time constant was very close to the activation time constant. B, Na activation curve measured in vitro
(dots, digitized from [86]) and Boltzmann fits over the entire voltage range (dashed curve) and over the hyperpolarized range (V<−40 mV, solid curve).
Finally, our analysis relies on a single compartment model. In the compartmental model, we found that between spikes, the membrane potential was 1.8±0.6 mV more depolarized at the soma than at the
AIS. This is small compared to the slopes of all activation and inactivation curves in this model (5–9 mV). This agrees with our analysis of the electrotonic length in the subthreshold range, which
is much larger than the distance between the soma and the AIS, although very fast synaptic inputs or proximal axonal inhibition could produce larger voltage gradients. Thus, our analysis should
remain valid if the compartment represents both the soma and initiation site (and also proximal dendrites). However, that approximation is not valid anymore after spike initiation (see below).
Sharpness of spikes and threshold variability
Spikes look sharper in the soma than in the AIS, presumably because they are initiated in the AIS and back-propagated to the soma [10], [13], [14]. That property is also seen in numerical simulations
of multicompartmental models [34], [70]. Yet, linear cable theory predicts the opposite property: the voltage at the soma is a low-pass filtered version of the voltage at the AIS, therefore spikes
should look less sharp in the soma. Thus, increased sharpness must be due to active backpropagation of the action potential, which cannot be seen in a two compartment model (such as described in Text
S1). From a theoretical point of view, the sharpening effect of backpropagation can be intuitively understood from the cable equation:It appears that the membrane equation is augmented by a diffusion
term, which is positive and large in the rising phase of the action potential between the initiation site and the soma. Thus, for the same membrane potential V, the time derivative gets larger as
this diffusion term increases, which sharpens action potentials.
Sharpness can be measured in numerical simulations by plotting dV/dt vs. V in response to a suprathreshold DC current, and fitting it to an exponential model (). In the model of Hu et al. [54], we
found that the slope factor, characterizing spike sharpness, was =1.6 mV in the AIS and only 0.8 mV in the soma. This is in approximate agreement with empirical fits of exponential
integrate-and-fire models to cortical neurons stimulated with fluctuating inputs, which reveal a surprisingly small slope factor , slightly above 1 mV [22]. Thus, in the multicompartmental model,
active backpropagation did increase spike sharpness in the soma, but also in the AIS, since the slope factor was about twice smaller than predicted from fitting the Na activation curve to a Boltzmann
function (3.6 mV). This increased sharpness did not affect the magnitude of threshold modulation. In single-compartment models, sharpness of spikes and threshold modulation are determined by the same
quantity, related to the sharpness of the Na activation curve (k[a]). It appears that this link does not hold anymore when active backpropagation is considered (in multicompartmental models). Thus,
in the threshold equation, the modulating factor is indeed k[a] (from the Na activation curve) rather than (from spike sharpness, measured in the phase plot (dV/dt, V)). This explains that Na
inactivation can produce large threshold variability (10 mV in our simulations) even though spikes are very sharp.
Materials and Methods
Membrane equation
We consider a single-compartment neuron model with voltage-gated sodium channels and other ion channels (voltage-gated or synaptic), driven by a current I. The membrane potential V is governed by the
membrane equation:where C is the membrane capacitance, g[L] (resp. E[L]) is the leak conductance (resp. reversal potential), g[i] (resp. E[i]) is the conductance (resp. reversal potential) of channel
i, g[Na] (resp. E[Na]) the maximum conductance (resp. reversal potential) of sodium channels, P[Na] is the proportion of open Na channels and I is the input current. In this article, we used the
following convention for conductances: lower case (g) for the total conductance over the surface of a compartment (typically in units of nS) and upper case (G) for conductances per unit area (in
units of S/cm^2).
We assume that sodium channel activation and inactivation are independent, as in the Hodgkin-Huxley model [3], i.e., , where P[a] is the probability that activation gates are open and P[i] is the
probability that a channel is inactivated. Following the Hodgkin-Huxley formalism, we define . The steady-state activation curve can be empirically described as a Boltzmann function [51]:where is the
half-activation voltage () and the activation slope factor (). We make the approximation that Na activation is instantaneous and we replace P[a] by its equilibrium value, so that .
Exponential approximation
With instantaneous activation, the sodium current is:Action potentials are initiated well below V[a] (about −30 mV, Angelino and Brenner, 2007 [51]), so that except during the spike. Similarly, E[Na]
is very high (about 55 mV), so that is not very variable below threshold. We make the approximation and we obtain:where . This approximation is meaningful for spike initiation but not for spike
shape. With a reset (ignoring inactivation and other ionic channels), we obtain the exponential integrate-and-fire model [48], which predicts the response of cortical neurons to somatic injection
with good accuracy, in terms of spike timings [22], [81], [82]. In this model, V[T] is the voltage threshold for constant input currents I and k[a] (originally denoted Δ[T]) is the slope factor,
which measures the sharpness of spikes: in the limit mV, the model becomes a standard integrate-and-fire model with threshold V[T] (although this is different in multicompartmental models, see
Discussion). The resulting approximated model is thus:It is convenient to sum all conductances (except for the Na channel), which gives a simpler expression:where is the total conductance and E* is
the effective reversal potential:Finally, the inactivation variable h can be inserted in the exponential function:whereis the voltage threshold if all other variables are constant, i.e., it is such
that F′()=0, where F is the current-voltage function.
Dynamic threshold
The effect of Na inactivation on the threshold can be seen in the exponential model above, neglecting other conductances (thus ). Assuming that inactivation is slow compared to spike initiation
(quasi-static approximation), the voltage threshold is now , and it changes with the inactivation variable h. We assume, as in the Hodgkin-Huxley model, that inactivation has first-order kinetics:The
steady-state value of the threshold is thus . We differentiate the threshold equation with respect to time:We now express h as a function of using the invert relationships: and :If the threshold
remains close to its steady-state value (), the equation simplifies to:with . The same method applies for voltage-gated conductances (e.g. Kv1).
Numerical simulations
We compared our theoretical predictions with numerical simulations of a previously published point-conductance model with fluctuating synaptic inputs [83]. The membrane equation is:where and n are
respectively the maximal conductance and the activation variable of thedelayed-rectifier potassium current, and and p are respectively the maximal conductance and the activation variable of the
non-inactivating K current. All channel variables have standard Hodgkin-Huxley type dynamics.
In Fig. 3A–C, only Na channel activation was considered, with instantaneous dynamics, i.e., , h=1, n=0, p=0, I=0:In Fig. 3D, the threshold equation was used to calculate V[T] for the Na
channel properties reported in Angelino and Brenner (2007) [51], since only the values of V[a] and k[a] were available.
To evaluate our threshold equation with time-varying inputs (Figs. 2C, 5 and 6), we simulated the full conductance-based model with fluctuating synaptic conductances (same parameters as in Destexhe
et al., 2001 [83]). In Fig. 6, we shifted the voltage dependence of Na inactivation toward hyperpolarized potentials by −12.5 mV so as to obtain more threshold variability. To measure the
time-varying threshold, we used a similar method as one previously used in vitro by Reyes and Fetz [84], [61]. We simulated the model for 200 ms and measured the instantaneous threshold at regular
time intervals T as follows. The model was simulated repeatedly with the same synaptic inputs (frozen noise). In each trial, the neuron was depolarized at time nT (only once per 100 ms run) to a
voltage value between −51 mV and −38 mV. With T=0.6 ms and 65 voltage values, we ran 22,000 trials. The threshold at a given time is defined as the minimal voltage value above which a spike is
elicited. The measured threshold was compared to the prediction obtained with the threshold equation (see Results), where V[T] and k[a] were obtained from a Boltzmann fit to the activation function
over the range −51 mV to −38 mV, giving V[T]=−68 mV and k[a]=3.7 mV (V[a]=−30.4 mV). The values of V[T] and k[a] depended on the fitting window (see Discussion and Fig. 9). In Fig. 7, the
voltage dependence of Na inactivation was shifted by −20 mV to induce more threshold variability (giving V[i]=−62 mV instead of −42 mV with the original parameter values) and the maximum Na
conductance was multiplied by 3 (to keep threshold values in the same range). The standard deviations of synaptic conductances were also increased.
In Fig. 8, we simulated a multicompartmental model of spike initiation recently published by Hu et al. (2009) [54], with fluctuating injected current modeled as an Ornstein-Uhlenbeck process (mean
0.7 nA, standard deviation 0.2 nA, time constant 10 ms). The model was otherwise unchanged. The spike threshold, both at the soma and AIS, is defined at the voltage value when dV/dt first exceeds 10
V.s^−1 preceding a spike. In some panels (Fig. 8C, D, H), we extracted spikes from voltage traces by removing parts between spike onsets and 7 ms later. We estimated the activation properties of the
Nav1.6 channel, which is responsible for spike initiation in this model, by fitting a Boltzmann function to the activation curve () in the spike initiation zone (−60 mV to −40 mV), which gave V[a]=
−33 mV and k[a]=3.6 mV. We then calculated the total maximal conductance of the Nav1.6 channel over the AIS, by integrating the channel density over the surface of the AIS (using the morphology
and channel density implemented in the published model code). We found g[Na]=236 nS. Calculating the total leak conductance in this way was more difficult because leak channels were uniformly
distributed on the whole morphology, including the dendrites, so that spatial attenuation should be taken into account. While this is theoretically possible using linear cable theory, we chose a
simpler approach by directly measuring the membrane resistance at the soma with a DC current injection, and we found g[L]=38 nS. With these values, the threshold equation predicted that the base
threshold is V[T]=−55.9 mV. The model had a slow K+ current (Im) with the same channel density as the leak channels. Therefore the maximum total conductance was estimated as g[Km]=g[L]=38 nS.
It also had a fast K+ current which was distributed inhomogeneously on the whole neuron morphology, including dendrites. We estimated its total maximum conductance as , where the effective dendritic
area was estimated from the ratio of total leak conductance over leak channel density, i.e., . We found g[Kv]=906 nS. We then calculated the theoretical threshold using these parameters and the
instantaneous values of the relevant channel variables (h, n[Km], n[Kv]^4).
In Fig. 10A, we simulated a voltage clamp experiment in a simplified model with only Na channels, assuming the leak current was subtracted, where both activation and inactivation curves ( and ) were
Boltzmann functions, with parameters V[a]=−30 mV, k[a]=6 mV, V[i]=−65 mV and k[i]=6 mV. The activation and inactivation time constant were fixed (τ[m]=0.3 ms and τ[h] between 0.3 and 3 ms).
The conductance was measured at the current peak after the clamp voltage was switched from a fixed initial voltage V[0]=−70 mV to a test voltage V, which was varied between −100 and 50 mV (the
current was divided by V-E[Na] to obtain the conductance, and we assumed that E[Na] was known - in an experiment it would be obtained from a linear fit to the highest voltage region). The conductance
was normalized by the maximal conductance over the tested voltage range and the resulting curve was fit to a Boltzmann function in the hyperpolarized region V<−40 mV.
All simulations were written with the Brian simulator [85] on a standard desktop PC, except the simulation of the multicompartmental model of Hu et al. [54], for which we used Neuron.
We thank Nicolas Brunel, Michael Brenner and Alain Destexhe for fruitful discussions and Dan Goodman for assistance with the numerical simulations.
Author Contributions
Conceived and designed the experiments: JP RB. Performed the experiments: JP. Analyzed the data: JP RB. Wrote the paper: JP RB. | {"url":"https://journals.plos.org:443/ploscompbiol/article?id=10.1371/journal.pcbi.1000850","timestamp":"2024-11-13T08:02:46Z","content_type":"text/html","content_length":"335984","record_id":"<urn:uuid:91d09ee0-1ac7-4b99-91ab-0687653ec13a>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00490.warc.gz"} |
This chapter explains the meaning of the elements of expressions in Python.
Syntax Notes: In this and the following chapters, extended BNF notation will be used to describe syntax, not lexical analysis. When (one alternative of) a syntax rule has the form
name ::= othername
and no semantics are given, the semantics of this form of name are the same as for othername.
Arithmetic conversions
When a description of an arithmetic operator below uses the phrase “the numeric arguments are converted to a common type,” the arguments are coerced using the coercion rules listed at Coercion rules.
If both arguments are standard numeric types, the following coercions are applied:
• If either argument is a complex number, the other is converted to complex;
• otherwise, if either argument is a floating point number, the other is converted to floating point;
• otherwise, if either argument is a long integer, the other is converted to long integer;
• otherwise, both must be plain integers and no conversion is necessary.
Some additional rules apply for certain operators (e.g., a string left argument to the ‘%’ operator). Extensions can define their own coercions.
Atoms are the most basic elements of expressions. The simplest atoms are identifiers or literals. Forms enclosed in reverse quotes or in parentheses, brackets or braces are also categorized
syntactically as atoms. The syntax for atoms is:
atom ::= identifier | literal | enclosure
enclosure ::= parenth_form | list_display
| generator_expression | dict_display
| string_conversion | yield_atom
Identifiers (Names)
An identifier occurring as an atom is a name. See section Identifiers and keywords for lexical definition and section Naming and binding for documentation of naming and binding.
When the name is bound to an object, evaluation of the atom yields that object. When a name is not bound, an attempt to evaluate it raises a NameError exception.
Private name mangling: When an identifier that textually occurs in a class definition begins with two or more underscore characters and does not end in two or more underscores, it is considered a
private name of that class. Private names are transformed to a longer form before code is generated for them. The transformation inserts the class name in front of the name, with leading underscores
removed, and a single underscore inserted in front of the class name. For example, the identifier __spam occurring in a class named Ham will be transformed to _Ham__spam. This transformation is
independent of the syntactical context in which the identifier is used. If the transformed name is extremely long (longer than 255 characters), implementation defined truncation may happen. If the
class name consists only of underscores, no transformation is done.
Python supports string literals and various numeric literals:
literal ::= stringliteral | integer | longinteger
| floatnumber | imagnumber
Evaluation of a literal yields an object of the given type (string, integer, long integer, floating point number, complex number) with the given value. The value may be approximated in the case of
floating point and imaginary (complex) literals. See section Literals for details.
All literals correspond to immutable data types, and hence the object’s identity is less important than its value. Multiple evaluations of literals with the same value (either the same occurrence in
the program text or a different occurrence) may obtain the same object or a different object with the same value.
Parenthesized forms
A parenthesized form is an optional expression list enclosed in parentheses:
parenth_form ::= "(" [expression_list] ")"
A parenthesized expression list yields whatever that expression list yields: if the list contains at least one comma, it yields a tuple; otherwise, it yields the single expression that makes up the
expression list.
An empty pair of parentheses yields an empty tuple object. Since tuples are immutable, the rules for literals apply (i.e., two occurrences of the empty tuple may or may not yield the same object).
Note that tuples are not formed by the parentheses, but rather by use of the comma operator. The exception is the empty tuple, for which parentheses are required — allowing unparenthesized “nothing”
in expressions would cause ambiguities and allow common typos to pass uncaught.
List displays
A list display is a possibly empty series of expressions enclosed in square brackets:
list_display ::= "[" [expression_list | list_comprehension] "]"
list_comprehension ::= expression list_for
list_for ::= "for" target_list "in" old_expression_list [list_iter]
old_expression_list ::= old_expression [("," old_expression)+ [","]]
list_iter ::= list_for | list_if
list_if ::= "if" old_expression [list_iter]
A list display yields a new list object. Its contents are specified by providing either a list of expressions or a list comprehension. When a comma-separated list of expressions is supplied, its
elements are evaluated from left to right and placed into the list object in that order. When a list comprehension is supplied, it consists of a single expression followed by at least one for clause
and zero or more for or if clauses. In this case, the elements of the new list are those that would be produced by considering each of the for or if clauses a block, nesting from left to right, and
evaluating the expression to produce a list element each time the innermost block is reached .
Generator expressions
A generator expression is a compact generator notation in parentheses:
generator_expression ::= "(" expression genexpr_for ")"
genexpr_for ::= "for" target_list "in" or_test [genexpr_iter]
genexpr_iter ::= genexpr_for | genexpr_if
genexpr_if ::= "if" old_expression [genexpr_iter]
A generator expression yields a new generator object. It consists of a single expression followed by at least one for clause and zero or more for or if clauses. The iterating values of the new
generator are those that would be produced by considering each of the for or if clauses a block, nesting from left to right, and evaluating the expression to yield a value that is reached the
innermost block for each iteration.
Variables used in the generator expression are evaluated lazily in a separate scope when the next() method is called for the generator object (in the same fashion as for normal generators). However,
the in expression of the leftmost for clause is immediately evaluated in the current scope so that an error produced by it can be seen before any other possible error in the code that handles the
generator expression. Subsequent for and if clauses cannot be evaluated immediately since they may depend on the previous for loop. For example: (x*y for x in range(10) for y in bar(x)).
The parentheses can be omitted on calls with only one argument. See section Calls for the detail.
Dictionary displays
A dictionary display is a possibly empty series of key/datum pairs enclosed in curly braces:
dict_display ::= "{" [key_datum_list] "}"
key_datum_list ::= key_datum ("," key_datum)* [","]
key_datum ::= expression ":" expression
A dictionary display yields a new dictionary object.
The key/datum pairs are evaluated from left to right to define the entries of the dictionary: each key object is used as a key into the dictionary to store the corresponding datum.
Restrictions on the types of the key values are listed earlier in section The standard type hierarchy. (To summarize, the key type should be hashable, which excludes all mutable objects.) Clashes
between duplicate keys are not detected; the last datum (textually rightmost in the display) stored for a given key value prevails.
String conversions
A string conversion is an expression list enclosed in reverse (a.k.a. backward) quotes:
string_conversion ::= "'" expression_list "'"
A string conversion evaluates the contained expression list and converts the resulting object into a string according to rules specific to its type.
If the object is a string, a number, None, or a tuple, list or dictionary containing only objects whose type is one of these, the resulting string is a valid Python expression which can be passed to
the built-in function eval() to yield an expression with the same value (or an approximation, if floating point numbers are involved).
(In particular, converting a string adds quotes around it and converts “funny” characters to escape sequences that are safe to print.)
Recursive objects (for example, lists or dictionaries that contain a reference to themselves, directly or indirectly) use ... to indicate a recursive reference, and the result cannot be passed to
eval() to get an equal value (SyntaxError will be raised instead).
The built-in function repr() performs exactly the same conversion in its argument as enclosing it in parentheses and reverse quotes does. The built-in function str() performs a similar but more
user-friendly conversion.
Yield expressions
yield_atom ::= "(" yield_expression ")"
yield_expression ::= "yield" [expression_list]
New in version 2.5.
The yield expression is only used when defining a generator function, and can only be used in the body of a function definition. Using a yield expression in a function definition is sufficient to
cause that definition to create a generator function instead of a normal function.
When a generator function is called, it returns an iterator known as a generator. That generator then controls the execution of a generator function. The execution starts when one of the generator’s
methods is called. At that time, the execution proceeds to the first yield expression, where it is suspended again, returning the value of expression_list to generator’s caller. By suspended we mean
that all local state is retained, including the current bindings of local variables, the instruction pointer, and the internal evaluation stack. When the execution is resumed by calling one of the
generator’s methods, the function can proceed exactly as if the yield expression was just another external call. The value of the yield expression after resuming depends on the method which resumed
the execution.
All of this makes generator functions quite similar to coroutines; they yield multiple times, they have more than one entry point and their execution can be suspended. The only difference is that a
generator function cannot control where should the execution continue after it yields; the control is always transfered to the generator’s caller.
The following generator’s methods can be used to control the execution of a generator function:
Starts the execution of a generator function or resumes it at the last executed yield expression. When a generator function is resumed with a next() method, the current yield expression always
evaluates to None. The execution then continues to the next yield expression, where the generator is suspended again, and the value of the expression_list is returned to next()‘s caller. If the
generator exits without yielding another value, a StopIteration exception is raised.
Resumes the execution and “sends” a value into the generator function. The value argument becomes the result of the current yield expression. The send() method returns the next value yielded by
the generator, or raises StopIteration if the generator exits without yielding another value. When send() is called to start the generator, it must be called with None as the argument, because
there is no yield expression that could receive the value.
generator.throw(type[, value[, traceback]])
Raises an exception of type type at the point where generator was paused, and returns the next value yielded by the generator function. If the generator exits without yielding another value, a
StopIteration exception is raised. If the generator function does not catch the passed-in exception, or raises a different exception, then that exception propagates to the caller.
Raises a GeneratorExit at the point where the generator function was paused. If the generator function then raises StopIteration (by exiting normally, or due to already being closed) or
GeneratorExit (by not catching the exception), close returns to its caller. If the generator yields a value, a RuntimeError is raised. If the generator raises any other exception, it is
propagated to the caller. close() does nothing if the generator has already exited due to an exception or normal exit.
Here is a simple example that demonstrates the behavior of generators and generator functions:
>>> def echo(value=None):
... print "Execution starts when 'next()' is called for the first time."
... try:
... while True:
... try:
... value = (yield value)
... except Exception, e:
... value = e
... finally:
... print "Don't forget to clean up when 'close()' is called."
>>> generator = echo(1)
>>> print generator.next()
Execution starts when 'next()' is called for the first time.
>>> print generator.next()
>>> print generator.send(2)
>>> generator.throw(TypeError, "spam")
>>> generator.close()
Don't forget to clean up when 'close()' is called.
See also
PEP 0342 - Coroutines via Enhanced Generators
The proposal to enhance the API and syntax of generators, making them usable as simple coroutines.
Primaries represent the most tightly bound operations of the language. Their syntax is:
primary ::= atom | attributeref | subscription | slicing | call
Attribute references
An attribute reference is a primary followed by a period and a name:
attributeref ::= primary "." identifier
The primary must evaluate to an object of a type that supports attribute references, e.g., a module, list, or an instance. This object is then asked to produce the attribute whose name is the
identifier. If this attribute is not available, the exception AttributeError is raised. Otherwise, the type and value of the object produced is determined by the object. Multiple evaluations of the
same attribute reference may yield different objects.
A subscription selects an item of a sequence (string, tuple or list) or mapping (dictionary) object:
subscription ::= primary "[" expression_list "]"
The primary must evaluate to an object of a sequence or mapping type.
If the primary is a mapping, the expression list must evaluate to an object whose value is one of the keys of the mapping, and the subscription selects the value in the mapping that corresponds to
that key. (The expression list is a tuple except if it has exactly one item.)
If the primary is a sequence, the expression (list) must evaluate to a plain integer. If this value is negative, the length of the sequence is added to it (so that, e.g., x[-1] selects the last item
of x.) The resulting value must be a nonnegative integer less than the number of items in the sequence, and the subscription selects the item whose index is that value (counting from zero).
A string’s items are characters. A character is not a separate data type but a string of exactly one character.
A slicing selects a range of items in a sequence object (e.g., a string, tuple or list). Slicings may be used as expressions or as targets in assignment or del statements. The syntax for a slicing:
slicing ::= simple_slicing | extended_slicing
simple_slicing ::= primary "[" short_slice "]"
extended_slicing ::= primary "[" slice_list "]"
slice_list ::= slice_item ("," slice_item)* [","]
slice_item ::= expression | proper_slice | ellipsis
proper_slice ::= short_slice | long_slice
short_slice ::= [lower_bound] ":" [upper_bound]
long_slice ::= short_slice ":" [stride]
lower_bound ::= expression
upper_bound ::= expression
stride ::= expression
ellipsis ::= "..."
There is ambiguity in the formal syntax here: anything that looks like an expression list also looks like a slice list, so any subscription can be interpreted as a slicing. Rather than further
complicating the syntax, this is disambiguated by defining that in this case the interpretation as a subscription takes priority over the interpretation as a slicing (this is the case if the slice
list contains no proper slice nor ellipses). Similarly, when the slice list has exactly one short slice and no trailing comma, the interpretation as a simple slicing takes priority over that as an
extended slicing.
The semantics for a simple slicing are as follows. The primary must evaluate to a sequence object. The lower and upper bound expressions, if present, must evaluate to plain integers; defaults are
zero and the sys.maxint, respectively. If either bound is negative, the sequence’s length is added to it. The slicing now selects all items with index k such that i <= k < j where i and j are the
specified lower and upper bounds. This may be an empty sequence. It is not an error if i or j lie outside the range of valid indexes (such items don’t exist so they aren’t selected).
The semantics for an extended slicing are as follows. The primary must evaluate to a mapping object, and it is indexed with a key that is constructed from the slice list, as follows. If the slice
list contains at least one comma, the key is a tuple containing the conversion of the slice items; otherwise, the conversion of the lone slice item is the key. The conversion of a slice item that is
an expression is that expression. The conversion of an ellipsis slice item is the built-in Ellipsis object. The conversion of a proper slice is a slice object (see section The standard type hierarchy
) whose start, stop and step attributes are the values of the expressions given as lower bound, upper bound and stride, respectively, substituting None for missing expressions.
A call calls a callable object (e.g., a function) with a possibly empty series of arguments:
call ::= primary "(" [argument_list [","]
| expression genexpr_for] ")"
argument_list ::= positional_arguments ["," keyword_arguments]
["," "*" expression] ["," keyword_arguments]
["," "**" expression]
| keyword_arguments ["," "*" expression]
["," "**" expression]
| "*" expression ["," "*" expression] ["," "**" expression]
| "**" expression
positional_arguments ::= expression ("," expression)*
keyword_arguments ::= keyword_item ("," keyword_item)*
keyword_item ::= identifier "=" expression
A trailing comma may be present after the positional and keyword arguments but does not affect the semantics.
The primary must evaluate to a callable object (user-defined functions, built-in functions, methods of built-in objects, class objects, methods of class instances, and certain class instances
themselves are callable; extensions may define additional callable object types). All argument expressions are evaluated before the call is attempted. Please refer to section Function definitions for
the syntax of formal parameter lists.
If keyword arguments are present, they are first converted to positional arguments, as follows. First, a list of unfilled slots is created for the formal parameters. If there are N positional
arguments, they are placed in the first N slots. Next, for each keyword argument, the identifier is used to determine the corresponding slot (if the identifier is the same as the first formal
parameter name, the first slot is used, and so on). If the slot is already filled, a TypeError exception is raised. Otherwise, the value of the argument is placed in the slot, filling it (even if the
expression is None, it fills the slot). When all arguments have been processed, the slots that are still unfilled are filled with the corresponding default value from the function definition.
(Default values are calculated, once, when the function is defined; thus, a mutable object such as a list or dictionary used as default value will be shared by all calls that don’t specify an
argument value for the corresponding slot; this should usually be avoided.) If there are any unfilled slots for which no default value is specified, a TypeError exception is raised. Otherwise, the
list of filled slots is used as the argument list for the call.
An implementation may provide builtin functions whose positional parameters do not have names, even if they are ‘named’ for the purpose of documentation, and which therefore cannot be supplied by
keyword. In CPython, this is the case for functions implemented in C that use PyArg_ParseTuple to parse their arguments.
If there are more positional arguments than there are formal parameter slots, a TypeError exception is raised, unless a formal parameter using the syntax *identifier is present; in this case, that
formal parameter receives a tuple containing the excess positional arguments (or an empty tuple if there were no excess positional arguments).
If any keyword argument does not correspond to a formal parameter name, a TypeError exception is raised, unless a formal parameter using the syntax **identifier is present; in this case, that formal
parameter receives a dictionary containing the excess keyword arguments (using the keywords as keys and the argument values as corresponding values), or a (new) empty dictionary if there were no
excess keyword arguments.
If the syntax *expression appears in the function call, expression must evaluate to a sequence. Elements from this sequence are treated as if they were additional positional arguments; if there are
positional arguments x1,..., xN, and expression evaluates to a sequence y1, ..., yM, this is equivalent to a call with M+N positional arguments x1, ..., xN, y1, ..., yM.
A consequence of this is that although the *expression syntax may appear after some keyword arguments, it is processed before the keyword arguments (and the **expression argument, if any – see
below). So:
>>> def f(a, b):
... print a, b
>>> f(b=1, *(2,))
>>> f(a=1, *(2,))
Traceback (most recent call last):
File "<stdin>", line 1, in ?
TypeError: f() got multiple values for keyword argument 'a'
>>> f(1, *(2,))
It is unusual for both keyword arguments and the *expression syntax to be used in the same call, so in practice this confusion does not arise.
If the syntax **expression appears in the function call, expression must evaluate to a mapping, the contents of which are treated as additional keyword arguments. In the case of a keyword appearing
in both expression and as an explicit keyword argument, a TypeError exception is raised.
Formal parameters using the syntax *identifier or **identifier cannot be used as positional argument slots or as keyword argument names. Formal parameters using the syntax (sublist) cannot be used as
keyword argument names; the outermost sublist corresponds to a single unnamed argument slot, and the argument value is assigned to the sublist using the usual tuple assignment rules after all other
parameter processing is done.
A call always returns some value, possibly None, unless it raises an exception. How this value is computed depends on the type of the callable object.
If it is—
a user-defined function:
The code block for the function is executed, passing it the argument list. The first thing the code block will do is bind the formal parameters to the arguments; this is described in section
Function definitions. When the code block executes a return statement, this specifies the return value of the function call.
a built-in function or method:
The result is up to the interpreter; see Built-in Functions for the descriptions of built-in functions and methods.
a class object:
A new instance of that class is returned.
a class instance method:
The corresponding user-defined function is called, with an argument list that is one longer than the argument list of the call: the instance becomes the first argument.
a class instance:
The class must define a __call__() method; the effect is then the same as if that method was called.
The power operator
The power operator binds more tightly than unary operators on its left; it binds less tightly than unary operators on its right. The syntax is:
power ::= primary ["**" u_expr]
Thus, in an unparenthesized sequence of power and unary operators, the operators are evaluated from right to left (this does not constrain the evaluation order for the operands): -1**2 results in -1.
The power operator has the same semantics as the built-in pow() function, when called with two arguments: it yields its left argument raised to the power of its right argument. The numeric arguments
are first converted to a common type. The result type is that of the arguments after coercion.
With mixed operand types, the coercion rules for binary arithmetic operators apply. For int and long int operands, the result has the same type as the operands (after coercion) unless the second
argument is negative; in that case, all arguments are converted to float and a float result is delivered. For example, 10**2 returns 100, but 10**-2 returns 0.01. (This last feature was added in
Python 2.2. In Python 2.1 and before, if both arguments were of integer types and the second argument was negative, an exception was raised).
Raising 0.0 to a negative power results in a ZeroDivisionError. Raising a negative number to a fractional power results in a ValueError.
Unary arithmetic operations
All unary arithmetic (and bitwise) operations have the same priority:
u_expr ::= power | "-" u_expr | "+" u_expr | "~" u_expr
The unary - (minus) operator yields the negation of its numeric argument.
The unary + (plus) operator yields its numeric argument unchanged.
The unary ~ (invert) operator yields the bitwise inversion of its plain or long integer argument. The bitwise inversion of x is defined as -(x+1). It only applies to integral numbers.
In all three cases, if the argument does not have the proper type, a TypeError exception is raised.
Binary arithmetic operations
The binary arithmetic operations have the conventional priority levels. Note that some of these operations also apply to certain non-numeric types. Apart from the power operator, there are only two
levels, one for multiplicative operators and one for additive operators:
m_expr ::= u_expr | m_expr "*" u_expr | m_expr "//" u_expr | m_expr "/" u_expr
| m_expr "%" u_expr
a_expr ::= m_expr | a_expr "+" m_expr | a_expr "-" m_expr
The * (multiplication) operator yields the product of its arguments. The arguments must either both be numbers, or one argument must be an integer (plain or long) and the other must be a sequence. In
the former case, the numbers are converted to a common type and then multiplied together. In the latter case, sequence repetition is performed; a negative repetition factor yields an empty sequence.
The / (division) and // (floor division) operators yield the quotient of their arguments. The numeric arguments are first converted to a common type. Plain or long integer division yields an integer
of the same type; the result is that of mathematical division with the ‘floor’ function applied to the result. Division by zero raises the ZeroDivisionError exception.
The % (modulo) operator yields the remainder from the division of the first argument by the second. The numeric arguments are first converted to a common type. A zero right argument raises the
ZeroDivisionError exception. The arguments may be floating point numbers, e.g., 3.14%0.7 equals 0.34 (since 3.14 equals 4*0.7 + 0.34.) The modulo operator always yields a result with the same sign as
its second operand (or zero); the absolute value of the result is strictly smaller than the absolute value of the second operand .
The integer division and modulo operators are connected by the following identity: x == (x/y)*y + (x%y). Integer division and modulo are also connected with the built-in function divmod(): divmod(x,
y) == (x/y, x%y). These identities don’t hold for floating point numbers; there similar identities hold approximately where x/y is replaced by floor(x/y) or floor(x/y) - 1 .
In addition to performing the modulo operation on numbers, the % operator is also overloaded by string and unicode objects to perform string formatting (also known as interpolation). The syntax for
string formatting is described in the Python Library Reference, section String Formatting Operations.
Deprecated since version 2.3: The floor division operator, the modulo operator, and the divmod() function are no longer defined for complex numbers. Instead, convert to a floating point number using
the abs() function if appropriate.
The + (addition) operator yields the sum of its arguments. The arguments must either both be numbers or both sequences of the same type. In the former case, the numbers are converted to a common type
and then added together. In the latter case, the sequences are concatenated.
The - (subtraction) operator yields the difference of its arguments. The numeric arguments are first converted to a common type.
Shifting operations
The shifting operations have lower priority than the arithmetic operations:
shift_expr ::= a_expr | shift_expr ( "<<" | ">>" ) a_expr
These operators accept plain or long integers as arguments. The arguments are converted to a common type. They shift the first argument to the left or right by the number of bits given by the second
A right shift by n bits is defined as division by pow(2, n). A left shift by n bits is defined as multiplication with pow(2, n). Negative shift counts raise a ValueError exception.
Binary bitwise operations
Each of the three bitwise operations has a different priority level:
and_expr ::= shift_expr | and_expr "&" shift_expr
xor_expr ::= and_expr | xor_expr "^" and_expr
or_expr ::= xor_expr | or_expr "|" xor_expr
The & operator yields the bitwise AND of its arguments, which must be plain or long integers. The arguments are converted to a common type.
The ^ operator yields the bitwise XOR (exclusive OR) of its arguments, which must be plain or long integers. The arguments are converted to a common type.
The | operator yields the bitwise (inclusive) OR of its arguments, which must be plain or long integers. The arguments are converted to a common type.
Unlike C, all comparison operations in Python have the same priority, which is lower than that of any arithmetic, shifting or bitwise operation. Also unlike C, expressions like a < b < c have the
interpretation that is conventional in mathematics:
comparison ::= or_expr ( comp_operator or_expr )*
comp_operator ::= "<" | ">" | "==" | ">=" | "<=" | "<>" | "!="
| "is" ["not"] | ["not"] "in"
Comparisons yield boolean values: True or False.
Comparisons can be chained arbitrarily, e.g., x < y <= z is equivalent to x < y and y <= z, except that y is evaluated only once (but in both cases z is not evaluated at all when x < y is found to be
Formally, if a, b, c, ..., y, z are expressions and op1, op2, ..., opN are comparison operators, then a op1 b op2 c ... y opN z is equivalent to a op1 b and b op2 c and ... y opN z, except that each
expression is evaluated at most once.
Note that a op1 b op2 c doesn’t imply any kind of comparison between a and c, so that, e.g., x < y > z is perfectly legal (though perhaps not pretty).
The forms <> and != are equivalent; for consistency with C, != is preferred; where != is mentioned below <> is also accepted. The <> spelling is considered obsolescent.
The operators <, >, ==, >=, <=, and != compare the values of two objects. The objects need not have the same type. If both are numbers, they are converted to a common type. Otherwise, objects of
different types always compare unequal, and are ordered consistently but arbitrarily. You can control comparison behavior of objects of non-builtin types by defining a __cmp__ method or rich
comparison methods like __gt__, described in section Special method names.
(This unusual definition of comparison was used to simplify the definition of operations like sorting and the in and not in operators. In the future, the comparison rules for objects of different
types are likely to change.)
Comparison of objects of the same type depends on the type:
• Numbers are compared arithmetically.
• Strings are compared lexicographically using the numeric equivalents (the result of the built-in function ord()) of their characters. Unicode and 8-bit strings are fully interoperable in this
• Tuples and lists are compared lexicographically using comparison of corresponding elements. This means that to compare equal, each element must compare equal and the two sequences must be of the
same type and have the same length.
If not equal, the sequences are ordered the same as their first differing elements. For example, cmp([1,2,x], [1,2,y]) returns the same as cmp(x,y). If the corresponding element does not exist,
the shorter sequence is ordered first (for example, [1,2] < [1,2,3]).
• Mappings (dictionaries) compare equal if and only if their sorted (key, value) lists compare equal. Outcomes other than equality are resolved consistently, but are not otherwise defined.
• Most other objects of builtin types compare unequal unless they are the same object; the choice whether one object is considered smaller or larger than another one is made arbitrarily but
consistently within one execution of a program.
The operators in and not in test for collection membership. x in s evaluates to true if x is a member of the collection s, and false otherwise. x not in s returns the negation of x in s. The
collection membership test has traditionally been bound to sequences; an object is a member of a collection if the collection is a sequence and contains an element equal to that object. However, it
make sense for many other object types to support membership tests without being a sequence. In particular, dictionaries (for keys) and sets support membership testing.
For the list and tuple types, x in y is true if and only if there exists an index i such that x == y[i] is true.
For the Unicode and string types, x in y is true if and only if x is a substring of y. An equivalent test is y.find(x) != -1. Note, x and y need not be the same type; consequently, u'ab' in 'abc'
will return True. Empty strings are always considered to be a substring of any other string, so "" in "abc" will return True.
Changed in version 2.3: Previously, x was required to be a string of length 1.
For user-defined classes which define the __contains__() method, x in y is true if and only if y.__contains__(x) is true.
For user-defined classes which do not define __contains__() and do define __getitem__(), x in y is true if and only if there is a non-negative integer index i such that x == y[i], and all lower
integer indices do not raise IndexError exception. (If any other exception is raised, it is as if in raised that exception).
The operator not in is defined to have the inverse true value of in.
The operators is and is not test for object identity: x is y is true if and only if x and y are the same object. x is not y yields the inverse truth value.
Boolean operations
Boolean operations have the lowest priority of all Python operations:
expression ::= conditional_expression | lambda_form
old_expression ::= or_test | old_lambda_form
conditional_expression ::= or_test ["if" or_test "else" expression]
or_test ::= and_test | or_test "or" and_test
and_test ::= not_test | and_test "and" not_test
not_test ::= comparison | "not" not_test
In the context of Boolean operations, and also when expressions are used by control flow statements, the following values are interpreted as false: False, None, numeric zero of all types, and empty
strings and containers (including strings, tuples, lists, dictionaries, sets and frozensets). All other values are interpreted as true. (See the __nonzero__() special method for a way to change
The operator not yields True if its argument is false, False otherwise.
The expression x if C else y first evaluates C (not x); if C is true, x is evaluated and its value is returned; otherwise, y is evaluated and its value is returned.
New in version 2.5.
The expression x and y first evaluates x; if x is false, its value is returned; otherwise, y is evaluated and the resulting value is returned.
The expression x or y first evaluates x; if x is true, its value is returned; otherwise, y is evaluated and the resulting value is returned.
(Note that neither and nor or restrict the value and type they return to False and True, but rather return the last evaluated argument. This is sometimes useful, e.g., if s is a string that should be
replaced by a default value if it is empty, the expression s or 'foo' yields the desired value. Because not has to invent a value anyway, it does not bother to return a value of the same type as its
argument, so e.g., not 'foo' yields False, not ''.)
lambda_form ::= "lambda" [parameter_list]: expression
old_lambda_form ::= "lambda" [parameter_list]: old_expression
Lambda forms (lambda expressions) have the same syntactic position as expressions. They are a shorthand to create anonymous functions; the expression lambda arguments: expression yields a function
object. The unnamed object behaves like a function object defined with
def name(arguments):
return expression
See section Function definitions for the syntax of parameter lists. Note that functions created with lambda forms cannot contain statements.
Expression lists
expression_list ::= expression ( "," expression )* [","]
An expression list containing at least one comma yields a tuple. The length of the tuple is the number of expressions in the list. The expressions are evaluated from left to right.
The trailing comma is required only to create a single tuple (a.k.a. a singleton); it is optional in all other cases. A single expression without a trailing comma doesn’t create a tuple, but rather
yields the value of that expression. (To create an empty tuple, use an empty pair of parentheses: ().)
Evaluation order
Python evaluates expressions from left to right. Notice that while evaluating an assignment, the right-hand side is evaluated before the left-hand side.
In the following lines, expressions will be evaluated in the arithmetic order of their suffixes:
expr1, expr2, expr3, expr4
(expr1, expr2, expr3, expr4)
{expr1: expr2, expr3: expr4}
expr1 + expr2 * (expr3 - expr4)
expr1(expr2, expr3, *expr4, **expr5)
expr3, expr4 = expr1, expr2
The following table summarizes the operator precedences in Python, from lowest precedence (least binding) to highest precedence (most binding). Operators in the same box have the same precedence.
Unless the syntax is explicitly given, operators are binary. Operators in the same box group left to right (except for comparisons, including tests, which all have the same precedence and chain from
left to right — see section Comparisons — and exponentiation, which groups from right to left).
│ Operator │ Description │
│lambda │Lambda expression │
│or │Boolean OR │
│and │Boolean AND │
│not x │Boolean NOT │
│in, not in │Membership tests │
│is, is not │Identity tests │
│<, <=, >, >=, <>, !=, ==│Comparisons │
│| │Bitwise OR │
│^ │Bitwise XOR │
│& │Bitwise AND │
│<<, >> │Shifts │
│+, - │Addition and subtraction │
│*, /, % │Multiplication, division, remainder │
│+x, -x │Positive, negative │
│~x │Bitwise not │
│** │Exponentiation │
│x[index] │Subscription │
│x[index:index] │Slicing │
│x(arguments...) │Call │
│x.attribute │Attribute reference │
│(expressions...) │Binding or tuple display │
│[expressions...] │List display │
│{key:datum...} │Dictionary display │
│`expressions...` │String conversion │ | {"url":"http://www.doc.crossplatform.ru/python/2.6/reference/expressions.html","timestamp":"2024-11-10T19:11:05Z","content_type":"text/html","content_length":"119811","record_id":"<urn:uuid:99e355be-c72a-4fc5-b07d-124ad43e947f>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00807.warc.gz"} |
PROC SGPLOT and ODS PDF errors with extremely large values
I'm having trouble graphing data with extremely large values.
Here is an example that reproduces the problem:
data dummy;
do i = 1 to 15;
y = 10**i;
ods pdf file="problem.pdf";
proc sgplot data=dummy;
series x=i y=y / markers;
yaxis min=0 max=100;
ods pdf close;
Everything looks as expected in results viewer: I see the first two points and the rest is blank.
But the PDF does not look the same. Here is a screenshot of the graph:
There are 3 vertical lines on the right that should not be there.
What's strange is that when I zoom out (e.g. yaxis min=0 max=100000), the location and number of these lines change. And if I do not restrict the zoom at all, they disappear.
I think the large numbers are somehow interacting with the upper yaxis max value.
Any ideas how I can get the pdf output to match the desired output in the results viewer?
System Info: SAS 9.4 TS Level 1M7 on Windows.
08-09-2023 04:49 PM | {"url":"https://communities.sas.com/t5/Graphics-Programming/PROC-SGPLOT-and-ODS-PDF-errors-with-extremely-large-values/td-p/888679","timestamp":"2024-11-03T04:39:26Z","content_type":"text/html","content_length":"211182","record_id":"<urn:uuid:456819e4-9ec4-47e0-be09-61d14f0ff224>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00483.warc.gz"} |
Function Callbacks
Function Callbacks¶
For input file types that support functions, e.g., Lua, functions can also be read from the input file into a std::function, the wrapper for callables provided by the C++ standard library.
Defining And Storing¶
This is accomplished by calling addFunction on an Inlet or Container object.
Consider the following Lua function that accepts a vector in R^2 or R^3 and returns a double:
coef = function (v)
if v.dim == 2 then
return v.x + (v.y * 0.5)
return v.x + (v.y * 0.5) + (v.z * 0.25)
The schema for this function would be defined as follows:
{inlet::FunctionTag::Vector, inlet::FunctionTag::Double},
"The function representing the BC coefficient")
The return type and argument types are described with the inlet::FunctionTag enumeration, which has the following members:
□ Double - corresponds to a C++ double
□ String - corresponds to a C++ std::string
□ Vector - corresponds to a C++ inlet::InletVector
□ Void - corresponds to C++ void, should only be used for functions that don’t return a value
Note that a single type tag is passed for the return type, while a vector of tags is passed for the argument types. Currently a maximum of two arguments are supported. To declare a function with no
arguments, simply leave the list of argument types empty.
The InletVector type (and its Lua representation) are statically-sized vectors with a maximum dimension of three. That is, they can also be used to represent two-dimensional vectors.
In Lua, the following operations on the Vector type are supported (for Vector s u, v, and w):
1. Construction of a 3D vector: u = Vector.new(1, 2, 3)
2. Construction of a 2D vector: u = Vector.new(1, 2)
3. Construction of an empty vector (default dimension is 3): u = Vector.new()
4. Vector addition and subtraction: w = u + v, w = u - v
5. Vector negation: v = -u
6. Scalar multiplication: v = u * 0.5, v = 0.5 * u
7. Indexing (1-indexed for consistency with Lua): d = u[1], u[1] = 0.5
8. L2 norm and its square: d = u:norm(), d = u:squared_norm()
9. Normalization: v = u:unitVector()
10. Dot and cross products: d = u:dot(v), w = u:cross(v)
11. Dimension retrieval: d = u.dim
12. Component retrieval: d = u.x, d = u.y, d = u.z
To retrieve a function, both the implicit conversion and get<T> syntax is supported. For example, a function can be retrieved as follows:
// Retrieve the function (makes a copy) to be moved into the lambda
auto func =
base["coef"].get<std::function<double(FunctionType::Vector, double)>>();
It can also be assigned directly to a std::function without the need to use get<T>:
std::function<double(FunctionType::Vector)> coef = inlet["coef"];
Additionally, if a function does not need to be stored, the overhead of a copy can be eliminated by calling it directly:
double result = inlet["coef"].call<double>(axom::inlet::FunctionType::Vector{3, 5, 7});
Using call<ReturnType>(ArgType1, ArgType2, ...) requires both that the return type be explicitly specified and that argument types be passed with the exact type as used in the signature defined as
part of the schema. This is because the arguments do not participate in overload resolution. | {"url":"https://axom.readthedocs.io/en/develop/axom/inlet/docs/sphinx/functions.html","timestamp":"2024-11-03T04:16:23Z","content_type":"text/html","content_length":"22357","record_id":"<urn:uuid:8311005b-ebb3-4e93-9121-61b43ded3ee9>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00092.warc.gz"} |
5.8 Graphing Functions - Contemporary Mathematics | OpenStax
Learning Objectives
After completing this module, you should be able to:
1. Graph functions using intercepts.
2. Compute slope.
3. Graph functions using slope and $yy$-intercept.
4. Graph horizontal and vertical lines.
5. Interpret graphs of functions.
6. Model applications using slope and $yy$-intercept.
In this section, we will expand our knowledge of graphing by graphing linear functions. There are many real-world scenarios that can be represented by graphs of linear functions. Imagine a chairlift
going up at a ski resort. The journey a skier takes travelling up the chairlift could be represented as a linear function with a positive slope. The journey a skier takes down the slopes could be
represented by a linear function with a negative slope.
Graphing Functions Using Intercepts
Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to
graph. This means that two people graphing the line might use different sets of three points. At first glance, their two lines might not appear to be the same, since they would have different points
labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the $xx$-axis and
the $yy$-axis. These points are called the intercepts of a line. Let us review the graphs of the lines in Figure 5.65.
The table below lists where each of these lines crosses the $xx$- and $yy$-axis. Do you see a pattern? For each line, the $yy$-coordinate of the point where the line crosses the $xx$-axis is zero.
The point where the line crosses the $xx$-axis has the form $(a, 0)(a, 0)$ and is called the $xx$-intercept of the line. The $xx$-intercept occurs when $yy$ is zero. In each line, the $xx$-coordinate
of the point where the line crosses the $yy$-axis is zero. The point where the line crosses the $yy$-axis has the form $(0,b)(0,b)$ and is called the $yy$-intercept of the line. The $yy$-intercept
occurs when $xx$ is zero.
Figure The line crosses the $x-axisx-axis$ at: Ordered Pair for this Point The line crosses the $y-axisy-axis$ at: Ordered Pair for This Point
Figure (a) 3 $(3,0)(3,0)$ 6 $(0,6)(0,6)$
Figure (b) 4 $(4,0)(4,0)$ $−3−3$ $(0,−3)(0,−3)$
Figure (c) 5 $(5,0)(5,0)$ $−5−5$ $(0,−5)(0,−5)$
Figure (d) 0 $(0,0)(0,0)$ 0 $(0,0)(0,0)$
General Figure $aa$ $(a,0)(a,0)$ $bb$ $(0,b)(0,b)$
Finding $xx$- and $yy$-Intercepts
Find the $xx$-intercept and $yy$-intercept on the (a) and (b) graphs in Figure 5.66.
In Figure 5.66, the graph crosses the $xx$-axis at the point $(4,0)(4,0)$. The $xx$-intercept is $(4,0)(4,0)$. The graph crosses the $yy$-axis at the point $(0,2)(0,2)$. The $yy$-intercept is $(0,2)
(0,2)$. In Figure 5.66, the graph crosses the $xx$-axis at the point $(2,0)(2,0)$. The $xx$-intercept is $(2,0)(2,0)$. The graph crosses the $yy$-axis at the point $(0,−6)(0,−6)$. The $yy$-intercept
is $(0,−6)(0,−6)$.
Graphing a Function Using Intercepts
Find the intercepts of $2x+y=82x+y=8$. Then graph the function using the intercepts.
Let $y=0y=0$ to find the $xx$-intercept, and let $x=0x=0$ to find the $yy$-intercept.
$2x+y=82x+y=8$ $2x+y=82x+y=8$
To find the $xx$-intercept, let $y=0y=0$. $2x+0=82x+0=8$ To find the $yy$-intercept, let $x=0x=0$. $2(0)+y=82(0)+y=8$
Simplify. $2x=8x=42x=8x=4$ Simplify. $y=8y=8$
The $xx$-intercept is: $(4,0)(4,0)$ The $yy$-intercept is: $(0,8)(0,8)$
Plot the intercepts to get the graph in Figure 5.67.
Computing Slope
When graphing linear equations, you may notice that some lines tilt up as they go from left to right and some lines tilt down. Some lines are very steep and some lines are flatter. In mathematics,
the measure of the steepness of a line is called the slope of the line. To find the slope of a line, we locate two points on the line whose coordinates are integers. Then we sketch a right triangle
where the two points are vertices of the triangle and one side is horizontal and one side is vertical. Next, we measure or calculate the distance along the vertical and horizontal sides of the
triangle. The vertical distance is called the rise and the horizontal distance is called the run.
We can assign a numerical value to the slope of a line by finding the ratio of the rise and run. The rise is the amount the vertical distance changes while the run measures the horizontal change, as
shown in this illustration. Slope (Figure 5.68) is a rate of change.
To calculate slope $(m)(m)$, use the formula
where the rise measures the vertical change and the run measures the horizontal change.
The concept of slope has many applications in the real world. In construction, the pitch of a roof, the slant of plumbing pipes, and the steepness of stairs are all applications of slope. As you ski
or jog down a hill, you definitely experience slope.
Finding the Slope from a Graph
Step 1: Locate two points on the graph whose coordinates are integers, such as $(0,5)(0,5)$ and $(3,3)(3,3)$. Starting at $(0,5)(0,5)$, sketch a right triangle to $(3,3)(3,3)$ as shown in Figure 5.70
Step 2: Count the rise; since it goes down, it is negative. The rise is −2.
Step 3: Count the run. The run is 3.
Step 4: Use the slope formula $m=riserunm=riserun$ substitute the values of the rise and run. $m=−23m=−23$
The slope of the line is $−23−23$.
The solution is $yy$ decreases by 2 units as $xx$ increases by 3 units.
Sometimes we will need to find the slope of a line between two points when we don’t have a graph to measure the rise and the run. We could plot the points on grid paper, then count out the rise and
the run, but there is a way to find the slope without graphing. First, we need to introduce some algebraic notation.
We have seen that an ordered pair ($xx$, $yy$) gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol ($xx$, $yy$) be used to represent two
different points? Mathematicians use subscripts to distinguish such points. For example, ($x1x1$, $y1y1$) would be said aloud as “$xx$ sub 1, $yy$ sub 1” and ($x2x2$, $y2y2$) read “$xx$ sub 2, $yy$
sub 2.” The “sub” is a short way of saying “subscript.” We will use ($x1x1$, $y1y1$) to identify the first point and ($x2x2$, $y2y2$) to identify the second point in our slope equation. If we had
more than two points, (if we were finding more than one slope), we could use ($x3x3$, $y3y3$), ($x4x4$, $y4y4$), and so on.
Let’s review how the rise and run relate to the coordinates of the two points by taking another look at the slope of the line between the points $(2,3)(2,3)$ and $(7,6)(7,6)$, as shown in Figure 5.71
On the graph, we count the rise of 3 and the run of 5. Notice on the graph that that ($x1x1$, $y1y1$) is the point $(2,3)(2,3)$ and ($x2x2$, $y2y2$) is the point $(7,6)(7,6)$. The rise can be found
by subtracting the $yy$-coordinates, 6 and 3, and the run can be found by subtracting the $xx$-coordinates 7 and 2.
We have shown that $m=y2−y1x2−x1m=y2−y1x2−x1$ is really another version of $m=riserunm=riserun$. We can use this formula to find the slope of a line.
To find the slope of the line between two points ($x1x1$, $y1y1$) and ($x2x2$, $y2y2$), use the formula
Finding the Slope of the Line Using Points
Use the slope formula to find the slope of the line through the points (−2, −3) and (−7, 4).
We’ll call (−2, −3) point 1 and (−7, 4) point 2.
Step 1: Use the slope formula: $m=y2−y1x2−x1m=y2−y1x2−x1$
Step 2: Substitute the values: $m=4−(−3)−7−(−2)m=4−(−3)−7−(−2)$
Step 3: Simplify: $m=7−5=−75m=7−5=−75$
Step 4: Verify the slope on the graph shown in Figure 5.72.
Graphing Functions Using Slope and $yy$-Intercept
We have graphed linear equations by plotting points and using intercepts. Once we see how an equation in slope-intercept form and its graph are related, we will have one more method we can use to
graph lines. Review the graph of the equation $y=12x+3y=12x+3$ in Figure 5.73 and find its slope and $yy$-intercept.
The vertical and horizontal lines in the graph show us the rise is 1 and the run is 2, respectively.
Substituting into the slope formula: $m=riserun=12m=riserun=12$
The $yy$-intercept is $(0,3)(0,3)$. Look at the equation of this line.
Look at the slope and $yy$-intercept.
When a linear equation is solved for $yy$, the coefficient of the $xx$ term is the slope and the constant term is the $yy$-coordinate of the $yy$-intercept. We say that the equation $y=12x+3y=12x+3$
is in slope-intercept form. Sometimes the slope-intercept form is called the $yy$-form.
Finding the Slope and $yy$-Intercept of a Line
Identify the slope and $yy$-intercept of the line from the equation:
1. $y=−47x−2y=−47x−2$
2. $x+3y=9x+3y=9$
1. We compare our equation to the slope-intercept form of the equation.
Step 1: Write the slope-intercept form of the equation of the line.
Step 2: Write the equation of the line.
Step 3: Identify the slope.
Step 4: Identify the $yy$-intercept.
$y-intercept is (0,−2)y-intercept is (0,−2)$
2. When an equation of a line is not given in slope-intercept form, our first step will be to solve the equation for $yy$.
Step 1: Solve for $yy$.
Step 2: Subtract $xx$ from each side.
Step 3: Divide both sides by 3.
Step 4: Simplify.
Step 5: Write the slope-intercept form of the equation of the line.
Step 6: Write the equation of the line.
Step 7: Identify the slope.
Step 8: Identify the $yy$-intercept.
$y-intercept is(0,3)y-intercept is(0,3)$.
Graphing the Slope and $yy$-Intercept
Graph the line of the equation $y=−x+4y=−x+4$ using its slope and $yy$-intercept.
The equation is in slope-intercept form $y=mx+by=mx+b$.
$y = − x + 4 . y = − x + 4 .$
Step 1: Identify the slope and $yy$-intercept.
$m=−1m=−1$, $yy$-intercept is $(0,4)(0,4)$.
Step 2: Plot the $yy$-intercept on the coordinate system (Figure 5.74).
1. Identify the rise over the run.
2. Count out the rise and run to mark the second point.
rise $−1−1$, run 1
Graphing Horizontal and Vertical Lines
Some linear equations have only one variable. They may have just $xx$ without the $yy$, or just $yy$ without an $xx$. This changes how we make a table of values to get the points to plot. Let us
consider the equation $x=−3x=−3$. This equation has only one variable, $xx$. The equation says that $xx$ is always equal to $−3−3$, so its value does not depend on $yy$. No matter what the value of
$yy$ is, the value of $xx$ is always $−3−3$. To make a table of values, write $−3−3$ in for all the $xx$-values. Then choose any values for $yy$. Since $xx$ does not depend on $yy$, you can choose
any numbers you like. But to fit the points on our coordinate graph, we will use 1, 2, and 3 for the $yy$-coordinates in the table below.
$xx$ $yy$ ($xx$, $yy$)
−3 1 $(−3,1)(−3,1)$
$−3−3$ 2 $(−3,2)(−3,2)$
$−3−3$ 3 $(−3,3)(−3,3)$
Plot the points from the table and connect them with a straight line (Figure 5.75). Notice that we have graphed a vertical line.
What is the slope? If we take the two points $(−3,3)(−3,3)$ and $(−3,1)(−3,1)$ then the rise is 2 and the run is 0.
Using the slope formula we get: $m=riserun=20m=riserun=20$
The slope is undefined since division by zero is undefined. We say that the slope of the vertical line $x=−3x=−3$ is undefined. The slope of any vertical line $x=ax=a$ (where $aa$ is any number) will
be undefined.
What if the equation has $yy$ but no $xx$? Let’s graph the equation $y=4y=4$. This time the $yy$-value is a constant, so in this equation, $yy$ does not depend on $xx$. Fill in 4 for all the $yy$
values in the table below and then choose any values for $xx$. We will use 0, 2, and 4 for the $xx$-coordinates.
$xx$ $yy$ ($xx$, $yy$)
0 4 $(0,4)(0,4)$
2 4 $(2,4)(2,4)$
4 4 $(4,4)(4,4)$
In Figure 5.76, we have graphed a horizontal line passing through the $yy$-axis at 4.
What is the slope? If we take the two points $(2,4)(2,4)$ and $(4,4)(4,4)$ then the rise is 0 and the run is 2. Using the slope formula, we get $m=riserun=02=0m=riserun=02=0$. The slope of the
horizontal line $y=4y=4$ is 0. The slope of any horizontal line $y=by=b$ (where $bb$ is any number) will be 0. When the $yy$-coordinates are the same, the rise is 0.
Graphing A Vertical Line
The equation has only one variable, $xx$, and $xx$ is always equal to 2. We create a table where $xx$ is always 2 and then put in any values for $yy$. The graph is a vertical line passing through the
$xx$-axis at 2 (Figure 5.77).
$xx$ $yy$ ($xx$, $yy$)
2 1 $(2,1)(2,1)$
2 2 $(2,2)(2,2)$
2 3 $(2,3)(2,3)$
Graphing A Horizontal Line
The equation $y=−1y=−1$ has only one variable, $yy$. The value of $yy$ is constant. All the ordered pairs in the next table have the same $yy$-coordinate. The graph is a horizontal line passing
through the $yy$-axis at −1 (Figure 5.78).
$xx$ $yy$ ($xx$, $yy$)
0 $−1−1$ $(0,−1)(0,−1)$
3 $−1−1$ $(3,−1)(3,−1)$
$−3−3$ $−1−1$ $(−3,−1)(−3,−1)$
The table below summarizes all the methods we have used to graph lines.
Interpreting Graphs of Functions
An important yet often overlooked area in algebra involves interpreting graphs. Oftentimes in math classes, students are given mathematical functions and can make graphs to represent them. But the
interpretation of graphs is a more applicable skill to the real world. Being able to “read” a graph—understanding its domain and range, what the intercepts mean, and what the slope (or curve) means—
that's a real-world skill.
Interpreting a Graph
In Figure 5.79 the $xx$-axis on the graph represents the 120-minute bike ride Juan went on. The $yy$-axis represents how far away he was from his home.
1. Interpret the $xx$- and $yy$-intercept.
2. For each segment, find the slope.
3. Create an interpretation of this graph (i.e., make up a story that goes with it).
1. $(0,0)(0,0)$ is the $xx$- and $yy$-intercept and represents Juan at home before his bike ride. The distance from home is 0 miles and 0 minutes have passed.
2. In the first 30 minutes, the slope is $1515$ and indicates Juan is traveling 1 mile for every 5 minutes. Between 30 and 60 minutes, the slope is 0 and indicates that he’s not riding the bike (the
distance is not increasing). Then between 60 and 90 minutes, the slope is $1515$ again. Finally, after 90 minutes the slope is $−415,−415,$ meaning Juan is getting 4 miles closer to home every 15
3. Answers will vary. Juan left his house for a bike ride. After 30 minutes, he was 6 miles from home and he stopped for ice cream at his local ice-cream truck. He enjoyed his ice cream for 30
minutes. He then jumped back on his bike and rode to his friend’s house. He arrived there 30 minutes later. His friend’s house was 12 miles from his home. His friend was not home so he
immediately turned around and quickly rode home in 45 minutes.
Modeling Applications Using Slope and $yy$-Intercept
Many real-world applications are modeled by linear equations. We will review a few applications here so you can understand how equations written in slope-intercept form relate to real-world
situations. Usually when a linear equation model uses real-world data, different letters are used for the variables instead of using only $xx$ and $yy$. The variable names often remind us of what
quantities are being measured. Also, we often need to extend the axes in our rectangular coordinate system to bigger positive and negative numbers to accommodate the data in the application.
Converting Temperature
The equation $F=95C+32F=95C+32$ is used to convert temperatures from degrees Celsius ($CC$) to degrees Fahrenheit ($FF$).
1. Find the Fahrenheit temperature for a Celsius temperature of 0°.
2. Find the Fahrenheit temperature for a Celsius temperature of 20°.
3. Interpret the slope and $FF$-intercept of the equation.
4. Graph the equation.
1. Find the Fahrenheit temperature for a Celsius temperature of 0°.
Find $FF$ when $C=0C=0$. $F=95(0)+32F=95(0)+32$
Simplify. $F=32F=32$
2. Find the Fahrenheit temperature for a Celsius temperature of 20°.
Find $FF$ when $C=20C=20$. $F=95(20)+32F=95(20)+32$
Simplify. $F=36+32F=36+32$
Simplify. $F=68F=68$
3. Interpret the slope and $FF$-intercept of the equation.
Even though this equation uses $FF$ and $CC$, it is still in slope-intercept form.
The slope $9595$ means that the temperature Fahrenheit ($FF$) increases 9 degrees when the temperature Celsius ($CC$) increases 5 degrees.
The $FF$-intercept means that when the temperature is 0° on the Celsius scale, it is 32° on the Fahrenheit scale.
4. Graph the equation.
We will need to use a larger scale than our usual. Start at the $FF$-intercept $(0,32)(0,32)$, and then count out the rise of 9 and the run of 5 to get a second point as shown in Figure 5.80.
Calculating Driving Costs
Sam drives a delivery van. The equation $C=0.5d+60C=0.5d+60$ models the relation between his weekly cost, $CC$, in dollars and the number of miles, $dd$, that he drives.
1. Find Sam’s cost for a week when he drives 0 miles.
2. Find the cost for a week when he drives 250 miles.
3. Interpret the slope and $CC$-intercept of the equation.
4. Graph the equation.
1. Find Sam’s cost for a week when he drives 0 miles.
Find $CC$ when $dd$ = 0. $C=0.5(0)+60C=0.5(0)+60$
Simplify. $C=60C=60$
Sam’s costs are $60 when he drives 0 miles.
2. Find the cost for a week when he drives 250 miles.
Find $CC$ when $d=250d=250$. $C=0.5(250)+60C=0.5(250)+60$
Simplify. $C=185C=185$
Sam’s costs are $185 when he drives 250 miles.
3. Interpret the slope and $CC$-intercept of the equation.
The slope, 0.5, means that the weekly cost, $CC$, increases by $0.50 when the number of miles driven, $dd$, increases by 1. The $CC$-intercept means that when the number of miles driven is 0, the
weekly cost is $60.
4. Graph the equation (Figure 5.81).
We’ll need to use a larger scale than usual. Start at the $CC$-intercept (0, 60). To count out the slope $m=0.5m=0.5$, we rewrite it as an equivalent fraction that will make our graphing easier.
So to graph the next point go up 50 from the intercept of 60 and then to the right 100. The second point will be (100, 110).
Section 5.8 Exercises | {"url":"https://openstax.org/books/contemporary-mathematics/pages/5-8-graphing-functions","timestamp":"2024-11-11T03:55:16Z","content_type":"text/html","content_length":"607383","record_id":"<urn:uuid:716fa8f6-6592-4a44-9abf-75ba59d8a477>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00505.warc.gz"} |
Weekend Nerd Diversions
by Mark Meyer · Posted in: nerdiness
Math and visual representations of mathematical ideas is something of a hobby for me. One of my favorite tools for this is Nodebox, which is a Mac OS X application that gives you a basic canvas and
some simple python classes to make it easy to create vector images. To make a Nodebox rendering you simply write a python script. The possibilities are limited only by your imagination and
programming skill. The following script was inspired by the book Indra's Pearls: The Vision of Felix Klein by David Mumford, Caroline Series, and David Wright. It's a gorgeous book investigating
limit sets of discrete groups on the complex plane. Because python has excellent support for complex numbers, it is a natural choice for this sort of work. If you have some skills in scripting or
programming, the pseudo-code throughout the books provides some useful guidance in recreating many of the graphic images in the book.
The following is a simple python script that generates the above image.
from cmath import *
background(.96, .98, 1)
class cCircle():
'''A simple circle on the complex plane'''
def __init__(self, center, radius, draw=True):
self.center = center
self.radius = radius
if draw:
def draw(self):
oval(self.center.real-self.radius, self.center.imag-self.radius, self.radius*2, self.radius*2)
def marker(point, stroke=(.2), strokewidth=(.25), fill=None):
return oval(point[0]-2, point[1]-2, 4, 4, stroke=stroke, strokewidth=strokewidth, fill=fill)
def cmarker(z, stroke=(.2), strokewidth=(.25), fill=None):
return marker((z.real, z.imag), stroke=stroke, strokewidth=strokewidth, fill=fill)
def cpolyline(Z, draw=True):
beginpath(Z[0].real, Z[0].imag)
for z in Z[1:]:
lineto(z.real, z.imag)
return endpath(draw=draw)
def mobius_on_point(t, z):
return (t[0] * z +t[1])/(t[2]*z + t[3])
def mobius_on_circle(t, c):
'''t should be 1x4 array, c is a cCirlce'''
z = c.center -(c.radius**2/(t[3]/t[2] + c.center).conjugate())
center = mobius_on_point(t, z)
radius = abs(center - mobius_on_point(t, c.center+c.radius))
return cCircle(center, radius)
translate(408, 410)
fill(.98,.99,.9, .035)
a = complex(1., -5.)
b = 0.04
c = 0.39
d = complex(0.99, -5.)
cent = complex(1.16,0.91)
C = cCircle(cent,1.)
t = [a, b, c, d]
C2 = cCircle(-cent, 1)
t2 = [a, -b, -c, d]
for i in range(600):
C = mobius_on_circle(t, C)
C2 = mobius_on_circle(t2, C2) | {"url":"https://www.photo-mark.com/notes/weekend-nerd-diversions/","timestamp":"2024-11-11T19:59:27Z","content_type":"text/html","content_length":"12375","record_id":"<urn:uuid:d3b89401-5151-4d88-b433-9b6344a41245>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00036.warc.gz"} |
Rethinking Weight Initialization
Table of Contents
Rethinking Weight Initialization
Authored by: Timothy Letkeman
Published: 2021-10-16
Weight Initialization 101
What and Why?
Neural networks are trained using a technique called gradient descent. First, the network makes a prediction, that prediction is then evaluated, and the loss (error) of the prediction is then
backpropagated through the network in order to calculate the gradients of the weights in your network. Those gradients are then applied to the weights and biases of the network, changing them such
that they should then make better predictions. This process is run many, many times until the network is capable of making useful predictions. However, in order to perform this gradient descent, the
weights of the network must first be initialized so that the gradient descent process has a starting point.
Perhaps unsurprisingly from the name, weight initialization is the act of setting the starting weights and biases for your neural network, such that your network trains as efficiently as possible. It
is quite easy to see how using a poor weight initialization strategy could drastically slow down your system's learning or cause it not to converge at all. This is not a problem you can avoid, in
order to start training your model; you must first initialize its weights and biases. As such, it is important to think and learn about weight initialization and about which weight initialization
strategies can help you maximize your model's performance.
The main problem that weight initialization is attempting to solve is setting the weights of the network such that the gradients of the network don't grow too small or too big when propagated through
the layers of the network; this is known as the vanishing gradient problem and the exploding gradient problem.
Say you initialize your weights far too small, such that the gradients decrease by an order of magnitude for each layer they backpropagate through. This would cause the layers closer to the start of
the network to train much slower than those near the end. Overall this could drastically slow down the time it takes for your network to train, wasting your valuable time and resources. This would be
an example of the vanishing gradient problem.
On the other hand, say you initialize your weights far too large, such that the gradients increase by an order of magnitude for each layer they backpropagate through. This would cause the earlier
layers of the network to have gradients that are far larger than is optimal. This can cause multiple problems:
1) the large gradients may cause the weights to swing past their optimal value (similarly to an excessively large learning rate).
2) in extreme cases, this can cause the gradients to overflow into NaN values, possibly ruining the entire network.
These would be examples of the exploding gradient problem.
To avoid these situations and to maximize network performance, we need to find a weight initialization strategy that avoids both the problems of vanishing gradients and exploding gradients.
Doing it Wrong
One naive approach to weight initialization is to set all the weights and biases of the network to 0. However, as is quite evident, this is a very poor strategy. The reason this performs so poorly is
that the gradient vanishes.
As a refresher, the gradient of a single weight is calculated as: \(\mathrm{weight}_i,_j = \mathrm{input}_i * \mathrm{error}_j\). In code it would look something like this:
weightGradients[getWeightIdx(inputIdx, outputIdx)] = inputs[inputIdx] * errors[outputIdx];
Also, to backprogate the error to the previous layer: \(\mathrm{inputError}_i = \mathrm{activation}'(\mathrm{inputNodes}_i) * \sum_{j=0}^{outputCount} \mathrm{weight}_i,_j * \mathrm{outputError}_j\).
In pseudo-code:
inputErrors[inputIdx] = 0.0f;
for (int outputIdx = 0; outputIdx < outputCount; outputIdx++)
inputErrors[inputIdx] += weights[getWeightIdx(inputIdx, outputIdx)] * outputErrors[outputIdx];
inputErrors[inputIdx] *= activation_prime(inputNodes[inputIdx]);
You can see how with a weight initialization scheme of setting all weights to zero, the network wouldn't initially be able to get any training done. This is because the error would not be able to be
propagated through all the layers of the network. The error of the output layer would be calculated, but nothing would be propagated. This would slow training down to a crawl, as the weights would
very slowly get adjusted from back to front, eventually allowing the error to be properly propagated. This is a prime example of a vanishing gradient problem.
If setting all the weights to zero caused a vanishing gradient, then the logical solution is to set all the weights to one instead; however, this can be significantly worse than setting everything to
zero! Let us illustrate this scheme graphically; we will create a three-layer network with sigmoid activation functions that learns to count in binary. We will do this by feeding it a five binary
digit number (0-31) and expecting it to predict that number plus one. For example: when fed the input 1 0 0 1 1 [19] the expected output is 1 0 1 0 0 [20].
The network weights are represented by the lines; if they are green, it is a positive connection; if it is red, it is a negative connection. The thicker the line, the stronger the connection. For
now, all the weights are initialized to one, so they are all equal. Biases are also represented by the outlines around the circle, if it is white, there is no bias, if it is green, there is a
positive bias, and if it is red, there is a negative bias.
Now we are going to attempt to train this network; however, instead of training it on the set of all binary numbers, let us just attempt to train it to understand this single situation. We will do
this by getting the network to predict using the input 1 0 0 1 1, then using 1 0 1 0 0 as the expected output, applying standard backpropagation in order to update the weights and biases. Let us
apply this algorithm until the model accurately predicts the output for this situation. We will also use a quite high learning rate of 1.0 for these tests, as that should allow the network to train
on this particular input VERY quickly.
Despite our very large learning rate, it still took exactly 337 iterations for the network to finally predict correctly. This is FAR too slow to be acceptable performance, as this should've been
accomplished in 1 or 2 iterations. This also demonstrates the first problem with this kind of initialization scheme: for some activation functions, predicting too extreme values produces a low
derivative, which results in slow learning. You'll notice that the vast majority of the actual weight adjustments occur in the last few iterations; this is because, in the first iterations, the
derivative is far too low, causing incredibly low gradients. Once the network has been slightly adjusted such that the derivatives become larger, more change can occur.
In this network, we are using the sigmoid activation function for every layer. The sigmoid function softly clamps the given value into the range (0, 1). Its formula and derivative is:
\(f(x) = \frac{1}{1 + e^{-x}}\)
\(f'(x) = f(x) * (1 - f(x))\)
Sigmoid is a very useful function; however, it can cause vanishing gradients in some situations, one of those situations we just saw. When the sigmoid function is pushed to its extremes, the
derivative of the function is low at that point and results in lower gradients and slower learning.
So the first problem with the all 1s initialization scheme was it pushed clamping activation functions to their extremes, causing low derivatives and a vanishing gradient. Non-clamping activation
functions (such as ReLU) don't have this problem; instead, they can suffer from exploding gradients because their value is uncapped. Now, let us view the second problem with this example by training
on the entire 0-31 binary number dataset. We will define one epoch as performing the gradient descent algorithm once on each item in the dataset. Note: the order in which the dataset was iterated is
random (this is because it increases convergence).
Obviously, the model is not converging; in fact, it never got more than 6/32 correct predictions on any epoch. This is the second problem with this initialization scheme: when the weights are too
similar, the model has trouble creating useful decision boundaries. However, this is a gross exaggeration of the problem, as even a small amount of variance will quickly cascade into proper learning.
Doing it Right
Given the two problems discussed, let us attempt to think of possible solutions to them. In order to reduce the activation functions being pushed to their extremes, we can set the weights such that
the starting activation will never be too high or too low (i.e., the magnitude of the nodes should remain low, but not too low or else no learning will occur). However, setting all of our weights to
some pre-computed ideal value for the given activation function would not allow the model to create useful decision boundaries; therefore we should incorporate some kind of pseudo-randomness into our
initialization scheme.
Xavier Initialization (named after one of the authors of the paper that pioneered the technique) addresses both of these concerns. There are many different adaptations, interpretations, and
modifications of Xavier Initialization; we will only discuss the two mentioned in the paper: Standard Xavier Initialization and Normalized Xavier Initialization. The algorithm for Standard Xavier
Initialization is to initialize all of your weights to be from a uniform distribution with the range being \((-\frac{1}{\sqrt{x}}, \frac{1}{\sqrt{x}})\) where \(x\) is the number of input nodes for
the given layer (also known as fan-out; fan-in is the amount of output nodes for the given layer). Implemented in pseudo-code:
float bound = 1.0f / sqrt(inputCount);
for (int weightIdx = 0; weightIdx < inputCount * outputCount; weightIdx++)
weights[weightIdx] = random.uniformDistribution(-bound, bound);
This solves both problems, the range of the distribution is designed to maintain the variance of the values when forward propagating, and it is also randomized so it can quickly produce valuable
decision boundaries. Let us test this initialization scheme using the same method before, but instead of All 1s Initialization, we will use Standard Xavier Initialization.
Evidently, this is far improved from All 1s Initialization, the values of the nodes are initially kept to a small value where the derivative at that point is still a large enough number for efficient
learning, and the values are randomized effectively so that decision boundaries can form. However, Standard Xavier Initialization was only designed to regulate the variance when feeding forward.
Backpropagation is just as valuable to a model as forward-propagation. For this purpose, there is Normalized Xavier Initialization, where the algorithm is incredibly similar to Standard Xavier
Initialization; however, it attempts to regulate both the variance moving forward and the variance moving backward. It also utilizes a uniform distribution ; but with a range of \((-\frac{\sqrt{6}}{\
sqrt{x + y}}, \frac{\sqrt{6}}{\sqrt{x + y}})\) where \(x\) is the number of input nodes and \(y\) is the number of output nodes.
Uniform distributions are not the only method of applying randomness; a common approach is using a normal distribution. One approach that uses a normal distribution is Kaiming Initialization (we will
address a slightly simplified version, for more information, see this paper). Kaiming Initialization uses a standard deviation with a mean of \(0\) and a standard deviation of \(\sqrt{\frac{2}{x}}\)
where \(x\) can either be the number of input nodes, or the number of output nodes (depends on if you choose to preserve the magnitude while forward-propagating [using input node count/fan-out], or
while backpropagating [using output node count/fan-in]).
These initialization schemes are both among the leading modern approaches to weight initialization. They are typically used for different purposes; Xavier is generally used for networks with
sigmoidal activation functions (Sigmoid, TanH, etc.), whereas Kaiming is generally used for large networks (20+ layers) with a lot of linear activation functions (ReLU, LeakyReLU, etc.).
What can be improved?
Variance Preservation v.s. Magnitude Preservation
Xavier and Kaiming initialization have one main feature in common; they are designed to preserve the variance of the values while being propagated. This means that when backpropagating through the
network, the magnitude of the gradients may differ from layer to layer, but the variance (square of the standard deviation) would stay consistent. Instead of preserving the variance of the gradients
while backpropagating, wouldn't it make more sense to preserve the magnitude of the gradients? If the magnitude of the gradients were conserved while backpropagating, the gradients of the layers
earlier in the network would be roughly equivalent to the layers closer to the output, possibly allowing for more even training. This is all speculation; the only thing left to do is to test our
Let us start by testing how well Xavier preserves the magnitude of its input. We can test this by writing a Python program to emulate a Xavier initialization many MANY times for many different sized
layers, then evaluating the average magnitude of the outputs. The Python code for it is:
import random
import math
def test_uniform_magnitude_preservation(input_count, bound):
""" Simulates propagating with an input of all 1s, with weights initialized using a uniform distribution
then returns the absolute value of the output."""
value = 0.0
for _ in range(input_count):
value += random.uniform(-bound, bound) # assume every input is 1, therefore we can just add the weight
return abs(value)
iterations = 1000000
layer_sizes = [1, 5, 25, 100, 300, 1000]
for size in layer_sizes:
xavier_bound = 1.0 / math.sqrt(size)
total = 0.0
for i in range(iterations):
total += test_uniform_magnitude_preservation(size, xavier_bound)
print("Layer size: " + str(size) + " | magnitude: " + str(total / iterations))
The code produced this output:
Layer size: 1 | magnitude: 0.500474810495207
Layer size: 5 | magnitude: 0.4658366066117104
Layer size: 25 | magnitude: 0.4617913077193642
Layer size: 100 | magnitude: 0.4611953525250335
Layer size: 300 | magnitude: 0.4603023799913436
Layer size: 1000 | magnitude: 0.4605755033078747
Interesting, so it appears that at least experimentally, Xavier Initialization does not preserve magnitude. Now let us try to recreate these values theoretically by finding a formula for how much
magnitude was preserved given the layer size.
Standard Letkeman Initialization
Currently, we calculate the magnitude preservation by doing a sum of uniform distributions several times. Luckily, there is a formula for exactly this, it is called the Irwin-Hall Distribution, it
creates a probability curve for the sum of \(n\) uniform distributions that each have the range [0, 1]. It's formula is:
Despite being a slightly difficult formula to work with, this is great! However, Xavier weight initializations are not in the range of \([0, 1]\) therefore we need to manually tune this function
slightly to emulate our uniform distributions being in the range of \((-\frac{1}{\sqrt{n}}, \frac{1}{\sqrt{n}})\).
This gives us the probability of each weight value, now we just need to multiply by the absolute value of \(x\) to get our actual expected value from the weight.
Finally, we can intergrate from \((-\sqrt{n}, \sqrt{n})\) (the minimum and maximum possible ranges of our sum of Xavier uniform distributions) to get our final expected magnitude preservation.
Written out in its entirety, we get this behemoth:
This is quite a beast to calculate whenever you need to initialize a layer (remember you would need to re-integrate this and substitute \(n\) with the amount of input nodes. To solve this problem, we
can use an approximation function instead:
def approximate(n):
# Hardcode the first few values, then use an approximation function after that
if n == 1:
return 0.5
if n == 2:
return 0.471404520791
if n == 3:
return 0.469097093716
a = 0.039
return (0.5 - a) + ((1.0 / math.sqrt(2.0 * n - 1.0)) ** 2) * a
Now, if we were to take our Xavier bounds and divide by this approximated integral, we should get a magnitude very close to 1. Let us test that:
Layer size: 1 | magnitude: 0.9997531021498397
Layer size: 5 | magnitude: 1.0006271684273118
Layer size: 25 | magnitude: 0.9997050090618478
Layer size: 100 | magnitude: 0.9996631602474665
Layer size: 300 | magnitude: 1.0000115563283445
Layer size: 1000 | magnitude: 1.0003759838261814
Therefore, if we initialize our weights using a uniform distribution with the range \((-\frac{\frac{1}{\sqrt{x}}}{\int_{-\sqrt{n}}^{\sqrt{n}}\frac{\frac{n}{2(n-1)!}\sum_{k=0}^{n}(-1)^{k}{{n}\choose
{k}}(\frac{{x}\sqrt{n}}{2}+\frac{n}{2}-k)^{n-1}{\operatorname{sgn}}(\frac{{x}\sqrt{n}}{2}+\frac{n}{2}-k)}{{2}\sqrt{n}}|x|dx}, \frac{\frac{1}{\sqrt{x}}}{\int_{-\sqrt{n}}^{\sqrt{n}}\frac{\frac{n}{2
(n-1)!}\sum_{k=0}^{n}(-1)^{k}{{n}\choose{k}}(\frac{{x}\sqrt{n}}{2}+\frac{n}{2}-k)^{n-1}{\operatorname{sgn}}(\frac{{x}\sqrt{n}}{2}+\frac{n}{2}-k)}{{2}\sqrt{n}}|x|dx})\) (where \(n\) is the number of
nodes in the input layer) the magnitude should, on average, be the same as the magnitude of the inputs. For the sake of brevity, I'm going to refer to this as Standard Letkeman Initialization.
Normalized Letkeman Initialization
There is a problem with Standard Letkeman Initialization; it only ensures that the magnitude is maintained during the forward pass but does nothing for backpropagation. Just like how Standard Xavier
Initialization has its normalized counter-part that is designed to regulate both the forward pass and the backward pass. However, this introduces some complications; the only situation in which the
magnitude is perfectly preserved in both the forward pass and the backward pass is if the number of input and output nodes for that layer is equal. If the numbers are unequal, the magnitude is not
going to be perfectly preserved in at least one of the directions; our solution to this is just to make the average magnitude preservation across all nodes one (for both the forward and backward
pass); this will require changing our experimental testing program slightly:
import random
import math
def test_normalized_uniform_magnitude_preservation(input_count, output_count, bound):
""" Simulates propagating with an input of all 1s, with weights initialized using a uniform distribution
then returns the absolute value of the output for both forward and backward propagation."""
# for this version of the function we'll have to simulate the entirety of the forward and backward passes
# as well as creating the weight matrix
weights = [[random.uniform(-bound, bound) for __ in range(output_count)] for _ in range(input_count)]
total_forward_magnitudes = 0.0
for i in range(input_count):
value = 0.0
for o in range(output_count):
value += weights[i][o] # assume every input is 1, therefore we can just add the weight
total_forward_magnitudes += abs(value)
total_backward_magnitudes = 0.0
for o in range(output_count):
value = 0.0
for i in range(input_count):
value += weights[i][o] # assume every input is 1, therefore we can just add the weight
total_backward_magnitudes += abs(value)
return total_forward_magnitudes / float(output_count), total_backward_magnitudes / float(input_count), \
(total_forward_magnitudes + total_backward_magnitudes) / float(input_count + output_count)
iterations = 5000
layer_sizes = [(1, 1), (5, 5), (100, 1), (15, 25), (10, 100), (100, 50), (100, 300), (1000, 10), (1, 10000), (3, 10000)]
for size in layer_sizes:
normalized_xavier_bound = math.sqrt(6.0) / math.sqrt(size[0] + size[1])
forward_total = 0.0
backward_total = 0.0
avg_total = 0.0
for i in range(iterations):
forward, backward, avg = test_normalized_uniform_magnitude_preservation(size[0], size[1],
forward_total += forward
backward_total += backward
avg_total += avg
print("Layer size: " + str(size) +
" | forward magnitude: " + str(forward_total / iterations) +
" | backward magnitude: " + str(backward_total / iterations) +
" | avg magnitude: " + str(avg_total / iterations))
Which when ran produced these results:
Layer size: (1, 1) | forward magnitude: 0.8594973435645746 | backward magnitude: 0.8594973435645746 | avg magnitude: 0.8594973435645746
Layer size: (5, 5) | forward magnitude: 0.8084311839378276 | backward magnitude: 0.8045734522931961 | avg magnitude: 0.806502318115512
Layer size: (100, 1) | forward magnitude: 12.187413775506647 | backward magnitude: 0.011053896083235643 | avg magnitude: 0.1316119146913889
Layer size: (15, 25) | forward magnitude: 0.5368980806669289 | backward magnitude: 1.1514005259906934 | avg magnitude: 0.7673364976633413
Layer size: (10, 100) | forward magnitude: 0.1077280335302238 | backward magnitude: 3.4205235980572755 | avg magnitude: 0.4088912666690458
Layer size: (100, 50) | forward magnitude: 1.3046534263608478 | backward magnitude: 0.46020072148131086 | avg magnitude: 0.7416849564411578
Layer size: (100, 300) | forward magnitude: 0.3257606846562078 | backward magnitude: 1.6939843417948686 | avg magnitude: 0.6678165989408731
Layer size: (1000, 10) | forward magnitude: 11.277801840070467 | backward magnitude: 0.011307052186118244 | avg magnitude: 0.1228565055315076
Layer size: (1, 10000) | forward magnitude: 0.00011277174801493292 | backward magnitude: 122.47346786077979 | avg magnitude: 0.01235888264582831
Layer size: (3, 10000) | forward magnitude: 0.0003375413496494392 | backward magnitude: 66.31341829144523 | avg magnitude: 0.020225499187326784
Just like what we did with Standard Letkeman Initialization, let us try to recreate these values theoretically by finding a formula for how much magnitude was preserved given the input layers size as
\(n\) and the output layers size as \(m\). We can use the same Irwin-Hall distribution; however, we will need one distribution for the forward pass, and one for the backward pass.
Again, this represents the uniform distributions as having a range of \([0, 1]\), so we need functions to convert their range to Normalized Xavier ranges of \((-\frac{\sqrt{6}}{\sqrt{n+m}}, \frac{\
This gives us the probability of each weight value for the forward and backward passes, now we need to multiply by \(|x|\) to get our actual expected value from the weight.
Now we integrate from \((-\frac{n\sqrt{6}}{\sqrt{n+m}},\frac{n\sqrt{6}}{\sqrt{n+m}})\) for our forward pass, and \((-\frac{m\sqrt{6}}{\sqrt{n+m}},\frac{m\sqrt{6}}{\sqrt{n+m}})\) for our backward
Finally, we can sum our results together for our final calculation.
Written as a whole (the faint of heart may want to avert their eyes away from this transgression against God and Man):
Again, it is incredibly expensive to compute this at run-time, so we can use an approximation function that utilizes our previous approximation function:
def approximate_a(n):
a = 0.073
return (0.866 - a) + (1.0 / math.sqrt(6.0 * n - 5.0)) * a
def approximate_b(n):
b = -0.08
return (1.0 - b) + (1.0 / math.sqrt(6.0 * n - 5.0)) * b
def approximate_c(n):
return 1 / math.sqrt(0.5 * n + 0.5)
def approximate_normalized(n, m):
highest = n if n >= m else m
lowest = m if n >= m else n
return approximate_a(lowest) * approximate_b(highest - lowest + 1) * approximate_c(highest / lowest)
Now we take the Normalized Xavier bounds, divide by it using this approximation function. When we run this we get this output:
Layer size: (1, 1) | forward magnitude: 0.9943010700551295 | backward magnitude: 0.9943010700551295 | avg magnitude: 0.9943010700551295
Layer size: (5, 5) | forward magnitude: 0.9993559541424449 | backward magnitude: 1.0016561989451123 | avg magnitude: 1.0005060765437785
Layer size: (100, 1) | forward magnitude: 92.90548094688566 | backward magnitude: 0.08605654405489507 | avg magnitude: 1.0050607460631158
Layer size: (15, 25) | forward magnitude: 0.7219516088992118 | backward magnitude: 1.552360250077088 | avg magnitude: 1.0333548493409184
Layer size: (10, 100) | forward magnitude: 0.29088207337482874 | backward magnitude: 9.277565035000634 | avg magnitude: 1.1078532517044446
Layer size: (100, 50) | forward magnitude: 1.8668152727502358 | backward magnitude: 0.657427648552596 | avg magnitude: 1.0605568566184786
Layer size: (100, 300) | forward magnitude: 0.5372220506522588 | backward magnitude: 2.790470951744516 | avg magnitude: 1.1005342759253183
Layer size: (1000, 10) | forward magnitude: 92.54224171674791 | backward magnitude: 0.09236415107640739 | avg magnitude: 1.0077094735088017
Layer size: (1, 10000) | forward magnitude: 0.008441322648025064 | backward magnitude: 9262.953664512319 | avg magnitude: 0.93464322477678
Layer size: (3, 10000) | forward magnitude: 0.015693092438484952 | backward magnitude: 3084.4314192973225 | avg magnitude: 0.9407402961388404
Therefore, if we initialize our weights using a uniform distribution with the range: \(\begin{split}(-\frac{\frac{\sqrt{6}}{\sqrt{n+m}}}{\frac{n\left(\int_{-\frac{n\sqrt{6}}{\sqrt{n+m}}}^{\frac{n\
frac{m}{2}-k\right)\left|x\right|}{\frac{2m\sqrt{6}}{\sqrt{n+m}}}dx\right)}{n+m}},\\ \frac{\frac{\sqrt{6}}{\sqrt{n+m}}}{\frac{n\left(\int_{-\frac{n\sqrt{6}}{\sqrt{n+m}}}^{\frac{n\sqrt{6}}{\sqrt
That should make the average magnitude preservation of both the forward and backward pass to be equal to 1. I am going to refer to this as Normalized Letkeman Initialization.
We are also going to do one thing differently compared to Xavier and Kaiming; there are some situations in which the number of nodes in the input layer isn't necessarily the number of active nodes in
that layer. Namely when our input is either a one-hot vector or the product of a softmax activation. As a refresher, a one-hot vector is when you have a vector of all 0s except for one entry which is
a 1, and a softmax activation clamps the total activated output to 1 according to this formula: \(f(x)_{i} = \frac{e^{x}}{\sum_{j=0}^{n}e^{j}}\) (essentially set each element of the vector to \(e\)
raised to that element, then divide by the sum of all elements in the vector). In both of these situations, the guaranteed magnitude of the layer is 1; therefore, we can treat the entire layer as a
single input node. As the magnitude of that layer is always 1, the actual size of that layer is irrelevant and we can always treat it as 1.
This is something that Xavier and Kaiming don't account for, and we will see in the following results section that this can have a substantial impact on the speed of our learning. I will refer to
this method of determining the input nodes as the Active Node Method. As we will see in the results section, adding this slight change can have drastic improvements compared to the traditional fan-in
Overall Results
I compared these five initialization methods (Standard Xavier, Normalized Xavier, Standard Letkeman, Normalize Letkeman, and Kaiming) against each other using four different networks. For each
network and initialization method, I ran the simulation five different times using the seeds 1-5, just to get an average. I also ensured that the randomness was consistent (for all the initialization
methods that utilized a uniform distribution, the distribution always gave a similar number for each weight, the only thing that changed was the initialization method modifying the range of the
weights). I then created graphs of their training loss versus epochs which can be used to compare how efficiently the networks trained.
The four networks I used were: the binary counting network presented earlier in this article (the only difference being a learning rate of 0.1 instead), a convolutional network for the MNIST digit
dataset, a large dense network that utilizes sigmoid activation functions for the MNIST digit dataset, and a recurrent neural network trained to replicate Shakespeare's Hamlet. These networks were
not selected because they are the best at accomplishing their given task (they all perform to varying degrees of success), but because they represent a wide variety of different networks and tasks.
All the data can be downloaded here.
Binary Counting Network: Standard Letkeman seems to have a noticeable advantage for this network; something to note is that on the 5th seed, Kaiming slowed down quite considerably closer to the end
of the training, causing its average to fall noticeably behind, despite Kaiming performing quite comparable to Standard Letkeman on the other seeds.
MNIST Convolutional Network: Standard Letkeman appears to train significantly more effectively than the other initilization algorithms; much like in the Binary Counting Network, Kaiming started
training incredibly slow on the 3rd seed, which brought it's average much lower. Normalized Letkeman failed to converge in 2/5 seeds for this network architecture, and trained incredibly slow on the
other seeds. This is most likely due to the later layers in the network having far fewer nodes than the previous, causing those layers' activation values to skyrocket, causing a vanishing gradient.
Standard Xavier's average loss at the end of the 75 epochs was 0.007568, Standard Letkeman was able to surpass that in 43 epochs; meaning that Standard Letkeman trained at 1.744x the speed of
Standard Xavier (the next fastest method).
MNIST Dense Network: Kaiming was able to escape the plateau first, followed by Standard Letkeman, then Kaiming and both forms of Letkeman approached roughly the same values, with Kaiming having a
very slight lead. Standard Xavier was unable to converge in this architecture, likely due to weights being initilized too low, resulting in the weights having little effect near the end of the
network, causing it to be unable to train.
Hamlet RNN: Standard and Normalized Letkeman clearly trained far faster than any of the alternatives, with Normalized Letkeman being slightly faster than Standard letkeman. It is also worthy to note
that the data for these tests was incredibly regular and consistent, with minimal irregularities between the seeds. Kaiming's average loss at the end of the 25 epochs was 2.024412, Standard Letkeman
surpassed that loss in 10 epochs, while Normalized Letkeman only took 8 epochs to surpass Kaiming; meaning that Standard Letkeman trained at 2.5x the speed of Kaiming, while Normalized Letkeman
trained at 3.125x speed (although as you will see in the next section, the majority of that change is from Letkeman methods utilizing the active node method of counting input nodes, which is
something that other methods can utilize with only small adjustments).
Active Node Method
The only one of these networks to utilize the active node method was the RNN used to predict Shakespeare's Hamlet, it utilized a one-hot vector of all the possible characters in the input layer (ex.
if the current character was an 'a', then only the node corresponding to 'a' would be set to 1, all others would be 0). I ran two sets of tests for this, one test where the Xavier and Kaiming
initialization methods utilized the Active Node Method (ANM), and another test where they didn't; I then compared those to Letkeman methods (which always use ANM).
As you can see with the Active Node Method, Kaiming and Letkeman train at nearly the exact same rate (in fact, Kaiming is very slightly faster). It is quite evident that using the ANM greatly
improves training speed.
Given the results of the tests, Standard Letkeman initialization appears to be a more consistent and effective method for initializing weights compared to Xavier and Kaiming; in some cases even
cutting training time by 68%. However, these tests were not exhaustive and are not indicative of all possible machine learning models you may want to utilize. As such, it is important to do your own
tests and judge which method performs best for your network architecture. However, testing has shown that Standard Letkeman initialization is an effective initialization algorithm for most machine
learning models. | {"url":"https://tigpan.com/articles/rethinking-weight-initialization/article.html","timestamp":"2024-11-10T04:57:40Z","content_type":"text/html","content_length":"52314","record_id":"<urn:uuid:18e4d45a-daed-49da-9fce-25773b32327f>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00078.warc.gz"} |
DOI Number
Recently, a new geometric approach called the fixed-circle problem has been introduced to fixed-point theory. The problem has been studied using different techniques on metric spaces. In this paper,
we consider the fixed-circle problem on S-metric spaces. We investigate existence and uniqueness conditions for fixed circles of self-mappings on an <em>S</em>-metric space. Some examples of
self-mappings having fixed circles are also given.
fixed-circle problem; self-mapping; S-metric space.
bibitem{Banach} S. Banach, Sur les operations dans les ensembles abstraits
et leur application aux equations integrals, textit{Fund. Math.} 2 (1922),
bibitem{Ciesielski-2007} K. Ciesielski, On Stefan Banach and some of his
results, textit{Banach J. Math. Anal.} 1 (2007), no.1, 1-10.
bibitem{Hieu} N. T. Hieu, N. T. Ly and N. V. Dung, A Generalization of
Ciric Quasi-Contractions for Maps on $S$-Metric Spaces, textit{Thai Journal
of Mathematics} 13 (2015), no.2, 369-380.
bibitem{Ozdemir-2011} N. "{O}zdemir, B. B. .{I}skender and N. Y. "{O}zg%
"{u}r, Complex valued neural network with M"{o}bius activation function,
textit{Commun. Nonlinear Sci. Numer. Simul.} 16 (2011), no.12, 4698-4703.
bibitem{nihal} N. Y. "{O}zg"{u}r and N. Tac{s}, Some fixed point
theorems on $S$-metric spaces{,} textit{Mat. Vesnik}{ 69} (2017), no.1,
bibitem{nihal2} N. Y. "{O}zg"{u}r and N. Tac{s}, Some new contractive
mappings on $S$-metric spaces and their relationships with the mapping $%
(S25) $, textit{Math. Sci.} 11 (2017), no.1, 7-16.
bibitem{nihal3} N. Y. "{O}zg"{u}r and N. Tac{s}, Some generalizations of
fixed point theorems on $S$-metric spaces, textit{Essays in Mathematics and
Its Applications in Honor of Vladimir Arnold}, New York, Springer, 2016.
bibitem{nihal4} N. Y. "{O}zg"{u}r and N. Tac{s}, Some fixed-circle
theorems on metric spaces, Bull. Malays. Math. Sci. Soc., (2017).
bibitem{Ozgur-Aip} N. Y. "{O}zg"{u}r and N. Tac{s}, Some fixed-circle
theorems and discontinuity at fixed circle, AIP Conference Proceedings 1926,
bibitem{Rhoades} B. E. Rhoades, A comparison of various definitions of
contractive mappings, textit{Trans. Amer. Math. Soc.} 226 (1977), 257-290.
bibitem{Sedghi-2012} S. Sedghi, N. Shobe and A. Aliouche, A Generalization
of Fixed Point Theorems in $S$-Metric Spaces, textit{Mat. Vesnik} 64
(2012), no.3, 258-266.
bibitem{Sedghi-2014} S. Sedghi and N. V. Dung, Fixed Point Theorems on $S$%
-Metric Spaces{,} textit{Mat. Vesnik}{ 66} (2014), no.1, 113-124.
bibitem{tez} N. Tac{s}, Fixed point theorems and their various
applications, Ph. D. Thesis, 2017.
• There are currently no refbacks.
© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online) | {"url":"https://casopisi.junis.ni.ac.rs/index.php/FUMathInf/article/view/4530/0","timestamp":"2024-11-08T18:22:26Z","content_type":"application/xhtml+xml","content_length":"23417","record_id":"<urn:uuid:dd0532d8-628d-44c4-8209-ce090e4332c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00237.warc.gz"} |
Find The Number Of Solutions To A Linear Equation Worksheets [PDF]: Algebra 1 Math
How Will This Worksheet on "Find the Number of Solutions to a Linear Equation" Benefit Your Students' Learning?
• It enhances the ability to approach mathematical problems systematically and develop effective problem-solving skills.
• Helps to develop a better understanding of system of linear equation by identifying the number of solutions.
• It helps in laying the foundation for more advanced mathematical topics, including systems of equations, matrices, vectors, and linear transformations
How to Find the Number of Solutions to a Linear Equation?
• One Solution: If solving an equation results in a specific value, then the equation has one solution.
• No Solution: If solving an equation results in a false statement, then the equation has no solution.
• Infinite Solutions: If solving an equation leads to a statement that is always true, then the equation has infinitely many solutions. | {"url":"https://www.bytelearn.com/math-algebra-1/worksheet/find-the-number-of-solutions-to-a-linear-equation","timestamp":"2024-11-10T05:02:52Z","content_type":"text/html","content_length":"112446","record_id":"<urn:uuid:d1cc17a3-b645-490f-9d85-5aef92aca91a>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00457.warc.gz"} |
Flotation Principle and Explanation - Archimedes
Ever wondered how huge ships manage to stay afloat in water, while a small iron nail sinks? Puzzling as it may appear, you can easily
Principle of Flotation: Definition
The Archimedes’ principle states that any object, wholly or partially immersed in a fluid, experiences an upward force equal to the weight of the fluid displaced by it.
Here the term ‘fluid’ refers to all liquids and gases. For an object that is completely submerged in a fluid, the weight of the fluid displaced by it is less than its own weight. On the other hand,
for an object that floats on the surface of the fluid, the weight of the fluid displaced by it, is equal to the weight of the object. Now, the upward force experienced by the body is termed as the
buoyant force. Thus,
Buoyant force = weight of the fluid displaced by the body
Now, the weight of the fluid displaced by the body is directly proportional to the volume of the displaced fluid, since the density of the fluid is constant. This can be illustrated by the following
Weight = Mass x g (where g is the acceleration due to gravity and is a constant)
Mass = Density x Volume
Thus, we can say
Weight = Density x Volume x g
From the above equation, we can conclude that a body shall float in a fluid under any of the two conditions.
The density of the body is less than the density of the fluid.
The volume of the fluid displaced by the immersed part of the body is such that its weight is equal to the weight of the body.
The following two examples will help you understand this:
1. Firstly, consider two cubes of the same dimensions, one made of cork and the other made of solid iron. If you place them on the surface of the water, what will happen? Well, the iron cube would
sink while the cube made of cork would easily float on water.
2. Now, take the example of a nail and a ship, both made of iron. While the nail sinks, the ship floats on water carrying several passengers and cargo.
In the first example, the cork cube sinks while the iron cube does not, because the density of iron is higher but the density of cork is lower than that of water. It is the same reason we find it
easier to swim in sea water than in river water, as the density of sea water is high due to the dissolved salts present in it.
Now, let’s consider the example of the nail and the ship. When you place a nail in water, the weight of the water displaced by the nail is less than the weight of the nail itself. In other words, the
buoyant force experienced by the nail (which is equal to the weight of the water displaced by it) is less than its weight, and the nail sinks in water. However, when you observe a huge ship that
floats on water, you’ll see that the ship is hollow, which means it is filled with air. This makes the average density of the ship lower than that of water. Thus, with only a small part of it
submerged in water, the weight of the water displaced by the ship, becomes equal to the buoyant force, and the ship floats on water. From this, we can conclude that, for a body to float in water or
any other fluid, the weight of the fluid displaced by the body should be equal to the weight of the body. In other words, more the weight of a body, more the volume of fluid it needs to displace, in
order to float in the fluid.
Now, consider this. Suppose an iron ball weighs 20 kg. When a string is tied to the ball and it is submerged in water, the weight of water displaced by the ball is, say, 7 kg. Therefore, the ball
would experience an upward force equal to 7 kg. This means that the net downward force experienced by the string would be equal to 13 kg (20 – 7 = 13). Thus, it can be concluded that the weight of
the ball decreases when it is immersed in water. This reduced weight of the ball is termed as the apparent weight. Hence, the Archimedes’ principle can be restated as follows.
Reduced weight of the body in water (Apparent weight) = Weight of the body – weight of the fluid displaced
The Story Behind the Principle
Most of the inventions of Archimedes were made to help his country during the time of war. However, the story behind the discovery of the principle of flotation is an interesting one. Briefly put, it
goes this way. The king of the land had got a golden crown made, to be offered to the deity of a temple. However, he doubted the honesty of the goldsmith, due to which he wanted to make sure that it
Archimedes was one of the greatest mathematicians of all times. He was a Greek mathematician who was also a physicist, scientist and a great inventor. Born in 287 B.C. in Sicily, Archimedes had many
great inventions to his credit before his death in 212 B.C. His most famous inventions include the screw propeller and the principle of flotation, among others. His mathematical works include
inventing infinitesimals and formulas on the measurement of a circle, parabolas, spheres, cylinders and cones. The one theorem that he considered to be his most prized and valuable achievement, is
the one which states that if you have a sphere and a cylinder of the same height and diameter, then the volume and the surface area of the sphere will be 2/3 of that of the cylinder, with the surface
area of the cylinder inclusive of the surface areas of its bases. However, the principle of flotation remains one of his most popular inventions.
Credits- M. Mukherjee | {"url":"https://www.cultofsea.com/general/flotation-principle-and-explanation-archimedes/","timestamp":"2024-11-14T11:48:03Z","content_type":"text/html","content_length":"136768","record_id":"<urn:uuid:bee95583-7481-44e8-b363-58ba8b300ac9>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00518.warc.gz"} |
Question: 87
Since we are investing double amount in the portfolio and half of the amount is borrowing on which we have to pay interest that risk free rate.
• We will use "wb" for borrowings and "wi" for investment proportion.
• Define a function "expected_return" in python which will depend on investement proportion
def return_on_portfolio(investment_proportion,interest_rate,expected_return,borrowings)`:
rs = expected_return wi = investment_proportion wb = borrowings rf = interest_rate return result | {"url":"https://cloudxlab.com/assessment/displayslide/8100/question-87?playlist_id=1537","timestamp":"2024-11-10T07:41:12Z","content_type":"text/html","content_length":"74078","record_id":"<urn:uuid:6f7c7b1a-7b83-4b05-b112-b571852a1840>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00447.warc.gz"} |
How do you find the derivative of y= e^sqrt(x) ? | HIX Tutor
How do you find the derivative of #y= e^sqrt(x)# ?
Answer 1
In this problem we have to use the chain rule.
#y=e^(sqrt(x))=e^(x^(1/2))#, convert the square root to its rational power
Apply the chain rule and begin to simplify
Convert the exponents to positive numbers
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
Not the question you need?
HIX Tutor
Solve ANY homework problem with a smart AI
• 98% accuracy study help
• Covers math, physics, chemistry, biology, and more
• Step-by-step, in-depth guides
• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/how-do-you-find-the-derivative-of-y-e-sqrt-x-8f9af9db70","timestamp":"2024-11-08T04:37:29Z","content_type":"text/html","content_length":"571903","record_id":"<urn:uuid:b67b5c8a-9a6f-422b-a7e9-335017b120f4>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00132.warc.gz"} |