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# The 1876 Centennial Exhibition, Part I

##### Sponsored
Figure 5: Otis Brothers “Universal” elevator, safety drum, 1876

International exhibitions and trade shows are common aspects of the 21st-century commercial landscape, and the frequency with which these events occur easily allows companies – and countries – to maintain a global perspective and awareness of industry trends. The presence and importance of national interests in the development of new technology has been a critical aspect of industrial exhibitions since the first such event was held (the 1851 Crystal Palace Exhibition in London). In the 19th century, the expression of these interests is found in the official reports produced by visiting national delegations. These reports were intended as a means of industrial and cultural education and are important because their subject matter was observed, understood and interpreted from a different national perspective than that of the host country. The report published by the Austrian Commission to the U.S. Centennial Exhibition of 1876 illustrates this process of investigation and interpretation. The Centennial Exhibition, held in Philadelphia, featured the largest collection of American elevator machinery assembled in one location and included exhibits by 19 companies from across the country, as well as exhibits by Canadian and French firms.

The Austrian delegation report, titled Passenger Elevators, Freight Elevators and Mine Hoists was written by Alois Riedler (1850-1936). Riedler was a mechanical engineer and, at the time the report was written, he was on the faculty of the Technical University of Vienna. His detailed report consisted of 81 pages of text, nine line drawings embedded in the text and 13 plates of illustrations that contained 97 individual line drawings. Riedler’s report represents, in essence, the first attempt to write a comprehensive summary of vertical transportation in the U.S. His efforts to be as comprehensive as possible were evidenced by the fact that, in addition to thoroughly examining the machines exhibited in Philadelphia, he also researched and collected drawings of elevators not featured in the exhibition. The latter group included hydraulic elevators designed by Lane & Bodley Co. (Cincinnati), William B. Le Van (Philadelphia) and William E. Hale (New York); and blast-furnace hoists designed by P.L. Weimer & Brothers (Lebanon, Pennsylvania) and Taws & Hartman Machinists and Brass Founders (Philadelphia). Riedler also provided detailed descriptions of mine hoists in Nevada and Colorado and of the elevator designed by Mégy, Echeverria & Bazan (Paris).

The report opened with a general introduction and discussion of what Riedler characterized as the widespread use of elevators in American cities. He claimed that elevators in industrial settings and warehouses were viewed as a necessity due to their use as “labor saving” devices, which were critical aspects of efficient and economical business operations. He also claimed that the presence of elevators in hotels, retail stores and public buildings was so pervasive that stairs were now perceived as a “secondary” means of vertical movement in buildings. He stated that the extensive use of elevators resulted from an American desire for personal comfort and efficient business operations, and the availability of a cheap and dependable power source – steam.

Riedler claimed that “more than 90% of American elevators are operated by steam engines.” The widespread availability of steam power was seen as uniquely American and as contributing to the ease with which elevators could be installed and economically operated. According to Riedler this was possible because “no law in America prohibits the installation of steam boilers, even in the centers of large cities.” He also cited the availability of an ample supply of anthracite coal, which he described as burning “completely smokeless,” as promoting the use of steam power. He reported that these circumstances were “distinctly American” and implied that they would have been “impossible” to replicate in Austria. In other words, differing national resources and building laws had a direct impact on the application and availability of elevator technology. The introduction concluded with a general overview of technical details and elevator operation. Perhaps the most interesting aspect of this summary is the references to the apparently common use of electric signal systems to call elevators to a given floor.

Riedler began his detailed account of elevator systems with a discussion of the machines displayed by Otis Brothers, who offered visitors a large exhibit that consisted of a “complete collection” of their steam-powered elevators. The machines described in the report included the “New York” safety passenger elevator, the “Metropolitan” elevator and the “Universal” elevator. The “New York” was designed solely for passenger use and equipped with Otis Brothers’ reversible double-cylinder steam engine (located in the cellar or basement), safety drum (located at the top of the shaft) and numerous patented safety devices, including a version of Elisha Graves Otis’ 1861 ratchet safety (Figures 1 and 2). One of the most interesting drawings published in the report was a section that depicted a complete elevator installation (Figure 3). This drawing is one of the few published images that provided a clear understanding of how these machines were typically placed in a building (from basement to attic). Riedler provided a comprehensive report of the entire system, which included such details as the fact that the shape of the counterweights depended on their location: “in most cases,” the counterweights were composed of flat cast-iron plates; however, when they were required to run inside enclosed chases, they employed a cylindrical profile.

The report included a table of technical data and prices, which appears to have been (in part) a translation of information found in a typical Otis Brothers sales brochure (Table 1). The table described the volume of space occupied by the steam engine (length, width and height), typical shaft dimensions (ranging 1.52-2.26 m2 in size), the prices of the various models shipped from Otis Brothers’ factory in Yonkers, New York, and the prices of these same elevators installed by Otis in or near New York City (NYC). The price for the installed system included the steam engine, safety drum, counterweights, ropes, car and all wages associated with the complete assembly “up to the starting of the elevator.” While, perhaps logically, the price excluded the cost of the steam boiler and associated piping, it is of interest that the price also excluded all elevator shaft work (guide rails, doors, etc.). The two sets of prices (shipped versus installed) allow a rough calculation of typical installation costs, which equaled approximately 13.5% of the total cost of an installed system. The prices were also predicated on a total run that did not exceed 25 m. For longer installations, Otis added an additional cost that varied depending on the model number. (This figure was not supplied for the “New York” elevator; for the “Metropolitan” elevator, this cost varied from US$3 to US$6 per foot). Riedler also reported that systems that utilized a “simple car design” would have a US$250-500 price reduction. The “New York” elevator’s typical operating speed was 30-60 mpm. The description of Otis’ “Metropolitan” elevator was less comprehensive than that provided for the “New York”: because it included the same safeties, steam-engine type, etc. used in the previous system, Riedler doubtless felt it was unnecessary to repeat this information. Riedler also chose not to include illustrations of this system in his report. Although the “Metropolitan” was described as having been designed for either passenger or freight use, its base price included an open platform instead of an enclosed car, a clear indication that its primary use was intended as a freight elevator. A table of technical data and prices for this elevator was also provided (Table 2). Apparently, it could be purchased without the safety drum, and the corresponding price reductions were included in the table. This is somewhat surprising given Otis’ reputation for safety. Riedler also implied that these machines typically shipped with only one hoisting rope. If a second rope was desired, there was an additional cost of US$400-600. The base price of the “Metropolitan” was predicated on a total run height that did not exceed 30 m.

“Riedler’s report represents, in essence, the first attempt to write a comprehensive summary of vertical transportation in the U.S.”

Riedler’s description of the “Universal” elevator included the same high level of detail and number of illustrations that accompanied his examination of the “New York” elevator. This was because this system was designed solely as a freight elevator and was a belt- rather than steam-powered machine. The use of two leather belts to power the elevator “engine” meant that it followed the paradigm of factory machinery involving a system of continuously rotating leather belts attached to pulleys on line shafting that extended throughout the building. The illustrations included a section that depicted the complete elevator installation (which provided a clear understanding of how these machines were typically placed in a building), as well as detailed drawings of the belt-powered engine and safety-drum system (Figures 4-6). Riedler once again provided a detailed table of prices and technical data on the elevator’s various configurations (Table 3). The information found in this table included standard platform sizes, costs for platforms accessible on either four or two sides, and packing charges. Larger platforms could be ordered with an additional US\$32-43-per-m2 cost. The fact that no installation costs were provided indicates these machines were typically installed by local mechanics at the factory site.

Part two of this article will continue this examination of the American elevator industry in 1876 as seen through the lens of Riedler’s report for the Austrian delegation. Its focus will be on hydraulic elevators.

### Lee Freeland

Elevator World Senior Associate Editor

Flipbook

Flipbook


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# Gravitational Force HELP

1. Jun 18, 2007

1. The problem statement, all variables and given/known data

Three uniform spheres are located at the corners of an equilateral triangle. Each side of the triangle has a length of 3.80 m. Two of the spheres have a mass of 3.30 kg each. The third sphere (mass unknown) is released from rest. Considering only the gravitational forces that the spheres exert on each other, what is the magnitude of the initial acceleration of the third sphere?

2. Relevant equations

F=Gm1m2/d^2
F=ma

3. The attempt at a solution

I tried doing it by setting the 2 above equations = to each and so I got:

G(distance b/w 2 spheres)/d^2=a

and I don't know if I'm almost there or not....

2. Jun 19, 2007

### G01

Yes, you are eventually going to have to set two force equations equal to each other, but you have to remember there is a force from EACH of the other spheres, meaning two forces in total.

Try starting here:

Remember that force is a vector.
What do you know about the X components of the two forces on the 3rd sphere?

After you can answer these, you should be able to find the sum of the forces and then like you did above, remember that the sum of the forces is:

$$\Sigma \vec F=m\vec a$$

See how far you can get now. Good Luck!

Last edited: Jun 19, 2007
3. Jun 19, 2007

format: plain


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# Math Overflow Recent Questions

## Primary tabs

most recent 30 from mathoverflow.net 2019-05-17T22:14:05Z

### Does an asymptotic component with large size in a minimal subshift always exist?

Wed, 05/15/2019 - 22:00

Let $(X, T)$ be a minimal subshift, i.e. $X$ is a closed $T$-invariant subset of $A^\mathbb{Z}$, where $T$ is the shift. A pair $x,y\in X$ is asymptotic if $d(T^nx, T^ny)$ goes to zero as $n\to\infty$. Always exists such a pair when $X$ is infinite: for every $n\geq1$ there exists $x^{(n)}, y^{(n)} \in X$ such that $x^{(n)}_0\not= y^{(n)}_0$ and $x^{(n)}_{[1,N]}= y^{(n)}_{[1,N]}$ (if not, for some $n$, $x_{[1,n]} = y_{[1,n]}$ implies $x_0=y_0$, i.e., $x_{[1,n]}$ determines $x_0$, and this forces $X$ to be periodic), and any pair of convergent subsequences of $x^{(n)}$ and $y^{(n)}$ will do the trick.

My question is: do exist $k$-tuples of asymptotic points, for every $k\geq1$? More precisely, is it true that for every $k\geq1$ there exists $x_1,\dots,x_k\in X$ such that $$\lim_{n\to\infty}d(T^nx_i, T^nx_j) = 0\ \forall i,j$$

### Residually finite-dimensional $C^*$-algebra

Wed, 05/15/2019 - 17:51

Suppose $A$ is a non-unital residually finite-dimensional (RFD) $C^*$-algebra, then the multiplier algebra $M(A)$ is also RFD. I wonder whether there exists a trace on the corona algebra $M(A)/A$?

### Regular, algebraic and cuspidal for automorphic representations

Wed, 05/15/2019 - 07:25

Given an automorphic representation of the general linear group over a totally real number field, there are 3 properties it might have: regular, algebraic (or "of algebraic type"?) and cuspidal. There are $2^3=8$ possible combinations of which properties are true for a given representation. Can you give explicit examples for each of the 8 combinations?

Can you give examples for the general linear group over $\mathbb{Q}$? If yes, what is the minimum $n$ (as in $\mathrm{GL}_n$) such that all combinations arise?

### Edge coloring a cycle plus triangles graph

Wed, 05/15/2019 - 03:14

Let $G$ be a simple “cycle plus triangles” graph, that is, a graph with $3k$ vertices, $k>1$, the edges of which can be partitioned into a set that induces a $3k$-circuit, together with sets that induce disjoint triangles (3-circuits). Note $G$ is 4-regular.

UPDATED QUESTION (2019 may 17):

QUESTION: Is $G$ class 1, that is, can it be edge-4-colored, if it has an even number of triangles?

(SIDE) FACT : If $G$ has an odd number of triangles then it is class 2. This is because a cycle plus triangles graph with an odd number of triangles is a regular graph of odd order ... all such graphs are class 2 (suppose it is class 1 ... then each vertex is incident with an edge of each color. But the edges of a given color are independent, so they are incident with an even number of vertices, while there is an odd number of vertices, contradiction).

The hypothesis that C+T graphs which have an even number of triangles are class 1 is supported by computer experiments.

Cycle plus triangles graphs became well known when Erdös posed the “cycle plus triangles problem” (whether such graphs are vertex-3-colorable). This was solved affirmatively by Fleishner and Stiebitz using the Alon-Tarsi theorem, and later Sachs, inductively.

Perhaps the answer to my question above is known, but I am unaware of it.

In the generalization of this question to “cycle plus even $k$-cliques” (I hope it is clear what this means) the answer is that they are all class 1, as you can edge color the edges of the $k$-cliques with $k$-1 colors, and use two remaining colors for the edges of the even cycle in the ($k$+1)-regular graph. It is tantalizing to speculate that a similar answer as the answer for the “cycles plus triangles” case would hold for the general “cycle plus odd cliques” situation, of which the cycle plus triangles is a special case. It would be nice to settle at least this special case with proofs, if it is not settled already.

### Finding the distance between two beads connected by a spring

Tue, 05/14/2019 - 15:12

Imagine two beads each on parallel rods. The beads are connected to each other by a spring of stiffness $k$. The beads each exert an equal and opposite force $F$ to each other. The distance between the rods is $l$. If the beads are initially both at height zero and released they will begin to move in opposite directions before eventually coming to an equilibrium position.

The equilibrium position is the point at which the vertical component of the tension in the spring is equal to the force exerted by the beads.

There is a non-linear relationship between the force exerted by the beads and the final vertical distance between them. The same goes for the final angle.

If either the height or the angle between the beads at equilibrium was known the problem would be straight forward, but they are not known.

The Physics of Springs provides a method to solve more complex spring systems as long as the angle between the springs is assumed fixed as this allows a linear relation between distance and force. However, the angle is not fixed in this case.

This problem can be solved using numerical methods but takes time to converge and for a single node pair isn't worth it.

My question is: What analytical method can I use to find the distance between the two beads?

Note: This question was asked on Physics overflow but was closed and deleted as being an off-topic/homework. It is also similar to this question, which was also closed. I would argue however that on the basis that for more than two nodes the system may not have an analytical solution, the analytical solution to the special case of 2 nodes is of general interest and non-trivial due to the non-linear relationship. The analytical solution of the two-node case speeds up finding the equilibria position of many network configurations.

### Help with studying for upcoming exam [on hold]

Tue, 05/14/2019 - 14:41

can i get some help with solving those questions? i am not good at math and got an exam coming up so i am solving older exams as a reference but i don't have the answers for it to be sure that i solved it correctly. thanks in advance!

questions are;
mathematical induction
binomial expansion
partial fraction
Taylor's expansion
integration
differentiation
newton forward formula
matrix
Fourier series expansion
examples

### Positivity and zeros of Heun's function

Tue, 05/14/2019 - 14:21

I am interested in understanding where in the complex plane a Heun function might vanish, or where it (or its real part) is positive. I have looked in the book of Ronveaux, as well as the NIST Handbook of mathematical functions and could not find any information on this problem. Is there anything known in that direction? I could find some information on where this happens for a hypergeometric function.

### How does $E$ closed follow from the upper semicontinuity of the spectrum?

Tue, 05/14/2019 - 14:10

Let $f$ be an analytic function for a domain $D$ of $\mathbb{C}$ into a Banach algebra $A$. Suppose that, for all $\lambda \in D$, $\text{Sp}f(\lambda)$ is finite or a sequence converging to $0$. Suppose that $\mu \neq 0$ and $\mu \in \text{Sp}f(\lambda_0)$ for some $\lambda_0 \in D$.

Consider the set $E = \{ \lambda \in D: \mu \in \text{Sp}f(\lambda) \}$.

B. Aupetit mentions in a proof he writes for Theorem 3.4.26 in his book A Primer on Spectral Theory, that this set $E$ is closed by the upper semicontinuity of the spectrum. I am struggling to see why this is true.

Can anyone please point me in the right direction as to how I can show that $E$ is closed by the upper semicontinuity of the spectrum?

### Constrained optimization of sum of squares polynomials

Tue, 05/14/2019 - 13:17

Consider the problem $$\min p(x) \text{ subject to } g_j(x)\le 0 \quad p,g_j\in\text{SOS}, \qquad (*)$$

i.e. $p,g_j$ ($j=1,\ldots,m$) are sum of squares (SOS) polynomials. Can this problem be solved efficiently?

The paper [1] shows that unconstrained minimization of SOS polynomials can be reduced to a convex program. Surprisingly, the paper goes onto consider general (non-SOS) constrained polynomial optimization to derive the Lasserre hierarchy, but never explicitly discusses the special case $(*)$ above.

[1] Lasserre, Jean B., Global optimization with polynomials and the problem of moments, SIAM J. Optim. 11, No. 3, 796-817 (2001). ZBL1010.90061.

### How long before we spend our entire lives in grad school? [on hold]

Tue, 05/14/2019 - 13:11

In every field of pure mathematics, there are several layers of abstraction a person must absorb before they can contribute anything of note.

To absorb each layer a person needs to spend some effort (and thus some time). Say, for quite a few people, the first time they see the definition of a topological manifold it feels "weird" or somehow unintuitive. After they do some exercises, they typically understand that the definition is actually fairly natural and captures what it is supposed to capture reasonably well.

For example, in arithmetic geometry you need to understand the definition of a scheme (takes quite some time, doesn't it), a site & a topos, also the standard package of homological algebra and some other stuff too. I think an average person who decides to go into this field learns this stuff by the time they are 24-25 years old.

The life expectancy of a human being is not infinite, however. I think it is not significantly above 90 years in any given country, as of May 2019. From what I know, after we max out the advancements in medicine, the life expectancy would be 120 years (assuming we do not use replacement organs en masse).

Was there any research making an educated guess about how much time is left before a sufficient number of layers of abstraction accumulate so that a person would have to spend their entire life in the graduate school to learn all of them (and then die, having no time left to apply them to anything)?

### Effects of the first algebraic K-theory on the higher algebraic K-theory

Tue, 05/14/2019 - 13:06

Is there any counterexamples known to the following statements? ($A$ a commutative algebra)

If $K_1(A)$ is finitely generated then $K_n(A)$ is finitely generated for any $n\geq 2$

If $K_1(A)$ is torsion then $K_n(A)$ is torsion for any $n\geq 2$

### Representation of $4\times4$ matrices in the form of $\sum B_i\otimes C_i$

Tue, 05/14/2019 - 13:05

Every matrix $A\in M_4(\mathbb{R})$ can be represented in the form of $$A=\sum_{i=1}^{n(A)} B_i\otimes C_i$$ for $B_i,C_i\in M_2(\mathbb{R})$.

What is the least uniform upper bound $M$ for such $n(A)$?In the other words, what is the least integer $M$ such that every $A$ admit such a representation with $n(A)\leq M$?

Is this least upper bound equal to the corresponding least upper bound for all matrices $A$ which are a matrix representation of quaternions $a+bi+cj+dk$?

As another question about tensor product representation: What is a sufficients condition for a $4\times 4$ matrix $A$ to be represented in the form of $A=B\otimes C -C\otimes B$?

### $L^2$ norm of fractional Laplacian

Tue, 05/14/2019 - 12:23

Is it possible to calculate the norm associated to fractional Laplacian of $u$ and $s\in (0, 1).$

$$\|(-\Delta)^{s/2} u\|_2^2=\int_{\mathbb R^N}|(-\Delta)^{s/2} u|^2dx=C_{N,s}\int_{\mathbb R^{2N}} \frac{(u(x)-u(y))^2}{|x-y|^{N+2s}}dxdy$$

For example, $u(x)=(1+ |x|^2)^{-\frac{N-2s}{2}},$ can we calculate $\|(-\Delta)^{s/2} u\|_2$ just by using the above formula. I am aware $u$ is the some type of solution to the fractional Yamabe problem but I want a result independent proof. Any reference is welcome.

### Certain urn models

Tue, 05/14/2019 - 12:06

This is my first question here - so if i'm out of format - let me know :)

The problem is this:

• $m$ identical urns

• $k$ identical cells in each urn

• $n$ identical balls

The balls are distributed among the urns, with no restriction other than the maximal capacity of each urn (up to $k$ balls - one in each cell). The question at hand is: What is the probability that exactly $t$ urns are occupied by $r$ balls.

I thought of approaching it the same way Fang does (here: https://www.researchgate.net/publication/305977113_A_restricted_occupancy_problem): $P(M_t=r)=N(m,k,n,r,t)/N(m,k,n)$

Where $N(m,k,n,r,t)$ is the number of ways of distributing the $n$ balls into the $m$ urns, such that exactly $r$ of them have exactly $t$ balls.

And where $N(m,k,n)$ is the number of ways of distributing the balls into the urns.

The latter is simple to calculatr (if I'm not mistaken: $N(m,k,n)=mk(mk-1)\cdot...\cdot(mk-n+1)$), but the former seems quite hard. I have trouble using the generating functions to solve $N(m,k,n,r,t)$, because I can't find the independent variables to use (so that the generating functions can be multiplied).

I'd love it if someone could explain clearly how he would count $N(m,k,n,r,t)$. Thanks in advance:)

### Does this ideal in $B(L_1)$ have a (bounded) right approximate identity?

Tue, 05/14/2019 - 11:18

I will take a roundabout way to defining this ideal, because (a) this route is how my collaborators and I came to it (b) this alternative definition, rather than the standard one, may suggest a direct attack on the question stated in the title.

Notational conventions: $L_p$ is short-hand for $L_p([0,1])$ with the Lebesgue measure on $[0,1]$. If $E$ is a Banach space then $L_\infty(E)$ denotes the space of essentially bounded, strongly measurable functions $[0,1]\to E$ (modulo equivalence a.e.).

In the case where $E=L_1$, we shall regard elements of the space $A=L_\infty(L_1)$ as functions of two variables which are $L_\infty$ in the second variable and $L_1$ in the first variable. So here the norm would be given by $$\Vert f \Vert = \operatorname{ess.sup}_{t\in [0,1]} \Vert f( \cdot, t)\Vert_1$$ This convention has the advantage that given $f\in A$ and $\xi \in L_1$ we can define $T_f(\xi) \in L_1$ by $$T_f(\xi)(s) = \int_0^1 f(s,y)\xi(y)\,dy$$ The map $f\mapsto T_f$ is an isometric embedding of $A$ as a closed subalgebra of $B(L_1)$. For given $f,g\in A$, we may define $$(f\bullet g)(s,t) = \int_0^1 f(s,x) g(x,t)\,dt$$ Then $f\bullet g\in A$ and $T_fT_g = T_{f\bullet g}$. The image of $A$ in $B(L_1)$ under this embedding, which we will denote by $J$, turns out to be a much studied object in the theory of operators on $L_1$: it is the set of representable operators from $L_1$ to itself.

QUESTION: does $J$ (or equivalently $(A,\bullet)$) have a bounded right approximate identity? What if we drop the requirement of boundedness?

Note that the naive attempt of taking simple functions in $L_\infty(L_1)$ that approximate the "Dirac" measure concentrated on the diagonal in $[0,1]^2$ won't work, because such functions get sent by our embedding to elements of $K(L_1)$, which is properly contained in $J$ (see below).

Some remarks for background context, which might be relevant to a solution.

It can be shown that $J$ has no left approximate identity (bounded or otherwise). I would like to thank W. B. Johnson for indicating why this is the case; an expanded and paraphrased version of his explanation is given below.

From some vector measure theory, we know that $J$ contains the ideal $W(L_1)$ of weakly compact operators. Moreover, $J$ is contained in the ideal $CC(L_1)$ of completely continuous operators; "completely continuous" means that weakly convergent sequences are mapped to norm convergent sequences).

(The containment $J\subseteq CC(L_1)$ follows from a theorem of Lewis and Stegall, which characterizes operators in $J$ as those which factor through $\ell_1$. This also shows that $J$ is a $2$-sided ideal in $B(L_1)$, not just a subalgebra.)

It follows that if $S\in W(L_1)$ and $T\in J$, then $TS\in K(L_1)$. Since there exist weakly compact operators $S$ on $L_1$ which are not compact (e.g. take any non-compact map $L_1\to \ell_2$ and then compose with an isometric embedding $\ell_2\to L_1$), it follows that there is no net $(T_\alpha)$ in $J$ such that $\Vert T_\alpha S - S \Vert \to 0$.

### Given three circles with non-empty two-way intersections, what is the condition that their three-way intersection is empty?

Tue, 05/14/2019 - 11:08

Given the center coordinates $\{x_i,y_i\}$ and radii ($r_i$) of three circles ($i=1,2,3$) (the areas of which are certain probabilities), the three pairwise intersections being all non-empty, what is the condition that their three-way intersection is, on the other hand, empty?

In particular, I have in mind three circles $A,B,C$ of radii $\frac{8 \pi}{27 \sqrt{3}}$ and $\frac{1}{6}$ and $\frac{1}{6}$. The intersection of $B$ and $C$ is $\frac{1}{9}$ and $A \land B$ and $A \land C$ are both $-\frac{4}{9}+\frac{4 \pi }{27 \sqrt{3}}+\frac{\log (3)}{6} \approx 0.00736862$, while as indicated, $A \land B \land C$ is $\varnothing$.

This problem pertains to my efforts (https://mathematica.stackexchange.com/questions/198019/create-a-venn-diagram-showing-the-relations-of-three-sets-of-quantum-states/198091#198091) to represent--via a Venn or related diagram--the Hilbert-Schmidt probabilities of certain quantum ("positive-partial-transpose" and "bound-entangled") "two-qutrit" states (representable by $9 \times 9$ density matrices).

So, how to locate the three centers--subject to the three circles all being contained in a circle (arbitrarily centered at the origin) of area/probability 1--is also in question (https://math.stackexchange.com/questions/3224845/given-a-circle-a-of-area-1-centered-at-0-0-give-conditions-that-another).

I also how further information as to the probabilities assigned to various unions and intersections of $A,B,C$ and their negations--which might be helpful.

### Turn a coalgebra to an algebra by passing to right adjoint

Tue, 05/14/2019 - 11:00

Suppose I have a coalgebra $\mathcal{C}$ in the $\infty$-category of presentable stable $\infty$-categories (with continuous functors). If the structure functors for $\mathcal{C}$ all have continuous right adjoints, then passing to right adjoints I get an algebra structure on $\mathcal{C}$. However, in the general situation, I only have a full stable subcategory $\mathcal{D}$ of $\mathcal{C}$ such that the right adjoints preserve $\mathcal{D}$, i.e. they induce $\mathcal{D}\otimes \mathcal{D}\rightarrow\mathcal{D}$, and restricting to $\mathcal{D}$ they are continuous. Note that $\mathcal{D}$ is not necessarily a sub-coalgebra of $\mathcal{C}$.

Is there a clean way to show that in the above situation, $\mathcal{D}$ naturally induces an algebra structure? All I can say is that all the homotopical data needed are contained in the coalgebra structure of $\mathcal{C}$, but this seems not so clean.

### Global solution of nonlinear Schrödinger equation via blow-up argument

Tue, 05/14/2019 - 10:43

Set $u_0\in H^1 (\mathbb{R} ^N)$ and $1<\alpha < \frac{N+2}{N-2}$. I want to show that there exists $\varepsilon > 0$ s.t. if $\Vert u _0 \Vert _ {H^1} < \varepsilon$, then there is global solution (defined for all $t\in\mathbb{R})$ of $$u_t = i (\Delta u + \vert u \vert ^{\alpha -1} u)$$ $$u(\cdot,0)=u_0$$

I have proved the existence of solution (local). Now I want to prove that is in fact global. In order to do that, I thought to use the blow-up alternative, that is, I just have to show that $\Vert u \Vert _{H^1(\mathbb{R}^N)}$ is bounded, where $u$ is a $H^1-$solution. It is clear that $\Vert u \Vert _{H^1(\mathbb{R}^N)} = \Vert u \Vert _{L^2(\mathbb{R}^N)} + \Vert \nabla u \Vert _{L^2\mathbb{R}^N)}$.

By the mass conservation, $\Vert u \Vert _{L^2(\mathbb{R}^N)} = \Vert u _0\Vert _{L^2(\mathbb{R}^N)}$. By the energy conservation, $\frac{1}{2}\Vert \nabla u\Vert_{L^2} ^2 - \frac{1}{\alpha + 1} \Vert u \Vert_{L^{\alpha+1}} ^{\alpha +1} \equiv E\in\mathbb{R}$ therefore, by Sobolev embedding, $$\Vert \nabla u\Vert_{L^2} ^2\leq 2E + \frac{2}{\alpha + 1} \Vert u \Vert_{L^{\alpha+1}} ^{\alpha +1}\leq 2E + \frac{2C}{\alpha + 1} \Vert u \Vert_{H^1} ^{\alpha +1}$$

If I show that $\Vert \nabla u\Vert_{L^2} ^2 \leq 2E + \frac{2C}{\alpha + 1} \Vert u \Vert_{H^1} ^{\alpha +1} \leq \text{constant}$ I finish. I have the assumption that $\Vert u _ 0 \Vert _{H^1}$ is small, does it implies that $\Vert u \Vert _{H^1}$ is also small? As I am reading on Internet, I found something called Lyapunov stability. Maybe is useful here, can anyone explain how could be used in this situation? Thanks in advance. Any idea is welcome!

### Possible research with rigid motions in the plane and space

Tue, 05/14/2019 - 10:23

What are the existing studies and future directions of work to be done with constructing Mobius transformations with sterogrpahic projections from Riemann Sphere to the complex plane, rigid motions taking one Riemann sphere to another, and mapping that second sphere back to the complex plane ?

Even if not, what are some possible directions of research that can be done using Mobius transformations in the plane and rigid motions (or order preserving isometries) in the space ?

Thank you for you time and suggestions.

### Examples of automorphic representations to keep in mind

Tue, 05/14/2019 - 10:13

I have recently started studying the automorphic science and find it somewhat hard to form intuition. Can we have a list of examples of automorphic representations that you usually use to test a new idea, or that are counterexamples to a statement that might look reasonable to a beginner but is actually false? Inspired by this question.


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# Методические указания по английскому языку для студентов 5-го курса исторического факультета. Коныгина Г.И. - 6 стр.

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• ## Иностранный язык

6
equipped with the necessary language related to their needs and problems as learners
which would allow them to take part in the lesson. By practising the phrases and then
insisting on their use, the teacher is increasing the pupils’ opportunities for using the
language communicatively. Even at an elementary level pupils can acquire classroom
phrases as self-contained unchanging unites, e.g
. I’m sorry I’m late; Could you repeat
that; What’s the answer to number 1?
The phrases used to talk about the language
itself and learning it
Can you say that? ; What’s the English for this word? Is there
; etc. , are particularly useful.
In addition, the work includes notes, giving information about certain features of
the British educational practice from a lexical point of view. Some notes contain
comments and remarks related to language use, grammar, vocabulary and
pronunciation. Certain common errors are also listed ( marked
) with their correct
form.
                                             6

equipped with the necessary language related to their needs and problems as learners
which would allow them to take part in the lesson. By practising the phrases and then
insisting on their use, the teacher is increasing the pupils’ opportunities for using the
language communicatively. Even at an elementary level pupils can acquire classroom
phrases as self-contained unchanging unites, e.g. I’m sorry I’m late; Could you repeat
that; What’s the answer to number 1? The phrases used to talk about the language
itself and learning it Can you say that? ; What’s the English for this word? Is there
a corresponding adjective? ; etc. , are particularly useful.
In addition, the work includes notes, giving information about certain features of
the British educational practice from a lexical point of view. Some notes contain
comments and remarks related to language use, grammar, vocabulary and
pronunciation. Certain common errors are also listed ( marked • ) with their correct
form.


text:
# Math Help - function of expectation

1. ## function of expectation

Hello,

I have to show the following assertion:
$\sup\limits_{a \in [0,1]} E[log(1+a(exp(Z)-1))]=0$
I know that
$E[Z]\leq 0$ and $\sup\limits_{a \in [0,1]} E[log(1+a(exp(Z)-1))]\geq 0$
and for $a=1$ we have $\sup\limits_{a \in [0,1]} E[log(1+a(exp(Z)-1))]=E[Z]$

Unfortunately I have no idea how to show it.
Does anyone have a hint?

2. ## Re: function of expectation

Hey Juju.

One idea I have is to use Jensens Inequality since you are dealing with a convex function log(x):

Jensen's inequality - Wikipedia, the free encyclopedia

3. ## Re: function of expectation

Thanks Chiro.

But I don't think that Jensen's inequality helps (or at least I don't know how):

$E[\log(1+\alpha(\exp(Z)-1))]\geq \log E[1+\alpha(\exp(Z)-1)]\geq \log(1+\alpha(\exp(E(Z))-1))\geq 0$

since $(1+\alpha(\exp(E(Z))-1)) \leq 1.$ And that is what I already know.

Or did I anything wrong?

4. ## Re: function of expectation

Doesn't that mean that log of that term is <= 0 though?

5. ## Re: function of expectation

Of course you are right. But anyway I think that I cannot conclude that
$E[\log(1+\alpha(e^Z-1))]=0$

since the inequality does not imply
$E\log(1+\alpha(e^Z-1))\leq 0.$

Am I right?

6. ## Re: function of expectation

It is just an inequality: If you know that the value is zero then it is zero.

The inequality just gives a possible range without knowing any more information: now you can add your other piece of information to reduce the inequality even further.

7. ## Re: function of expectation

It is just an inequality: If you know that the value is zero then it is zero.
?? But I do not know that the value is zero. This is what I want to show.

But I think I can show it quite easily; just using
$E[\log(1+\alpha(e^Z-1))]\leq E[\alpha(e^Z-1)]$

$(\log(1+x))\leq x)$

Thank you.


text:
# Are all solutions to an ordinary differential equation continuous solutions to the corresponding implied differential equation and vice versa? [duplicate]

Regarding the duplicate. Yes, I know the other one has a lot of shared text, but those were just definitions/setup and I was being lazy. The core questions are still different unless you believe derivatives are weak derivatives in which case you might need to read up on them. I don't know the exact definition, but I do know they aren't the same...

Now I have to heavily emphasize the fact that I have never studied differential algebra or the concept of other types of differentiation (which is what I believe is the concept behind a differential algebra). So, if I am abusing the terminology a little bit, please forgive me.

Let us define a differential algebra known as implied differentiation. It actually does not have a unique value. Let us denote the implied derivative operator as $I(f)$, where $f$ is any function being implied differentiated. This is of course, nonstandard terminology.

We define the operator I to be $I(f)(x)(g)$ to be:

$$I(f)(x)(g) := g(x) \left(\lim_{h\to 0^+} \frac{f(x+h)-f(x)}{h} \right) + (1-g(x)) \left(\lim_{h\to 0^-} \frac{f(x+h)-f(x)}{h} \right)$$

Where $g(x)$ is an arbitrary characteristic/indicator function.

By this I mean that the evaluation of $I(f)$ at $x$ is either one of the one-sided limits and the choice of which one to pick comes from your particular choice of $g$.

The general crux of this question is that I wish to determine whether or not the following statement is true. I'm pretty sure it is, though that's just intuition.

Is a function a solution to an ordinary differential equation if and only if it is a continuous solution to the corresponding implied differential equation?

By corresponding equations, I just mean that they are corresponding if they are the same except with all of the derivative operators replaced with the implied derivative operator. So, the equation $D(y) = e^x$ has a corresponding equation of $I(y) = e^x$.

I do not know for sure whether or not anyone will actually be able to prove this statement. I think it is a bit tricky, but even just some advice on how to approach this or how to begin would be greatly appreciated. It isn't for homework or anything like that. It's just a statement and concept I've developed in my head over the years and I want to determine its truthfulness.

Note: if something equivalent to this or very similar that just makes this a special case has been proven in the past feel free to use that as an answer. I'm going under the (possibly mistaken) impression that this hasn't actually been proven before.

UPDATE:

I believe that a possible route to proving this statement might come by proving the following propositions.

The solution sets of sub-differential equations are a super set of the solution sets of the corresponding differential equations

The solution sets of sub-differential equations are a super set of the solution sets of the corresponding implied differential equations

If a solution to a sub-differential equation is continuous then it is a solution to the corresponding differential equation

If a solution to a sub-differential equation is continuous then it is a solution to the corresponding implied differential equation

I believe that the the first two propositions might follow trivially from the definition of the sub-derivative. The third one might have been proven in the past. I am unsure. The fourth one would then be the true meat and bones of the proof.

The subderivative is defined here: https://en.wikipedia.org/wiki/Subderivative

Reasons for wanting this statement proved:

The implied differential operator is defined via the limits; however, the purpose of that is to emulate a differential algebra wherein all step functions have a derivative of 0 and all the other normal rules are preserved. If it wasn't apparent, because of this any implied differential equation involving step functions is trivial to solve (at least in terms of the step functions themselves providing difficulty). If the statement were true, it would provide a new avenue by which to attack differential equations, some of which might be made trivial to solve via this method.

• But $I(y) = e^x$ doesn't make sense unless $y$ is differentiable, in which case, $I(y) = D(y)$ and it is just the same equation. If $y$ is not differentiable, but does have left and right derivatives, then $I(y)$ is a set of two real numbers for each $x$, whereas $e^x$ is just a single real number. – Paul Sinclair Apr 25 '17 at 3:53
• You might want to look at different definitions of sub-differentials, en.wikipedia.org/wiki/Subderivative. Consider also the equivalence of the IVP to the Picard integral equation. – Lutz Lehmann Apr 25 '17 at 10:46
• IVP = Initial value problem. If you look for extended frameworks for non-continuous ODE, look for A.F. Filippov, see for instance books.google.de/books?isbn=9401577935. Essentially, you get to look at all mollifications of the right side and then examine if they have limit points when the mollification parameter goes to zero. – Lutz Lehmann Apr 25 '17 at 15:47
• I was aware that was an example problem, and I was giving an example of what needs to be clarified about your concept. Whether you consider it a set of possible functions (which might get messy as there is no relation between choices made at different points) or a function returning sets is immaterial. In either case, you have to replace $e^x =$ with $e^x \in$, and deal with the consequences of this change. But fortunately Lutzl is more informed on the subject. – Paul Sinclair Apr 25 '17 at 16:27

One direction is trivial: a solution to any ODE is clearly a continuous solution to the corresponding IDE (implied differential equation).

The opposite direction seems to be false. Consider $I(f)=2\lfloor x\rfloor -1$ on the interval $[-\frac12, \frac12]$. This is solved by $f(x)=|x|$, which is continuous. But $f'=2\lfloor x\rfloor -1$ has no solutions: since RHS has a nonessential discontinuity at 0, it cannot be the derivative of any function.

The solution below doesn't work because Darboux's theorem assumes $f$ is differentiable. I've kept it for archival purposes.

When we restrict our attention to functions defined on a closed interval, we get a simple solution thanks to Darboux's theorem. This theorem implies, in particular, that $f'$ has only essential discontinuities (this fact is mentioned on the Wiki page currently). But the fact that $f$ is a solution to an IDE on the interval implies that it has only nonessential discontinuities.

So in fact, $f'$ is continuous on the whole interval, which means that $f$ is differentiable, which means that $I(f)=D(f)$ (or, more pedantically, $I(f)(a)=\{D(f)(a)\}$ for all $a$ in the interval). This is certainly strong enough to prove the desired result :P

I will spend some time thinking about more exotic scenarios. In particular, I have reason to believe that OP would be particularly interested in the case of half-open intervals (we talked in chat; this isn't true). I am not quite sure how derivatives at endpoints work, so if there anyone can shore up that bit in a line or two, please mention it in the comments :)

• Just for the record you probably meant the function $2\left \lfloor{x}\right \rfloor +1$. – lcv Apr 3 '20 at 2:58


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### Mathematics in Engineering

2019, Issue 2: 359-377. doi: 10.3934/mine.2019.2.359
Research article Special Issues

# Stages of dynamics in the Fermi-Pasta-Ulam system as probed by the first Toda integral

• Received: 30 November 2018 Accepted: 26 February 2019 Published: 12 April 2019
• We investigate the long term evolution of trajectories in the Fermi-Pasta-Ulam (FPU) system, using as a probe the first non-trivial integral $J$ in the hierarchy of integrals of the corresponding Toda lattice model. To this end we perform simulations of FPU-trajectories for various classes of initial conditions produced by the excitation of isolated modes, packets, as well as 'generic' (random) initial data. For initial conditions corresponding to localized energy excitations, $J$ exhibits variations yielding 'sigmoid' curves similar to observables used in literature, e.g., the 'spectral entropy' or various types of 'correlation functions'. However, $J(t)$ is free of fluctuations inherent in such observables, hence it constitutes an ideal observable for probing the timescales involved in the stages of FPU dynamics. We observe two fundamental timescales: i) the 'time of stability' (in which, roughly, FPU trajectories behave like Toda), and ii) the 'time to equilibrium' (beyond which energy equipartition is reached). Below a specific energy crossover, both times are found to scale exponentially as an inverse power of the specific energy. However, this crossover goes to zero with increasing the degrees of freedom $N$ as $\varepsilon _c \sim N^{-b}$, with $b \in [1.5, 2.5]$. For 'generic data' initial conditions, instead, $J(t)$ allows to quantify the continuous in time slow diffusion of the FPU trajectories in a direction transverse to the Toda tori.

Citation: Helen Christodoulidi, Christos Efthymiopoulos. Stages of dynamics in the Fermi-Pasta-Ulam system as probed by the first Toda integral[J]. Mathematics in Engineering, 2019, 1(2): 359-377. doi: 10.3934/mine.2019.2.359

### Related Papers:

• We investigate the long term evolution of trajectories in the Fermi-Pasta-Ulam (FPU) system, using as a probe the first non-trivial integral $J$ in the hierarchy of integrals of the corresponding Toda lattice model. To this end we perform simulations of FPU-trajectories for various classes of initial conditions produced by the excitation of isolated modes, packets, as well as 'generic' (random) initial data. For initial conditions corresponding to localized energy excitations, $J$ exhibits variations yielding 'sigmoid' curves similar to observables used in literature, e.g., the 'spectral entropy' or various types of 'correlation functions'. However, $J(t)$ is free of fluctuations inherent in such observables, hence it constitutes an ideal observable for probing the timescales involved in the stages of FPU dynamics. We observe two fundamental timescales: i) the 'time of stability' (in which, roughly, FPU trajectories behave like Toda), and ii) the 'time to equilibrium' (beyond which energy equipartition is reached). Below a specific energy crossover, both times are found to scale exponentially as an inverse power of the specific energy. However, this crossover goes to zero with increasing the degrees of freedom $N$ as $\varepsilon _c \sim N^{-b}$, with $b \in [1.5, 2.5]$. For 'generic data' initial conditions, instead, $J(t)$ allows to quantify the continuous in time slow diffusion of the FPU trajectories in a direction transverse to the Toda tori.

 [1] Bambusi D, Ponno A (2006) On metastability in FPU. Commun Math Phys 264: 539–561. doi: 10.1007/s00220-005-1488-1 [2] Berman GP, Izrailev FM (2005) The Fermi-Pasta-Ulam problem: 50 years of progress. Chaos 15: 015104. doi: 10.1063/1.1855036 [3] Benettin G, Livi R, Ponno A (2009) The Fermi-Pasta-Ulam problem: Scaling laws vs. initial conditions. J Stat Phys 135: 873–893. [4] Benettin G, Ponno A (2011) Time-scales to equipartition in the Fermi-Pasta-Ulam problem: Finite-size effects and thermodynamic limit. J Stat Phys 144: 793–812. doi: 10.1007/s10955-011-0277-9 [5] Benettin G, Christodoulidi H, Ponno A (2013) The Fermi-Pasta-Ulam problem and its underlying integrable dynamics. J Stat Phys 152: 195–212. doi: 10.1007/s10955-013-0760-6 [6] Benettin G, Pasquali S, Ponno A (2018) The Fermi-Pasta-Ulam problem and its underlying integrable dynamics. J Stat Phys 171: 521–542. doi: 10.1007/s10955-018-2017-x [7] Carati A, Galgani L, Giorgilli A, et al. (2007) Fermi-Pasta-Ulam phenomenon for generic initial data. Phys Rev E 76: 022104. [8] Carati A, Galgani L (1999) On the specific heat of Fermi-Pasta-Ulam Systems and their glassy behavior. J Stat Phys 94: 859–869. doi: 10.1023/A:1004531032623 [9] Carati A, Maiocchi A, Galgani L, et al. (2015) The Fermi-Pasta-Ulam system as a model for glasses. Math Phys Anal Geom 18: 31. doi: 10.1007/s11040-015-9201-x [10] Carati A, Ponno A (2018) Chopping time of the FPU $\alpha$-model. J Stat Phys 170: 883–894. doi: 10.1007/s10955-018-1962-8 [11] Casetti L, Cerruti-Sola M, Pettini M, et al. (1997) The Fermi-Pasta-Ulam problem revisited: Stochasticity thresholds in nonlinear Hamiltonian systems. Phys Rev E 55: 6566–6574. doi: 10.1103/PhysRevE.55.6566 [12] Christodoulidi H, Efthymiopoulos C, Bountis T (2010) Energy localization on q-tori, long-term stability, and the interpretation of Fermi-Pasta-Ulam recurrences. Phys Rev E 81: 016210/1–16. [13] Christodoulidi H, Efthymiopoulos C (2013) Low-dimensional q-Tori in FPU lattices: Dynamics and localization properties. Phys D 261: 92–113. doi: 10.1016/j.physd.2013.07.007 [14] Christodoulidi H (2017) Extensive packet excitations in FPU and Toda lattices. EPL 119: 40005. doi: 10.1209/0295-5075/119/40005 [15] Chirikov BV (1960) Resonance processes in magnetic traps. Soviet J At Energy 6: 464–470. doi: 10.1007/BF01483352 [16] Chirikov BV (1979) A universal instability of many-dimensional oscillator systems. Phys Rep 52: 263–379. doi: 10.1016/0370-1573(79)90023-1 [17] Danieli C, Campbell DK, Flach S (2017) Intermittent many-body dynamics at equilibrium. Phys Rev E 95: 060202(R). [18] Danieli C, Mithun T, Kati Y, et al. (2018) Dynamical glass in weakly non-integrable many-body systems. [arxiv:1811.10832]. [19] Dauxois T (2008) Fermi, Pasta, Ulam, and a mysterious lady. Phys Today 61: 55–57. [20] Ferguson WE Jr, Flaschka H, McLaughlin DW (1982) Nonlinear normal modes for the Toda Chain. J Comput Phys 45: 157–209. doi: 10.1016/0021-9991(82)90116-4 [21] Fermi E, Pasta J, Ulam S (1995) Studies of non linear problems. Los Alamos report No LA-1940. [22] Flach S, Ivanchenko MV, Kanakov OI (2005) q-Breathers and the Fermi-Pasta-Ulam problem. Phys Rev Lett 95: 064102. doi: 10.1103/PhysRevLett.95.064102 [23] Flach S, IvanchenkoMV, Kanakov OI (2006) q-Breathers in Fermi-Pasta-Ulam chains: Existence, localization, and stability. Phys Rev E 73: 036618. doi: 10.1103/PhysRevE.73.036618 [24] Flach S, Ponno A (2008) The Fermi-Pasta-Ulam problem Periodic orbits, normal forms and resonance overlap criteria. Phys D 237: 908–917. doi: 10.1016/j.physd.2007.11.017 [25] Flaschka H (1974) The Toda lattice. II. Existence of integrals. Phys Rev B 9: 1924–1925. [26] Goldfriend T, Kurchan T (2019) Equilibration of Quasi-Integrable Systems. Phys Rev E 99: 022146. doi: 10.1103/PhysRevE.99.022146 [27] Izrailev FM, Chirikov BV (1966) Statistical properties of a nonlinear string. Dokl Akad Nauk SSSR 166: 57–59. [28] Gallavotti G (2008) The Fermi-Pasta-Ulam Problem: A Status Report. Springer, Berlin- Heidelberg, vol. 728. [29] Genta T, Giorgilli A, Paleari S, et al. (2012) Packets of resonant modes in the Fermi-Pasta-Ulam system. Phys Lett A 376: 2038–2044. doi: 10.1016/j.physleta.2012.05.006 [30] Hénon M (1974) Integrals of the Toda lattice. Phys Rev B 9: 1921–1923. doi: 10.1103/PhysRevB.9.1921 [31] Kantz H, Livi R, Ruffo S (1994) Equipartition thresholds in chains of anharmonic oscillators. J Stat Phys 76: 627–643. doi: 10.1007/BF02188678 [32] Kantz H (1989) Vanishing stability thresholds in the thermodynamic limit of nonintegrable conservative systems. Phys D 39: 322–335. doi: 10.1016/0167-2789(89)90014-6 [33] Kruskal MD, Zabusky NJ (1965) Interaction of "solitons" in a collisionless plasma and the recurrence of initial states. Phys Rev Lett 15: 240–243. [34] Parisi G (1997) On the approach to equilibrium of a Hamiltonian chain of anharmonic oscillators. EPL 40: 357–362. doi: 10.1209/epl/i1997-00471-9 [35] Penati T, Flach S (2007) Tail resonances of Fermi-Pasta-Ulam q-breathers and their impact on the pathway to equipartition. Chaos 17: 023102/1-16. [36] Paleari S, Penati T (2005) Equipartition times in a Fermi-Pasta-Ulam system. Discrete Contin Dyn S 2005: 710–719. [37] Ponno A, Bambusi D (2005) Korteweg-de Vries equation and energy sharing in Fermi-Pasta- Ulam. Chaos 15: 015107. doi: 10.1063/1.1832772 [38] Ponno A, Christodoulidi H, Skokos Ch, et al. (2011) The two-stage dynamics in the Fermi-Pasta- Ulam problem: From regular to diffusive behavior. Chaos 21: 043127. doi: 10.1063/1.3658620 [39] Livi R, Pettini M, Ruffo S, et al. (1985) Equipartition threshold in nonlinear large Hamiltonian systems The Fermi-Pasta-Ulam model. Phys Rev A 31: 1039–1045. [40] Lvov YV, Onorato M (2018) Double scaling in the relaxation time in the $\beta$-Fermi-Pasta-Ulam- Tsingou model. Phys Rev Lett 120: 144301. doi: 10.1103/PhysRevLett.120.144301 [41] Shepelyansky DL (1997) Low-energy chaos in the Fermi-Pasta-Ulam problem. Nonlinearity 10: 1331–1338. doi: 10.1088/0951-7715/10/5/017 [42] Toda M (1970) Waves in Nonlinear Lattice. Prog Theor Phys Suppl 45: 174–200. doi: 10.1143/PTPS.45.174
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# Thomas Bond Sprague

### Quick Info

Born
29 March 1830
London, England
Died
29 November 1920
Edinburgh, Scotland

Summary
Thomas Bond Sprague became the most important actuary of the late 19th Century. He wrote more than 100 papers.

### Biography

Thomas Bond Sprague's father was Thomas Sprague (born about 1801) who was a stationer. His mother was Ellen Sprague (born about 1801). Thomas Bond Sprague had six younger siblings: Mary (born about 1831), William (born about 1833), Robert (born about 1834), John (born about 1835), Frances (born about 1837) and Caroline (born about 1839). Thomas Bond Sprague married Margaret Vaughan Steains, daughter of James Steains, at St Mary, Edge Hill, Lancashire on 13 April 1859. They had six children (Mary, Margaret, James, Charles, William and Alfred)

Thomas Sprague was the dominant actuary of the second half of the 19th century, his supremacy unrivalled.

Dr Sprague was the main person behind the International Actuarial Notation which was adopted unanimously at the 2nd International Congress of Actuaries in 1898 [1]. This established an international actuarial notation.

Dr Sprague was the main person behind a mortality study covering the experience of twenty U. K. life offices. This study resulted in the Institute of Actuaries Life Tables (the so-called Twenty Offices Table) which was published in 1869 [2]. From this study, he produced, in 1879, the first Select Tables of Mortality [3] which were the first two-dimensional mortality tables ever published (the two dimensions being 'insured duration' i.e. the 'select period' and 'age attained'). The 'select period' was five years.

Dr Sprague pioneered the important 1870 Life Insurance Companies Act [4] which was introduced following the notorious insolvencies of both the Albert and the European life assurance companies. The 1870 Act required:-
... an investigation into the financial condition of a life insurance company to be made regularly by an actuary,
required a separate "long-term fund" and required the:-
... preparation of a revenue account and balance sheet every year in prescribed form to be filed with the Board of Trade,
the latter being a public document. Dr Sprague was one of the foremost advocates of the principle of 'Freedom with Publicity' (i.e. documents available to the public) and was opposed to there being any Government regulation prescribing the manner of valuation of policy liabilities. He wrote the major 19th century work on the preparation of life office accounts in conformity with the 1870 Act [5].

Dr Sprague published some eighty papers in the Journal of the Institute of Actuaries and, including articles, Presidential Addresses, translations, comments and reviews for this and other actuarial journals (Transactions of the Faculty of Actuaries, Transactions of the Actuarial Society of Edinburgh), his published contributions to actuarial science must number over one hundred. In addition, he published eight pure mathematical papers: [16] and [17] in the Transactions of the Royal Society of Edinburgh and six papers in the Proceedings of the Edinburgh Mathematical Society: [18],[19], [20], [21], [22] and [23].

A particularly memorable translation, from Danish, is Sprague's translation of Dr T N Thiele's 1871 paper on mortality [6]. Thiele was from 1872 to 1910, among other appointments, the chief actuary of the Danish insurance company Hafnia. Thiele stated, for the first time, a mathematical formula expressing the force of mortality throughout the whole of life. The Thiele 1871 mathematical formula was not bettered until the 1980s [7]. Thiele's law, where $x$ is the age and $\mu_{x}$ is the force of mortality, is:-
$\mu_{x} = Ae^{-Bx} + Ce^{-D(x-E)^{2}} + FG^{x}$
Sprague was elected a Fellow of St John's College following his attainment, in 1853, of the position of Senior Wrangler and First Smith's Prizeman at Cambridge University (first place among those attaining first class honours in mathematics) [8]. The year before Sprague, in 1852, the Senior Wrangler and First Smith's Prizeman had been P G Tait (as well as the Professor of Natural Philosophy at Edinburgh University and the founder of the mathematical theory of knots, Tait was a director of the Scottish Provident Institution, a life insurance company in Edinburgh) and the year after, in 1854, the world-famous physicist, Clerk Maxwell had been second Wrangler (Routh being the Senior), the two Smith's Prizes of 1854 being shared equally between these two eminent men. Dr Sprague qualified as a barrister prior to becoming an actuary and knew both Tait and Maxwell.

Dr Sprague was the only person to have been President of both the Institute of Actuaries (1882-86) in London and the Faculty of Actuaries (1894-96) in Edinburgh. He was also President of the Actuarial Society of Edinburgh on three occasions (1874, 1882 and 1891) and did much to mould the futures of all these bodies.

He was elected a Fellow of the Royal Society of Edinburgh (FRSE) in 1874 [9] and of the Edinburgh Mathematical Society (EMS). The University of Aberdeen made him an LL.D. in 1893.

Dr Sprague was the actuary to the Equity and Law life insurance company from 1861-1873 when he was appointed to be Chief Executive (1873-1900) of the Scottish Equitable Life Assurance Society in Edinburgh (not to be confused with the Equitable Life Assurance Society in London) retiring from business life at age 70. When President of the Institute of Actuaries, from 1882-86, he was resident in Edinburgh but travelled to London regularly to fulfil his role as President of the Institute. He was on the Council of the Institute of Actuaries for thirty-eight years (1863-1900) and editor of the Journal of the Institute of Actuaries for 16 years (1867-83) [10].

Dr Sprague was invited by Clerk Maxwell (the latter being scientific editor, along with T H Huxley, of the 9th Edition of the Encyclopaedia Britannica) to write the article on 'Annuities' [11]. This article was a most thorough analysis of the contributions, including those of de Witt, Halley, and de Moivre, to the advancement of actuarial science.

At the 1874 meeting of the British Association in Belfast (at which occasion Clerk Maxwell wrote his poems entitled 'Notes of the President's Address' and 'Molecular Evolution' [12]), Sprague gave a paper on the solvency of life offices [13] which the 1870 Life Insurance Companies Act (see above) did much to ensure.

Dr Sprague's one peculiarity was his advocacy of spelling reform which caused some difficulties with the Editorial Board of the Journal of the Institute of Actuaries as to whether his special brand of phonetic spelling should be permitted in the papers which he submitted e.g. 'treatis', 'anser', 'articl', 'coud', 'giv', 'conclusiv', 'hav', 'ov' (e.g. the wording in reference [5] is not a mis-spelling). However, such was the eminence of Dr Sprague that the Journal had no alternative but to permit it!

In Maclaurin House (the home of the Faculty of Actuaries, named after Maclaurin, who, in 1744, acted as consultant to the first pension fund in the world established on sound actuarial principles, namely the pension fund for the widows and children of the 'Ministers of the Church of Scotland and the Professors at the Scottish Universities') his bust [14] stands alongside that of William T Thomson (one of the founders of the Institute of Actuaries in 1848 and the Faculty of Actuaries in 1856). A bust [15] of him also stands in the main hall of Staple Inn, the home of the Institute of Actuaries.

A prize for the best performance in actuarial science, finance, insurance, operational research, probability and statistics in Part III of the Cambridge Mathematical Tripos is awarded each year in honour of Dr. Sprague.

His son, Alfred Ernest Sprague, MA, DSc, FFA, FIA was also President of the Faculty of Actuaries from 1919-21.

### References (show)

1. T B Sprague, G King, J Karup, M Am Bégault and M L Marie, The Universal Notation, Transactions of the 2nd International Actuarial Congress (1898), 618.
2. T B Sprague, Select life tables deduced from the Institute of Actuaries' experience (healthy males), with commutation columns and monetary values at 2 1/2, 3, 3 1/2 and 4 per cent (1896). Shelved at: VXL/73 rsl [IF archive] (Lon) [Faculty: Edinburgh University Library, Special Collections Department 072.1942 SPR] [RKN: 4587]
3. D O Forfar, Early Mortality Tables, Encyclopedia of Actuarial Science, (eds. J L Teugels and B Sundt), Vol. 2 (John Wiley, 2004), 597.
4. T B Sprague, On legislation as to life insurance and life insurance companies (c1870). [Faculty: Edinburgh University Library, Special Collections Department Pamphlet file, vol 13] [RKN: 70761]
5. T B Sprague, Treatise on Life Office Accounts: Showing in particular how the annual revenue account and balance sheet ov a company shoud be drawn up, so as to be in strict conformity with the schedules ov the Life Assurance Companies Act 1870; and containing a complete examination ov the accounts deposited with the Board of Trade under that Act : Forming Part II ov 'Life insurance in 1872', (Charles and Edwin Layton, 1874).
6. T N Thiele, On a mathematical formula to express the rate of mortality throughout the whole span of life (translation by T B Sprague in Journal of the Institute of Actuaries 16 (1871), 313-329.
7. D O Forfar, Mortality Laws, Encyclopedia of Actuarial Science (eds. J L Teugels and B Sundt) Vol. 2 (John Wiley, 2004), 1141,
8. University of Cambridge, Historical Register of the University of Cambridge up to 1910.
9. Royal Society of Edinburgh, Archive of Fellows, http://www.royalsoced.org.uk
10. G King, Memoir of Dr Thomas Bond Sprague, Journal of the Institute of Actuaries, 52 (1921), 241-259. Also Obituary Notice, Transactions of the Faculty of Actuaries 9 (1921), 40; 379.
11. T B Sprague, Annuities, article in 9th edition of the Encyclopaedia Britannica (scientific eds, J Clerk Maxwell and T H Huxley) (A and C Black, Edinburgh, 1875).
12. L Campbell and W Garnett, Life of James Clerk Maxwell (1882) page 326. http://www.sonnetusa.com/bio/maxbio.pdf
13. T B Sprague, On the causes of insolvency in life assurance companies and the best means of detecting, exposing and preventing it (1874): [Paper read at the Belfast meeting in 1874 of the British Association]. [Faculty: Edinburgh University Library, Special Collections Department 368.3 SPR Pamphlet file, vol 13 Pamphlet file, vol 44] [RKN: 70199]
14. Bust placed in the Faculty of Actuaries, Transactions of the Faculty of Actuaries 1, 15.
15. Bust placed in the Institute of Actuaries, Journal of the Institute of Actuaries 37, 111.
16. T B Sprague, On the nature of the curves whose intersections give the imaginary roots of an algebraic equation, Transactions of Royal Society of Edinburgh 30 (1882), 467-480.
17. T B Sprague, On a new algebra, by means of which permutations can be transformed in a variety of ways, and their properties investigated, Transactions of Royal Society of Edinburgh 37 (1893), 399-411.
18. T B Sprague, Note on the evaluation of functions of the form 00, Edinburgh Mathematical Society (1884).
19. T B Sprague, On the different possible non-linear arrangements of eight men on a chess board, Edinburgh Mathematical Society (1889).
20. T B Sprague, On the transformation and classification of permutations, Edinburgh Mathematical Society (1890).
21. T B Sprague, On the geometrical interpretation of ii, Edinburgh Mathematical Society (1893).
22. T B Sprague, On the eight Queens' problem, Edinburgh Mathematical Society (1899).
23. T B Sprague, On the singular points of plane curves, Edinburgh Mathematical Society (1902).
24. T B Sprague, Presidential Address to Institute of Actuaries, Journal of the Institute of Actuaries, 24 (1882), 1; 24 (1883), 229; 25 (1884), 65; 25 (1885), 293.
25. T B Sprague, Presidential Address to the Actuarial Society of Edinburgh and to the Faculty of Actuaries, Transactions of the Actuarial Society of Edinburgh, 1, 269-287.
26. Dr. Sprague wrote actuarial papers on mortality, graduation, interpolation, life interests and reversionary interests, combined marriage and mortality tables, vaccination against smallpox, premium rates, distribution of life office surplus, valuation of life office liabilities, life office expenses, legislation of life companies, causes of life office insolvencies, transfer of business from one life office to another, use of probability in actuarial studies. http://ioa.soutron.com/ioa/library/default.htm

Other pages about Thomas Bond Sprague:

Other websites about Thomas Bond Sprague:

### Honours (show)

Honours awarded to Thomas Bond Sprague

### Cross-references (show)

Written by David O Forfar Heriot-Watt University.
Last Update August 2006


text:
# Group Equivariant Convolutional Networks in Medical Image Analysis

## Introduction

AI has been one of the most heated topic in this planet recently and greatly changed people's life. Past years has witnessed the widely applications of convolutional neural networks (CNNs) in medical image analysis, including several tasks such as classification, segmentation, which greatly advanced diagnosis and treatment in clinical practice.
CNNs' success rely on reasonable architectures of networks, and strong computational power, as well as large scale datasets. But in some scenarios like healthcare system, large scale data is hard to obtain. Thus extensive data augmentation should be implemented to achieve better performance.
Different from natural images, some medical images exhibit not only translational symmetry but rotation and reflection symmetry. However, CNNs failed to make good use of these symmetries.
Group equivariant convolutional networks (G-CNNs) [1] was proposed in 2016 as a generalization of CNNs, using G-convolutions to enjoy a substantially higher degree of weight sharing, which can be implemented with negligible computational overhead for discrete groups generated by translations, reflections and rotations. This architecture has been used in medical image analysis and proved that it can improve the performance in multi-tasks
This paper will give a brief review for the applications of G-CNNs in medical image analysis and show some details about how it improves the models.

## Prelimiaries

### Symmetries

Cohen et al. [1:1] explained the concept of symmetry as transformation that maintains the object invariant or equivalent. Cohen's PhD thesis [2] further explained that the set of all symmetries can be considered as a label function where the equivalence classes induced by the group are the same as the classes.
Figure 1: Example of a label function
Figure 1 [2:1] is an example of a label function $L:\mathcal{X}\to \mathcal{C}$$L:\mathcal{X}\to \mathcal{C}$L:XrarrCL: \mathcal{X} \rightarrow \mathcal{C} (in green) from input space $\mathcal{X}$$\mathcal{X}$X\mathcal{X} to label space $\mathcal{C}$$\mathcal{C}$C\mathcal{C}. A symmetry transformation $g:\mathcal{X}\to \mathcal{X}$$g:\mathcal{X}\to \mathcal{X}$g:XrarrXg: \mathcal{X} \rightarrow \mathcal{X} of $L$$L$LL is drawn in blue.
As $g$$g$gg is a symmetry of $L$$L$LL and each class is a union of orbits of $g$$g$gg (connected components of the blue graph). That is, ${L}^{-1}\left({c}_{1}\right)=\left\{{x}_{1},{x}_{2}\right\}\cup \left\{{x}_{3},{x}_{4}\right\}$${L}^{-1}\left({c}_{1}\right)=\left\{{x}_{1},{x}_{2}\right\}\cup \left\{{x}_{3},{x}_{4}\right\}$L^(-1)(c_(1))={x_(1),x_(2)}uu{x_(3),x_(4)}L^{-1}\left(c_{1}\right)=\left\{x_{1}, x_{2}\right\} \cup\left\{x_{3}, x_{4}\right\} and ${L}^{-1}\left({c}_{2}\right)=$${L}^{-1}\left({c}_{2}\right)=$L^(-1)(c_(2))=L^{-1}\left(c_{2}\right)= $\left\{{x}_{5},{x}_{6},{x}_{7}\right\}$$\left\{{x}_{5},{x}_{6},{x}_{7}\right\}${x_(5),x_(6),x_(7)}\left\{x_{5}, x_{6}, x_{7}\right\}. In this case, we can reduce the minimum number of labels required from 7 to 3 in the noiseless setting if we know that $g$$g$gg is a symmetry of $L$$L$LL.
And [1:2] also takes sampling grid of images as an example, which is easier to understand: for a sampling grid of images ${\mathbb{Z}}^{2}$${\mathbb{Z}}^{2}$Z^(2)\mathbb{Z}^{2}, giving an operation of flipping: $-{\mathbb{Z}}^{2}=$$-{\mathbb{Z}}^{2}=$-Z^(2)=-\mathbb{Z}^{2}= $\left\{\left(-n,-m\right)\mid \left(n,m\right)\in {\mathbb{Z}}^{2}\right\}={\mathbb{Z}}^{2}$$\left\{\left(-n,-m\right)\mid \left(n,m\right)\in {\mathbb{Z}}^{2}\right\}={\mathbb{Z}}^{2}${(-n,-m)∣(n,m)inZ^(2)}=Z^(2)\left\{(-n,-m) \mid(n, m) \in \mathbb{Z}^{2}\right\}=\mathbb{Z}^{2}. So the flipping operation is with prosperity of symmetry of the sampling grid of images.
We call a set of transformations with properties of symmetry as symmetry group.

### Representations

After the introduction symmetry, let us go to "representations". When the symmetry transformations act on vectors, we have a representation of the group. In [2:2], group representation is defined as the appropriate notion of group.
To be easier, a group action $\rho \left(g\right)$$\rho \left(g\right)$rho(g)\rho(g) that acts on $\mathcal{X}$$\mathcal{X}$X\mathcal{X} is the representation of group element $g\in G$$g\in G$g in Gg \in G.
Feature maps of CNNs as spaces of functions can also be regarded as one kind of representation.

### Equivariance

Equivariance is a kind of important symmetry.
Define $\mathrm{\Phi }$$\mathrm{\Phi }$Phi\Phi as an action, which maps one representation to another and is structure preserving.
For $G$$G$GG-spaces, $\mathrm{\Phi }$$\mathrm{\Phi }$Phi\Phi has to be equivariant if:
$\mathrm{\Phi }\left({T}_{g}x\right)={T}_{g}^{\mathrm{\prime }}\mathrm{\Phi }\left(x\right)$$\mathrm{\Phi }\left({T}_{g}x\right)={T}_{g}^{\mathrm{\prime }}\mathrm{\Phi }\left(x\right)$Phi(T_(g)x)=T_(g)^(')Phi(x)\Phi\left(T_{g} x\right)=T_{g}^{\prime} \Phi(x)
where ${T}_{g}$${T}_{g}$T_(g)T_{g} is transformation of $g$$g$gg on $x$$x$xx, then input the result to $\mathrm{\Phi }$$\mathrm{\Phi }$Phi\Phi,while ${T}_{g}^{\mathrm{\prime }}$${T}_{g}^{\mathrm{\prime }}$T_(g)^(')T_{g}^{\prime} is a transformation of $g$$g$gg on $\mathrm{\Phi }\left(x\right)$$\mathrm{\Phi }\left(x\right)$Phi(x)\Phi(x). Equivariance means that the above two transformations are equal.
[2:3] also pointed out that equivariance is transitive: if ${\mathrm{\Phi }}_{l}:{\mathcal{X}}_{l}\to {\mathcal{X}}_{l+1}$${\mathrm{\Phi }}_{l}:{\mathcal{X}}_{l}\to {\mathcal{X}}_{l+1}$Phi_(l):X_(l)rarrX_(l+1)\Phi_{l}: \mathcal{X}_{l} \rightarrow \mathcal{X}_{l+1} is equivariant with respect to ${\rho }_{l}$${\rho }_{l}$rho_(l)\rho_{l} and ${\rho }_{l+1}$${\rho }_{l+1}$rho_(l+1)\rho_{l+1}, and ${\mathrm{\Phi }}_{l+1}:{\mathcal{X}}_{l+1}\to {\mathcal{X}}_{l+2}$${\mathrm{\Phi }}_{l+1}:{\mathcal{X}}_{l+1}\to {\mathcal{X}}_{l+2}$Phi_(l+1):X_(l+1)rarrX_(l+2)\Phi_{l+1}: \mathcal{X}_{l+1} \rightarrow \mathcal{X}_{l+2} is equivariant with respect to ${\rho }_{l+1}$${\rho }_{l+1}$rho_(l+1)\rho_{l+1} and ${\rho }_{l+2}$${\rho }_{l+2}$rho_(l+2)\rho_{l+2}, then their composition ${\mathrm{\Phi }}_{l+1}\circ {\mathrm{\Phi }}_{l}:{\mathcal{X}}_{l}\to {\mathcal{X}}_{l+2}$${\mathrm{\Phi }}_{l+1}\circ {\mathrm{\Phi }}_{l}:{\mathcal{X}}_{l}\to {\mathcal{X}}_{l+2}$Phi_(l+1)@Phi_(l):X_(l)rarrX_(l+2)\Phi_{l+1} \circ \Phi_{l}: \mathcal{X}_{l} \rightarrow \mathcal{X}_{l+2} is equivariant with respect to ${\rho }_{l}$${\rho }_{l}$rho_(l)\rho_{l} and ${\rho }_{l+2}$${\rho }_{l+2}$rho_(l+2)\rho_{l+2}. Take CNNs for an example, if each layer of CNNs is of equivariance, the whole neural network is equivariant.
In machine learning, generally, equivariance is of more importance than invariance, as we can judge spatial relationships based on equivariant features rather than invariant features.

### The group $p4$$p4$p4p 4$p4$ and $p4m$$p4m$p4mp 4m$p4m$

[1:3] proposed the concept of two types of groups, $p4$$p4$p4p 4 and $p4m$$p4m$p4mp 4m. For a sampling grid of images ${\mathbb{Z}}^{2}$${\mathbb{Z}}^{2}$Z^(2)\mathbb{Z}^{2}, the group $p4$$p4$p4p 4 consists of translations and rotations by 90 degrees about any center of rotation in a square grid. The parameterization of this group can be represented in terms of three integers $r,u,v$$r,u,v$r,u,vr, u, v is
$g\left(r,u,v\right)=\left[\begin{array}{ccc}\mathrm{cos}\left(r\pi /2\right)& -\mathrm{sin}\left(r\pi /2\right)& u\\ \mathrm{sin}\left(r\pi /2\right)& \mathrm{cos}\left(r\pi /2\right)& v\\ 0& 0& 1\end{array}\right]$$g\left(r,u,v\right)=\left[\begin{array}{ccc}\mathrm{cos}\left(r\pi /2\right)& -\mathrm{sin}\left(r\pi /2\right)& u\\ \mathrm{sin}\left(r\pi /2\right)& \mathrm{cos}\left(r\pi /2\right)& v\\ 0& 0& 1\end{array}\right]$g(r,u,v)=[[cos(r pi//2),-sin(r pi//2),u],[sin(r pi//2),cos(r pi//2),v],[0,0,1]]g(r, u, v)=\left[\begin{array}{ccc} \cos (r \pi / 2) & -\sin (r \pi / 2) & u \\ \sin (r \pi / 2) & \cos (r \pi / 2) & v \\ 0 & 0 & 1 \end{array}\right]
where $0\le r<4$$0\le r<4$0 <= r < 40 \leq r<4 and $\left(u,v\right)\in {\mathbb{Z}}^{2}$$\left(u,v\right)\in {\mathbb{Z}}^{2}$(u,v)inZ^(2)(u, v) \in \mathbb{Z}^{2}. The group operation is given by matrix multiplication.
The group $p4m$$p4m$p4mp 4 m consists of all compositions of translations, mirror reflections, and rotations by 90 degrees about any center of rotation in the grid. The parameterization of this group can be represented in terms of three integers $m,r,u,v$$m,r,u,v$m,r,u,vm, r, u, v is
$g\left(m,r,u,v\right)=\left[\begin{array}{ccc}\left(-1{\right)}^{m}\mathrm{cos}\left(\frac{r\pi }{2}\right)& -\left(-1{\right)}^{m}\mathrm{sin}\left(\frac{r\pi }{2}\right)& u\\ \mathrm{sin}\left(\frac{r\pi }{2}\right)& \mathrm{cos}\left(\frac{r\pi }{2}\right)& v\\ 0& 0& 1\end{array}\right]$$g\left(m,r,u,v\right)=\left[\begin{array}{ccc}\left(-1{\right)}^{m}\mathrm{cos}\left(\frac{r\pi }{2}\right)& -\left(-1{\right)}^{m}\mathrm{sin}\left(\frac{r\pi }{2}\right)& u\\ \mathrm{sin}\left(\frac{r\pi }{2}\right)& \mathrm{cos}\left(\frac{r\pi }{2}\right)& v\\ 0& 0& 1\end{array}\right]$g(m,r,u,v)=[[(-1)^(m)cos((r pi)/(2)),-(-1)^(m)sin((r pi)/(2)),u],[sin((r pi)/(2)),cos((r pi)/(2)),v],[0,0,1]]g(m, r, u, v)=\left[\begin{array}{ccc} (-1)^{m} \cos \left(\frac{r \pi}{2}\right) & -(-1)^{m} \sin \left(\frac{r \pi}{2}\right) & u \\ \sin \left(\frac{r \pi}{2}\right) & \cos \left(\frac{r \pi}{2}\right) & v \\ 0 & 0 & 1 \end{array}\right]
where $m\in \left\{0,1\right\},0\le r<4$$m\in \left\{0,1\right\},0\le r<4$m in{0,1},0 <= r < 4m \in\{0,1\}, 0 \leq r<4 and $\left(u,v\right)\in {\mathbb{Z}}^{2}$$\left(u,v\right)\in {\mathbb{Z}}^{2}$(u,v)inZ^(2)(u, v) \in \mathbb{Z}^{2}.

## Group Equivariant Convolutional Networks

### Structure of Symmetric Feature Maps

The map from images to stacks of feature maps with $K$$K$KK channels in a CNN can be modeled as $f:{\mathbb{Z}}^{2}\to {\mathbb{R}}^{K}$$f:{\mathbb{Z}}^{2}\to {\mathbb{R}}^{K}$f:Z^(2)rarrR^(K)f: \mathbb{Z}^{2} \rightarrow \mathbb{R}^{K} supported on a bounded domain. Every pixel coordinate $\left(x,y\right)\in {\mathbb{Z}}^{2}$$\left(x,y\right)\in {\mathbb{Z}}^{2}$(x,y)inZ^(2)(x, y) \in \mathbb{Z}^{2}, the stack of feature maps returns a $K$$K$KK-dimensional vector $f\left(x,y\right)$$f\left(x,y\right)$f(x,y)f(x, y).
A transformation $g$$g$gg acting on a set of feature maps is shown as below:
$\left[{L}_{g}f\right]\left(x\right)=\left[f\circ {g}^{-1}\right]\left(x\right)=f\left({g}^{-1}x\right)$$\left[{L}_{g}f\right]\left(x\right)=\left[f\circ {g}^{-1}\right]\left(x\right)=f\left({g}^{-1}x\right)$[L_(g)f](x)=[f@g^(-1)](x)=f(g^(-1)x)\left[L_{g} f\right](x)=\left[f \circ g^{-1}\right](x)=f\left(g^{-1} x\right)
Feature maps in a $G$$G$GG-CNN are functions on the group $G$$G$GG rather than on the group ${\mathbb{Z}}^{2}$${\mathbb{Z}}^{2}$Z^(2)\mathbb{Z}^{2}. And $x$$x$xx, the element of ${\mathbb{Z}}^{2}$${\mathbb{Z}}^{2}$Z^(2)\mathbb{Z}^{2} can also be replaced by $h$$h$hh, an element of $G$$G$GG .,.
And the following two figures[1:4] are feature maps of $p4$$p4$p4p4 and $p4m$$p4m$p4mp4m and their rotations by $m$$m$mm:
Figure 2: A p4 feature map and its rotation by r
Figure 3: A p4m feature map and its rotation by r

### The Structure of Group Equivariant Convolutional Networks

The first layer of Group Equivariant Convolutional Networks is used to lift the images (a sampling grid of images ${\mathbb{Z}}^{2}$${\mathbb{Z}}^{2}$Z^(2)\mathbb{Z}^{2}) to feature maps, which can be defined as: $\left[f\star \psi \right]\left(g\right)=\sum _{y\in {\mathbb{Z}}^{2}}\sum _{k}{f}_{k}\left(y\right){\psi }_{k}\left({g}^{-1}y\right)$$\left[f\star \psi \right]\left(g\right)=\sum _{y\in {\mathbb{Z}}^{2}} \sum _{k} {f}_{k}\left(y\right){\psi }_{k}\left({g}^{-1}y\right)$[f***psi](g)=sum_(y inZ^(2))sum_(k)f_(k)(y)psi_(k)(g^(-1)y)[f \star \psi](g)=\sum_{y \in \mathbb{Z}^{2}} \sum_{k} f_{k}(y) \psi_{k}\left(g^{-1} y\right)
And all layers after the first can be defined as: $\left[f\star \psi \right]\left(g\right)=\sum _{h\in G}\sum _{k}{f}_{k}\left(h\right){\psi }_{k}\left({g}^{-1}h\right)$$\left[f\star \psi \right]\left(g\right)=\sum _{h\in G} \sum _{k} {f}_{k}\left(h\right){\psi }_{k}\left({g}^{-1}h\right)$[f***psi](g)=sum_(h in G)sum_(k)f_(k)(h)psi_(k)(g^(-1)h)[f \star \psi](g)=\sum_{h \in G} \sum_{k} f_{k}(h) \psi_{k}\left(g^{-1} h\right)
The difference is that the function is on the plane ${\mathbb{Z}}^{2}$${\mathbb{Z}}^{2}$Z^(2)\mathbb{Z}^{2} or on the discrete group $G$$G$GG.
Both these two functions are proved to be equivariant in [1:5].

### 3D Group Convolutions

[3] proposed a 3D Group Convolution to solve 3D images' analysis like CT scans. The 3D group convolution is implemented as $\mathrm{GConv}3d\left(\mathrm{\Psi },f\right)=\mathrm{Conv}3d\left({\mathrm{\Psi }}^{+},f\right)$$\mathrm{GConv}3d\left(\mathrm{\Psi },f\right)=\mathrm{Conv}3d\left({\mathrm{\Psi }}^{+},f\right)$GConv 3d(Psi,f)=Conv 3d(Psi^(+),f)\operatorname{GConv} 3 d(\Psi, f)=\operatorname{Conv} 3 d\left(\Psi^{+}, f\right)
where $\mathrm{\Psi }$$\mathrm{\Psi }$Psi\Psi is the filter bank.
The $\mathrm{map}\mathrm{\Psi }↦{\mathrm{\Psi }}^{+}$$\mathrm{map}\mathrm{\Psi }↦{\mathrm{\Psi }}^{+}$map Psi|->Psi^(+)\operatorname{map} \Psi \mapsto \Psi^{+}can be defined as an indexing operation of $\mathrm{\Psi }$$\mathrm{\Psi }$Psi\Psi using a precomputed array of indices. This review will not include the details of mathematics of 3D Group Convolution.

## Applications in Medical Image Analysis

### Classification

General process of classification consists of three steps by using G-CNNs:
1. Convert an image to feature maps
2. Convert feature maps to feature maps
3. Convert feature maps to a label.
In the final layer, a group-pooling layer is used to ensure that the output is invariant as a function on the plan for classification tasks.[4]
Example1: [4:1] proposed equivariant DenseNet architecture for the $p4$$p4$p4p 4 group, consisting of 5 Dense Blocks (D.B.) alternated with Transition Blocks (T.B.). The main process is as below:
1. Define the G-convolution on image $\left({\mathbb{Z}}^{2}\to \mathbit{G}\right)$$\left({\mathbb{Z}}^{2}\to \mathbit{G}\right)$(Z^(2)rarr G)\left(\mathbb{Z}^{2} \rightarrow \boldsymbol{G}\right)-convolution:
$\left[f\ast \psi \right]\left(g\right)=\sum _{y\in {\mathbb{Z}}^{2}}\sum _{k=1}^{K}{f}_{k}\left(y\right){\psi }_{k}\left({g}^{-1}y\right)$$\left[f\ast \psi \right]\left(g\right)=\sum _{y\in {\mathbb{Z}}^{2}} \sum _{k=1}^{K} {f}_{k}\left(y\right){\psi }_{k}\left({g}^{-1}y\right)$[f**psi](g)=sum_(y inZ^(2))sum_(k=1)^(K)f_(k)(y)psi_(k)(g^(-1)y)[f * \psi](g)=\sum_{y \in \mathbb{Z}^{2}} \sum_{k=1}^{K} f_{k}(y) \psi_{k}\left(g^{-1} y\right)
where $g=\left(r,t\right)$$g=\left(r,t\right)$g=(r,t)g=(r, t) is a roto-translation (in case $G=p4$$G=p4$G=p4G=p 4 ) or roto-reflection-translation (in case $G=p4m\right)$$G=p4m\right)$G=p4m)G=p 4 m).
1. Define the $\left(G\to G\right)$$\left(G\to G\right)$(G rarr G)(G \rightarrow G)-convolution:
$\left[f\ast \psi \right]\left(g\right)=\sum _{h\in G}\sum _{k=1}^{K}{f}_{k}\left(h\right){\psi }_{k}\left({g}^{-1}h\right)$$\left[f\ast \psi \right]\left(g\right)=\sum _{h\in G} \sum _{k=1}^{K} {f}_{k}\left(h\right){\psi }_{k}\left({g}^{-1}h\right)$[f**psi](g)=sum_(h in G)sum_(k=1)^(K)f_(k)(h)psi_(k)(g^(-1)h)[f * \psi](g)=\sum_{h \in G} \sum_{k=1}^{K} f_{k}(h) \psi_{k}\left(g^{-1} h\right)
1. Define a projection layer: $G\to {\mathbb{Z}}^{2}$$G\to {\mathbb{Z}}^{2}$G rarrZ^(2)G \rightarrow \mathbb{Z}^{2}
The final layer of the model is a $p4\to {\mathbb{Z}}^{2}$$p4\to {\mathbb{Z}}^{2}$p4rarrZ^(2)p 4 \rightarrow \mathbb{Z}^{2} group pooling layer followed by a sigmoid activation. The four orientations in $p4$$p4$p4p 4 are illustrated through primary colors. A ${\mathbb{Z}}^{2}\to p4$${\mathbb{Z}}^{2}\to p4$Z^(2)rarr p4\mathbb{Z}^{2} \rightarrow p 4 kernel (left), $p4\to p4$$p4\to p4$p4rarr p4p 4 \rightarrow p 4 kernel and $p4\to {\mathbb{Z}}^{2}$$p4\to {\mathbb{Z}}^{2}$p4rarrZ^(2)p 4 \rightarrow \mathbb{Z}^{2} kernel illustrate how equivariance arises in the model.
Figure 4: Equivariant DenseNet architecture, the figure is from [^5]
Example2: [5] proposed a kind of dynamic group convolution to improve the performance of breast tumor classification.The dynamic group convolutional is parameterized as a dynamic combination rather than a fixed one of multiple kernels. And the operation was generalized to an equivariant one.
For $\mathcal{K}:\mathcal{X}\to \mathcal{Y}$$\mathcal{K}:\mathcal{X}\to \mathcal{Y}$K:XrarrY\mathcal{K}: \mathcal{X} \rightarrow \mathcal{Y} and $F:\mathcal{X}\to \mathcal{Y}$$F:\mathcal{X}\to \mathcal{Y}$F:XrarrYF: \mathcal{X} \rightarrow \mathcal{Y} defined on $G$$G$GG, the dynamic group equivariant convolution is defined as follows:
$\left(\mathcal{K}F\right)\left(g\right)={\int }_{G}\mathcal{A}\left(g,\stackrel{~}{g}\right)K\left({g}^{-1}\stackrel{~}{g}\right)\left(\stackrel{~}{g}\right)F\left(\stackrel{~}{g}\right)d{\mu }_{\overline{g}}^{-}$$\left(\mathcal{K}F\right)\left(g\right)={\int }_{G} \mathcal{A}\left(g,\stackrel{~}{g}\right)K\left({g}^{-1}\stackrel{~}{g}\right)\left(\stackrel{~}{g}\right)F\left(\stackrel{~}{g}\right)d{\mu }_{\overline{g}}^{-}$(KF)(g)=int_(G)A(g, tilde(g))K(g^(-1)( tilde(g)))( tilde(g))F( tilde(g))dmu_( bar(g))^(-)(\mathcal{K} F)(g)=\int_{G} \mathcal{A}(g, \tilde{g}) K\left(g^{-1} \tilde{g}\right)(\tilde{g}) F(\tilde{g}) d \mu_{\bar{g}}^{-}
where $\mathcal{A}$$\mathcal{A}$A\mathcal{A} is an attention map obtained from dynamic attention operator $\mathcal{R}\left[F\right]$$\mathcal{R}\left[F\right]$R[F]\mathcal{R}[F], $K\left({g}^{-1}\stackrel{~}{g}\right)$$K\left({g}^{-1}\stackrel{~}{g}\right)$K(g^(-1)( tilde(g)))K\left(g^{-1} \tilde{g}\right) is a stack of kernels ${\left\{{K}_{i}\left({g}^{-1}\stackrel{~}{g}\right)\right\}}_{i\in k}$${\left\{{K}_{i}\left({g}^{-1}\stackrel{~}{g}\right)\right\}}_{i\in k}${K_(i)(g^(-1)( tilde(g)))}_(i in k)\left\{K_{i}\left(g^{-1} \tilde{g}\right)\right\}_{i \in k} concatenated by matrix form: $\left[{K}_{1}\left({g}^{-1}\stackrel{~}{g}\right){K}_{2}\left({g}^{-1}\stackrel{~}{g}\right)\cdots {K}_{k}\left({g}^{-1}\stackrel{~}{g}\right)\right]$$\left[{K}_{1}\left({g}^{-1}\stackrel{~}{g}\right){K}_{2}\left({g}^{-1}\stackrel{~}{g}\right)\cdots {K}_{k}\left({g}^{-1}\stackrel{~}{g}\right)\right]$[K_(1)(g^(-1)( tilde(g)))K_(2)(g^(-1)( tilde(g)))cdotsK_(k)(g^(-1)( tilde(g)))]\left[K_{1}\left(g^{-1} \tilde{g}\right) K_{2}\left(g^{-1} \tilde{g}\right) \cdots K_{k}\left(g^{-1} \tilde{g}\right)\right].
The paper proved that the dynamic group convolution operator's equivariance can be maintained by giving an equivariant constraint on $\mathcal{R}$$\mathcal{R}$R\mathcal{R}.
The results and comparisons on PCAM Dataset is shown as below, which proved that dynamic group convolution can led to improvement in breast tumor classification:
 Method Group $\mathrm{#}$$\mathrm{#}$#\#$\mathrm{#}$ Params AUC DenseNet ${\mathbb{Z}}^{2}$${\mathbb{Z}}^{2}$Z^(2)\mathbb{Z}^{2}${\mathbb{Z}}^{2}$ $128\text{}\mathrm{K}$$128\text{}\mathrm{K}$128K128 \mathrm{~K} $95.5$$95.5$95.595.5$95.5$ DenseNet+ ${\mathbb{Z}}^{2}$${\mathbb{Z}}^{2}$Z^(2)\mathbb{Z}^{2}${\mathbb{Z}}^{2}$ $128\text{}\mathrm{K}$$128\text{}\mathrm{K}$128K128 \mathrm{~K} $95.1$$95.1$95.195.1$95.1$ P4-DenseNet $p4$$p4$p4p 4$p4$ $125\text{}\mathrm{K}$$125\text{}\mathrm{K}$125K125 \mathrm{~K} $94.5$$94.5$94.594.5$94.5$ P4M-DenseNet $p4m$$p4m$p4mp 4 m$p4m$ $119\text{}\mathrm{K}$$119\text{}\mathrm{K}$119K119 \mathrm{~K} $\mathbf{9}\mathbf{6}\mathbf{.}\mathbf{3}$$\mathbf{9}\mathbf{6}\mathbf{.}\mathbf{3}$96.3\mathbf{9 6 . 3}$\mathbf{9}\mathbf{6}\mathbf{.}\mathbf{3}$ Dy-DenseNet ${\mathbb{Z}}^{2}$${\mathbb{Z}}^{2}$Z^(2)\mathbb{Z}^{2}${\mathbb{Z}}^{2}$ $221\text{}\mathrm{K}$$221\text{}\mathrm{K}$221K221 \mathrm{~K} $95.8$$95.8$95.895.8$95.8$ Dy-DenseNet+ ${\mathbb{Z}}^{2}$${\mathbb{Z}}^{2}$Z^(2)\mathbb{Z}^{2}${\mathbb{Z}}^{2}$ $221\text{}\mathrm{K}$$221\text{}\mathrm{K}$221K221 \mathrm{~K} $95.6$$95.6$95.695.6$95.6$ Dy-P4-DenseNet $p4$$p4$p4p 4$p4$ $202\text{}\mathrm{K}$$202\text{}\mathrm{K}$202K202 \mathrm{~K} $95.1$$95.1$95.195.1$95.1$ Dy-P4M-DenseNet $p4m$$p4m$p4mp 4 m$p4m$ $198\text{}\mathrm{K}$$198\text{}\mathrm{K}$198K198 \mathrm{~K} $\mathbf{9}\mathbf{6}\mathbf{.}\mathbf{7}$$\mathbf{9}\mathbf{6}\mathbf{.}\mathbf{7}$96.7\mathbf{9 6 . 7}$\mathbf{9}\mathbf{6}\mathbf{.}\mathbf{7}$
Example3: [3:1] applied 3D GCNNs on Nodule classification and evaluated performance by using Free-Response Operating Characteristic (FROC)[]. ${D}_{4},{D}_{4h},O,{O}_{h}$${D}_{4},{D}_{4h},O,{O}_{h}$D_(4),D_(4h),O,O_(h)D_{4}, D_{4 h}, O, O_{h} are different types of 3D groups.
The results of and comparison is shown as below, from which we can see the 3D G-CNNs can improve classification performance:
 $\mathbf{N}$$\mathbf{N}$N\mathbf{N}$\mathbf{N}$ ${\mathbb{Z}}^{3}$${\mathbb{Z}}^{3}$Z^(3)\mathbb{Z}^{3}${\mathbb{Z}}^{3}$ ${\mathrm{D}}_{4}$${\mathrm{D}}_{4}$D_(4)\mathrm{D}_{4}${\mathrm{D}}_{4}$ ${\mathrm{D}}_{4h}$${\mathrm{D}}_{4h}$D_(4h)\mathrm{D}_{4 h}${\mathrm{D}}_{4h}$ $\mathbf{O}$$\mathbf{O}$O\mathbf{O}$\mathbf{O}$ ${\mathbf{O}}_{h}$${\mathbf{O}}_{h}$O_(h)\mathbf{O}_{h}${\mathbf{O}}_{h}$ 30 $0.252$$0.252$0.2520.252$0.252$ $0.398$$0.398$0.3980.398$0.398$ $0.382$$0.382$0.3820.382$0.382$ $0.562$$0.562$0.5620.562$0.562$ $0.514$$0.514$0.5140.514$0.514$ 300 $0.550$$0.550$0.5500.550$0.550$ $0.765$$0.765$0.7650.765$0.765$ $0.759$$0.759$0.7590.759$0.759$ $0.767$$0.767$0.7670.767$0.767$ $0.733$$0.733$0.7330.733$0.733$ 3,000 $0.791$$0.791$0.7910.791$0.791$ $0.849$$0.849$0.8490.849$0.849$ $0.844$$0.844$0.8440.844$0.844$ $0.830$$0.830$0.8300.830$0.830$ $0.850$$0.850$0.8500.850$0.850$ 30,000 $0.843$$0.843$0.8430.843$0.843$ $0.867$$0.867$0.8670.867$0.867$ $0.880$$0.880$0.8800.880$0.880$ $0.873$$0.873$0.8730.873$0.873$ $0.869$$0.869$0.8690.869$0.869$

### Segmentation

General process of segmentation consists of three steps by using G-CNNs:
1. Convert an image to feature maps,
2. Convert feature maps to feature maps
3. Convert feature maps to a segmentation.
In the final layer, a group-pooling layer is used to ensure equivariant as a function on the plane (for segmentation tasks, where the output is supposed to transform together with the input).[4:2]
Example1: [6] proposed a Rota-Net to segment simultaneous gland and lumen segmentation in colon histology images, incorporating the inherent rotational symmetry within histology images into an encoder-decoder based network by utilizing G-CNNs, specifically using the symmetry group of rotations by multiples of 90∘.
The main process is as below:
1. Define the $\left({\mathbb{Z}}^{2}\to G\right)$$\left({\mathbb{Z}}^{2}\to G\right)$(Z^(2)rarr G)\left(\mathbb{Z}^{2} \rightarrow G\right)-convolution on image $f:{\mathbb{Z}}^{2}\to {\mathbb{R}}^{K}$$f:{\mathbb{Z}}^{2}\to {\mathbb{R}}^{K}$f:Z^(2)rarrR^(K)f: \mathbb{Z}^{2} \rightarrow \mathbb{R}^{K} $\left[f\star w\right]\left(g\right)=\sum _{y\in {\mathbb{Z}}^{2}}\sum _{k=1}^{K}{f}_{k}\left(y\right){w}_{k}\left({g}^{-1}y\right)$$\left[f\star w\right]\left(g\right)=\sum _{y\in {\mathbb{Z}}^{2}} \sum _{k=1}^{K} {f}_{k}\left(y\right){w}_{k}\left({g}^{-1}y\right)$[f***w](g)=sum_(y inZ^(2))sum_(k=1)^(K)f_(k)(y)w_(k)(g^(-1)y){[f \star w](g)=\sum_{y \in \mathbb{Z}^{2}} \sum_{k=1}^{K} f_{k}(y) w_{k}\left(g^{-1} y\right) }
where ${w}_{k}$${w}_{k}$w_(k)w_{k} denotes kernel $k$$k$kk, with corresponding input channel ${f}_{k}$${f}_{k}$f_(k)f_{k} and $g$$g$gg is a roto-translation.
1. Define the $\left(G\to G\right)$$\left(G\to G\right)$(G rarr G)(G \rightarrow G)-convolution on feature maps $f:\mathbb{G}\to {\mathbb{R}}^{K}$$f:\mathbb{G}\to {\mathbb{R}}^{K}$f:GrarrR^(K)f: \mathbb{G} \rightarrow \mathbb{R}^{K} as:
$\left[f\star w\right]\left(g\right)=\sum _{h\in \mathbb{G}}\sum _{k=1}^{K}{f}_{k}\left(h\right){w}_{k}\left({g}^{-1}h\right)$$\left[f\star w\right]\left(g\right)=\sum _{h\in \mathbb{G}} \sum _{k=1}^{K} {f}_{k}\left(h\right){w}_{k}\left({g}^{-1}h\right)$[f***w](g)=sum_(h inG)sum_(k=1)^(K)f_(k)(h)w_(k)(g^(-1)h)[f \star w](g)=\sum_{h \in \mathbb{G}} \sum_{k=1}^{K} f_{k}(h) w_{k}\left(g^{-1} h\right)
1. Define the projection layer: $G\to {\mathbb{Z}}^{2}$$G\to {\mathbb{Z}}^{2}$G rarrZ^(2)G \rightarrow \mathbb{Z}^{2}
The architecture of Rota-Net is as the figure [6:1]:
Figure 5: The architecture of Rota-Net: the yellow box within the input means the part of the image considered at the output. The number at the top of each operation means the number of feature maps produced.
Example2: [7] propose a 3D left-right-reflection equivariant network to segment the anatomical structures of the brain, exploiting the left and right symmetry property of the brain. The segmentation model was 3D U-Net[8], which is a classical segmentation model and it incorporated RE convolutions.
For a multi-channel function $\mathbit{f}$$\mathbit{f}$f\boldsymbol{f} of spatial location $\mathbit{x}\in {\mathbb{Z}}^{3}$$\mathbit{x}\in {\mathbb{Z}}^{3}$x inZ^(3)\boldsymbol{x} \in \mathbb{Z}^{3}, i.e., a $3\mathrm{D}$$3\mathrm{D}$3D3 \mathrm{D} image or feature maps, suppose $R$$R$RR is left-right reflection and $R\mathbit{f}=\mathbit{f}\left({R}^{-1}\mathbit{x}\right)=\mathbit{f}\left(R\mathbit{x}\right)$$R\mathbit{f}=\mathbit{f}\left({R}^{-1}\mathbit{x}\right)=\mathbit{f}\left(R\mathbit{x}\right)$Rf=f(R^(-1)x)=f(Rx)R \boldsymbol{f}=\boldsymbol{f}\left(R^{-1} \boldsymbol{x}\right)=\boldsymbol{f}(R \boldsymbol{x}) where ${R}^{-1}$${R}^{-1}$R^(-1)R^{-1} is the inverse of $R$$R$RR and ${R}^{-1}=R$${R}^{-1}=R$R^(-1)=RR^{-1}=R.
The difference between in G-CNNs between example 1 and example 2 is that the feature map is 3D, and the transformation is left-right-reflection. The comparison of result is shown as below [7:1]:
Figure 6: Example brain tissue segmentation. (A), (E): original and reflected testing images. (B), (F): manual delineations. (C), (G): results of the conventional U-Net trained with reflection augmentation. (D), (H): the results of the RE U-Net.
Example3: [9] proposed a new model of two-pathway-group CNN architecture for brain tumor segmentation, making use of local features and global contextual features at the same time. Equivariance has been maintained to reduce instabilities and overfitting parameter sharing, which could improve the segmentation performance. The following figure is from [9:1]
Figure 7: Two-Pathway-Group CNN architecture (2PG-CNN) showing that the input patch is processed by two different group CNNs. The four blocks in a feature map of both CNNs show a p4 group features map that inherits group CNN properties. The upper CNN represents a local feature map and the lower CNN shows a global feature map.

## Conclusions and Future Opportunities

This paper reviews the basic theory of G-CNNs, and gives some typical examples of its application on medical images. Due to the characteristics of lesions, tissues or structures, medical images are very suitable for using G-CNNs. It can be seen from many examples that compared with the standard CNN, G-CNNs can effectively improve the performance of the model as a more generalized one.
In future studies, we can design the combination of G-CNNs and classification or segmentation models according to the specific characteristics of lesions, tissues, structures due to the more complex symmetries in 3D images. In addition, we can also analyze the latest state-of-the-arts models and explore the possibility of combining with G-CNNs.
Besides, the examples in this review focus on classification and segmentation, but there are also tasks of generation and reconstruction in medical image analysis. Equavariant neural networks can also combine with methods in these tasks. Additionally, frame equavariance may be used in clinical videos analysis.
And the equavariance not only can be explored in CNNs, but also in Transformers. As the great success of transformers has achieved these days, there would also are opportunities for the research of Transformers and their exploiting with equavariance, as well as potential applications in medical image analysis.

## Reference

1. Cohen, T., & Welling, M. (2016, June). Group equivariant convolutional networks. In International conference on machine learning (pp. 2990-2999). PMLR.(http://proceedings.mlr.press/v48/cohenc16.html) ↩ ↩ ↩ ↩ ↩ ↩
2. Van Ginneken, B., Armato III, S. G., de Hoop, B., van Amelsvoort-van de Vorst, S., Duindam, T., Niemeijer, M., ... & Prokop, M. (2010). Comparing and combining algorithms for computer-aided detection of pulmonary nodules in computed tomography scans: the ANODE09 study. Medical image analysis, 14(6), 707-722.(https://www.sciencedirect.com/science/article/pii/S1361841510000587) ↩ ↩
3. Veeling, B. S., Linmans, J., Winkens, J., Cohen, T., & Welling, M. (2018, September). Rotation equivariant CNNs for digital pathology. In International Conference on Medical image computing and computer-assisted intervention (pp. 210-218). Springer, Cham (https://link.springer.com/chapter/10.1007/978-3-030-00934-2_24) ↩ ↩ ↩
4. Li, Y., Cao, G., & Cao, W. (2020, December). A dynamic group equivariant convolutional networks for medical image analysis. In 2020 IEEE International Conference on Bioinformatics and Biomedicine (BIBM) (pp. 1056-1062). IEEE.(https://ieeexplore.ieee.org/abstract/document/9313601) (https://www.sciencedirect.com/science/article/pii/S1361841510000587) ↩
5. Graham, Simon, David Epstein, and Nasir Rajpoot. "Rota-net: Rotation equivariant network for simultaneous gland and lumen segmentation in colon histology images." European Congress on Digital Pathology. Springer, Cham, 2019. (https://link.springer.com/chapter/10.1007/978-3-030-23937-4_13) ↩ ↩
6. Han, Shuo, Jerry L. Prince, and Aaron Carass. Reflection-equivariant convolutional neural networks improve segmentation over reflection augmentation.Medical Imaging 2020: Image Processing. Vol. 11313. International Society for Optics and Photonics, 2020. (https://www.spiedigitallibrary.org/conference-proceedings-of-spie/11313/1131337/Reflection-equivariant-convolutional-neural-networks-improve-segmentation-over-reflection-augmentation/10.1117/12.2549399.full) ↩ ↩
7. Çiçek, Ö., Abdulkadir, A., Lienkamp, S. S., Brox, T., & Ronneberger, O. (2016, October). 3D U-Net: learning dense volumetric segmentation from sparse annotation. In International conference on medical image computing and computer-assisted intervention (pp. 424-432). Springer, Cham. (https://link.springer.com/chapter/10.1007/978-3-319-46723-8_49) ↩
8. Razzak, M. I., Imran, M., & Xu, G. (2018). Efficient brain tumor segmentation with multiscale two-pathway-group conventional neural networks. IEEE journal of biomedical and health informatics, 23(5), 1911-1919.（https://ieeexplore.ieee.org/abstract/document/8481481） ↩ ↩

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