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A Simple Proof of Berry–Esséen Bounds for the Quadratic Variation of the Subfractional Brownian Motion
{L}_{q,p}
-Cohomology of Some Twisted Products
Vladimir Golʼdshtein; Yaroslav Kopylov
Existence of a unique invariant measure for a class of equicontinuous Markov operators with application to a stochastic model for an autoregulated gene
Sander Hille; Katarzyna Horbacz; Tomasz Szarek
Quantum isometry group of dual of finitely generated discrete groups - II
Harmonic functions on Manifolds whose large spheres are small.
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Many programming languages have both terms and types. Terms are also sometimes called expressions.
Terms, like 3 or false, denote the data being manipulated.
Types, like u32 (aka "unsigned 32-bit integer") or bool, describe what operations are permitted on terms.
For instance, if we have a term of type u32, we might be able to do things like add or subtract with other u32s. And if we have a term of type bool (aka true or false), we could do things like negate it or use it to branch with an if construct.
Many programming languages also have functions. For example, we could define the function is_zero, which takes a term of type u32 (like 3) and returns a term of type bool (like false).
is_zero is a function from terms to terms. However, given the existence of both types and terms, there are
2^2 = 4
distinct varieties of function to consider:
From terms to terms
From types to terms
From types to types
From terms to types
Let us examine some examples of each kind of function.
Terms to terms
As mentioned, the most common kind of function is the one that takes a term and returns a term, like is_zero.
For example, in Rust:
fn is_zero(n: u32) -> bool {
Types to terms
Consider the identity function, which is the function that returns its argument unchanged.
The implementation of the identity function is identical for any choice of parameter/return type: just return the term passed in. So, it would be convenient if, instead of having to define a "different" identity function for every type, we could have a single identity function that would work for any type. This is sometimes called a generic function.
What we can do is allow the identity function to take a type argument. Let us call it T. We then take a term argument whose type is T and return it.
In a sense, then, the identity "function" is actually defined with two functions. First is a function that takes a type T and then returns a term. That term is also a function. It takes a term x of type T, and then returns that term x.
In Rust, the identity function is:
Types to types
Many programming languages provide a list type, which is a ordered sequence of elements that can be dynamically added to and removed from. Different programming languages call this type different things: list, array, vector, sequence, and so on, but the general idea is the same.
We would like a list type to permit the elements stored to be any fixed type. That is, instead of separately defining ListOfU32 and ListOfBool, we would like to just define List, and have it work for any element type. This is called a generic type.
But note that List itself is not a type. Rather, it is a function that takes a type (the type of the elements) and returns a type (the type of lists of that element type).
In Rust, the list type is called Vec. So, if T is a Rust type, then Vec<T> is the type of a vector of Ts.
We can combine type-to-term and type-to-type functions to write highly generic, reusable code. For instance, we could write a Rust function push that takes a vector and an element to add ("push") to the end of the vector, then returns the new vector.
fn push<T>(xs: Vec<T>, x: T) -> Vec<T>
In fn push<T>, the T is a type parameter. This is an example of a type-to-term function.
In Vec<T>, the T is a type argument. It is being passed to the type-to-type function Vec.
Terms to types
Some languages have a fixed-length array type. This is a type which is a bit like a list, but its length is fixed, and thus part of the type itself. Languages like C and Rust permit types like this.
For instance, in Rust, the definition
const A: [u32; 3] = [2, 4, 6];
defines A to be an array, with a fixed length of 3, of 32-bit unsigned integers. Note that the term 3 appears in the type [u32; 3].
As we've seen, Rust allows type-to-term and type-to-type functions via generic type arguments. Rust also allows for term-to-type functions with a feature called const generics:
type U32Array<const N: usize> = [u32; N];
Here, U32Array is a function from a term to a type. The input term N has type usize, and the output type is the type of arrays of unsigned 32-bit integers with length N.
const A: U32Array<3> = [2, 4, 6];
Types like [u32; N] that contain, or "depend on", terms, are called dependent types. Not many programming languages fully support dependent types, likely due to their incredible expressive power.
Notably, Rust only permits using term-to-type functions when the terms are known before the program actually runs (aka, const).
To reiterate, the four varieties of functions are:
Most languages have term-term functions, but choose to allow or disallow the other three varieties of functions. There are three yes-or-no choices to make, and thus
2^3 = 8
possible configurations.
We may visualize the three choices as dimensions, and thus organize the possibilities into a cube. The vertices of the cube represent languages that arise from choosing combinations of allowing or disallowing the three varieties of function. All vertices on the cube allow for term-to-term functions.
Some commonly-known vertices on the cube are shown below. Columns 1-4 correspond to the 4 varieties of function discussed.
\lambda\!\rightarrow
Simply typed lambda calculus ✓ × × ×
\lambda 2
F
✓ ✓ × ×
\lambda \omega
F\omega
✓ ✓ ✓ ×
\lambda C
Calculus of constructions ✓ ✓ ✓ ✓
Once we reach the calculus of constructions (CoC), the distinction between types and terms somewhat disappears, since each may freely appear in both themselves and the other. Indeed, as powerful as the CoC is, it has a very sparse syntax of terms, fully described by the following context-free grammar:
\begin{aligned} t ::= \ & \mathsf{Prop} && \text{base type} \\ | \ & \mathsf{Type} && \text{type of $\mathsf{Prop}$} \\ | \ & x && \text{variable} \\ | \ & t(t') && \text{application} \\ | \ & \lambda (x: t) \ t' && \text{abstraction} \\ | \ & \Pi (x: t) \ t' && \text{forall} \end{aligned}
There is no separate syntax for types in the CoC: all terms and types are represented with just the above syntax.
I wrote up an implementation of the CoC in Rust for edification.
The calculus of constructions serves as the foundation for many dependently-typed programming languages, like Coq. Using the CoC as a foundation, Coq is able to express and prove mathematical theorems like the four-color theorem.
It's rather remarkable to me that functions and variables, the most basic realization of the concept of "abstraction", can be so powerful in allowing all different types of language features. In the words of jez, on variables:
I think variables are just so cool!
I think it's straight-up amazing that something so simple can at the same time be that powerful. Functions!
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torch.special — PyTorch 1.11.0 documentation
torch.special¶
The torch.special module, modeled after SciPy’s special module.
torch.special.entr(input, *, out=None) → Tensor¶
Computes the entropy on input (as defined below), elementwise.
\begin{align} \text{entr(x)} = \begin{cases} -x * \ln(x) & x > 0 \\ 0 & x = 0.0 \\ -\infty & x < 0 \end{cases} \end{align}
>>> a = torch.arange(-0.5, 1, 0.5)
>>> torch.special.entr(a)
tensor([ -inf, 0.0000, 0.3466])
torch.special.erf(input, *, out=None) → Tensor¶
Computes the error function of input. The error function is defined as follows:
\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt
>>> torch.special.erf(torch.tensor([0, -1., 10.]))
torch.special.erfc(input, *, out=None) → Tensor¶
Computes the complementary error function of input. The complementary error function is defined as follows:
\mathrm{erfc}(x) = 1 - \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt
>>> torch.special.erfc(torch.tensor([0, -1., 10.]))
tensor([ 1.0000, 1.8427, 0.0000])
torch.special.erfcx(input, *, out=None) → Tensor¶
Computes the scaled complementary error function for each element of input. The scaled complementary error function is defined as follows:
\mathrm{erfcx}(x) = e^{x^2} \mathrm{erfc}(x)
>>> torch.special.erfcx(torch.tensor([0, -1., 10.]))
tensor([ 1.0000, 5.0090, 0.0561])
torch.special.erfinv(input, *, out=None) → Tensor¶
Computes the inverse error function of input. The inverse error function is defined in the range
(-1, 1)
\mathrm{erfinv}(\mathrm{erf}(x)) = x
>>> torch.special.erfinv(torch.tensor([0, 0.5, -1.]))
tensor([ 0.0000, 0.4769, -inf])
torch.special.expit(input, *, out=None) → Tensor¶
Computes the expit (also known as the logistic sigmoid function) of the elements of input.
\text{out}_{i} = \frac{1}{1 + e^{-\text{input}_{i}}}
>>> t = torch.randn(4)
>>> torch.special.expit(t)
torch.special.expm1(input, *, out=None) → Tensor¶
Computes the exponential of the elements minus 1 of input.
y_{i} = e^{x_{i}} - 1
>>> torch.special.expm1(torch.tensor([0, math.log(2.)]))
torch.special.exp2(input, *, out=None) → Tensor¶
Computes the base two exponential function of input.
y_{i} = 2^{x_{i}}
>>> torch.special.exp2(torch.tensor([0, math.log2(2.), 3, 4]))
torch.special.gammaln(input, *, out=None) → Tensor¶
\text{out}_{i} = \ln \Gamma(|\text{input}_{i}|)
>>> torch.special.gammaln(a)
torch.special.gammainc(input, other, *, out=None) → Tensor¶
Computes the regularized lower incomplete gamma function:
\text{out}_{i} = \frac{1}{\Gamma(\text{input}_i)} \int_0^{\text{other}_i} t^{\text{input}_i-1} e^{-t} dt
\text{input}_i
\text{other}_i
are weakly positive and at least one is strictly positive. If both are zero or either is negative then
\text{out}_i=\text{nan}
\Gamma(\cdot)
in the equation above is the gamma function,
\Gamma(\text{input}_i) = \int_0^\infty t^{(\text{input}_i-1)} e^{-t} dt.
See torch.special.gammaincc() and torch.special.gammaln() for related functions.
Supports broadcasting to a common shape and float inputs.
The backward pass with respect to input is not yet supported. Please open an issue on PyTorch’s Github to request it.
input (Tensor) – the first non-negative input tensor
other (Tensor) – the second non-negative input tensor
>>> a1 = torch.tensor([4.0])
>>> a2 = torch.tensor([3.0, 4.0, 5.0])
>>> a = torch.special.gammaincc(a1, a2)
>>> b = torch.special.gammainc(a1, a2) + torch.special.gammaincc(a1, a2)
torch.special.gammaincc(input, other, *, out=None) → Tensor¶
Computes the regularized upper incomplete gamma function:
\text{out}_{i} = \frac{1}{\Gamma(\text{input}_i)} \int_{\text{other}_i}^{\infty} t^{\text{input}_i-1} e^{-t} dt
\text{input}_i
\text{other}_i
\text{out}_i=\text{nan}
\Gamma(\cdot)
\Gamma(\text{input}_i) = \int_0^\infty t^{(\text{input}_i-1)} e^{-t} dt.
See torch.special.gammainc() and torch.special.gammaln() for related functions.
torch.special.polygamma(n, input, *, out=None) → Tensor¶
Computes the
n^{th}
derivative of the digamma function on input.
n \geq 0
is called the order of the polygamma function.
\psi^{(n)}(x) = \frac{d^{(n)}}{dx^{(n)}} \psi(x)
This function is implemented only for nonnegative integers
n \geq 0
n (int) – the order of the polygamma function
>>> a = torch.tensor([1, 0.5])
>>> torch.special.polygamma(1, a)
tensor([1.64493, 4.9348])
tensor([ -2.4041, -16.8288])
tensor([ -24.8863, -771.4742])
torch.special.digamma(input, *, out=None) → Tensor¶
Computes the logarithmic derivative of the gamma function on input .
\digamma(x) = \frac{d}{dx} \ln\left(\Gamma\left(x\right)\right) = \frac{\Gamma'(x)}{\Gamma(x)}
input (Tensor) – the tensor to compute the digamma function on
This function is similar to SciPy’s scipy.special.digamma .
From PyTorch 1.8 onwards, the digamma function returns -Inf for 0 . Previously it returned NaN for 0 .
>>> torch.special.digamma(a)
torch.special.psi(input, *, out=None) → Tensor¶
torch.special.i0(input, *, out=None) → Tensor¶
Computes the zeroth order modified Bessel function of the first kind for each element of input.
\text{out}_{i} = I_0(\text{input}_{i}) = \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!)^2}
>>> torch.i0(torch.arange(5, dtype=torch.float32))
tensor([ 1.0000, 1.2661, 2.2796, 4.8808, 11.3019])
torch.special.i0e(input, *, out=None) → Tensor¶
Computes the exponentially scaled zeroth order modified Bessel function of the first kind (as defined below) for each element of input.
\text{out}_{i} = \exp(-|x|) * i0(x) = \exp(-|x|) * \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!)^2}
>>> torch.special.i0e(torch.arange(5, dtype=torch.float32))
Computes the first order modified Bessel function of the first kind (as defined below) for each element of input.
\text{out}_{i} = \frac{(\text{input}_{i})}{2} * \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!) * (k+1)!}
>>> torch.special.i1(torch.arange(5, dtype=torch.float32))
Computes the exponentially scaled first order modified Bessel function of the first kind (as defined below) for each element of input.
\text{out}_{i} = \exp(-|x|) * i1(x) = \exp(-|x|) * \frac{(\text{input}_{i})}{2} * \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!) * (k+1)!}
torch.special.logit(input, eps=None, *, out=None) → Tensor¶
Returns a new tensor with the logit of the elements of input. input is clamped to [eps, 1 - eps] when eps is not None. When eps is None and input < 0 or input > 1, the function will yields NaN.
\begin{align} y_{i} &= \ln(\frac{z_{i}}{1 - z_{i}}) \\ z_{i} &= \begin{cases} x_{i} & \text{if eps is None} \\ \text{eps} & \text{if } x_{i} < \text{eps} \\ x_{i} & \text{if } \text{eps} \leq x_{i} \leq 1 - \text{eps} \\ 1 - \text{eps} & \text{if } x_{i} > 1 - \text{eps} \end{cases} \end{align}
eps (float, optional) – the epsilon for input clamp bound. Default: None
>>> torch.special.logit(a, eps=1e-6)
tensor([-0.9466, 2.6352, 0.6131, -1.7169, 0.6261])
torch.special.logsumexp(input, dim, keepdim=False, *, out=None)¶
Alias for torch.logsumexp().
torch.special.log1p(input, *, out=None) → Tensor¶
Alias for torch.log1p().
torch.special.log_softmax(input, dim, *, dtype=None) → Tensor¶
Computes softmax followed by a logarithm.
While mathematically equivalent to log(softmax(x)), doing these two operations separately is slower and numerically unstable. This function is computed as:
\text{log\_softmax}(x_{i}) = \log\left(\frac{\exp(x_i) }{ \sum_j \exp(x_j)} \right)
input (Tensor) – input
dtype (torch.dtype, optional) – the desired data type of returned tensor. If specified, the input tensor is cast to dtype before the operation is performed. This is useful for preventing data type overflows. Default: None.
>>> t = torch.ones(2, 2)
>>> torch.special.log_softmax(t, 0)
torch.special.multigammaln(input, p, *, out=None) → Tensor¶
Computes the multivariate log-gamma function with dimension
p
element-wise, given by
\log(\Gamma_{p}(a)) = C + \displaystyle \sum_{i=1}^{p} \log\left(\Gamma\left(a - \frac{i - 1}{2}\right)\right)
C = \log(\pi) \times \frac{p (p - 1)}{4}
\Gamma(\cdot)
All elements must be greater than
\frac{p - 1}{2}
, otherwise an error would be thrown.
input (Tensor) – the tensor to compute the multivariate log-gamma function
p (int) – the number of dimensions
>>> a = torch.empty(2, 3).uniform_(1, 2)
>>> torch.special.multigammaln(a, 2)
torch.special.ndtr(input, *, out=None) → Tensor¶
Computes the area under the standard Gaussian probability density function, integrated from minus infinity to input, elementwise.
\text{ndtr}(x) = \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{x} e^{-\frac{1}{2}t^2} dt
>>> torch.special.ndtr(torch.tensor([-3., -2, -1, 0, 1, 2, 3]))
torch.special.ndtri(input, *, out=None) → Tensor¶
Computes the argument, x, for which the area under the Gaussian probability density function (integrated from minus infinity to x) is equal to input, elementwise.
\text{ndtri}(p) = \sqrt{2}\text{erf}^{-1}(2p - 1)
Also known as quantile function for Normal Distribution.
>>> torch.special.ndtri(torch.tensor([0, 0.25, 0.5, 0.75, 1]))
tensor([ -inf, -0.6745, 0.0000, 0.6745, inf])
torch.special.round(input, *, out=None) → Tensor¶
Alias for torch.round().
torch.special.sinc(input, *, out=None) → Tensor¶
Computes the normalized sinc of input.
\text{out}_{i} = \begin{cases} 1, & \text{if}\ \text{input}_{i}=0 \\ \sin(\pi \text{input}_{i}) / (\pi \text{input}_{i}), & \text{otherwise} \end{cases}
>>> torch.special.sinc(t)
torch.special.softmax(input, dim, *, dtype=None) → Tensor¶
Computes the softmax function.
\text{Softmax}(x_{i}) = \frac{\exp(x_i)}{\sum_j \exp(x_j)}
It is applied to all slices along dim, and will re-scale them so that the elements lie in the range [0, 1] and sum to 1.
dim (int) – A dimension along which softmax will be computed.
>>> torch.special.softmax(t, 0)
torch.special.xlog1py(input, other, *, out=None) → Tensor¶
Computes input * log1p(other) with the following cases.
\text{out}_{i} = \begin{cases} \text{NaN} & \text{if } \text{other}_{i} = \text{NaN} \\ 0 & \text{if } \text{input}_{i} = 0.0 \text{ and } \text{other}_{i} != \text{NaN} \\ \text{input}_{i} * \text{log1p}(\text{other}_{i})& \text{otherwise} \end{cases}
Similar to SciPy’s scipy.special.xlog1py .
input (Number or Tensor) – Multiplier
other (Number or Tensor) – Argument
At least one of input or other must be a tensor.
>>> x = torch.zeros(5,)
>>> y = torch.tensor([-1, 0, 1, float('inf'), float('nan')])
>>> torch.special.xlog1py(x, y)
tensor([0., 0., 0., 0., nan])
>>> torch.special.xlog1py(x, 4)
>>> torch.special.xlog1py(2, y)
torch.special.xlogy(input, other, *, out=None) → Tensor¶
Computes input * log(other) with the following cases.
\text{out}_{i} = \begin{cases} \text{NaN} & \text{if } \text{other}_{i} = \text{NaN} \\ 0 & \text{if } \text{input}_{i} = 0.0 \\ \text{input}_{i} * \log{(\text{other}_{i})} & \text{otherwise} \end{cases}
Similar to SciPy’s scipy.special.xlogy .
>>> torch.special.xlogy(x, y)
>>> torch.special.xlogy(x, 4)
>>> torch.special.xlogy(2, y)
torch.special.zeta(input, other, *, out=None) → Tensor¶
Computes the Hurwitz zeta function, elementwise.
\zeta(x, q) = \sum_{k=0}^{\infty} \frac{1}{(k + q)^x}
input (Tensor) – the input tensor corresponding to x .
other (Tensor) – the input tensor corresponding to q .
The Riemann zeta function corresponds to the case when q = 1
>>> x = torch.tensor([2., 4.])
>>> torch.special.zeta(x, 1)
>>> torch.special.zeta(x, torch.tensor([1., 2.]))
>>> torch.special.zeta(2, torch.tensor([1., 2.]))
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Assertion: The work done by an external unbalanced force during a round trip is not zero Reason : No - Science - Work and Energy - 16911755 | Meritnation.com
Assertion: The work done by an external unbalanced force during a round trip is not zero.
Reason : No force is required to move a body in its round trip.
Work done = Force×displacement\phantom{\rule{0ex}{0ex}}Since, the displacement is 0 so the net work done in a round trip is 0.\phantom{\rule{0ex}{0ex}}
Thus, the Assertion, "The work done by an external unbalanced force during a round trip is not zero." is TRUE.
However, there is a force acting on a body in its round trip since starts from a point and ends its journey to that point. So, the body accelerates and decelerates and hence, force can't be 0.
The reason, "No force is required to move a body in its round trip." is FALSE.
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EUDML | A functional calculus for Rockland operators on nilpotent Lie groups EuDML | A functional calculus for Rockland operators on nilpotent Lie groups
A functional calculus for Rockland operators on nilpotent Lie groups
Hulanicki, Andrzej. "A functional calculus for Rockland operators on nilpotent Lie groups." Studia Mathematica 78.3 (1984): 253-266. <http://eudml.org/doc/218577>.
@article{Hulanicki1984,
author = {Hulanicki, Andrzej},
keywords = {Schwartz function; Rockland operator; hypoelliptic differential; operator; homogeneous Lie group; spectral resolution},
title = {A functional calculus for Rockland operators on nilpotent Lie groups},
AU - Hulanicki, Andrzej
TI - A functional calculus for Rockland operators on nilpotent Lie groups
KW - Schwartz function; Rockland operator; hypoelliptic differential; operator; homogeneous Lie group; spectral resolution
Véronique Fischer, Fulvio Ricci, Gelfand transforms of
SO\left(3\right)
{N}_{3,2}
Alessandro Veneruso, Schwartz kernels on the Heisenberg group
Guorong Hu, Littlewood-Paley characterization of Hölder-Zygmund spaces on stratified Lie groups
Jacek Dziubański, Schwartz spaces associated with some non-differential convolution operators on homogeneous groups
J. Ludwig, Hull-minimal ideals in the Schwartz algebra of the Heisenberg group
Jacek Dziubański, On semigroups generated by subelliptic operators on homogeneous groups
S. Thangavelu, Some remarks on Bochner-Riesz means
Alessio Martini, Analysis of joint spectral multipliers on Lie groups of polynomial growth
Schwartz function, Rockland operator, hypoelliptic differential, operator, homogeneous Lie group, spectral resolution
Analysis on other specific Lie groups
Articles by Andrzej Hulanicki
|
DEX 1.0 and Rented Liquidity - RadioShack Swap
DEX 1.0 and the double-edged sword of rented liquidity pools
For RadioShack to win in the current decentralized swap game (henceforth referred to as DEX 1.0), it must convince users & aggregators to use the RadioShack swap.
To win that, customer slippage must be extremely low.
Slippage is the price difference between the submitted transaction and the actual completed transaction on the blockchain. Swap users find high slippage to be anathema.
Two dynamics are responsible for slippage:
High Swap Volume
RadioShack has an elegant solution to minimize slippage.
The logical place to start is to own extremely deep liquidity pools.
In general, the deeper the liquidity pool, the lower the slippage. Therefore, users (and DEX aggregators) route swaps through the deepest liquidity pools to achieve the optimal outcome for their swap.
But there is a problem with the current DEX 1.0 solution. The current major players do NOT own their own liquidity - they rent it - meaning other users provide liquidity to the protocol in return for rewards (normally minted by the DEX in a yield farm).
This has multiple negative consequences/externalities:
A. As the DEX mints its tokens to continuously encourage users to provide liquidity, it dilutes itself (see the 1920's Weimar German republic). Yet despite incurring the damage of dilution from new tokens minted, the protocol still doesn’t own the liquidity. It’s just paying the users to park their liquidity in the pool while the rewards are flowing. The moment the rewards stop flowing, the users flee and so does that rented liquidity. A quick spiral to the bottom ensues.
B. Regardless of the reward rate, there’s always churn in ‘rented’ liquidity pools. Someone may remove their liquidity to deploy in another more lucrative opportunity, or might even panic sell in a downward market. For each person removing their liquidity from a pool, the DEX has to find someone else to replace that liquidity to achieve the same level of slippage. This constant churn is coined the ‘leaky bucket’ problem and defined as, "How a bucket with a constant leak will shrink if...the average rate at which water leaks out exceeds the rate at which the bucket is filled." For all current DEX 1.0 a steady-state point is reached where the churn eventually catches up with the speed of adding new liquidity as formulated below:
SpeedOfLiquidity_{addition} = SpeedOfLiquidity_{removal}
When the saturation condition is satisfied, the pool no longer grows - a cap on how deep the liquidity is reached (which annoyingly keeps the DEX from offering low slippage to its customers). This graph is the horrific destiny for all liquidity pools in all DEX 1.0's:
And even further than this graph above illustrated, the dynamics get even worse because the only way to achieve deeper liquidity (once saturation is reached), is to increase the reward rate to attract more liquidity providers. However, as more liquidity accumulates in the pool, the churn increases until Equation 1 is satisfied again, at which point the liquidity growth stalls yet again. This is similar to a drug addict who sadly needs more and more to get their fix.
And to complete the above death spiral, the swap fee dynamics now deteriorate because the larger the pool, the lower the fee per pool participant (same swap fee spread thinner among more pool participants).
Enter RadioShack Swap with the solution.
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Liste des citations dans Numdam pour : A Harnack inequality approach to the regularity of free boundaries. Part III : existence theory, compactness, and dependence on
X
Regularity of flat free boundaries in two-phase problems for the p-Laplace operator
De Silva, D. ; Roquejoffre, J.M.
Regularity in a one-phase free boundary problem for the fractional Laplacian
A variational treatment for general elliptic equations of the flame propagation type : regularity of the free boundary
p
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(Redirected from Cross-section (geometry))
Not to be confused with Cross section (drawing).
2 Mathematical examples of cross sections and plane sections
2.1 In related subjects
5 Examples in science
Plane sectionsEdit
Mathematical examples of cross sections and plane sectionsEdit
In related subjectsEdit
The cross-sectional area (
{\displaystyle A'}
) of an object when viewed from a particular angle is the total area of the orthographic projection of the object from that angle. For example, a cylinder of height h and radius r has
{\displaystyle A'=\pi r^{2}}
when viewed along its central axis, and
{\displaystyle A'=2rh}
when viewed from an orthogonal direction. A sphere of radius r has
{\displaystyle A'=\pi r^{2}}
when viewed from any angle. More generically,
{\displaystyle A'}
can be calculated by evaluating the following surface integral:
{\displaystyle A'=\iint \limits _{\mathrm {top} }d\mathbf {A} \cdot \mathbf {\hat {r}} ,}
{\displaystyle \mathbf {\hat {r}} }
is the unit vector pointing along the viewing direction toward the viewer,
{\displaystyle d\mathbf {A} }
is a surface element with an outward-pointing normal, and the integral is taken only over the top-most surface, that part of the surface that is "visible" from the perspective of the viewer. For a convex body, each ray through the object from the viewer's perspective crosses just two surfaces. For such objects, the integral may be taken over the entire surface (
{\displaystyle A}
) by taking the absolute value of the integrand (so that the "top" and "bottom" of the object do not subtract away, as would be required by the Divergence Theorem applied to the constant vector field
{\displaystyle \mathbf {\hat {r}} }
) and dividing by two:
{\displaystyle A'={\frac {1}{2}}\iint \limits _{A}|d\mathbf {A} \cdot \mathbf {\hat {r}} |}
Examples in scienceEdit
Retrieved from "https://en.wikipedia.org/w/index.php?title=Cross_section_(geometry)&oldid=1075824773"
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EUDML | On common fixed points of pairs of a single and a multivalued coincidentally commuting mappings in -metric spaces. EuDML | On common fixed points of pairs of a single and a multivalued coincidentally commuting mappings in -metric spaces.
On common fixed points of pairs of a single and a multivalued coincidentally commuting mappings in
D
-metric spaces.
Dhage, B. C.; Asha, A. Jennifer; Kang, S. M.
Dhage, B. C., Asha, A. Jennifer, and Kang, S. M.. "On common fixed points of pairs of a single and a multivalued coincidentally commuting mappings in -metric spaces.." International Journal of Mathematics and Mathematical Sciences 2003.40 (2003): 2519-2539. <http://eudml.org/doc/51235>.
@article{Dhage2003,
author = {Dhage, B. C., Asha, A. Jennifer, Kang, S. M.},
keywords = {common fixed points; coincidently commuting mappings; -metric spaces; generalized contraction conditions; -metric spaces},
title = {On common fixed points of pairs of a single and a multivalued coincidentally commuting mappings in -metric spaces.},
AU - Dhage, B. C.
AU - Asha, A. Jennifer
TI - On common fixed points of pairs of a single and a multivalued coincidentally commuting mappings in -metric spaces.
KW - common fixed points; coincidently commuting mappings; -metric spaces; generalized contraction conditions; -metric spaces
common fixed points, coincidently commuting mappings,
D
-metric spaces, generalized contraction conditions,
D
-metric spaces
Articles by Dhage
|
EUDML | Local compactness in approach spaces. II. EuDML | Local compactness in approach spaces. II.
Local compactness in approach spaces. II.
Lowen, R.; Verbeeck, C.
Lowen, R., and Verbeeck, C.. "Local compactness in approach spaces. II.." International Journal of Mathematics and Mathematical Sciences 2003.2 (2003): 109-117. <http://eudml.org/doc/53017>.
@article{Lowen2003,
author = {Lowen, R., Verbeeck, C.},
keywords = {locally compact; approach space; open map; product},
title = {Local compactness in approach spaces. II.},
AU - Lowen, R.
AU - Verbeeck, C.
TI - Local compactness in approach spaces. II.
KW - locally compact; approach space; open map; product
locally compact, approach space, open map, product
Local compactness,
\sigma
Articles by Lowen
Articles by Verbeeck
|
Mnemonic Keys - Maple Help
Home : Support : Online Help : Programming : Maplets : Mnemonic Keys
The mnemonic key (or access key) allows the user to activate buttons, check boxes, menus, menu items, radio buttons, and toggle buttons by using the keyboard.
The mnemonic key is specified in an element caption option. The Button, CheckBox, CheckBoxMenuItem, Menu, MenuItem, RadioButton, RadioButtonMenuItem, and ToggleButton elements accept a mnemonic key.
To specify a mnemonic key for an element caption, precede the character that is to be the mnemonic key with the & character. The mnemonic appears as an underlined character.
For example, to specify the K of the caption OK to be the mnemonic, use "O&K". The caption appears as OK.
To display an ampersand (&) in the caption, use two ampersands (&&). For example, "A&&B" displays as "A&B" without a mnemonic key.
The following table contains examples of possible captions that include a mnemonic key, where the Specification column contains examples of the Maplet application element caption text and Caption Displayed column contains the corresponding Maplet application caption appearance.
Note that "_character" indicates that the "character" is underlined in the Maplet application.
Caption Displayed
A&&&B
A&_B
If you specify more than one mnemonic key for a single caption, the first & specification in the caption is the mnemonic key. The other single ampersands are ignored. Only one mnemonic key can be specified for each caption.
If a character that appears multiple times in a caption is specified to be a mnemonic key, the first instance is underlined as the mnemonic. The mnemonic key is case-insensitive.
The mnemonic key is restricted to characters that can be entered by pressing a single key. Invalid character specifications for the mnemonic key are characters that are SHIFT key combinations or require other modifiers to enter the character.
If an invalid character is specified, it is displayed underlined but it does not activate the element.
To activate the element using the mnemonic key, press Alt and the underlined character. For example, if the underlined key is K for the OK button, press Alt + k to activate the button.
Note: On some platforms and different keyboards, another key is used instead of Alt.
\mathrm{with}\left(\mathrm{Maplets}[\mathrm{Elements}]\right):
\mathrm{maplet}≔\mathrm{Maplet}\left(["Select a Button:",[\mathrm{Button}\left("O&K",\mathrm{Shutdown}\left("true"\right)\right),\mathrm{Button}\left("&Cancel",\mathrm{Shutdown}\left(\right)\right)]]\right):
\mathrm{result}≔\mathrm{Maplets}[\mathrm{Display}]\left(\mathrm{maplet}\right)
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RFID Attendance System (2022) - Price, Disadvantages, Advantages
RFID Attendance System - Disadvantages, Advantages, Price, Demo and more
There are many methods to track the attendance of an individual. A few years ago, when Radio Frequency Identification (RFID) electronics were still at their adolescence stages, the time stamping technique was greatly followed. As software programming and technology progressed, especially in electronics, they gave way to a much more secure and reliable arrangement known as RFID Attendance System.
Try for Free - RFID Attendance System and buy online
Why Implement the RFID System?
Most educational institutions still utilize the traditional paper method to mark attendance. This is both paper and time-consuming. By utilizing RFID Systems in the daily activities of students, data can be well organized and recorded.
The RFID method has enormous benefits. With it maintaining daily attendance on Excel sheets could be avoided, labor can be reduced, and the monthly attendance record of staff and students can be assembled in a single place and format.
Let’s Discuss Disadvantages and Advantages of RFID Attendance System
Disadvantages of RFID Attendance System
RFID is being used actively in retail, healthcare, and other sectors to monitor workers. Since the workers in these sectors are large in number, hard to handle and their work can be performed by others in case of absenteeism; there the attendance mechanism is of trivial significance.
In the case of educational institutions, where monitoring students’ regularity is of utmost importance, this attendance management system comes in handy but with some minor drawbacks and pitfalls.
The System is expensive because a lot of technology goes into making it
In case of a large strength of students, purchasing tags for everyone is costly
Replacing the microchip, radio transceiver, antenna, and battery in the system is tiresome and costs money
Since it is not as secure as biometric, the system is prone to manipulation
Advantages of RFID for Attendance Tracking
RFID card-based time attendance system was an abstract concept that craved path to biometric fingerprint systems. The latter is a topic of discussion for another day; this article elaborated on how the former can be implemented in educational institutions to monitor and track students, teachers, and staff whereabouts, their absenteeism, and regularity.
The System provides a more accurate identification
Quick and Rapid: Identifies candidates in seconds
The System is less tedious, cost-efficient, easy to use, hard to adulterate, and modest
Educational institutions like schools, colleges, and universities can precisely screen the regularity of students, teachers, and working staff and counteract proxy marking and error
RFID Machine: Card and Reader
A rectangular Plastic Card sometimes called Tag, and Reader devices are part and parcel of the RFID machine. The card serves as an identity mark of an individual and contains essential identification data of a person. The Tag is incorporated with a chip that stores a unique serial number – EPC – Electronic Product Code. Every individual is issued a card with a different unique code.
The reader is the brains of the RFID Attendance System as it stores data of every person in the organization and only records people’s attendance, for a respective day, when the Smart Card is displayed or put under it. This is a wireless system that can be integrated with SMS notifications too.
EduSys cloud-based Enterprise Resource Planning (ERP) Software is programmed with 21st-century codes to take in and track attendance, and datasheets in any format. Attendance Management is one of the many modules in the application.
The application can be integrated with any RFID and Biometric Systems. From managing staff data to tracking school buses, everything and anything can be performed online simply on EduSys.
What is the cost of the RFID Attendance System?
RFID Attendance System includes hardware and software. Generally, the hardware, Biometric Fingerprint & RFID ID card reader cost between
40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to40to
40 to
200 (India: Rs.2,800 to 14,000) and RFID Walk Through Attendance Gate costs between
300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to300to
300 to
1200 (India: Rs.21,000 - 84,000). And software costs
5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto5/yearto
5/year to
100/year per user or student.
Note: The cost of the Card is excluded from the cost of the system.
Request a Demo and Buy Online.
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Sorting Algorithms Algorithms bucket sort
Bucket sort is a comparison sort algorithm that works by distributing the elements of an array into a number of buckets and then each bucket is sorted individually using a separate sorting algorithm or by applying the bucket sort algorithm recursively.This algorithm is mainly useful when the input is uniformly distributed over a range.
Let's take an array A[]={0.78,0.26,0.72,0.17,0.39}. Total number of elements are n=10. So we need to create 5 buckets.
maximum element(max)=0.78
Using the formula: bi = (n)*arr[i]/(max+1), we find out bucket index of each element of the array A[].
For eg: bucket index of 0th element is bi= floor((5)(0.78)/(0.78+1))= 2
Similarly,bucket index of 1st element is bi=0
bucket index of 2nd element is bi=1
bucket index of 3rd element is bi=0
bucket index of 4th element is bi=1
Now, since buckets have been created we need to concatenate all the buckets in the order to get the sorted array.
Step 1: function bucketSort(array, n) is
Step 1: buckets ← new array of n empty lists
Step 2: M ← the maximum key value in the array
Step 3: for i = 1 to length(array) do
insert array[i] into buckets[floor(array[i] / M * n)]
Step 4: for i = 1 to n do
sort(buckets[i])
Step 5: return the concatenation of buckets[1], ...., buckets[k]
Worst case time complexity:Θ(n^2)
where n is the number of elements to be sorted and k is the number of buckets
For an upper bound on the worst-case cost, it's sufficient to show that it can't be worse. Assuming that insertion sort takes ≤cn2 steps on n elements, consider the sum
\phantom{\rule{2em}{0ex}}\sum _{i=1}^{n}c|{B}_{i}{|}^{2}
it is an upper bound on the cost of sorting all the buckets. For an upper bound on the worst case for bucket sort, maximize this function subject to ∑|Bi|=n (and add the remaining cost, which is O(n) for all inputs).
For a lower bound on the worst-case cost, we have to find an infinite class of actual inputs and show that their cost behaves as claimed. [0,…,0] serves to show an Ω(n2) lower bound.
float findMax(float A[], int n)
return max(A[n-1], findMax(A, n-1));
float max=findMax(arr,n);
int bi = n*arr[i]/(max+1); // Index in bucket
Constructing Histograms
One common computation in data visualization and analysis is computing a histogram. For example, n students might be assigned integer scores in some range, such as 0 to 100, and are then placed into ranges or “buckets” based on these scores.
Is Bucket sort an in-place algorithm?
Student at Kalinga Institute of Industrial Technology, Bhubaneswar
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Estimate Capital Asset Pricing Model Using SUR - MATLAB & Simulink - MathWorks Australia
Create Multivariate Time Series Model
Estimate Multivariate Time Series Model
Analyze Coefficient Estimates
This example shows how to implement the capital asset pricing model (CAPM) using the Econometrics Toolbox™ VAR model framework.
The CAPM model characterizes comovements between asset and market prices. Under this framework, individual asset returns are linearly associated with the return of the whole market (for details, see [92], [139], and [181]). That is, given the return series of all stocks in a market (
{M}_{t}
) and the return of a riskless asset (
{C}_{t}
), the CAPM model for return series
j
{R}_{j}
{R}_{jt}-{C}_{t}={a}_{j}+{b}_{j}\left({M}_{t}-{C}_{t}\right)+{\epsilon }_{jt}
for all assets
j=1,...,n
in the market.
a=\left[{a}_{1}\phantom{\rule{0.2777777777777778em}{0ex}}...\phantom{\rule{0.2777777777777778em}{0ex}}{a}_{n}{\right]}^{\prime }
is an
-by-1 vector of asset alphas that should be zero, and it is of interest to investigate assets whose asset alphas are significantly away from zero.
b=\left[{b}_{1}\phantom{\rule{0.2777777777777778em}{0ex}}...\phantom{\rule{0.2777777777777778em}{0ex}}{b}_{n}{\right]}^{\prime }
n
-by-1 vector of asset betas that specify the degree of comovement between the asset being modeled and the market. An interpretation of element
j
b
{b}_{j}=1
, then asset
j
moves in the same direction and with the same volatility as the market, i.e., is positively correlated with the market .
{b}_{j}=-1
j
moves in the opposite direction, but with the same volatility as the market, i.e., is negatively correlated with the market.
{b}_{j}=0
j
is uncorrelated with the market.
sign\left({b}_{j}\right)
determines the direction that the asset is moving relative to the market as described in the previous bullets.
|{b}_{j}|
is the factor that determines how much more or less volatile asset
j
is relative to the market. For example, if
|{b}_{j}|=10
j
is 10 times as volatile as the market.
Load the CAPM data set included in the Financial Toolbox™.
varWithNaNs = Assets(any(isnan(Data),1))
varWithNaNs = 1x2 cell
{'AMZN'} {'GOOG'}
dateRange = datestr([Dates(1) Dates(end)])
dateRange = 2x11 char array
The variable Data is a 1471-by-14 numeric matrix containing the daily returns of a set of 12 stocks (columns 1 through 12), one riskless asset (column 13), and the return of the whole market (column 14). The returns were measured from 03Jan2000 through 07Nov2005. AMZN and GOOG had their IPO during sampling, and so they have missing values.
Assign variables for the response and predictor series.
Y = bsxfun(@minus,Data(:,1:12),Data(:,14));
X = Data(:,13) - Data(:,14);
[T,n] = size(Y)
Y is a 1471-by-12 matrix of the returns adjusted by the riskless return. X is a 1471-by-1 vector of the market return adjusted by the riskless return.
Create a varm model object that characterizes the CAPM model. You must specify the number of response series and degree of the autoregressive polynomial.
Mdl = varm(n,0);
Mdl is a varm model object that characterizes the desired CAPM model.
Pass the CAPM model specification (Mdl), the response series (Y), and the predictor data (X) to estimate. Request to return the estimated multivariate time series model and the estimated coefficient standard errors. estimate maximizes the likelihood using the expectation-conditional-maximization (ECM) algorithm.
[EstMdl,EstCoeffSEMdl] = estimate(Mdl,Y,'X',X);
EstMdl has the same structure as Mdl, but EstMdl contains the parameter estimates. EstCoeffSEMdl is a structure array containing the estimated standard errors of the parameter estimates. EstCoeffSEMdl:
Contains the biased maximum likelihood standard errors.
Does not include the estimated standard errors of the intra-period covariances.
Display the regression estimates, their standard errors, their t statistics, and p-values. By default, the software estimates, stores, and displays standard errors from maximum likelihood.
results.Table
Constant(1) 0.0044305 0.0013709 3.2319 0.0012298
Constant(2) 0.00016934 0.0012625 0.13413 0.8933
Constant(3) -0.00039977 0.00072318 -0.5528 0.5804
Constant(4) -0.00067309 0.00070971 -0.9484 0.34293
Constant(5) 0.00018643 0.001389 0.13421 0.89324
Constant(7) 0.0015126 0.00088576 1.7077 0.087697
Constant(8) -0.00022511 0.00050184 -0.44856 0.65375
Constant(9) 0.00020429 0.00072638 0.28124 0.77853
Constant(10) 0.00016834 0.00042152 0.39937 0.68962
Constant(11) 0.0004766 0.00086392 0.55167 0.58118
Constant(12) 0.00083861 0.00093527 0.89665 0.3699
Beta(1,1) 1.385 0.20647 6.708 1.9727e-11
Beta(2,1) 1.4067 0.19016 7.3974 1.3886e-13
Beta(3,1) 1.0482 0.10892 9.6237 6.353e-22
Beta(4,1) 0.84687 0.10689 7.9226 2.3256e-15
Response series 6 has a significant asset alpha.
sigASymbol = Assets(6)
sigASymbol = 1x1 cell array
{'GOOG'}
As a result, GOOG has exploitable economic properties.
All asset betas are greater than 3. This indicates that all assets are significantly correlated with the market.
However, GOOG has an asset beta of approximately 0.37, whereas all other asset betas are greater than or close to 1. This indicates that the magnitude of the volatility of GOOG is approximately 37% of the market volatility. The reason for this is that GOOG steadily and almost consistently appreciated in value while the market experienced volatile horizontal movements.
For more details and an alternative analysis, see Capital Asset Pricing Model with Missing Data.
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Atomic formula in first-order logic
{\displaystyle t\equiv c\mid x\mid f(t_{1},\dotsc ,t_{n})}
{\displaystyle A,B,...\equiv P(t_{1},\dotsc ,t_{n})\mid A\wedge B\mid \top \mid A\vee B\mid \bot \mid A\supset B\mid \forall x.\ A\mid \exists x.\ A}
{\displaystyle P(x)}
{\displaystyle Q(y,f(x))}
{\displaystyle R(z)}
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Classification loss for multiclass error-correcting output codes (ECOC) model - MATLAB loss - MathWorks Nordic
Determine Test-Sample Loss of ECOC Model
Determine ECOC Model Quality Using Custom Loss
Classification loss for multiclass error-correcting output codes (ECOC) model
L = loss(Mdl,tbl,ResponseVarName) returns the classification loss (L), a scalar representing how well the trained multiclass error-correcting output codes (ECOC) model Mdl classifies the predictor data in tbl compared to the true class labels in tbl.ResponseVarName. By default, loss uses the classification error to compute L.
L = loss(Mdl,tbl,Y) returns the classification loss for the predictor data in table tbl and the true class labels in Y.
L = loss(Mdl,X,Y) returns the classification loss for the predictor data in matrix X and the true class labels in Y.
L = loss(___,Name,Value) specifies options using one or more name-value pair arguments in addition to any of the input argument combinations in previous syntaxes. For example, you can specify a decoding scheme, classification loss function, and verbosity level.
Estimate the test-sample classification error, which is the default classification loss.
The ECOC model correctly classifies all irises in the test sample.
classOrder = unique(Y); % Class order
Train an ECOC model using SVM binary classifiers. Specify a 15% holdout sample, standardize the predictors using an SVM template, and define the class order.
Create a function that takes the minimal loss for each observation, then averages the minimal losses for all observations. S corresponds to the NegLoss output of predict.
Compute the test-sample custom loss.
loss(Mdl,XTest,YTest,'LossFun',lossfun)
The average minimal binary loss for the test-sample observations is 0.0033.
Example: loss(Mdl,X,Y,'BinaryLoss','hinge','LossFun',@lossfun) specifies 'hinge' as the binary learner loss function and the custom function handle @lossfun as the overall loss function.
S is an n-by-K numeric matrix of negated loss values for the classes. Each row corresponds to an observation. The column order corresponds to the class order in Mdl.ClassNames. The input S resembles the output argument NegLoss of predict.
Observation weights, specified as the comma-separated pair consisting of 'Weights' and a numeric vector or the name of a variable in tbl. If you supply weights, then loss computes the weighted loss.
If you do not specify your own loss function (using LossFun), then the software normalizes Weights to sum up to the value of the prior probability in the respective class.
Classification loss, returned as a numeric scalar or row vector. L is a generalization or resubstitution quality measure. Its interpretation depends on the loss function and weighting scheme, but in general, better classifiers yield smaller classification loss values.
If Mdl.BinaryLearners contains ClassificationLinear models, then L is a 1-by-ℓ vector, where ℓ is the number of regularization strengths in the linear classification models (numel(Mdl.BinaryLearners{1}.Lambda)). The value L(j) is the loss for the model trained using regularization strength Mdl.BinaryLearners{1}.Lambda(j).
Otherwise, L is a scalar value.
L=\sum _{j=1}^{n}{w}_{j}{e}_{j},
L=\sum _{j=1}^{n}{w}_{j}{c}_{{y}_{j}{\stackrel{^}{y}}_{j}},
{c}_{{y}_{j}{\stackrel{^}{y}}_{j}}
{\stackrel{^}{y}}_{j}
\stackrel{^}{k}
\stackrel{^}{k}=\underset{k}{\text{argmin}}\frac{1}{B}\sum _{j=1}^{B}|{m}_{kj}|g\left({m}_{kj},{s}_{j}\right).
\stackrel{^}{k}=\underset{k}{\text{argmin}}\frac{\sum _{j=1}^{B}|{m}_{kj}|g\left({m}_{kj},{s}_{j}\right)}{\sum _{j=1}^{B}|{m}_{kj}|}.
loss does not support tall table data when Mdl contains kernel or linear binary learners.
ClassificationECOC | CompactClassificationECOC | predict | resubLoss | fitcecoc
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Bijective function - zxc.wiki
Bijectivity (the adjective bijective comprising about, clearly reversibly on 'is - hence the term one-one or substantively corresponding to one correspondence ) is a mathematical term in the field of set theory . It describes a special property of images and functions . Bijective images and functions are also called bijections . Bijections associated with a mathematical structure often have their own names such as isomorphism , diffeomorphism , homeomorphism , mirroring or the like. As a rule, additional requirements have to be met here with regard to the maintenance of the structure under consideration.
To illustrate, one can say that in a bijection a complete pairing takes place between the elements of the definition set and the target set . Bijections treat their domain and its range of values so symmetrical ; therefore a bijective function always has an inverse function .
In the case of a bijection , the definition set and the target set always have the same thickness . In the case that there is a bijection between two finite sets , this common cardinality is a natural number , namely exactly the number of elements in each of the two sets .
The bijection of a set onto itself is also called permutation . Here, too, there are many names of their own in mathematical structures. If the bijection has structure-preserving properties beyond that, it is called an automorphism .
A bijection between two sets is sometimes called a bijective correspondence .
2 graphic illustrations
Be and sets and be a function that maps from to , that is . Then called bijective if for just one with exists.
{\ displaystyle X}
{\ displaystyle Y}
{\ displaystyle f}
{\ displaystyle X}
{\ displaystyle Y}
{\ displaystyle f \ colon X \ to Y}
{\ displaystyle f}
{\ displaystyle y \ in Y}
{\ displaystyle x \ in X}
{\ displaystyle f \ left (x \ right) = y}
That means: is bijective if and only if both
{\ displaystyle f}
{\ displaystyle f}
(1) Injective is:
No value of the target quantity is assumed more than once . In other words: the archetype of each element of the target set consists of at most one element of . It therefore always follows from .
{\ displaystyle Y}
{\ displaystyle Y}
{\ displaystyle X}
{\ displaystyle f (x_ {1}) = f (x_ {2})}
{\ displaystyle x_ {1} = x_ {2}}
(2) is surjective :
Every element of the target set is accepted. In other words: the target set and the image set match, that is . For each out there is (at least) one out with .
{\ displaystyle Y}
{\ displaystyle Y}
{\ displaystyle f (X)}
{\ displaystyle f \ left (X \ right) = Y}
{\ displaystyle y}
{\ displaystyle Y}
{\ displaystyle x}
{\ displaystyle X}
{\ displaystyle f (x) = y}
The principle of bijectivity: each point in the target set (Y) is hit exactly once.
Four bijective, strictly monotonically increasing, real continuous functions.
Four bijective strictly monotonically decreasing real continuous functions.
The set of real numbers is denoted by, the set of non-negative real numbers by .
{\ displaystyle \ mathbb {R}}
{\ displaystyle \ mathbb {R} _ {0} ^ {+}}
The function is bijective with the inverse function .
{\ displaystyle f \ colon \ mathbb {R} \ to \ mathbb {R}, x \ mapsto x + a}
{\ displaystyle f ^ {- 1} \ colon \ mathbb {R} \ to \ mathbb {R}, x \ mapsto xa}
Likewise for the function is bijective with the inverse function .
{\ displaystyle a \ neq 0}
{\ displaystyle g \ colon \ mathbb {R} \ to \ mathbb {R}, x \ mapsto ax}
{\ displaystyle g ^ {- 1} \ colon \ mathbb {R} \ to \ mathbb {R}, x \ mapsto {\ frac {x} {a}}}
Example: If one assigns each ( monogamously ) married person to his or her spouse, this is a bijection of the set of all married people onto himself. This is even an example of a self-inverse mapping .
The following four square functions only differ in their definition or value sets:
{\ displaystyle f_ {1} \ colon \ mathbb {R} \ \ \ rightarrow \ mathbb {R}, \ \ \ x \ mapsto x ^ {2}}
{\ displaystyle f_ {2} \ colon \ mathbb {R} _ {0} ^ {+} \ rightarrow \ mathbb {R}, \ \ \ x \ mapsto x ^ {2}}
{\ displaystyle f_ {3} \ colon \ mathbb {R} \ \ \ rightarrow \ mathbb {R} _ {0} ^ {+}, \ x \ mapsto x ^ {2}}
{\ displaystyle f_ {4} \ colon \ mathbb {R} _ {0} ^ {+} \ rightarrow \ mathbb {R} _ {0} ^ {+}, \ x \ mapsto x ^ {2}}
With these definitions is
{\ displaystyle f_ {1}}
not injective, not surjective, not bijective
{\ displaystyle f_ {2}}
injective, not surjective, not bijective
{\ displaystyle f_ {3}}
not injective, surjective, not bijective
{\ displaystyle f_ {4}}
injective, surjective, bijective
If and are finite sets with the same number of elements and is a function, then:
{\ displaystyle A}
{\ displaystyle B}
{\ displaystyle f \ colon A \ to B}
Is injective, then it is already bijective.
{\ displaystyle f}
{\ displaystyle f}
If surjective is already bijective.
{\ displaystyle f}
{\ displaystyle f}
In particular, the following holds for functions of a finite set in themselves:
{\ displaystyle f \ colon A \ to A}
{\ displaystyle A}
{\ displaystyle f}
is injective ⇔ is surjective ⇔ is bijective.
{\ displaystyle f}
{\ displaystyle f}
This is generally wrong for infinite sets. These can be mapped injectively onto real subsets; there are also surjective mappings of an infinite set onto themselves that are not bijections. Such surprises are described in more detail in the Hilbert's Hotel article, see also Dedekind infinity .
If the functions and are bijective, then this also applies to the concatenation . The inverse function of is then .
{\ displaystyle f \ colon A \ to B}
{\ displaystyle g \ colon B \ to C}
{\ displaystyle g \ circ f \ colon A \ to C}
{\ displaystyle g \ circ f}
{\ displaystyle f ^ {- 1} \ circ g ^ {- 1}}
Is bijective, then is injective and surjective.
{\ displaystyle g \ circ f}
{\ displaystyle f}
{\ displaystyle g}
Is a function and is there a function that the two equations
{\ displaystyle f \ colon A \ to B}
{\ displaystyle g \ colon B \ to A}
{\ displaystyle g \ circ f = \ operatorname {id} _ {A}}
( = Identity on the crowd )
{\ displaystyle \ operatorname {id} _ {A}}
{\ displaystyle A}
{\ displaystyle f \ circ g = \ operatorname {id} _ {B}}
{\ displaystyle \ operatorname {id} _ {B}}
{\ displaystyle B}
is fulfilled, then is bijective, and is the inverse function of , thus .
{\ displaystyle f}
{\ displaystyle g}
{\ displaystyle f}
{\ displaystyle g = f ^ {- 1}}
The set of permutations of a given basic set , together with the composition as a link, forms a group , the so-called symmetrical group of .
{\ displaystyle A}
{\ displaystyle A}
After using formulations such as “one-to-one” for a long time, the need for a more concise description finally arose in the middle of the 20th century in the course of the consistent set-theoretical representation of all mathematical sub-areas. The terms bijective , injective and surjective were coined in the 1950s by the Nicolas Bourbaki group of authors .
Heinz-Dieter Ebbinghaus: Introduction to set theory . 4th edition. Spectrum Academic Publishing House, Heidelberg [u. a.] 2003, ISBN 3-8274-1411-3 .
Gerd Fischer: Linear Algebra . 17th edition. Vieweg + Teubner, Wiesbaden 2010, ISBN 978-3-8348-0996-4 .
Wikibooks: Evidence archive: set theory - learning and teaching materials
↑ Don Zagier : Zeta functions and quadratic fields: An introduction to higher number theory . Springer, 1981, ISBN 3-540-10603-0 , here p. 94 ( limited preview in Google book search [accessed June 7, 2017]).
↑ Gernot Stroth : Algebra: Introduction to Galois Theory . de Gruyter, Berlin 1998, ISBN 3-11-015534-6 , here p. 100 ( limited preview in Google book search [accessed June 7, 2017]).
^ Earliest Known Uses of Some of the Words of Mathematics.
This page is based on the copyrighted Wikipedia article "Bijektive_Funktion" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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Read the passage given below and answer the following questions (3) After the rejection of Rutherford's model of an - Science - Atoms and Molecules - 16911767 | Meritnation.com
. Read the passage given below and answer the following questions. (3) After the rejection of Rutherford's model of an atom, Neils Bohr put forward his model of atom which was widely accepted. He put forward certain postulates about the model of the atom which are as follows: • Only certain special orbits, known as discrete orbits of electrons, are allowed inside the atom. • While revolving in discrete orbits, the electrons do not radiate energy. Bohr named these orbits as energy levels. These orbits (or shells) are represented by the letters K, L, M, N,. or the numbers n = 1, 2, 3, 4,.... He also suggested that the maximum number of electrons in a given shell can be calculated as 2n 2 , where ‘n’ represents the orbit number or energy level index, 1,2,3,….
(a) Calculate the maximum number of electrons that can be accommodated in an M shell.
(b) Write the distribution of electrons in a sodium atom.
(c) If K and L shells of an atom are completely filled, calculate the total number of electrons in the atom.
a) The maximum number of electrons which can be accommodated in a shell of an atom = 2n2
Where, n is the number of the shell.
For M shell, n=3
Maximum number of electrons =
2×3×3 = 18 electrons
(b) Sodium atoms has 11 electrons, the distribution of electrons is as follows- 2,8,1
(c) K shell can accommodate 2 electrons and L shell can accommodate 8 electrons, so total of 10 electrons will be present in the atoms.
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RGB / HEX / HSV / HSL converter - Hirota Yano
RGB / HEX / HSV / HSL converter
You can also enter HEX in a 3-digit abbreviation.
e.g. #3F9 -> #33FF99
* Conversion from RGB to HSV and HSL is irreversible. Because RGB can express more colors and will be rounded during conversion.
RGB: 256 x 256 x 256 = 16,777,216 ways
HSV: 360 x 101 x 101 = 3,672,360 ways
How to convert from RGB to HSV
The maximum value of R, G, and B is MAX, and the minimum value is MIN.
MAX=\max \{ R, G, B \}
MIN=\min \{ R, G, B \}
The calculation method changes depending on whether MAX is R, G, or B.
H = \begin{cases} {\dfrac {G-B}{MAX-MIN}} \times 60 &\text{, } MAX=R \\ \\ {\dfrac {B-R}{MAX-MIN}} \times 60+120 &\text{, } MAX=G \\ \\ {\dfrac {R-G}{MAX-MIN}} \times 60+240 &\text{, } MAX=B \end{cases}
S={\dfrac {MAX-MIN}{MAX}} \times 100
S={\dfrac {MAX}{255}} \times 100
How to convert from RGB to HSL
The definitions of MAX and MIN are the same as HSV.
Same as HSV.
L={\dfrac {MAX+MIN}{2}} \times {\dfrac {100}{255}}
The calculation method changes depending on the Lightness.
S = \begin{cases} {\dfrac {MAX-MIN}{MAX+MIN}} \times 100 &\text{, } 0 \leqq L \leqq 50 \\ \\ {\dfrac {MAX-MIN}{510-(MAX+MIN)}} \times 100 &\text{, } 51 \leqq L \leqq 100 \end{cases}
How to convert from HSV to RGB
H
is 0 for 360.
H = \begin{cases} H &\text{, } H \neq 360 \\ 0 &\text{, } H = 360 \end{cases}
Find the remainder (= decimal part) by dividing H / 60 by 1.
e.g. When H is 90:
{\dfrac {90}{60}} \bmod 1=1.5 \bmod 1=0.5
H'={\dfrac {H}{60}} \bmod 1
Convert S and V from percentages to decimals.
S'={\dfrac {S}{100}}
V'={\dfrac {V}{100}}
The value of H determines the solution. The exception is achromatic color (S = 0).
A=V' \times 255
B=V' \times (1-S') \times 255
C=V' \times (1-S' \times H') \times 255
D=V' \times (1-S' \times (1-H')) \times 255
(R,G,B) = \begin{cases} (A,A,A) &\text{, } S = 0 \\ (A,D,B) &\text{, } 0 \leqq H < 60 \\ (C,A,B) &\text{, } 60 \leqq H < 120 \\ (B,A,D) &\text{, } 120 \leqq H < 180 \\ (B,C,A) &\text{, } 180 \leqq H < 240 \\ (D,B,A) &\text{, } 240 \leqq H < 300 \\ (A,B,C) &\text{, } 300 \leqq H < 360 \end{cases}
How to convert from HSL to RGB
H
H = \begin{cases} H &\text{, } H \neq 360 \\ 0 &\text{, } H = 360 \end{cases}
Apply magic to L.
L' = \begin{cases} L &\text{, } 0 \leqq L < 50 \\ 100-L &\text{, } 50 \leqq L \leqq 100 \end{cases}
The value of H determines the solution.
MAX=2.55 \times (L+L' \times {\dfrac {S}{100}})
MIN=2.55 \times (L-L' \times {\dfrac {S}{100}})
f(x)={\dfrac {x}{60}} \times (MAX-MIN)+MIN
(R,G,B) = \begin{cases} (MAX,\ f(H),\ MIN) &\text{, } 0 \leqq H < 60 \\ (f(120 - H),\ MAX,\ MIN) &\text{, } 60 \leqq H < 120 \\ (MIN,\ MAX,\ f(H - 120)) &\text{, } 120 \leqq H < 180 \\ (MIN,\ f(240 - H),\ MAX) &\text{, } 180 \leqq H < 240 \\ (f(H - 240),\ MIN,\ MAX) &\text{, } 240 \leqq H < 300 \\ (MAX,\ MIN,\ f(360 - H)) &\text{, } 300 \leqq H < 360 \end{cases}
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A Composite number is a number greater than 1 with more than two factors.
A Divisor is a number that divides another number either completely or with a remainder
So, given a number N, we have to find:
Sum of Divisors of N
1. Number of divisors
n = 4, divisors are 1, 2, 4
n = 18, divisors are 1, 2, 3, 6, 9, 18
n = 36, divisors are 1, 2, 3, 4, 6, 9, 12, 18, 36
We need a formula to find the number of divisors of a composite number without computing the factors of that number.
The theorem States that every positive integer greater than 1 can be represented uniquely as a product of its primes not considering the arrangement of the prime factors.
{2}^{1}·{3}^{2}
Prime factoriazation 90
{2}^{1}·{3}^{2}·{5}^{1}
Notice that we get a unique prime factorization always if we do not change the order of the factors.
Finding number of divisors
Assume n = 18
Let the prime factorization of n be,
n=p{1}^{{\alpha }_{1}}+p{2}^{{\alpha }_{2}}+...+p{k}^{{\alpha }_{k}}
Number of divisors will be,
divisors=\left({\alpha }_{1}+1\right)·\left({\alpha }_{2}+1\right)·...·\left({\alpha }_{k}+1\right)
{2}^{1}·{3}^{2}
Divisors of
{2}^{1}
{2}^{1}
{2}^{0}
{3}^{2}
{3}^{2}
{3}^{1}
{3}^{0}
Therefore the divisors of 18 are
\left({2}^{0}·{3}^{0}\right),\left({2}^{0}·{3}^{1}\right),\left({2}^{0}·{3}^{2}\right),\left({2}^{1}·{3}^{0}\right),\left({2}^{1}·{3}^{1}\right),\left({2}^{1}·{3}^{2}\right)
making a total of 6 divisors which is 3 * 2.
In this approach we would iterate over all the numbers from 1 to the square root of n checking the divisibility of an element to n while keeping count of the number of divisors.
The time complexity is proportional to the square root of n, that is
O\left(\sqrt{n}\right)
The space complexity is constant O(1), no extra space is needed.
In this optimized solution we will try to reduce the
O\left(\sqrt{n}\right)
complexity to
O\left({n}^{\frac{1}{3}}\right)
We write n as a product of 3 numbers, p, q, r that is p * q * r = N where p <= q <= r hence the maximum value for p is
{N}^{\frac{1}{3}}
We now loop over all prime numbers in the range of [2,
{N}^{\frac{1}{3}}
] and try to reduce n to its prime factorization while counting the number of factors of n.
We split n into two number x and y such that x * y = n.
x contains prime factors in range of [2,
{N}^{\frac{1}{3}}
] and y deals with larger prime factors (>
{N}^{\frac{1}{3}}
Thus the gcd of x,y is 1.
We let count of divisors be a function f(n).
f(mn) = f(m) * f(n) if gcd(m, n) == 1
Therefore if we can find f(x) and f(y), we can also find f(x * y) which is the required number of divisors.
finding f(x)
Here we use division to prime factorize x and calculate the number of factors. After this procedure we will remain with y=
\frac{n}{x}
which is to be factorized.
Possibilities of y are,
Y is prime number f(y) = 2
y is a square of a prime number f(y) = 3
y is a product of two distinct prime numbers f(y) = 4
If we found f(x) and f(y) we are done because f(n) = f(x * y);
N is input.
initialize primes array to store primes up to 10^6.
initialize result to 1
for all p in primes:
while N is divisible by p:
result = result * count
else if N is not equal to 1
Code: (naive approach included)
class Divisors{
void eratosthenesSieve(ll n, bool prime[], bool primeSquare[], ll arr[]){
//boolean array with true entries
//boolean array with false entries
for(int i = 0; i <= (n*n+1); i++)
primeSquare[i] = false;
for(int p = 2; p*p <= n; p++){
//update multiples of p from p*p
for(int i = p*p; i <= n; i+= p)
for(int p = 2; p <= n; p++){
if(prime[p]){
arr[j] = p;
//if p is prime, update value in primesquare array
primeSquare[p*p] = true;
//naive approach
ll numDivisorsNaive(ll n){
for(int i = 1; i <= sqrt(n); i++){
//if divisible
//if equal divisors, increment count by 1
if(n / i == i)
//not equal divisors, increment count twice
//efficient approach using sieve
ll numDivisorsOptimal(ll n){
//1 factor
//init prime and primesquares arrays
bool prime[n+1], primeSquare[n*n+1];
//store primes
//store results in prime and primesqaures arrays
eratosthenesSieve(n, prime, primeSquare, arr);
//count distinct divisors
for(int i = 0;; i++){
//terminates loop if condition is encountered
if(arr[i] * arr[i] * arr[i] > n)
while(n % arr[i] == 0){
n = n / arr[i];
//increment power
//if n = a^p * b^q, total divisors are (p+1)*(q+1)
res = res * count;
//loop terminates
//check cases
else if(primeSquare[n])
Divisors div;
cout << "num " << endl;
cout << "Number of divisors: " << div.numDivisorsNaive(n) << endl;
cout << "Number of divisors: " << div.numDivisorsOptimal(n) << endl;
Note: In the above implementation we used Sieve of Eratosthenes which is an algorithm for finding primes up to a given limit.
In large datasets > 500k we may use miller-rabin's primality test to check for primes which will be more efficient.
We check primality using Sieve of Eratosthenes whose time complexity is O(n log log n) for 3 cases, total time complexity is
O\left({n}^{\frac{1}{3}}\right)
The space complexity is and O(n).
2. Sum of divisors
We shall use the previous example of n = 18.
We have the following divisors
\left({2}^{0}·{3}^{0}\right),\left({2}^{0}·{3}^{1}\right),\left({2}^{0}·{3}^{2}\right),\left({2}^{1}·{3}^{0}\right),\left({2}^{1}·{3}^{1}\right),\left({2}^{1}·{3}^{2}\right)
\left({2}^{0}+{2}^{1}\right)·\left({3}^{0}+{3}^{1}+{3}^{2}\right)
The above is a geometric progression.
The geometric progression formula is
\frac{a\left({r}^{n}-1\right)}{r-1}
Where a = first term, r = common ratio, n = number of terms.
Therefore the sum of divisors is
\frac{1\left({2}^{2}-1\right)}{2-1}·\frac{1\left({3}^{3}-1\right)}{3-1}=39
In this approach we would iterate over all the numbers from 1 to the square root of n checking the divisibility of an element and add it to a variable result which we would return when the loop terminates.
ll sumDivisors(ll n){
int sumDivisors = 0;
return sumDivisors;
//iterate looking for numbers than can divide n
//add once of divisors are same
if(i == (n / i))
sumDivisors += i;
//add once
sumDivisors += (i + n / i);
//add 1, loop starts at 2 and add n
return sumDivisors+1+n;
cout << "num ", cin >> n;
cout << "Sum of divisors: " << sumDivisors(n) << endl;
The time complexity is proportional to the square root n,
O\left(n\sqrt{n}\right)
The space complexity is O(1) constant space.
In this optimal approach we use Sieve of Eratosthenes algorithm for finding the prime factors and implement an algorithm that solves the problem in (logn) time for n using prime factors generated by the sieve algorithm.
class SumDivisors{
bitset<N+10> flag;
void eratosthenesSieve(){
for(ll i = 3; i <= N; i+=2){
if(flag[i] == 0){
if(i*i <= N){
for(ll j = i*i; j <= N; j += i*2){
for(ll i = 0; p[i]*p[i] <= n; i++){
if(n % p[i] == 0){
while(n % p[i] == 0){
n = n / p[i];
result *= (pow(p[i], count) - 1) / (p[i] - 1);
result *= (pow(n, 2) - 1) / (n - 1);
cout << "num : ", cin >> n;
eratosthenesSieve();
cout << sumDivisors(n) << endl;
SumDivisors sd;
Sieve of Eratosthenes will find powers in O(nlog(log(n))) time for a integer n. The sumDivisors algorithm works in O(logn) time after the vector is filled. Therefore the total time complexity is O(nlog(log(n)) which is better than
O\left(n\sqrt{n}\right)
The space complexity if O(n), we use extra space to store the prime factors of n.
Note: This is a good case of a trade off between space and time complexity, We degraded the previous constant space complexity to achieve a better time complexity.
With this article at OpenGenus, you must have the complete idea of finding Sum and Number of Divisors of a Number.
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PandaX 4ton
PandaX is a series of experimental projects that utilizes Xenon detectors to search for elusive dark matter particles and to understand the fundamental properties of neutrinos. The PandaX collaboration has now entered into the multi-ton stage of the project, PandaX-4T.
PandaX II
PandaX-II is a dark matter direct detection experiment equipped with a half-ton scale dual-phase time projection chamber (TPC), operated in CJPL between Oct. 2014 and June 2019. In 2016 and 2017, PandaX-II produced the world leading constraints to dark matter-nucleon interactions. See a list of scientific publications here.
PandaX-4T
With 6-ton of total Xenon and 4-ton sensitive target, PandaX-4T aims to improve the dark matter sensitivity by one order of magnitude in comparison to PandaX-II. PandaX-4T also plans to make sensitive searches on neutrinoless double beta decay of
{}^{136}Xe
, and other signals from new physics. This project is expected to commence data taking in 2021.
PandaX III
PandaX-III searches for the possible neutrinoless double beta decay with 200 kg to one ton of 90% enriched
{}^{136}Xe
in a high pressure gaseous Xenon TPC.
PandaX-4T published the first result in Physics Review Letters, giving the strongest WIMP-nucleon interaction constraint!
On December 24, 2021, the first dark matter search result from PandaX-4T, a liquid xenon detection experiment, was published in Physical Review Letters as the "Editor's Suggestion" [1]. The American Physical Society 《Physics》 reported this PandaX-4T latest result together with another result from the Axion Dark Matter Experiment with a title of "Tightening the Net on Two Kinds of Dark Matter"
PandaX collaboration published a new result of light dark matter searching in Physics Review Letters
A new result on searching of light dark matter (DM) with PandaX-II experimental data was published online in Physical Review Letters on May 28, 2021 with a title of "Search for Light Dark Matter–Electron Scattering in the PandaX-II Experiment "[1]. This is the 6th PRL paper published with PandaX-II data.
Latest progress report and future plan
The PandaX-II experiment is completed and the PandaX-4T experiment is under installation. The future PandaX program will focus on the following two main directions:
Develop PandaX-4T into a multi-purpose liquid xenon experiment, to push further the dark matter search and other physics topics;
Develop and operate a 100-kg high pressure gas TPC (HpgTPC), PandaX-III, as a pathfinder for a tracking calorimeter to search for $0\nu\beta\beta$ in ${}^{136}Xe$.
The full International Advisory Committee 2020 Report.
New WIMP and axion search results from complete exposure of PandaX-II released
The PandaX collaboration released the latest results on the search of WIMPs and axions based on the full exposure of data at the International High Energy Physics Conference (ichep2020) on July 30, 2020. Detailed analyses were reported with an online seminar on August 20. The most stringent constraint is set for WIMPs around 10 GeV/c2 using the nuclear recoil events. The intriguing electron recoil event excess observed by XENON1T [1] was tested with the electron recoil data, and was found to be within the constraint of the PandaX-II data.
PandaX-II published new results on the search of axions
New results on the detection of axions and galaxy axion-like particles(ALPs) with 80 day of PandaX-II data were published online in Physical Review Letters on Nov 1, 2017 (on the same issue with another PandaX-II paper on 54 ton-day WIMP search results).
上海物理学会粒子物理与核物理专委会及上海核学会核物理专委会暨2020年度联合学术交流会
T. D. Lee Library
上海市物理学会粒子物理与核物理专委会及上海市核学会核物理专委会计划于2021年1月8日-10日在上海夏阳湖皇冠假日酒店举行2020年度联合学术交流会暨粒子物理与核物理上海研讨会，欢迎两个专委会的同事以及相关团队成员参会，开展学术交流；会议也将邀请国内有关专家参会交流。
PandaX Collaboration Annual Meeting 2019
PandaX 2019 Annual Report and Project Discussion Meeting
PandaX Annual Report 2021
webmaster@pandax.sjtu.edu.cn
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Comment #761 by Antonio Ruiz on October 21, 2020 at 22:03
In definition 5.4.1.1, left-degenerate and right-degenerate are phrased in such a way that it is impossible to be simultaneously left-degenerate and right-degenerate, contradicting the second bullet of the following remark.
Looks fine to me. Constant maps factor through everything.
My bad, I saw
\sigma^0(1) = 0
\sigma^0(1) = 1
and it distraught me a bit. But now I see it's perfectly reasonable.
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Implement parallel RLC branch - Simulink - MathWorks Nordic
Parallel RLC Branch
Implement parallel RLC branch
The Parallel RLC Branch block implements a single resistor, inductor, and capacitor or a parallel combination of these. Use the Branch type parameter to select elements you want to include in the branch.
The initial inductor current used at the start of the simulation. Default is 0. This parameter is not visible and has no effect on the block if the inductor is not modeled and if the Set the initial inductor current parameter is not selected.
Select Branch voltage to measure the voltage across the Parallel RLC Branch block terminals.
Select Branch current to measure the total current (sum of R, L, C currents) flowing through the Parallel RLC Branch block.
Select Branch voltage and current to measure the voltage and the current of the Parallel RLC Branch block.
The power_paralbranch example is used to obtain the frequency response of an eleventh-harmonic filter (tuned frequency at 660 Hz) connected on a 60 Hz power system:
Z\left(s\right)=\frac{V\left(s\right)}{I\left(s\right)}=\frac{RLC{s}^{2}+Ls+R}{LC{s}^{2}+RCs}.
This system is a one input (Is) and one output (Vs) system.
If you have Control System Toolbox™ software installed, you can get the transfer function Z(s) from the state-space matrices and the bode function.
[A,B,C,D] = power_analyze('power_paralbranch');
[Zmag,Zphase] = bode(A,B,C,D,1,w);
title('11th harmonic filter')
ylabel('Impedance Z')
You can also use the Impedance Measurement block and the Powergui block to plot the impedance as a function of frequency.
Multimeter, Parallel RLC Load, powergui, Series RLC Branch, Series RLC Load
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Quantifier (logic) - Wikipedia
Operator specifying how many individuals satisfy an open formula
For other uses, see Quantifier (disambiguation).
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier
{\displaystyle \forall }
in the first order formula
{\displaystyle \forall xP(x)}
expresses that everything in the domain satisfies the property denoted by
{\displaystyle P}
. On the other hand, the existential quantifier
{\displaystyle \exists }
{\displaystyle \exists xP(x)}
expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable.
The mostly commonly used quantifiers are
{\displaystyle \forall }
{\displaystyle \exists }
. These quantifiers are standardly defined as duals and are thus interdefinable using negation. They can also be used to define more complex quantifiers, as in the formula
{\displaystyle \neg \exists xP(x)}
which expresses that nothing has the property
{\displaystyle P}
. Other quantifiers are only definable within second order logic or higher order logics. Quantifiers have been generalized beginning with the work of Mostowski and Lindström.
1 Relations to logical conjunction and disjunction
1.1 Infinite domain of discourse
4 Order of quantifiers (nesting)
Relations to logical conjunction and disjunction[edit]
For a finite domain of discourse D = {a1,...an}, the universal quantifier is equivalent to a logical conjunction of propositions with singular terms ai (having the form Pai for monadic predicates).
Infinite domain of discourse[edit]
Algebraic approaches to quantification[edit]
An example of translating a quantified statement in a natural language such as English would be as follows. Given the statement, "Each of Peter's friends either likes to dance or likes to go to the beach (or both)", key aspects can be identified and rewritten using symbols including quantifiers. So, let X be the set of all Peter's friends, P(x) the predicate "x likes to dance", and Q(x) the predicate "x likes to go to the beach". Then the above sentence can be written in formal notation as
{\displaystyle \forall {x}{\in }X,P(x)\lor Q(x)}
, which is read, "for every x that is a member of X, P applies to x or Q applies to x".
{\displaystyle \exists {x}\,P}
{\displaystyle \forall {x}\,P}
{\displaystyle \bigvee _{x}P}
{\displaystyle (\exists {x})P}
{\displaystyle (\exists x\ .\ P)}
{\displaystyle \exists x\ \cdot \ P}
{\displaystyle (\exists x:P)}
{\displaystyle \exists {x}(P)}
{\displaystyle \exists _{x}\,P}
{\displaystyle \exists {x}{,}\,P}
{\displaystyle \exists {x}{\in }X\,P}
{\displaystyle \exists \,x{:}X\,P}
{\displaystyle \bigwedge _{x}P}
{\displaystyle \bigwedge xP}
{\displaystyle (x)\,P}
Mention explicitly the range of quantification, perhaps using a symbol for the set of all objects in that domain (or the type of the objects in that domain).
For every natural number x, ...
There exists an x such that ...
For at least one x, ....
For exactly one natural number x, ...
Order of quantifiers (nesting)[edit]
See also: Quantifier shift
Pointwise continuous if
{\displaystyle \forall \varepsilon >0\;\forall x\in \mathbb {R} \;\exists \delta >0\;\forall h\in \mathbb {R} \;(|h|<\delta \,\Rightarrow \,|f(x)-f(x+h)|<\varepsilon )}
Uniformly continuous if
{\displaystyle \forall \varepsilon >0\;\exists \delta >0\;\forall x\in \mathbb {R} \;\forall h\in \mathbb {R} \;(|h|<\delta \,\Rightarrow \,|f(x)-f(x+h)|<\varepsilon )}
Equivalent expressions[edit]
{\displaystyle \forall x\!\in \!D\;P(x).}
{\displaystyle \forall x\;(x\!\in \!D\to P(x)).}
{\displaystyle \exists x\!\in \!D\;P(x),}
{\displaystyle \exists x\;(x\!\in \!\!D\land P(x)).}
{\displaystyle \neg (\forall x\!\in \!D\;P(x))\equiv \exists x\!\in \!D\;\neg P(x),}
{\displaystyle \neg (\exists x\!\in \!D\;P(x))\equiv \forall x\!\in \!D\;\neg P(x),}
Range of quantification[edit]
A universally quantified formula over an empty range (like
{\displaystyle \forall x\!\in \!\varnothing \;x\neq x}
) is always vacuously true. Conversely, an existentially quantified formula over an empty range (like
{\displaystyle \exists x\!\in \!\varnothing \;x=x}
) is always false.
in Zermelo–Fraenkel set theory, one would write
{\displaystyle \forall x(\exists yB(x,y))\vee C(y,x)}
Syntax tree of the formula
{\displaystyle \forall x(\exists yB(x,y))\vee C(y,x)}
, illustrating scope and variable capture. Bound and free variable occurrences are colored in red and green, respectively.
{\displaystyle \forall x_{n}A(x_{1},\ldots ,x_{n})}
{\displaystyle \exists x_{n}A(x_{1},\ldots ,x_{n})}
{\displaystyle \exists !x_{n}A(x_{1},\ldots ,x_{n})}
{\displaystyle \exists x_{n}A(x_{1},\ldots ,x_{n})}
{\displaystyle \forall y,z{\big (}A(x_{1},\ldots ,x_{n-1},y)\wedge A(x_{1},\ldots ,x_{n-1},z)\implies y=z{\big )}.}
Paucal, multal and other degree quantifiers[edit]
{\displaystyle \exists ^{\mathrm {many} }x_{n}A(x_{1},\ldots ,x_{n-1},x_{n})}
{\displaystyle \operatorname {P} \{w:F(v_{1},\ldots ,v_{n-1},w)=\mathbf {T} \}\geq b}
{\displaystyle \exists ^{\mathrm {few} }x_{n}A(x_{1},\ldots ,x_{n-1},x_{n})}
{\displaystyle 0<\operatorname {P} \{w:F(v_{1},\ldots ,v_{n-1},w)=\mathbf {T} \}\leq a}
Other quantifiers[edit]
{\displaystyle \left[\S n\in \mathbb {N} \quad n^{2}\leq 4\right]=\{0,1,2\}}
is read "those n in N such that n2 ≤ 4 are in {0,1,2}." The same construct is expressible in set-builder notation as
{\displaystyle \{n\in \mathbb {N} :n^{2}\leq 4\}=\{0,1,2\}.}
There are uncountably many elements such that...
In 1827, George Bentham published his Outline of a new system of logic, with a critical examination of Dr Whately's Elements of Logic, describing the principle of the quantifier, but the book was not widely circulated.[11]
Counting quantification
^ "Predicates and Quantifiers". www.csm.ornl.gov. Retrieved 2020-09-04.
^ "1.2 Quantifiers". www.whitman.edu. Retrieved 2020-09-04.
^ K.R. Apt (1990). "Logic Programming". In Jan van Leeuwen (ed.). Formal Models and Semantics. Handbook of Theoretical Computer Science. Vol. B. Elsevier. pp. 493–574. ISBN 0-444-88074-7. Here: p.497
^ Schwichtenberg, Helmut; Wainer, Stanley S. (2009). Proofs and Computations. Cambridge: Cambridge University Press. ISBN 978-1-139-03190-5.
^ John E. Hopcroft and Jeffrey D. Ullman (1979). Introduction to Automata Theory, Languages, and Computation. Reading/MA: Addison-Wesley. ISBN 0-201-02988-X. Here: p.p.344
^ Hans Hermes (1973). Introduction to Mathematical Logic. Hochschultext (Springer-Verlag). London: Springer. ISBN 3540058192. ISSN 1431-4657. Here: Def. II.1.5
^ Glebskii, Yu. V.; Kogan, D. I.; Liogon'kii, M. I.; Talanov, V. A. (1972). "Range and degree of realizability of formulas in the restricted predicate calculus". Cybernetics. 5 (2): 142–154. doi:10.1007/bf01071084. ISSN 0011-4235.
^ in general, for a quantifer Q, closure makes sense only if the order of Q quantification does not matter, i.e. if Qx Qy p(x,y) is equivalent to Qy Qx p(x,y). This is satisfied for Q ∈ {∀,∃}, cf. #Order of quantifiers (nesting) above.
^ Hehner, Eric C. R., 2004, Practical Theory of Programming, 2nd edition, p. 28
^ Hehner (2004) uses the term "quantifier" in a very general sense, also including e.g. summation.
^ Peters, Stanley; Westerståhl, Dag (2006-04-27). Quantifiers in Language and Logic. Clarendon Press. pp. 34–. ISBN 978-0-19-929125-0.
"Quantifier", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
""For all" and "there exists" topical phrases, sentences and expressions". Archived from the original on March 1, 2000. . From College of Natural Sciences, University of Hawaii at Manoa.
Shapiro, Stewart (2000). "Classical Logic" (Covers syntax, model theory, and metatheory for first order logic in the natural deduction style.)
Westerståhl, Dag (2005). "Generalized quantifiers"
Peters, Stanley; Westerståhl, Dag (2002). "Quantifiers"
Retrieved from "https://en.wikipedia.org/w/index.php?title=Quantifier_(logic)&oldid=1084083273"
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Detect errors in input data using CRC - Simulink - MathWorks í•œêµ
General CRC Syndrome Detector HDL Optimized
The General CRC Syndrome Detector HDL Optimized block performs a cyclic redundancy check (CRC) on data and compares the resulting checksum with the appended checksum. The General CRC Syndrome Detector HDL Optimized block processing is optimized for HDL code generation. If the two checksums do not match, the block reports an error. Instead of processing an entire frame at once, the block accepts and returns a data sample stream with accompanying control signals. The control signals indicate the validity of the samples and the boundaries of the frame. To achieve higher throughput, the block accepts vector data up to the CRC length and implements a parallel architecture.
This a control signal that indicates if the data on the dataIn port is valid.
Output data, returned as a scalar or vector. The output data type and size are the same as the input data.
This is a control signal that indicates if the data on the dataOut port is valid.
err — Error indicator
Error indicator for the corruption of the received data, returned as a Boolean scalar.
When this value is 1, the message contains at least one error. When this value is 0, the message contains zero errors.
Specify the method of calculating checksum as a Boolean scalar.
Select this parameter to use the direct algorithm for CRC checksum calculations.
Clear this parameter to use the nondirect algorithm for CRC checksum calculations.
To learn about the direct and non-direct algorithms, see Cyclic Redundancy Check Codes.
When you use vector or integer input, the block implements a parallel CRC algorithm [1].
{X}^{\text{'}}={F}_{W}\left(Ã\right)X\left(+\right)D.
FW is an M-by-M matrix that selects elements of the current state for the polynomial calculation with the new input bits. D is an M-element vector that provides the new input bits, ordered in relation to the generator polynomial and padded with zeros. The block implements the (×) with logical AND and (+) with logical XOR.
This waveform shows streaming data and the accompanying control signals for a CRC16 with 8-bit binary vector input. The input frames are contiguous. The output frames include space between them because the detector block removes the checksum word.
This waveform diagram shows continuous input data. Non-continuous data is also supported.
The General CRC Syndrome Detector HDL Optimized block introduces a latency on the output. This latency can be computed with the following equation, assuming the input data is continuous:
initialdelay = 3 * (CRC length/input data width) + 2.
[1] Campobello, G., G. Patane, and M. Russo. “Parallel Crc Realization.†IEEE Transactions on Computers 52, no. 10 (October 2003): 1312–19. https://doi.org/10.1109/TC.2003.1234528.
General CRC Generator HDL Optimized | General CRC Syndrome Detector | comm.HDLCRCDetector
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pi - Simple English Wiktionary
Οο
{\displaystyle \longleftarrow }
{\displaystyle \longrightarrow }
Ρρ
enPR: pī, IPA (key): /paɪ/
SAMPA: /paI/
Homophone(s): pie
Lower case pi
Pi is equal to a circle's circumference divided by its diameter
Pi is a Greek letter written "π".
(mathematics) Pi is a number that is found by dividing the circumference of a circle by its diameter. Pi is constant, it is always the same (close to 3.14159 , also accurate approx as
{\displaystyle {\tfrac {355}{113}}}
, or rough approx as
{\displaystyle {\tfrac {22}{7}}}
). It is usually written as "π"
The distance around a circle is twice the radius times pi; C = 2πr.
Retrieved from "https://simple.wiktionary.org/w/index.php?title=pi&oldid=484785"
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Mianningite, (☐,Pb,Ce,Na) (U4+,Mn,U6+) Fe3+2(Ti,Fe3+)18O38, a new member of the crichtonite group from Maoniuping REE deposit, Mianning county, southwest Sichuan, China | European Journal of Mineralogy | GeoScienceWorld
Mianningite, (☐,Pb,Ce,Na) (U4+,Mn,U6+) Fe3+2(Ti,Fe3+)18O38, a new member of the crichtonite group from Maoniuping REE deposit, Mianning county, southwest Sichuan, China
Xiangkun Ge;
Xiangkun Ge
Beijing Research Institute of Uranium Geology, 100029 Beijing, China
Corresponding author, e-mail: gxk0621@163.com
Guang Fan;
Guowu Li;
Laboratory of Crystal Structure, China University of Geosciences (Beijing), 100083 Beijing, China
Ganfu Shen;
Ganfu Shen
Chengdu Institute of Geology and Mineral Resources, 610082 Chengdu, China
Zhangru Chen;
Zhangru Chen
Xiangkun Ge, Guang Fan, Guowu Li, Ganfu Shen, Zhangru Chen, Yujie Ai; Mianningite, (☐,Pb,Ce,Na) (U4+,Mn,U6+) Fe3+2(Ti,Fe3+)18O38, a new member of the crichtonite group from Maoniuping REE deposit, Mianning county, southwest Sichuan, China. European Journal of Mineralogy 2017;; 29 (2): 331–338. doi: https://doi.org/10.1127/ejm/2017/0029-2600
Mianningite (IMA 2014-072), ideally (☐,Pb,Ce,Na)(U4+,Mn,U6+)Fe3+2(Ti,Fe3+)18O38, is a new member of the crichtonite group from the Maoniuping REE deposit, Mianning county, Sichuan province, China. It was found in fractures of lamprophyre veins and in the contact between lamprophyre and a later quartz–alkali feldspar syenite dyke with REE mineralization, and is named after its type locality. Associated minerals are microcline, albite, quartz, iron-rich phlogopite, augite, muscovite, calcite, baryte, fluorite, epidote, pyrite, magnetite, hematite, galena, hydroxylapatite, titanite, ilmenite, rutile, garnet-group minerals, zircon, allanite-(Ce), monazite-(Ce), bastnäsite-(Ce), parisite-(Ce), maoniupingite-(Ce), thorite, pyrochlore-group minerals and chlorite. Mianningite occurs as opaque subhedral to euhedral tabular crystals, up to 1–2 mm in size, black in color and streak, and with a submetallic luster. Mianningite is brittle, with a conchoidal fracture. Its average micro-indentation hardness is 83.8 kg/mm2 (load 0.2 kg), which is equivalent to ~6 on the Mohs hardness scale. Its measured and calculated densities are 4.62 (8) g/cm3 and 4.77 g/cm3, respectively. Under reflected light, mianningite is grayish white, with no internal reflections. It appears isotropic and exhibits neither bireflectance nor pleochroism. The empirical formula, calculated on the basis of 38 O atoms per formula unit (apfu), is [☐0.322(Pb0.215Ba0.037Sr0.036Ca0.010)Σ0.298(Ce0.128La0.077Nd0.012)Σ0.217 (Na0.127K0.036)Σ0.163]Σ1.000(U4+0.447Mn0.293U6+0.112Y0.091Zr0.023Th0.011)Σ0.977(Fe3+1.224Fe2+0.243Mg0.023P0.008Si0.006☐0.496)Σ2.000(Ti12.464Fe3+5.292V5+0.118Nb0.083Al0.026Cr3+0.017)Σ18.000O38. Mianningite is trigonal, belongs to the space group
R3¯
, and has unit-cell parameters a = 10.3462(5) Å, c = 20.837(2) Å, V = 1931.65 (20) Å3, and Z = 3. The structure was solved (R1 = 0.070) using reflections with I > 2σ(I) on a heated crystal; it is isostructural with the other members of the crichtonite group. Mianningite can be considered as the analogue of mapiquiroite with the M0 site preferentially vacant. Its eight strongest X-ray powder-diffraction lines [d in Å(I/I0)(hkl)] are 2.627(100) (125), 2.144 (100) (135), 3.065(75) (025), 2.254 (70) (028), 1.545(60) (336), 2.883(55) (116), 2.476 (55) (027), and 1.705 (55) (146).
Maoniuping Deposit
Mianning China
mianningite
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Leverage your crypto on Fantom - Mai Finance - Tutorials
This guide is proposing a complete analysis of the different leverage options proposed by Mai Finance on Fantom, using Yearn vaults and Beefy vaults.
Mai Finance has launched its lending platform on Fantom with many different vault types, enabling the possibility to mint the MAI stable coin based on the assets you will deposit in a vault. The idea is that you will be able to keep your crypto currencies and benefit from their price appreciation, while still being able to buy other coins and farm yields with high APRs. If you use your loan to buy more of the same asset you already deposited, this is what is called leveraging your tokens. We will show you the benefits of this strategy using 2 different lending platforms on Fantom to leverage our DAI tokens.
Leverage your Yearn Vault tokens
Deposit your assets on Yearn Finance
Yearn Finance is a group of protocols running on the Ethereum Mainnet and other blockchains that allow users to optimize their earnings on crypto assets through lending and trading services. On Fantom, the product that we will be using is the vaults on yearn finance. This is a tool that will accept single token deposits and will make you earn yields on this deposit. As a proof of deposit, you will receive a yvToken. In our case, we will deposit DAI and will get yvDAI in exchange.
yearn vaults on Fantom network
The yearn finance website is still in beta mode on Fantom. The team is still working on the platform and APRs/APYs aren't showing. If you head to the Iron Bank tab, which is the lending/borrowing protocol on yearn platform, you'll see that lending DAI is getting ~8% APR. Please invest at your own risk.
Deposit your yvToken on Mai Finance
Once you deposited your DAI on yearn finance, you should have yvDAI in your wallet. This is what we call a yield bearing token: it's a token that doesn't have any value per se, but represents your share of a pool where your assets are earning yields and in which rewards are automatically compounded. In other words, if your DAI doesn't change in value because the DAI is pegged to the US dollar, the underlying value of your yvDAI token increases anyway.
Mai Finance accepts a lot of different yield bearing tokens as collateral, including yvDAI. You can now deposit this token and borrow MAI against it.
The yvDAI vault has a liquidation threshold of 110%, this means that you can borrow MAI so that the ratio between your collateral value and the debt value is 110%. Be careful that 110% is actually the ratio at which your vault will be liquidated. You need to keep the ratio above this minimum threshold. Since DAI doesn't vary much in price (less than a few cents up or down) it's possible to keep a "safe" CDR (Collateral to Debt Ratio) of 115%, but feel free to keep something higher.
As always, to calculate the loan value we can get based on the value of our collateral and the target CDR we want to get, we will use the following formula:
MAI_{available} = \frac{Collateral_{value} - Debt_{value} * Target_{CDR}}{Target_{CDR}}
With a collateral value of $100 and no debt, if we want to keep a healthy CDR of 115% we can borrow up to
MAI_{available}=\frac{100-0*1.15}{1.15}=86.95
You are now in a position where you have your DAI earning yields in a Yearn vault, and you also have some MAI stable coin ready to use. Since we want to leverage our DAI position, we will now swap our MAI for more DAI.
Swapping your MAI on BeethovenX
On Fantom, the main source of liquidity for MAI is BeethovenX. This is the main place where you will be able to swap your MAI tokens for more DAI for our strategy.
Swapping MAI for more DAI
This is the last step of our loop. Now that you have more DAI you can deposit them in a Yearn vault and repeat the loop. Doing so increases the amount of assets you have in the Yearn vault, meaning that you will collect more rewards by lending your DAI on that platform. The APR/APY remains the same, but because you have more assets, you earn more yield, and if you compare to your initial investment, it's your APR that increases. If you want to get more examples on what APR you can achieve using the yvDAI loops, please go read our camDAI token guide for Polygon that uses the exact same strategy but different tools.
BeethovenX is actually a fantastic opportunity to farm yields with your borrowed MAI. Simply deposit your MAI in the MAI-DAI-USDC pool (APR of ~30% as of November 2021) if you cannot achieve a better APR using leveraged loops.
Leverage your mooScreamTokens on Mai Finance
Deposit your assets on Beefy Finance
Beefy Finance is a Decentralized, Multi-Chain Yield Optimizer platform that allows its users to earn compound interest on their crypto holdings. In other words, you can deposit some assets or LP tokens from other platforms on Beefy Finance and let the auto-compounder harvest farm tokens and compound them into more of your deposited asset / LP token. For our exemple, we will use single DAI deposits on Beefy and use Scream as the underlying platform. Scream is a Compound fork on the Fantom network on which you will be able to lend your assets and collect SCREAM tokens. Beefy will then sell the SCREAM tokens for more DAI.
To deposit our DAI, we will visit the Beefy Finance app and select Scream as the platform on which we will farm yields. You can also add the DAI filter in order to get the direct DAI deposit.
Deposit your DAI on Beefy using Scream
As you can see, Beefy is already giving an unbelievable APY on DAI single deposits. Once you have your DAI deposited on Beefy, you should have a proof of deposit in your wallet under the form of mooScreamDAI tokens. As for the yvDAI token, the mooScreamDAI token is a yield bearing deposit, meaning that you asset is still used on Scream and compounded on Beefy, earning yields. But you will be able to use this token on Mai Finance to borrow MAI against them.
Deposit your mooScreamToken on Mai Finance
Once you deposited your DAI on yearn finance, you should have mooScreamDAI in your wallet. You can use the exact same steps as for the Yearn Vault strategy above, the only difference is that the mooScreamDAI liquidation ratio is 135%. Since DAI is a stable coin, it's still possible to borrow MAI and keep a CDR very close to the liquidation ration. For our exemple, we will aim at a 140% CDR, and with the same formula as above, we can calculate the amount of MAI we can mint with 100$ worth of DAI.
MAI_{available}=\frac{100-0*1.4}{1.4}=71.43
Since we are borrowing less, we will be able to perform less loops and the final equivalent APY will also be lower, however this is still a pretty good beginner strategy.
The rest of the loop is the same as for yvDAI, meaning you will have to swap your MAI for DAI on BeethovenX and repeat until you're satisfied.
Some notes on leveraging strategies
Leverage DAI is considered a beginner strategy in the sense that it presents very little risk (you are working with stable coins) and you can get some nice yields using at most 3 protocols. However, there's still some risk.
The more loops you will perform, the higher the liquidation risk. Indeed, even a small variation of the DAI price will be magnified by the leverage you applied, and even if you keep a CDR 5 points above the liquidation ratio, your vault can be at risk. It's always a good idea to stop the leverage loops at the step where you deposit your assets on MAI finance and don't borrow additional MAI in order to keep a better CDR.
Also, in case of a liquidation, because your vault on MAI finance contains a lot more assets, a liquidation will also have a bigger impact than if you didn't levered your position, simply because the debt you have to repay is also much bigger.
If you use a lot of protocols for your investment legos, you need to make sure that these protocols are safe. Indeed, in our leveraging strategy, if a single protocol gets hacked, the entire strategy may collapse. Make sure you do your due diligence before investing in DeFi projects.
Hitting debt ceilings
Because these strategies are easy to set and present low risks, there's a very high demand for them. However, you certainly noticed that in the leverage process, borrowed MAI is swapped for DAI (or other tokens). If too much MAI is sold on Beethoven, its price will decrease slowly and there is a risk for MAI to lose its peg, which is pretty bad for a stable coin. In order to let time for the price to stabilize, Mai Finance has security mechanisms in place, and the most important one is a debt ceiling for each vault.
A debt ceiling represents the maximum number of MAI that can be minted for a given vault. Once the ceiling is reached, no more MAI can be borrowed. Then the core team in charge of MAI finance can decide to increase the ceiling or wait a little more for a better price for MAI.
You can at all time verify the amount of MAI that can be minted on the vault creation page, but you will usually notice that there aren't any more MAI if you get the following error message:
Error message received when debt ceiling is reached
This error message will appear even if your health factor is correct. In most cases, waiting for the ceiling to be increased is the only solution. Keep an eye on twitter or on Discord to know when this happen.
This guide presented some of the ways you can use your assets on Fantom and include Mai Finance to your strategy in order to increase your gains. However, as usual, this tutorial isn't a financial advice and you should always DYOR before applying an investment strategy, and invest in a responsible manner.
Keep also in mind that this solution may not be the best strategy depending on when you plan to use it. We just highlighted that BeethovenX has pretty interesting APRs too for your MAI, and you can also use Beefy Finance to compound the BEETS rewards into more stable coins.
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SimsonLine - Maple Help
Home : Support : Online Help : Mathematics : Geometry : 2-D Euclidean : Triangle Geometry : SimsonLine
find the Simson line of a given triangle with respect to a given point on the circumcircle of the triangle
SimsonLine(sl, N, T)
the name of the Simson line
The feet of the perpendiculars from any point N on the circumcircle of a triangle T to the sides of the triangle are collinear. The line of collinearity is called the Simson line of the point N for the triangle T
For a detailed description of the Simson line sl, use the routine detail (i.e., detail(sl))
The command with(geometry,SimsonLine) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{geometry}\right):
\mathrm{triangle}\left(T,[\mathrm{point}\left(A,-1,0\right),\mathrm{point}\left(B,1,0\right),\mathrm{point}\left(C,0,1\right)]\right):
\mathrm{point}\left(N,\frac{1}{\mathrm{sqrt}\left(2\right)},\frac{1}{\mathrm{sqrt}\left(2\right)}\right):
\mathrm{SimsonLine}\left(\mathrm{sl},N,T\right)
\textcolor[rgb]{0,0,1}{\mathrm{sl}}
\mathrm{detail}\left(\mathrm{sl}\right)
\begin{array}{ll}\textcolor[rgb]{0,0,1}{\text{name of the object}}& \textcolor[rgb]{0,0,1}{\mathrm{sl}}\\ \textcolor[rgb]{0,0,1}{\text{form of the object}}& \textcolor[rgb]{0,0,1}{\mathrm{line2d}}\\ \textcolor[rgb]{0,0,1}{\text{equation of the line}}& \textcolor[rgb]{0,0,1}{\mathrm{_x}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\frac{\sqrt{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{2}}\right)\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{\mathrm{_y}}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\left(\frac{\sqrt{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\right)\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\end{array}
\mathrm{draw}\left({N,T,\mathrm{sl}}\right)
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Application of Bayesian Forecasting to Change Detection and Prognosis of Gas Turbine Performance | J. Eng. Gas Turbines Power | ASME Digital Collection
Holger Lipowsky,
Institute of Aircraft Propulsion Systems (ILA),
, Pfaffenwaldring 6, 70569 Stuttgart, Germany
e-mail: lipowsky@ila.uni-stuttgart.de
e-mail: staudacher@ila.uni-stuttgart.de
Department of Performance, TEAP,
e-mail: michael.bauer@mtu.de
Klaus-Juergen Schmidt
e-mail: klaus-juergen.schmidt@mtu.de
Lipowsky, H., Staudacher, S., Bauer, M., and Schmidt, K. (December 3, 2009). "Application of Bayesian Forecasting to Change Detection and Prognosis of Gas Turbine Performance." ASME. J. Eng. Gas Turbines Power. March 2010; 132(3): 031602. https://doi.org/10.1115/1.3159367
The performance of gas turbines degrades over time due to deterioration mechanisms and single fault events. While deterioration mechanisms occur gradually, single fault events are characterized by occurring accidentally. In the case of single events, abrupt changes in the engine parameters are expected. Identifying these changes as soon as possible is referred to as detection. State-of-the-art detection algorithms are based on expert systems, neural networks, special filters, or fuzzy logic. This paper presents a novel detection technique, which is based on Bayesian forecasting and dynamic linear models (DLMs). Bayesian forecasting enables the calculation of conditional probabilities, whereas DLMs are a mathematical tool for time series analysis. The combination of the two methods can be used to calculate probability density functions prior to the next observation, or the so called forecast distributions. The change detection is carried out by comparing the current model with an alternative model, where the mean value is shifted by a prescribed offset. If the forecast distribution of the alternative model better fits the actual observation, a potential change is detected. To determine whether the respective observation is a single outlier or the first observation of a significant change, a special logic is developed. In addition to change detection, the proposed technique has the ability to perform a prognosis of measurement values. The developed method was run through an extensive test program. Detection rates
>92%
have been achieved for changed heights, as small as 1.5 times the standard deviation of the observed signal (sigma). For changed heights greater than 2 sigma, the detection rates have proven to be 100%. It could also be shown that a high detection rate is gained by a high false detection rate
(∼2%)
. An optimum must be chosen between a high detection rate and a low false detection rate, by choosing an appropriate uncertainty limit for the detection. Increasing the uncertainty limit decreases both detection rate and false detection rate. In terms of prognostic abilities, the proposed technique not only estimates the point of time of a potential limit exceedance of respective parameters, but also calculates confidence bounds, as well as probability density and cumulative distribution functions for the prognosis. The conflictive requirements of a high degree of smoothing and a quick reaction to changes are fulfilled in parallel by combining two different detection conditions.
aerospace engines, Bayes methods, diagnostic expert systems, failure analysis, flaw detection, fuzzy logic, gas turbines, neural nets, probability, time series, engine diagnostics, failure detection, gas turbine performance
Algorithms, Gas turbines, Probability, Uncertainty, Time series, Engines, Artificial neural networks, Fuzzy logic, Expert systems, Cycles, Filters
Method and Apparatus for Gas Turbine Monitoring
,” German Patent No. DPMA 10 2008 022 459.6.
Anomaly Detection for Advance Military Aircraft Using Neural Networks
Diagnostics of a Small Jet Engine-Neural Networks Approach
Fuzzy Logic Intelligent System for Gas Turbine Module and System Fault Isolation
A Comparative Study of Optimization Methods for Jet Engine Condition Diagnosis
16th International Symposium on Air-Breathing Engines
, Cleveland, OH, Aug. 31–Sept. 5.
Multiple Operating Point Analysis Using Genetic Algorithm Optimisation for Gas Turbine Diagnostics
, Bangalore, India, Sept. 3–7.
Identification of Sensor Faults on Turbofan Engines Using Pattern Recognition Techniques
Optimizing Automated Gas Turbine Fault Detection Using Statistical Pattern Recognition
An Integrated Fault Diagnostics Model Using Genetic Algorithm and Neural Networks
Gas Turbine Fault Diagnostics Using a Fusion of Least Squares Estimations and Fuzzy Logic Rules
Fully Automated Model Based Performance Analysis Procedure for Online and Offline Applications
Kalman Filtering Applied to Time Series Analysis
VKI Lecture Series, Gas Turbine Condition Monitoring and Fault Diagnosis
, Brussels, Belgium, Jan. 13–17, Paper No. LS-2003-01.
A System Theory Based Solution of the Failure Diagnosis Problem Applied to a Jet Engine
,” Doctoral thesis, Institute of Automation Engineering, University of the German Federal Armed Forces, Hamburg, Germany.
HCCI Engine Combustion Phasing Prediction Using a Symbolic-Statistics Approach
Identification Technique of Aircraft Gas Turbine Engine’s Health
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3-D superpixel oversegmentation of 3-D image - MATLAB superpixels3 - MathWorks India
[L,NumLabels] = superpixels3(A,N)
[L,NumLabels] = superpixels3(___,Name,Value)
[L,NumLabels] = superpixels3(A,N) computes 3-D superpixels of the 3-D image A. N specifies the number of superpixels you want to create. The function returns L, a 3-D label matrix, and NumLabels, the actual number of superpixels returned.
[L,NumLabels] = superpixels3(___,Name,Value) computes superpixels of image A using name-value pairs to control aspects of the segmentation.
Load 3-D MRI data, remove any singleton dimensions, and convert the data into a grayscale intensity image.
A = ind2gray(D,map);
Calculate the 3-D superpixels. Form an output image where each pixel is set to the mean color of its corresponding superpixel region.
[L,N] = superpixels3(A,34);
Show all xy-planes progressively with superpixel boundaries.
imSize = size(A);
Create a stack of RGB images to display the boundaries in color.
imPlusBoundaries = zeros(imSize(1),imSize(2),3,imSize(3),'uint8');
for plane = 1:imSize(3)
BW = boundarymask(L(:, :, plane));
% Create an RGB representation of this plane with boundary shown
% in cyan.
imPlusBoundaries(:, :, :, plane) = imoverlay(A(:, :, plane), BW, 'cyan');
implay(imPlusBoundaries,5)
Set the color of each pixel in output image to the mean intensity of the superpixel region. Show the mean image next to the original. If you run this code, you can use implay to view each slice of the MRI data.
meanA = zeros(size(A),'like',D);
for superpixel = 1:N
memberPixelIdx = pixelIdxList{superpixel};
meanA(memberPixelIdx) = mean(A(memberPixelIdx));
implay([A meanA],5);
A — Volume to segment
Volume to segment, specified as a 3-D numeric array.
Example: B = superpixels3(A,100,'NumIterations', 20);
0.001 if method is slic0 and 0.05 if method is slic (default) | numeric scalar
Shape of superpixels, specified as a numeric scalar. The compactness parameter of the SLIC algorithm controls the shape of the superpixels. A higher value makes the superpixels more regularly shaped, that is, a square. A lower value makes the superpixels adhere to boundaries better, making them irregularly shaped. You can specify any value in the range [0 Inf) but typical values are in the range [0.01,0.1].
If you specify the 'slic0' method, you typically do not need to adjust the 'Compactness' parameter. With the 'slic0' method, superpixel3 adaptively refines the 'Compactness' parameter automatically, thus eliminating the need to determine a good value.
Algorithm used to compute the superpixels, specified as one of the following values. For more information, see Algorithms.
superpixels3 uses the SLIC0 algorithm to refine 'Compactness' adaptively after the first iteration. This is the default.
3-D array of positive integers
Label matrix, returned as a 3-D array of positive integers. The value 1 indicates the first region, 2 the second region, and so on for each superpixel region in the image.
Number of superpixels computed, returned as a positive number.
The algorithm used in superpixels3 is a modified version of the Simple Linear Iterative Clustering (SLIC) algorithm used by superpixels. At a high level, it creates cluster centers and then iteratively alternates between assigning pixels to the closest cluster center and updating the locations of the cluster centers. superpixels3 uses a distance metric to determine the closest cluster center for each pixel. This distance metric combines intensity distance and spatial distance.
The function's Compactness argument comes from the mathematical form of the distance metric. The compactness parameter of the algorithm is a scalar value that controls the shape of the superpixels. The distance between two pixels i and j, where m is the compactness value, is:
\begin{array}{l}{d}_{\mathrm{int}ensity}=\sqrt{{\left({l}_{i}-{l}_{j}\right)}^{2}}\\ {d}_{spatial}=\sqrt{{\left({x}_{i}-{x}_{j}\right)}^{2}+{\left({y}_{i}-{y}_{j}\right)}^{2}+{\left({z}_{i}-{z}_{j}\right)}^{2}}\\ D=\sqrt{{\left(\frac{{d}_{\mathrm{int}ensity}}{m}\right)}^{2}+{\left(\frac{{d}_{spatial}}{S}\right)}^{2}}\end{array}
Compactness has the same meaning as in the 2-D superpixels function: It determines the relative importance of the intensity distance and the spatial distance in the overall distance metric. A lower value makes the superpixels adhere to boundaries better, making them irregularly shaped. A higher value makes the superpixels more regularly shaped. The allowable range for compactness is (0 Inf), as in the 2-D function. The typical range has been found through experimentation to be [0.01 0.1]. The dynamic range of input images is normalized within the algorithm to be from 0 to 1. This enables a consistent meaning of compactness values across images.
superpixels | boundarymask | imoverlay | label2idx | label2rgb
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Discharging method (discrete mathematics) - Wikipedia
Discharging method (discrete mathematics)
Technique used to Prove Lemmas in Structural Graph Theory
The discharging method is a technique used to prove lemmas in structural graph theory. Discharging is most well known for its central role in the proof of the four color theorem. The discharging method is used to prove that every graph in a certain class contains some subgraph from a specified list. The presence of the desired subgraph is then often used to prove a coloring result.
Most commonly, discharging is applied to planar graphs. Initially, a charge is assigned to each face and each vertex of the graph. The charges are assigned so that they sum to a small positive number. During the Discharging Phase the charge at each face or vertex may be redistributed to nearby faces and vertices, as required by a set of discharging rules. However, each discharging rule maintains the sum of the charges. The rules are designed so that after the discharging phase each face or vertex with positive charge lies in one of the desired subgraphs. Since the sum of the charges is positive, some face or vertex must have a positive charge. Many discharging arguments use one of a few standard initial charge functions (these are listed below). Successful application of the discharging method requires creative design of discharging rules.
In 1904, Wernicke introduced the discharging method to prove the following theorem, which was part of an attempt to prove the four color theorem.
Theorem: If a planar graph has minimum degree 5, then it either has an edge with endpoints both of degree 5 or one with endpoints of degrees 5 and 6.
Proof: We use
{\displaystyle V}
{\displaystyle F}
{\displaystyle E}
to denote the sets of vertices, faces, and edges, respectively. We call an edge light if its endpoints are both of degree 5 or are of degrees 5 and 6. Embed the graph in the plane. To prove the theorem, it is sufficient to only consider planar triangulations (because, if it holds on a triangulation, when removing nodes to return to the original graph, neither node on either side of the desired edge can be removed without reducing the minimum degree of the graph below 5). We arbitrarily add edges to the graph until it is a triangulation. Since the original graph had minimum degree 5, each endpoint of a new edge has degree at least 6. So, none of the new edges are light. Thus, if the triangulation contains a light edge, then that edge must have been in the original graph.
We give the charge
{\displaystyle 6-d(v)}
to each vertex
{\displaystyle v}
and the charge
{\displaystyle 6-2d(f)}
to each face
{\displaystyle f}
{\displaystyle d(x)}
denotes the degree of a vertex and the length of a face. (Since the graph is a triangulation, the charge on each face is 0.) Recall that the sum of all the degrees in the graph is equal to twice the number of edges; similarly, the sum of all the face lengths equals twice the number of edges. Using Euler's Formula, it's easy to see that the sum of all the charges is 12:
{\displaystyle {\begin{aligned}\sum _{f\in F}6-2d(f)+\sum _{v\in V}6-d(v)=&\\6|F|-2(2|E|)+6|V|-2|E|=&\\6(|F|-|E|+|V|)=&&12.\end{aligned}}}
We use only a single discharging rule:
Each degree 5 vertex gives a charge of 1/5 to each neighbor.
We consider which vertices could have positive final charge. The only vertices with positive initial charge are vertices of degree 5. Each degree 5 vertex gives a charge of 1/5 to each neighbor. So, each vertex is given a total charge of at most
{\displaystyle d(v)/5}
. The initial charge of each vertex v is
{\displaystyle 6-d(v)}
. So, the final charge of each vertex is at most
{\displaystyle 6-4d(v)/5}
. Hence, a vertex can only have positive final charge if it has degree at most 7. Now we show that each vertex with positive final charge is adjacent to an endpoint of a light edge.
If a vertex
{\displaystyle v}
has degree 5 or 6 and has positive final charge, then
{\displaystyle v}
received charge from an adjacent degree 5 vertex
{\displaystyle u}
, so edge
{\displaystyle uv}
is light. If a vertex
{\displaystyle v}
has degree 7 and has positive final charge, then
{\displaystyle v}
received charge from at least 6 adjacent degree 5 vertices. Since the graph is a triangulation, the vertices adjacent to
{\displaystyle v}
must form a cycle, and since it has only degree 7, the degree 5 neighbors cannot be all separated by vertices of higher degree; at least two of the degree 5 neighbors of
{\displaystyle v}
must be adjacent to each other on this cycle. This yields the light edge.
Appel, Kenneth; Haken, Wolfgang (1977), "Every planar map is four colorable. I. Discharging", Illinois Journal of Mathematics, 21: 429–490, doi:10.1215/ijm/1256049011 .
Appel, Kenneth; Haken, Wolfgang (1977), "Every planar map is four colorable. II. Reducibility", Illinois Journal of Mathematics, 21: 491–567, doi:10.1215/ijm/1256049012 .
Hliněný, Petr (2000), Discharging technique in practice . (Lecture text for Spring School on Combinatorics).
Robertson, Neil; Sanders, Daniel P.; Seymour, Paul; Thomas, Robin (1997), "The four-color theorem", Journal of Combinatorial Theory, Series B, 70: 2–44, doi:10.1006/jctb.1997.1750 .
Wernicke, P. (1904), "Über den kartographischen Vierfarbensatz" (PDF), Math. Ann. (in German), 58 (3): 413–426, doi:10.1007/bf01444968 .
Retrieved from "https://en.wikipedia.org/w/index.php?title=Discharging_method_(discrete_mathematics)&oldid=1032259596"
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Physics - Spin-Orbit Coupling Comes in From the Cold
Spin-Orbit Coupling Comes in From the Cold
Experimentalists simulate the effects of spin-orbit coupling in ultracold Fermi gases, paving the way for the creation of new exotic phases of matter.
APS/Erich J. Mueller
Figure 1: Scheme for generating spin-orbit coupling in a neutral, ultracold atomic gas. Two counterpropagating laser beams couple two spin states by a resonant stimulated two-photon Raman transition: an atom in a spin-up (
↑
) state is excited to a virtual level by absorbing a photon from the left beam, then flips to the spin-down (
↓
) state by emitting another photon into the right beam. The lasers are detuned by a frequency
\delta
from an excited multiplet. This stimulated Raman process results in a momentum kick to the atom, leading to single-particle eigenstates where spin and momentum are entangled.Scheme for generating spin-orbit coupling in a neutral, ultracold atomic gas. Two counterpropagating laser beams couple two spin states by a resonant stimulated two-photon Raman transition: an atom in a spin-up (
↑
) state is excited to a virtual level... Show more
One thriving area of cold-atom research is the development of techniques allowing dilute gases at nanokelvin temperatures to reproduce phenomena central to other fields, such as solid-state or nuclear physics. By precisely tuning properties such as density, temperature, and interaction strength, one can gain unprecedented quantitative insights into many physical processes. As reported in Physical Review Letters, two groups of researchers, Pengjun Wang at Shanxi University in China and colleagues [1], and Lawrence Cheuk from the Massachusetts Institute of Technology, Cambridge, and colleagues [2], have expanded the cold-atom experimental toolbox by engineering a system of fermionic atoms in which lasers induce strong spin-orbit coupling. One can envision that this technique may be combined with Feshbach resonances [3] (which control the interatomic interactions) and optical lattices (which mimic the lattices in real materials), enabling the production of exotic states found in condensed-matter systems such as topological insulators. Even more importantly, one hopes to realize novel states of matter (e.g., “fractional topological insulators”), which are anticipated by many theoretical studies but are hard to create and analyze experimentally.
Spin-orbit coupling refers to the interaction between the spin and motion degrees of freedom of an electron. A simple illustrative model is a 2D electron gas in the presence of a uniform electric field perpendicular to the plane. According to special relativity, the electric field is seen as a magnetic field in the moving electrons’ frame of reference. The magnetic field’s strength and direction depend on the velocity of the electron, producing a correlation between the electrons’ momenta and their spin states. At field strengths available in the laboratory this correlation can generally be neglected for any reasonable electronic velocities. However, strong spin-orbit coupling can be found in materials that contain heavier elements and lack inversion symmetry: the electron motion becomes relativistic near the ion cores, and the local electric field can be strong.
While these relativistic effects can couple an electron’s spin to its motion, coupling the spin and the center-of-mass motion of a neutral atom presents a challenge. In order to introduce spin-orbit coupling in neutral atomic gases, both Wang et al. and Cheuk et al. turned to a technique pioneered with bosonic Rubidium atoms [4]. While spin-orbit-coupled Bose gases have generated much excitement, Fermi gases hold even more promise. Due to the Pauli exclusion principle, fermions occupy a large number of momentum states, and are therefore sensitive to global (topological) features of the band structure. Conversely, Bose-Einstein condensation typically occurs in one or two single-particle states. Spin-orbit-coupled fermionic gases would thus provide ways to explore a much richer phenomenology.
The basic principles of the technique are illustrated in Fig. 1. The experimentalists single out two among the many internal hyperfine atomic states, labeled and in analogy with the electronic spin. Two counterpropagating laser beams are introduced to couple these states by a resonant stimulated two-photon Raman transition: by absorbing a photon from the left beam, and emitting it into the right beam, a atom will flip into the state. Since in the process the atom receives a momentum kick, this provides a mechanism by which spin and momentum become coupled. In these experiments, the symmetries are different from our 2D example of electrons in a field and from spin-orbit coupling in solids: here, a unidirectional spin-orbit coupling is realized, where only the component of the momentum is coupled to spin.
Such symmetry differences can be viewed as either a blessing (allowing one to study novel physics) or a curse (making it difficult to model the physics most relevant for solid-state physics). Nevertheless, the current geometry can lead to exciting effects. For example, near a Feshbach resonance the fermions should pair to form a superconducting state similar to that seen in spin-orbit-coupled semiconducting wires [5]. The analogy becomes even stronger if an optical lattice is used to break the atomic cloud into an array of wires [6]. Such wires are expected to exhibit Majorana edge modes, which can be used for quantum computation (see 15 March 2010 Viewpoint).
There are a number of technical difficulties in using lasers to couple atomic motion and hyperfine states: photons can only flip electronic or nuclear spins when they are coupled to electronic motion. Thus, to allow the transition from to , experimentalists must tune the lasers near a multiplet of excited states (see Fig. 1). The detuning from resonance ( ), must be of the same order of magnitude as the fine structure splitting of that multiplet. Unfortunately, close to an optically allowed transition, one must contend with resonant absorption, which heats the gas. The crucial figure of merit is the ratio of the linewidth of the resonance to the fine structure splitting. This ratio is larger for lighter elements, which will therefore suffer more heating: bosonic rubidium-87 is more favorable than fermionic potassium-40, which in turn is more favorable than fermionic lithium-6.
Wang et al. carried out their studies using potassium-40. They looked at equilibrium properties (measuring the thermal occupation of the various momentum and spin states) as well as dynamics (monitoring the time evolution of the spin populations starting from a polarized state) and measured the energy-momentum dispersion relation (using momentum-resolved radio-frequency spectroscopy, a technique related to angle-resolved photoemission spectroscopy). Through these studies they deliver a quantitative, coherent, and very intuitive picture of spin-orbit coupling in this noninteracting Fermi gas.
Cheuk et al. instead focused on lithium-6. This was a bold choice: as previously mentioned, lithium has an unfavorable ratio of fine structure splitting to linewidth. In fact, prior to this study the common wisdom was that inelastic light scattering would make such a study impractical. The authors came up with an ingenious approach to circumvent this problem: they worked with four atomic states, and used radio waves to drive transitions from two “reservoir” states into two spin-orbit-coupled states. They controlled the heating by keeping the population of the spin-orbit-coupled states low. By monitoring the transition rate as a function of the radio frequency, they were able to map out the dispersion of the spin-orbit-coupled states. Their study provided a comprehensive view of spin-orbit coupling in the absence of interparticle interactions. It remains to be seen if a similar strategy will allow the study of physics where such interparticle interactions are important.
The approach of Wang et al. and Cheuk et al. may prove valuable for using cold gases to recreate and explore the physics of topological insulators and superconductors. By providing new ways to investigate spin-orbit physics, these studies could also have a profound impact on the development of spintronic devices operated by spin instead of charge (see 15 June 2009 Trend).
P. Wang, Z-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai, and J. Zhang, ”Spin-Orbit Coupled Degenerate Fermi Gases,” Phys. Rev. Lett. 109, 095301 (2012)
L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, ”Spin-Injection Spectroscopy of a Spin-Orbit Coupled Fermi Gas,” Phys. Rev. Lett. 109, 095302 (2012)
C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, “Feshbach Resonances in Ultracold Gases,” Rev. Mod. Phys. 82, 1225 (2010)
Y.-J. Lin, K. Jimnez-Garca, and I. B. Spielman, “Spin Orbit-Coupled Bose Einstein Condensates,” Nature 471, 83 (2011)
V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven, “Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices,” Science 336, 1003 (2012); J. R. Williams et al., “Signatures of Majorana Fermions in Hybrid Superconductor-Topological Insulator Devices,” arXiv:1202.2323 (2012); M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson, P. Caro, and H. Q. Xu, “Observation of Majorana Fermions in a Nb-InSb Nanowire-Nb Hybrid Quantum Device,” arXiv:1204.4130 (2012); A. Das. Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, and H. Shtrikman, “Evidence of Majorana Fermions in an Al-InAs Nanowire Topological Superconductor, arXiv:1205.7073 (2012)
Y. Liao, A. S. C. Rittner, T. Paprotta, W. Li, G. B. Partridge, R. G. Hulet, S. K. Baur, and E. J. Mueller,“Spin-Imbalance in a One-Dimensional Fermi Gas,” Nature 467, 567 (2010)
Pengjun Wang, Zeng-Qiang Yu, Zhengkun Fu, Jiao Miao, Lianghui Huang, Shijie Chai, Hui Zhai, and Jing Zhang
Lawrence W. Cheuk, Ariel T. Sommer, Zoran Hadzibabic, Tarik Yefsah, Waseem S. Bakr, and Martin W. Zwierlein
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FAQ - The Twilight ZONE
(3,3) is the idea that, if everyone cooperated in Twilight, it would generate the greatest gain for everyone (from a game theory standpoint).
Staking and bonding are considered beneficial to the protocol, while selling is considered detrimental. Staking and selling will also cause a price move, while bonding does not (we consider buying ZONE from the market as a prerequisite of staking, thus causing a price move). If both actions are beneficial, the actor who moves price also gets half of the benefit (+1). If both actions are contradictory, the bad actor who moves price gets half of the benefit (+1), while the good actor who moves price gets half of the downside (-1).
If both actions are detrimental, which implies both actors are selling, they both get half of the downside (-1). Thus, given two actors, all scenarios of what they could do and the effect on the protocol are shown here:
If one of us stakes and the other one bonds, it is also great because staking takes ZONE off the market and put it into the protocol, while bonding provides liquidity and DAI for the treasury (3 + 1 = 4).
As the protocol controls the funds in its treasury, ZONE can only be minted or burned by the protocol. This also guarantees that the protocol can always back 1 ZONE with 1 DAI. You can easily define the risk of your investment because you can be confident that the protocol will indefinitely buy ZONE below 1 DAI with the treasury assets until no one is left to sell. You can't trust the FED but you can trust the code. As the protocol accumulates more PCV, more runway is guaranteed for the stakers. This means the stakers can be confident that the current staking APY can be sustained for a longer term because more funds are available in the treasury.
Twilight owns most of its liquidity thanks to its bond mechanism. This has several benefits: Twilight does not have to pay out high farming rewards to incentivize liquidity providers a.k.a renting liquidity. Twilight guarantees the market that the liquidity is always there to facilitate sell or buy transaction. By being the largest LP (liquidity provider), it earns most of the LP fees which represents another source of income to the treasury. All POL can be used to back ZONE. The LP tokens are marked down to their risk-free value for this purpose.
Why is the market price of ZONE so volatile?
It is extremely important to understand how early in development the Twilight protocol is. A large amount of discussion has centered around the current price and expected a stable value moving forward. The reality is that these characteristics are not yet determined. The network is currently tuned for expansion of ZONE supply, which when paired with the staking, bonding, and yield mechanics of Twilight, result in a fair amount of volatility. ZONE could trade at a very high price because the market is ready to pay a hefty premium to capture a percentage of the current market capitalization. However, the price of ZONE could also drop to a large degree if the market sentiment turns bearish. We would expect significant price volatility during our growth phase so please do your own research whether this project suits your goals.
What is the point of buying it now when ZONE trades at a very high premium?
When you buy and stake ZONE, you capture a percentage of the supply (market cap) which will remain close to a constant. This is because your staked ZONE balance also increases along with the circulating supply. The implication is that if you buy ZONE when the market cap is low, you would be capturing a larger percentage of the market cap.
Rebase is a mechanism by which your staked ZONE balance increases automatically. When new ZONE are minted by the protocol, a large portion of it goes to the stakers. Because stakers only see staked ZONE balance instead of ZONE, the protocol utilizes the rebase mechanism to increase the staked ZONE balance so that 1 staked ZONE is always redeemable for 1 ZONE.
Reward yield is the percentage by which your staked ZONE balance increases on the next epoch. It is also known as rebase rate. You can find this number on the Twilight staking page.
APY stands for annual percentage yield. It measures the real rate of return on your principal by taking into account the effect of compounding interest. In the case of Twilight ZONE, your staked ZONE represents your principal, and the compound interest is added periodically on every epoch (8 hours) thanks to the rebase mechanism.
One interesting fact about APY is that your balance will grow not linearly but exponentially over time! Assuming a daily compound interest of 2%, if you start with a balance of 1 ZONE on day 1, after a year, your balance will grow to about 1377.
APY = (1 + rewardYield)1095
rewardYield = ZONEdistributed/ZONEtotalStaked
The number of ZONE distributed to the staking contract is calculated from ZONE total supply using the following equation:
ZONEdistributed = ZONEtotalSupply X rewardRate
Note that the reward rate is subject to change.
Why does the price of ZONE become irrelevant in long term?
As illustrated above, your ZONE balance will grow exponentially over time thanks to the power of compounding. Let's say you buy a ZONE for $400 now and the market decides that in 1 year time, the intrinsic value of ZONE will be $2. Assuming a daily compound interest rate of 2%, your balance would grow to about 1377 ZONE by the end of the year, which is worth around $2754. That is a cool $2354 profit! By now, you should understand that you are paying a premium for ZONE now in exchange for a long-term benefit. Thus, you should have a long-time horizon to allow your ZONE balance to grow exponentially and make this a worthwhile investment.
What will be ZONE's intrinsic value in the future?
There is no clear answer for this, but the intrinsic value can be determined by the treasury performance. For example, if the treasury could guarantee to back every ZONE with 100 DAI, the intrinsic value will be 100 DAI. It can also be decided by the future DAO. For example, if the DAO decides to raise the price floor of ZONE, its intrinsic value will rise accordingly.
If there are 100,000 ZONE tokens staked right now, the protocol would need to mint an additional 2,000 ZONE to achieve this daily growth. This is achievable if the protocol can bring in at least 20,000 DAI daily from bond sales. If the protocol fails to achieve this, the APY of 100,000% cannot be guaranteed.
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Atomic structure: Rutherford's Model — lesson. Science CBSE, Class 9.
J. J. Thomson introduced an atom model, which helped in laying the foundation for further research in the atomic model. Significantly the research on the atomic model was done by Rutherford.
Ernest Rutherford was curious about the arrangement of electrons in an atom. He designed an experiment in which alpha particles were used to fall on a thin gold foil. This experiment is also known as the Alpha particle scattering experiment.
Rutherford used alpha particles (also called alpha rays or alpha radiation) as the source.
Alpha particles are helium ions with two charges
Fast-moving particles
Heavier than protons
Symbol is or or
{\mathit{He}}^{2+}
The radioactive source of alpha particles were kept in a lead box with a small hole. Hitting particles were stocked by the box and passed through the hole inside the box.
Why Rutherford used alpha particles for this experiment?
As Thomson proposed, if the atom is a pudding of positive charge with electrons embedded in it, the particles pass straight through it because heavier particles will pass through a lighter pudding structure of the atom.
He chose gold foil because he wanted the layer to be as thin as possible. The thickness of this gold foil was about \(1000 \)atoms.
The detector is a device that allows scientists to see what is happening in an experiment.
Rutherford used a circular fluorescent screen (coated with zinc sulphide, \(ZnS\)) as the detector.
When particles strike it, this detector glows or emits fluorescent light.
The alpha particles that pass through the box's hole are constrained to a straight line.
They hit the gold foil.
The scattered alpha particles were detected by the detector.
Figure \(1\): Scattering of \(α\)-particles by a gold foil
Since most alpha particles passed through the gold foil without being deflected, most of the space inside the atom is empty.
Only a few particles were deflected from their direction, suggesting that the atom's positive charge uses very little space.
Just a small percentage of alpha particles were deflected by \(180°\), showing that the gold atom's positive charge and mass were concentrated in a very small volume within it.
He proposed the nuclear model of the atom based on these observations. This experiment results in a more detailed definition of an atom.
Feature of the atom:
The positive centre of the atom is known as the nucleus.
All the mass is concentrated on the nucleus, around which the electrons circulate in the well-defined orbit, much like planets revolving around the Sun.
The size of the nucleus is less than the atom.
He was known as the ‘Father’ of nuclear physics.
For this theory, he was awarded the Nobel prize for chemistry in the year \(1908\).
Drawbacks of Rutherford's atomic model:
As electrons revolve in orbit, they accelerate and lose energy. After that, they fall into the nucleus. If this happened, the atom would no longer be stable, and the matter would no longer exist in the way we know it. Atoms are known to be very stable. Thus, Rutherford's model failed to explain the stability of the atom.
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One goal of this course will be to review and enhance your algebra skills. Read the Math Notes box for this lesson. Then solve for
x
in each equation below, show all steps leading to your solution, and check your answer.
34x-18=10x-9
\begin{array} \; 34x-18=10x-9\\ {\underline{\; \; 10x \quad \; \; \, = -10x \quad}} \quad \text{Subtract }10x\ \text{from each sides}\\ 24x-18= -9 \end{array}
24x−18=-9\\ {\underline{\qquad +18 = +18}} \quad \text{Add 18 to both sides}\\ \qquad \; 24x=9
24x=9\quad\text{ (Divide both sides by 24)}
x=\frac{9}{24} = \frac{3}{8} = 0.375
4x-5=4x+10
3(x-5)+2(3x+1)=45
3x-15+6x+2=45
-2(x+4)+6=-3
Remember to combine only like terms.
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Comments on tag 00PK—Kerodon
Subsection 2.5.3: The Differential Graded Nerve (cite)
Comment #287 by Markus Zetto on February 25, 2020 at 15:05
There's a small typo in Construction 2.5.3.1: In the formula at the bottom, in the second index, there should be a
i_{k-1}
i_{m-1}
Comment #861 by Wei Wang on February 09, 2021 at 07:05
Possible typo in the proof of Proposition 2.5.3.5. (tag 00PQ) :
Equation (2.25) (tag 00PR) : possible duplicate
(-1)^a
, and the summation might be within the range
0<a<k
1<a<k
In the item "Suppose that the restrction
\alpha\mid_J
is injective ..." : similarly for range of summation
In the next item the tag number of the reference of an equation might be 00PR instead of 00PK
Comment #862 by Kerodon on February 09, 2021 at 19:28
4 comment(s) on Section 2.5: Differential Graded Categories
In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00PK. The letter 'O' is never used.
The tag you filled in for the captcha is wrong. You need to write 00PK, in case you are confused.
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AddPaletteEntry - Maple Help
Home : Support : Online Help : Programming : Document Tools : AddPaletteEntry
add a task to a custom Snippets palette
AddPaletteEntry(entry, palette=palette_name, icon=icon_name)
string ; the name of the task to add to the custom Snippets palette
string ; the name of the Snippets palette that entry will be added to
(optional) string ; the name of the icon to associate with entry
The AddPaletteEntry command adds a task to a Snippets palette. If the palette does not already exist, it will be created using the default option values. See DocumentTools[AddPalette] for information on creating a Snippets palette.
The task added to the palette must first be created and saved as a Task Template.
If the optional icon=icon_name parameter is not provided, a default text icon is used. See DocumentTools[AddIcon] for information on how to create icons for this purpose.
After the task is added to a palette, the associated task template can be inserted into a worksheet by clicking its icon in the palette.
To remove an entry from a palette, use DocumentTools[RemovePaletteEntry].
\mathrm{with}\left(\mathrm{DocumentTools}\right):
The palette entry in this first example will have a "text" icon with the name "Task_1". If the palette "My first palette" does not exist, it will be created and added to the top of the palette dock.
\mathrm{AddPaletteEntry}\left("Task_1",\mathrm{palette}="My first palette"\right)
For the next example, we first create a palette and store an icon for the task, and then add the task with its icon to the palette.
\mathrm{AddPalette}\left("My second palette",\mathrm{position}="bottom"\right)
\mathrm{AddIcon}\left("Task 2 icon",\mathrm{path}="/where/the/icon/file/is.png"\right)
\mathrm{AddPaletteEntry}\left("Task_2",\mathrm{palette}="My second palette",\mathrm{icon}="Task 2 icon"\right)
The DocumentTools[AddPaletteEntry] command was introduced in Maple 16.
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\mathrm{Matrix}\left([[5.,3.,2.],[2.,Float\left(\mathrm{undefined}\right),4.],[Float\left(\mathrm{undefined}\right),5.,1.],[6.,2.,6.],[4.,4.,4.],[1.,3.,2.]]\right)
then the 4th through 6th row of data will be returned. The value
Float\left(\mathrm{undefined}\right)
is used to represent missing data. (Data for time series is converted to floating point data, so any input of type undefined is converted to
Float\left(\mathrm{undefined}\right)
and subsequently considered missing.)
\mathrm{with}\left(\mathrm{TimeSeriesAnalysis}\right):
\mathrm{ts}≔\mathrm{TimeSeries}\left(\mathrm{Matrix}\left([[5.,3.,2.],[2.,Float\left(\mathrm{undefined}\right),4.],[Float\left(\mathrm{undefined}\right),5.,1.],[6.,2.,6.],[4.,4.,4.],[1.,3.,2.]]\right),\mathrm{headers}=["A","B","C"],\mathrm{frequency}=\mathrm{annual}\right)
\textcolor[rgb]{0,0,1}{\mathrm{ts}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{Time series}}\\ \textcolor[rgb]{0,0,1}{\mathrm{A, B, C}}\\ \textcolor[rgb]{0,0,1}{\mathrm{6 rows of data:}}\\ \textcolor[rgb]{0,0,1}{\mathrm{2015 - 2020}}\end{array}]
\mathrm{GetData}\left(\mathrm{ts}\right)
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{5.}& \textcolor[rgb]{0,0,1}{3.}& \textcolor[rgb]{0,0,1}{2.}\\ \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{Float}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{undefined}}\right)& \textcolor[rgb]{0,0,1}{4.}\\ \textcolor[rgb]{0,0,1}{Float}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{undefined}}\right)& \textcolor[rgb]{0,0,1}{5.}& \textcolor[rgb]{0,0,1}{1.}\\ \textcolor[rgb]{0,0,1}{6.}& \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{6.}\\ \textcolor[rgb]{0,0,1}{4.}& \textcolor[rgb]{0,0,1}{4.}& \textcolor[rgb]{0,0,1}{4.}\\ \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{3.}& \textcolor[rgb]{0,0,1}{2.}\end{array}]
\mathrm{ldts}≔\mathrm{LongestDefinedSubsequence}\left(\mathrm{ts}\right)
\textcolor[rgb]{0,0,1}{\mathrm{ldts}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{Time series}}\\ \textcolor[rgb]{0,0,1}{\mathrm{A, B, C}}\\ \textcolor[rgb]{0,0,1}{\mathrm{3 rows of data:}}\\ \textcolor[rgb]{0,0,1}{\mathrm{2018 - 2020}}\end{array}]
\mathrm{GetData}\left(\mathrm{ldts}\right)
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{6.}& \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{6.}\\ \textcolor[rgb]{0,0,1}{4.}& \textcolor[rgb]{0,0,1}{4.}& \textcolor[rgb]{0,0,1}{4.}\\ \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{3.}& \textcolor[rgb]{0,0,1}{2.}\end{array}]
\mathrm{ts2}≔\mathrm{TimeSeries}\left([2.1,\mathrm{undefined},2.5,2.4,\mathrm{undefined},\mathrm{undefined},3.2,2.4]\right)
\textcolor[rgb]{0,0,1}{\mathrm{ts2}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{Time series}}\\ \textcolor[rgb]{0,0,1}{\mathrm{data set}}\\ \textcolor[rgb]{0,0,1}{\mathrm{8 rows of data:}}\\ \textcolor[rgb]{0,0,1}{\mathrm{2013 - 2020}}\end{array}]
\mathrm{ldts2}≔\mathrm{LongestDefinedSubsequence}\left(\mathrm{ts2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{ldts2}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{Time series}}\\ \textcolor[rgb]{0,0,1}{\mathrm{data set}}\\ \textcolor[rgb]{0,0,1}{\mathrm{2 rows of data:}}\\ \textcolor[rgb]{0,0,1}{\mathrm{2015 - 2016}}\end{array}]
\mathrm{GetData}\left(\mathrm{ldts2}\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{2.50000000000000}\\ \textcolor[rgb]{0,0,1}{2.40000000000000}\end{array}]
|
Fejér and Hermite-Hadamard Type Inequalities for Harmonically Convex Functions
Feixiang Chen, Shanhe Wu, "Fejér and Hermite-Hadamard Type Inequalities for Harmonically Convex Functions", Journal of Applied Mathematics, vol. 2014, Article ID 386806, 6 pages, 2014. https://doi.org/10.1155/2014/386806
Feixiang Chen 1 and Shanhe Wu 2
1School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou, Chongqing 404000, China
Academic Editor: Yu-Ming Chu
We establish a Fejér type inequality for harmonically convex functions. Our results are the generalizations of some known results. Moreover, some properties of the mappings in connection with Hermite-Hadamard and Fejér type inequalities for harmonically convex functions are also considered.
Let be a convex function and with ; then
Inequality (1) is known in the literature as the Hermite-Hadamard inequality. Fejér [1] established the following weighted generalization of inequality (1).
Theorem 1. If is a convex function, then the following inequality holds: where is positive, integrable, and symmetric with respect to .
Some generalizations, refinements, variations, and improvements of inequalities (1) and (2) were investigated by Wu [2], Chen and Liu [3], Sarikaya and Ogunmez [4], and Xiao et al. [5], respectively.
In [6], Dragomir proposed an interesting Hermite-Hadamard type inequality which refines the left hand side of inequality of (1) as follows.
Theorem 2 (see [6]). Let be a convex function defined on . Then is convex, increasing on , and for all , one has where
An analogous result for convex functions which refines the right hand side of inequality (1) was obtained by Yang and Hong in [7] as follows.
Yang and Tseng in [8] established the following Fejér type inequalities, which is the generalization of inequalities (3) and (5) as well as the refinement of the Fejér inequality (2).
Theorem 4 (see [8]). If is convex on , is positive, integrable, and symmetric about . Then and are convex, increasing on , and for all , one has where
In [9, 10], İşcan and Wu gave the definition of harmonic convexity as follows.
Definition 5. Let be a real interval. A function is said to be harmonically convex if for all and . If the inequality in (10) is reversed, then is said to be harmonically concave.
The following Hermite-Hadamard inequality for harmonically convex functions holds true.
Theorem 6 (see [9]). Let be a harmonically convex function and with . If , then one has
In [10], İşcan and Wu established the following Hermite-Hadamard inequalities for harmonically convex functions via the Riemann-Liouville fractional integral.
Theorem 7 (see [10]). Let be a function such that , where with . If is a harmonically convex function on , then the following inequalities for fractional integrals hold: where and .
The Riemann-Liouville fractional integrals and of order with are defined by where is the Gamma function defined by .
In this paper, we establish a Fejér type inequality for harmonically convex functions; our main result includes, as special cases, the inequalities given by Theorems 6 and 7. Moreover, we investigate some properties of the mappings in connection to Hermite-Hadamard and Fejér type inequalities for harmonically convex functions.
2. Fejér Type Inequality for Harmonically Convex Functions
The following Fejér inequality for harmonically convex functions holds true.
Theorem 8. Let be a harmonically convex function and with . If , then one has where is nonnegative and integrable and satisfies
Proof. Since is a harmonically convex function on , we have, for all , Choosing and , we have Since is nonnegative and satisfies the condition of (15), we obtain Integrating both sides of the above inequalities with respect to over , we obtain The proof of Theorem 8 is completed.
Remark 9. Putting in Theorem 8, we obtain inequality (11).
Remark 10. Choosing in Theorem 8, it is easy to observe that .
Since where , which implies that inequality (14) can be transformed to inequality (12) under an appropriate selection of .
Remark 11. In Theorem 8, taking , where , is nonnegative, integrable, and symmetric with respect to . Then inequality (14) becomes
3. Some Mappings in connection with Hermite-Hadamard and Fejér Inequalities for Harmonically Convex Functions
Lemma 12. Let be a harmonically convex function and with , and let . Then is convex, increasing on , and for all ,
Proof. Firstly, for , we have and hence is convex on .
Next, if , it follows from the harmonic convexity of that
It is easy to observe that
Thus inequality (24) holds.
Finally, for , since is convex, it follows from (24) that and hence, , which means that is increasing on . This completes the proof of Lemma 12.
Theorem 13. Let be a harmonically convex function and with . If and is defined by then is convex and increasing on , and
Proof. It follows from Lemma 12 that is convex and increasing on . Hence is convex and increasing on . Further, inequality (30) can be deduced from (24). Theorem 13 is proved.
Proof. We note that if is convex and is linear, then the composition is convex. It follows from Lemma 12 that and are increasing on and , respectively. Hence, is convex and increasing on . We infer that is convex and increasing on . Furthermore, inequality (33) follows directly from (24). The proof of Theorem 14 is completed.
Theorem 15. Let be a harmonically convex function and with . If and is defined by where is nonnegative and integrable and satisfies the condition of (15), then is convex and increasing on , and
Proof. From Lemma 12 we obtain that is convex and increasing on . Since is nonnegative and satisfies , it follows that is convex and increasing on , while inequality (37) can be deduced from (24). Theorem 15 is proved.
Proof. By using the same method as in the proof of Theorem 14, we obtain from Lemma 12 that is convex and increasing on . Since is nonnegative and satisfies , we deduce that is convex and increasing on . Inequality (40) follows from (24) and the identity
Remark 17. If we put in inequalities (37) and (40), respectively, we obtain the refined versions of inequality (12).
The present investigation was supported, in part, by the Youth Project of Chongqing Three Gorges University of China (no. 13QN11) and, in part, by the Foundation of Scientific Research Project of Fujian Province Education Department of China (no. JK2012049).
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-convex functions,” Journal of Applied Mathematics, vol. 2013, Article ID 978493, 5 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet
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Copyright © 2014 Feixiang Chen and Shanhe Wu. 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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A Note on Lax Pairs of the Sawada-Kotera Equation
Sergei Sakovich, "A Note on Lax Pairs of the Sawada-Kotera Equation", Journal of Mathematics, vol. 2014, Article ID 906165, 4 pages, 2014. https://doi.org/10.1155/2014/906165
Sergei Sakovich 1
1Institute of Physics, National Academy of Sciences of Belarus, 220072 Minsk, Belarus
We prove that the new Lax pair of the Sawada-Kotera equation, discovered recently by Hickman, Hereman, Larue, and Göktaş, and the well-known old Lax pair of this equation, considered in the form of zero-curvature representations, are gauge equivalent to each other if and only if the spectral parameter is nonzero, while for zero spectral parameter a nongauge transformation is required.
Recently, the following interesting result was obtained by Hickman et al. [1]. It turned out that the Sawada-Kotera equation [2, 3] possesses two different Lax representations in the operator form where subscripts of the scalar functions and denote respective derivatives, and are linear differential operators expressed in powers of the derivative operator , and is the spectral parameter. The first Lax pair, given by the operators is well known [4, 5]. The second Lax pair, given by the operators is new, in the sense that it appeared in [1] for the first time in the literature.
Many experts, according to their private communications, noticed that the second Lax pair (4) is related to the first Lax pair (3) by the transformation where the dagger denotes the Hermitian conjugate. This transformation (5) always turns a Lax pair of an integrable equation into a Lax pair of the same equation, but usually the resulting Lax pair has essentially the same form as the original one (we believe that for this reason no second Lax pair was discovered in [1] for the Kaup-Kupershmidt equation, in particular). Let us note, however, that the Lax pairs (3) and (4) are different in form. Some other experts, also according to their private communications, noticed that the old Lax pair (3) and the new one (4) are related to each other by the transformation which corresponds to the transformation made in (2). Thus, there exist (at least) two different ways to relate the Lax pairs (3) and (4) to each other, and we believe that this point deserves further investigation using more general description of Lax pairs than their operator form.
In the present paper, we study these two Lax pairs of the Sawada-Kotera equation (1)—the old one, (2) with (3), and the new one, (2) with (4)—in the matrix form or, what is the same, in the form of zero-curvature representations (ZCRs) where is a three-component column vector, and are matrices, and the square brackets denote the matrix commutator. In Section 2, we show that, for any nonzero value of the spectral parameter, the new Lax pair of the Sawada-Kotera equation and the old one are related to each other by a gauge transformation of ZCRs where is a matrix. In Section 3, we show that, for any value of the spectral parameter including zero, the new Lax pair and the old one are related to each other by a gauge transformation (9) combined with a different type of equivalence transformations of ZCRs (8); namely, where the tilde denotes the matrix transpose. Section 4 contains concluding remarks.
We use computationally effective techniques, such as the method of gauge-invariant description of ZCRs, developed in [6, 7] independently, and the method of cyclic bases of ZCRs [7, 8], and follow the terminology and notations adopted in [8].
2. Nonzero Spectral Parameter
Introducing the three-component column vector we can rewrite the Lax pairs (2) with the operators (3) and (4) in their matrix form (7). The old Lax pair of the Sawada-Kotera equation, determined by the operators (3), corresponds to the ZCR (8) with the matrices where denotes the spectral parameter, and The new Lax pair of the Sawada-Kotera equation, (2) with (4), corresponds to the ZCR (8) with the matrices where stands for the spectral parameter, is given by (13), and We have changed the notation for the spectral parameter in (14) because no relation between the parameters of (12) and (14) is assumed initially.
Let us compute the cyclic bases [7, 8] of the ZCRs (8) with the matrices (12) and (14), in order to see if there are any obstacles to relate these two ZCRs by a gauge transformation
For the matrix given by (12) with a nonzero spectral parameter , we find that the cyclic basis is eight-dimensional, consisting of the matrices , where is the characteristic matrix, and the covariant derivative is defined by the relation with any matrix . The closure equation of the cyclic basis, has the following coefficients in this case: where is given by (13).
For the matrix (12) with , we get quite a different situation. In this case, the dimension of the cyclic basis is five, not eight. The closure equation has the coefficients
For the matrix (14), which contains , the characteristic matrix is computed in the following, more general, way: where the covariant derivative is defined by the relation with any matrix . The cyclic basis for the matrix has the dimension if and if —the same dimensions as for the matrix . The coefficients of closure equations in the case of are given by the expressions (19), after the replacement , for , and by the expressions (21) for —the same expressions as for the matrix . Taking into account that the dimensions of cyclic bases and the coefficients of closure equations are gauge invariants, we see that the only obstacle for the existence of a gauge transformation (16) we have found so far is the condition . This makes sense to try to find the matrix of (16) explicitly.
It is very convenient to make use of the fact that, under the gauge transformation (16), the characteristic matrix and its covariant derivatives transform as tensors [6, 7]; namely, Denoting the elements of the matrix as , , we find from the relation that Next, we find from the relation that Then, the relation leads us to At this point, we can immediately conclude that the conditions and hold necessarily because . Finally, we get directly from (16), that is, with any nonzero constant , and obtain With the natural choice of in (31), we have , and the inverse matrix does not exist for . Of course, one can take and get , but in this case the matrix does not exist for . As we have already pointed out above, the condition is necessary for the existence of the gauge transformation sought.
Consequently, the two considered ZCRs with the matrices and given by (12) and (14) are related to each other by the gauge transformation (16) if and only if , and the corresponding matrix is given by (31), where one can take without loss of generality. One can see easily from (9), (11), and (31) that this gauge transformation corresponds to the transformation (6) between the Lax pairs considered in their operator form. Another way to see this consists in taking into account that in (31) with is identical to in (12), and therefore we have in (9) owing to (7).
Let us note that it is a new, interesting, and quite surprising phenomenon that two ZCRs containing an essential parameter are related to each other by a gauge transformation for all values of the parameter except one value and no gauge transformation exists between those ZCRs for that single value of the parameter.
3. Arbitrary Spectral Parameter
Besides gauge transformations (9), there is a different—quite evident but rarely mentioned in the literature—nongauge type of equivalence transformations of ZCRs (8), namely, the transformation (10). Let us try to make use of a combination of transformations (9) and (10) to relate the two ZCRs given by (12) and (14) to each other.
The problem is to find a matrix such that where Since the gauge invariants of the cyclic basis in the case of coincide with the ones of , we omit their consideration and proceed directly to the analysis of the relations , , where is defined by for any matrix , and . From the relation , we find for the elements of the matrix the following: and . Next, we find from the relation that and . Then, the relation leads us to and , where in order to have . Finally, we get directly from (32), set without loss of generality, and obtain
Consequently, the two considered ZCRs with the matrices and given by (12) and (14) are related to each other by the combination of transformations (32) and (33) if and only if , and the corresponding matrix is given by (34). The case of zero spectral parameter is included now. Let us note that we were forced to use the nongauge transformation (10), which is evidently a counterpart of the transformation (5), in order to cover the case of zero spectral parameter, because the two studied ZCRs belong to two distinct classes of gauge equivalence if the spectral parameter is zero.
In this paper, using the method of gauge-invariant description of zero-curvature representations (ZCRs) and the method of cyclic bases of ZCRs, we have shown that the new Lax pair of the Sawada-Kotera equation, discovered recently by Hickman, Hereman, Larue, and Göktas, and the well-known old Lax pair of this equation, considered in the form of ZCRs, are gauge equivalent to each other if and only if the spectral parameter is nonzero, while for zero spectral parameter a nongauge transformation is required. As a by-product, we have obtained an interesting example of two ZCRs which share the same set of gauge invariants but cannot be related to each other by a gauge transformation.
The author is grateful to Ziemowit Popowicz, Takayuki Tsuchida, Allan Fordy, and anonymous reviewers for valuable comments.
M. Hickman, W. Hereman, J. Larue, and Ü. Göktaş, “Scaling invariant Lax pairs of nonlinear evolution equations,” Applicable Analysis, vol. 91, no. 2, pp. 381–402, 2012. View at: Publisher Site | Google Scholar | MathSciNet
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Copyright © 2014 Sergei Sakovich. 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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How We Proved the Eth2 Deposit Contract Is Free of Runtime Errors | ConsenSys
In the near future, Ethereum will transition to a new Proof of Stake consensus protocol. Instead of the miners used today, Ethereum 2.0, relies on validators to create and validate the blocks of transactions that are added to the distributed ledger. The protocol is designed to be fault-tolerant to up to
1/3
of Byzantine (i.e. malicious or dishonest) validators. Economic staking mechanisms are in place to try and keep dishonest validators under
1/3
: validators have to stake some ETH, and if they are dishonest they may be slashed and lose their stake. The process of staking ETH is handled by the Beacon Chain staking deposit contract on the Ethereum network. A validator sends a transaction to “deposit some Ether” by calling the deposit function of the contract. The deposit contract was deployed in November 2020 and, at the time of writing, has topped 7,157,820 ETH ($25 billion USD) in deposits.
This blog post is based on this research paper, an extended abstract of which is to be published at the 24th international symposium on Formal Methods 2021.
Why should we formally verify the Deposit Contract?
The Eth2 deposit contract is a mission-critical component of the Ethereum 2.0 consensus mechanism, also known as the Beacon Chain. Any bugs in the deposit contract could result in inaccurate tracking of deposits, missing registered validators, or cause crashes and downtimes. Any of these compromise the integrity and availability of Ethereum.
Bugs can trigger runtime errors like division-by-zero or array-out-of-bounds. In a networked environment these vulnerabilities can be exploited by attackers to disrupt or take control of the computer system. Other types of bugs can also compromise the business logic of a system e.g., an implementation may contain subtle errors that make the system deviate from its intended specifications (for instance using
\text{\verb|+=|}
in C/C++ instead of
\text{\verb|=+|}
; this PVS-studio link has more examples in C++).
It is hard to guarantee that programs and smart contracts implement their intended business logic, have no common runtime errors, and terminate properly. There are notorious examples of smart contract vulnerabilities that have been exploited and publicly reported. In 2016, a reentrance vulnerability in the Decentralized Autonomous Organization (DAO) smart contract was exploited to steal more than $50 Million.
The deposit contract implements the functions, deposit and get_deposit_root. These rely on sophisticated data structures and algorithms to record the list of deposits so that they can be communicated efficiently over the network. The history of deposits is summarized as a unique number, a hash, computed using a Merkle (or Hash) tree. The tree is built (or updated) incrementally after each deposit using the elegant incremental Merkle tree algorithm (Progressive merkle tree, Vitalik Buterin). The algorithm is efficient and concise but as pointed out previously:
The efficient incremental algorithm leads to the deposit contract implementation being unintuitive, and makes it non-trivial to ensure its correctness.
Park, D., Zhang, Y., Rosu, G.: End-to-end formal verification of ethereum 2.0 deposit smart contract. CAV 2020.
Why not write tests?
To catch bugs in the deposit contract, testing is probably the first thing to do. While testing the contract (e.g. with random inputs, or property-based testing) is useful and can detect bugs, it has some severe limitations:
Unlike testing, program verification aims at proving the absence of bugs. The idea is to use rigorous mathematical and logical reasoning about programs, more precisely about implementations vs specifications. One such logical framework was proposed by Floyd and Hoare in the 1960s. It creates a distinction between how the program computes the result (its implementation) and what the expected result of the program is (the specification). Program, implementations are typically written in a standard programming language e.g., a list-sorting algorithm in Java, and the specification as a logical (first-order logic) statement e.g., the list is sorted.
Automated program verification with Dafny
Program correctness can be precisely stated using so-called Hoare triples of the form:
\{ P \} \quad \textit{Prog} \quad \{ Q \}
Such a triple is valid if and only if: assuming the pre-condition
P
holds before program
\textit{Prog}
is executed, then the post-condition
Q
is guaranteed to hold when
\textit{Prog}
terminates. Floyd logic provides logical rules that can be used to prove that a Hoare triple is valid. Note that there is a termination condition in the previous statement. A triple does not capture the property that a program necessarily terminates. If we want to prove termination, say for a program with a single while loop, there are ranking functions. A ranking function
f
is a measure computed using the program variables, that strictly decreases every time the loop body is executed. If the function
f
takes its values in a well-founded set (in which there are no infinite strictly decreasing sequences of values), the existence of a ranking function is a proof of termination for the while loop.
Proving correctness and termination of programs with Floyd logic and ranking functions used to be a long and tedious process: one had to follow Floyd logic rules and write lengthy (ironically error-prone) pen and paper proofs.
Recently, a lot of progress has been made to improve the verification experience and its soundness. Dafny is a verification-friendly programming language in which you can write logical pre- and post-conditions, functional or object-oriented programs. Dafny automates the verification of Hoare triples: it checks every single step of a proof which greatly reduces the likelihood of reasoning flaws. For example, in a proof by induction the Dafny verifier checks that 1) the base case is valid, 2) the induction step is valid and 3) the induction is well-founded. In Dafny, the verifier also checks the programs for standard runtime errors like division-by-zero, array-out-of-bounds.
This blog post by my colleague Joanne Fuller provides a quick introduction to what you can do with Dafny.
So what does the proof look like?
The algorithms in the deposit contract simulate the computation of a synthesised attribute, hash, on a binary tree (Merkle tree) without building the tree. As Merkle trees are perfect (all levels are full), computing a synthesised attribute on such a tree takes exponential time in the height
h
of the tree: all the nodes of the tree have to be processed and a Merkle tree has
2^{h + 1} - 1
In our proof of the deposit contract, we followed a standard dynamic programming technique to derive recursive linear-time and space algorithms to compute the attribute on a tree. We prove that these recursive algorithms are correct in the sense that they compute the same value as if we had built a Merkle tree.
The details of the proofs (including accompanying paper and video presentations) can be found in this GitHub repository. At a high-level, the proof shows that some invariants are preserved by the functions of the deposit contract. These invariants guarantee:
The absence of runtime errors: like division-by-zero, over/underflows and array-out-of-bounds (accessing elements in an array outside of the range of indices of the array).
Functional correctness: the values returned by the functions in the contract are the same as the values that would be returned if we had built a Merkle tree.
Finally, we show that the functions implemented in the deposit contract (proposed in Progressive merkle tree, Vitalik Buterin) compute the same values as the recursive algorithms. By (provable and Dafny-checked) transitivity this shows that the deposit contract is correct.
Writing a Dafny proof
The sample code below (Algo 1) is a slightly simplified version of the Dafny source code of get_deposit_root . It illustrates how correctness and termination can be specified in Dafny. The pre-conditions are written using requires and the post-conditions using ensures. The predicate Valid is an invariant (it is a pre- and post-condition of get_deposit_root) and captures some essential properties that are needed to prove the contract correctness. The post-condition at line 4 specifies that the result computed by get_deposit_root is the same as the one that would be obtained if we had build a Merkle tree. The absence of array-out-of-bounds errors in dereferencing the array branch (lines 19 and 21) are proved using the loop invariants (lines 12 and 13): the Dafny verifier checks that these are indeed loop invariants and this in turn ensures that in the loop body, the value of h is always a valid array index.
Algo 1: The get_deposit_root function is Dafny.
The ranking function (line 15) uses the fact that the variable h strictly increases in the loop body (line 24) and, as TREE_HEIGHT is constant, the difference TREE_HEIGHT - h strictly decreases. Together with the invariant at line 12 that proves TREE_HEIGHT - h >= 0, the value of TREE_HEIGHT - h:
strictly decreases at each iteration of the loop and
is bounded from below
which imply that the while loop at line 10 always terminates.
How hard is it to design a proof?
Program verification is challenging because it is not always possible to automatically synthesise a proof for a given Hoare triple. I had to create many loop invariants and other lemmas to design a proof of correctness (and termination) for the deposit contract. The nice thing is they are all written in Dafny!
In a single language we can write programs, specifications, and proofs (as programs). You only need the Dafny verifier and the deposit contract Dafny source code to check any of the proofs and to reproduce the verification results.
It took me roughly 12 weeks to first write a sketch pen and paper proof and then write all the details and proofs in Dafny. The code base has approximately 3500 lines of code, 90% of which is proofs. The proofs are written as (functional) programs, loop invariants or theorems (lemmas in Dafny). The Dafny verifier checks the proofs but can also figure out some steps automatically. There are several strategies or heuristics that can be applied by the verifier to synthesise tedious steps of a proof for instance induction, or reasoning about integers.
We have designed a machine-checkable correctness (and termination) proof for the Solidity version of the deposit contract. The proof ensures that the deposit contract is correct and free of runtime errors. We have also identified some (provably correct) new optimizations of some of the functions of the contract (see this explanation). We hope that this work can serve as a guide for new smart contract verification projects.
If you are interested in this project, feel free to contact us to discuss how the Trustworthy Smart Contracts’s team at ConsenSys might help you to design efficient, bug-free reliable smart contracts, email myself and Joanne Fuller.
Eth2Ethereum 2.0R&D
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The day before Gerardo returned from a two-week trip, he wondered if he left his plants inside his apartment or outside on his deck. He knows these facts:
If his plants are indoors, he must water them at least once a week or they will die.
If he leaves his plants outdoors and it rains, then he does not have to water them. Otherwise, he must water them at least once a week or they will die.
It has not rained in his town for
2
When Gerardo returns, will his plants be dead? Explain your reasoning.
First consider each scenario separately.
If he left his plants indoors for two weeks what would happen?
If he left his plants outdoors for two weeks, and it did not rain, what would happen?
Reasoning and Answer:
If his plants are indoors... They will be dead because he did not water them once a week.
If his plants are outdoors... They will be dead because it did not rain.
Yes, Gerardo's plants will be dead.
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Subtropics - Wikipedia
Areas of the world with subtropical climates according to Köppen climate classification
The subtropics and tropics
Subtropical climates are often characterized by hot summers and mild winters with infrequent frost. Most subtropical climates fall into two basic types: humid subtropical (Koppen climate Cfa), where rainfall is often concentrated in the warmest months, but not particularly, for example Southeast China and the Southeastern United States, and dry summer or Mediterranean climate (Koppen climate Csa/Csb), where seasonal rainfall is concentrated in the cooler months, such as the Mediterranean Basin or Southern California.
Subtropical climates can also occur at high elevations within the tropics, such as in the southern end of the Mexican Plateau and in the Vietnamese Highlands. The six climate classifications use the term to help define the various temperature and precipitation regimes for planet Earth.
A great portion of the world's deserts are located within the subtropics, due to the development of the subtropical ridge from the 30s latitudes. Areas bordering warm oceans (typically on the southeast sides of continents) are prone to locally heavy rainfall in the summers from tropical cyclones, which can contribute a significant percentage of the annual rainfall. Areas bordering cool oceans (typically on the southwest sides of continents) are prone to fog, aridity, and dry summers. Plants such as palms, citrus, mango, pistachio, lychee, and avocado are grown in the subtropics.
4.1 Humid subtropical climate
4.2 Mediterranean climate
4.3 Semi-desert/desert climate
See also: List of locations with a subtropical climate
The tropics have been historically defined as lying between the Tropic of Cancer and Tropic of Capricorn, located at latitudes 23°26′11.0″ (or 23.43638°) north and south, respectively.[1] According to the American Meteorological Society, the poleward fringe of the subtropics is located at latitudes approximately 35° north and south, respectively.[2]
Homes in Charleston, South Carolina along The Battery
Several methods have been used to define the subtropical climate depending on the climate system used.
The most well known[3] is the Trewartha climate classification, which defines a subtropical region as one that has at least eight months with a mean temperature greater than 10 °C (50.0 °F) and at least one month with a mean temperature under 18 °C (64.4 °F).[4] In most regions in this climate zone the coldest month has a mean temperature of above 7 °C (45 °F) and the hottest month has a mean temperature of above 24 °C (75 °F) . In the Trewartha climate classification, most of these climates are located in the southernmost portions of the temperate zone (latitudes between 23.5° and 35° north and south), aka the subtropics.
German climatologists Carl Troll and Karlheinz Paffen defined warm temperate zones as plain and hilly lands having an average temperature of the coldest month between 2 °C (35.6 °F) and 13 °C (55.4 °F) in the Northern Hemisphere and between 6 °C (42.8 °F) and 13 °C (55.4 °F) in the Southern Hemisphere, excluding oceanic and continental climates. According to the Troll-Paffen climate classification, there generally exists one large subtropical zone named the warm-temperate subtropical zone,[5] which is subdivided into seven smaller areas.[6]
According to the E. Neef climate classification, the subtropical zone is divided into two parts: rainy winters of the west sides and eastern subtropical climate.[7] According to the Wilhelm Lauer & Peter Frankenberg climate classification, the subtropical zone is divided into three parts: high-continental, continental, and maritime.[8] According to the Siegmund/Frankenberg climate classification, subtropical is one of six climate zones in the world.[9]
Leslie Holdridge defined the subtropical climates as having a mean annual biotemperature between the frost line or critical temperature line, 16 °C to 18 °C (depending on locations in the world), and 24 °C.[10] The frost line separates the warm temperate region from the subtropical region. It represents the dividing line between two major physiological groups of evolved plants. On the warmer side of the line, the majority of the plants are sensitive to low temperatures. They can be killed back by frosts as they have not evolved to withstand periods of cold. On the colder temperate side of the line, the total flora is adapted to survive periods of variable length of low temperatures, whether as seeds in the case of the annuals or as perennial plants that can withstand the cold. The 16 °C–18 °C segment is often "simplified" as 17 °C
{\textstyle {\bigl (}2^{(\log _{2}12\ +\ 0.5)}\ ^{\circ }\!\mathrm {C} \approx 16.97\ ^{\circ }\!\mathrm {C} {\bigr )}}
The Holdridge subtropical climates straddle more or less the warmest subtropical climates and the less warm tropical climates as defined by the Köppen-Geiger or Trewartha climate classifications.
However Wladimir Köppen has distinguished the hot or subtropical and tropical (semi-)arid climates (BWh or BSh) having an average annual temperature greater than or equal to 18 °C (64.4 °F) from the cold or temperate (semi-)arid climates (BWk or BSk) whose annual temperature average is lower.[12] This definition, though restricted to dry regions, is almost similar to Holdridge's.
See also: Earth rainfall climatology, Subtropical ridge, Tropical cyclone, and Wet season
Hadley cells located on the Earth's atmospheric circulation.
Heating of the earth by the sun near the equator leads to large amounts of upward motion and convection winds along the monsoon trough or intertropical convergence zone. The upper-level divergence over the near-equatorial trough leads to air rising and moving away from the equator aloft. As the air moves towards the mid-latitudes, it cools, gets denser and sinks, which leads to subsidence near the 30th parallel of both hemispheres. This circulation is known as the Hadley cell and leads to the formation of the subtropical ridge.[13] Many of the world's deserts are caused by these climatological high-pressure areas,[14] located within the subtropics. This regime is known as a semiarid/arid subtropical climate, which is generally located in areas adjacent to powerful cold ocean currents. Examples of this climate are the coastal areas of Southern Africa and the west coast of South America.[15]
The humid subtropical climate is often located on the western side of the subtropical high. Here, unstable tropical airmasses in summer bring convective overturning and frequent tropical downpours, and summer is normally the season of peak annual rainfall. In the winter (dry season) the monsoon retreats, and the drier trade winds bring more stable airmass and often dry weather, and frequent sunny skies. Areas that have this type of subtropical climate include Australia, Southeast Asia, and parts of South America.[16][17][18] In areas bounded by warm ocean like the southeastern United States and East Asia, tropical cyclones can contribute significantly to local rainfall within the subtropics.[19] Japan receives over half of its rainfall from typhoons.[20]
The Mediterranean climate is a subtropical climate with a wet season in winter and a dry season in the summer. Regions with this type of climate include the rim lands of the Mediterranean Sea, southwestern Australia, parts of the west coast of South America around Santiago, the coastal areas of the lower west coast of the United States. [21][22][23][24]
Live oak with araucarias in Curitiba, Brazil
These climates do not routinely see hard freezes or snow due to winter on average being above freezing, which allows plants such as palms and citrus to flourish.[25][26] As one moves toward the tropical side the slight winter cool season disappears, while at the poleward threshold of the subtropics the winters become cooler. Some crops which have been traditionally farmed in tropical climates, such as mango, litchi, and avocado, are also cultivated in the subtropics. Pest control of the crops is easier than in the tropics, due to the cooler winters.[27]
Tree ferns (pteridophytes) are grown in subtropical areas, as are dracaena and yucca, and trees in the Taxaceae. Apple, pear and pomegranate also grow well in the subtropics.[28]
Humid subtropical climateEdit
Wetland Park in Hong Kong.
Main article: Humid subtropical climate
The humid subtropical climate is a subtropical climate type characterized by hot, humid summers and generally warm to cool winters, though climates at the edges of the zone can feature cold winters. In summer, the subtropical high pressure cells provide a sultry flow of tropical air with high dew points, and thundershowers are typical, though brief. Normally, rainfall is concentrated in the warmest months of the year. With decreasing latitude most humid subtropical climates typically have drier winters and wetter summers. Tropical lows and weakening tropical storms often contribute to seasonal rainfall in most humid subtropical climates.
Mediterranean climateEdit
Main article: Mediterranean climate
The Mediterranean climate regime resembles the climate of the lands in the Mediterranean Basin, parts of lower West Coast of the United States, parts of Western and South Australia, in southwestern South Africa and in parts of central Chile. The climate is characterized by hot dry summers and cooler winters with rainfall.[30] In Europe, the northernmost mediterranean climates are found along the Italian Riviera, located at 44° latitude. Parts of southwestern Australia around Perth have a Mediterranean climate as do areas around coastal South Africa.
Semi-desert/desert climateEdit
Acacia in HaMakhtesh HaGadol, Negev Desert
Main articles: Desert climate and Semi-arid climate
According to Köppen, arid subtropical climates are characterized by an annual average temperature above 18 °C (64.4 °F), the absence of regular rainfall, and high humidity.[15]
Source: AEdM
Wikimedia Commons has media related to Subtropics.
^ I. G. Sitnikov. "1" (PDF). Principal Weather Systems in Subtropical and Tropical Zones. Vol. 1. Encyclopedia of Life Support Systems.
^ Glossary of Meteorology (25 April 2012). "Subtropics". American Meteorological Society. Retrieved 24 March 2013.
^ Arise, Lotus (27 January 2021). "Trewartha Climatic Classification - UPSC (Climatology)". Retrieved 26 March 2022.
^ Belda et al. Climate classification revisited: from Köppen to Trewartha. In: Climate Research Vol. 59: 1–13, 2014.
^ Climatic map by Istituto Geografico De Agostini, according to Troll-Paffen climate classification Archived 4 October 2012 at the Wayback Machine
^ Die Klimaklassifikation nach Troll / Paffen – klimadiagramme.de
^ Dr. Owen E. Thompson (1996). Hadley Circulation Cell. Archived 5 March 2009 at the Wayback Machine Channel Video Productions. Retrieved on 11 February 2007.
^ ThinkQuest team 26634 (1999). The Formation of Deserts. Archived 17 October 2012 at the Wayback Machine Oracle ThinkQuest Education Foundation. Retrieved on 16 February 2009.
^ a b "Tropical and subtropical desert climate".
^ Susan Woodward (2 February 2005). "Tropical Savannas". Radford University. Archived from the original on 25 February 2008. Retrieved 16 March 2008.
^ Randy Lascody (2008). The Florida Rain Machine. National Weather Service. Retrieved on 6 February 2009.
^ John J. Stransky (1 January 1960). "Site Treatments Have Little Effect During Wet Season in Texas". Tree Planters' Notes. 10 (2).
^ Geoffrey John Cary; David B. Lindenmayer; Stephen Dovers (2003). Australia Burning: Fire Ecology, Policy and Management Issues. Csiro Publishing. p. 33. ISBN 978-0-643-06926-8.
^ Whipple, Addison (1982). Storm. Alexandria, VA: Time Life Books. p. 54. ISBN 978-0-8094-4312-3.
^ Remote Sensing for Migratory Creatures (2002). Phenology and Creature Migration: Dry season and wet season in West Mexico. Arizona Remote Sensing Center. Retrieved on 6 February 2009.
^ J. Horel (2006). Normal Monthly Precipitation, Inches. Archived 13 November 2006 at the Wayback Machine University of Utah. Retrieved on 19 March 2008.
^ D. Bozkurt, O.L. Sen and M. Karaca (2008). Wet season evaluation of RegCM3 performance for Eastern Mediterranean. EGU General Assembly. Retrieved on 6 February 2009.
^ Ron Kahana; Baruch Ziv; Yehouda Enzel & Uri Dayan (2002). "Synoptic Climatology of Major Floods in the Negev Desert, Israel" (PDF). International Journal of Climatology. 22 (7): 869. Bibcode:2002IJCli..22..867K. doi:10.1002/joc.766. Archived from the original (PDF) on 19 July 2011.
^ Walter Tennyson Swingle (1904). The Date Palm and its Utilization in the Southwestern States. United States Government Printing Office. p. 11.
^ Wilson Popenoe (1920). "Manual of Tropical and Subtropical Fruits: Excluding the Banana, Coconut, Pineapple, Citrus Fruits, Olive, and Fig". Nature. 108 (2715): 7. Bibcode:1921Natur.108Q.334.. doi:10.1038/108334a0. hdl:2027/hvd.32044106386147. Retrieved 24 March 2013.
^ Galán Saúco, V. Robinson, J. C., Tomer, E., Daniells, J. (2010). "S18.001: Current Situation and Challenges of Cultivating Banana and other Tropical Fruits in the Subtropics" (PDF). 28th International Horticultural Congress. Archived from the original (PDF) on 1 May 2013. Retrieved 24 March 2013. {{cite web}}: CS1 maint: multiple names: authors list (link)
^ R. K. Kholi; D. R. Batish & H. B. SIngh. "Forests and Forest Plants Volume II – Important Tree Species" (PDF). Encyclopedia of Life Support Systems. Retrieved 9 April 2013.
^ Michael Ritter (24 December 2008). "Mediterranean or Dry Summer Subtropical Climate". University of Wisconsin–Stevens Point. Archived from the original on 5 August 2009. Retrieved 17 July 2009.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Subtropics&oldid=1088439740"
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Reciprocity law - Wikipedia
(Redirected from Reciprocity law (mathematics))
Mathematical law, a generalization of quadratic reciprocity
For various other concepts of reciprocity and reciprocity laws, see Reciprocity (disambiguation).
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials
{\displaystyle f(x)}
with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an irreducible polynomial
{\displaystyle f(x)=x^{2}+ax+b}
splits into linear terms when reduced mod
{\displaystyle p}
. That is, it determines for which prime numbers the relation
{\displaystyle f(x)\equiv f_{p}(x)=(x-n_{p})(x-m_{p}){\text{ }}({\text{mod }}p)}
holds. For a general reciprocity law[1]pg 3, it is defined as the rule determining which primes
{\displaystyle p}
{\displaystyle f_{p}}
splits into linear factors, denoted
{\displaystyle {\text{Spl}}\{f(x)\}}
There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a power residue symbol (p/q) generalizing the quadratic reciprocity symbol, that describes when a prime number is an nth power residue modulo another prime, and gave a relation between (p/q) and (q/p). Hilbert reformulated the reciprocity laws as saying that a product over p of Hilbert norm residue symbols (a,b/p), taking values in roots of unity, is equal to 1. Artin reformulated the reciprocity laws as a statement that the Artin symbol from ideals (or ideles) to elements of a Galois group is trivial on a certain subgroup. Several more recent generalizations express reciprocity laws using cohomology of groups or representations of adelic groups or algebraic K-groups, and their relationship with the original quadratic reciprocity law can be hard to see.
2 Cubic reciprocity
3 Quartic reciprocity
4 Octic reciprocity
5 Eisenstein reciprocity
6 Kummer reciprocity
7 Hilbert reciprocity
8 Artin reciprocity
9 Local reciprocity
10 Explicit reciprocity laws
11 Power reciprocity laws
12 Rational reciprocity laws
13 Scholz's reciprocity law
14 Shimura reciprocity
15 Weil reciprocity law
16 Langlands reciprocity
17 Yamamoto's reciprocity law
19.1 Survey articles
Quadratic reciprocity[edit]
Main article: quadratic reciprocity
{\displaystyle \left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}.}
Cubic reciprocity[edit]
Main article: cubic reciprocity
The law of cubic reciprocity for Eisenstein integers states that if α and β are primary (primes congruent to 2 mod 3) then
{\displaystyle {\Bigg (}{\frac {\alpha }{\beta }}{\Bigg )}_{3}={\Bigg (}{\frac {\beta }{\alpha }}{\Bigg )}_{3}.}
Quartic reciprocity[edit]
Main article: quartic reciprocity
In terms of the quartic residue symbol, the law of quartic reciprocity for Gaussian integers states that if π and θ are primary (congruent to 1 mod (1+i)3) Gaussian primes then
{\displaystyle {\Bigg [}{\frac {\pi }{\theta }}{\Bigg ]}\left[{\frac {\theta }{\pi }}\right]^{-1}=(-1)^{{\frac {N\pi -1}{4}}{\frac {N\theta -1}{4}}}.}
Octic reciprocity[edit]
Main article: Octic reciprocity
Eisenstein reciprocity[edit]
Main article: Eisenstein reciprocity
Suppose that ζ is an
{\displaystyle l}
th root of unity for some odd prime
{\displaystyle l}
. The power character is the power of ζ such that
{\displaystyle \left({\frac {\alpha }{\mathfrak {p}}}\right)_{l}\equiv \alpha ^{\frac {N({\mathfrak {p}})-1}{l}}{\pmod {\mathfrak {p}}}}
for any prime ideal
{\displaystyle {\mathfrak {p}}}
of Z[ζ]. It is extended to other ideals by multiplicativity. The Eisenstein reciprocity law states that
{\displaystyle \left({\frac {a}{\alpha }}\right)_{l}=\left({\frac {\alpha }{a}}\right)_{l}}
for a any rational integer coprime to
{\displaystyle l}
and α any element of Z[ζ] that is coprime to a and
{\displaystyle l}
and congruent to a rational integer modulo (1–ζ)2.
Kummer reciprocity[edit]
Suppose that ζ is an lth root of unity for some odd regular prime l. Since l is regular, we can extend the symbol {} to ideals in a unique way such that
{\displaystyle \left\{{\frac {p}{q}}\right\}^{n}=\left\{{\frac {p^{n}}{q}}\right\}}
where n is some integer prime to l such that pn is principal.
The Kummer reciprocity law states that
{\displaystyle \left\{{\frac {p}{q}}\right\}=\left\{{\frac {q}{p}}\right\}}
for p and q any distinct prime ideals of Z[ζ] other than (1–ζ).
Hilbert reciprocity[edit]
Main article: Hilbert symbol
In terms of the Hilbert symbol, Hilbert's reciprocity law for an algebraic number field states that
{\displaystyle \prod _{v}(a,b)_{v}=1}
where the product is over all finite and infinite places. Over the rational numbers this is equivalent to the law of quadratic reciprocity. To see this take a and b to be distinct odd primes. Then Hilbert's law becomes
{\displaystyle (p,q)_{\infty }(p,q)_{2}(p,q)_{p}(p,q)_{q}=1}
But (p,q)p is equal to the Legendre symbol, (p,q)∞ is 1 if one of p and q is positive and –1 otherwise, and (p,q)2 is (–1)(p–1)(q–1)/4. So for p and q positive odd primes Hilbert's law is the law of quadratic reciprocity.
Artin reciprocity[edit]
Main article: Artin reciprocity law
In the language of ideles, the Artin reciprocity law for a finite extension L/K states that the Artin map from the idele class group CK to the abelianization Gal(L/K)ab of the Galois group vanishes on NL/K(CL), and induces an isomorphism
{\displaystyle \theta :C_{K}/{N_{L/K}(C_{L})}\to {\text{Gal}}(L/K)^{\text{ab}}.}
Although it is not immediately obvious, the Artin reciprocity law easily implies all the previously discovered reciprocity laws, by applying it to suitable extensions L/K. For example, in the special case when K contains the nth roots of unity and L=K[a1/n] is a Kummer extension of K, the fact that the Artin map vanishes on NL/K(CL) implies Hilbert's reciprocity law for the Hilbert symbol.
Local reciprocity[edit]
Hasse introduced a local analogue of the Artin reciprocity law, called the local reciprocity law. One form of it states that for a finite abelian extension of L/K of local fields, the Artin map is an isomorphism from
{\displaystyle K^{\times }/N_{L/K}(L^{\times })}
onto the Galois group
{\displaystyle Gal(L/K)}
Explicit reciprocity laws[edit]
Main article: Explicit reciprocity law
In order to get a classical style reciprocity law from the Hilbert reciprocity law Π(a,b)p=1, one needs to know the values of (a,b)p for p dividing n. Explicit formulas for this are sometimes called explicit reciprocity laws.
Power reciprocity laws[edit]
Main article: Power reciprocity law
A power reciprocity law may be formulated as an analogue of the law of quadratic reciprocity in terms of the Hilbert symbols as[2]
{\displaystyle \left({\frac {\alpha }{\beta }}\right)_{n}\left({\frac {\beta }{\alpha }}\right)_{n}^{-1}=\prod _{{\mathfrak {p}}|n\infty }(\alpha ,\beta )_{\mathfrak {p}}\ .}
Rational reciprocity laws[edit]
Main article: Rational reciprocity law
A rational reciprocity law is one stated in terms of rational integers without the use of roots of unity.
Scholz's reciprocity law[edit]
Main article: Scholz's reciprocity law
Shimura reciprocity[edit]
Main article: Shimura's reciprocity law
Weil reciprocity law[edit]
Main article: Weil reciprocity law
Langlands reciprocity[edit]
Further information: Langlands program § Reciprocity
The Langlands program includes several conjectures for general reductive algebraic groups, which for the special of the group GL1 imply the Artin reciprocity law.
Yamamoto's reciprocity law[edit]
Main article: Yamamoto's reciprocity law
Yamamoto's reciprocity law is a reciprocity law related to class numbers of quadratic number fields.
Stanley's reciprocity theorem
^ Hiramatsu, Toyokazu; Saito, Seiken (2016-05-04). An Introduction to Non-Abelian Class Field Theory. Series on Number Theory and Its Applications. WORLD SCIENTIFIC. doi:10.1142/10096. ISBN 978-981-314-226-8.
^ Neukirch (1999) p.415
Frei, Günther (1994), "The reciprocity law from Euler to Eisenstein", in Chikara, Sasaki (ed.), The intersection of history and mathematics. Papers presented at the history of mathematics symposium, held in Tokyo, Japan, August 31 - September 1, 1990, Sci. Networks Hist. Stud., vol. 15, Basel: Birkhäuser, pp. 67–90, doi:10.1090/S0002-9904-1972-12997-5, ISBN 9780817650292, MR 0308080, Zbl 0818.01002
Hilbert, David (1897), "Die Theorie der algebraischen Zahlkörper", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 4: 175–546, ISSN 0012-0456
Hilbert, David (1998), The theory of algebraic number fields, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-662-03545-0, ISBN 978-3-540-62779-1, MR 1646901
Lemmermeyer, Franz (2000), Reciprocity laws. From Euler to Eisenstein, Springer Monographs in Mathematics, Berlin: Springer-Verlag, doi:10.1007/978-3-662-12893-0, ISBN 3-540-66957-4, MR 1761696, Zbl 0949.11002
Lemmermeyer, Franz, Reciprocity laws. From Kummer to Hilbert
Neukirch, Jürgen (1999), Algebraic number theory, Grundlehren der Mathematischen Wissenschaften, vol. 322, Translated from the German by Norbert Schappacher, Berlin: Springer-Verlag, ISBN 3-540-65399-6, Zbl 0956.11021
Stepanov, S. A. (2001) [1994], "Reciprocity laws", Encyclopedia of Mathematics, EMS Press
Wyman, B. F. (1972), "What is a reciprocity law?", Amer. Math. Monthly, 79 (6): 571–586, doi:10.2307/2317083, JSTOR 2317083, MR 0308084 . Correction, ibid. 80 (1973), 281.
Survey articles[edit]
Reciprocity laws and Galois representations: recent breakthroughs
Retrieved from "https://en.wikipedia.org/w/index.php?title=Reciprocity_law&oldid=1081043188"
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Part 1 – partial derivatives – ebvalaim.log
Part 1 - partial derivatives
The series' table of contents
As I mentioned in the introduction, I assume that the reader knows what a derivative of a function is. It is a good foundation, but to get our hands wet in relativity, we need to expand that concept a bit. Let's then get to know the partial derivative. What is it?
Let's remember the ordinary derivatives first. We denote a derivative of a function
f(x)
f'(x)
\frac{df}{dx}
. It means, basically, how fast the value of the function changes while we change the argument x. For example, when
f(x) = x^2
\frac{df}{dx} = 2x
But what if the function depends on more than one variable? Like if we have a function
f(x,y) = x^2 + y^2
that assigns to each point of the plane the square of its distance from the origin. How do we even define the derivative of such a function?
This problem is what partial derivatives solve. A partial derivative can be calculated with respect to any of the variables, so in this case we have two possibilities:
\frac{\partial f}{\partial x}
\frac{\partial f}{\partial y}
(for the sake of simplicity those are sometimes denoted as
f_{,x}
f_{,y}
\partial_x f
\partial_y f
). To calculate a partial derivative one assumes that only the variable with respect to which we differentiate is a variable, the rest is treated as constants.
To present what this means we will use a linear function
f(x) = ax
a
is some constant. The derivative of this function is
f'(x) = a
a
was a variable from the beginning - this would be precisely the partial derivative with respect to x! Taking a function
f(a,x) = ax
and treating
as a constant, we get exactly the described situation. Thus,
\frac{\partial f(a,x)}{\partial x} = a
. If we changed the symbols a bit and wrote
y
a
, we would get:
f(x, y) = xy
\frac{\partial f}{\partial x} = y
On the other hand, we can treat
x
as a constant, and
y
as a variable and calculate
\frac{\partial f}{\partial y}
- it is exactly analogous and in this case we get
x
Let's go back to our initial function, the square of the distance. To calculate
\partial_x f
y
to be constant - which means that the whole
y^2
part is constant and will vanish upon differentiation. So we only have to calculate the derivative of
x^2
\partial_x f = 2x
. Voila.
What about the derivative with respect to
y
? It's the same, but now it is
x^2
that is constant and we get
\partial_y f = 2y
As is the case with ordinary derivatives, also with partial derivatives it is possible to have higher-order derivatives (second derivative, third derivative etc.). They are calculated exactly the same way as the ordinary ones - by differentiating the function, then again differentiating the result etc. The only difference is that in the case of partial derivatives there are more possible higher-order derivatives. The reason is simple - each time we can choose one of the many variables.
Let's say then that we have a function of
n
f(x_1, x_2, ..., x_n)
n
possible first derivatives:
\frac{\partial f}{\partial x_1}
\frac{\partial f}{\partial x_2}
\frac{\partial f}{\partial x_n}
The number of second derivatives is then
n^2
\frac{\partial^2 f}{\partial x_1^2}
\frac{\partial^2 f}{\partial x_1 \partial x_2}
\frac{\partial^2 f}{\partial x_1 \partial x_n}
\frac{\partial^2 f}{\partial x_2 \partial x_1}
\frac{\partial^2 f}{\partial x_2^2}
\frac{\partial^2 f}{\partial x_n^2}
The number of third derivatives would be
n^3
It is not the whole truth, though. Not all the derivatives differ. Actually, differentiating with respect to different variables is commutative, that is, it doesn't matter if we first differentiate with respect to
x_i
x_j
, or the opposite:
\frac{\partial^2 f}{\partial x_i \partial x_j} = \frac{\partial^2 f}{\partial x_j \partial x_i}
One more thing worth mentioning is that often expressions like
\frac{\partial}{\partial x}
\partial_x
are treated as separate objects - differential operators. A differential operator is then just something that applied to a function will differentiate it. Differential operators can also be "multiplied", yielding higher-order operators:
\frac{\partial}{\partial x} \frac{\partial}{\partial y} = \frac{\partial^2}{\partial x \partial y}
(now you can see why the higher-order derivatives are written as they are - why only the derivative symbol has a "power" in the "numerator", and the whole expression like
\partial x
in the "denominator"). Some other objects can also be created, but I will explain that in the next part.
I recommend calculating some partial derivatives for yourself in order to acquire some familiarity with them. Example functions:
f(x, y) = x \sin y
g(u,v) = u\left(1 - \frac{2}{v}\right)
The task - calculate all their first and second derivatives. I will check the solutions for the readers who would like that ;)
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G
{}^{c}
G
A bumpy metric theorem and the Poisson relation for generic strictly convex domains.
Luchezar N. Stojanov (1990)
A characterization of Gromov hyperbolicity of surfaces with variable negative curvature.
A. Portilla, E. Tourís (2009)
A characterization of shortest geodesics on surfaces.
Neumann-Coto, Max (2001)
Salvai, Marcos (2005)
Ernst Heintze, Hermann Karcher (1978)
A geometric space without conjugate points.
Bucataru, Ioan, Dahl, Matias F. (2010)
A lossless reduction of geodesics on supermanifolds to non-graded differential geometry
Stéphane Garnier, Matthias Kalus (2014)
ℳ=\left(M,{𝒪}_{ℳ}\right)
be a smooth supermanifold with connection
\nabla
and Batchelor model
{𝒪}_{ℳ}\cong {\Gamma }_{\Lambda {E}^{*}}
\left(ℳ,\nabla \right)
we construct a connection on the total space of the vector bundle
E\to M
. This reduction of
\nabla
is well-defined independently of the isomorphism
{𝒪}_{ℳ}\cong {\Gamma }_{\Lambda {E}^{*}}
. It erases information, but however it turns out that the natural identification of supercurves in
ℳ
(as maps from
{ℝ}^{1|1}
ℳ
) with curves in
E
restricts to a 1 to 1 correspondence on geodesics. This bijection is induced by a natural identification of initial conditions for geodesics...
A new curvature invariant and entropy of geodesic flow.
P. Sarnak, R. Osserman (1984)
A note on geodesic mappings of pseudosymmetric Riemannian manifolds
Filip Defever, Ryszard Deszcz (1991)
A note on the volume of balls on Riemannian manifolds of non-negative curvature
Paweł Grzegorz Walczak (1984)
A property on geodesic mappings of pseudo-symmetric Riemannian manifolds.
Fu, Fengyun, Zhao, Peibiao (2010)
A typical convex surface contains no closed geodesic.
Abnormal sub-riemannian geodesics : Morse index and rigidity
A. A. Agrachev, A. V. Sarychev (1996)
Abnormality of trajectory in sub-Riemannian structure
F. Pelletier, L. Bouche (1995)
In the sub-Riemannian framework, we give geometric necessary and sufficient conditions for the existence of abnormal extremals of the Maximum Principle. We give relations between abnormality,
{C}^{1}
-rigidity and length minimizing. In particular, in the case of three dimensional manifolds we show that, if there exist abnormal extremals, generically, they are locally length minimizing and in the case of four dimensional manifolds we exhibit abnormal extremals which are not
{C}^{1}
-rigid and which can be minimizing...
An algorithm based on rolling to generate smooth interpolating curves on ellipsoids
Krzysztof Krakowski, Fátima Silva Leite (2014)
We present an algorithm to generate a smooth curve interpolating a set of data on an
-dimensional ellipsoid, which is given in closed form. This is inspired by an algorithm based on a rolling and wrapping technique, described in [11] for data on a general manifold embedded in Euclidean space. Since the ellipsoid can be embedded in an Euclidean space, this algorithm can be implemented, at least theoretically. However, one of the basic steps of that algorithm consists in rolling the ellipsoid, over...
An elementary formula for the Fenchel-Nielsen twist.
Scott Wolpert (1981)
An integrable flow on a family of Hilbert Grassmannians.
Gomez, Rodrigo P. (1996)
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The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of septua-, a Latin-derived numerical prefix, rather than hepta-, a Greek-derived numerical prefix; both are cognate) together with the Greek suffix "-agon" meaning angle.
1 Regular heptagon
1.5 Diagonals and heptagonal triangle
1.6 In polyhedra
2 Star heptagons
A regular heptagon, in which all sides and all angles are equal, has internal angles of 5π/7 radians (1284⁄7 degrees). Its Schläfli symbol is {7}.
{\displaystyle A={\frac {7}{4}}a^{2}\cot {\frac {\pi }{7}}\simeq 3.634a^{2}.}
This can be seen by subdividing the unit-sided heptagon into seven triangular "pie slices" with vertices at the center and at the heptagon's vertices, and then halving each triangle using the apothem as the common side. The apothem is half the cotangent of
{\displaystyle \pi /7,}
and the area of each of the 14 small triangles is one-fourth of the apothem.
The area of a regular heptagon inscribed in a circle of radius R is
{\displaystyle {\tfrac {7R^{2}}{2}}\sin {\tfrac {2\pi }{7}},}
while the area of the circle itself is
{\displaystyle \pi R^{2};}
thus the regular heptagon fills approximately 0.8710 of its circumscribed circle.
As 7 is a Pierpont prime but not a Fermat prime, the regular heptagon is not constructible with compass and straightedge but is constructible with a marked ruler and compass. It is the smallest regular polygon with this property. This type of construction is called a neusis construction. It is also constructible with compass, straightedge and angle trisector. The impossibility of straightedge and compass construction follows from the observation that
{\displaystyle \scriptstyle {2\cos {\tfrac {2\pi }{7}}\approx 1.247}}
is a zero of the irreducible cubic x3 + x2 − 2x − 1. Consequently, this polynomial is the minimal polynomial of 2cos(2π⁄7), whereas the degree of the minimal polynomial for a constructible number must be a power of 2.
An animation from a neusis construction with radius of circumcircle
{\displaystyle {\overline {OA}}=6}
, according to Andrew M. Gleason[1] based on the angle trisection by means of the Tomahawk. This construction relies on the fact that
{\displaystyle 6\cos \left({\frac {2\pi }{7}}\right)=2{\sqrt {7}}\cos \left({\frac {1}{3}}\arctan \left(3{\sqrt {3}}\right)\right)-1.}
An animation from a neusis construction with marked ruler, according to David Johnson Leisk (Crockett Johnson).
An approximation for practical use with an error of about 0.2% is shown in the drawing. It is attributed to Albrecht Dürer.[2] Let A lie on the circumference of the circumcircle. Draw arc BOC. Then
{\displaystyle \scriptstyle {BD={1 \over 2}BC}}
gives an approximation for the edge of the heptagon.
This approximation uses
{\displaystyle \scriptstyle {{\sqrt {3}} \over 2}\approx 0.86603}
for the side of the heptagon inscribed in the unit circle while the exact value is
{\displaystyle \scriptstyle 2\sin {\pi \over 7}\approx 0.86777}
Symmetries of a regular heptagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edges. Gyration orders are given in the center.[3]
The regular heptagon belongs to the D7h point group (Schoenflies notation), order 28. The symmetry elements are: a 7-fold proper rotation axis C7, a 7-fold improper rotation axis, S7, 7 vertical mirror planes, σv, 7 2-fold rotation axes, C2, in the plane of the heptagon and a horizontal mirror plane, σh, also in the heptagon's plane.[4]
Diagonals and heptagonal triangleEdit
Main article: Heptagonal triangle
a=red, b=blue, c=green lines
The regular heptagon's side a, shorter diagonal b, and longer diagonal c, with a<b<c, satisfy[5]: Lemma 1
{\displaystyle a^{2}=c(c-b),}
{\displaystyle b^{2}=a(c+a),}
{\displaystyle c^{2}=b(a+b),}
{\displaystyle {\frac {1}{a}}={\frac {1}{b}}+{\frac {1}{c}}}
{\displaystyle ab+ac=bc,}
and[5]: Coro. 2
{\displaystyle b^{3}+2b^{2}c-bc^{2}-c^{3}=0,}
{\displaystyle c^{3}-2c^{2}a-ca^{2}+a^{3}=0,}
{\displaystyle a^{3}-2a^{2}b-ab^{2}+b^{3}=0,}
Thus –b/c, c/a, and a/b all satisfy the cubic equation
{\displaystyle t^{3}-2t^{2}-t+1=0.}
However, no algebraic expressions with purely real terms exist for the solutions of this equation, because it is an example of casus irreducibilis.
The approximate lengths of the diagonals in terms of the side of the regular heptagon are given by
{\displaystyle b\approx 1.80193\cdot a,\qquad c\approx 2.24698\cdot a.}
{\displaystyle b^{2}-a^{2}=ac,}
{\displaystyle c^{2}-b^{2}=ab,}
{\displaystyle a^{2}-c^{2}=-bc,}
{\displaystyle {\frac {b^{2}}{a^{2}}}+{\frac {c^{2}}{b^{2}}}+{\frac {a^{2}}{c^{2}}}=5.}
A heptagonal triangle has vertices coinciding with the first, second, and fourth vertices of a regular heptagon (from an arbitrary starting vertex) and angles
{\displaystyle \pi /7,2\pi /7,}
{\displaystyle 4\pi /7.}
Thus its sides coincide with one side and two particular diagonals of the regular heptagon.[5]
In polyhedraEdit
Star heptagonsEdit
Two kinds of star heptagons (heptagrams) can be constructed from regular heptagons, labeled by Schläfli symbols {7/2}, and {7/3}, with the divisor being the interval of connection.
A regular triangle, heptagon, and 42-gon can completely fill a plane vertex. Together, they can tessellate space if irregular polygons are admitted.[7]
In the hyperbolic plane, regular heptagons can tile space. This heptagonal tiling is shown in a Poincaré disk model projection:
Geometry problem of the surface of a heptagon divided into triangles, on a clay tablet belonging to a school for scribes; Susa, the first half of the 2nd millennium BCE
The United Kingdom currently, as of 2022, has two heptagonal coins, the 50p and 20p pieces, and the Barbados Dollar are also heptagonal. The 20-eurocent coin has cavities placed similarly. Strictly, the shape of the coins is a Reuleaux heptagon, a curvilinear heptagon which has curves of constant width; the sides are curved outwards to allow the coins to roll smoothly when they are inserted into a vending machine. Botswana pula coins in the denominations of 2 Pula, 1 Pula, 50 Thebe and 5 Thebe are also shaped as equilateral-curve heptagons. Coins in the shape of Reuleaux heptagons are also in circulation in Mauritius, U.A.E., Tanzania, Samoa, Papua New Guinea, São Tomé and Príncipe, Haiti, Jamaica, Liberia, Ghana, the Gambia, Jordan, Jersey, Guernsey, Isle of Man, Gibraltar, Guyana, Solomon Islands, Falkland Islands and Saint Helena. The 1000 Kwacha coin of Zambia is a true heptagon.
Many police badges in the US have a {7/2} heptagram outline.
^ Gleason, Andrew Mattei (March 1988). "Angle trisection, the heptagon, and the triskaidecagon p. 186 (Fig.1) –187" (PDF). The American Mathematical Monthly. 95 (3): 185–194. doi:10.2307/2323624. Archived from the original (PDF) on 19 December 2015.
^ G.H. Hughes, "The Polygons of Albrecht Dürer-1525, The Regular Heptagon", Fig. 11 the side of the Heptagon (7) Fig. 15, image on the left side, retrieved on 4 December 2015
^ Salthouse, J.A; Ware, M.J. (1972). Point group character tables and related data. Cambridge: Cambridge University Press. ISBN 0 521 08139 4.
^ a b c Abdilkadir Altintas, "Some Collinearities in the Heptagonal Triangle", Forum Geometricorum 16, 2016, 249–256.http://forumgeom.fau.edu/FG2016volume16/FG201630.pdf
^ Leon Bankoff and Jack Garfunkel, "The heptagonal triangle", Mathematics Magazine 46 (1), January 1973, 7–19.
^ "Shield - a 3.7.42 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 7 March 2022.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Heptagon&oldid=1088596874"
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Detect ARCH Effects - MATLAB & Simulink - MathWorks India
The null hypothesis is soundly rejected (h = 1, p = 0) in favor of the ARCH(2) alternative. The F statistic for the test is 399.97, much larger than the critical value from the
{\chi }^{2}
distribution with two degrees of freedom, 5.99.
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How to Multiply Binomials Using the FOIL Method: 9 Steps
2 Multiplying Binomials
When multiplying two binomials you must use the distributive property to ensure that each term is multiplied by every other term. This can sometimes be a confusing process, as it is easy to lose track of which terms you have already multiplied together. You can use FOIL to multiply binomials using the distributive property in an organized way.[1] X Research source By simply remembering the words in the acronym, this method will help you multiply binomials quickly.
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Write the two binomials side-by-side in parentheses. This setup helps you easily keep track of operations when using the foil method.
{\displaystyle 2x-7}
{\displaystyle 5x+3}
, you would set up the problem like this:
{\displaystyle (2x-7)(5x+3)}
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Ensure you are multiplying two binomials. A binomial is an algebraic expression with two terms.[2] X Research source The FOIL method does not work when multiplying trinomials, or a binomial by a trinomial.
A term is a single number or variable, such as
{\displaystyle 3}
{\displaystyle x}
, or it could be a multiplied number and variable, such as
{\displaystyle 3x}
Read Multiply Polynomials for instructions on multiplying other types of polynomials.
For example, you could NOT multiply
{\displaystyle (2x-4)(3x^{2}-2x+8)}
using the FOIL method, because the second expression is a trinomial, with three terms.
You could multiply
{\displaystyle (2x-7)(5x+3)}
, because both expressions are binomials, with two terms each.
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Arrange the binomials by terms. Most algebra problems will already be arranged this way, but if not, make sure the first term in each expression contains the variable, and the second term in each expression contains the coefficient.
Setting up the problem this way makes simplifying easier.
A coefficient is a number without a variable.
For example, you would change
{\displaystyle (2x-7)(3+5x)}
{\displaystyle (2x-7)(5x+3)}
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Multiply the first terms in each expression. The F in FOIL stands for "first."
Remember when multiplying a variable by itself, such as
{\displaystyle x\times x}
, the result is a squared variable (
{\displaystyle x^{2}}
{\displaystyle (2x-7)(5x+3)}
, you would first calculate:
{\displaystyle (2x)(5x)}
{\displaystyle =10x^{2}}
Multiply the outside terms in each expression. The O in FOIL stands for "outside," or "outer." The outside terms are the first term of the first expression, and the last term of the second expression.
Pay close attention to addition and subtraction. If the second binomial is a subtraction expression, that means in this step you will be multiplying a negative number.
For example, for the problem
{\displaystyle (2x-7)(5x+3)}
, you would next calculate:
{\displaystyle (2x)(3)}
{\displaystyle =6x}
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Multiply the inside terms in each expression. The I in FOIL stands for "inside," or "inner." The inner terms are the last term of the first expression, and the first term of the second expression.
Pay close attention to addition and subtraction. If the first binomial is a subtraction expressions, that means in this step you will be multiplying a negative number.
{\displaystyle (2x-7)(5x+3)}
{\displaystyle (-7)(5x)}
{\displaystyle =-35x}
Multiply the last terms in each expression. The L in FOIL stands for "last."
Pay close attention to addition and subtraction. If either binomial is a subtraction expression, that means in this step you will be multiplying a negative number.
{\displaystyle (2x-7)(5x+3)}
{\displaystyle (-7)(3)}
{\displaystyle =-21}
Write the new expression. To do this, write out the new terms you created during the FOIL process. You should have four new terms.
For example, after multiplying
{\displaystyle (2x-7)(5x+3)}
, your new expression is
{\displaystyle 10x^{2}+6x-35x-21}
Simplify the expression. To do this, combine like terms. Usually you will have two terms with the
{\displaystyle x}
variable that need to be combined.
Pay close attention to positive and negative signs as you add or subtract.
{\displaystyle 10x^{2}+6x-35x-21}
, you would simplify by combining
{\displaystyle 6x-35x}
. Thus, the expression simplifies to
{\displaystyle 10x^{2}-29x-21}
Does this work If I am multiplying binomials with different variables?
Yes, you can use the FOIL method if the binomials have different variables, such as x and y. In this case, after you complete the steps, you will not have any like terms to combine, so your final expression will have four terms. For example, (2x -7)(5y + 3) would simplify to 10xy + 6x -35y - 21.
What determines where I put the addition and subtraction signs?
The sign of each term of the expansion is the product of the signs of the terms you multiplied to get it. Meaning if you're doing (y-4)(5-2n), the term corresponding to inner is (-4)(5) = -20 inheriting the negative sign from the - in -4, and the term corresponding to last is (-4)(-2n) = +8n because both terms are negative and the product of two negatives is positive.
How do I solve (x - 4)(x + 2) = 3(x - 1)?
(x - 4)(x + 2) = x² - 2x - 8. 3(x - 1) = 3x - 3. Therefore x² - 2x - 8 = 3x - 3. Then x² - 5 = 5x, and x² - 5x - 5 = 0. The left side of that equation cannot be factored, so you'd have to use the quadratic formula to solve for x. Thus, x = {5 +/- √[25 - (4)(1)(-5)]} ÷ (2)(1) = {5 +/- �√[25 + 20]} ÷ 2 = (5 +/- √45) ÷ 2 = (5 +/- 6.7) ÷ 2 = 5.85 or -0.85 (two values for x, which is normal with quadratic equations).
John goes on a diet and loses one ninth of his weight, n. What is his weight after going on a diet?
If he loses one-ninth of this weight, that means his new weight is 8/9 of his original weight. So his new weight is (8n/9).
You can think of this as two separate distributions: (2x)(5x + 3) added with (-7)(5x + 3)
To know how to multiply, add, and subtract
↑ http://www.coolmath.com/prealgebra/15-intro-to-polynomials/06-polynomials-multiplying-foil-01-108
Русский:перемножить два бинома
Español:multiplicar binomios utilizando el método FOIL
"I had forgotten what FOIL was, so I had to go through the steps."
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Physics - Analysis Finds <i>B</i> Meson Behaves Itself
Analysis Finds B Meson Behaves Itself
A new analysis of Large Hadron Collider data measures rare decays of the B meson that behave according to the standard model.
The standard model of particle physics predicts precisely the various decay rates of the B meson, an unstable neutral or charged particle consisting of a bottom antiquark and another quark. By experimentally verifying that the B meson’s rare decays conform to theory, particle physicists aim to pinpoint any deviations from predictions that would indicate the existence of new particles or processes beyond the standard model. Now, the Large Hadron Collider beauty (LHCb) Collaboration has released new measurements relating to three rare decay channels that are consistent with theoretical predictions [1, 2]. The result constrains parameters such as the mass of potential Higgs bosons in proposed models beyond the standard model.
The team’s eponymous LHCb detector is designed to search for signatures of bottom quarks in proton-proton collisions at the Large Hadron Collider. In this analysis, the researchers used data from two different experimental runs. The center-of-mass collision energies vary among 7, 8, and 13 TeV.
The new analyses yielded the most precise measurements yet of three rare decay channels in which the B meson’s decay products include two muons. These decays are rare because they require the exchange of multiple virtual particles. The researchers found that the strange B meson, which consists of a bottom antiquark and a strange quark, decays into a pair of muons three times every billion decays, consistent with their previous measurement.
The researchers did not find statistically significant signals of the other two decay channels. Still, they were also able to put upper limits on the likelihood of two processes: With 95% confidence, the researchers determined that less than 26 in 100 billion neutral B mesons (a bottom quark and a down quark) decay into a pair of muons, and less than 2 in a billion strange B mesons decay into two muons (with a combined mass above 4.9 ) and a photon. This was also LHCb’s first search for the strange B meson decay, which has been difficult to observe because its selection is affected by a larger background. The collaboration plans to collect more data in this region in the LHC’s third run, to begin at the end of 2022, to further study how these B meson decay channels could probe new physics.
R. Aaij et al., “Analysis of neutral B-meson decays into two muons,” Phys. Rev. Lett. 128, 041801 (2022).
R. Aaij et al., “Measurement of the
{B}_{s}^{0}\to {𝜇}^{+}{𝜇}^{-}
decay properties and search for the
{B}^{0}\to {𝜇}^{+}{𝜇}^{-}
{B}_{s}^{0}\to {𝜇}^{+}{𝜇}^{-}𝛾
decays,” Phys. Rev. D 105, 012010 (2022).
{B}_{s}^{0}\to {\mu }^{+}{\mu }^{-}
{B}^{0}\to {\mu }^{+}{\mu }^{-}
{B}_{s}^{0}\to {\mu }^{+}{\mu }^{-}\gamma
decays
Analysis of Neutral
B
-Meson Decays into Two Muons
{B}_{s}^{0}\to {\mu }^{+}{\mu }^{-}
{B}^{0}\to {\mu }^{+}{\mu }^{-}
{B}_{s}^{0}\to {\mu }^{+}{\mu }^{-}\gamma
B
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Comparing Nations' PPP
Pairing PPP and GDP
Drawbacks of PPP
One popular macroeconomic analysis metric to compare economic productivity and standards of living between countries is purchasing power parity (PPP). PPP is an economic theory that compares different countries' currencies through a "basket of goods" approach, not to be confused with the Paycheck Protection Program created by the CARES Act.
Purchasing power parity (PPP) is a popular metric used by macroeconomic analysts that compares different countries' currencies through a "basket of goods" approach.
Purchasing power parity (PPP) allows for economists to compare economic productivity and standards of living between countries.
Some countries adjust their gross domestic product (GDP) figures to reflect PPP.
Click Play to Learn How to Calculate Purchasing Power Parity
The relative version of PPP is calculated with the following formula:
\begin{aligned} &S=\frac{P_1}{P_2}\\ &\textbf{where:}\\ &S=\text{ Exchange rate of currency }1\text{ to currency }2\\ &P_1=\text{ Cost of good }X\text{ in currency }1\\ &P_2=\text{ Cost of good }X\text{ in currency }2 \end{aligned}
S=P2P1where:S= Exchange rate of currency 1 to currency 2P1= Cost of good X in currency 1
Comparing Nations' Purchasing Power Parity
To make a meaningful comparison of prices across countries, a wide range of goods and services must be considered. However, this one-to-one comparison is difficult to achieve due to the sheer amount of data that must be collected and the complexity of the comparisons that must be drawn. To help facilitate this comparison, the University of Pennsylvania and the United Nations joined forces to establish the International Comparison Program (ICP) in 1968.
With this program, the PPPs generated by the ICP have a basis from a worldwide price survey that compares the prices of hundreds of various goods and services. The program helps international macroeconomists estimate global productivity and growth.
Every few years, the World Bank releases a report that compares the productivity and growth of various countries in terms of PPP and U.S. dollars. Both the International Monetary Fund (IMF) and the Organization for Economic Cooperation and Development (OECD) use weights based on PPP metrics to make predictions and recommend economic policy. The recommended economic policies can have an immediate short-term impact on financial markets.
Also, some forex traders use PPP to find potentially overvalued or undervalued currencies. Investors who hold stock or bonds of foreign companies may use the survey's PPP figures to predict the impact of exchange-rate fluctuations on a country's economy, and thus the impact on their investment.
Pairing Purchasing Power Parity With Gross Domestic Product
In contemporary macroeconomics, gross domestic product (GDP) refers to the total monetary value of the goods and services produced within one country. Nominal GDP calculates the monetary value in current, absolute terms. Real GDP adjusts the nominal gross domestic product for inflation.
However, some accounting goes even further, adjusting GDP for the PPP value. This adjustment attempts to convert nominal GDP into a number more easily comparable between countries with different currencies.
To better understand how GDP paired with purchase power parity works, suppose it costs $10 to buy a shirt in the U.S., and it costs €8.00 to buy an identical shirt in Germany. To make an apples-to-apples comparison, we must first convert the €8.00 into U.S. dollars. If the exchange rate was such that the shirt in Germany costs $15.00, the PPP would, therefore, be 15/10, or 1.5.
In other words, for every $1.00 spent on the shirt in the U.S., it takes $1.50 to obtain the same shirt in Germany buying it with the euro.
GDP by Purchasing Power Parity vs Nominal GDP
Since 1986, The Economist has playfully tracked the price of McDonald's Corp.’s (MCD) Big Mac hamburger across many countries. Their study results in the famed "Big Mac Index". In "Burgernomics"—a prominent 2003 paper that explores the Big Mac Index and PPP—authors Michael R. Pakko and Patricia S. Pollard cited the following factors to explain why the purchasing power parity theory is not a good reflection of reality.
Goods that are unavailable locally must be imported, resulting in transport costs. These costs include not only fuel but import duties as well. Imported goods will consequently sell at a relatively higher price than do identical locally sourced goods.
Government sales taxes such as the value-added tax (VAT) can spike prices in one country, relative to another.
Tariffs can dramatically augment the price of imported goods, where the same products in other countries will be comparatively cheaper.
Non-Traded Services
The Big Mac's price factors input costs that are not traded. These factors include such items as insurance, utility costs, and labor costs. Therefore, those expenses are unlikely to be at parity internationally.
Goods might be deliberately priced higher in a country. In some cases, higher prices are because a company may have a competitive advantage over other sellers. The company may have a monopoly or be part of a cartel of companies that manipulate prices, keeping them artificially high.
While it's not a perfect measurement metric, purchase power parity does allow for the possibility of comparing pricing between countries that have differing currencies.
Congress.gov. "H.R. 266 - Paycheck Protection Program and Health Care Enhancement Act."
World Bank. "International Comparison Program (ICP): History."
World Bank. "International Comparison Program (ICP): Uses."
World Bank. "International Comparison Program (ICP): Overview."
World Bank. "Who uses PPPs – Examples of Uses by International Organizations."
St. Louis Federal Reserve Bank. "Burgernomics: A Big Mac Guide to Purchasing Power Parity," Page 1.
St. Louis Federal Reserve Bank. "Burgernomics: A Big Mac Guide to Purchasing Power Parity," Pages 16-17.
St. Louis Federal Reserve Bank. "Burgernomics: A Big Mac Guide to Purchasing Power Parity," Page 21.
Balassa-Samuelson Effect Definition
The Balassa-Samuelson Effect is a pattern wherein countries with high productivity and wage growth also experience higher real exchange rates.
Parity price is a term used to explain when two assets are equal in value.
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82C24 Interface problems; diffusion-limited aggregation
82C26 Dynamic and nonequilibrium phase transitions (general)
82C80 Numerical methods (Monte Carlo, series resummation, etc.)
A deterministic displacement theorem for Poisson processes.
Knill, Oliver (1997)
A kinetic approach to the study of opinion formation
Laurent Boudin, Francesco Salvarani (2009)
In this work, we use the methods of nonequilibrium statistical mechanics in order to derive an equation which models some mechanisms of opinion formation. After proving the main mathematical properties of the model, we provide some numerical results.
A lattice gas model for the incompressible Navier–Stokes equation
J. Beltrán, C. Landim (2008)
We recover the Navier–Stokes equation as the incompressible limit of a stochastic lattice gas in which particles are allowed to jump over a mesoscopic scale. The result holds in any dimension assuming the existence of a smooth solution of the Navier–Stokes equation in a fixed time interval. The proof does not use nongradient methods or the multi-scale analysis due to the long range jumps.
A microscopic model for the Burgers equation and longest increasing subsequences.
Seppäläinen, Timo (1996)
Anne-Laure Basdevant, Philippe Laurençot, James R. Norris, Clément Rau (2011)
A stochastic system of particles is considered in which the sizes of the particles increase by successive binary mergers with the constraint that each coagulation event involves a particle with minimal size. Convergence of a suitably renormalized version of this process to a deterministic hydrodynamical limit is shown and the time evolution of the minimal size is studied for both deterministic and stochastic models.
Thierry Bodineau, Alice Guionnet (1999)
Amir Dembo, Jean-Dominique Deuschel (2007)
Amine Asselah (2011)
We study the upper tails for the energy of a randomly charged symmetric and transient random walk. We assume that only charges on the same site interact pairwise. We consider annealed estimates, that is when we average over both randomness, in dimension three or more. We obtain a large deviation principle, and an explicit rate function for a large class of charge distributions.
Characterization of equilibrium measures for critical reversible Nearest Particle Systems
Thomas Mountford, Li Wu (2008)
We show that for critical reversible attractive Nearest Particle Systems all equilibrium measures are convex combinations of the upper invariant equilibrium measure and the point mass at all zeros, provided the underlying renewal sequence possesses moments of order strictly greater than
\frac{7+\sqrt{41}}{2}
and obeys some natural regularity conditions.
Competing particle systems evolving by I.I.D. Increments.
Shkolnikov, Mykhaylo (2009)
Mustapha Mourragui (1996)
Convergence of coalescing nonsimple random walks to the Brownian web.
Newman, Charles M., Ravishankar, Krishnamurthi, Sun, Rongfeng (2005)
Thierry Gobron, Ellen Saada (2010)
Attractiveness is a fundamental tool to study interacting particle systems and the basic coupling construction is a usual route to prove this property, as for instance in simple exclusion. The derived markovian coupled process (ξt, ζt)t≥0 satisfies: (A) if ξ0≤ζ0 (coordinate-wise), then for all t≥0, ξt≤ζt a.s. In this paper, we consider generalized misanthrope models which are conservative particle systems on ℤd such that, in each transition, k particles may jump from a site x to another site y,...
Diffusion and scattering of shocks in the partially asymmetric simple exclusion process.
Belitsky, Vladimir, Schütz, Gunter M. (2002)
Diffusive long-time behavior of Kawasaki dynamics.
Cancrini, Nicoletta, Cesi, Filippo, Roberto, Cyril (2005)
M. Escobedo, S. Mischler (2006)
Equilibrium fluctuations for a one-dimensional interface in the solid on solid approximation.
Posta, Gustavo (2005)
O Benois, R Esposito, R Marra (2003)
Equilibrium states for the Landau-Fermi-Dirac equation
Véronique Bagland, Mohammed Lemou (2004)
A kinetic collision operator of Landau type for Fermi-Dirac particles is considered. Equilibrium states are rigorously determined under minimal assumptions on the distribution function of the particles. The particular structure of the considered operator (strong non-linearity and degeneracy) requires a special investigation compared to the classical Boltzmann or Landau operator.
G. Maillard, T. S. Mountford (2013)
We answer some questions raised by Gantert, Löwe and Steif (Ann. Inst. Henri Poincaré Probab. Stat.41(2005) 767–780) concerning “signed” voter models on locally finite graphs. These are voter model like processes with the difference that the edges are considered to be either positive or negative. If an edge between a site
x
and a site
y
is negative (respectively positive) the site
y
will contribute towards the flip rate of
x
if and only if the two current spin values are equal (respectively opposed)....
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On the Three-Dimensional Correlation Between Myofibroblast Shape and Contraction | J. Biomech Eng. | ASME Digital Collection
Alex Khang,
James T. Willerson Center for Cardiovascular Modeling and Simulation, The Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin
Emma Lejeune,
Department of Biomedical Engineering, The University of Texas at Austin, Austin
Department of Mechanical Engineering, Boston University
Ali Abbaspour,
Daniel P. Howsmon,
Daniel P. Howsmon
e-mail: msacks@oden.utexas.edu
A. Khang and E. Lejeune contributed equally to this work.
Khang, A., Lejeune, E., Abbaspour, A., Howsmon, D. P., and Sacks, M. S. (May 18, 2021). "On the Three-Dimensional Correlation Between Myofibroblast Shape and Contraction." ASME. J Biomech Eng. September 2021; 143(9): 094503. https://doi.org/10.1115/1.4050915
Myofibroblasts are responsible for wound healing and tissue repair across all organ systems. In periods of growth and disease, myofibroblasts can undergo a phenotypic transition characterized by an increase in extracellular matrix (ECM) deposition rate, changes in various protein expression (e.g., alpha-smooth muscle actin (αSMA)), and elevated contractility. Cell shape is known to correlate closely with stress-fiber geometry and function and is thus a critical feature of cell biophysical state. However, the relationship between myofibroblast shape and contraction is complex, even as well in regards to steady-state contractile level (basal tonus). At present, the relationship between myofibroblast shape and basal tonus in three-dimensional (3D) environments is poorly understood. Herein, we utilize the aortic valve interstitial cell (AVIC) as a representative myofibroblast to investigate the relationship between basal tonus and overall cell shape. AVICs were embedded within 3D poly(ethylene glycol) (PEG) hydrogels containing degradable peptide crosslinkers, adhesive peptide sequences, and submicron fluorescent microspheres to track the local displacement field. We then developed a methodology to evaluate the correlation between overall AVIC shape and basal tonus induced contraction. We computed a volume averaged stretch tensor
⟨U⟩
for the volume occupied by the AVIC, which had three distinct eigenvalues (
λ1,2,3=1.08,0.99, and 0.89
), suggesting that AVIC shape is a result of anisotropic contraction. Furthermore, the direction of maximum contraction correlated closely with the longest axis of a bounding ellipsoid enclosing the AVIC. As gel-imbedded AVICs are known to be in a stable state by 3 days of incubation used herein, this finding suggests that the overall quiescent AVIC shape is driven by the underlying stress-fiber directional structure and potentially contraction level.
Shapes, Valves, Hydrogels
Mechanoregulation of the Myofibroblast in Wound Contraction, Scarring, and Fibrosis: Opportunities for New Therapeutic Intervention
.10.1089/wound.2012.0393
Fibroblasts and Myofibroblasts in Wound Healing: Force Generation and Measurement
.10.1016/j.jtv.2009.11.004
The Concept of Cellular Tone: Reflections on the Endothelium, Fibroblasts, and Smooth Muscle Cells
.10.1353/pbm.1993.0008
Specific Regional and Directional Contractile Responses of Aortic Cusp Tissue
. https://www.researchgate.net/profile/Adrian-Chester/publication/8243281_Specific_regional_and_directional_contractile_responses_of_aortic_cusp_tissue/links/0deec52d3ad9bb0f7a000000/Specific-regional-and-directional-contractile-responses-of-aortic-cusp-tissue.pdf
.10.1039/C6IB00120C
Valve Interstitial Cell Shape Modulates Cell Contractility Independent of Cell Phenotype
Cell Shape: Effects on Gene Expression and Signaling
How Cells Sense Their Own Shape—Mechanisms to Probe Cell Geometry and Their Implications in Cellular Organization and Function
Lipshtat
Decoding Information in Cell Shape
Dupuytren's Contracture Unfolded
Quantifying Heart Valve Interstitial Cell Contractile State Using Highly Tunable Poly(Ethylene Glycol) Hydrogels
Characterization of Valvular Interstitial Cell Function in Three Dimensional Matrix Metalloproteinase Degradable PEG Hydrogels
Dynamic Stiffening of Poly(Ethylene Glycol)-Based Hydrogels to Direct Valvular Interstitial Cell Phenotype in a Three-Dimensional Environment
Small Peptide Functionalized Thiol–Ene Hydrogels as Culture Substrates for Understanding Valvular Interstitial Cell Activation and de Novo Tissue Deposition
Three-Dimensional Highthroughput Cell Encapsulation Platform to Study Changes in Cell-Matrix Interactions
Appl. Mater. Interfaces
Measurement of Mechanical Tractions Exerted by Cells in Three-Dimensional Matrices
Porcine Cardiac Valvular Subendothelial Cells in Culture: Cell Isolation and Growth Characteristics
FM-Track: A Fiducial Marker Tracking Software for Studying Cell Mechanics in a Three-Dimensional Environment
.10.1016/j.softx.2020.100417
.https://jmlr.org/papers/volume12/pedregosa11a/pedregosa11a.pdf
.10.7717/peerj.453
Minimum Volume Enclosing Ellipsoid
,” MATLAB Central File Exchange, accessed Apr. 24, 2021, https://www.mathworks.com/matlabcentral/fileexchange/9542-minimum-volume-enclosing-ellipsoid
Polynomial Algorithms in Linear Programming
Toyjanova
Mean Deformation Metrics for Quantifying 3D Cell–Matrix Interactions Without Requiring Information About Matrix Material Properties
.10.1039/c3ib40230d
.10.1039/c1ib00061f
Loughian
Sanchez-Adams
On Intrinsic Stress Fiber Contractile Forces in Semilunar Heart Valve Interstitial Cells Using a Continuum Mixture Model
Deconstructing the Third Dimension—How 3D Culture Microenvironments Alter Cellular Cues
A Stochastic Model for Chemotaxis Based on the Ordered Extension of Pseudopods
What Is the Diameter of the Actin Filament?
A Perturbation Analysis Approach for Studying the Effect of Swelling Kinetics on Instabilities in Hydrogel Plates
A Variational Approach and Finite Element Implementation for Swelling of Polymeric Hydrogels Under Geometric Constraints
Stereolithography of PEG Hydrogel Multi-Lumen Nerve Regeneration Conduits
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Can zero be represented by any number of tiles? Using only the unit tiles (in other words, only the
1
-1
tiles), determine whether you can represent zero on an Expression Mat with the number of tiles below. If you can, draw an Expression Mat demonstrating that it is possible. If it is not possible, explain why not.
2
6
Use the same ideas as in part (a).
3
Use the expression mat in the eTool below to determine if zero can be represented.
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G
{}^{c}
G
2-Killing vector fields on Riemannian manifolds.
Oprea, Teodor (2008)
{C}^{0}
-theory for the blow-up of second order elliptic equations of critical Sobolev growth.
Druet, Olivier, Hebey, Emmanuel, Robert, Frédéric (2003)
A Canonical Form for Compact Nonpositively Curved Manifolds Whose Fundamental Groups Have Nontrivial Center.
Patrick Eberlein (1982)
A canonical metric for Möbius structures and its applications.
Ravi S. Kulkarni, Ulrich Pinkall (1994)
A Characterization of the Cannonical Spheres by the Spectrum.
Shukichi Tanno (1980)
A class of discriminant varieties in the conformal 3-sphere.
Baird, Paul, Gallardo, Luis (2002)
A class of self-concordant functions on Riemannian manifolds.
Bercu, Gabriel, Postolache, Mihai (2009)
Atsushi Kasue (1988)
A comparison-estimate of Topogonov type for Ricci curvature.
Xianzhe Dai, Guofang Wei (1995)
Sun-Yung A. Chang, Matthew J. Gursky, Paul C. Yang (2003)
A Conjecture of Besse on Harmonic Manifolds.
Lieven Vanhecke (1981)
A Construction of Non-Flat, Compact Irreducible Riemannian Manifolds Which are Isospectral But Not Isometric.
Norio Ejiri (1979)
A curvature identity on a 6-dimensional Riemannian manifold and its applications
Yunhee Euh, Jeong Hyeong Park, Kouei Sekigawa (2017)
We derive a curvature identity that holds on any 6-dimensional Riemannian manifold, from the Chern-Gauss-Bonnet theorem for a 6-dimensional closed Riemannian manifold. Moreover, some applications of the curvature identity are given. We also define a generalization of harmonic manifolds to study the Lichnerowicz conjecture for a harmonic manifold “a harmonic manifold is locally symmetric” and provide another proof of the Lichnerowicz conjecture refined by Ledger for the 4-dimensional case under a...
A. D. Alexandrov's length manifolds with one-sided bounded curvature
Patrick Eberlein, Jens Heber (1990)
A Direct Approach to the Determination of Gaussian and Scalar Curvature Functions.
Jerry L. Kazdan, F.W. Warner (1975)
Paweł G. Walczak (1992)
A General Schwarz Lemma for Riemannian-Manifolds
Samuel I. Goldberg, Zvi Har΄ El (1977)
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Liquidity Pools - TOAD Wiki
Liquidity pools are a place to pool tokens (which we sometimes call liquidity) so that users can use them to make trades in a decentralized way. These pools are created by users and decentralized apps (or Dapps, for short) who want to profit from their usage. To pool liquidity, the amounts a user supplies must be equally divided between two coins: the primary token (sometimes called the quote token) and the base token (usually BNB, MOVR or a stable coin).
The existence of this pooled liquidity gives other traders access to the underlying tokens in exchange for a small fee, which is distributed proportionately to all the liquidity providers. In this sense, PADSwap is also an “automated market maker” (or AMM, for short).
PADSwap's liquidity pools allow anyone to provide liquidity here.
Once a user provides liquidity, they will receive PADSwap liquidity provider (LP) tokens that represent their share of the pooled liquidity for that token pair. If a user deposited $TOAD and $BNB into a pool, they would receive TOAD-BNB LP tokens. These LP tokens represent a proportional share of the pooled assets, allowing a user to reclaim their provided liquidity at any point. Every time another user uses the pool to trade between $TOAD and $BNB, a 0.3% fee is taken on the trade. 0.25% of that trade goes back to the LP pool. .05% Is sent to the The Vault as index backing for $PAD. The value of the LP tokens (which represent the shares of the total liquidity in each pool) is updated with each trade to add their value relative to the tokens the pool uses to trade. If previously there were 100 LP tokens representing 100 BNB and 100 TOAD, each token would be worth 1 BNB & 1 TOAD (note in this example, BNB and TOAD are the same relative value). If a user were then to trade 10 BNB for 10 TOAD in that pool, and another user were to trade 10 TOAD for 10 BNB, then there would now be 100.025 BNB and 100.025 TOAD. This means each LP token would be worth 1.0025 BNB and 1.00025 TOAD now when it is withdrawn.
Instructions for adding liquidity
The following guide walks you through the process of providing liquidity:
If the pool you wish to provide liquidity to does not exist, you can create it! Simply provide the tokens, and off you go. As the first liquidity provider, you set the initial exchange ratio (price). This often quickly corrects itself through arbitrage and by more liquidity providers adding to the pool.
The arbitrage is to your disadvantage. We therefore recommend you to set the initial exchange ratio (price) according to the current market price.
For the creation of the liquidity pool (contract) the gas price will be a bit higher.
In order for a swap service like PADSwap to function, the system must have some funds to operate with. After all, when you swap one token for another, where are the tokens coming from?
These funds are added by ordinary users in exchange for earning transaction fees and staking their liquidity in a wide variety of farms. The funds added to the swap are usually referred to as liquidity, which is a measure of how easily you can buy or sell an asset — or how much of it you can buy/sell.
Liquidity is always stored in pairs, such as TOAD-BNB or PAD-BTC, with each pair being held in a separate liquidity pool. For example, when someone swaps BNB for TOAD, the user's BNB is added to the TOAD-BNB pool, and the equivalent amount of TOAD is taken from the pool and given to the user.
Generally, more liquidity is better, as it allows users to easily buy and sell large amounts of tokens without causing equally large price impacts.
There are two main reasons to provide liquidity:
0.25% of every transaction on PADSwap goes to the liquidity pool rewards drip as a way to reward liquidity providers. Since the supply of LP tokens stays the same, the value of your LP tokens increases with every transaction on that liquidity pair.
While some tokens can be staked separately, most of our farms are using LP tokens and are much more lucrative. Keep in mind that while your LP tokens are staked, you are also earning LP fees mentioned above!
Providing liquidity on PadSwap
Liquidity can be added and removed by any user on the liquidity tab on PadSwap. Since liquidity is always added in pairs, you will have to choose two tokens and add them in equal parts (e.g. $50 worth of token A and $50 worth of token B). We encourage you to take a look at our farms before deciding which token pair you wish to provide.
After adding liquidity, you will receive LP tokens that you can now stake in corresponding farms. If at any point you wish to take out your liquidity and exchange it for the original tokens, you can do it on the same liquidity tab.
In case you don't see your liquidity on the liquidity tab:
Only LP tokens not staked in any farm are shown
On the liquidity page, click "Import it" and select your liquidity pair.
As you might know, the price of a token is directly related to the ratio between tokens in the pool. For example, if 1 TOAD is worth 100 BUSD, it means that the TOAD-BUSD pool has 100 BUSD tokens for every TOAD token.
When the price of a token changes, so does the ratio of tokens in the pool. The total value of the pool remains constant, and you are still entitled to the exact same percentage of the pool - however, you will receive more of token A and less of token B when withdrawing your liquidity, which is slightly less value than if you held the tokens separately.
The adjustment of the token ratio conforms to the equation
x * y = k
x
y
are the quantities of the two paired tokens, and
k
is constant. This means that even though you supply equal parts of two tokens to the pool, the quantities you receive when you reclaim your liquidity will change relative to the difference in the change in price of the two tokens when you remove the liquidity. If the price of
x
token goes up, and
y
token goes down, you will have less of
y
y
, and vice versa. If the price of both tokens goes up, or the price of both goes down, you will nonetheless have relative quantities of each token proportionately to the difference in the change of the price of x and y.
Ratio adjustment between two tokens in a pool due to price changes
Usually, the LP rewards combined with farming are more than enough to counter the effects of impermanent loss, but it could be problematic in case of extreme price fluctuations.
To see an example of how price fluctuations can cause impermanent loss, you can use our impermanent loss calculator in TOAD Toolbox
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To measure the dispersion of the MFI interest rates of the individual euro area countries around the euro area interest rate, coefficients of variation are calculated for each euro area MFI interest rate. The coefficient of variation is computed as the standard deviation divided by the euro area interest rate, thus adjusting for the fact that the standard deviation is influenced by the level of the euro area rate. By definition, the coefficient of variation is unit-free.
Coefficients of variation for MFI interest rates:
on new euro-denominated loans to euro area non-financial corporations.
on new euro-denominated loans involving collateral and/or guarantees to euro area non-financial corporations.
for MFI interest rates on new euro-denominated loans to euro area households.
for MFI interest rates on new euro-denominated deposits from euro area residents.
on outstanding amounts of euro-denominated loans to, and deposits from, euro area residents.
All coefficients
All coefficients of cross-country variation time series
The standard deviation is computed as the square root of the weighted variance of the national MFI interest rates (MIR) with respect to the euro area interest rate. The national business volumes serve as country weights.
CV=\frac{\sqrt{\mathit{\text{weighted_variance}}}}{\mathit{\text{euro_area_MIR}}}
the weighted variance is obtained as
\sum _{k}w\left(k{\right)}_{t}{\left(i\left(k{\right)}_{t}-{\mathit{\text{euro_area_MIR}}}_{t}\right)}^{2}
the euro area MIR is obtained as
\sum _{k}w\left(k{\right)}_{t}\text{ }\ast w\left(k{\right)}_{t}i\left(k\right){t}_{w}{\left(}_{k}{\right)}_{t}
{}_{i}{\left(}_{k}{\right)}_{t}
= national MIR level of euro area country k at month t
{}_{w}{\left(}_{k}{\right)}_{t}
= national weight of euro area country k at month t, i.e. volume of national business in relation to euro area total.
The weighted variance is the squared deviations between the national and euro area MFI interest rates, calculated according to the national share in the total euro area business volume for a given instrument category.
By measuring, on a monthly basis, the variation of national interest rates around the euro area MFI interest rate (“euro area MIR”) adjusted for the level of the euro area rate, the coefficient of variation allows further a comparison of different MIR indicators.
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Train Reinforcement Learning Agent with Constraint Enforcement - MATLAB & Simulink - MathWorks India
The dynamics for the green ball from velocity
\mathit{v}
\mathit{x}
are governed by Newton's law with a small damping coefficient
\tau
\frac{1}{s\left(\tau s+1\right)}
The feasible region for the ball position
0\le \mathit{x}\le 1
and the velocity of the green ball is limited to the range
\left[-1,1\right]
The position of the target red ball is uniformly random across the range
\left[0,1\right]
. The agent can observe only a noisy estimate of this target position.
Computes the training reward
\mathit{r}={\left[1-10{\left(\mathit{x}-{\mathit{x}}_{\mathit{r}}\right)}^{2}\right]}^{+}
{\mathit{x}}_{\mathit{r}}
denotes the position of the red ball
Sets the termination signal isDone to true if the ball position violates the constraint
0\le \mathit{x}\le 1
In this example, the ball position signal
{\mathit{x}}_{\mathit{k}+1}
0\le
{\text{\hspace{0.17em}}\mathit{x}}_{\mathit{k}+1}\le 1
. To allow for some slack, the constraint is set to be
{0.1\le \text{\hspace{0.17em}}\mathit{x}}_{\mathit{k}+1}\le 0.9.
The dynamic model from velocity to position has a very small damping constant, thus it can be approximated by
{\mathit{x}}_{\mathit{k}+1}\approx {\mathit{x}}_{\mathit{k}}+\mathit{h}\left({\mathit{x}}_{\mathit{k}}\right){\mathit{u}}_{\mathit{k}}
. Therefore, the constraints for green ball are given by the following equation.
\left[\begin{array}{c}{\mathit{x}}_{\mathit{k}}\\ {-\mathit{x}}_{\mathit{k}}\end{array}\right]+\left[\begin{array}{c}\mathit{h}\left({\mathit{x}}_{\mathit{k}}\right)\\ -\mathit{h}\left({\mathit{x}}_{\mathit{k}}\right)\end{array}\right]{\mathit{u}}_{\mathit{k}}\le \left[\begin{array}{c}0.9\\ -0.1\end{array}\right]
{\mathit{f}}_{\mathit{x}}+{\mathit{g}}_{\mathit{x}}\mathit{u}\le \mathit{c}
. For the above equation, the coefficients of this constraint function are as follows.
{\mathit{f}}_{\mathit{x}}=\left[\begin{array}{c}{\mathit{x}}_{\mathit{k}}\\ {-\mathit{x}}_{\mathit{k}}\end{array}\right],{\mathit{g}}_{\mathit{x}}=\left[\begin{array}{c}\mathit{h}\left({\mathit{x}}_{\mathit{k}}\right)\\ -\mathit{h}\left({\mathit{x}}_{\mathit{k}}\right)\end{array}\right],\mathit{c}=\left[\begin{array}{c}0.9\\ -0.1\end{array}\right]
\mathit{h}\left({\mathit{x}}_{\mathit{k}}\right)
is approximated by a deep neural network that is trained on the data collected by simulating the RL agent within the environment. To learn the unknown function
\mathit{h}\left({\mathit{x}}_{\mathit{k}}\right)
, the RL agent passes a random external action to the environment that is uniformly distributed in the range
\left[-1,1\right]
To collect data, use the collectDataBall helper function. This function simulates the environment and agent and collects the resulting input and output data. The resulting training data has three columns:
{\mathit{x}}_{\mathit{k}}
{\mathit{u}}_{\mathit{k}}
{\mathit{x}}_{\mathit{k}+1}
\mathit{h}\left({\mathit{x}}_{\mathit{k}}\right)
, and the Constraint Enforcement block enforces the constraint function and velocity bounds.
Since Total Number of Steps equals the product of Episode Number and Episode Steps, each training episode runs to the end without early termination. Therefore, the Constraint Enforcement block ensures that the ball position
\mathit{x}
never violates the constraint
0\le \mathit{x}\le 1
RL Agent | Constraint Enforcement (Simulink Control Design)
Constraint Enforcement for Control Design (Simulink Control Design)
Train RL Agent for Adaptive Cruise Control with Constraint Enforcement (Simulink Control Design)
Train RL Agent for Lane Keeping Assist with Constraint Enforcement (Simulink Control Design)
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James noticed that his ruler includes eighths and sixteenths of an inch. Help him to make the following conversions.
\frac { 2 } { 16 } = \frac { \square } { 8 }
You may have noticed that
8
16
\text{ It will help to know that it is also true that the number of }\frac{1}{8}\text{'s is equal to half the number of }\frac{1}{16}\text{'s}.
\frac { 4 } { 16 } = \frac { \square } { 8 }
\text{Remember that the number of }\frac{1}{8}\text{'s is half the number of } \frac{1}{16}\text{'s. What is half of four?}
\frac { 10 } { 16 } = \frac { \square } { 8 }
5
\frac { 18 } { 16 } = \frac { \square } { 8 }
9
In general, if you have any number of sixteenths, how can you figure out how many eighths you have?
How were you able to solve parts (a) through (d)?
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Submersion (mathematics) - Wikipedia
"Regular point" redirects here. For "regular point of an algebraic variety", see Singular point of an algebraic variety.
In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion.
2 Submersion theorem
3.1 Maps between spheres
3.2 Families of algebraic varieties
4 Local normal form
5 Topological manifold submersions
Let M and N be differentiable manifolds and
{\displaystyle f\colon M\to N}
be a differentiable map between them. The map f is a submersion at a point
{\displaystyle p\in M}
if its differential
{\displaystyle Df_{p}\colon T_{p}M\to T_{f(p)}N}
is a surjective linear map.[1] In this case p is called a regular point of the map f, otherwise, p is a critical point. A point
{\displaystyle q\in N}
is a regular value of f if all points p in the preimage
{\displaystyle f^{-1}(q)}
are regular points. A differentiable map f that is a submersion at each point
{\displaystyle p\in M}
is called a submersion. Equivalently, f is a submersion if its differential
{\displaystyle Df_{p}}
has constant rank equal to the dimension of N.
A word of warning: some authors use the term critical point to describe a point where the rank of the Jacobian matrix of f at p is not maximal.[2] Indeed, this is the more useful notion in singularity theory. If the dimension of M is greater than or equal to the dimension of N then these two notions of critical point coincide. But if the dimension of M is less than the dimension of N, all points are critical according to the definition above (the differential cannot be surjective) but the rank of the Jacobian may still be maximal (if it is equal to dim M). The definition given above is the more commonly used; e.g., in the formulation of Sard's theorem.
Submersion theoremEdit
Given a submersion between smooth manifolds
{\displaystyle f\colon M\to N}
of dimensions
{\displaystyle m}
{\displaystyle n}
{\displaystyle x\in M}
there are surjective charts
{\displaystyle \phi :U\to \mathbb {R} ^{m}}
{\displaystyle M}
{\displaystyle x}
{\displaystyle \psi :V\to \mathbb {R} ^{n}}
{\displaystyle N}
{\displaystyle f(x)}
{\displaystyle f}
restricts to a submersion
{\displaystyle f\colon U\to V}
which, when expressed in coordinates as
{\displaystyle \psi \circ f\circ \phi ^{-1}:\mathbb {R} ^{m}\to \mathbb {R} ^{n}}
, becomes an ordinary orthogonal projection. As an application, for each
{\displaystyle p\in N}
the corresponding fiber o{\displaystyle f}
{\displaystyle M_{p}=f^{-1}(\{p\})}
can be equipped with the structure of a smooth submanifold of
{\displaystyle M}
whose dimension is equal to the difference of the dimensions of
{\displaystyle N}
{\displaystyle M}
{\displaystyle f\colon \mathbb {R} ^{3}\to \mathbb {R} }
{\displaystyle f(x,y,z)=x^{4}+y^{4}+z^{4}.}
{\displaystyle {\begin{bmatrix}{\frac {\partial f}{\partial x}}&{\frac {\partial f}{\partial y}}&{\frac {\partial f}{\partial z}}\end{bmatrix}}={\begin{bmatrix}4x^{3}&4y^{3}&4z^{3}\end{bmatrix}}.}
This has maximal rank at every point except for
{\displaystyle (0,0,0)}
. Also, the fibers
{\displaystyle f^{-1}(\{t\})=\left\{(a,b,c)\in \mathbb {R} ^{3}:a^{4}+b^{4}+c^{4}=t\right\}}
are empty for
{\displaystyle t<0}
, and equal to a point when
{\displaystyle t=0}
. Hence we only have a smooth submersion
{\displaystyle f\colon \mathbb {R} ^{3}\setminus \{(0,0,0)\}\to \mathbb {R} _{>0},}
{\displaystyle M_{t}=\left\{(a,b,c)\in \mathbb {R} ^{3}:a^{4}+b^{4}+c^{4}=t\right\}}
are two-dimensional smooth manifolds for
{\displaystyle t>0}
{\displaystyle \pi \colon \mathbb {R} ^{m+n}\rightarrow \mathbb {R} ^{n}\subset \mathbb {R} ^{m+n}}
Local diffeomorphisms
The projection in a smooth vector bundle or a more general smooth fibration. The surjectivity of the differential is a necessary condition for the existence of a local trivialization.
Maps between spheresEdit
One large class of examples of submersions are submersions between spheres of higher dimension, such as
{\displaystyle f:S^{n+k}\to S^{k}}
whose fibers have dimensio{\displaystyle n}
. This is because the fibers (inverse images of elements
{\displaystyle p\in S^{k}}
) are smooth manifolds of dimensio{\displaystyle n}
. Then, if we take a path
{\displaystyle \gamma :I\to S^{k}}
and take the pullback
{\displaystyle {\begin{matrix}M_{I}&\to &S^{n+k}\\\downarrow &&\downarrow f\\I&\xrightarrow {\gamma } &S^{k}\end{matrix}}}
we get an example of a special kind of bordism, called a framed bordism. In fact, the framed cobordism groups
{\displaystyle \Omega _{n}^{fr}}
are intimately related to the stable homotopy groups.
Families of algebraic varietiesEdit
Another large class of submersions are given by families of algebraic varieties
{\displaystyle \pi :{\mathfrak {X}}\to S}
whose fibers are smooth algebraic varieties. If we consider the underlying manifolds of these varieties, we get smooth manifolds. For example, the Weierstauss family
{\displaystyle \pi :{\mathcal {W}}\to \mathbb {A} ^{1}}
of elliptic curves is a widely studied submersion because it includes many technical complexities used to demonstrate more complex theory, such as intersection homology and perverse sheaves. This family is given by
{\displaystyle {\mathcal {W}}=\{(t,x,y)\in \mathbb {A} ^{1}\times \mathbb {A} ^{2}:y^{2}=x(x-1)(x-t)\}}
{\displaystyle \mathbb {A} ^{1}}
is the affine line and
{\displaystyle \mathbb {A} ^{2}}
is the affine plane. Since we are considering complex varieties, these are equivalently the spaces
{\displaystyle \mathbb {C} ,\mathbb {C} ^{2}}
of the complex line and the complex plane. Note that we should actually remove the points
{\displaystyle t=0,1}
because there are singularities (since there is a double root).
Local normal formEdit
If f: M → N is a submersion at p and f(p) = q ∈ N, then there exists an open neighborhood U of p in M, an open neighborhood V of q in N, and local coordinates (x1, …, xm) at p and (x1, …, xn) at q such that f(U) = V, and the map f in these local coordinates is the standard projection
{\displaystyle f(x_{1},\ldots ,x_{n},x_{n+1},\ldots ,x_{m})=(x_{1},\ldots ,x_{n}).}
It follows that the full preimage f−1(q) in M of a regular value q in N under a differentiable map f: M → N is either empty or is a differentiable manifold of dimension dim M − dim N, possibly disconnected. This is the content of the regular value theorem (also known as the submersion theorem). In particular, the conclusion holds for all q in N if the map f is a submersion.
Topological manifold submersionsEdit
Submersions are also well-defined for general topological manifolds.[3] A topological manifold submersion is a continuous surjection f : M → N such that for all p in M, for some continuous charts ψ at p and φ at f(p), the map ψ−1 ∘ f ∘ φ is equal to the projection map from Rm to Rn, where m = dim(M) ≥ n = dim(N).
Ehresmann's fibration theorem
^ Crampin & Pirani 1994, p. 243. do Carmo 1994, p. 185. Frankel 1997, p. 181. Gallot, Hulin & Lafontaine 2004, p. 12. Kosinski 2007, p. 27. Lang 1999, p. 27. Sternberg 2012, p. 378.
^ Arnold, Gusein-Zade & Varchenko 1985.
Arnold, Vladimir I.; Gusein-Zade, Sabir M.; Varchenko, Alexander N. (1985). Singularities of Differentiable Maps: Volume 1. Birkhäuser. ISBN 0-8176-3187-9.
Bruce, James W.; Giblin, Peter J. (1984). Curves and Singularities. Cambridge University Press. ISBN 0-521-42999-4. MR 0774048.
Frankel, Theodore (1997). The Geometry of Physics. Cambridge: Cambridge University Press. ISBN 0-521-38753-1. MR 1481707.
Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques (2004). Riemannian Geometry (3rd ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-20493-0.
Sternberg, Shlomo Zvi (2012). Curvature in Mathematics and Physics. Mineola, New York: Dover Publications. ISBN 978-0-486-47855-5.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Submersion_(mathematics)&oldid=1077520750"
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Independence (probability theory) - Wikipedia
Fundamental concept in probability theory
1.1 For events
1.1.1 Two events
1.1.2 Log probability and information content
1.1.4 More than two events
1.2 For real valued random variables
1.2.1 Two random variables
1.2.2 More than two random variables
1.3 For real valued random vectors
1.4 For stochastic processes
1.4.1 For one stochastic process
1.4.2 For two stochastic processes
1.5 Independent σ-algebras
2.1 Self-independence
3.4 Triple-independence but no pairwise-independence
4 Conditional independence
4.2 For random variables
{\displaystyle A}
{\displaystyle B}
{\displaystyle A\perp B}
{\displaystyle A\perp \!\!\!\perp B}
) if and only if their joint probability equals the product of their probabilities:[2]: p. 29 [3]: p. 10
{\displaystyle \mathrm {P} (A\cap B)=\mathrm {P} (A)\mathrm {P} (B)}
{\displaystyle A}
{\displaystyle B}
{\displaystyle A\cap B=\emptyset }
{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}}
{\displaystyle A}
{\displaystyle B}
{\displaystyle \mathrm {P} (A\cap B)=\mathrm {P} (A)\mathrm {P} (B)\iff \mathrm {P} (A\mid B)={\frac {\mathrm {P} (A\cap B)}{\mathrm {P} (B)}}=\mathrm {P} (A).}
{\displaystyle \mathrm {P} (A\cap B)=\mathrm {P} (A)\mathrm {P} (B)\iff \mathrm {P} (B\mid A)={\frac {\mathrm {P} (A\cap B)}{\mathrm {P} (A)}}=\mathrm {P} (B).}
{\displaystyle B}
{\displaystyle A}
{\displaystyle A}
{\displaystyle B}
{\displaystyle \mathrm {P} (A)}
{\displaystyle \mathrm {P} (B)}
{\displaystyle A}
{\displaystyle B}
{\displaystyle B}
{\displaystyle A}
Log probability and information content[edit]
{\displaystyle \log \mathrm {P} (A\cap B)=\log \mathrm {P} (A)+\log \mathrm {P} (B)}
{\displaystyle \mathrm {I} (A\cap B)=\mathrm {I} (A)+\mathrm {I} (B)}
Odds[edit]
{\displaystyle A}
{\displaystyle B}
{\displaystyle O(A\mid B)=O(A){\text{ and }}O(B\mid A)=O(B),}
{\displaystyle O(A\mid B)=O(A\mid \neg B){\text{ and }}O(B\mid A)=O(B\mid \neg A).}
{\displaystyle O(A\mid B):O(A\mid \neg B),}
{\displaystyle B}
{\displaystyle A}
More than two events[edit]
{\displaystyle \{A_{i}\}_{i=1}^{n}}
{\displaystyle m,k}
{\displaystyle \mathrm {P} (A_{m}\cap A_{k})=\mathrm {P} (A_{m})\mathrm {P} (A_{k})}
{\displaystyle k\leq n}
{\displaystyle 1\leq i_{1}<\dots <i_{k}\leq n}
{\displaystyle \mathrm {P} \left(\bigcap _{j=1}^{k}A_{i_{j}}\right)=\prod _{j=1}^{k}\mathrm {P} (A_{i_{j}})}
For real valued random variables[edit]
Two random variables[edit]
{\displaystyle X}
{\displaystyle Y}
{\displaystyle x}
{\displaystyle y}
{\displaystyle \{X\leq x\}}
{\displaystyle \{Y\leq y\}}
{\displaystyle X}
{\displaystyle Y}
{\displaystyle F_{X}(x)}
{\displaystyle F_{Y}(y)}
{\displaystyle (X,Y)}
{\displaystyle F_{X,Y}(x,y)=F_{X}(x)F_{Y}(y)\quad {\text{for all }}x,y}
{\displaystyle f_{X}(x)}
{\displaystyle f_{Y}(y)}
{\displaystyle f_{X,Y}(x,y)}
{\displaystyle f_{X,Y}(x,y)=f_{X}(x)f_{Y}(y)\quad {\text{for all }}x,y.}
More than two random variables[edit]
{\displaystyle n}
{\displaystyle \{X_{1},\ldots ,X_{n}\}}
{\displaystyle n}
{\displaystyle \{X_{1},\ldots ,X_{n}\}}
{\displaystyle \{x_{1},\ldots ,x_{n}\}}
{\displaystyle \{X_{1}\leq x_{1}\},\ldots ,\{X_{n}\leq x_{n}\}}
{\displaystyle F_{X_{1},\ldots ,X_{n}}(x_{1},\ldots ,x_{n})}
{\displaystyle n}
{\displaystyle \{X_{1},\ldots ,X_{n}\}}
{\displaystyle F_{X_{1},\ldots ,X_{n}}(x_{1},\ldots ,x_{n})=F_{X_{1}}(x_{1})\cdot \ldots \cdot F_{X_{n}}(x_{n})\quad {\text{for all }}x_{1},\ldots ,x_{n}}
{\displaystyle k}
{\displaystyle n}
{\displaystyle F_{X_{1},X_{2},X_{3}}(x_{1},x_{2},x_{3})=F_{X_{1}}(x_{1})\cdot F_{X_{2}}(x_{2})\cdot F_{X_{3}}(x_{3})}
{\displaystyle F_{X_{1},X_{3}}(x_{1},x_{3})=F_{X_{1}}(x_{1})\cdot F_{X_{3}}(x_{3})}
{\displaystyle \{X\in A\}}
{\displaystyle \{X\leq x\}}
{\displaystyle A}
For real valued random vectors[edit]
{\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{m})^{\mathrm {T} }}
{\displaystyle \mathbf {Y} =(Y_{1},\ldots ,Y_{n})^{\mathrm {T} }}
{\displaystyle F_{\mathbf {X,Y} }(\mathbf {x,y} )=F_{\mathbf {X} }(\mathbf {x} )\cdot F_{\mathbf {Y} }(\mathbf {y} )\quad {\text{for all }}\mathbf {x} ,\mathbf {y} }
{\displaystyle F_{\mathbf {X} }(\mathbf {x} )}
{\displaystyle F_{\mathbf {Y} }(\mathbf {y} )}
{\displaystyle \mathbf {X} }
{\displaystyle \mathbf {Y} }
{\displaystyle F_{\mathbf {X,Y} }(\mathbf {x,y} )}
{\displaystyle \mathbf {X} }
{\displaystyle \mathbf {Y} }
{\displaystyle \mathbf {X} \perp \!\!\!\perp \mathbf {Y} }
{\displaystyle \mathbf {X} }
{\displaystyle \mathbf {Y} }
{\displaystyle F_{X_{1},\ldots ,X_{m},Y_{1},\ldots ,Y_{n}}(x_{1},\ldots ,x_{m},y_{1},\ldots ,y_{n})=F_{X_{1},\ldots ,X_{m}}(x_{1},\ldots ,x_{m})\cdot F_{Y_{1},\ldots ,Y_{n}}(y_{1},\ldots ,y_{n})\quad {\text{for all }}x_{1},\ldots ,x_{m},y_{1},\ldots ,y_{n}.}
For stochastic processes[edit]
For one stochastic process[edit]
{\displaystyle n}
{\displaystyle t_{1},\ldots ,t_{n}}
{\displaystyle n}
{\displaystyle \left\{X_{t}\right\}_{t\in {\mathcal {T}}}}
{\displaystyle n\in \mathbb {N} }
{\displaystyle t_{1},\ldots ,t_{n}\in {\mathcal {T}}}
{\displaystyle F_{X_{t_{1}},\ldots ,X_{t_{n}}}(x_{1},\ldots ,x_{n})=F_{X_{t_{1}}}(x_{1})\cdot \ldots \cdot F_{X_{t_{n}}}(x_{n})\quad {\text{for all }}x_{1},\ldots ,x_{n}}
{\displaystyle F_{X_{t_{1}},\ldots ,X_{t_{n}}}(x_{1},\ldots ,x_{n})=\mathrm {P} (X(t_{1})\leq x_{1},\ldots ,X(t_{n})\leq x_{n})}
For two stochastic processes[edit]
{\displaystyle \left\{X_{t}\right\}_{t\in {\mathcal {T}}}}
{\displaystyle \left\{Y_{t}\right\}_{t\in {\mathcal {T}}}}
{\displaystyle (\Omega ,{\mathcal {F}},P)}
{\displaystyle \left\{X_{t}\right\}_{t\in {\mathcal {T}}}}
{\displaystyle \left\{Y_{t}\right\}_{t\in {\mathcal {T}}}}
{\displaystyle n\in \mathbb {N} }
{\displaystyle t_{1},\ldots ,t_{n}\in {\mathcal {T}}}
{\displaystyle (X(t_{1}),\ldots ,X(t_{n}))}
{\displaystyle (Y(t_{1}),\ldots ,Y(t_{n}))}
{\displaystyle F_{X_{t_{1}},\ldots ,X_{t_{n}},Y_{t_{1}},\ldots ,Y_{t_{n}}}(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n})=F_{X_{t_{1}},\ldots ,X_{t_{n}}}(x_{1},\ldots ,x_{n})\cdot F_{Y_{t_{1}},\ldots ,Y_{t_{n}}}(y_{1},\ldots ,y_{n})\quad {\text{for all }}x_{1},\ldots ,x_{n}}
Independent σ-algebras[edit]
{\displaystyle (\Omega ,\Sigma ,\mathrm {P} )}
{\displaystyle {\mathcal {A}}}
{\displaystyle {\mathcal {B}}}
{\displaystyle \Sigma }
{\displaystyle {\mathcal {A}}}
{\displaystyle {\mathcal {B}}}
{\displaystyle A\in {\mathcal {A}}}
{\displaystyle B\in {\mathcal {B}}}
{\displaystyle \mathrm {P} (A\cap B)=\mathrm {P} (A)\mathrm {P} (B).}
{\displaystyle (\tau _{i})_{i\in I}}
{\displaystyle I}
{\displaystyle \forall \left(A_{i}\right)_{i\in I}\in \prod \nolimits _{i\in I}\tau _{i}\ :\ \mathrm {P} \left(\bigcap \nolimits _{i\in I}A_{i}\right)=\prod \nolimits _{i\in I}\mathrm {P} \left(A_{i}\right)}
{\displaystyle E\in \Sigma }
{\displaystyle \sigma (\{E\})=\{\emptyset ,E,\Omega \setminus E,\Omega \}.}
{\displaystyle X}
{\displaystyle Y}
{\displaystyle \Omega }
{\displaystyle X}
{\displaystyle S}
{\displaystyle \Omega }
{\displaystyle X^{-1}(U)}
{\displaystyle U}
{\displaystyle S}
{\displaystyle X}
{\displaystyle Y}
{\displaystyle Y}
{\displaystyle X}
{\displaystyle Y}
{\displaystyle \{\varnothing ,\Omega \}}
{\displaystyle Y}
Self-independence[edit]
{\displaystyle \mathrm {P} (A)=\mathrm {P} (A\cap A)=\mathrm {P} (A)\cdot \mathrm {P} (A)\iff \mathrm {P} (A)=0{\text{ or }}\mathrm {P} (A)=1.}
Expectation and covariance[edit]
{\displaystyle X}
{\displaystyle Y}
{\displaystyle \operatorname {E} }
{\displaystyle \operatorname {E} [XY]=\operatorname {E} [X]\operatorname {E} [Y],}
{\displaystyle \operatorname {cov} [X,Y]}
{\displaystyle \operatorname {cov} [X,Y]=\operatorname {E} [XY]-\operatorname {E} [X]\operatorname {E} [Y].}
{\displaystyle \left\{X_{t}\right\}_{t\in {\mathcal {T}}}}
{\displaystyle \left\{Y_{t}\right\}_{t\in {\mathcal {T}}}}
Characteristic function[edit]
{\displaystyle X}
{\displaystyle Y}
{\displaystyle (X,Y)}
{\displaystyle \varphi _{(X,Y)}(t,s)=\varphi _{X}(t)\cdot \varphi _{Y}(s).}
{\displaystyle \varphi _{X+Y}(t)=\varphi _{X}(t)\cdot \varphi _{Y}(t),}
Rolling dice[edit]
Drawing cards[edit]
Pairwise and mutual independence[edit]
{\displaystyle \mathrm {P} (A)=\mathrm {P} (B)=1/2}
{\displaystyle \mathrm {P} (C)=1/4}
{\displaystyle \mathrm {P} (A|B)=\mathrm {P} (A|C)=1/2=\mathrm {P} (A)}
{\displaystyle \mathrm {P} (B|A)=\mathrm {P} (B|C)=1/2=\mathrm {P} (B)}
{\displaystyle \mathrm {P} (C|A)=\mathrm {P} (C|B)=1/4=\mathrm {P} (C)}
{\displaystyle \mathrm {P} (A|BC)={\frac {\frac {4}{40}}{{\frac {4}{40}}+{\frac {1}{40}}}}={\tfrac {4}{5}}\neq \mathrm {P} (A)}
{\displaystyle \mathrm {P} (B|AC)={\frac {\frac {4}{40}}{{\frac {4}{40}}+{\frac {1}{40}}}}={\tfrac {4}{5}}\neq \mathrm {P} (B)}
{\displaystyle \mathrm {P} (C|AB)={\frac {\frac {4}{40}}{{\frac {4}{40}}+{\frac {6}{40}}}}={\tfrac {2}{5}}\neq \mathrm {P} (C)}
{\displaystyle \mathrm {P} (A|BC)={\frac {\frac {1}{16}}{{\frac {1}{16}}+{\frac {1}{16}}}}={\tfrac {1}{2}}=\mathrm {P} (A)}
{\displaystyle \mathrm {P} (B|AC)={\frac {\frac {1}{16}}{{\frac {1}{16}}+{\frac {1}{16}}}}={\tfrac {1}{2}}=\mathrm {P} (B)}
{\displaystyle \mathrm {P} (C|AB)={\frac {\frac {1}{16}}{{\frac {1}{16}}+{\frac {3}{16}}}}={\tfrac {1}{4}}=\mathrm {P} (C)}
Triple-independence but no pairwise-independence[edit]
{\displaystyle \mathrm {P} (A\cap B\cap C)=\mathrm {P} (A)\mathrm {P} (B)\mathrm {P} (C),}
Conditional independence[edit]
{\displaystyle A}
{\displaystyle B}
{\displaystyle C}
{\displaystyle \mathrm {P} (A\cap B\mid C)=\mathrm {P} (A\mid C)\cdot \mathrm {P} (B\mid C)}
For random variables[edit]
{\displaystyle X}
{\displaystyle Y}
{\displaystyle Z}
{\displaystyle Z}
{\displaystyle Y}
{\displaystyle X}
{\displaystyle X}
{\displaystyle Y}
{\displaystyle Z}
{\displaystyle Z}
{\displaystyle X}
{\displaystyle Y}
{\displaystyle Z}
{\displaystyle X}
{\displaystyle Y}
{\displaystyle Z}
{\displaystyle \mathrm {P} (X\leq x,Y\leq y\;|\;Z=z)=\mathrm {P} (X\leq x\;|\;Z=z)\cdot \mathrm {P} (Y\leq y\;|\;Z=z)}
{\displaystyle x}
{\displaystyle y}
{\displaystyle z}
{\displaystyle \mathrm {P} (Z=z)>0}
{\displaystyle f_{XYZ}(x,y,z)}
{\displaystyle X}
{\displaystyle Y}
{\displaystyle Z}
{\displaystyle f_{XY|Z}(x,y|z)=f_{X|Z}(x|z)\cdot f_{Y|Z}(y|z)}
{\displaystyle x}
{\displaystyle y}
{\displaystyle z}
{\displaystyle f_{Z}(z)>0}
{\displaystyle X}
{\displaystyle Y}
{\displaystyle Z}
{\displaystyle \mathrm {P} (X=x|Y=y,Z=z)=\mathrm {P} (X=x|Z=z)}
{\displaystyle x}
{\displaystyle y}
{\displaystyle z}
{\displaystyle \mathrm {P} (Z=z)>0}
{\displaystyle X}
{\displaystyle Y}
{\displaystyle Z}
{\displaystyle Z}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Independence_(probability_theory)&oldid=1088571443"
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Physics - Surfing in an Atom’s Wake
Surfing in an Atom’s Wake
September 26, 2000 • Phys. Rev. Focus 6, 14
The wavelike surface electrons on certain metal surfaces give rise to an unusual, long-range interatomic force that has now been directly measured.
Gerhard Meyer/Free Univ. of Berlin
Atoms making waves. Wavelike surface electrons scatter from a few copper atoms on a copper surface. These electrons generate a long-range force between atoms that has now been directly measured.
In 1993, physicists at IBM created the picturesque “quantum corral” by placing 48 iron atoms in a circle on a copper surface. The famous images dramatically displayed the standing waves made by surface electrons inside the corral. Now, in the 2 October PRL, a team shows that these same waves allow atoms dropped on the surface to interact with one another over long distances. Their scanning tunneling microscope (STM) data show that this electron-mediated force is oscillatory in space–alternately attractive and repulsive as one atom “rides” the electron waves produced by the other. The interaction leads to rings of attraction and repulsion surrounding each atom, so the results may improve understanding of the formation of atomic-scale structures on surfaces.
Theories dating back to 1967 have suggested that electrons in a metal generate so-called indirect interactions between adatoms–atoms sitting on a surface that aren’t part of the solid’s crystal structure. A 1978 paper by Nobel Laureate Walter Kohn of the University of California, Santa Barbara, and K. H. Lau predicted that, if the electrons are in specific quantum states at the surface, the force diminishes with the inverse square of the distance between the adatoms–a much longer-range interaction than exists otherwise. Lau and Kohn also expected the force to be oscillatory (similar to other indirect interactions), with a period related to the surface electrons’ wavelength. Under these conditions, the potential energy surface surrounding each adatom looks something like a still picture of the circular waves around a stone thrown in a pond. The wavelike surface electrons scattering from the adatom create ring-shaped troughs of attraction in the potential energy function, and neighboring adatoms are most likely to collect at these troughs. Until now, no one had directly measured this unusual long-range, oscillatory interaction predicted by Lau and Kohn.
The problem, explains Gerhard Meyer of the Free University of Berlin (FUB), is that these long-range forces between adatoms are so weak; the corresponding energies are less than 1 meV. Meyer is part of a team that captured the oscillatory interaction by taking 3400 STM images of copper adatoms on a copper surface at temperatures between 9 and 21 K. They waited 30 seconds between images to allow the adatoms to hop to new positions. For each image the team measured the distances between isolated pairs of adatoms and collected a large histogram showing the likelihood of each separation distance, from 0 to 7.5 nm. “What you see is that certain distances are preferred,” says Meyer.
The team, led by FUB’s Karl-Heinz Rieder, found an oscillatory potential energy function as one moves away from an adatom, with a period (1.5 nm) and decay rate (inverse square) in agreement with the 1978 predictions. To further verify their results, the team used the STM to directly image the electron waves surrounding pairs of adatoms. The properties of the scattered electron waves were in rough agreement with the authors’ theoretical predictions.
“I was thrilled” to see the paper, says Ted Einstein of the University of Maryland in College Park. Although there have been other hints of the effect, he says, “this is really clear-cut and beautiful.” Einstein points out that it is unusual for an oscillatory interaction to be so long-ranged and to have circular symmetry and says it might have practical consequences for interactions between single-atom-high steps in atomic-scale devices.
Substrate Mediated Long-Range Oscillatory Interaction between Adatoms: Cu
/
Cu(111)
Jascha Repp, Francesca Moresco, Gerhard Meyer, Karl-Heinz Rieder, Per Hyldgaard, and Mats Persson
/
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Jorge was thinking about using variables to represent lengths of a tightrope walker’s many tricks.
Jorge wrote the expression
x+x+x+x+3
to represent the sequence shown in the diagram at right. Does his expression make sense? Explain.
Try thinking about how you might write this expression first. Is it different from Jorge's?
Yes, this expression does make sense. Why do you think so?
Jorge explained why he chose this expression and said, ''I chose this expression because the tightrope is divided into separate parts, and I wanted to write one expression that could combine all parts into one. The
x
3
are all different lengths that make the tightrope a whole. So, by adding the lengths together, the total length could be found.''
Write an expression to represent the sequence shown in the diagram at right.
Together, all of these letters represent the entire length of the rope. How can you combine them into one expression?
In part (a), if
x=5
feet, how long is the tightrope?
Use Jorge's expression and substitute
5
x
's. Can you now find the value of the expression?
In part (b), if
j=3
k=2
Using the expression you wrote in part (b), try substituting the
j
3
k
2
Your expression should have looked something like this:
j+j+k+k+k
2j+3k
. Can you find the length now?
The tightrope is
12
feet long.
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The Carbon - Cycle — lesson. Science CBSE, Class 9.
The circulation of carbon from the atmosphere to the earth is known as the carbon cycle. This is a biogeochemical process.
The amount of carbon present in the atmosphere is \(0.04\%\). This is also released by the respiration process and on the complete combustion process. Carbon occurs in its elemental forms as graphite and diamond in nature.
The following is the process involved in the carbon cycle
Plants take in atmospheric
{\mathit{CO}}_{2}
for photosynthesis.
Animals feed on primary producers, and carbon is accumulated in them.
The carbon is released back to the atmosphere on the decomposition of dead plants and animals.
Some amount of carbon is retained in the soil and becomes fossil fuel.
The combustion of fossil fuels by human activities releases carbon back to the atmosphere.
The intake of
{\mathit{CO}}_{2}
from the atmosphere for preparing food with the help of sunlight is known as photosynthesis.
The carbon is passed from producers to consumers in the ecosystem from this process.
Fact: \(18.5\%\) of carbon is present in the human body.
The carbon present in the human body is not in pure form. It is found in proteins, carbohydrates, fats etc., which is the building blocks for these compounds. When plants and animals die, the carbon content is released back to the atmosphere by decomposition.
Carbon sink: The intake of
{\mathit{CO}}_{2}
is more compared to the release of
{\mathit{CO}}_{2}
The plants, soil and ocean act as a natural carbon sink. We know the shells of marine organisms are made of calcium carbonates. So when the marine organisms die, they get accumulated in the seafloor, and decomposition takes place.
Apart from the natural release, there is also an additional amount of
{\mathit{CO}}_{2}
which is released into the atmosphere due to modernisation techniques, burning, industrialisation etc., which can alter the carbon cycle. If the carbon cycle is altered, it will have a drastic effect on earth, which concerns the life on earth and other conditions such as global warming, climate change, etc.
Harry C, CC BY-SA 3.0 <https://creativecommons.org/licenses/by-sa/3.0>, via Wikimedia Commons
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Get Started with Cascade Object Detector - MATLAB & Simulink - MathWorks América Latina
Why Train a Detector?
What Kinds of Objects Can You Detect?
How Does the Cascade Classifier Work?
Create a Cascade Classifier Using the trainCascadeObjectDetector
Considerations when Setting Parameters
Feature Types Available for Training
Supply Positive Samples
Supply Negative Images
Training Time of Detector
What if you run out of positive samples?
What to do if you run out of negative samples?
The vision.CascadeObjectDetector System object comes with several pretrained classifiers for detecting frontal faces, profile faces, noses, eyes, and the upper body. However, these classifiers are not always sufficient for a particular application. Computer Vision Toolbox™ provides the trainCascadeObjectDetector function to train a custom classifier.
The Computer Vision Toolbox cascade object detector can detect object categories whose aspect ratio does not vary significantly. Objects whose aspect ratio remains fixed include faces, stop signs, and cars viewed from one side.
The vision.CascadeObjectDetector System object detects objects in images by sliding a window over the image. The detector then uses a cascade classifier to decide whether the window contains the object of interest. The size of the window varies to detect objects at different scales, but its aspect ratio remains fixed. The detector is very sensitive to out-of-plane rotation, because the aspect ratio changes for most 3-D objects. Thus, you need to train a detector for each orientation of the object. Training a single detector to handle all orientations will not work.
The cascade classifier consists of stages, where each stage is an ensemble of weak learners. The weak learners are simple classifiers called decision stumps. Each stage is trained using a technique called boosting. Boosting provides the ability to train a highly accurate classifier by taking a weighted average of the decisions made by the weak learners.
Each stage of the classifier labels the region defined by the current location of the sliding window as either positive or negative. Positive indicates that an object was found and negative indicates no objects were found. If the label is negative, the classification of this region is complete, and the detector slides the window to the next location. If the label is positive, the classifier passes the region to the next stage. The detector reports an object found at the current window location when the final stage classifies the region as positive.
The stages are designed to reject negative samples as fast as possible. The assumption is that the vast majority of windows do not contain the object of interest. Conversely, true positives are rare and worth taking the time to verify.
A true positive occurs when a positive sample is correctly classified.
A false positive occurs when a negative sample is mistakenly classified as positive.
A false negative occurs when a positive sample is mistakenly classified as negative.
To work well, each stage in the cascade must have a low false negative rate. If a stage incorrectly labels an object as negative, the classification stops, and you cannot correct the mistake. However, each stage can have a high false positive rate. Even if the detector incorrectly labels a nonobject as positive, you can correct the mistake in subsequent stages.
The overall false positive rate of the cascade classifier is
{f}^{s}
f
is the false positive rate per stage in the range (0 1), and
s
is the number of stages. Similarly, the overall true positive rate is
{t}^{s}
t
is the true positive rate per stage in the range (0 1]. Thus, adding more stages reduces the overall false positive rate, but it also reduces the overall true positive rate.
Cascade classifier training requires a set of positive samples and a set of negative images. You must provide a set of positive images with regions of interest specified to be used as positive samples. You can use the Image Labeler to label objects of interest with bounding boxes. The Image Labeler outputs a table to use for positive samples. You also must provide a set of negative images from which the function generates negative samples automatically. To achieve acceptable detector accuracy, set the number of stages, feature type, and other function parameters.
Select the function parameters to optimize the number of stages, the false positive rate, the true positive rate, and the type of features to use for training. When you set the parameters, consider these tradeoffs.
A large training set (in the thousands). Increase the number of stages and set a higher false positive rate for each stage.
A small training set. Decrease the number of stages and set a lower false positive rate for each stage.
To reduce the probability of missing an object. Increase the true positive rate. However, a high true positive rate can prevent you from achieving the desired false positive rate per stage, making the detector more likely to produce false detections.
To reduce the number of false detections. Increase the number of stages or decrease the false alarm rate per stage.
Choose the feature that suits the type of object detection you need. The trainCascadeObjectDetector supports three types of features: Haar, local binary patterns (LBP), and histograms of oriented gradients (HOG). Haar and LBP features are often used to detect faces because they work well for representing fine-scale textures. The HOG features are often used to detect objects such as people and cars. They are useful for capturing the overall shape of an object. For example, in the following visualization of the HOG features, you can see the outline of the bicycle.
You might need to run the trainCascadeObjectDetector function multiple times to tune the parameters. To save time, you can use LBP or HOG features on a small subset of your data. Training a detector using Haar features takes much longer. After that, you can run the Haar features to see if the accuracy improves.
To create positive samples easily, you can use the Image Labeler app. The Image Labeler provides an easy way to label positive samples by interactively specifying rectangular regions of interest (ROIs).
You can also specify positive samples manually in one of two ways. One way is to specify rectangular regions in a larger image. The regions contain the objects of interest. The other approach is to crop out the object of interest from the image and save it as a separate image. Then, you can specify the region to be the entire image. You can also generate more positive samples from existing ones by adding rotation or noise, or by varying brightness or contrast.
Negative samples are not specified explicitly. Instead, the trainCascadeObjectDetector function automatically generates negative samples from user-supplied negative images that do not contain objects of interest. Before training each new stage, the function runs the detector consisting of the stages already trained on the negative images. Any objects detected from these image are false positives, which are used as negative samples. In this way, each new stage of the cascade is trained to correct mistakes made by previous stages.
As more stages are added, the detector's overall false positive rate decreases, causing generation of negative samples to be more difficult. For this reason, it is helpful to supply as many negative images as possible. To improve training accuracy, supply negative images that contain backgrounds typically associated with the objects of interest. Also, include negative images that contain nonobjects similar in appearance to the objects of interest. For example, if you are training a stop-sign detector, include negative images that contain road signs and shapes similar to a stop sign.
There is a trade-off between fewer stages with a lower false positive rate per stage or more stages with a higher false positive rate per stage. Stages with a lower false positive rate are more complex because they contain a greater number of weak learners. Stages with a higher false positive rate contain fewer weak learners. Generally, it is better to have a greater number of simple stages because at each stage the overall false positive rate decreases exponentially. For example, if the false positive rate at each stage is 50%, then the overall false positive rate of a cascade classifier with two stages is 25%. With three stages, it becomes 12.5%, and so on. However, the greater the number of stages, the greater the amount of training data the classifier requires. Also, increasing the number of stages increases the false negative rate. This increase results in a greater chance of rejecting a positive sample by mistake. Set the false positive rate (FalseAlarmRate) and the number of stages, (NumCascadeStages) to yield an acceptable overall false positive rate. Then you can tune these two parameters experimentally.
Training can sometimes terminate early. For example, suppose that training stops after seven stages, even though you set the number of stages parameter to 20. It is possible that the function cannot generate enough negative samples. If you run the function again and set the number of stages to seven, you do not get the same result. The results between stages differ because the number of positive and negative samples to use for each stage is recalculated for the new number of stages.
Training a good detector requires thousands of training samples. Large amounts of training data can take hours or even days to process. During training, the function displays the time it took to train each stage in the MATLAB® Command Window. Training time depends on the type of feature you specify. Using Haar features takes much longer than using LBP or HOG features.
The trainCascadeObjectDetector function automatically determines the number of positive samples to use to train each stage. The number is based on the total number of positive samples supplied by the user and the values of the TruePositiveRate and NumCascadeStages parameters.
The number of available positive samples used to train each stage depends on the true positive rate. The rate specifies what percentage of positive samples the function can classify as negative. If a sample is classified as a negative by any stage, it never reaches subsequent stages. For example, suppose you set the TruePositiveRate to 0.9, and all of the available samples are used to train the first stage. In this case, 10% of the positive samples are rejected as negatives, and only 90% of the total positive samples are available for training the second stage. If training continues, then each stage is trained with fewer and fewer samples. Each subsequent stage must solve an increasingly more difficult classification problem with fewer positive samples. With each stage getting fewer samples, the later stages are likely to overfit the data.
Ideally, use the same number of samples to train each stage. To do so, the number of positive samples used to train each stage must be less than the total number of available positive samples. The only exception is that when the value of TruePositiveRate times the total number of positive samples is less than 1, no positive samples are rejected as negatives.
The function calculates the number of positive samples to use at each stage using the following formula:
This calculation does not guarantee that the same number of positive samples are available for each stage. The reason is that it is impossible to predict with certainty how many positive samples will be rejected as negatives. The training continues as long as the number of positive samples available to train a stage is greater than 10% of the number of samples the function determined automatically using the preceding formula. If there are not enough positive samples the training stops and the function issues a warning. The function also outputs a classifier consisting of the stages that it had trained up to that point. If the training stops, you can add more positive samples. Alternatively, you can increase TruePositiveRate. Reducing the number of stages can also work, but such reduction can also result in a higher overall false alarm rate.
The function calculates the number of negative samples used at each stage. This calculation is done by multiplying the number of positive samples used at each stage by the value of NegativeSamplesFactor.
Just as with positive samples, there is no guarantee that the calculated number of negative samples are always available for a particular stage. The trainCascadeObjectDetector function generates negative samples from the negative images. However, with each new stage, the overall false alarm rate of the cascade classifier decreases, making it less likely to find the negative samples.
The training continues as long as the number of negative samples available to train a stage is greater than 10% of the calculated number of negative samples. If there are not enough negative samples, the training stops and the function issues a warning. It outputs a classifier consisting of the stages that it had trained up to that point. When the training stops, the best approach is to add more negative images. Alternatively, you can reduce the number of stages or increase the false positive rate.
Train a Five-Stage Stop-Sign Detector
This example shows you how to set up and train a five-stage, stop-sign detector, using 86 positive samples. The default value for TruePositiveRate is 0.995.
Step 1: Load the positive samples data from a MAT-file. In this example, file names and bounding boxes are contained in the array of structures labeled 'data'.
Step 2: Add the image directory to the MATLAB path.
Step 3: Specify the folder with negative images.
Step 4: Train the detector.
Computer Vision Toolbox software returns the following message:
All 86 positive samples were used to train each stage. This high rate occurs because the true positive rate is very high relative to the number of positive samples.
Train a Five-Stage Stop-Sign Detector with a Decreased True Positive Rate
This example shows you how to train a stop-sign detector on the same data set as the first example, (steps 1–3), but with the TruePositiveRate decreased to 0.98.
Only 79 of the total 86 positive samples were used to train each stage. This lowered rate occurs because the true positive rate was low enough for the function to start rejecting some of the positive samples as false negatives.
Train a Ten-Stage Stop-Sign Detector
This example shows you how to train a stop-sign detector on the same data set as the first example, (steps 1–3), but with the number of stages increased to 10.
In this case, NegativeSamplesFactor was set to 2, therefore the number of negative samples used to train each stage was 172. Notice that the function generated only 33 negative samples for stage 6 and was not able to train stage 7 at all. This condition occurs because the number of negatives in stage 7 was less than 17, (roughly half of the previous number of negative samples). The function produced a stop-sign detector with 6 stages, instead of the 10 previously specified. The resulting overall false alarm rate is 0.27=1.28e-05, while the expected false alarm rate is 1.024e-07.
At this point, you can add more negative images, reduce the number of stages, or increase the false positive rate. For example, you can increase the false positive rate, FalseAlarmRate, to 0.5. The expected overall false-positive rate in this case is 0.0039.
This time the function trains eight stages before the threshold reaches the overall false alarm rate of 0.000587108 and training stops.
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Non-deterministic criterion extension Non-deterministic criterion extension | Technically Exists
Non-deterministic criterion extension
Non-deterministic criterion extension is a means of taking a criterion meant only for deterministic voting methods and creating a new version of it that can also apply to non-deterministic voting methods. This may be useful for making criteria robust to otherwise-deterministic voting methods that must occasionally resort to breaking ties randomly, as is standard behavior for many real-world elections. It is unclear to what extent this technique is useful for extending criteria that assume determinism to voting methods designed around randomness, such as random ballot.
Before this extension can be defined, some notation needs to be specified. In the following definition,
\{0, 1\}^\infty
refers to the set of all possible infinite bitstrings,
E
refers to the set of all elections (where an election is just a list of ballots), and
C
refers to the set of all candidates.
For a given criterion, a voting method
m
passes the extended version of that criterion if there exists a function
f : \{0, 1\}^\infty \times E \rightarrow C
such that both of the following hold:
r \in \{0, 1\}^\infty
chosen uniformly at random,
\Pr[f(r, e) = c] = \Pr[m(e) = c]
e \in E
c \in C
r \in \{0, 1\}^\infty
, the deterministic voting method
m_r = f(r, \cdot)
passes the original criterion.
The idea here is that
f
is essentially the same as
m
except with the randomness isolated to a single variable
r
. This makes it possible to ignore that randomness and compare the original criterion to
f
m_r
. At the same time, the extension only requires some function
f
with the right properties to exist in order to prevent the extended criterion from being dependent on the internal workings of the voting method, specifically how it uses its randomness.
r
is chosen from the set of infinite bitstrings in order to ensure that
f
has an arbitrarily large number of random bits at its disposal. If for some reason this is insufficient,
r
could instead be drawn from a set with a greater cardinality, such as the power set of
\{0, 1\}^\infty
It is trivial to show that any deterministic method
m_d
that passes the original criterion will also pass the extended version. Consider the function
f_d(r, e) = m_d(e)
\Pr[f_d(r, e) = c] = \Pr[m_d(e) = c]
e
c
, and it is also clear that for all values of
r
, the voting method
m_r = f_d(r, \cdot) = m_d
will pass any criterion that
m_d
passes. Thus,
f_d
meets both conditions and
m_d
passes the extended criterion.
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The Works of Lord Byron (ed. Coleridge, Prothero)/Poetry/Volume 4/Francesca of Rimini - Wikisource, the free online library
The Works of Lord Byron (ed. Coleridge, Prothero)/Poetry/Volume 4/Francesca of Rimini
< The Works of Lord Byron (ed. Coleridge, Prothero) | Poetry | Volume 4(Redirected from Francesca of Rimini)
1414000The Works of Lord Byron — Francesca of RiminiGeorge Gordon Byron
The MS. of "a literal translation, word for word (versed like the original), of the episode of Francesca of Rimini" (Letter March 23, 1820, Letters, 1900, iv. 421), was sent to Murray from Ravenna, March 20, 1820 (ibid., p. 419), a week after Byron had forwarded the MS. of the Prophecy of Dante. Presumably the translation had been made in the interval by way of illustrating and justifying the unfamiliar metre of the "Dante Imitation." In the letter which accompanied the translation he writes, "Enclosed you will find, line for line, in third rhyme (terza rima,) of which your British Blackguard reader as yet understands nothing, Fanny of Rimini. You know that she was born here, and married, and slain, from Cary, Boyd, such people already. I have done it into cramp English, line for line, and rhyme for rhyme, to try the possibility. You had best append it to the poems already sent by last three posts."
In the matter of the "British Blackguard," that is, the general reader, Byron spoke by the card. Hayley's excellent translation of the three first cantos of the Inferno (vide ante, "Introduction to the Prophecy of Dante," p. 237), which must have been known to a previous generation, was forgotten, and with earlier experiments in terza rima, by Chaucer and the sixteenth and seventeenth century poets, neither Byron nor the British Public had any familiar or definite acquaintance. But of late some interest had been awakened or revived in Dante and the Divina Commedia.
Cary's translation—begun in 1796, but not published as a whole till 1814—had met with a sudden and remarkable success. "The work, which had been published four years, but had remained in utter obscurity, was at once eagerly sought after. About a thousand copies of the first edition, that remained on hand, were immediately disposed of; in less than three months a new edition was called for." Moreover, the Quarterly and Edinburgh Reviews were loud in its praises (Memoir of H. F. Cary, 1847, ii. 28). Byron seems to have thought that a fragment of the Inferno, "versed like the original," would challenge comparison with Cary's rendering in blank verse, and would lend an additional interest to the "Pulci Translations, and the Dante Imitation." Dîs aliter visum and Byron's translation of the episode of Franasca of Rimini, remained unpublished till it appeared in the pages of The Letters and Journals of Lord Byron, 1830, ii. 309-311. (For separate translations of the episode, see Stories of the Italian Poets, by Leigh Hunt, 1846, i. 393-395, and for a rendering in blank verse by Lord [John] Russell, see Literary Souvenir, 1830, pp. 285-287.)
DANTE, L'INFERNO.
'Siede la terra dove nata fui
Sulla marina, dove il Po discende
Per aver pace co' seguaci sui.
Prese costui della bella persona
Che mi fu tolta, e il modo ancor m' offende.
Amor, che a nullo amato amar perdona,
Mi prese del costui piacer sì forte,
Che, come vedi, ancor non mi abbandona.
Amor condusse noi ad una morte:10
Caino attende chi vita ci spense.'
Queste parole da lor ci fur porte.
Da che io intesi quelle anime offense
Chinai 'l viso, e tanto il tenni basso,
Finchè il Poeta mi disse: 'Che pense?'
Quando risposi, cominciai: 'O lasso!
Quanti dolci pensier, quanto disio
Menò costoro al doloroso passo!'
Poi mi rivolsi a loro, e parla' io,
E cominiciai: 'Francesca, i tuoi martiri20
A lagrimar mi fanno tristo e pio.
Ma dimmi: al tempo de' dolci sospiri
A che e come concedette Amore,
Che conoscesti i dubbiosi desiri?'
Ed ella a me: 'Nessun maggior dolore
Che ricordarsi del tempo felice
Nella miseria; e ciò sa il tuo dottore.
Del nostro amor tu hai cotanto affetto
Farò come colui che piange e dice.30
Noi leggevamo un giorno per diletto
Di Lancelotto, come Amor lo strinse:
Soli eravamo, e senza alcun sospetto.
Quella lettura, e scolorocci il viso:
Ma solo un punto fu quel che ci vinse.
Esser baciato da cotanto amante,
Questi, che mai da me non fia diviso,
La bocca mi baciò tutto tremante:40
Galeotto fu il libro, e chi lo scrisse—
Quel giorno più non vi leggemmo avante
L' altro piangeva sì che di pietade
Io venni meno così com' io morisse:
FRANCESCA OF RIMINI.[1]
"The Land where I was born[2] sits by the Seas
Upon that shore to which the Po descends,
With all his followers, in search of peace.
Seized him for the fair person which was ta'en
From me[3], and me even yet the mode offends.
Love, who to none beloved to love again
Remits, seized me with wish to please, so strong,[4]
That, as thou see'st, yet, yet it doth remain.
Love to one death conducted us along,10
But Caina[5] waits for him our life who ended:"
These were the accents uttered by her tongue.—
I bowed my visage, and so kept it till—
'What think'st thou?' said the bard;[6] when I unbended,
How many sweet thoughts, what strong ecstacies,
Led these their evil fortune to fulfill!'
And said, 'Francesca, thy sad destinies20
Have made me sorrow till the tears arise.
By what and how thy Love to Passion rose,
So as his dim desires to recognize?'
Is to remind us of our happy days[7][8]
In misery, and that thy teacher knows.
Upon thy spirit with such Sympathy,
I will do even as he who weeps and says.[9][10]30
We read one day for pastime, seated nigh,
Of Lancilot, how Love enchained him too.
We were alone, quite unsuspiciously.
All o'er discoloured by that reading were;
But one point only wholly us o'erthrew;[11]
When we read the long-sighed-for smile of her,[12]
To be thus kissed by such devoted lover,[13]
He, who from me can be divided ne'er,
Kissed my mouth, trembling in the act all over:40
Accurséd was the book and he who wrote![14]
That day no further leaf we did uncover.'
The other wept, so that with Pity's thralls
I swooned, as if by Death I had been smote,[15]
And fell down even as a dead body falls."[16]
↑ [Dante, in his Inferno (Canto V. lines 97-142), places Francesca and her lover Paolo among the lustful in the second circle of Hell. Francesca, daughter of Guido Vecchio da Polenta, Lord of Ravenna, married (circ. 1275) Gianciotto, second son of Malatesta da Verucchio, Lord of Rimini. According to Boccaccio (Il Comento sopra la Commedia, 1863, i. 476, sq.), Gianciotto was "hideously deformed in countenance and figure," and determined to woo and marry Francesca by proxy. He accordingly "sent, as his representative, his younger brother Paolo, the handsomest and most accomplished man in all Italy. Francesca saw Paolo arrive, and imagined she beheld her future husband. That mistake was the commencement of her passion." A day came when the lovers were surprised together, and Gianciotto slew both his brother and his wife.]
↑ ["On arrive à Ravenne en longeant une forêt de pins qui a sept lieues de long, et qui me semblait un immense bois funèbre servant d'avenue au sépulcre commun de ces deux grandes puissances. A peine y a-t-il place pour d'autres souvenirs à côté de leur mémoire. Cependant d'autres noms poétiques sont attachés à la Pineta de Ravenne. Naguère lord Byron y évoquait les fantastiques récits empruntés par Dryden à Boccace, et lui-même est maintenant une figure du passé, errante dans ce lieu mélancolique. Je songeais, en le traversant, que le chantre du désespoir avait chevauché sur cette plage lugubre, fouiée avant lui par le pas grave et lent du poëte de l'Enfer....
"Il suffit de Jeter les yeux sur une carte pour reconnaitre l'exactitude topographique de cette dernière expression. En effet, dans toute la partie supérieure de son cours, le Po reçoit une foule d'affluents qui convergent vers son lit; ce sont le Tésin, l'Adda, l'Olio, le Mincio, la Trebbia, la Bormida, le Taro...."—La Grèce, Rome, et Dante ("Voyage Dantesque"), par M. J. J. Ampère, 1850, pp. 311-313.]
↑ [The meaning is that she was despoiled of her beauty by death, and that the manner of her death excites her indignation still.
"Among Lord Byron's unpublished letters we find the following varied readings of the translation from Dante:—
Bloom was ta'en from me, yet the mode offends.
Seized me
{\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}}
with mutual wish to please
with wish of pleasing him
with the desire to please
{\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}}
so strong,
That, as thou see'st, not yet that passion quits, etc.
You will find these readings vary from the MS. I sent you. They are closer, but rougher: take which is liked best; or, if you like, print them as variations. They are all close to the text."—Works of Lord Byron, 1832, xii. 5, note 2.]
↑ ["The man's desire is for the woman; but the woman's desire is rarely other than for the desire of the man."—S. T. Coleridge, Table Talk, July 23, 1827.]
↑ [Caïna is the first belt of Cocytus, that is, circle ix. of the Inferno, in which fratricides and betrayers of their kindred are immersed up to the neck.]
↑ [Virgil.]
Is to recall to mind our happy days.
In misery, and this thy teacher knows.—[MS.]
↑ [The sentiment is derived from Boethius: "In omni adversitate fortunæ infelicissimum genus est infortunii, fuisse felicem."—De Consolat, Philos. Lib. II. Prosa 4. The earlier commentators (e.g. Venturi and Biagioli), relying on a passage in the Convito (ii. 16), assume that the "teacher" (line 27) is the author of the sentence, but later authorities point out that "mio dottore" can only apply to Virgil (v. 70), who then and there in the world of shades was suffering the bitter experience of having "known better days." Compare—
"For of fortunes sharp adversitee
A man to have ben in prosperitee,
And it remembren whan it passéd is."
Troilus and Criseyde, Bk. III. stanza ccxxxiii. lines 1-4.
"E perché rimembrare il ben perduto
Fa più meschino lo stato presente."
Fortiguerra's Ricciardetto, Canto XI. stanza Ixxxiii.
Compare, too—
"A sorrow's crown of sorrow is remembering happier things."
Tennyson's Locksley Hall.]
I will relate as he who weeps and says.—[MS.]
(The sense is, I will do even as one who relates while weeping.)
↑ Byron affixed the following note to line 126 of the Italian: "In some of the editions it is 'dirò,' in others 'faro;'—an essential difference between 'saying' and 'doing' which I know not how to decide—Ask Foscolo—the damned editions drive me mad." In La Divina Commedia, Firenze, 1892. and the Opere de Dante, Oxford, 1897, the reading is faro.]
↑ —— wholly overthrew.—[MS.]
↑ When we read the desired-for smile of her,—[MS. Alternative reading.]
↑ —— by such a fervent lover.—[MS.]
↑ ["A Gallehault was the book and he who wrote it" (A. J. Butler). "Writer and book were Gallehault to our will" (E. J. Plumptre). The book which the lovers were reading is entitled L'Illustre et Famosa Historia di Lancilotto del Lago. The "one point" of the original runs thus: "Et la reina ... lo piglia per il mento, et lo bacia davanti a Gallehault, assai lungamente."—Venice, 1558, Lib. Prim, cap. lxvi. vol. i. p. 229. The Gallehault of the Lancilotto, the shameless "purveyor," must not be confounded with the stainless Galahad of the Morte d' Arthur.]
↑ [Dante was in his twentieth, or twenty-first year when the tragedy of Francesca and Paolo was enacted, not at Rimini, but at Pesaro. Some acquaintance he may have had with her, through his friend Guido (not her father, but probably her nephew), enough to account for the peculiar emotion caused by her sangiunary doom.]
Alternative Versions transcribed by Mrs. Shelley.
line 4: Love, which too soon the soft heart apprehends,
Seized him for the fair form, the which was there
Torn from me, and even yet the mode offends.
line 8: Remits, seized him for me with joy so strong—
line 12: These were the words then uttered—
Since I had first perceived these souls offended,
I bowed my visage and so kept it till—
"What think'st thou?" said the bard, whom I (sic)
And then commenced—"Alas unto such ill—
line 18: Led these?" and then I turned me to them still
And spoke, "Francesca, thy sad destinies
Have made me sad and tender even to tears,
But tell me, in the season of sweet sighs,
By what and how Love overcame your fears,
So ye might recognize his dim desires?"
Then she to me, "No greater grief appears
Than, when the time of happiness expires,
To recollect, and this your teacher knows,
But if to find the first root of our——
Thou seek'st with such a sympathy in woes,
I will do even as he who weeps and speaks.
We read one day for pleasure, sitting close,
Of Launcelot, where forth his passion breaks.
We were alone and we suspected nought,
But oft our eyes exchanged, and changed our cheeks.
When we read the desiring smile of her
Who to be kissed by such true lover sought,
All tremulously kissed my trembling mouth.
Accursed the book and he who wrote it were—
That day no further did we read in sooth."
While the one spirit in this manner spoke
The other wept, so that, for very ruth,
I felt as if my trembling heart had broke,
To see the misery which both enthralls:
So that I swooned as dying with the stroke,—
And fell down even as a dead body falls.
line 21: Have made me sad even until the tears arise—
line 27: In wretchedness, and that your teacher knows.
line 31: We read one day for pleasure—
Of Launcelot, how passion shook his frame.
We were alone all unsuspiciously.
But oft our eyes met and our cheeks the same,
Pale and discoloured by that reading were;
But one part only wholly overcame;
When we read the desiring smile of her
Who sought the kiss of such devoted lover;
He who from me can be divided ne'er
Kissed my mouth, trembling to that kiss all over!
Accurséd was that book and he who wrote—
That day we did no further page uncover."
While thus— etc
line 45: I swooned to death with sympathetic thought—
[Another version.]
line 33: We were alone, and we suspected nought.
But oft our meeting eyes made pale our cheeks,
Urged by that reading for our ruin wrought;
But one point only wholly overcame:
When we read the desiring smile which sought
By such true lover to be kissed—the same
Who from my side can be divided ne'er
Kissed my mouth, trembling o'er all his frame!
Accurst the book, etc., etc.
line 33: We were alone and—etc.
But one point only 'twas our ruin wrought.
Who to be kissed of such true lover sought;
He who for me, etc., etc.
Retrieved from "https://en.wikisource.org/w/index.php?title=The_Works_of_Lord_Byron_(ed._Coleridge,_Prothero)/Poetry/Volume_4/Francesca_of_Rimini&oldid=4183584"
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57R45 Singularities of differentiable mappings
57R10 Smoothing
57R12 Smooth approximations
57R27 Controllability of vector fields on
{C}^{\infty }
and real-analytic manifolds
57R35 Differentiable mappings
57R40 Embeddings
57R42 Immersions
57R50 Diffeomorphisms
57R52 Isotopy
57R70 Critical points and critical submanifolds
\mathrm{O}
\mathrm{SO}
57R77 Complex cobordism (
\mathrm{U}
\mathrm{SU}
-cobordism)
h
s
A generalization of Thom’s transversality theorem
Lukáš Vokřínek (2008)
We prove a generalization of Thom’s transversality theorem. It gives conditions under which the jet map
{f}_{*}{|}_{Y}:Y\subseteq {J}^{r}\left(D,M\right)\to {J}^{r}\left(D,N\right)
is generically (for
f:M\to N
) transverse to a submanifold
Z\subseteq {J}^{r}\left(D,N\right)
. We apply this to study transversality properties of a restriction of a fixed map
g:M\to P
to the preimage
{\left({j}^{s}f\right)}^{-1}\left(A\right)
of a submanifold
A\subseteq {J}^{s}\left(M,N\right)
in terms of transversality properties of the original map
f
. Our main result is that for a reasonable class of submanifolds
A
and a generic map
f
{g|}_{{\left({j}^{s}f\right)}^{-1}\left(A\right)}
is also generic. We also present an example of
A
where the...
A note on octahedral spherical foldings.
d'Azevedo Breda, A.M. (1996)
A Topological Invariant for Stable Map Germs.
Addendum to: A Bound for the Fixed-Point Index of an Area-Preserving Map with Applications to Mechanics.
{C}^{\infty }
Olivier Le Gal, Jean-Philippe Rolin (2009)
We present an example of an o-minimal structure which does not admit
{C}^{\infty }
cellular decomposition. To this end, we construct a function
H
whose germ at the origin admits a
{C}^{k}
representative for each integer
k
, but no
{C}^{\infty }
representative. A number theoretic condition on the coefficients of the Taylor series of
H
then insures the quasianalyticity of some differential algebras
{𝒜}_{n}\left(H\right)
H
. The o-minimality of the structure generated by
H
is deduced from this quasianalyticity property.
Branched Immersions of Surfaces and Reduction of Topological Type, I.
Robert Gulliver (1975)
Branched Immersions of Surfaces and Reduction of Topological Type. II.
Bernard Morin (1975)
Calculation of the avoiding ideal for
{\Sigma }^{1,1}
Tamás Terpai (2009)
We calculate the mapping
H*\left(BO;ℤ₂\right)\to H*\left({K}^{1,0};ℤ₂\right)
and obtain a generating system of its kernel. As a corollary, bounds on the codimension of fold maps from real projective spaces to Euclidean space are calculated and the rank of a singular bordism group is determined.
Shyuichi Izumiya, Masatomo Takahashi (2008)
Classification des germes à point critique isolé et à nombres de modules 0 ou 1
Michel Demazure (1973/1974)
Classification of singularities with compact abelian symmetry
Gordon Wassermann (1988)
Cobordism of immersions and singular maps, loop spaces and multiple points
András Szücs (1986)
Cobordism of Morse functions on surfaces, the universal complex of singular fibers and their application to map germs.
Saeki, Osamu (2006)
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Hesse_normal_form Knowpia
The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in
{\displaystyle \mathbb {R} ^{2}}
or a plane in Euclidean space
{\displaystyle \mathbb {R} ^{3}}
or a hyperplane in higher dimensions.[1][2] It is primarily used for calculating distances (see point-plane distance and point-line distance).
Distance from the origin O to the line E calculated with the Hesse normal form. Normal vector in red, line in green, point O shown in blue.
It is written in vector notation as
{\displaystyle {\vec {r}}\cdot {\vec {n}}_{0}-d=0.\,}
The dot
{\displaystyle \cdot }
indicates the scalar product or dot product. Vector
{\displaystyle {\vec {r}}}
points from the origin of the coordinate system, O, to any point P that lies precisely in plane or on line E. The vector
{\displaystyle {\vec {n}}_{0}}
represents the unit normal vector of plane or line E. The distance
{\displaystyle d\geq 0}
is the shortest distance from the origin O to the plane or line.
Derivation/Calculation from the normal formEdit
Note: For simplicity, the following derivation discusses the 3D case. However, it is also applicable in 2D.
In the normal form,
{\displaystyle ({\vec {r}}-{\vec {a}})\cdot {\vec {n}}=0\,}
a plane is given by a normal vector
{\displaystyle {\vec {n}}}
as well as an arbitrary position vector
{\displaystyle {\vec {a}}}
{\displaystyle A\in E}
{\displaystyle {\vec {n}}}
is chosen to satisfy the following inequality
{\displaystyle {\vec {a}}\cdot {\vec {n}}\geq 0\,}
By dividing the normal vector
{\displaystyle {\vec {n}}}
by its magnitude
{\displaystyle |{\vec {n}}|}
, we obtain the unit (or normalized) normal vector
{\displaystyle {\vec {n}}_{0}={{\vec {n}} \over {|{\vec {n}}|}}\,}
and the above equation can be rewritten as
{\displaystyle ({\vec {r}}-{\vec {a}})\cdot {\vec {n}}_{0}=0.\,}
{\displaystyle d={\vec {a}}\cdot {\vec {n}}_{0}\geq 0\,}
we obtain the Hesse normal form
{\displaystyle {\vec {r}}\cdot {\vec {n}}_{0}-d=0.\,}
In this diagram, d is the distance from the origin. Because
{\displaystyle {\vec {r}}\cdot {\vec {n}}_{0}=d}
holds for every point in the plane, it is also true at point Q (the point where the vector from the origin meets the plane E), with
{\displaystyle {\vec {r}}={\vec {r}}_{s}}
, per the definition of the Scalar product
{\displaystyle d={\vec {r}}_{s}\cdot {\vec {n}}_{0}=|{\vec {r}}_{s}|\cdot |{\vec {n}}_{0}|\cdot \cos(0^{\circ })=|{\vec {r}}_{s}|\cdot 1=|{\vec {r}}_{s}|.\,}
{\displaystyle |{\vec {r}}_{s}|}
{\displaystyle {{\vec {r}}_{s}}}
is the shortest distance from the origin to the plane.
^ Bôcher, Maxime (1915), Plane Analytic Geometry: With Introductory Chapters on the Differential Calculus, H. Holt, p. 44 .
^ John Vince: Geometry for Computer Graphics. Springer, 2005, ISBN 9781852338343, pp. 42, 58, 135, 273
Weisstein, Eric W. "Hessian Normal Form". MathWorld.
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p
-adic representations of finite groups
20C35 Applications of group representations to physics
A characterization of characters which arise from ...-normalizers.
Dilip S. Gajendragadkar (1980)
{L}_{2}\left({2}^{f}\right)
in terms of the number of character zeros.
Qian, Guohua, Shi, Wujie (2009)
A Frobenius formula for the characters of the Hecke algebras.
Arun Ram (1991)
{B}_{n}
Araujo, J.O. (2003)
A local method in group cohomology.
A new characterization for the simple group
\mathrm{PSL}\left(2,{p}^{2}\right)
by order and some character degrees
Behrooz Khosravi, Behnam Khosravi, Bahman Khosravi, Zahra Momen (2015)
G
be a finite group and
p
a prime number. We prove that if
G
is a finite group of order
|\mathrm{PSL}\left(2,{p}^{2}\right)|
G
has an irreducible character of degree
{p}^{2}
G
has no irreducible character
\theta
2p\mid \theta \left(1\right)
G
\mathrm{PSL}\left(2,{p}^{2}\right)
. As a consequence of our result we prove that
\mathrm{PSL}\left(2,{p}^{2}\right)
is uniquely determined by the structure of its complex group algebra.
A note on finite groups with few values in a column of the character table
Mariagrazia Bianchi, David Chillag, Emanuele Pacifici (2006)
A note on the converse of the Clifford's theorem and some consequences
G. Navarro (1987)
A note on the subclass algebra.
Karlof, John (1980)
The notion of age of elements of complex linear groups was introduced by M. Reid and is of importance in algebraic geometry, in particular in the study of crepant resolutions and of quotients of Calabi–Yau varieties. In this paper, we solve a problem raised by J. Kollár and M. Larsen on the structure of finite irreducible linear groups generated by elements of age
\le 1
. More generally, we bound the dimension of finite irreducible linear groups generated by elements of bounded deviation. As a consequence...
A remark on homogeneous sets in finite groups. (Eine Bemerkung über homogene Mengen in endlichen Gruppen.)
Kerber, Adalbert (1984)
A remark on the irreducible characters and fake degrees of finite real reflection groups.
E.M. Opdam (1995)
A splitting principle for group representations.
Peter Symonds (1991)
A theorem on restricted group representations.
Jan Myrheim (1975)
A Theorem on the Restriction of Group Characters, and Its Application to the Character Theory of SL (n, q).
M.T. Karkar, J.A. Green (1975)
A version of Brauer’s theorem for integer central functions
Fedor Bogomolov, Jorge Maciel (2009)
In this article we prove an effective version of the classical Brauer’s Theorem for integer class functions on finite groups.
\lambda
G\wr {S}_{n}
Mendes, Anthony, Remmel, Jeffrey, Wagner, Jennifer (2004)
Abelian Normal Subgroups of M-Groups.
I. Martin Isaacs (1983)
Ion Armeanu (1996)
|
Physics - Revised Prediction for Mercury’s Orbit
Revised Prediction for Mercury’s Orbit
May 8, 2018 • Physics 11, s54
Mercury’s orbital ellipse is predicted to shift an additional
{1}^{\circ }
every two billion years as a result of previously unaccounted for effects of general relativity.
Mercury’s orbit of the Sun isn’t fixed in space. Every 625 years, the ellipse shifts by because of its gravitational interactions with the planets and the Sun. Now Clifford Will from the University of Florida, Gainesville, has used the general theory of relativity to calculate the impact of indirect gravitational forces—such as the pull between the Sun and Jupiter—on Mercury’s orbit. He predicts that these additional forces add of rotation to Mercury’s orbit every two billion years. Though tiny, and currently not measureable, this correction should be detectable by BepiColombo, a European and Japanese mission to Mercury scheduled to launch at the end of this year.
If Mercury were the only planet in the Solar System, its path around the Sun would stay fixed in space, according to Newtonian physics. But Mercury isn’t alone, and its Newtonian gravitational interactions with the other planets shift its orbit by 0.15 degrees per century (deg/cy). In addition, as Einstein famously predicted, general relativity affects the Sun-Mercury attraction and adds another 0.01 deg/cy to the planet’s orbital precession.
But the influence of the Sun doesn’t stop at Mercury. General relativistic effects of the Sun extend throughout the Solar System and alter the tug each planet exerts on Mercury. The theory also modifies the direct gravitational attraction between the planets and Mercury. Finally, the so-called gravitomagnetic force—a general relativistic effect of moving masses that is analogous to the magnetic force—also perturbs Mercury. Will predicts that these previously ignored effects should increase Mercury’s precession rate by about deg/cy.
New General Relativistic Contribution to Mercury’s Perihelion Advance
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Creating Constraint Violating Initial Data
Date
The IDConstraintViolate thorn creates initial data which purposefully violates the constraint equations for testing purposes.
Choosing the ADMBase parameter initial_data to have the value Constraint violating gaussian creates initial data with the form
\begin{array}{rcll}{g}_{xx}={g}_{yy}={g}_{zz}& =& 1+A\mathrm{exp}\left(\frac{-{\left(r-{r}_{0}\right)}^{2}}{{\sigma }^{2}}\right)& \text{}\\ {g}_{xy}={g}_{xz}={g}_{yz}& =& 0& \text{}\\ {K}_{ij}& =& 0& \text{}\end{array}
where the size and shape of the Gaussian is specified by parameters,
A
=amplitude,
{r}_{0}
=radius and
\sigma
=sigma.
Description: The radial position of the Gaussian wave
Constraint violating Gaussian
constraint violating gaussian
This section lists all the variables which are assigned storage by thorn EinsteinInitialData/IDConstraintViolate. Storage can either last for the duration of the run (Always means that if this thorn is activated storage will be assigned, Conditional means that if this thorn is activated storage will be assigned for the duration of the run if some condition is met), or can be turned on for the duration of a schedule function.
idconstraintviolate_paramchecker
idconstraintviolate_initial
set up constraint violating initial data
|
{C}^{*}
{W}^{*}
{C}^{*}
{C}^{*}
{W}^{*}
{C}^{*}
{C}^{*}
{C}^{*}
{C}^{*}
{C}^{*}
{C}^{*}
{C}^{*}
K
{C}^{*}
{C}^{*}
-algebra on Schur algebras.
Chaisuriya, Pachara (2011)
A characterization of completely bounded multipliers of Fourier algebras
Paul Jolissaint (1992)
A class of simple tracially AF
{C}^{*}
A direct approach to co-universal algebras associated to directed graphs.
Sims, Aidan, Webster, Samuel B.G. (2010)
A Duality Between Hilbert Modules And Fields Of Hilbert Spaces.
Alonso Takahashi (1979)
A fixed point approach to the stability of a quadratic functional equation in
{C}^{*}
Moghimi, Mohammad B., Najati, Abbas, Park, Choonkil (2009)
A functional inequality in restricted domains of Banach modules.
Moghimi, M.B., Najati, Abbas, Park, Choonkil (2009)
A geometry on the space of probabilities (II). Projective spaces and exponential families.
In this note we continue a theme taken up in part I, see [Gzyl and Recht: The geometry on the class of probabilities (I). The finite dimensional case. Rev. Mat. Iberoamericana 22 (2006), 545-558], namely to provide a geometric interpretation of exponential families as end points of geodesics of a non-metric connection in a function space. For that we characterize the space of probability densities as a projective space in the class of strictly positive functions, and these will be regarded as a...
A Holomorphic Characterization of Jordan C*-Algebras.
Wilhelm Kaup, Harald Upmeier, Robert Braun (1978)
A Korovkin Theorem for Schwarz Maps on C*-Algebras.
A. Guyan Robertson (1977)
A local version of the Dauns-Hofmann theorem.
Martin Mathieu, Pere Ara (1991)
A maximal Abelian subalgebra of the Calkin algebra with the extension property.
Joel Anderson (1978)
A new proof of the noncommutative Banach-Stone theorem
David Sherman (2006)
Surjective isometries between unital C*-algebras were classified in 1951 by Kadison [K]. In 1972 Paterson and Sinclair [PS] handled the nonunital case by assuming Kadison’s theorem and supplying some supplementary lemmas. Here we combine an observation of Paterson and Sinclair with variations on the methods of Yeadon [Y] and the author [S1], producing a fundamentally new proof of the structure of surjective isometries between (nonunital) C*-algebras. In the final section we indicate how our techniques...
A New Technique to Deal With Cuntz-Algebras.
A. van Daele, D. de Schreye (1981)
A note on complemented Banach *-algebras.
B.D. Malviya (1975)
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"Chromatic diesis" redirects here. For 27/26, see Comma (music).
In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma[2] is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125) (around 21.51 cents). Two notes that differ by this interval would sound different from each other even to untrained ears,[3] but would be close enough that they would be more likely interpreted as out-of-tune versions of the same note than as different notes. The comma is also referred to as a Didymean comma because it is the amount by which Didymus corrected the Pythagorean major third (81:64, around 407.82 cents)[4] to a just major third (5:4, around 386.31 cents).
Syntonic comma (81:80) on C
Just perfect fifth on D The perfect fifth above D (A+) is a syntonic comma higher than the (A♮) that is a just major sixth above C, assuming C and D are 9/8 apart.[1]
3-limit 9:8 major tone
5-limit 10:9 minor tone
The word "comma" came via Latin from Greek κόμμα, from earlier *κοπ-μα = "a thing cut off".
2 Syntonic comma in the history of music
3 Comma pump
The prime factors of the just interval 81/80 known as the syntonic comma can be separated out and reconstituted into various sequences of two or more intervals that arrive at the comma, such as 81/1 * 1/80 or (fully expanded and sorted by prime) 1/2 * 1/2 * 1/2 * 1/2 * 3/1 * 3/1 * 3/1 *3/1 * 1/5. All sequences are mathematically valid, but some of the more musical sequences people use to remember and explain the comma's composition, occurrence, and usage are listed below:
The difference in size between a Pythagorean ditone (frequency ratio 81:64, or about 407.82 cents) and a just major third (5:4, or about 386.31 cents). Namely, 81:64 ÷ 5:4 = 81:80.
The difference between four justly tuned perfect fifths, and two octaves plus a justly tuned major third. A just perfect fifth has a size of 3:2 (about 701.96 cents), and four of them are equal to 81:16 (about 2807.82 cents). A just major third has a size of 5:4 (about 386.31 cents), and one of them plus two octaves (4:1 or exactly 2400 cents) is equal to 5:1 (about 2786.31 cents). The difference between these is the syntonic comma. Namely, 81:16 ÷ 5:1 = 81:80.
The difference between one octave plus a justly tuned minor third (12:5, about 1515.64 cents), and three justly tuned perfect fourths (64:27, about 1494.13 cents). Namely, 12:5 ÷ 64:27 = 81:80.
The difference between the two kinds of major second which occur in 5-limit tuning: major tone (9:8, about 203.91 cents) and minor tone (10:9, about 182.40 cents). Namely, 9:8 ÷ 10:9 = 81:80.[4]
The difference between a Pythagorean major sixth (27:16, about 905.87 cents) and a justly tuned or "pure" major sixth (5:3, about 884.36 cents). Namely, 27:16 ÷ 5:3 = 81:80.[4]
On a piano keyboard (typically tuned with 12-tone equal temperament) a stack of four fifths (700 * 4 = 2800 cents) is exactly equal to two octaves (1200 * 2 = 2400 cents) plus a major third (400 cents). In other words, starting from a C, both combinations of intervals will end up at E. Using justly tuned octaves (2:1), fifths (3:2), and thirds (5:4), however, yields two slightly different notes. The ratio between their frequencies, as explained above, is a syntonic comma (81:80). Pythagorean tuning uses justly tuned fifths (3:2) as well, but uses the relatively complex ratio of 81:64 for major thirds. Quarter-comma meantone uses justly tuned major thirds (5:4), but flattens each of the fifths by a quarter of a syntonic comma, relative to their just size (3:2). Other systems use different compromises. This is one of the reasons why 12-tone equal temperament is currently the preferred system for tuning most musical instruments.
Mathematically, by Størmer's theorem, 81:80 is the closest superparticular ratio possible with regular numbers as numerator and denominator. A superparticular ratio is one whose numerator is 1 greater than its denominator, such as 5:4, and a regular number is one whose prime factors are limited to 2, 3, and 5. Thus, although smaller intervals can be described within 5-limit tunings, they cannot be described as superparticular ratios.
Syntonic comma in the history of musicEdit
Syntonic comma (the mismatch at the top)
is tempered out in 12TET (bottom)
Syntonic comma, such as between the 9/8 (203.91 approximate cents) and 10/9 (182.40 approximate cents) major and minor tones (top), is tempered out in 12TET, leaving one 200 cent tone (bottom).
The syntonic comma has a crucial role in the history of music. It is the amount by which some of the notes produced in Pythagorean tuning were flattened or sharpened to produce just minor and major thirds. In Pythagorean tuning, the only highly consonant intervals were the perfect fifth and its inversion, the perfect fourth. The Pythagorean major third (81:64) and minor third (32:27) were dissonant, and this prevented musicians from using triads and chords, forcing them for centuries to write music with relatively simple texture. In late Middle Ages, musicians realized that by slightly tempering the pitch of some notes, the Pythagorean thirds could be made consonant. For instance, if the frequency of E is decreased by a syntonic comma (81:80), C-E (a major third), and E-G (a minor third) become just. Namely, C-E is narrowed to a justly intonated ratio of
{\displaystyle {81 \over 64}\cdot {80 \over 81}={{1\cdot 5} \over {4\cdot 1}}={5 \over 4}}
and at the same time E-G is widened to the just ratio of
{\displaystyle {32 \over 27}\cdot {81 \over 80}={{2\cdot 3} \over {1\cdot 5}}={6 \over 5}}
The drawback is that the fifths A-E and E-B, by flattening E, become almost as dissonant as the Pythagorean wolf fifth. But the fifth C-G stays consonant, since only E has been flattened (C-E * E-G = 5/4 * 6/5 = 3/2), and can be used together with C-E to produce a C-major triad (C-E-G). These experiments eventually brought to the creation of a new tuning system, known as quarter-comma meantone, in which the number of major thirds was maximized, and most minor thirds were tuned to a ratio which was very close to the just 6:5. This result was obtained by narrowing each fifth by a quarter of a syntonic comma, an amount which was considered negligible, and permitted the full development of music with complex texture, such as polyphonic music, or melody with instrumental accompaniment. Since then, other tuning systems were developed, and the syntonic comma was used as a reference value to temper the perfect fifths in an entire family of them. Namely, in the family belonging to the syntonic temperament continuum, including meantone temperaments.
Comma pumpEdit
Giovanni Benedetti's 1563 example of a comma "pump" or drift by a comma during a progression.[5]
Play (help·info) Common tones between chords are the same pitch, with the other notes tuned in pure intervals to the common tones.
Play first and last chords (help·info)
The syntonic comma arises in "comma pump" (comma drift) sequences such as C G D A E C, when each interval from one note to the next is played with certain specific intervals in just intonation tuning. If we use the frequency ratio 3/2 for the perfect fifths (C-G and D-A), 3/4 for the descending perfect fourths (G-D and A-E), and 4/5 for the descending major third (E-C), then the sequence of intervals from one note to the next in that sequence goes 3/2, 3/4, 3/2, 3/4, 4/5. These multiply together to give
{\displaystyle {3 \over 2}\cdot {3 \over 4}\cdot {3 \over 2}\cdot {3 \over 4}\cdot {4 \over 5}={81 \over 80}}
which is the syntonic comma (musical intervals stacked in this way are multiplied together). The "drift" is created by the combination of Pythagorean and 5-limit intervals in just intonation, and would not occur in Pythagorean tuning due to the use only of the Pythagorean major third (64/81) which would thus return the last step of the sequence to the original pitch.
So in that sequence, the second C is sharper than the first C by a syntonic comma
Play (help·info). That sequence, or any transposition of it, is known as the comma pump. If a line of music follows that sequence, and if each of the intervals between adjacent notes is justly tuned, then every time the sequence is followed, the pitch of the piece rises by a syntonic comma (about a fifth of a semitone).
Study of the comma pump dates back at least to the sixteenth century when the Italian scientist Giovanni Battista Benedetti composed a piece of music to illustrate syntonic comma drift.[5]
Note that a descending perfect fourth (3/4) is the same as a descending octave (1/2) followed by an ascending perfect fifth (3/2). Namely, (3/4)=(1/2)*(3/2). Similarly, a descending major third (4/5) is the same as a descending octave (1/2) followed by an ascending minor sixth (8/5). Namely, (4/5)=(1/2)*(8/5). Therefore, the above-mentioned sequence is equivalent to:
{\displaystyle {3 \over 2}\cdot {1 \over 2}\cdot {3 \over 2}\cdot {3 \over 2}\cdot {1 \over 2}\cdot {3 \over 2}\cdot {1 \over 2}\cdot {8 \over 5}={81 \over 80}}
or, by grouping together similar intervals,
{\displaystyle {3 \over 2}\cdot {3 \over 2}\cdot {3 \over 2}\cdot {3 \over 2}\cdot {8 \over 5}\cdot {1 \over 2}\cdot {1 \over 2}\cdot {1 \over 2}={81 \over 80}}
This means that, if all intervals are justly tuned, a syntonic comma can be obtained with a stack of four perfect fifths plus one minor sixth, followed by three descending octaves (in other words, four P5 plus one m6 minus three P8).
Just major chord on C in Ben Johnston's notation.
Play (help·info) Pythagorean major chord on C in Helmholtz-Ellis notation.
Pythagorean major chord, Ben Johnston's notation.
Just major chord, in Helmholtz-Ellis notation.
Moritz Hauptmann developed a method of notation used by Hermann von Helmholtz. Based on Pythagorean tuning, subscript numbers are then added to indicate the number of syntonic commas to lower a note by. Thus a Pythagorean scale is C D E F G A B, while a just scale is C D E1 F G A1 B1. Carl Eitz developed a similar system used by J. Murray Barbour. Superscript positive and negative numbers are added, indicating the number of syntonic commas to raise or lower from Pythagorean tuning. Thus a Pythagorean scale is C D E F G A B, while the 5-limit Ptolemaic scale is C D E−1 F G A−1 B−1.
In Helmholtz-Ellis notation, a syntonic comma is indicated with up and down arrows added to the traditional accidentals. Thus a Pythagorean scale is C D E F G A B, while the 5-limit Ptolemaic scale is C D E
Composer Ben Johnston uses a "−" as an accidental to indicate a note is lowered by a syntonic comma, or a "+" to indicate a note is raised by a syntonic comma.[1] Thus a Pythagorean scale is C D E+ F G A+ B+, while the 5-limit Ptolemaic scale is C D E F G A B.
HE C D E
Johnston C D E F G A B C D E+ F G A+ B+
^ a b John Fonville. "Ben Johnston's Extended Just Intonation – A Guide for Interpreters", p. 109, Perspectives of New Music, vol. 29, no. 2 (Summer 1991), pp. 106-137. and Johnston, Ben and Gilmore, Bob (2006). "A Notation System for Extended Just Intonation" (2003), "Maximum clarity" and Other Writings on Music, p. 78. ISBN 978-0-252-03098-7.
^ Johnston B. (2006). "Maximum Clarity" and Other Writings on Music, edited by Bob Gilmore. Urbana: University of Illinois Press. ISBN 0-252-03098-2.
^ "Sol-Fa – The Key to Temperament" Archived 2005-02-08 at the Wayback Machine, BBC.
^ a b c Llewelyn Southworth Lloyd (1937). Music and Sound, p. 12. ISBN 0-8369-5188-3.
^ a b Wild, Jonathan; Schubert, Peter (Spring–Fall 2008), "Historically Informed Retuning of Polyphonic Vocal Performance" (PDF), Journal of Interdisciplinary Music Studies, 2 (1&2): 121–139 [127], archived from the original (PDF) on September 11, 2010, retrieved April 5, 2013 , art. #0821208.
Indiana University School of Music: Piano Repair Shop: Harpsichord Tuning, Repair, and Temperaments: "What is the Syntonic Comma?"
Tonalsoft: "Syntonic-comma"
Explanation of comma drift
Retrieved from "https://en.wikipedia.org/w/index.php?title=Syntonic_comma&oldid=1063135265"
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<
goodale@cct.lsu.edu
>
This section lists all the variables which are assigned storage by thorn CactusTest/TestSchedule. Storage can either last for the duration of the run (Always means that if this thorn is activated storage will be assigned, Conditional means that if this thorn is activated storage will be assigned for the duration of the run if some condition is met), or can be turned on for the duration of a schedule function.
testschedule_cctk_startup
test to see if function is placed in schedule and run at cctk_startup
CCTK_RECOVER_PARAMETERS
testschedule_cctk_recover_parameters
test to see if function is placed in schedule and run at cctk_recover_parameters
testschedule_cctk_checkpoint
test to see if function is placed in schedule and run at cctk_checkpoint
testschedule_cctk_prestep
test to see if function is placed in schedule and run at cctk_prestep
testschedule_cctk_evol
test to see if function is placed in schedule and run at cctk_evol
testschedule_cctk_poststep
test to see if function is placed in schedule and run at cctk_poststep
testschedule_cctk_postrestrict
test to see if function is placed in schedule and run at cctk_postrestrict
testschedule_cctk_postregrid
test to see if function is placed in schedule and run at cctk_postregrid
testschedule_cctk_analysis
test to see if function is placed in schedule and run at cctk_analysis
testschedule_cctk_terminate
test to see if function is placed in schedule and run at cctk_terminate
testschedule_cctk_shutdown
test to see if function is placed in schedule and run at cctk_shutdown
testschedule_cctk_wragh
test to see if function is placed in schedule and run at cctk_wragh
testschedule_cctk_paramcheck
test to see if function is placed in schedule and run at cctk_paramcheck
testschedule_cctk_basegrid
test to see if function is placed in schedule and run at cctk_basegrid
testschedule_cctk_initial
test to see if function is placed in schedule and run at cctk_initial
testschedule_cctk_postinitial
test to see if function is placed in schedule and run at cctk_postinitial
CCTK_RECOVER_VARIABLES
testschedule_cctk_recover_variables
test to see if function is placed in schedule and run at cctk_recover_variables
testschedule_cctk_post_recover_variables
test to see if function is placed in schedule and run at cctk_post_recover_variables
testschedule_cctk_cpinitial
test to see if function is placed in schedule and run at cctk_cpinitial
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Physics - A New Way to Slow Down Complex Molecules
A New Way to Slow Down Complex Molecules
Mirco Siercke and Silke Ospelkaus
Leibniz University Hannover, Hanover, Germany
A magnetic-field based method can slow molecular beams that cannot be slowed using other techniques, unlocking the door to ultracold polyatomic molecular physics.
Figure 1: Fast-moving molecules in a “low-field-seeking” state (green circles) lose energy and slow down as they climb a magnetic-field gradient to a region of high field strength. When the molecules reach the top of a magnetic “hill,” a laser induces a transition (pink arrows) to a “high-field-seeking” state (blue circles). Molecules in this state lose energy as they move down a magnetic-field gradient. By changing the state of the molecules before they can “roll down” the potential-energy gradient, they can be made to slow down continuously.Fast-moving molecules in a “low-field-seeking” state (green circles) lose energy and slow down as they climb a magnetic-field gradient to a region of high field strength. When the molecules reach the top of a magnetic “hill,” a laser induces a transi... Show more
Studying polyatomic molecules trapped at microkelvin temperatures promises to revolutionize precision metrology and to shed light on the fundamental quantum basis of chemistry. It could even lead to never-before-seen states of matter. To get the molecules trapped, however, researchers first have to slow down a molecular beam nearly to a standstill. While such “trappable” velocities have been achieved for a few diatomic molecules using laser cooling, applying the technique to polyatomic molecules seems out of reach. Benjamin Augenbraun and colleagues, at Harvard University, have now shown that an alternative cooling technique called Zeeman-Sisyphus deceleration—first proposed by researchers at Imperial College London [1]—can be used instead [2]. The demonstration paves the way to trapping gases of ultracold polyatomic molecules and all the rich physics that comes with them.
Research in ultracold atomic physics has been incredibly fruitful over the last few decades, resulting in multiple Nobel Prizes and countless new applications. The comparatively new field of ultracold diatomic molecular physics promises an equally rich landscape of novel phenomena. So far, progress in this field has mainly been in the preparation of diatomic species at ultracold temperatures, which can be achieved either through the controlled association of atoms into molecules [3] or through the direct laser cooling to microkelvin temperatures of certain “ready-made” diatomic molecules [4–7].
Against the background of these advances, some groups have already set their eyes on polyatomic molecules, which, because of the additional degrees of freedom afforded by their rotational and vibrational states, promise an even greater leap in progress than their diatomic counterparts [8, 9]. Certain polyatomic molecules, for example, have excitation modes that make them very sensitive to the electron’s electric dipole moment, so trapping them could lead to the discovery of physics beyond the standard model. Furthermore, as molecules become more complex, the possibility opens up for studying novel ultracold chemistry and gaining an understanding of chirality in molecules. But while ultracold polyatomic molecules might offer greater experimental potential, they also come with greater experimental challenges.
Typically, optical techniques for cooling and slowing molecules get exponentially harder when more atoms are added to the molecule. Laser slowing any particles—whether atoms or molecules—requires thousands of photons to be scattered. While this is trivial for atoms because of their simple energy structures, it can only be achieved for a select handful of diatomic molecules. That’s because molecules have extra degrees of freedom, which increase the number of the ways in which they can interact with photons. Some of these interactions can excite the molecules to “dark” vibrational or rotational energy sublevels that make further cooling impossible. Polyatomic molecules have even more of these dark states than diatomic molecules do, and they typically switch to a dark state after just a few photon interactions. Laser cooling is even more likely, therefore, to induce transitions that make polyatomic molecules experimentally inaccessible. As a result of this limitation, trapping and cooling of polyatomics has so far been restricted to molecules that can be decelerated using electric fields rather than photon scattering (see Viewpoint: Slowing Continuous Molecular Beams in a Rotating Spiral) [10]. The scheme demonstrated by Augenbraun and colleagues does not suffer from the limitations of laser cooling or electric-field-induced deceleration: It can be applied to any molecule that is paramagnetic, that is, molecules that have the inherent advantage of being magnetically trappable.
Zeeman-Sisyphus cooling relies on the fact that all paramagnetic molecules behave similarly in the “Paschen-Back” high-magnetic-field regime. In this regime, the molecules’ ground-state structures have two energy levels: one with a low energy in high fields (high-field seeker) and one with a low energy in low fields (low-field seeker). Because of these states, a molecule in the low-field-seeking state loses energy as it “rolls up” a magnetic-field-strength gradient, while a high-field-seeking molecule loses energy as it “rolls down” this gradient. By using a laser to change them from the high-field-seeking state to the low-field-seeking state at the right times, molecules can be made to roll continuously up a potential-energy “hill” as they pass through regions of high and low magnetic-field strength, losing kinetic energy as they go (Fig. 1). This behavior likens the scheme to the famous myth of Sisyphus, who was condemned to roll a boulder up a steep hill over and over again.
The Harvard team’s scheme differs from the original proposal for Zeeman-Sisyphus cooling [1] in a few key aspects [2]. Whereas the original Imperial group’s scheme specified about a hundred permanent magnets to produce about a hundred magnetic hills and valleys (therefore requiring hundreds of transitions between the low-field-seeking and high-field-seeking states), the new scheme instead uses two sets of superconducting magnetic coils to implement just two hills. These hills are of such a height (2.8 T, corresponding to an energy loss of 1.9 K per ascent) that the molecules lose all their kinetic energy in just two stages. As such, while the original proposal requires the scattering of hundreds of photons from each molecule, the new experiment manages to bring the molecules to rest by scattering, on average, just seven photons each.
This low number of scattered photons is key to the significance of the new experiment. It allows Augenbraun and colleagues to circumvent the problem of polyatomic molecules decaying into random, experimentally inaccessible energy states after scattering just a handful of photons, making it possible to trap and investigate a plethora of paramagnetic polyatomic molecules in a new temperature regime and for much longer periods than ever before. The exact set of molecules that can be addressed by this scheme remains unknown, but the diversity of polyatomic molecules promises rich prospects for applications in physics and chemistry, with every species offering something new compared with others. Some of the first targets for investigation are likely to be barium monohydroxide and radium monohydroxide as these molecules are very similar to the calcium monohydroxide used in the new demonstration. These molecules are exciting, as their high mass makes them extremely sensitive probes of the electron’s electric dipole moment, measurements of which could lead to beyond-standard-model physics.
N. J. Fitch and M. R. Tarbutt, “Principles and design of a Zeeman–Sisyphus decelerator for molecular beams,” ChemPhysChem 17, 3609 (2016).
B. L. Augenbraun et al., “Zeeman-Sisyphus deceleration of molecular beams,” Phys. Rev. Lett. 127, 263002 (2021).
K.-K. Ni et al., “A high phase-space-density gas of polar molecules,” Science 322, 231 (2008).
J. F. Barry et al., “Magneto-optical trapping of a diatomic molecule,” Nature 512, 286 (2014).
L. W. Cheuk et al., “
\Lambda
-enhanced imaging of molecules in an optical trap,” Phys. Rev. Lett. 121, 083201 (2018).
L. Caldwell et al., “Deep laser cooling and efficient magnetic compression of molecules,” Phys. Rev. Lett. 123, 033202 (2019).
S. Ding et al., “Sub-doppler cooling and compressed trapping of YO molecules at
𝜇\text{K}
temperatures,” Phys. Rev. X 10, 021049 (2020).
M. Zeppenfeld et al., “Sisyphus cooling of electrically trapped polyatomic molecules,” Nature 491, 570 (2012).
D. Mitra et al., “Direct laser cooling of a symmetric top molecule,” Science 369, 1366 (2020).
S. Chervenkov et al., “Continuous centrifuge decelerator for polar molecules,” Phys. Rev. Lett. 112, 013001 (2014).
Mirco Siercke is a postdoctoral researcher at the Leibniz University Hannover, Germany. After receiving his Ph.D. in cold-atom physics in 2010 from the University of Toronto, Canada, he moved to the Nanyang Technical University, Singapore, where he worked on a broad spectrum of atomic physics experiments including Rydberg atoms, matter-wave interferometers, and superconducting atom chips. He currently works in the group of Silke Ospelkaus on laser cooling of molecular gases.
Silke Ospelkaus works in experimental atomic and molecular physics. She received a diploma in physics from the University of Bonn, Germany, and a Ph.D. from the University of Hamburg, Germany. After a postdoctoral stay at JILA in Colorado, she moved back to Germany. Currently she is a professor for experimental physics at the Leibniz University Hannover, Germany, developing techniques to cool and control molecules.
Zeeman-Sisyphus Deceleration of Molecular Beams
Benjamin L. Augenbraun, Alexander Frenett, Hiromitsu Sawaoka, Christian Hallas, Nathaniel B. Vilas, Abdullah Nasir, Zack D. Lasner, and John M. Doyle
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As shown in the diagram at right, the two numbers to the left and right of the diamond (marked with a
\#
symbol) can be used to get the numbers on the top and bottom of the diamond. The number on the top of the diamond is the product of (answer you get when you multiply) the side numbers. The number on the bottom is the sum of (answer you get when you add) the side numbers.
When you multiply the right side number by
2
6
! The bottom number is the sum of the two side numbers.
In the diamond above, the product (c) and number to the right (d) are missing. Use the numbers that are already filled in to find the missing pieces.
Try finding the difference of the sum (
6.2
) and the number to the left (
5.2
). This value will be the number to the right (d).
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Physics - Heavy into Stability
Heavy into Stability
Frankfurt Institute for Advanced Studies, J. W. Goethe-Universität, 60438 Frankfurt am Main, Germany
The discoverers of element 117 have followed through with a year-long study of the lifetime and decay products of the newest superheavy element.
Wikimedia Commons/InvaderXan; Image on homepage: Dusan Petricic
Figure 1: Three-dimensional representation of the theoretical island of stability in nuclear physics. The half-life of nuclides is plotted as a function of proton (
Z
) and neutron (
N
) number. The continent of stable elements ends at the lead–bismuth cape, and a region of relative stability appears around the isotopes of thorium and uranium (
Z=90,92
). In the region of the heaviest (superheavy) elements, theory predicts an “island of stability” around proton number 114 or 120 and neutron number 184.Three-dimensional representation of the theoretical island of stability in nuclear physics. The half-life of nuclides is plotted as a function of proton (
Z
N
) number. The continent of stable elements ends at the lead–bismuth cape, and a... Show more
In 1940, the first synthetic element heavier than uranium—neptunium-239—was produced by bombarding uranium with neutrons. Since then, nuclear scientists have ventured into the search for new heavy elements, expanding the frontiers of the physical world. The creation of elements with atomic number beyond that of uranium is challenging, as the half-life of elements decreases with increasing atomic number. However, nuclear theories have predicted that a so-called “island of stability” exists for certain superheavy elements of the nuclide chart, which should have half-lives ranging from minutes to many years.
The search for this island of stability has led to the creation of elements with up to 118 protons. The last element to be discovered was 117 [1] (see 9 April, 2010 Viewpoint), filling in the final gap on the list of observed elements up to element 118. Now, writing in Physical Review Letters, Yuri Oganessian at the Joint Institute for Nuclear Research (JINR), Russia, and colleagues report on a second production campaign for element 117 [2], which verifies their initial findings and provides a new comprehensive characterization of the decay chains of two isotopes of the 117 element. Their results confirm that we are indeed approaching the shores of the island of stability.
The stability of nuclides is a function of proton ( ) and neutron ( ) number, as illustrated in Fig. 1. A connected region (“continent”) of stable elements is found for lighter elements, ending at the lead–bismuth “cape.” All elements with an atomic number exceeding 82 (lead) are unstable, with decreasing half-life for higher atomic numbers. However, a first region of relative stability appears around the isotopes of thorium and uranium ( equal to 90 and 92, respectively) whose lifetimes are comparable with the age of the universe. Elements with atomic number greater than that of uranium (transuranium elements) have only been produced in laboratory experiments (see the historical review in Ref. [3]). The progress in this field is impressive: 26 new, manmade heavy elements have been synthesized within 60 years. Some of these elements (up to californium) can be produced in macroscopic quantities in nuclear reactors, using neutron capture processes to form heavier elements from actinides.
Elements beyond uranium should become more and more unstable as they get heavier, as Coulomb repulsion starts to be stronger than the strong force that holds the nucleus together. But in the late sixties, Glenn T. Seaborg postulated the existence of a relatively stable region of superheavy elements, an island of stability. This idea is based on the nuclear shell model, which describes the atomic nucleus as made of shells, similar to the well-known electronic shell model for atoms. Nuclear theorists, including myself [4,5], predicted that the stability of nuclei with so-called closed proton and neutron shells should counteract the repelling Coulomb forces. In isotopes with so-called “magic” proton and neutron numbers, neutrons and protons completely fill the energy levels of a given shell in the nucleus. Those particular isotopes will have a longer lifetime than nearby ones. According to theory, this second island of stability should be located around proton number or and neutron number . Reaching this island of stability would open new horizons in nuclear physics and technology, enabling the production of superheavy nuclides in macroscopic quantities and with sufficiently long half-life to carry out actual experiments. This would allow us to test our understanding of nuclear matter and to possibly exploit such long-lived elements for applications in medicine or chemistry.
Substantial progress in the synthesis of superheavy nuclei was achieved at the GSI Helmholtz Centre for Heavy Ion Research in Germany, where the elements with to have been synthesized for the first time in fusion reactions of heavy projectiles (from iron to zinc) with lead and bismuth targets [6]. Unfortunately, in these projectile–target combinations only the proton-rich isotopes of superheavy elements with very short half-lives can be produced, as they lie outside the island of nuclear stability. Within the last ten years, researchers at JINR have successfully synthesized six new heavy elements with – by following a different approach: instead of a heavy projectile, a high-intensity beam of lighter atoms (calcium-48) is aimed at heavy actinide targets made of uranium or transuranium elements. The use of neutron richer calcium-48 allows the synthesis of nuclides with neutron number closer to that needed for stability.
Up until 2010, there was a gap between elements 116 and 118. The obstacle towards the production of element 117 was that the appropriate target material, berkelium-249 ( ) with 97 protons, has a short half-life of only 330 days. In 2009, several milligrams of were produced at Oak Ridge National Laboratory in the US—enough to prepare a target and to perform the first experiment for the synthesis of element 117 at JINR [1]. In early March of 2012, a new portion of , , was shipped again from Oak Ridge to JINR, where physicists started the second production campaign for the synthesis of element 117.
The results of this campaign, reported in the paper by Oganessian et al. [2], confirm that a reliable method for the production of 117 exists. The authors can now state with confidence that two isotopes of this element, and , have been synthesized and provide a comprehensive characterization of their decay properties. Two decay chains of and five decay chains of were detected. Oganessian et al. also observe a concomitant decay chain of element 118. This occurs because, at the time of the experiment, part of the target material had already decayed into californium-249, which can generate element 118 in a fusion reaction with calcium-48. The measured lifetimes of the 117 isotopes and other elements along its decay chain are long, lying in the millisecond-to-second range. This is consistent with shell-model predictions, confirming that these elements are indeed located at the southwest shores of the island of stability. The consistent results emerging from the two productions campaigns at JINR may get the authors close to laying claim on naming the new element.
What are the prospects of reaching deeper into the center of the island of stability? Although fairly long lived, the isotopes of superheavy elements produced in the experiments with calcium-48 are still neutron deficient: each isotope needs six to eight more neutrons to lie within the island. This occurs because heavier stable atoms must have a larger neutron/proton ratio that lighter atoms. Thus creating a heavy atom by fusion of two lighter ones inevitably leads to an atom that has too few neutrons and too many protons to be stable. One would then deduce that there is no way to the island of stability. However, pathways towards the center of the island of stability may exist. Recent theoretical studies carried out in my research group suggest that superheavy nuclei located at the top left side of the island of stability, formed in ordinary fusion reactions, could get rid of excess protons via decay [7]. Other alternatives to get to the right neutron number might exploit neutron capture, rather than fusion: such techniques would require the exposure of heavy elements, such as uranium, to very high neutron fluxes. Theory shows that this could be achieved in hypothetical small-scale underground nuclear explosions [8] or by using pulsed nuclear reactors of the next generation, if their neutron fluence per pulse is increased by about three orders of magnitude.
While the island of stability is now more firmly in sight, the jury is still out on what navigation plan will turn out to be successful.
Yu.Ts. Oganessian et al., “Synthesis of a New Element with Atomic Number Z=117,” Phys. Rev. Lett. 104, 142502 (2010)
Y. T. Oganessian et al., “Production and Decay of the Heaviest Nuclei 293,294 117 and 294 118,” Phys. Rev. Lett. 109, 162501 (2012)
G. T. Seaborg and W. D. Loveland, The Elements Beyond Uranium (John Wiley and Sons, New York, 1990)[Amazon][WorldCat]
S. G. Nilsson, S. G. Thompson, and C. F. Tsang, “Stability of Superheavy Nuclei and Their Possible Occurrence in Nature,” Phys. Lett. 28B, 458 (1969)
U. Mosel and W. Greiner, “On the Stability of Superheavy Nuclei Against Fission,” Z. Phys. A 222, 261 (1969); Also in the Proposal for the Establishment of GSI: Frankfurt-Darmstadt-Marburg (1967)
S. Hofmann and G. Munzenberg, “The Discovery of the Heaviest Elements,” Rev. Mod. Phys. 72, 733 (2000)
V. I. Zagrebaev, A. V. Karpov, and W. Greiner, “Possibilities for Synthesis of New Isotopes of Superheavy Elements in Fusion Reactions,” Phys. Rev. C 85, 014608 (2012)
V. I. Zagrebaev, A. V. Karpov, I. N. Mishustin, and W. Greiner, “Production of Heavy and Superheavy Neutron-Rich Nuclei in Neutron Capture Processes,” Phys. Rev. C 84, 044617 (2011)
Walter Greiner is full professor of Physics and Founding Director of the Frankfurt Institute for Advanced Studies at the Johann Wolfgang Goethe-Universität Frankfurt/Main, Germany. He received his Ph.D. in Physics in 1961 from Freiburg University and has honorary doctorate degrees from fourteen other Universities. He is a recipient of the Max Born Prize, the Otto Hahn Prize, and the Alexander von Humboldt Medal. He has carried out research on a wide spectrum of topics in Nuclear, Atomic, and Particle Physics, including studies on the prediction of superheavy nuclei as well as on the extension of the periodic system into new directions (strange matter and antimatter).
Production and Decay of the Heaviest Nuclei
{}^{293,294}117
{}^{294}118
{}^{293,294}117
{}^{294}118
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49N15 Duality theory
49N05 Linear optimal control problems
49N10 Linear-quadratic problems
49N20 Periodic optimization
49N30 Problems with incomplete information
49N35 Optimal feedback synthesis
49N45 Inverse problems
49N60 Regularity of solutions
49N70 Differential games
A.M. Вершик (1984)
A new method for solving monotone generalized variational inequalities.
Anh, Pham Ngoc, Kim, Jong Kyu (2010)
Christian Léonard (2011)
The Monge-Kantorovich problem is revisited by means of a variant of the saddle-point method without appealing to c-conjugates. A new abstract characterization of the optimal plans is obtained in the case where the cost function takes infinite values. It leads us to new explicit sufficient and necessary optimality conditions. As by-products, we obtain a new proof of the well-known Kantorovich dual equality and an improvement of the convergence of the minimizing sequences.
A simple proof in Monge-Kantorovich duality theory
D. A. Edwards (2010)
A simple proof is given of a Monge-Kantorovich duality theorem for a lower bounded lower semicontinuous cost function on the product of two completely regular spaces. The proof uses only the Hahn-Banach theorem and some properties of Radon measures, and allows the case of a bounded continuous cost function on a product of completely regular spaces to be treated directly, without the need to consider intermediate cases. Duality for a semicontinuous cost function is then deduced via the use of an...
Adherence of minimizers for dual convergences
Szymon Dolecki (2007)
Alcuni problemi matematici legati alla gestione ottima di un portafoglio
Maurizio Pratelli (2004)
In questa conferenza, vengono esposte le idee essenziali che stanno alla base del classico problema di gestire un portafoglio in modo da rendere massima l'utilità media. I metodi tipici del controllo stocastico sono confrontati con le idee della dualità convessa infinito-dimensionale.
Antihomogeneous conjugacy operators in convex analysis.
Rubinov, Alexander (1995)
Caracterización algebraica de las aristas infinitas en el conjunto dual factible de un PSI-lineal.
Jesús T. Pastor Ciurana (1987)
Las propiedades geométricas del conjunto factible del dual de un problema semiinfinito lineal son análogas a las correspondientes para el caso finito. En este trabajo mostramos cómo, a partir de la caracterización algebraica de vértices y direcciones extremas, se consigue la correspondiente para aristas infinitas, estableciéndose así las bases para una extensión del método simplex a programas semiinfinitos lineales.
Characterization of the stability of a minimization problem associated with a particular perturbation function. (Caractérisation de la stabilité d'un problème de minimisation associé à une fonction de perturbation particulière.)
Mentagui, D. (1996)
Characterizations of ɛ-duality gap statements for constrained optimization problems
Horaţiu-Vasile Boncea, Sorin-Mihai Grad (2013)
In this paper we present different regularity conditions that equivalently characterize various ɛ-duality gap statements (with ɛ ≥ 0) for constrained optimization problems and their Lagrange and Fenchel-Lagrange duals in separated locally convex spaces, respectively. These regularity conditions are formulated by using epigraphs and ɛ-subdifferentials. When ɛ = 0 we rediscover recent results on stable strong and total duality and zero duality gap from the literature.
Comparison between different duals in multiobjective fractional programming
Radu Boţ, Robert Chares, Gert Wanka (2007)
The present paper is a continuation of [2] where we deal with the duality for a multiobjective fractional optimization problem. The basic idea in [2] consists in attaching an intermediate multiobjective convex optimization problem to the primal fractional problem, using an approach due to Dinkelbach ([6]), for which we construct then a dual problem expressed in terms of the conjugates of the functions involved. The weak, strong and converse duality statements for the intermediate problems allow...
Computation of the distance to semi-algebraic sets
Christophe Ferrier (2000)
This paper is devoted to the computation of distance to set, called S, defined by polynomial equations. First we consider the case of quadratic systems. Then, application of results stated for quadratic systems to the quadratic equivalent of polynomial systems (see [5]), allows us to compute distance to semi-algebraic sets. Problem of computing distance can be viewed as non convex minimization problem:
d\left(u,S\right)={inf}_{x\in S}{\parallel x-u\parallel }^{2}
, where u is in
{ℛ}^{n}
. To have, at least, lower approximation of distance, we consider the dual...
Convex Duality in Problems of the Bolza type with time-delay
G. I. Tsoutsinos (1992)
Das Umkugelproblem und lineare semiinfinite Optimierung
F. Juhnke (1989)
Dual dynamic approach to shape optimization
Piotr Fulmański, Andrzej Nowakowski (2006)
Dual extremum principles for a homogeneous Dirichlet problem for a parabolic equation.
Wyborski, Jerzy (1993)
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Kármán vortex street - Wikipedia
Repeating pattern of swirling vortices caused by the unsteady separation of flow of a fluid around blunt bodies
Visualisation of the vortex street behind a circular cylinder in air; the flow is made visible through release of glycerol vapour in the air near the cylinder
In fluid dynamics, a Kármán vortex street (or a von Kármán vortex street) is a repeating pattern of swirling vortices, caused by a process known as vortex shedding, which is responsible for the unsteady separation of flow of a fluid around blunt bodies.
It is named after the engineer and fluid dynamicist Theodore von Kármán,[1] and is responsible for such phenomena as the "singing" of suspended telephone or power lines and the vibration of a car antenna at certain speeds. Mathematical modeling of von Kármán vortex street can be performed using different techniques including but not limited to solving the full Navier-Stokes equations with k-epsilon, SST, k-omega and Reynolds stress, and large eddy simulation (LES) turbulence models,[2][3] or by numerically solving some dynamic equations such as the Ginzburg-Landau equation.[4][5][6]
A look at the Kármán vortex street effect from ground level, as air flows quickly from the Pacific ocean eastward over Mojave desert mountains. This phenomenon observed from ground level is extremely rare, as most cloud-related Kármán vortex street activity is viewed from space.
A vortex street in a 2D liquid of hard disks
A vortex street will form only at a certain range of flow velocities, specified by a range of Reynolds numbers (Re), typically above a limiting Re value of about 90. The (global) Reynolds number for a flow is a measure of the ratio of inertial to viscous forces in the flow of a fluid around a body or in a channel, and may be defined as a nondimensional parameter of the global speed of the whole fluid flow:
{\displaystyle \mathrm {Re} _{L}={\frac {UL}{\nu _{0}}}}
{\displaystyle U}
= the free stream flow speed (i.e. the flow speed far from the fluid boundaries
{\displaystyle U_{\infty }}
like the body speed relative to the fluid at rest, or an inviscid flow speed, computed through the Bernoulli equation), which is the original global flow parameter, i.e. the target to be non-dimensionalised.
{\displaystyle L}
= a characteristic length parameter of the body or channel
{\displaystyle \nu _{0}}
= the free stream kinematic viscosity parameter of the fluid, which in turn is the ratio:
{\displaystyle \nu _{0}={\frac {\mu _{0}}{\rho _{0}}}}
{\displaystyle \rho _{0}}
= the reference fluid density.
{\displaystyle \mu _{0}}
= the free stream fluid dynamic viscosity
For common flows (the ones which can usually be considered as incompressible or isothermal), the kinematic viscosity is everywhere uniform over all the flow field and constant in time, so there is no choice on the viscosity parameter, which becomes naturally the kinematic viscosity of the fluid being considered at the temperature being considered. On the other hand, the reference length is always an arbitrary parameter, so particular attention should be put when comparing flows around different obstacles or in channels of different shapes: the global Reynolds numbers should be referred to the same reference length. This is actually the reason for which the most precise sources for airfoil and channel flow data specify the reference length at the Reynolds number. The reference length can vary depending on the analysis to be performed: for a body with circle sections such as circular cylinders or spheres, one usually chooses the diameter; for an airfoil, a generic non-circular cylinder or a bluff body or a revolution body like a fuselage or a submarine, it is usually the profile chord or the profile thickness, or some other given widths that are in fact stable design inputs; for flow channels usually the hydraulic diameter about which the fluid is flowing.
For an aerodynamic profile the reference length depends on the analysis. In fact, the profile chord is usually chosen as the reference length also for aerodynamic coefficient for wing sections and thin profiles in which the primary target is to maximize the lift coefficient or the lift/drag ratio (i.e. as usual in thin airfoil theory, one would employ the chord Reynolds as the flow speed parameter for comparing different profiles). On the other hand, for fairings and struts the given parameter is usually the dimension of internal structure to be streamlined (let us think for simplicity it is a beam with circular section), and the main target is to minimize the drag coefficient or the drag/lift ratio. The main design parameter which becomes naturally also a reference length is therefore the profile thickness (the profile dimension or area perpendicular to the flow direction), rather than the profile chord.
The range of Re values varies with the size and shape of the body from which the eddies are shed, as well as with the kinematic viscosity of the fluid. For the wake of a circular cylinder, for which the reference length is conventionally the diameter d of the circular cylinder, the lower limit of this range is Re ≈ 47.[7] Eddies are shed continuously from each side of the circle boundary, forming rows of vortices in its wake. The alternation leads to the core of a vortex in one row being opposite the point midway between two vortex cores in the other row, giving rise to the distinctive pattern shown in the picture. Ultimately, the energy of the vortices is consumed by viscosity as they move further down stream, and the regular pattern disappears. Above the Re value of 188.5, the flow becomes three-dimensional, with periodic variation along the cylinder.[8] Above Re on the order of 105 at the drag crisis, vortex shedding becomes irregular and turbulence sets in.
In meteorology[edit]
The flow of atmospheric air over obstacles such as islands or isolated mountains sometimes gives birth to von Kármán vortex streets. When a cloud layer is present at the relevant altitude, the streets become visible. Such cloud layer vortex streets have been photographed from satellites.[9] The vortex street can reach over 400 km from the obstacle and the diameter of the vortices are normally 20–40 km.[10]
Engineering problems[edit]
Chimneys with strakes fitted to break up vortices
In low turbulence, tall buildings can produce a Kármán street, so long as the structure is uniform along its height. In urban areas where there are many other tall structures nearby, the turbulence produced by these prevents the formation of coherent vortices.[11] Periodic crosswind forces set up by vortices along object's sides can be highly undesirable, due to the vortex-induced vibrations caused, which can damage the structure, hence it is important for engineers to account for the possible effects of vortex shedding when designing a wide range of structures, from submarine periscopes to industrial chimneys and skyscrapers. For monitoring such engineering structures, the efficient measurements of von Kármán streets can be performed using smart sensing algorithms such as compressive sensing.[2]
In order to prevent the unwanted vibration of such cylindrical bodies, a longitudinal fin can be fitted on the downstream side, which, provided it is longer than the diameter of the cylinder, will prevent the eddies from interacting, and consequently they remain attached. Obviously, for a tall building or mast, the relative wind could come from any direction. For this reason, helical projections resembling large screw threads are sometimes placed at the top, which effectively create asymmetric three-dimensional flow, thereby discouraging the alternate shedding of vortices; this is also found in some car antennas. Another countermeasure with tall buildings is using variation in the diameter with height, such as tapering - that prevents the entire building being driven at the same frequency.
Even more serious instability can be created in concrete cooling towers, especially when built together in clusters. Vortex shedding caused the collapse of three towers at Ferrybridge Power Station C in 1965 during high winds.
The failure of the original Tacoma Narrows Bridge was originally attributed to excessive vibration due to vortex shedding, but was actually caused by aeroelastic flutter.
Kármán turbulence is also a problem for airplanes, especially when landing.[12][13]
This formula will generally hold true for the range 250 < Red < 200000:
{\displaystyle {\text{St}}=0.198\left(1-{\frac {19.7}{{\text{Re}}_{d}}}\right)\ }
{\displaystyle {\text{St}}={\frac {fd}{U}}}
U = flow velocity.
This dimensionless parameter St is known as the Strouhal number and is named after the Czech physicist, Vincenc Strouhal (1850–1922) who first investigated the steady humming or singing of telegraph wires in 1878.
Although named after Theodore von Kármán,[14][15] he acknowledged[16] that the vortex street had been studied earlier by Arnulph Mallock[17] and Henri Bénard.[18] Kármán tells the story in his book Aerodynamics:[19]
...Prandtl had a doctoral candidate, Karl Hiemenz, to whom he gave the task of constructing a water channel in which he could observe the separation of the flow behind a cylinder. The object was to check experimentally the separation point calculated by means of the boundary-layer theory. For this purpose, it was first necessary to know the pressure distribution around the cylinder in a steady flow. Much to his surprise, Hiemenz found that the flow in his channel oscillated violently. When he reported this to Prandtl, the latter told him: 'Obviously your cylinder is not circular.' However, even after very careful machining of the cylinder, the flow continued to oscillate. Then Hiemenz was told that possibly the channel was not symmetric, and he started to adjust it. I was not concerned with this problem, but every morning when I came in the laboratory I asked him, 'Herr Hiemenz, is the flow steady now?' He answered very sadly, 'It always oscillates.'
In his autobiography, von Kármán described how his discovery was inspired by an Italian painting of St Christopher carrying the child Jesus whilst wading through water. Vortices could be seen in the water, and von Kármán noted that "The problem for historians may have been why Christopher was carrying Jesus through the water. For me it was why the vortices". It has been suggested by researchers that the painting is one from the 14th century that can be found in the museum of the San Domenico church in Bologna.[20]
Eddy (fluid dynamics) – Swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime
Kelvin–Helmholtz instability – Phenomenon of fluid mechanics
Reynolds number – Dimensionless quantity used to help predict fluid flow patterns
Vortex shedding – Oscillating flow effect resulting from fluid passing over a blunt body
Coandă effect – Tendency of a fluid jet to stay attached to a convex surface
^ Theodore von Kármán, Aerodynamics. McGraw-Hill (1963): ISBN 978-0-07-067602-2. Dover (1994): ISBN 978-0-486-43485-8.
^ a b Bayindir, C. and Namli, B., Efficient sensing of von Kármán vortices using compressive sensing. Computers & Fluids, volume 226, 104975, 2021.
^ Amalia, E., Moelyadi, M. A. Ihsan, M., Effects of Turbulence Model and Numerical Time Steps on Von Karman Flow Behavior and Drag Accuracy of Circular Cylinder, Journal of Physics: Conference Series, 1005, 012012, 2018.
^ Albarède, P., & Provansal, M. Quasi-periodic cylinder wakes and the Ginzburg–Landau model. Journal of Fluid Mechanics, 291, 191-222, 1995.
^ Farazande, S. and Bayindir, C., The Interaction of Von Kármán Vortices with the Solitons of the Complex GinzburgLandau Equation. International Conference on Applied Mathematics in Engineering (ICAME) September 1–3, 2021 - Balikesir, Turkey
^ Monkewitz, P. A., Williamson, C. H. K. and Miller, G. D., Phase dynamics of Kármán vortices in cylinder wakes. Physics of Fluids, 8, 1, 1996.
^ Jackson, C.P. (1987). "A finite-element study of the onset of vortex shedding in flow past variously shaped bodies". Journal of Fluid Mechanics. 182: 23–45. doi:10.1017/S0022112087002234. ; Provansal, M.; Mathis, C.; Boyer, L. (1987). "Bénard-von Kármán instability: transient and forced regimes". Journal of Fluid Mechanics. 182: 1–22. doi:10.1017/S002211208700223.
^ Barkley, D.; Henderson, R.D. (1996). "Three-dimensional Floquet stability analysis of the wake of a circular cylinder". Journal of Fluid Mechanics. 322: 215–241. doi:10.1017/S0022112096002777.
^ "Rapid Response - LANCE - Terra/MODIS 2010/226 14:55 UTC". Rapidfire.sci.gsfc.nasa.gov. Retrieved 2013-12-20.
^ Etling, D. (1990-03-01). "Mesoscale vortex shedding from large islands: A comparison with laboratory experiments of rotating stratified flows". Meteorology and Atmospheric Physics. 43 (1): 145–151. Bibcode:1990MAP....43..145E. doi:10.1007/BF01028117. ISSN 1436-5065. S2CID 122276209.
^ Irwin, Peter A. (September 2010). "Vortices and tall buildings: A recipe for resonance". Physics Today. American Institute of Physics. 63 (9): 68–69. Bibcode:2010PhT....63i..68I. doi:10.1063/1.3490510. ISSN 0031-9228.
^ Wake turbulence
^ "Airport Opening Ceremony Postponed". Archived from the original on 2016-07-26. Retrieved 2016-10-18.
^ T. von Kármán: Nachr. Ges. Wissenschaft. Göttingen Math. Phys. Klasse pp. 509–517 (1911) and pp. 547–556 (1912).
^ T. von Kármán: and H. Rubach, 1912: Phys. Z.", vol. 13, pp. 49–59.
^ T. Kármán, 1954. Aerodynamics: Selected Topics in the Light of Their Historical Development (Cornell University Press, Ithaca), pp. 68–69.
^ A. Mallock, 1907: On the resistance of air. Proc. Royal Soc., A79, pp. 262–265.
^ H. Bénard, 1908: Comptes Rendus de l'Académie des Sciences (Paris), vol. 147, pp. 839–842, 970–972.
^ Von Kármán, T. (1954). Aerodynamics (Vol. 203). Columbus: McGraw-Hill.
^ Mizota, Taketo; Zdravkovich, Mickey; Graw, Kai-U.; Leder, Alfred (March 2000). "Science in culture". Nature. 404 (6775): 226–226. doi:10.1038/35005158. ISSN 1476-4687.
"von Karman vortex shedding". Encyclopedia of Mathematics.
"Flow visualisation of the vortex shedding mechanism on circular cylinder using hydrogen bubbles illuminated by a laser sheet in a water channel". Archived from the original on 2021-12-22 – via YouTube.
"Guadalupe Island Produces von Kármán Vortices". NOAASatellites. Archived from the original on 2021-12-22 – via YouTube.
"Various Views of von Karman Vortices" (PDF). NASA page. Archived from the original (PDF) on March 12, 2016.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Kármán_vortex_street&oldid=1084597697"
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A class of orthogonal polynomials of a new type.
Dutta, M., Manocha, Kanchan Prabha (1983)
A Collocation Method for Boundary Value Problems.
R.D. Russell, L.F. Shampine (1972)
A contribution to the phase theory of a linear second-order differential equation in the Jacobian form
A derivative array approach for linear second order differential-algebraic systems.
Scholz, Lena (2011)
A Family of Solutions of the IVP for the Equation x'(t) = ax(...), ... >>1
MELVIN HEARD (1973)
A general form of the product integral and linear ordinary differential equations
Jiří Jarník, Jaroslav Kurzweil (1987)
A generalization of a theorem of Mammana
Roberto Camporesi, Antonio J. Di Scala (2011)
We prove that any linear ordinary differential operator with complex-valued coefficients continuous in an interval I can be factored into a product of first-order operators globally defined on I. This generalizes a theorem of Mammana for the case of real-valued coefficients.
A generalization of rotations and hyperbolic matrices and its applications.
Bayat, M., Teimoori, H., Mehri, B. (2007)
A Jacobi dual-Petrov-Galerkin method for solving some odd-order ordinary differential equations.
Doha, E.H., Bhrawy, A.H., Hafez, R.M. (2011)
A method for determining constants in the linear combination of exponentials
Jiří Cerha (1996)
Shifting a numerically given function
{b}_{1}exp{a}_{1}t+\cdots +{b}_{n}exp{a}_{n}t
we obtain a fundamental matrix of the linear differential system
\stackrel{˙}{y}=Ay
with a constant matrix
A
. Using the fundamental matrix we calculate
A
, calculating the eigenvalues of
A
{a}_{1},\cdots ,{a}_{n}
and using the least square method we determine
{b}_{1},\cdots ,{b}_{n}
A method in diophantine approximation (VI)
A modification of the Sturm's theorem on separating zeros of solutions of a linear differential equation of the 2nd order
A note on differential equations with periodic solutions
František Neuman (1970)
A note on disconjugate linear differential equations of the second order with periodic coefficients
A note on functional equations of the
p
-adic polylogarithms
Zdzisław Wojtkowiak (1991)
A note on higher monotonicity properties of certain Sturm-Liouville functions. III
Rudolf Blaško, Miloš Háčik (1985)
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A stochastic oscillator is a popular technical indicator for generating overbought and oversold signals.
It is a popular momentum indicator, first developed in the 1950s.
Stochastic oscillators tend to vary around some mean price level, since they rely on an asset's price history.
\begin{aligned} &\text{\%K}=\left(\frac{\text{C} - \text{L14}}{\text{H14} - \text{L14}}\right)\times100\\ &\textbf{where:}\\ &\text{C = The most recent closing price}\\ &\text{L14 = The lowest price traded of the 14 previous}\\ &\text{trading sessions}\\ &\text{H14 = The highest price traded during the same}\\ &\text{14-day period}\\ &\text{\%K = The current value of the stochastic indicator}\\ \end{aligned}
%K=(H14−L14C−L14)×100where:C = The most recent closing priceL14 = The lowest price traded of the 14 previoustrading sessionsH14 = The highest price traded during the same14-day period%K = The current value of the stochastic indicator
Notably, %K is referred to sometimes as the fast stochastic indicator. The "slow" stochastic indicator is taken as %D = 3-period moving average of %K.
The difference between the slow and fast Stochastic Oscillator is the Slow %K incorporates a %K slowing period of 3 that controls the internal smoothing of %K. Setting the smoothing period to 1 is equivalent to plotting the Fast Stochastic Oscillator.
The stochastic oscillator was developed in the late 1950s by George Lane. As designed by Lane, the stochastic oscillator presents the location of the closing price of a stock in relation to the high and low range of the price of a stock over a period of time, typically a 14-day period. Lane, over the course of numerous interviews, has said that the stochastic oscillator does not follow price or volume or anything similar. He indicates that the oscillator follows the speed or momentum of price.
Lane also reveals in interviews that, as a rule, the momentum or speed of the price of a stock changes before the price changes itself. In this way, the stochastic oscillator can be used to foreshadow reversals when the indicator reveals bullish or bearish divergences. This signal is the first, and arguably the most important, trading signal Lane identified.
Example of How to Use the Stochastic Oscillator
The stochastic oscillator is included in most charting tools and can be easily employed in practice. The standard time period used is 14 days, though this can be adjusted to meet specific analytical needs. The stochastic oscillator is calculated by subtracting the low for the period from the current closing price, dividing by the total range for the period and multiplying by 100. As a hypothetical example, if the 14-day high is $150, the low is $125 and the current close is $145, then the reading for the current session would be: (145-125) / (150 - 125) * 100, or 80.
By comparing the current price to the range over time, the stochastic oscillator reflects the consistency with which the price closes near its recent high or low. A reading of 80 would indicate that the asset is on the verge of being overbought.
The Difference Between The Relative Strength Index (RSI) and The Stochastic Oscillator
The relative strength index (RSI) and stochastic oscillator are both price momentum oscillators that are widely used in technical analysis. While often used in tandem, they each have different underlying theories and methods. The stochastic oscillator is predicated on the assumption that closing prices should close near the same direction as the current trend.
Meanwhile, the RSI tracks overbought and oversold levels by measuring the velocity of price movements. In other words, the RSI was designed to measure the speed of price movements, while the stochastic oscillator formula works best in consistent trading ranges.
In general, the RSI is more useful during trending markets, and stochastics more so in sideways or choppy markets.
The primary limitation of the stochastic oscillator is that it has been known to produce false signals. This is when a trading signal is generated by the indicator, yet the price does not actually follow through, which can end up as a losing trade. During volatile market conditions, this can happen quite regularly. One way to help with this is to take the price trend as a filter, where signals are only taken if they are in the same direction as the trend.
https://www.fidelity.com/learning-center/trading-investing/technical-analysis/technical-indicator-guide/slow-stochastic
George Pruitt. "The Ultimate Algorithmic Trading System Toolbox + Website: Using Today's Technology To Help You Become A Better Trader." John Wiley & Sons, 2016.
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Seed_agnetic_ields: A Module for Seeding a Matter Distribution with Magnetic Fields
Seed_Magnetic_Fields: A Module for Seeding a Matter Distribution with Magnetic Fields
The Seed_Magnetic_Fields thorn seeds magnetic fields into an initial hydrodynamic configuration. Currently seeding into TOV stars is supported, according to the poloidal magnetic field prescription:
\begin{array}{rcll}{A}_{x}& =& -y\ast {A}_{b}\ast pow\left(MAX\left(P-{P}_{c}ut,0.0\right),{n}_{s}\right)& \text{(1)}\text{}\text{}\\ {A}_{y}& =& x\ast {A}_{b}\ast pow\left(MAX\left(P-{P}_{c}ut,0.0\right),{n}_{s}\right)& \text{(2)}\text{}\text{}\\ {A}_{z}& =& 0& \text{(3)}\text{}\text{}\\ Phi& =& 0& \text{(4)}\text{}\text{}\end{array}
as specified in Appendix B of the IllinoisGRMHD code announcement paper:
Note that we must be careful if
{A}_{i}
’s are staggered. In this case, the pressure must be interpolated to the staggered point, and the values of
x
y
must also be shifted.
Both staggered and unstaggered vector potential fields are currently supported in this thorn.
Description: Magnetic field strength parameter.
afield_type
Description: A-field prescription
Range Default: Pressure_prescription
A_phi propto (P - P_cut)n^_s
A_phi propto rho
Pressure\_prescription
Density\_prescription
enable_illinoisgrmhd_staggered_a_fields
Description: Define A fields on an IllinoisGRMHD staggered grid
enable_varpi_squared_multiplication
Description: Multiply A_phi by varpi2^?
Description: Magnetic field strength pressure exponent.
p_cut
Description: Cutoff pressure, below which vector potential is set to zero. Typically set to 4% of the maximum initial pressure.
rho_cut
Description: Cutoff density, below which vector potential is set to zero. Typically set to 20% of the maximum initial density.
This section lists all the variables which are assigned storage by thorn WVUThorns/Seed_Magnetic_Fields. Storage can either last for the duration of the run (Always means that if this thorn is activated storage will be assigned, Conditional means that if this thorn is activated storage will be assigned for the duration of the run if some condition is met), or can be turned on for the duration of a schedule function.
HydroBase::rho[1] HydroBase::press[1] HydroBase::eps[1] HydroBase::vel[1] HydroBase::Bvec[1] HydroBase::Avec[1] HydroBase::Aphi[1]
set up seed magnetic field configuration.
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Polynomial degree and algebraic expression — lesson. Mathematics State Board, Class 9.
The polynomial degree is the highest variable power in a polynomial.
. In this polynomial, the highest variable power is \(3\).
Polynomial classification based on degree:
Linear Polynomial: A polynomial of degree \(1\) — .
Quadratic Polynomial: A polynomial of degree \(2\) — .
Cubic Polynomial: A polynomial of degree \(3\) — .
It must be noted that there will be a maximum of \(2\) terms in a linear polynomial, \(3\) terms in quadratic polynomials and \(4\) terms in the cubic polynomial of polynomials in one variable.
General form of polynomials of different degrees:
Linear Polynomial: A polynomial in one variable with degree one is called a linear polynomial. It can be denoted as
p\left(x\right)=\mathit{ax}+b
Quadratic Polynomial: A polynomial in one variable with degree two is called a quadratic polynomial. It can be denoted as .
Cubic Polynomial: A polynomial in one variable with degree three is called a cubic polynomial. It is denoted as .
It's not defined the degree of zero polynomial. There can be any degree.
p\left(x\right)=0
can be substituted as — where '\(n\)' can be any number.
For example: \(p(x) = 0 × x^6 = 0\).
The constant polynomial is the form \(p(x) = c\), where \(c\) is the actual number. This means that it is constant for all possible values of \(x\), \(p(x) = c\).
For example: \(p(x) = 6 = 6 x^0\) [where \(x^0 = 1\)]
Note that the highest power of the '\(x\)' is zero.
Therefore, the degree of the non-zero constant polynomial is zero.
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ebvalaim.log – A log of my ideas, plans, creations
When Special Relativity is being introduced at school (if it is at all - the curriculum might depend on your country and it can change in time), one of the notions being discussed is so called "relativistic mass".
One of the consequences of relativity is that faster moving objects are harder to accelerate, which means that their inertia increases. And since it is being said from the beginning of the physics lessons that mass is the measure of inertia, it is tempting to try to explain this effect with an increase in mass. So, the notion of mass is being split into "rest mass" - the mass an object has at rest - and a "relativistic mass" - the mass of the object in motion, larger than the rest mass. The equations also become prettier right away, since if we denote the relativistic mass by
m
E = mc^2
, and momentum can be expressed using the formula known from classical physics
p = mv
(versions with the rest mass also have an ugly square root in the denominator - we'll see it later). This is the life!
If you are following articles or discussions about relativity on the internet, you probably noticed relativistic mass being mentioned in multiple contexts. It is often used to explain the impossibility of reaching the speed of light ("because the mass would grow to infinity"), or sometimes someone will ask whether an object can become a black hole by going fast enough (it can't). The relativistic increase in mass is being treated as fact in such situations, as something certain.
Well, I'd like to disturb this state of affairs slightly with this article ;) Because, as it turns out, the notion of relativistic mass loses a lot of its appeal upon closer scrutiny. As a result, relativistic mass is rarely being used in academia and you can encounter it pretty much only at school, in discussions on the internet and in popular science publications. Let's take a closer look at the reasons behind that.
Articles, Physics, Relativity
The simplest kind of geometry, taught in schools, is the so called Euclidean geometry - named after an ancient Greek mathematician, Euclid, who described its basics in the 4th century BC in his "Elements". It is based on the notions of points, straight lines and planes and it seems to correspond perfectly to our everyday experiences with various shapes. However, we can notice problems for which Euclidean geometry is insufficient even in our immediate surroundings.
Let's imagine, for example, that we are airline pilots and our task is to fly as quickly as possible from Warsaw, Poland to San Francisco. We take a world map and knowing from Euclidean geometry that a straight line is the shortest path between two points, we draw such a line from Warsaw to San Francisco. We're getting ready to depart and fly along the course we plotted... but fortunately, our navigator friend tells us that we fell into a trap.
The trap is that the surface of the Earth isn't flat! The map we used to plot our straight line course is just a projection of a surface that is close to spherical in reality. Because of that, the red line on the map below is not the shortest path - the purple line is:
Red line - a straight line between Warsaw and San Francisco on the map. The purple line is the actual shortest path.
Articles, Physics, Physics for everyone
c
c
MaidSafe and PARSEC - a new distributed consensus algorithm
I'll write something about my work for a change.
I've been employed at MaidSafe for over a year and a half now. It's a small, Scottish company working on creating a fully distributed Internet. Sounds a bit weird - the Internet is already distributed, isn't it? Well, it isn't completely - every website in the Internet exists on some servers belonging to some single company. All data in the Internet is controlled by the owners of the servers that host it, and not necessarily the actual owners of the data itself. This leads to situations in which our data is sometimes used in ways we don't like (GDPR, which came into force recently, is supposed to improve the state of affairs, but I wouldn't expect too much...).
MaidSafe aims to change all of this. It counters the centralised servers with the SAFE Network - a distributed network, in which everyone controls their data. When we upload a file to this network, we aren't putting it on a specific server. Instead, the file is sliced into multiple pieces, encrypted and distributed in multiple copies among the computers of the network's users. Every user shares a part of their hard drive, but only controls their own data - the rest is unreadable to them thanks to encryption. What's more, in order to prevent spam and incentivise the users to share their space, SAFE Network is going to have its own native cryptocurrency - Safecoin - but it won't be blockchain-based, unlike the other cryptocurrencies.
I'm slowly preparing a new post for the category "Physics for everyone". The post will describe the Lorentz transformation a bit more in depth, and say something about the consequences of it being the correct description of reality. I prepared two GIFs for this purpose:
The transformation of the coordinate system by a rotation (click for an animation)
The transformation of a coordinate system by a Lorentz transformation (click for an animation)
A more detailed description of those GIFs will be a part of the new post in the category Physics for everyone :)
I published the code I used to generate them on GitHub: https://github.com/fizyk20/spacetime-graph/tree/blog-post
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A-level Mathematics/AQA/MPC3 - Wikibooks, open books for an open world
A-level Mathematics/AQA/MPC3
1.1 Mappings and functions
1.2 Domain and range of a function
1.3 Modulus function
2.4 x as a function of y
3.1 The functions cosec θ, sec θ and cot θ
3.2 Standard trigonometric identities
3.3 Differentiation of sin x, cos x and tan x
3.4 Integration of sin(kx) and cos(kx)
Mappings and functions[edit | edit source]
We think of a function as an operation that takes one number and transforms it into another number. A mapping is a more general type of function. It is simply a way to relate a number in one set, to a number in another set. Let us look at three different types of mappings:
one-to-one - this mapping gives one unique output for each input.
many-to-one - this type of mapping will produce the same output for more than one value of
{\displaystyle x}
one-to-many - this mapping produces more than one output for each input.
Only the first two of these mappings are functions. An example of a mapping which is not a function is
{\displaystyle f(x)=\pm {\sqrt {x}}}
Domain and range of a function[edit | edit source]
{\displaystyle f(x)}
{\displaystyle x}
The set of permitted
{\displaystyle x}
values is called the domain of the function
The set of all images is called the range of the function
Modulus function[edit | edit source]
{\displaystyle x}
{\displaystyle |x|}
{\displaystyle |x|={\begin{cases}x&{\mbox{for }}x\geq 0\\-x&{\mbox{for }}x<0\end{cases}}}
Chain rule[edit | edit source]
{\displaystyle y}
{\displaystyle u}
{\displaystyle u}
{\displaystyle x}
{\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}{\frac {du}{dx}}}
As you can see from above, the first step is to notice that we have a function that we can break down into two, each of which we know how to differentiate. Also, the function is of the form
{\displaystyle f(g(x))}
. The process is then to assign a variable to the inner function, usually
{\displaystyle u}
, and use the rule above;
{\displaystyle y=2(x-1)^{3}}
We can see that this is of the correct form, and we know how to differentiate each bit.
{\displaystyle u=x-1}
Now we can rewrite the original function,
{\displaystyle y=2u^{3}}
We can now differentiate each part;
{\displaystyle {\frac {dy}{du}}=6u^{2}}
{\displaystyle {\frac {du}{dx}}=1}
Now applying the rule above;
{\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}*{\frac {du}{dx}}=6u^{2}*1=6u^{2}=6(x-1)^{2}}
Product rule[edit | edit source]
{\displaystyle y=uv}
{\displaystyle u}
{\displaystyle v}
{\displaystyle x}
{\displaystyle {\frac {d}{dx}}(uv)=u{\frac {dv}{dx}}+v{\frac {du}{dx}}}
An alternative way of writing the product rule is:
{\displaystyle (uv)'=uv'+u'v\,\!}
Or in Lagrange notation:
{\displaystyle k(x)=f(x)g(x)}
{\displaystyle k'(x)=f'(x)g(x)+f(x)g'(x)}
Quotient rule[edit | edit source]
{\displaystyle y={\frac {u}{v}}}
{\displaystyle u}
{\displaystyle v}
{\displaystyle x}
{\displaystyle {\frac {d}{dx}}\left({\frac {u}{v}}\right)={\frac {v{\frac {du}{dx}}-u{\frac {dv}{dx}}}{v^{2}}}}
An alternative way of writing the quotient rule is:
{\displaystyle \left({\frac {u}{v}}\right)'={\frac {u'v-uv'}{v^{2}}}}
x as a function of y[edit | edit source]
{\displaystyle {\frac {dy}{dx}}={\frac {1}{\frac {dx}{dy}}}}
Trigonometric functions[edit | edit source]
The functions cosec θ, sec θ and cot θ[edit | edit source]
{\displaystyle \operatorname {cosec} {\theta }={\frac {1}{\sin {\theta }}}}
{\displaystyle \sec {\theta }={\frac {1}{\cos {\theta }}}}
{\displaystyle \cot {\theta }={\frac {1}{\tan {\theta }}}}
Standard trigonometric identities[edit | edit source]
{\displaystyle \cot {\theta }={\frac {\cos {\theta }}{\sin {\theta }}}}
{\displaystyle \sec ^{2}{\theta }=1+\tan ^{2}{\theta }\,\!}
{\displaystyle \operatorname {cosec} ^{2}{\theta }=1+\cot ^{2}{\theta }}
Differentiation of sin x, cos x and tan x[edit | edit source]
{\displaystyle {\frac {d}{dx}}\left(\sin {x}\right)=\cos {x}}
{\displaystyle {\frac {d}{dx}}\left(\cos {x}\right)=-\sin {x}}
{\displaystyle {\frac {d}{dx}}\left(\tan {x}\right)=\sec ^{2}{x}}
Integration of sin(kx) and cos(kx)[edit | edit source]
{\displaystyle \int \cos {kx}\ dx={\frac {1}{k}}\sin {kx}+c}
{\displaystyle \int \sin {kx}\ dx=-{\frac {1}{k}}\cos {kx}+c}
Exponentials and logarithms[edit | edit source]
Differentiating exponentials and logarithms[edit | edit source]
{\displaystyle {\mbox{when}}\ y=e^{kx},\ {\frac {dy}{dx}}=ke^{kx}}
{\displaystyle \int e^{kx}\ dx={\frac {1}{k}}e^{kx}+c}
Natural logarithms[edit | edit source]
{\displaystyle y=\ln {x}}
{\displaystyle {\frac {dy}{dx}}={\frac {1}{x}}}
{\displaystyle \int {\frac {1}{x}}\ dx=\ln {x}+c}
{\displaystyle \int {\frac {f'(x)}{f(x)}}\ dx=\ln {f(x)}+c,\ {\mbox{provided}}\ f(x)>0}
Integration by parts[edit | edit source]
{\displaystyle \int u{\frac {dv}{dx}}\ dx=uv-\int v{\frac {du}{dx}}\ dx}
Standard integrals[edit | edit source]
{\displaystyle \int {\frac {dx}{a^{2}+x^{2}}}={\frac {1}{a}}\tan ^{-1}{\left({\frac {x}{a}}\right)}+c}
{\displaystyle \int {\frac {dx}{\sqrt {(a^{2}-x^{2})}}}=\sin ^{-1}{\left({\frac {x}{a}}\right)}+c}
Volumes of revolution[edit | edit source]
The volume of the solid formed when the area under the curve
{\displaystyle y=f(x)}
{\displaystyle x=a}
{\displaystyle x=b}
, is rotated through 360° about the
{\displaystyle x}
-axis is given by:
{\displaystyle V=\pi \int _{a}^{b}y^{2}\ dx}
{\displaystyle y=f(x)}
{\displaystyle y=a}
{\displaystyle y=b}
{\displaystyle y}
{\displaystyle V=\pi \int _{a}^{b}x^{2}\ dy}
Numerical methods[edit | edit source]
Iterative methods[edit | edit source]
An iterative method is a process that is repeated to produce a sequence of approximations to the required solution.
Mid ordinate rule
{\displaystyle \int _{a}^{b}y\ dx\approx h\lbrack y_{\frac {1}{2}}+y_{\frac {3}{2}}+\ldots +y_{n-{\frac {3}{2}}}+y_{n-{\frac {1}{2}}}\rbrack }
{\displaystyle {\mbox{where}}\ h={\frac {b-a}{n}}}
{\displaystyle \int _{a}^{b}y\ dx\approx {\frac {h}{3}}\lbrack \left(y_{0}+y_{n}\right)+4\left(y_{1}+y_{3}\ldots +y_{n-1}\right)+2\left(y_{2}+y_{4}+\ldots +y_{n-2}\right)\rbrack }
{\displaystyle {\mbox{where}}\ h={\frac {b-a}{n}}\ {\mbox{and}}\ n\ {\mbox{is even}}}
Retrieved from "https://en.wikibooks.org/w/index.php?title=A-level_Mathematics/AQA/MPC3&oldid=3934358"
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G
{}^{c}
G
Fabio Podestà (1989)
\left({\overline{\nabla }}_{\stackrel{˙}{\gamma }}R\right)\left(·,\stackrel{˙}{\gamma }\right)\stackrel{˙}{\gamma }=0
\overline{\nabla }
\gamma
\overline{\nabla }
\overline{\nabla }
\gamma
\xi
k
\overline{\nabla }
\gamma
G={K}^{ℂ}
\Omega \subset G
K
G
M:=G/K
\Omega
{\Omega }_{M}\subset M
S:=\partial {\Omega }_{M}
{K}^{S}\left(E\right)\ge {K}^{M}\left(E\right)+k\left(E,n\right)
k\left(E,n\right)
A Foliated Metric Rigidity Theorem for Higher Rank Irreducible Symmetric Spaces.
S. Adams, L. Hernández (1994)
Olivier Biquard, Rafe Mazzeo (2011)
M
M
A short topological proof for the symmetry of 2 point homogeneous spaces.
Z.I. Szabó (1991)
A unified approach to compact symmetric spaces of rank one
Adam Korányi, Fulvio Ricci (2010)
A relatively simple algebraic framework is given, in which all the compact symmetric spaces can be described and handled without distinguishing cases. We also give some applications and further results.
Addendum to "Existence of Hermitian n-Symmetric Spaces and of Non-commutative Naturally Reductive Spaces".
J. Alfredo Jiménez (1988)
Admissible distributions on p-adic symmetric spaces.
Jeffrey Hakim (1994)
Algebraically independent generators of invariant differential operators on a symmetric cone.
Takaaki Nomura (1989)
Almost contact metric submersions and curvature tensors.
Tshikunguila Tshikuna-Matamba (2005)
It is known that L. Vanhecke, among other geometers, has studied curvature properties both on almost Hermitian and almost contact metric manifolds.The purpose of this paper is to interrelate these properties within the theory of almost contact metric submersions. So, we examine the following problem: Let f: M → B be an almost contact metric submersion. Suppose that the total space is a C(α)-manifold. What curvature properties do have the fibres or the base space?
Almost Hermitian manifolds with constant holomorphic sectional curvature
Alfred Gray, Lieven Vanhecke (1979)
An exhaustion of locally symmetric spaces by compact submanifolds with corners.
Enrico Leuzinger (1995)
Associated families of pluriharmonic maps and isotropy.
R. Tribuzy, J.-H. Eschenburg (1998)
Asymptotic behaviour of generalized Poisson integrals in rank one symmetric spaces and in trees
Peter Sjögren (1988)
Asymptotic geometry of arithmetic quotients of symmetric spaces.
Toshiaki Hattori (1996)
Ball-Homogeneous and Disk-Homogeneous Riemannian Manifolds.
Oldrich Kowalski, Lieven Vanhecke (1982)
Piotr Graczyk, Jean-Jacques Lœb (1994)
Boundaries for left-invariant sub-elliptic operators on semidirect products of nilpotent and abelian groups.
A. Hulanicki, Eva Damek (1990)
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A class of differential equations similar to linear equations
A contribution to Runge-Kutta formulas of the 7th order with rational coefficients for systems of differential equations of the first order
Anton Huťa, Vladimír Penjak (1984)
The purpose of this article is to find the 7th order formulas with rational parameters. The formulas are of the 11th stage. If we compare the coefficients of the development
{\sum }_{i=1}^{\infty }\frac{{h}^{i}}{i!}\frac{{d}^{i-1}}{d{x}^{i-1}}𝐟\left[x,𝐲\left(x\right)\right]
{h}^{7}
with the development given by successive insertion into the formula
h.{f}_{i}\left({k}_{0},{k}_{1},...,{k}_{i-1}\right)
i=1,2,...,10
k={\sum }_{i=0}^{10}{p}_{i},{k}_{i}
we obtain a system of 59 condition equations with 65 unknowns (except, the 1st one, all equations are nonlinear). As the solution of this system we get the parameters of the 7th order Runge-Kutta formulas as rational numbers.
Krystyna Szafraniec (1989)
A differential equation related to the
{l}^{p}
-norms
Jacek Bojarski, Tomasz Małolepszy, Janusz Matkowski (2011)
Let p ∈ (1,∞). The question of existence of a curve in ℝ₊² starting at (0,0) and such that at every point (x,y) of this curve, the
{l}^{p}
-distance of the points (x,y) and (0,0) is equal to the Euclidean length of the arc of this curve between these points is considered. This problem reduces to a nonlinear differential equation. The existence and uniqueness of solutions is proved and nonelementary explicit solutions are given.
A family of multistep methods to integrate orbits on spheres.
J.M. Ferrándiz, M. Teresa Pérez (1993)
A fixed point method to compute solvents of matrix polynomials
Fernando Marcos, Edgar Pereira (2010)
A generalization of Tichonov theorem
Jaromír Šiška, Ivan Dvořák (1985)
A geometric approach to integrability conditions for Riccati equations.
Carinena, Jose F., de Lucas, Javier, Ramos, Arturo (2007)
A high accuracy method for solving ODEs with continuous right-hand side.
David Stewart (1990/1991)
A method for finding the step size of integration of a system of ordinary differential equations
J. S. Chomicz, A. Olejniczak, M. Szyszkowicz (1983)
A modified version of explicit Runge-Kutta methods for energy-preserving
Guang-Da Hu (2014)
In this paper, Runge-Kutta methods are discussed for numerical solutions of conservative systems. For the energy of conservative systems being as close to the initial energy as possible, a modified version of explicit Runge-Kutta methods is presented. The order of the modified Runge-Kutta method is the same as the standard Runge-Kutta method, but it is superior in energy-preserving to the standard one. Comparing the modified Runge-Kutta method with the standard Runge-Kutta method, numerical experiments...
A multiplicity result for periodic solutions to higher-order ordinary differential equations via the method of upper and lower solutions.
Lee, Yong-Hoon (1998)
A new approach to solve systems of second order non-linear ordinary differential equations.
Rafiullah, Muhammad, Rafiq, Arif (2010)
A new method for the explicit integration of Lotka-Volterra equations.
Mingari Scarpello, Giovanni, Ritelli, Daniele (2003)
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Confidence intervals for coefficients of generalized linear mixed-effects model - MATLAB - MathWorks Australia
95% Confidence Intervals for Fixed Effects
99% Confidence Intervals for Random Effects
Confidence intervals for coefficients of generalized linear mixed-effects model
feCI = coefCI(glme)
feCI = coefCI(glme,Name,Value)
feCI = coefCI(glme) returns the 95% confidence intervals for the fixed-effects coefficients in the generalized linear mixed-effects model glme.
feCI = coefCI(glme,Name,Value) returns the confidence intervals using additional options specified by one or more Name,Value pair arguments. For example, you can specify a different confidence level or the method used to compute the approximate degrees of freedom.
[feCI,reCI] = coefCI(___) also returns the confidence intervals for the random-effects coefficients using any of the previous syntaxes.
Fixed-effects confidence intervals, returned as a p-by-2 matrix. feCI contains the confidence limits that correspond to the p-by-1 fixed-effects vector returned by the fixedEffects method. The first column of feCI contains the lower confidence limits and the second column contains the upper confidence limits.
When fitting a GLME model using fitglme and one of the maximum likelihood fit methods ('Laplace' or 'ApproximateLaplace'):
If you specify the 'CovarianceMethod' name-value pair argument as 'conditional', then the confidence intervals are conditional on the estimated covariance parameters.
If you specify the 'CovarianceMethod' name-value pair argument as 'JointHessian', then the confidence intervals account for the uncertainty in the estimated covariance parameters.
When fitting a GLME model using fitglme and one of the pseudo likelihood fit methods ('MPL' or 'REMPL'), coefci uses the fitted linear mixed effects model from the final pseudo likelihood iteration to compute confidence intervals on the fixed effects.
Random-effects confidence intervals, returned as a q-by-2 matrix. reCI contains the confidence limits corresponding to the q-by-1 random-effects vector B returned by the randomEffects method. The first column of reCI contains the lower confidence limits, and the second column contains the upper confidence limits.
When fitting a GLME model using fitglme and one of the maximum likelihood fit methods ('Laplace' or 'ApproximateLaplace'), coefCI computes the confidence intervals using the conditional mean squared error of prediction (CMSEP) approach conditional on the estimated covariance parameters and the observed response. Alternatively, you can interpret the confidence intervals from coefCI as approximate Bayesian credible intervals conditional on the estimated covariance parameters and the observed response.
When fitting a GLME model using fitglme and one of the pseudo likelihood fit methods ('MPL' or 'REMPL'), coefci uses the fitted linear mixed effects model from the final pseudo likelihood iteration to compute confidence intervals on the random effects.
{\text{defects}}_{ij}\sim \text{Poisson}\left({\mu }_{ij}\right)
\mathrm{log}\left({\mu }_{ij}\right)={\beta }_{0}+{\beta }_{1}{\text{newprocess}}_{ij}+{\beta }_{2}{\text{time}\text{_}\text{dev}}_{ij}+{\beta }_{3}{\text{temp}\text{_}\text{dev}}_{ij}+{\beta }_{4}{\text{supplier}\text{_}\text{C}}_{ij}+{\beta }_{5}{\text{supplier}\text{_}\text{B}}_{ij}+{b}_{i},
{\text{defects}}_{ij}
i
j
{\mu }_{ij}
i
i=1,2,...,20
j
j=1,2,...,5
{\text{newprocess}}_{ij}
{\text{time}\text{_}\text{dev}}_{ij}
{\text{temp}\text{_}\text{dev}}_{ij}
i
j
{\text{newprocess}}_{ij}
i
j
{\text{supplier}\text{_}\text{C}}_{ij}
{\text{supplier}\text{_}\text{B}}_{ij}
i
j
{b}_{i}\sim N\left(0,{\sigma }_{b}^{2}\right)
i
Use fixedEffects to display the estimates and names of the fixed-effects coefficients in glme.
betanames=6×1 table
{'(Intercept)'}
{'newprocess' }
{'time_dev' }
{'temp_dev' }
{'supplier_C' }
{'supplier_B' }
Each row of beta contains the estimated value for the coefficient named in the corresponding row of betanames. For example, the value –0.0945 in row 3 of beta is the estimated coefficient for the predictor variable time_dev.
Column 1 of feCI contains the lower bound of the 95% confidence interval. Column 2 contains the upper bound. Row 1 corresponds to the intercept term. Rows 2, 3, and 4 correspond to newprocess, time_dev, and temp_dev, respectively. Rows 5 and 6 correspond to the indicator variables supplier_C and supplier_B, respectively. For example, the 95% confidence interval for the coefficient for time_dev is [-1.7395 , 1.5505]. Some of the confidence intervals include 0, which indicates that those predictors are not significant at the 5% significance level. To obtain specific
p
-values for each fixed-effects term, use fixedEffects. To test significance for entire terms, use anova.
Fit a generalized linear mixed-effects model using newprocess, time_dev, temp_dev, and supplier as fixed-effects predictors. Include a random-effects intercept grouped by factory, to account for quality differences that might exist due to factory-specific variations. The response variable defects has a Poisson distribution, and the appropriate link function for this model is log. Use the Laplace fit method to estimate the coefficients.
{\text{defects}}_{ij}\sim \text{Poisson}\left({\mu }_{ij}\right)
\mathrm{log}\left({\mu }_{ij}\right)={\beta }_{0}+{\beta }_{1}{\text{newprocess}}_{ij}+{\beta }_{2}{\text{time}\text{_}\text{dev}}_{ij}+{\beta }_{3}{\text{temp}\text{_}\text{dev}}_{ij}+{\beta }_{4}{\text{supplier}\text{_}\text{C}}_{ij}+{\beta }_{5}{\text{supplier}\text{_}\text{B}}_{ij}+{b}_{i},
{\text{defects}}_{ij}
i
j
{\mu }_{ij}
i
i=1,2,...,20
j
j=1,2,...,5
{\text{newprocess}}_{ij}
{\text{time}\text{_}\text{dev}}_{ij}
{\text{temp}\text{_}\text{dev}}_{ij}
i
j
{\text{newprocess}}_{ij}
i
j
{\text{supplier}\text{_}\text{C}}_{ij}
{\text{supplier}\text{_}\text{B}}_{ij}
i
j
{b}_{i}\sim N\left(0,{\sigma }_{b}^{2}\right)
i
Use randomEffects to compute and display the estimates of the empirical Bayes predictors (EBPs) for the random effects associated with factory.
Each row of B contains the estimated EBPs for the random-effects coefficient named in the corresponding row of Bnames. For example, the value -0.2633 in row 3 of B is the estimated coefficient of '(Intercept)' for level '3' of factory.
Compute the 99% confidence intervals of the EBPs for the random effects.
[feCI,reCI] = coefCI(glme,'Alpha',0.01);
reCI = 20×2
Column 1 of reCI contains the lower bound of the 99% confidence interval. Column 2 contains the upper bound. Each row corresponds to a level of factory, in the order shown in Bnames. For example, row 3 corresponds to the coefficient of '(Intercept)' for level '3' of factory, which has a 99% confidence interval of [-0.8219 , 0.2954]. For additional statistics related to each random-effects term, use randomEffects.
[1] Booth, J.G., and J.P. Hobert. “Standard Errors of Prediction in Generalized Linear Mixed Models.” Journal of the American Statistical Association. Vol. 93, 1998, pp. 262–272.
GeneralizedLinearMixedModel | anova | coefTest | covarianceParameters | fixedEffects | randomEffects
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Permanent Magnet Synchronous Machine (PMSM) - SIMBA Documentation
Electrical model and equations
Model of a three-phase Permanent Magnet Synchronous Machine (PMSM) with sinusoidal back Electro-Motive Force (EMF). In motor operation torque and speed have the same sign. In this model, the flux linking each winding is assumed to depend linearly on all stator winding currents and it is assumed that the permanent magnet flux linkage is sinusoidal.
\phi_d = L_d i_d + \phi_m
\phi_q = L_q i_q
where \phi_m = \frac{K_e}{N_{pp}}
\phi_m = \frac{K_e}{N_{pp}}
is the permanent magnet flux linkage, and \omega_r = N_{pp} \Omega
\omega_r = N_{pp} \Omega
is the electrical speed of the rotor field.
Electro-magnetic torque:
T_e = 1.5 * N_{pp} * (i_q * \phi_d - id * \phi_q)
Mechanical rotational speed \Omega
\Omega
J \frac{d\Omega}{dt} = T_e - B \Omega
Ld Direct Axis Inductance [H]
Lq Quadratic Axis Inductance [H]
Ke Back-EMF Constant
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Physics - Ultrafast Switching in a Phase-Change Material
Ultrafast Switching in a Phase-Change Material
August 5, 2016 • Physics 9, s85
New experiments show that picosecond pulses of light can effectively switch off the resistance in phase-change materials that are used for storing computer information.
Peter Zalden/SLAC
Chalcogenide compounds are phase-change materials used in some types of rewriteable DVDs. The materials’ phase change—between an amorphous glass and an ordered crystal—relies on a separate electrical change: a field-induced drop in resistance, called “threshold switching.” To explore how fast this switching occurs, researchers exposed a chalcogenide compound to electrical pulses that were shorter than those previously tested. They found that threshold switching occurs on subpicosecond scales, suggesting chalcogenides have a future in ultrafast memory devices and switches.
In memory applications, chalcogenide compounds are transformed between their glass and crystalline phases using electric fields. The fields induce currents that heat the material to certain transition temperatures. Threshold switching plays a role in reducing the resistance (by as much as a factor of a thousand) in the chalcogenide glass phase so that sufficient current can flow through the material.
Some recent data have suggested that threshold switching occurs at nanosecond scales, which is not fast enough for certain proposed memory applications. To test the switching speed, Peter Zalden and Michael Shu, from the SLAC National Accelerator Laboratory in California, and colleagues fired a train of picosecond light pulses in the terahertz frequency range at a common chalcogenide. The team deposited the chalcogenide between gold electrodes to boost the electric field amplitude. Following the pulse exposure, crystalline filaments appeared in the sample, as confirmed with x-ray diffraction. The presence of crystallization implies that threshold switching takes less than a picosecond to occur. This could prove important for the development of phase-change memory, which would use chalcogenides for random access memory (like in a flash drive).
Peter Zalden, Michael J. Shu, Frank Chen, Xiaoxi Wu, Yi Zhu, Haidan Wen, Scott Johnston, Zhi-Xun Shen, Patrick Landreman, Mark Brongersma, Scott W. Fong, H.-S. Philip Wong, Meng-Ju Sher, Peter Jost, Matthias Kaes, Martin Salinga, Alexander von Hoegen, Matthias Wuttig, and Aaron M. Lindenberg
{\text{CaH}}_{6}
|
Babadjian, Jean-François ; Francfort, Gilles A.
A justification of heterogeneous membrane models as zero-thickness limits of a cylindral three-dimensional heterogeneous nonlinear hyperelastic body is proposed in the spirit of Le Dret (1995). Specific characterizations of the 2D elastic energy are produced. As a generalization of Bouchitté et al. (2002), the case where external loads induce a density of bending moment that produces a Cosserat vector field is also investigated. Throughout, the 3D-2D dimensional reduction is viewed as a problem of
\Gamma
-convergence of the elastic energy, as the thickness tends to zero.
Classification : 49J45, 74B20, 74G65, 74K15, 74K35
Mots clés : dimension reduction,
\Gamma
-convergence, equi-integrability, quasiconvexity, relaxation
author = {Babadjian, Jean-Fran\c{c}ois and Francfort, Gilles A.},
title = {Spatial heterogeneity in {3D-2D} dimensional reduction},
AU - Babadjian, Jean-François
AU - Francfort, Gilles A.
TI - Spatial heterogeneity in 3D-2D dimensional reduction
Babadjian, Jean-François; Francfort, Gilles A. Spatial heterogeneity in 3D-2D dimensional reduction. ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 1, pp. 139-160. doi : 10.1051/cocv:2004031. http://www.numdam.org/articles/10.1051/cocv:2004031/
[1] E. Acerbi and N. Fusco, Semicontinuity results in the calculus of variations. Arch. Rat. Mech. Anal. 86 (1984) 125-145. | Zbl 0565.49010
[2] M. Bocea and I. Fonseca, Equi-integrability results for 3D-2D dimension reduction problems. ESAIM: COCV 7 (2002) 443-470. | Numdam | Zbl 1044.49010
[3] G. Bouchitté, I. Fonseca and M.L. Mascarenhas, Bending moment in membrane theory. J. Elasticity 73 (2003) 75-99. | Zbl 1059.74034
[4] A. Braides, personal communication.
[5] A. Braides and A. Defranceschi, Homogenization of multiple integrals. Oxford lectures Ser. Math. Appl. Clarendon Press, Oxford (1998). | MR 1684713 | Zbl 0911.49010
[6] A. Braides, I. Fonseca and G. Francfort, 3D-2D asymptotic analysis for inhomogeneous thin films. Indiana Univ. Math. J. 49 (2000) 1367-1404. | Zbl 0987.35020
[7] B. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag, Berlin (1988). | MR 2361288 | Zbl 0703.49001
\Gamma
-convergence. Birkhaüser, Boston (1993). | MR 1201152 | Zbl 0816.49001
[9] I. Ekeland and R. Temam, Analyse convexe et problèmes variationnels. Dunod, Gauthiers-Villars, Paris (1974). | MR 463993 | Zbl 0281.49001
[10] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Boca Raton, CRC Press (1992). | MR 1158660 | Zbl 0804.28001
[11] D. Fox, A. Raoult and J.C. Simo, A justification of nonlinear properly invariant plate theories. Arch. Rat. Mech. Anal. 25 (1992) 157-199. | Zbl 0789.73039
[12] G. Friesecke, R.D. James and S. Müller, Rigorous derivation of nonlinear plate theory and geometric rigidity. C.R. Acad. Sci. Paris, Série I 334 (2001) 173-178. | Zbl 1012.74043
[13] G. Friesecke, R.D. James and S. Müller, A Theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity. Comm. Pure Appl. Math. 55 (2002) 1461-1506. | Zbl 1021.74024
[14] G. Friesecke, R.D. James and S. Müller, The Föppl-von Kármán plate theory as a low energy
\Gamma
-limit of nonlinear elasticity. C.R. Acad. Sci. Paris, Série I 335 (2002) 201-206. | Zbl 1041.74043
[15] H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 74 (1995) 549-578. | Zbl 0847.73025
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El Houcein El Abdalaoui (2000)
A Criterion for a Process to Be Prime.
William A. Veech (1982)
A criterion for Toeplitz flows to be topologically isomorphic and applications
T. Downarowicz, J. Kwiatkowski, Y. Lacroix (1995)
A dynamical system is said to be coalescent if its only endomorphisms are automorphisms. The question whether there exist coalescent ergodic dynamical systems with positive entropy has not been solved so far and it seems to be difficult. The analogous problem in topological dynamics has been solved by Walters ([W]). His example, however, is not minimal. In [B-K2], a class of strictly ergodic (hence minimal) Toeplitz flows is presented, which have positive entropy and trivial topological centralizers...
A cut salad of cocycles
Jon Aaronson, Mariusz Lemańczyk, Dalibor Volný (1998)
We study the centraliser of locally compact group extensions of ergodic probability preserving transformations. New methods establishing ergodicity of group extensions are introduced, and new examples of squashable and non-coalescent group extensions are constructed.
A cylinder flow arising from irregularity of distribution
K. Schmidt (1978)
A description of stochastic systems using chaotic maps.
Boyarski, Abraham, Góra, Pawełl (2004)
A Differentiation Theorem for Additive Processes.
Mustafa A. Akcoglu, Ulrich Krengel (1978)
A dominated ergodic estimate for
{L}_{p}
spaces with weights
E. Atencia, A. de la Torre (1982)
Davide Barilari, Luca Rizzi (2013)
For an equiregular sub-Riemannian manifold M, Popp’s volume is a smooth volume which is canonically associated with the sub-Riemannian structure, and it is a natural generalization of the Riemannian one. In this paper we prove a general formula for Popp’s volume, written in terms of a frame adapted to the sub-Riemannian distribution. As a first application of this result, we prove an explicit formula for the canonical sub- Laplacian, namely the one associated with Popp’s volume. Finally, we discuss...
A Function with Countably Many Ergodic Equilibrium States.
F. Hofbauer (1977)
A generalization of Steinhaus' theorem to coordinatewise measure preserving binary transformations
Marcin E. Kuczma (1976)
A generalization of the individual ergodic theorem
Radko Mesiar (1980)
A generalized skew product
Zbigniew Kowalski (1987)
A Horseshoe with Positive Measure.
Rufus Bowen (1975)
A joint limit theorem for compactly regenerative ergodic transformations
David Kocheim, Roland Zweimüller (2011)
We study conservative ergodic infinite measure preserving transformations satisfying a compact regeneration property introduced by the second-named author in J. Anal. Math. 103 (2007). Assuming regular variation of the wandering rate, we clarify the asymptotic distributional behaviour of the random vector (Zₙ,Sₙ), where Zₙ and Sₙ are respectively the time of the last visit before time n to, and the occupation time of, a suitable set Y of finite measure.
A large set containing few orbits of measure preserving transformations.
A. Iwanik (1992)
A large set containing few orbits of measure preserving transformations. (Summary).
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A characterization of complex
{L}_{1}
-preduals via a complex barycentric mapping
Petr Petráček, Jiří Spurný (2016)
We provide a complex version of a theorem due to Bednar and Lacey characterizing real
{L}_{1}
-preduals. Hence we prove a characterization of complex
{L}_{1}
-preduals via a complex barycentric mapping.
A C(K) Banach space which does not have the Schroeder-Bernstein property
Piotr Koszmider (2012)
We construct a totally disconnected compact Hausdorff space K₊ which has clopen subsets K₊” ⊆ K₊’ ⊆ K₊ such that K₊” is homeomorphic to K₊ and hence C(K₊”) is isometric as a Banach space to C(K₊) but C(K₊’) is not isomorphic to C(K₊). This gives two nonisomorphic Banach spaces (necessarily nonseparable) of the form C(K) which are isomorphic to complemented subspaces of each other (even in the above strong isometric sense), providing a solution to the Schroeder-Bernstein problem for Banach spaces...
{l}_{1}
{l}_{p}
1\le p<\infty
{l}_{p}\left({c}_{0}\right)
p=1
{l}_{1}
A geometrical/combinatorical question with implications for the John-Nirenberg inequality for BMO functions
Michael Cwikel, Yoram Sagher, Pavel Shvartsman (2011)
The first and last sections of this paper are intended for a general mathematical audience. In addition to some very brief remarks of a somewhat historical nature, we pose a rather simply formulated question in the realm of (discrete) geometry. This question has arisen in connection with a recently developed approach for studying various versions of the function space BMO. We describe that approach and the results that it gives. Special cases of one of our results give alternative proofs of the...
A nondentable set without the tree property
A normalized weakly null sequence with no shrinking subsequence in a Banach space not containing
{\ell }_{1}
E. Odell (1980)
A Note on compactness in Banach spaces.
Heinz Cremers, Dieter Kadelka (1982)
A note on Dunford-Pettis like properties and complemented spaces of operators
Ioana Ghenciu (2018)
Equivalent formulations of the Dunford-Pettis property of order
p
{DPP}_{p}
1<p<\infty
, are studied. Let
L\left(X,Y\right)
W\left(X,Y\right)
K\left(X,Y\right)
U\left(X,Y\right)
{C}_{p}\left(X,Y\right)
denote respectively the sets of all bounded linear, weakly compact, compact, unconditionally converging, and
p
-convergent operators from
X
Y
. Classical results of Kalton are used to study the complementability of the spaces
W\left(X,Y\right)
K\left(X,Y\right)
{C}_{p}\left(X,Y\right)
{C}_{p}\left(X,Y\right)
U\left(X,Y\right)
L\left(X,Y\right)
{\ell }_{p}
A note on isomorphisms between powers of Banach spaces.
J. C. Díaz (1987)
{c}_{0}
J. A. López Molina, M. J. Rivera (1996)
A note on norm-attaining functionals
A quasi-dichotomy for C(α,X) spaces, α < ω₁
Elói Medina Galego, Maurício Zahn (2015)
We prove the following quasi-dichotomy involving the Banach spaces C(α,X) of all X-valued continuous functions defined on the interval [0,α] of ordinals and endowed with the supremum norm. Suppose that X and Y are arbitrary Banach spaces of finite cotype. Then at least one of the following statements is true. (1) There exists a finite ordinal n such that either C(n,X) contains a copy of Y, or C(n,Y) contains a copy of X. (2) For any infinite countable...
|
ε-Coverings of Hölder-Zygmund Type Spaces on Data-Defined Manifolds
\mathbit{\epsilon }
-Coverings of Hölder-Zygmund Type Spaces on Data-Defined Manifolds
Martin Ehler, Frank Filbir
We first determine the asymptotes of the
\epsilon
-covering numbers of Hölder-Zygmund type spaces on data-defined manifolds. Secondly, a fully discrete and finite algorithmic scheme is developed providing explicit
\epsilon
-coverings whose cardinality is asymptotically near the
\epsilon
-covering number. Given an arbitrary Hölder-Zygmund type function, the nearby center of a ball in the
\epsilon
-covering can also be computed in a discrete finite fashion.
Martin Ehler. Frank Filbir. "
\mathbit{\epsilon }
-Coverings of Hölder-Zygmund Type Spaces on Data-Defined Manifolds." Abstr. Appl. Anal. 2014 (SI64) 1 - 6, 2014. https://doi.org/10.1155/2014/402918
Martin Ehler, Frank Filbir "
\mathbit{\epsilon }
-Coverings of Hölder-Zygmund Type Spaces on Data-Defined Manifolds," Abstract and Applied Analysis, Abstr. Appl. Anal. 2014(SI64), 1-6, (2014)
|
A Computational Study of the Boundary Value Methods and the Block Unification Methods for y″=f(x,y,y′)
2016 A Computational Study of the Boundary Value Methods and the Block Unification Methods for
{y}^{″}=f\left(x,y,{y}^{\prime }\right)
T. A. Biala
We derive a new class of linear multistep methods (LMMs) via the interpolation and collocation technique. We discuss the use of these methods as boundary value methods and block unification methods for the numerical approximation of the general second-order initial and boundary value problems. The convergence of these families of methods is also established. Several test problems are given to show a computational comparison of these methods in terms of accuracy and the computational efficiency.
T. A. Biala. "A Computational Study of the Boundary Value Methods and the Block Unification Methods for
{y}^{″}=f\left(x,y,{y}^{\prime }\right)
." Abstr. Appl. Anal. 2016 1 - 14, 2016. https://doi.org/10.1155/2016/8465103
Received: 13 November 2015; Revised: 7 January 2016; Accepted: 18 January 2016; Published: 2016
T. A. Biala "A Computational Study of the Boundary Value Methods and the Block Unification Methods for
{y}^{″}=f\left(x,y,{y}^{\prime }\right)
," Abstract and Applied Analysis, Abstr. Appl. Anal. 2016(none), 1-14, (2016)
|
GANs, mutual information, and possibly algorithm selection? – 2Cents
GANs, mutual information, and possibly algorithm selection?
If you ask deep learning people "what is the best image generation model now", many of them would probably say "generative adversarial networks" (GAN). The original paper describes an adversarial process that asks the generator and the discriminator to fight against each other, and many people including myself like this intuition. What lessons can we learn from GAN for better approximate inference (which is my thesis topic)? I need to rewrite the framework in the language I'm comfortable with, hence I decided to put it on this blog post as a research note.
Suppose there's a machine that can show you some images. This machine flips a coin to determine what it will show to you. For heads, the machine shows you a gorgeous paint from human artists. For tails, it shows you a forged famous paint. Your job is to figure out whether the shown image is "real" or not.
Let me be precise about it using math language. Assume
s \in \{0, 1\}
denotes the outcome of that coin flip, where 0 represents tails and 1 represents heads. The coin could possibly be a bent one so let's say your prior belief of the outcome is
\tilde{p}(s = 0) = \pi
. Then the above process is summarized as the following:
Here I use
p_D(x)
to describe the unknown data distribution and
p(x)
as the actual generative model we want to learn. As discussed, the generative model don't want you to know about
\pi
, in math this is achieved by minimizing the mutual information
This is quite intuitive:
\mathrm{I}[s; x] = 0
iff.
p_D(x) = p(x)
, and in this case observing an image
x
tells you nothing about the outcome of the coin flip
s
. However computing this mutual information requires evaluating
p(x)
which is generally intractable. Fortunately we can use variational approximation similar to the variational information maximization algorithm. In detail we construct a lower-bound by subtracting the KL divergence:
Let's have a closer look at the equations. From the KL term we can interpret
q(s|x)
as the approximated posterior of
s
under the augmented model
\tilde{p}
. But more interestingly
q
can also be viewed as the discriminator in the GAN framework. To see this, we expand the second term as
Notice something familiar with? Indeed when we pick the prior as
\pi = 0.5
, the lower-bound
\mathcal{L}[p; q]
is exactly GAN's objective function (up to scaling and adding constants), and the mutual information
\mathrm{I}[s; x]
becomes the Jensen-Shannon divergence
\mathrm{JS}[p(x)||p_D(x)]
, which GAN is actually minimizing.
To summarize, GAN can be viewed as an augmented generative model which is trained by minimizing mutual information. This augmentation is smart in the sense that it uses label-like information
s
that can be obtained for free, which introduces supervision signal to help unsupervised learning.
How useful is this interpretation? Well, in principle we can carefully design an augmented model and learn it by minimizing L2 loss/KL divergence/mutual information... you name it. But more interestingly, we can then do (automatic) divergence/algorithm selection as model selection in the "augmented" space. In the GAN example setting
\pi
determines which JS divergence variant we use in training, e.g. see this paper. I'm not sure how useful it would be, but we can also learn
\pi
by say maximum likelihood, or even treat
\pi
as a latent variable and put a Beta prior on it.
Recently I started to think about automatic algorithm selection. Probably because I'm tired of my reviewers complaining on my alpha-divergence papers "I don't know how to choose alpha and you should give us a guidance". Tom Minka gives an empirical guidance in his divergence measure tech report, and same for us in recent papers. I know this is an important but very difficult topic for research, but at least not only myself have thought about it, e.g. in this paper the authors connected beta-divergences to tweedie distributions and performed approximate maximum likelihood to select beta. Another interesting paper in this line is the "variational tempering" paper which models the annealing temperature in a probabilistic model as well. I like these papers as the core idea is very simple: we should also use probabilistic modeling for algorithmic parameters. Perhaps this also connects to Bayesian optimization but I'm gonna stop here as the note is already a bit too long.
I briefly presented this MI interpretation (also extended to ALI) to the MLG summer reading group and you can see some notes here.
Ferenc also posted a discussion of infoGAN with mutual information interpretation which is very nice. He also has a series of great blog posts on GANs that I wish I could have read them earlier!
My paper list on deep generative models – 2Cents
[…] another powerful discriminator that "the image I'm showing you is from the real data". I have a post that discussed the maths in detail and I also recommend reading a series of blog post by Ferenc […]
NIPS 2016: random thoughts
My paper list on deep generative models
A general example of the failure of the 1-step learning algorithm
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स्वकुं - Wikipedia
स्वकुं
स्वकुं धाःगु स्वंगु कुं छ्यला कुनातःगु, द्विआयामिक ख्यः ख। थ्व छगु आधारभूत रेखागणितीय पोलिगन खः। थ्व पोलिगनय् स्वंगु भर्टेक्स स्वंगु साइड व स्वंगु कोण दै।
२ साधारण ज्याखंतः
स्वकुंया ल्हा कथं थुकित ३गु भायय् बाय् छिं
समबाहू [१]
समद्विबाहू [२]
विषमबाहू [३]
समबाहू समद्विबाहू विषमबाहू
कोणयागु कथं त्रिभूजयात स्वंगु भागय् बाय् छिं-
राइट त्रिकोण
अब्ट्युज त्रिकोण .
एक्युट त्रिकोण .
राइट त्रिकोण अब्ट्युज त्रिकोण एक्युट त्रिकोण
साधारण ज्याखंतः[सम्पादन]
त्रिकोणयागु छुं खंतेत युक्लिडनं वेकयागु युक्लिडयागु एलेमेन्टतः सफूयु भाग १-४य् थ्यं-मथ्यं ३०० इ पूय् च्वयादिगु दु।
A triangle is a polygon and a 2-simplex (see polytope). All triangles are two-dimensional.
In Euclidean geometry, the sum of the internal angles α + β + γ is equal to two right angles (180° or π radians). This allows determination of the third angle of any triangle as soon as two angles are known.
पाइथागोरस थियोरम
A central theorem is the Pythagorean theorem stating that in any right triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides. If side C is the hypotenuse, we can write this as
{\displaystyle c^{2}=a^{2}+b^{2}\,}
{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \gamma \,}
साइनयु लः कथं :
{\displaystyle {\frac {\sin \alpha }{a}}={\frac {\sin \beta }{b}}={\frac {\sin \gamma }{c}}={\frac {1}{d}}}
where d is the diameter of the circumcircle (the circle which passes through all three points of the triangle). The law of sines can be used to compute the side lengths for a triangle as soon as two angles and one side are known. If two sides and an unenclosed angle is known, the law of sines may also be used; however, in this case there may be zero, one or two solutions.
There are two special right triangles that appear commonly in geometry. The so-called "45-45-90 triangle" has angles with those angle measures and the ratio of its sides is :
{\displaystyle 1:1:{\sqrt {2}}}
. The "30-60-90 triangle" has sides in the ratio of
{\displaystyle 1:{\sqrt {3}}:2}
Points, lines and circles associated with a triangle[सम्पादन]
There are hundreds of different constructions that find a special point inside a triangle, satisfying some unique property: see the references section for a catalogue of them. Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point: an important tool for proving the existence of these is Ceva's theorem, which gives a criterion for determining when three such lines are concurrent. Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are collinear: here Menelaus' theorem gives a useful general criterion. In this section just a few of the most commonly-encountered constructions are explained.
An altitude of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle with) the opposite side. This opposite side is called the base of the altitude, and the point where the altitude intersects the base (or its extension) is called the foot of the altitude. The length of the altitude is the distance between the base and the vertex. The three altitudes intersect in a single point, called the orthocenter of the triangle. The orthocenter lies inside the triangle if and only if the triangle is acute. The three vertices together with the orthocenter are said to form an orthocentric system.
An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half. The three angle bisectors intersect in a single point, the incenter, the center of the triangle's incircle. The incircle is the circle which lies inside the triangle and touches all three sides. There are three other important circles, the excircles; they lie outside the triangle and touch one side as well as the extensions of the other two. The centers of the in- and excircles form an orthocentric system.
Nine-point circle demonstrates a symmetry where six points lie on the same circle.
Computing the area of a triangle[सम्पादन]
Calculating the area of a triangle is an elementary problem encountered often in many different situations. Various approaches exist, depending on what is known about the triangle. What follows is a selection of frequently used formulae for the area of a triangle.[४]
Using geometry[सम्पादन]
The area of the parallelogram is the magnitude of the cross product of the two vectors.
The product of the inradius and the semiperimeter of a triangle also gives its area.
Using vectors[सम्पादन]
The area of triangle ABC can also be expressed in term of dot products as follows:
{\displaystyle {\frac {1}{2}}{\sqrt {(\mathbf {AB} \cdot \mathbf {AB} )(\mathbf {AC} \cdot \mathbf {AC} )-(\mathbf {AB} \cdot \mathbf {AC} )^{2}}}={\frac {1}{2}}{\sqrt {|\mathbf {AB} |^{2}|\mathbf {AC} |^{2}-(\mathbf {AB} \cdot \mathbf {AC} )^{2}}}}
Applying trigonometry to find the altitude h.
Using trigonometry[सम्पादन]
The altitude of a triangle can be found through an application of trigonometry. Using the labelling as in the image on the left, the altitude is h = a sin γ. Substituting this in the formula S = ½bh derived above, the area of the triangle can be expressed as S = ½ab sin γ.
{\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}}
{\displaystyle \sin C={\sqrt {1-\cos ^{2}C}}}
and also the formula shown above, then one arrives at the following formula for area
{\displaystyle {\frac {1}{4}}{\sqrt {2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})}}}
[Note that, this is a multiplied out form of Heron's formula]
Using coordinates[सम्पादन]
If vertex A is located at the origin (0, 0) of a Cartesian coordinate system and the coordinates of the other two vertices are given by B = (xB, yB) and C = (xC, yC), then the area S can be computed as ½ times the absolute value of the determinant
{\displaystyle S={\frac {1}{2}}\left|\det {\begin{pmatrix}x_{B}&x_{C}\\y_{B}&y_{C}\end{pmatrix}}\right|={\frac {1}{2}}|x_{B}y_{C}-x_{C}y_{B}|.}
{\displaystyle S={\frac {1}{2}}\left|\det {\begin{pmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{pmatrix}}\right|={\frac {1}{2}}{\big |}x_{A}y_{C}-x_{A}y_{B}+x_{B}y_{A}-x_{B}y_{C}+x_{C}y_{B}-x_{C}y_{A}{\big |}.}
In three dimensions, the area of a general triangle {A = (xA, yA, zA), B = (xB, yB, zB) and C = (xC, yC, zC)} is the 'Pythagorean' sum of the areas of the respective projections on the three principal planes (i.e. x=0, y=0 and z=0):
{\displaystyle S={\frac {1}{2}}{\sqrt {\left(\det {\begin{pmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{pmatrix}}\right)^{2}+\left(\det {\begin{pmatrix}y_{A}&y_{B}&y_{C}\\z_{A}&z_{B}&z_{C}\\1&1&1\end{pmatrix}}\right)^{2}+\left(\det {\begin{pmatrix}z_{A}&z_{B}&z_{C}\\x_{A}&x_{B}&x_{C}\\1&1&1\end{pmatrix}}\right)^{2}}}.}
Using Heron's formula[सम्पादन]
{\displaystyle S={\sqrt {s(s-a)(s-b)(s-c)}}}
where s = ½ (a + b + c) is the semiperimeter, or half of the triangle's perimeter.
Multiplied out form of Heron's formula (see above for proof)
{\displaystyle {\frac {1}{4}}{\sqrt {2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})}}}
Non-planar triangles[सम्पादन]
A non-planar triangle is a triangle which is not contained in a (flat) plane. Examples of non-planar triangles in noneuclidean geometries are spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry.
While all regular, planar (two dimensional) triangles contain angles that add up to 180°, there are cases in which the angles of a triangle can be greater than or less than 180°. In curved figures, a triangle on a negatively curved figure ("saddle") will have its angles add up to less than 180° while a triangle on a positively curved figure ("sphere") will have its angles add up to more than 180°. Thus, if one were to draw a giant triangle on the surface of the Earth, one would find that the sum of its angles were greater than 180°.
↑ http://mathworld.wolfram.com/ScaleneTriangle.html
↑ http://mathworld.wolfram.com/TriangleArea.html
Triangle Calculator - solves for remaining sides and angles when given three sides or angles, supports degrees and radians.
Napoleon's theorem A triangle with three equilateral triangles. A purely geometric proof. It uses the Fermat point to prove Napoleon's theorem without transformations by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
Proof that the sum of the angles in a triangle is 180 degrees
Area of a triangle - 7 different ways
Triangle definition pages with interactive applets that are also useful in a classroom setting. Math Open Reference
Constructing an equilateral triangle , Isosceles triangle and Copying a Triangle with only a compass and straightedge, interactive animation.
विकिमिडिया मंका य् थ्व विषय नाप स्वापु दुगु मिडिया दु: Triangles
Retrieved from "https://new.wikipedia.org/w/index.php?title=स्वकुं&oldid=840194"
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ℂ
-convexity in infinite-dimensional Banach spaces and applications to Kergin interpolation.
t
p
A Banach space that is MLUR but not HR.
Mark A. Smith (1981)
A Banach space with a symmetric basis which contains no
{\ell }_{p}
{c}_{0}
, and all its symmetric basic sequences are equivalent
Z. Altshuler (1977)
B
-convexity and
J
-convexity of Banach spaces.
Mitani, Ken-Ichi, Saito, Kichi-Suke (2007)
Banach Journal of Mathematical Analysis [electronic only]
{l}_{1}
{l}_{p}
1\le p<\infty
{l}_{p}\left({c}_{0}\right)
p=1
{l}_{1}
A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces
Benoit Bossard (2002)
When the set of closed subspaces of C(Δ), where Δ is the Cantor set, is equipped with the standard Effros-Borel structure, the graph of the basic relations between Banach spaces (isomorphism, being isomorphic to a subspace, quotient, direct sum,...) is analytic non-Borel. Many natural families of Banach spaces (such as reflexive spaces, spaces not containing ℓ₁(ω),...) are coanalytic non-Borel. Some natural ranks (rank of embedding, Szlenk indices) are shown to be coanalytic ranks. Applications...
A coefficient related to some geometric properties of a Banach space.
Zuo, Zhanfei, Cui, Yunan (2009)
A conjecture about the Dunford-Pettis property
Núnez, Carmelo (1989)
A continuum of totally incomparable hereditarily indecomposable Banach spaces
I. Gasparis (2002)
A family is constructed of cardinality equal to the continuum, whose members are totally incomparable hereditarily indecomposable Banach spaces.
A contribution to a theorem of Ulam and Mazur.
Walter Benz, Hubert Berens (1987)
A curious generalization of local uniform rotundity
A dichotomy on Schreier sets
Robert Judd (1999)
We show that the Schreier sets
{S}_{\alpha }\left(\alpha <{\omega }_{1}\right)
have the following dichotomy property. For every hereditary collection ℱ of finite subsets of ℱ, either there exists infinite
M={\left({m}_{i}\right)}_{i=1}^{\infty }\subseteq ℕ
{S}_{\alpha }\left(M\right)={m}_{i}:i\in E:E\in {S}_{\alpha }\subseteq ℱ
, or there exist infinite
M={\left({m}_{i}\right)}_{i=1}^{\infty },N\subseteq ℕ
ℱ\left[N\right]\left(M\right)={m}_{i}:i\in F:F\in ℱandF\subset N\subseteq {S}_{\alpha }
A generalization of an Ekeland-Lebourg theorem and the differentiability of distance functions
Zajíček, L. (1984)
A generalized projection decomposition in Orlicz-Bochner spaces
Henryk Hudzik, Ryszard Płuciennik, Yuwen Wang (2005)
In this paper, a precise projection decomposition in reflexive, smooth and strictly convex Orlicz-Bochner spaces is given by the representation of the duality mapping. As an application, a representation of the metric projection operator on a closed hyperplane is presented.
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