text:
# Fitting subgroup

In mathematics, especially in the area of algebra known as group theory, the Fitting subgroup F of a finite group G, named after Hans Fitting, is the unique largest normal nilpotent subgroup of G. Intuitively, it represents the smallest subgroup which "controls" the structure of G when G is solvable. When G is not solvable, a similar role is played by the generalized Fitting subgroup F*, which is generated by the Fitting subgroup and the components of G.

For an arbitrary (not necessarily finite) group G, the Fitting subgroup is defined to be the subgroup generated by the nilpotent normal subgroups of G. For infinite groups, the Fitting subgroup is not always nilpotent.

## The Fitting subgroup

The nilpotency of the Fitting subgroup of a finite group is guaranteed by Fitting's theorem which says that the product of a finite collection of normal nilpotent subgroups of G is again a normal nilpotent subgroup. It may also be explicitly constructed as the product of the p-cores of G over all of the primes p dividing the order of G.

If G is a finite non-trivial solvable group then the Fitting subgroup is always non-trivial, i.e. if G≠1 is finite solvable, then F(G)≠1. Similarly the Fitting subgroup of G/F(G) will be nontrivial if G is not itself nilpotent, giving rise to the concept of Fitting length. Since the Fitting subgroup of a finite solvable group contains its own centralizer, this gives a method of understanding finite solvable groups as extensions of nilpotent groups by faithful automorphism groups of nilpotent groups.

In a nilpotent group, every chief factor is centralized by every element. Relaxing the condition somewhat, and taking the subgroup of elements of a general finite group which centralize every chief factor, one simply gets the Fitting subgroup again (Huppert 1967, Kap.VI, Satz 5.4, p.686):

${\displaystyle \operatorname {Fit} (G)=\bigcap \{C_{G}(H/K):H/K{\text{ a chief factor of }}G\}.}$

The generalization to p-nilpotent groups is similar.

## The generalized Fitting subgroup

A component of a group is a subnormal quasisimple subgroup. (A group is quasisimple if it is a perfect central extension of a simple group.) The layer E(G) or L(G) of a group is the subgroup generated by all components. Any two components of a group commute, so the layer is a perfect central extension of a product of simple groups, and is the largest normal subgroup of G with this structure. The generalized Fitting subgroup F*(G) is the subgroup generated by the layer and the Fitting subgroup. The layer commutes with the Fitting subgroup, so the generalized Fitting subgroup is a central extension of a product of p-groups and simple groups.

The layer is also the maximal normal semisimple subgroup, where a group is called semisimple if it is a perfect central extension of a product of simple groups.

The definition of the generalized Fitting subgroup looks a little strange at first. To motivate it, consider the problem of trying to find a normal subgroup H of G that contains its own centralizer and the Fitting group. If C is the centralizer of H we want to prove that C is contained in H. If not, pick a minimal characteristic subgroup M/Z(H) of C/Z(H), where Z(H) is the center of H, which is the same as the intersection of C and H. Then M/Z(H) is a product of simple or cyclic groups as it is characteristically simple. If M/Z(H) is a product of cyclic groups then M must be in the Fitting subgroup. If M/Z(H) is a product of non-abelian simple groups then the derived subgroup of M is a normal semisimple subgroup mapping onto M/Z(H). So if H contains the Fitting subgroup and all normal semisimple subgroups, then M/Z(H) must be trivial, so H contains its own centralizer. The generalized Fitting subgroup is the smallest subgroup that contains the Fitting subgroup and all normal semisimple subgroups.

The generalized Fitting subgroup can also be viewed as a generalized centralizer of chief factors. A nonabelian semisimple group cannot centralize itself, but it does act one itself as inner automorphisms. A group is said to be quasi-nilpotent if every element acts as an inner automorphism on every chief factor. The generalized Fitting subgroup is the unique largest subnormal quasi-nilpotent subgroup, and is equal to the set of all elements which act as inner automorphisms on every chief factor of the whole group (Huppert & Blackburn 1982, Chapter X, Theorem 5.4, p. 126):

${\displaystyle \operatorname {Fit} ^{*}(G)=\bigcap \{HC_{G}(H/K):H/K{\text{ a chief factor of }}G\}.}$

Here an element g is in HCG(H/K) if and only if there is some h in H such that for every x in H, xgxh mod K.

## Properties

If G is a finite solvable group, then the Fitting subgroup contains its own centralizer. The centralizer of the Fitting subgroup is the center of the Fitting subgroup. In this case, the generalized Fitting subgroup is equal to the Fitting subgroup. More generally, if G is any finite group, the generalized Fitting subgroup contains its own centralizer. This means that in some sense the generalized Fitting subgroup controls G, because G modulo the centralizer of F*(G) is contained in the automorphism group of F*(G), and the centralizer of F*(G) is contained in F*(G). In particular there are only a finite number of groups with given generalized Fitting subgroup.

## Applications

The normalizers of nontrivial p-subgroups of a finite group are called the p-local subgroups and exert a great deal of control over the structure of the group (allowing what is called local analysis). A finite group is said to be of characteristic p type if F*(G) is a p-group for every p-local subgroup, because any group of Lie type defined over a field of characteristic p has this property. In the classification of finite simple groups, this allows one to guess over which field a simple group should be defined. Note that a few groups are of characteristic p type for more than one p.

If a simple group is not of Lie type over a field of given characteristic p, then the p-local subgroups usually have components in the generalized Fitting subgroup, though there are many exceptions for groups that have small rank, are defined over small fields, or are sporadic. This is used to classify the finite simple groups, because if a p-local subgroup has a known component, it is often possible to identify the whole group (Aschbacher & Seitz 1976).

The analysis of finite simple groups by means of the structure and embedding of the generalized Fitting subgroups of their maximal subgroups was originated by Helmut Bender (Bender 1970) and has come to be known as Bender's method. It is especially effective in the exceptional cases where components or signalizer functors are not applicable.


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Nuclear Physics Seminars at BNL

# Physics of STAR Forward pi^0 Transverse Single Spin Asymmetries (AN) with pA and pp Beams

## by Steven Heppelmann (PSU)

US/Eastern
Description

The STAR collaboration reports a measurement of the transverse single-spin asymmetries, AN, for neutral pions produced in polarized proton collisions with protons (pp), with aluminum nuclei (pAl) and with gold nuclei (pAu) at a nucleon-nucleon center-of-mass energy of 200 GeV. Neutral pions are observed in the forward direction relative to the transversely polarized proton beam, in the pseudorapidity region 2.7<η<3.8. Results are presented for $\pi^{0}$s observed in the STAR forward meson spectrometer electromagnetic calorimeter in narrow Feynman x ($x_F$) and transverse momentum ($p_{T}$) bins, spanning the range $0.17<x_F<0.81$ and $1.7<p_{T}<6.0$  GeV/c. For fixed $x_F<0.47$, the asymmetries are found to rise with increasing transverse momentum. For larger $x_F$, the asymmetry flattens or falls as $p_T$ increases. Parametrizing the ratio $r(A)≡A_{N}(pA)/A_{N}(pp)=A^{P}$ over the kinematic range, the ratio $r(A)$ is found to depend only weakly on $A$, with $⟨P⟩=−0.027\pm0.005$. No significant difference in $P$ is observed between the low-$p_T$ region, $p_T<2.5$  GeV/c, where gluon saturation effects may play a role, and the high-$p_T$ region, $p_T>2.5$  GeV/c. It is further observed that the value of $A_{N}$ is significantly larger for events with a large-$p_T$ isolated $\pi^{0}$ than for events with a nonisolated $\pi^{0}$ accompanied by additional jetlike fragments. The nuclear dependence $r(A)$ is similar for isolated and nonisolated $\pi^{0}$ events.

Organized by

Rongrong Ma

format: plain


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In editing the algebraic geometry notes posted here, prompted by Brian Conrad, I am introducing the notion of radicial morphism. This seems to me to not be a notion that absolutely everyone should see in a first serious schemes course, given the volume of definitions a student must digest (even if this idea is wafer-thin). So I would like to make clear to the learner who this notion is for. But the problem is: I don't know. I have two bits of data: I've never used it professionally, and I know some arithmetic geometers (e.g. in my department) have used it. Thus I ask:

In the answers, I expect to hear multiple interesting answers to the implicit question:

What are radicial morphisms good for?

• Your "This seems to me to not be a notion that absolutely everyone should see" had me rereading the phrase fve times until it made sense. Good work :) Dec 20, 2010 at 0:09
• I don't know, but I'm sure I'd prefer the term "universally injective" or even "geometrically injective" over radicial anything. Dec 20, 2010 at 1:47
• Anyone who uses etale topology or characteristic-$p$ geometry or char-$p$ alg. groups (e.g., ab. varieties or linear algebraic groups) in a serious way will find it useful. For example, "etale + radicial = open immersion" and "radicial integral surjections have no effect on the etale topos" (e.g., passing from sep. closed ground field to an alg. closed ground field when doing an etale cohom. or fundamental gp calculation). As Allen notes, it's often called "universally injective" (never heard of "geom. injective"), though if you do then you need to prove the lemma that justifies the name... Dec 20, 2010 at 1:58
• Allow me to add an example to BCnrd's comment (which I learned from Olsson). Let G be a group scheme of finite type over a field k of char. p, but not smooth over k. Let BG be the classifying stack of G. To compute the etale cohom of BG we'd like to use cohom descent, but pt --> BG is not a presentation (not smooth). The max reduced G_{red} is a group scheme, and is more likely to be smooth (and will be if we make a finite extension of k). The morphism BG_{red} --> BG is rep and radicial, so they have the same cohom, and we can apply descent. This works over a general base, using devissage. Dec 20, 2010 at 11:15
• Dear shenghao: As tiny correction, for imperfect $k$, $G_{\rm{red}}$ may not be subgp; see Ex. A.8.3 in "Pseudo-reductive groups" for an example over any imperfect field (with $G$ finite). As you say, if we first make a suitable finite purely insep. (!) extn on $k$ and then kill the nilpotents we get a smooth subgp. That's good enough for purposes of etale cohomology, but for other purposes can be a nuisance. Here's a version avoiding change of $k$: the quotient of $G$ by inf'tml kernel of (perhaps not-flat!) $n$-fold rel. Frob. is smooth for suff. large $n$; see SGA3, VII_A, 8.3. Dec 20, 2010 at 15:53

Grothendieck's version of the Ax-Grothendieck theorem is that if you have an $S$-scheme $X$ of finite type, then any radicial $S$-endomorphism $f: X\to X$ is surjective: EGA IV_3, Prop. (10.4.11).
Such a wonderfully unexpected theorem certainly justifies the investment in learning what radicial means...

• Thanks for the answer! I checked the Wikipedia page for the Ax-Grothendieck theorem which led to the confirmation of my speculation that polynomial algebras over a commutative ring $R$ are Hopfian (any $R$-endomorphism is an automorphism). Feb 15, 2017 at 6:54

A reasonable prerequisite for a course in Algebraic Geometry is a course in Galois Theory including some characteristic $p$ results. Every finite algebraic extension of fields $E|K$ (meaning $K \subset E$) factors trough an intermediate separable extension $E_s|K$ such that $E|E_s$ is purely inseparable. This theorem sheds light on the structure of field extensions.

I consider finite separable extensions of fields as the one important idea behind the concept of etale map. An etale map is a flat family of finite separable extensions. The counterpart of purely inseparable extensions in geometry are radical morphisms. The topological characterization as universally injective morphisms makes them interesting also. I would not discuss etale maps without treating radical morphisms.

Another reason is that the Frobenius maps (relative, absolute...) are the essential tools for understanding the cohomology of varieties in characteristic $p$ and this is probably the main example of radical morphisms. For these two reasons I would say that radical morphisms belong in an introductory course on schemes.

This isn't exactly the same thing, but it's obviously related (and its a special case of radicial), so it might be a place to look. Brian Conrad pointed this relation out to me several months ago (here on mathoverflow). Radicial morphisms are closely related to (but slightly weaker than) the following notion which appears in algebraic geometry and especially in commutative algebra:

An extension of rings $R \subseteq S$ is called weakly subintegral if the induced map on Spec is a bijection (this is where it differs from radicial) and if the extensions of residue fields is inseparable. This is the same thing as being a universal homeomorphism if I recall correctly.

Having no finite bijective (on Spec) birational radicial extensions is called being weakly normal. Weak normality probably has a lot of papers written about it over the years.

Especially in equal-characteristic zero, being weakly normal is also sometimes called being semi-normal (although semi-normality is a distinct notion in general). Both these conditions are used in the study of various moduli spaces of algebraic varieties.

Here is a fun example of a place where radicial morphisms get used: in the valuative criterion for when a map of schemes is a locally closed embedding (i.e. the composition of a closed embedding and an open embedding), due to Mochizuki.

Roughly speaking, this valuative criterion * says that if we are given a map $$X \to Y$$ and a map from the spectrum of a DVR $$T$$ to $$Y$$ such that both points of the DVR map to $$X$$ with the natural diagrams commuting, then we have a map from $$T$$ to $$X$$ making the diagrams commute.

Finite type maps of noetherian schemes satisfy * if and only if they can be expressed as the composition of a radicial map and an open immersion. From this, one deduces that finite type maps of noetherian schemes are locally closed immersions if and only if they are monomorphisms and satisfy *.

The only proof I know for this valuative criterion for locally closed immersions, which seems like a natural thing to want, goes by first proving the statement about radicial maps.

Of course, none of this should go in a first course on schemes, but I just thought it was a fun application of radicial maps.


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# zbMATH — the first resource for mathematics

The Ricci flow on the 2-sphere. (English) Zbl 0734.53033
Let $$(M,g)$$ be a compact oriented Riemannian surface. Hamilton’s Ricci flow is defined as the solution $$g(x,t)$$ of the differential equation $$(\partial /\partial t)g(x,t)=(r-R(x,t))g(x,t)$$ $$(x\in M,\;t>0)$$, where $$g$$ is the metric, $$R$$ is the scalar curvature of $$g$$, and $$r$$ is the average of $$R$$. Generalizing a result of R. S. Hamilton [Mathematics and general relativity, Proc. AMS-IMS-SIAM Jt. Summer Res. Conf., Santa Cruz/Calif. 1986, Contemp. Math. 71, 237–262 (1988; Zbl 0663.53031)], the author proves the following theorem: If $$g$$ is any metric on $$S^ 2$$, then under Hamilton’s Ricci flow, the Gaussian curvature becomes positive in finite time.
As a corollary the author obtains the following: If $$g$$ is any metric on a Riemannian surface, then under Hamilton’s Ricci flow, $$g$$ converges to a metric of constant curvature.

##### MSC:
 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 35K55 Nonlinear parabolic equations 58J35 Heat and other parabolic equation methods for PDEs on manifolds 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
##### Keywords:
Ricci flow; scalar curvature; Gaussian curvature
Full Text:


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# Magnetic helicity

In plasma physics, magnetic helicity is the extent to which a magnetic field "wraps around itself". It is a generalization of the topological concept of linking number to the differential quantities required to describe the magnetic field. As with many quantities in electromagnetism, magnetic helicity (which describes magnetic field lines) is closely related to fluid mechanical helicity (which describes fluid flow lines).

If magnetic field lines follow the strands of a twisted rope, this configuration would have nonzero magnetic helicity; left-handed ropes would have negative values and right-handed ropes would have positive values.[clarification needed]

Formally,

$H = \int {\mathbf A}\cdot{\mathbf B}\,d^3{\mathbf r}$

where

${\mathbf B}$ is the magnetic field strength
${\mathbf B}=\nabla\times{\mathbf A}$;
$\mathbf A$ is the vector potential of ${\mathbf B}$

Magnetic helicity has units of Wb2 (webers squared). It is a conserved quantity in electromagnetic fields, even when magnetic reconnection dissipates energy. The concept is useful in solar dynamics and in hydromagnetic dynamo theory.[1] Magnetic helicity is a gauge-dependent quantity, because $\mathbf A$ can be redefined by adding a gradient to it (gauge transformation). However, for perfectly conducting boundaries or periodic systems without a net magnetic flux, the magnetic helicity is gauge invariant. A gauge-invariant relative helicity has been defined for volumes with non-zero magnetic flux on their boundary surfaces[2] If the magnetic field is turbulent and weakly inhomogeneous a magnetic helicity density and its associated flux can be defined in terms of the density of field line linkages.[3]

## References

1. ^ Brandenburg, A. (2009). "Hydromagnetic Dynamo Theory". Scholarpedia 2 (3): 2309. Bibcode:2007SchpJ...2.2309B. doi:10.4249/scholarpedia.2309. rev #73469.
2. ^ Berger, M.A. (1999). "Introduction to magnetic helicity". Plasma Physics and Controlled Fusion 41 (12B): 167. Bibcode:1999PPCF...41..167B. doi:10.1088/0741-3335/41/12B/312.
3. ^ Subramanian, K.; Brandenburg, A. (2006). "Magnetic helicity density and its flux in weakly inhomogeneous turbulence". The Astrophysical Journal Letters 648: L71–L74. arXiv:astro-ph/0509392. Bibcode:2006ApJ...648L..71S. doi:10.1086/507828.


text:
Volume 33, Issue 2
Monomial Derivations Without Darboux Polynomials

Commun. Math. Res., 33 (2017), pp. 185-192.

Published online: 2019-11

Cited by

Export citation
• Abstract

In this paper, it is proved that a monomial derivation d of $k[x, y, z]$ has no Darboux polynomials if and only if $d$ is a strict derivation with a trivial ring of constants, and we give the specific conditions when it has no Darboux polynomials.

• Keywords

derivation, monomial derivation, Darboux polynomial, ring of constants

13N15, 12H05

jtlimath@qq.com (Jiantao Li)

• BibTex
• RIS
• TXT
@Article{CMR-33-185, author = {Jiantao and Li and jtlimath@qq.com and 5482 and School of Mathematics, Liaoning University, Shenyang, 110036 and Jiantao Li}, title = {Monomial Derivations Without Darboux Polynomials}, journal = {Communications in Mathematical Research }, year = {2019}, volume = {33}, number = {2}, pages = {185--192}, abstract = {

In this paper, it is proved that a monomial derivation d of $k[x, y, z]$ has no Darboux polynomials if and only if $d$ is a strict derivation with a trivial ring of constants, and we give the specific conditions when it has no Darboux polynomials.

}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2017.02.10}, url = {http://global-sci.org/intro/article_detail/cmr/13398.html} }
TY - JOUR T1 - Monomial Derivations Without Darboux Polynomials AU - Li , Jiantao JO - Communications in Mathematical Research VL - 2 SP - 185 EP - 192 PY - 2019 DA - 2019/11 SN - 33 DO - http://doi.org/10.13447/j.1674-5647.2017.02.10 UR - https://global-sci.org/intro/article_detail/cmr/13398.html KW - derivation, monomial derivation, Darboux polynomial, ring of constants AB -

In this paper, it is proved that a monomial derivation d of $k[x, y, z]$ has no Darboux polynomials if and only if $d$ is a strict derivation with a trivial ring of constants, and we give the specific conditions when it has no Darboux polynomials.

Jiantao Li. (2019). Monomial Derivations Without Darboux Polynomials. Communications in Mathematical Research . 33 (2). 185-192. doi:10.13447/j.1674-5647.2017.02.10
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Evaluaten’th derivative at x = a

I’ve found a few more problems about derivatives online and I found the following problem:

Evaluate $\frac{d^n}{dx^n}x^n$ at x=a.

At first I thought that this would be the same as the power rule, but I think I’ve been thinking about this wrong so I might be wrong. I have some difficulties in solving this derivative problem, even though you can always use some derivatives calculator or even $\frac{d^n}{dx^n}x^n=\frac{d^n}{dx^n}(ax^n)$ and so on.
Since I really can’t think of anything that seems to work, I’d like to ask you how to solve this derivative problem.

A:

Use the chain rule:
\begin{align}
\frac{d^n}{dx^n}(x^n) &= (x^n) \frac{d^n}{dx^n}1 & &\text{(since we are not dividing by 0)} \\
&= n!x^{n-1} \\
&= n!x^{ -1} \\
&= n!x \cdot x^{n-1} \\
&= (n! \cdot x) \cdot x^{n-1} \\
&= (n \cdot x^{n-1})(n \cdot x) \\
&= n \cdot n \cdot x^{n-1} \cdot
37a470d65a

#### Frank Castle

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text:
# Wilcoxon rank sum test methodology

I'm comparing the performance of two algorihtms $A$ and $B$ based on a metric.

For each algorithm, I have $31$ independent samples that represent its performance. This samples are grouped in two sets: $X$ and $Y$ (algoritmh $A$ and algorithm $B$, respectively).

For this task, I'm using a Wilcoxon rank-sum test with a level of significance $\alpha=0.05$ on R

I have selected this nonparametric test because it makes no assumption about data distribution.

This is my methodology:

1. Hypothesis

Let $E(X)$ and $E(Y)$ be the means of $X$ and $Y$ respectively. Then, the one-tailed test is defined as follows:

-$H_0$: $E(X) = E(Y)$
(the performance of both algorithms is similar)

-$H_1$: $E(X) > E(Y)$
(the performance of $A$ is better than $B$)

2. Means

$E(X) = 36.87548$, $E(Y)=37.72585$

> summary(X) Min. 1st Qu. Median Mean 3rd Qu. Max. 0.00 35.09 45.34 36.88 46.63 48.05

> summary(Y) Min. 1st Qu. Median Mean 3rd Qu. Max. 9.332 34.860 41.120 37.730 42.510 48.110

3. Compare $A$ vs $B$

p = wilcox.test(X, Y, alternative="greater", conf.level=0.95)$p.value If$p=0.0170 < \alpha = 0.05$, then$E(X)$should be greater than$E(Y)$(It's what I think, but it's not greater!), We can reject$H_0$, so we can conclude that: algorithm$A$outperforms algorithm$B$with a significance level of$\alpha=0.05$. 4. Compare$B$vs$A$p = wilcox.test(Y, X, alternative="greater", conf.level=0.95)$p.value

If $p=0.9835 > \alpha$, then $E(X)$ and $E(Y)$ are similar (Again, It's my guess). We can not reject $H_0$, therefore we can conclude that: $A$ and $B$ have similar performance.

Data

X = c(  45.51885768, 35.65081119, 44.60124311, 15.39979541, 48.05143243, 47.90604081,
7.58163868,  0.00000000, 40.94718019, 45.34194687, 28.55451125, 46.15113458,
48.03542321, 47.91413840, 45.38912357, 47.10730083, 47.42726563, 47.80316539,
34.51956662,  0.05853162, 45.29245167, 48.00199937, 45.28839538, 44.89017125,
45.47222435,  7.32111177, 43.35755055, 45.52413737, 45.53528261, 45.45233121,
3.04515436)

Y = c(  41.603296, 43.005620, 38.345339, 28.483733, 41.342548, 44.344933, 34.030309,
42.012604, 35.718175, 45.203532, 40.482022, 45.345594, 41.155187, 41.141522,
48.111677, 41.117034, 34.713158, 44.972073, 35.091889, 34.206018,  9.332199,
39.776291, 28.236449, 13.792789, 35.016681, 41.699073, 41.203090, 47.988765,
36.496740, 45.346652, 30.186485)

Questions

1. is The Wilcoxon rank sum test based on means (as I think in steps 3 and 4) or is based on medians?

2. Is it possible that $A$ can be better than $B$ and at the same time $E(X) < E(Y)$?

3. Is the following methodology correct?

for x in (X, Y)

for y in (X, Y)

if x == y: continue

p = wilcox.test(x, y, alternative="greater", conf.level=0.95)

if p < alpha:
x is better than y

else
if p > alpha:
p = wilcox.test(x, y, alternative="less", conf.level=0.95)

if p < alpha:
x is worse than y

else
x and y have similar performance

4. If I wish to test $A$ against several algorithms $B$, $C$, $...$, what could be the best approach to take?

I have selected this nonparametric test because it makes no assumption about data distribution.

This is not quite the case; it makes some assumptions (such as continuity), it just doesn't assume a specific functional form.

is The Wilcoxon rank sum test based on means (as I think in steps 3 and 4) or is based on medians?

Neither. It's the median of pairwise differences (two sample Hodges-Lehmann difference) - that we're dealing with.

See this post for some discussion on that point (near the top of the post).

As whuber quite rightly points out below, under the location-shift alternative, it's a difference in means or medians as much as it is a median of pairwise differences.

See this post for a discussion of both the location-shift alternative and the more general alternative that the Wilcoxon-Mann-Whitney is sensitive to; there's some more discussion at the end of the post here

Is it possible that A can be better than B and at the same time $E(X)<E(Y)$?

Certainly, if by 'better' you mean "has a high median pairwise difference".

Note that your density displays show a roughly similar asymmetric shape but quite different spread; that's one way (of a number of ways) you might see it. Different shapes but similar spread can also produce it. If there's only a shift in location, the difference in population means and population median-pairwise-difference will be the same - but even with a pure location-shift in the populations, the samples might show opposite shifts.

Is the following methodology correct?

As expressed I don't understand it. For example, the comparison "if x==y" doesn't make sense - why would the samples be identical, and if they were, what would be the point in proceeding, since no test can find a difference?

If I wish to test A against several algorithms B, C, ..., what could be the best approach to take?

What would be best depends on many things which I don't have the information to answer (if you want a nonparametric test I'd suggest considering permutation tests with good power against whatever alternative is of primary interest). The $k$-sample equivalent of the Wilcoxon-Mann-Whitney would be the Kruskal-Wallis test, so if you're happy with the WMW, you might consider the KW.

• Another way to view this test is that it assumes the two (continuous) distributions have identical shapes (after possibly undergoing a common monotonic transformation) but may have been location-shifted (post-transformation). Although this is a non-trivial distributional assumption it is still nonparametric. With it, one may indeed interpret the test as comparing medians. (The equal-shape assumption appears to be at least approximately true of the data shown in the question's graphic. A location shift in times could arise, say, by removing a $O(1)$ bottleneck in the algorithm.) – whuber Apr 10 '14 at 21:56
• @whuber A 'same shape' assumption might be plausible, but the spreads don't seem to be: the leftmost lower peak and the rightmost higher peak are both from the same sample. That (and the related reversal in means vs median pairwise difference) suggests a straight location-shift may not be the case; it looks to me likely to be more complicated than a simple removal of an O(1) operation. – Glen_b -Reinstate Monica Apr 10 '14 at 22:49
• You're right: I simply misread the graph! My eyes focused on common patterns and neglected the colors altogether. Clearly the spreads differ. What could rescue that analysis would be to identify a simple monotonic transformation after which there is a change of location. For instance, replacement of a $O(2)$ algorithm by a $O(1)$ algorithm could result in an apparent location shift upon taking logarithms. But, like you, I am not too optimistic about that (based on a more careful look at the graphics!). – whuber Apr 11 '14 at 13:46
• @whuber My eyes did the same as yours the first few times I looked at it. – Glen_b -Reinstate Monica Apr 11 '14 at 14:54


text:
Documentation/UserGuide/Restart Files

Restart files (sometimes called checkpoints) are useful when a calculation must be continued from a previous point. The files contain enough information, and with the necessary accuracy, that a restart calculation generates identical data (to all significant digits) to a calculation run continuously. Athena defines its own format for restart files. The file restart.c contains all the functions needed to read and write these files.

The following example shows an <output> block in an input file that generates restart files:

    <output2>
out_fmt = rst            # Restart dump
dt      = 1.0            # time increment between outputs

The time increment <output>/dt is measured in problem time, and should be set to give the desired output frequency of files (usually writing one restart dump every 6 hours of wall clock time is useful). For jobs run in parallel with MPI, there will be one restart file per process, and the restarted job must use the same number of processors as there are restart files.

If the problem contains special, user-defined data, these must be added to the restart dumps. Athena provides a mechanism for automatically adding such data. In the problem generator, two functions are provided:

    void problem_write_restart(MeshS *pM, FILE *fp)
{
return;
}

{
return;
}

Generally these functions are empty, but if necessary they can be used to read and write extra parameters using fwrite and fread statements, and the file pointer *fp passed as the second argument to these functions. They can also set problem-specific boundary conditions on restart, and enroll user-defined outputs and variables. The problem generator src/prob/rt.c contains an example of usage. See the Programmer Guide for more information about the structures in the argument list to these functions.

To read a restart file, the -r command is used on the command line:

    % athena -r myfile.rst

Note that an input file, specified by -i myinput, is not needed for restarts. This is because the restart file contains the original input file, in ASCII format, at the beginning, from which all the necessary parameters are read by par.c on restart.
This also makes restart files self-documenting: the values of input parameters used in the calculation that generated the restart file can be read with an editor. If an input file is specified along with a restart,

    % athena -r myfile.rst -i myinput

then the values in myinput overwrite the values stored in the restart file itself. Alternatively, values in the input file can be overwritten using the command line, for example:

    % athena -r myfile.rst time/tlim=20.0

Usually the time/tlim parameter needs to be modified on restart. For parallel jobs run with MPI, only the name of the restart file for the root (rank 0) process needs to be specified, all other processes will create their own appropriate restart filename based on this name. See Running Athena on Parallel Processors.


text:
# Condition for two conics to intersect in four concyclic points

I am facing trouble with the following question

What is the condition that the two conic sections $ax_1^2+2h_1xy+b_1y^2+2g_1x+2f_1y+c_1=0$ and $ax_2^2+2h_2xy+b_2y^2+2g_2x+2f_2y+c_2=0$ intersects each other in four concyclic points.

I took the four points as unknown terms and since the points satisfies both the equations,I tried to put the points in each of the equations and tried to solve for the coordinates but it became useless and I got nothing.

Is there some way to arrive at the condition for the concyclicity of the intersecting points?Any help would be appreciated.Thanks.

Eliminate the cross term:

$$a(h_2-h_1)x^2+(b_1 h_2-b_2 h_1)y^2+2(g_1h_2-g_2h_1)x+2(f_1h_2-f_2h_1)y+(c_1h_2-c_2h_1)=0$$

Equating $x^2$ and $y^2$ terms: \begin{array}{rclll} a(h_2-h_1) &=& b_1 h_2-b_2 h_1 && \\ h_2 &=& \dfrac{h_1(a-b_2)}{a-b_1} &,& (a\ne b_1) \\ b_2 &=& a &,& (a= b_1) \\ \end{array}

• I didnot get how you arrived at the equation of the circle.Can you please elaborate it.Thanks. – Navin Dec 1 '16 at 14:08
• The quadratic terms in circles must be in the form of $A(x^2+y^2)$. That's no cross terms $xy$ and same coefficients in $x^2$ and $y^2$. By the way, the two conics satisfy the condition not necessarily intersecting at four points. – Ng Chung Tak Dec 1 '16 at 14:09
• Yes,I know that coefficient of $x^2$ and $y^2$ are the same.but how did you arrived at the equation of the circle passing through the four points of intersection. – Navin Dec 1 '16 at 14:11
• For intersection, $(x,y)$ satisfies both $C_1=0$ and $C_2=0$ and hence their linear combination $\lambda C_1+\mu C_2=0$ (which is a family of conics also passing through those intersections). If there exists $\lambda$, $\mu$ such that $\lambda C_1+\mu C_2=0$ is a circle, then the points are concyclic. – Ng Chung Tak Dec 1 '16 at 14:15

Given $4$ points, the family of conics $ax^2+hxy+by^2+gx+fy+c = 0$ going through them is given by a projective line (it gives $4$ linear equations on the coefficients of the conic).

(and conversely, given $2$ conics in generic position, they meet in $4$ points, and the projective line joining those two conics give you equations of conics that also go through those points, so every projective line of conics correspond to $4$ points)

And such a conic is a circle when $h=0$ and $a=b$.

Thus, if you have two conics, they intersect at $4$ cocyclic points if and only if there is an equation of a circle in the projective line joining the two equations.

This means there exists $s,t$ not both zero such that $sh_1+th_2 = 0$ and $sa_1+ta_2 = sb_1+tb_2$.

If for example $h_1$ or $h_2$ is nonzero we have to pick $t = -h_1$ and $s=h_2$ (up to a multiplicative constant), and so we need $h_2a_1-h_1a_2 = h_2b_1-h_1b_2$


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# Why do the Lagrangian and Hamiltonian formulations give the same conserved quantities for the same symmetries?

The connection between symmetries and conservation laws can be viewed through the lens of both Lagrangian and Hamiltonian mechanics. In the Lagrangian picture we have Noether's theorem. In the Hamiltonian picture we have the so-called "moment map." When we consider the same "symmetry" in both viewpoints, we get the exact same conserved quantities. Why is that?

I'll give an example. For a 2D particle moving in a central potential, the action is

$$S = \int dt \Bigl(\frac{m}{2} ( \dot q_1^2 + \dot q_2^2) - V(q_1^2 + q_2^2)\Bigr).$$

We can then consider the $$SO(2)$$ rotational symmetry that leaves this action invariant. When we vary the path by a infinitesimal time dependent rotation,

$$\delta q_1(t) = - \varepsilon(t) q_2(t)$$ $$\delta q_2(t) = \varepsilon(t) q_1(t)$$ we find that the change in the action is

$$\delta S = \int dt \Bigl( m ( \dot q_1 \delta \dot q_1 + \dot q_2 \delta \dot q_2) - \delta V \Bigr)$$ $$= \int dt m (q_1 \dot q_2 - q_2 \dot q_1)\dot \varepsilon(t)$$

As $$\delta S = 0$$ for tiny perturbations from the actual path of the particle, an integration by parts yields

$$\frac{d}{dt} (m q_1 \dot q_2 - m q_2 \dot q_1) = \frac{d}{dt}L = 0$$ and angular momentum is conserved.

In the Hamiltonian picture, when we rotate points in phase space by $$SO(2)$$, we find that $$L(q,p) = q_1 p_2 - q_2p_1$$ remains constant under rotation. As the Hamiltonian is $$H$$, we have

$$\{ H, L\} = 0$$ implying that angular momentum is conserved under time evolution.

In the Lagrangian picture, our $$SO(2)$$ symmetry acted on paths in configuration space, while in the Hamiltonian picture our symmetry acted on points in phase space. Nevertheless, the conserved quantity from both is the same angular momentum. In other words, our small perturbation to the extremal path turned out to be the one found by taking the Poisson bracket with the derived conserved quantity:

$$\delta q_i = \varepsilon(t) \{ q_i, L \}$$

Is there a way to show this to be true in general, that the conserved quantity derived via Noether's theorem, when put into the Poisson bracket, re-generates the original symmetry? Is it even true in general? Is it only true for conserved quantities that are at most degree 2 polynomials?

Edit (Jan 23, 2019): A while ago I accepted QMechanic's answer, but since then I figured out a rather short proof that shows that, in the "Hamiltonian Lagrangian" framework, the conserved quantity does generate the original symmetry from Noether's theorem.

Say that $$Q$$ is a conserved quantity:

$$\{ Q, H \} = 0.$$ Consider the following transformation parameterized by the tiny function $$\varepsilon(t)$$: $$\delta q_i = \varepsilon(t)\frac{\partial Q}{\partial p_i} \\ \delta p_i = -\varepsilon(t)\frac{\partial Q}{\partial q_i}$$ Note that $$\delta H = \varepsilon(t) \{ H, Q\} = 0$$. We then have \begin{align*} \delta L &= \delta(p_i \dot q_i - H )\\ &= -\varepsilon\frac{\partial Q}{\partial q_i} \dot q_i - p_i \frac{d}{dt} \Big( \varepsilon\frac{\partial Q}{\partial p_i} \Big) \\ &= -\varepsilon\frac{\partial Q}{\partial q_i} \dot q_i - \dot p_i \varepsilon\frac{\partial Q}{\partial p_i} + \frac{d}{dt} \Big( \varepsilon p_i \frac{\partial Q}{\partial p_i}\Big) \\ &= - \varepsilon \dot Q + \frac{d}{dt} \Big( \varepsilon p_i \frac{\partial Q}{\partial p_i}\Big) \\ \end{align*}

(Note that we did not use the equations of motion yet.) Now, on stationary paths, $$\delta S = 0$$ for any tiny variation. For the above variation in particular, assuming $$\varepsilon(t_1) = \varepsilon(t_2) = 0$$,

$$\delta S = -\int_{t_1}^{t_2} \varepsilon \dot Q dt$$

implying that $$Q$$ is conserved.

Therefore, $$Q$$ "generates" the very symmetry which you can use to derive its conservation law via Noether's theorem (as hoped).

• What of symplectic invariants, like the area $dq\wedge dp$, which have no analogue in Lagrangian mechanics? – ZeroTheHero Aug 17 '17 at 19:25
• The conserved quantity obtained through Noether's theorem is actually the generator of infinitesimal canonical transformations and these transformations form a symmetry group. For example, rotations can be viewed as active canonical transformations generated by angular momentum. Of course there is a lot to cover here, but I think the starting point is the theory of infinitesimal canonical transformations. – Diracology Aug 17 '17 at 19:27
• I know that conserved quantities given by the "moment map" can be found for symplectomorphisms of the phase space, and the Lie algebra of the symplectic group corresponds to degree 2 polynomials on phase space. However, it seems that the answer to my question shouldn't involve any sophisticated mathematical machinery because it is such a basic question. – user1379857 Aug 17 '17 at 19:45
• Can you explain how you rotate points in phase space by SO(2) to arrive at your conserved L? – lalala Aug 17 '17 at 20:01
• For a rotation $R \in SO(2)$, act on both $\begin{pmatrix} q_1 \\ q_2 \end{pmatrix}$ and $\begin{pmatrix} p_1 \\ p_2 \end{pmatrix}$ by $R$. The quantity $L$ is not changed by such a rotation. – user1379857 Aug 17 '17 at 20:05

In this answer let us for simplicity restrict to the case of a regular Legendre transformation in a point mechanical setting, cf. this related Phys.SE post. (Generalizations to field theory and gauge theory are in principle possible, with appropriate modifications of conclusions.)

1. On one hand, the action principle for a Hamiltonian system is given by the Hamiltonian action $$S_H[q,p] ~:= \int \! dt ~ L_H(q,\dot{q},p,t).\tag{1}$$ Here $L_H$ is the so-called Hamiltonian Lagrangian $$L_H(q,\dot{q},p,t) ~:=~\sum_{i=1}^n p_i \dot{q}^i - H(q,p,t). \tag{2}$$ In the Hamiltonian formulation there is a bijective correspondence between conserved quantities $Q_H$ and infinitesimal (vertical) quasi-symmetry transformations $\delta$, as showed in my Phys.SE answers here & here. It turns out that a quasi-symmetry transformation $\delta$ is a Hamiltonian vector field generated by a conserved quantity $Q_H$: $$\delta z^I~=~ \{z^I,Q_H\}\varepsilon,\qquad I~\in~\{1, \ldots, 2n\}, \qquad \delta t~=~0,$$ $$\delta q^i~=~\frac{\partial Q_H}{\partial p_i}\varepsilon, \qquad \delta p_i~=~ -\frac{\partial Q_H}{\partial q^i}\varepsilon, \qquad i~\in~\{1, \ldots, n\},\tag{3}$$

2. On the other hand, if we integrate out the momenta $p_i$, we get the corresponding Lagrangian action $$S[q] ~= \int \! dt ~ L(q,\dot{q},t),\tag{4}$$ cf. this related Phys.SE post. The Hamiltonian eqs. $$0~\approx~\frac{\delta S_H}{\delta p_i} ~=~\dot{q}^i-\frac{\partial H}{\partial p_i} \tag{5}$$ for the momenta $p_i$ yield via the Legendre transformation the defining relation $$p_i~\approx~ \frac{\partial L}{\partial \dot{q}^i}\tag{6}$$ of Lagrangian momenta. Eqs. (5) & (6) establish a bijective correspondence between velocities and momenta.

3. If we take this bijective correspondence $\dot{q} \leftrightarrow p$ into account it is clear that Hamiltonian and Lagrangian conserved charges $$Q_H(q,p,t)~\approx~Q_L(q,\dot{q},t) \tag{7}$$ are in bijective correspondence. Below we will argue that the same is true for (vertical) infinitesimal quasi-symmetries on both sides.

4. On one hand, if we start with a (vertical) infinitesimal quasi-symmetry in (Hamiltonian) phase space $$\varepsilon \frac{df^0_H}{dt}~=~\delta L_H ~=~\sum_{i=1}^n\frac{\delta S_H}{\delta p_i}\delta p_i + \sum_{i=1}^n\frac{\delta S_H}{\delta q^i}\delta q^i + \frac{d}{dt}\sum_{i=1}^n p_i~\delta q^i ,\tag{8}$$ it can with the help of eq. (5) be restricted to a (vertical) infinitesimal quasi-symmetry within the (Lagrangian) configuration space: $$\varepsilon \frac{df^0_L}{dt}~=~\delta L ~=~ \sum_{i=1}^n\frac{\delta S}{\delta q^i}\delta q^i + \frac{d}{dt}\sum_{i=1}^n p_i~\delta q^i ,\tag{9}$$ In fact we may take $$f^0_L(q,\dot{q},t)~\approx~f^0_H(q,p,t) \tag{10}$$ the same. The restriction procedure also means that the bare Noether charges $$Q^0_H(q,p,t)~\approx~Q^0_L(q,\dot{q},t) \tag{11}$$ are the same, since there are no $\dot{p}_i$ appearance.

5. Conversely, if we start with an infinitesimal quasi-symmetry in (Lagrangian) configuration space, we can use Noether's theorem to generate a conserved quantity $Q_L$, and in this way close the circle.

6. Example: Consider $n$ harmonic oscillators with Lagrangian $$L~=~\frac{1}{2}\sum_{k,\ell=1}^n \left(\dot{q}^k g_{k\ell}\dot{q}^{\ell} - q^k g_{k\ell} q^{\ell}\right),\tag{12}$$ where $g_{k\ell}$ is a metric, i.e. a non-degenerate real symmetric matrix. The Hamiltonian reads $$H~=~\frac{1}{2}\sum_{k,\ell=1}^n \left( p_k g^{k\ell} p_{\ell} + q^k g_{k\ell} q^{\ell}\right) ~=~\sum_{k,\ell=1}^n z^{k \ast} g_{k\ell} z^{\ell},\tag{13}$$ with complex coordinates $$z^k~:=~\frac{1}{\sqrt{2}}(q^k+ip^k), \qquad p^k~:=~\sum_{\ell=1}^ng^{k\ell}p_{\ell}, \qquad \{z^{k \ast},z^{\ell}\}~=~ig^{k\ell}. \tag{14}$$ The Hamiltonian Lagrangian (2) reads $$L_H~=~\sum_{k=1}^n p_k \dot{q}^k - H ~=~\frac{i}{2}\sum_{k,\ell=1}^n \left( z^{k \ast} g_{k\ell} \dot{z}^{\ell} - z^{k} g_{k\ell} \dot{z}^{\ell\ast} \right) - H, \tag{15}$$ Hamilton's eqs. are $$\dot{z}^k~\approx~-iz^k, \qquad \dot{q}^k~\approx~p^k, \qquad \dot{p}^k~\approx~-q^k. \tag{16}$$ Some conserved charges are $$Q_H ~=~ \sum_{k,\ell=1}^n z^{k \ast} H_{k\ell} z^{\ell} ~=~\sum_{k,\ell=1}^n \left( \frac{1}{2}q^k S_{k\ell} q^{\ell} +\frac{1}{2}p^k S_{k\ell} p^{\ell}+ p^k A_{k\ell} q^{\ell}\right), \tag{17}$$ where $$H_{k\ell}~:=~S_{k\ell}+i A_{k\ell}~=~H_{\ell k}^{\ast} \tag{18}$$ is an Hermitian $n\times n$ matrix, which consists of a symmetric and an antisymmetric real matrix, $S_{k\ell}$ and $A_{k\ell}$, respectively. The conserved charges (17) generate an infinitesimal $u(n)$ quasi-symmetry of the Hamiltonian action $$\delta z_k~=~ \varepsilon\{z_k , Q_H\} ~=~-i \varepsilon\sum_{\ell=1}^n H_{k\ell} z^{\ell},$$ $$\delta q_k ~=~ \varepsilon\sum_{\ell=1}^n \left( A_{k\ell} q^{\ell} +S_{k\ell} p^{\ell} \right), \qquad \delta p_k ~=~ \varepsilon\sum_{\ell=1}^n \left( -S_{k\ell} q^{\ell} +A_{k\ell} p^{\ell} \right). \tag{19}$$ The bare Noether charges are $$Q^0_H ~=~\sum_{k,\ell=1}^n p^k \left( A_{k\ell} q^{\ell} +S_{k\ell} p^{\ell} \right). \tag{20}$$ Also $$f^0_H~=~\frac{1}{2}\sum_{k,\ell=1}^n \left( \frac{1}{2}p^k S_{k\ell} p^{\ell}- q^k S_{k\ell} q^{\ell}\right). \tag{21}$$ The corresponding infinitesimal $u(n)$ quasi-symmetry of the Lagrangian action (1) is $$\delta q_k ~=~ \varepsilon\sum_{\ell=1}^n \left( A_{k\ell} q^{\ell} +S_{k\ell} \dot{q}^{\ell} \right), \tag{22}$$ as one may easily verify.

Another example of a symmetry of the Hamiltonian not present in the Lagrangian formulation is provided by the isotropic harmonic oscillator.

Suppose $$L=\frac{1}{2}\dot q^T \cdot \dot{q} - \frac{1}{2}q^T\cdot q$$ where $q^T=(q_1,q_2,\ldots,q_n)$ and likewise $\dot{q}^T=(\dot q_1,\dot q_2,\ldots ,\dot{q}_n)$. Obviously the Lagrangian is invariant under $O(n)$, i.e. under (real) rotations $O$ of the coordinates so that $O^T O=\hat 1$.

The corresponding Hamiltonian is $$H=\frac{1}{2}p^T\cdot p +\frac{1}{2}q^T\cdot q$$ but if we now introduce the normal coordinates $$\alpha_k = (p_k+iq_k)\, ,\qquad \alpha_k^*=(p_k-iq_k)$$ the Hamiltonian takes the form $$H=\frac{1}{2}(\alpha^*)^T\cdot \alpha$$ and is now invariant under the larger group $U(n)$ of complex transformations that satisfy $U^\dagger U=\hat 1$ since, under this transformation: $$\alpha\to \beta = U\alpha\, ,\qquad (\alpha^*)^\dagger \to (\beta^*)^T =(\alpha^*)^T U^\dagger$$ so that $(\alpha^*)^T\cdot\alpha = (\beta^*)^T\cdot \beta$, leaving the Hamiltonian unchanged.

Of course since $O(n)$ is a subgroup of $U(n)$ it follows that the Hamiltonian has symmetries not possible with the Lagrangian.


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# Key encapsulation

Key encapsulation mechanisms (KEMs) are a class of encryption techniques designed to secure symmetric cryptographic key material for transmission using asymmetric (public-key) algorithms. In practice, public key systems are clumsy to use in transmitting long messages. Instead they are often used to exchange symmetric keys, which are relatively short. The symmetric key is then used to encrypt the longer message.

The traditional approach to sending a symmetric key with public key systems is to first generate a random symmetric key and then encrypt it using the chosen public key algorithm. The recipient then decrypts the public key message to recover the symmetric key. As the symmetric key is generally short, padding is required for full security and proofs of security for padding schemes are often less than complete.[1] KEMs simplify the process by generating a random element in the finite group underlying the public key system and deriving the symmetric key by hashing that element, eliminating the need for padding.

## Example using RSA encryption

Using the same notation employed in the RSA system article, say Alice has transmitted her public key ${\displaystyle (n,e)}$ to Bob, while keeping her private key secret, as usual. Bob then wishes to send symmetric key M to Alice. M might be a 128 or 256-bit AES key, for example. Note that the public key n is typically 1024-bits or even longer, thus much larger than typical symmetric keys. If e is small enough that ${\displaystyle M^{e}, then the encryption can be quickly broken using ordinary integer arithmetic.[2]

To avoid such potential weakness, Bob first turns M into a larger integer ${\displaystyle 1 by using an agreed-upon reversible protocol known as a padding scheme, such as OAEP. He then computes the ciphertext ${\displaystyle c}$ corresponding to:

${\displaystyle c\equiv m^{e}{\pmod {n}}.}$

Alice can recover ${\displaystyle m}$ from ${\displaystyle c}$ by using her private key exponent ${\displaystyle d}$ by the following computation:

${\displaystyle m\equiv c^{d}{\pmod {n}}.}$

Given ${\displaystyle m}$, she recovers the original message M by reversing the padding scheme.

With KEM the process is simplified as follows:[3]

Instead of generating a random symmetric key M, Bob first generates a random m, ${\displaystyle 1. He derives his symmetric key M by ${\displaystyle M=KDF(m)}$, where KDF is a key derivation function, such as a cryptographic hash. He then computes the ciphertext ${\displaystyle c}$ corresponding to m:

${\displaystyle c\equiv m^{e}{\pmod {n}}.}$

Alice then recovers ${\displaystyle m}$ from ${\displaystyle c}$ by using her private key exponent ${\displaystyle d}$ by the same method as above:

${\displaystyle m\equiv c^{d}{\pmod {n}}.}$

Given ${\displaystyle m}$, she can recover the symmetric key M by ${\displaystyle M=KDF(m)}$.

The KEM eliminates the complexity of the padding scheme and the proofs needed to show the padding is secure.[1] p. 4 Note that while M can be calculated from m in the KEM approach, the reverse is not possible, assuming the key derivation function is one-way. An attacker who somehow recovers M cannot get the plaintext m. With the padding approach, he can. Thus KEM is said to encapsulate the key.

Note that if the same m is used to encapsulate keys for e or more recipients, and the receivers share the same exponent e, but different p, q, and n, then one can recover m via the Chinese remainder theorem. Thus, if key encapsulations for several recipients need to be computed, independent values m should be used.

Similar techniques are available for Diffie-Hellman encryption and other public key methods.[4]


text:
(1) We have a new paper accepted in IEEE Communications Letters:

Ce Zheng, Malcolm Egan, Laurent Clavier, Anders Kalør and Petar Popovski, “Stochastic Resource Optimization of Random Access for Transmitters with Correlated Activation,” IEEE Communications Letters, (2021).

(2) On 10th June, Prof. Dejan Vukobratovic (University of Novi Sad) will present in our seminar:

Speaker: Prof. Dejan Vukobratovic (University of Novi Sad)

Date: 2pm (CET), 10th June 2021

Title: Designing Unequal Error Protection Codes Using Deep Autoencoders

Abstract: In this talk, we will discuss an autoencoder-based approach for designing codes that provide unequal error protection (UEP) capabilities. The proposed design accommodates both message-wise and bit-wise UEP scenarios. For both scenarios, we present the design method for the proposed autoencoder-based UEP codes and compare them with classical UEP code designs available in the literature.

Bio: Dejan Vukobratovic received a PhD degree in electrical engineering from the University of Novi Sad, Serbia, in 2008, where he is now a full Professor from 2019. During 2009 and 2010, he was Marie Curie Intra-European Fellow at the University of Strathclyde, Glasgow, UK. He published about 40 journal and 80 conference papers mainly in top-tier IEEE journals and conference venues. He received the best paper award at IEEE MMSP 2010 and his PhD student received the best student paper award at IEEE SmartGridComm 2017. His research interests are in the broad area of information and coding theory, wireless communications, distributed signal and information processing in Smart Grids, and massive machine-type communications in mobile cellular networks.

(3) On 17th June, Dr Xuewen Qian (Centrale Supelec) will present in our seminar:

Speaker: Dr. Xuewen Qian (Centrale Supelec)

Date: 2pm (CET), 17th June 2021

Title: Advanced Detection Schemes for Molecular Communications based on K-Means Clustering Approach

Abstract: We consider non-coherent detection (without channel information) for molecular communication systems in the presence of inter-symbol-interference. In particular, we study non-coherent detectors based on memory-bits-based thresholds in order to achieve low bit-error-ratio (BER) transmission. The main challenge of realizing detectors based on memory-bits-based thresholds is to obtain the channel state information based only on the received signals. We tackle this issue by reformulating the thresholds through intermediate variables, which can be obtained by clustering multi-dimensional data from the received signals, and by using the K-means clustering algorithm. In addition to estimating the thresholds, we show that the transmitted bits can be retrieved from the clustered data. To reduce clustering errors, we propose iterative clustering methods from one-dimensional to multi-dimensional data, which are shown to reduce the BER. Simulation results are presented to verify the effectiveness of the proposed methods.

Bio: Xuewen Qian received the B.Sc. and M.S. degrees with distinction in Electronic Science and Technology from Central South University, Changsha, China in 2014 and 2017, respectively. He obtained the Ph.D. degree from Paris-Saclay University, Paris, France, in 2020. In 2021, he was awarded the NEC Student Research Fellowship Award. Currently, he is a Research Fellow at CentraleSupelec, Paris-Saclay University, Paris, France. His current research interests include wireless communications, molecular communications, machine learning, deep learning, and reconfigurable intelligent surfaces.

(4) On 24th June, Dr. Vyacheslav Kungurtsev (Czech Technical University in Prague) will present in our seminar:

Speaker: Dr. Vyacheslav Kungurtsev (Czech Technical University in Prague)

Date: 2pm (CET), 24th June 2021

Title: Levenberg Marquardt Algorithms for Nonlinear Inverse Least Squares

Abstract: Levenberg Marquardt (LM) algorithms are a class of methods that add a regularization term to a Gauss Newton method to promote better convergence properties. This talk presents three works on this class of methods. The first discusses a new method that simultaneously achieves all types of state of the art convergence guarantees for unconstrained problems. Stochastic LM is discussed next, which is an algorithm to handle noisy data. An example is presented on data assimilation. Finally, a LM method is presented to handle equality constraints, with examples from inverse problems in PDEs.

Bio: Dr. Vyacheslav Kungurtsev is a researcher in the Department of Computer Science, Czech Technical University in Prague, Department of Com- puter Science, Faculty of Electrical Engineering, Prague, Czech Republic.

(1) With Bayram Akdeniz and Bao Tang, we have a new paper accepted:

Malcolm Egan, Bayram Akdeniz and Bao Quoc Tang, “Stochastic Reaction and Diffusion Systems in Molecular Communications: Recent Results and Open Problems.” Digital Signal Processing, (2021).

(1) On the 1st April, Michael Barros (Univ. Essex, UK) presented in our seminar on information theory and signal processing:

Title: Molecular Communications using Astrocytes for Boolean logic gates implementation in mammalian cells.

Abstract: In this talk we will show the use of astrocytes to realize Boolean logic gates, through manipulation of the threshold of Ca2+ ion fows between the cells based on the input signals. Through wet-lab experiments that engineer the astrocytes cells with pcDNA3.1-hGPR17 genes as well as chemical compounds, we show that both AND and OR gates can be implemented by controlling Ca2+ signals that fow through the population. A reinforced learning platform is also presented in the paper to optimize the Ca2+ activated level and time slot of input signals Tb into the gate. This design platform caters for any size and connectivity of the cell population, by taking into consideration the delay and noise produced from the signalling between the cells. To validate the efectiveness of the reinforced learning platform, a Ca2+ signalling simulator was used to simulate the signalling between the astrocyte cells. The results from the simulation show that an optimum value for both the Ca2+ activated level and time slot of input signals Tb is required to achieve up to 90% accuracy for both the AND and OR gates. Our method can be used as the basis for future Neural–Molecular Computing chips, constructed from engineered astrocyte cells, which can form the basis for a new generation of brain implants.

Bio: Dr Barros is an Assistant Professor (Lecturer) since June 2020 in the School of Computer Science and Electronic Engineering at the University of Essex, UK. He is also a MSCA-IF Research Fellow (part-time) at the Tampere University, Finland. He received the PhD in Computer Science at the Waterford Institute of Technology in 2016. He previously held multiple academic positions as a Research Fellow in the Waterford Institute of Technology, Ireland.

(2) On the 9th April, El Houcine Bergou (INRAE, France) presented in our seminar on information theory and signal processing:

Title: Stochastic Three Points Method For Unconstrained Smooth Minimization

Abstract:In this work, we consider the unconstrained minimization problem of a smooth function in a setting where only function evaluations are possible. We design a novel randomized derivative-free algorithm—the stochastic three points (STP) method—and analyze its iteration complexity. At each iteration, STP generates a random search direction according to a certain fixed probability law. Our assumptions on this law are very mild: roughly speaking, all laws which do not concentrate all measure on any halfspace passing through the origin will work. Although, our approach is designed to not explicitly use derivatives, it covers some first order methods. For instance, if the probability law is chosen to be the Dirac distribution concentrated on the sign of the gradient, then STP recovers the Signed Gradient Descent method. If the probability law is the uniform distribution on the coordinates of the gradient, then STP recovers the Randomized Coordinate Descent Method.
The complexity of STP depends on the probability law via a simple characteristic closely related to the cosine measure which is used in the analysis of deterministic direct search (DDS) methods. Unlike in DDS, where $O(n)$ ($n$ is the dimension of the problem) function evaluations must be performed in each iteration in the worst case, our method only requires two new function evaluations per iteration. Consequently, while the complexity of DDS depends quadratically on $n$, our method depends linearly on $n$.

Bio: I am a research scientist in INRAE. My research interests are in all areas that intersect with optimization, including algorithms, machine learning, statistics, and operations research. I am particularly interested in algorithms for large scale optimization including randomised and distributed optimization methods.

(3) I have a new paper to appear in the IEEE International Symposium on Information Theory (ISIT):

Malcolm Egan, “Dependence Testing via Extremes for Regularly Varying Models,” Proc. IEEE International Symposium on Information Theory (ISIT)(2021).

(1) With Bayram Akdeniz, we have a new paper accepted:

Bayram Akdeniz and Malcolm Egan, “Molecular Communication for Equilibrium State Estimation in Biochemical Processes on a Lab-on-a-Chip,” accepted for publication in IEEE Transactions on NanoBioscience, (2021).

(2) On 11th of February, Daryus Chandra (Southampton) presented in our seminar on Information Theory and Related Topics.

Speaker: Dr Daryus Chandra (University of Southampton, UK)

Title: Quantum Communications over Noisy Entanglement

Abstract: Within the Quantum Internet framework, multiple quantum devices are interconnected via pre-shared maximally-entangled quantum states for enabling various applications, including the on-demand classical and quantum communication. Hence, the pre-shared entanglement, which is constituted by the EPR pair, can be viewed as the primary consumable resources within the Quantum Internet. However, the generation and the distribution of the EPR pairs are subject to quantum decoherence imposed by the quantum channels, which will manifest as quantum errors. Similar to the classical domain, the quantum errors imposed by the quantum channels can be mitigated using quantum error-correction codes. In this talk, we will explore two approaches for achieving reliable quantum communication over noisy entanglement by incorporating quantum error-correction codes. More specifically, the first approach is constituted by the consecutive steps of quantum entanglement distillation followed by quantum teleportation, while the second approach can be viewed as the direct quantum communication over noisy entanglement. We will also discuss the pros and the cons of each approach while examining their compatibilities for a broader range of applications for the Quantum Internet framework.

Bio: Daryus Chandra is a research fellow at the Next-Generation Wireless Research Group, School of Electronics and Computer Science, University of Southampton, UK. He received the B.Eng. and M.Eng. degree from the Department of Electrical Engineering and Information Technology, Universitas Gadjah Mada, Indonesia, in 2013 and 2014, respectively. He obtained his PhD with the Next-Generation Wireless Research Group, School of Electronics and Computer Science, University of Southampton, UK, in 2020. He returned to Southampton in 2021 after completing a one-year postdoctoral research fellowship at Quantum Internet Research Group, University of Naples Federico II, Italy.

Time: 14h on the 11th February 2021.

(3) On 18th February, Prof. Datta (Univ. Idaho) presented in our seminar on Information Theory and Related Topics:

Speaker: Prof. Somantika Datta

Title: Construction and properties of certain real multi-angle tight frames

Abstract:

Frames are now standard tools in signal processing, and have applications ranging from compressed sensing, to communication systems and quantum sensing. Designing frames with some special structure such as equiangularity and tightness is highly desirable in applications. However, constructing equiangular tight frames (ETFs) with a given size in a specific dimension can be difficult or impossible in some cases. This leads one to consider the construction of frames with few distinct angles among pairs of frame vectors. In the special case of d+1 vectors in a d-dimensional space, it is well known that the vertices of a regular simplex will give an ETF. Using this, we will show a specific construction which, for a given dimension d and integer 1 < k ≤ d, gives a real unit norm tight frame such that the number of distinct angles among the vectors is bounded above by k. We will present several properties of this multi-angle tight frame. We also show how one can strategically choose subsets of such a multi-angle tight frame that will be equiangular or orthogonal. This property is meaningful in the context of erasures. We will also discuss a connection between certain unit norm tight frames with three angles and adjacency matrices of regular graphs.

Bio: Somantika Datta is an associate professor of mathematics at the University of Idaho. She received a Ph.D. in mathematics from the University of Maryland, College Park. This was followed by postdoctoral positions at Arizona State University and Princeton University. Her research interests lie in the area of applied harmonic analysis with focus on frame theory and applications in signal processing.

(4) On the 25th February, Bapi Chatterjee (IST Austria) will present in our seminar on Information Theory and Related Topics.

Speaker: Dr Bapi Chatterjee (IST Austria)

Title: Elastic Consistency: A General Consistency Model for Distributed Stochastic Gradient Descent

Abstract: One key element behind the progress of machine learning in recent years has been the ability to train machine learning models in large-scale distributed shared-memory and message-passing environments. Most of these models are trained employing variants of stochastic gradient descent (SGD) based optimization. In this work, we introduce a general consistency condition covering communication-reduced and asynchronous distributed SGD implementations. Our framework, called elastic consistency, decouples the system-specific aspects of the implementation from the SGD convergence requirements, giving a general way to obtain convergence bounds for a wide variety of distributed SGD methods used in practice. Elastic consistency can be used to re-derive or improve several previous convergence bounds in message-passing and shared-memory settings, but also to analyze new models and distribution schemes. In particular, we propose and analyze a new  synchronization-avoiding scheme for distributed SGD, and show that it can be used to efficiently train deep convolutional models for image classification.

Bio: Bapi Chatterjee is a Postdoc Fellow at IST Austria working in the area of Distributed Machine Learning. He is also a visiting scientist at EmbeDL AB, Gothenburg, Sweden. Prior to joining IST Austria he worked as a Research Staff Member with IBM India Research Lab, where he worked on Blockchains and related technologies. He obtained a PhD in Computer Science and Engineering from Chalmers University of Technology, Sweden. His current research interests includes Distributed Machine Learning, Neural Architecture Search, Concurrent Data Structures and Learned Index Structures.

(1) With Bayram Akdeniz (Univ. Oslo) and Bao Tang (Univ. Graz), we have a new paper accepted in IEEE Transactions on Molecular, Biological and Multi-Scale Communications:

Malcolm Egan, Bayram Akdeniz and Bao Quoc Tang, “Equilibrium signaling in spatially inhomogeneous diffusion and external forces,” accepted for publication in IEEE Transactions on Molecular, Biological and Multi-Scale Communications, (2021) [HAL]

(2) I will be presenting a lecture in the International Symposium on Molecular and Biological Communications, held online 1-5th February 2021:

Title: Stochastic Reaction-Diffusion Systems in Molecular Communication: Channel Models, Inference, and Coexistence

Abstract: A fundamental aspect of molecular communication is the motion of colloidal information-carrying molecules. To capture interactions between these molecules and other comprising the fluid, stochastic models play a key role. While the simplest family of these models—namely the Wiener process—-is well known, it is also possible to account for inhomogeneous diffusion, external forces, and general chemical reactions. In this talk, I will begin by overviewing this general family of models (including Langevin diffusions and the reaction-diffusion master equation) and present a recently proposed approach for developing near-optimal detection rules, called equilibrium signaling. I will then turn to other applications of stochastic reaction-diffusion systems in molecular communication related to molecular circuits for inference and modeling interactions with external biological systems to support coexistence, in the process identifying open challenges.

(1) On the 8th of December, Ce Zheng, a PhD student I am co-supervising with Prof. Laurent Clavier (IMT Lille-Douai, France) and Jean-Marie Gorce (INSA-Lyon, France), defended his thesis entitled: “Impulsive and Dependent Interference in IoT Networks”. The jury consisted of:

Prof. Claude Oestges (Ecole Polytechnique de Louvain, Belgium)
Assis. Prof. Lina Mroueh (Institut Supérieur d’Electronique de Paris, France)
Assoc. Prof. Mylene Pischella (Conservatoire National des Arts et Métiers, France)
Prof. Jean-François Hélard (INSA Rennes)
Assoc. Prof. Troels Pedersen (Univ. Aalborg, Denmark)
Prof. Gareth Peters (Heriot-Watt, UK)

Congratulations Chris!

(2) On the 10th of December, Ignacio Rodriguez (Aalborg University) will be presenting in our (currently online) seminar on information theory and related topics. Details are below:

Title: Experimental Research on Wireless Systems for Industrial Automation

Abstract: The fourth industrial revolution – or Industry 4.0 (I4.0), will introduce major shifts in the way that products will be manufactured in the future. By integrating different cyberphysical systems (CPS), Internet-of-Things (IoT) technologies and cloud computing; the factories of the future will be equipped with highly flexible manufacturing equipment offering also a high reliability, thereby increasing the overall production throughput. One of the key enablers for such revolution is wireless communication. By replacing existing wirelines in the current industrial equipment with wireless technologies, the overall cost of deployment will be reduced, while at the same time a faster re-configuration of the smart production facilities will be  enabled. Moreover, the use of wireless technologies will also allow for new industrial use cases requiring full mobility support such as autonomous robots moving items over different workstations in the factory for the sake of manufacturing customized products.
During this talk, the AAU Industrial Automation Applied Research Flow will be introduced and illustrated with application examples detailing the different steps from understanding the needs of a factory and the specific communication requirements of industrial use cases; to the final deployment and optimization of the wireless solutions.

Bio: Ignacio Rodriguez received the B.Sc. and M.Sc. degrees in Telecommunication Engineering from University of Oviedo, Spain, and the M.Sc. degree in Mobile Communications and the Ph.D. degree in Wireless Communications from Aalborg University, Denmark. Since December 2016, he has been a Postdoctoral Researcher at the same institution, where he is currently coordinating the Industry 4.0 experimental research activities at the Wireless Communication Networks Section and the AAU 5G Smart Production Lab in collaboration with the Department of Materials and Production. Ignacio is also an External Research Engineer with Nokia Bell Labs, where he is involved in 3GPP and ITU-R standardization activities. His research interests are mainly related to radio propagation, channel modeling, radio network planning and optimization, machine-to-machine communications, ultra-reliable and low-latency communications, 5G and Industrial IoT. He is a co-recipient of the IEEE VTS 2017 Neal Shepherd Memorial Best Propagation Paper Award, and in 2019, he was awarded with the 5G-prize by the Danish Energy Agency and the Danish Society of Telecommunication Engineers.

(3) With Vyacheslav Kungurtsev, Bapi Chatterjee and Dan Alistarh, we have a new paper accepted to AAAI 2021:

Kungurtsev, V., Egan, M., Chatterjee, B. and Alistarh, D., “Asynchronous Optimization Methods for Efficient Training of Deep Neural Networks with Guarantees”, AAAI Conference on Artificial Intelligence, (2021).

Abstract: Asynchronous distributed algorithms are a popular way to reduce synchronization costs in large-scale optimization, and in particular for neural network training. However, for nonsmooth and nonconvex objectives, few convergence guarantees exist beyond cases where closed-form proximal operator solutions are available.
As training most popular deep neural networks corresponds to optimizing nonsmooth and nonconvex objectives, there is a pressing need for such convergence guarantees. In this paper, we analyze for the first time the convergence of stochastic \emph{asynchronous} optimization for this general class of objectives. In particular, we focus on stochastic subgradient methods allowing for block variable partitioning, where the shared model is asynchronously updated by concurrent processes. To this end, we use a probabilistic model which captures key features of real asynchronous scheduling between concurrent processes. Under this model, we establish convergence with probability one to an invariant set for stochastic subgradient methods with momentum.
From a practical perspective, one issue with the family of algorithms that we consider is that they are not efficiently supported by machine learning frameworks, which mostly focus on distributed data-parallel strategies.
To address this, we propose a new implementation strategy for shared-memory based training of deep neural networks for a partitioned but shared model in single- and multi-GPU settings.
Based on this implementation, we achieve on average ~1.2x speed-up in comparison to state-of-the-art training methods for popular image classification tasks, without compromising accuracy.

(4) On the 17th of December, Lucien Etienne (IMT Lille-Douai) will be presenting in our (currently online) seminar on information theory and related topics. Details are below:

Speaker: Prof. Lucien Etienne (IMT Lille-Douai, France)

Date: 2pm (CET), 17th December 2020

Title: Self trigger co-design using LASSO regression

Abstract: Networked systems have become more and more pervasive in many modern industrial application.
A good justification for their deployment is that they can be cheaper/faster to set in place as well are scalable while also enabling lower maintenance cost. In the past decade a new paradigm has been developed where the controller is not sampled periodically (i.e. with a time–triggered policy), but rather sampled when some condition has been met (Usually a stability or performance criterion being violated). After recalling some general element on classical control scheme ( Linear Quadratic regulator and model predictive control)
In this talk, the control of a linear time invariant system with self triggered sampling is considered .
In order to address  the controller computation and the future sampling schedule a sparse optimization problem will be considered. A relaxation of the optimal self triggered control can be formulated as a LASSO regression. Using the properties of the solution of the Lasso regression it is shown how to obtain a controller ensuring  practical or asymptotic stability while reducing sampling of the control action.

Bio: Dr. Lucien Etienne received a M.Sc. Degree in applied mathematics at the INSA Rouen in 2012 and a joint Ph.D. in automatic control from the university of L’aquila and the university of Cergy-Pontoise in 2016.  From 2016 to 2017 he was a post doctoral researcher at Inria Lille-Nord Europe. Since 2017 He is  an associate professor at Institut Mines-Télécom Lille Douai. His research interests include switched and hybrid systems, observer synthesis and sampled data systems.

(1) On 17th November, Ido Nevat (TUMCREATE) presented in our seminar on information theory and related topics:

Title: Spatial Field Reconstruction and Sensor Selection in Heterogeneous Sensor Networks with Stochastic Energy Harvesting

Abstract:

We address the two fundamental problems of spatial field reconstruction and sensor selection in heterogeneous sensor networks.
We consider the case where two types of sensors are deployed: the first consists of expensive, high quality sensors; and the second, of cheap low quality sensors, which are activated only if the intensity of the spatial field exceeds a pre-defined activation threshold (e.g., wind sensors). In addition, these sensors are powered by means of energy harvesting and their time varying energy status impacts on the accuracy of the measurement that may be obtained.
We then address the following two important problems: (i) how to efficiently perform spatial field reconstruction based on measurements obtained simultaneously from both networks; and (ii) how to perform query based sensor set selection with predictive MSE performance guarantee.

To overcome this problem, we solve the first problem by developing a low complexity algorithm based on the spatial best linear unbiased estimator (S-BLUE).  Next, building on the S-BLUE, we address the second problem, and develop an efficient algorithm for query based sensor set selection with performance guarantee. Our algorithm is based on the Cross Entropy method which solves the combinatorial optimization problem in an efficient manner.
We present a comprehensive study of the performance gain that can be obtained by augmenting the high-quality sensors with low-quality sensors using both synthetic and real insurance storm surge database known as the Extreme Wind Storms Catalogue.

Bio: Ido Nevat received the B.Sc. degree in electrical engineering from the Technion-Israel Institute of Technology, Haifa, Israel, in 1998 and the Ph.D. degree in electrical engineering from the University of New South Wales, Sydney, NSW, Australia, in 2010.
Between 2010 and 2013, he was a Postdoctoral Research Fellow with the Wireless and Networking Technologies Laboratory at CSIRO, Australia.
Between 2013 and 2016, he was a Scientist with the Institute for Infocomm Research (I2R), Singapore.
Since 2017, he has been a team leader and PI of the Cooling Singapore project at TUMCREATE.
His main research interests include statistical signal processing, machine learning, and Bayesian statistics

(2) On 16th November, I presented an invited paper with Bayram Akdeniz in the 1st ACM International Workshop on Nanoscale Computing, Communication, and Applications.

(3) With Bayram Akdeniz and Bao Tang, we have a new paper accepted in IEEE Transactions on Communications:

Abstract: A basic property of any diffusion-based molecular communication system is the geometry of the enclosing container. In particular, the geometry influences the system’s behavior near the boundary and in all existing modulation schemes governs receiver design. However, it is not always straightforward to characterize the geometry of the system. This is particularly the case when the molecular communication system operates in vitro, where the geometry may be complex or dynamic. In this paper, we propose a new scheme-called equilibrium signaling-which is robust to uncertainties in the container geometry. In particular, receiver design only depends on the relative volumes of the transmitter or receiver, and the entire container. Our scheme relies on reversible reactions in the transmitter and the receiver, which ensure the existence of an equilibrium state into which information is encoded. In this case, we derive near optimal detection rules and develop a simple and effective estimation method to obtain the container volume. We also show that equilibrium signaling can outperform classical modulation schemes, such as concentration shift keying, under practical sampling constraints imposed by biological oscillators.

(4) With Laurent Clavier, Troels Pedersen, Ignacio Rodriguez, Mads Lauridsen, we have a new paper accepted in IEEE Communications Letters:

Abstract: 5G and beyond sees an ever increasing density of connected things. As not all devices are coordinated, there are limited opportunities to mitigate interference. As such, it is crucial to characterize the interference in order to understand its impact on coding, waveform and receiver design. While a number of theoretical models have been developed for the interference statistics in communications for the IoT, there is very little experimental validation. In this paper, we address this key gap in understanding by performing statistical analysis on recent measurements in the unlicensed 863 MHz to 870 MHz band in different regions of Aalborg, Denmark. In particular, we show that the measurement data suggests the distribution of the interference power is heavy tailed, confirming predictions from theoretical models.

(5) On Thursday 25 November, Howard Yang (Zhejiang University) will present in our seminar on information theory and related topics:

Title: Spatiotemporal Modeling of Wireless Networks

Abstract: The rapid growth of wireless applications has brought along new challenges for the next generation network, which is expected to manage a massive number of devices in real-time under a highly dynamic environment. To give an adequate response, it is of necessity to develop an analytical model with which designers can build intuitions, grasp insights, and identify critical issues.

In this talk, I will describe a framework for the analysis of large-scale wireless networks in which the transceivers interact with each other through the interference they caused and hence are correlated in both space and time. The analysis straddles stochastic geometry and queueing theory to cope with the issues of spatially interacting queues, and arrive at handy expressions for the SINR distribution. As a result, a wide variety of systems/architecture can be devised based on this theoretical foundation. Specifically, I will demonstrate how to adopt such a mathematical model to the analysis of two particular network metrics, i.e., the packet delay and age of information, and the subsequent network deployment guidelines based on the analytical results.

(1) I am pleased to announce that with Klemens Fellner and Bao Tang at the University of Graz I am co-organizing the workshop “Reaction, diffusion and molecular communication” on the 25th May. Our speakers are:

Eugenio Cinquemani (Inria Grenoble)
Werner Haselmayr (JKU Linz)
Stefan Müller (Uni Wien)
Maximilian Schäfer (FAU Erlangen-Nürnberg)
Florian Wodlei (Living Systems Research, Klagenfurt)

Full details can be found at the workshop website here. Due to COVID-19 complications, it will be held as an online workshop. Participation is open to all for free, with registration information (for access to the videoconferencing software) here.

(2) I am currently looking for PhD candidates to work on problems at the interface of information theory, statistical signal processing and control theory. If you have a background in at least one of these subjects (e.g., undergraduate studies in electrical engineering, mathematics or computer science), I can be contacted here. For further information on the kind of research that I do, my recent publications can be found here.

(1) We have a new paper lead by my student Ce Zheng accepted for IEEE International Conference on Communications (ICC) 2020:

Ce Zheng, Malcolm Egan, Laurent Clavier, Troels Pedersen and Jean-Marie Gorce, “Linear combining in dependent alpha-stable noise,” Proc. IEEE ICC, 2020. Available here.

(2) We also have two new preprints continuing our work impulsive interference modelling in the IoT.

The first paper is concerned with statistical analysis of a measurement campaign in Aalborg, where we show strong evidence for heavy tails. While this has long been expected from theory, it appears to be the first empirical study of this phenomenon.

Laurent Clavier, Troels Pedersen, Ignacio Rodriguez, Mads Lauridsen and Malcolm Egan, “Experimental evidence for heavy tailed interference in the IoT,” 2020. Available here.

The second paper is concerned with interference modelling in NB-IoT type scenarios, where devices transmit on multiple subbands. This leads to multivariate statistical models with nontrivial dependence structures. We deal with this problem using a combination of alpha-stable and copula models.

Ce Zheng, Malcolm Egan, Laurent Clavier, Gareth Peters and Jean-Marie Gorce, “Impulsive interference models for IoT networks,” 2020. Available here.

This work builds on our previous paper from ICC 2019:

Ce Zheng, Malcolm Egan, Laurent Clavier, Gareth Peters and Jean-Marie Gorce, “Copula-based interference models for IoT wireless networks,” Proc. IEEE ICC, 2019. Available here.

(3) With my postdoc Bayram Akdeniz, we have a new paper accepted in IEEE Transactions on Molecular, Biological and Multi-Scale Communications:

Bayram Akdeniz and Malcolm Egan, “A reactive signalling approach to ensure coexistence between molecular communication and external biochemical systems,” IEEE Transactions on Molecular, Biological and Multi-Scale Communications, In press, 2020. Available here.

This work studies the problem of coexistence in molecular communication from a more practical signalling point of view. Our previous work introducing the problem of coexistence can be found in:

Malcolm Egan, Valeria Loscri, Trung Q. Duong and Marco Di Renzo, “Strategies for coexistence in molecular communication,” IEEE Transactions on NanoBioscience, vol. 19, no. 1, pp. 51-60, 2019. Available here.

Malcolm Egan, Trang Mai, Trung Q. Duong and Marco Di Renzo, “Coexistence in molecular communications,” Nano Communication Networks, vol. 16, pp. 37-44, 2018. Available here.

(4) With Bayram Akdeniz and Bao Tang, we have also introduced a new signalling scheme for molecular communications. This scheme has a desirable property of being highly robust to uncertainty in the geometry of the system, exploiting the equilibrium theory of stochastic reaction-diffusion systems. A preprint of this work is now online:

Bayram Akdeniz, Malcolm Egan and Bao Tang, “Equilibrium signaling: molecular communication robust to geometry uncertainties,” 2020. Available here.


text:
# 5x + 1 - 4 (3) Compute lim 8-3 x _3(2+h)2 (4) Compute lim h-+0 h

###### Question:

5x + 1 - 4 (3) Compute lim 8-3 x _3 (2+h)2 (4) Compute lim h-+0 h

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Explain in detail how the human body destroys the wear UHMWPE particles/debris in a total joint replacemen...
##### Let f be a function whose domain is the closed interval [0.5]: The graph of f is shown below.Let hl) =f (t)dtOrrh ol f(a) Find the domain of h;(b) Find h (2)At what x is h(x) a minimum? Show the analysis that leads to your conclusion:
Let f be a function whose domain is the closed interval [0.5]: The graph of f is shown below. Let hl) = f (t)dt Orrh ol f (a) Find the domain of h; (b) Find h (2) At what x is h(x) a minimum? Show the analysis that leads to your conclusion:...
##### QuestionG04 pts 0 2DetailsCan Cramer's rule be used to solve this system of equations?YesIf Yes write the determinants used to computedet(det(Compute311212
Question G04 pts 0 2 Details Can Cramer's rule be used to solve this system of equations? Yes If Yes write the determinants used to compute det( det( Compute 311 212...
##### (_1)"nl for n = 0,1,2, 5" (n + 4)f() (4)then the Taylor series for f centered at 4 isf(z)((-14n)/(5"n(n+4)))(xf(z)1/43+_
(_1)"nl for n = 0,1,2, 5" (n + 4) f() (4) then the Taylor series for f centered at 4 is f(z) ((-14n)/(5"n(n+4))) (x f(z) 1/4 3+_...
##### Question 8Arrange in order by increasing boiling point:CH3CHOH; HOCH2CH2OH; C4H1oHOCH2CH2OH; CH3CHZOH; C4H1o C4H1o CH3CHzOH HOCH2CHZOH HOCH2CH2OH C4H1o CH3CH2OH CH3CHzOH: C4H1o HOCHZCHZOH C4H1o HOCH2CHzOH CH3CHzOH
Question 8 Arrange in order by increasing boiling point: CH3CHOH; HOCH2CH2OH; C4H1o HOCH2CH2OH; CH3CHZOH; C4H1o C4H1o CH3CHzOH HOCH2CHZOH HOCH2CH2OH C4H1o CH3CH2OH CH3CHzOH: C4H1o HOCHZCHZOH C4H1o HOCH2CHzOH CH3CHzOH...
##### Medical Terminology Week 3 Word: Angiogenesis Prefix: Suffix: Root: Definition: The process of developing new blood...
Medical Terminology Week 3 Word: Angiogenesis Prefix: Suffix: Root: Definition: The process of developing new blood vessels. Angiogenesis is critically importante normal development of the embryo and fetus. It also appears to be important during tumor fo Word: Aneurysm Prefix: Suffix: Root: Definiti...
##### Both 5 and 6 please 5) What is the H30* concentration to the correct number of...
both 5 and 6 please 5) What is the H30* concentration to the correct number of significant figures for solutions with the following pH values? (a) 0.00 (b) 7.00 (c) 2.63 (d) -0.42 (e) 11.21 6) (a) Calculate the pH of a buffer solution that is 0.45 M in NH4Cl and 0.15 M in NH3. (b) How would you p...
##### Jestion 39 In STR, stuttering is an example of which of the following: O A. Amplification...
Jestion 39 In STR, stuttering is an example of which of the following: O A. Amplification artifacts B. Allelic dropout O C. Spikes OD. All of the above...
##### QUESTION 9polntaBalance this reaction Meny [ ofthe coefficient; 502 -WntTatherinan nothingH2o12504
QUESTION 9 polnta Balance this reaction Meny [ ofthe coefficient; 502 - Wnt Tatherinan nothing H2o 12504...
##### A radioactive source contains an activity of 1 MBq. After aperiod of 20 weeks, the remaining activity turns out to be 1 kBq.Calculate the half-life? *A. About 1 weekB. About 10 daysC. About 2 weeksD. About 20 weeksLow dose irradiation has a potential to extend the shelf-life ofcertain food items but very low doses shows predominant moldformation. *a) Trueb) False
A radioactive source contains an activity of 1 MBq. After a period of 20 weeks, the remaining activity turns out to be 1 kBq. Calculate the half-life? * A. About 1 week B. About 10 days C. About 2 weeks D. About 20 weeks Low dose irradiation has a potential to extend the shelf-life of certain food i...
##### A shopper standing 1.60 m from a convex security mirror sees his image with a magnification...
A shopper standing 1.60 m from a convex security mirror sees his image with a magnification of 0.300. (a) Where is his image (in m)? (Use the correct sign.) m ---Select-- (b) What is the focal length (in m) of the mirror? m (c) What is its radius of curvature (in m)? m...
##### Evaluate the limit. lim (sin St, cos St)
Evaluate the limit. lim (sin St, cos St)...