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Global Constraint Catalog: Cderangement
<< 5.113. deepest_valley5.115. differ_from_at_least_k_pos >>
\mathrm{𝚌𝚢𝚌𝚕𝚎}
\mathrm{𝚍𝚎𝚛𝚊𝚗𝚐𝚎𝚖𝚎𝚗𝚝}\left(\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚜𝚞𝚌𝚌}-\mathrm{𝚍𝚟𝚊𝚛}\right)
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|>1
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝙽𝙾𝙳𝙴𝚂},\left[\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚜𝚞𝚌𝚌}\right]\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\ge 1
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝙽𝙾𝙳𝙴𝚂},\mathrm{𝚒𝚗𝚍𝚎𝚡}\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚜𝚞𝚌𝚌}\ge 1
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚜𝚞𝚌𝚌}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
Enforce to have a permutation with no cycle of length one. The permutation is depicted by the
\mathrm{𝚜𝚞𝚌𝚌}
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\left(\begin{array}{c}〈\begin{array}{cc}\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\hfill & \mathrm{𝚜𝚞𝚌𝚌}-2,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-2\hfill & \mathrm{𝚜𝚞𝚌𝚌}-1,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-5,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-4\hfill & \mathrm{𝚜𝚞𝚌𝚌}-3,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-5\hfill & \mathrm{𝚜𝚞𝚌𝚌}-4\hfill \end{array}〉\hfill \end{array}\right)
In the permutation of the example we have the following 2 cycles:
1\to 2\to 1
3\to 5\to 4\to 3
. Since these cycles have both a length strictly greater than one the corresponding
\mathrm{𝚍𝚎𝚛𝚊𝚗𝚐𝚎𝚖𝚎𝚗𝚝}
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|>2
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\left(\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚜𝚞𝚌𝚌}\right)
A special case of the
\mathrm{𝚌𝚢𝚌𝚕𝚎}
[BeldiceanuContejean94] constraint.
n
Solutions 1 2 9 44 265 1854 14833 133496 1334961
\mathrm{𝚍𝚎𝚛𝚊𝚗𝚐𝚎𝚖𝚎𝚗𝚝}
0..n
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚌𝚢𝚌𝚕𝚎}
\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚝𝚠𝚒𝚗}
𝚔_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
filtering: arc-consistency, DFS-bottleneck.
\mathrm{𝚍𝚎𝚛𝚊𝚗𝚐𝚎𝚖𝚎𝚗𝚝}\left(\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
\mathrm{𝚙𝚎𝚛𝚖𝚞𝚝𝚊𝚝𝚒𝚘𝚗}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}:\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1},\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}\right)
•\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚜𝚞𝚌𝚌}=\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡}
•\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚜𝚞𝚌𝚌}\ne \mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚒𝚗𝚍𝚎𝚡}
\mathrm{𝐍𝐓𝐑𝐄𝐄}
=0
\mathrm{𝙾𝙽𝙴}_\mathrm{𝚂𝚄𝙲𝙲}
\mathrm{𝚍𝚎𝚛𝚊𝚗𝚐𝚎𝚖𝚎𝚗𝚝}
constraint holds since the final graph does not contain any vertex that does not belong to a circuit (i.e.,
\mathrm{𝐍𝐓𝐑𝐄𝐄}
=
\mathrm{𝚍𝚎𝚛𝚊𝚗𝚐𝚎𝚖𝚎𝚗𝚝}
In order to express the binary constraint that links two vertices of the
\mathrm{𝙽𝙾𝙳𝙴𝚂}
collection one has to make explicit the index value of the vertices. This is why the
\mathrm{𝚍𝚎𝚛𝚊𝚗𝚐𝚎𝚖𝚎𝚗𝚝}
constraint considers objects that have two attributes:
\mathrm{𝚒𝚗𝚍𝚎𝚡}
\mathrm{𝚜𝚞𝚌𝚌}
that is the successor of the vertex.
Forbidding cycles of length one is achieved by the second condition of the arc constraint.
Since 0 is the smallest possible value of
\mathrm{𝐍𝐓𝐑𝐄𝐄}
we can rewrite the graph property
\mathrm{𝐍𝐓𝐑𝐄𝐄}
=
\mathrm{𝐍𝐓𝐑𝐄𝐄}
\le
\underline{\overline{\mathrm{𝐍𝐓𝐑𝐄𝐄}}}
\underline{\mathrm{𝐍𝐓𝐑𝐄𝐄}}
|
Vector and matrix norms - MATLAB norm
\begin{array}{l}a=0\underset{}{\overset{ˆ}{i}}+3\underset{}{\overset{ˆ}{j}}\\ b=-2\underset{}{\overset{ˆ}{i}}+1\underset{}{\overset{ˆ}{j}}\\ \\ \begin{array}{rl}{d}_{\left(a,b\right)}& =||b-a||\\ & =\sqrt{\left(-2-0{\right)}^{2}+\left(1-3{\right)}^{2}}\\ & =\sqrt{8}\end{array}\end{array}
‖v‖=\sqrt{\sum _{k=1}^{N}{|{v}_{k}|}^{2}}\text{\hspace{0.17em}}.
{‖v‖}_{p}={\left[\sum _{k=1}^{N}{|{v}_{k}|}^{p}\right]}^{\text{\hspace{0.17em}}1/p}\text{\hspace{0.17em}},
{‖v‖}_{\infty }={\mathrm{max}}_{i}\left(|v\left(i\right)|\right)
{‖v‖}_{-\infty }={\mathrm{min}}_{i}\left(|v\left(i\right)|\right)
{‖X‖}_{1}=\underset{1\le j\le n}{\mathrm{max}}\left(\sum _{i=1}^{m}|{a}_{ij}|\right).
{‖X‖}_{\infty }=\underset{1\le i\le m}{\mathrm{max}}\left(\sum _{j=1}^{n}|{a}_{ij}|\right)\text{\hspace{0.17em}}.
{‖X‖}_{F}=\sqrt{\sum _{i=1}^{m}\sum _{j=1}^{n}{|{a}_{ij}|}^{2}}=\sqrt{\text{trace}\left({X}^{†}X\right)}\text{\hspace{0.17em}}.
{‖X‖}_{F}=\sqrt{\sum _{i=1}^{m}\sum _{j=1}^{n}\sum _{k=1}^{p}...\sum _{w=1}^{q}{|{a}_{ijk...w}|}^{2}}.
|
Chaos Theory | Brilliant Math & Science Wiki
Matt DeCross, Satyabrata Dash, Christopher Williams, and
Chaos theory is the study of a particular type of systems that evolved from some initial conditions. A small perturbation in the initial setup of a chaotic system may lead to drastically different behavior, a concept popularly referred to as the butterfly effect from the idea that the actions of a butterfly may dramatically alter the physical state of the rest of the world. Although the behavior of chaotic systems may seem scattered and random, chaotic systems are strictly defined to be deterministic, meaning that a particular set of initial conditions always evolves in the same way.
An uncoupled map lattice, one type of chaotic map. By Travdog8 - Own work, CC BY-SA 3.0, https://en.wikipedia.org/w/index.php?curid=22892411
Chaotic maps can be either discrete or continuous functions where slightly different initial values are gradually mapped further and further apart over time. Typically, they are given either in terms of recurrence relations for discrete maps, or in the time domain for continuous maps.
Applications of chaos theory are widespread across biology, chemistry, physics, economics, and mathematics, among other fields. Often, systems with a large number of coupled variables exhibit chaotic behavior, including weather systems, job markets, population dynamics, and celestial mechanics.
There are three required mathematical properties for a system to be classified as chaotic:[1]
density of periodic orbits.
In certain cases, the second two imply the first mathematically. However, as discussed below, each of the three conditions captures different qualitative aspects of chaotic systems in general. Each is defined as a condition on the phase space of a dynamical system. In one dimension, the phase space is the two-dimensional space whose axes are the position
x
\dot{x}
of a point; in higher dimensions, the axes are the positions and velocities in each possible direction.
Suppose one has two sets of initial conditions
z_0
z_0^{\prime}
for a dynamical system, separated by a distance of
\Delta z (t)
in phase space, which may increase or decrease as the system evolves in
t
. As the system evolves, the separation between the two initial states evolves in time as
\Delta z(t) = e^{\lambda t} \Delta z(0).
\lambda
is called a Lyapunov exponent. In
d
spatial dimensions, the phase space is
2d
-dimensional, so there are
2d
Lyapunov exponents: one for each direction of separation of the states in phase space. Systems are often characterized by the largest of these exponents; if the largest exponent is positive, the separation grows exponentially in time (at least locally) and the system is chaotic. If the largest exponent is negative, the trajectories of the initial conditions
z_0
z_0^{\prime}
stay close in phase space, so closely separated trajectories are good approximations for each other and the system is not chaotic.
Sensitivity to initial conditions alone is not enough to make a map chaotic. For instance, consider the dynamical system generated by the map:
z_{n+1} = f(z_n) = 1.5z_n+1.
Consider two starting points
z_0 = 0
z_0^{\prime} = 0.1
. The evolution of each point is displayed in the diagram below:
Gradual separation of the points
z_0
z_0^{\prime}
over repeated iteration of the map
f(z_n)
Since the map generating the system multiplies by
1.5
and then adds one, any small difference between starting points is magnified by a factor of
1.5
at each step. However, the system is not chaotic: regardless of the starting point, every point approaches positive or negative infinity, so the asymptotic behavior given a set of initial conditions is very predictable.
The topological mixing condition is designed to exclude such cases. It essentially says that given any possible set of states for the dynamical system, a given set of initial conditions for the system will eventually evolve to at least some of the states in the set. Topologically, this can be stated as follows: any open set in the phase space of the dynamical system eventually intersects any other given open set in the phase space.
Density of periodic orbits in a dynamical system means that any given point in phase space is arbitrarily close to a set of initial conditions that leads to a periodic orbit. This is an interesting condition because combined with topological mixing it implies sensitivity to initial conditions. Take two close initial conditions, and draw open sets around each initial condition in phase space such that the two open sets are disjoint. By topological mixing, these open sets eventually evolve to intersect any other given open set, i.e. they "smear out" over the rest of phase space over time. But if there are arbitrarily many periodic orbits within each of these open sets, this "smearing out" is only possible if the periodic orbits look drastically different. One way of looking at this is that any given point looks like it ought to be well-approximated by a nearby periodic orbit, but it is not, because the full set of periodic orbits must evolve to intersect any open set in phase space. If this is true of arbitrarily close initial conditions, the trajectories in phase space must diverge, since the nearby periodic orbits don't converge to the trajectories of the initial conditions.
Consider the dynamical system generated by the discrete recurrence
z_{n+1} = f(z_n) = 2z_n.
Is this system chaotic?
Probabilistic behavior Evolving with a complicated trajectory An infinite number of orbits in phase space A large number of coupled variables
Which condition is most likely to be a quality of only dynamical systems that are chaotic?
Complex quadratic polynomials
A complex quadratic polynomial is a standard quadratic equation where the variable involved can be a complex number. A particularly simple example of this is the polynomial
f(z) = z^2 + c
c
. One can define a dynamical system from this map via the recursion
z_{n+1} = f(z_n)
. In this case, the dynamical system defined is chaotic. In fact, given
z_0 = 0
, the system diverges as
n \to \infty
c
takes values on a fractal set, the Mandlebrot set.
The fractal known as the Mandlebrot set. The complex quadratic map converges wherever the plot is black, and diverges at a rate according to the intensity of color elsewhere. By Wolfgang Beyer, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=321973
The Rössler attractor refers to a dynamical system given by the following set of first-order ODEs in three dimensions:
\begin{array}{c}&\frac{dx}{dt} = -y-z, &\frac{dy}{dt} &=x + ay, &\frac{dz}{dt} &= b+z(x-c)\end{array}
a,b,c
real parameters. It is similar to the well-known chaotic map called the Lorenz attractor, although mathematically simpler. Notably, the Rössler attractor has been used to study equilibria in reaction chemistry.
A set of orbits of the Rössler attractor given some set of parameters
a,b,c
. By Wolfl - Own work, CC BY-SA 2.5, https://commons.wikimedia.org/w/index.php?curid=346875
The double pendulum is one of the simplest scenarios in physics where chaotic behavior is manifest. Hamilton's equations of motion for the double pendulum yield four coupled first-order ordinary differential equations, which is a sufficient condition for chaos. The below animation shows the highly unpredictable evolution of the double pendulum given a particular initial configuration.
Chaotic evolution of the double pendulum over time. By Catslash - Own work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=10404903
The Hénon map is a discrete map on the plane given by the set of recursion relations
\begin{aligned} x_{n+1} &= 1 - ax_n^2+y_n \\ y_{n+1} &= bx_n \end{aligned}
for some parameters
and
b
. The choice of parameters
(a,b) = (1.4,0.3)
is called the classical Hénon map, which exhibits chaotic behavior.
Evolution of the Hénon map for a given choice of initial conditions. By Akarpe - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=18263865
In a map lattice, a discrete array of points are indexed and arranged in a lattice, and each site is able to evolve according to a particular type of recursion relation. In an uncoupled map lattice, each site evolves independently, e.g. by
x_{n+1} = rx_n ( 1-x_n),
r
is some parameter. In a coupled map lattice, the recursion for the
i^\text{th}
site depends not only on the previous value at that site but also the adjacent site:
(x_{n+1})_i = \epsilon \big(rx_n ( 1-x_n)\big)_i + (1 -\epsilon)\big(rx_n ( 1-x_n)\big)_{i-1}.
\epsilon
gives the degree of the coupling; as
\epsilon \to 1
the map lattice becomes uncoupled, and as
\epsilon \to 0
the map is maximally coupled. Map lattices are interesting first because both uncoupled and coupled map lattices are chaotic even though coupled map lattices display much richer structure. Secondly, however, they have been used to model interactions between adjacent chemicals in space as well as electrical circuits.
Forty different seedings of an uncoupled map lattice, each with a different value of
r
. Color indicates the value of the point in the array indexed by a particular point on the lattice. The map is chaotic and no particular structure exists in any given iteration. By Travdog8 - Own work, CC BY-SA 3.0, https://en.wikipedia.org/w/index.php?curid=22892411
In chaotic mixing, quantities such as density, viscosity, or temperature that track the flow of a fluid mix in a fractal-like way. Such fluids are governed by a system of first-order ODEs. For a fluid governed by the Navier-Stokes equations in three dimensions, there are sufficient degrees of freedom for the fluid flow to be chaotic.
Filaments in a fluid, indicated in red, mixing in a chaotic way as the fluid flows. By Peteymills - Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=18966717
The Ikeda map is a discrete complex map given by the recursion
z_{n+1} = A + Bz_n e^{i\left(|z_n|^2 + C\right)},
A,B
C
. It is a model used in physics for a series of pulses of laser light interacting in a nonlinear medium called an optical resonator. The parameter
B
characterizes how lossy the resonator is.
Trajectories of random initial conditions for the Ikeda map. By Accelerometer (Own work) [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons
The standard map is a dynamical system defined as a recursion relation on the square, that is
\begin{aligned} p_{n+1} &= p_n + K \sin (\theta_n) \\ \theta_{n+1} &= \theta_n + p_{n+1} \end{aligned}
p_n
\theta_n
2\pi
K
some constant. It describes the momentum and angle of a particle constrained to a ring which experiences periodic kicks in a particular direction of strength
K
, that is, a particle obeying the Hamiltonian
H = \frac{p^2}{2} + K\cos (x) \sum_n \delta (t - n)
with the extra constraint that the momentum be periodic. This system is called the kicked rotator and the constant
K
is called the kicking strength. It arises in the study of any periodically kicked systems and is thus particularly useful in physics with particles confined to a ring, such as accelerator or plasma physics.
Set of orbits of the standard map in phase space for the value
K = 0.6
. By Linas - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=12053416
Hasselblatt, B., & Katok, A. (2003). A First Course in Dynamics: With a Panorama of Recent Developments. Cambridge University Press.
Cite as: Chaos Theory. Brilliant.org. Retrieved from https://brilliant.org/wiki/chaos-theory/
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Global Constraint Catalog: Cglobal_cardinality_no_loop
<< 5.165. global_cardinality_low_up_no_loop5.167. global_cardinality_with_costs >>
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
\mathrm{𝚝𝚛𝚎𝚎}
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚗𝚘}_\mathrm{𝚕𝚘𝚘𝚙}\left(\mathrm{𝙽𝙻𝙾𝙾𝙿},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\right)
\mathrm{𝚐𝚌𝚌}_\mathrm{𝚗𝚘}_\mathrm{𝚕𝚘𝚘𝚙}
\mathrm{𝙽𝙻𝙾𝙾𝙿}
\mathrm{𝚍𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚕}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝙽𝙻𝙾𝙾𝙿}\ge 0
\mathrm{𝙽𝙻𝙾𝙾𝙿}\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)
|\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|>0
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂},\left[\mathrm{𝚟𝚊𝚕},\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}\right]\right)
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂},\mathrm{𝚟𝚊𝚕}\right)
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}\ge 0
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\left[i\right].\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}
\left(1\le i\le |\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|\right)
is equal to the number of variables
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[j\right].\mathrm{𝚟𝚊𝚛}
\left(j\ne i,1\le j\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|\right)
that are assigned value
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\left[i\right].\mathrm{𝚟𝚊𝚕}
The number of assignments of the form
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[i\right].\mathrm{𝚟𝚊𝚛}=i
i\in \left[1,|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|\right]
) is equal to
\mathrm{𝙽𝙻𝙾𝙾𝙿}
\left(\begin{array}{c}1,〈1,1,8,6〉,\hfill \\ 〈\begin{array}{cc}\mathrm{𝚟𝚊𝚕}-1\hfill & \mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}-1,\hfill \\ \mathrm{𝚟𝚊𝚕}-5\hfill & \mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}-0,\hfill \\ \mathrm{𝚟𝚊𝚕}-6\hfill & \mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}-1\hfill \end{array}〉\hfill \end{array}\right)
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚗𝚘}_\mathrm{𝚕𝚘𝚘𝚙}
Values 1, 5 and 6 are respectively assigned to the set of variables
\left\{\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[2\right].\mathrm{𝚟𝚊𝚛}\right\}
(i.e., 1 occurrence of value 1),
\left\{\right\}
(i.e., no occurrence of value 5) and
\left\{\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[4\right].\mathrm{𝚟𝚊𝚛}\right\}
(i.e., 1 occurrence of value 6). Note that, due to the definition of the constraint, the fact that
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[1\right].\mathrm{𝚟𝚊𝚛}
is assigned to 1 is not counted.
In addition the number of assignments of the form
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[i\right].\mathrm{𝚟𝚊𝚛}=i
i\in \left[1,4\right]
\mathrm{𝙽𝙻𝙾𝙾𝙿}=1
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)>1
|\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|>1
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>|\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝙽𝙻𝙾𝙾𝙿}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
Within the context of the
\mathrm{𝚝𝚛𝚎𝚎}
constraint the
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚗𝚘}_\mathrm{𝚕𝚘𝚘𝚙}
constraint allows to model a minimum and maximum degree constraint on each vertex of our trees.
The flow algorithm that handles the original
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
constraint [Regin96] can be adapted to the context of the
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚗𝚘}_\mathrm{𝚕𝚘𝚘𝚙}
constraint. This is done by creating an extra value node representing the loops corresponding to the roots of the trees.
\mathrm{𝚝𝚛𝚎𝚎}
(graph partitioning by a set of trees with degree restrictions).
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}_\mathrm{𝚗𝚘}_\mathrm{𝚕𝚘𝚘𝚙}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}
\mathrm{𝚏𝚒𝚡𝚎𝚍}
\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝑆𝐸𝐿𝐹}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\right)
•\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}.\mathrm{𝚟𝚊𝚛}=\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
•\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}.\mathrm{𝚔𝚎𝚢}\ne \mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}
=\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝑆𝐸𝐿𝐹}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\right)
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}.\mathrm{𝚟𝚊𝚛}=\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}.\mathrm{𝚔𝚎𝚢}
\mathrm{𝐍𝐀𝐑𝐂}
=\mathrm{𝙽𝙻𝙾𝙾𝙿}
Since, within the context of the first graph constraint, we want to express one unary constraint for each value we use the “For all items of
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚗𝚘}_\mathrm{𝚕𝚘𝚘𝚙}
|
A light body and a heavy body have the same momentum.
Which of the two bodies will have greater kinetic energy?
Light body Heavy body
A sphere of mass 1 kg is traveling with velocity 100 m/s toward a stationary sphere of mass 1000 kg. The two spheres collide elastically. What is the total kinetic energy (in Joules) of the spheres after the collision?
Assume that both spheres have the same radius. All surfaces are smooth. There is no rolling motion.
A pendulum with string of length 0.1 m is raised to an angle of
30^\circ
below the horizontal, as shown below, and then released. What is the velocity (in m/s) of the pendulum when it reaches the bottom ?
g=10 \text{ m/s}^2
This is a problem of Energy Tranfers .
Scenario A: Bob attaches a mass
m
at the end of a long string (length
L
) and holds the other end of the string, allowing the mass to swing like a pendulum, with amplitude
A
. When the mass reaches a turning point, Bob quickly pulls on the string so that the swinging part of the string shortens to
\tfrac12 L
. The pendulum now swings with a higher frequency.
Scenario B: Bob repeats the exact same situation: a pendulum length
L
m
is swinging with amplitude
A
. This time, Bob quickly pulls on the string when the mass passes through the equilibrium position. Again, the string is shortened to
\tfrac12 L
, and the pendulum swings with a higher frequency.
In which scenario, if any, will the pendulum have a greater amplitude after Bob shortens the string?
Assumptions: The size of the swinging mass is negligible. When Bob pulls on the string, he does this in a negligible amount of time. During the pull, the force is directed upward along the string.
The amplitude is greater in scenario A. The amplitude is greater in scenario B. The amplitudes will be the same. It depends on how exactly Bob applies the force.
The momentum of a bullet of mass 20 g fired from a gun is
10 \text{ kg m/s}
The kinetic energy of this bullet expressed in kJ will be:
|
Global Constraint Catalog: Ccycle
<< 5.102. cutset5.104. cycle_card_on_path >>
\mathrm{𝚌𝚢𝚌𝚕𝚎}\left(\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴},\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}
\mathrm{𝚍𝚟𝚊𝚛}
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚜𝚞𝚌𝚌}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}\ge 1
\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝙽𝙾𝙳𝙴𝚂},\left[\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚜𝚞𝚌𝚌}\right]\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\ge 1
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝙽𝙾𝙳𝙴𝚂},\mathrm{𝚒𝚗𝚍𝚎𝚡}\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚜𝚞𝚌𝚌}\ge 1
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚜𝚞𝚌𝚌}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
G
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}
is equal to the number of circuits for covering
G
in such a way that each vertex of
G
\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}
can also be interpreted as the number of cycles of the permutation associated with the successor variables of the
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\left(\begin{array}{c}2,〈\begin{array}{cc}\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\hfill & \mathrm{𝚜𝚞𝚌𝚌}-2,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-2\hfill & \mathrm{𝚜𝚞𝚌𝚌}-1,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-5,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-4\hfill & \mathrm{𝚜𝚞𝚌𝚌}-3,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-5\hfill & \mathrm{𝚜𝚞𝚌𝚌}-4\hfill \end{array}〉\hfill \end{array}\right)
\left(\begin{array}{c}1,〈\begin{array}{cc}\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\hfill & \mathrm{𝚜𝚞𝚌𝚌}-2,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-2\hfill & \mathrm{𝚜𝚞𝚌𝚌}-5,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-1,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-4\hfill & \mathrm{𝚜𝚞𝚌𝚌}-3,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-5\hfill & \mathrm{𝚜𝚞𝚌𝚌}-4\hfill \end{array}〉\hfill \end{array}\right)
\left(\begin{array}{c}5,〈\begin{array}{cc}\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\hfill & \mathrm{𝚜𝚞𝚌𝚌}-1,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-2\hfill & \mathrm{𝚜𝚞𝚌𝚌}-2,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-3,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-4\hfill & \mathrm{𝚜𝚞𝚌𝚌}-4,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-5\hfill & \mathrm{𝚜𝚞𝚌𝚌}-5\hfill \end{array}〉\hfill \end{array}\right)
In the first example we have the following 2 (
\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}=2
) cycles:
1↦2↦1
3↦5↦4↦3
. Consequently, the corresponding
\mathrm{𝚌𝚢𝚌𝚕𝚎}
In the second example we have 1 (
\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}=1
) cycle:
1↦2↦5↦4↦3↦1
In the third example we have the following 5 (
\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}=5
1↦1
2↦2
3↦3
4↦4
5↦5
\mathrm{𝚌𝚢𝚌𝚕𝚎}
N\in \left[1,2\right]
{V}_{1}\in \left[2,4\right]
{V}_{2}\in \left[2,3\right]
{V}_{3}\in \left[1,6\right]
{V}_{4}\in \left[2,5\right]
{V}_{5}\in \left[2,3\right]
{V}_{6}\in \left[1,6\right]
\mathrm{𝚌𝚢𝚌𝚕𝚎}
\left(N,〈1{V}_{1},2{V}_{2},3{V}_{3},4{V}_{4},5{V}_{5},6{V}_{6}〉\right)
\mathrm{𝚌𝚢𝚌𝚕𝚎}
\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}<|\mathrm{𝙽𝙾𝙳𝙴𝚂}|
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|>2
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\left(\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚜𝚞𝚌𝚌}\right)
\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}
\mathrm{𝙽𝙾𝙳𝙴𝚂}
The PhD thesis of Éric Bourreau [Bourreau99] mentions the following applications of extensions of the
\mathrm{𝚌𝚢𝚌𝚕𝚎}
The balanced Euler knight problem where one tries to cover a rectangular chessboard of size
N·M
C
knights that all have to visit between
2·⌊⌊\left(N·M\right)/C⌋/2⌋
2·⌈⌈\left(N·M\right)/C⌉/2⌉
distinct locations. For some values of
N
M
C
there does not exist any solution to the previous problem. This is for instance the case when
N=M=C=6
. Figure 5.103.2 depicts the graph associated with the
6×6
chessboard as well as examples of balanced solutions with respectively 1, 2, 3, 4 and 5 knights.
Some pick-up delivery problems where a fleet of vehicles has to transport a set of orders. Each order is characterised by its initial location, its final destination and its weight. In addition one also has to take into account the capacity of the different vehicles.
Figure 5.103.2. Graph of potential moves of a
6×6
chessboard, corresponding balanced knight's tours with 1 up to 5 knights, and collection of nodes passed to the
\mathrm{𝚌𝚢𝚌𝚕𝚎}
constraint corresponding to the solution with 5 knights; note that their is no balanced knight's tour on a
6×6
chessboard where each knight exactly performs 6 moves.
\mathrm{𝚌𝚢𝚌𝚕𝚎}
\mathrm{𝚒𝚗𝚍𝚎𝚡}
In an early version of the CHIP there was a constraint named
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}
that, from a declarative point of view, was equivalent to
\mathrm{𝚌𝚢𝚌𝚕𝚎}\left(1,\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
. In ALICE [Lauriere78] the
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}
constraint was also present.
Given a complete digraph of
n
vertices as well as an unrestricted number of circuits
\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}
\mathrm{𝚌𝚢𝚌𝚕𝚎}
constraint corresponds to the sequence A000142 of the On-Line Encyclopaedia of Integer Sequences [Sloane10]. Given a complete digraph of
n
vertices as well as a fixed number of circuits
\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}
n
\mathrm{𝚌𝚢𝚌𝚕𝚎}
constraint corresponds to the so called Stirling number of first kind.
\mathrm{𝚜𝚞𝚌𝚌}
variables have to take distinct values one can reuse the algorithms associated with the
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
constraint. A second necessary condition is to have no more than
\overline{\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}}
strongly connected components. Pruning for enforcing this condition, as soon as we have
\overline{\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}}
strongly connected components, can be done by forcing all strong bridges to belong to the final solution, since otherwise we would have more than
\overline{\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}}
strongly connected components. Since all the vertices of a circuit belong to the same strongly connected component an arc going from one strongly connected component to another strongly connected component has to be removed.
n
{s}_{1},{s}_{2},\cdots ,{s}_{n}
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|
1,2,\cdots ,n
\mathrm{𝚌𝚢𝚌𝚕𝚎}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
n·\left(n-1\right)
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
constraints,
n
\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}
constraints, and one
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\left(〈{s}_{1},{s}_{2},\cdots ,{s}_{n}〉\right)
Second, the key idea is to extract for each vertex
i
i\in \left[1,n\right]
) all the vertices that belong to the same cycle. This is done by stating a conjunction of
n-1
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
\left(i,〈{s}_{1},{s}_{2},\cdots ,{s}_{n}〉,{s}_{i,1}\right)
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
\left({s}_{i,1},〈{s}_{1},{s}_{2},\cdots ,{s}_{n}〉,{s}_{i,2}\right)
\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
\left({s}_{i,n-2},〈{s}_{1},{s}_{2},\cdots ,{s}_{n}〉,{s}_{i,n-1}\right)
Then, using a
\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}
\left({m}_{i},
〈i,{s}_{i,1},{s}_{i,2},\cdots ,{s}_{i,n-1}〉\right)
constraint, we get a unique representative for the cycle containing vertex
i
Third, using a
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\left(\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴},
〈{m}_{1},{m}_{2},\cdots ,{m}_{n}〉\right)
constraint, we get the number of distinct cycles.
n
Solutions 2 6 24 120 720 5040 40320 362880 3628800
\mathrm{𝚌𝚢𝚌𝚕𝚎}
0..n
n
Total 2 6 24 120 720 5040 40320 362880 3628800
2 1 3 11 50 274 1764 13068 109584 1026576
3 - 1 6 35 225 1624 13132 118124 1172700
4 - - 1 10 85 735 6769 67284 723680
5 - - - 1 15 175 1960 22449 269325
6 - - - - 1 21 322 4536 63273
7 - - - - - 1 28 546 9450
8 - - - - - - 1 36 870
9 - - - - - - - 1 45
\mathrm{𝚌𝚢𝚌𝚕𝚎}
0..n
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}_\mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}
\mathrm{𝚌𝚢𝚌𝚕𝚎}_\mathrm{𝚌𝚊𝚛𝚍}_\mathrm{𝚘𝚗}_\mathrm{𝚙𝚊𝚝𝚑}
(permutation,graph partitioning constraint),
\mathrm{𝚌𝚢𝚌𝚕𝚎}_\mathrm{𝚘𝚛}_\mathrm{𝚊𝚌𝚌𝚎𝚜𝚜𝚒𝚋𝚒𝚕𝚒𝚝𝚢}
\mathrm{𝚌𝚢𝚌𝚕𝚎}_\mathrm{𝚛𝚎𝚜𝚘𝚞𝚛𝚌𝚎}
(graph partitioning constraint),
\mathrm{𝚍𝚎𝚛𝚊𝚗𝚐𝚎𝚖𝚎𝚗𝚝}
\mathrm{𝚐𝚛𝚊𝚙𝚑}_\mathrm{𝚌𝚛𝚘𝚜𝚜𝚒𝚗𝚐}
(graph constraint,graph partitioning constraint),
\mathrm{𝚒𝚗𝚟𝚎𝚛𝚜𝚎}
\mathrm{𝚖𝚊𝚙}
\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚝𝚘𝚞𝚛}
\mathrm{𝚝𝚛𝚎𝚎}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚊𝚝𝚕𝚎𝚊𝚜𝚝}_\mathrm{𝚗𝚟𝚎𝚌𝚝𝚘𝚛}
\mathrm{𝚋𝚊𝚕𝚊𝚗𝚌𝚎}_\mathrm{𝚌𝚢𝚌𝚕𝚎}
(counting number of cycles versus controlling how balanced the cycles are).
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}
\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}
set to 1).
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
filtering: strong bridge, DFS-bottleneck.
final graph structure: circuit, connected component, strongly connected component, one_succ.
modelling: cycle, functional dependency.
problems: pick-up delivery.
puzzles: Euler knight.
•
\mathrm{𝚌𝚢𝚌𝚕𝚎}\left(\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴},\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}=1
\mathrm{𝚋𝚊𝚕𝚊𝚗𝚌𝚎}_\mathrm{𝚌𝚢𝚌𝚕𝚎}
\left(\mathrm{𝙱𝙰𝙻𝙰𝙽𝙲𝙴},\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
\mathrm{𝙱𝙰𝙻𝙰𝙽𝙲𝙴}=0
•
\mathrm{𝚌𝚢𝚌𝚕𝚎}\left(\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴},\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
\mathrm{𝚙𝚎𝚛𝚖𝚞𝚝𝚊𝚝𝚒𝚘𝚗}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}:\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1},\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}\right)
\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚜𝚞𝚌𝚌}=\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡}
•
\mathrm{𝐍𝐓𝐑𝐄𝐄}
=0
•
\mathrm{𝐍𝐂𝐂}
=\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}
\mathrm{𝙾𝙽𝙴}_\mathrm{𝚂𝚄𝙲𝙲}
From the restrictions and from the arc constraint, we deduce that we have a bijection from the successor variables to the values of interval
\left[1,|\mathrm{𝙽𝙾𝙳𝙴𝚂}|\right]
. With no explicit restrictions it would have been impossible to derive this property.
\mathrm{𝚌𝚢𝚌𝚕𝚎}
\mathrm{𝚒𝚗𝚍𝚎𝚡}
\mathrm{𝚜𝚞𝚌𝚌}
The graph property
\mathrm{𝐍𝐓𝐑𝐄𝐄}
=
0 is used in order to avoid having vertices that both do not belong to a circuit and have at least one successor located on a circuit. This concretely means that all vertices of the final graph should belong to a circuit.
\mathrm{𝐍𝐂𝐂}
graph property, we show the two connected components of the final graph. The constraint holds since all the vertices belong to a circuit (i.e.,
\mathrm{𝐍𝐓𝐑𝐄𝐄}
=
0) and since
\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}
=
\mathrm{𝐍𝐂𝐂}
=
\mathrm{𝚌𝚢𝚌𝚕𝚎}
\mathrm{𝚌𝚢𝚌𝚕𝚎}
\mathrm{𝚌𝚢𝚌𝚕𝚎}
\mathrm{𝚌𝚢𝚌𝚕𝚎}
\mathrm{𝚌𝚢𝚌𝚕𝚎}
\mathrm{𝚌𝚢𝚌𝚕𝚎}
: De Bruijn sequence
|
Practice Limits of Functions | Brilliant
x
gets closer and closer and closer, but never quite reaches its goal, how does your function behave?
(Be careful, most people who try this problem don’t get it right and/or get it right, but without understanding the full picture.)
Limits are used to describe the most extreme cases of a function’s behavior. They also allow you to use the “smoothness” of a function as a tool for understanding how a function behaves around strange singularities.
Being able to solve limit problems requires a combination of both knowing the essential strategies for simplifying functions and also understanding the rules for when limits do and don’t exist. For example, in the problem above, the limit does not exist because as x approaches 0 from the left, the values of
\frac{1}{x}
-\infty
vs. as x approaches 0 from the right, the values of
\frac{1}{x}
\infty.
In other cases, such as the
\lim_{x\to 0} \left|\frac{1}{x}\right|,
there is no real, numerical limit as
\infty
is not a real number, however, conceptually, we say that
\lim_{x\to 0} \left|\frac{1}{x}\right| = \infty,
since, in this case, the left-hand and right-hand limits both equal
+\infty.
Even the simplest-seeming limits questions trick many people because of how nuanced these tools and techniques are!
|
Search results for: Xin-Fa Deng
The environmental dependence of the stellar mass of active galactic nucleus host galaxies and dependence of the clustering properties of active galactic nucleus host galaxies on the stellar mass
Xin‐Fa Deng, Xiao‐Qing Wen
We used two volume‐limited active galactic nucleus (AGN) host galaxy samples constructed by Deng & Wen (2020, RMxAA, 56, 87), and explored the environmental dependence of the stellar mass of AGN host galaxies. In the luminous volume‐limited AGN host galaxy sample, the stellar mass of AGN host galaxies apparently depends on local environments: high mass AGN host galaxies exist preferentially in...
Comparisons of the galaxy age, stellar velocity dispersion and K-band luminosity distributions between grouped galaxies and isolated ones
Ping Wu, Xin-Fa Deng
Astrophysics and Space Science > 2016 > 361 > 2 > 1-5
In two volume-limited Main galaxy samples of the Sloan Digital Sky Survey Data Release 10 (SDSS DR10), we compare the age, stellar velocity dispersion and K-band luminosity distributions of grouped galaxies with those of isolated galaxies, to explore the environmental dependence of these properties of galaxies. It is found that grouped galaxies have preferentially larger stellar velocity dispersions...
Xin-Fa Deng, Jun Song, Yi-Qing Chen, Peng Jiang, more
Using two volume-limited Main galaxy samples of the Sloan Digital Sky Survey Data Release 10 (SDSS DR10), we examine the environmental dependence of galaxy age at fixed parameters or for different galaxy families. Statistical results show that the environmental dependence of galaxy age is stronger for late type galaxies, but can be still observed for the early types: the age of galaxies in the densest...
u - r Color Dependence of Galaxy Clustering in the Main Galaxy Sample of SDSS DR10
Fuyang Zhang, Xin-Fa Deng
Astrophysics > 2015 > 58 > 1 > 21-28
Using two volume-limited Main galaxy samples of the Sloan Digital Sky Survey Data Release 10 (SDSS DR10), we investigate u - r color dependence of clustering properties of galaxies. We can get the same statistical conclusion in two volume-limited Main galaxy samples: blue galaxies preferentially form isolated galaxies, close pairs, and small groups at all scales, whereas red galaxies preferentially...
Correlations Between the Morphological Type and Concentration Indices in Different Photometric Bands
Xin-Fa Deng, Guisheng Yu
Astrophysics > 2015 > 58 > 2 > 250-257
In this work, we continue to examine whether the u-, g-, i- and z-band concentration indices are a good morphological classification tool. Our statistical results demonstrate that compared with the r-band concentration index, the g-band concentration index may be a better choice for use as a parameter in automated morphological classification schemes. The u-band concentration index is the worst choice...
The influence of environment on galaxy age, stellar velocity dispersion, and stellar mass in the LOWZ sample of the SDSS-III
Astronomy Letters > 2015 > 41 > 6 > 252-259
In this work, I examine the environmental dependence of galaxy age, stellar velocity dispersion and stellar mass in the LOWZ sample of the Sloan Digital Sky Survey Data Release 10 (SDSS DR10). I measure the projected local density Σ5, divide the LOWZ sample into subsamples with a redshift binning size of Δz = 0.02 and analyze the environmental dependence of galaxy age, stellar velocity dispersion...
Dependence of galaxy clustering on K-band luminosity
Xin-Fa Deng, Xiao-Ping Qi, Ping Wu, Peng Jiang, more
Using two volume-limited Main galaxy samples of the Sloan Digital Sky Survey Data Release 10 (SDSS DR10), we investigate K-band luminosity dependence of galaxy clustering by cluster analysis. It is found that low-luminosity galaxies are preferentially isolated or form close pairs and small groups at all scales, whereas high-luminosity galaxies preferentially inhabit dense groups and clusters.
Astrophysics and Space Science > 2014 > 352 > 2 > 833-838
Using two volume-limited Main galaxy samples of the Sloan Digital Sky Survey Data Release 10 (SDSS DR10), we investigate the dependence of the clustering properties of galaxies on stellar velocity dispersion by cluster analysis. It is found that in the luminous volume-limited Main galaxy sample, except at r=1.2, richer and larger systems can be more easily formed in the large stellar velocity dispersion...
Environmental Dependence of Five Photometric Band Structural Parameters of Main Galaxies
Using two volume-limited Main galaxy samples of the Sloan Digital Sky Survey Data Release 7 above and below the value of
M_r^{*}
we explore the environmental dependence of five photometric band concentration indexes. It is found that all the five band concentration indexes strongly correlate with local environment for all galaxies above and below the value of
M_r^{*}
high concentration...
Dependence of Some Properties of Groups on Group Local Number Density
Xin-Fa Deng, Ping Wu
In this study we investigate the dependence of projected size Sizesky, and rms deviation σR of projected distance in the sky from the group center, rms velocities σV , and virial radius RVir of groups on group local number density. In the volume-limited group samples, it is found that groups in high density regions preferentially have larger Sizesky, σR , σV , and RVir than ones in low density regions.
U-band luminosity dependence of galaxy clustering in the main galaxy sample of SDSS DR10
Astronomy Letters > 2014 > 40 > 12 > 753-758
Using two volume-limited Main galaxy samples of the Sloan Digital Sky Survey Data Release 10 (SDSS DR10), we study u-band luminosity dependence of galaxy clustering by cluster analysis. Statistical results of two volume-limited Main galaxy samples are the same: luminous galaxies in M u preferentially form isolated galaxies, close pairs and small groups at all scales, whereas faint galaxies in M...
Correlations between morphology and other galaxy parameters at different environmental density levels
Xin-Fa Deng, Cheng-Hong Luo, Peng Jiang, Ying-Ping Ding
Using two volume-limited samples above and below the value of
M_{r}^{ *}
constructed from the Main galaxy sample of the Sloan Digital Sky Survey Data Release 8 (SDSS DR8), we investigate correlations between galaxy morphology and star formation rate (SFR), specific star formation rate (SSFR) and stellar mass at different environmental density levels. For each sample, three subsamples at both...
The environmental dependence of all the five band luminosities in the volume-limited main galaxy catalogs of the SDSS DR7
Xin-Fa Deng, Yong Xin, Cheng-Hong Luo, Ping Wu, more
Astroparticle Physics > 2012 > 36 > 1 > 1-6
In this study, we performed a comparative study among all the five band luminosity distributions of galaxies in groups and isolated, to explore the environmental dependence of u-, g-, r-, i- and z-band luminosities. It is found that for r, i and z bands, isolated galaxies have a higher proportion of faint galaxies and a lower proportion of luminous galaxies than galaxy members of groups, for u-band...
Investigation of the correlation between morphology and luminosity for two classes of main galaxies
Xin-Fa Deng, Xiao-Xia Qian, Cheng-Hong Luo, Ping Wu
Using an apparent-magnitude limited Main galaxy sample of the Sloan Digital Sky Survey Data Release 7(SDSS DR7), we investigate the correlation between morphologies and luminosity for the Main galaxy sample. Our Main galaxy sample is divided into two classes: Main galaxies only with TARGET_GALAXY flag (bestPrimtarget = 64), and ones also with other flags. It is found that for the second class Main...
Environmental dependence of star formation rate, specific star formation rate and stellar mass for blue and red galaxies
Xin‐Fa Deng, Yi‐Qing Chen, Peng Jiang
Monthly Notices of the Royal Astronomical Society > 417 > 1 > 453 - 457
Using the two volume‐limited main galaxy samples of the Sloan Digital Sky Survey Data Release 7, we explore the environmental dependence of the star formation rate (SFR), the specific star formation rate (SSFR) and the stellar mass for blue and red galaxies. It is found that the environmental dependence of the SFR, the SSFR and the stellar mass for red galaxies is still fairly strong, but the environmental...
Environmental dependence of other properties of main galaxies at fixed luminosity
Xin-Fa Deng, Yong Xin, Cheng-Hong Luo, Ping Wu
Using three volume-limited samples of the Sloan Digital Sky Survey Data Release 6 (SDSS DR6), we have investigated how other properties of galaxies depend on the environment at fixed luminosity. At fixed luminosity, we still observe strong environmental dependence of g - r color, concentration index, and morphology of galaxies: red, highly concentrated, and early type galaxies exist preferentially...
Dependence of the holmberg effect on separations between paired galaxies
Xin-Fa Deng, Yong Xin, Jiang Peng, Ping Wu
To investigate the dependence of the Holmberg effect of paired galaxies on three-dimensional separations, we have constructed four subsamples characterized by three-dimensional separations s ≤ 50 kpc, 50 kpc < s ≤ 100 kpc , 100kpc < s ≤150kpc , and 150 kpc < s ≤ 200 kpc , respectively. In this study, linear correlation coefficients and surplus standard deviations of color indices, luminosity,...
Correlations between environment and other properties of galaxies from the Sloan Digital Sky Survey at fixed color
Xin-Fa Deng, Si-Yu Zou
Astroparticle Physics > 2009 > 32 > 2 > 129-135
From the volume-limited Main galaxy sample of the Sloan Digital Sky Survey Data Release 6 (SDSS DR6), we construct three samples with g–r color bins 0.4⩽g–r<0.6,0.6⩽g–r<0.8,0.8⩽g–r<1.0, labeled S1–S3, to investigate how other properties of galaxies depend on environment at fixed color. For each sample, we measure the local three-dimensional galaxy density in a comoving sphere with radius...
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LARGE SCALE STRUCTURE (4)
GALAXY: LARGE SCALE STRUCTURE (3)
FUNDAMENTAL PARAMETERS-GALAXIES (2)
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GALAXIES:STATISTICS (2)
PUBLICATIONS IN ELECTRONIC MEDIA (1)
COSMOLOGY, LARGE (1)
COSMOLOGY: LARGE-SCALE STRUCTURE OF UNIVERSE (1)
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Chinese Astronomy and Astrophysics (3)
|
An electro-optic phase modulator for free-space beams
An optical intensity modulator for optical telecommunications
An electro-optic modulator (EOM) is an optical device in which a signal-controlled element exhibiting an electro-optic effect is used to modulate a beam of light. The modulation may be imposed on the phase, frequency, amplitude, or polarization of the beam. Modulation bandwidths extending into the gigahertz range are possible with the use of laser-controlled modulators.
The electro-optic effect is the change in the refractive index of a material resulting from the application of a DC or low-frequency electric field. This is caused by forces that distort the position, orientation, or shape of the molecules constituting the material. Generally, a nonlinear optical material (organic polymers have the fastest response rates, and thus are best for this application) with an incident static or low frequency optical field will see a modulation of its refractive index.
The simplest kind of EOM consists of a crystal, such as lithium niobate, whose refractive index is a function of the strength of the local electric field. That means that if lithium niobate is exposed to an electric field, light will travel more slowly through it. But the phase of the light leaving the crystal is directly proportional to the length of time it takes that light to pass through it. Therefore, the phase of the laser light exiting an EOM can be controlled by changing the electric field in the crystal.
Note that the electric field can be created by placing a parallel plate capacitor across the crystal. Since the field inside a parallel plate capacitor depends linearly on the potential, the index of refraction depends linearly on the field (for crystals where Pockels effect dominates), and the phase depends linearly on the index of refraction, the phase modulation must depend linearly on the potential applied to the EOM.
The voltage required for inducing a phase change of
{\displaystyle \pi }
is called the half-wave voltage (
{\displaystyle V_{\pi }}
). For a Pockels cell, it is usually hundreds or even thousands of volts, so that a high-voltage amplifier is required. Suitable electronic circuits can switch such large voltages within a few nanoseconds, allowing the use of EOMs as fast optical switches.
Liquid crystal devices are electro-optical phase modulators if no polarizers are used.
3 Polarization modulation
Phase modulation (PM) is a modulation pattern that encodes information as variations in the instantaneous phase of a carrier wave.
The phase of a carrier signal is modulated to follow the changing voltage level (amplitude) of modulation signal. The peak amplitude and frequency of the carrier signal remain constant, but as the amplitude of the information signal changes, the phase of the carrier changes correspondingly. The analysis and the final result (modulated signal) are similar to those of frequency modulation.
A very common application of EOMs is for creating sidebands in a monochromatic laser beam. To see how this works, first imagine that the strength of a laser beam with frequency
{\displaystyle \omega }
entering the EOM is given by
{\displaystyle Ae^{i\omega t}.}
Now suppose we apply a sinusoidally varying potential voltage to the EOM with frequency
{\displaystyle \Omega }
and small amplitude
{\displaystyle \beta }
. This adds a time dependent phase to the above expression,
{\displaystyle Ae^{i\omega t+i\beta \sin(\Omega t)}.}
{\displaystyle \beta }
is small, we can use the Taylor expansion for the exponential
{\displaystyle Ae^{i\omega t}\left(1+i\beta \sin(\Omega t)\right),}
to which we apply a simple identity for sine,
{\displaystyle Ae^{i\omega t}\left(1+{\frac {\beta }{2}}\left(e^{i\Omega t}-e^{-i\Omega t}\right)\right)=A\left(e^{i\omega t}+{\frac {\beta }{2}}e^{i(\omega +\Omega )t}-{\frac {\beta }{2}}e^{i(\omega -\Omega )t}\right).}
This expression we interpret to mean that we have the original carrier signal plus two small sidebands, one at
{\displaystyle \omega +\Omega }
and another at
{\displaystyle \omega -\Omega }
. Notice however that we only used the first term in the Taylor expansion – in truth there are an infinite number of sidebands. There is a useful identity involving Bessel functions called the Jacobi–Anger expansion which can be used to derive
{\displaystyle Ae^{i\omega t+i\beta \sin(\Omega t)}=Ae^{i\omega t}\left(J_{0}(\beta )+\sum _{k=1}^{\infty }J_{k}(\beta )e^{ik\Omega t}+\sum _{k=1}^{\infty }(-1)^{k}J_{k}(\beta )e^{-ik\Omega t}\right),}
which gives the amplitudes of all the sidebands. Notice that if one modulates the amplitude instead of the phase, one gets only the first set of sidebands,
{\displaystyle \left(1+\beta \sin(\Omega t)\right)Ae^{i\omega t}=Ae^{i\omega t}+{\frac {A\beta }{2i}}\left(e^{i(\omega +\Omega )t}-e^{i(\omega -\Omega )t}\right).}
A phase modulating EOM can also be used as an amplitude modulator by using a Mach–Zehnder interferometer. This alternative technique is often used in integrated optics where the requirements of phase stability is more easily achieved. The beam splitter divides the laser light into two paths, one of which has a phase modulator as described above. The beams are then recombined. Changing the electric field on the phase modulating path will then determine whether the two beams interfere constructively or destructively at the output, and thereby control the amplitude or intensity of the exiting light. This device is called a Mach–Zehnder modulator.
Polarization modulation[edit]
Depending on the type and orientation of the nonlinear crystal, and on the direction of the applied electric field, the phase delay can depend on the polarization direction. A Pockels cell can thus be seen as a voltage-controlled waveplate, and it can be used for modulating the polarization state. For a linear input polarization (often oriented at 45° to the crystal axis), the output polarization will in general be elliptical, rather than simply a linear polarization state with a rotated direction.
Polarization modulation in electro-optic crystals can also be used as a technique for time-resolved measurement of unknown electric fields. [1][2] Compared to conventional techniques using conductive field probes and cabling for signal transport to read-out systems, electro-optical measurement is inherently noise resistant as signals are carried by fiber-optics, preventing distortion of the signal by electrical noise sources. The polarization change measured by such techniques is linearly dependent on the electric field applied to the crystal, hence providing absolute measurements of the field, without the need for numerical integration of voltage traces, as is the case for conductive probes sensitive to the time-derivative of the electric field.
Karna, Shashi and Yeates, Alan (ed.) (1996). Nonlinear Optical Materials: Theory and Modeling. Washington, DC: American Chemical Society. pp. 2–3. ISBN 0-8412-3401-9. {{cite book}}: |author= has generic name (help)CS1 maint: multiple names: authors list (link)
Saleh, Teich (first ed.) (1991). Fundamentals of Photonics. New York: Wiley-Interscience Publications. p. 697. ISBN 0-471-83965-5.
^ Consoli, F.; De Angelis, R.; Duvillaret, L.; Andreoli, P. L.; Cipriani, M.; Cristofari, G.; Di Giorgio, G.; Ingenito, F.; Verona, C. (15 June 2016). "Time-resolved absolute measurements by electro-optic effect of giant electromagnetic pulses due to laser-plasma interaction in nanosecond regime". Scientific Reports. 6 (1): 27889. Bibcode:2016NatSR...627889C. doi:10.1038/srep27889. PMC 4908660. PMID 27301704.
^ Robinson, T. S.; Consoli, F.; Giltrap, S.; Eardley, S. J.; Hicks, G. S.; Ditter, E. J.; Ettlinger, O.; Stuart, N. H.; Notley, M.; De Angelis, R.; Najmudin, Z.; Smith, R. A. (20 April 2017). "Low-noise time-resolved optical sensing of electromagnetic pulses from petawatt laser-matter interactions". Scientific Reports. 7 (1): 983. Bibcode:2017NatSR...7..983R. doi:10.1038/s41598-017-01063-1. PMC 5430545. PMID 28428549.
AdvR – Research and custom EO phase and amplitude modulators
Interactive visualization of the transfer characteristic of a Mach–Zehnder modulator for phase and amplitude modulation
Retrieved from "https://en.wikipedia.org/w/index.php?title=Electro-optic_modulator&oldid=1071536547"
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Global Constraint Catalog: Cminimum_weight_alldifferent
<< 5.265. minimum_modulo5.267. multi_global_contiguity >>
[FocacciLodiMilano99]
\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}_\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇},\mathrm{𝙲𝙾𝚂𝚃}\right)
\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}_\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}
\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}_\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\mathrm{𝚖𝚒𝚗}_\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}
\mathrm{𝚖𝚒𝚗}_\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚖𝚒𝚗}_\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(𝚒-\mathrm{𝚒𝚗𝚝},𝚓-\mathrm{𝚒𝚗𝚝},𝚌-\mathrm{𝚒𝚗𝚝}\right)
\mathrm{𝙲𝙾𝚂𝚃}
\mathrm{𝚍𝚟𝚊𝚛}
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>0
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\ge 1
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇},\left[𝚒,𝚓,𝚌\right]\right)
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚜𝚎𝚚}
\left(\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇},\left[𝚒,𝚓\right]\right)
\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}.𝚒\ge 1
\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}.𝚒\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}.𝚓\ge 1
\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}.𝚓\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
|\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}|=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|*|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
collection should take a distinct value located within interval
\left[1,|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|\right]
. In addition
\mathrm{𝙲𝙾𝚂𝚃}
is equal to the sum of the costs associated with the fact that we assign value
i
j
. These costs are given by the matrix
\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}
\left(\begin{array}{c}〈2,3,1,4〉,\hfill \\ 〈\begin{array}{ccc}𝚒-1\hfill & 𝚓-1\hfill & 𝚌-4,\hfill \\ 𝚒-1\hfill & 𝚓-2\hfill & 𝚌-1,\hfill \\ 𝚒-1\hfill & 𝚓-3\hfill & 𝚌-7,\hfill \\ 𝚒-1\hfill & 𝚓-4\hfill & 𝚌-0,\hfill \\ 𝚒-2\hfill & 𝚓-1\hfill & 𝚌-1,\hfill \\ 𝚒-2\hfill & 𝚓-2\hfill & 𝚌-0,\hfill \\ 𝚒-2\hfill & 𝚓-3\hfill & 𝚌-8,\hfill \\ 𝚒-2\hfill & 𝚓-4\hfill & 𝚌-2,\hfill \\ 𝚒-3\hfill & 𝚓-1\hfill & 𝚌-3,\hfill \\ 𝚒-3\hfill & 𝚓-2\hfill & 𝚌-2,\hfill \\ 𝚒-3\hfill & 𝚓-3\hfill & 𝚌-1,\hfill \\ 𝚒-3\hfill & 𝚓-4\hfill & 𝚌-6,\hfill \\ 𝚒-4\hfill & 𝚓-1\hfill & 𝚌-0,\hfill \\ 𝚒-4\hfill & 𝚓-2\hfill & 𝚌-0,\hfill \\ 𝚒-4\hfill & 𝚓-3\hfill & 𝚌-6,\hfill \\ 𝚒-4\hfill & 𝚓-4\hfill & 𝚌-5\hfill \end{array}〉,17\hfill \end{array}\right)
\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}_\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
constraint holds since the cost 17 corresponds to the sum
\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}\left[\left(1-1\right)·4+2\right].𝚌+\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}\left[\left(2-1\right)·4+3\right].𝚌+\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}\left[\left(3-1\right)·4+1\right].𝚌+\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}\left[\left(4-1\right)·4+4\right].𝚌=\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}\left[2\right].𝚌+\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}\left[7\right].𝚌+\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}\left[9\right].𝚌+\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}\left[16\right].𝚌=1+8+3+5
\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}_\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
{V}_{1}\in \left[2,4\right]
{V}_{2}\in \left[2,3\right]
{V}_{3}\in \left[1,6\right]
{V}_{4}\in \left[2,5\right]
{V}_{5}\in \left[2,3\right]
{V}_{6}\in \left[1,6\right]
C\in \left[0,25\right]
\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}_\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\left(〈{V}_{1},{V}_{2},{V}_{3},{V}_{4},{V}_{5},{V}_{6}〉
〈115,120,131,141,153,160
212,227,230,242,255,261
313,323,336,346,350,369
414,423,430,440,450,462
512,520,536,543,557,562
615,624,635,644,655,664〉,C\right)
\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}_\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}.𝚌\right)>1
\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}.𝚌>0
\mathrm{𝙲𝙾𝚂𝚃}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}
The Hungarian method for the assignment problem [Kuhn55] can be used for evaluating the bounds of the
\mathrm{𝙲𝙾𝚂𝚃}
variable. A filtering algorithm is described in [Sellmann02]. It can be used for handling both side of the
\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}_\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
Evaluating a lower bound of the
\mathrm{𝙲𝙾𝚂𝚃}
variable and pruning the variables of the
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
collection in order to not exceed the maximum value of
\mathrm{𝙲𝙾𝚂𝚃}
Evaluating an upper bound of the
\mathrm{𝙲𝙾𝚂𝚃}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
collection in order to not be under the minimum value of
\mathrm{𝙲𝙾𝚂𝚃}
all_different in SICStus, all_distinct in SICStus.
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚠𝚒𝚝𝚑}_\mathrm{𝚌𝚘𝚜𝚝𝚜}
\mathrm{𝚜𝚞𝚖}_\mathrm{𝚘𝚏}_\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚜}_\mathrm{𝚘𝚏}_\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}_\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}
(weighted assignment),
\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚎𝚍}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚊𝚕}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}
(cost filtering constraint,weighted assignment).
filtering: cost filtering constraint, Hungarian method for the assignment problem.
modelling: cost matrix, functional dependency.
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1},\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}\right)
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1}.\mathrm{𝚟𝚊𝚛}=\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}.\mathrm{𝚔𝚎𝚢}
•
\mathrm{𝐍𝐓𝐑𝐄𝐄}
=0
•\mathrm{𝐒𝐔𝐌}_\mathrm{𝐖𝐄𝐈𝐆𝐇𝐓}_\mathrm{𝐀𝐑𝐂}\left(\begin{array}{c}\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}\left[\sum \left(\begin{array}{c}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1}.\mathrm{𝚔𝚎𝚢}-1\right)*|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|,\hfill \\ \mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1}.\mathrm{𝚟𝚊𝚛}\hfill \end{array}\right)\right].𝚌\hfill \end{array}\right)=\mathrm{𝙲𝙾𝚂𝚃}
Since each variable takes one value, and because of the arc constraint
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1}=\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}.\mathrm{𝚔𝚎𝚢}
, each vertex of the initial graph belongs to the final graph and has exactly one successor. Therefore the sum of the out-degrees of the vertices of the final graph is equal to the number of vertices of the final graph. Since the sum of the in-degrees is equal to the sum of the out-degrees, it is also equal to the number of vertices of the final graph. Since
\mathrm{𝐍𝐓𝐑𝐄𝐄}=0
, each vertex of the final graph belongs to a circuit. Therefore each vertex of the final graph has at least one predecessor. Since we saw that the sum of the in-degrees is equal to the number of vertices of the final graph, each vertex of the final graph has exactly one predecessor. We conclude that the final graph consists of a set of vertex-disjoint elementary circuits.
Finally the graph constraint expresses that the
\mathrm{𝙲𝙾𝚂𝚃}
\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}
{c}_{ij}
𝚌
{\left(\left(i-1\right)·|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)|+j\right)}^{th}
\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
restriction that enforces that the items of the
\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}
𝚒
𝚓
\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}_\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝐒𝐔𝐌}_\mathrm{𝐖𝐄𝐈𝐆𝐇𝐓}_\mathrm{𝐀𝐑𝐂}
graph property, the arcs of the final graph are stressed in bold. We also indicate their corresponding weight.
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Notes for MIT 6.031 Concurrency | upupming 的博客
2. Two modules for concurrent programming
2.2. Memory passing
3. Processes, threads, time-slicing
4. Starting a thread in Java
4.1. Using Thread
5. Shared memory example
8. Tweaking the code won’t help
9.1.1. Interleaving 1
9.1.3. Race conditions
10. Message passing example
11. Concurrency is hard to test and debug
This is an easy-to-understand note for [Reading 19: Concurrency][1].
Multiple computers in a network
Multiple applications running on one computer
Multiple processors in a computer(today, often multiple processor cores on a single chip)
In modern programming:
Websites must handle multiple simultaneous users.
Mobile apps need to do some of their processing on servers(“in the cloud”).
Graphical user interfaces almost always require background work that does not interrupt the user. (For example, Eclipse compiles your Java code while you’re still editing it.)
Processor clock speeds are no longer increasing. Instead, we’re getting more cores with each new generation of chips.
Two modules for concurrent programming
two processors(or processor cores) the same computer the same physical memory
two programs the same computer a common filesystem
two threads the same Java program the same Java objects
Memory passing
Concurrent modules send off messages through a communication channel, and incoming messages to each module are queued up for handling. Examples include:
two computers network network connections
web browser and web server network A opens a connection; B sends the web page data to A’s asks
instant messaging client and server
\vert
grep terminal pipe
If you have read [CSAPP][2], this section can be ignored.
Modules A and B above can be: processes and threads.
A process is an instance of a running program that is isolated from other processes on the same machine. In particular, it has its own private section of the machine’s memory.
A thread is a locus of control inside a running program. Think of it as a place in the program that is being run, plus the stack of method calls that led to that place (so the thread can go back up the stack when it reaches return statements).
How can I have many concurrent threads with only one or two processors in my computer? When there are more threads than processors, concurrency is simulated by time slicing, which means that the processor switches between threads. The figure above shows how three threads T1, T2, and T3 might be time-sliced on a machine that has only two actual processors. In the figure, time proceeds downward, so at first, one processor is running thread T1 and the other is running thread T2, and then the second processor switches to run thread T3. Thread T2 simply pauses, until its next time slice on the same processor or another processor.
On most systems, time slicing happens unpredictably and nondeterministically, meaning that a thread may be paused or resumed at any time.
// ... in the main method:
Using anonymous class:
If you’re feeling clever, you can go one step further with Java’s [lambda expressions][3]:
new Thread(() -> System.out.println("Hello from a thread!")).start();
Runnable is a functional interface, the interface that contains one and only one abstract method run().
The Arrow operator -> divides the lambda expressions in two parts:
() -> System.out.println("Hello from a thread!")
The left side specifies the parameters required by the expressions, which could also be empty if no parameters are required.
The right side is the lambda body which specifies the actions of the lambda expression.
It might be helpful to think about -> as “becomes”.
interface MyGreeting {
String processName(String str);
MyGreeting morningGreeting = (str) -> "Good Morning " + str + "!";
MyGreeting eveningGreeting = (str) -> "Good Evening " + str + "!";
// Output: Good Morning Luis!
System.out.println(morningGreeting.processName("Luis"));
// Output: Good Evening Jessica!
System.out.println(eveningGreeting.processName("Jessica"));
More: [parameter type, blocks, generic, etc][3].
The point of this example is to show that concurrent programming is hard because it can have subtle bugs.
Simplify the bank down to a single account, with a dollar balance stored in balance variable, and two operations deposit and withdraw that simply add or remove a dollar:
// suppose all the cash machines share a single bank account
private static int balance = 0;
private static void deposit() {
private static void withdraw() {
balance = balance - 1;
Each transaction is just a one dollar deposit followed by a one-dollar withdrawal, so it should leave the balance in the account unchanged. Throughout the day, each cash machine in our network is processing a sequence of deposit/withdraw transactions:
// each ATM does a bunch of transactions that
// modify balance, but leave it unchanged afterward
private static void cashMachine() {
for (int i = 0; i < TRANSACTIONS_PER_MACHINE; ++i) {
deposit(); // put a dollar in
withdraw(); // take it back out
So at the end of the day, regardless of how many cash machines were running, or how many transactions we processed, we should expect the account balance to still be 0.
But if we run this code, we discover frequently that the balance at the end of the day is not 0. If more than one cashMachine() call is running at the same time – say, on separate processors in the same computer – then balance may not be zero at the end of the day. Why not?
Suppose two cash machines, A and B, are both working on a deposit at the same time. Here’s how the deposit() step typically breaks down into low-level processor instructions:
A get balance (balance=0)
A write back the result (balance=1)
B get balance (balance=1)
B write back the result (balance=2)
This interleaving is fine. We end up with balance 2, so both A and B successfully put in a dollar.
This balance is now 1. A’s dollar was lost! A and B both read the balance at the same time, computed separate final balances, and then raced to store back the new balance, which failed to take the other’s deposit into account.
A race condition means that the correctness of the program(the satisfaction od postconditions and invariants) depends on the relative timing of events in concurrent computations A and B. When this happens, we say “A is in a race with B”.
Some interleavings of events may be OK, in the sense that they are consistent with what a single, nonconcurrent process would produce; but other interleavings produce wrong answers, which violates postconditions or invariants.
All these versions of the bank-account code exhibit the same race condition.
private static void deposit() { balance = balance + 1; }
private static void withdraw() { balance = balance - 1; }
private static void deposit() { balance += 1; }
private static void withdraw() { balance -= 1; }
private static void deposit() { ++balance; }
private static void withdraw() { --balance; }
Even simple statements can translate into multiple steps by the virtual machine.
The key lesson is that you can’t tell by looking at an expression whether it will be safe from race conditions.
Here’s an example. Note that it uses a loop that continuously checks for a concurrent condition; this is called busy waiting and it is not a good pattern. In this case, the code is also broken:
private int answer = 0;
// computeAnswer runs in one thread
private void computeAnswer() {
// useAnswer runs in a different thread
private void useAnswer() {
if (answer == 0) throw new RuntimeException("answer wasn't ready!");
Looking at the code, answer is set before ready is set, so once useAnswer sees ready as true, then it seems reasonable that it can assume that the answer will be 42, right? Not so.
The problem is that modern compilers and processors do a lot of things to make the code fast. One of those things is making temporary copies of variables like answer and ready in faster storage (registers or caches on a processor), and working with them temporarily before eventually storing them back to their official location in memory. The storeback may occur in a different order than the variables were manipulated in your code. Here’s what might be going on under the covers (but expressed in Java syntax to make it clear). The processor is effectively creating two temporary variables, tmpr and tmpa, to manipulate the fields ready and answer:
boolean tmpr = ready;
int tmpa = answer;
tmpa = 42;
tmpr = true;
ready = tmpr;
// <-- what happens if useAnswer() interleaves here?
// ready is set, but answer isn't.
answer = tmpa;
public class Moirai {
Thread clotho = new Thread(new Runnable() {
public void run() { System.out.println("spinning"); };
clotho.start();
public void run() { System.out.println("measuring"); };
public void run() { System.out.println("cutting"); };
// bug! never started
Possible running results
The third thread is never started, so cutting will never be printed.
The order of the other outputs depends on whether the first thread runs println before or after the second.
public class Parcae {
Thread nona = new Thread(new Runnable() {
nona.run(); // bug! called run instead of start
Runnable decima = new Runnable() {
decima.run(); // bug? maybe meant to create a Thread?
There is only one thread running in this program, and only one possible output.
Now suppose methodA and methodB run concurrently, so that their instructions might interleave arbitrarily. Which of the following are possible final values of x?
Possible running results 5 6 10 30
Message passing example
Accounts are modules just like machine modules. Incoming requests form a queue waiting for handling.
Message passing doesn’t eliminate the possibility of race conditions. Suppose each account supports get-balance and withdraw operations with corresponding messages.
As shown above, two users at machines A and B, are both trying to withdraw a dollar if the account holds more than a dollar.
get-balance()
Interleaving: the messages sent to the bank account rather than the instructions executed by A and B.
Interleaving of messages will fool A and B into thinking they can both withdraw a dollar, thereby overdrawing the account
A sends a message and gets the reply: “balance >= $1, it’s OK to withdraw 1 dollar”
B sends a message and gets the reply: “balance >= $1, it’s OK to withdraw 1 dollar”
A start withdraw
B start withdraw
Oops, both A and B are fooled!
It’s very hard to discover and localize race conditions using testing.
Bugs are heisenbugs(non-deterministic and hard to reproduce, as opposed to bohrbugs)
It may even disappear when you try to look it with println or debugger!
Reason: Printing and debugging are 100-1000x slower than other operations so that they dynamically change the timing of operations, and the interleaving.
For example, insert println in the cashMachine():
System.out.println(balance); // makes the bug disappear!
… and suddenly the balance is always 0, as desired.
Over the next several readings, we’ll see principled ways to design concurrent programs so that they are safer from there kinds of bugs.
Concurrency: multiple computations running simultaneously
Shared-memory & message-passing paradigms
Process is like a virtual computer; thread is like a virtual processor
When correctness of result (postconditions and invariants) depends on the relative timing of events
Recall three key properties of good software, we need to fix these problems:
Safety from bugs: Concurrency bugs are some of the hardest bugs to find and fix, and repair careful design to avoid.
Easy to understand: Predicting how concurrent code might interleave with other concurrent code is very hard for programmers to do. It’s best to design your code in such a way that programmers don’t have to think about interleaving as all.
Ready for change: Not particularly relevant here.
[Reading 19: Concurrency | 6.031: Software Construction][1]
[CSAPP][2]
[How to start working with Lambda Expressions in Java][3]
[1]: http://web.mit.edu/6.031/www/sp17/classes/19-concurrency/
[2]: http://csapp.cs.cmu.edu/
[3]: https://medium.freecodecamp.org/learn-these-4-things-and-working-with-lambda-expressions-b0ab36e0fffc
文章链接: https://upupming.site/2018/06/14/6.031-concurrency/
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Collateral Framework - Zeta
Zeta currently only accepts USDC as collateral. So users can only deposit and withdraw USDC from the system. All trades are settled and margined in USDC.
The following specifications are used in the margin system:
Account Balance (AB)
USDC deposited in account.
Changes on deposit, withdrawal, position close and settlement
Unrealized PnL (UP)
Profit and loss from existing open positions
Margin level required to open new orders
Margin level required to maintain positions before liquidation
When a user places an order that adds to their existing position the initial margin requirements are checked. This is done to ensure that opening the position will not push the user into a state of bankruptcy - if this condition is violated users will (1) not be able to place an order, and (2) their existing open orders will be cancelled trustlessly. (3) Liquidators, when taking over a position must also pass this check:
AB + UP - IM +min(0, IPnL) > 0
UP = Unrealized PnL
IM = Initial Margin x (Opening Orders + Positions)
IPnL = instantaneous PnL from execution
When a user places an order that closes an existing position the maintenance margin requirements (including open orders) are checked. This is done to ensure that the user has sufficient funds in their account to place an order and is not currently being liquidated.
AB + UP - MM(O) +min(0, IPnL) > 0
MM (including open orders) = Maintenance Margin x (Opening Orders + Positions)
The margin system also monitors a user's existing positions and orders to ensure that the user does not enter a state of bankruptcy:
AB + UP - MM > 0
MM = Maintenance Margin x (Positions)
User withdrawals are also considered by the Zeta margin system ensuring that a withdrawal does not push a user into bankruptcy. As such withdrawals are limited to:
AB + min(0, UPnL) - IM
Orders that close a position are not charged margin.
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This problem is a checkpoint for solving equations with fractions and decimals (Fraction Busters). It will be referred to as Checkpoint 7. Solve each equation or system of equations.
\frac { 1 } { 5 } x + \frac { 1 } { 3 } x = 2
x+0.15x=\$2
\frac { x + 2 } { 3 } = \frac { x - 2 } { 7 }
\left. \begin{array} [t]{ l } { y = \frac { 2 } { 3 } x + 8 } \\ { y = \frac { 1 } { 2 } x + 10 } \end{array} \right.
Check your answers by referring to the Checkpoint 7 materials materials located at the back of your book.
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Bowling average - WikiMili, The Best Wikipedia Reader
Statistic used to compare cricket bowlers
For information on average in Ten-pin bowling, see Glossary of bowling § Other bowling terms and jargon.
When a bowler has taken only a small number of wickets, their bowling average can be artificially high or low, and unstable, with further wickets taken or runs conceded resulting in large changes to their bowling average. Due to this, qualification restrictions are generally applied when determining which players have the best bowling averages. After applying these criteria, George Lohmann holds the record for the lowest average in Test cricket, having claimed 112 wickets at an average of 10.75 runs per wicket.
A cricketer's bowling average is calculated by dividing the numbers of runs they have conceded by the number of wickets they have taken. [2] The number of runs conceded by a bowler is determined as the total number of runs that the opposing side have scored while the bowler was bowling, excluding any byes, leg byes, [3] or penalty runs. [4] The bowler receives credit for any wickets taken during their bowling that are either bowled, caught, hit wicket, leg before wicket or stumped. [5]
{\displaystyle \mathrm {Bowling~average} ={\frac {\mathrm {Runs~conceded} }{\mathrm {Wickets~taken} }}}
A number of flaws have been identified for the statistic, most notable among these the fact that a bowler who has taken no wickets cannot have a bowling average, as dividing by zero does not give a result. The effect of this is that the bowling average cannot distinguish between a bowler who has taken no wickets and conceded one run, and a bowler who has taken no wickets and conceded one hundred runs. The bowling average also does not tend to give a true reflection of the bowler's ability when the number of wickets they have taken is small, especially in comparison to the number of runs they have conceded. [6] In his paper proposing an alternative method of judging batsmen and bowlers, Paul van Staden gives an example of this:
Suppose a bowler has bowled a total of 80 balls, conceded 60 runs and has taken only 2 wickets so that.. [their average is] 30. If the bowler takes a wicket with the next ball bowled (no runs obviously conceded), then [their average is] 20. [6]
Due to this, when establishing records for bowling averages, qualification criteria are generally set. For Test cricket, the Wisden Cricketers' Almanack sets this as 75 wickets, [7] while ESPNcricinfo requires 2,000 deliveries. [8] Similar restrictions are set for one-day cricket. [9] [10]
A number of factors other than purely the ability level of the bowler have an effect on a player's bowling average. Most significant among these are the different eras in which cricket has been played. The bowling average tables in Test and first-class cricket are headed by players who competed in the nineteenth century, [11] a period when pitches were uncovered and some were so badly looked after that they had rocks on them. The bowlers competing in the Howa Bowl, a competition played in South African during the apartheid-era, restricted to non-white players, [12] during which time, according to Vincent Barnes: "Most of the wickets we played on were underprepared. For me, as a bowler, it was great." [13] Other factors which provided an advantage to bowlers in that era was the lack of significant safety equipment; batting gloves and helmets were not worn, and batsmen had to be warier. Other variations are caused by frequent matches against stronger or weaker opposition, changes in the laws of cricket and the length of matches. [14]
A. N. Hornby is one of three players to have a bowling average of zero in Test cricket.
Due to the varying qualifying restrictions placed on the records by different statisticians, the record for the lowest career bowling average can be different from publication to publication.
In Test cricket, George Lohmann is listed as having the superior average by each of the Wisden Cricketers' Almanack , ESPNcricinfo and CricketArchive. Though all three use different restrictions, Lohmann's average of 10.75 is considered the best. [1] [7] [8] If no qualification criteria were applied at all, three players—Wilf Barber, A. N. Hornby and Bruce Murray—would tie for the best average, all having claimed just one wicket in Test matches, without conceding any runs, thus averaging zero. [15]
ESPNcricinfo list Betty Wilson as having the best Women's Test cricket average with 11.80, [16] while CricketArchive accept Mary Spear's average of 5.78. [17]
In One Day Internationals, the varying criteria set by ESPNcricinfo and CricketArchive result in different players being listed as holding the record. ESPNcricinfo has the stricter restriction, requiring 1,000 deliveries: by this measure, Joel Garner is the record-holder, having claimed his wickets at an average of 18.84. [9] By CricketArchive's more relaxed requirement of 400 deliveries, John Snow leads the way, with an average of 16.57. [18]
In women's One Day International cricket, Caroline Barrs tops the CricketArchive list with an average of 9.52, [19] but by ESPNcricinfo's stricter guidelines, the record is instead held by Gill Smith's 12.53. [20]
The record is again split for the two websites for Twenty20 International cricket; in this situation ESPNcricinfo has the lower boundary, requiring just 30 balls to have been bowled. George O'Brien's average of 8.20 holds the record using those criteria, but the stricter 200 deliveries required by CricketArchive results in Andre Botha being listed as the superior, averaging 8.76. [10] [21]
Domestically, the records for first-class cricket are dominated by players from the nineteenth century, who make up sixteen of the top twenty by ESPNcricinfo's criteria of 5,000 deliveries. William Lillywhite, who was active from 1825 to 1853 has the lowest average, claiming his 1,576 wickets at an average of just 1.54. The leading players from the twentieth century are Stephen Draai and Vincent Barnes with averages of just under twelve, [11] both of whom claimed the majority of their wickets in the South African Howa Bowl tournament during the apartheid era. [22] [23]
Courtney Andrew Walsh OJ is a former Jamaican cricketer who represented the West Indies from 1984 to 2001, captaining the West Indies in 22 Test matches. He is a fast bowler, and best known for a remarkable opening bowling partnership along with fellow West Indian Curtly Ambrose for several years. Walsh played 132 Tests and 205 ODIs for the West Indies and took 519 and 227 wickets respectively. He shared 421 Test wickets with Ambrose in 49 matches. He held the record of most Test wickets from 2000, after he broke the record of Kapil Dev. This record was later broken in 2004 by Shane Warne. He was the first bowler to reach 500 wickets in Test cricket. His autobiography is entitled "Heart of the Lion". Walsh was named one of the Wisden Cricketers of the Year in 1987. In October 2010, he was inducted into the ICC Cricket Hall of Fame. He was appointed as the Specialist Bowling Coach of Bangladesh Cricket Team in August 2016.
Anil Kumble is an Indian cricket coach, captain, former cricketer and commentator who played Test and One Day International cricket for the national team for 18 years. Widely regarded as one of the best leg spin bowlers in Test cricket history, he took 619 wickets in Test cricket and is the fourth highest wicket taker of all time as of 2021. In 1999 while playing against Pakistan, Kumble dismissed all ten batsmen in a Test match innings, joining England's Jim Laker as the only other player to achieve the feat. Unlike his contemporaries, Kumble was not a big turner of the ball, but relied primarily on pace, bounce, and accuracy. He was nicknamed "Apple" and "Jumbo". Kumble was selected as the Cricketer of the Year in 1993 Indian Cricket, and one of the Wisden Cricketers of the Year three years later.
Wasim Akram is a Pakistani cricket commentator, coach, and former cricketer and captain of the Pakistan national cricket team. A left-arm fast bowler who could bowl with significant pace, he is known as the "King of Reverse Swing". In October 2013, Wasim Akram was the only Pakistani cricketer to be named in an all-time Test World XI to mark the 150th anniversary of Wisden Cricketers' Almanack.
Abdul Qadir Khan was an international cricketer who bowled leg spin for Pakistan. Qadir is widely regarded as one of the best leg spinners of the 1970s and 1980s and was a role model for up and coming leg spinners. Later he was a commentator and Chief Selector of the Pakistan Cricket Board, from which he resigned due to differences of opinion with leading Pakistan cricket administrators.
Hedley Verity was a professional cricketer who played for Yorkshire and England between 1930 and 1939. A slow left-arm orthodox bowler, he took 1,956 wickets in first-class cricket at an average of 14.90 and 144 wickets in 40 Tests at an average of 24.37. Named as one of the Wisden Cricketers of the Year in 1932, he is regarded as one of the most effective slow left-arm bowlers to have played cricket. Never someone who spun the ball sharply, he achieved success through the accuracy of his bowling. On pitches which made batting difficult, particularly ones affected by rain, he could be almost impossible to bat against.
1 2 "Test Lowest Career Bowling Average". CricketArchive. Retrieved 6 January 2013.
↑ "Understanding byes and leg byes". BBC Sport. Retrieved 6 January 2013.
↑ "Law 42 (Fair and unfair play)". Marylebone Cricket Club. 2010. Archived from the original on 5 January 2013. Retrieved 6 January 2013.
↑ "The Laws of Cricket (2000 Code 4th Edition – 2010)" (PDF). Marylebone Cricket Club. 2010. pp. 42–49. Archived from the original (PDF) on 23 September 2010. Retrieved 6 January 2013.
1 2 van Staden (2008), p. 3.
1 2 Berry, Scyld, ed. (2011). Wisden Cricketers' Almanack 2011 (148 ed.). Alton, Hampshire: John Wisden & Co. Ltd. p. 1358. ISBN 978-1-4081-3130-5.
1 2 "Records / Test matches / Bowling records / Best career bowling average". ESPNcricinfo. Retrieved 6 January 2013.
1 2 "Records / One-Day Internationals / Bowling records / Best career bowling average". ESPNcricinfo. Retrieved 6 January 2013.
1 2 "Records / Twenty20 Internationals / Bowling records / Best career bowling average". ESPNcricinfo. Retrieved 6 January 2013.
1 2 "Records / First-class matches / Bowling records / Best career bowling average". ESPNcricinfo. Retrieved 6 January 2013.
↑ "Player Profile: Vincent Barnes". ESPNcricinfo. Retrieved 6 January 2013.
↑ Odendaal, Andre; Reddy, Krish; Samson, Andrew (2012). The Blue Book: History of Western Province Cricket: 1890–2011. Johannesburg: Fanele. p. 185. ISBN 978-1-920196-40-0 . Retrieved 6 January 2013.
↑ Boycott, Geoffrey (19 July 2011). "Geoffrey Boycott: ICC's Dream XI is a joke – it has no credibility". The Daily Telegraph . London: Telegraph Media Group . Retrieved 6 January 2013.
↑ "Records / Test matches / Bowling records / Best career bowling average (without qualification)". ESPNcricinfo. Retrieved 6 January 2013.
↑ "Records / Women's Test matches / Bowling records / Best career bowling average". ESPNcricinfo. Retrieved 6 January 2013.
↑ "Women's Test Lowest Career Bowling Average". CricketArchive. Retrieved 6 January 2013.
↑ "ODI Lowest Career Bowling Average". CricketArchive. Retrieved 6 January 2013.
↑ "Women's ODI Lowest Career Bowling Average". CricketArchive. Retrieved 6 January 2013.
↑ "Records / Women's One-Day Internationals / Bowling records / Best career bowling average". ESPNcricinfo. Retrieved 6 January 2013.
↑ "International Twenty20 Lowest Career Bowling Average". CricketArchive. Retrieved 6 January 2013.
↑ "First-Class Matches played by Stephen Draai (48)". CricketArchive. Retrieved 6 January 2013.
↑ "First-Class Matches played by Vince Barnes (68)". CricketArchive. Retrieved 6 January 2013.
van Staden, Paul J. (January 2008). Comparison of bowlers, batsmen and all-rounders in cricket using graphical displays (PDF). Pretoria: University of Pretoria, Faculty of Natural and Agricultural Sciences, Department of Statistics. ISBN 978-1-86854-733-3. Archived from the original (PDF) on 1 July 2014. Retrieved 6 January 2013.
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Arrivals - Maple Help
Home : Support : Online Help : Mathematics : Discrete Mathematics : Graph Theory : GraphTheory Package : Arrivals
vertices which are tails of arcs inbound to vertex
vertices which are heads of arcs outbound from vertex
neighbors of vertex
Arrivals(G, v)
Departures(G,v)
Neighbors(G, v)
(optional) vertex of the graph
Neighbors returns a list of lists if the input is just a graph. The ith list is the list of neighbors of the ith vertex of the graph. Neighbors(G, v) returns the list of neighbors of vertex v in the graph G.
Arrivals returns a list of the lists of vertices which are at the tail of arcs directed into vertex i. Undirected edges are treated as if they were bidirectional. If a vertex v is specified, the output is only the list of vertices which are at the tail of arcs directed into vertex v.
Departures is similar to Arrivals, but returns a list of the lists of vertices which are at the head of edges directed out of vertex i. If a vertex v is specified, the output is only the list of vertices which are at the head of edge directed out of vertex v.
\mathrm{with}\left(\mathrm{GraphTheory}\right):
G≔\mathrm{Digraph}\left(\mathrm{Trail}\left(1,2,3,4,5,6,4,7,8,2\right)\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 1: a directed unweighted graph with 8 vertices and 9 arc\left(s\right)}}
\mathrm{DrawGraph}\left(G\right)
\mathrm{Neighbors}\left(G,4\right)
[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]
\mathrm{Arrivals}\left(G,4\right)
[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}]
\mathrm{Departures}\left(G,4\right)
[\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]
\mathrm{Neighbors}\left(G\right)
[[\textcolor[rgb]{0,0,1}{2}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]]
\mathrm{Arrivals}\left(G\right)
[[]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{7}]]
\mathrm{Departures}\left(G\right)
[[\textcolor[rgb]{0,0,1}{2}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{6}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{8}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}]]
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Order-5 cubic honeycomb - Wikipedia
Order-5 cubic honeycomb
Poincaré disk models
Type Hyperbolic regular honeycomb
Uniform hyperbolic honeycomb
Edge figure pentagon {5}
{\displaystyle {\overline {BH}}_{3}}
Dual Order-4 dodecahedral honeycomb
Properties Regular
The order-5 cubic honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol {4,3,5}, it has five cubes {4,3} around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb.
3 Related polytopes and honeycombs
3.1 Rectified order-5 cubic honeycomb
3.1.1 Related honeycomb
3.2 Truncated order-5 cubic honeycomb
3.2.1 Related honeycombs
3.3 Bitruncated order-5 cubic honeycomb
3.4 Cantellated order-5 cubic honeycomb
3.5 Cantitruncated order-5 cubic honeycomb
3.6 Runcinated order-5 cubic honeycomb
3.7 Runcitruncated order-5 cubic honeycomb
3.8 Runcicantellated order-5 cubic honeycomb
3.9 Omnitruncated order-5 cubic honeycomb
3.10 Alternated order-5 cubic honeycomb
3.10.1 Related honeycombs
3.11 Cantic order-5 cubic honeycomb
3.12 Runcic order-5 cubic honeycomb
3.13 Runcicantic order-5 cubic honeycomb
It is analogous to the 2D hyperbolic order-5 square tiling, {4,5}
One cell, centered in Poincare ball model
Cells with extended edges to ideal boundary
It has a radial subgroup symmetry construction with dodecahedral fundamental domains: Coxeter notation: [4,(3,5)*], index 120.
Related polytopes and honeycombs[edit]
The order-5 cubic honeycomb has a related alternated honeycomb, ↔ , with icosahedron and tetrahedron cells.
The honeycomb is also one of four regular compact honeycombs in 3D hyperbolic space:
Four regular compact honeycombs in H3
There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including the order-5 cubic honeycomb as the regular form:
[5,3,4] family honeycombs
The order-5 cubic honeycomb is in a sequence of regular polychora and honeycombs with icosahedral vertex figures.
It is also in a sequence of regular polychora and honeycombs with cubic cells. The first polytope in the sequence is the tesseract, and the second is the Euclidean cubic honeycomb.
Rectified order-5 cubic honeycomb[edit]
Rectified order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol r{4,3,5} or 2r{5,3,4}
{\displaystyle {\overline {BH}}_{3}}
{\displaystyle {\overline {DH}}_{3}}
, [5,31,1]
The rectified order-5 cubic honeycomb, , has alternating icosahedron and cuboctahedron cells, with a pentagonal prism vertex figure.
Related honeycomb[edit]
It can be seen as analogous to the 2D hyperbolic tetrapentagonal tiling, r{4,5} with square and pentagonal faces
There are four rectified compact regular honeycombs:
Four rectified regular compact honeycombs in H3
r{p,3,5}
... r{∞,3,5}
r{∞,3}
Truncated order-5 cubic honeycomb[edit]
Truncated order-5 cubic honeycomb
{\displaystyle {\overline {BH}}_{3}}
The truncated order-5 cubic honeycomb, , has truncated cube and icosahedron cells, with a pentagonal pyramid vertex figure.
It can be seen as analogous to the 2D hyperbolic truncated order-5 square tiling, t{4,5}, with truncated square and pentagonal faces:
It is similar to the Euclidean (order-4) truncated cubic honeycomb, t{4,3,4}, which has octahedral cells at the truncated vertices.
Related honeycombs[edit]
Four truncated regular compact honeycombs in H3
Bitruncated order-5 cubic honeycomb[edit]
The bitruncated order-5 cubic honeycomb is the same as the bitruncated order-4 dodecahedral honeycomb.
Cantellated order-5 cubic honeycomb[edit]
Cantellated order-5 cubic honeycomb
Schläfli symbol rr{4,3,5}
pentagon {5}
{\displaystyle {\overline {BH}}_{3}}
The cantellated order-5 cubic honeycomb, , has rhombicuboctahedron, icosidodecahedron, and pentagonal prism cells, with a wedge vertex figure.
It is similar to the Euclidean (order-4) cantellated cubic honeycomb, rr{4,3,4}:
Four cantellated regular compact honeycombs in H3
Cantitruncated order-5 cubic honeycomb[edit]
Cantitruncated order-5 cubic honeycomb
Schläfli symbol tr{4,3,5}
{\displaystyle {\overline {BH}}_{3}}
The cantitruncated order-5 cubic honeycomb, , has truncated cuboctahedron, truncated icosahedron, and pentagonal prism cells, with a mirrored sphenoid vertex figure.
It is similar to the Euclidean (order-4) cantitruncated cubic honeycomb, tr{4,3,4}:
Four cantitruncated regular compact honeycombs in H3
Runcinated order-5 cubic honeycomb[edit]
Runcinated order-5 cubic honeycomb
Semiregular honeycomb
Schläfli symbol t0,3{4,3,5}
irregular triangular antiprism
{\displaystyle {\overline {BH}}_{3}}
The runcinated order-5 cubic honeycomb or runcinated order-4 dodecahedral honeycomb , has cube, dodecahedron, and pentagonal prism cells, with an irregular triangular antiprism vertex figure.
It is analogous to the 2D hyperbolic rhombitetrapentagonal tiling, rr{4,5}, with square and pentagonal faces:
It is similar to the Euclidean (order-4) runcinated cubic honeycomb, t0,3{4,3,4}:
Three runcinated regular compact honeycombs in H3
Runcitruncated order-5 cubic honeycomb[edit]
Runctruncated order-5 cubic honeycomb
Runcicantellated order-4 dodecahedral honeycomb
{\displaystyle {\overline {BH}}_{3}}
The runcitruncated order-5 cubic honeycomb or runcicantellated order-4 dodecahedral honeycomb, , has truncated cube, rhombicosidodecahedron, pentagonal prism, and octagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.
It is similar to the Euclidean (order-4) runcitruncated cubic honeycomb, t0,1,3{4,3,4}:
Four runcitruncated regular compact honeycombs in H3
Runcicantellated order-5 cubic honeycomb[edit]
The runcicantellated order-5 cubic honeycomb is the same as the runcitruncated order-4 dodecahedral honeycomb.
Omnitruncated order-5 cubic honeycomb[edit]
Omnitruncated order-5 cubic honeycomb
{10}x{}
{8}x{}
decagon {10}
{\displaystyle {\overline {BH}}_{3}}
The omnitruncated order-5 cubic honeycomb or omnitruncated order-4 dodecahedral honeycomb, , has truncated icosidodecahedron, truncated cuboctahedron, decagonal prism, and octagonal prism cells, with an irregular tetrahedral vertex figure.
It is similar to the Euclidean (order-4) omnitruncated cubic honeycomb, t0,1,2,3{4,3,4}:
Three omnitruncated regular compact honeycombs in H3
Alternated order-5 cubic honeycomb[edit]
Alternated order-5 cubic honeycomb
Schläfli symbol h{4,3,5}
Coxeter diagram ↔
{\displaystyle {\overline {DH}}_{3}}
Properties Vertex-transitive, edge-transitive, quasiregular
In 3-dimensional hyperbolic geometry, the alternated order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb). With Schläfli symbol h{4,3,5}, it can be considered a quasiregular honeycomb, alternating icosahedra and tetrahedra around each vertex in an icosidodecahedron vertex figure.
It has 3 related forms: the cantic order-5 cubic honeycomb, , the runcic order-5 cubic honeycomb, , and the runcicantic order-5 cubic honeycomb, .
Cantic order-5 cubic honeycomb[edit]
Cantic order-5 cubic honeycomb
Schläfli symbol h2{4,3,5}
{\displaystyle {\overline {DH}}_{3}}
The cantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h2{4,3,5}. It has icosidodecahedron, truncated icosahedron, and truncated tetrahedron cells, with a rectangular pyramid vertex figure.
Runcic order-5 cubic honeycomb[edit]
Runcic order-5 cubic honeycomb
triangular frustum
{\displaystyle {\overline {DH}}_{3}}
The runcic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h3{4,3,5}. It has dodecahedron, rhombicosidodecahedron, and tetrahedron cells, with a triangular frustum vertex figure.
Runcicantic order-5 cubic honeycomb[edit]
Runcicantic order-5 cubic honeycomb
Schläfli symbol h2,3{4,3,5}
{\displaystyle {\overline {DH}}_{3}}
The runcicantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h2,3{4,3,5}. It has truncated dodecahedron, truncated icosidodecahedron, and truncated tetrahedron cells, with an irregular tetrahedron vertex figure.
Regular tessellations of hyperbolic 3-space
Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294-296)
Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
Norman Johnson Uniform Polytopes, Manuscript
N.W. Johnson: Geometries and Transformations, (2015) Chapter 13: Hyperbolic Coxeter groups
Retrieved from "https://en.wikipedia.org/w/index.php?title=Order-5_cubic_honeycomb&oldid=962576910"
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Completing the Square: Level 5 Challenges Practice Problems Online | Brilliant
16x^{4} + 8x^{3} - 12x^{2} - 4x +1 =0
a,b,c
and
are the real roots of the above equation such that
a<b<c<d
\left\lfloor \dfrac{100ab}{cd} \right\rfloor
by U Z
x, y, z
3x^2+12y^2+27z^2-4xy-12yz-6xz-8y-24z+100 ?
An ambitious king plans to build a number of new cities in the wilderness, connected by a network of roads, so that any city can be reached from any other. He expects the annual tax revenues from each city to be numerically equal to the square of the population. Road maintenance will be expensive, though; the annual cost for each road is expected to be numerically equal to the product of the populations of the two cities that road connects. The project is considered viable as long as the tax revenues exceed the cost, regardless of the populations of the various cities (as long as at least one city has any inhabitants at all). The court engineer submits some proposals (attached), but the king deems them "boring" and asks for other options. How many other graphs are there (up to isomorphism) that make this project viable?
\begin{cases}x^4+2x^3-y=-\frac{1}{4}+\sqrt{3}\\ y^4+2y^3-x=-\frac{1}{4}-\sqrt{3} \end{cases}
All the ordered pairs of real numbers that satisfy the system of equations above are
(x_1,y_1),(x_2,y_2),...(x_n,y_n)
x_1+x_2+...+x_n+y_1+...y_n
correct upto two decimal places.
If you think there are infinite solutions then answer 777 and if you think no real solutions answer 666.
Ordered pair means (11,12),(12,11) are considered different.
P(x)=(x^2+5000x-1)^2+(2x+5000)^2
P(x)
be a polynomial, then find the sum of all the real solutions to the equation
P(x)=\text{min}(P(x)).
Image Credit: Wikimedia Five Thousand
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Global Constraint Catalog: Kmaximum_clique
<< 3.7.147. Maximum3.7.149. Maximum number of occurrences >>
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚍𝚒𝚜𝚝}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚌𝚕𝚒𝚚𝚞𝚎}
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
\mathrm{𝚌𝚕𝚒𝚚𝚞𝚎}
) that can be used for searching for a maximum clique in a graph, or a constraint (i.e.,
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚍𝚒𝚜𝚝}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
) that can be stated by extracting a large clique [BronKerbosch73] from a specific graph of elementary constraints.
A maximum clique is a clique of maximum size, a clique being a subset of vertices such that each vertex is connected to all other vertices of the clique.
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Home : Support : Online Help : Mathematics : Discrete Mathematics : Graph Theory : GraphTheory Package : Radius
find the minimum eccentricity of a graph
Radius returns the minimum eccentricity over all vertices in the graph G.
If G is disconnected, then the output is infinity.
\mathrm{with}\left(\mathrm{GraphTheory}\right):
\mathrm{with}\left(\mathrm{SpecialGraphs}\right):
P≔\mathrm{PetersenGraph}\left(\right)
\textcolor[rgb]{0,0,1}{P}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 1: an undirected unweighted graph with 10 vertices and 15 edge\left(s\right)}}
\mathrm{Radius}\left(P\right)
\textcolor[rgb]{0,0,1}{2}
C≔\mathrm{CycleGraph}\left(19\right)
\textcolor[rgb]{0,0,1}{C}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 2: an undirected unweighted graph with 19 vertices and 19 edge\left(s\right)}}
\mathrm{Radius}\left(C\right)
\textcolor[rgb]{0,0,1}{9}
G≔\mathrm{Graph}\left({[{1,2},0.2],[{1,4},1.1],[{2,3},0.3],[{3,4},0.4]}\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 3: an undirected weighted graph with 4 vertices and 4 edge\left(s\right)}}
\mathrm{DMrawGraph}\left(G\right)
\textcolor[rgb]{0,0,1}{\mathrm{DMrawGraph}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{Graph 3: an undirected weighted graph with 4 vertices and 4 edge\left(s\right)}}\right)
\mathrm{Radius}\left(G\right)
\textcolor[rgb]{0,0,1}{0.5}
\mathrm{DijkstrasAlgorithm}\left(G,1,4\right)
[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.9}]
The GraphTheory[Radius] command was introduced in Maple 2017.
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Global Constraint Catalog: Csequence_folding
<< 5.342. scalar_product5.344. set_value_precede >>
\mathrm{𝚜𝚎𝚚𝚞𝚎𝚗𝚌𝚎}_\mathrm{𝚏𝚘𝚕𝚍𝚒𝚗𝚐}\left(\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}\right)
\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚗𝚎𝚡𝚝}-\mathrm{𝚍𝚟𝚊𝚛}\right)
|\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}|\ge 1
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂},\left[\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚗𝚎𝚡𝚝}\right]\right)
\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\ge 1
\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\le |\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}|
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚜𝚎𝚚}
\left(\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂},\mathrm{𝚒𝚗𝚍𝚎𝚡}\right)
\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}.\mathrm{𝚗𝚎𝚡𝚝}\ge 1
\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}.\mathrm{𝚗𝚎𝚡𝚝}\le |\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}|
Express the fact that a sequence is folded in a way that no crossing occurs. A sequence is modelled by a collection of letters. For each letter
{l}_{1}
of a sequence, we indicate the next letter
{l}_{2}
located after
{l}_{1}
that is directly in contact with
{l}_{1}
{l}_{1}
itself if such a letter does not exist).
\left(\begin{array}{c}〈\begin{array}{cc}\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\hfill & \mathrm{𝚗𝚎𝚡𝚝}-1,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-2\hfill & \mathrm{𝚗𝚎𝚡𝚝}-8,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-3\hfill & \mathrm{𝚗𝚎𝚡𝚝}-3,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-4\hfill & \mathrm{𝚗𝚎𝚡𝚝}-5,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-5\hfill & \mathrm{𝚗𝚎𝚡𝚝}-5,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-6\hfill & \mathrm{𝚗𝚎𝚡𝚝}-7,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-7\hfill & \mathrm{𝚗𝚎𝚡𝚝}-7,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-8\hfill & \mathrm{𝚗𝚎𝚡𝚝}-8,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-9\hfill & \mathrm{𝚗𝚎𝚡𝚝}-9\hfill \end{array}〉\hfill \end{array}\right)
Figure 5.343.1 gives the folded sequence associated with the previous example. Each number represents the index of an item. The
\mathrm{𝚜𝚎𝚚𝚞𝚎𝚗𝚌𝚎}_\mathrm{𝚏𝚘𝚕𝚍𝚒𝚗𝚐}
constraint holds since no crossing occurs.
Figure 5.343.1. Folded sequence (in blue) of the Example slot: links from a letter to a distinct letter are represented by a dashed arc, while self-loops are not drawn
|\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}|>2
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}.\mathrm{𝚗𝚎𝚡𝚝}\right)>1
Motivated by RNA folding [FlammHofackerStadler99].
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜}
\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}
\mathrm{𝑆𝐸𝐿𝐹}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚕𝚎𝚝𝚝𝚎𝚛𝚜}\right)
\mathrm{𝚕𝚎𝚝𝚝𝚎𝚛𝚜}.\mathrm{𝚗𝚎𝚡𝚝}\ge \mathrm{𝚕𝚎𝚝𝚝𝚎𝚛𝚜}.\mathrm{𝚒𝚗𝚍𝚎𝚡}
\mathrm{𝐍𝐀𝐑𝐂}
=|\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}|
\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
\left(<\right)↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚕𝚎𝚝𝚝𝚎𝚛𝚜}\mathtt{1},\mathrm{𝚕𝚎𝚝𝚝𝚎𝚛𝚜}\mathtt{2}\right)
\bigvee \left(\begin{array}{c}\mathrm{𝚕𝚎𝚝𝚝𝚎𝚛𝚜}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\ge \mathrm{𝚕𝚎𝚝𝚝𝚎𝚛𝚜}\mathtt{1}.\mathrm{𝚗𝚎𝚡𝚝},\hfill \\ \mathrm{𝚕𝚎𝚝𝚝𝚎𝚛𝚜}\mathtt{2}.\mathrm{𝚗𝚎𝚡𝚝}\le \mathrm{𝚕𝚎𝚝𝚝𝚎𝚛𝚜}\mathtt{1}.\mathrm{𝚗𝚎𝚡𝚝}\hfill \end{array}\right)
\mathrm{𝐍𝐀𝐑𝐂}
=|\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}|*\left(|\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}|-1\right)/2
\mathrm{𝐍𝐀𝐑𝐂}
\mathrm{𝚜𝚎𝚚𝚞𝚎𝚗𝚌𝚎}_\mathrm{𝚏𝚘𝚕𝚍𝚒𝚗𝚐}
Consider the first graph constraint. Since we use the
\mathrm{𝑆𝐸𝐿𝐹}
\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}
collection the maximum number of arcs of the final graph is equal to
|\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}|
\mathrm{𝐍𝐀𝐑𝐂}=|\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}|
\mathrm{𝐍𝐀𝐑𝐂}\ge |\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}|
\underline{\overline{\mathrm{𝐍𝐀𝐑𝐂}}}
\overline{\mathrm{𝐍𝐀𝐑𝐂}}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}\left(<\right)
\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}
|\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}|·\left(|\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}|-1\right)/2
\mathrm{𝐍𝐀𝐑𝐂}=|\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}|·\left(|\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}|-1\right)/2
\mathrm{𝐍𝐀𝐑𝐂}\ge |\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}|·\left(|\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}|-1\right)/2
\underline{\overline{\mathrm{𝐍𝐀𝐑𝐂}}}
\overline{\mathrm{𝐍𝐀𝐑𝐂}}
\mathrm{𝚜𝚎𝚚𝚞𝚎𝚗𝚌𝚎}_\mathrm{𝚏𝚘𝚕𝚍𝚒𝚗𝚐}
constraint. Consider the
{i}^{th}
{j}^{th}
\left(i<j\right)
\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}
{\mathrm{𝙸𝙽𝙳𝙴𝚇}}_{i}
{\mathrm{𝙽𝙴𝚇𝚃}}_{i}
\mathrm{𝚒𝚗𝚍𝚎𝚡}
\mathrm{𝚗𝚎𝚡𝚝}
{i}^{th}
\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}
{\mathrm{𝙸𝙽𝙳𝙴𝚇}}_{j}
{\mathrm{𝙽𝙴𝚇𝚃}}_{j}
\mathrm{𝚒𝚗𝚍𝚎𝚡}
\mathrm{𝚗𝚎𝚡𝚝}
{j}^{th}
\mathrm{𝙻𝙴𝚃𝚃𝙴𝚁𝚂}
. To each quadruple
\left({\mathrm{𝙸𝙽𝙳𝙴𝚇}}_{i},{\mathrm{𝙽𝙴𝚇𝚃}}_{i},{\mathrm{𝙸𝙽𝙳𝙴𝚇}}_{j},{\mathrm{𝙽𝙴𝚇𝚃}}_{j}\right)
{S}_{i,j}
, which takes its value in
\left\{0,1,2\right\}
, as well as the following signature constraint:
\left({\mathrm{𝙸𝙽𝙳𝙴𝚇}}_{i}\le {\mathrm{𝙽𝙴𝚇𝚃}}_{i}\right)\wedge \left({\mathrm{𝙸𝙽𝙳𝙴𝚇}}_{j}\le {\mathrm{𝙽𝙴𝚇𝚃}}_{j}\right)\wedge \left({\mathrm{𝙽𝙴𝚇𝚃}}_{i}\le {\mathrm{𝙽𝙴𝚇𝚃}}_{j}\right)⇔{S}_{i,j}=0\wedge
\left({\mathrm{𝙸𝙽𝙳𝙴𝚇}}_{i}\le {\mathrm{𝙽𝙴𝚇𝚃}}_{i}\right)\wedge \left({\mathrm{𝙸𝙽𝙳𝙴𝚇}}_{j}\le {\mathrm{𝙽𝙴𝚇𝚃}}_{j}\right)\wedge \left({\mathrm{𝙽𝙴𝚇𝚃}}_{i}>{\mathrm{𝙸𝙽𝙳𝙴𝚇}}_{j}\right)\wedge \left({\mathrm{𝙽𝙴𝚇𝚃}}_{j}\le {\mathrm{𝙽𝙴𝚇𝚃}}_{i}\right)⇔{S}_{i,j}=1
\mathrm{𝚜𝚎𝚚𝚞𝚎𝚗𝚌𝚎}_\mathrm{𝚏𝚘𝚕𝚍𝚒𝚗𝚐}
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Introduction to Anomaly Detection - Fizzy
Machine Learning Math Data
Anomaly detection has two basic assumptions:
Anomalies only occur very rarely in the data;
(Susan Li, 2019)
Different anomaly detection approaches shall be applied based on the characteristics of dataset and the purpose of the analysis. There are mainly two ways to deal with outliers in statistics and machine learning, namely unsupervised learning and supervised learning. Also the algorithms often been modified and integrated with other techniques to achieve specific goals in production environment, for example time series analysis.
3 Sigma Rule
For one dimension outlier detection we can simply use the
3\sigma
-Rule.
Given the dataset
X = \lbrace x_1,x_2,...,x_n \rbrace
X
follows normal distribution, then its probability density function is:
p(x,\mu,\sigma^2)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}
from maximal likelihood, we know that:
\hat{\mu}=\overline{x}=\frac{\sum_{i=1}^{n} x_{i}}{n}
\hat{\sigma}^2=\frac{\sum_{i=1}^{n} (x_{i}-\overline{x})^2}{n}
Then we create a
\mu \pm 3\sigma
window contains 99.73% of the data. If a new point falls outside of the window (i.e.
x_k \notin (\hat{\mu}-3\hat{\sigma},\hat{\mu}+3\hat{\sigma})
), then such new data point can be perceived as an outlier.
The boxplot can be applied to any form of distribution of the data. The define the boundary of the outlier detection, we first find the Q1 and Q3 then calculate the Interquartile Range (IQR = Q3 - Q1). The 2 outlier boundaries then defines as: $ Q1- \lambda * IQR $, $ Q3+\lambda * IQR $. (usually we take
\lambda = 1.5
There are many other statistical models could perform the tasks of anomaly detection. I will probably introduce those in future posts.
Distance Based Models
Angle-Based model is not suitable for large datasets. But it is a basic way to detect outliers by simply comparing the angles between pairs of distance vectors to other points.
Nearest Neighbour Approaches
For each data point
x_i
, compute the distance to the
k
-th nerest neighbour
d_{k}(x_i)
. Then for all data points, we diagnose the outliers are the points that have the larger distance. Therefore, the outliers are located in the more sparse neighbourhoods. However, this methods is not suitable for dataset that have modes with low density.
The LOF score is equal to ratio of average local density of the
k
nearest neighbours of the instance and the local density of the data instance itself.
Connectivity Outlier Factor (COF)
Outliers are points
p
where average chaining distance
ac
dist_{kNN(p)}(p)
is larger than the average chaining distance (
ac
dist
) of their
k
-nearest neighbourhood
kNN(p)
COF identifies outliers as points whose neighourhoods are sparser than the neighbourhoods of their neighbours.
Principal Componenet Analysis
I will not try to explain what PCA is in here. The major argument here is that since the variances of the transformed data along the eigenvectors with small eigenvalues are low, significant deviations of the transformed data from the mean values along these directions may represent outliers.
R
p \times p
sample correlation matrix computed from
n
observations on each of
p
X_1,...,X_p
(\lambda_1,e_1),...,(\lambda_p,e_p)
p
eigenvalue-eigenvector pairs of
R
\lambda_1 \geq ... \geq \lambda_p \geq 0
i
-th sample principal component of an observation vector
\pmb{x} = (x_1,...,x_p)
y_{i} = < \pmb{e}\_i,\pmb{z} > = \sum_{k=1}^{p} e_{ik} \cdot z_k \quad \text{for i = 1,2,...,p}
e_i = (e_{i1},...,e_{ip})^{T}
i
-th eigenvector. That means
\pmb{y} = (y_1,...,y_p)^{T} = (e_{i1},...,e_{ip})^{T}\pmb{z}
is the principal component of vector
\pmb{x}
\pmb{z} = (z_1,...,z_p)^{T}
is the vector of standardized observations defined as
z_k = \frac{x_k - \overline{x}\_k}{\sqrt{s_{kk}}} \quad \text{k=1,...,p}
where $ \overline{x}_{k} $ and $ s_{kk} $ are the sample mean and the sample variance of the variable
X_k
Consider the sample principal components
y_1,...,y_p
of and observation
\pmb{x}
. The sum of the squares of the standardized principal component values
Score(\pmb{x}) = \sum_{i=1}^{p} \frac{y_{i}^2}{\lambda_i} = \frac{y_{1}^2}{\lambda_1} + \cdots + \frac{y_{p}^2}{\lambda_p}
is equivalent to the Mahalanobis distance of the observation
\pmb{x}
from the mean of the sample. An observation is outlier if fro a given significance level
\sum_{i=1}^{p} \frac{y_{i}^{2}}{\lambda_{i}}>\chi_{q}^{2} (\alpha) \quad \text { where } 1 \leq q \leq p
Replicator Neural Networks
The RNN in here is not referring the commonly known Recurrent Neural Networks. The Replicator Neural Networks, or autoencoders, are multi-layer feed-forward neural networks.
The input and output layers have the same number of nodes. Then the network is trained to learn identifying-mapping from inputs to outputs. Given a trained RNN, the reconstruction error is used as the outlier score: test instances incurring a high reconstruction error are considered outliers.
Staistical Methods based OD
Normal data instances occur in high probability regions of a stochastic model, while anomalies occur in the low probability regions of the stochastic model.
Classification based OD
A classifier that can distinguish between normal and anomalous classes can be learnt in the given feature space.
Nearest Neighbour based OD
Normal data instances occur in dense neigbourhoods, while anomalies occur far from their closest neighbours.
Clustering based OD
Normal data instances belong to a cluster in the data, while anomalies either do not belong to any cluster.
Normal data instances lie close to their closest cluster centroid, while anomalies are far away from their closest cluster centroid.
Normal data instances belong to large and dense clusters, while anomalies either belong to small or sparse clusters.
PCA based OD
Data can be embedded into a lower dimensional subspace in which normal instances and anomalies appear significantly different.
Charu C.Aggarwal (2013). Outlier Analysis. Springer.
Jiawei Han (2000). Data Mining: Concepts and Techniques.
Susan Li (2019). Anomaly Detection for Dummies. Retrieved from https://towardsdatascience.com/anomaly-detection-for-dummies-15f148e559c1.
Anomaly Detection Learning Resources
|
Design of a Novel Electrode of Radiofrequency Ablation for Large Tumors: In Vitro Validation and Evaluation | J. Biomech Eng. | ASME Digital Collection
Zheng Fang,
e-mail: jasonfang2014@hotmail.com
Micheal A. J. Moser,
Micheal A. J. Moser
Saskatoon SK S7N 0W8, Canada
e-mail: mam305@mail.usask.ca
Edwin Zhang,
e-mail: ezhang@ualberta.ca
e-mail: chris.zhang@usask.ca
Tumor Ablation Group,
Biomedical Science and
Technology Research Center,
School of Mechatronic
e-mail: bingzhang84@shu.edu.cn
Manuscript received August 28, 2018; final manuscript received November 13, 2018; published online January 18, 2019. Assoc. Editor: Ram Devireddy.
Fang, Z., Moser, M. A. J., Zhang, E., Zhang, W., and Zhang, B. (January 18, 2019). "Design of a Novel Electrode of Radiofrequency Ablation for Large Tumors: In Vitro Validation and Evaluation." ASME. J Biomech Eng. March 2019; 141(3): 031007. https://doi.org/10.1115/1.4042179
In a prior study, we proposed a novel monopolar expandable electrode (MEE) for use in radiofrequency ablation (RFA). The purpose of our work was to now validate and evaluate this electrode using on in vitro experimental model and computer simulation. Two commercially available RF electrodes (conventional electrode (CE) and umbrella electrode (UE)) were used to compare the ablation results with the novel MEE using an in vitro egg white model and in vivo liver tumor model to verify the efficacy of MEE in the large tumor ablation, respectively. The sharp increase in impedance during RFA procedures was taken as the termination of RFA protocols. In the in vitro egg white experiment, the ablation volume of MEE, CE, and UE was 75.3
±
1.6 cm3, 2.7
±
0.4 cm3, and 12.4
±
1.8 cm3 (P < 0.001), respectively. Correspondingly, the sphericity was 88.1
±
0.9%, 12.9
±
1.3%, and 62.0
±
3.0% (P < 0.001), respectively. A similar result was obtained in the in vitro egg white computer simulation. In the liver tumor computer simulation, the volume and sphericity of ablation zone generated by MEE, CE, and UE were 36.6 cm3 and 93.6%, 3.82 cm3 and 16.9%, and 13.5 cm3 and 56.7%, respectively. In summary, MEE has the potential to achieve complete ablation in the treatment of large tumors (>3 cm in diameter) compared to CE and UE due to the larger electrode–tissue interface and more round shape of hooks.
computer modeling, egg white, tumor ablation, radiofrequency ablation, RF electrode
Ablation (Vaporization technology), Biological tissues, Electrodes, Liver, Radiofrequency ablation, Tumors, Computer simulation, Shapes
A Review of Radiofrequency Ablation: Large Target Tissue Necrosis and Mathematical Modelling
Principles of Radiofrequency Ablation
Evaluation of the Current Radiofrequency Ablation Systems Using Axiomatic Design Theory
., International Working Group on Image-Guided Tumor Ablation, Interventional Oncology Sans Frontieres Expert Panel, Technology Assessment Committee of the Society of Interventional Radiology, Standard of Practice Committee of the Cardiovascular and Interventional Radiological Society of Europe,
Randomized Clinical Trial of Hepatic Resection Versus Radiofrequency Ablation for Early-Stage Hepatocellular Carcinoma
Comparative Effectiveness of First-Line Radiofrequency Ablation Versus Surgical Resection and Transplantation for Patients With Early Hepatocellular Carcinoma
Comparison of Laparoscopic Microwave to Radiofrequency Ablation of Small Hepatocellular Carcinoma (≤3 cm)
Thermal Tumour Ablation: Devices, Clinical Applications and Future Directions
Tissue Ablation With Radiofrequency: Effect of Probe Size, Gauge, Duration, and Temperature on Lesion Volume
Electrodes and Multiple Electrode Systems for Radio Frequency Ablation: A Proposal for Updated Terminology
CT Findings After Radiofrequency Ablation in Rabbit Livers: Comparison of Internally Cooled Electrodes, Perfusion Electrodes, and Internally Cooled Perfusion Electrodes
Radio-Frequency Thermal Ablation With NaCl Solution Injection: Effect of Electrical Conductivity on Tissue Heating and Coagulation—Phantom and Porcine Liver Study
Comparison Between Different Thickness Umbrella-Shaped Expandable Radiofrequency Electrodes (SuperSlim and CoAccess): Experimental and Clinical Study
ASME J. Med. Devices, 13
J. Eng. Sci. Med. Diagn. Ther.
Mihalef
Comprehensive Preclinical Evaluation of a Multi-Physics Model of Liver Tumor Radiofrequency Ablation
(9), pp 1543–1559.https://www.ncbi.nlm.nih.gov/pubmed/28097603
Parametric Study of Radiofrequency Ablation in the Clinical Practice With the Use of Two-Compartment Numerical Models
Electromagn. Biol. Med.
Numerical Modeling of Heat Transfer and Pasteurizing Value During Thermal Processing of Intact Egg
Review of the Mathematical Functions Used to Model the Temperature Dependence of Electrical and Thermal Conductivities of Biological Tissue in Radiofrequency Ablation
Experimental Cookery: From the Chemical and Physical Standpoint
Simulation of Radiofrequency Ablation in Real Human Anatomy
Numerical Analysis of the Relationship Between the Area of Target Tissue Necrosis and the Size of Target Tissue in Liver Tumours With Pulsed Radiofrequency Ablation
Electrical Conductivity Measurement of Excised Human Metastatic Liver Tumours Before and After Thermal Ablation
RF Ablation at Low Frequencies for Targeted Tumor Heating: In Vitro and Computational Modeling Results
Morphologic Analysis of Bipolar Radiofrequency Lesions: Implications for Treatment of the Sacroiliac Joint
Reg. Anesth. Pain Med.
Automatic Liver Tumor Detection Using EM/MPM Algorithm and Shape Information
IEEE Second International Conference on Software Engineering and Data Mining (SEDM),
Chengdu, China, June 23–25, pp.
Hepatocellular Carcinoma: Stiffness Value and Ratio to Discriminate Malignant From Benign Focal Liver Lesions
Topology Optimization of Efficient and Strong Hybrid Compliant Mechanisms Using a Mixed Mesh of Beams and Flexure Hinges With Strength Control
|
Discussion: “Kinematics of a New High-Precision Three-Degree-of-Freedom Parallel Manipulator” (Tahmasebi, F., 2007, ASME J. Mech. Des., 129, pp. 320–325) | J. Mech. Des. | ASME Digital Collection
Discussion: “Kinematics of a New High-Precision Three-Degree-of-Freedom Parallel Manipulator” (Tahmasebi, F., 2007, ASME J. Mech. Des., 129, pp. 320–325)
Juan A. Carretero,
Juan A. Carretero
, Fredericton, NB, E3B 5A3, Canada
e-mail: juan.carretero@unb.ca
Meyer A. Nahon,
, Montreal, QC, H3A 2T5, Canada
, Victoria, BC, V8W 3P6, Canada
A commentary has been published: Closure to “Discussion of ‘Kinematics of a New High-Precision Three-Degree-of-Freedom Parallel Manipulator’ ” (2008, ASME J. Mech. Des., 130, p. 035501)
This is a correction to: Kinematics of a New High-Precision Three-Degree-of-Freedom Parallel Manipulator
Carretero, J. A., Nahon, M. A., and Podhorodeski, R. P. (February 5, 2008). "Discussion: “Kinematics of a New High-Precision Three-Degree-of-Freedom Parallel Manipulator” (Tahmasebi, F., 2007, ASME J. Mech. Des., 129, pp. 320–325)." ASME. J. Mech. Des. March 2008; 130(3): 035501. https://doi.org/10.1115/1.2829981
end effectors, manipulator kinematics
Kinematics, Manipulators, End effectors, Displacement, Manipulator kinematics
The author of Ref. 1 proposes a “new” high-precision three-degree-of-freedom parallel manipulator. The paper discusses the inverse and forward displacement solutions for the 3-PRS manipulator, which is formed by three identical limbs, each containing an actuated prismatic joint (P) followed by a passive perpendicular revolute joint (R), a fixed-length leg, and a passive spherical joint (S), which attaches to the moving platform. In Ref. 1, the three active prismatic joints intersect at the center of the base platform and are contained within the base plane while being separated by
120deg
from one another.
We would like to bring to the readers’ attention that although Tahmasebi (1) presents some interesting implementation issues, the same 3-PRS manipulator was first presented in 1997 by Carretero et al. (2). In March 2000, an improved version of that paper was published in ASME Journal of Mechanical Design (3). Both papers present the inverse displacement solution of the manipulator where the lengths of the active prismatic joints are obtained as a function of the pose of the end effector. The pose is defined by three variables: two angles around mutually perpendicular inertial axes parallel to the base plane (tip and tilt) and a displacement normal to the base platform—just as they are in Ref. 1.
Also worth noting is the fact that the forward displacement problem (FDP) for this manipulator was solved by Tsai et al. in Ref. 4 for a variation of the 3-PRS whose prismatic joints are all perpendicular to the base platform. The formulation in Ref. 4 is easily modified to obtain the forward displacement solution of the version where all prismatic joints are parallel to the base. This equivalent formulation of the FDP is presented in Ref. 1 but the work in Ref. 4 is only mentioned in passing in the Introduction.
With respect to the inverse displacement problem, Tahmasebi (1) considered a less general version of the 3-PRS manipulator. That is, in Ref. 3, the angles between the three branches were left as design variables whereas in Tahmasebi’s work, these angles are fixed at
120deg
−120deg
, respectively. Note that the applications considered in Refs. 2,3,5 are also high-precision tasks, just as those claimed in Ref. 1. More specifically, in Refs. 2,3, the authors suggested that the 3-PRS manipulator could be used near its singular configuration for high-precision applications such as telescope image correction. This configuration, reached when all three fixed-length legs are close to being parallel, gives the manipulator higher resolution in that region of the workspace.
As pointed out in Refs. 3,6, motions in the tip and tilt directions come with unavoidable motions in the other three directions. That is, when the platform is tipped and∕or tilted, small translations occur along two noncollinear axes coplanar to the base plane as well as a rotation around an axis perpendicular to this plane. These extraneous motions, independent of the platform’s elevation, were deemed parasitic and minimized in Refs. 3,6. Although these motions are noted in Ref. 1 as Eqs. (4) and (5), they are not recognized as having an adverse effect on the payload’s location, particularly for the high-precision applications claimed by the author.
In addition to the aforementioned seminal work by Carretero et al., a number of documents have been published since 1997 discussing different aspects and variations of the 3-PRS manipulator as well as analyzing the manipulators for a number of different applications. Following is a list of some of the works presented to date on the 3-PRS manipulator, grouped by the general topic. These important works on the 3-PRS manipulator were also overlooked in Ref. 1.
Kinematics: Inverse displacement problem (2,3,7) and later generalized in Ref. 8 for any orientation of the prismatic joints relative to the base. FDP (4). Reduction of parasitic motions (3,6). Kinematic calibration (9).
Workspace size and quality: Dexterity (10,11). Stiffness (12).
Design and applications: Medical assistants (13). Also, Tsai et al. (4) presented a slight variation of the version by Merlet presented in Ref. 13. Telescope image correction (2). Machine tool applications (14). Micromanipulation (5).
Kinematics of a New High-Precision Three-Degree-of-Freedom Parallel Manipulator
Direct Kinematic Analysis of a 3-PRS Parallel Manipulator
Study of an Intelligent Micro-Manipulator
Architecture Optimization of a Three DOF Parallel Manipulator
Ebrahimi Moghaddam
A Numerical Algorithm for Solving Displacement Kinematics of Parallel Manipulators
IECON 2005: 31st Annual Conference of the IEEE Industrial Electronics Society
Rakigh, NC
Kinematic Analysis and Workspace Determination of the Inclined PRS Parallel Manipulator
Proceedings of the 15th CISM-IFToMM Symposium on Robot Design, Dynamics and Control (RoManSy 2004)
Saint-Hubert (Montreal), Quebec, Canada
Calibration of the 3-PRS Parallel Manipulator Using a Motion Capture System
CSME Transactions
Kinematic and Stiffness Analysis for a General 3-PRS Spatial Parallel Manipulator
Micro Parallel Robot MIPS for Medical Applications
Proceedings of the Eighth International Conference on Emerging Technologies and Factory Automation (ETFA 2001)
|
Global Constraint Catalog: Ck_same
<< 5.206. k_disjoint5.208. k_same_interval >>
𝚔_\mathrm{𝚜𝚊𝚖𝚎}\left(\mathrm{𝚂𝙴𝚃𝚂}\right)
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚂𝙴𝚃𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚜𝚎𝚝}-\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|\ge 1
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚂𝙴𝚃𝚂},\mathrm{𝚜𝚎𝚝}\right)
|\mathrm{𝚂𝙴𝚃𝚂}|>1
\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚜𝚒𝚣𝚎}
\left(\mathrm{𝚂𝙴𝚃𝚂},\mathrm{𝚜𝚎𝚝}\right)
|\mathrm{𝚂𝙴𝚃𝚂}|
sets, each containing the same number of domain variables, the
𝚔_\mathrm{𝚜𝚊𝚖𝚎}
constraint forces that the multisets of values assigned to each set are all identical.
\left(\begin{array}{c}〈\begin{array}{c}\mathrm{𝚜𝚎𝚝}-〈1,9,1,5,2,1〉,\hfill \\ \mathrm{𝚜𝚎𝚝}-〈9,1,1,1,2,5〉,\hfill \\ \mathrm{𝚜𝚎𝚝}-〈5,2,1,1,9,1〉\hfill \end{array}〉\hfill \end{array}\right)
𝚔_\mathrm{𝚜𝚊𝚖𝚎}
The first and second collections of variables are assigned to the same multiset.
The second and third collections of variables are also assigned to the same multiset.
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>1
\mathrm{𝚂𝙴𝚃𝚂}
\mathrm{𝚂𝙴𝚃𝚂}.\mathrm{𝚜𝚎𝚝}
\mathrm{𝚂𝙴𝚃𝚂}.\mathrm{𝚜𝚎𝚝}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚂𝙴𝚃𝚂}.\mathrm{𝚜𝚎𝚝}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚂𝙴𝚃𝚂}
It was shown in [ElbassioniKatrielKutzMahajan05] that, finding out whether the
𝚔_\mathrm{𝚜𝚊𝚖𝚎}
constraint has a solution or not is NP-hard when we have more than one
\mathrm{𝚜𝚊𝚖𝚎}
constraint. This was achieved by reduction from 3-dimensional-matching in the context where we have 2
\mathrm{𝚜𝚊𝚖𝚎}
𝚔_\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}
𝚔_\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚖𝚘𝚍𝚞𝚕𝚘}
𝚔_\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗}
𝚔_\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}
\mathrm{𝚜𝚊𝚖𝚎}
\mathrm{𝚜𝚊𝚖𝚎}
\mathrm{𝚂𝙴𝚃𝚂}
\mathrm{𝑃𝐴𝑇𝐻}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚜𝚎𝚝}\mathtt{1},\mathrm{𝚜𝚎𝚝}\mathtt{2}\right)
\mathrm{𝚜𝚊𝚖𝚎}
\left(\mathrm{𝚜𝚎𝚝}\mathtt{1}.\mathrm{𝚜𝚎𝚝},\mathrm{𝚜𝚎𝚝}\mathtt{2}.\mathrm{𝚜𝚎𝚝}\right)
\mathrm{𝐍𝐀𝐑𝐂}
=|\mathrm{𝚂𝙴𝚃𝚂}|-1
Parts (A) and (B) of Figure 5.207.1 respectively show the initial and final graph associated with the Example slot. To each vertex corresponds a collection of variables, while to each arc corresponds a
\mathrm{𝚜𝚊𝚖𝚎}
𝚔_\mathrm{𝚜𝚊𝚖𝚎}
|
NDF_TUNE
The routine sets a new value for an NDF_ system internal tuning parameter.
CALL NDF_TUNE( VALUE, TPAR, STATUS )
VALUE = INTEGER (Given)
New value for the tuning parameter.
\ast
\ast
Name of the parameter to be set (case insensitive). This name may be abbreviated, to no less than 3 characters.
The following tuning parameters are currently available:
’AUTO_HISTORY’; Controls whether to include an empty History component in NDFs created using NDF_NEW or NDF_CREAT. If the tuning parameter is zet to zero (the default), no History component will be included in the new NDFs. If the tuning parameter is zet non-zero, a History component will be added automatically to the new NDFs.
’DOCVT’: Controls whether to convert foreign format data files to and from native NDF format for access (using the facilities described in SSN/20). If DOCVT is set to 1 (the default), and the other necessary steps described in SSN/20 have been taken, then such conversions will be performed whenever they are necessary to gain access to data stored in a foreign format. If DOCVT is set to 0, no such conversions will be attempted and all data will be accessed in native NDF format only. The value of DOCVT may be changed at any time. It is the value current when a dataset is first accessed by the NDF_ library which is significant.
’KEEP’: Controls whether to retain a native format NDF copy of any foreign format data files which are accessed by the NDF_ library (and automatically converted using the facilities described in SSN/20). If KEEP is set to 0 (the default), then the results of converting foreign format data files will be stored in scratch filespace and deleted when no longer required. If KEEP is set to 1, the results of the conversion will instead be stored in permanent NDF data files in the default directory (such files will have the same name as the foreign file from which they are derived and a file type of ’.sdf’). Setting KEEP to 1 may be useful if the same datasets are to be re-used, as it avoids having to convert them on each occasion. The value of KEEP may be changed at any time. It is the value current when a foreign format file is first accessed by the NDF_ library which is significant.
’ROUND’: Specifies the way in which floating-point values should be converted to integer during automatic type conversion. If ROUND is set to 1, then floating-point values are rounded to the nearest integer value. If ROUND is set to 0 (the default), floating-point values are truncated towards zero.
’SHCVT’: Controls whether diagnostic information is displayed to show the actions being taken to convert to and from foreign data formats (using the facilities described in SSN/20). If SHCVT is set to 1, then this information is displayed to assist in debugging external format conversion software whenever a foreign format file is accessed. If SHCVT is set to 0 (the default), this information does not appear and format conversion proceeds silently unless an error occurs.
’TRACE’: Controls the reporting of additional error messages which may occasionally be useful for diagnosing internal problems within the NDF_ library. If TRACE is set to 1, then any error occurring within the NDF_ system will be accompanied by error messages indicating which internal routines have exited prematurely as a result. If TRACE is set to 0 (the default), this internal diagnostic information will not appear and only standard error messages will be produced.
’WARN’: Controls the issuing of warning messages when certain non-fatal errors in the structure of NDF data objects are detected. If WARN is set to 1 (the default), then a warning message is issued. If WARN is set to 0, then no message is issued. In both cases normal execution continues and no STATUS value is set.
’PXT...’: Controls whether a named NDF extension should be propagated by default when NDF_PROP or NDF_SCOPY is called. The name of the extension should be appended to the string "PXT" to form the complete tuning parameter name. Thus the tuning parameter PXTFITS would control whether the FITS extension is propagated by default. If the value for the parameter is non-zero, then the extension will be propagated by default. If the value for the parameter is zero, then the extension will not be propagated by default. The default established by this tuning parameter can be over-ridden by specifying the extension explicitly within the CLIST argument when calling NDF_PROP or NDF_SCOPY. The default value for all "PXT..." tuning parameters is 1, meaning that all extensions are propagated by default.
|
Global Constraint Catalog: Ccutset
<< 5.101. cumulatives5.103. cycle >>
[FagesLal03]
\mathrm{𝚌𝚞𝚝𝚜𝚎𝚝}\left(\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝚄𝚃𝚂𝙴𝚃},\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝚄𝚃𝚂𝙴𝚃}
\mathrm{𝚍𝚟𝚊𝚛}
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚜𝚞𝚌𝚌}-\mathrm{𝚜𝚒𝚗𝚝},\mathrm{𝚋𝚘𝚘𝚕}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝚄𝚃𝚂𝙴𝚃}\ge 0
\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝚄𝚃𝚂𝙴𝚃}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝙽𝙾𝙳𝙴𝚂},\left[\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚜𝚞𝚌𝚌},\mathrm{𝚋𝚘𝚘𝚕}\right]\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\ge 1
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝙽𝙾𝙳𝙴𝚂},\mathrm{𝚒𝚗𝚍𝚎𝚡}\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚋𝚘𝚘𝚕}\ge 0
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚋𝚘𝚘𝚕}\le 1
G
n
\mathrm{𝙽𝙾𝙳𝙴𝚂}
collection. Enforces that the subset of kept vertices of cardinality
n-\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝚄𝚃𝚂𝙴𝚃}
and their corresponding arcs form a graph without circuit.
\left(\begin{array}{c}1,〈\begin{array}{ccc}\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{2,3,4\right\}\hfill & \mathrm{𝚋𝚘𝚘𝚕}-1,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-2\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{3\right\}\hfill & \mathrm{𝚋𝚘𝚘𝚕}-1,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{4\right\}\hfill & \mathrm{𝚋𝚘𝚘𝚕}-1,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-4\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{1\right\}\hfill & \mathrm{𝚋𝚘𝚘𝚕}-0\hfill \end{array}〉\hfill \end{array}\right)
\mathrm{𝚌𝚞𝚝𝚜𝚎𝚝}
constraint holds since the vertices of the
\mathrm{𝙽𝙾𝙳𝙴𝚂}
collection for which the
\mathrm{𝚋𝚘𝚘𝚕}
attribute is set to 1 correspond to a graph without circuit and since exactly one (
\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝚄𝚃𝚂𝙴𝚃}=1
) vertex has its
\mathrm{𝚋𝚘𝚘𝚕}
attribute set to 0.
\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝚄𝚃𝚂𝙴𝚃}>0
\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝚄𝚃𝚂𝙴𝚃}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|>1
\mathrm{𝙽𝙾𝙳𝙴𝚂}
The article [FagesLal03] introducing the
\mathrm{𝚌𝚞𝚝𝚜𝚎𝚝}
constraint mentions applications from various areas such that deadlock breaking or program verification.
The undirected version of the
\mathrm{𝚌𝚞𝚝𝚜𝚎𝚝}
constraint corresponds to the minimum feedback vertex set problem.
The filtering algorithm presented in [FagesLal03] uses graph reduction techniques inspired from Levy and Low [LevyLow88] as well as from Lloyd, Soffa and Wang [LloydSoffaWang88].
application area: deadlock breaking, program verification.
final graph structure: circuit, directed acyclic graph, acyclic, no loop.
problems: minimum feedback vertex set.
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1},\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}\right)
•
\mathrm{𝚒𝚗}_\mathrm{𝚜𝚎𝚝}
\left(\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚜𝚞𝚌𝚌}\right)
•\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚋𝚘𝚘𝚕}=1
•\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}.\mathrm{𝚋𝚘𝚘𝚕}=1
•
\mathrm{𝐌𝐀𝐗}_\mathrm{𝐍𝐒𝐂𝐂}
\le 1
•
\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}
=|\mathrm{𝙽𝙾𝙳𝙴𝚂}|-\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝚄𝚃𝚂𝙴𝚃}
•
\mathrm{𝙰𝙲𝚈𝙲𝙻𝙸𝙲}
•
\mathrm{𝙽𝙾}_\mathrm{𝙻𝙾𝙾𝙿}
We use a set of integers for representing the successors of each vertex. Because of the arc constraint, all arcs such that the
\mathrm{𝚋𝚘𝚘𝚕}
attribute of one extremity is equal to 0 are eliminated; Therefore all vertices for which the
\mathrm{𝚋𝚘𝚘𝚕}
attribute is equal to 0 are also eliminated (since they will correspond to isolated vertices). The graph property
\mathrm{𝐌𝐀𝐗}_\mathrm{𝐍𝐒𝐂𝐂}
\le
1 enforces the size of the largest strongly connected component to not exceed 1; Therefore, the final graph cannot contain any circuit.
Part (A) of Figure 5.102.1 shows the initial graph from which we have chosen to start. It is derived from the set associated with each vertex. Each set describes the potential values of the
\mathrm{𝚜𝚞𝚌𝚌}
\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}
graph property, the vertices of the final graph are stressed in bold. The
\mathrm{𝚌𝚞𝚝𝚜𝚎𝚝}
constraint holds since the final graph does not contain any circuit and since the number of removed vertices
\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝚄𝚃𝚂𝙴𝚃}
\mathrm{𝚌𝚞𝚝𝚜𝚎𝚝}
|
Gammatone filter bank - Simulink - MathWorks í•œêµ
Bands as separate output port
The Gammatone Filter Bank block decomposes a signal by passing it through a bank of gammatone filters equally spaced on the equivalent rectangular bandwidth (ERB) scale. Gammatone filter banks are designed to model the human auditory system.
Port_1 — Audio input to filter bank
Audio input to the filter bank, specified as a scalar, vector, or matrix. If you specify the input as a matrix, the block treats the columns as independent audio channels. If you specify the input as a vector, the block treats the input as containing a single channel.
Port_1 — Audio output from filter bank
Audio output from the filter bank, returned as a scalar, vector, matrix, or 3-D array. The shape of output signal depends on the shape of input signal and Number of filters. If input is an M-by-N matrix, then output is an M-by-Number of filters-by-N array. If N is 1, then output is a matrix.
Frequency range (Hz) — Frequency range of filter bank
[50 8000] (default) | two-element row vector of monotonically increasing values
Frequency range of the filter bank, specified as a two-element row vector of monotonically increasing values in Hz.
Number of filters — Number of filters
Number of filters in the filter bank, specified as a positive integer.
Select this parameter to specify the sample rate from the input port.
Input sample rate, specified as a positive integer in Hz.
Bands as separate output port — Separate ports for each filter output
Select this parameter to separate ports for each filter output.
View Filter Response — Visualize filter bank responses
This button uses the fvtool function to visualize gammatone filter bank responses.
Name of the variable in the base workspace to contain the filter bank when it is exported. The name must be a valid MATLAB® variable name.
Use the Gammatone Filter Bank block to decompose a signal by passing it through a bank of gammatone filters.
A gammatone filter bank is often used as the front end of a cochlea simulation. A cochlea simulation transforms complex sounds into a multichannel activity pattern like the one observed in the auditory nerve [2] .The Gammatone Filter Bank block follows the algorithm described in [1]. The algorithm is an implementation of an idea proposed in [2]. The design of the gammatone filter bank can be described in two parts: the filter shape (gammatone) and the frequency scale. The equivalent rectangular bandwidth (ERB) scale defines the relative spacing and bandwidth of the gammatone filters. The derivation of the ERB scale also provides an estimate of the auditory filter response that closely resembles the gammatone filter.
The block determines the ERB scale using the notched-noise masking method. This method involves a listening test wherein notched noise is centered on a tone. The power of the tone is tuned, and the audible threshold (the power required for the tone to be heard) is recorded. The experiment is repeated for different notch widths and center frequencies.
The notched-noise method assumes that the audible threshold corresponds to a constant signal-to-masker ratio at the output of the theoretical auditory filter. That is, the ratio of the power of the fc tone and the shaded area is constant. Therefore, the relationship between the audible threshold and 2Δf (the notch bandwidth) is linearly related to the relationship between the noise passed through the filter and 2Δf.
The derivative of the function relating Δf to the noise passed through the filter estimates the shape of the auditory filter. Because Δf has an inverse relationship with the noise power passed through the filter, the derivative of the function must be multiplied by –1. The resulting shape of the auditory filter is usually approximated as a roex filter.
The equivalent rectangular bandwidth of the auditory filter is defined as the width of a rectangular filter required to pass the same noise power as the auditory filter.
[4] defines ERB as a function of center frequency for young listeners with normal hearing and a moderate noise level:
\text{ERB}=24.7\left(0.00437{f}_{c}+1\right)
The ERB scale (ERBs) is an extension of the relationship between ERB and the center frequency, derived by integrating the reciprocal of the ERB function:
\text{ERBs}=21.4{\mathrm{log}}_{10}\left(0.00437f+1\right)
To design a gammatone filter bank, [2] suggests distributing the center frequencies of the filters in proportion to their bandwidth. To accomplish this, Gammatone Filter Bank block defines the center frequencies as linearly spaced on the ERB scale, covering the specified frequency range with the desired number of filters. You can specify the frequency range and desired number of filters using the Frequency range (Hz) and Number of filters parameters.
The gammatone filter was introduced in [3]. The continuous impulse response is:
g\left(t\right)=a{t}^{nâ1}{e}^{â2\mathrm{Ï}bt}\mathrm{cos}\left(2\mathrm{Ï}{f}_{\text{c}}t+\mathrm{Ï}\right)
a –– amplitude factor
n –– filter order (set to four to model human hearing)
fc–– center frequency
b –– bandwidth, set to 1.019*hz2erb(fc).
Ï• –– phase factor
The gammatone filter is similar to the roex filter derived from the notched-noise experiment. The Gammatone Filter Bank block implements the digital filter as a cascade of four second-order sections, as described in [1].
[2] Patterson, R.D., K. Robinson, J. Holdsworth, D. McKeown, C. Zhang, and M. Allerhand. "Complex Sounds and Auditory Images." Auditory Physiology and Perception. 1992, pp. 429–446.
[3] Aertsen, A. M. H. J., and P. I. M. Johannesma. "Spectro-Temporal Receptive Fields of Auditory Neurons in the Grassfrog." Biological Cybernetics. Vol. 38, Issue 4, 1980, pp. 223–234.
[4] Glasberg, Brian R., and Brian C. J. Moore. "Derivation of Auditory Filter Shapes from Notched-Noise Data." Hearing Research. Vol. 47. Issue 1-2, 1990, pp. 103–138.
gammatoneFilterBank | Multiband Parametric EQ | Octave Filter Bank
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Global Existence of Solution to Initial Boundary Value Problem for Bipolar Navier-Stokes-Poisson System
Jian Liu, Haidong Liu, "Global Existence of Solution to Initial Boundary Value Problem for Bipolar Navier-Stokes-Poisson System", Abstract and Applied Analysis, vol. 2014, Article ID 214546, 8 pages, 2014. https://doi.org/10.1155/2014/214546
Jian Liu 1 and Haidong Liu2
Academic Editor: Xiaohong Qin
This paper concerns initial boundary value problem for 3-dimensional compressible bipolar Navier-Stokes-Poisson equations with density-dependent viscosities. When the initial data is large, discontinuous, and spherically symmetric, we prove the global existence of the weak solution.
Bipolar Navier-Stokes-Poisson (BNSP) has been used to simulate the transport of charged particles under the influence of electrostatic force governed by the self-consistent Poisson equation. In this paper, we consider the initial boundary value problem (IBVP) for 3-dimensional isentropic compressible BNSP with density-dependent viscosities: where the unknown functions are the charges densities , , the velocities , , the pressure functions , , and the electrostatic potential . In (1), the strain tensors and are defined by , , and the Lamé viscosity coefficients satisfying , , , .
There have been extensive studies on the global existence and asymptotic behavior of weak solution to the unipolar Navier-Stokes-Poisson system (NSP). The global existence of weak solution to NSP with general initial data was proved in [1, 2]. The quasineutral and some related asymptotic limits were studied in [3–5]. In the case when the Poisson equation describes the self-gravitational force for stellar gases, the global existence of weak solution and asymptotic behavior were also investigated together with the stability analysis; refer to [6, 7] and the references therein. In addition, Hao and Li [8] proved the global well-posedness of NSP in the Besov space. Li et al. in [9] proved the global existence and the optimal time convergence rates of the classical solution.
For bipolar Navier-Stokes-Poisson system, there are also abundant results on the existence and asymptotic behavior of the global solution. Li et al. [10] proved optimal time convergence rate for the global classical solution for a small initial perturbation of the constant equilibrium state. The optimal time decay rate of global strong solution was established in [11, 12]. Liu and Lian in [13] proved global existence of weak solution to free boundary value problem. Liu et al. [14] established global existence and asymptotic behavior of weak solution to initial boundary value problem in one-dimensional case. Lin et al. [15] studied the global existence and uniqueness of the strong solution in hybrid Besov spaces with the initial data close to an equilibrium state. As a continuation of the study in this direction, in this paper, we will deal with the initial boundary value problem for BNSP.
The rest of this paper is as follows. In Section 2, we state the main results of this paper. In Section 3, we give the entropy estimates and the pointwise bounds of the density of the smooth approximate solution. In Section 4, we prove the global existence of weak solution.
For the sake of simplicity, the viscosity terms are assumed to satisfy , , , and , and the strain tensors are given by , . Then (1) is reduced to for with and being the unit ball in .
The boundary condition is taken as where is outward pointing unit normal vector of .
Definition 1. is said to be a weak solution to the initial boundary value problem (2)–(4) on , provided that and the equations are satisfied in the sense of distributions. Namely, it holds for any and that and for satisfying on and that
Before stating the main result, we make the following assumptions on the initial data (4): where is small enough. It follows from (9) that
Then, we have the following results for global weak solution.
Theorem 2. Let . If the initial data satisfies (9), then the initial boundary value problem (2)–(4) has a global spherically symmetric weak solution which satisfies, for all , Moreover, where is a constant.
3. Approximate Solutions and Their Estimates
The key point of the proof of Theorem 2 is to construct smooth approximate solution satisfying the a priori estimates required in the stability analysis. The crucial issue is to obtain lower and upper bounds of the density. To this end, we study the following approximate system of (2): where is a constant.
Set , , , , and , and rewrite (15) in the form for . We will first construct the smooth solution of (16) in the truncated region with the initial condition and the boundary condition
For the approximate solution which will have lower bound of the density (see Lemma 8), the boundary condition of (18) is equivalent to , .
We assume that the initial data is smooth and satisfies (9) with constants independent of .
In the following, we will state the energy and entropy estimates for approximate solution . First, making use of similar arguments as in [16] with modifications, we can establish the following Lemma 3, of which we omit the details.
Lemma 3. Let be smooth solution of (16) defined on with boundary conditions (18) such that , . Then there exists a constant , independent of , such that
Lemma 4. Given , there is an absolute constant , which is independent of , such that for and .
Proof. Define characteristic line: . Then, along the particle path, (16)1 can be solved to obtain which implies that provided that .
It follows from (20) and (21) that for some absolute constant independent of .
Then, it follows from (19) and (24) that for and .
Similarly, we also have The proof of the lemma is finished.
To derive a priori estimates about the velocity of the approximate solution, the crucial step is to obtain lower bounds of the density. For this purpose and for simplicity, we solve the IBVP (16) in Lagrangian coordinates. Since the process is the same, we just deal with (16)1-(16)2.
Let be fixed and define Without loss of generality, we set . Then, and (16)1-(16)2 becomes for and .
The corresponding initial data is and the boundary condition is
For this system, the following a priori estimates hold.
Lemma 5. For all , it holds that
Proof. Multiplying (29)2 by , using (29)1 and integration by parts, we can get Since then from (36), we get Thus (32) holds.
Next, (33) follows from Lemma 4 and (34) holds trivially.
Now, we prove (35). In fact, multiplying (29)2 by , using (29)1 and integration by parts, we get Thus Using Hölder inequality, Young’s inequality, and Lemma 4, we estimate the right hand side of (40) as follows: Putting the above estimates into (40) and using (32), one gets This proves (35).
Remark 6. Consider
Lemma 7. There is a positive constant such that
Proof. We rewrite (29)1 in the form Then substituting (45) into (29)2, one gets that is, Since , the above equation can be rewritten as Integrating it over , one gets Multiplying (49) by and integrating over with respect to , one gets in which Using Lemma 5 and Young’s inequality, we deduce from (50) that there is a positive constant , depending on , , , and , such that that is, Applying Gronwall’s inequality to (53), we have This proves (44).
Now we can obtain the lower bound of the density .
Proof. Set and . Equation (29)1 can be written as , which implies that . Then it follows from Sobolev’s embedding that, for any , Choosing small enough, which may depend on and , we obtain where . The proof of the lemma is completed.
Proof. With the estimates obtained in Section 3, we can apply the method in [16] and references therein with modifications to prove the existence of weak solution to the IBVP (2). The details are omitted.
The authors are grateful to Professor Hai-Liang Li for his helpful discussions and suggestions about the problem. The research of Jian Liu is supported by NNSFC no. 11326140 and the Doctoral Starting up Foundation of Quzhou University no. BSYJ201314 and no. XNZQN201313.
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Copyright © 2014 Jian Liu and Haidong Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Guanwei Chen, "Quasilinear Elliptic Equations with Hardy-Sobolev Critical Exponents: Existence and Multiplicity of Nontrivial Solutions", Journal of Applied Mathematics, vol. 2014, Article ID 482740, 6 pages, 2014. https://doi.org/10.1155/2014/482740
Guanwei Chen1
We study the existence of positive solutions and multiplicity of nontrivial solutions for a class of quasilinear elliptic equations by using variational methods. Our obtained results extend some existing ones.
Let us consider the following problem: where denotes the -Laplacian differential operator, is an open bounded domain in with smooth boundary and , , is the Hardy-Sobolev critical exponent, and is the Sobolev critical exponent. Here, we let which is equivalent to the usual norm of Sobolev space due to the Poincaré inequality. Let which is the best Hardy-Sobolev constant.
In the case where and hold, then (1) reduces to the quasilinear elliptic problem: Gonçalves and Alves [1] have studied (4) in involving , and to obtain existence of positive solutions where , , or and a suitable . We should mention that problem (4) with has been widely studied since Brézis and Nirenberg; see [2–4] and the references therein.
Ghoussoub and Yuan [5] have studied (1) with , where . They obtained a positive solution in the case where , , and (in particular if ) hold. They also obtained a sign-changing solution in the case where , , and (in particular if ) hold. For other relevant papers, see [6–12] and the references therein.
We should mention that the energy functional associated with (1) is defined on , which is not a Hilbert space for . Due to the lack of compactness of the embedding in and , we cannot use the standard variational argument directly. The corresponding energy functional fails to satisfy the classical Palais-Smale ( for short) condition in . However, a local () condition can be established in a suitable range. Then the existence result is obtained via constructing a minimax level within this range and the Mountain Pass Lemma due to Ambrosetti et al. [2] and Rabinowitz [13].
In this paper, we study (1) with a general nonlinearity by using a variational method; besides, we also considerably generalize the results obtained in [5]. In what follows, we always assume that the nonlinearity satisfies . Let , . To state our main results, we still need the following assumptions., , and uniformly for .There exists a constant with such that , , and uniformly for .There exists a constant with such that
Now, our main results read as follows.
Theorem 1. Suppose that , , , and hold. If , , and hold, then (1) has at least one positive solution.
Theorem 2. Suppose that , , , and hold. If , , and (7) hold, then (1) has at least two distinct nontrivial solutions.
Noting that and imply that (7) holds, therefore, we have the following corollaries.
Corollary 3. Suppose that , , , and . Moreover, and hold; then (1) has at least one positive solution.
Corollary 4. Suppose that , , , and . Moreover, and hold; then (1) has at least two distinct nontrivial solutions.
Remark 5. Theorem 1 generalizes Theorem 1.3 in [5], where the author only studied the special situation that , . There are functions satisfying the assumptions of our Theorem 1 and not satisfying those in [5]. Let where , , , and . Obviously, satisfies all the conditions of Theorem 1 in this paper, while it does not satisfy the conditions of Theorem 1.3 in [5].
The rest of this paper is organized as follows. In Section 2, we give some preliminary lemmas, which are useful in the proofs of our main results. In Section 3, we give the detailed proofs of our main results.
In what follows, we let denote the norm in . It is obvious that the values of for are irrelevant in Theorem 1, so we may define We firstly consider the existence of nontrivial solutions to the problem: The energy functional corresponding to (10) is given by By Hardy-Sobolev inequalities (see [5, 14]) and , we know . Now it is well known that there exists a one-to-one correspondence between the weak solutions of (10) and the critical points of on . More precisely we say that is a weak solution of (10), if, for any , there holds
Lemma 6 (see [15]). If a.e. in and for all and some , then
Lemma 7. For any , , and , we have .
Proof. Let Clearly, , , so .
Lemma 8 (see [5]). If hold, then we have(i) is independent of , and will henceforth be denoted by ;(ii) is attained when by the functions for some . Moreover, the functions are the only positive radial solutions of in , and satisfy
Lemma 9. If , , and hold, then satisfies condition.
Proof. Suppose that is a sequence in . By , we have where . Hence we conclude that is a bounded sequence in . So there exists ; going if necessary to a subsequence, we have By the continuity of embedding, we have . From [5], going if necessary to a subsequence, one can get that as . By , we know that for any there exists such that Set . When , , we get It follows from Vitali’s theorem that Similarly, we can also get Since , we have Let , which together with Lemma 6 implies From (20), we can obtain Note that as , which together with Lemma 6 implies Therefore, one gets that From (26) and (27), we have then as . Otherwise, there exists a subsequence (still denoted by ) such that By (3), we have then . That is, . It follows from (29) and that . However, we have by (27) and (). We get a contradiction. Therefore, we can obtain From the discussion above, satisfies condition.
In the following, we shall give some estimates for the extremal functions. Let Define a function such that for , for , , where . Set so that . Then, by using the argument as used in [5], we can get the following results: Moreover, by using the Sobolev embedding theorem and (36), one can deduce
Lemma 10. Suppose that . If , and (7) hold, then there exists , , such that
Proof. We consider the functions Since , , and for small enough, is attained for some . Therefore, we have and hence Therefore, we obtain By (), we can easily get Hence, we can get By (36)–(38), when is small enough, we conclude that On the one hand, from Lemma 7 and (36), it follows that On the other hand, the function attains its maximum at and is increasing in the interval . Note that () implies , which together with (36), (46), and (47) implies that Furthermore, from (7) and (37), we get By (7), we have , which implies Therefore, by choosing small enough, we have Hence, the proof of the lemma is completed by taking .
Proof of Theorem 1. Let . From the Sobolev and Hardy-Sobolev inequalities, we can easily get It follows from () that uniformly for all . Therefore, we deduce that for all and for . Then one gets for all and for . By (52) and (55) we have for small enough. So there exists such that By Lemma 10, there exists with such that It follows from the nonnegativity of that Therefore, , so we can choose such that By virtue of the Mountain Pass Lemma in [16], there is a sequence satisfying where Note that By Lemma 9 we can assume that in . From the continuity of , we know that is a weak solution of problem (10). Then , where . Thus . Therefore, is a nonnegative solution of (1). By the Strong Maximum Principle [17], is a positive solution of problem (1). Therefore, Theorem 1 holds.
Proof of Theorem 2. By Theorem 1, we know that (1) has a positive solution . Set It follows from Theorem 1 that the equation has at least one positive solution . Let ; then is a solution of It is obvious that , , and . So (1) has at least two nontrivial solutions. Therefore, Theorem 2 holds.
The author thanks the referees and the editors for their helpful comments and suggestions. Research is supported by the Tianyuan Fund for Mathematics of NSFC (Grant no. 11326113) and the Key Project of Natural Science Foundation of Educational Committee of Henan Province of China (Grant no. 13A110015).
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Copyright © 2014 Guanwei Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Global Constraint Catalog: Ck_used_by
<< 5.210. k_same_partition5.212. k_used_by_interval >>
\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}
𝚔_\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}\left(\mathrm{𝚂𝙴𝚃𝚂}\right)
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚂𝙴𝚃𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚜𝚎𝚝}-\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|\ge 1
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚂𝙴𝚃𝚂},\mathrm{𝚜𝚎𝚝}\right)
|\mathrm{𝚂𝙴𝚃𝚂}|>1
\mathrm{𝚗𝚘𝚗}_\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚜𝚒𝚣𝚎}
\left(\mathrm{𝚂𝙴𝚃𝚂},\mathrm{𝚜𝚎𝚝}\right)
|\mathrm{𝚂𝙴𝚃𝚂}|
sets of domain variables, the
𝚔_\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}
constraint forces a
\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}
constraint between each pair of consecutive sets.
\left(\begin{array}{c}〈\begin{array}{c}\mathrm{𝚜𝚎𝚝}-〈1,9,1,5,2,1〉,\hfill \\ \mathrm{𝚜𝚎𝚝}-〈9,1,1,1,2,5〉,\hfill \\ \mathrm{𝚜𝚎𝚝}-〈1,1,2,5〉\hfill \end{array}〉\hfill \end{array}\right)
𝚔_\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}
The multiset of values
\left\{\left\{1,1,1,2,5,9\right\}\right\}
associated with the second collection of variables is included into the multiset
\left\{\left\{1,1,1,2,5,9\right\}\right\}
associated with the first collection of variables.
\left\{\left\{1,1,2,5\right\}\right\}
associated with the third collection of variables is included into the multiset
\left\{\left\{1,1,1,2,5,9\right\}\right\}
associated with the second collection of variables.
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>1
\mathrm{𝚂𝙴𝚃𝚂}
\mathrm{𝚂𝙴𝚃𝚂}.\mathrm{𝚜𝚎𝚝}
\mathrm{𝚂𝙴𝚃𝚂}.\mathrm{𝚜𝚎𝚝}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚂𝙴𝚃𝚂}.\mathrm{𝚜𝚎𝚝}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚂𝙴𝚃𝚂}
Similarly to the
𝚔_\mathrm{𝚜𝚊𝚖𝚎}
constraint [ElbassioniKatrielKutzMahajan05], finding out whether the
𝚔_\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}
\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}
𝚔_\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}
𝚔_\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}_\mathrm{𝚖𝚘𝚍𝚞𝚕𝚘}
𝚔_\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗}
𝚔_\mathrm{𝚜𝚊𝚖𝚎}
\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}
\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}
\mathrm{𝚂𝙴𝚃𝚂}
\mathrm{𝑃𝐴𝑇𝐻}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚜𝚎𝚝}\mathtt{1},\mathrm{𝚜𝚎𝚝}\mathtt{2}\right)
\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}
\left(\mathrm{𝚜𝚎𝚝}\mathtt{1}.\mathrm{𝚜𝚎𝚝},\mathrm{𝚜𝚎𝚝}\mathtt{2}.\mathrm{𝚜𝚎𝚝}\right)
\mathrm{𝐍𝐀𝐑𝐂}
=|\mathrm{𝚂𝙴𝚃𝚂}|-1
\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}
𝚔_\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}
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Base-2 logarithm of symbolic input - MATLAB log2 - MathWorks Benelux
Base-2 Logarithm of Numeric and Symbolic Input
Find Mantissa and Exponent of Base-2 Logarithm
Base-2 logarithm of symbolic input
Y = log2(X) returns the logarithm to the base 2 of X such that 2Y = X. If X is an array, then log2 acts element-wise on X.
[F,E] = log2(X) returns arrays of mantissas and exponents, F and E, such that
X=F\cdot {2}^{E}
. The values returned in F are in the range 0.5 <= abs(F) < 1. Any zeros in X return F = 0 and E = 0.
Compute the base-2 logarithm of a numeric input.
y = log2(4^(1/3))
Compute the base-2 logarithm of a symbolic input. The result is in terms of the natural logarithm log function.
ySym = log2(x^(1/3))
ySym =
\frac{\mathrm{log}\left({x}^{1/3}\right)}{\mathrm{log}\left(2\right)}
Substitute the symbolic variable x with a number by using subs. Simplify the result by using simplify.
yVal = subs(ySym,x,4)
yVal =
\frac{\mathrm{log}\left({4}^{1/3}\right)}{\mathrm{log}\left(2\right)}
simplify(yVal)
\frac{2}{3}
Find the mantissa and exponent of a base-2 logarithm of an input
X
. The mantissa
F
E
X=F\cdot {2}^{E}
Create a symbolic variable a and assume that it is real. Create a symbolic vector X that contains symbolic numbers and expressions. Find the exponent and mantissa for each element of X.
syms a real;
X = [1 0.5*2^a 5/7]
\left(\begin{array}{ccc}1& \frac{{2}^{a}}{2}& \frac{5}{7}\end{array}\right)
\left(\begin{array}{ccc}\frac{1}{2}& \frac{\frac{1}{{2}^{⌊\frac{\mathrm{log}\left(\frac{{2}^{a}}{2}\right)}{\mathrm{log}\left(2\right)}⌋+1}} {2}^{a}}{2}& \frac{5}{7}\end{array}\right)
\left(\begin{array}{ccc}1& ⌊\frac{\mathrm{log}\left(\frac{{2}^{a}}{2}\right)}{\mathrm{log}\left(2\right)}⌋+1& 0\end{array}\right)
The values returned in F have magnitudes in the range 0.5 <= abs(F) < 1.
Simplify the results using simplify.
\left(\begin{array}{ccc}\frac{1}{2}& {2}^{a-⌊a⌋-1}& \frac{5}{7}\end{array}\right)
\left(\begin{array}{ccc}1& ⌊a⌋& 0\end{array}\right)
symbolic number | symbolic array | symbolic variable | symbolic function | symbolic expression
Input array, specified as a symbolic number, array, variable, function, or expression.
When computing the base-2 logarithms of complex elements in X, log2 ignores their imaginary parts.
For the syntax [F,E] = log2(X), any zeros in X produce F = 0 and E = 0. Input values of Inf, -Inf, or NaN are returned unchanged in F with a corresponding exponent of E = 0.
Y — Base-2 logarithm values
Base-2 logarithm values, returned as a symbolic number, vector, matrix, or array of the same size as X.
Mantissa values, returned as a symbolic scalar, vector, matrix, or array of the same size as X. The values in F and E satisfy X = F.*2.^E.
Exponent values, returned as a symbolic scalar, vector, matrix, or array of the same size as X. The values in F and E satisfy X = F.*2.^E.
For floating-point input, the syntax [F,E] = log2(X) corresponds to the ANSI® C function frexp() and the IEEE® standard function logb(). Any zeros in X produce F = 0 and E = 0.
log | log10 | power
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Hybrid MoM-PO Method for Metal Antennas with Large Scatterers - MATLAB & Simulink
{\stackrel{\to }{f}}_{n}\left(\stackrel{\to }{r}\right)=\left\{\begin{array}{l}\frac{{l}_{n}}{2{A}_{n}^{+}}{\stackrel{\to }{\rho }}_{n}^{+}\text{ }\stackrel{\to }{r}\text{ in }t{r}_{n}^{+}\\ \frac{{l}_{n}}{2{A}_{n}^{-}}{\stackrel{\to }{\rho }}_{n}^{-}\text{ }\stackrel{\to }{r}\text{ in }t{r}_{n}^{-}\end{array}\right\}\text{ (1)}
{\stackrel{\to }{\rho }}_{n}^{+}=\stackrel{\to }{r}-{\stackrel{\to }{r}}_{n}^{-}
\stackrel{\to }{r}
{\stackrel{\to }{\rho }}_{n}^{-}={\stackrel{\to }{r}}_{n}^{-}-\stackrel{\to }{r}
{\stackrel{\to }{n}}_{n}^{±}
{\stackrel{\to }{t}}_{n}^{±}
{\stackrel{\to }{t}}_{n}^{+}
{\stackrel{\to }{r}}_{n}
{\stackrel{\to }{t}}_{n}^{±}
{\stackrel{\to }{n}}_{n}^{±}
{\stackrel{\to }{l}}_{n}^{±}
{\stackrel{\to }{l}}_{n}^{±}={\stackrel{\to }{t}}_{n}^{±}×{\stackrel{\to }{n}}_{n}^{±}\text{ (2)}
{\stackrel{\to }{l}}_{n}^{±}={\stackrel{\to }{l}}_{n}^{-}={\stackrel{\to }{l}}_{n}\text{ (3)}
{\stackrel{\to }{l}}_{n}
\stackrel{\to }{J}\left(\stackrel{\to }{r}\right)
\left({N}_{PO}+{N}_{MoM}=N\right)
\stackrel{\to }{J}\left(\stackrel{\to }{r}\right)=\sum _{n=1}^{{N}_{MoM}}{I}_{n}^{MoM}{\stackrel{\to }{f}}_{n}\left(\stackrel{\to }{r}\right),\text{\hspace{0.17em}}\text{ }\stackrel{\to }{J}\left(\stackrel{\to }{r}\right)=\sum _{n=1}^{{N}_{PO}}{I}_{n}^{PO}{\stackrel{\to }{f}}_{n+{N}_{MoM}}\left(\stackrel{\to }{r}\right)\text{ }\text{ (4)}
\stackrel{^}{Z}
\stackrel{^}{Z}=\left(\begin{array}{cc}{\stackrel{^}{Z}}_{11}& {\stackrel{^}{Z}}_{12}\\ {\stackrel{^}{Z}}_{21}& {\stackrel{^}{Z}}_{22}\end{array}\right),\mathrm{dim}\left({\stackrel{^}{Z}}_{11}\right)={N}_{MoM}×{N}_{MoM},\mathrm{dim}\left({\stackrel{^}{Z}}_{12}\right)={N}_{MoM}×{N}_{PO}\text{ (5)}
\stackrel{\to }{V}
{\stackrel{^}{Z}}_{11}
{\stackrel{^}{Z}}_{12}
{\stackrel{^}{Z}}_{12}
{\stackrel{^}{Z}}_{22}
\stackrel{\to }{H}\left(\stackrel{\to }{r}\right)
{\stackrel{^}{Z}}_{PO}
{\stackrel{^}{Z}}_{22}
\stackrel{\to }{J}\left(\stackrel{\to }{r}\right)=2\delta \left(\stackrel{\to }{r}\right)\left[\stackrel{\to }{n}\left(\stackrel{\to }{r}\right)×\stackrel{\to }{H}\left(\stackrel{\to }{r}\right)\right]\text{ (6)}
where δ accounts for the shadowing effects. If the observation point lies in the shadow region, δ must be zero. Otherwise it equals ±1 depending on the direction of incidence with respect to the orientation normal vector
\stackrel{\to }{n}\left(\stackrel{\to }{r}\right)
\sum _{n=1}^{{N}_{PO}}{I}_{n}^{PO}{\stackrel{\to }{f}}_{n+{N}_{MoM}}\left(\stackrel{\to }{r}\right)=2\delta \left(\stackrel{\to }{r}\right)\left[\stackrel{\to }{n}\left(\stackrel{\to }{r}\right)×\stackrel{\to }{H}\left(\stackrel{\to }{r}\right)\right]\text{ (7)}
{\stackrel{\to }{r}}_{n+{N}_{MoM}}
{\stackrel{\to }{f}}_{n+{N}_{MoM}}\left(\stackrel{\to }{r}\right)
{\stackrel{\to }{t}}_{n+{N}_{MoM}}^{+}
{I}_{n}^{PO}=2\delta \left({\stackrel{\to }{r}}_{n+{N}_{MoM}}\right){\stackrel{\to }{t}}_{n+{N}_{MoM}}^{+}\cdot ⌊{\stackrel{\to }{n}}_{n+{N}_{MoM}}^{+}×\stackrel{\to }{H}\left({\stackrel{\to }{r}}_{n+{N}_{MoM}}\right)⌋\text{ }\text{ }\text{ }\text{ (8a)}
{I}_{n}^{PO}=2\delta \left({\stackrel{\to }{r}}_{n+{N}_{MoM}}\right){\stackrel{\to }{t}}_{n+{N}_{MoM}}^{-}\cdot ⌊{\stackrel{\to }{n}}_{n+{N}_{MoM}}^{-}×\stackrel{\to }{H}\left({\stackrel{\to }{r}}_{n+{N}_{MoM}}\right)⌋\text{ }\text{ }\text{ }\text{ (8b) }
{I}_{n}^{PO}=2\delta \left({\stackrel{\to }{r}}_{n+{N}_{MoM}}\right)\stackrel{\to }{H}\left({\stackrel{\to }{r}}_{n+{N}_{MoM}}\right)\cdot \left(⌊{\stackrel{\to }{t}}_{n+{N}_{MoM}}^{+}×{\stackrel{\to }{n}}_{n+{N}_{MoM}}^{+}⌋+⌊{\stackrel{\to }{t}}_{n+{N}_{MoM}}^{-}×{\stackrel{\to }{n}}_{n+{N}_{MoM}}^{-}⌋\right)/2\text{ }\text{ }\text{ (9)}
{I}_{n}^{PO}=2\delta \left({\stackrel{\to }{r}}_{n+{N}_{MoM}}\right)\stackrel{\to }{H}\left({\stackrel{\to }{r}}_{n+{N}_{MoM}}\right)\cdot {\stackrel{\to }{l}}_{n+{N}_{MoM}}\text{ }\text{ }\text{ }\left(10\right)
\stackrel{\to }{H}\left(\stackrel{\to }{r}\right)=\sum _{n=1}^{{N}_{MoM}}{\stackrel{\to }{C}}_{n}\left(\stackrel{\to }{r}\right){I}_{n}^{MoM}\text{ (11)}
{\stackrel{\to }{C}}_{n}\left(r\right)
\begin{array}{l}{I}_{n}^{PO}=\sum _{n=1}^{{N}_{MoM}}{\stackrel{^}{Z}}_{POmn}{I}_{n}^{MoM}\text{ }\text{ (12)}\\ {\stackrel{^}{Z}}_{POmn}=2\delta \left({\stackrel{\to }{r}}_{n+{N}_{MoM}}\right)\stackrel{\to }{C}\left({\stackrel{\to }{r}}_{n+{N}_{MoM}}\right)\cdot {\stackrel{\to }{l}}_{m+{N}_{MoM}}\text{ }\text{ }m=1,....,{N}_{PO},\text{ }n=1,...,{N}_{MoM}\end{array}
\begin{array}{l}{\stackrel{^}{Z}}_{11}{\stackrel{\to }{I}}^{MoM}+{\stackrel{^}{Z}}_{12}{\stackrel{\to }{I}}^{PO}=\stackrel{\to }{V}\text{ (13)}\\ {\stackrel{\to }{I}}^{PO}={\stackrel{^}{Z}}_{PO}{\stackrel{\to }{I}}^{MoM}\end{array}
\left({\stackrel{^}{Z}}_{11}+{\stackrel{^}{Z}}_{12}{\stackrel{^}{Z}}_{PO}\right){\stackrel{\to }{I}}^{MoM}=\stackrel{\to }{V}\text{ (14)}
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Global Constraint Catalog: Clex_greatereq
<< 5.229. lex_greater5.231. lex_less >>
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1},\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}\right)
\mathrm{𝚕𝚎𝚡𝚎𝚚}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}
\mathrm{𝚛𝚎𝚕}
\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
\mathrm{𝚐𝚎𝚚}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚎𝚚}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1},\mathrm{𝚟𝚊𝚛}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2},\mathrm{𝚟𝚊𝚛}\right)
|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|=|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}|
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}
\stackrel{\to }{X}
\stackrel{\to }{Y}
n
〈{X}_{0},\cdots ,{X}_{n-1}〉
〈{Y}_{0},\cdots ,{Y}_{n-1}〉
\stackrel{\to }{X}
\stackrel{\to }{Y}
n=0
{X}_{0}>{Y}_{0}
{X}_{0}={Y}_{0}
〈{X}_{1},\cdots ,{X}_{n-1}〉
〈{Y}_{1},\cdots ,{Y}_{n-1}〉
\left(〈5,2,8,9〉,〈5,2,6,2〉\right)
\left(〈5,2,3,9〉,〈5,2,3,9〉\right)
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}=〈5,2,8,9〉
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}=〈5,2,6,2〉
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}=〈5,2,3,9〉
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}=〈5,2,3,9〉
|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|>1
\bigvee \left(\begin{array}{c}|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|<5,\hfill \\ \mathrm{𝚗𝚟𝚊𝚕}\left(\left[\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}.\mathrm{𝚟𝚊𝚛},\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}.\mathrm{𝚟𝚊𝚛}\right]\right)<2*|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|\hfill \end{array}\right)
\bigvee \left(\begin{array}{c}\mathrm{𝚖𝚊𝚡𝚟𝚊𝚕}\left(\left[\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}.\mathrm{𝚟𝚊𝚛},\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}.\mathrm{𝚟𝚊𝚛}\right]\right)\le 1,\hfill \\ 2*|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|-\mathrm{𝚖𝚊𝚡}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}\left(\left[\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}.\mathrm{𝚟𝚊𝚛},\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}.\mathrm{𝚟𝚊𝚛}\right]\right)>2\hfill \end{array}\right)
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}
The following reformulations in term of arithmetic and/or logical expressions exist for enforcing the lexicographically greater than or equal to constraint. The first one converts
\stackrel{\to }{X}
\stackrel{\to }{Y}
\stackrel{\to }{X}
\stackrel{\to }{Y}
\left[0,a-1\right]
{a}^{n-1}{Y}_{0}+{a}^{n-2}{Y}_{1}+\cdots +{a}^{0}{Y}_{n-1}\le {a}^{n-1}{X}_{0}+{a}^{n-2}{X}_{1}+\cdots +{a}^{0}{X}_{n-1}
n
a
\left({Y}_{0}<{X}_{0}+\left({Y}_{1}<{X}_{1}+\left(\cdots +\left({Y}_{n-1}<{X}_{n-1}+1\right)\cdots \right)\right)\right)=1
Finally, the lexicographically greater than or equal to constraint can be expressed as a conjunction or a disjunction of constraints:
\begin{array}{cc}\hfill {Y}_{0}\le {X}_{0}& \hfill \wedge \\ \hfill \left({Y}_{0}={X}_{0}\right)⇒{Y}_{1}\le {X}_{1}& \hfill \wedge \\ \hfill \left({Y}_{0}={X}_{0}\wedge {Y}_{1}={X}_{1}\right)⇒{Y}_{2}\le {X}_{2}& \hfill \wedge \\ \hfill ⋮& \\ \hfill \left({Y}_{0}={X}_{0}\wedge {Y}_{1}={X}_{1}\wedge \cdots \wedge {Y}_{n-2}={X}_{n-2}\right)⇒{Y}_{n-1}\le {X}_{n-1}& \\ & \\ \hfill {Y}_{0}<{X}_{0}& \hfill \vee \\ \hfill {Y}_{0}={X}_{0}\wedge {Y}_{1}<{X}_{1}& \hfill \vee \\ \hfill {Y}_{0}={X}_{0}\wedge {Y}_{1}={X}_{1}\wedge {Y}_{2}<{X}_{2}& \hfill \vee \\ \hfill ⋮& \\ \hfill {Y}_{0}={X}_{0}\wedge {Y}_{1}={X}_{1}\wedge \cdots \wedge {Y}_{n-2}={X}_{n-2}\wedge {Y}_{n-1}\le {X}_{n-1}\end{array}
lexEq in Choco, rel in Gecode, lex_greatereq in MiniZinc, lex_chain in SICStus.
\mathrm{𝚌𝚘𝚗𝚍}_\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚋𝚎𝚝𝚠𝚎𝚎𝚗}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚎𝚚𝚞𝚊𝚕}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛}
\mathrm{𝚜𝚘𝚛𝚝}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚕𝚎𝚜𝚜}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
\mathrm{𝚌𝚘𝚕}\left(\begin{array}{c}\mathrm{𝙳𝙴𝚂𝚃𝙸𝙽𝙰𝚃𝙸𝙾𝙽}-\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},𝚡-\mathrm{𝚒𝚗𝚝},𝚢-\mathrm{𝚒𝚗𝚝}\right),\hfill \\ \mathrm{𝚒𝚝𝚎𝚖}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-0,𝚡-0,𝚢-0\right)\right]\hfill \end{array}\right)
\mathrm{𝚌𝚘𝚕}\left(\begin{array}{c}\mathrm{𝙲𝙾𝙼𝙿𝙾𝙽𝙴𝙽𝚃𝚂}-\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},𝚡-\mathrm{𝚍𝚟𝚊𝚛},𝚢-\mathrm{𝚍𝚟𝚊𝚛}\right),\hfill \\ \left[\begin{array}{c}\mathrm{𝚒𝚝𝚎𝚖}\left(\begin{array}{c}\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}.\mathrm{𝚔𝚎𝚢},\hfill \\ 𝚡-\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}.\mathrm{𝚟𝚊𝚛},\hfill \\ 𝚢-\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}.\mathrm{𝚟𝚊𝚛}\hfill \end{array}\right)\hfill \end{array}\right]\hfill \end{array}\right)
\mathrm{𝙲𝙾𝙼𝙿𝙾𝙽𝙴𝙽𝚃𝚂}
\mathrm{𝙳𝙴𝚂𝚃𝙸𝙽𝙰𝚃𝙸𝙾𝙽}
\mathrm{𝑃𝑅𝑂𝐷𝑈𝐶𝑇}
\left(
\mathrm{𝑃𝐴𝑇𝐻}
,
\mathrm{𝑉𝑂𝐼𝐷}
\right)↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚝𝚎𝚖}\mathtt{1},\mathrm{𝚒𝚝𝚎𝚖}\mathtt{2}\right)
\bigvee \left(\begin{array}{c}\mathrm{𝚒𝚝𝚎𝚖}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡}>0\wedge \mathrm{𝚒𝚝𝚎𝚖}\mathtt{1}.𝚡=\mathrm{𝚒𝚝𝚎𝚖}\mathtt{1}.𝚢,\hfill \\ \bigwedge \left(\begin{array}{c}\mathrm{𝚒𝚝𝚎𝚖}\mathtt{1}.\mathrm{𝚒𝚗𝚍𝚎𝚡}<|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|,\hfill \\ \mathrm{𝚒𝚝𝚎𝚖}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡}=0,\hfill \\ \mathrm{𝚒𝚝𝚎𝚖}\mathtt{1}.𝚡>\mathrm{𝚒𝚝𝚎𝚖}\mathtt{1}.𝚢\hfill \end{array}\right),\hfill \\ \bigwedge \left(\begin{array}{c}\mathrm{𝚒𝚝𝚎𝚖}\mathtt{1}.\mathrm{𝚒𝚗𝚍𝚎𝚡}=|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|,\hfill \\ \mathrm{𝚒𝚝𝚎𝚖}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡}=0,\hfill \\ \mathrm{𝚒𝚝𝚎𝚖}\mathtt{1}.𝚡\ge \mathrm{𝚒𝚝𝚎𝚖}\mathtt{1}.𝚢\hfill \end{array}\right)\hfill \end{array}\right)
\mathrm{𝐏𝐀𝐓𝐇}_\mathrm{𝐅𝐑𝐎𝐌}_\mathrm{𝐓𝐎}
\left(\mathrm{𝚒𝚗𝚍𝚎𝚡},1,0\right)=1
\mathrm{𝐏𝐀𝐓𝐇}_\mathrm{𝐅𝐑𝐎𝐌}_\mathrm{𝐓𝐎}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
{c}_{i}
i
We create an additional dummy vertex called
{c}_{i}
and
{c}_{i}
{\mathrm{𝚒𝚝𝚎𝚖}}_{1}.x\ge {\mathrm{𝚒𝚝𝚎𝚖}}_{2}.y
{\mathrm{𝚒𝚝𝚎𝚖}}_{1}.x>{\mathrm{𝚒𝚝𝚎𝚖}}_{2}.y
{c}_{i}
{c}_{i+1}
{\mathrm{𝚒𝚝𝚎𝚖}}_{1}.x={\mathrm{𝚒𝚝𝚎𝚖}}_{2}.y
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
{c}_{1}
d
. This path can be interpreted as a maximum sequence of equality constraints on the prefix of both vectors, possibly followed by a greater than constraint.
\mathrm{𝐏𝐀𝐓𝐇}_\mathrm{𝐅𝐑𝐎𝐌}_\mathrm{𝐓𝐎}
\mathrm{𝐏𝐀𝐓𝐇}_\mathrm{𝐅𝐑𝐎𝐌}_\mathrm{𝐓𝐎}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡},1,0\right)=1
\mathrm{𝐏𝐀𝐓𝐇}_\mathrm{𝐅𝐑𝐎𝐌}_\mathrm{𝐓𝐎}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡},1,0\right)\ge 1
\underline{\overline{\mathrm{𝐏𝐀𝐓𝐇}_\mathrm{𝐅𝐑𝐎𝐌}_\mathrm{𝐓𝐎}}}
\overline{\mathrm{𝐏𝐀𝐓𝐇}_\mathrm{𝐅𝐑𝐎𝐌}_\mathrm{𝐓𝐎}}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
\mathrm{𝚅𝙰𝚁}{\mathtt{1}}_{i}
\mathrm{𝚅𝙰𝚁}{\mathtt{2}}_{i}
\mathrm{𝚟𝚊𝚛}
{i}^{th}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}
\left(\mathrm{𝚅𝙰𝚁}{\mathtt{1}}_{i},\mathrm{𝚅𝙰𝚁}{\mathtt{2}}_{i}\right)
{S}_{i}
\left(\mathrm{𝚅𝙰𝚁}{\mathtt{1}}_{i}<\mathrm{𝚅𝙰𝚁}{\mathtt{2}}_{i}⇔{S}_{i}=1\right)\wedge \left(\mathrm{𝚅𝙰𝚁}{\mathtt{1}}_{i}=\mathrm{𝚅𝙰𝚁}{\mathtt{2}}_{i}⇔{S}_{i}=2\right)\wedge \left(\mathrm{𝚅𝙰𝚁}{\mathtt{1}}_{i}>\mathrm{𝚅𝙰𝚁}{\mathtt{2}}_{i}⇔{S}_{i}=3\right)
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
|
Towards a dynamic clamp for neuro-chemical modalities | BMC Neuroscience | Full Text
Towards a dynamic clamp for neuro-chemical modalities
Catalina Maria Rivera1,
Jihen Zhao2 &
The classic dynamic clamp technique uses a real-time electrical interface between living cells and neural simulations in order to investigate hypotheses about neural function and structure [3]. However, it has two major drawback [2]: the electrodes can clamp electrically only a small section of a cell, and hence frequently do not control the behavior of whole cells; and all control to-date has been concentrated on the electric properties of neurons, neglecting their chemical state. As noted in [2], the latter has been done by necessity, since until recently registering or controlling the chemical state of either neurons or the chemical environment in which they reside has been essentially impossible with the level of precision needed to simulate the appropriate dynamics of the various extracellular chemical players. In this manuscript we present an expansion of the dynamic clamp method to include simulation and control of the effects of a variety of chemicals associated with neural signaling. To achieve that, we use the emergent discipline of microfluidics [4], which deals specifically with the behavior, precise control and manipulation of fluids at the microscale. We use a novel combination of microfluidic and nanosensor technology to add sensing and control of chemical concentrations to the dynamic clamp technique. Specifically, we use a microfluidic chip to generate distinct chemical concentration gradients (ions or neuromodulators), register the concentrations with embedded nanosensors, and use the processed signals as an input to simulations of a neural cell. The ultimate goal of this project is to close the loop, and provide control signals to the microfluidic lab to mimic the interaction of the simulated cell with other cells in its chemical environment, by modifying the chemical concentrations in the microfluidic lab environment to reflect simulated outputs of the model neurons. Here we used Hodgkin-Huxley type neurons,
C\frac{dV}{dt}={I}_{o}+{I}_{HH}+{I}_{Ca}
with added calcium-gated currents,
{I}_{Ca}={m}_{\infty }{\left(V\right)}^{2}h{P}_{max}\frac{{z}^{2}{F}^{2}}{RT}V\left(\frac{{\left[Ca\right]}_{in}-{\left[Ca\right]}_{out}{e}^{\frac{-zVF}{RT}}}{1-{e}^{\frac{-zVF}{RT}}}\right)
In order to predict the distribution of specific chemicals inside a microfluidic chamber, 2-D Computational Fluid Dynamics technique was used. In this study, the commercial software FLUENT® 6.3 was used to build the computational domain, the models for a microfluidic mixer and a neuron chamber by using finite element methods.
Shows the simulated microfluidic device (A) and simulated square pulse flow at pts C1 and C2 (B). The responses of the model neuron to changes in Ca++ concentration are shown in panel C. The device in (A) was realized physically, using standard lithography and bonding methods as described in [1]. The Ca++ concentration was recorded and the record was used as an input to the same model neuron, with results indistinguishable from (C).
Li P, Lei N, Sheadel DA, Xu J, Xue W: Integration of nanosensors into a sealed microchannel in a hybrid lab-on-a-chip device. Sensors and Actuators B: Chemical. 2012, 166-167: 751-9.
Prinz AA, Cudmore RH: Dynamic clamp. Scholarpedia. 2011, 6 (5): 1470-10.4249/scholarpedia.1470.
Sharp AA, O'Neil MB, Abbott LF, Marder E: Dynamic clamp: Computer-generated conductances in real neurons. J Neurophysiol. 1993, 69: 992-5.
Squires TM, Quake SR: Microfluidics: Fluid physics at the nanoliter scale. Rev Mod Phy. 2005, 77: 977-1026. 10.1103/RevModPhys.77.977.
Alexander G Dimitrov & Catalina Maria Rivera
Department of Mechanical Engineering, Washington State University, Vancouver, WA, 98686, USA
Wei Xue, Jie Xu, Gan Yu, Jihen Zhao & Hyuck-Jin Kwon
Catalina Maria Rivera
Jihen Zhao
Dimitrov, A.G., Xue, W., Xu, J. et al. Towards a dynamic clamp for neuro-chemical modalities. BMC Neurosci 14, P232 (2013). https://doi.org/10.1186/1471-2202-14-S1-P232
Distinct Chemical
|
Argument (complex analysis) - Wikipedia
For the argument principle, see argument principle.
Figure 1. This Argand diagram represents the complex number lying on a plane. For each point on the plane, arg is the function which returns the angle
{\displaystyle \varphi }
In mathematics (particularly in complex analysis), the argument of a complex number z, denoted arg(z), is the angle between the positive real axis and the line joining the origin and z, represented as a point in the complex plane, shown as
{\displaystyle \varphi }
in Figure 1. It is a multi-valued function operating on the nonzero complex numbers. To define a single-valued function, the principal value of the argument (sometimes denoted Arg z) is used. It is often chosen to be the unique value of the argument that lies within the interval (−π, π].[1][2]
2 Principal value
3 Computing from the real and imaginary part
4.2 Using the complex logarithm
5 Extended Argument
Figure 2. Two choices for the argument
{\displaystyle \varphi }
Geometrically, in the complex plane, as the 2D polar angle
{\displaystyle \varphi }
from the positive real axis to the vector representing z. The numeric value is given by the angle in radians, and is positive if measured counterclockwise.
Algebraically, as any real quantity
{\displaystyle \varphi }
{\displaystyle z=r(\cos \varphi +i\sin \varphi )=re^{i\varphi }}
for some positive real r (see Euler's formula). The quantity r is the modulus (or absolute value) of z, denoted |z|:
{\displaystyle r={\sqrt {x^{2}+y^{2}}}.}
The names magnitude, for the modulus, and phase,[3][1] for the argument, are sometimes used equivalently.
Under both definitions, it can be seen that the argument of any non-zero complex number has many possible values: firstly, as a geometrical angle, it is clear that whole circle rotations do not change the point, so angles differing by an integer multiple of 2π radians (a complete circle) are the same, as reflected by figure 2 on the right. Similarly, from the periodicity of sin and cos, the second definition also has this property. The argument of zero is usually left undefined.
The complex argument can also be defined algebraically in terms of complex roots as:
{\displaystyle \arg(z)=\lim _{n\to \infty }n\cdot \operatorname {Im} {\sqrt[{n}]{z/|z|}}}
This definition removes reliance on other difficult-to-compute functions such as arctangent as well as eliminating the need for the piecewise definition. Because it's defined in terms of roots, it also inherits the principal branch of square root as its own principle branch. The normalization of
{\displaystyle z}
{\displaystyle |z|}
isn't necessary for convergence to the correct value, but it does speed up convergence and ensures that
{\displaystyle \arg(0)}
is left undefined.
Principal value[edit]
Because a complete rotation around the origin leaves a complex number unchanged, there are many choices which could be made for
{\displaystyle \varphi }
by circling the origin any number of times. This is shown in figure 2, a representation of the multi-valued (set-valued) function
{\displaystyle f(x,y)=\arg(x+iy)}
, where a vertical line (not shown in the figure) cuts the surface at heights representing all the possible choices of angle for that point.
When a well-defined function is required, then the usual choice, known as the principal value, is the value in the open-closed interval (−π rad, π rad], that is from −π to π radians, excluding −π rad itself (equiv., from −180 to +180 degrees, excluding −180° itself). This represents an angle of up to half a complete circle from the positive real axis in either direction.
The principal value sometimes has the initial letter capitalized, as in Arg z, especially when a general version of the argument is also being considered. Note that notation varies, so arg and Arg may be interchanged in different texts.
{\displaystyle \arg(z)=\{\operatorname {Arg} (z)+2\pi n\mid n\in \mathbb {Z} \}.}
Computing from the real and imaginary part[edit]
Main article: atan2
If a complex number is known in terms of its real and imaginary parts, then the function that calculates the principal value Arg is called the two-argument arctangent function atan2:
{\displaystyle \operatorname {Arg} (x+iy)=\operatorname {atan2} (y,\,x)}
The atan2 function (also called arctan2 or other synonyms) is available in the math libraries of many programming languages, and usually returns a value in the range (−π, π].[1]
{\displaystyle \operatorname {Arg} (x+iy)=\operatorname {atan2} (y,\,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&{\text{if }}x>0,\\\arctan \left({\frac {y}{x}}\right)+\pi &{\text{if }}x<0{\text{ and }}y\geq 0,\\\arctan \left({\frac {y}{x}}\right)-\pi &{\text{if }}x<0{\text{ and }}y<0,\\+{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y>0,\\-{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y<0,\\{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}}
A compact expression with 4 overlapping half-planes is
{\displaystyle \operatorname {Arg} (x+iy)=\operatorname {atan2} (y,\,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&{\text{if }}x>0,\\{\frac {\pi }{2}}-\arctan \left({\frac {x}{y}}\right)&{\text{if }}y>0,\\-{\frac {\pi }{2}}-\arctan \left({\frac {x}{y}}\right)&{\text{if }}y<0,\\\arctan \left({\frac {y}{x}}\right)\pm \pi &{\text{if }}x<0,\\{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}}
{\displaystyle \operatorname {Arg} (x+iy)={\begin{cases}\displaystyle 2\arctan \left({\frac {y}{{\sqrt {x^{2}+y^{2}}}+x}}\right)&{\text{if }}x>0{\text{ or }}y\neq 0,\\\pi &{\text{if }}x<0{\text{ and }}y=0,\\{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}}
This is based on a parametrization of the circle (except for the negative x-axis) by rational functions. This version of Arg is not stable enough for floating point computational use (as it may overflow near the region x < 0, y = 0), but can be used in symbolic calculation.
{\displaystyle \operatorname {Arg} (x+iy)={\begin{cases}\displaystyle 2\arctan \left({\frac {{\sqrt {x^{2}+y^{2}}}-x}{y}}\right)&{\text{if }}y\neq 0,\\0&{\text{if }}x>0{\text{ and }}y=0,\\\pi &{\text{if }}x<0{\text{ and }}y=0,\\{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}}
{\displaystyle z=\left|z\right|e^{i\operatorname {Arg} z}.}
This is only really valid if z is non-zero, but can be considered valid for z = 0 if Arg(0) is considered as an indeterminate form—rather than as being undefined.
{\displaystyle {\begin{aligned}\operatorname {Arg} (z_{1}z_{2})&\equiv \operatorname {Arg} (z_{1})+\operatorname {Arg} (z_{2}){\pmod {\mathbb {R} /2\pi \mathbb {Z} }},\\\operatorname {Arg} \left({\frac {z_{1}}{z_{2}}}\right)&\equiv \operatorname {Arg} (z_{1})-\operatorname {Arg} (z_{2}){\pmod {\mathbb {R} /2\pi \mathbb {Z} }}.\end{aligned}}}
If z ≠ 0 and n is any integer, then[1]
{\displaystyle \operatorname {Arg} \left(z^{n}\right)\equiv n\operatorname {Arg} (z){\pmod {\mathbb {R} /2\pi \mathbb {Z} }}.}
{\displaystyle \operatorname {Arg} {\biggl (}{\frac {-1-i}{i}}{\biggr )}=\operatorname {Arg} (-1-i)-\operatorname {Arg} (i)=-{\frac {3\pi }{4}}-{\frac {\pi }{2}}=-{\frac {5\pi }{4}}}
Using the complex logarithm[edit]
{\displaystyle z=|z|e^{i\operatorname {Arg} (z)}}
{\displaystyle \operatorname {Arg} (z)=-i\ln {\frac {z}{|z|}}}
. This is useful when one has the complex logarithm available.
Extended Argument[edit]
Extended argument of a number z (denoted as
{\displaystyle {\overline {\arg }}(z)}
) is the set of all real numbers congruent to
{\displaystyle \arg(z)}
{\displaystyle \pi }
{\displaystyle {\overline {\arg }}(z)=\arg(z)+2k\pi ,\forall k\in \mathbb {Z} }
^ a b c d Weisstein, Eric W. "Complex Argument". mathworld.wolfram.com. Retrieved 2020-08-31.
^ "Pure Maths". internal.ncl.ac.uk. Retrieved 2020-08-31.
^ Dictionary of Mathematics (2002). phase.
^ "Algebraic Structure of Complex Numbers". www.cut-the-knot.org. Retrieved 2021-08-29.
Argument at Encyclopedia of Mathematics.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Argument_(complex_analysis)&oldid=1081379087"
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Global Constraint Catalog: Cdiffer_from_at_least_k_pos
<< 5.114. derangement5.116. differ_from_at_most_k_pos >>
\mathrm{𝚍𝚒𝚏𝚏𝚎𝚛}_\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚊𝚝}_\mathrm{𝚕𝚎𝚊𝚜𝚝}_𝚔_\mathrm{𝚙𝚘𝚜}\left(𝙺,\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1},\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}\right)
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
𝙺
\mathrm{𝚒𝚗𝚝}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}
|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}|\ge 1
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁},\mathrm{𝚟𝚊𝚛}\right)
𝙺\ge 0
𝙺\le |\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|
|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|=|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}|
Enforce two vectors
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}
to differ from at least
𝙺
\left(2,〈2,5,2,0〉,〈3,6,2,1〉\right)
\mathrm{𝚍𝚒𝚏𝚏𝚎𝚛}_\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚊𝚝}_\mathrm{𝚕𝚎𝚊𝚜𝚝}_𝚔_\mathrm{𝚙𝚘𝚜}
constraint holds since the first and second vectors differ from 3 positions, which is greater than or equal to
𝙺=2
𝙺>0
𝙺<|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|
|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|>1
\left(𝙺\right)
\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1},\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}\right)
𝙺
\ge 0
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}
Used in the Arc constraint(s) slot of the
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚍𝚒𝚏𝚏𝚎𝚛}_\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚊𝚝}_\mathrm{𝚕𝚎𝚊𝚜𝚝}_𝚔_\mathrm{𝚙𝚘𝚜}
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚍𝚒𝚏𝚏𝚎𝚛}_\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚊𝚝}_\mathrm{𝚕𝚎𝚊𝚜𝚝}_𝚔_\mathrm{𝚙𝚘𝚜}
\mathrm{𝚍𝚒𝚏𝚏𝚎𝚛}_\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚎𝚡𝚊𝚌𝚝𝚕𝚢}_𝚔_\mathrm{𝚙𝚘𝚜}
\ge 𝙺
=𝙺
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚍𝚒𝚏𝚏𝚎𝚛}_\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚊𝚝}_\mathrm{𝚕𝚎𝚊𝚜𝚝}_𝚔_\mathrm{𝚙𝚘𝚜}
characteristic of a constraint: vector, automaton, automaton with counters.
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}
\mathrm{𝑃𝑅𝑂𝐷𝑈𝐶𝑇}
\left(=\right)↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚎𝚌𝚝𝚘𝚛}\mathtt{1},\mathrm{𝚟𝚎𝚌𝚝𝚘𝚛}\mathtt{2}\right)
\mathrm{𝚟𝚎𝚌𝚝𝚘𝚛}\mathtt{1}.\mathrm{𝚟𝚊𝚛}\ne \mathrm{𝚟𝚎𝚌𝚝𝚘𝚛}\mathtt{2}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝐍𝐀𝐑𝐂}
\ge 𝙺
\mathrm{𝐍𝐀𝐑𝐂}
\mathrm{𝚍𝚒𝚏𝚏𝚎𝚛}_\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚊𝚝}_\mathrm{𝚕𝚎𝚊𝚜𝚝}_𝚔_\mathrm{𝚙𝚘𝚜}
\mathrm{𝚍𝚒𝚏𝚏𝚎𝚛}_\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚊𝚝}_\mathrm{𝚕𝚎𝚊𝚜𝚝}_𝚔_\mathrm{𝚙𝚘𝚜}
\mathrm{𝚅𝙰𝚁}{\mathtt{1}}_{i}
\mathrm{𝚅𝙰𝚁}{\mathtt{2}}_{i}
{i}^{th}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}
collections. To each pair of variables
\left(\mathrm{𝚅𝙰𝚁}{\mathtt{1}}_{i},\mathrm{𝚅𝙰𝚁}{\mathtt{2}}_{i}\right)
{S}_{i}
\mathrm{𝚅𝙰𝚁}{\mathtt{1}}_{i}
\mathrm{𝚅𝙰𝚁}{\mathtt{2}}_{i}
{S}_{i}
\mathrm{𝚅𝙰𝚁}{\mathtt{1}}_{i}=\mathrm{𝚅𝙰𝚁}{\mathtt{2}}_{i}⇔{S}_{i}
\mathrm{𝚍𝚒𝚏𝚏𝚎𝚛}_\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚊𝚝}_\mathrm{𝚕𝚎𝚊𝚜𝚝}_𝚔_\mathrm{𝚙𝚘𝚜}
\mathrm{𝚍𝚒𝚏𝚏𝚎𝚛}_\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚊𝚝}_\mathrm{𝚕𝚎𝚊𝚜𝚝}_𝚔_\mathrm{𝚙𝚘𝚜}
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If A: The quotient of two integers is always a rational numbers and R:is not rational, then which of the - Maths - Rational Numbers - 9179573 | Meritnation.com
If A: The quotient of two integers is always a rational numbers and
R: is not rational, then which of the following statements is true?
(a) A is true and R is correct explanation of A.
(b) A is false and R is correct explanation of A.
(c) A is true and R is false.
(d) Both A and R is false.
The correct answer is B . Please explain How?
\mathrm{Yes}, \mathrm{the} \mathrm{correct} \mathrm{answer} \mathrm{is} \left(\mathrm{B}\right).\phantom{\rule{0ex}{0ex}}\mathrm{as} \mathrm{the} \mathrm{statement} \left(\mathrm{A}\right) \mathrm{is} \mathrm{false} \phantom{\rule{0ex}{0ex}}\mathrm{as} \mathrm{quotient} \mathrm{of} \mathrm{two} \mathrm{integer} \mathrm{is} \mathrm{not} \mathrm{always} \mathrm{a} \mathrm{rational} \mathrm{number} . \mathrm{as} \mathrm{when} \mathrm{there} \mathrm{is} \mathrm{a} \mathrm{zero} \mathrm{in} \mathrm{denominator}.\phantom{\rule{0ex}{0ex}}\mathrm{It} \mathrm{is} \mathrm{not} \mathrm{rational} \mathrm{number}.\phantom{\rule{0ex}{0ex}}\mathrm{In} \mathrm{statement} \left(\mathrm{R}\right) \mathrm{correct} \mathrm{reason} \mathrm{is} \mathrm{given} . \mathrm{why} \mathrm{statement} \left(\mathrm{A}\right) \mathrm{is} \mathrm{incorrect}.\phantom{\rule{0ex}{0ex}}
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The table shows the prices for various-sized bags of rice offered at a grocery store.
0.5
\$0.89
1
\$1.29
2
\$1.89
5
\$4.60
10
\$8.95
20
\$17.80
What are the costs per pound for each bag?
Divide the cost by the bag size for each row.
The answers for part (a) are average rates of change. What other data point is being used, but not shown, in these rate calculations?
The data point is missing from the top of the table.
Review your calculations from part (a). As you move down the table are your values for cost per pound increasing or decreasing?
Sketching a graph of rate with respect to bag size may help.
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\psi
\psi(z) = \dfrac{\mathrm{d}}{\mathrm{d}z} \ln \Gamma(z) = \dfrac{\Gamma^\prime (z)}{\Gamma(z)}
\psi(s+1)=\psi(s)+\dfrac{1}{s}
\Gamma(s+1)=s\Gamma(s).
\ln\big(\Gamma(s+1)\big)=\ln\big(\Gamma(s)\big)+\ln(s).
s,
\psi(s+1)=\psi(s)+\dfrac{1}{s}.\ _\square
\psi(s+1)=-\gamma+\sum_{k=1}^\infty \left(\dfrac{1}{k}-\dfrac{1}{k+s}\right)
\Gamma(s)=\dfrac{e^{-\gamma s}}{s} \prod_{k=1}^\infty e^{s/k} \left(1+\dfrac{s}{k}\right)^{-1}.
\ln\big(\Gamma(s)\big)=-\gamma s-\ln(s)+\sum_{k=1}^\infty \left(\dfrac{s}{k}-\ln \Big(1+\dfrac{s}{k}\Big)\right).
s,
\psi(s)=-\gamma-\dfrac{1}{s}+\sum_{k=1}^\infty \left(\dfrac{1}{k}-\dfrac{1}{k+s}\right)=-\gamma+\sum_{k=1}^\infty \left(\dfrac{1}{k}-\dfrac{1}{k+s-1}\right).
s
s+1
\psi(s+1)=-\gamma+\sum_{k=1}^\infty \left(\dfrac{1}{k}-\dfrac{1}{k+s}\right).\ _\square
\psi(s+1) = -\gamma + \int_0^1 \dfrac{1-x^s}{1-x} dx
\begin{aligned} \psi(s+1) &=-\gamma+\sum_{n=1}^\infty \left(\dfrac{1}{n}-\dfrac{1}{n+s}\right)\\ &=-\gamma+\sum_{n=1}^\infty \int_0^1 \big(x^{n-1}-x^{n+s-1}\big)dx\\ &=-\gamma+ \int_0^1\sum_{n=1}^\infty \big(x^{n-1}-x^{n+s-1}\big)dx. \end{aligned}
\psi(s+1) = -\gamma + \int_0^1 \dfrac{1-x^s}{1-x} dx.\ _\square
s=0,
\psi(1)=-\gamma.
\psi(s+1) = -\gamma + H_s.
\psi(1-z) - \psi(z) = \pi \cot \pi z.
\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin \pi z}.
\ln\big(\Gamma(z)\big)+\ln\big(\Gamma(1-z)\big) = \log \pi - \log \sin \pi z.
\begin{aligned} \dfrac{\Gamma^{\prime}(z)}{\Gamma(z)}-\dfrac{\Gamma^{\prime}(1-z)}{\Gamma(1-z)} &= - \dfrac{\pi \cos \pi z}{\sin \pi z}\\\\ \psi(z) - \psi(1-z) &= - \dfrac{\pi \cos \pi z}{\sin \pi z}\\\\ \psi(1-z) - \psi(z) &= \pi \cot \pi z, \end{aligned}
\psi
_\square
2\psi(2s)=2\ln(2)+\psi(s)+\psi\left(s+\frac12\right)
\sqrt{\pi} \ \Gamma(2s)=2^{2s-1} \Gamma(s)\Gamma\left(s+\frac12\right).
\ln\big(\sqrt{\pi}\big)+\ln\big(\Gamma(2s)\big)=(2s-1)\ln(2)+\ln\big(\Gamma(s)\big)+\ln\left(\Gamma\Big(s+{\small\frac12}\Big)\right).
s,
2\psi(2s)=2\ln(2)+\psi(s)+\psi\left(s+\frac12\right).\ _\square
\sum_{n=0}^\infty \dfrac{1}{n^2+1} =\dfrac{\pi+1}{2} + \dfrac{\pi}{e^{2\pi}-1}.
S = \sum_{n=0}^{\infty} \frac{1}{n^2+1} = \frac{1}{2i} \sum_{n=0}^{\infty} \left( \frac{1}{n-i} - \frac{1}{n+i} \right).
\begin{aligned} 2iS &= \sum_{n=1}^{\infty} \left( \frac{1}{n-1-i} - \frac{1}{n-1+i} \right)\\ &=\sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n-1+i} \right)-\sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n-1-i} \right). \end{aligned}
\begin{aligned} 2iS &=\psi(i)-\psi(-i)\\ &=\psi(i)-\psi(1-i)-\dfrac{1}{i}\\ &=-\pi \cot(i\pi)+i\\ &=\pi i \coth(\pi)+i. \end{aligned}
S= \dfrac{1+\pi\coth(\pi)}{2}\implies S=\dfrac{\pi+1}{2} + \dfrac{\pi}{e^{2\pi}-1}.\ _\square
n^\text{th}
\psi_n (s)= \dfrac{d^n}{ds^n} \psi(s)=\psi^{(n)}(s).
We can get many properties from this; for example, by differentiating the series representation
\psi_n(s)=(-1)^{n+1}n! \sum_{k=1}^\infty \dfrac{1}{(k+s-1)^{n+1}}=(-1)^{n+1}n! \sum_{k=0}^\infty \dfrac{1}{(k+s)^{n+1}}=(-1)^{n+1} n!\zeta(n+1,s),
\zeta(n+1,s)
s=1,
\psi_n(1)=(-1)^{n+1} n!\zeta(n+1).
\begin{aligned} \psi(s)&=\sum_{n=0}^\infty \dfrac{\psi^{(n)}(1)(s-1)^n}{n!}\\ &=-\gamma+\sum_{n=1}^\infty \dfrac{\psi_n(1)(s-1)^n}{n!}\\ &=-\gamma-\sum_{n=1}^\infty (-1)^n\zeta(n+1)(s-1)^n\\ \psi(s+1)&=-\gamma-\sum_{n=1}^\infty \zeta(n+1)(-s)^n. \end{aligned}
We can differentiate the integral representation
\psi_n(s+1)=\int_0^1 \dfrac{\ln^n(x) x^s}{x-1}dx.
\psi_n(s+1)=\psi_n(s)+(-1)^nn! z^{-n-1}.
\sum _{ k=1 }^{ \infty }{ \dfrac { { \psi }^{ 1 }\left( k \right) }{ k } } =\sum _{ n=1 }^{ \infty }{ \dfrac { A }{ { n }^{ B } } }
A
B
A+B
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Given the graph of the piecewise-defined function at right, complete the following limit statements or state that the limit does not exist.
\lim\limits _ { x \rightarrow - \infty } f ( x ) =
What happens to the curve on the far left side?
\lim\limits _ { x \rightarrow - 2 ^ { - } } f ( x ) =
What is the height of the function if you approach
x=-2
from the left?
\lim\limits _ { x \rightarrow - 2 ^ { + } } f ( x ) =
x=-2
from the right?
\lim\limits _ { x \rightarrow - 2 } f ( x ) =
Do your answers to parts (b) and (c) agree?
\lim\limits_ { x \rightarrow 2 } f ( x ) =
Since the curve is continuous at
x=2
, this is just
f(2)
\lim\limits _ { x \rightarrow \infty } f ( x ) =
What happens to the curve on the far right side?
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Golygon - Wikipedia
The smallest golygon has 8-sides. It is the only solution with fewer than 16 sides. It contains two concave corners, and fits on an 8×10 grid. It is also a spirolateral, 890°1,5.
A golygon, or more generally a serial isogon of 90°, is any polygon with all right angles (a rectilinear polygon) whose sides are consecutive integer lengths. Golygons were invented and named by Lee Sallows, and popularized by A.K. Dewdney in a 1990 Scientific American column (Smith).[1] Variations on the definition of golygons involve allowing edges to cross, using sequences of edge lengths other than the consecutive integers, and considering turn angles other than 90°.[2]
3.1 Golyhedron
In any golygon, all horizontal edges have the same parity as each other, as do all vertical edges. Therefore, the number n of sides must allow the solution of the system of equations
{\displaystyle \pm 1\pm 3\pm \cdots \pm (n-1)=0}
{\displaystyle \pm 2\pm 4\pm \cdots \pm n=0.}
It follows from this that n must be a multiple of 8. For example, in the figure we have
{\displaystyle -1+3+5-7=0}
{\displaystyle 2-4-6+8=0}
The number of golygons for a given permissible value of n may be computed efficiently using generating functions (sequence A007219 in the OEIS). The number of golygons for permissible values of n is 4, 112, 8432, 909288, etc.[3] Finding the number of solutions that correspond to non-crossing golygons seems to be significantly more difficult.
There is a unique eight-sided golygon (shown in the figure); it can tile the plane by 180-degree rotation using the Conway criterion.
16-sided golygon. Spirolateral 1690°1,3,6,8,11
32-sided golygon. Spirolateral 3290°1,3,5,7,11,12,14,17,19,21,23,26,29,31
A serial-sided isogon of order n is a closed polygon with a constant angle at each vertex and having consecutive sides of length 1, 2, ..., n units. The polygon may be self-crossing.[4] Golygons are a special case of serial-sided isogons.[5]
A spirolateral is similar construction, notationally nθi1,i2,...,ik which sequences lengths 1,2,3,...,n with internal angles θ, with option of repeating until it returns to close with the original vertex. The i1,i2,...,ik superscripts list edges that follow opposite turn directions.
A serial-sided isogon order 9, internal angle 60°.[5]
Spirolateral 60°91,4,7.
A serial-sided isogon order 11, internal angle 60°.[5]
Spirolateral 60°114,5,7,8.
A serial-sided isogon order 12, internal angle 120°.[5]
Spirolateral 120°121,4,8.
A serial-sided isogon order 5, internal angles 60° and 120°.[5]
Golyhedron[edit]
The three-dimensional generalization of a golygon is called a golyhedron – a closed simply-connected solid figure confined to the faces of a cubical lattice and having face areas in the sequence 1, 2, ..., n, for some integer n, first introduced in a MathOverflow question.[6][7]
Golyhedrons have been found with values of n equal to 32, 15, 12, and 11 (the minimum possible).[8]
^ Dewdney, A.K. (1990). "An odd journey along even roads leads to home in Golygon City". Scientific American. 263: 118–121. doi:10.1038/scientificamerican0790-118.
^ Harry J. Smith. "What is a Golygon?". Archived from the original on 2009-10-27.
^ Weisstein, Eric W. "Golygon". MathWorld.
^ Sallows, Lee (1992). "New pathways in serial isogons". The Mathematical Intelligencer. 14 (2): 55–67. doi:10.1007/BF03025216. S2CID 121493484.
^ a b c d e Sallows, Lee; Gardner, Martin; Guy, Richard K.; Knuth, Donald (1991). "Serial isogons of 90 degrees". Mathematics Magazine. 64 (5): 315–324. doi:10.2307/2690648. JSTOR 2690648.
^ "Can we find lattice polyhedra with faces of area 1,2,3,…?"
^ Golygons and golyhedra
^ Golyhedron update
Golygons at the On-Line Encyclopedia of Integer Sequences
Retrieved from "https://en.wikipedia.org/w/index.php?title=Golygon&oldid=1055915301"
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Floor Function | Brilliant Math & Science Wiki
Patrick Corn, Thaddeus Abiy, Jubayer Nirjhor, and
The floor function (also known as the greatest integer function)
\lfloor\cdot\rfloor: \mathbb{R} \to \mathbb{Z}
of a real number
x
denotes the greatest integer less than or equal to
x
\lfloor 5\rfloor=5, ~\lfloor 6.359\rfloor =6, ~\left\lfloor \sqrt{7}\right\rfloor=2, ~\lfloor \pi\rfloor = 3, ~\lfloor -13.42\rfloor = -14.
\lfloor x \rfloor
is the unique integer satisfying
\lfloor x\rfloor\le x<\lfloor x\rfloor +1
\{x\}
x
0\le \{x\}<1
\{2.137\}=0.137.
x=\lfloor x\rfloor+\{x\}
x
3.1416=3+0.1416,
\lfloor x\rfloor =3
\{x\}=0.1416
The floor function is discontinuous at every integer. [1]
Applications of Floor Function to Calculus
Other Applications of the Floor Function
The key fact that
\lfloor x \rfloor \le x < \lfloor x \rfloor +1
is often enough to solve basic problems involving the floor function.
x
\big\lfloor 0.5 + \lfloor x \rfloor \big\rfloor = 20 .
\lfloor x \rfloor= y.
\lfloor 0.5 + y \rfloor = 20 .
20\le y + 0.5 < 21,
19.5\le y < 20.5 .
y
y = 20
is the only integer in that interval, this becomes
y = 20 = \lfloor x \rfloor.
Any value less than
21
20
will satisfy this equation. Thus, the answer is all the real numbers
x
20\le x<21 . \ _\square
\big\lfloor 5 - \lfloor x \rfloor \big\rfloor = 15
If all the values of
x
that satisfy the equation above are in the interval
a \le x < b
, find the product
ab
\lfloor x \rfloor
is the floor function, or the greatest integer function.
\left \lfloor \dfrac{10^n}{x} \right \rfloor=1989
n \in \mathbb{N}
such that the equation above has an integer solution
x.
\lfloor x+n \rfloor = \lfloor x \rfloor + n
n.
\lfloor x \rfloor + \lfloor -x \rfloor = \begin{cases} -1&\text{if } x \notin {\mathbb Z} \\ 0&\text{if } x\in {\mathbb Z}. \end{cases}
\lfloor x+y \rfloor = \lfloor x \rfloor + \lfloor y \rfloor
\lfloor x \rfloor + \lfloor y \rfloor + 1.
The proofs of these are straightforward. To illustrate, here is a proof of (2). If
x
\lfloor x \rfloor + \lfloor -x \rfloor = x+(-x) = 0.
x
\lfloor x \rfloor < x < \lfloor x \rfloor + 1.
-\lfloor x \rfloor -1 < -x < -\lfloor x \rfloor,
and the outsides of the inequality are consecutive integers, so the left side of the inequality must equal
\lfloor -x \rfloor,
by the characterization of the greatest integer function given in the introduction.
\lfloor -x \rfloor = -\lfloor x \rfloor - 1,
\lfloor x \rfloor + \lfloor -x \rfloor = -1.
_\square
Problems involving the floor function of
x
are often simplified by writing
x = n+r
n = \lfloor x \rfloor
r = \{x\}
0\le r <1.
\lfloor x \rfloor\{x\} = 1
\lfloor x \rfloor^2 - \lfloor x \rfloor(1+x) + 4 = 0
x
x = n+r
n = \lfloor x \rfloor
r = \{ x \}
as suggested above. Then the first equation becomes
nr=1.
Expanding and rearranging the second equation,
\begin{aligned} n^2 - n(1+n+r) + 4 &= 0\\ -n-nr+4&=0\\ -n+3&=0, \end{aligned}
n=3.
r=\frac13
x = n+r = \frac{10}3.
_\square
Find the smallest positive real
x
\big\lfloor x^2 \big\rfloor-x\lfloor x \rfloor=6.
If your answer is in the form
\frac{a}{b}
and
b
a+b.
\lfloor \cdot \rfloor
Definite integrals and sums involving the floor function are quite common in problems and applications. The best strategy is to break up the interval of integration (or summation) into pieces on which the floor function is constant.
\int\limits_0^\infty \lfloor x \rfloor e^{-x} \, dx.
Break the integral up into pieces of the form
\begin{aligned} \int\limits_n^{n+1} \lfloor x \rfloor e^{-x} \, dx &= \int\limits_n^{n+1} ne^{-x} \, dx \\ &= -ne^{-x}\Big|_n^{n+1} \\ &= n\left(e^{-n}-e^{-(n+1)}\right) \\ &= ne^{-(n+1)}(e-1). \end{aligned}
So the integral is the sum of these pieces over all
n
\begin{aligned} \int_0^\infty \lfloor x \rfloor e^{-x} \, dx &= \sum_{n=0}^\infty \int_n^{n+1} \lfloor x \rfloor e^{-x} \, dx \\ &= \sum_{n=0}^\infty ne^{-(n+1)}(e-1) \\ &= (e-1)\sum_{n=0}^\infty \frac{n}{e^{n+1}}, \end{aligned}
\sum\limits_{n=0}^\infty nx^{n+1} = x^2\sum\limits_{n=0}^\infty nx^{n-1} = \frac{x^2}{(1-x)^2}
by differentiating the geometric series, so the answer is
(e-1)\frac{\left(\frac1e\right)^2}{\left(1-\frac1e\right)^2} = (e-1)\frac1{(e-1)^2} = \frac1{e-1}.\ _\square
\ln 2
\frac2e
e^{2}
\int_0^\infty \! \left\lfloor 2e^{-x} \right\rfloor dx,
\lfloor \cdot \rfloor
S = \left\lfloor \sqrt{1} \right\rfloor +\left\lfloor \sqrt{2} \right\rfloor +\left\lfloor \sqrt{3} \right\rfloor +\cdots +\left\lfloor \sqrt{1988} \right\rfloor
S
One common application of the floor function is finding the largest power of a prime dividing a factorial.
p
n
a positive integer. The largest power of
p
n!
p^k,
k = \left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \cdots = \sum_{i=1}^\infty \left\lfloor \frac{n}{p^i} \right\rfloor.
n.
n =1
is clear (both sides are 0), and if it is true for
n-1,
then the largest power of
p
n! = (n-1)! \cdot n
p^\ell,
\ell = v_p(n) + \sum_{i=1}^\infty \left\lfloor \frac{n-1}{p^i} \right\rfloor,
v_p(n)
k
p^k|n.
\left\lfloor \frac{n}{p^i} \right\rfloor - \left\lfloor \frac{n-1}{p^i} \right\rfloor = 1
p^i
n,
0
otherwise. The number of
i \ge 1
p^i
n
v_p(n),
\begin{aligned} \sum_{i=1}^\infty \left\lfloor \frac{n}{p^i} \right\rfloor - \sum_{i=1}^\infty \left\lfloor \frac{n-1}{p^i} \right\rfloor &= v_p(n) \\ \sum_{i=1}^\infty \left\lfloor \frac{n}{p^i} \right\rfloor &= v_p(n) + \sum_{i=1}^\infty \left\lfloor \frac{n-1}{p^i} \right\rfloor, \end{aligned}
\ell,
_\square
8000!
n! = n \times (n-1) \times (n-2) \times \cdots \times 3 \times 2 \times 1.
Omegatron, . Floor function. Retrieved March 30, 2006, from https://commons.wikimedia.org/wiki/File:Floor_function.svg
Cite as: Floor Function. Brilliant.org. Retrieved from https://brilliant.org/wiki/floor-function/
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IsKei - Maple Help
Home : Support : Online Help : Mathematics : Algebra : Magma : IsKei
test whether a magma is a kei (involutary quandle)
IsKei( m )
The IsKei command returns true if the given magma is a kei. It returns false otherwise.
A kei, also called an involutary quandle, is a quandle that satisfies the right involutary law (XY)Y = X.
\mathrm{with}\left(\mathrm{Magma}\right):
m≔〈〈〈1,3,2〉|〈3,2,1〉|〈2,1,3〉〉〉
\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{3}\end{array}]
\mathrm{IsKei}\left(m\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
m≔〈〈〈1,2,3〉|〈2,3,3〉|〈3,1,2〉〉〉
\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}\end{array}]
\mathrm{IsKei}\left(m\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
The Magma[IsKei] command was introduced in Maple 15.
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Global Constraint Catalog: Clex_greater
<< 5.228. lex_equal5.230. lex_greatereq >>
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛}\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1},\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}\right)
\mathrm{𝚕𝚎𝚡}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}
\mathrm{𝚛𝚎𝚕}
\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛}
\mathrm{𝚐𝚝}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1},\mathrm{𝚟𝚊𝚛}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2},\mathrm{𝚟𝚊𝚛}\right)
|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|=|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}|
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}
\stackrel{\to }{X}
\stackrel{\to }{Y}
n
〈{X}_{0},\cdots ,{X}_{n-1}〉
〈{Y}_{0},\cdots ,{Y}_{n-1}〉
\stackrel{\to }{X}
\stackrel{\to }{Y}
{X}_{0}>{Y}_{0}
{X}_{0}={Y}_{0}
〈{X}_{1},\cdots ,{X}_{n-1}〉
〈{Y}_{1},\cdots ,{Y}_{n-1}〉
\left(〈5,2,7,1〉,〈5,2,6,2〉\right)
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}=〈5,2,7,1〉
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}=〈5,2,6,2〉
|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|>1
\bigvee \left(\begin{array}{c}|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|<5,\hfill \\ \mathrm{𝚗𝚟𝚊𝚕}\left(\left[\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}.\mathrm{𝚟𝚊𝚛},\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}.\mathrm{𝚟𝚊𝚛}\right]\right)<2*|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|\hfill \end{array}\right)
\bigvee \left(\begin{array}{c}\mathrm{𝚖𝚊𝚡𝚟𝚊𝚕}\left(\left[\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}.\mathrm{𝚟𝚊𝚛},\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}.\mathrm{𝚟𝚊𝚛}\right]\right)\le 1,\hfill \\ 2*|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}|-\mathrm{𝚖𝚊𝚡}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}\left(\left[\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}.\mathrm{𝚟𝚊𝚛},\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}.\mathrm{𝚟𝚊𝚛}\right]\right)>2\hfill \end{array}\right)
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}
The following reformulations in term of arithmetic and/or logical expressions exist for enforcing the lexicographically strictly greater than constraint. The first one converts
\stackrel{\to }{X}
\stackrel{\to }{Y}
\stackrel{\to }{X}
\stackrel{\to }{Y}
\left[0,a-1\right]
{a}^{n-1}{Y}_{0}+{a}^{n-2}{Y}_{1}+\cdots +{a}^{0}{Y}_{n-1}<{a}^{n-1}{X}_{0}+{a}^{n-2}{X}_{1}+\cdots +{a}^{0}{X}_{n-1}
n
a
\left({Y}_{0}<{X}_{0}+\left({Y}_{1}<{X}_{1}+\left(\cdots +\left({Y}_{n-1}<{X}_{n-1}+0\right)\cdots \right)\right)\right)=1
Finally, the lexicographically strictly greater than constraint can be expressed as a conjunction or a disjunction of constraints:
\begin{array}{cc}\hfill {Y}_{0}\le {X}_{0}& \hfill \wedge \\ \hfill \left({Y}_{0}={X}_{0}\right)⇒{Y}_{1}\le {X}_{1}& \hfill \wedge \\ \hfill \left({Y}_{0}={X}_{0}\wedge {Y}_{1}={X}_{1}\right)⇒{Y}_{2}\le {X}_{2}& \hfill \wedge \\ \hfill ⋮& \\ \hfill \left({Y}_{0}={X}_{0}\wedge {Y}_{1}={X}_{1}\wedge \cdots \wedge {Y}_{n-2}={X}_{n-2}\right)⇒{Y}_{n-1}<{X}_{n-1}& \\ & \\ \hfill {Y}_{0}<{X}_{0}& \hfill \vee \\ \hfill {Y}_{0}={X}_{0}\wedge {Y}_{1}<{X}_{1}& \hfill \vee \\ \hfill {Y}_{0}={X}_{0}\wedge {Y}_{1}={X}_{1}\wedge {Y}_{2}<{X}_{2}& \hfill \vee \\ \hfill ⋮& \\ \hfill {Y}_{0}={X}_{0}\wedge {Y}_{1}={X}_{1}\wedge \cdots \wedge {Y}_{n-2}={X}_{n-2}\wedge {Y}_{n-1}<{X}_{n-1}\end{array}
lex in Choco, rel in Gecode, lex_greater in MiniZinc, lex_chain in SICStus.
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛}
\mathrm{𝚌𝚘𝚗𝚍}_\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚋𝚎𝚝𝚠𝚎𝚎𝚗}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚕𝚎𝚜𝚜}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛}
\mathrm{𝚌𝚘𝚕}\left(\begin{array}{c}\mathrm{𝙳𝙴𝚂𝚃𝙸𝙽𝙰𝚃𝙸𝙾𝙽}-\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},𝚡-\mathrm{𝚒𝚗𝚝},𝚢-\mathrm{𝚒𝚗𝚝}\right),\hfill \\ \mathrm{𝚒𝚝𝚎𝚖}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-0,𝚡-0,𝚢-0\right)\right]\hfill \end{array}\right)
\mathrm{𝚌𝚘𝚕}\left(\begin{array}{c}\mathrm{𝙲𝙾𝙼𝙿𝙾𝙽𝙴𝙽𝚃𝚂}-\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},𝚡-\mathrm{𝚍𝚟𝚊𝚛},𝚢-\mathrm{𝚍𝚟𝚊𝚛}\right),\hfill \\ \left[\begin{array}{c}\mathrm{𝚒𝚝𝚎𝚖}\left(\begin{array}{c}\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}.\mathrm{𝚔𝚎𝚢},\hfill \\ 𝚡-\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}.\mathrm{𝚟𝚊𝚛},\hfill \\ 𝚢-\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}.\mathrm{𝚟𝚊𝚛}\hfill \end{array}\right)\hfill \end{array}\right]\hfill \end{array}\right)
\mathrm{𝙲𝙾𝙼𝙿𝙾𝙽𝙴𝙽𝚃𝚂}
\mathrm{𝙳𝙴𝚂𝚃𝙸𝙽𝙰𝚃𝙸𝙾𝙽}
\mathrm{𝑃𝑅𝑂𝐷𝑈𝐶𝑇}
\left(
\mathrm{𝑃𝐴𝑇𝐻}
,
\mathrm{𝑉𝑂𝐼𝐷}
\right)↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚝𝚎𝚖}\mathtt{1},\mathrm{𝚒𝚝𝚎𝚖}\mathtt{2}\right)
\bigvee \left(\begin{array}{c}\mathrm{𝚒𝚝𝚎𝚖}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡}>0\wedge \mathrm{𝚒𝚝𝚎𝚖}\mathtt{1}.𝚡=\mathrm{𝚒𝚝𝚎𝚖}\mathtt{1}.𝚢,\hfill \\ \mathrm{𝚒𝚝𝚎𝚖}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡}=0\wedge \mathrm{𝚒𝚝𝚎𝚖}\mathtt{1}.𝚡>\mathrm{𝚒𝚝𝚎𝚖}\mathtt{1}.𝚢\hfill \end{array}\right)
\mathrm{𝐏𝐀𝐓𝐇}_\mathrm{𝐅𝐑𝐎𝐌}_\mathrm{𝐓𝐎}
\left(\mathrm{𝚒𝚗𝚍𝚎𝚡},1,0\right)=1
\mathrm{𝐏𝐀𝐓𝐇}_\mathrm{𝐅𝐑𝐎𝐌}_\mathrm{𝐓𝐎}
graph property we show the following information on the final graph:
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛}
{c}_{i}
i
We create an additional dummy vertex called
{c}_{i}
and
{\mathrm{𝚒𝚝𝚎𝚖}}_{1}.x>{\mathrm{𝚒𝚝𝚎𝚖}}_{2}.y
{c}_{i}
{c}_{i+1}
{\mathrm{𝚒𝚝𝚎𝚖}}_{1}.x={\mathrm{𝚒𝚝𝚎𝚖}}_{2}.y
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛}
{c}_{1}
d
. This path can be interpreted as a sequence of equality constraints on the prefix of both vectors, immediately followed by a greater than constraint.
\mathrm{𝐏𝐀𝐓𝐇}_\mathrm{𝐅𝐑𝐎𝐌}_\mathrm{𝐓𝐎}
\mathrm{𝐏𝐀𝐓𝐇}_\mathrm{𝐅𝐑𝐎𝐌}_\mathrm{𝐓𝐎}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡},1,0\right)=1
\mathrm{𝐏𝐀𝐓𝐇}_\mathrm{𝐅𝐑𝐎𝐌}_\mathrm{𝐓𝐎}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡},1,0\right)\ge 1
\underline{\overline{\mathrm{𝐏𝐀𝐓𝐇}_\mathrm{𝐅𝐑𝐎𝐌}_\mathrm{𝐓𝐎}}}
\overline{\mathrm{𝐏𝐀𝐓𝐇}_\mathrm{𝐅𝐑𝐎𝐌}_\mathrm{𝐓𝐎}}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛}
\mathrm{𝚅𝙰𝚁}{\mathtt{1}}_{i}
\mathrm{𝚅𝙰𝚁}{\mathtt{2}}_{i}
\mathrm{𝚟𝚊𝚛}
{i}^{th}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{1}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\mathtt{2}
\left(\mathrm{𝚅𝙰𝚁}{\mathtt{1}}_{i},\mathrm{𝚅𝙰𝚁}{\mathtt{2}}_{i}\right)
{S}_{i}
\left(\mathrm{𝚅𝙰𝚁}{\mathtt{1}}_{i}<\mathrm{𝚅𝙰𝚁}{\mathtt{2}}_{i}⇔{S}_{i}=1\right)\wedge \left(\mathrm{𝚅𝙰𝚁}{\mathtt{1}}_{i}=\mathrm{𝚅𝙰𝚁}{\mathtt{2}}_{i}⇔{S}_{i}=2\right)\wedge \left(\mathrm{𝚅𝙰𝚁}{\mathtt{1}}_{i}>\mathrm{𝚅𝙰𝚁}{\mathtt{2}}_{i}⇔{S}_{i}=3\right)
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛}
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Early Warning Effect of “Wearing Cap” and “Catching Cap” on the Company’s Risk Structure
—Empirical Research Based on Breakpoint Regression Design
―Empirical Research Based on Breakpoint Regression Design
{Y}_{i}={\alpha }_{0}+{\alpha }_{1}D+{\alpha }_{2}{Z}_{i}+{\epsilon }_{i}
{Y}_{i}
{\alpha }_{1}
{Z}_{i}
{\alpha }_{0}
{\epsilon }_{i}
{\alpha }_{1}
{\alpha }_{1}
Ruan, S. (2019) Early Warning Effect of “Wearing Cap” and “Catching Cap” on the Company’s Risk Structure. Modern Economy, 10, 1018-1032. https://doi.org/10.4236/me.2019.103068
1. Servaes, J. (2017) A Matter of Trust? The Bond Market Benefits of Corporate Social Capital during the Financial Crisis. Accounting Research, No. 4, 23-27.
2. Cai, W.C. (2016) Empirical Analysis of Financial Deterioration Prediction of Listed Companies. Accounting Research, No. 4, 31-38.
3. Feng, Y. (2009) A Study on the Usefulness of A Share Surplus Report—An Empirical Study from Shanghai and Shenzhen Stock Markets. Economic Research, No. 6, 21-28.
4. Lv, C. (2007) Information Advantage, Timing Behavior and Insider Trading of Major Shareholders. Financial Research, No. 5, 179-192.
5. Li, Z. (2006) An Empirical Study on the Performance of ST Company’s Cap Removal under the Pressure of Delisting. Friends of Accounting, No. 2, 53-59.
6. Wu, J.H. (2013) An Empirical Study on the Prediction of ST Companies in China’s Securities Market. Economic Science, No. 6, 57-67.
7. Tang, Q.M. (2006) Information Content and Influencing Factors of Listed Companies Being Specially Processed (ST) Announcement. Financial Research, No. 9, 61-71.
8. Peng, J. (2011) CEO Change and Financial Distress Recovery—An Empirical Study Based on ST’s Listed Companies “Caps”. Journal of Capital University of Economics and Business, 14, 47-54.
9. Cheng, J. (2010) Study on the Status Quo of Executive Compensation Incentive Mechanism of ST Company—Based on Empirical Evidence of Listed Companies. Friends of Accounting, No. 30, 64-66.
10. Mahenthiran, J. (2009) Analysis of Financial Statement Analysis and ST Forecast of Listed Companies. Audit Research, No. 6, 60-63.
11. Green, J. (2009) Should a Listed Company Lose Its “ST” for Two Consecutive Years? Economic Research, No. 3, 100-107.
12. Yang, J. (2006) Research on the Role of “Special Treatment” in Listed Companies. Accounting Research, No. 8, 15-21.
13. Bai, J. (2004) New CEO Characteristics, Management Team Adjustment and Corporate Performance—Based on Data Empirical Data of ST Listed Companies. China Management Science, 22, 47-55.
14. Zhu, T. (2012) Who Can Exempt Salary Punishment?—Based on the Research of ST Company. Accounting Research, No. 12, 46-51.
15. Zhu, D.S. (2006) Research on Cross-Layer Governance of ST Risk in Chinese Listed Companies: Based on Regional Differences. Economic Geography, 34, 23-30.
16. Gong, D.L. (2012) Study on Strategic Reorganization of Financial Crisis of Listed ST Companies in China. Management Modernization, No. 3, 53-57.
17. Zhang, X.M. (2016) Supporting, Restructuring and ST Company’s “Uncap” Road. Nankai Management Review, 9, 39-44.
18. Jiang, G.H. (2004) Backdoor Listing, Insider Trading and Stock Price Change— Based on the Study of ST Companies. Financial Research, No. 5, 126-142.
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AddCoordinates - Maple Help
Home : Support : Online Help : Mathematics : Vector Calculus : AddCoordinates
AddCoordinates(newsys, eqns, owrite, global)
symbol[name, name, ...]; specify the name of the new coordinate system indexed by the names of the new coordinates
list(algebraic); specify the expressions relating the new coordinate system to cartesian coordinates
(optional) equation of the form overwrite=t where t is either true or false; specify whether to overwrite an existing system in the coordinate tables
(optional) equation of the form addtoglobal=t, where t is either true or false; specify whether to add coordinate system to the global coordinate system tables
The AddCoordinates(newsys, eqns, owrite, global) command adds a new orthogonally curvilinear coordinate system to the VectorCalculus package. This new system can be used in the same way as the other built-in coordinate systems.
If the owrite argument is specified, it must be the name overwrite or an equation of type identical(overwrite)=truefalse. The default is overwrite=false.
If a coordinate system already exists with the same name as in newsys, the owrite parameter determines whether the old system is overwritten. If the old system is not overwritten, an error is raised.
If the global argument is specified, it must be an equation of type identical(addtoglobal)=truefalse. The default is addtoglobal=true.
If addtoglobal=true, then the specified coordinate system is also added to the global coordinate system tables and in consequence will be available in the plots package.
Note that the global coordinate system table only supports 2 or 3-dimensional coordinate systems with a maximum of 3 parameters.
Before calling AddCoordinates, place assumptions on the variables or use the assuming keyword. This assists computations, as more information is recognized for the possible values of the coordinates of the new system.
\mathrm{with}\left(\mathrm{VectorCalculus}\right):
\mathrm{assume}\left(0<r,0\le \mathrm{\theta },\mathrm{\theta }<2\mathrm{\pi }\right)
\mathrm{AddCoordinates}\left('\mathrm{mypolar}'[r,\mathrm{\theta }],[r\mathrm{cos}\left(\mathrm{\theta }\right),r\mathrm{sin}\left(\mathrm{\theta }\right)]\right)
\textcolor[rgb]{0,0,1}{\mathrm{mypolar}}
\mathrm{Laplacian}\left(f\left(r,\mathrm{\theta }\right),'\mathrm{mypolar}'[r,\mathrm{\theta }]\right)
\frac{\frac{\textcolor[rgb]{0,0,1}{∂}}{\textcolor[rgb]{0,0,1}{∂}\textcolor[rgb]{0,0,1}{\mathrm{r~}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{r~}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{θ~}}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{r~}}\textcolor[rgb]{0,0,1}{}\left(\frac{{\textcolor[rgb]{0,0,1}{∂}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{∂}{\textcolor[rgb]{0,0,1}{\mathrm{r~}}}^{\textcolor[rgb]{0,0,1}{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{r~}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{θ~}}\right)\right)\textcolor[rgb]{0,0,1}{+}\frac{\frac{{\textcolor[rgb]{0,0,1}{∂}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{∂}{\textcolor[rgb]{0,0,1}{\mathrm{θ~}}}^{\textcolor[rgb]{0,0,1}{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{r~}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{θ~}}\right)}{\textcolor[rgb]{0,0,1}{\mathrm{r~}}}}{\textcolor[rgb]{0,0,1}{\mathrm{r~}}}
\mathrm{assume}\left(0\le u,0\le v\right)
\mathrm{AddCoordinates}\left('\mathrm{foo}'[u,v],[{u}^{2}+{v}^{2},{u}^{2}-{v}^{2}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{foo}}
F≔\mathrm{VectorField}\left(〈f\left(u,v\right),g\left(u,v\right)〉,'\mathrm{foo}'[u,v]\right)
\textcolor[rgb]{0,0,1}{F}\textcolor[rgb]{0,0,1}{≔}\left(\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{u~}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{v~}}\right)\right){\stackrel{\textcolor[rgb]{0,0,1}{_}}{\textcolor[rgb]{0,0,1}{e}}}_{\textcolor[rgb]{0,0,1}{u}}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{u~}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{v~}}\right)\right){\stackrel{\textcolor[rgb]{0,0,1}{_}}{\textcolor[rgb]{0,0,1}{e}}}_{\textcolor[rgb]{0,0,1}{v}}
\mathrm{Divergence}\left(F\right)
\frac{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{v~}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{∂}}{\textcolor[rgb]{0,0,1}{∂}\textcolor[rgb]{0,0,1}{\mathrm{u~}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{u~}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{v~}}\right)\right)\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{u~}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{∂}}{\textcolor[rgb]{0,0,1}{∂}\textcolor[rgb]{0,0,1}{\mathrm{v~}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{u~}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{v~}}\right)\right)\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{u~}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{v~}}}
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'Physicality of Matter, foundations' in Po8 - johnvalentine.co.uk
Physicality of Matter, foundations
This article is an easy-reading version of our Physicality article.
After working on some problems in mathematical physics, we saw an opportunity to design a way for the foundations of physics to be expressed as minimally as possible.
Consistency with accepted science
Although this sounds pretentious, it is respectful to the work done so far on the Standard Model, which has proven correct, for the areas it is intended. Indeed, we reconcile with standard physics, and our purpose is to find a way of originating the assumptions of the Standard Model from some first principles, without necessarily using the Standard Model constructs and free parameters. The relevant standard details emerge as our configurations are grown and explored for their observable effects.
This article describes physicality, or the things that make up what is physical, what those things are doing when they are not physical (when they are ‘hiding’!), how things can behave fairly consistently, and then seem to randomly interact.
Quantum field theory is a statistical method, which does not usually tell you want happens in an exact experimental configuration. Instead, it quantifies the possible results you might get from similar configurations including some unknowns. It uses fields to contain energy and states, optionally-charged fermions, force-carrying particles to communicate between fermions, and coupling constants to define how much of one quantity transfers to another entity. Quantum mechanics, as used in this way, with the unknowns included, is non-deterministic, in that it does not tell you exactly what nor when your result will be, for any given input.
How are we thinking differently?
Our design is deterministic, and it claims to be able to hold all information that can define a system, such that its future states can be computed from a complete description, if you’re lucky enough to have one. Analytically, we can rebuild the constructs that QM enjoys, like fields, vacuum, wavefunctions, etc.
Our 'trick' is to go smaller than a fermion; we break it up, and find that a relatively simple construction can lead us to an expression of fermions, bosons, their interactions, and their observable effects, without needing to convert between incompatible algebras or pictures. We do this without discarding quantum mechanics, and continue to find ways that the existing standard work can be correct in its designed context.
We’ll start with the foundations. There are just six rules, which are enough to describe a dynamic framework for matter and energy. This first step is to define a basic unit of information.
1. Waves and bosons
Waves are scalar, bound in pairs as oscillating bosons
We’ve defined a boson, which has two waves, and only two waves. These waves cannot be separated, and they will always be bound together in the boson. No other waves can merge with these in the boson.
Fig.1: Boson propagation. See also: time.
This unit does some things that are important for us. First, it is an entity. It is not a physical entity yet, because we later define what ‘physical’ means, and it needs a few more conditions to be satisfied.
This entity has a mass-energy value, encoded by the two component waves. Later, we’ll see that this value is conserved, and influences interactions.
If you know a bit of Newtonian physics, you might have spotted that this resembles a harmonic oscillator with the same two components. Each wave is simply a sine wave, so you can address its properties either by its phase angle
\varphi
, or, more ambiguously, the value of the phase angle (some value on the
b
axis).
Unlike an oscillator, we don’t privilege either wave, so the boson is a superposition of states. Depending which wave is chosen as the reference, we can obtain spin-up or spin-down (spin being the sign of the oscillator’s angular momentum).
The rules below will read into some of the properties of this boson, and show how they express themselves in physical systems.
d\varphi =ds=dt
These bosons (waves) propagate at light speed, and only at light speed. The wave phase progresses as it extends from its source, and it does so at a constant rate. You could say that the oscillator itself defines time and distance, as it changes phase.
We do not assume any background space for this boson to propagate. Fundamentally, it’s just a changing phase value, extending in one spatial dimension, marking time implicitly. However, it might be easier to visualise this as a bubble in 3D space, whose surface is expanding outwards at the speed of light. Having zero thickness, the bubble is infinitesimally thin, and if you’ve defined a space, it sweeps over any defined points in that space. Nothing really exists inside or outside of this surface, and nothing, not even space itself, can be meaningfully addressed unless it is a boson surface.
3. Quantization and localization
\left({A}_{1}or{A}_{2}\right)=\left({B}_{1}or{B}_{2}\right)=-b
Bosons collapse into a fermion where waves from two different bosons have value
-b
at a unique point:
Fig.2: Fermion Event.
This provides opportunities for the continuous waves to generate a discrete point. It’s a quantization condition. Importantly, this is the only point where the two bosons share this condition, so they are coupled uniquely at this point. Immediately before and after this point, the bosons had no unique position, only a phase value, offset from their respective sources. We may construct an uncertainty principle from this behaviour.
It remains as a fermion for precisely zero time, before resuming as bosons, radiating from that point. The waves emerging from that point are entangled quanta.
Waves having the same phase and source are excluded
Fig.3: Conserved re-constitution
of an electron-type fermion.
Waves having the same phase and source are non-unique, so are excluded from interactions. This occurs immediately after every fermion, because two of the waves will have the same phase, and they originated from the same source point. There is no way to distinguish them. The exclusion is only removed when one of the two bosons is removed from the radiating shell, which happens when the one of the non-excluded waves couples to another boson (fig.3:
{t}_{2}
). We map this to electro-weak symmetry breaking.
5. Mass-energy
\rho =-b\phantom{\rule{mediummathspace}{0ex}}{e}^{-i\left({\varphi }_{B}-{\varphi }_{A}\right)}
We define mass-energy as a function of the phases of the waves in a boson. This means that the boson carries the mass-energy as it radiates. When the boson collapses (rule 3), the mass-energy is relocated and conserved.
You might have noticed that the oscillator of fig.1 is not ideal; one of its phases is offset slightly, so that it does not have a circular phase picture. This phase offset gives the boson mass-energy. Next, we’ll see how this helps collapse bosons, and generate gravitational fields.
6. Phase operator
{\varphi }_{\mathrm{modulated}}={\varphi }_{\mathrm{carrier}+\rho }\phantom{\rule{0.6em}{0ex}};\phantom{\rule{0.6em}{0ex}}\rho =\sum _{\mathrm{i=1}}^{n}{\rho }_{i}
Fig.4: Phase Modulation from ρ, on introduction of a shell to wave Z.
While propagating, mass-energy is a phase operator where it overlaps other bosons. If bosons overlap, they will co-modulate at the overlap, and enable a wider phase window for collapse. The implied behaviour is that large masses will collapse at smaller radius than smaller masses, where an environment of bosons is present.
The modulation advances or retards the collapse by a fraction of Planck length, which looks like a curvature of space.
We assume that the vacuum is simply the bosons radiated from other fermions, rather than a continuous field.
We’ve only just mentioned the vacuum, which is where it gets interesting, and we see lots of physics emerging, like the gravitational field, a context for charge, implicit Compton radius, and a picture of coherence that applies to trivial particles and black holes alike. You can find further reading in our 2014 paper, and a concise list of emergent properties in our introduction.
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In geometry, a nonagon /ˈnɒnəɡɒn/ (or enneagon /ˈɛniːəɡɒn/) is a nine-sidit regular polygon.
The name "nonagon" is a prefix hybrid formation, frae Laitin (nonus, "ninth" + gonon), uised equivalently, attestit already in the 16t century in French nonogone an in Inglis frae the 17t century. The name "enneagon" comes frae Greek enneagonon (εννεα, "nine" + γωνον (from γωνία = "corner")), an is arguably mair correct, tho somewhit less common nor "nonagon".
A regular nonagon haes internal angles o 140°. The aurie o a regular nonagon o side length a is gien bi
{\displaystyle A={\frac {9}{4}}a^{2}\cot {\frac {\pi }{9}}\simeq 6.18182\,a^{2}.}
Taen frae "https://sco.wikipedia.org/w/index.php?title=Nonagon&oldid=785896"
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@include( ABSPATH . WPINC . '/client.php'); Mathematica, 2.5 hours. Matlab, 47 seconds. Not impressed, WRI, not impressed. – Posts technical---or quite simplistic
Mathematica, 2.5 hours. Matlab, 47 seconds. Not impressed, WRI, not impressed.
The code is more or less identical (Mathematica version is necessarily more verbose, because, well, the syntax is more verbose. Unless you are drawing a clock applet.)
By the way, since people or than me might be looking at this code, here's a brief synopsis of what it does: we are trying to optimize the transmission function of a certain planar structure over two parameters. The cost function we chose to use in this optimization is
\int | F^{-1}\{T(k)\} | /\mathrm{max}[|F^{-1}\{T(k)\}|]dx
, where T(k) depends on the optimization parameters. So the code simply iterates through a 2D matrix of parameters computing the cost function integral and recording the result.
Mathematica: Matlab:
(* Fourier transform function definitions: *)
FftShift1D[x_] :=
Module[{n = Ceiling[Length[x]/2]},
Join[Take[x, n - Length[x]],
Take[x, n]]]; (* operates just like matlab fftshift function *)
(* wrapper over the fft function. Return values are in a replacement \
list, so usage is as follows:
{a,b,c,d}={transform,kpoints,klim,smpRateK} \
/.PerformDft1[foo,xlim,samprate]
PerformDft1[func_, xLim_, smpRateX_] :=
Module[{kLim, step, kstep, Nw, data, dataTransform, xvec, kvec},
step = 1/smpRateX;
Nw = 2*xLim/step; kLim = smpRateX; kstep = 1/(2*xLim);
xvec = Table[x, {x, -xLim, xLim - step, step}];
kvec = Table[k, {k, -kLim/2, kLim/2 - kstep, kstep}];
data = FftShift1D[func /@ xvec];
dataTransform =
smpRateX^-1*Fourier[data, FourierParameters -> {1, -1}];
{transform -> dataTransform, kpoints -> kvec, klim -> kLim,
smpRateK -> 1/kstep}
TESlabTransmissionNice[kx_, mu11_, d_] :=
Module[{m0 = 1, mu1, kz0, kz1, epar = 1, mperp = 1, eps = 10^-20},
m0 = m0 + I*eps; mu1 = mu11 + I*eps; epar = epar + I*eps;
kz0 = Sqrt[m0 - (kx^2) ]; kz1 = Sqrt[mu1 (epar - kx^2/mperp)];
1 / (Cos[d kz1] -
I/2 ((mu1 kz0)/(m0 kz1) + (m0 kz1)/(mu1 kz0)) Sin[d kz1])
MagneticSlabTransmission[kx_, mupp_, d_] :=
Module[{mu1 = -1 + I*mupp, h = 0.5},
Exp[-2 Pi 2 h Sqrt[kx^2 - 1]] TESlabTransmissionNice[kx, mu1,
2 Pi d]];
CostFunctionIntegrand[fn_, print_: False] :=
Module[{xLim = 100, xSmpRate = 100, xform, kvec, srk},
{xform, kvec, srk} = {transform, kpoints, smpRateK} /.
PerformDft1[fn, xLim, xSmpRate];
xform = FftShift1D@xform;
If[print,
Print[ListPlot[Re[Transpose[{kvec, xform}]],
PlotRange -> {{-1, 1}, All}]];
Print["k sampling rate: ", srk];
Transpose[{kvec, Abs[xform]/Max[Abs[xform]]}]
ComputeCostFunction[fn_] := Module[{integrand, min, max},
integrand = CostFunctionIntegrand[fn];
min = Min[integrand[[All, 1]]];
max = Max[integrand[[All, 1]]];
Interpolation[integrand, InterpolationOrder -> 2][x], {x, min, max}]
ComputeCostAndOutput[d_, mpp_,
fname_] := PutAppend[{d, mpp,
ComputeCostFunction[MagneticSlabTransmission[#, mpp, d] &]},fname];
(* can be read in using perl -pe 's/{(.+)}$/$1/' to strip the braces,
(nb: here + should really be *, just can't easily put it within mma
comment); then: Import["param_optim_numerics\\par_opt1.dat", "CSV"] *)
Map[((ComputeCostAndOutput[#1, #2, "costfn1.dat"] &) @@ #) &,
Table[{d, mpp}, {d, 0.05, 1, 0.05}, {mpp, 0.05, 1, 0.05}], {2}]
%% ComputeCostAndOutput.m
outfile = fopen('par_opt1_matlab.dat','w');
for d=0.05:0.05:1
for mpp=0.05:0.05:1
integrand = CostFunctionIntegrand(...
@(x)(MagneticSlabTransmission(x,mpp,d)));
costfn=trapz(integrand.kvec,integrand.int);
fprintf(outfile,'%f,%f,%f\n',d,mpp,costfn);
% CostFunctionIntegrand.m
function out=CostFunctionIntegrand(fh)
xLim=100; xSmpRate=100;
dftStruct = PerformDft1(fh,xLim,xSmpRate);
xform = fftshift(dftStruct.transform);
out.kvec = dftStruct.kpoints;
out.int = abs(xform)./max(abs(xform));
% MagneticSlabTransmission.m
function out = MagneticSlabTransmission(kx,mu1,d,h)
out=exp(-2*pi*2*h*sqrt(kx.^2-1)).*...
TESlabTransmissionNice(kx,mu1,2*pi*d);
% TESlabTransmissionNice.m
function out = TESlabTransmissionNice(kx,mu11,d)
m0=1; epar=1; mperp=1;
m0 = m0 + i*eps; mu1 = mu11 + i*eps;
epar = epar + i*eps; %#ok
kz0 = sqrt(m0 - (kx.^2) );
kz1 = sqrt(mu1.*(epar - kx.^2/mperp));
out=(cos(d.*kz1)+(sqrt(-1)*(-1/2)).*(kz0.^(-1).*...
kz1.*m0.*mu1.^(-1)+ ...
kz0.*kz1.^(-1).*m0.^(-1).*mu1).*sin(d.*kz1)).^(-1);
% PerformDft1.m
function out = PerformDft1(fh,xLim,smpRateX)
Nw = 2*xLim/step; kLim = smpRateX;
kstep = 1/(2*xLim);
xvec = linspace(-xLim,xLim-step,Nw);
kvec = linspace(-kLim/2,kLim/2-kstep,Nw);
data = fftshift(fh(xvec));
dataTransform = smpRateX^-1*fft(data);
out.transform = dataTransform;
out.kpoints = kvec;
out.klim = kLim;
out.smpRateK = 1/kstep;
JonyEpsilon said
FWIW, I had a crack at speeding up the Mathematica code. You can more than double the speed by adding periods to the numeric constants, which forces Mathematica into doing approximate rather than exact numerics. You can get about another factor of two by replacing the core functions in the integrand with pure functions and removing the modules. (Mathematica is almost always fastest with pure functions as it then doesn't have to invoke the pattern matcher on the arguments. Modules involve some significant symbolic name-mangling and are pretty slow.)
It's still nowhere near 47 seconds, but I thought you might be interested anyway!
dnquark said
Interesting. I figured there'd be room for some optimization, and I'm really glad somebody looked into it. Doesn't make me feel much better about Mathematica, though. If I don't use Modules[] when developing code, my namespace is littered with globals, which eventually blows up in my face. And wrapping every possible numeric constant in N[] (or using decimals) gets tiring after a while.
Daniel Lichtblau said
The bottlenecks are:
(1) Use of exact inputs that will slow some functions (trigs, Sqrt[], and so on).
(2) Repeated omputation of a function that is mapped over a list. Much faster to have the function take a list argument and compute a list result. This is in part because we can now work with the vector as a packed array, and do not need to break it up element-by-element.
Here is the faster code. It runs in about 80 seconds on my machine.
FftShift1D[x_] := Module[{n = Ceiling[Length[x]/2]},
RotateRight[x, n]]
Module[{kLim = smpRateX, step = 1/smpRateX,
kstep = 1/(2*xLim), data, dataTransform, xvec, kvec},
xvec = Range[-xLim, xLim - step, step];
kvec = Range[-kLim/2, kLim/2 - kstep, kstep];
data = FftShift1D[func[N[xvec]]];
dataTransform = Fourier[data, FourierParameters -> {1, -1}]/smpRateX;
{dataTransform, kvec, 1/kstep}]
Module[{m0, mu1, kz0, kz1, epar, mperp = 1., eps = 10.^(-20)},
m0 = 1. + I*eps;
mu1 = mu11 + I*eps;
epar = 1. + I*eps;
kz0 = Sqrt[m0 - kx^2];
kz1 = Sqrt[mu1*(epar - kx^2/mperp)];
1/(Cos[d*kz1] -
I/2.*((mu1*kz0)/(m0*kz1) + (m0*kz1)/(mu1*kz0))*Sin[d*kz1])]
Module[{mu1 = -1. + I*mupp, h = 0.5},
Exp[-4*Pi*h*Sqrt[kx^2 - 1.]]*
TESlabTransmissionNice[kx, mu1, 2.*Pi*d]];
Module[{xLim = 100, xSmpRate = 100, xform, kvec,
srk}, {xform, kvec, srk} = PerformDft1[fn, N[xLim], N[xSmpRate]];
Print["k sampling rate: ", srk];];
Transpose[{N[kvec], Abs[xform]/Max[Abs[xform]]}]]
ComputeCostFunction[fn_] :=
Module[{integrand, min, max}, integrand = CostFunctionIntegrand[fn];
Integrate[Interpolation[integrand,InterpolationOrder->2][x],{x,
min,max}]
ComputeCostAndOutput[d_, mpp_, fname_] :=
PutAppend[{d, mpp,
ComputeCostFunction[MagneticSlabTransmission[#, mpp, d] &]},
fname];
Timing[Map[((ComputeCostAndOutput[#1, #2,
"/tmp/costfn1.dat"] &) @@ #) &,
Table[{d, mpp}, {d, 0.05, 1, 0.05}, {mpp, 0.05, 1, 0.05}], {2}];]
We can furthermore replace Integrate[...] by
Interpolation[integrand, InterpolationOrder -> 2][x], {x, min,
max}, PrecisionGoal -> 2, AccuracyGoal -> 2, MaxRecursion -> 6,
MaxPoints -> Ceiling[Length[integrand[[All, 1]]]]/2,
Method -> {"Trapezoidal", "SymbolicProcessing" -> None}]
This variant runs in under a minute on my machine (3 Ghz, Linux, version 7.0.1 Mathematica).
Map[((ComputeCostAndOutput[#1, #2, "/tmp/costfn1.dat"] &) @@ #) &,
Locating the bottlenecks was in part knowing to look for (1) above, coupled with use of Print[] statements to isolate (2).
My timing, on Solaris 1.5 GHz server, same version of Mathematica:
posted code, 8 mins 7 seconds;
replace Integrate with NIntegrate: 5m35s
Matlab? 48 seconds. still.
Now, I have to disqualify the NIntegrate results, because for several values of parameters, they seem to be numerically different from the Integrate[] or from matlab's trapz by about 5%. (Matlab agrees with the Integrate[] approach to within 0.0003%)
I translated MATLAB code to Mathematica code directly.
A bottleneck is in MagneticSlabTransmission.
If xLim=54, underflow occurs in MagneticSlabTransmission,
and Mathematica will use arbitrary precision arithmetic, not FPU.
trapz[x_, y_] := (x[[2 ;;]] - x[[;; -2]]).(y[[2 ;;]] + y[[;; -2]])/2;
fftshift[x_] := RotateRight[x, Quotient[Length[x], 2]];
PerformDft1[fh_, xLim_, smpRateX_] :=
Module[{step, Nw, kLim, kstep, xvec, kvec, data, dataTransform},
step = smpRateX^-1;
Nw = 2*xLim*smpRateX;
kLim = smpRateX;
kstep = (2*xLim)^-1;
data = fftshift[fh[xvec]];
dataTransform = step*Fourier[data, FourierParameters -> {1, -1}];
{dataTransform, kvec}
Module[{eps, m0, epar, mperp, mu1, kz0, kz1, kz2},
eps = $MachineEpsilon;
m0 = 1. + eps*I;
epar = 1. + eps*I;
mperp = 1.;
mu1 = mu11 + eps*I;
kz2 = kz1*m0/(kz0*mu1);
(Cos[d*kz1] - (kz2 + kz2^(-1))*Sin[d*kz1]*I/2)^(-1)
MagneticSlabTransmission[kx_, mu1_, d_] :=
With[{h = 0.5},
Exp[-4*Pi*h*Sqrt[kx^2 - 1 + 0.*I]]*
TESlabTransmissionNice[kx, mu1, 2*Pi*d](*
unpacking occurs . bottleneck *)
CostFunctionIntegrand[fh_] :=
Module[{xLim, xSmpRate, dftStruct, absxform},
xLim = 100.;
xSmpRate = 100.;
dftStruct = PerformDft1[fh, xLim, xSmpRate];
absxform = Abs[fftshift[dftStruct[[1]]]];
{dftStruct[[2]], absxform/Max[absxform]}
outfile = OpenWrite["costfn1.dat"];
Write[outfile, {d, mpp,
trapz @@
CostFunctionIntegrand[MagneticSlabTransmission[#, mpp, d] &]}],
{d, 0.05, 1, 0.05}, {mpp, 0.05, 1, 0.05}];
I found a solution to remove the bottleneck.
prepend this to my previous code.
Greg Klopper said
ALWAYS compile numeric functions. The performance boost is phenomenal to say the least. I'm not even talking about C compiling in version 8, just using the built-in Compile function to get away from symbolic/interpreted code. Also, give numeric attribute to your numeric functions. That way the interpreter knows that given a set of numbers, the result will never be symbolic, but also a number.
But comparing a true symbolic calculation engine against a purely numeric one is rather unfair. You can add extra effort to ask Mathematica to treat your input as purely numeric (like NIntegrate), but good luck getting Matlab to do some of the fascinating things that Mathematica can do out of the box (even if they take a bit of time, because a symbol's representation and checking its various values under various conditions takes longer).
attempt_to_blog++ : The nice thing about Wolfram Research is… linked to this post on July 8, 2009
[...] that the product of his life’s work, basically, blows, the good folks at Wolfram actually went and optimized my code, making it run an order of magnitude faster (although still an order of magnitude slower than [...]
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Polarity of a Molecule | Brilliant Math & Science Wiki
The polarity of a molecule tells whether the electron cloud is equally distributed across the atoms within the molecule, or whether an electronegative atom is affecting the electron density. The distribution of the electrons will affect the behavior and reactivity of the molecule. For example, you can predict which solvents will be most effective with a given chemical if you know its polarity.
Water and oil do not mix well, despite the fact that they are both homogenous solutions on their own. This behavior can be explained through a deeper look at the polar forces affecting the two solutions.
When atoms with differing electronegativity are bonded together, the electrons may spend more time around one atom than the other, creating an unequal distribution of charge and a polar bond. In a polar bond, the electron-rich atom has a partial negative charge (
\delta^-
) and the electron-poor atom has a partial positive charge(
\delta^+
The dipole moment is the result of asymmetrical charge distribution in a polar substance. Mathematically, it is the product of the partial charge on the bonded atoms and the distance between them.
The vector sum of the dipole moments determines the polarity of a molecule.
The vector sum of the dipole moments of a non-polar molecule will be zero. It is possible for a molecule to have polar bonds, but be a non-polar molecule. This occurs when the three-dimensional shape of the molecule is symmetric. For example, carbon dioxide
\left(\ce{CO2}\right)
and methane
\left(\ce{CH4}\right)
are non-polar because their symmetric shapes cancel out the dipole moments to zero, as shown in the figure below.
On the other hand, water
\left(\ce{H2O}\right)
is a polar molecule because the overall dipole moment points toward the oxygen atom (indicating the oxygen atom is the most electron-rich).
Understanding the electron pair and molecular geometries described in the valence shell electron pair repulsion (vsper) model is useful in predicting the polarity of molecules.
Polarity is important because it determines whether a molecule is hydrophilic (from the Greek for water-loving) or hydrophobic (from the Greek for water-fearing or water-averse).
Molecules with high polarity are hydrophilic, and mix well with other polar compounds such as water. Molecules that are non-polar or have very low polarity tend to be hydrophobic, and mix well with other non-polar (or nearly non-polar) compounds such as oil.
Polarity also affects the strength of intermolecular forces. The only intermolecular force that non-polar molecules exhibit is the van der Waals force. Polar molecules can bond with each other via dipole-dipole interactions, which are generally stronger than van der Waals forces. Thus, if two molecules are similar in size and one is polar while the other is non-polar, the polar molecule will have higher melting and boiling points compared to non-polar molecule.
Determine the polarity of a boron trifluoride molecule
\left(\ce{BF3}\right)
The above left shows the Lewis dot structure of a
\ce{BF3}
molecule. Boron trifluoride is an exception of the octet rule, where the boron atom only has 3 electron pairs. For this reason, boron trifluoride has a trigonal planar shape, which is symmetric. Thus, although the
\ce{B-F}
bond is polar, the dipole moments cancel out and the overall dipole moment sums to zero, as shown in the above right figure. Therefore, boron trifluoride is nonpolar. \
Oil or water would work the same Water Oil
Nonane (
\ce{C9H20}
) is a saturated hydrocarbon. Would it be better to make a nonane solution in oil or water?
Cite as: Polarity of a Molecule. Brilliant.org. Retrieved from https://brilliant.org/wiki/polarity-of-a-molecule/
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The forces and the velocities acting in a Darrieus turbine are depicted in figure 1. The resultant velocity vector,
{\displaystyle {\vec {W}}}
, is the vectorial sum of the undisturbed upstream air velocity,
{\displaystyle {\vec {U}}}
, and the velocity vector of the advancing blade,
{\displaystyle -{\vec {\omega }}\times {\vec {R}}}
{\displaystyle {\vec {W}}={\vec {U}}+\left(-{\vec {\omega }}\times {\vec {R}}\right)}
Thus the oncoming fluid velocity varies during each cycle. Maximum velocity is found for
{\displaystyle \theta =0{}^{\circ }}
and the minimum is found for
{\displaystyle \theta =180{}^{\circ }}
{\displaystyle \theta }
is the azimuthal or orbital blade position. The angle of attack,
{\displaystyle \alpha }
, is the angle between the oncoming air speed, W, and the blade's chord. The resultant airflow creates a varying, positive angle of attack to the blade in the upstream zone of the machine, switching sign in the downstream zone of the machine.
{\displaystyle V_{t}=R\omega +U\cos(\theta )}
{\displaystyle V_{n}=U\sin(\theta )}
{\displaystyle W={\sqrt {V_{t}^{2}+V_{n}^{2}}}}
Thus, combining the above with the definitions for the tip speed ratio
{\displaystyle \lambda =(\omega R)/U}
yields the following expression for the resultant velocity:
{\displaystyle W=U{\sqrt {1+2\lambda \cos \theta +\lambda ^{2}}}}
{\displaystyle \alpha =\tan ^{-1}\left({\frac {V_{n}}{V_{t}}}\right)}
{\displaystyle \alpha =\tan ^{-1}\left({\frac {\sin \theta }{\cos \theta +\lambda }}\right)}
{\displaystyle C_{L}={\frac {F_{L}}{{1}/{2}\;\rho AW^{2}}}{\text{ }};{\text{ }}C_{D}={\frac {D}{{1}/{2}\;\rho AW^{2}}}{\text{ }};{\text{ }}C_{T}={\frac {T}{{1}/{2}\;\rho AU^{2}R}}{\text{ }};{\text{ }}C_{N}={\frac {N}{{1}/{2}\;\rho AU^{2}}}}
{\displaystyle P={\frac {1}{2}}C_{p}\rho A\nu ^{3}}
{\displaystyle C_{p}}
{\displaystyle \rho }
is air density,
{\displaystyle A}
is the swept area of the turbine, and
{\displaystyle \nu }
is the wind speed.[7]
환경공학Heat recovery ventilationHydraulic engineering재생 가능 에너지풍력 터빈Wind turbine designDarrieus wind turbine베스타스풍력 발전바람Wind farm가스 터빈Renewable resource지속 가능한 발전Environmental protectionGlobal Wind Day사막화기후 변화가뭄자연재해
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Does there exist a random variable such that the probability of it exceeding or equaling its expected value is 0?
Suppose you are sitting in an exam hall for an IQ-test containing 5 True/False questions and 5 multiple-choice questions, each having 4 different choices such that only one of the choices is right.
The probability of scoring 100% in the test by selecting options randomly is
\frac{p}{q}
p
q
\left\lfloor \log _{ 2 }{ (p+q) } \right\rfloor \quad =\quad ?
by Clive Chen
S
H
S+H?
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Hessian Matrix - Fizzy
\mathbf{H} f=\left[ \begin{array}{cccc}{\frac{\partial^{2} f}{\partial x^{2}}} & {\frac{\partial^{2} f}{\partial x \partial y}} & {\frac{\partial^{2} f}{\partial x \partial z}} & {\cdots} \\ {\frac{\partial^{2} f}{\partial y \partial x}} & {\frac{\partial^{2} f}{\partial y^{2}}} & {\frac{\partial^{2} f}{\partial y \partial z}} & {\cdots} \\ {\frac{\partial^{2} f}{\partial z \partial x}} & {\frac{\partial^{2} f}{\partial z \partial y}} & {\frac{\partial^{2} f}{\partial z^{2}}} & {\cdots} \\ {\vdots} & {\vdots} & {\vdots} & {\ddots}\end{array}\right]
Example 1: Computing a Hessian
Problem: Compute the Hessian of
f(x, y)=x^{3}-2 x y-y^{6}
First compute both partial derivatives:
f_{x}(x, y)=\frac{\partial}{\partial x}\left(x^{3}-2 x y-y^{6}\right)=3 x^{2}-2 y
f_{y}(x, y)=\frac{\partial}{\partial y}\left(x^{3}-2 x y-y^{6}\right)=-2 x-6 y^{5}
With these, we compute all four second partial derivatives:
f_{x x}(x, y)=\frac{\partial}{\partial x}\left(3 x^{2}-2 y\right)=6 x
{f_{x y}(x, y)=\frac{\partial}{\partial y}\left(3 x^{2}-2 y\right)=-2}
{f_{y x}(x, y)=\frac{\partial}{\partial x}\left(-2 x-6 y^{5}\right)=-2}
f_{y y}(x, y)=\frac{\partial}{\partial y}\left(-2 x-6 y^{5}\right)=-30 y^{4}
The Hessian matrix in this case is a $ 2\times 2$ matrix with these functions as entries:
\mathbf{H} f(x, y)=\left[ \begin{array}{cc}{f_{x x}(x, y)} & {f_{x y}(x, y)} \\ {f_{y x}(x, y)} & {f_{y y}(x, y)}\end{array}\right]=\left[ \begin{array}{cc}{6 x} & {-2} \\ {-2} & {-30 y^{4}}\end{array}\right]
Problem: the function
f(x)=x^{\top} A x+b^{\top} x+c
A
n \times n
b
n
c
Determine the gradient of
\nabla f(x)
Determine the Hessian of
H_{f}(x)
compute the gradient
\nabla f(x)
\begin{aligned} \nabla f(x)&=\underbrace{\frac{\partial x^{T}}{\partial x}\cdot (Ax)+x^{T}\cdot \frac{\partial (Ax)}{\partial x}}_{product-rule}+\frac{\partial b^Tx}{\partial x}+\frac{\partial c}{\partial x}\\ &= Ax + x^{T}\cdot A+b \\ &= Ax + x\cdot A^{T} + b \\ &= (A+A^{T})x + b \end{aligned}
compute the Hessian
H_{f}(x)
H_{f}(x) = \frac{\partial \nabla f(x)}{\partial x} = A + A^{T}
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Thrust - Wikipedia
Find sources: "Thrust" – news · newspapers · books · scholar · JSTOR (December 2017) (Learn how and when to remove this template message)
A Pratt & Whitney F100 jet engine being tested. This engine produces a jet of gas to generate thrust. Its purpose is to propel a jet airplane. This particular model turbofan engine powers McDonnell Douglas F-15 and General Dynamics F-16 fighters both.
Thrust is a reaction force described quantitatively by Newton's third law. When a system expels or accelerates mass in one direction, the accelerated mass will cause a force of equal magnitude but opposite direction to be applied to that system.[1] The force applied on a surface in a direction perpendicular or normal to the surface is also called thrust. Force, and thus thrust, is measured using the International System of Units (SI) in newtons (symbol: N), and represents the amount needed to accelerate 1 kilogram of mass at the rate of 1 meter per second per second. In mechanical engineering, force orthogonal to the main load (such as in parallel helical gears) is referred to as static thrust.
2.4 Thrust axis
A fixed-wing aircraft propulsion system generates forward thrust when air is pushed in the direction opposite to flight. This can be done by different means such as the spinning blades of a propeller, the propelling jet of a jet engine, or by ejecting hot gases from a rocket engine.[2] Reverse thrust can be generated to aid braking after landing by reversing the pitch of variable-pitch propeller blades, or using a thrust reverser on a jet engine. Rotary wing aircraft use rotors and thrust vectoring V/STOL aircraft use propellers or engine thrust to support the weight of the aircraft and to provide forward propulsion.
A motorboat propeller generates thrust when it rotates and forces water backwards.
A rocket is propelled forward by a thrust equal in magnitude, but opposite in direction, to the time-rate of momentum change of the exhaust gas accelerated from the combustion chamber through the rocket engine nozzle. This is the exhaust velocity with respect to the rocket, times the time-rate at which the mass is expelled, or in mathematical terms:
{\displaystyle \mathbf {T} =\mathbf {v} {\frac {\mathrm {d} m}{\mathrm {d} t}}}
Where T is the thrust generated (force),
{\displaystyle {\frac {\mathrm {d} m}{\mathrm {d} t}}}
is the rate of change of mass with respect to time (mass flow rate of exhaust), and v is the velocity of the exhaust gases measured relative to the rocket.
Each of the three Space Shuttle Main Engines could produce a thrust of 1.8 meganewton, and each of the Space Shuttle's two Solid Rocket Boosters 14.7 MN (3,300,000 lbf), together 29.4 MN.[3]
By contrast, the simplified Aid For EVA Rescue (SAFER) has 24 thrusters of 3.56 N (0.80 lbf) each.[citation needed]
In the air-breathing category, the AMT-USA AT-180 jet engine developed for radio-controlled aircraft produce 90 N (20 lbf) of thrust.[4] The GE90-115B engine fitted on the Boeing 777-300ER, recognized by the Guinness Book of World Records as the "World's Most Powerful Commercial Jet Engine," has a thrust of 569 kN (127,900 lbf) until it was surpassed by the GE9X, fitted on the upcoming Boeing 777X, at 609 kN (134,300 lbf).
Thrust to power[edit]
The power needed to generate thrust and the force of the thrust can be related in a non-linear way. In general,
{\displaystyle \mathbf {P} ^{2}\propto \mathbf {T} ^{3}}
. The proportionality constant varies, and can be solved for a uniform flow:
{\displaystyle {\frac {\mathrm {d} m}{\mathrm {d} t}}=\rho A{v}}
{\displaystyle \mathbf {T} ={\frac {\mathrm {d} m}{\mathrm {d} t}}{v},\mathbf {P} ={\frac {1}{2}}{\frac {\mathrm {d} m}{\mathrm {d} t}}{v}^{2}}
{\displaystyle \mathbf {T} =\rho A{v}^{2},\mathbf {P} ={\frac {1}{2}}\rho A{v}^{3}}
{\displaystyle \mathbf {P} ^{2}={\frac {\mathbf {T} ^{3}}{4\rho A}}}
The inverse of the proportionality constant, the "efficiency" of an otherwise-perfect thruster, is proportional to the area of the cross section of the propelled volume of fluid (
{\displaystyle A}
) and the density of the fluid (
{\displaystyle \rho }
). This helps to explain why moving through water is easier and why aircraft have much larger propellers than watercraft.
Thrust to propulsive power[edit]
A very common question is how to compare the thrust rating of a jet engine with the power rating of a piston engine. Such comparison is difficult, as these quantities are not equivalent. A piston engine does not move the aircraft by itself (the propeller does that), so piston engines are usually rated by how much power they deliver to the propeller. Except for changes in temperature and air pressure, this quantity depends basically on the throttle setting.
{\displaystyle \mathbf {P} =\mathbf {F} {\frac {d}{t}}}
{\displaystyle \mathbf {P} =\mathbf {T} {v}}
This formula looks very surprising, but it is correct: the propulsive power (or power available [7]) of a jet engine increases with its speed. If the speed is zero, then the propulsive power is zero. If a jet aircraft is at full throttle but attached to a static test stand, then the jet engine produces no propulsive power, however thrust is still produced. The combination piston engine–propeller also has a propulsive power with exactly the same formula, and it will also be zero at zero speed – but that is for the engine–propeller set. The engine alone will continue to produce its rated power at a constant rate, whether the aircraft is moving or not.
Excess thrust[edit]
Thrust axis[edit]
The thrust axis for an airplane is the line of action of the total thrust at any instant. It depends on the location, number, and characteristics of the jet engines or propellers. It usually differs from the drag axis. If so, the distance between the thrust axis and the drag axis will cause a moment that must be resisted by a change in the aerodynamic force on the horizontal stabiliser.[8] Notably, the Boeing 737 MAX, with larger, lower-slung engines than previous 737 models, had a greater distance between the thrust axis and the drag axis, causing the nose to rise up in some flight regimes, necessitating a pitch-control system, MCAS. Early versions of MCAS malfunctioned in flight with catastrophic consequences, leading to the deaths of over 300 people in 2018 and 2019.[9][10]
Pound of thrust (same as pound (force))
^ "What is Thrust?". www.grc.nasa.gov. Archived from the original on 14 February 2020. Retrieved 2 April 2020.
^ "Newton's Third Law of Motion". www.grc.nasa.gov. Archived from the original on 3 February 2020. Retrieved 2 April 2020.
^ "Space Launchers - Space Shuttle". www.braeunig.us. Archived from the original on 6 April 2018. Retrieved 16 February 2018.
^ "AMT-USA jet engine product information". Archived from the original on 10 November 2006. Retrieved 13 December 2006.
^ Yoon, Joe. "Convert Thrust to Horsepower". Archived from the original on 13 June 2010. Retrieved 1 May 2009.
^ Yechout, Thomas; Morris, Steven. Introduction to Aircraft Flight Mechanics. ISBN 1-56347-577-4.
^ Anderson, David; Eberhardt, Scott (2001). Understanding Flight. McGraw-Hill. ISBN 0-07-138666-1.
^ Kermode, A.C. (1972) Mechanics of Flight, Chapter 5, 8th edition. Pitman Publishing. ISBN 0273316230
^ "Control system under scrutiny after Ethiopian Airlines crash". Al Jazeera. Archived from the original on 28 April 2019. Retrieved 7 April 2019.
^ "What is the Boeing 737 Max Maneuvering Characteristics Augmentation System?". The Air Current. 14 November 2018. Archived from the original on 7 April 2019. Retrieved 7 April 2019.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Thrust&oldid=1088439222"
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File Dialog - Maple Help
Home : Support : Online Help : Programming : Maplets : Elements : Dialog Elements : File Dialog
define a file chooser dialog
FileDialog(opts)
FileDialog[refID](opts)
equation(s) of the form option=value where option is one of approvecaption, closeonapprove, directory, filefilter, filename, fileselectionmode, filterdescription, height, onapprove, oncancel, reference, resizable, title, value, or width; specify options for the FileDialog element
The FileDialog dialog element defines a file chooser dialog.
The FileDialog element features can be modified by using options. To simplify specifying options in the Maplets package, certain options and contents can be set without using an equation. The following table lists elements, symbols, and types (in the left column) and the corresponding option or content (in the right column) to which inputs of this type are, by default, assigned.
A FileDialog element can contain Action or command elements to specify the onapprove and oncancel options.
A FileDialog element can be contained in a Maplet element.
The following table describes the control and use of the FileDialog element options.
closeonapprove
The text that appears on the approval button. By default, the text is Open.
closeonapprove = true or false
Indicates whether the file dialog should close when the approval button is clicked. Setting this to false leaves the dialog open after the approval button is clicked, and it must be manually closed after validation is performed from within the opapprove action. The default is true.
directory = string
The current directory for the file dialog.
filefilter = string
Searches for the file extension specified. The default is all files, which is equivalent to a filefilter of "*". To specify only files with a non-empty extension, use "*.*". To specify more than one file extension, separate by a comma, as in "*.gif,*.jpg".
The name of the currently selected file for the file dialog.
fileselectionmode = filesonly, directoriesonly, filesanddirectories
The type of files to allow for selection. By default this value is filesonly, which means that only files can be selected.
filterdescription = string
This is a description of the filter provided by the option filefilter. The default description is "All Files".
The action that occurs when the approval button is clicked if a file (not a directory) is selected. By default, the action shuts down the dialog.
A reference for the FileDialog element.
If the reference is specified by both an index, for example, FileDialog[refID], and a reference in the calling sequence, the index reference takes precedence.
The text that appears in the title bar of the file dialog. The default title is Select File.
* value = string
The string that represents the file that is currently selected, for example, "C:\\myfile.mpl". It is generally advisable to use the directory and filename options for setting an initial directory/file, and using value for output.
A Maplet application that has one file dialog, which returns the file selected in a list or NULL:
\mathrm{with}\left(\mathrm{Maplets}[\mathrm{Elements}]\right):
\mathrm{maplet}≔\mathrm{Maplet}\left(\mathrm{FileDialog}['\mathrm{FD1}']\left('\mathrm{filefilter}'="mpl",'\mathrm{filterdescription}'="Maple Source Files",'\mathrm{onapprove}'=\mathrm{Shutdown}\left(['\mathrm{FD1}']\right),'\mathrm{oncancel}'=\mathrm{Shutdown}\left(\right)\right)\right):
\mathrm{Maplets}[\mathrm{Display}]\left(\mathrm{maplet}\right)
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NDF_CINP
The routine obtains a new value for a character component of an NDF via the ADAM parameter system and uses it to replace any pre-existing value of that component in the NDF.
CALL NDF_CINP( PARAM, INDF, COMP, STATUS )
\ast
\ast
\ast
\ast
Name of the character component for which a value is to be obtained: ’LABEL’, ’TITLE’ or ’UNITS’.
A "null" parameter value is interpreted as indicating that no new value should be set for the character component. In this event, the routine will return without action (and without setting a STATUS value). A suitable default value for the character component should therefore be established before this routine is called.
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Simulating Amplitude Modulation using Python | Nabanita Sarkar
Simulating Amplitude Modulation using Python
In communication labs in our colleges we all generate amplitude modulated signals using CROs. But the same can be performed by using Python and few of its additional libraries and the end result can be equally dope. At first let’s revise the formulas needed to generate amplitude modulated signal.
m(t)=A_mcos(2\pi f_mt)
c(t)=A_ccos(2\pi f_ct)
s(t)=A_c[1+\mu cos(2\pi f_m t)] cos(2\pi f_ct)
Now to simulate the signals in python first we have to import 2 python libraries: numpy and matplotlib
Then we have to take carrier amplitude, carrier frequency, message amplitude, message frequency and modulation index as inputs.
A_c = float(input('Enter carrier amplitude: '))
f_c = float(input('Enter carrier frquency: '))
A_m = float(input('Enter message amplitude: '))
f_m = float(input('Enter message frquency: '))
modulation_index = float(input('Enter modulation index: ')
The built-in input function in python returns string value. So we have to convert then to integer or float. Now float is preferred because amplitude or frequencies can be in decimal too. The time function in analog communication is continuous function. To replicate its behavior we will use linspace function which will provide large number of discreet points which will act almost similar to continuous function. The linspace function will generate evenly spaced number within a given interval. The first argument of linspace is starting point, second argument is ending point and third argument is number of breakpoints between the given interval.
Now we will create our carrier, modulator(message), and product function. Here we will use sin , cos and pi from numpy.
carrier = A_c*np.cos(2*np.pi*f_c*t)
modulator = A_m*np.cos(2*np.pi*f_m*t)
product = A_c*(1+modulation_index*np.cos(2*np.pi*f_m*t))*np.cos(2*np.pi*f_c*t)
Here comes the plotting part. We will utilize matplotlib library functions. The subplot function creates more than one plot in the canvas. The plot function plots the given function. In plot function we can pass color names or their acronyms to plot the graph in any color we wish. The ‘g’ , ‘r’ stands for green and red respectively. The xlabel and ylabel prints the x-axis and y-axis variable names. And the title function prints the title of the over all plot.
plt.title('Amplitude Modulation')
plt.plot(modulator,'g')
plt.xlabel('Message signal')
plt.plot(carrier, 'r')
plt.xlabel('Carrier signal')
plt.plot(product, color="purple")
plt.xlabel('AM signal')
Now we can customize the plot as much we want. We can change the space between plots, font, size etc. And if we want to save the picture we can easily do so by savefig function where dpi means dots per inch which in today’s world, can be inter-changeably used with ppi(pixel per inch).
fig.savefig('Amplitude Modulation.png', dpi=100)
Thus we can create simple amplitude modulation plot in python just using two external libraries only. The article has been published in Medium Simulating Amplitude Modulation using Python
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Global Constraint Catalog: Kstrip_packing
<< 3.7.244. Statistics3.7.246. Strong articulation point >>
\mathrm{𝚍𝚒𝚏𝚏𝚗}
\mathrm{𝚐𝚎𝚘𝚜𝚝}
A constraint that can be used to model the strip packing problem: Given a set of rectangles pack them into an open ended strip of given width in order to minimise the total overall height. Borders of the rectangles to pack should be parallel to the borders of the strip and rectangles should not overlap. Some variants of strip packing allow to rotate rectangles from 90 degrees. Benchmarks with known optima can be obtained from Hopper's PhD thesis [Hopper00].
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Social Integrity and the Cost of Equity Capital
After the establishment of the new China and the course of the Cultural Revolution, under the general trend of reform and opening up, China’s market economy has developed rapidly, and the company’s credit has become more and more important. Since ancient times, honesty has always been the basic morality of human beings, and integrity has always been respected by literati. Integrity is the spiritual leader of a person, the soul of a company, and integrity is also the connotation of a country. And social integrity has high requirements for the integrity of individuals and companies and even the country because it is related to the interests of everyone. Social integrity is the cornerstone of the market economy’s continuous development in a good direction. It is also the foundation of the company’s sustainable management and development. A good social integrity environment can bring hope to individuals, bring opportunities to enterprises, and bring endless power to the country. Based on the importance of social integrity to the market economy, this paper explores the impact of social integrity on the company’s business development by searching and reviewing the literature on social integrity and the cost of equity capital. By analyzing the impact of social integrity on the company’s business development, this paper puts forward a viewpoint: In other words, the cost of equity capital is affected by social integrity and is negatively correlated. And by designing an empirical solution, the data are compiled to prove this point.
\text{COE}={\beta }_{0}+{\beta }_{1}\text{Honesty}+{\sum }^{\text{}}\left(\lambda \text{Control_Variable}\right)+\epsilon
{M}_{t}
{B}_{t}
{E}_{t}
{\text{ROE}}_{t+k}
{E}_{t+k}
{B}_{t+k-1}
{B}_{t+k}
{B}_{t+k}={B}_{t+k-1}+{E}_{t+k}-{D}_{t+k}
{D}_{t+k}
{D}_{t+k}=r\ast {E}_{t+k}
\text{COE}={\beta }_{0}+{\beta }_{1}\text{Honesty}+{\sum }^{\text{}}\left(\lambda \text{Control_Variable}\right)+\epsilon
{\beta }_{0}-{\beta }_{2}
Wang, M. (2019) Social Integrity and the Cost of Equity Capital. Open Journal of Business and Management, 7, 229-244. https://doi.org/10.4236/ojbm.2019.71016
1. Long, J.Y. (2002) Integrity: The Moral Soul of the Healthy Development of the Market Economy. Philosophical Research, 8, 27-34.
2. Yue, S.Z. (2005) Thoughts on the Construction of Accounting Integrity Evaluation System. Accounting Research, 4, 73-76.
3. Wang, L.L. (2005) Social Integrity Is the Basis for Sustainable Economic Development. Academic Exchange, 7, 60-63.
4. Ma, C., Shen, T. and Yan, H.P. (2010) The Roots of China’s Social Integrity Loss and Its Countermeasures. Development Research, 2, 138-141.
5. Hu, B.F. (2003) On the Reconstruction of Social Integrity. Enterprise Economy, 8, 14-15.
6. Xia, W.D. (2003) On the Relationship between Honesty and Market Economy. Teaching and Research, 4, 8-14.
7. Xu, Y.M. and Shi, L. (2014) Analysis of Social Integrity Costs and Benefits. Modern Business, 12, 24-25.
8. Zhao, X. (2011) Empirical Research on the Integrity and Enterprise Value of Listed Companies. Journal of Shanxi University of Finance and Economics, 1, 94-100.
9. Yang, X.S. (2002) Rational Thinking on Accounting Integrity Problem. Accounting Research, 3, 6-12.
10. Ping, X.Z. and Zhou, J.J. (2012) Institutional Environment and the Cost of Equity Capital—A Comparison between Chinese Provincial Data. Securities Market Herald, 8, 19-27.
11. Xie, F.H. and Bao, G.M. (2005) Research on the Relationship between Corporate Integrity and Competitive Advantage—Based on the Empirical Investigation of 188 Enterprises in Suzhou and Other Places. Nankai Management Review, 4, 21-22.
13. Ross, S.A. (1976) The Arbitrage Theory of Capital Asset Pricing. Journal of Economic Theory, 13, 341-360. https://doi.org/10.1016/0022-0531(76)90046-6
14. Fama, E.F. and French, K.R. (1993) Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics, 33, 3-56. https://doi.org/10.1016/0304-405X(93)90023-5
15. Williams, B. (1938) The Theory of Investment Value. Harvard University Press, Cambridge, 238-240.
16. Edwards, E. and Bell, P. (1961) The Theory and Measurement of Business Income. University of California Press, Berkeley, 176-179.
17. Gebhardt, W.R., Lee, C. and Swaminathan, B. (2001) Toward an Implied Cost of Capital. Journal of Accounting Research, 39, 135-176. https://doi.org/10.1111/1475-679X.00007
18. Kong, W.C. and Xue, H. (2005) Empirical Research on Corporate Governance, Investor Protection, and Equity Capital Costs. Business Economy, 5, 77-79.
19. Yan, H.H. (2011) Research on the Relationship between Internal Governance and Capital Cost. Financial Research, 6, 54-58.
20. Wang, Y.M. and Gong, Y.C. (2015) Research on the Relationship between Transparency of Information Disclosure and Cost of Equity Capital of Listed Companies. Journal of Xi’an University of Arts and Science (Social Science Edition), 2, 86-89.
21. Wu, S.Y. and Xu, J.H. (2001) Integrity: The Moral Basis of the Effective Operation of Modern Market Economy. Journal of Fudan University (Social Science Edition), 5, 1-6.
22. Gordon, J.R. and Gordon, M.J. (1997) The Finite Horizon Expected Return Model. Financial Analysts Journal, 53, 52-61. https://doi.org/10.2469/faj.v53.n3.2084
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Global Constraint Catalog: Ccumulative_two_d
<< 5.98. cumulative_product5.100. cumulative_with_level_of_priority >>
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
\mathrm{𝚍𝚒𝚏𝚏𝚗}
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚝𝚠𝚘}_𝚍\left(\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂},\mathrm{𝙻𝙸𝙼𝙸𝚃}\right)
\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\begin{array}{c}\mathrm{𝚜𝚝𝚊𝚛𝚝}\mathtt{1}-\mathrm{𝚍𝚟𝚊𝚛},\hfill \\ \mathrm{𝚜𝚒𝚣𝚎}\mathtt{1}-\mathrm{𝚍𝚟𝚊𝚛},\hfill \\ \mathrm{𝚕𝚊𝚜𝚝}\mathtt{1}-\mathrm{𝚍𝚟𝚊𝚛},\hfill \\ \mathrm{𝚜𝚝𝚊𝚛𝚝}\mathtt{2}-\mathrm{𝚍𝚟𝚊𝚛},\hfill \\ \mathrm{𝚜𝚒𝚣𝚎}\mathtt{2}-\mathrm{𝚍𝚟𝚊𝚛},\hfill \\ \mathrm{𝚕𝚊𝚜𝚝}\mathtt{2}-\mathrm{𝚍𝚟𝚊𝚛},\hfill \\ \mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}-\mathrm{𝚍𝚟𝚊𝚛}\hfill \end{array}\right)
\mathrm{𝙻𝙸𝙼𝙸𝚃}
\mathrm{𝚒𝚗𝚝}
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎}_\mathrm{𝚊𝚝}_\mathrm{𝚕𝚎𝚊𝚜𝚝}
\left(2,\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂},\left[\mathrm{𝚜𝚝𝚊𝚛𝚝}\mathtt{1},\mathrm{𝚜𝚒𝚣𝚎}\mathtt{1},\mathrm{𝚕𝚊𝚜𝚝}\mathtt{1}\right]\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎}_\mathrm{𝚊𝚝}_\mathrm{𝚕𝚎𝚊𝚜𝚝}
\left(2,\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂},\left[\mathrm{𝚜𝚝𝚊𝚛𝚝}\mathtt{2},\mathrm{𝚜𝚒𝚣𝚎}\mathtt{2},\mathrm{𝚕𝚊𝚜𝚝}\mathtt{2}\right]\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂},\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}\right)
\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂}.\mathrm{𝚜𝚒𝚣𝚎}\mathtt{1}\ge 0
\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂}.\mathrm{𝚜𝚒𝚣𝚎}\mathtt{2}\ge 0
\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂}.\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}\ge 0
\mathrm{𝙻𝙸𝙼𝙸𝚃}\ge 0
ℛ
of rectangles described by the
\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂}
collection. Enforces that at each point of the plane, the cumulated height of the set of rectangles that overlap that point, does not exceed a given limit.
\left(\begin{array}{c}〈\begin{array}{ccccccc}\mathrm{𝚜𝚝𝚊𝚛𝚝}\mathtt{1}-1\hfill & \mathrm{𝚜𝚒𝚣𝚎}\mathtt{1}-4\hfill & \mathrm{𝚕𝚊𝚜𝚝}\mathtt{1}-4\hfill & \mathrm{𝚜𝚝𝚊𝚛𝚝}\mathtt{2}-3\hfill & \mathrm{𝚜𝚒𝚣𝚎}\mathtt{2}-3\hfill & \mathrm{𝚕𝚊𝚜𝚝}\mathtt{2}-5\hfill & \mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}-4,\hfill \\ \mathrm{𝚜𝚝𝚊𝚛𝚝}\mathtt{1}-3\hfill & \mathrm{𝚜𝚒𝚣𝚎}\mathtt{1}-2\hfill & \mathrm{𝚕𝚊𝚜𝚝}\mathtt{1}-4\hfill & \mathrm{𝚜𝚝𝚊𝚛𝚝}\mathtt{2}-1\hfill & \mathrm{𝚜𝚒𝚣𝚎}\mathtt{2}-2\hfill & \mathrm{𝚕𝚊𝚜𝚝}\mathtt{2}-2\hfill & \mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}-2,\hfill \\ \mathrm{𝚜𝚝𝚊𝚛𝚝}\mathtt{1}-1\hfill & \mathrm{𝚜𝚒𝚣𝚎}\mathtt{1}-2\hfill & \mathrm{𝚕𝚊𝚜𝚝}\mathtt{1}-2\hfill & \mathrm{𝚜𝚝𝚊𝚛𝚝}\mathtt{2}-1\hfill & \mathrm{𝚜𝚒𝚣𝚎}\mathtt{2}-2\hfill & \mathrm{𝚕𝚊𝚜𝚝}\mathtt{2}-2\hfill & \mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}-3,\hfill \\ \mathrm{𝚜𝚝𝚊𝚛𝚝}\mathtt{1}-4\hfill & \mathrm{𝚜𝚒𝚣𝚎}\mathtt{1}-1\hfill & \mathrm{𝚕𝚊𝚜𝚝}\mathtt{1}-4\hfill & \mathrm{𝚜𝚝𝚊𝚛𝚝}\mathtt{2}-1\hfill & \mathrm{𝚜𝚒𝚣𝚎}\mathtt{2}-1\hfill & \mathrm{𝚕𝚊𝚜𝚝}\mathtt{2}-1\hfill & \mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}-1\hfill \end{array}〉,4\hfill \end{array}\right)
Part (A) of Figure 5.99.1 shows the 4 parallelepipeds of height 4, 2, 3 and 1 associated with the items of the
\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂}
collection (parallelepipeds since each rectangle also has a height). Part (B) gives the corresponding cumulated 2-dimensional profile, where each number is the cumulated height of all the rectangles that contain the corresponding region. The
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚝𝚠𝚘}_𝚍
constraint holds since the highest peak of the cumulated 2-dimensional profile does not exceed the upper limit 4 imposed by the last argument of the
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚝𝚠𝚘}_𝚍
Figure 5.99.1. Two representations of a 2-dimensional cumulative profile of the Example slot (where the profile provides for each point of coordinates
\left({c}_{x},{c}_{y}\right)
the corresponding sum of the heights of the items intersecting that point): (A) a three dimensional representation and (B) a two dimensional representation from above with the height of the profile in red; as for the
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
constraint the position of an item on the
z
axis does not matter, i.e. only its height matters.
|\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂}|>1
\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂}.\mathrm{𝚜𝚒𝚣𝚎}\mathtt{1}>0
\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂}.\mathrm{𝚜𝚒𝚣𝚎}\mathtt{2}>0
\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂}.\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}>0
\mathrm{𝙻𝙸𝙼𝙸𝚃}<
\mathrm{𝚜𝚞𝚖}
\left(\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂}.\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}\right)
\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂}
\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂}
\left(\mathrm{𝚜𝚝𝚊𝚛𝚝}\mathtt{1},\mathrm{𝚜𝚝𝚊𝚛𝚝}\mathtt{2}\right)
\left(\mathrm{𝚜𝚒𝚣𝚎}\mathtt{1},\mathrm{𝚜𝚒𝚣𝚎}\mathtt{2}\right)
\left(\mathrm{𝚕𝚊𝚜𝚝}\mathtt{1},\mathrm{𝚕𝚊𝚜𝚝}\mathtt{2}\right)
\left(\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}\right)
\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂}.\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}
\ge 0
\mathrm{𝚜𝚝𝚊𝚛𝚝}\mathtt{1}
\mathrm{𝚕𝚊𝚜𝚝}\mathtt{1}
\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂}
\mathrm{𝚜𝚝𝚊𝚛𝚝}\mathtt{2}
\mathrm{𝚕𝚊𝚜𝚝}\mathtt{2}
\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂}
\mathrm{𝙻𝙸𝙼𝙸𝚃}
\mathrm{𝚁𝙴𝙲𝚃𝙰𝙽𝙶𝙻𝙴𝚂}
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚝𝚠𝚘}_𝚍
constraint is a necessary condition for the
\mathrm{𝚍𝚒𝚏𝚏𝚗}
constraint in 3 dimensions (i.e., the placement of parallelepipeds in such a way that they do not pairwise overlap and that each parallelepiped has his sides parallel to the sides of the placement space).
A first natural way to handle this constraint would be to accumulate the compulsory part [Lahrichi82] of the different rectangles in a quadtree [Samet89]. To each leave of the quadtree we associate the cumulated height of the rectangles containing the corresponding region.
geost in Choco.
\mathrm{𝚍𝚒𝚏𝚏𝚗}
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚝𝚠𝚘}_𝚍
\mathrm{𝚍𝚒𝚏𝚏𝚗}
: forget one dimension when the number of dimensions is equal to 3).
\mathrm{𝚋𝚒𝚗}_\mathrm{𝚙𝚊𝚌𝚔𝚒𝚗𝚐}
\mathrm{𝚜𝚚𝚞𝚊𝚛𝚎}
of size 1 with a
\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}
\mathrm{𝚝𝚊𝚜𝚔}
\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
\mathrm{𝚛𝚎𝚌𝚝𝚊𝚗𝚐𝚕𝚎}
\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}
\mathrm{𝚝𝚊𝚜𝚔}
with same
\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}
filtering: quadtree, compulsory part.
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ABCD is a parallelogram,P and Q are points on DC and AB respectively,such that angle DAP=angleBCQ Show that Acqp is - Maths - Quadrilaterals - 7150241 | Meritnation.com
ABCD is a parallelogram,P and Q are points on DC and AB respectively,such that angle DAP=angleBCQ.Show that Acqp is a parallelogram.
Given: A paralle\mathrm{log}ram ABCD\phantom{\rule{0ex}{0ex}}To Prove: APCQ is paralle\mathrm{log}ram.\phantom{\rule{0ex}{0ex}}Proof: \angle A = \angle C\phantom{\rule{0ex}{0ex}}\left\{Opposite angle of paralle\mathrm{log}ram are equal\right\}\phantom{\rule{0ex}{0ex}}\angle DAP +\angle PAQ = \angle PCQ+\angle QCB\phantom{\rule{0ex}{0ex}}\angle PAQ = \angle PCQ \phantom{\rule{0ex}{0ex}}Which is a pair of opposite angles\phantom{\rule{0ex}{0ex}}so, APCQ is paralle\mathrm{log}ram.
Aishwarya Suresh answered this
It is given that angDAP= angBCQ.
But angDAP= ang APC ( alternate interior angles) and
angBCQ= ang AQC (alternate interior angles)
Therefore angAPC= ang AQC
Consider Quadrilateral AQCP,
Opposite angles i.e. ang APC and ang AQC are equal,
AQCP is a parallelogram
Thumbs up if you find it useful!
1s2, 2s2, 2p6, 3s2, 3p6, 4s2, 3d10, 4p6, 5s2, 4d10, 5p2
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Halogen lamp (11375 views - Electronics & PCB)
{\displaystyle V^{3}}
{\displaystyle V^{1.3}}
{\displaystyle V^{-14}}
Electrical wiringFluorescent lampHigh-voltage cableLED DisplayLightLight switchLight tableSwitchElectrical ballastCompact fluorescent lampTwo-port networkSafety lamp
This article uses material from the Wikipedia article "Halogen lamp", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia
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∇×\mathbf{F}
\mathbf{F}=\left[\begin{array}{c}u(\mathrm{ρ},\mathrm{φ},\mathrm{θ})\\ v(\mathrm{ρ},\mathrm{φ},\mathrm{θ})\\ w(\mathrm{ρ},\mathrm{φ},\mathrm{θ})\end{array}\right]
be a vector field in cylindrical coordinates.
From Table 9.3.3, the curl of F is given by
∇×\mathbf{F}=\left|\begin{array}{ccc}\frac{{\mathbit{e}}_{\mathbf{\rho }}}{{\mathrm{\rho }}^{2}\mathrm{sin}\left(\mathrm{\phi }\right)}& \frac{{\mathbit{e}}_{\mathbf{\phi }}}{\mathrm{\rho } \mathrm{sin}\left(\mathrm{\phi }\right)}& \frac{{\mathbit{e}}_{\mathbf{\theta }}}{\mathrm{\rho }}\\ {∂}_{\mathrm{\rho }}& {∂}_{\mathrm{\phi }}& {∂}_{\mathrm{\theta }}\\ u& \mathrm{\rho } v& \mathrm{\rho } w \mathrm{sin}\left(\mathrm{\theta }\right)\end{array}\right|=\left[\begin{array}{c}\frac{w \mathrm{cos}(\mathrm{\phi })}{\mathrm{\rho } \mathrm{sin}(\mathrm{\phi })}+\frac{{w}_{\mathrm{\phi }}}{\mathrm{\rho }}-\frac{{v}_{\mathrm{θ}}}{\mathrm{\rho } \mathrm{sin}(\mathrm{\phi })}\\ \frac{{u}_{\mathrm{\theta }}}{\mathrm{\rho } \mathrm{sin}(\mathrm{\phi })}-\frac{w}{\mathrm{\rho }}-{w}_{\mathrm{\rho }}\\ \frac{v}{\mathrm{\rho }}+{v}_{\mathrm{\rho }}-\frac{{u}_{\mathrm{\phi }}}{\mathrm{\rho }}\end{array}\right]
\left[\begin{array}{c}U\\ V\\ W\end{array}\right]
From Table 9.3.2, the divergence of
∇×\mathbf{F}
\frac{{\left({\mathrm{\rho }}^{2}U\right)}_{\mathrm{\rho }}}{{\mathrm{\rho }}^{2}}+\frac{{\left(V \mathrm{sin}\left(\mathrm{\phi }\right)\right)}_{\mathrm{\phi }}}{\mathrm{\rho } \mathrm{sin}\left(\mathrm{\phi }\right)}+\frac{{W}_{\mathrm{\theta }}}{\mathrm{\rho } \mathrm{sin}\left(\mathrm{\phi }\right)}
. Table 9.4.5(a) takes each term separately.
\frac{{\left({\mathrm{\rho }}^{2}U\right)}_{\mathrm{\rho }}}{{\mathrm{\rho }}^{2}}
=\frac{{∂}_{\mathrm{\rho }}\left({\mathrm{\rho }}^{2}\left(\frac{w \mathrm{cos}\left(\mathrm{\phi }\right)}{\mathrm{\rho } \mathrm{sin}\left(\mathrm{\phi }\right)}+\frac{{w}_{\mathrm{\phi }}}{\mathrm{\rho }}-\frac{{v}_{\mathrm{θ}}}{\mathrm{\rho } \mathrm{sin}\left(\mathrm{\phi }\right)}\right)\right)}{{\mathrm{\rho }}^{2}}
=\frac{{∂}_{\mathrm{\rho }}\left(\mathrm{\rho } w \frac{\mathrm{cos}\left(\mathrm{\phi }\right)}{\mathrm{sin}\left(\mathrm{\phi }\right)}+\mathrm{\rho } {w}_{\mathrm{\phi }}-\frac{\mathrm{\rho } {v}_{\mathrm{θ}}}{\mathrm{sin}\left(\mathrm{\phi }\right)}\right)}{{\mathrm{\rho }}^{2}}
=\frac{w\frac{\mathrm{cos}\left(\mathrm{\phi }\right)}{\mathrm{sin}\left(\mathrm{\phi }\right)}+\mathrm{\rho } {w}_{\mathrm{\rho }}\frac{\mathrm{cos}\left(\mathrm{\phi }\right)}{\mathrm{sin}\left(\mathrm{\phi }\right)}+{w}_{\mathrm{\phi }}+\mathrm{\rho } {w}_{\mathrm{\phi } \mathrm{\rho }}-\frac{{v}_{\mathrm{θ}}}{\mathrm{sin}\left(\mathrm{\phi }\right)}-\frac{\mathrm{\rho } {v}_{\mathrm{θ} \mathrm{ρ}}}{\mathrm{sin}\left(\mathrm{\phi }\right)} }{{\mathrm{\rho }}^{2}}
=\frac{w}{{\mathrm{\rho }}^{2}}\frac{\mathrm{cos}\left(\mathrm{\phi }\right)}{\mathrm{sin}\left(\mathrm{\phi }\right)}+\frac{{w}_{\mathrm{\rho }}}{\mathrm{\rho }}\frac{\mathrm{cos}\left(\mathrm{\phi }\right)}{\mathrm{sin}\left(\mathrm{\phi }\right)}+\frac{{w}_{\mathrm{\phi }}}{{\mathrm{\rho }}^{2}}+\frac{{w}_{\mathrm{\phi } \mathrm{\rho }}}{\mathrm{\rho }}-\frac{{v}_{\mathrm{\phi }}}{{\mathrm{\rho }}^{2}\mathrm{sin}\left(\mathrm{\phi }\right)}-\frac{{v}_{\mathrm{\phi } \mathrm{\rho }}}{\mathrm{\rho } \mathrm{sin}\left(\mathrm{\phi }\right)}
\frac{{\left(V \mathrm{sin}\left(\mathrm{\phi }\right)\right)}_{\mathrm{\phi }}}{\mathrm{\rho } \mathrm{sin}\left(\mathrm{\phi }\right)}
=\frac{{∂}_{\mathrm{\phi }}\left(\left(\frac{{u}_{\mathrm{\theta }}}{\mathrm{\rho } \mathrm{sin}\left(\mathrm{\phi }\right)}-\frac{w}{\mathrm{\rho }}-{w}_{\mathrm{\rho }}\right)\mathrm{sin}\left(\mathrm{\phi }\right)\right)}{\mathrm{\rho } \mathrm{sin}\left(\mathrm{\phi }\right)}c
=\frac{{∂}_{\mathrm{\phi }}\left(\frac{{u}_{\mathrm{\theta }}}{\mathrm{\rho }}-\frac{w \mathrm{sin}\left(\mathrm{\phi }\right)}{\mathrm{\rho }}-{w}_{\mathrm{\rho }}\mathrm{sin}\left(\mathrm{\phi }\right)\right)}{\mathrm{\rho } \mathrm{sin}\left(\mathrm{\phi }\right)}
=\frac{\frac{{u}_{\mathrm{\theta } \mathrm{\phi }}}{\mathrm{\rho }}-\frac{{w}_{\mathrm{\phi }} \mathrm{sin}\left(\mathrm{\phi }\right)+w \mathrm{cos}\left(\mathrm{\phi }\right)}{\mathrm{\rho }}-{w}_{\mathrm{\rho }}\mathrm{cos}\left(\mathrm{\phi }\right)-{w}_{\mathrm{\rho } \mathrm{\phi }}\mathrm{sin}\left(\mathrm{\phi }\right)}{\mathrm{\rho } \mathrm{sin}\left(\mathrm{\phi }\right)}
=\frac{{u}_{\mathrm{θ} \mathrm{φ}}}{{\mathrm{ρ}}^{2}\mathrm{sin}\left(\mathrm{φ}\right)}-\frac{{w}_{\mathrm{φ}}}{{\mathrm{ρ}}^{2}}-\frac{w}{{\mathrm{ρ}}^{2}}\frac{\mathrm{cos}\left(\mathrm{φ}\right)}{\mathrm{sin}\left(\mathrm{φ}\right)}-\frac{{w}_{\mathrm{ρ}}}{\mathrm{ρ}}\frac{\mathrm{cos}\left(\mathrm{φ}\right)}{\mathrm{sin}\left(\mathrm{φ}\right)}-\frac{{w}_{\mathrm{ρ} \mathrm{φ}}}{\mathrm{ρ}}
\frac{{W}_{\mathrm{\theta }}}{\mathrm{\rho } \mathrm{sin}\left(\mathrm{\phi }\right)}
=\frac{{∂}_{\mathrm{\theta }}\left(\frac{v}{\mathrm{\rho }}+{v}_{\mathrm{\rho }}-\frac{{u}_{\mathrm{\phi }}}{\mathrm{\rho }}\right)}{ \mathrm{\rho } \mathrm{sin}\left(\mathrm{\phi }\right)}
=\frac{\frac{{v}_{\mathrm{\theta }}}{\mathrm{\rho }}+{v}_{\mathrm{\rho } \mathrm{\theta }}-\frac{{u}_{\mathrm{\phi } \mathrm{\theta }}}{\mathrm{\rho }}}{\mathrm{\rho } \mathrm{sin}\left(\mathrm{\phi }\right)}
=\frac{{v}_{\mathrm{θ}}}{{\mathrm{ρ}}^{2}\mathrm{sin}\left(\mathrm{φ}\right)}+\frac{{v}_{\mathrm{ρ} \mathrm{θ}}}{\mathrm{ρ} \mathrm{sin}\left(\mathrm{φ}\right)}-\frac{{u}_{\mathrm{φ} \mathrm{θ}}}{{\mathrm{ρ}}^{2}\mathrm{sin}\left(\mathrm{φ}\right)}
Table 9.4.5(a) Divergence of the curl in spherical coordinates
Table 9.4.5(b) gathers up the three sets of terms that are to be summed.
\frac{w}{{\mathrm{\rho }}^{2}}\frac{\mathrm{cos}\left(\mathrm{\phi }\right)}{\mathrm{sin}\left(\mathrm{\phi }\right)}+\frac{{w}_{\mathrm{\rho }}}{\mathrm{\rho }}\frac{\mathrm{cos}\left(\mathrm{\phi }\right)}{\mathrm{sin}\left(\mathrm{\phi }\right)}+\frac{{w}_{\mathrm{\phi }}}{{\mathrm{\rho }}^{2}}+\frac{{w}_{\mathrm{\phi } \mathrm{\rho }}}{\mathrm{\rho }}-\frac{{v}_{\mathrm{θ}}}{{\mathrm{\rho }}^{2}\mathrm{sin}\left(\mathrm{\phi }\right)}-\frac{{v}_{\mathrm{θ} \mathrm{ρ}}}{\mathrm{\rho } \mathrm{sin}\left(\mathrm{\phi }\right)}
\frac{{u}_{\mathrm{\theta } \mathrm{\phi }}}{{\mathrm{\rho }}^{2}\mathrm{sin}\left(\mathrm{\phi }\right)}-\frac{{w}_{\mathrm{\phi }}}{{\mathrm{\rho }}^{2}}-\frac{w}{{\mathrm{\rho }}^{2}}\frac{\mathrm{cos}\left(\mathrm{\phi }\right)}{\mathrm{sin}\left(\mathrm{\phi }\right)}-\frac{{w}_{\mathrm{\rho }}}{\mathrm{\rho }}\frac{\mathrm{cos}\left(\mathrm{\phi }\right)}{\mathrm{sin}\left(\mathrm{\phi }\right)}-\frac{{w}_{\mathrm{\rho } \mathrm{\phi }}}{\mathrm{ρ}}
\frac{{v}_{\mathrm{\theta }}}{{\mathrm{\rho }}^{2}\mathrm{sin}\left(\mathrm{\phi }\right)}+\frac{{v}_{\mathrm{\rho } \mathrm{\theta }}}{\mathrm{\rho } \mathrm{sin}\left(\mathrm{\phi }\right)}-\frac{{u}_{\mathrm{\phi } \mathrm{\theta }}}{{\mathrm{\rho }}^{2}\mathrm{sin}\left(\mathrm{\phi }\right)}
Table 9.4.5(b) The terms in
∇·\left(∇×\mathbf{F}\right)
that are to be combined
Careful matching of terms and the equality of mixed partials leads to the vanishing of the sum of terms in Table 9.4.5(b).
Define the spherical vector field F
Write the vector field as a free vector.
Context Panel: Student Vector Calculus≻Conversions≻Apply Co-ordinate System (Complete dialog as per figure on right.)
〈u\left(\mathrm{ρ},\mathrm{φ},\mathrm{θ}\right),v\left(\mathrm{ρ},\mathrm{φ},\mathrm{θ}\right),w\left(\mathrm{ρ},\mathrm{φ},\mathrm{θ}\right)〉
\left[\begin{array}{c}u\textcolor[rgb]{0,0,1}{}\left(\mathrm{ρ},\mathrm{φ},\mathrm{θ}\right)\\ v\textcolor[rgb]{0,0,1}{}\left(\mathrm{ρ},\mathrm{φ},\mathrm{θ}\right)\\ w\textcolor[rgb]{0,0,1}{}\left(\mathrm{ρ},\mathrm{φ},\mathrm{θ}\right)\end{array}\right]
\stackrel{\text{apply coordinates}}{\to }
\left[\begin{array}{c}u\textcolor[rgb]{0,0,1}{}\left(\mathrm{ρ},\mathrm{φ},\mathrm{θ}\right)\\ v\textcolor[rgb]{0,0,1}{}\left(\mathrm{ρ},\mathrm{φ},\mathrm{θ}\right)\\ w\textcolor[rgb]{0,0,1}{}\left(\mathrm{ρ},\mathrm{φ},\mathrm{θ}\right)\end{array}\right]
\stackrel{\text{to Vector Field}}{\to }
\left[\begin{array}{c}u\textcolor[rgb]{0,0,1}{}\left(\mathrm{ρ},\mathrm{φ},\mathrm{θ}\right)\\ v\textcolor[rgb]{0,0,1}{}\left(\mathrm{ρ},\mathrm{φ},\mathrm{θ}\right)\\ w\textcolor[rgb]{0,0,1}{}\left(\mathrm{ρ},\mathrm{φ},\mathrm{θ}\right)\end{array}\right]
\stackrel{\text{assign to a name}}{\to }
\textcolor[rgb]{0,0,1}{F}
Compute the divergence of the curl of F
Del, dot product,and cross product operators
∇·\left(∇×\mathbf{F}\right)
\textcolor[rgb]{0,0,1}{0}
\mathrm{with}\left(\mathrm{Student}:-\mathrm{VectorCalculus}\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{BasisFormat}\left(\mathrm{false}\right):
Implement notational simplifications with the declare command in the PDEtools package
\mathrm{PDEtools}:-\mathrm{declare}\left(u\left(\mathrm{ρ},\mathrm{φ},\mathrm{θ}\right),v\left(\mathrm{ρ},\mathrm{φ},\mathrm{θ}\right),w\left(\mathrm{ρ},\mathrm{φ},\mathrm{θ}\right),\mathrm{quiet}\right)
Use the VectorField command in the Student VectorCalculus package to define F
\mathbf{F}≔\mathrm{VectorField}\left(〈u\left(\mathrm{ρ},\mathrm{φ},\mathrm{θ}\right),v\left(\mathrm{ρ},\mathrm{φ},\mathrm{θ}\right),w\left(\mathrm{ρ},\mathrm{φ},\mathrm{θ}\right)〉,\mathrm{spherical}\left[\mathrm{ρ},\mathrm{φ},\mathrm{θ}\right]\right)
\left[\begin{array}{c}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{\rho }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\phi }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\theta }}\right)\\ \textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{\rho }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\phi }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\theta }}\right)\\ \textcolor[rgb]{0,0,1}{w}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{\rho }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\phi }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\theta }}\right)\end{array}\right]
∇·\left(∇×\mathbf{F}\right)=0
Apply the Curl and Divergence commands from the Student VectorCalculus package.
\mathrm{Divergence}\left(\mathrm{Curl}\left(\mathbf{F}\right)\right)
\textcolor[rgb]{0,0,1}{0}
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How to create a Liquidity Pool on Aldrin AMM - Aldrin FAQ
1) Visit Aldrin Liquidity Pools page and press the "Create Pool" button.
2) Select the two tokens that you want to have in your liquidity pool from your wallet.
3) Decide if you want to lock your liquidity for a while and if so, how long. Pools with the locked liquidity of the founder will be crown-labeled and will cause more trust among users.
4) Specify the price of the base token relative to the quote token.
Keep in mind that by listing a price that does not correspond to the market, you open up a lot of opportunities for arbitrage, which will quickly equalize the price and cause your impermanent loss.
5) And specify how many tokens you want to deposit.
We recommend depositing at least $5,000 worth of liquidity, so the maximum pool size will be high enough that the pool will never hit the ceiling.
6) Set up a Farming. You can skip this step, but without farming, APR in your pool will be very low at the start, which will affect the motivation of people to deposit exactly in your pool.
We recommend allocating a significant amount to farming so that your pool attracts enough TVL.
Select the token you want to giveaway for farming.
Specify the number of tokens you want to giveaway to farming.
On the balance of your wallet must be at least an amount equal to the amount you want to give in farming. It will be transferred to the pool. Keep in mind that it is impossible to withdraw tokens from farming.
Specify the farming period. Tokens will be distributed equally among the participants during this period. In the field on the right, you can see how many tokens will be distributed daily.
Optional: Setup vesting. With the vesting, only 33% of the daily allocations specified in the field will be distributed daily. The remaining 77% will be distributed at the specified interval from each day. Vesting helps to reduce inflation.
For example: If your daily rewards are 1000 RIN and your vesting period is 7 days, then in the first 7 days of farming participants will receive 333 RIN, and from the 7th day 333 RIN + 767 RIN, and also 767 RIN for the week after the end of farming.
How to estimate how many tokens should I giveaway for farming?
You can approximate your pool's APR with the next formula:
APR = ((Rewards USD Value + (Volume * 0.02))/(TVL USD Value/Participants Number) ∗ 12
7) Make sure your settings are correct and confirm the creation of the pool by pressing the "Create Pool" button.
Make sure you have at least 0.5 SOL to proceed with all necessary transactions and approve the transactions in your wallet. Pool creation may take up to 5 minutes.
8) You can find the pool in the "Your Liquidity" tab. You will be able to extend your farming 1 hour before it ends.
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Suppose that we drop a stone, which falls freely with no air resistance. Experiments show that, under that assumption of negligible air resistance, the acceleration
{y}^{\prime\prime}=\frac { {d}^{2}y }{ {d}^{2}t }
of this motion is constant, i.e. equal to the so-called acceleration of gravity
g=9.8 \text{ m/sec}^2.
State this as an ordinary differential equation for
y(t),
the distance fallen as a function of time
t.
{y}^{\prime\prime}=gt
{y}^{\prime\prime}=\frac{1}{2}g{t}^{2}
{y}^{\prime\prime}=g
{y}^{\prime\prime}=gt+\frac{1}{2}g{t}^{2}
Because of limited food and space, a squirrel population cannot exceed
2900
squirrels. The population grows at a rate proportional to the product of the existing population and the attainable additional population. If
P
denotes the squirrel population at time
t,
which of the following equations represents the population growth rate for
k>0?
\frac{dP(t)}{dt}=k(2900+P)
\frac{dP(t)}{dt}=kP(2900+P)
\frac{dP(t)}{dt}=k(2900-P)
\frac{dP(t)}{dt}=kP(2900-P)
In psychology, a stimulus-response situation is a situation in which the response
y=f(x)
changes at a rate inversely proportional to the strength of the stimulus
x
. Which of the following equations represents this?
f'(x)=\sqrt{x}
f'(x)=k\ln{x}
f'(x)=kx^2
f'(x)=\frac{k}{x}
For a body moving along a straight line, let
y(t)
denote its distance from a fixed point
O
t.
Given that velocity plus distance is equal to square of time, find the ordinary differential equation of the motion.
{y}^{\prime\prime}+ty={t}^{2}
{ty}^{\prime}+y={t}^{2}
{y}^{\prime\prime}+y={t}^{2}
{y}^{\prime}+y={t}^{2}
Jack has a magic beanstalk, whose rate of growth is directly proportional to its height
H
. Which of the following equations represents his beanstalk's height as a function of time
t
\frac{dH}{dt}=k
\frac{dH}{dt}=kt
\frac{dH}{dt}=kH
\frac{dH}{dt}=ke^H
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Template:Arithmetic operations - Wikipedia
(Redirected from Template:Calculation results)
{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}
{\displaystyle \scriptstyle {\text{sum}}}
{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}
{\displaystyle \scriptstyle {\text{difference}}}
{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}
{\displaystyle \scriptstyle {\text{product}}}
{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\\scriptstyle {\text{ }}\\\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}
{\displaystyle {\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}}
{\displaystyle \scriptstyle {\text{base}}^{\text{exponent}}\,=\,}
{\displaystyle \scriptstyle {\text{power}}}
{\displaystyle \scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,}
{\displaystyle \scriptstyle {\text{root}}}
{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}
{\displaystyle \scriptstyle {\text{logarithm}}}
This template lists various calculations and the oul' names of their results.
Commons:Template:Calculation results
Retrieved from "https://en.wikipedia.org/w/index.php?title=Template:Arithmetic_operations&oldid=1069185439"
Mathematics sidebar templates
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Dyson series - Wikipedia
1 The Dyson operator
2 Derivation of the Dyson series
The Dyson operator[edit]
{\displaystyle V_{I}(t)=\mathrm {e} ^{\mathrm {i} H_{0}(t-t_{0})}V_{S}(t)\mathrm {e} ^{-\mathrm {i} H_{0}(t-t_{0})},}
{\displaystyle V_{S}(t)}
{\displaystyle V(t)}
{\displaystyle V_{\text{I}}(t)}
{\displaystyle \Psi (t)=U(t,t_{0})\Psi (t_{0})}
{\displaystyle U(t,t)=I,}
{\displaystyle U(t,t_{0})=U(t,t_{1})U(t_{1},t_{0}),}
{\displaystyle U^{-1}(t,t_{0})=U(t_{0},t),}
{\displaystyle i{\frac {d}{dt}}U(t,t_{0})\Psi (t_{0})=V(t)U(t,t_{0})\Psi (t_{0}).}
{\displaystyle U(t,t_{0})=1-i\int _{t_{0}}^{t}{dt_{1}\ V(t_{1})U(t_{1},t_{0})}.}
Derivation of the Dyson series[edit]
{\displaystyle {\begin{aligned}U(t,t_{0})={}&1-i\int _{t_{0}}^{t}dt_{1}V(t_{1})+(-i)^{2}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}\,dt_{2}V(t_{1})V(t_{2})+\cdots \\&{}+(-i)^{n}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}dt_{2}\cdots \int _{t_{0}}^{t_{n-1}}dt_{n}V(t_{1})V(t_{2})\cdots V(t_{n})+\cdots .\end{aligned}}}
{\displaystyle t_{1}>t_{2}>\cdots >t_{n}}
{\displaystyle {\mathcal {T}}}
{\displaystyle U_{n}(t,t_{0})=(-i)^{n}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}dt_{2}\cdots \int _{t_{0}}^{t_{n-1}}dt_{n}\,{\mathcal {T}}V(t_{1})V(t_{2})\cdots V(t_{n}).}
{\displaystyle S_{n}=\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}dt_{2}\cdots \int _{t_{0}}^{t_{n-1}}dt_{n}\,K(t_{1},t_{2},\dots ,t_{n}).}
{\displaystyle I_{n}=\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t}dt_{2}\cdots \int _{t_{0}}^{t}dt_{n}K(t_{1},t_{2},\dots ,t_{n}).}
{\displaystyle n!}
{\displaystyle t_{1}>t_{2}>\cdots >t_{n}}
{\displaystyle t_{2}>t_{1}>\cdots >t_{n}}
{\displaystyle S_{n}}
{\displaystyle S_{n}={\frac {1}{n!}}I_{n}.}
{\displaystyle U_{n}={\frac {(-i)^{n}}{n!}}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t}dt_{2}\cdots \int _{t_{0}}^{t}dt_{n}\,{\mathcal {T}}V(t_{1})V(t_{2})\cdots V(t_{n}).}
{\displaystyle U(t,t_{0})=\sum _{n=0}^{\infty }U_{n}(t,t_{0})={\mathcal {T}}e^{-i\int _{t_{0}}^{t}{d\tau V(\tau )}}.}
Wavefunctions[edit]
{\displaystyle |\Psi (t)\rangle =\sum _{n=0}^{\infty }{(-i)^{n} \over n!}\left(\prod _{k=1}^{n}\int _{t_{0}}^{t}dt_{k}\right){\mathcal {T}}\left\{\prod _{k=1}^{n}e^{iH_{0}t_{k}}V(t_{k})e^{-iH_{0}t_{k}}\right\}|\Psi (t_{0})\rangle .}
{\displaystyle \langle \psi _{\rm {f}};t_{\rm {f}}\mid \psi _{\rm {i}};t_{\rm {i}}\rangle =\sum _{n=0}^{\infty }(-i)^{n}\underbrace {\int dt_{1}\cdots dt_{n}} _{t_{\rm {f}}\,\geq \,t_{1}\,\geq \,\cdots \,\geq \,t_{n}\,\geq \,t_{\rm {i}}}\,\langle \psi _{\rm {f}};t_{\rm {f}}\mid e^{-iH_{0}(t_{\rm {f}}-t_{1})}V_{S}(t_{1})e^{-iH_{0}(t_{1}-t_{2})}\cdots V_{S}(t_{n})e^{-iH_{0}(t_{n}-t_{\rm {i}})}\mid \psi _{\rm {i}};t_{\rm {i}}\rangle .}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Dyson_series&oldid=1086374828"
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Global Constraint Catalog: Kcumulative_longest_hole_problems
<< 3.7.70. Counting constraint3.7.72. Cycle >>
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
A constraint that can use some filtering based on the longest closed and open hole problems [BeldiceanuCarlssonPoder08]. We follow the presentation from the previous paper. Before presenting the longest closed open hole scheduling problems, let us first introduce some notation related to the
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂},\mathrm{𝙻𝙸𝙼𝙸𝚃}\right)
constraint that will be used within the context of the longest closed and open hole problems.
\mathrm{𝚃𝙰𝚂𝙺𝚂}
is a collection of tasks, and for a task
t\in \mathrm{𝚃𝙰𝚂𝙺𝚂}
t.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}
t.\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}
t.\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}
denote respectively its start, duration and height, while
\mathrm{𝙻𝙸𝙼𝙸𝚃}\in {ℤ}^{+}
is the height of the resource. The constraint is equivalent to finding an assignment
s:\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}\to {ℤ}^{+}
Without loss of generality we assume the earliest start of each task to be greater than or equal to 0. that solves the cumulative placement of
\mathrm{𝚃𝙰𝚂𝙺𝚂}
of maximum height
\mathrm{𝙻𝙸𝙼𝙸𝚃}
\forall i\in ℤ:{\sigma }_{s}\left(i\right)=\mathrm{𝙻𝙸𝙼𝙸𝚃}-P\left(\mathrm{𝚃𝙰𝚂𝙺𝚂},i\right)\ge 0
where the coverage
P\left(\mathrm{𝚃𝙰𝚂𝙺𝚂},i\right)
\mathrm{𝚃𝙰𝚂𝙺𝚂}
of instant
i\in ℤ
P\left(\mathrm{𝚃𝙰𝚂𝙺𝚂},i\right)=\sum _{t\in \mathrm{𝚃𝙰𝚂𝙺𝚂}\mid t.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}\le i<t.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}+t.\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}}t.\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}
We are now in position to define the longest closed and open hole problems. Given a quantity
\sigma \in {ℤ}^{+}
of slack (i.e. the difference between the available space and the total area of the tasks to place), the longest closed hole problem is to find the largest integer
{\mathrm{𝑙𝑐𝑚𝑎𝑥}}_{\sigma }^{\mathrm{𝙻𝙸𝙼𝙸𝚃}}\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}\right)
for which there exists a cumulative placement
s
of a subset of tasks
{\mathrm{𝚃𝙰𝚂𝙺𝚂}}^{\text{'}}\subseteq \mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝙻𝙸𝙼𝙸𝚃}
, such that the resource area that is not occupied by
s
\left[0,{\mathrm{𝑙𝑐𝑚𝑎𝑥}}_{\sigma }^{\mathrm{𝙻𝙸𝙼𝙸𝚃}}\right)
does not exceed the maximum allowed slack value
\sigma
\sum _{i=0}^{{\mathrm{𝑙𝑐𝑚𝑎𝑥}}_{\sigma }^{\mathrm{𝙻𝙸𝙼𝙸𝚃}}-1}{\sigma }_{s}\left(i\right)\le \sigma .
The longest open hole problem is to find the largest integer
{\mathrm{𝑙𝑚𝑎𝑥}}_{\sigma }^{\mathrm{𝙻𝙸𝙼𝙸𝚃}}\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}\right)
s
{\mathrm{𝚃𝙰𝚂𝙺𝚂}}^{\text{'}}\subseteq \mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝙻𝙸𝙼𝙸𝚃}
\left[{i}^{\text{'}},{i}^{\text{'}}+{\mathrm{𝑙𝑚𝑎𝑥}}_{\sigma }^{\mathrm{𝙻𝙸𝙼𝙸𝚃}}\right)\subset ℤ
{\mathrm{𝑙𝑚𝑎𝑥}}_{\sigma }^{\mathrm{𝙻𝙸𝙼𝙸𝚃}}
s
\left[{i}^{\text{'}},{i}^{\text{'}}+{\mathrm{𝑙𝑚𝑎𝑥}}_{\sigma }^{\mathrm{𝙻𝙸𝙼𝙸𝚃}}\right)
\sigma
\sum _{i={i}^{\text{'}}}^{{i}^{\text{'}}+{\mathrm{𝑙𝑚𝑎𝑥}}_{\sigma }^{\mathrm{𝙻𝙸𝙼𝙸𝚃}}-1}{\sigma }_{s}\left(i\right)\le \sigma .
As an example, consider seven tasks of respective size
11×11
9×9
8×8
7×7
6×6
4×4
2×2
. Part (A) of Figure 3.7.20 provides a cumulative placement corresponding to the longest open hole problem according to
\mathrm{𝙻𝙸𝙼𝙸𝚃}=11
\sigma =0
. The longest open hole
{\mathrm{𝑙𝑚𝑎𝑥}}_{0}^{11}\left(\left\{11×11,9×9,8×8,7×7,6×6,4×4,2×2\right\}\right)=17
8×8
cannot contribute since a gap of 3 cannot be filled by the unique candidate the task
2×2
6×6
can also not contribute since a gap of 5 cannot be completely filled by the candidates
4×4
2×2
The longest close hole
{\mathrm{𝑙𝑐𝑚𝑎𝑥}}_{0}^{11}\left(\left\{11×11,9×9,8×8,7×7,6×6,4×4,2×2\right\}\right)=15
: it corresponds to the longest time interval on which the resource is saturated by the illustrated placement and such that one bound of the interval does not intersect any tasks.
Second, consider a task of size
3×2
. Part (B) of Figure 3.7.20 provides a cumulative placement corresponding to the longest open hole problem according to
ϵ=11
\sigma =20
{\mathrm{𝑙𝑚𝑎𝑥}}_{20}^{11}\left(\left\{3×2\right\}\right)=2
Figure 3.7.20. Two examples for illustrating the longest open hole problem: (A) a first instance with seven tasks of size
11×11
9×9
8×8
7×7
6×6
4×4
2×2
with a slack
\sigma =0
and a gap of 11, (B) a second instance with a single task of size
3×2
\sigma =20
and a gap of 11.
Figure 3.7.21 provides examples of the longest closed hole when we have 15 squares of sizes
1,2,\cdots ,15
and a zero slack. Parts (A), (B),...,(O) respectively give a solution achieving the longest closed hole for a gap of
1,2,\cdots ,15
. For comparison, Figure 3.7.22 provides the same examples of the longest open hole with zero slack.
Figure 3.7.21. Given 15 tasks of sizes
1×1,2×2,...,15×15
and a slack
\sigma =0
, examples of longest closed holes (in red) for a gap of
1,2,...,15
1×1,2×2,...,15×15
\sigma =0
, examples of longest open holes (in red) for a gap of
1,2,...,15
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A Haptic Simulator for Training the Application of Range of Motion Exercise to Premature Infants | J. Med. Devices | ASME Digital Collection
Kareem N. Adnan,
Kareem N. Adnan
, 4200 Engineering Gateway, Irvine, CA 92697-3975
e-mail: kareem.adnan@gmail.com
Irfan Ahmad,
, 4200 Engineering Gateway, Irvine, CA 92697-3975 91440
e-mail: iahmad@uci.edu
Maria Coussens,
Maria Coussens
e-mail: macousse@uci.edu
Alon Eliakim,
e-mail: aeliaki2@uci.edu
Susan Gallitto,
e-mail: sagallit@uci.edu
Donna Grochow,
Donna Grochow
e-mail: dmgrocho@uci.edu
Robin Koeppel,
e-mail: rkoeppel@uci.edu
Dan Nemet,
Child Health and Sports Center,
Pediatrics Meir Medical Center
, 44281, Israel
e-mail: dnemet@uci.edu
Julia Rich,
e-mail: jkrich@uci.edu
Feizal Waffarn,
e-mail: fwaffarn@uci.edu
Dan M. Cooper,
e-mail: dcooper@uci.edu
Department of Biomedical Engineering, Department of Mechanical and Aerospace Engineering,
, 4200 Engineering Gateway, Irvine, CA 92697-3975;
e-mail: dreinken@uci.edu
Adnan, K. N., Ahmad, I., Coussens, M., Eliakim, A., Gallitto, S., Grochow, D., Koeppel, R., Nemet, D., Rich, J., Waffarn, F., Cooper, D. M., and Reinkensmeyer, D. J. (December 9, 2009). "A Haptic Simulator for Training the Application of Range of Motion Exercise to Premature Infants." ASME. J. Med. Devices. December 2009; 3(4): 041008. https://doi.org/10.1115/1.4000430
The range of motion exercise is an experimental therapy for improving bone and muscle growth in premature infants but little is known about the magnitude of pressures that must be applied to the limbs during this exercise to elicit a physiological benefit and novice caregivers currently must rely on subjective instruction to learn to apply appropriate pressures. The goal of this study was to quantify the pressures applied by experienced caregivers during application of this exercise and to create a haptic simulator that could be used to train novice caregivers such as parents to apply the same pressures. We quantified the pressure applied by two neonatal intensive care nurses (“experts”) to the wrists of nine newborn, premature infants of varying gestational ages using an infant blood pressure cuff modified to act as a finger pressure sensor. The experts applied statistically significant different pressures depending on gestational age but did not differ significantly between themselves in the pressure they applied. We then created a robotic simulator of the premature infant wrist and programmed it to respond with the measured pressure-angle properties of the actual infants’ wrists. The novice adult participants
(n=19)
used the simulator to learn to apply target pressures for simulated wrists that corresponded to three different gestational ages. Training with the simulator for 30 min allowed the participants to learn to apply pressures significantly more like those of the experts. The performance improvement persisted at a retention test several days later. These results quantify for the first time the pressures applied during assisted exercise, include novel observations about joint flexibility and maturation early in life and suggest a strategy for teaching exercise intervention teams to provide assisted exercise within a more reproducible range using haptic simulation technology.
biomedical education, educational aids, paediatrics, patient care, patient treatment, pressure measurement, joint flexibility, muscle and bone maturation, robotic simulation
Haptics, Pressure, Robotics, Muscle, Bone, Patient treatment, Simulation
The Effects of Exercise on Body Weight and Circulating Leptin in Premature Infants
Assisted Exercise and Bone Strength in Preterm Infants
Litmanowitz
Evidence for Exercise-Induced Bone Formation in Premature Infants
Surgical Simulation: A Current Review
Elastic Properties of an Intact and ACL-Ruptured Knee Joint: Measurement, Mathematical Modelling, and Haptic Rendering
Measurement of Rigidity in Parkinson’s Disease
Neonatal Hip Instability: A Prospective Comparison of Clinical Examination and Anterior Dynamic Ultrasound
A Recurring FBN1 Gene Mutation in Neonatal Marfan Syndrome
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(Redirected from Natual Language Processing)
Symbolic NLP (1950s – early 1990s)[edit]
Statistical NLP (1990s–2010s)[edit]
Neural NLP (present)[edit]
Methods: Rules, statistics, neural networks[edit]
Common NLP tasks[edit]
Text and speech processing[edit]
Morphological analysis[edit]
Syntactic analysis[edit]
Lexical semantics (of individual words in context)[edit]
Relational semantics (semantics of individual sentences)[edit]
Discourse (semantics beyond individual sentences)[edit]
Higher-level NLP applications[edit]
General tendencies and (possible) future directions[edit]
Cognition and NLP[edit]
{\displaystyle {RMM(token_{N})}={PMM(token_{N})}\times {\frac {1}{2d}}\left(\sum _{i=-d}^{d}{((PMM(token_{N-1})}\times {PF(token_{N},token_{N-1}))_{i}}\right)}
|
f(x) = \left\{ \begin{array} { l } { x ^ { 2 } - 3 \quad\text {for } x \leq 2 } \\ { 5 - x \quad\ \text { for } x > 2 } \end{array} \right.
. Evaluate the following limits. If the limit does not exist, explain why it does not exist.
\lim\limits _ { x \rightarrow 2 ^ { + } } f ( x )
x→2
from the right (positive) side, what height does the graph approach? Which part of the function should you use to determine your answer?
\lim\limits _ { x \rightarrow 2 ^ { - } } f ( x )
x→2
from the left (negative) side, what height does the graph approach? Which part of the function should you use to determine your answer?
\lim\limits _ { x \rightarrow 2 } f ( x )
|
ConvertVersion - Maple Help
Home : Support : Online Help : Programming : Input and Output : File Manipulation : LibraryTools : ConvertVersion
converts back and forth between ".mla" and ".lib" Maple repositories
ConvertVersion( archive )
string denoting the Maple library file name
The ConvertVersion( ) command converts back and forth between the .mla format and .lib format of Maple repository files.
When the archive is a .mla file, a file of the same name with the .lib format and extension is created.
If the file is a .lib file, a .mla file is created.
If the archive is a directory, all libraries contained in that directory are converted.
The order of first conversion is unspecified when a .lib and .mla file with the same name already exist in the directory being converted.
\mathrm{with}\left(\mathrm{LibraryTools}\right):
\mathrm{LibLocation}≔\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{mapledir}\right),"/lib/myLib.mla"\right)
\mathrm{Create}\left(\mathrm{LibLocation}\right)
\mathrm{ConvertVersion}\left(\mathrm{LibLocation}\right)
\mathrm{FileTools}[\mathrm{Exists}]\left(\mathrm{LibLocation}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
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Global Constraint Catalog: Cweighted_partial_alldiff
<< 5.419. visible5.421. xor >>
[Thiel04]
\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚎𝚍}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚊𝚕}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳},\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂},\mathrm{𝙲𝙾𝚂𝚃}\right)
\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚎𝚍}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚊𝚕}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚎𝚍}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚊𝚕}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\mathrm{𝚠𝚙𝚊}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}
\mathrm{𝚒𝚗𝚝}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚕}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}-\mathrm{𝚒𝚗𝚝}\right)
\mathrm{𝙲𝙾𝚂𝚃}
\mathrm{𝚍𝚟𝚊𝚛}
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)
|\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|>0
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂},\left[\mathrm{𝚟𝚊𝚕},\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}\right]\right)
\mathrm{𝚒𝚗}_\mathrm{𝚊𝚝𝚝𝚛}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛},\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂},\mathrm{𝚟𝚊𝚕}\right)
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂},\mathrm{𝚟𝚊𝚕}\right)
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
collection that are not assigned to value
\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}
must have pairwise distinct values from the
\mathrm{𝚟𝚊𝚕}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝙲𝙾𝚂𝚃}
\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}
attributes associated with the values assigned to the variables of
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
. Within the
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
collection, value
\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}
must be explicitly defined with a weight of 0.
\left(\begin{array}{c}〈4,0,1,2,0,0〉,0,\hfill \\ 〈\begin{array}{cc}\mathrm{𝚟𝚊𝚕}-0\hfill & \mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}-0,\hfill \\ \mathrm{𝚟𝚊𝚕}-1\hfill & \mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}-2,\hfill \\ \mathrm{𝚟𝚊𝚕}-2\hfill & \mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}--1,\hfill \\ \mathrm{𝚟𝚊𝚕}-4\hfill & \mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}-7,\hfill \\ \mathrm{𝚟𝚊𝚕}-5\hfill & \mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}--8,\hfill \\ \mathrm{𝚟𝚊𝚕}-6\hfill & \mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}-2\hfill \end{array}〉,8\hfill \end{array}\right)
\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚎𝚍}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚊𝚕}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}
No value, except value
\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}=0
, is used more than once.
\mathrm{𝙲𝙾𝚂𝚃}=8
is equal to the sum of the weights 2,
-1
and 7 of the values 1, 2 and 4 assigned to the variables of
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}=〈4,0,1,2,0,0〉
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>0
\mathrm{𝚊𝚝𝚕𝚎𝚊𝚜𝚝}
\left(1,\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}\right)
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|\le |\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|+2
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
that are both different from
\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}
can be renamed to any unused value that is also different from
\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}
\mathrm{𝙲𝙾𝚂𝚃}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
In his PhD thesis [Thiel04], Sven Thiel describes the following three potential scenarios of the
\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚎𝚍}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚊𝚕}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}
Given a set of tasks (i.e., the items of the
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
collection), assign to each task a resource (i.e., an item of the
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
collection). Except for the resource associated with value
\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}
, every resource can be used at most once. The cost of a resource is independent from the task to which the resource is assigned. The cost of value
\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}
is equal to 0. The total cost
\mathrm{𝙲𝙾𝚂𝚃}
of an assignment corresponds to the sum of the costs of the resources effectively assigned to the tasks. Finally we impose an upper bound on the total cost.
Given a set of persons (i.e., the items of the
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
collection), select for each person an offer (i.e., an item of the
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
collection). Except for the offer associated with value
\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}
, every offer should be selected at most once. The profit associated with an offer is independent from the person that selects the offer. The profit of value
\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}
is equal to 0. The total benefit
\mathrm{𝙲𝙾𝚂𝚃}
is equal to the sum of the profits of the offers effectively selected. In addition we impose a lower bound on the total benefit.
The last scenario deals with an application to an over-constraint problem involving the
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
constraint. Allowing some variables to take an "undefined" value is done by setting all weights of all the values different from
\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}
to 1. As a consequence all variables assigned to a value different from
\mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}
will have to take distinct values. The
\mathrm{𝙲𝙾𝚂𝚃}
variable allows to control the number of such variables.
It was shown in [Thiel04] that, finding out whether the
\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚎𝚍}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚊𝚕}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}
constraint has a solution or not is NP-hard. This was achieved by reduction from subset sum.
A filtering algorithm is given in [Thiel04]. After showing that, deciding whether the
\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚎𝚍}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚊𝚕}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}
has a solution is NP-complete, [Thiel04] gives the following results of his filtering algorithm with respect to consistency under the 3 scenarios previously described:
For scenario 1, if there is no restriction of the lower bound of the
\mathrm{𝙲𝙾𝚂𝚃}
variable, the filtering algorithm achieves arc-consistency for all variables of the
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
collection (but not for the
\mathrm{𝙲𝙾𝚂𝚃}
variable itself).
For scenario 2, if there is no restriction of the upper bound of the
\mathrm{𝙲𝙾𝚂𝚃}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝙲𝙾𝚂𝚃}
Finally, for scenario 3, the filtering algorithm achieves arc-consistency for all variables of the
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
collection as well as for the
\mathrm{𝙲𝙾𝚂𝚃}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚎𝚡𝚌𝚎𝚙𝚝}_\mathtt{0}
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚠𝚒𝚝𝚑}_\mathrm{𝚌𝚘𝚜𝚝𝚜}
\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}_\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚟𝚊𝚛}
(soft constraint),
\mathrm{𝚜𝚞𝚖}_\mathrm{𝚘𝚏}_\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚜}_\mathrm{𝚘𝚏}_\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}_\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}
characteristic of a constraint: all different, joker value.
complexity: subset sum.
constraint type: soft constraint, relaxation.
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝑃𝑅𝑂𝐷𝑈𝐶𝑇}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜},\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}\right)
•\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}.\mathrm{𝚟𝚊𝚛}\ne \mathrm{𝚄𝙽𝙳𝙴𝙵𝙸𝙽𝙴𝙳}
•\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}.\mathrm{𝚟𝚊𝚛}=\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}.\mathrm{𝚟𝚊𝚕}
•
\mathrm{𝐌𝐀𝐗}_\mathrm{𝐈𝐃}
\le 1
•
\mathrm{𝐒𝐔𝐌}
\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂},\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}\right)=\mathrm{𝙲𝙾𝚂𝚃}
Parts (A) and (B) of Figure 5.420.1 respectively show the initial and final graph associated with the Example slot. Since we also use the
\mathrm{𝐒𝐔𝐌}
graph property we show the vertices of the final graph from which we compute the total cost in a box.
\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚎𝚍}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚊𝚕}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}
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Derivatives of Polynomials | Brilliant Math & Science Wiki
Mei Li, Abhineet Goel, Ram Mohith, and
Polynomials are one of the simplest functions to differentiate. When taking derivatives of polynomials, we primarily make use of the power rule.
n
f(x)= x^n
\frac{d}{dx} f(x) = n x ^{n-1}.
Derivatives of Polynomials - Basic
Derivatives of Polynomials - Intermediate
Given a linear function
f(x) = ax+b
, we use the property that differentiation is linear to show
\frac{d}{dx} f(x) = \frac{d}{dx} ( ax + b ) = \frac{d}{dx} ( ax ) + \frac{d}{dx} (b) = a.
Thus, the derivative of a linear function is a constant.
f(x) = 3x + 2
f'(x)?
f'(x) = 3,
the slope of the line.
_\square
f(x)
is a linear function such that
f(1) = 2
f'(3) = 4
f(x)?
f(x)
be the linear function
f(x)= ax + b
f'(3) = 4
a = 4
f(1) = 2
2 = 4 \times 1 + b
b = - 2
f(x) = 4x - 2
_\square
f(x)
f(2)=10
f'(1)=4,
f(0)+f'(0)?
f(x)
f(x) = ax + b
a=f'(1)=4
f(x) = 4x + b
10 = f(2) = 4(2) + b
b=2
f(0) = 4(0) + 2 = 2
f'(0) = a = 4
f(0) + f'(0) = 2 + 4 = 6. \ _\square
f(x) = 3x^2 + 4x + 5
f'(1)?
f'(x) = 2 \times 3 x + 1 \times 4 = 6x + 4
f'(1) = 10
_\square
f(x) = -3x^4 + 7x^3 - 6x^2 - \pi x + 5
\frac{d}{dx} f(x)?
\begin{aligned} \frac{d}{dx} \big( -3x^4 + 7x^3 - 6x^2 - \pi x + 5 \big) &= (-3)(4) x^{4-1} + (7)(3) x^{3-1} - (6)(2) x^{2-1} - (\pi) (1) x^{1-1} + 0\\ &= -12x^3 + 21x^2 - 12x -\pi. \ _\square \end{aligned}
f(x) = (3x-2) ^4
\frac{d}{dx} f(x)?
f(x)
is a polynomial but is not in the form given in the summary above. We will later see methods to differentiate this function directly, but to use the tools we have so far, we first expand the polynomial by the binomial theorem. This gives
\begin{aligned} f(x) &= (3x-2)^4 \\ &= (3x)^4 + 4(3x)^3(-2) + \frac{4 \cdot 3}{2} (3x)^2 (-2)^2 + \frac{4\cdot 3 \cdot 2}{1 \cdot 2 \cdot 3} (3x)(-2)^3 + (-2)^4\\ &= 81 x^4 - 216 x^3 + 216 x^2 - 96 x + 16, \end{aligned}
\begin{aligned} \frac{d}{dx} f(x) &= (81)(4)x^3 - (216)(3)x^2 + (216)(2) x - 96 \\ & = 324 x^3 - 648x^2 + 432 x - 96. \ _\square \end{aligned}
f(x) = x^3
{\displaystyle \lim_ { h \rightarrow 0 }}\frac{ f(2+h) - f(2) } { h }?
\begin{aligned} \lim_ { h \rightarrow 0 } \frac{ f(2+h) - f(2) } { h } &= \lim_ { h \rightarrow 0 } \frac{ (2+h)^3 - 2^3 } { h }\\ &= 3 \times 2^{3-1}\\ &= 12. \ _\square \end{aligned}
{\displaystyle\lim_ { h \rightarrow 0 }} \frac{ f(x+h) - f(x) } { h } = 4x^2
f'(2)?
Using the above formula, we know
{\displaystyle \lim_ { h \rightarrow 0 }} \frac{ f(x+h) - f(x) } { h } = \frac{d}{dx}f(x) = f'(x) = 4x^2
\begin{aligned} f'(2) &=\lim_ { h \rightarrow 0 } \frac{ f(2+h) - f(2) } { h } \\ &= 4\times 2^2 \\ &= 16. \ _\square \end{aligned}
f(x) = x^2 + 4x
{\displaystyle\lim_ { h \rightarrow 3 }} \frac{ f(h) - f(3) } { h -3}?
{\displaystyle\lim_ { h \rightarrow 3 }} \frac{ f(h) - f(3) } { h -3}
is not in the form given in the formula above. Thus, we use the substitution method in order to get a general formula. Substituting
k
h - 3,
\lim_ { h \rightarrow 3 } \frac{ f(h) - f(3) } { h -3} = \lim_ { k \rightarrow 0 } \frac{ f(3+k) - f(3) } {k}.
{\displaystyle\lim_ { h \rightarrow 0 }} \frac{ f(x+h) - f(x) } { h } = \frac{d}{dx}f(x) = f'(x) = 2x + 4
\lim_ { k \rightarrow 0 } \frac{ f(3+k) - f(3) } {k}=f'(3) = 2 \times 3 + 4 = 10. \ _\square
f(x) = (ax + b)^n
x > -\frac{b}{a}.
u = ax + b \implies u' = a
f(x) = u^n.
f'(x) = \frac{d}{dx} (u^n) = nu^{n - 1} \cdot u' = an(ax + b)^{n - 1} .\ _\square
x^n
n
is a non-zero real number, is
n x ^{n-1}
n
, we can prove this by first principles, using the binomial theorem:
\begin{aligned} \lim_ { h \rightarrow 0 } \frac{ ( x+h)^n - x^n } { h } & = \lim_{ h \rightarrow 0 } \frac{ \left[ x^n + n x^{n-1} h + n(n-1) x^{n-2} h^2 + \cdots + h^n \right] - x^n } { h} \\ & = \lim_{h \rightarrow 0 } n x^{n-1} + n(n-1) x^{n-2} h + \cdots + h^{n-1}\\ & = n x^{n-1}. \end{aligned}
By linearity of differentiation we conclude that if
f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_ 0
\frac{ d}{dx} f(x) = n a_n x^{n-1} + (n-1) a_{n-1} x^{n-1} + \cdots + 2 a_2 x + 1 a_1 x^0.
Compute the derivative of the function
f(x)=2x^{2}
x = 2048
f(x) = (x^2 - 1 ) ( x^2 - 2x + 3)
f'(1)?
Multiplying out, we get
f(x) = x^4 - 2x^3 + 2x^2 + 2x - 3
f'(x) = 4x^3 - 6x^2 + 4x + 2
f'(1) = 4 - 6 + 4 + 2 = 4. \ _\square
Note: We can also use the product rule to differentiate this function, to conclude that
f'(x) = (x^2 - 1 ) ( 2x - 2 ) + ( 2x) ( x^2 - 2x + 3 ).
f(x) = (ax + b)^n(cx + d)^m.
\begin{aligned} f(x) & = \underbrace{(ax + b)^n}_{u(x)} \cdot \underbrace{(cx + d)^m}_{v(x)} \\\\ f'(x) & = u'(x)v(x) + u(x)v'(x) \\ & = \underbrace{n(ax + b)^{n - 1}(a)}_{u'(x)}(cx + d)^m + (ax + b)^n\underbrace{m(cx + d)^{m - 1}(c)}_{v'(x)} \\ & = (ax + b)^n(cx + d)^m\big(na(ax + d)^{-1} + mc(ax + b)^{-1}\big) \\ & = (ax + b)^n(cx + d)^m \left(\dfrac{na}{ax + b} + \dfrac{mc}{cx + d}\right).\ _\square \\ \end{aligned}
f(x) = \sqrt{x} + 2x^\frac34 + 3x^\frac56.
\begin{aligned} f(x) & = x^\frac12 + 2x^\frac34 + 3x^\frac56 \\\\ f'(x) & = \dfrac12 x^{\frac12 - 1} + 2\cdot \dfrac34 x^{\frac34 - 1} + 3 \cdot \dfrac56 x^{\frac56 - 1} \\ & = \dfrac{1}{2\sqrt{x}} + \dfrac{3}{2 \sqrt[4]{x}} + \dfrac{5}{2 \sqrt[6]{x}}.\ _\square \\ \end{aligned}
Cite as: Derivatives of Polynomials. Brilliant.org. Retrieved from https://brilliant.org/wiki/derivatives-of-polynomials/
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Since the use of a linguistic expression is only possible if the speaker who uses it understands its meaning, one of the central problems for analytic philosophers has always been the question of meaning. What is it? Where does it come from? How is it communicated? And, among these questions, what is the smallest unit of meaning, the smallest fragment of language with which it is possible to communicate something? At the end of the 19th and beginning of the 20th century, Gottlob Frege and his followers abandoned the view, common at the time, that a word gets its meaning in isolation, independently from all the rest of the words in a language. Frege, as an alternative, formulated his famous context principle, according to which it is only within the context of an entire sentence that a word acquires its meaning. In the 1950s, the agreement that seemed to have been reached regarding the primacy of sentences in semantic questions began to unravel with the collapse of the movement of logical positivism and the powerful influence exercised by the later Ludwig Wittgenstein. Wittgenstein wrote in the Philosophical Investigations that "comprehending a proposition means comprehending a language". About the same time or shortly after, W. V. O. Quine wrote that "the unit of measure of empirical meaning is all of science in its globality"; and Donald Davidson, in 1967, put it even more sharply by saying that "a sentence (and therefore a word) has meaning only in the context of a (whole) language".
Holism of mental content
The key to answering this question lies in going back to Quine and his attack on logical positivism. The logical positivists, who dominated the philosophical scene for almost the entire first half of the twentieth century, maintained that genuine knowledge consisted in all and only such knowledge as was capable of manifesting a strict relationship with empirical experience. Therefore, they believed, the only linguistic expressions (manifestations of knowledge) that had meaning were those that either directly referred to observable entities, or that could be reduced to a vocabulary that directly referred to such entities. A sentence S contained knowledge only if it possessed a meaning, and it possessed a meaning only if it was possible to refer to a set of experiences that could, at least potentially, verify it and to another set that could potentially falsify it. Underlying all this, there is an implicit and powerful connection between epistemological and semantic questions. This connection carries over into the work of Quine in Two Dogmas of Empiricism.
All of our so-called knowledge or convictions, from questions of geography and history to the most profound laws of atomic physics or even mathematics and logic, are an edifice made by man that touches experience only at the margins. Or, to change images, science in its globality is like a force field whose limit points are experiences...a particular experience is never tied to any proposition inside the field except indirectly, for the needs of equilibrium which affect the field in its globality.
{\displaystyle \forall p\exists q\neq p\Box (B(x,p)\rightarrow B(x,q))}
{\displaystyle \Box (B(x,p)\rightarrow \exists q\neq p(B(x,q))}
The first statement asserts that there are other propositions, besides p, that one must believe in order to believe p. The second says that one cannot believe p unless there are other propositions in which one believes. If one accepts the first reading, then one must accept the existence of a set of sentences that are necessarily believed and hence fall into the analytic/synthetic distinction. The second reading is useless (too weak) to serve the molecularist's needs since it only requires that if, say, two people believe the same proposition p, they also believe in at least one other proposition. But, in this way, each one will connect to p his own inferences and communication will remain impossible.
{\displaystyle \Box (B(x,p)\land B(y,p)\rightarrow \exists q\neq p(B(x,q)\land B(y,q))}
This says that two people cannot believe the same proposition unless they also both believe a proposition different from p. This helps to some extent but there is still a problem in terms of identifying how the different propositions shared by the two speakers are specifically related to each other. Dummett's proposal is based on an analogy from logic. To understand a logically complex sentence it is necessary to understand one that is logically less complex. In this manner, the distinction between logically less complex sentences that are constitutive of the meaning of a logical constant and logically more complex sentences that are not takes on the role of the old analytic/synthetic distinction. "The comprehension of a sentence in which the logical constant does not figure as a principal operator depends on the comprehension of the constant, but does not contribute to its constitution." For example, one can explain the use of the conditional in
{\displaystyle (a\lor \lnot b)\rightarrow c}
by stating that the whole sentence is false if the part before the arrow is true and c is false. But to understand
{\displaystyle a\lor \lnot b}
one must already know the meaning of "not" and "or." This is, in turn, explained by giving the rules of introduction for simple schemes such as
{\displaystyle P\lor Q}
{\displaystyle \lnot Q}
. To comprehend a sentence is to comprehend all and only the sentences of less logical complexity than the sentence that one is trying to comprehend. However, there is still a problem with extending this approach to natural languages. If I understand the word "hot" because I have understood the phrase "this stove is hot", it seems that I am defining the term by reference to a set of stereotypical objects with the property of being hot. If I don't know what it means for these objects to be "hot", such a set or listing of objects is not helpful.
{\displaystyle L(\alpha ,\beta ,\gamma )\leftrightarrow P(\alpha ,\beta )\land I(\beta ,\gamma )}
{\displaystyle R(\alpha ,\beta ,\gamma )\leftrightarrow P(\alpha ,\gamma )\land I(\beta ,\gamma )}
{\displaystyle G(\alpha )=([(\beta ,\gamma ):L(\alpha ,\beta ,\gamma )],[(\beta ,\gamma ):R(\alpha ,\beta ,\gamma )])\;}
The global role of
{\displaystyle \alpha }
consists in a pair of sets, each one composed of a pair of sets of expressions. If F accepts the inference from
{\displaystyle \beta }
{\displaystyle \gamma }
{\displaystyle \alpha }
{\displaystyle \gamma }
, then the couple
{\displaystyle (\beta ,\gamma )}
is an element of the set which is an element of the right side of the Global Role of α. This makes Global Roles for simple expressions sensitive to changes in the acceptance of inferences by F. The Global Role for complex expressions can be defined as:
{\displaystyle G(\beta )=[G(\alpha _{1})...G(\alpha _{n})]\;}
{\displaystyle h(G(\beta ))=h([G(\alpha _{1})...G(\alpha _{n})])=F(h(G(\alpha _{1}))...h(G(\alpha _{n}))).\;}
This function is one to one in that it assigns exactly one meaning to every Global Role. According to Fodor and Lepore, holistic inferential role semantics leads to the absurd conclusion that part of the meaning of "brown cow" is constituted by the inference "Brown cow implies dangerous." This is true if the function from meanings to Global Roles is one to one. In this case, in fact, the meanings of "brown", "cow" and "dangerous" all contain the inference "Brown cows are dangerous"!! But this is only true if the relation is one to one. Since it is one to one, "brown" would not have the meaning it has unless it had the global role that it has. If we change the relation so that it is many to one (h*), many global roles can share the same meaning. So suppose that the meaning of "brown "is given by M("brown"). It does not follow from this that L("brown", "brown cow", "dangerous") is true unless all of the global roles that h* assigns to M("brown") contain ("brown cow", "dangerous"). And this is not necessary for holism. In fact, with this many to one relation from Global Roles to meanings, it is possible to change opinions with respect to an inference consistently. Suppose that B and C initially accept all of the same inferences, speak the same language and they both accept that "brown cows imply dangerous." Suddenly, B changes his mind and rejects the inference. If the function from meanings to Global Role is one to one, then many of B's Global Roles have changed and therefore their meanings. But if there is no one to one assignment, then B's change in belief in the inference about brown cows does not necessarily imply a difference in the meanings of the terms he uses. Therefore, it is not intrinsic to holism that communication or change of opinion is impossible.
Since the concept of semantic holism, as explained above, is often used to refer to not just theories of meaning in natural languages but also to theories of mental content such as the hypothesis of a language of thought, the question often arises as to how to reconcile the idea of semantic holism (in the sense of the meanings of expressions in mental languages) with the phenomenon called externalism in philosophy of mind. Externalism is the thesis that the propositional attitudes of an individual are determined, at least in part, by her relations with her environment (both social and natural). Hilary Putnam formulated the thesis of the natural externalism of mental states in his The Meaning of "Meaning". In it, he described his famous thought experiment involving Twin Earths: two individuals, Calvin and Carvin, live, respectively, on the real earth (E) of our everyday experience and on an exact copy (E') with the only difference being that on E "water" stands for the substance
{\displaystyle H_{2}O}
while on E' it stands for some substance macroscopically identical to water but which is actually composed of XYZ. According to Putnam, only Calvin has genuine experiences that involve water, so only his term "water" really refers to water.
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3Blue1Brown - Gradient descent, how neural networks learn
Chapter 2Gradient descent, how neural networks learn
In the last lesson we explored the structure of a neural network. Now, let’s talk about how the network learns by seeing many labeled training data. The core idea is a method known as gradient descent, which underlies not only how neural networks learn, but a lot of other machine learning as well.
As a reminder, our goal is to recognize handwritten digits. It’s a classic example—the “hello world” of neural networks.
Our goal is to recognize these handwritten digits.
These digits are rendered onto a 28x28 pixel grid, where each pixel has some grayscale value between 0.0 and 1.0. Those 784 values determine the activations of the neurons in the input layer of the network.
Each pixel value in the image becomes the activation of a particular neuron in the first layer of the network.
The activation for each neuron in the following layers is based on a weighted sum of all activations from the previous layer, plus some number called the “bias.” We then wrap a nonlinear function around this sum, such as a sigmoid or a ReLU.
a_{0}^{(1)} = \textcolor{#d69d00}{ \sigma\left( \textcolor{black}{ \textcolor{green}{w_{0,0}} a_{0}^{(0)} + \textcolor{green}{w_{0,1}} a_{1}^{(0)} + \cdots + \textcolor{green}{w_{0,n}} a_{n}^{(0)} \textcolor{blue}{+ b_{0}} } \right) }
In this way, values propagate from one layer to the next, entirely determined by the weights and biases. The brightest of the ten neurons in the output layer is the network’s choice for which digit the image represents.
The input layer of neurons has activations based on the pixel values of an image. The rest of the layers have activations depending on each previous layer.
In total, given the somewhat arbitrary choice of 2 hidden layers with 16 neurons each, this network has 13,002 weights and biases that we can adjust, and it’s these values that determine what exactly the network does.
Remember, the motivation we had in mind for this layered structure was that maybe the second layer could pick up on edges, the third on patterns like loops and lines, and the last one pieces together those patterns to recognize digits:
We hope that the layered structure might allow the problem to be broken into smaller steps, from pixels to edges to loops and lines, and finally to digits.
How the Network Learns
What makes machine learning different from other computer science is that you don’t explicitly write instructions for doing a particular task. In this case, you never actually write an algorithm for recognizing digits. Instead, you write an algorithm that can take in a bunch of example images of handwritten digits along with labels for what they are and adjust those 13,002 weights and biases of the network to make it perform better on those examples.
The labeled images we feed in are collectively known as the "training data".
The network learns by looking at examples of correctly identified digits.
Hopefully, the layered structure will mean that what it learns generalizes to images beyond the training data. To test this, after training the network, we can show it more labeled data that it’s never seen before and see how accurately it classifies them.
We use some of the data for training the network and some other data it’s never seen before for testing it after it’s trained.
This process requires a lot of data. Fortunately, the good people behind the MNIST database have put together a collection of tens of thousands of handwritten digit images, each labeled with the number they are supposed to be, that we can use for free.
I should point out that although it can be provocative to describe a machine as "learning", once you see how it works, it feels a lot less like some crazy sci-fi premise and a lot more like a calculus exercise. It really just comes down to finding the minimum of a specific function.
The process of learning is essentially just finding lower and lower points on a specific function.
Remember, our network's behavior is determined by all of its weights and biases. The weights represent the strength of connections between each neuron in one layer and each neuron in the next. And each bias is an indication of whether its neuron tends to be active or inactive.
The behavior of the network depends on its weights and biases.
To start things off, initialize all the weights and biases to be random numbers. This network will perform horribly on the given training example since it’s just doing something random.
For example, if you feed in this image of a 3, the output layer just looks like a mess:
Initialized with totally random weights and biases, the network is terrible at identifying digits.
So how do you programmatically identify that the computer is doing a lousy job and then help it improve?
What you do is define a cost function, which is a way of telling the computer, “No, bad computer, that output layer should have activations which are 0 for most neurons, but with a 1 for the third neuron. What you gave me is utter trash!”
We measure how badly the network is doing by comparing its results to the correct answer.
To say that a little more mathematically, add up the squares of the differences between each of these trash output activations and the values you want them to have. We’ll call this the "cost" of a single training example.
The "cost" is calculated by adding up the squares of the differences between what we got and what we want.
The cost is small when the network confidently classifies this image correctly but large when it doesn’t know what it’s doing.
Suppose the network is shown an image of a 4. Rank the costs associated with each of the possible outputs shown above.
D > A > C > B
The Cost Over Many Examples
But we aren’t just interested in how the network performs on a single image. To really measure its performance, we need to consider the average cost over all the tens of thousands of training examples. This average is our measure of how lousy the network is and how bad the computer should feel.
This is, to put it lightly, a complicated function. Remember how the network itself is a function? It has 784 inputs (the pixel values), 10 outputs, and 13,002 parameters.
The neural network is a function that takes in images and spits out digit predictions based on the weights and biases.
The cost function is a layer of complexity on top of that. The inputs of the cost function are those 13,002 weights and biases, and it spits out a single number describing how bad those weights and biases are. It’s defined in terms of the network’s behavior on all the tens of thousands of pieces of labeled training data. In other words, this training data is a massive set of parameters to the cost function.
The cost function takes in the weights and biases of a network and uses the training images to compute a “cost,” measuring how bad the network is at classifying those training images.
Just telling the computer what a terrible job it’s doing isn’t very helpful. You want to tell it how it should change those 13,002 weights and biases so as to improve.
We can do better! Growth mindset!
To make it easier, rather than struggling to imagine this cost function with 13,002 inputs, let’s imagine a simple function with one number as an input, and one number as an output.
In this simple example, we’ll imagine that the cost function takes just one input (although, in reality, ours takes 13,002 inputs). How do we find an input that minimizes the cost?
How do you find an input that minimizes the value of this cost function?
Calculus students will know you can sometimes figure out that minimum explicitly by solving for when the slope is zero. However, that’s not always feasible for really complicated functions, and it certainly won’t be for our 13,002-input function defined in terms of an ungodly number of parameters.
For a complicated cost function, computing the exact minimum directly isn’t going to work.
A more flexible tactic is to start at a random input and figure out which direction to step to make the output lower. Specifically, find the slope of the function where you are. If the slope is negative, shift to the right. If the slope is positive, shift to the left.
By following the slope (moving in the downhill direction), we approach a local minimum.
Checking the new slope at each point and doing this repeatedly, you approach some local minimum of the function.
The image to have in mind is a ball rolling down a hill:
Moving the input position according to the slope is a lot like a ball rolling down a hill.
Notice, even for this simplified single-input cost function, there are many possible valleys you might land in. It depends on which random input you start at, and there’s no guarantee that the local minimum you land in will be the smallest possible value for the cost function.
Also, notice how if you make your step sizes proportional to the slope itself when the slope flattens out towards a minimum, your steps will get smaller and smaller, and that keeps you from overshooting.
As the slope gets shallower, take smaller steps to avoid overshooting the minimum.
Bumping up the complexity a bit, imagine a function with two inputs and one output. You might think of the input space as the xy-plane, with the cost function graphed as a surface above it.
We can imagine minimizing a function that takes two inputs (which is still not 13,002 inputs, but it’s one step closer). Starting with a random value, look for the downhill direction and repeatedly move that way.
Instead of asking about the slope of the function, you ask which direction you should step in this input space to decrease the output of the function most quickly. Again, think of a ball rolling down a hill.
Beyond Slope: Using The Gradient
In this higher-dimensional space, it doesn’t really make sense to talk about the “slope” as a single number. Instead, we need to use a vector to represent the direction of steepest ascent.
Those of you familiar with multivariable calculus will know that this vector is called the “gradient”, and it tells you which direction you should step to increase the function most quickly.
Naturally enough, taking the negative of that vector gives you the direction to step which decreases the function most quickly. What’s more, the length of that gradient vector is an indication for just how steep that steepest slope is.
The gradient,
\nabla C
, gives the uphill direction, so the negative of the gradient,
-\nabla C
, gives the downhill direction.
To those of you unfamiliar with multivariable calculus, check out some of the work I did for Khan Academy on the topic. Honestly though? All that matters for you and me right now is that in principle, there exists a way to compute this vector, which tells you both what the “downhill” direction is, and how steep it is. You’ll be okay even if you’re not rock solid on the details.
So the algorithm for minimizing this function is to compute this gradient direction, take a step downhill, and repeat that over and over. This process is called “gradient descent.”
Gradient descent just means walking in the downhill direction to minimize the cost function.
And remember, our cost function is specially designed to measure how bad the network is at classifying the training examples. So changing the weights and biases to decrease the cost will make the network’s performance better!
In practice, each step will look like
-\eta \nabla C
\eta
is known as the learning rate. The larger it is, the bigger your steps, which means you might approach the minimum faster, but there’s a risk of overshooting and oscillating a lot around that minimum.
It’s the same basic idea for a function with 13,002 inputs instead of two inputs. I’m still showing the picture of a function with two inputs because nudges to a 13,002-dimensional input are a little hard to wrap your mind around. But let me give you a non-spatial way to think about it.
Another Way to Think About The Gradient
Imagine organizing all the weights and biases for the entire network into one big column vector with 13,002 entries. Then the negative gradient of the cost function will also be a vector with 13,002 entries.
The vector on the left is a list of all the weights and biases for the entire network. The vector on the right is the negative gradient, which is just a list of all the little nudges you should make to the weights and biases (on the left) in order to move downhill.
The negative gradient is a vector direction in this insanely huge input space that tells you what nudges to those 13,002 numbers would cause the most rapid decrease to the cost function.
Each component of the negative gradient vector tells us two things. The sign, of course, tells us whether the corresponding component of the input vector should be nudged up or down. But importantly, the relative magnitudes of the components in this gradient tells us which of those changes matters more.
You see, in our network, an adjustment to one weight might have a greater impact on the cost function than an adjustment to some other weight. Consider the image below, and suppose that the network is supposed to be classifying its input as a “3”.
Some weights have a larger impact on the cost than others.
Some of these connections just matter more for our training data than others, and a way you can think about the gradient of our massive cost function is that it encodes the relative importance of each weight and bias. That is, which changes will carry the most bang for your buck.
So if the cost function is a layer of complexity on top of the original neural network function, its gradient is one more layer of complexity still, telling us what nudge to all these weights and biases causes the fastest change to the value of the cost function.
The negative gradient tells us how to change the weights and biases to decrease the cost most effectively.
The algorithm for computing the value of this gradient vector efficiently is called backpropagation, which we’ll dig into in the next lesson. There, I want to take the time to really walk through what happens to each weight and bias for a given piece of training data, trying to give an intuitive feel for what’s happening beyond the pile of relevant formulas.
The main thing I want you to understand here, independent of implementation details, is that when we refer to a network “learning”, we mean changing the weights and biases to minimize a cost function, which improves the network's performance on the training data.
But before we continue on to learn the details of computing this gradient, let’s take a brief sidestep to dig into what this network looks like after it’s trained.
Analyzing our neural network
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Erratum to “Theoretical Study of the Reaction of (2, 2)-Dichloro (Ethyl) Arylphosphine with Bis (2, 2)-Dichloro (Ethyl) Arylphosphine by Hydrophosphination Regioselective by the DFT Method” [Computational Chemistry 5 (2017) 113-128]
Kouadio Valery Bohoussou1, Anoubilé Benié2, Mamadou Guy-Richard Koné1, Affi Baudelaire Kakou2, Kafoumba Bamba1, Nahossé Ziao1
1 Laboratoire de Thermodynamique et de Physico-Chimie du Milieu, UFR SFA, Universite Nangui Abrogoua, Abidjan, Republique de Cote-d’Ivoire.
2 Laboratoire de Chimie Bio-Organique et de Substances Naturelles (LCBOSN), UFR SFA, Universite Nangui Abrogoua, Abidjan,Republique de Cote-d’Ivoire.
Abstract: The original online version of this article (Kouadio Valery Bohoussou1, Anoubilé Benié2, Mamadou Guy-Richard Koné1, Affi Baudelaire Kakou2, Kafoumba Bamba1, Nahossé Ziao1) Theoretical Study of the Reaction of (2, 2)-Dichloro (Ethyl) Arylphosphine with Bis (2, 2)-Dichloro (Ethyl) Arylphosphine by Hydrophosphination Regio selective by the DFT Method. Computational Chemistry 5 (2017) 113-128. DOI: 10.4236/cc.2017.53010) unfortunately contains a mistake. The author wishes to correct the errors from Table 3 to Table 4, on pages 121 and the beginning of page 122.
On analysis of the values in Table 3, phosphines 1a and 1b have the highest values of the local nucleophilic indices Nk. Similarly, the carbon C1 of the compound R2 has the highest value of the local electrophilic index (ωk). This shows that the most favored interaction takes place between the P1 atom of the compound 1a and the C1 atom of the compound R2 for the first reaction, and between the P9 and C1 atoms for the second reaction. Therefore, the formation of experimentally observed P1-C1 and P2-C1 bonds are correctly predicted by the Domingo model with the Mulliken and NPA approaches.
Table 3. Local reactivity descriptors on the P1, P2, C1 and C2 atoms of reactants 1a, 1b and R2 using NPA and Mulliken population analyzes at B3LYP/6-311 + G (d, p).
3.3.2. Prediction Using the Gazquez-Mendez Model
The prediction according to the Gazquez-Mendez model presents values of the Fukui function (
{f}_{k}^{+}
{f}_{k}^{-}
), local softness
{S}_{k}^{+}
for reactants R2 and local softness
{S}_{k}^{-}
for reactants 1a or 1b. These values of the local descriptors on the atoms P1, P2, C1 and C2 of the reactants 1a, 1b and R2 were calculated according to the Gazquez-Mendez model with the NPA population analyzes and MK at the B3LYP 6-311+G level (d, p) are given in Table 4.
{f}_{k}^{+}
{f}_{k}^{-}
), local softness k S+ for reactants R2 and local softness kS—for reactants 1a or 1b. These values of the local descriptors on the atoms P1, P2, C1 and C2 of the reactants 1a, 1b and R2 were calculated according to the Gazquez-Mendez model with the NPA population analyzes and MK at the B3LYP 6-311+G level (d, p) are given in Table 4. Examination of the values in Table 4 indicates that the phosphines 1a, and dichloroethylene R2 have similar values of local softnesses (
{S}_{k}^{+}
{S}_{k}^{-}
) by the approach of Mulliken. This observation shows that the most favored interaction takes place between the P1 atom of the compound 1a and the C1 atom of the dichloroethylene.
Table 4. Values of Fukui Functions (
{f}_{k}^{+}
{f}_{k}^{-}
), local softnesses,
{S}_{k}^{+}
for reactants R2 and local softness,
{S}_{k}^{-}
for reactants 1a and 1b calculated by NPA, MK.
That the most favored interaction takes place between the P1 atom of the compound 1a and the C1 atom of the dichloroethylene for the first reaction and the P2 atom of the compound 1b and the C1 atom of the dichloroethylene for the second reaction.
Cite this paper: Bohoussou, K. , Benié, A. , Koné, M. , Kakou, A. , Bamba, K. and Ziao, N. (2020) Erratum to “Theoretical Study of the Reaction of (2, 2)-Dichloro (Ethyl) Arylphosphine with Bis (2, 2)-Dichloro (Ethyl) Arylphosphine by Hydrophosphination Regioselective by the DFT Method” [Computational Chemistry 5 (2017) 113-128]. Computational Chemistry, 8, 14-16. doi: 10.4236/cc.2020.81002.
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Implement αβ0 to abc transform - Simulink - MathWorks Nordic
Implement αβ0 to abc transform
The Inverse Clarke Transform block converts the time-domain alpha, beta, and zero components in a stationary reference frame to three-phase components in an abc reference frame. The block can preserve the active and reactive powers with the powers of the system in the stationary reference frame by implementing an invariant power version of the inverse Clarke transform. If the zero component is zero, the components in the three-phase system are balanced.
Balanced ɑ, β, and zero components in a stationary reference frame
The direction of the magnetic axes of the stator windings in the stationary ɑβ0 reference frame and the abc reference frame
The time-response of the individual components of equivalent balanced ɑβ0 and abc systems
The block implements the inverse Clarke transform as
\left[\begin{array}{c}a\\ b\\ c\end{array}\right]=\left[\begin{array}{ccc}1& 0& 1\\ â\frac{1}{2}& \frac{\sqrt{3}}{2}& 1\\ â\frac{1}{2}& â\frac{\sqrt{3}}{2}& 1\end{array}\right]\left[\begin{array}{c}\mathrm{α}\\ \mathrm{β}\\ 0\end{array}\right]
α and β are the components in the stationary reference frame.
0 is the zero component in the stationary reference frame.
The block implements this power invariant version of the inverse Clarke transform as
\left[\begin{array}{c}a\\ b\\ c\end{array}\right]=\sqrt{\frac{2}{3}}\left[\begin{array}{ccc}1& 0& \frac{1}{\sqrt{2}}\\ â\frac{1}{2}& \frac{\sqrt{3}}{2}& \frac{1}{\sqrt{2}}\\ â\frac{1}{2}& â\frac{\sqrt{3}}{2}& \frac{1}{\sqrt{2}}\end{array}\right]\left[\begin{array}{c}\mathrm{α}\\ \mathrm{β}\\ 0\end{array}\right]
Preserve the active and reactive power of the system in the rotating reference frame.
Clarke Transform | Clarke to Park Angle Transform | Inverse Park Transform | Park Transform | Park to Clarke Angle Transform
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Feature Scaling in Machine Learning - Fizzy
Machine Learning algorithms don't perform well when the input numerical attributes have different scales significantly (Ge), i.e. some features may range from 0 to 10 while another range from 1,000 to 10,000. This may cause some algorithms to assign these two variables with different importance.
For example, in the k-nearest neighbor algorithm, the classifier mainly calculates the Euclidean distance between two points. If a feature has a larger range value than other features, the distance will be dominated by this eigenvalue. Therefore each feature should be normalized, such as processing the range of values between 0 and 1.
In addition, feature scaling would make the algorithms (gradient descent) to converge faster.
Decision tree like algorithms are usually not sensitive to feature scales.
Methods of Feature Scaling
1. Rescaling/Min-Max Scaling
In this approach, the data is scaled to a fixed range - usually [0,1] or [-1,1]. The cost of having this bounded range is that we will end up with smaller standard deviations, which can suppress the effect of outliers.
x^{\prime}=\frac{x-\min (x)}{\max (x)-\min (x)}
enhances the stability of attributes with small variance
keeps the 0s in a sparse matrix
Feature standardization makes the values of each feature in the data have zero-mean (when subtracting the mean in the numerator) and unit-variance (1).
This method is widely used in machine learning algorithms, e.g. SVM, logistic regression and neural networks.
x^{\prime}=\frac{x-\overline{x}}{\sigma}
\sigma=\sqrt{\frac{\sum(x-\operatorname{mean}(x))^{2}}{n}}
3. Mean normalisation
x^{\prime}=\frac{x-\operatorname{mean}(x)}{\max (x)-\min (x)}
4. Scaling to unit length
Devided by the Euclidean length of the vector, two norm.
x^{\prime}=\frac{x}{\|x\|}
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Train support vector machine (SVM) classifier for one-class and binary classification - MATLAB fitcsvm - MathWorks France
G\left({x}_{j},{x}_{k}\right)=\mathrm{exp}\left(-{‖{x}_{j}-{x}_{k}‖}^{2}\right)
G\left({x}_{j},{x}_{k}\right)={x}_{j}\prime {x}_{k}
G\left({x}_{j},{x}_{k}\right)={\left(1+{x}_{j}\prime {x}_{k}\right)}^{q}
\left\{\begin{array}{l}{\alpha }_{j}\left[{y}_{j}f\left({x}_{j}\right)-1+{\xi }_{j}\right]=0\\ {\xi }_{j}\left(C-{\alpha }_{j}\right)=0\end{array}
f\left({x}_{j}\right)=\varphi \left({x}_{j}\right)\prime \beta +b,
0.5\sum _{jk}{\alpha }_{j}{\alpha }_{k}G\left({x}_{j},{x}_{k}\right)
{\alpha }_{1},...,{\alpha }_{n}
\sum {\alpha }_{j}=n\nu
0\le {\alpha }_{j}\le 1
f\left(x\right)=x\prime \beta +b,
2/‖\beta ‖.
‖\beta ‖
0.5{‖\beta ‖}^{2}+C\sum {\xi }_{j}
{y}_{j}f\left({x}_{j}\right)\ge 1-{\xi }_{j}
{\xi }_{j}\ge 0
0.5\sum _{j=1}^{n}\sum _{k=1}^{n}{\alpha }_{j}{\alpha }_{k}{y}_{j}{y}_{k}{x}_{j}\prime {x}_{k}-\sum _{j=1}^{n}{\alpha }_{j}
\sum {\alpha }_{j}{y}_{j}=0
0\le {\alpha }_{j}\le C
\stackrel{^}{f}\left(x\right)=\sum _{j=1}^{n}{\stackrel{^}{\alpha }}_{j}{y}_{j}x\prime {x}_{j}+\stackrel{^}{b}.
\stackrel{^}{b}
{\stackrel{^}{\alpha }}_{j}
\stackrel{^}{\alpha }
\text{sign}\left(\stackrel{^}{f}\left(z\right)\right).
0.5\sum _{j=1}^{n}\sum _{k=1}^{n}{\alpha }_{j}{\alpha }_{k}{y}_{j}{y}_{k}G\left({x}_{j},{x}_{k}\right)-\sum _{j=1}^{n}{\alpha }_{j}
\sum {\alpha }_{j}{y}_{j}=0
0\le {\alpha }_{j}\le C
\stackrel{^}{f}\left(x\right)=\sum _{j=1}^{n}{\stackrel{^}{\alpha }}_{j}{y}_{j}G\left(x,{x}_{j}\right)+\stackrel{^}{b}.
{C}_{j}=n{C}_{0}{w}_{j}^{\ast },
{x}_{j}^{\ast }=\frac{{x}_{j}-{\mu }_{j}^{\ast }}{{\sigma }_{j}^{\ast }},
\begin{array}{c}{\mu }_{j}^{\ast }=\frac{1}{\sum _{k}{w}_{k}^{*}}\sum _{k}{w}_{k}^{*}{x}_{jk},\\ {\left({\sigma }_{j}^{\ast }\right)}^{2}=\frac{{v}_{1}}{{v}_{1}^{2}-{v}_{2}}\sum _{k}{w}_{k}^{*}{\left({x}_{jk}-{\mu }_{j}^{\ast }\right)}^{2},\\ {v}_{1}=\sum _{j}{w}_{j}^{*},\\ {v}_{2}=\sum _{j}{\left({w}_{j}^{*}\right)}^{2}.\end{array}
\sum _{j=1}^{n}{\alpha }_{j}=n\nu .
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Global Constraint Catalog: Cincreasing_sum
<< 5.189. increasing_peak5.191. increasing_valley >>
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
\mathrm{𝚜𝚞𝚖}_\mathrm{𝚌𝚝𝚛}
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚜𝚞𝚖}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},𝚂\right)
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚜𝚞𝚖}_\mathrm{𝚌𝚝𝚛}
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚜𝚞𝚖}_\mathrm{𝚎𝚚}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
𝚂
\mathrm{𝚍𝚟𝚊𝚛}
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
𝚂
is the sum of the variables of the collection
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\left(〈3,3,6,8〉,20\right)
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚜𝚞𝚖}
〈3,3,6,8〉
𝚂=20
is set to the sum
〈3+3+6+8〉
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)>1
𝚂
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚜𝚞𝚖}
constraint can be used for breaking some symmetries in bin packing problems. Given a set of
n
bins with the same maximum capacity, and a set of items each of them with a specific height, the problem is to pack all items in the bins. To break symmetry we order bins by increasing use. This is done by introducing a variable
{x}_{i}
\left(0\le i<n\right)
for each bin
i
giving its use, i.e., the sum of items heights assigned to bin
i
, and by posting the following
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚜𝚞𝚖}
\left(〈{x}_{0},{x}_{1},\cdots ,{x}_{n-1}〉,s\right)
s
denotes the sum of the heights of all the items to pack.
A linear time filtering algorithm achieving bound-consistency for the
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚜𝚞𝚖}
constraint is described in [PetitReginBeldiceanu11]. This algorithm was motivated by the fact that achieving bound-consistency on the inequality constraints and on the sum constraint independently hinders propagation, as illustrated by the following small example, where the maximum value of
{x}_{1}
is not reduced to 2:
{x}_{1}\in \left[1,3\right]
{x}_{2}\in \left[2,5\right]
s\in \left[5,6\right]
{x}_{1}<{x}_{2}
{x}_{1}+{x}_{2}=s
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚜𝚞𝚖}
\left(〈{x}_{0},{x}_{1},\cdots ,{x}_{n-1}〉,s\right)
constraint, the bound-consistency algorithm consists of three phases:
A normalisation phase adjusts the minimum and maximum value of variables
{x}_{0},{x}_{1},\cdots ,{x}_{n-1}
with respect to the chain of inequalities
{x}_{0}\le {x}_{1}\le \cdots \le {x}_{n-1}
. A forward phase adjusts the minimum value of
{x}_{1},{x}_{2},\cdots ,{x}_{n-1}
\underline{{x}_{i+1}}\ge \underline{{x}_{i}}
), while a backward phase adjusts the maximum value of
{x}_{n-2},{x}_{n-1},\cdots ,{x}_{0}
\overline{{x}_{i-1}}\le \overline{{x}_{i}}
A phase restricts the minimum and maximum value of the sum variable
s
{x}_{0}\le {x}_{1}\le \cdots \le {x}_{n-1}
\underline{s}\ge {\sum }_{0\le i<n}\underline{{x}_{i}}
\overline{s}\le {\sum }_{0\le i<n}\overline{{x}_{i}}
A final phase reduces the minimum and maximum value of variables
{x}_{0},{x}_{1},\cdots ,{x}_{n-1}
both from the bounds of
s
and from the chain of inequalities. Without loss of generality we now focus on the pruning of the maximum value of variables
{x}_{0},{x}_{1},\cdots ,{x}_{n-1}
. For this purpose we first need to introduce the notion of last intersecting index of a variable
{x}_{i}
{\mathrm{𝑙𝑎𝑠𝑡}}_{i}
. This corresponds to the greatest index in
\left[i+1,n-1\right]
\overline{{x}_{i}}>\underline{{x}_{{\mathrm{𝑙𝑎𝑠𝑡}}_{i}}}
if no such integer exists. Then the increase of the minimum value of
s
{x}_{i}
\overline{{x}_{i}}
{\sum }_{k\in \left[i,{\mathrm{𝑙𝑎𝑠𝑡}}_{i}\right]}\left(\overline{{x}_{i}}-\underline{{x}_{k}}\right)
. When this increase exceeds the available margin, i.e.
\overline{s}-{\sum }_{0\le i<n}\underline{{x}_{i}}
, we update the maximum value of
{x}_{i}
We illustrate a part of the final phase on the following example
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚜𝚞𝚖}
\left(〈{x}_{0},{x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5}〉,s\right)
{x}_{0}\in \left[2,6\right]
{x}_{1}\in \left[4,7\right]
{x}_{2}\in \left[4,7\right]
{x}_{3}\in \left[5,7\right]
{x}_{4}\in \left[6,9\right]
{x}_{5}\in \left[7,9\right]
s\in \left[28,29\right]
. Observe that the domains are consistent with the first two phases of the algorithm, since,
the minimum (and maximum) values of variables
{x}_{0},{x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5}
are increasing,
the sum of the minimum of the variables
{x}_{0},{x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5}
, i.e., 28 is less than or equal to the maximum value of
s
the sum of the maximum of the variables
{x}_{0},{x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5}
, i.e., 45 is greater than or equal to the minimum value of
s
Now, assume we want to know the increase of the minimum value of
s
{x}_{0}
is set to its maximum value 6. First we compute the last intersecting index of variable
{x}_{0}
{x}_{4}
is the last variable for which the minimum value is less than or equal to maximum value of
{x}_{0}
{\mathrm{𝑙𝑎𝑠𝑡}}_{0}=4
. The increase is equal to
{\sum }_{k\in \left[0,4\right]}\left(\overline{{x}_{0}}-\underline{{x}_{k}}\right)=\left(6-2\right)+\left(6-4\right)+\left(6-4\right)+\left(6-5\right)+\left(6-6\right)=9
. Since it exceeds the margin
29-\left(2+4+4+5+6+7\right)=1
we have to reduce the maximum value of
{x}_{0}
. How to do this incrementally is described in [PetitReginBeldiceanu11].
n
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚜𝚞𝚖}
0..n
n
6 - 3 7 9 11 11 11
7 - 2 7 11 13 15 15
10 - - 7 18 28 34 38
15 - - 1 18 51 87 116
16 - - 1 16 55 100 141
17 - - - 14 55 112 164
19 - - - 9 55 136 221
26 - - - - 28 166 440
31 - - - - 7 125 519
34 - - - - 2 87 515
44 - - - - - 7 255
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚜𝚞𝚖}
0..n
\mathrm{𝚜𝚞𝚖}_\mathrm{𝚌𝚝𝚛}
(sum).
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
constraint type: predefined constraint, order constraint, arithmetic constraint.
•
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚜𝚞𝚖}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},𝚂\right)
\mathrm{𝚖𝚒𝚗𝚟𝚊𝚕}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)>0
\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\left(𝚂,\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
•
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚜𝚞𝚖}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},𝚂\right)
\mathrm{𝚖𝚒𝚗𝚟𝚊𝚕}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)>0
\mathrm{𝚜𝚞𝚖}_\mathrm{𝚘𝚏}_\mathrm{𝚒𝚗𝚌𝚛𝚎𝚖𝚎𝚗𝚝𝚜}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝙻𝙸𝙼𝙸𝚃}\right)
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Global Constraint Catalog: Ccircuit_cluster
<< 5.66. circuit5.68. circular_change >>
Inspired by [LaporteAsefVaziriSriskandarajah96].
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}_\mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}\left(\mathrm{𝙽𝙲𝙸𝚁𝙲𝚄𝙸𝚃},\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
\mathrm{𝙽𝙲𝙸𝚁𝙲𝚄𝙸𝚃}
\mathrm{𝚍𝚟𝚊𝚛}
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚜𝚞𝚌𝚌}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝙽𝙲𝙸𝚁𝙲𝚄𝙸𝚃}\ge 1
\mathrm{𝙽𝙲𝙸𝚁𝙲𝚄𝙸𝚃}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝙽𝙾𝙳𝙴𝚂},\left[\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛},\mathrm{𝚜𝚞𝚌𝚌}\right]\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\ge 1
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝙽𝙾𝙳𝙴𝚂},\mathrm{𝚒𝚗𝚍𝚎𝚡}\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚜𝚞𝚌𝚌}\ge 1
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚜𝚞𝚌𝚌}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
G
, described by the
\mathrm{𝙽𝙾𝙳𝙴𝚂}
collection, such that its vertices are partitioned among several clusters.
\mathrm{𝙽𝙲𝙸𝚁𝙲𝚄𝙸𝚃}
is the number of circuits containing more than one vertex used for covering
G
in such a way that each cluster is visited by exactly one circuit of length greater than 1.
\left(\begin{array}{c}1,〈\begin{array}{ccc}\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\hfill & \mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}-1\hfill & \mathrm{𝚜𝚞𝚌𝚌}-1,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-2\hfill & \mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}-1\hfill & \mathrm{𝚜𝚞𝚌𝚌}-4,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-3\hfill & \mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}-2\hfill & \mathrm{𝚜𝚞𝚌𝚌}-3,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-4\hfill & \mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}-2\hfill & \mathrm{𝚜𝚞𝚌𝚌}-5,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-5\hfill & \mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-8,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-6\hfill & \mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-6,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-7\hfill & \mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-7,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-8\hfill & \mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}-4\hfill & \mathrm{𝚜𝚞𝚌𝚌}-2,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-9\hfill & \mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}-4\hfill & \mathrm{𝚜𝚞𝚌𝚌}-9\hfill \end{array}〉\hfill \end{array}\right)
\left(\begin{array}{c}2,〈\begin{array}{ccc}\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\hfill & \mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}-1\hfill & \mathrm{𝚜𝚞𝚌𝚌}-1,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-2\hfill & \mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}-1\hfill & \mathrm{𝚜𝚞𝚌𝚌}-4,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-3\hfill & \mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}-2\hfill & \mathrm{𝚜𝚞𝚌𝚌}-3,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-4\hfill & \mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}-2\hfill & \mathrm{𝚜𝚞𝚌𝚌}-2,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-5\hfill & \mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-5,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-6\hfill & \mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-9,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-7\hfill & \mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-7,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-8\hfill & \mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}-4\hfill & \mathrm{𝚜𝚞𝚌𝚌}-8,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-9\hfill & \mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}-4\hfill & \mathrm{𝚜𝚞𝚌𝚌}-6\hfill \end{array}〉\hfill \end{array}\right)
Figure 5.67.1. Four clusters and a covering with one circuit corresponding to the first example of the Example slot
Both examples involve 9 vertices
1,2,\cdots ,9
such that vertices 1 and 2 belong to cluster number 1, vertices 3 and 4 belong to cluster number 2, vertices 5, 6 and 7 belong to cluster number 3, and vertices 8 and 9 belong to cluster number 4.
The first example involves only a single circuit containing more than one vertex (i.e., see in Figure 5.67.1 the circuit
2\to 4\to 5\to 8\to 2
). The corresponding
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}_\mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}
constraint holds since exactly one vertex of each cluster (i.e., vertex 2 for cluster 1, vertex 4 for cluster 2, vertex 5 for cluster 3, vertex 8 for cluster 4) belongs to this circuit.
The second example contains the two circuits
2\to 4\to 2
6\to 9\to 6
that both involve more than one vertex. The corresponding
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}_\mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}
constraint holds since exactly one vertex of each cluster (i.e., see in Figure 5.67.2 vertex 2 in
2\to 4\to 2
for cluster 1, vertex 4 in
2\to 4\to 2
6\to 9\to 6
6\to 9\to 6
for cluster 4) belongs to these two circuits.
Figure 5.67.2. The same clusters as in the first example of the Example slot and a covering with two circuits corresponding to the second example of the Example slot
\mathrm{𝙽𝙲𝙸𝚁𝙲𝚄𝙸𝚃}<|\mathrm{𝙽𝙾𝙳𝙴𝚂}|
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|>2
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}\right)>1
\mathrm{𝙽𝙾𝙳𝙴𝚂}
A related abstraction in Operations Research was introduced in [LaporteAsefVaziriSriskandarajah96]. It was reported as the Generalised Travelling Salesman Problem (GTSP). The
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}_\mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}
constraint differs from the GTSP because of the two following points:
Each node of our graph belongs to a single cluster,
We do not constrain the number of circuits to be equal to 1: The number of circuits should be equal to one of the values of the domain of the variable
\mathrm{𝙽𝙲𝙸𝚁𝙲𝚄𝙸𝚃}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}
\mathrm{𝚌𝚢𝚌𝚕𝚎}
(graph constraint, one_succ).
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎𝚜}
final graph structure: strongly connected component, one_succ.
modelling: cluster.
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1},\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}\right)
•\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚜𝚞𝚌𝚌}\ne \mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚒𝚗𝚍𝚎𝚡}
•\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚜𝚞𝚌𝚌}=\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡}
•
\mathrm{𝐍𝐓𝐑𝐄𝐄}
=0
•
\mathrm{𝐍𝐒𝐂𝐂}
=\mathrm{𝙽𝙲𝙸𝚁𝙲𝚄𝙸𝚃}
\mathrm{𝙾𝙽𝙴}_\mathrm{𝚂𝚄𝙲𝙲}
\begin{array}{c}\mathrm{𝖠𝖫𝖫}_\mathrm{𝖵𝖤𝖱𝖳𝖨𝖢𝖤𝖲}↦\hfill \\ \left[\begin{array}{c}\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}-\mathrm{𝚌𝚘𝚕}\left(\begin{array}{c}\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}-\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right),\hfill \\ \mathrm{𝚒𝚝𝚎𝚖}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}\right)\right]\hfill \end{array}\right)\hfill \end{array}\right]\hfill \end{array}
•
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\right)
•
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎𝚜}
\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜},=,\mathrm{𝚜𝚒𝚣𝚎}\left(\mathrm{𝙽𝙾𝙳𝙴𝚂},\mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}\right)\right)
In order to express the binary constraint linking two vertices one has to make explicit the identifier of each vertex as well as the cluster to which belongs each vertex. This is why the
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}_\mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}
constraint considers objects that have the following three attributes:
The attribute
\mathrm{𝚒𝚗𝚍𝚎𝚡}
that is the identifier of a vertex.
\mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}
that is the cluster to which belongs a vertex.
\mathrm{𝚜𝚞𝚌𝚌}
that is the unique successor of a vertex.
The partitioning of the clusters by different circuits is expressed in the following way:
First note the condition
\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚜𝚞𝚌𝚌}\ne \mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚒𝚗𝚍𝚎𝚡}
prevents the final graph of containing any loop. Moreover the condition
\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚜𝚞𝚌𝚌}=\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡}
imposes no more than one successor for each vertex of the final graph.
\mathrm{𝐍𝐓𝐑𝐄𝐄}
=
0 enforces that all vertices of the final graph belong to one circuit.
\mathrm{𝐍𝐒𝐂𝐂}
=
\mathrm{𝙽𝙲𝙸𝚁𝙲𝚄𝙸𝚃}
express the fact that the number of strongly connected components of the final graph is equal to
\mathrm{𝙽𝙲𝙸𝚁𝙲𝚄𝙸𝚃}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\right)
\mathrm{𝖠𝖫𝖫}_\mathrm{𝖵𝖤𝖱𝖳𝖨𝖢𝖤𝖲}
(i.e., all the vertices of the final graph) states that the cluster attributes of the vertices of the final graph should be pairwise distinct. This concretely means that no cluster should be visited more than once.
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎𝚜}
\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜},=,\mathrm{𝚜𝚒𝚣𝚎}\left(\mathrm{𝙽𝙾𝙳𝙴𝚂},\mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}\right)\right)
\mathrm{𝖠𝖫𝖫}_\mathrm{𝖵𝖤𝖱𝖳𝖨𝖢𝖤𝖲}
conveys the fact that the number of distinct values of the cluster attribute of the vertices of the final graph should be equal to the total number of clusters. This implies that each cluster is visited at least one time.
Parts (A) and (B) of Figure 5.67.3 respectively show the initial and final graph associated with the second example of the Example slot. Since we use the
\mathrm{𝐍𝐒𝐂𝐂}
graph property, we show the two strongly connected components of the final graph. They respectively correspond to the two circuits
2\to 4\to 2
6\to 9\to 6
. Since all the vertices belongs to a circuit we have that
\mathrm{𝐍𝐓𝐑𝐄𝐄}
=
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}_\mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}
|
Change in Mass of Spinning Black Holes w.r.t. the Angular Momentum
Change in Mass of Spinning Black Holes w.r.t. the Angular Momentum ()
Mahto et al. have shown δS≥0 and M2=a*J by using the first law of the black hole mechanics in the vacuum and Einstein mass-energy equivalence relation specially for spinning black holes. In the present paper, this work is extended to propose a model for the change in mass of the spinning black holes due to corresponding change in the angular momentum for maximum & half spin parameter of black holes (a* = 1 & 1/2) and calculated their values for different test black holes in XRBs and AGN. We have also shown that the change in mass of the spinning black holes due to corresponding change in the angular momentum for maximum spinning rate of black holes (a* = 1) is double to that of the spinning black holes having spinning parameter (a* = 1/2).
Spinning Black Holes, Angular Momentum, Spinning Parameter
Mahto, D. and Kumari, A. (2018) Change in Mass of Spinning Black Holes w.r.t. the Angular Momentum. Journal of Modern Physics, 9, 1037-1042. doi: 10.4236/jmp.2018.95065.
Classically black holes are perfect absorbers, but do not emit anything; their physical temperature is absolute zero [1] . Quantum mechanically, however, there is a possibility that one of a particle production pairs in a black hole is able to tunnel the gravitational barrier and escapes the black hole’s horizon. Thus it can radiate or evaporate particles [2] . The mass, area and surface gravity of the black hole mechanics have the same role as the energy, entropy and temperature in the ordinary laws of thermodynamics [3] . In 2006, B. Aschenbach has shown that the orbital velocity of a test particle is no longer a monotonic function of the orbit radius when the spin of the black hole is greater than 0.9953 , but displays a local minimum-maximum structure for radii smaller than 1.8 gravitational radii [4] . In 2007, Adel Bouchareb and Gerard Clement extended the Abbott-Deser-Tekin approach to the Computation of the Killing charge for a solution of topologically massive gravity (TMG) linearized around an arbitrary background and then applied to evaluate the mass and angular momentum of black hole solutions of TMG with non-constant curvature asymptotic [5] . In 2009, Richard B Larson suggested that in all cases, gravitational interactions with other stars or mass concentrations in a forming system play an important role in redistributing angular momentum and thereby enabling the formation of a compact object [6] . In 2013, Mahto et al. have shown
\delta S\ge 0
to use the first law of the black hole mechanics in the vacuum and Einstein mass-energy equivalence relation specially for spinning black holes and established the relation
{M}^{2}={a}^{\ast }J
by entirely new methods [7] . All the works done as discussed above do not give the comparative study for the change in mass of the spinning black holes due to corresponding change in the angular momentum for half spin and maximum spin of the black holes.
In the present work, a model for the change in mass of the spinning black holes due to corresponding change in the angular momentum is proposed for maximum spinning rate of black holes (a* = 1/2 and 1).
The mass (M), angular momentum (J) and spin parameter (a*) of black holes are co-related by the following equation [7] .
{M}^{2}=a\ast J
where a* is a constant called spin parameter of spinning black holes lying between −1 to +1 including zero of different test black holes [8] .
Now the equation is differentiated , we have
2M\delta M={a}^{\ast }\delta J
\frac{\delta M}{\delta J}=\frac{{a}^{\ast }}{2M}
For spinning black holes having maximum spin
{a}^{\ast }=1
[8] . Hence Equation (2) leads to
\frac{\delta M}{\delta J}=\frac{1}{2M}
The above equation shows that the change in mass of the spinning black holes due to corresponding change in the angular momentum for maximum spinning rate of black holes.
For half spin parameter (a* = 1/2), the Equation (2) becomes [9]
\frac{\delta M}{\delta J}=\frac{1}{4M}
The above equation shows that the change in mass of the spinning black holes due to corresponding change in the angular momentum for half spin parameter of black holes.
3. Data in the Support for Mass of Black Holes and Sun
There are two categories of black holes classified on the basis of their masses clearly very distinct from each other, with very different masses M ~ 5 20Mʘ for stellar-mass Black holes in X-ray binaries and M ~106 - 109.5Mʘ for super massive black holes in Active Galactic Nuclei [8] . Mass of sun (Mʘ) = 1.99 × 1030 kg [8] .
On the basis of the data mentioned in the Section 3, the change in mass of different test spinning black holes due to corresponding change in angular momentum are calculated in XRBs and AGN to plot the graphs with the help of Equation (3) & (4) as shown in Figures 1-3 of the Section 4.
To obtain the change in mass of the spinning black holes due to corresponding change in the angular momentum, the equation (1) is differentiated for maximum spinning rate of black holes (a* = 1) and half spin parameter (a* = 1/2) given by the Equation (3) & (4) respectively.
After this, we have calculated the change in mass of the spinning black holes due to corresponding change in the angular momentum for maximum spinning rate of black holes (a* = 1 & a* = 1/2) in XRBs and AGN with the help of (3) & (4).
Figure 1. The graph plotted between the mass of spinning black holes and corresponding change in their angular momentum in XRBs with spin parameter a* = 1/2 and 1.
Figure 2. The graph plotted between the mass of spinning black holes and corresponding change in their angular momentum in AGN with spin parameter a*=1/2 using logarithmic scale.
We have plotted the graph between the change in mass of the spinning black holes w.r.t. the change in the angular momentum for maximum spinning rate of black holes (a* = 1) and spinning parameter (a* = 1/2)
\left(\delta M/\delta J\right)
and corresponding value of the mass of black holes (M). We also have shown that the change in mass of the spinning black holes due to corresponding change in the angular momentum for maximum spinning rate of black holes (a* = 1) is greater than to that of the spinning black holes having spinning parameter (a* = 1/2) in XRBs and shows that the change in mass of the spinning black holes w.r.t. the change in the angular momentum for maximum spinning rate of black holes (a* = 1) is twice times to that of spinning of black holes with spinning parameter (a* = 1/2). This fact is also clear from the comparative study from the Figure 1 for XRBs, while in the case of AGN, this change is quite different regarding the maximum and half integral spin as clear from the Figure 2 & Figure 3.
Figure 2 & Figure 3 show the graph plotted between the mass of spinning black holes and corresponding change in their angular momentum in AGN with spin parameter a* = 1/2 and a* = 1 respectively. From both the graphs, it is clear that the change in mass of the spinning black holes w.r.t. the change in the angular momentum for maximum spinning rate of black holes (a* = 1) fluctuates with certain values on increasing the value of the mass of the spinning black holes, while for the same mass of black holes with spin parameter (a* = 1/2), there is uniform variation in mass of the spinning black holes w.r.t. the change in the angular momentum on increasing the value of the mass of the spinning black holes.
From the graph in the Figure 3, it is obvious that the change in mass of
Figure 3. The graph plotted between the mass of spinning black holes and corresponding change in their angular momentum in AGN using logarithmic scale explaining complex nature of spinning black holes with spin parameter a* = 1.
spinning black holes and corresponding change in their angular momentum in AGN with spin parameter a* = 1 decreases up to a certain value of the mass with the increasing the mass of the spinning black holes and then rapidly increases with a certain value These variations are repeated in periodic manner as shown in Figure 3. On the basis of observation of the graph in the Figure 3, the black holes can be categorized regarding their same order of mass or radius of event horizon as discussed below:
The spinning black holes of mass:
1) (1 × 106Mʘ, 1 × 107Mʘ, 1 ×108Mʘ, 1 × 109Mʘ)―line a
2) (2 × 106Mʘ, 2 × 107Mʘ, 2 ×108Mʘ, 2 × 109Mʘ)―line b
3) (3 × 106Mʘ, 3 × 107Mʘ, 3 ×108Mʘ, 3 × 109Mʘ)―line c
4) (4 × 106Mʘ, 4 × 107Mʘ, 4 ×108Mʘ, 4 × 109Mʘ)―line d
5) (5 × 106Mʘ, 5 × 107Mʘ, 5 ×108Mʘ, 5 × 109Mʘ)―line e
6) (6 × 106Mʘ, 6 × 107Mʘ, 6 ×108Mʘ, 6 × 109Mʘ)―line f
7) (7 × 106Mʘ, 7 × 107Mʘ, 7 ×108Mʘ, 7 × 109Mʘ)―line g
8) (8 × 106Mʘ, 8 × 107Mʘ, 8 ×108Mʘ, 8 × 109Mʘ)―line h
9) (9 × 106Mʘ, 9 × 107Mʘ, 9 ×108Mʘ, 0 × 109Mʘ)―line i
When the graph is plotted for each category in the same graph paper, nine parallel lines are obtained. When the results obtained from our research work is compared with that of the spinning black holes with spin a* = 1/2, we see that the change in mass of the spinning black holes w.r.t. the change in the angular momentum for maximum spinning rate of black holes (a* = 1) is quite different to that of the spinning black holes with spin parameter (a* = 1/2). Hence, we may conclude that the spinning parameters of black holes are mainly responsible for the identification and characterization of black holes.
In the study of present research paper, we can draw the following conclusions:
1) The change in mass of the spinning black holes due to corresponding change in the angular momentum for the black holes of maximum spin is double to that of the spinning black holes of half spin parameter and decreases with increasing the mass of black holes in XRBs.
2) The change in mass of the spinning black holes w.r.t. the change in the angular momentum for the black holes of maximum spin is quite different to that of the spinning black holes with half spin parameter in AGN.
3) The spinning parameter is mainly responsible for the identification and characterization of black holes.
[1] Wald, R.M. (2001) Living Reviews in Relativity, 4, 6.
[2] Triyanta, T. and Bowaire, A.N. (2013) Journal of Mathematical and Fundamental Sciences, 45, 114-123.
[4] Aschenbach, B. (2006) Chinese Journal of Astronomy and Astrophysics, 10, 24.
[5] Adel, B. and Gerard, C. (2007) Classical and Quantum Gravity, 24, 5581-5594.
[6] Richard, B.L (2010) Reports on Progress in Physics, 73, Article ID: 014901, 14 p.
[7] Mahto, D., Kumari, A. and Singh, K.M. (2013) International Journal of Engineering and Innovative Technology, 3, 270-273.
[8] Narayan, R. (2005) Black Holes in Astrophysics. arXiv:gr-qc/0506078 V1.
[9] Mahto, D., Singh, A.K. and Kumari, N. (2016) International Journal of Astronomy and Astrophysics, 6, 328-333.
|
Global Constraint Catalog: Cnvalue
<< 5.285. nset_of_consecutive_values5.287. nvalue_on_intersection >>
[PachetRoy99]
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}\left(\mathrm{𝙽𝚅𝙰𝙻},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚘𝚗}_\mathrm{𝚊𝚝𝚝𝚛𝚒𝚋𝚞𝚝𝚎𝚜}_\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}
\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}
\mathrm{𝙽𝚅𝙰𝙻}
\mathrm{𝚍𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)
\mathrm{𝙽𝚅𝙰𝙻}\ge \mathrm{𝚖𝚒𝚗}\left(1,|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|\right)
\mathrm{𝙽𝚅𝙰𝙻}\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
\mathrm{𝙽𝚅𝙰𝙻}\le
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)
\mathrm{𝙽𝚅𝙰𝙻}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\left(4,〈3,1,7,1,6〉\right)
\left(1,〈6,6,6,6,6〉\right)
\left(5,〈6,3,0,2,9〉\right)
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝙽𝚅𝙰𝙻}=4
〈3,1,7,1,6〉
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝙽𝚅𝙰𝙻}=1
〈6,6,6,6,6〉
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝙽𝚅𝙰𝙻}=5
〈6,3,0,2,9〉
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
N\in \left[1,2\right]
{V}_{1}\in \left[2,4\right]
{V}_{2}\in \left[1,2\right]
{V}_{3}\in \left[2,4\right]
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\left(N,〈{V}_{1},{V}_{2},{V}_{3}〉\right)
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝙽𝚅𝙰𝙻}>1
\mathrm{𝙽𝚅𝙰𝙻}<|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>1
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝙽𝚅𝙰𝙻}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝙽𝚅𝙰𝙻}=1
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>0
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝙽𝚅𝙰𝙻}=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
constraint allows relaxing the
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
constraint by restricting its first argument
\mathrm{𝙽𝚅𝙰𝙻}
to be close, but not necessarily equal, to the number of variables of the
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
A classical example from the early 1850s is the dominating queens chess puzzle problem: Place a number of queens on an
n
chessboard in such a way that all cells of the chessboard are either attacked by a queen or are occupied by a queen. A queen can attack all cells located on the same column, on the same row or on the same diagonal. Part (A) of Figure 5.286.2 illustrates a set of five queens which together attack all of the cells of an 8 by 8 chessboard. The dominating queens problem can be modelled by just one
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
We first label the different cells of the chessboard from 1 to
{n}^{2}
We then associate to each cell
c
of the chessboard a domain variable. Its initial domain is set to the labels of the cells that can attack cell
c
. For instance, in the context of an 8 by 8 chessboard, the initial domain of
{V}_{29}
will be set to {2,5,8,11,13,15,20..22,25..32,36..38,43,45,47,50,53,56,57,61} (see the green cells of part (B) of Figure 5.286.2).
Finally, we post the constraint
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\left(Q,〈\mathrm{𝚟𝚊𝚛}-{V}_{1},\mathrm{𝚟𝚊𝚛}-{V}_{2},\cdots ,\mathrm{𝚟𝚊𝚛}-{V}_{{n}^{2}}〉\right)
Q
is a domain variable in
\left[1,{n}^{2}\right]
that gives the total number of queens used for controlling all cells of the chessboard. For the solution depicted by Part (A) of Figure 5.286.2, the label in each cell of Part (C) of Figure 5.286.2 gives the value assigned to the corresponding variable. Note that, since a given cell can be attacked by several queens, we have also other assignments corresponding to the solution depicted by Part (A) of Figure 5.286.2.
Figure 5.286.2. Modelling the dominating queens problem with a single
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
constraint; (A) a solution to the dominating queens problem, (B) the initial domain (in bold) of the variable associated with cell 29: in a solution the value
j
assigned to the variable associated with cell
i
represents the label of the cell attacking cell
i
(i.e. in a solution one of the selected queens is located on cell
j
), (C) the value of each cell in the model with one single
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
constraint corresponding to the solution depicted in (A).
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
constraint occurs also in many practical applications. In the context of timetabling one wants to set up a limit on the maximum number of activity types it is possible to perform. For frequency allocation problems, one optimisation criterion is to minimise the number of distinct frequencies that you use all over the entire network.
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
constraint generalises several constraints like:
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
: in order to get the
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
constraint, one has to set
\mathrm{𝙽𝚅𝙰𝙻}
to the total number of variables.
\mathrm{𝚗𝚘𝚝}_\mathrm{𝚊𝚕𝚕}_\mathrm{𝚎𝚚𝚞𝚊𝚕}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝚗𝚘𝚝}_\mathrm{𝚊𝚕𝚕}_\mathrm{𝚎𝚚𝚞𝚊𝚕}
constraint, one has to set the minimum value of
\mathrm{𝙽𝚅𝙰𝙻}
This constraint appears in [PachetRoy99] under the name of Cardinality on Attributes Values. The
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}
in JaCoP (http://www.jacop.eu/). A constraint called
𝚔_\mathrm{𝚍𝚒𝚏𝚏}
enforcing that a set of variables takes at least
k
distinct values appears in the PhD thesis of J.-C. Régin [Regin95].
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
constraint has a solution or not is NP-hard. This was achieved by reduction from 3-SAT. In the same article, it is also shown, by reduction from minimum hitting set cardinality, that computing a sharp lower bound on
\mathrm{𝙽𝚅𝙰𝙻}
is NP-hard.
Both reformulations of the
\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎𝚜}
constraint use the
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
A first filtering algorithm for the
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
constraint was described in [Beldiceanu01]. Assuming that the minimum value of variable
\mathrm{𝙽𝚅𝙰𝙻}
is not constrained at all, two algorithms that both achieve bound-consistency were provided one year later in [BeldiceanuCarlssonThiel02]. Under the same assumption, algorithms that partially take into account holes in the domains of the variables of the
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
collection are described in [BeldiceanuCarlssonThiel02], [BessiereHebrardHnichKiziltanWalsh05].
A model, involving linear inequalities constraints, preserving bound-consistency was introduced in [BessiereKatsirelosNarodytskaQuimperWalsh10CP].
n
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
0..n
n
5 - - - 720 37800 940800 15876000
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
0..n
nvalues in Gecode, nvalue in MiniZinc, nvalue in SICStus.
\mathrm{𝚝𝚛𝚊𝚌𝚔}
\mathrm{𝚊𝚜𝚜𝚒𝚐𝚗}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎𝚜}
\mathrm{𝚊𝚖𝚘𝚗𝚐}
\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚍𝚒𝚏𝚏}_\mathtt{0}
\mathrm{𝚌𝚘𝚞𝚗𝚝}
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
\mathrm{𝚖𝚊𝚡}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚖𝚒𝚗}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎𝚜}_\mathrm{𝚎𝚡𝚌𝚎𝚙𝚝}_\mathtt{0}
(counting constraint,number of distinct values).
\mathrm{𝚜𝚞𝚖}_\mathrm{𝚘𝚏}_\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚜}_\mathrm{𝚘𝚏}_\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}_\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}
(introduce a weight for each value and replace number of distinct values by sum of weights associated with distinct values).
\mathrm{𝚗𝚌𝚕𝚊𝚜𝚜}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}\in \mathrm{𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗}
\mathrm{𝚗𝚎𝚚𝚞𝚒𝚟𝚊𝚕𝚎𝚗𝚌𝚎}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}\mathrm{mod}\mathrm{𝚌𝚘𝚗𝚜𝚝𝚊𝚗𝚝}
\mathrm{𝚗𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}/\mathrm{𝚌𝚘𝚗𝚜𝚝𝚊𝚗𝚝}
\mathrm{𝚗𝚙𝚊𝚒𝚛}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}
\mathrm{𝚙𝚊𝚒𝚛}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎𝚜}
(replace an equality with the number of distinct values by a comparison with the number of distinct values),
\mathrm{𝚗𝚟𝚎𝚌𝚝𝚘𝚛}
(variable replaced by vector).
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚊𝚝𝚕𝚎𝚊𝚜𝚝}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
=\mathrm{𝙽𝚅𝙰𝙻}
\ge \mathrm{𝙽𝚅𝙰𝙻}
\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
=\mathrm{𝙽𝚅𝙰𝙻}
\le \mathrm{𝙽𝚅𝙰𝙻}
\mathrm{𝚋𝚊𝚕𝚊𝚗𝚌𝚎}
(restriction on how balanced an assignment is),
\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
(restrict number of distinct colours on each maximum clique of the interval graph associated with the tasks),
\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎𝚜}
(restrict number of distinct colours on each maximum clique of the interval graph associated with the tasks assigned to the same machine),
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}_\mathrm{𝚌𝚑𝚊𝚒𝚗}
𝚔_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
(necessary condition for two overlapping
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
constraints),
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚟𝚊𝚛}
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}_\mathrm{𝚘𝚗}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚜𝚎𝚌𝚝𝚒𝚘𝚗}
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎𝚜}_\mathrm{𝚎𝚡𝚌𝚎𝚙𝚝}_\mathtt{0}
(value 0 is ignored).
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚎𝚚𝚞𝚊𝚕}
(enforce to have one single value),
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
(enforce a number of distinct values equal to the number of variables),
\mathrm{𝚗𝚘𝚝}_\mathrm{𝚊𝚕𝚕}_\mathrm{𝚎𝚚𝚞𝚊𝚕}
(enforce to have at least two distinct values).
\mathrm{𝚌𝚘𝚗𝚜𝚎𝚌𝚞𝚝𝚒𝚟𝚎}_\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}
\mathrm{𝚌𝚢𝚌𝚕𝚎}
\mathrm{𝚖𝚒𝚗}_𝚗
complexity: 3-SAT, minimum hitting set cardinality.
filtering: bound-consistency, convex bipartite graph.
puzzles: dominating queens.
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}\left(\mathrm{𝙽𝚅𝙰𝙻},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\left(\mathrm{𝙽𝚅𝙰𝙻},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1},\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}\right)
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1}.\mathrm{𝚟𝚊𝚛}=\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝐍𝐒𝐂𝐂}
=\mathrm{𝙽𝚅𝙰𝙻}
\mathrm{𝙴𝚀𝚄𝙸𝚅𝙰𝙻𝙴𝙽𝙲𝙴}
\mathrm{𝐍𝐒𝐂𝐂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
{S}_{i}
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
: identifying infeasible values wrt the at most side
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
: identifying infeasible variable-value pairs wrt the at least side
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
: variations of dominating knights
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Proving \frac{1}{(\tan(\frac x2)+1)^2 \cos^2(\frac x2)}=\frac{1}{1+\sin x}
PEEWSRIGWETRYqx 2022-01-16 Answered
\frac{1}{{\left(\mathrm{tan}\left(\frac{x}{2}\right)+1\right)}^{2}{\mathrm{cos}}^{2}\left(\frac{x}{2}\right)}=\frac{1}{1+\mathrm{sin}x}
\frac{1}{{\left(\mathrm{tan}\frac{x}{2}+1\right)}^{2}{\mathrm{cos}}^{2}\frac{x}{2}}=\frac{1}{\left({\mathrm{tan}}^{2}\frac{x}{2}+2\mathrm{tan}\frac{x}{2}+1\right){\mathrm{cos}}^{2}\frac{x}{2}}
=\frac{1}{{\mathrm{sin}}^{2}\frac{x}{2}+2\mathrm{sin}\frac{x}{2}\mathrm{cos}\frac{x}{2}+{\mathrm{cos}}^{2}\frac{x}{2}}=\frac{1}{1+2\mathrm{sin}\frac{x}{2}\mathrm{cos}\frac{x}{2}}=\frac{1}{1+\mathrm{sin}x}
\mathrm{tan}\frac{x}{2}=t
\frac{1+{t}^{2}}{{\left(t+1\right)}^{2}}=\dots =\frac{1}{1+\frac{2t}{1+{t}^{2}}}
Now use Weierstrass substitution
\mathrm{sin}x+\mathrm{sin}y=a\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\mathrm{cos}x+\mathrm{cos}y=b
\mathrm{tan}\left(x-\frac{y}{2}\right)
use the given function value(s), and trigonometric identities (including the cofunction identities), to find the indicated trigonometric functions.
\mathrm{cos}\theta =\frac{1}{3}
\text{(a)}\mathrm{sin}\theta \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{(b)}\mathrm{tan}\theta
\text{(c)}\mathrm{sec}\theta \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{(d)}\mathrm{csc}\left(90-\theta \right)
Maximum and minimum of
f\left(x\right)=2{\mathrm{sin}}^{2}x+2{\mathrm{cos}}^{4}x
\text{tanh}\left(x\right)
Using Quotient and Reciprocal Identities:Given
\mathrm{sin}t=\frac{2}{5}
\mathrm{cos}t=\frac{\sqrt{21}}{5}
, find the value of each of the four remaining trigonometric functions.
\mathrm{sin}0
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Practice Euler's Theorem | Brilliant
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that if two numbers
and
n
are relatively prime (if they share no common factors apart from 1) then:
a^{\phi(n)} \equiv 1 \pmod n,
\phi(n)
is Euler's totient function, which counts the number of positive integers
\le n
n.
Euler’s Theorem is a generalization of Fermat's little theorem. It arises in many applications of elementary number theory, including calculating the last digits of large powers and, relatedly, it is part of the theoretical foundation for the RSA cryptosystem (online security).
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See Keong Lee, V. Ravichandran, Shamani Supramaniam, "Initial Coefficients of Biunivalent Functions", Abstract and Applied Analysis, vol. 2014, Article ID 640856, 6 pages, 2014. https://doi.org/10.1155/2014/640856
See Keong Lee ,1 V. Ravichandran,2 and Shamani Supramaniam 1
1School of Mathematical Sciences, Universiti Sains Malaysia (USM), 11800 Penang, Malaysia
An analytic function defined on the open unit disk is biunivalent if the function and its inverse are univalent in . Estimates for the initial coefficients of biunivalent functions are investigated when and , respectively, belong to some subclasses of univalent functions. Some earlier results are shown to be special cases of our results.
Let be the class of all univalent analytic functions in the open unit disk and normalized by the conditions and . For , it is well known that the th coefficient is bounded by . The bounds for the coefficients give information about the geometric properties of these functions. Indeed, the bound for the second coefficient of functions in the class gives rise to the growth, distortion and covering theorems for univalent functions. In view of the influence of the second coefficient in the geometric properties of univalent functions, it is important to know the bounds for the (initial) coefficients of functions belonging to various subclasses of univalent functions. In this paper, we investigate this coefficient problem for certain subclasses of biunivalent functions.
Recall that the Koebe one-quarter theorem [1] ensures that the image of under every univalent function contains a disk of radius 1/4. Thus, every univalent function has an inverse satisfying , , and A function is biunivalent in if both and are univalent in . Let denote the class of biunivalent functions defined in the unit disk . Lewin [2] investigated this class and obtained the bound for the second coefficient of the biunivalent functions. Several authors subsequently studied similar problems in this direction (see [3, 4]). A function is bistarlike or strongly bistarlike or biconvex of order if and are both starlike, strongly starlike, or convex of order , respectively. Brannan and Taha [5] obtained estimates for the initial coefficients of bistarlike, strongly bistarlike, and biconvex functions. Bounds for the initial coefficients of several classes of functions were also investigated in [6–24].
An analytic function is subordinate to an analytic function , written , if there is an analytic function with satisfying . Ma and Minda [25] unified various subclasses of starlike () and convex functions () by requiring that either the quantity or is subordinate to a more general superordinate function with positive real part in the unit disk , , , maps onto a region starlike with respect to and symmetric with respect to the real axis. The class of Ma-Minda starlike functions with respect to consists of functions satisfying the subordination . Similarly, the class of Ma-Minda convex functions consists of functions satisfying the subordination . Ma and Minda investigated growth and distortion properties of functions in and as well as Fekete-Szegö inequalities for and . Their proof of Fekete-Szegö inequalities requires the univalence of . Ali et al. [7] investigated Fekete-Szegö problems for various other classes and their proof does not require the univalence or starlikeness of . In particular, their results are valid even if one just assumes the function to have a series expansion of the form , . So, in this paper, we assume that has series expansion , , are real, and . A function is Ma-Minda bistarlike or Ma-Minda biconvex if both and are, respectively, Ma-Minda starlike or convex. Motivated by the Fekete-Szegö problem for the classes of Ma-Minda starlike and Ma-Minda convex functions [25], Ali et al. [26] recently obtained estimates of the initial coefficients for biunivalent Ma-Minda starlike and Ma-Minda convex functions.
The present work is motivated by the results of Kędzierawski [27] who considered functions belonging to certain subclasses of univalent functions while their inverses belong to some other subclasses of univalent functions. Among other results, he obtained the following coefficient estimates.
Theorem 1 (see [27]). Let with Taylor series and . Then,
We need the following classes investigated in [6, 7, 26].
Definition 2. Let be analytic and with and . For , let
In this paper, we obtain the estimates for the second and third coefficients of functions when(i) and , or , or ,(ii) and , or ,(iii) and .
In the sequel, it is assumed that and are analytic functions of the form
Theorem 3. Let and . If , and is of the form then where .
Proof. Since and , , then there exist analytic functions , with , satisfying Define the functions and by or, equivalently, Then, and are analytic in with . Since , the functions and have positive real part in , and and . In view of (8) and (10), it is clear that Using (10) together with (4), it is evident that Since has the Maclaurin series given by (5), a computation shows that its inverse has the expansion Since it follows from (11) and (12) that It follows from (15) and (17) that Equations (15), (16), (18), and (19) lead to where , which, in view of and , gives us the desired estimate on as asserted in (6).
By using (16), (18), and (19), we get and this yields the estimate given in (7).
Remark 4. When and , , then (6) reduces to Theorem 1. When and , Theorem 3 reduces to [26, Theorem 2.2].
Theorem 5. Let and . If and , then where .
Proof. Let and , . Then, there exist analytic functions , with , such that Since (12) and (24) yield It follows from (26) and (28) that Hence, (26), (27), (29), and (30) lead to which gives us the desired estimate on as asserted in (22) when and .
Further, (27), (29), and (30) give and this yields the estimate given in (23).
Proof. Let and , . Then, there are analytic functions , with , satisfying Using and (12) and (34) will yield Further implication of (36) and applying the fact that and give the estimates in (33).
Theorem 7. Let and . If , , then where .
Proof. For and , , there exist analytic functions , with , satisfying Since then (12) and (38) yield Further implication of (40) and applying the fact that and give the estimates in (37).
Remark 8. When and , Theorem 7 reduces to [26, Theorem 2.3].
The following theorems give the estimates for the second and third coefficients of functions when (i) and and (ii) and . The proofs are similar as for the theorems above; hence, they are omitted here.
Theorem 10. Let and . If and , then where .
Remark 11. When and , Theorem 10 reduces to [26, Theorem 2.4].
The research of the first and last authors is supported, respectively, by FRGS Grant and MyBrain MyPhD Programme of the Ministry of Higher Education, Malaysia.
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Copyright © 2014 See Keong Lee et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Evaluate the laplace transform for this (a)L\{4x^2-3\cos(2x)+5e^x\} (b)L\{4+\frac{1}{2}\sin(x)-e^{4x}\}
L\left\{4{x}^{2}-3\mathrm{cos}\left(2x\right)+5{e}^{x}\right\}
L\left\{4+\frac{1}{2}\mathrm{sin}\left(x\right)-{e}^{4x}\right\}
Use following laplace formulas in solving following problems
L\left\{1\right\}=\frac{1}{s}
L\left\{{x}^{n}\right\}=\frac{n!}{{s}^{n+1}}
L\left\{\mathrm{cos}ax\right\}=\frac{s}{{s}^{2}+{a}^{2}}
L\left\{\mathrm{sin}ax\right\}=\frac{a}{{s}^{2}+{a}^{2}}
L\left\{{e}^{ax}\right\}=\frac{1}{s-a}
Step 2 Step-by-Step Explanation
L\left\{4{x}^{2}-3\mathrm{cos}\left(2x\right)+5{e}^{x}\right\}
by the linearity property
=4L\left\{{x}^{2}\right\}-3L\left\{\mathrm{cos}\left(2x\right)\right\}+5L\left\{{e}^{x}\right\}
=4\left(\frac{2!}{{s}^{3}}\right)-3\left(\frac{s}{{s}^{2}+{2}^{2}}\right)+5\left(\frac{1}{s-1}\right)
=\frac{8}{{s}^{3}}-\frac{3s}{{s}^{2}+4}+\frac{5}{s-1}
L\left\{4+\frac{1}{2}\mathrm{sin}\left(x\right)-{e}^{4x}\right\}
=4L\left\{1\right\}+\frac{1}{2}L\left\{\mathrm{sin}\left(x\right)\right\}-L\left\{{e}^{4x}\right\}
=4\left(\frac{1}{s}\right)+\frac{1}{2}\left(\frac{1}{{s}^{2}+1}\right)-\frac{1}{s-4}
=\frac{4}{s}+\frac{1}{2\left({s}^{2}+1\right)}-\frac{1}{s-4}
L\left\{4{x}^{2}-3\mathrm{cos}\left(2x\right)+5{e}^{x}\right\}=\frac{8}{{s}^{3}}-\frac{3s}{{s}^{2}+4}+\frac{5}{s-1}
L\left\{4+\frac{1}{2}\mathrm{sin}\left(x\right)-{e}^{4x}\right\}=\frac{4}{s}+\frac{1}{2\left({s}^{2}+1\right)}-\frac{1}{s-4}
\frac{dy}{dx}+5y=15
Find the Laplace transforms of the given functions.
g\left(t\right)=4\mathrm{cos}\left(4t\right)-9\mathrm{sin}\left(4t\right)+2\mathrm{cos}\left(10t\right)
Find the Laplace transform of the following:
h\left(t\right)={t}^{2}{e}^{-t}h\left(t-2\right)
f\left(t\right)={\int }_{0}^{t}\left(t-w\right)\mathrm{cos}\left(2w\right)dw
Obtain Laplace transforms for the function
{t}^{2}
step(t-2)
f\left(t\right)={\mathrm{cos}}^{2}at
{\int }_{\partial \mathrm{\Omega }}\frac{\partial u}{\left[\partial N\right\}d\sigma \text{ }\text{if}\text{ }\mathrm{\Delta }u=0\text{ }\text{in}\text{ }\mathrm{\Omega }}
Is it always possible to solve an equation
u{}^{″}=0
in [0,1] if
{u}^{\prime }\left(0\right)\text{ }\text{and}\text{ }{u}^{\prime }\left(1\right)
are given?(What is connection to first question?) Is the solution - unique?
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A random sample of 2,500 people was selected, and the people were asked to give
A random sample of 2,500 people was selected, and the people were asked to give their favorite season. Their responses, along with their age group, ar
A random sample of 2,500 people was selected, and the people were asked to give their favorite season. Their responses, along with their age group, are summarized in the two-way table below.
\begin{array}{cccccc}& \text{Winter}& \text{Spring}& \text{Summer }& \text{Fall}& \text{Total}\\ \text{Children}& 30& 0& 170& 0& 200\\ \text{Teens}& 150& 75& 250& 25& 500\\ \text{Adults }& 250& 250& 250& 250& 1000\\ \text{Seniors}& 300& 150& 50& 300& 800\\ \text{Total}& 730& 475& 720& 575& 2500\end{array}
Among those whose favorite season is spring, what proportion are adults?
a\right)\frac{250}{1000}
b\right)\frac{250}{2500}
c\right)\frac{475}{2500}
d\right)\frac{250}{475}
e\right)\frac{225}{475}
Obtain the proportion of adults whose favourite season is spring:
The value of probability is obtained by:
P\left(E\right)=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}
P\left(\text{Adults whose favorite season is spring}\right)=\frac{\text{Number of adults whose favorite season is spring}}{\text{Total number of outcomes}}
=\frac{250}{2500}
Therefore, the correct answer is “Option B.)
\frac{250}{2500}
The following two-way contingency table gives the breakdown of the population of adults in a town according to their highest level of education and whether or not they regularly take vitamins:
\begin{array}{|ccc|}\hline \text{Education}& \text{Use of vitamins takes}& \text{Does not take}\\ \text{No High School Diploma}& 0.03& 0.07\\ \text{High School Diploma}& 0.11& 0.39\\ \text{Undergraduate Degree}& 0.09& 0.27\\ \text{Graduate Degree}& 0.02& 0.02\\ \hline\end{array}
You select a person at random. What is the probability the person does not take vitamins regularly?
You randomly survey shoppers at a supermarket about whether they use reusable bags. Of 60 male shoppers 15 use reusable bags. Of 110 female shoppers, 60 use reusable bags. Organize your results in a two-way table. Include the marginal frequencies.
A group of children and adults were polled about whether they watch a particular TV show. The survey results, showing the joint relative frequencies and marginal relative frequencies, are shown in the two-way table. What is the value of x? YesNoTotal Children0.30.40.7 Adults0.25x0.3 Total0.550.451
Answer the questions using the table below:
P\left(A\right)=?
P\left(\frac{A}{B}\right)=?
P\left(X\right)=?
P\left(\frac{A}{X}\right)=?
Find the probability that a randomly selected student is in sports using the to-way table below.
\begin{array}{|cccc|}\hline & \text{Yes}& \text{No}& \text{Total}\\ \text{Group 1}& 710& 277& 987\\ \text{Group 2}& 1175& 323& 1498\\ \text{ }\text{Total}& 1885& 600& 2485\\ \hline\end{array}
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Attribute Dependence Graph and Evaluation in Semantic Analysis
In this article we discuss attribute dependency graphs and attribute evaluation during semantic analysis phase in compiler design.
Attribute evaluation.
Given a parse tree and an Syntax Directed Definition, we draw edges among attribute instances that are associated with each node in the parse tree to denote that the value of the attribute at the head of the edge is dependent upon the value of the tail of the edge, this is referred to as a dependency graph.
These are used to show the flow of information among attribute instances within a parse tree.
In the graph, an edge from one attribute instance to another means that the value of the first attribute is required to compute the value of the second.
Edges are used to express constraints that are implied by the language semantic rules.
For each node in the parse tree, for example a node X, the dependency graph will have a node associated with X.
If a semantic rule is associated with a production P defines the value of a synthesized attribute A.b in terms of the value of X.c, then the dependency graph is said to have an edge from X.c to A.b. In more detail, at every node N labeled A where this production P is applied, we create an edge to attribute b at N from attribute c at the child of N that corresponds to this instance of symbol X in the body of the production.
If a semantic rule associated with a production P defines a value of an inherited attribute B.c in terms of the value X.a, the dependency graph contains an edge from X.a to B.c. For each node N that is labeled B and corresponds to an occurrence of B in the body of the production P, we create an edge to and attribute c at N from the attribute a at node M that corresponds to this occurrence of X. Keep in mind that M should be either the parent or a sibling.
We have the production;
E →
{\mathrm{E}}_{1}
+ T
And the semantic rule;
E.val =
{\mathrm{E}}_{1}
.val + T.val.
For every node N that is labeled E, that has children corresponding to the body of the production, the synthesized attribute val at N is computed using values of val at the two children E and T.
Therefore a section of the dependency graph for every parse tree where this production is used will look like the image below.
As a common convention we use dotted lines to represent the edges of the parse tree and solid lines to represent edges of the dependency graph.
We have the annotated parse tree for 3 * 5.
and the SDD:
T → F
{\mathrm{T}}^{\text{'}}
{\mathrm{T}}^{\text{'}}
.inh = F.val
{\mathrm{T}}^{\text{'}}
→ * F
{\mathrm{T}}_{1}^{\text{'}}
{\mathrm{T}}_{1}^{\text{'}}
.inh =
{\mathrm{T}}^{\text{'}}
.inh x F.val,
{\mathrm{T}}^{\text{'}}
{\mathrm{T}}_{1}^{\text{'}}
.syn
{\mathrm{T}}^{\text{'}}
→ ϵ
{\mathrm{T}}^{\text{'}}
{\mathrm{T}}^{\text{'}}
.inh
F → digit F.val = digit.lexval
Now its complete dependency graph.
Numbers 1-9 represent nodes of the graph and correspond to attributes in the annotated parse tree for 3 * 5.
Nodes 1 and 2 represent the attribute lexval that is associated with the two leaves that are labeled digit.
Nodes 3 and 4 represent the attribute val that is associated with nodes labeled F.
Edges to node 3 from 1 and those from node 4 from 2 result from the semantic rule defining F.val in terms of digit.lexval.
As a matter of fact, F.val is equal to digit.lexval although the edges represent dependence and not equality.
5 and 6, represent the inherited attribute
{\mathrm{T}}^{\text{'}}
.inh that is associated with every occurrence of the nonterminal
{\mathrm{T}}^{\text{'}}
Edge(5, 3) is a result of the rule
{\mathrm{T}}^{\text{'}}
.inh = F.val that defines
{\mathrm{T}}^{\text{'}}
.inh at the right child of the root from F.val at the left child.
Edges from node 5 to 6 for
{\mathrm{T}}^{\text{'}}
.inh and from node 4 for F.val since these value are multiplied to evaluate the attribute inh at node 6.
Nodes 7 and 8 represent a synthesized attribute syn associated with the occurrences of
{\mathrm{T}}^{\text{'}}
Edge from 6 to 7 is a result of the semantic rule 3 associated with production 3 from the SDD.
Edge from 7 to 8 is a result of a semantic rule associated with the production 2 from the SDD.
Node 9 represents attribute T.val and the edge from 8 to 9 is a result of the semantic rule associated with production 1 from the SDD.
A dependency graph characterizes all possible orderings in which we can use to evaluate attributes at various nodes of a parse tree.
If the graph has an edge (M, N), then the attribute that corresponds to M is evaluated before attribute of N.
Therefore the only allowed orderings for an evaluation are the sequences of nodes
{\mathrm{N}}_{1}
{\mathrm{N}}_{2}
{\mathrm{N}}_{k}
such that if there is an edge of the dependency graph from
{\mathrm{N}}_{i}
{\mathrm{N}}_{j}
then i < j.
This ordering embeds a directed graph into a linear ordering referred to as the topological sort of the graph.
If the graph has no cycles, then there exists at least one topological sort otherwise there are no topological sorts, that is, there is no way to evaluate the syntax directed definition(SDD) on the parse tree.
We can see why there are no cycles by finding a node with node entering edge, if there was such a node, we proceed from predecessor to predecessor until we return to the same node we have already been at, this results in a cycle. Then we mark it as the first in a topological ordering and remove it from the graph and repeat this for the remaining nodes.
Looking at the dependency graph from the previous section, we can see there are no cycles, a topological sort ordering would be in the way the nodes are numbered, that is, 1, 2, 3, ..., 9.
See how every edge goes from one node to a higher-numbered node - this is surely a topological sort.
Another topological sort would be 1, 3, 5, 2, 4, 6, 7, 8, 9.
Dependency graphs show the flow of information among attribute instances in a parse tree.
In problematic SDDs, there are parse trees for which it is impossible to find an ordering that can be used to compute all attributes at all nodes since they have cycles.
Modern Compiler Design, Dick Grune, Kees van Reeuwijk, Henri E. Bal
Compilers Principles, Techniques, & Tools, Alfred V. Aho, Monica S. Lam
|
A motor having an efficiency Of 90 % operates a crane having an efficiency of 40% With what constant - Physics - Work Energy And Power - 7916437 | Meritnation.com
Given, input power of motor = 5 kW = 5000 W
Efficiency of motor = 90 %
efficiency = \frac{output power}{input power}
Therefore, output power of motor = 5000 X 90/100 = 4500 W
This output power of motor will act as input power for the crane.
Therefore, input power of crane = 4500 W
Given, efficiency of crane = 40%
Hence, output power of crane = 4500 X 40/100 = 1800 W (using eq. 1)
Therefore, 1800 = mg X velocity
1800 = 500 X 9.8 X velocity
velocity = 0.36 m/s.
|
3Blue1Brown - e^(iπ) in 3.14 minutes, using dynamics
Chapter 5e^(iπ) in 3.14 minutes, using dynamics
One way to think about the function
e^t
is to ask what properties it has. Probably the most important one, from some points of view the defining property, is that it’s a function which is equal to its own derivative. Together with the added condition that inputting
0
1
, it’s the only function with this property.
You can illustrate what that means with a physical model: Think of
e^t
as your position on a number line as a function of time. The condition
e^0 = 1
means you start at
1
. The equation above is saying that your velocity, the derivative of position, is always equal to your position. In other words, the farther away from 0 you are, the faster you move, with the very specific constraint that your velocity vector is perpetually identical to your position vector.
So even before knowing how to compute
e^t
exactly, this ability to associate each position with the velocity you must have at that position paints a very strong intuitive picture of how the function must grow. You know you’ll be accelerating, at an accelerating rate, with an all-around feeling of things getting out of hand quickly.
If we add a constant to this exponent, like
e^{2t}
, the chain rule tells us the derivative of our function is now
2
times the function itself.
So at every point on the number line, rather than attaching a vector corresponding to the number itself, first double the magnitude, then attach it. Moving so that your position is always
e^{2t}
is the same thing as moving in such a way that your velocity is always twice your position. The implication of that
2
is that our runaway growth feels all the more out of control.
If that constant was negative, say -0.5, then your velocity vector is always -0.5 times your position vector, meaning you flip it around 180-degrees, and scale its length by a half.
Moving in such a way that your velocity always matches this flipped and squished copy of the position vector, you’d go the other direction, slowing down in exponential decay towards 0.
What about if the constant was
i
, the imaginary unit? If your position was always
e^{it}
, how would you move as that time,
t
, ticks forward?
Well, assuming there’s any way to make sense out of
e^{it}
, and assuming that derivatives still work the way we’d expect when extending to complex numbers, the derivative of your position
e^{it}
would now always be
i
times itself. Multiplying by
i
has the effect of rotating numbers 90-degrees, and as you might expect, things only make sense here if we start thinking beyond the number line and in the complex plane.
Geometrically, which of the following describes the point
i \cdot (a + bi)
on the complex plane?
A 90-degree clockwise rotation of the point
a + bi
A 90-degree counterclockwise rotation of the point
a + bi
A reflection of the point
a + bi
about the real axis.
a + bi
about the imaginary axis.
So what does
e^{it}
So even before you know how to compute
e^{it}
, you know that for any position this might give for some value of t, the velocity at that time will be a 90-degree rotation of that position.
Each blue arrow above pointing outward from the origin represents an example value of
e^{it}
, thought of as a position vector. The corresponding green arrow at its tip shows what the velocity, equal to the position rotated 90 degrees counterclockwise, would have to be for that particular position.
Drawing this for all possible positions you might come across, we get a vector field, where, usually with vector fields we shrink things down to avoid clutter.
t=0
e^{it}
will be 1. There’s only one trajectory starting from that position where your velocity is always matching the vector it’s passing through, a 90-degree rotation of position. It’s when you go around the unit circle at a speed of 1 unit per second.
So after
\pi
seconds, you’ve traced a distance of
\pi
around;
e^{i\pi} = -1
\tau = 2\pi
seconds, you’ve gone full circle;
e^{i\tau} = 1
e^{it}
equals a number
t
radians [1] around this circle.
Nevertheless, something might still feel immoral about putting an imaginary number up in that exponent. And you’d be right to question that! For values of
x
which are not real numbers, what we write as
e^x
has very little to do with repeated multiplication, and its relation to the number
e
feels frankly more incidental than definitional. To dig in a little bit deeper, the next lesson covers more general exponents, with a focus on matrices.
[1] - radian(s), SI unit of angle, 1 radian is equal to an angle at the center of a circle where the arc length of the circular sector is equal to the radius of the circle
But what is a Fourier series? From heat flow to circle drawings
|
Determine whether the integral is divergent or convergent. If it
Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as "divergent
Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as "divergent".
{\int }_{3}^{\mathrm{\infty }}\left(\frac{8}{{\left(x+4\right)}^{3}}/2\right)
{\int }_{3}^{\mathrm{\infty }}\frac{4}{{\left(x+4\right)}^{3}}dx
Factor out constants:
=4{\int }_{3}^{\mathrm{\infty }}\frac{1}{{\left(x+4\right)}^{3}}dx
For the integrand
\frac{1}{{\left(x+4\right)}^{3}}
, substitute u = x + 4 and du = dx.
This gives a new lower bound
u=4+3=7
u=\infty
=4{\int }_{7}^{\mathrm{\infty }}\frac{1}{{u}^{3}}du
\frac{1}{{u}^{3}}is-\frac{1}{2{u}^{2}}
=\underset{b\to \mathrm{\infty }}{lim}\left(-\frac{2}{{u}^{2}}\right){|}_{7}^{b}
Evaluate the antiderivative at the limits and subtract.
\left(-\frac{2}{{u}^{2}}\right){|}_{7}^{\mathrm{\infty }}=\left(-\frac{2}{{\mathrm{\infty }}^{2}}\right)-\left(-\frac{2}{{7}^{2}}\right)=\frac{2}{49}
=\frac{2}{49}
{\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{\mathrm{sin}y}{\sqrt{\mathrm{sin}y+c}}dy
{\int }_{0}^{1}\frac{1}{{e}^{x}+1}dx
{\int }_{0}^{\frac{\pi }{2}}\mathrm{arccos}\left(\mathrm{sin}x\right)dx
-d\frac{{\left\{\pi \right\}}^{2}}{8}
{t}^{7}-4
{\int }_{0}^{\frac{\pi }{2}}\frac{{\mathrm{sin}}^{3}x}{{\mathrm{sin}}^{3}x+{\mathrm{cos}}^{3}x}dx
\mathrm{ln}x
|
Solve for x :- ➩ mod(x - 2) is greater then or equal to 5 How to do the - Maths - Linear Inequalities - 10209551 | Meritnation.com
➩ mod(x - 2) is greater then or equal to 5.
How to do the last step, please explain sir.
\left|\mathrm{x}-2\right|\ge 5\phantom{\rule{0ex}{0ex}}⇒\left(\mathrm{x}-2\right)\le -5 \mathrm{or} \left(\mathrm{x}-2\right)\ge 5 \left[\mathrm{When} \left|\mathrm{x}\right|\ge \mathrm{a}, \mathrm{then} \mathrm{x}\le -\mathrm{a} \mathrm{or} \mathrm{x}\ge \mathrm{a}\right]\phantom{\rule{0ex}{0ex}}⇒\mathrm{x}\le -5+2 \mathrm{or} \mathrm{x}\ge 5+2\phantom{\rule{0ex}{0ex}}⇒\mathrm{x}\le -3 \mathrm{or} \mathrm{x}\ge 7\phantom{\rule{0ex}{0ex}}\therefore \mathrm{x}\in \left(-\infty ,-3\right]\cup \left[7,\infty \right)
|
Then the game stops. All bills are kept by the
Then the game stops. All bills are kept by the player. Determine the probability of winning $12.
An urn contains 3 one-dollar bills, 1 five-dollar bill and 1 ten-dollar bill. A player draws bills one at a time without replacement from the urn until a ten-dollar bill is drawn. Then the game stops. All bills are kept by the player. Determine the probability of winning $12.
Answer the following conceptual questions about probability.
a) Suppose your friend Rudy says that the probability it rains tomorrow is 125 percent. Is Rudy’s statement coherent? Explain your answer.
b) Suppose Rudy assigns probability 0.3 to the sentence
\left(P\to Q\right)
What probability should Rudy assign to the sentence
\left(\sim P\vee Q\right)?
c) Suppose Rudy buys a lottery ticket. You say, “Don’t you know how unlikely it is that you’ll win?” Rudy replies, “Either I’ll win or I won’t, so there’s a 50% chance!” Is Rudy correct?
What might you say to Rudy in this case?
A financial services committee had 60 members, of which 10 were women. If 8 members are selected at random, find the probability that composed as the following.
Find the probability that the group will consist of 5 men and 3 women
|
K-means Clustering - Fizzy
K-means clustering is a type of unsupervised learning, which is used for unlabeled data (i.e., data without defined categories or groups). The goal of this algorithm is to find groups in the data, with the number of groups represented by the variable K (defined manually as an input).
The algorithm works iteratively as following:
randomly assign K prototype vectors to the axis
x_i, i \in\{1, \ldots, s\}
find its nearest centroid (i.e. prototype vector)
[\widehat{c_{j}}|x_{i}]=\arg \min _{j} Dist\left(x_{i}, c_{j}\right)
x_i
to this cluster j
for each cluster j = 1...K:
recompute its centoid
c_j
= mean of all data points assigned to it previously
c_{j}(a)=\frac{1}{n}\sum x_{i}(a) \quad \text { for } a=1 \ldots d
a
is a particular attribute value. It can only take numerical values, since here we are computing the average. It cannot be categorical or ordinal values.)
stop when none of the cluster assignments change (i.e. no data points change cluster memerships anymore)
Compare to all other clustering algorithms, K-means is blazingly fast. The computation required for K-mean clustering can be illustrated as:
iterations \times clusters(K) \times instances(x_i) \times dimensions
Properties of K-means
Every member of a cluster is closer to its cluster than any other cluster
Some Features of K-means
The variable K indicates the number of groups is predetermined manually, also called a hyper-parameter. Hence, it could be helpful to do feature analysis prior to decide what is the K.
The position of starting prototype vectors are chosen randomly. The algorithm will converge slower if they are close to each other. There are ways to optimize it, such as K-means++.
K-means++ is a strategy to choose the initial centroids:
randomly choose one initial centroid
c_1
for all data points, calculate the distance between them and
c_1
choose a new centroid: the longer the distance calculated in the previous step, the higher the chance this point is selected as the new centroid
repeat step 2 and 3 until number of prototype vectors (K) are picked
then do the standard K-means iteration algorithm
Other Optimized Algorithms
K-centroids
|
$\mathbb{Z}_2$–Thurston norm and complexity of $3$–manifolds, II
William Jaco, J Hyam Rubinstein, Jonathan Spreer and Stephan Tillmann
In this sequel to earlier papers by three of the authors, we obtain a new bound on the complexity of a closed
3
–manifold, as well as a characterisation of manifolds realising our complexity bounds. As an application, we obtain the first infinite families of minimal triangulations of Seifert fibred spaces modelled on Thurston’s geometry
{\stackrel{˜}{SL}}_{2}\left(ℝ\right)
3–manifold, minimal triangulation, layered triangulation, efficient triangulation, complexity, Seifert fibred space, lens space
|
Find the second partial derivatives. f(x,y)=x^{4}y-3x^{5}y^{2}f_{xx}(x,y)=f_{xy}(x,y)=f_{yx}(x,y)=f_{yy}(x,y)
f\left(x,y\right)={x}^{4}y-3{x}^{5}{y}^{2}
{f}_{xx}\left(x,y\right)=
{f}_{xy}\left(x,y\right)=
{f}_{yx}\left(x,y\right)=
{f}_{yy}\left(x,y\right)=
f\left(x,y\right)={x}^{4}y-3{x}^{5}{y}^{2}
f\left(x,y\right)={x}^{4}y-3{x}^{5}{y}^{2}
{f}_{x}\left(x,y\right)=\frac{\partial }{\partial x}\left(f\left(x,y\right)\right)=4{x}^{3}-15{x}^{4}{y}^{2}
{f}_{xy}\left(x,y\right)=\frac{\partial }{\partial y}\left(\frac{\partial f}{\partial x}\right)
=\frac{\partial }{\partial y}\left(4{x}^{3}-15{x}^{4}{y}^{2}\right)
=\frac{\partial }{\partial y}\left(4{x}^{3}\right)-\frac{\partial }{\partial y}\left(15{x}^{4}{y}^{2}\right)
=0-15{x}^{4}\left(2y\right)
{f}_{xy}\left(x,y\right)=-30{x}^{4}y
{f}_{xx}=\frac{\partial }{\partial x}\left(\frac{\partial f}{\partial x}\right)
=\frac{\partial }{\partial x}\left(4{x}^{3}-15{x}^{4}{y}^{2}\right)
=\frac{\partial }{\partial x}\left(4{x}^{3}\right)-\frac{\partial }{\partial x}\left(15{x}^{4}{y}^{2}\right)
{f}_{xx}=12{x}^{2}-60{x}^{3}{y}^{2}
{f}_{y}=\frac{\partial }{\partial y}\left({x}^{4}y-3{x}^{5}{y}^{2}\right)
=\frac{\partial }{\partial y}\left({x}^{4}y\right)-\frac{\partial }{\partial y}\left(3{x}^{5}{y}^{2}\right)
={x}^{4}-3{x}^{5}\left(2y\right)
{f}_{y}={x}^{4}-6{x}^{5}y
{f}_{yx}=\frac{\partial }{\partial x}\left(\frac{\partial f}{\partial y}\right)
The magnetic field 10 cm from a wire carrying a 1 A current is
2\mu T
. What is the field 4 cm from the wire? Express your answer with the appropriate units.
Raindrops make an angle theta with the vertical when viewed through a moving train window. If the speed of the train is v (subscript t), what is the speed of the raindrops in the reference frame of the Earth in which they are assumed to fall vertically?
f\left(x\right)={x}^{2}-1x-6
The line of symmetry has the equation:
|
Evaluate the indefinite integral: \int \sec^{2}x\tan^{4}xdx
\int {\mathrm{sec}}^{2}x{\mathrm{tan}}^{4}xdx
\int {\mathrm{sec}}^{2}x{\mathrm{tan}}^{4}xdx
\mathrm{tan}\left(x\right)=t
{\mathrm{sec}}^{2}\left(x\right)dx=dt
\int {\mathrm{sec}}^{2}\left(x\right){\mathrm{tan}}^{4}\left(x\right)dx=\int {t}^{4}dt
=\frac{{t}^{5}}{5}+c
=\frac{{\mathrm{tan}}^{5}\left(x\right)}{5}+c
Step 1: Use Integration by Substitution.
u=\mathrm{tan}x,du={\mathrm{sec}}^{2}xdx
Step 2: Using u and du above, rewrite
\int {\mathrm{sec}}^{2}x{\mathrm{tan}}^{4}xdx
\int {u}^{4}du
\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+C
\frac{{u}^{5}}{5}
u=\mathrm{tan}x
back into the original integral.
\frac{{\mathrm{tan}}^{5}x}{5}
\frac{{\mathrm{tan}}^{5}x}{5}+C
{\mathrm{tan}}^{-1}\left(\frac{1}{2\mathrm{sin}\left(x\right)}\right)
I want to calculate the following integral :
{\int }_{\left\{0\right\}}^{\frac{\pi }{2}}{\mathrm{tan}}^{-1}\left(\frac{1}{2\mathrm{sin}\left(x\right)}\right)dx
But I don't how; I tried by subsituting
u=\frac{1}{2\mathrm{sin}\left(x\right)}
u={\mathrm{tan}}^{-1}\left(\frac{1}{2\mathrm{sin}\left(x\right)}\right)
but it doesn't lead me anywhere.
\int \frac{3{x}^{2}+\sqrt{x}}{\sqrt{x}}dx
\int \frac{{z}^{2}}{\sqrt[3]{1+{z}^{3}}}dz
{\int }_{0}^{\frac{\pi }{4}}\frac{\mathrm{sin}x+\mathrm{cos}x}{{\mathrm{sin}}^{4}x+{\mathrm{cos}}^{2}x}dx
Evaluate the indefinate integral.
\int {x}^{2}\mathrm{sin}\left({x}^{3}\right)dx
{\int }_{0}^{3}\mathrm{sin}\left({x}^{2}\right)dx={\int }_{0}^{5}\mathrm{sin}\left({x}^{2}\right)dx+{\int }_{5}^{3}\mathrm{sin}\left({x}^{2}\right)dx
{\int }_{\alpha }^{\beta }\sqrt{\frac{\mathrm{arctan}h\left(r\right)}{r}}dr
-1\le \alpha \le \beta \le 1
|
Combinatorics/Bounds for Ramsey numbers - Wikibooks, open books for an open world
Combinatorics/Bounds for Ramsey numbers
The numbers R(r,s) in Ramsey's theorem (and their extensions to more than two colours) are known as Ramsey numbers. A major research problem in Ramsey theory is to find out Ramsey numbers for various values of r and s. We will derive the classical bounds here for any general Ramsey number R(r,s). This will just mean the exact value of that R(r,s) lies between the two professed bounds, the lower bound and the upper bound.
3 Known Ramsey numbers
4 A Multicolour Example: R(3,3,3) = 17
Upper bound[edit | edit source]
The upper bound is actually weaned off from the proof of Ramsey's theorem itself. Since R(r, s) ≤ R(r − 1, s) + R(r, s − 1) so this automatically gives an upper bound, although not in the closed form that we expect. The closed form expression is
{\displaystyle R(r,s)\leq {\binom {r+s-2}{r-1}}}
. To derive this use double induction on r and s. The base case r = s = 2 is easily established as
{\displaystyle R(2,2)=2\leq {\binom {2+2-2}{2-1}}=2}
. Now assume the expression holds for R(r − 1, s) and R(r, s − 1). Then
{\displaystyle R(r,s)\leq R(r-1,s)+R(r,s-1)\leq {\binom {r+s-3}{r-2}}+{\binom {r+s-3}{r-1}}={\binom {r+s-2}{r-1}}}
gives us our upper bound. Note that we have used Pascal's relation in the last equivalence.
A slight improvement is possible if both R(r − 1, s) and R(r, s − 1) are even numbers. In that case the inequality is strict.
Result: If both R(r − 1, s) + R(r, s − 1) are even numbers then R(r, s) < R(r − 1, s) + R(r, s − 1).
Proof: Suppose R(r, s) = R(r − 1, s) + R(r, s − 1) where both R(r − 1, s) and R(r, s − 1) are even numbers. Let n = R(r − 1, s) + R(r, s − 1) − 1. Now by our choice of n there exists a complete graph on n vertices, i.e. a Kn, with neither a red Kr nor a blue Ks. Fix this Kn in the discussion. Also fix a vertex v in Kn.
Now clearly if the number of red edges from v is less than R(r − 1, s) − 1 and the number of blue edges is less than R(r, s − 1) − 1 then the total edges from v would be less than R(r − 1, s) + R(r, s − 1) − 2, a contradiction as v is connected with exactly R(r − 1, s) + R(r, s − 1) − 2 vertices. So either the number of red edges from v is at least R(r − 1, s) − 1 or the number of blue edges from v is at least R(r, s − 1) − 1. WLOG assume the first. Now we shall show that the number of red edges from v cannot exceed R(r − 1, s) − 1. Suppose they do. Now, pick any R(r − 1, s) of the vertices adjoined to v by red edges and consider the KR(r − 1, s) formed by these vertices. By Ramsey's theorem it should contain either a red Kr-1 or a blue Ks. It cannot contain the latter by our choice of n. Hence it contains a red Kr-1. But adjoining the vertices of this red Kr-1 to v gives us a red Kr, which is also not possible. Hence, the number of red edges cannot exceed R(r − 1, s) − 1. So they must be precisely R(r − 1, s) − 1. Then the remaining number of blue edges are R(r, s − 1) − 1. In summary we can say that each vertex in our Kn is connected to the other n − 1 vertices by precisely R(r − 1, s) − 1 red edges and R(r, s − 1) − 1 blue edges.
The total number of red edges in Kn is therefore
{\displaystyle {\frac {n(R(r-1,s)-1)}{2}}}
which must be an integer. However as both R(r − 1, s) and R(r, s − 1) are even numbers this is impossible. Hence the initial equality was impossible. So R(r, s) < R(r − 1, s) + R(r, s − 1).
Lower bound[edit | edit source]
The classical lower bound, by an argument given essentially by Paul Erdős, is developed below. The argument is referred to as the probabilistic method. It provides a lower bound for R(r,r), which is abbreviated to R(r).
To begin note that if we are given a set of r vertices in any Kn whose edges are randomly colored, red or blue, then the probability that Kr is monochromatic is
{\displaystyle 2^{1-{\binom {r}{2}}}}
. To see this note that there are 2 ways to color each of the
{\displaystyle {\binom {r}{2}}}
edges and so the total possibilities for the colorings are
{\displaystyle 2^{\binom {r}{2}}}
. Out of these only 2 are favorable: a red Kr or a blue Kr. Hence the probability that our given Kr is monochromatic is
{\displaystyle 2^{1-{\binom {r}{2}}}}
Now what is the probability that there will be a monochromatic Kr if a set of r vertices is not fixed in the begining? There are
{\displaystyle {\binom {n}{r}}}
ways to choose r vertices out of n. By the addition law of probability we could just take any r-vertex set, get its probability as above to be
{\displaystyle 2^{1-{\binom {r}{2}}}}
, then move to another r-set, get its probability as
{\displaystyle 2^{1-{\binom {r}{2}}}}
and so on; then we could add up all the probabilities to get the probability of getting a monochromatic Kr. But it may happen that two of the chosen batches of r-sets both give monochromatic Kr's. This is another way of saying that the events are not necessarily mutually exclusive. So the addition theorem doesn't apply and the total probability is definitely not guaranteed to be
{\displaystyle {\binom {n}{r}}2^{1-{\binom {r}{2}}}}
. However by the Boole's inequality we can at least conclude that the probability that there is a monochromatic Kr in Kn can't exceed
{\displaystyle {\binom {n}{r}}2^{1-{\binom {r}{2}}}}
{\displaystyle {\binom {n}{r}}2^{1-{\binom {r}{2}}}<1}
then this tells us that the probability of a monochromatic Kr in Kn is less then 1 and so a monochromatic Kr in Kn is not guaranteed. This is just another way of saying that
{\displaystyle R(r)>n}
Now fix r and let N be the minimum value of n satisfying
{\displaystyle {\binom {n}{r}}2^{1-{\binom {r}{2}}}\geq 1}
. Since the collection of numbers satisfying this inequality is non-empty; R(r) itself is an example, and is bounded below; r and any number less then r are lower bounds not belonging to this collection, so the minimum value is guaranteed to exist.
{\displaystyle R(r)\geq N}
{\displaystyle R(r)<N}
{\displaystyle R(r)\leq N-1}
{\displaystyle R(r)>N-1}
{\displaystyle {\binom {N-1}{r}}2^{1-{\binom {r}{2}}}<1}
{\displaystyle R(r)\geq N}
{\displaystyle R(r)\geq N=(N^{r})^{\frac {1}{r}}>\left({\binom {N}{r}}r!\right)^{\frac {1}{r}}\geq (2^{{\binom {r}{2}}-1}r!)^{\frac {1}{r}}=2^{{\frac {r}{2}}-{\frac {1}{2}}-{\frac {1}{r}}}(r!)^{\frac {1}{r}}}
Now use Stirlings approximation for the factorial, valid for large values of r; that is
{\displaystyle r!\approx {\sqrt {2\pi r}}\,\left({\frac {r}{e}}\right)^{r}}
. Hence for large r, on simplification we get,
{\displaystyle R(r)>{\frac {r2^{\frac {r}{2}}}{e{\sqrt {2}}}}(\left({\frac {\pi }{2}}\right)^{\frac {1}{2r}}r^{\frac {1}{2r}})}
Since the terms in the brackets on the right hand side always exceed 1 so we can conclude by saying that for large r,
{\displaystyle R(r)>{\frac {r2^{\frac {r}{2}}}{e{\sqrt {2}}}}}
Known Ramsey numbers[edit | edit source]
R(r,s) for values of r and s up to 10 are shown in the table below. Where the exact value is unknown, the table lists the best known bounds. R(r,s) for values of r and s less than 3 are given by R(1,s) = 1 and R(2,s) = s for all values of s. The standard survey on the development of Ramsey number research has been written by Stanisław Radziszowski, who also found the exact value of R(4,5) (with Brendan McKay).
1 3 6 9 14 18 23 28 36 40–42
1 4 9 18 25 36–41 49–61 59–84 73–115 92–149
1 5 14 25 43–48 58–87 80–143 101–216 133–316 149–442
1 6 18 36–41 58–87 102–165 115–298 134–495 183–780 204–1171
1 7 23 49–61 80–143 115–298 205–540 217–1031 252–1713 292–2826
1 8 28 56–84 101–216 134–495 217–1031 282–1870 329–3583 343-6090
1 9 36 73–115 133–316 183–780 252–1713 329–3583 565–6588 581–12677
1 10 40–42 92–149 149–442 204–1171 292–2826 343-6090 581–12677 798–23556
There is a trivial symmetry across the diagonal.
This table is extracted from a larger table compiled by Stanisław Radziszowski.
A Multicolour Example: R(3,3,3) = 17[edit | edit source]
The only two 3-colourings of K16 with no monochromatic K3.
A multicolour Ramsey number is a Ramsey number using 3 or more colours. There is only one nontrivial multicolour Ramsey number for which the exact value is known, namely R(3,3,3) = 17.
Suppose that you have an edge colouring of a complete graph using 3 colours, red, yellow and green. Suppose further that the edge colouring has no monochromatic triangles. Select a vertex v. Consider the set of vertices which have a green edge to the vertex v. This is called the green neighborhood of v. The green neighborhood of v cannot contain any green edges, since otherwise there would be a green triangle consisting of the two endpoints of that green edge and the vertex v. Thus, the induced edge colouring on the green neighborhood of v has edges coloured with only two colours, namely yellow and red. Since R(3,3) = 6, the green neighborhood of v can contain at most 5 vertices. Similarly, the red and yellow neighborhoods of v can contain at most 5 vertices each. Since every vertex, except for v itself, is in one of the green, red or yellow neighborhoods of v, the entire complete graph can have at most 1 + 5 + 5 + 5 = 16 vertices. Thus, we have R(3,3,3) ≤ 17.
To see that R(3,3,3) ≥ 17, it suffices to draw an edge colouring on the complete graph on 16 vertices with 3 colours, which avoids monochromatic triangles. It turns out that there are exactly two such colourings on K16, the so-called untwisted and twisted colourings. Both colourings are shown in the figure to the right, with the untwisted colouring on the top, and the twisted colouring on the bottom. In both colourings in the figure, note that the vertices are labeled, and that the vertices v11 through v15 are drawn twice, on both the left and the right, in order to simplify the drawings.
Thus, R(3,3,3) = 17.
It is known that there are exactly two edge colourings with 3 colours on K15 which avoid monochromatic triangles, which can be constructed by deleting any vertex from the untwisted and twisted colourings on K16, respectively.
It is also known that there are exactly 115 edge colourings with 3 colours on K14 which avoid monochromatic triangles, provided that we consider edge colourings which differ by a permutation of the colours as being the same.
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