text:
# A cycle which is the product of disjoint $m$-cycles does not necessarily have order $m$?

Suppose a cycle $\sigma\in S_n$ has decomposition a product of disjoint $p$-cycles, for $p$ a prime. So each cycle has order $p$, and thus $\sigma^p=id$, so $\sigma$ has order dividing $p$, and thus order $p$ when $\sigma\neq id$.

What if $\sigma$ has a decomposition into a product of disjoint $m$-cycles, for $m$ composite. I know $\sigma^m=id$, and thus has order dividing $m$. In every example I've tried, it seems $\sigma$ still has order $m$. Is it possible for $\sigma$ to have order smaller than $m$ (when not the identity of course)? If so, is there an example?

Much thanks,

-
exercise: the order of a product of disjoint cycles is the lcm of the cycle lengths – yoyo Aug 31 '11 at 23:05

A permutation that is the product of disjoint permutations that all have the same order, has the same order as its factors. So your $\sigma$ will have order $m$, guaranteed.
(It must have order at most $m$ because the disjoint factors commute, so $\sigma^m = (xy\cdots z)^m = x^m y^m \cdots z^m = 1\cdot 1\cdots 1 = 1$. On the other hand, by the same argument, if it had order less than $m$, each factor would also have that order, because the only way a product of disjoint permutations can be the identity is if each of the factors are).
(By the way $\sigma$ is not itself a "cycle" if it is the product of two or more disjoint permutations).


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# Derivative of a vector valued function

Let $f(\vec{x})=|\vec{x}|^2 \vec{x}$ be a function from $\mathbb{R}^n$ to $\mathbb{R}^n$. How does one differentiate such a function? I want to use the "abstract definition" and consider the $|\cdot|^2$ to be the dot product, and look at the sums of the squares of the components, but this seems unwieldy. What's the right approach here?

-

## 1 Answer

Suppose you want to calculate $grad f(\vec{a})$. Use the "Taylor expansion",

$f(\vec{a} + \vec{h}) = \langle \vec{a} + \vec{h}, \vec{a} + \vec{h} \rangle (\vec{a} + \vec{h}) = \| \vec{a} \|^2 \vec{a} + \langle \vec{a}, \vec{a} \rangle \vec{h} + 2 \langle \vec{a}, \vec{h} \rangle \vec{a} + O(\| \vec{h} \|^2) = f(\vec{a}) + \langle \vec{a}, \vec{a} \rangle \vec{h} + 2 \langle \vec{a}, \vec{h} \rangle \vec{a} + O(\| \vec{h} \|^2)$

The derivative means the linear term, thus in this case, it is the linear transformation that sends $\vec{h}$ to $\|\vec{a} \|^2 \vec{h} + 2 \langle \vec{a}, \vec{h} \rangle \vec{a}$.

-
Can you explain this Taylor expansion? This seems to come out of nowhere. This is what is lacking in my approach, such an understanding of the f(a+h) equality. – Wouter Zeldenthuis Nov 8 '12 at 2:26
I just expanded the inner product and the bracket. – user27126 Nov 8 '12 at 2:28

format: plain


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# Absolute Galois group

Absolute Galois group

In mathematics, the absolute Galois group "GK" of a field "K" is the Galois group of "K"sep over "K", where "K"sep is a separable closure of "K". Alternatively it is the group of all automorphisms of the algebraic closure of "K" that fix "K". The absolute Galois group is unique up to isomorphism. It is a profinite group.

(When "K" is a perfect field, "K"sep is the same as an algebraic closure "K"alg of "K". This holds e.g. for "K" of characteristic zero, or "K" a finite field.)

Examples

* The absolute Galois group of an algebraic closed field is trivial.
* The absolute Galois group of the real numbers is a cyclic group of two elements (complex conjugation and the identity map), since $Bbb\left\{C\right\}$ is the separable closure of $Bbb\left\{R\right\}$ and $\left[ Bbb\left\{C\right\} : Bbb\left\{R\right\} \right] =2$.
* The absolute Galois group of a finite field "K" is isomorphic to the group ::$hat\left\{mathbb\left\{Z = lim_\left\{leftarrow\right\}mathbb\left\{Z\right\}/nmathbb\left\{Z\right\}$. The Frobenius automorphism Fr is a canonical generator of "GK". (Recall that Fr("x") = "xq" for all "x" in "K"alg, where "q" is the number of elements in "K".)
* The absolute Galois group of the field of rational functions with complex coefficients is free (as a profinite group). This result is due to Adrien Douady and has its origins in Riemann's existence theorem.
* More generally, Let "C" be an algebraically closed field and "x" a variable. Then the absolute Galois group of "K"="C"("x") is free of rank equal to the cardinality of "C". This result is due to David Harbater and Florian Pop, and was also proved later by Dan Haran and Moshe Jarden.
* Let "K" be a "p"-adic field. Then its absolute Galois group is finitely generated and has an explicit description by generators and relations.

Problems

* No direct description is known for the absolute Galois group of the rational numbers. In this case, it follows from Belyi's theorem that the absolute Galois group has a faithful action on the "dessins d'enfants" of Grothendieck (maps on surfaces), enabling us to "see" the Galois theory of algebraic number fields.

* Let "K" be the maximal abelian extension of the rational numbers. Then Shafarevich's conjecture asserts that the absolute Galois group of "GK" is the free profinite group of countable rank.

Some general results

* Every profinite group occurs as a Galois group of some Galois extension, however not every profinite group occurs as an absolute Galois group. For example Artin-Schreier Theorem asserts that the only finite absolute Galois groups are the trivial one and the cyclic group of order 2.

* Every projective profinite group can be realized as a absolute Galois group of a Pseudo algebraically closed field. This result is due to Alexander Lubotzky and Lou van den Dries.

References

* M. D. Fried and M. Jarden, Field Arithmetic, Second Edition, revised and enlarged by Moshe Jarden, Ergebnisse der Mathematik (3) 11, Springer, Heidelberg, 2004.

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### Look at other dictionaries:

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text:
BREAKING NEWS
Imaginary time

## Summary

Imaginary time is a mathematical representation of time which appears in some approaches to special relativity and quantum mechanics. It finds uses in connecting quantum mechanics with statistical mechanics and in certain cosmological theories.

Mathematically, imaginary time is real time which has undergone a Wick rotation so that its coordinates are multiplied by the imaginary unit i. Imaginary time is not imaginary in the sense that it is unreal or made-up (any more than, say, irrational numbers defy logic), it is simply expressed in terms of what mathematicians call imaginary numbers.

## Origins

Mathematically, imaginary time ${\textstyle \tau }$ may be obtained from real time ${\textstyle t}$ via a Wick rotation by ${\textstyle \pi /2}$ in the complex plane: ${\textstyle \tau =it}$, where ${\displaystyle i^{2}}$ is defined to be ${\displaystyle -1}$, and ${\displaystyle i}$ is known as the imaginary unit.

Stephen Hawking popularized the concept of imaginary time in his book The Universe in a Nutshell.

One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?

In fact, the names "real" and "imaginary" for numbers are just a historical accident, much like the names "rational" and "irrational":

...the words real and imaginary are picturesque relics of an age when the nature of complex numbers was not properly understood.

## In cosmology

In the Minkowski spacetime model adopted by the theory of relativity, spacetime is represented as a four-dimensional surface or manifold. Its four-dimensional equivalent of a distance in three-dimensional space is called an interval. Assuming that a specific time period is represented as a real number in the same way as a distance in space, an interval ${\displaystyle d}$ in relativistic spacetime is given by the usual formula but with time negated:

${\displaystyle d^{2}=x^{2}+y^{2}+z^{2}-t^{2}}$
where ${\displaystyle x}$, ${\displaystyle y}$ and ${\displaystyle z}$ are distances along each spatial axis and ${\displaystyle t}$ is a period of time or "distance" along the time axis (Strictly, the time coordinate is ${\displaystyle (ct)^{2}}$ where ${\displaystyle c}$ is the speed of light, however we conventionally choose units such that ${\displaystyle c=1}$).

Mathematically this is equivalent to writing

${\displaystyle d^{2}=x^{2}+y^{2}+z^{2}+(it)^{2}}$

In this context, ${\displaystyle i}$ may be either accepted as a feature of the relationship between space and real time, as above, or it may alternatively be incorporated into time itself, such that the value of time is itself an imaginary number, denoted by ${\displaystyle \tau }$, and the equation rewritten in normalised form:

${\displaystyle d^{2}=x^{2}+y^{2}+z^{2}+\tau ^{2}}$

Similarly its four vector may then be written as

${\displaystyle (x_{0},x_{1},x_{2},x_{3})}$
where distances are represented as ${\displaystyle x_{n}}$, ${\displaystyle c}$ is the velocity of light and ${\displaystyle x_{0}=ict}$.

Hawking noted the utility of rotating time intervals into an imaginary metric in certain situations, in 1971.[3]

In physical cosmology, imaginary time may be incorporated into certain models of the universe which are solutions to the equations of general relativity. In particular, imaginary time can help to smooth out gravitational singularities, where known physical laws break down, to remove the singularity and avoid such breakdowns (see Hartle–Hawking state). The Big Bang, for example, appears as a singularity in ordinary time but, when modelled with imaginary time, the singularity can be removed and the Big Bang functions like any other point in four-dimensional spacetime. Any boundary to spacetime is a form of singularity, where the smooth nature of spacetime breaks down.[4] With all such singularities removed from the Universe, it thus can have no boundary and Stephen Hawking speculated that "the boundary condition to the Universe is that it has no boundary".[5]

However, the unproven nature of the relationship between actual physical time and imaginary time incorporated into such models has raised criticisms.[6] Penrose has noted that there needs to be a transition from the Riemannian metric (often referred to as "Euclidean" in this context) with imaginary time at the Big Bang to a Lorenzian metric with real time for the evolving Universe. Also, modern observations suggest that the Universe is open and will never shrink back to a Big Crunch. If this proves true, then the end-of-time boundary still remains.[4]

## In quantum statistical mechanics

The equations of the quantum field can be obtained by taking the Fourier transform of the equations of statistical mechanics. Since the Fourier transform of a function typically shows up as its inverse, the point particles of statistical mechanics become, under a Fourier transform, the infinitely extended harmonic oscillators of quantum field theory.[7] The Green's function of an inhomogeneous linear differential operator, defined on a domain with specified initial conditions or boundary conditions, is its impulse response, and mathematically we define the point particles of statistical mechanics as Dirac delta functions, which is to say impulses. [8] At finite temperature ${\displaystyle T}$, the Green's functions are periodic in imaginary time with a period of ${\textstyle 2\beta =2/T}$. Therefore, their Fourier transforms contain only a discrete set of frequencies called Matsubara frequencies.

The connection between statistical mechanics and quantum field theory is also seen in the transition amplitude ${\textstyle \langle F\mid e^{-itH}\mid I\rangle }$ between an initial state I and a final state F, where H is the Hamiltonian of that system. If we compare this with the partition function ${\textstyle Z=\operatorname {Tr} e^{-\beta H}}$ we see that to get the partition function from the transition amplitudes we can replace ${\textstyle t=\beta /i}$, set F = I = n and sum over n. This way we don't have to do twice the work by evaluating both the statistical properties and the transition amplitudes.

Finally, by using a Wick rotation one can show that the Euclidean quantum field theory in (D + 1)-dimensional spacetime is nothing but quantum statistical mechanics in D-dimensional space.

## References

### Notes

1. ^ Hawking (2001), p.59.
2. ^ Coxeter, H.S.M.; The Real Projective Plane, 3rd Edn, Springer 1993, p. 210 (footnote).
3. ^ Hawking, Stephen (15 September 1978). "Quantum gravity and path integrals". Physical Review D. 18 (6): 1747–1753. Bibcode:1978PhRvD..18.1747H. doi:10.1103/PhysRevD.18.1747. "It is convenient to rotate the time interval on this timelike tube between the two surfaces into the complex plane so that it becomes purely imaginary."
4. ^ a b Roger Penrose. The Road to Reality. Vintage. 2005. pp.769-772.
5. ^ Hawking 2001, p.85.
6. ^ Deltete, Robert J.; Reed, A. Guy (1996). "Emerging from Imaginary Time". Synthese. 108 (2): 185–203. doi:10.1007/BF00413497. S2CID 44131608.
7. ^ Uwe-Jens Wiese, "Quantum Field Theory", Institute for Theoretical Physics, University of Bern, 21 August 2007, page 63.
8. ^ Andy Royston; "Notes on the Dirac Delta and Green Functions", 23 November 2008.

### Bibliography

• Hawking, Stephen W. (1998). A Brief History of Time (Tenth Anniversary Commemorative ed.). Bantam Books. p. 157. ISBN 978-0-553-10953-5.
• Hawking, Stephen W. (2001). The Universe in a Nutshell. United States & Canada: Bantam Books. pp. 58–61, 63, 82–85, 90–94, 99, 196. ISBN 0-553-80202-X.


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Direct Graphical Models  v.1.5.2
DirectGraphicalModels::fex Namespace Reference

## Namespaces

global
Global-features extraction.

## Classes

class  CCommonFeatureExtractor
Common class, which unites feature extraction algorithms. More...

class  CCoordinate
Coordinate feature extraction class. More...

class  CDistance
Distance feature extraction class. More...

class  CGlobalFeatureExtractor
Interface class for global feature extraction algorithms. More...

Gradient feature extraction class. More...

class  CHOG
HOG (histogram of oriented gradients) feature extraction class. More...

class  CHSV
Hue, Saturation and Value feature extraction class. More...

class  CIntensity
Intensity feature extraction class. More...

class  CNDVI
NDVI (normalized difference vegetation index) feature extraction class. More...

class  CScale
Scale feature extraction class. More...

class  CSIFT
SIFT (scale-invariant feature transform) feature extraction class. More...

class  CSparseCoding
Sparse Coding feature extraction class. More...

class  CSparseDictionary
Sparse Dictionary Learning class. More...

class  CVariance
Variance feature extraction class. More...

class  IFeatureExtractor
Interface class for feature extraction algorithms. More...

class  ILocalFeatureExtractor
Interface class for local feature extraction algorithms. More...

struct  SqNeighbourhood
Square neighborhood structure. More...

## Typedefs

typedef struct DirectGraphicalModels::fex::SqNeighbourhood SqNeighbourhood
Square neighborhood structure. More...

## Enumerations

enum  ChannelsRGB { CH_BLUE, CH_GREEN, CH_RED }
Channels in the BGR color space. More...

enum  ChannelsHSV { CH_HUE, CH_SATURATION, CH_VALUE }
Channels in the HSV color space. More...

enum  coordinateType { COORDINATE_ORDINATE, COORDINATE_ABSCISS, COORDINATE_RADIUS }
Types of the coordinate feature. More...

enum  BasePointLocation {
BP_CENTER, BP_LEFT, BP_RIGHT, BP_TOP,
BP_BOTTOM
}
Some special cases of the base point location inside the neighborhood. More...

## Functions

byte linear_mapper (float val, float min, float max)
Linear 1D mapping. More...

byte two_linear_mapper (float val, float min, float max, float mid, byte midPoint)
Two-linear 1D mapping. More...

SqNeighbourhood sqNeighbourhood (int leftGap, int rightGap, int upperGap, int lowerGap)
Initializes the square neighborhood structure. More...

SqNeighbourhood sqNeighbourhoodAll (int R)
Initializes the square neighborhood structure with all the same values (base point in the center) More...

SqNeighbourhood sqNeighbourhood (int R, BasePointLocation location=BP_CENTER)
Initializes the square neighborhood structure with a pre-define shape. More...

## Variables

const float SC_LRATE_W = 5e-2f
Learning rate (speed) for weights $$W$$. More...

const float SC_LRATE_D = 1e-2f
Learning rate (speed) for dictionary $$D$$. More...

## Typedef Documentation

Square neighborhood structure.

This structure defines rectangular neighborhood around its base point. The size of the neighborhoood is given via four gap values: $$(leftGap+rightGap+1)\times(upperGap+lowerGap+1)$$, and its shape is defines as depicted at Figure 1.

This definition of a neighbourhood extends the classical one, where the neighborhood is represented as a square with the base point in the center.

Fig. 1

## Enumeration Type Documentation

Some special cases of the base point location inside the neighborhood.

Enumerator
BP_CENTER

The base point is located in the neighborhood's center.

BP_LEFT

The base point is located at the neighborhood's left boundry.

BP_RIGHT

The base point is located in the neighborhood's right boundry.

BP_TOP

The base point is located in the neighborhood's upper boundry.

BP_BOTTOM

The base point is located in the neighborhood's lower boundry.

Definition at line 26 of file SquareNeighborhood.h.

Channels in the HSV color space.

Enumerator
CH_HUE

Hue channel.

CH_SATURATION

Saturation channel.

CH_VALUE

Value channel.

Definition at line 30 of file CommonFeatureExtractor.h.

Channels in the BGR color space.

Enumerator
CH_BLUE

Blue channel.

CH_GREEN

Green channel.

CH_RED

Red channel.

Definition at line 23 of file CommonFeatureExtractor.h.

Types of the coordinate feature.

Enumerator
COORDINATE_ORDINATE

Coordinate feature depend on the pixel's ordinate (y-coordinate).

COORDINATE_ABSCISS

Coordinate feature depend on the pixel's absciss (x-coordinate).

Coordinate feature depend on the pixel's distance to the image center.

Definition at line 12 of file Coordinate.h.

## Function Documentation

 byte DirectGraphicalModels::fex::linear_mapper ( float val, float min, float max )
inline

Linear 1D mapping.

This function perform linear mapping of the value val from one interval to another: $$val\in[min; max]\rightarrow res\in[0; 255]$$, such that:

\begin{eqnarray*} min&\rightarrow&0 \\ max&\rightarrow&255 \end{eqnarray*}

Parameters
 val The value to map. min The lower boundary of the val. max The higher bounday of the val.
Returns
The mapped value res.

Definition at line 19 of file LinearMapper.h.

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 SqNeighbourhood DirectGraphicalModels::fex::sqNeighbourhood ( int leftGap, int rightGap, int upperGap, int lowerGap )
inline

Initializes the square neighborhood structure.

Parameters
 leftGap Distance from the base point to the neighborhood's left boundary. rightGap Distance from the base point to the neighborhood's right boundary. upperGap Distance from the base point to the neighborhood's upper boundary. lowerGap Distance from the base point to the neighborhood's lower boundary.
Returns
Initialized square neighborhood of the size: $$(leftGap+rightGap+1)\times(upperGap+lowerGap+1)$$ (Ref. SqNeighbourhood).

Definition at line 42 of file SquareNeighborhood.h.

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 SqNeighbourhood DirectGraphicalModels::fex::sqNeighbourhood ( int R, BasePointLocation location = BP_CENTER )
inline

Initializes the square neighborhood structure with a pre-define shape.

Parameters
 R Distance from the base point to the neighborhood's boundaries location Flag describing the location of the base point (Ref. BasePointLocation and Figure 2) Fig. 2
Returns
Initialized square neighborhood of the size $$(2R + 1)\times(2R + 1)$$ (Ref. SqNeighbourhood)

Definition at line 64 of file SquareNeighborhood.h.

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 SqNeighbourhood DirectGraphicalModels::fex::sqNeighbourhoodAll ( int R )
inline

Initializes the square neighborhood structure with all the same values (base point in the center)

Parameters
 R Distance from the base point to the neighborhood's boundaries (radius)
Returns
Initialized square neighborhood of the size $$(2R + 1)\times(2R + 1)$$ (Ref. SqNeighbourhood)

Definition at line 56 of file SquareNeighborhood.h.

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 byte DirectGraphicalModels::fex::two_linear_mapper ( float val, float min, float max, float mid, byte midPoint )
inline

Two-linear 1D mapping.

This function perform linear mapping of the value val from one interval to another: $$val\in[min; max]\rightarrow res\in[0; 255]$$, such that:

\begin{eqnarray*} min&\rightarrow&0 \\ mid&\rightarrow&midPoint \\ max&\rightarrow&255 \end{eqnarray*}

For more detail please refer to the Figure 1.

Fig. 1
Parameters
 val The value to map. min The lower boundary of the val. max The higher bounday of the val. mid The x-coordinate of the intersection point, $$mid\in(min; max)$$ (Ref. Figure 1). midPoint The y-coordinate of the intersection point, $$midPoint\in[0; 255]$$ (Ref. Figure 1).
Returns
The mapped value res.

Definition at line 38 of file LinearMapper.h.

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## Variable Documentation

 const float DirectGraphicalModels::fex::SC_LRATE_D = 1e-2f

Learning rate (speed) for dictionary $$D$$.

Definition at line 10 of file SparseDictionary.h.

 const float DirectGraphicalModels::fex::SC_LRATE_W = 5e-2f

Learning rate (speed) for weights $$W$$.

Definition at line 9 of file SparseDictionary.h.


text:
Cross Products

In an equation of the form

$\frac{a}{b}=\frac{c}{d}$ ,

we can multiply both sides by $bd$ to get the equation

$ad=bc$ .

This is known by many names: cross-multiplying, taking cross-products, the means-extremes property, or the reciprocal property.

Example:

Solve for $x$ .

$\frac{3}{7}=\frac{x+1}{9}$

Use the means-extremes property to get a simpler equation.

$3\left(9\right)=7\left(x+1\right)$

Now simplify and solve.

$\begin{array}{l}27=7x+7\\ 7x=20\\ x=\frac{20}{7}\end{array}$


text:
Distance between permutations

Imagine a space whose points, or vertices, are the elements of ${S_n}$, and each vertex ${\sigma}$ is connected by an edge to each of its neighbors ${\tau \sigma}$, where ${\tau}$ is any transposition.

This picture gives a natural notion of distance between two permutations: it is the smallest number of edges that must be traversed to go from one permutation to the other. More formally,

${ \displaystyle d(\sigma,\pi)=k}$ means that ${k}$ is the smallest integer for which there exist transpositions ${\tau_1,\ldots,\tau_k}$ satisfying

${ \displaystyle \tau_1 \ldots \tau_k\,\sigma=\pi.}$

(Of course, we take ${k=0}$ to mean that ${\sigma=\pi}$.) A pleasant few moments’ thought convinces you that ${d(\cdot,\cdot)}$ is indeed a metric on ${S_n}$.

The definition of ${d}$ can be restated in terms of the length function:

${ \displaystyle d(\sigma, \pi)= l(\pi\sigma^{-1}).}$

A full cycle in ${S_n}$, that is a permutation with exactly one orbit, is at maximum distance from the identity ${\iota}$: ${ \displaystyle d\left(\iota, (1\,2\,\ldots\,n)\right)= l(1\,2\,\ldots\,n)=n-1.}$

Recall that for any transposition ${\tau}$, the length of ${\tau\sigma}$ is ${l(\sigma)+1}$ if ${\tau}$ transposes elements in different orbits of ${\sigma}$, and is ${l(\sigma)-1}$ if it transposes elements within one orbit of ${\sigma}$. Thus, ${\tau\sigma}$ is either ${1}$ unit distance further from ${\iota}$ or ${1}$ unit distance closer to $\iota$.

Signature of a permutation

We have seen that a permutation ${\sigma\in S_n}$ can always be expressed as a product of transpositions. Moreover, for any transposition ${\tau}$ and permutation ${\pi}$ the length of ${\tau \pi}$ is either ${1}$ more or ${1}$ less than the length of ${\pi}$:

${ \displaystyle l(\tau\pi) =l(\pi)\pm 1.}$

Hence

${ \displaystyle (-1)^{l(\pi\tau)}=(-1)\cdot (-1)^{l(\pi)}.}$

Hence writing ${\sigma}$ as a product

${ \displaystyle \sigma=\tau_1\ldots\tau_k,}$

where each ${\tau_j}$ is a transposition, we have

${ \displaystyle (-1)^{l(\sigma)}= (-1) (-1)^{l(\tau_2\ldots\tau_k)}=\underbrace{(-1)(-1)\ldots(-1)}_{k\, {\rm terms}} = (-1)^k.}$

Being ${+1}$ if ${k}$ is even and is ${-1}$ if ${k}$ is odd, it is just the signature of ${\sigma}$.

Thus, the signature of ${\sigma}$ is equal to

${ \displaystyle \epsilon(\sigma)=(-1)^{l(\sigma)}.}$

Displaying permutations ${\sigma}$ and ${\pi}$ each as a product of transpositions produces a display of ${\sigma\pi}$ as a product of transpositions; from this we see that ${ \displaystyle \epsilon(\sigma\pi)=\epsilon(\sigma)\epsilon(\pi).}$ We can understand the signature further by considering the relationship between permutations and reflections. For any ${\sigma\in S_n}$ let ${R(\sigma)}$ be the linear map on ${{\mathbf F}^n}$ (with ${{\mathbf F}}$ being some field) given by ${ \displaystyle R(\sigma)(x_1,\ldots, x_n)=\Bigl(x_{\sigma^{-1}(1)},\ldots, x_{\sigma^{-1}(n)}\Bigr).}$ For a transposition ${ (j\,k)}$ we see that ${R(j\,k)}$ is reflection across the hyperplane ${x_j=x_k}$. Thus, taking a field ${{\mathbf F}}$ in which ${2\neq 0}$, we see that

${ \displaystyle \epsilon(\sigma) =\det R(\sigma),}$

essentially detecting whether ${R(\sigma)}$ is a `rotation’ or a reflection.

Length of a permutation

Introductory texts that discuss permutations don’t seem to stress the notion of length of a permutation. In my book I tried to stress this useful notion, but the treatment in my book is confusing at points, so let me try here to give a very short and hopefully clearer account.

Fix an integer ${ n\geq 1}$ and let ${ S_n}$ be the set of all bijections of the set ${ [n]=\{1,2,3,\ldots, n\}}$. Since the composition of bijections is a bijection, and the inverse of a bijection is also a bijection, ${ S_n}$ is a group under composition. The elements of ${ S_n}$ are called permutations of ${ [n]}$. Sometimes it is convenient to display a permutation by listing the elements ${ 1,2, \ldots, n}$ in a row and the images ${ \sigma(1),\sigma(2),\ldots, \sigma(n)}$ below:

${ \displaystyle \left(\begin{matrix} 1 & 2 & \ldots & n \\ \sigma(1) & \sigma(2) &\ldots &\sigma(n) \end{matrix}\right) }$

For any ${ \sigma\in S_n}$ and ${ x\in [n]}$, the orbit of ${ x}$ is the set

${ \displaystyle \{x, \sigma(x),\sigma^2(x),\ldots\}.}$

For example, the orbits of

${ \displaystyle \left( \begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 3& 1 & 2 &5 &6 &4&7\end{matrix}\right)}$

are

${ \displaystyle \{1,2,3\}, \{4,5,6\}, \{7\}.}$

Let

${ \displaystyle {\rm Orb}(\sigma)}$

denote the set of all orbits of ${ \sigma}$. The length of ${ \sigma}$ is

${ \displaystyle l(\sigma)= n-|{\rm Orb}(\sigma)|.}$

Thus, in our example above,

${ \displaystyle l\left( \begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 3& 1 & 2 &5 &6 &4&7\end{matrix}\right)=7-3=4.}$

Notice that the length does not change if we view a permutation in ${ S_7}$ as a permutation of ${ [9]}$, keeping ${ 8}$ and ${ 9}$ fixed.

The notation

${ c=\displaystyle (a_1\,a_2\,\ldots, a_k),}$

means a permutation that takes ${ a_1}$ to ${ a_2}$, ${ a_2}$ to ${ a_3}$, and so on, with ${ a_k}$ being taken to ${ a_1}$, and all other elements are held fixed. Such a permutation is called a cycle. (Whether you view the identity permutation as a cycle or not is up to you; one could make it less ambiguous by using the term cycle only for those permutations that have exactly one orbit.) Thus,

${ (1\,3\,8)}$

is a cycle. The notation does not require we specify which set this permutation acts on (of course, the set must contain ${ 1,3, 8}$.) If ${ c}$ as above is in ${ S_n}$ then

${ \displaystyle l(c)= n-[1+\underbrace{1+\cdots +1}_{n-k \, {\rm terms}}]=k-1.}$

Notice that ${ n}$ dropped out of this calculation. The length of a cycle is just the number of elements in the orbit of this cycle minus ${ 1}$. (“The orbit” having the obvious interpretation.)

Every permutation is a product of the cycles that correspond to its orbits, the distinct orbits being disjoint.

transposition is a permutation that exchanges two elements, and holds all others fixed. For example,

${ (1\, 3)}$

is a transposition. Clearly, a permutation is a transposition if and only if its length is ${ 1}$. Moreover, every transposition is equal to its own inverse.

If we multiply a permutation ${ \sigma\in S_n}$ by a transposition ${ \tau=(a\,b) \in S_n}$ to form ${ \tau\sigma}$ then there are exactly two possibilities:

(1) if ${ a}$ and ${ b}$ are both in the same orbit of ${ \sigma}$ then that orbit is split into two orbits for ${ \tau\sigma}$;

(2) if ${ a}$ and ${ b}$ are in different orbits of ${ \sigma}$ then these orbits are combined into one orbit for ${ \tau\sigma}$.

All other orbits of ${ \sigma}$ remain orbits of ${ \tau\sigma}$.

For example,

${ \displaystyle (1\,2)\, (1\,3\,4\,5)(2\,6\,7)=(1\,3\,4\,5\,2\,6\,7)}$

and, multiplying on the left by ${ (1\,2)}$, we have

${ \displaystyle (1\,2)\, (1\,3\,4\,5\,2\,6\,7)=(1\,3\,4\,5)(2\,6\,7).}$

A consequence of this very nice fact is the effect on the length of a permutation under multiplication by a transposition:

${ \displaystyle l(\tau\sigma)=\begin{cases}l(\sigma)+1 & \hbox{if } \tau \hbox{ transposes elements in different orbits of} \sigma ; \\ l(\sigma)-1 & \hbox{if } \tau \hbox{ transposes elements in the same orbit of } \sigma \end{cases}}$

where the first case occurs when ${ \tau}$ transposes elements in different orbits of ${ \sigma}$, and the second case occurs when ${ \tau}$ transposes elements within one orbit of ${ \sigma}$.

Thus for any permutation ${ \sigma}$ we can systematically break down each of its cycles (orbits) by hitting them with transpositions: there is a sequence of transpositions ${ \tau_1,\ldots,\tau_k}$ such that the length of ${ \tau_k\ldots\tau_1\,\sigma}$ is ${ 0}$, which means that

${ \displaystyle \tau_k\ldots\tau_1\,\sigma=\iota,}$

the identity. Note that this is equivalent to

${ \displaystyle \sigma=\tau_1\ldots\tau_k,}$

because each ${\tau_j}$ is a transposition (hence equal to its inverse).

Thus, every permutation ${ \sigma}$ is a product of transpositions, and the smallest such number of transpositions is the length of ${ \sigma}$.

As consequence,

${ \displaystyle l(\sigma\pi) \leq l(\sigma) + l(\pi),}$

for all permutations ${\sigma}$ and ${\pi}$.

The orbits of ${\sigma^{-1}}$ are exactly the orbits of ${\sigma}$, and so both ${\sigma}$ and ${\sigma^{-1}}$ have the same number of orbits. Hence they have the same length:

${\displaystyle l(\sigma^{-1})=l(\sigma).}$

Next for any permutations ${\sigma}$ and ${\pi}$, the orbits of ${\pi\sigma\pi^{-1}}$ are exactly of the form ${\pi(O)}$ where ${O}$ runs over the orbits of ${\sigma}$. Hence

${\displaystyle l(\pi\sigma \pi^{-1})=l(\sigma).}$

Posted in Uncategorized | 1 Comment

Charlie Egedy

Last evening a celebration was held at the LSU mathematics department in honor of Charlie Egedy’s life. Charlie was kind enough to allow me to include his beautiful poem “A Reckoning” in my book.  I also had the opportunity to serve on his doctoral committee and recall how much I enjoyed his mathematical work and presentation. He was a man of many talents and interests, ranging from mathematics and chess to baseball and poetry. His warm personality was recalled through many stories and memories from those who were enriched by having known him.

Let me quote the first few lines from his poem:

A Reckoning

He sits in a seamless room

staring

into the depths

of a wall that is not a wall,

opaque,

unfathomable.

Though deep understanding

lies

just beyond that wall,

the vision he desires

can be seen

only from within the room.

Chapter on Clifford algebras

This is an old version of the manuscript. It was not  proof read but this version has a chapter 14 on Clifford algebras (part of which survived into the print version published in 2012).

Posted in Uncategorized | 1 Comment

Representing finite groups, semisimply

This blog is devoted to stray thoughts and comments on representations of finite groups, were finite may mean “not necessarily infinite”, to turn the tables on the more widely used convention.

Some of this blog relates to my book Representing Finite Groups: A Semisimple Introduction

It is rarely acknowledged how much of the theory of representations of finite groups, and so much of what flowed from this theory, was discovered by Frobenius, starting with his 1896 paper.  For example, Frobenius’s counting formulas for solutions of equations in finite groups contain the essence of the discrete analog of Witten’s formula for the volume of the moduli space of flat connections.


text:
B. Rooter's Song
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

Wherever the destination is, whoever we meet, let's render this song together.

On a Cartesian coordinate plane lies a rectangular stage of size w × h, represented by a rectangle with corners (0, 0), (w, 0), (w, h) and (0, h). It can be seen that no collisions will happen before one enters the stage.

On the sides of the stage stand n dancers. The i-th of them falls into one of the following groups:

• Vertical: stands at (x i, 0), moves in positive y direction (upwards);
• Horizontal: stands at (0, y i), moves in positive x direction (rightwards).

According to choreography, the i-th dancer should stand still for the first t i milliseconds, and then start moving in the specified direction at 1 unit per millisecond, until another border is reached. It is guaranteed that no two dancers have the same group, position and waiting time at the same time.

When two dancers collide (i.e. are on the same point at some time when both of them are moving), they immediately exchange their moving directions and go on.

Dancers stop when a border of the stage is reached. Find out every dancer's stopping position.

Input

The first line of input contains three space-separated positive integers n, w and h (1 ≤ n ≤ 100 000, 2 ≤ w, h ≤ 100 000) — the number of dancers and the width and height of the stage, respectively.

The following n lines each describes a dancer: the i-th among them contains three space-separated integers g i, p i, and t i (1 ≤ g i ≤ 2, 1 ≤ p i ≤ 99 999, 0 ≤ t i ≤ 100 000), describing a dancer's group g i ( g i = 1 — vertical, g i = 2 — horizontal), position, and waiting time. If g i = 1 then p i = x i; otherwise p i = y i. It's guaranteed that 1 ≤ x i ≤ w - 1 and 1 ≤ y i ≤ h - 1. It is guaranteed that no two dancers have the same group, position and waiting time at the same time.

Output

Output n lines, the i-th of which contains two space-separated integers (x i, y i) — the stopping position of the i-th dancer in the input.

Examples
Input
8 10 81 1 101 4 131 7 11 8 22 2 02 5 142 6 02 6 1
Output
4 810 58 810 610 21 87 810 6
Input
3 2 31 1 22 1 11 1 5
Output
1 32 11 3
Note

The first example corresponds to the initial setup in the legend, and the tracks of dancers are marked with different colours in the following figure.

In the second example, no dancers collide.


text:
1. ## Radius of Convergence, limits: Complex numbers.

I'm really shaky with radius of convergence and limits when complex numbers are thrown into the mix, so I would be really grateful if someone could check I've answered these questions correctly! (I can't find any examples online that are quite as complicated)
1. Find the limit of (2n*e(( ln(n^2) + i*pi*n )/(( 16(n^2) + 5i ))^0.5))/((4n2 + 3in)(1/2)) [From n=1 to infinity]
2. Compute the radius of convergence of the power series: (sum from n=1 to infinity) of anzn, where an = (2n + 1)!(n + 2i)n/(3n)!
My answers: 1. The limit does not exist. 2. R=27/4
Help would be veeeery much appreciated!

2. ## Re: Radius of Convergence, limits: Complex numbers.

Originally Posted by ebb
I'm really shaky with radius of convergence and limits when complex numbers are thrown into the mix, so I would be really grateful if someone could check I've answered these questions correctly! (I can't find any examples online that are quite as complicated)
1. Find the limit of (2n*e(( ln(n^2) + i*pi*n )/(( 16(n^2) + 5i ))^0.5))/((4n2 + 3in)(1/2)) [From n=1 to infinity]
2. Compute the radius of convergence of the power series: (sum from n=1 to infinity) of anzn, where an = (2n + 1)!(n + 2i)n/(3n)!
My answers: 1. The limit does not exist. 2. R=27/4
Help would be veeeery much appreciated!
For question 1, you need to find a limit to a particular value, not to a counter of values, which you have written.

For question 2, a series \displaystyle \begin{align*} \sum{t_n} \end{align*} is convergent when \displaystyle \begin{align*} \lim_{n \to \infty}\left|\frac{t_{n+1}}{t_n}\right| < 1 \end{align*}, so

\displaystyle \begin{align*} \lim_{n \to \infty}\left| \frac{ \frac{[2(n+1) + 1]!(n + 1 + 2i)^{n + 1}}{[3(n+1)]!} \, z^{n+1} }{ \frac{(2n+1)!(n + 2i)^n}{(3n)!}\,z^n } \right| &< 1 \\ \lim_{n \to \infty}\left| \frac{\frac{(2n + 3)!(n + 1 + 2i)(n + 1 + 2i)^n}{(3n + 3)!}}{\frac{(2n + 1)!(n + 2i)^n}{(3n)!}} \right||z| &< 1 \\ \lim_{n \to \infty}\left| \frac{(3n)!(2n+3)!(n + 1 + 2i)(n + 1 + 2i)^n}{(3n + 3)!(2n + 1)!(n + 2i)^n} \right||z| &< 1 \\ \lim_{n \to \infty}\left| \frac{(2n + 3)(2n + 2)(n + 1 + 2i)(n + 1 + 2i)^n}{(3n + 3)(3n + 2)(3n + 1)(n + 2i)^n} \right||z| &< 1 \\ \lim_{n \to \infty}\left| \frac{(2n + 3)(2n + 2)(n + 1 + 2i)}{(3n + 3)(3n + 2)(3n + 1)} \right| \lim_{n \to \infty}\left| \left(\frac{n + 1 + 2i}{n + 2i}\right)^n \right| |z| &< 1 \\ \lim_{n \to \infty} \left| \frac{4n^3 + (14 + 8i)n^2 + (16 + 20i)n + 6 + 12i}{27n^3 + 54n^2 + 33n + 6} \right| \lim_{n \to \infty}\left| e^{n\ln{\left(\frac{n + 1 + 2i}{n + 2i}\right)}} \right| |z| &< 1 \end{align*}

\displaystyle \begin{align*} \lim_{n \to \infty}\left| \frac{4 + \frac{14 + 8i}{n} + \frac{16 + 20i}{n^2} + \frac{6 + 12i}{n^3}}{27 + \frac{54}{n} + \frac{33}{n^2} + \frac{6}{n^3}} \right| \lim_{n \to \infty}\left|e^{\frac{\ln{\left( 1 + \frac{1}{n + 2i}\right)}}{\frac{1}{n}}}\right||z| &< 1 \\ \frac{4}{27}\lim_{n \to \infty}\left|e^{\frac{\frac{1}{n + 1 + 2i} - \frac{1}{n + 2i}}{-\frac{1}{n^2}}}\right||z| &< 1 \textrm{ by L'Hospital's Rule} \\ \frac{4}{27}\lim_{n \to \infty}\left| e^{\frac{3 - 4i}{n + 1 + 2i} - \frac{4}{n + 2i} + 1} \right||z| &< 1 \\ \frac{4e}{27} |z| &< 1 \\ |z| &< \frac{27}{4e} \end{align*}

So the radius of convergence is \displaystyle \begin{align*} \frac{27}{4e} \end{align*}.

3. ## Re: Radius of Convergence, limits: Complex numbers.

For 1. I'm not entirely sure what you mean by a 'counter of values'?
Thank you so much for your response!

4. ## Re: Radius of Convergence, limits: Complex numbers.

Also, in the module I'm doing we've not been taught L'Hopital's rule, although I've come across it in further reading, and the lecturer said we will not be allowed to answer questions using it in the exam...

5. ## Re: Radius of Convergence, limits: Complex numbers.

Originally Posted by ebb
Also, in the module I'm doing we've not been taught L'Hopital's rule, although I've come across it in further reading, and the lecturer said we will not be allowed to answer questions using it in the exam...
Well I can't think of another way to evaluate \displaystyle \begin{align*} \lim_{n \to \infty}\left|\left(\frac{n + 1 + 2i}{n + 2i}\right)^n\right| \end{align*}, so if that's the case you would probably have to use a CAS calculator.


text:
# zbMATH — the first resource for mathematics

Compute Distance To:
 Documents Indexed: 42 Publications since 1984, including 3 Books
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#### Co-Authors

 11 single-authored 4 Kowalczyk, Piotr 3 Standar, Christoffer 2 Beilina, Larisa 2 Gebäck, Tobias 2 Sopasakis, Alexandros 1 Brahme, Anders 1 Eriksson, Kenneth 1 Humbürg, Ale Jan 1 Jamalzadeh, Javad 1 Kazemi, Ehsan 1 Kumlin, Peter 1 Larsson, Stig 1 Rezaei, Gholamreza 1 Shahgholian, Henrik 1 Xin, Jiping
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#### Serials

 5 Transport Theory and Statistical Physics 4 SIAM Journal on Numerical Analysis 2 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 2 Kinetic and Related Models 1 Computer Methods in Applied Mechanics and Engineering 1 Inverse Problems 1 BIT 1 Bulletin of the Iranian Mathematical Society 1 RAIRO. Modélisation Mathématique et Analyse Numérique 1 Numerical Methods for Partial Differential Equations 1 Journal of Mathematical Sciences (New York) 1 Advances in Computational Mathematics 1 Scientific Bulletin. Series A. Applied Mathematics and Physics. Politehnica University of Bucharest 1 International Journal of Numerical Analysis and Modeling 1 Applied and Computational Mathematics
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#### Fields

 22 Numerical analysis (65-XX) 17 Statistical mechanics, structure of matter (82-XX) 14 Partial differential equations (35-XX) 9 Integral equations (45-XX) 5 Fluid mechanics (76-XX) 2 Optics, electromagnetic theory (78-XX) 1 General and overarching topics; collections (00-XX) 1 Mathematical logic and foundations (03-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 Number theory (11-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Integral transforms, operational calculus (44-XX) 1 Probability theory and stochastic processes (60-XX) 1 Mechanics of deformable solids (74-XX) 1 Quantum theory (81-XX)

#### Citations contained in zbMATH

27 Publications have been cited 148 times in 92 Documents Cited by Year
Convergence analysis of the streamline diffusion and discontinuous Galerkin methods for the Vlasov-Fokker-Planck system. Zbl 1072.82023
2005
Asymptotic error expansions for the finite element method for second order elliptic problems in $$R^N, N\geq 2$$. I: Local interior expansions. Zbl 1219.65125
2010
A new approach to Richardson extrapolation in the finite element method for second order elliptic problems. Zbl 1198.65216
2009
A finite element method for the neutron transport equation in an infinite cylindrical domain. Zbl 0918.65092
1998
Analysis of a fully discrete scheme for neutron transport in two- dimensional geometry. Zbl 0608.65098
1986
A posteriori error estimates for the Fokker-Planck and Fermi pencil beam equations. Zbl 1012.82008
2000
On adaptive finite element methods for Fredholm integral equations of the second kind. Zbl 0814.65134
1994
$$L_ p$$ and eigenvalue error estimates for the discrete ordinates method for two-dimensional neutron transport. Zbl 0668.65119
1989
Streamline diffusion methods for Fermi and Fokker-Planck equations. Zbl 0907.65143
1997
Chebyshev spectral-$$S_N$$ method for the neutron transport equation. Zbl 1137.65443
2006
Streamline diffusion methods for the Vlasov-Poisson equation. Zbl 0703.76106
1990
Discontinuous Galerkin and multiscale variational schemes for a coupled damped nonlinear system of Schrödinger equations. Zbl 1307.65129
2013
A posteriori error estimates for a coupled wave system with a local damping. Zbl 1283.65089
2011
The Fokker-Planck operator as an asymptotic limit in anisotropic media. Zbl 1053.35040
2002
The discrete ordinates method for the neutron transport equation in an infinite cylindrical domain. Zbl 0767.65095
1992
Discrete-ordinates and streamline diffusion methods for a flow described by BGK model. Zbl 1305.76091
2014
A posteriori error estimates in a globally convergent FEM for a hyperbolic coefficient inverse problem. Zbl 1222.65103
2010
On the stability of characteristic schemes for the Fermi equation. Zbl 1142.90318
2002
$$L_ 2$$-error estimates for the discrete ordinates method for three- dimensional neutron transport. Zbl 0652.65097
1988
Galerkin methods for primary ion transport in inhomogeneous media. Zbl 1217.82040
2010
Ion transport in inhomogeneous media based on the bipartition model for primary ions. Zbl 1205.78017
2010
An inverse problem of the synthesis of optimal output variable regulators. Zbl 1284.49040
2006
On fully discrete schemes for the Fermi pencil-beam equation. Zbl 1010.82002
2002
Convergence analysis for backward-Euler and mixed discontinuous Galerkin methods for the Vlasov-Poisson system. Zbl 1330.65146
2015
Convergence of finite volume scheme for a three-dimensional Poisson equation. Zbl 1315.65095
2014
Efficiency of combined Daubechies and statistical parameters applied in mammography. Zbl 06287356
2013
Linear transport equations in flatland with small angular diffusion and their finite element approximations. Zbl 1173.90322
2008
Convergence analysis for backward-Euler and mixed discontinuous Galerkin methods for the Vlasov-Poisson system. Zbl 1330.65146
2015
Discrete-ordinates and streamline diffusion methods for a flow described by BGK model. Zbl 1305.76091
2014
Convergence of finite volume scheme for a three-dimensional Poisson equation. Zbl 1315.65095
2014
Discontinuous Galerkin and multiscale variational schemes for a coupled damped nonlinear system of Schrödinger equations. Zbl 1307.65129
2013
Efficiency of combined Daubechies and statistical parameters applied in mammography. Zbl 06287356
2013
A posteriori error estimates for a coupled wave system with a local damping. Zbl 1283.65089
2011
Asymptotic error expansions for the finite element method for second order elliptic problems in $$R^N, N\geq 2$$. I: Local interior expansions. Zbl 1219.65125
2010
A posteriori error estimates in a globally convergent FEM for a hyperbolic coefficient inverse problem. Zbl 1222.65103
2010
Galerkin methods for primary ion transport in inhomogeneous media. Zbl 1217.82040
2010
Ion transport in inhomogeneous media based on the bipartition model for primary ions. Zbl 1205.78017
2010
A new approach to Richardson extrapolation in the finite element method for second order elliptic problems. Zbl 1198.65216
2009
Linear transport equations in flatland with small angular diffusion and their finite element approximations. Zbl 1173.90322
2008
Chebyshev spectral-$$S_N$$ method for the neutron transport equation. Zbl 1137.65443
2006
An inverse problem of the synthesis of optimal output variable regulators. Zbl 1284.49040
2006
Convergence analysis of the streamline diffusion and discontinuous Galerkin methods for the Vlasov-Fokker-Planck system. Zbl 1072.82023
2005
The Fokker-Planck operator as an asymptotic limit in anisotropic media. Zbl 1053.35040
2002
On the stability of characteristic schemes for the Fermi equation. Zbl 1142.90318
2002
On fully discrete schemes for the Fermi pencil-beam equation. Zbl 1010.82002
2002
A posteriori error estimates for the Fokker-Planck and Fermi pencil beam equations. Zbl 1012.82008
2000
A finite element method for the neutron transport equation in an infinite cylindrical domain. Zbl 0918.65092
1998
Streamline diffusion methods for Fermi and Fokker-Planck equations. Zbl 0907.65143
1997
On adaptive finite element methods for Fredholm integral equations of the second kind. Zbl 0814.65134
1994
The discrete ordinates method for the neutron transport equation in an infinite cylindrical domain. Zbl 0767.65095
1992
Streamline diffusion methods for the Vlasov-Poisson equation. Zbl 0703.76106
1990
$$L_ p$$ and eigenvalue error estimates for the discrete ordinates method for two-dimensional neutron transport. Zbl 0668.65119
1989
$$L_ 2$$-error estimates for the discrete ordinates method for three- dimensional neutron transport. Zbl 0652.65097
1988
Analysis of a fully discrete scheme for neutron transport in two- dimensional geometry. Zbl 0608.65098
1986
all top 5

#### Cited by 141 Authors

 8 Transport Theory and Statistical Physics 7 Computers & Mathematics with Applications 7 Journal of Computational Physics 7 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 5 Mathematics of Computation 4 Journal of Computational and Applied Mathematics 3 Computer Methods in Applied Mechanics and Engineering 3 BIT 3 Numerical Methods for Partial Differential Equations 3 Mathematical and Computer Modelling 3 Abstract and Applied Analysis 2 Applied Mathematics and Computation 2 Numerische Mathematik 2 SIAM Journal on Numerical Analysis 2 Journal of Scientific Computing 2 International Applied Mechanics 2 Journal of Mathematical Sciences (New York) 2 Communications in Nonlinear Science and Numerical Simulation 2 Engineering Computations 1 Applicable Analysis 1 Journal of Mathematical Analysis and Applications 1 Journal of Statistical Physics 1 Mathematical Methods in the Applied Sciences 1 Calcolo 1 Computing 1 International Journal of Mathematics and Mathematical Sciences 1 Kybernetes 1 Applied Numerical Mathematics 1 Sequential Analysis 1 Computational Mechanics 1 International Journal of Computer Mathematics 1 SIAM Journal on Mathematical Analysis 1 Advances in Computational Mathematics 1 Computational Methods in Applied Mathematics 1 Iranian Journal of Science and Technology. Transaction A: Science 1 Journal of Applied Mathematics and Computing 1 Central European Journal of Mathematics 1 Kinetic and Related Models 1 Asian-European Journal of Mathematics 1 Iranian Journal of Mathematical Chemistry 1 Stochastic and Partial Differential Equations. Analysis and Computations


text:
# How to solve these trigonometry equations?

I have to work with the following 5 equations:

1. $$(1-\tan^2x)(1-\tan^22x)(1-\tan^24x)=8$$
2. $$(2\cos 2x+1)(2\cos 6x+1)(2\cos 18x+1)=1$$
3. $$\dfrac{\cos 2x}{\sin 3x}+\dfrac{\cos 6x}{\sin 9x}+\dfrac{\cos 18x}{\sin 27x}=0$$
4. $$\dfrac{\cos x}{\sin 3x}+\dfrac{\cos 3x}{\sin 9x}+\dfrac{\cos 9x}{\sin 27x}=0$$
5. $$\dfrac{1}{\cos x\cos 2x}+\dfrac{1}{\cos 2x\cos 3x}+\dfrac{1}{\cos 3x\cos 4x}=0$$

These equations have patterns, and I know if we can use the pattern we will solve the equations very easily. I managed to use the pattern on the first equation to find a telescoping series and get this (it is not a full solution but it is the way to solve the first equation):

We have $$\frac{2\tan x}{1-\tan^2x}=\tan 2x$$ therefore

\begin{align*} (1)&\Leftrightarrow\dfrac{1}{(1-\tan^2x)(1-\tan^22x)(1-\tan^24x)}=\dfrac18\\ &\Leftrightarrow\left(2\tan x\cdot\dfrac{1}{1-\tan^2x}\right)\cdot\dfrac{1}{1-\tan^22x}\cdot\dfrac{1}{1-\tan^24x}=\dfrac14\tan x\\ &\Leftrightarrow\left(2\tan 2x\cdot\dfrac{1}{1-\tan^22x}\right)\cdot\dfrac{1}{1-\tan^24x}=\dfrac12\tan x\\ \end{align*}

and so on.

However, I can't manage to solve the last four. I can't find the key equalities like $$\frac{2\tan x}{1-\tan^2x}=\tan 2x$$ in the first equation. So here are my questions:

1. How to solve equations 2, 3, 4, and 5?
2. What is the strategy to find the key equalities like $$\frac{2\tan x}{1-\tan^2x}=\tan 2x$$ to get a telescoping series for each equation?

In the first you can get it by the following way: $$LS=\frac{2\tan{x}}{\tan{2x}}\cdot\frac{2\tan{2x}}{\tan{4x}}\frac{2\tan{4x}}{\tan{8x}}=\frac{8\tan{x}}{\tan{8x}}$$ and the rest is smooth.

$$2. 2\cos2y+1=2(1-2\sin^2y)+1=\dfrac{\sin3y}{\sin y}$$ for $$\sin y\ne0$$

$$3.\dfrac{\cos2y}{\sin3y}=\dfrac{2\cos2y\sin y}{2\sin3y\sin y}=?$$

$$4. \dfrac{\cos x}{\sin3x}=\dfrac{\sin(3x-x)}{2\sin3x\sin x}=?$$

$$5.\sin x=\sin((n+1)x-nx)=?$$

Set $$n=1,2,3$$

Using that $$\tan(2x)=2\,{\frac {\tan \left( x \right) }{1- \left( \tan \left( x \right) \right) ^{2}}}$$ $$\tan(4x)={\frac {4\,\tan \left( x \right) -4\, \left( \tan \left( x \right) \right) ^{3}}{1-6\, \left( \tan \left( x \right) \right) ^{2}+ \left( \tan \left( x \right) \right) ^{4}}}$$ we get for your first equation $$\left( 8\, \left( \cos \left( x \right) \right) ^{3}+4\, \left( \cos \left( x \right) \right) ^{2}-4\,\cos \left( x \right) -1 \right) \left( 8\, \left( \cos \left( x \right) \right) ^{3}-4\, \left( \cos \left( x \right) \right) ^{2}-4\,\cos \left( x \right) + 1 \right) =0$$ for #5: Your equation is equivalent to $$\cos \left( 3\,x \right) \cos \left( 4\,x \right) +\cos \left( 2\,x \right) \cos \left( x \right) +\cos \left( 4\,x \right) \cos \left( x \right) =0$$ and this can be written in the form $$\cos \left( x \right) \left( 32\, \left( \cos \left( x \right) \right) ^{6}-48\, \left( \cos \left( x \right) \right) ^{4}+22\, \left( \cos \left( x \right) \right) ^{2}-3 \right) =0$$


text:
## Introduction

In recent months, severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) variants associated with increased transmissibility have emerged and spread, one after another. The World Health Organization has designated a list of variants of concern (VOC), including Alpha (B.1.1.7), Beta (B.1.351), Gamma (P.1), Delta (B.1.617.2), and Omicron (B.1.1.529), as well as multiple variants of interest1. Viral lineages carrying the N501Y and/or E484K mutations, i.e., the Alpha, Beta, and Gamma variants, were first identified in Ontario, Canada in December 20202. These VOC quickly outcompeted earlier SARS-CoV-2 lineages and, as of late April 2021, accounted for all new infections in Ontario, with Alpha being the most prevalent lineage1. Around that time, the Delta variant was first reported in the province. By 3 June 2021, the Alpha variant has been reported from at least 160 countries, Beta from 113 countries, Gamma from 64 countries, and Delta from 62 countries. Since then, the Delta variant has rapidly become the dominant circulating lineage. As of July 2021, Delta represented the majority of infections in Ontario3. At the time of this study, Omicron has begun spreading.

Initially, no pharmaceutical agents are known to be safe and effective at preventing or treating coronavirus disease 2019 (COVID-19)4. This leaves public health authorities with nonpharmaceutical interventions (NPI) as the only recourse to reduce disease transmission; these measures include bans on public gatherings, compulsory stay-at-home policies, mandating closures of schools and nonessential businesses, face mask ordinances, and quarantine. Even though the WHO initially expected vaccine development to take 18 months5, dedicated global efforts resulted in the approval of vaccines first in China6 and Russia7 in the summer of 2020. Near the end of that year, major vaccines developed by Pfizer-BioNTech, Moderna, and Oxford-AstraZeneca were approved in North America. Since then, the deployment of vaccines has become the most important intervention for mitigating disease severity and spread.

Despite costly NPI and the availability of vaccines, SARS-CoV-2 continues to spread and wreak havoc on a global scale. A major challenge is the rapid emergence of VOC with higher transmissivity, as noted above, and vaccine breakthrough. While major vaccines are similarly effective against the SARS-CoV-2 wild type, their efficacy against VOC is significantly less and differs among vaccine types8. Large discrepancy exists in vaccine deployment among countries, not only in the fraction of the population vaccinated, but also in the types of vaccines distributed. Thus, a pressing question is: How ready are different regions of the world in stopping the spread of SARS-CoV-2 VOC? This question must be addressed not only for existing VOC, but also for future, potentially deadlier ones. Another challenge lies in the role of asymptomatic patients, including those who are vaccinated, in spreading the disease. Asymptomatic patients may account for more than half of the COVID-19 cases9, and non-severe COVID-19 patients can transmit the disease regardless of their symptomatic status10. Given the increasing prevalence of VOC vaccine breakthrough, many of these asymptomatic patients may have been vaccinated and underestimate the likelihood of themselves being COVID-19 carriers. Thus, another critical question is: What role do individuals with asymptomatic infections play in disease transmission?

To answer the above equations, we developed and applied a computational model of the spread of SARS-CoV-2 variants within a community. The model represents asymptomatic and symptomatic infections, and two-dose vaccinations with variable dosing intervals. Additionally, the effects of NPI are represented by scaling the disease infectivity. A schematic diagram of the model is shown in Fig. 1. A large number of epidemiological modeling studies have been published11, many of which are based on the Susceptible-Infected-Recovered (SIR) formulation (e.g., 12,13,14,15). A key feature that distinguishes the present study from existing SIR-type models in that it is the first model to represent multiple SARS-CoV-2 VOC and their competition. We refer to this model as a SV2(AIR)3 model, for its representation of two vaccine types (V2), asymptomatic infections (A), and three VOC (the power ‘3’ refers to the three instantiations of each infected and recovered class). The SV2(AIR)3 model can answer the question: To what extent do the interactions and competition among VOC impact the spread of COVID-19? An example of VOC competition at a population level is this: an individual having been infected and subsequently recovered from one strain may be partially but incompletely protected against other strains, and if infected again, they may be more likely to develop an asymptomatic infection. Given the key role of asymptomatic cases in disease transmission, a thorough understanding of VOC competition is important in determining which (costly) NPI are necessary to prevent and limit the spread of COVID-19. We apply the model to analyze VOC dynamics in the province of Ontario in Canada. Ontario has managed a reasonable deployment of vaccines, but is nonetheless on the cusp of another wave of infections due to a highly transmissible VOC such as Delta or Omicron. That said, at the time of this study, the number of Delta infections in Ontario remains substantially lower than much of the rest of the world, which makes Ontario an interesting case study.

## Results

### The dynamics of COVID-19 VOC in Ontario, Canada

Model solution was computed to simulate the spread of SARS-CoV-2 in Ontario for two years, from January 1 2020 to December 31 2021. The SV2(AIR)3 model represents the dynamics of and interactions among three SARS-CoV-2 strains: Wild type, Alpha, and Delta, denoted by superscripts ‘W’, ‘A’, and ‘D’, respectively. On January 1 2020, we assume that the entire Ontario population was susceptible, except for 15 infected symptomatic individuals (wild-type). The Alpha strain was introduced into Ontario via the entrance of 50 asymptomatic infected individuals per day for 1.5 months at the end of 2020; Delta was introduced similarly in February and March 2021 (25 asymptomatic infected individuals per day). These parameters were chosen to product simulation results that match the initial rise of each of the VOC, Vaccination became available on December 14 2020; details for vaccination rates can be found in the supplement. The Omicron strain is not represented in the model, because insufficient data is available for this strain at the time of this study. Nonetheless, a hypothetical, more transmissible VOC is simulated.

Key measures of the pandemic are illustrated in Fig. 2. The total number of new infections exhibits two peaks (Fig. 2a), with the first wave in December 2020 driven by the wild type, and the second around March 2021 driven by Alpha. By mid-June 2021, the new infections are dominated by the Delta variant. These predictions are consistent with Ontario statistics (Ontario COVID-19 Science Advisory Table). The consistency between model predictions and Ontario statistics before July 2021 can be attributed to the choice of model parameters, which were based on statistics available then. Model predictions for the second half of 2021 can be used to validation. The total number of infections, including asymptomatic and symptomatic infections, are shown in Fig. 2b. The model predicts that by the end of 2021, about 680 K people would have been infected, half of whom are asymptomatic. The cumulative infection curves exhibit the steepest rise in December 2020 and March 2021, corresponding to the two peaks in the new case counts. By the end of 2021, 71% of the Ontario population would have been vaccinated with at least one dose, with 57% fully vaccinated (Fig. 2c). At that time, 11 K would have died from COVID, with 36, 47, and 17% from wild type, Alpha, and Delta, respectively (Fig. 2d). The year-end infection and death statistics are within 10% of reported data in Ontario (750 K infections and 10.2 K deaths)16.

Predicted population dynamics are shown in Fig. 3. The drop in the susceptible population becomes significant after December 2020 (Fig. 3a) and is driven primarily by the vaccine deployment. By the end of June 2021, only 37% of the Ontario population are susceptible, with the majority having received at least one dose of vaccine (Fig. 3b,c). The number of active infections peaks at the end of 2020, followed by an even higher peak at the end of spring (Fig. 3d). These are the wild-type and Alpha-driven waves identified in Fig. 2a and described above. The Alpha-driven infection peak accounts for 0.7% of the total population, but subsides as the vaccination effort accelerates. The infections are split approximately half-half between asymptomatic and symptomatic ones, with the majority unvaccinated (Fig. 3d–g). As the infection spreads, the recovered population also increases, to 7% of the population at the end of 2021 (Fig. 3h). These dynamics are summarized as fractions of the population that are susceptible, vaccinated, infected, and recovered, as shown in Fig. 3i.

### Sensitivity analysis

To identify the strongest determinants of the pandemic severity, we assess the sensitivity of model predictions to variations in key model parameters. We first conduct a sensitivity analysis in a simplified model that represents only the wild-type SARS-CoV-2, for the year 2020, but with infectivity βW elevated to 5E−9. The Alpha and Delta strains are not included. We assume that NPI begin on March 1 2020, with the restrictiveness index λ taken to be uniformly 0.5 (i.e., infectivity β’s halved) for simplicity. Vaccination begins on April 1 2020 (not the end of 2020, for this simplified one-year simulation), with baseline dosing interval taken to be 4 weeks for Pfizer and 8 weeks for AstraZeneca (dosing interval is one of the parameters varied)17. Parameters that are varied include those that describe the clinical features of the viruses as well as provincial policy and pharmaceutical intervention.

Clinical features of the viruses that were considered include its infectivity (βW), the fraction of infections that are asymptomatic (σW), the infectivity ratio between asymptomatic and symptomatic individuals (αW), mortality rate (μW), and recovery rates (γWA and γWI, varied simultaneously). Recall the baseline assumption that asymptomatic individuals are assumed to be more active and 3 times more likely to spread the disease than symptomatic ones (αW = 3). Thus, the number of infections directly depends on, and is in fact highly sensitive to, βW, σW, αW, and also to the recovery rate γW, which impacts how many new infections an infected individual can cause. Small variations in these four parameters (5%) result in changes in the number of infections and deaths that are an order of magnitude larger; see Fig. 4. COVID-19 mortality rate μWI affects the number of infected individuals who are alive and can participate in the spread of the disease. Because COVID-19 mortality rate can vary substantially depending on health care infrastructure, we varied μWI by 50%. As expected, changes in μWI are immediately reflected in the number of deaths. However, compared to other clinical features of SARS-CoV-2, the mortality rate has a relatively small effect on the infected population, since < 5% of them die (Fig. 4).

Provincial policy and pharmaceutical interventions that were considered include NPI that reduce the effective disease infectivity (via scaling by λ), vaccination rate (ω1), delay between the first and second vaccine doses (ω2,PZ and ω2,AZ, varied simultaneously), and vaccine escape rate (βWV). λ and ω1 are varied by 5%. Since NPI are modelled by scaling βW directly, they have large effects on model predictions (Fig. 4). Vaccination rate has lesser but still significant effects on the spread of the disease. The effects of dosing interval (ω2) and vaccine escape rate (βWV) are much smaller. Because dosing interval can vary greatly within the population, and because vaccine escape rate is not sufficiently well characterized, these two parameters were varied by 50%. The vaccine-related results suggest that among these three factors, an effective vaccine deployment (captured in ω1) is the key in combating a pandemic, and success may be achieved even by means of a less-than-ideal vaccine with higher escape rate (Fig. 4). Since the predicted numbers of infections and deaths are substantially more sensitive to variations in ω1 than ω2, if vaccine supply is limited, prioritizing the first dose by lengthening the dosing interval would be advisable, as was done in Canada.

In a second sensitivity analysis, we considered the full model that represents all three variants, using the baseline parameter values (Table S1). Key model parameters were increased to determine the effects on the number of infections and deaths associated with each variant. In addition to parameters considered in the above analysis, we also considered two parameters that mediate VOC interactions: βXR, rate of infection of individuals with partial immunity from having recovered from a different variant, and ηR, rate of loss of immunity due to infection. Some of these parameters have distinct values for each of the variants. Simulations were conducted by varying each of the parameter values for individual variants separately. Specifically, βX (X = W, A, or D), σX, αX, γX, and ω1 were increased by 5%, λ by 10%, and μI, βXV, and ω2 by 50%. βXR was increased 4 folds to assess the scenario in which recovery from one variant offers only limited protected from other VOCs. ηXR was doubled to test the effect of a shorter immunity period. Sensitivity results are summarized in Fig. 5.

The effects of varying parameters associated with a given variant on the spread of that variant are similar to the single-variant analysis; compare Figs. 4 and 5. Thus, we will discuss cross-variant results. As can be seen in Fig. 5, most wild-type specific parameters have opposite effects on wild-type and on the other variants (βX, σX, αX, γX, βXV). This suggests that promoting the spread of the wild-type variant protects the population from Alpha and Delta, due to the partial natural immunity individuals acquire after recovering from wild-type infections. In contrast, the spread of Alpha or Delta has no effect on wild-type. That is because by the time either Alpha or Delta became rampant, the number of new wild-type cases has become negligible. βWR (natural immunity against the wild-type strain, acquired from a Alpha or Delta infection) has essentially no effect because during the spread of the wild-type virus, no one has recovered from a different variant. In contrast, lowering the natural immunity acquired from recovery from a different variant (i.e., increasing βAR or βDR) significantly increases the spread of Alpha or Delta. When an individually loses their naturally acquired immunity, they are susceptible to all variants. Thus, increasing the rate of loss of naturally acquired immunity (ηXR) increases the spread of all three variants.

Note that λ, ω1, and ω2 are not variant specific. Their effects on the spread of the wild-type virus have been discussed above in the single-variant analysis. Due to VOC interactions, the effects of varying these parameters on the spread of Alpha and Delta are less straightforward. Reducing the severity of the NPI (i.e., increasing λ) has competing effects on the spread of Alpha and Delta: one directly from the variation in NPI, the other from the enhanced spread of the wild-type virus. These competing effects result in opposite effects on Alpha and Delta; see Fig. 5. The opposite effects of increasing ω1, and ω2 on the spread of Alpha and Delta can be explained similarly. Competition among the VOC is further considered below.

### Competition among VOC

Individuals who have recovered from COVID-19 gain partial immunity not only to that particular strain, but other strains as well, albeit to a lesser extent. In that sense, the VOC don’t spread independently but may instead exhibit inhibitory effect on each other. To understand their interactions, we conduct simulations with some of the VOC eliminated. The predicted number of infections for each scenario is shown in Fig. 6. Results suggest that Alpha and Delta have minimal impact on the spread of the wild type. When Alpha and/or Delta were eliminated, the number of wild-type infections was barely affected. That insensitivity can be explained by the observation that the wild-type strain spread with little competition for almost a year. By the time the other VOC emerge, the wild type is already on its decline due to the NPI. In contrast, the wild type has a significant impact on both Alpha and Delta. In the absence of the wild type, fewer of the population would have acquired partial immunity to Alpha, and the number of Alpha infections would be 19% higher. Similarly, in the absence of both the wild type and Alpha, there would have been 35% more Delta infections. Similar trends are seen in the mortality counts (results not shown). These results point to the importance of taking VOC competition into account when analyzing and predicting COVID-19 dynamics.

### Spread of the Delta variant

The Delta variant, which compared to earlier strains exhibits substantially higher transmissivity and vaccines breakthrough rate, drove a surge in COVID-19 cases in much of the world. The baseline model simulation describes what happened in Ontario: that after its introduction at the beginning of 2021, the number of new Delta cases peaked in April, before it was suppressed by the provincial lockdown that began on April 13. Simulation predicts that provided that some significant NPI remain in place throughout the remainder of 2021 and 2022 (λ ≤ 0.4, except during the winter holidays), the Delta variant may remain under control. However, given the high price that society must pay for being insufficiently vigilant against COVID-19, it is prudent to explore all possibilities in which the pandemic can explode. Hence, we conduct simulations to analyze the spread of the Delta variant under alternative scenarios.

The baseline model assumes that 95% of the vaccines are Pfizer and 5% are AstraZeneca, consistent with Ontario vaccination statistics. What if a larger fraction of the vaccines were AstraZeneca, as is the case in the UK18. A notable difference between the two vaccine types is that the Pfizer vaccine offers stronger protection than AstraZeneca against all COVID variants, especially Delta, after either one or two doses. For instance, an individual having received both doses of the Pfizer vaccine is 88% protected against the Delta variant, whereas with the AstraZeneca vaccine the protection is only at 60%19. We conduct a simulation in which only 40% of the vaccines are Pfizer. With a larger portion of the vaccinated population now under reduced vaccine protection, the spread of the VOC is substantially accelerated. By the end of 2021, the model predicts 794 new Delta infections a day, more than three times higher than the baseline prediction of 234 new Delta infections a day (Fig. 7a). The total number of Delta infections and related deaths are also significantly higher, predicted to be 62 K and 882 at the end of 2021, respectively, compared to baseline values of 39 K and 689 (Fig. 7b,d).

If the provincial vaccination rate were lower, such that by September 1 2021 only 67% of the population are vaccinated with at least one dose, compared to the baseline vaccinated percentage of 74% (Fig. 7c), the total number of Delta infections and deaths would increase by 75% by the end of March 2022; see Fig. 7b,d.

Beginning mid-June 2021, Ontario began to gradually reopen, with in-person classes set to resume in September. The baseline model assumes that the NPI that remain will still reduce COVID-19 infectivity by 60%. What if that assumption is overly optimistic? Indeed, it is not uncommon to experience pandemic fatigue and crave personal interactions; also, there may be outbreaks in schools. Thus, we consider a scenario with less stringent NPI starting from the fall of 2021 (September 1 2021), with λ increased from its baseline value of 0.4 to 0.5 onwards. The model predicts the reduced NPI would accelerate the spread of Delta, such that by March 2022, the total number of Delta infections would be 50% higher than baseline (Fig. 7b).

Who are getting infected by the Delta variant, and who are the carriers? Interestingly, while the wild-type infections are about equally split between asymptomatic and symptomatic, ~ 60% of the Delta-infected population are asymptomatic. This difference may be attributable to the vaccination status of the infected population. Due to the larger vaccine breakthrough rate of the Delta variant, almost a quarter of the infected individuals have been vaccinated (Fig. 8a); in contrast, for the wild type, vaccinated individuals make up a negligible fraction of the infected population (Fig. 3). Because vaccination protects one from most major COVID-19 symptoms, most of the vaccinated infections are assumed to be asymptomatic (85% compared to 50% for the unvaccinated). Taken together, the asymptomatic patients, including those who are vaccinated, play a more significant role in spreading the Delta variant, compared to the wild type. And that difference is even larger in the alternative scenario where most of the vaccines are AstraZeneca (Fig. 8b).

Alpha is more infectious than the wild type; Delta is worse than Alpha; Lambda may be even worse. How bad can things get? What are the effective and reasonable measures against the challenges presented by an ever-evolving coronavirus? To gain insights, we simulate a dangerous hypothetical variant, which we refer to as “Neos.” Neos is designed to be deadlier by increasing its (i) infectivity (14% higher than the Delta variant), (ii) vaccine escape rate (70% for one dose of Pfizer or AstraZeneca, similar to Delta which is 67%; 25% and 50% for two doses of Pfizer and AstraZeneca, respectively; compared to 12% and 40% for Delta), and (iii) fraction of asymptomatic infections (55% instead of 50%). Recovery rate, mortality rate, and other clinical features are assumed to be the same as the wild type.

We assume that Neos emerges in the fall of 2021. The predicted population dynamics that describe its spread, from August 1 2021 to June 30 2022, is shown in Fig. 9. After its introduction, Neos quickly takes hold due to its high transmissivity, and by the end of 2021, it has overtaken Delta to account for the majority of new infections (Fig. 9c). By mid 2022, the infected individuals would account for 1% of the entire population (Fig. 9f), despite nearly 80% of the population having been vaccinated. More than 6.6 K would have died from Neos (Fig. 9g); that is likely an underestimate, since the number of severe cases might overwhelm the hospitals, elevating the mortality rate.

While those predicted numbers are alarming, the situation could be even more dire. How much more quickly would Neos spread under the three alternative scenarios we previously considered for Delta? That is, (i) if 60% of the vaccines are AstraZeneca, (ii) if vaccination deployment is slower, and (iii) if NPI are relaxed. If the vaccination rate is 20% lower, the model predicts that the total number of Neos infections would be 3 times the original value; with 60% AstraZeneca vaccines, 4.6 times; fewer restrictions (λ = 0.5 instead of 0.4), 5 times. See Fig. 10b. Similar trends are predicted for the vaccinated individuals infected by Neos (Fig. 10c), and for the total deaths from the Neos strain (Fig. 10d). In the case where NPI are relaxed, by the middle of 2022, 9% of the entire population would have been infected by Neos, and 0.3% would have died from the disease, not counting other strains.

What can the province to do combat the spread of Neos? Two potential measures are investigated. First, we consider more stringent NPI, perhaps a provincial lockdown. As noted above, the baseline model includes some NPI that are assumed to lower social contacts by at least 60%. We evaluate the effectiveness of additional NPI that further limit social contacts, and thus infectivity of all variants, by another 60%. These measures are assumed to commence on September 1 2021 and remain in place through the end of the simulation. The predicted number of symptomatic infections from Neos, Alpha, and Delta are shown in Fig. 11a1. We choose to highlight symptomatic infections because they strongly correlate with stress on the healthcare system. During the simulated period, the wild-type has essentially disappeared and is thus not shown. The model predicts that such stringent measures may slow the spread of Neos. The number of Neos infections still increases, but at a drastically lower rate than in the original setting (compare Figs. 9d and 11a1).

Many stringent NPI are associated with high economic costs. And the model predicts that even stringent NPI may not eradicate Neos. Thus, we consider a pharmaceutical measure: a vaccine booster. A third Pfizer shot has been reported to offer increased protection from the Delta variant and potentially other VOC20. In this simulation, we assume that a third dose of either the Pfizer or AstraZeneca vaccine would reduce vaccine breakthrough of Neos to 10% and 20%, respectively (compared to 25% and 50%, respectively, with two doses), and also double the protection against Alpha and Delta. To model a third vaccine dose, we add a new population V3M that has received three vaccine shots.

$$\frac{d{V}_{3}^{M}}{dt}={\omega }_{2}^{M}{V}_{2}^{M}-{V}_{3}^{M}\left(\sum_{X=W,A,D}{\beta }_{V3,M}^{X}\left({I}^{X}+{\alpha }^{X}{A}^{X}\right)\right)-\left({\eta }_{V3}+\mu \right){V}_{3}^{M}$$
(1)

The model predicts that the third vaccine dose, if indeed effective, would be successful in suppressing the spread of Neos; see Fig. 11b1. During 2022, the number of active COVID-19 cases steadily decreases, driven primarily by the decline in Alpha and Delta infections, even as Neos infection numbers remain relatively stable.

While neither NPI or a vaccine booster alone can rapidly eradicate Neos, implemented together that goal can be accomplished. A model that combines the two measures predicts that the number of symptomatic COVID-19 cases would decrease throughout the simulated period to fewer than 200 cases in mid 2022; see Fig. 11c1.

How effective are these measures in a region with 60% AstraZeneca vaccines? Model simulations predict that with NPI alone, there would still be widespread Neos infections, due to Neos’ higher vaccine breakthrough with the AstraZeneca vaccines (Fig. 11a2). With the dissemination of a third vaccine dose to increase vaccine protection, the number of Neos infections continues to increase, albeit at a slower rate, and the total symptomatic infections finally begin to decline by mid 2022 (Fig. 11b2), after exceeding 0.1% of the total population. These results suggest that neither NPI nor a vaccine booster would prevent the healthcare system from being overwhelmed. Implemented together, these two measures are sufficiently efficient to bring the total number of symptomatic COVID-19 infections to < 1000 by mid 2022 (Fig. 11c2).

## Discussion

The principal objective of the present study is to better understand the spread of the VOC of SARS-CoV-2, and how effective different interventions may be. To achieve that objective, we developed an SV2(AIR)3 model that represents factors key to the spread of COVID-19: (i) asymptomatic and symptomatic infections21,22, (ii) simultaneous spread of multiple VOC, (iii) two-dose vaccinations with variable dosing intervals23, (iv) two vaccines with different efficacy, and (v) the effects of NPI24. Main findings of this study include:

1. 1.

It is well understood that for any infectious disease, the major factors that determine how fast it spreads include infectivity, the extent of the NPI, and vaccination rate. For COVID-19, our sensitivity analysis reveals two additional factors, the prevalence of asymptomatic infections and the enhanced infectivity of asymptomatic patients (Fig. 4).

2. 2.

The present model is the first in capturing VOC interactions at a population level. Individuals having recovered from an infection by one strain acquire partial immunity against other strains, and if infected again, are more likely to develop an asymptomatic infection. Model simulations indicate that some VOC have an inhibitory effect on the spread of others (Figs. 5 and 6).

3. 3.

While the Delta strain is undoubtedly wreaking havoc globally, severity differs among countries. Our analysis suggests that how fast Delta spreads depends not only on the NPI and on the fraction of the population who are vaccinated, but also on the types of the vaccines distributed (Fig. 7).

4. 4.

Vaccinated individuals have a significant chance of suffering a vaccine breakthrough with Delta. Model analysis points to a significant role of the vaccinated individuals in the spread of Delta, particularly in a community where most of the vaccines are AstraZeneca (Fig. 8).

5. 5.

Given its current NPI and relatively successful vaccination rollout, is Ontario prepared for the emergence of a more dangerous VOC? A VOC that is more infectious, causes more asymptomatic infections, and has a higher vaccine breakthrough rate? Unfortunately, our model simulation suggests that the answer is no. Without additional interventions, such a VOC would infect many more and likely overwhelm the health care system (Fig. 9). The situation would likely be worse in another region that has a lower rate of vaccination or less effective vaccines (Fig. 10).

6. 6.

To stop such a dangerous VOC, one may need both simultaneous and rapid deployment of pharmaceutical and NPI (Fig. 11).

Results of this study point to the importance of sustained vigilance against SARS-CoV-2. As previously noted, a significant number of the Delta infections are caused by vaccinated individuals with asymptomatic infections. Consequently, even with > 70% of the Ontario population have received at least one vaccine dose, some NPI must remain to prevent the further spread of Delta, or the emergence and rapid spread of the next series of more dangerous VOC. When in-person classes resume in public schools and college in September, any Delta outbreak and the emergence of new VOC must be closely monitored, and if necessary, NPI must be reinstated. And for these NPI to be effective, they must begin sufficiently early.

The news that a third (or fourth) dose of the Pfizer vaccine enhances one’s protection against VOC is exciting25. Indeed, the combination of NPI and a booster shot may be necessary in our fight against a future, more transmissible and deadlier VOC. That said, given that much of the world have inadequate access to vaccines, the deployment of a third vaccine dose in the developed countries raises some moral questions.

### Limitations of the study

The SV2(AIR)3 model separates the population in terms of their health and vaccination status: susceptible, vaccinated (one dose or two, with Pfizer or AstraZeneca), infected (asymptomatic or symptomatic, infected by different VOC), and recovered. Each subpopulation is assumed to be homogeneous. One major limitation of the model is its lack of age structure. COVID-19 infection and mortality rates are known to exhibit distinct age and, to a lesser extent, gender specificity26,27,28, and those demographics characteristics differ among the VOC29. Also, age and gender determine one’s social behaviors, which would affect their susceptibility, or ability to infect others if they are sick. The impacts of NPI (e.g., school closure) would also differentially impact different age groups. The severity of COVID-19 sequela depends also on the infected individual’s health30,31,32. For more realistic simulations of COVID-19 dynamics, age, gender, and health specificity of COVID-19 infections should be incorporated into the present model by separating each subpopulation into key age groups, genders, and health conditions.

Furthermore, spatial heterogeneity likely has a significant impact on the spread of infections. As can be seen in the geographical heterogeneity in the number of COVID-19 cases in Ontario, socio-economic factors also play an important role in the local spread of COVID-19. These factors include the number of people per household, job locations, frequency of public transportation usage, regional vaccination rates, and social connectedness, etc. Different socio-economic groups are likely affected differently by the NPI. Thus, a worthwhile extension would be to divide Ontario into regions with distinct demographics, represent the spread of COVID-19 within each region using an incarnation of the SV2(AIR)3 model, and then loosely connect these sub-models based on the estimated amounts of communications mediated by job or social travels.

The present model is formulated using data from Ontario, Canada (Ontario COVID-19 Science Advisory Table). Geographical and socio-economic parameters can be refitted to simulate VOC dynamics in a different province in Canada or a different country (e.g., infectivity β, vaccine type ε, COVID-19 mortality rate μX, NPI λ). These models can be a valuable decision-support tool for public health.

## Methods

### Computational model description

The SV2(AIR)3 model simulates the spread of three SARS-CoV-2 strains in Ontario: wild type, Alpha, and Delta. Population in the province is divided into several classes based on their health and vaccination status. Two “susceptible” populations are tracked: the original susceptible individuals (denoted ‘S’), who are unvaccinated and have never been infected, and those who become susceptible again after losing their immunity from either vaccination or infection (denoted ‘SVR’; more about loss of immunity below). The original susceptible population (S, but not SVR) vaccinate at a rate of ω1 and enter the V1M class. Two vaccine types are represented, Pfizer-BioNTech (M = PZ) and AstraZeneca (M = AZ). Pfizer-BioNTech, which we refer to as “Pfizer,” also represents the Moderna vaccines. We denote the fraction of Pfizer vaccines by εPZ and the remainder εAZ = 1−εPZ are AstraZeneca. Those who received the first dose will receive their second dose and enter the V2M class at rate of ω2M, where M denotes either PZ or AZ, as dosing interval differs between vaccine types. Mixed vaccination is not represented. Individuals vaccinated with one or two doses lose their immunity at a rate of ηV1 and ηV2, respectively, and enter the SVR class. We assume that those who lost the immunity did so unwittingly so would not get vaccinated again.

$$\frac{dS}{dt}=-S\left(\sum_{X=W,A,D}{\beta }^{X}{I}_{\text{total}}^{X}+{\omega }_{1}\right)+\mu \left(N-S\right)$$
(2)
$$\frac{d{S}_{VR}}{dt}={\eta }_{V1}\sum_{M=PZ,AZ}{V}_{1}^{M}+{\eta }_{V2}\sum_{M=PZ,AZ}{V}_{2}^{M}+{\eta }_{R}\left({R}^{F}+\sum_{X=W,A,D}{R}^{X}\right)-{S}_{VR}\sum_{X=W,A,D}{\beta }^{X}{I}_{\text{total}}^{X}-\mu {S}_{VR}$$
(3)
$$\frac{d{V}_{1}^{M}}{dt}={\varepsilon }_{M}{\omega }_{1}\left(S+\sum_{X=W,A,D}{A}^{X}\right)-{V}_{1}^{M}\sum_{X=W,A,D}{\beta }_{V1,M}^{X}{I}_{\text{total}}^{X}-\left({\omega }_{2}^{M}+{\eta }_{V1}+\mu \right){V}_{1}^{M}$$
(4)
$$\frac{d{V}_{2}^{M}}{dt}={\omega }_{2}^{M}{V}_{1}^{M}-{V}_{2}^{M}\left(\sum_{X=W,A,D}{\beta }_{V2,M}^{X}{I}_{\text{total}}^{X}\right)-\left({\eta }_{V2}+\mu \right){V}_{2}^{M}$$
(5)

where $${I}_{\text{total}}^{X}={I}^{X}+{I}_{V}^{X}+{I}_{R}^{X}+{\alpha }^{X}\left({A}^{X}+{A}_{R}^{X}\right)$$. Except for the symptomatic patients (see below), all die naturally at a rate of μ. Births are added to S at a rate of μ.

Both susceptible classes (S and SVR) become infected by one of the viruses at rate βX, potentially by coming in contact with a symptomatic infected individual (denoted IX), where superscript ‘X’ denotes one of the SARS-CoV-2 viruses: ‘W’ for wild-type, ‘A’ for Alpha, and ‘D’ for Delta. An infection can also be caused by an asymptomatic infected individual (denoted AX), with a higher rate of αXβX, where αX > 1 with the assumption that asymptomatic individuals are more socially active. We assume that for a virus type X, a fraction σX of the infections are asymptomatic. Symptomatic infection is associated with a mortality rate (μX) higher than natural death (μ), but asymptomatic infection does not increase one’s mortality rate. Vaccinated individuals may be infected as well, at a reduced rate of βXVM (< βX), where V is either V1 or V2, and M denotes the vaccine type. Asymptomatic infections of vaccinated individuals enter AX class, same as infections of unvaccinated individuals. In contrast, symptomatic infections of vaccinated individuals are assumed to have a better survival rate (with a mortality rate μXV < μX), and are represented as a separate infected class IXV. Unvaccinated asymptomatic patients vaccinate at the same rate as the susceptible population (ω1).

$$\frac{d{I}^{X}}{dt}=\left({1-\sigma }^{X}\right)\left(S+{S}_{VR}\right)\left(\sum_{X=W,A,D}{\beta }^{X}{I}_{\text{total}}^{X}\right)-\left({\gamma }^{X}+{\mu }^{X}\right){I}^{X}$$
(6)
$$\frac{d{I}_{V}^{X}}{dt}=\left({1-\sigma }^{X}\right)\left(\sum_{M=PZ,AZ}\left({V}_{1}^{M}\sum_{X=W,A,D}{\beta }_{V1,M}^{X}{I}_{\text{total}}^{X}\right)+\sum_{M=PZ,AZ}\left({V}_{2}^{M}\sum_{X=W,A,D}{\beta }_{V2,M}^{X}{I}_{\text{total}}^{X}\right)\right)-\left({\gamma }^{X}+{\mu }_{V}^{X}\right){I}_{V}^{X}$$
(7)
$$\frac{d{I}_{R}^{X}}{dt}=\left({1-\sigma }^{X}\right){\beta }_{R}^{X}{I}_{\text{total}}^{X}\left(\sum_{Y\ne X}{R}^{Y}\right)-\left({\gamma }^{X}+{\mu }_{V}^{X}\right){I}_{R}^{X}$$
(8)
$$\frac{d{A}^{X}}{dt}={\sigma }^{X}\left(S+{S}_{VR}\right)\left(\sum_{X=W,A,D}{\beta }^{X}{I}_{\text{total}}^{X}\right)+{\sigma }_{V}^{X}\left(\sum_{X=W,A,D}\left({\beta }_{V1}^{X}{V}_{1}+{\beta }_{V2}^{X}{V}_{2}\right){I}_{\text{total}}^{X}\right)-\left({{\omega }_{1}+\gamma }^{X}+\mu \right){A}^{X}$$
(9)
$$\frac{d{A}_{R}^{X}}{dt}={\sigma }^{X}{\beta }_{R}^{X}{I}_{\text{total}}^{X}\sum_{Y\ne X}{R}^{Y}-\left({\gamma }^{X}+\mu \right){A}_{R}^{X}$$
(10)

The model assumes that an individuated infected by one variant cannot be infected by a second variant. Infected individuals recover and enter class RX at a rate of γX. Recovered individuals are entirely protected from the variant that they were infected with, but may be infected by a different variant at rate βYR, where Y ≠ X, and βYR < βY to represent some degree of protection. Infected recovered individuals are denoted IXR, which like IXV are associated with a better survival rate (μXR < μX). Recovered individuals lose their partial immunity and enter SVR at a rate of ηR. Four recovered classes are represented, one for each virus type (RW, RA, RD) and also RF who has recovered from two infections. For simplicity, we assume that RF is fully immune to all three viruses, until they lose their immunity and become fully susceptible.

$$\frac{d{R}^{X}}{dt}={\gamma }^{X}\left({I}^{X}+{I}_{V}^{X}+{A}^{X}\right)-{R}^{X}\left(\sum_{Y\ne X}{\beta }_{R}^{Y}{I}_{\text{total}}^{Y}\right)-\left({\eta }_{R}+\mu \right){R}^{X}$$
(11)
$$\frac{d{R}^{F}}{dt}=\sum_{X=W,A,D}{\gamma }^{X}\left({I}_{R}^{X}+{A}_{R}^{X}\right)-\left({\eta }_{R}+\mu \right){R}^{F}$$
(12)

### Model parameters

The SV2(AIR)3 model involves parameters that describe the clinical characteristics of SARS-CoV-2 VOC, which were estimated from published studies. Other parameters describe the demographics and social behaviors of the Ontario population, which were estimated from published provincial statistics. Parameters were then fine-tuned manually so that model predictions approximate the published number of new infections and deaths (Ontario COVID-19 Science Advisory Table). Baseline parameter values are shown in Table S1.

### Simulating NPI

As the number of COVID-19 rose, on March 23 2020 the Ontario government began a number of public health safety measures, which varied in severity over time, including stay-at-home orders, workplace safety measures, restrictions on public events and social gatherings, temporary shutdown of selected businesses, and mandatory mask wearing. These NPI reduce the transmissivity of COVID-19, and are represented by scaling all βX’s and βXV,M’s simultaneously by a NPI severity index λ, where 0 ≤ λ ≤ 1. The time-varying function λ is chosen in part based on Ontario COVID-19 lockdown timeline and Google mobility data, and in part to fit the predicted new case numbers against provincial data available prior to July 2021; the values of λ for 2020–2021 are shown in Fig. S1.