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Réunion d'ete SMC 2018

Fredericton, 1 - 4 juin 2018

Systèmes dynamiques et modèles spatiaux en écologie
Org: Amy Hurford (memorial University)
[PDF]

ALI GHAROUNI, Population Ecology Division, Fisheries and Oceans Canada
Larval versus adult dispersal: implications of two modes of dispersal on the spread rate and population structure at an invasion  [PDF]

Marine species can spread by both adult movement and through dispersal of pelagic larval stages or propagules. Optimal methods to control the spread of the species depend on which mechanism is dominant. However, measuring the scale of stage-specific dispersal is usually difficult, if not impossible. By comparison, sampling the population at the leading edge of an invasion is much easier logistically. Hence, we are interested in the relationship between the population structure at the leading edge of an invasion and the relative scales of dispersal between stages. To address this problem, we considered a deterministic, stage-structured integrodifference equation (IDE) model with dispersal by both larval and adult stages. The fractions of the population by stages at the wave front are predicted by the eigenvector corresponding to the principal eigenvalue of the linearized IDE model. Analysis of this eigenvector leads to a relationship between the fractions of the population by stages at the invasion wave front and the demographic parameters, the dispersal parameters and the spread rate.

AMY HURFORD, Memorial University
Skewed temperature dependence affects range and abundance in a warming world  [PDF]

In response to climate warming, range shifts have occurred in almost all major taxa, but it remains poorly understood why some species respond positively, for example increasing their range sizes, while others respond negatively. We hypothesize that skewness in the relationship between temperature and fitness will affect species’ responses to climate change. Asymmetric fitness curves arise due to the relative strength of temperature dependence in positive and negative components of fitness, and although empirical evidence suggests that cold-skewed fitness curves are common, the strength of the skew can vary dramatically between species. We formulate a moving habitat integrodifference model that combines net reproduction and dispersal to test if differences in fitness curve skewness may explain differences in range size, abundance, and lag arising due to climate warming. We find that climate warming will adversely affect populations with cold-skewed, positively affect populations with warm-skewed, and have relatively little or mixed effect on populations with symmetric fitness curves, and that the degree of the skew will further magnify each of these effects. These results are robust to different choices of growth and dispersal functions and parameters, and are a necessary step towards understanding species’ responses to climate change more generally. This research is in collaboration with Christina Cobbold and Peter Molnar.

THEODORE KOLOKOLKINOV, Dalhousie University
Agent-based models exhibiting spatial aggregation  [PDF]

We discuss two agent-based models which exhibit spatial aggregation. The first example is a simple stochastic model of bacterial aggregation which leads to a novel fourth-order nonlinear PDE in its continuum limit. This PDE admits soliton-type solutions corresponding to bacterial aggregation patterns, which we explicitly construct.

The second example models complex predator-swarm interactions. Here, each prey within a swarm is represented by a point particle, with near repulsion and far-field attraction, in addition to the repulsion from the predator. The resulting system of ODE's can be approximated by a nonlocal PDE whose analysis yields insight about whether swarming behaviour helpful in avoiding a predator.

MARK LEWIS, University of Alberta
Generational spreading speeds for integrodifference equations  [PDF]

Some of the most fundamental quantities in population ecology describe the growth and spread of populations. Population dynamics are often characterized by the annual rate of increase, lambda or the generational increase, $R_0$. Analyses involving $R_0$ have deepened our understanding of disease dynamics and life-history complexities beyond that afforded by analysis of annual growth alone. While range expansion is quantified by the annual spreading speed, a spatial analog of lambda, an $R_0$-like expression for the rate of spread is missing. Using integrodifference models, we derive the appropriate generational spreading speed for populations with complex stage-structured life histories. The resulting measure, relevant to locations near the expanding edge of a (re)colonizing population, incorporates both local population growth and explicit spatial dispersal. The calculations for generational spreading speed are often simpler than those for annual spreading speed, and analytic or partial analytic solutions can yield insight into the processes that facilitate or slow a population’s spatial spread. We analyze the spatial dynamics of teasel as an example to demonstrate the flexibility of our methods and the intuitive insights that they afford. This work is joint with Andrew Bateman, Marty Krkosek and Mike Neubert.

AO LI, University of Western Ontario
Modelling the fear effect on prey population in patchy environment  [PDF]

Empirical results and theoretical analyses reveal that fear alters the behaviours of prey then influences its population size. In most cases, animals are observed to reduce activities to avoid being captured by predators at the cost of growth, which also results in the reduction of movement. We formulate a two-patch model incorporated with dispersal. Survival of prey is possible in a poor environment if prey from both patches can move freely to the other at a low rate. Using methods of adaptive dynamics to study the evolution of anti-predation strategy, we show that in an isolated patch there may exist a non-zero optimal strategy which is both an evolutionary stable strategy (ESS) and a convergence stable strategy (CSS). Numerical simulations are given for the two-patch model, implying that anti-predation strategy plays an important role in the long-term population dynamics of prey and an optimal strategy may exist which is stable in evolutionary sense.

JOANY MARIÑO, Memorial University
Dynamic Energy Budget theory predicts smaller energy reserves in clams that harbour symbionts  [PDF]

Dynamic energy budget (DEB) theory describes the upktake and use of energy by individuals, and distinguishes between the biomass of an organism that function as an energy reserve and as a structure. Two related species of small-sized clams inhabit the same seasonal environment and have similar diets. However, one species harbours bacteria that are digested as an additional resource, and how this nutritional symbiosis affects the energy budget of the clam is unknown. We parameterized a DEB model for each species and we found that the symbiotic clam has a smaller fraction of its biomass as an energy reserve relative to the asymbiotic species. A smaller energy reserve, in turn, implies reduced energy assimilation and mobilization fluxes, lower structural maintenance costs and growth rate, and larger energy allocation to maturity and reproduction in the symbiotic clam. Together, these features may comprise an evolutionary strategy where the symbionts function as a partial energy reserve for the host, and are an adaptation to fluctuating food availability. Our results highlight how a nutritional symbiosis may alter the energy budget of a host clam.

This work is joint with Starrlight Augustine, Suzanne C. Dufour and Amy Hurford.

MICHAEL NEUBERT, Woods Hole Oceanographic Institution
Two-Sex Invasion Dynamics  [PDF]

A key descriptive statistic of an ecological invasion is the speed at which it spreads across the landscape---the invasion speed. In general, the invasion speed depends on the vital rates (growth, survival, fecundity, etc.) and dispersal rates of individuals. These rates vary from individual to individual and much of that variability can be attributed to the individual's life-history stage. Ecologists have incorporated these stage-specific differences into integrodifference matrix population models from which they can compute the invasion speed, and the sensitivity of the speed to changes in the vital and dispersal rates.

Vital rates and dispersal rates also vary according to an individual’s sex, and for many animal species, fertility depends on the formation of breeding pairs, which in turn depends on the relative frequency of the sexes. We have developed an invasion model that accounts for both the age- and the sex-structure of the population, and includes the pair formation process and sex bias in the vital and dispersal rates.

We have derived, and will present, a formula for the invasion speed obtained from the model using low-population-density approximations. Our comparison of this formula with the results of numerical simulations suggests that the formula is correct for a class of reasonable nonlinear models and initial conditions as well. Using the formula, we will show that the invasion speed depends in complex ways upon the model parameters and on the nature of the pair-formation process that governs mating.

ERIC PEDERSEN, Fisheries and Oceans Canada
Statistical estimation of the moment dynamics of ecological communities to detect regime changes  [PDF]

As anthropogenic changes increasingly shape the global ecosystem, more are more ecological communities are undergoing regime shifts (abrupt changes from one persistent ecological state to another). Current statistical techniques for detecting regime shifts have focused on detecting abrupt changes in the abundances of single species, ignoring the possibility of more complex community dynamics during shifts such as transient responses, the existence of slow variables, or compositional re-arrangements. Here I argue that regime shifts are more readily detectable from among-species statistical moments of rates of change; i.e. means and variances of instantaneous rates of change among species. I show how it is possible to estimate these rates of change from noisy community data by using a novel penalized spline / Markov Random Field tensor product smoother that fits species-specific time series while simultaneously penalizing species towards a common trend, using long-term monitoring data from the collapse of the Newfoundland Atlantic Cod stocks as an example.

OLGA VASILYEVA, Memorial University of Newfoundland - Grenfell Campus
Nonlinear Population Dynamics in River Networks  [PDF]

We study the population dynamics of aquatic organisms in a river network. The habitat is viewed as a tree-like metric graph with the population density satisfying a reaction-diffusion-advection equation on each segment along with the appropriate junction and boundary conditions. In the case of a linear reaction term, the question of persistence in such models was studied by Sarhad, Carlson and Anderson. We focus on the case of a nonlinear (logistic) reaction term. We obtain necessary and sufficient conditions for the existence and uniqueness of a positive steady state solution for an arbitrary river network.

YANG WANG, University of Western Ontario
A predator-prey system with digest delay and anti-predation strategy  [PDF]

Predator-prey system is one of the most fundamental topics in ecology and evolutionary biology. Classic point of view focuses on direct predation only. With theoretical and experimental studies focusing on indirect effect such as cost of fear, researchers proposed that indirect effect may even have stronger impact than direct killing. Recent works by Wang, Zanette and Zou (2016,2017) have proposed some mathematical models to discuss the influence of such kind of effect. In this project, we consider indirect effect represented by anti-predation strategy. The anti-predation strategy such as group defense and avoiding detection can help prey survive from hunting by predator. Meanwhile, as a cost of these anti-predation strategies, the reproduction rate decreases due to the lack of time, space and energy. We adopt differential equation models to describe the dynamics between two species. At the same time, we take digest delay of predator into account. That is, after consuming the prey, predators need time $\tau$ to convert the biomass to their own. By applying stability analysis for delay differential equations, we find that for specialist predator, if the prey species use high level anti-predation strategy, the predator will become extinct and the prey will survive after suffering from the cost of this kind of strategy. A mild anti-predation strategy can guarantee coexistence of two species. Periodic change of population is possible for low level anti-predation strategy and sufficient large delay. Numerical simulations are given to verify our theoretical results.

RUIWEN WU, Memorial University of Newfoundland
Propagation Dynamics for a Spatially Periodic Integrodifference Competition Model  [PDF]

In this talk, I will report our recent research on the propagation dynamics for a class of integrodifference competition models in a periodic habitat. An interesting feature of such a system is that multiple spreading speeds can be observed, which biologically means different species may have different spreading speeds. We show that the model system admits a single spreading speed, and it coincides with the minimal wave speed of the spatially periodic traveling waves. A set of sufficient conditions for linear determinacy of the spreading speed is also given. This talk is based on a joint work with Dr. Xiaoqiang Zhao.

ALI BEYK ZADEH, University of New Brunswick
Dispersal Redistribution Approximation  [PDF]

Integrodifference equations (IDEs) are often used as discrete-time models for populations with distinct growth and dispersal stages. Most studies of these systems simplify the model by assuming either homogeneous dispersal, isolated single patches, or spatial or temporal periodicity. However, patchy landscape with bias at patch boundaries are often of interest and can lead to asymmetric and discontinuous dispersal kernels. In this case, the non-trivial solution of the IDE is a piecewise continuous function. Using a partial integral of the dispersal kernel known as the redistribution function, I derive an approximation of the non-trivial steady-state solution of the IDE model with a discontinuous, asymmetric kernel modeling dispersal and growth on a patchy landscape. The error of the this Dispersal Redistribution Approximation (DRA) is a decreasing function of the bias at patch boundaries. This work is joint with Dr. James Watmough.


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HERE Lanes
HERE Lanes Data Specification

# LaneGroup

## Message Summary

message LaneGroup

A group of lanes running between the same pair of lane group connectors, potentially sharing adjacent lane boundaries. Lane groups are logically oriented from their start to end lane group connectors.

Include: lane_topology.proto

## Properties

Field Type Label Description
lane_group_id uint64

The external reference for a lane group

start_lane_group_connector_ref uint64

The external reference for the start lane group connector for the lane group

end_lane_group_connector_ref TiledLaneGroupConnectorReference

End lane group connector for the lane group, which may be in a different tile

lanes Lane repeated

Ordered list of lanes in this group, spanning between the start and end lane group connectors. Lanes are ordered and sequentially numbered within a lane group from left to right in the relative to the lane group orientation from start to end connector.

boundary_geometry LaneGroupReferenceGeometry

Bounding geometry for the lane group, representing it's 2D spatial extent. This is specified by the outermost left and right lane boundaries. Together with the lane group connector bounding_geometry this forms a polygon for the lane group.

intersects_tiles uint32 repeated

List of all tiles that this lane group intersects, including the current one. Intersection is defined by the geometry of the group's lane boundaries

lane_group_length_meters uint32

Length rounded to nearest meter

format: plain


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# Use of Nets (topological) in Rigged Hilbert Space

I have been reading about Rigged Hilbert Spaces (AKA Gel'Fand Triple). This is characterized by the relationship $\Phi \subset H \subset \Phi'$, where $\Phi$ is Schwartz space, $H$ is Hilbert Space, or $L^2$, and $\Phi'$ is the conjugate to the Schwartz space. $\Phi$ is the space of wave functions (or state vectors) of Quantum Mechanics. $\Phi'$ is the space of the functionals on $\Phi$ and contains the Dirac delta function $\delta (x)$ and plane wave function $e^{ipx}$, which are the eigenfunctions of the position and momentum operators of Quantum Mechanics. Of note, $\Phi'$ does not satisfy the first axiom of countability.

So, it would seem that the concept of Nets (https://en.wikipedia.org/wiki/Net_(mathematics), or something more general than sequences, would be applicable to analyzing $\Phi'$, but I have never seen that in the literature.

Does anyone have any pointers to nets, or something more general than sequences, being used for defining convergence in $\Phi'$, in Rigged Hilbert Space?

Thanks

• According to Gel'Fand, "Generalized Functions", V2, P 57 (paraphrasing): A sequence in $\Phi'_p$ converges if all the elements of the sequence are in one of the fixed $\Phi'_p$, and they converge in the norm of $\Phi'_p$, where $\Phi'_0 \subset...\subset \Phi'_p \subset...\subset \Phi'$. It seems restrictive to only talk about convergence of sequences where all the elements of the converging sequence have to be in the same $\Phi'_p$ subspace. Maybe if we talk about nets, instead of sequences, all the elements of the net do not have to be in the same $\Phi'_p$ and can still converge? – David Jan 12 '17 at 21:30
• The above comment should begin "A sequence in $\Phi'$ converges if all the elements of the sequence...". Given the above comment (and its correction just stated), it seems invalid to talk about the convergence of a sequence of delta functions in $\Phi'$, since all $\Phi'_p$ are normed and so the delta function could not be in any of the $\Phi'_p$, as it is not part of any Hilbert space. But, it seems, to me, like one should be able to talk about the convergence of a sequence of delta functions. Hence, I am confused and wondering if introducing things like nets in $\Phi'$ would help. – David Jan 14 '17 at 9:32


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# Computing

## Maths Linux System Password

Log into a Maths Linux computer and type passwd at the command line. If you have a Maths Windows account (all new users do), this will also change your Windows password.

## Maths Windows System Password

If you are sitting in front of a Windows computer you can change your password by pressing Ctrl-Alt-Delete and then selecting the 'change a password' link. This won't change your Linux password.

If you want to make your Windows password the same as your Linux password then run the command windows_password.

## To change your computing service passwords such as Email (Hermes), Raven (web) and Desktop Services

Consider setting up self-service password recovery in case you forget your UIS password.


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If a, b, c are consecutive positive integers and a < b < c, which of

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If a, b, c are consecutive positive integers and a < b < c, which of  [#permalink]

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Updated on: 18 Dec 2017, 23:06
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If a, b, c are consecutive positive integers and a < b < c, which of the following must be true?

I. c - a = 2
II. abc is an even integer
III. (a + b + c)/3 is an integer

A. I
B. II
C. I and II
D. II and III
E. I, II and III

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-a-b-and-c ... 44907.html

Originally posted by Lolaergasheva on 07 Feb 2011, 23:23.
Last edited by Bunuel on 18 Dec 2017, 23:06, edited 3 times in total.
Renamed the topic, edited the question and added the OA.
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Re: If a, b, c are consecutive positive integers and a < b < c, which of  [#permalink]

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07 Feb 2011, 23:58
1
If a, b, c are consequitive positive integers and a<b<c, which of the following must be true?
1. c-a=2
2. abc is an even integer
3. (a+b+c) /3 is an integer

1.
a=a
b=a+1
c=a+2

c-a=a+2-a=2.
Correct

2. abc
a can be either even or odd
if a=even; abc=even;
if a=odd,b=even; abc=even

Rule:
multiple of any number of integers if atleast once multiplied by an even number will result in even.

even*even=even
even*odd=even

Correct

3. (a+b+c) /3 is an integer
Again;

a=a
b=a+1
c=a+2

a+b+c=(a+a+1+a+2)=3a+3

3a is always divisible by 3
3 is divisible by 3.
Thus "3a+3" is also divisible by 3.

Rule:
if integer x is divisible by n
and
integer y is divisible by n
then; x+y must be divisible by n

Correct.

Ans : "E"
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Re: If a, b, c are consecutive positive integers and a < b < c, which of  [#permalink]

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08 Feb 2011, 01:22
The sum of three consecutive numbers will always be divisible by 3 <<< It's a helpful result. You might want to commit it to your memory
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Re: If a, b, c are consecutive positive integers and a < b < c, which of  [#permalink]

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08 Feb 2011, 03:48
Lolaergasheva wrote:
If a, b, and c are consecutive positive integers and a<b<c, which of the following must be true?
1. c-a=2
2. abc is an even integer
3. (a+b+c) /3 is an integer

A. 1
B. 2
c. 1 and 2
d. 2 and 3
e. 1, 2 and 3

1. c-a=2 --> a, b, c are consequitive positive integers and a<b<c then c=a+2 --> c-a=2. So this statement is always true;

2. abc is an even integer --> out of any 3 consecutive integers at least one must be even thus abc=even. So this statement is also always true;

3. (a+b+c)/3 is an integer --> the sum of odd number of consecutive integers is ALWAYS divisible by that odd number. So this statement is also always true. Or: (a+b+c)/3=(a+a+1+a+2)/3=(3a+3)/3=a+1=integer.

Answer: E.

AmrithS wrote:
The sum of three consecutive numbers will always be divisible by 3 <<< It's a helpful result. You might want to commit it to your memory

Not only that:

• If $$k$$ is odd, the sum of $$k$$ consecutive integers is always divisible by $$k$$. Given $$\{9,10,11\}$$, we have $$k=3$$ consecutive integers. The sum of 9+10+11=30, therefore, is divisible by 3.

• If $$k$$ is even, the sum of $$k$$ consecutive integers is never divisible by $$k$$. Given $$\{9,10,11,12\}$$, we have $$k=4$$ consecutive integers. The sum of 9+10+11+12=42, therefore, is not divisible by 4.

• The product of $$k$$ consecutive integers is always divisible by $$k!$$, so by $$k$$ too. Given $$k=4$$ consecutive integers: $$\{3,4,5,6\}$$. The product of 3*4*5*6 is 360, which is divisible by 4!=24.
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Re: If a, b, c are consecutive positive integers and a < b < c, which of  [#permalink]

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18 Dec 2017, 23:07
Lolaergasheva wrote:
If a, b, c are consecutive positive integers and a < b < c, which of the following must be true?

I. c - a = 2
II. abc is an even integer
III. (a + b + c)/3 is an integer

A. I
B. II
C. I and II
D. II and III
E. I, II and III

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-a-b-and-c ... 44907.html
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Re: If a, b, c are consecutive positive integers and a < b < c, which of &nbs [#permalink] 18 Dec 2017, 23:07
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## Half Precision Arithmetic: fp16 Versus bfloat16

The 2008 revision of the IEEE Standard for Floating-Point Arithmetic introduced a half precision 16-bit floating point format, known as fp16, as a storage format. Various manufacturers have adopted fp16 for computation, using the obvious extension of the rules for the fp32 (single precision) and fp64 (double precision) formats. For example, fp16 is supported by the NVIDIA P100 and V100 GPUs and the AMD Radeon Instinct MI25 GPU, as well as the A64FX Arm processor that will power the Fujitsu Post-K exascale computer.

## Bfloat16

Fp16 has the drawback for scientific computing of having a limited range, its largest positive number being $6.55 \times 10^4$. This has led to the development of an alternative 16-bit format that trades precision for range. The bfloat16 format is used by Google in its tensor processing units. Intel, which plans to support bfloat16 in its forthcoming Nervana Neural Network Processor, has recently (November 2018) published a white paper that gives a precise definition of the format.

The allocation of bits to the exponent and significand for bfloat16, fp16, and fp32 is shown in this table, where the implicit leading bit of a normalized number is counted in the significand.

Format Significand Exponent
bfloat16 8 bits 8 bits
fp16 11 bits 5 bits
fp32 24 bits 8 bits

Bfloat16 has three fewer bits in the significand than fp16, but three more in the exponent. And it has the same exponent size as fp32. Consequently, converting from fp32 to bfloat16 is easy: the exponent is kept the same and the significand is rounded or truncated from 24 bits to 8; hence overflow and underflow are not possible in the conversion.

On the other hand, when we convert from fp32 to the much narrower fp16 format overflow and underflow can readily happen, necessitating the development of techniques for rescaling before conversion—see the recent EPrint Squeezing a Matrix Into Half Precision, with an Application to Solving Linear Systems by me and Sri Pranesh.

The drawback of bfloat16 is its lesser precision: essentially 3 significant decimal digits versus 4 for fp16. The next table shows the unit roundoff $u$, smallest positive (subnormal) number xmins, smallest normalized positive number xmin, and largest finite number xmax for the three formats.

$u$ xmins xmin xmax
bfloat16 3.91e-03 (*) 1.18e-38 3.39e+38
fp16 4.88e-04 5.96e-08 6.10e-05 6.55e+04
fp32 5.96e-08 1.40e-45 1.18e-38 3.40e+38

(*) Unlike the fp16 format, Intel’s bfloat16 does not support subnormal numbers. If subnormal numbers were supported in the same way as in IEEE arithmetic, xmins would be 9.18e-41.

The values in this table (and those for fp64 and fp128) are generated by the MATLAB function float_params that I have made available on GitHub and at MathWorks File Exchange.

## Harmonic Series

An interesting way to compare these different precisions is in summation of the harmonic series $1 + 1/2 + 1/3 + \cdots$. The series diverges, but when summed in the natural order in floating-point arithmetic it converges, because the partial sums grow while the addends decrease and eventually the addend is small enough that it does not change the partial sum. Here is a table showing the computed sum of the harmonic series for different precisions, along with how many terms are added before the sum becomes constant.

Arithmetic Computed Sum Number of terms
bfloat16 $5.0625$ $65$
fp16 $7.0859$ $513$
fp32 $15.404$ $2097152$
fp64 $34.122$ $2.81\dots\times 10^{14}$

The differences are striking! I determined the first three values in MATLAB. The fp64 value is reported by Malone based on a computation that took 24 days, and he also gives analysis to estimate the limiting sum and corresponding number of terms for fp64.

The NVIDIA V100 has tensor cores that can carry out the computation D = C + A*B in one clock cycle for 4-by-4 matrices A, B, and C with only one rounding error; this is a 4-by-4 fused multiply-add (FMA) operation. Moreover, C and D can be in fp32. The benefits that the speed and accuracy of the tensor cores can bring over plain fp16 is demonstrated in Harnessing GPU Tensor Cores for Fast FP16 Arithmetic to Speed up Mixed-Precision Iterative Refinement Solvers.

Intel’s bfloat16 format supports a scalar FMA d = c + a*b, where c and d are in fp32.

## Conclusion

A few years ago we had just single precision and double precision arithmetic. With the introduction of fp16 and fp128 in the IEEE standard in 2008, and now bfloat16 by Google and Intel, the floating-point landscape is becoming much more interesting.

Posted in research | Tagged , , | 5 Comments

## What’s New in MATLAB R2018b?

The MATLAB R2018b release notes report a number of performance improvements, including faster startup and faster calls to built-in functions. I pick out here a few other highlights from the release (excluding the toolboxes) that are of interest from the numerical analysis point of view.

The new xline and yline functions add vertical or horizontal lines to a plot—something that I have needed from time to time and have previously had to produce with a line or a plot command, together with hold on. For example, xline(pi) plots a vertical line at x = pi.

The stackedplot function is mainly of interest for plotting multiple variables from a table in a single plot, but it can also plot the columns of a matrix. In this example, A is the symmetric eigenvector matrix for the second difference matrix:

A = gallery('orthog',10,1); stackedplot(A);

The resulting plot clearly shows that the number of sign changes increases with the column index.

String arrays, introduced in R2016b, are now supported throughout MATLAB. In the previous example I could have typed A = gallery("orthog",10,1).

A new option 'all' to the functions all, any, max, min, prod, sum (and a few others) makes them operate on all the dimensions of the array. For example:

>> A = pascal(3)
A =
1     1     1
1     2     3
1     3     6
>> max(A,[],'all')
ans =
6
>> [prod(A,'all'), sum(A,'all')]
ans =
108    19

The empty second argument in the max call is needed because max(x,y) is also a supported usage. This is a useful addition. I have often written norm(A(:),inf) to avoid max(max(abs(A))) (which does not work for arrays with more than two dimensions), but now I can write max(abs(A),[],'all') without incurring the overhead of forming A(:).

New functions sinpi and cospi plot the sine and cosine functions at the specified multiple of $\pi$. Thus sinpi(x) is the same as sin(pi*x) except that it does not explicitly compute pi*x and so returns exact answers for integer arguments:

>> [sin(pi)         sinpi(1)]
ans =
1.2246e-16            0

## Bohemian Matrices in Manchester

Bohemian matrices are families of matrices in which the entries are drawn from a fixed discrete set of small integers (or some other discrete set). The term is a contraction of BOunded HEight Matrix of Integers and was coined by Rob Corless and Steven Thornton of the University of Western Ontario. Such matrices arise in many situations:

• adjacency matrices of graphs have entries from $\{0, 1\}$;
• Bernoulli matrices, which occur in compressed sensing, have entries from $\{-1,1\}$;
• Hadamard matrices have entries from $\{-1,1\}$ and orthogonal columns; and
• matrices with elements from $\{-1, 0, 1\}$ provide worst case growth factors for Gaussian elimination with partial pivoting and yield the most ill conditioned triangular matrices with elements bounded by $1$.

Rob’s group have done extensive computations of eigenvalues and characteristic polynomials of Bohemian matrices, which have led to interesting observations and conjectures. Many beautiful visualizations are collected on the website http://www.bohemianmatrices.com as well as on the Bohemian Matrices Twitter feed.

In June 2018, Rob and I organized a 3-day workshop Bohemian Matrices and Applications, bringing together 16 people with an interest in the subject from a variety of backgrounds. The introductory talks by Rob, Steven, and I were videod (and are embedded below), and the slides from most of the talks are available on the conference website.

We scheduled plenty of time for discussion and working together. New collaborations were started, several open problems were solved and numerous new questions were posed.

The workshop has led to various strands of ongoing work. Steven has created the Characteristic Polynomial Database, which contains more than $10^9$ polynomials from more than $10^{12}$ Bohemian matrices and has led to a number of conjectures concerning matches of properties to sequences at the On-Line Encyclopedia of Integer Sequences. Three recent outputs are

Sponsorship of the workshop by the Heilbronn Institute for Mathematical Research, the Royal Society and the School of Mathematics at the University of Manchester, as well as support from Canada for some of the Canadian participants, is gratefully acknowledged.

Eunice Chan (University of Western Ontario)

## Conversations with Gil Strang

At JuliaCon 2018 in London, one of the keynote presentations was a conversation with Gil Strang led by Alan Edelman and Pontus Stenetorp. Gil is well known for his many books on linear algebra, applied mathematics and numerical analysis, as well as his research contributions.

Gil talked about his famous 18.06 linear algebra course at MIT, which in its YouTube form has had almost 3 million views. A number of people in the audience commented that they had learned linear algebra from Gil.

Gil also talked about his book Linear Algebra and Learning from Data, due to be published at the end of 2018, which includes chapters on deep neural networks and back propagation. Many people will want to read Gil’s insightful slant on these topics (see also my SIAM News article The World’s Most Fundamental Matrix Equation).

As well as the video of Gil’s interview embedded below, two written interviews will be of interest to Gil’s fans:

## Tricks and Tips in Numerical Computing

In a keynote talk at JuliaCon 2018 I described a variety of tricks, tips and techniques that I’ve found useful in my work in numerical computing.

The first part of the talk was about two aspects of complex arithmetic: the complex step method for approximating derivatives, and understanding multivalued functions with the help of the unwinding number. Then I talked about the role of the associativity of matrix multiplication, which turns out to be the key property that makes the Sherman-Morrison formula work (this formula gives the inverse of a matrix after a rank 1 update). I pointed out the role of associativity in backpropagation in neural networks and deep learning.

After giving an example of using randomization to avoid pathological cases, I discussed why low precision (specifically half precision) arithmetic is of growing interest and identified some issues that need to be overcome in order to make the best use of it.

Almost every talk at JuliaCon was livecast on YouTube, and these talks are available to watch on the Julia Language channel. The slides for my talk are available here.

Also at the conference, my PhD student Weijian Zhang spoke about the work he has been doing on evolving graphs in his PhD.

Posted in conferences | Tagged | 2 Comments

## What’s New in MATLAB R2018a?

MATLAB R2018a was released in March 2018. With each biannual release I try to give a brief overview of the changes in MATLAB (not the toolboxes) that are of most interest to me. These are not comprehensive summaries of what’s new and you should check the release notes for full details.

Complex empty arrays, such as that generated with complex([]) now have an (empty) imaginary part instead of being real.

% R2017b
>> complex([])
ans =
[]

% R2018a
>> complex([])
ans =
0×0 empty complex double matrix

This is a consequence of a change in the way MATLAB stores complex arrays. Prior to R2018a it stored the real parts together and the imaginary parts together. In R2081 it uses an interleaved format in which the real and imaginary parts of a number are stored together; this is the storage format used in the C and C++ languages. For more details see MATLAB Support for Interleaved Complex API in C MEX Functions. Unless you write MEX files or create complex empty arrays, this change should have no effect on you.

The Live Editor continues to gain improved functionality, including embedded sliders and drop-down menus.

Legends can now have multiple columns, specified with the NumColumns property for Legend objects. I must admit that I had not realized that is was already possible to arrange legends in a horizontal (1-by-n) rather than vertical orientation (n-by-1) using the Orientation property.

Tall arrays have several new capabilities, including the ability to compute a least squares solution to an overdetermined linear system Ax = b with a tall A, which is done by QR factorization.

MATLAB starts up faster and has further optimizations in the execution engine.

The GraphPlot Object has some additional options for the 'force', 'force3', and 'circle' layouts.

I noticed that the recent Windows 10 Spring Creators Update broke the Symbolic Toolbox. An update to fix this (and various other issues) is available at this page (MathWorks login required).

For a concise but wide-ranging introduction and reference to MATLAB, see MATLAB Guide (third edition, 2017)

Posted in software | Tagged | 3 Comments

## How to Program log z

While Fortran was the first high-level programming language used for scientific computing, Algol 60 was the vehicle for publishing mathematical software in the early 1960s. Algol 60 had real arithmetic, but complex arithmetic had to be programmed by working with real and imaginary parts. Functions of a complex variable were not built-in, and had to be written by the programmer.

I’ve written a number of papers on algorithms to compute the (principal) logarithm of a matrix. The problem of computing the logarithm of a complex scalar—given a library routine that handles real arguments—might appear trivial, by comparison. That it is not can be seen by looking at early attempts to provide such a function in Algol 60.

The paper

J. R. Herndon (1961). Algorithm 48: Logarithm of a complex number. Comm. ACM, 4(4), 179.

presents an Algol 60 code for computing $\log z$, for a complex number $z$. It uses the arctan function to obtain the argument of a complex number.

The paper

A. P. Relph (1962). Certification of Algorithm 48: Logarithm of a complex number. Comm. ACM, 5(6), 347.

notes three problems with Herndon’s code: it fails for $z$ with zero real part, the imaginary part of the logarithm is on the wrong range (it should be $(-\pi,\pi]$ for the principal value), and the code uses log (log to the base 10) instead of ln (log to the base $e$). The latter error suggests to me that the code had never actually been run, as for almost any argument it would produce an incorrect value. This is perhaps not surprising since Algol 60 compilers must have only just started to become available in 1961.

The paper

M. L. Johnson and W. Sangren, W. (1962). Remark on Algorithm 48: Logarithm of a complex number. Comm. CACM, 5(7), 391.

contains more discussion about avoiding division by zero and getting signs correct. In

D. S. Collens (1964). Remark on remarks on Algorithm 48: Logarithm of a complex number. Comm. ACM, 7(8), 485.

Collens notes that Johnson and Sangren’s code wrongly gives $\log 0 = 0$ and has a missing minus sign in one statement. Finally, Collens gives in

D. S. Collens (1964). Algorithm 243: Logarithm of a complex number: Rewrite of Algorithm 48. Comm. CACM, 7(11), 660.

a rewritten algorithm that fixes all the earlier errors.

So it took five papers over a three year period to produce a correct Algol 60 code for the complex logarithm! Had those authors had the benefit of today’s interactive computing environments that period could no doubt have been shortened, but working with multivalued complex functions is necessarily a tricky business, as I have explained in earlier posts here and here.

## Lectures on Multiprecision Algorithms in Kácov

At the end of May, I was one of four lecturers at the ESSAM school on Mathematical Modelling, Numerical Analysis and Scientific Computing, held in Kácov, about a hour’s drive south-east of Prague in the Czech Republic.

The event was superbly organized by Josef Malek, Miroslav Rozlozník, Zdenek Strakos and Miroslav Tuma. This was a relaxed and friendly event, and the excellent weather enabled most meals to be taken on the terrace of the family-run Sporthotel Kácov in which we were staying.

I gave three lectures of about one hour each on Multiprecision Algorithms. The slides are available from this link. Here is an abstract for the lectures:

Today’s computing environments offer multiple precisions of floating-point arithmetic, ranging from quarter precision (8 bits) and half precision (16 bits) to double precision (64 bits) and even quadruple precision (128 bits, available only in software), as well as arbitrary precision arithmetic (again in software). Exploiting the available precisions is essential in order to reduce the time to solution, minimize energy consumption, and (when necessary) solve ill-conditioned problems accurately.

In this course we will describe the precision landscape, explain how we can exploit different precisions in numerical linear algebra, and discuss how to analyze the accuracy and stability of multiprecision algorithms.

• Lecture 1. IEEE standard arithmetic and availability in hardware and software. Motivation for low precision from applications, including machine learning. Exploiting reduced communication cost of low precision. Issued relating to rounding error analyses in low precision. Simulating low precision for testing purposes. Challenges of implementing algorithms in low precision.
• Lecture 2. Basics of rounding error analysis, illustrated with summation. Why increasing precision is not a panacea. Software for high precision and its cost. Case study: the matrix logarithm in high precision.
• Lecture 3. Solving very linear systems (possibly very ill conditioned and/or sparse) using mixed precision: iterative refinement in three precisions. A hybrid direct-iterative method: GMRES-IR.

I gave an earlier version of these lectures in March 2018 at the EU Regional School held at the Aachen Institute for Advanced Study in Computational Engineering Science (AICES), Germany. This single two and a half hour lecture was recorded and can be viewed on YouTube. The slides are available here.

## Joan E. Walsh (1932–2017)

By Len Freeman and Nick Higham

Joan Eileen Walsh was born on 7 October 1932 and passed away on 30 December 2017 at the age of 85.

Joan obtained a First Class B.A. honours degree in Mathematics from the University of Oxford in 1954. She then spent three years working as an Assistant Mistress at Howell’s School in Denbigh, North Wales. In 1957 Joan left teaching and enrolled at the University of Cambridge to study for a Diploma in Numerical Analysis. This qualification was awarded, with Distinction, in 1958. At this point, Joan returned to the University of Oxford Computing Laboratory to study for a D.Phil. under the supervision of Professor Leslie Fox. She was Fox’s first doctoral student. Her D.Phil. was awarded in 1961.

Between October 1960 and March 1963, Joan worked as a Mathematical Programmer for the CEGB (Central Electricity Generating Board) Computing Department in London. In April 1963, she was appointed to a Lectureship in the Department of Mathematics at the University of Manchester. She progressed through the positions of Senior Lecturer (1966) and Reader (1971) before being appointed as Professor of Numerical Analysis at the University of Manchester in October 1974. For the academic year 1967-1968 Joan had leave of absence at the SRC Atlas Computer Laboratory—a joint appointment with St Hilda’s College, Oxford.

Joan led the Numerical Analysis group at the University of Manchester until 1985, after which Christopher Baker took over. This was a period of expansion both for the Numerical Analysis group at Manchester and, more generally, for numerical analysis in Britain. This expansion of British numerical analysis was supported by special grants from the SRC (Science Research Council) to provide additional funding for the subject at the Universities of Dundee, Manchester and Oxford, from 1973 until 1976. This funding supported one Senior Research Fellow and two Research Fellows at each Institution. Joan helped establish the Manchester group as one of the leading Numerical Analysis research centres in the United Kingdom (with eight permanent staff by 1987)—a position that is maintained to the present day.

Joan was Head of the Department of Mathematics between 1986 and 1989, and subsequently became Pro-Vice Chancellor of the University of Manchester in 1990. She held the latter role for four years, and was responsible for undergraduate affairs across the University. Joan’s tenure as Pro-Vice Chancellor coincided with substantial, and sometimes controversial, changes in undergraduate teaching—for example, the introduction of semesterisation and of credit-based degree programmes; Joan managed these major changes across the University with her customary tact, energy and determination. Joan was an efficient and effective administrator at a time when relatively few women occupied senior management roles in universities.

After 35 years’ service, Joan retired from the University in 1998 and was appointed Professor Emeritus.

In retirement, Joan returned to her studies; between 2000 and 2003 she studied for an MA in “Contemporary Theology in the Catholic Tradition” at Heythrop College of the University of London.

Over the years, and particularly during her tenure as Pro-Vice Chancellor, Joan sat on, and chaired, numerous University committees, far too many to list. She had a very long relationship with Allen Hall (a University Hall of Residence) where she was on the Hall Advisory Committee from 1975 until her retirement in 1998.

Joan served leadership roles nationally, as well as in the University. She was Vice President of the IMA (1992-1993) and a member of the Council of the IMA (1990-1991 and 1994-1995). She was elected Fellow of the Institute of Mathematics and its Applications (IMA) in 1984. She was a member of the Computer Board for Universities and Research Councils for several years from the late 1970s. She encouraged the creation of its Software Provision Committee, formally constituted in 1980 with Joan as its first Chairman, which she led until 1985. She was also President of the National Conference of University Professors (1993–1994). Further, she was a member of the Board of Governors at Withington Girls’ School, a leading independent school, for six years between 1993 and 1999.

Nowadays, all computational scientists take for granted the existence of software libraries such as the NAG Library. It is unimaginable to undertake major computational tasks without them. In 1970, Joan was one of a group of four academics who founded the Nottingham Algorithms Group with the aim of developing a comprehensive mathematical software library for use by the group of universities that were running ICL 1906A mainframe computers. Subsequently, the Nottingham Algorithms Group moved from the University of Nottingham to the University of Oxford and the project was incorporated as the Numerical Algorithms Group (NAG) Ltd. Joan became the Founding Chairman of NAG Ltd. in 1976, a position she held for the next ten years. She was subsequently a member of the Council of NAG Ltd. from 1992 until 1996. In recognition of her contribution to the NAG project Joan was elected as a Founding Member of the NAG Life Service Recognition Award in 2011.

Joan’s research interests focused on the numerical solution of ordinary differential equation boundary value problems and the numerical solution of partial differential equations. She conducted much of her research in collaboration with PhD students, supervising the following PhD students at the University of Manchester, who obtained their degrees in the years shown:

• Thomas Sag, 1966;
• Les Graney, 1973;
• David Sayers, 1973;
• Geoffrey McKeown, 1977;
• Roderick Cook, 1978;
• Patricia Hanson, 1979;
• Guy Lonsdale, 1985;
• Chan Basaruddin, 1990;
• Fathalla Rihan (supervised jointly with C. T. H. Baker), 2000.

Joan was an important figure in the development of Numerical Analysis and Scientific Computing at the University of Manchester and in the UK more generally. Her essay Numerical Analysis at the Victoria University of Manchester, 1957-1979 gives an interesting perspective on early developments at Manchester.

Brian Ford OBE, Founder Director of NAG, writes:

Joan had a brilliant career in Mathematics (particularly areas of Numerical Mathematics, Ordinary and Partial Differential Equations), Computing, University Education and Teaching, and was an excellent researcher, teacher, administrator, doctoral supervisor and colleague. But she was so much more than that!

Joan was invariably kind and thoughtful, intellectually gifted and generous with advice and guidance. Her profound Christian faith illuminated every aspect of her life. Joan’s deep reading and wide intellectual interests coupled with her prudence and clear thinking gave her profound knowledge and command. She was excellent company –amusing, modest, never belittling nor intimidating- and enjoyed fine wine and food in good company. She held firm beliefs, gently and persuasively seeking what she saw as the right way. Many people turned to her for help, advice and references and were grateful for her readily-offered help and support.

Joan was a private, even guarded, person. A devout Catholic, on her retirement she completed an MA in “Contemporary Theology in the Catholic Tradition” at Heythorp College, University of London. Fluent in Latin and reading regularly at services, she loved the traditional Tridentine Mass of the Church. Along with her local bishop in Salford and other like-minded Catholics, she therefore worked actively for the restitution of the Tridentine Mass to the liturgy of the world-wide Church (sidelined after Vatican II in favour of local languages), an involvement which culminated her joining high-level discussions at the Vatican. This bore fruit, the Tridentine Latin Mass being officially declared the extraordinary form of the Roman Rite of Mass a few years later: Joan was thrilled. Such was Joan’s commitment to things she believed in and her endless thought and work for others.

Joan was an excellent contributor to the NAG Library, believing strongly in collaboration and sharing, with high quality standards for all aspects of our work and thorough checking and testing. She was an excellent first Chairman of NAG and invaluable colleague and advisor. We thoroughly enjoyed working together, invariably in an excellent spirit. We achieved much.

Posted in people | 2 Comments

## Conference in Honour of Walter Gautschi

Last week I had the pleasure of attending and speaking at the Conference on Scientific Computing and Approximation (March 30-31, 2018) at Purdue University, held in honour of Walter Gautschi (Professor Emeritus of Computer Science and Mathematics at Purdue University) on the occasion of his 90th birthday.

The conference was expertly organized by Alex Pothen and Jie Shen. The attendees, numbering around 70, included many of Walter’s friends and colleagues.

The speakers made many references to Walter’s research contributions, particularly in the area of orthogonal polynomials. In my talk, Matrix Functions and their Sensitivity, I emphasized Walter’s work on conditioning of Vandermonde matrices.

A Vandermonde matrix $V_n$ is an $n\times n$ matrix depending on parameters $x_1,x_2,\ldots,x_n$ that has $j$ th column $[1, x_j, \ldots, x_j^{n-1}]^T$. It is nonsingular when the $x_i$ are distinct. This is a notoriously ill conditioned class of matrices. Walter said that he first experienced the ill conditioning when he computed Gaussian quadrature formulas from moments of a weight function.

Walter has written numerous papers on Vandermonde matrices that give much insight into their conditioning. Here is a very a brief selection of Walter’s results. For more, see my chapter Numerical Conditioning in Walter’s collected works.

In a 1962 paper he showed that

$\displaystyle\|V_n^{-1}\|_{\infty} \le \max_i \prod_{j\ne i}\frac{ 1+|x_j| }{ |x_i-x_j| }.$

In 1978 he obtained

$\displaystyle\|V_n^{-1}\|_{\infty} \ge \max_i \prod_{j\ne i} \frac{ \max(1,|x_j|) }{ |x_i-x_j| },$

which differs from the upper bound by at most a factor $2^{n-1}$. A 1975 result is that for $x_i$ equispaced on $[0,1]$,

$\displaystyle\kappa(V_n)_{\infty} \sim \frac{1}{\pi} e^{-\frac{\pi}{4}} (3.1)^n.$

A 1988 paper returns to lower bounds, showing that for $x_i \ge 0$ and $n\ge 2$,

$\displaystyle\kappa(V_n)_{\infty} > 2^{n-1}.$

When some of the $x_i$ coincide a confluent Vandermonde matrix can be defined, in which columns are “repeatedly differentiated”. Walter has obtained bounds for the confluent case, too.

These results quantify the extreme ill conditioning. I should note, though, that appropriate algorithms that exploit structure can nevertheless obtain accurate solutions to Vandermonde problems, as described in Chapter 22 of Accuracy and Stability of Numerical Algorithms.

Posted in conferences | Tagged | 1 Comment


text:
[<< wikibooks] On 2D Inverse Problems/Zolotarev problem
The elliptic function
can be defined by the following integral equation:

u
=

∫

0

ξ

d
t

(
1
−

t

2

)
(
1
−

κ

2

t

2

)

.

{\displaystyle u=\int _{0}^{\xi }{\frac {dt}{\sqrt {(1-t^{2})(1-\kappa ^{2}t^{2})}}}.}
The following rational approximation problem demonstrates how fast the considered discrete models converge to their continuous analogs. The problem of optimal rational approximation of the square root on an interval was solved by Russian mathematician Zolotarev with the closed form solutions in terms of  elliptic integrals, see [DIK] & [PP].
The optimal rational approximation function of order (k-1,k).
is given.
Exercise (**). Use the interlacing of zeros property to prove that the function

s
(
λ
)
=
λ

r
~

(

λ

2

)

{\displaystyle s(\lambda )=\lambda {\tilde {r}}(\lambda ^{2})}
can be written as a Stieltjes continued fraction.
The optimal approximation converges exponentially fast with the degree of the rational function.


text:
# Evaluate integral trouble

I'm having trouble evaluating this integral. I tried $u$-substitution and integration by parts but they didn't work. Any ideas guys?

Evaluate $$\int \frac{\ln(\sin x)}{\sin^2 x} dx$$

Thanks you for the help!

-
integration by parts works, pick f = ln term, so that f' can be found. Then g' is csc²x of which g is a known anti derivative –  imranfat Sep 27 '13 at 15:54
You r confusing me can u explain me more please? –  Morgan Stone Sep 27 '13 at 16:03
Follow the technique given in the comment by imranfat. –  Mhenni Benghorbal Sep 27 '13 at 16:03
I don't understand what he meant..... –  Morgan Stone Sep 27 '13 at 16:07

Here is a start. using integration by parts,

$$\int u dv = u v - \int v du .$$

Let

$$u=\ln(\sin(x)) \implies u'=\frac{\cos(x)}{\sin(x)}=\cot(x),\quad v=\int \frac{dx}{\sin^2 x}=-\cot(x).$$

Can you finish it now?

-
Ok I think I got it, thank you very much! –  Morgan Stone Sep 27 '13 at 16:16
@MorganStone: you are very welcome. –  Mhenni Benghorbal Sep 27 '13 at 18:22


text:
#### On Empirical Risk Minimization with Dependent and Heavy-Tailed Data

In this work, we establish risk bounds for Empirical Risk Minimization (ERM) with both dependent and heavy-tailed data-generating processes. We do so by extending the seminal works~\cite{pmlr-v35-mendelson14, mendelson2018learning} on the analysis of ERM with heavy-tailed but independent and identically distributed observations, to the strictly stationary exponentially $\beta$-mixing case. We allow for the interaction between the noise and inputs to be even polynomially heavy-tailed, which covers a significantly large class of heavy-tailed models beyond what is analyzed in the learning theory literature. We illustrate our theoretical results by obtaining rates of convergence for high-dimensional linear regression with dependent and heavy-tailed data.


text:
# Is a monoid finitely generated by periodic matrices periodic?

This question is somewhat linked to this one -- in fact, I meant to ask the following:

Let $\mathcal{M}_n$ be the multiplicative monoid of $n \times n$ matrices on $\mathbb{N}$ (including $0$). Let $M_1, \ldots, M_k \in \mathcal{M}_n$ and suppose the monoid generated by each one of them is finite. Is the monoid generated by all of them periodic (i.e. such that any of its elements generates a finite monoid)?

I hope I got it right this time. Sorry for the noise.
Thanks!

-
 Well, this is why I posted a new question: generating a finite monoid is different from being of finite order. For instance, $\left(\begin{array}{cc}1&0\\1&0\end{array}\right)$ or $\left(\begin{array}{cc}0&2\\0&0\end{array}\right)$ both generate finite monoids. – Michaël Cadilhac Jan 1 '11 at 17:02 Ah, right. My mistake. – Qiaochu Yuan Jan 1 '11 at 17:03

In fact, using the two examples in your comment provides a counterexample.

Let $A = \begin{bmatrix} 1 & 0\\1&0\end{bmatrix}$ and $B = \begin{bmatrix}0&2\\0&0 \end{bmatrix}$.

As you mention, $A^2 = \begin{bmatrix} 1 & 0 \\ 1 & 0\end{bmatrix}$ so $A$ generates a finite monoid. Likewise, $B^2$ is the $0$ matrix so this generates a finite monoid.

However, the monoid generated by $A$ and $B$ is not periodic: the element $BA = \begin{bmatrix} 2 & 0 \\ 0 & 0\end{bmatrix}$ doesn't generate a finite monoid since $(BA)^n = \begin{bmatrix} 2^n & 0\\ 0 & 0\end{bmatrix}$, which is distinct for each $n$.

-
 Haha, yes, I thought about that a few minutes later, while in the car :) That's really silly of me! Thank you, Jason! – Michaël Cadilhac Jan 2 '11 at 6:26

More generally, for any $n\gt 1$, the matrices $$A=\left(\begin{array}{cccc} 1 & 0 & \cdots & 0\\ 1 & 0 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 1 & 0 & \cdots & 0 \end{array}\right)\quad\text{and}\quad B=\left(\begin{array}{cccc} 1 & 1 & \cdots & 1\\ 0 & 0 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & 0 \end{array}\right)$$ are each idempotent, so they each generate a finite monoid, but neither $$AB = \left(\begin{array}{cccc} 1 & 1 & \cdots & 1\\ 1 & 1 & \cdots & 1\\ \vdots & \vdots & \ddots & \vdots\\ 1 & 1 & \cdots & 1 \end{array}\right)\quad\text{nor}\quad BA = \left(\begin{array}{cccc} n & 0 & \cdots & 0\\ 0 & 0 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & 0 \end{array}\right)$$ are periodic.

-
 Ah, right, thank you! – Michaël Cadilhac Jan 2 '11 at 6:26

Let $A$ be an element of $\mathcal{M}_n$ such that $A$ is periodic. Then $A^\mathrm{T}$ is also periodic. However, $AA^\mathrm{T}$ is periodic if and only if each nonzero entry of $A$ is $1$ and each row and column of $A$ has at most one nonzero entry.

Suppose $A=(a_{ij})$ and $a_{i_0j_0}\gt1$ for some $i_0$ and $j_0$. If $AA^\mathrm{T}=(b_{ij})$, then $b_{i_0i_0}\geq a_{i_0j_0}^2\gt1$. In particular this implies that the Euclidean norm of $AA^\mathrm{T}e_{i_0}$ is greater than $1$, where $e_{i_0}$ is the $i_0^\text{th}$ standard basis vector. Since $AA^\mathrm{T}$ is positive semidefinite, this implies that its spectral radius is greater than $1$, and therefore its powers have unbounded spectral radius.

Suppose that $a_{i_0j_0}\gt0$ and $a_{i_0j_1}\gt0$ for some $i_0$ and $j_0\neq j_1$. Then $b_{i_0i_0}\geq a_{i_0j_0}^2+a_{i_0j_1}^2\gt1$, which again implies that the powers of $AA^\mathrm{T}$ have unbounded spectral radius.

Suppose that $a_{i_0j_0}\gt0$ and $a_{i_1j_0}\gt0$ for some $i_0\neq i_1$ and $j_0$. Then $b_{i_0i_0}\geq a_{i_0j_0}^2\gt0$ and $b_{i_1i_0}\geq a_{i_1j_0}a_{i_0j_0}\gt0$. This implies that the Euclidean norm of $AA^\mathrm{T}e_{i_0}$ is at least $\sqrt{2}$. Thus, again, the powers of $AA^\mathrm{T}$ have unbounded spectral radius.

On the other hand, if each nonzero entry of $A$ is $1$ and each row and column of $A$ has at most one nonzero entry, then $AA^\mathrm{T}$ is idempotent.

-
 That's really nice, thank you very much! – Michaël Cadilhac Jan 2 '11 at 6:27


text:
### Give reason or explain: 1. Utility is a subjective concept.

Give reason or explain:
1. Utility is a subjective concept.


text:
1. ## linear transformations

consider the linear transformation T:R4--> R4 defined by
[w] [w+z]
T [x] = [y]
[y] [x-y]
[z] [w]

for each i have to justify the answer
a. is T injective?

b. is T surjective?

c.is T invertible?

2. Hi,
consider the linear transformation T:R4--> R4 defined by
[w] [w+z]
T [x] = [y]
[y] [x-y]
[z] [w]

for each i have to justify the answer
a. is T injective?
$T$ is injective if $\ker T = \{\vec{0}\}$. Is this true ?
b. is T surjective?
$T$ is surjective if, given any vector $v\in\mathbb{R}^4$, one can find $u\in\mathbb{R}^4$ such that $T(u)=v$. In other words, given $v=\begin{bmatrix}a\\b\\c\\d\end{bmatrix}\in\mathbb {R}^4$, can you find at least a 4-tuple $(w,x,y,z)$ such that $\begin{cases} w+z=a\\y=b\\x-y=c\\w=d\end{cases}$ ?

3. Ok I understand the definitions but am still struggling to decide if T is injective or not, for the kernel(T) obviously we know this equals the null space(T) so if I find one vector in R4 such that Tv=the zero vector does this mean it is injective.
eg. if I choose the vector
[1]
[0]
[0]
[-1]

Ok I understand the definitions but am still struggling to decide if T is injective or not, for the kernel(T) obviously we know this equals the null space(T) so if I find one vector in R4 such that Tv=the zero vector does this mean it is injective.
eg. if I choose the vector
[1]
[0]
[0]
[-1]
To show that $T$ is injective you have to show that the only vector $v$ such that $Tv=0$ is $v=0$; that's not what you're doing. Anyway, this idea is often used to show that a linear application is not injective. For if you find a non-zero vector $v$ such that $Tv=0$ then $\ker T$ contains both 0 and $v$ hence $\ker T\neq \{0\}$ and $T$ can't be injective.

EDIT: by the way, for $v=\begin{bmatrix}1\\0\\0\\-1\end{bmatrix},\, Tv\neq 0$

5. Oh yeah I see that that is incorrect. So T(0) = 0 so T is injective!? Is that what you're saying?

No! $T(0)=0$ is true for any linear application $T$, even if $T$ is not injective (this result comes from $T(0)=T(x-x)=T(x)+T(-x)=T(x)-T(x)=0$). A linear application is injective if $T(v)=0 \implies v=0$. You are saying that a linear application is injective if $v=0\implies T(v)=0$. These two sentences are very different and only the former is correct.
Now, how can one show that $T(v)=0\implies v=0$? Well, the only thing we have to do is solve $T(v)=0$ for $v$. Let $v=\begin{bmatrix}w\\x\\y\\z \end{bmatrix}$.
$T(v)=0\implies \begin{cases} w+z=0\\y=0\\x-y=0\\w=0\end{cases}\implies\begin{cases} z=-w=0\\y=0\\x=y=0\\w=0\end{cases}\implies v=0$
hence $T$ is injective.


text:
• Development of a silicon photodiode-based compact gamma spectrometer using a Gd$_3$Ga$_3$Al$_2$O$_{12}$:Ce,B single crystal scintillator

• # Fulltext

https://www.ias.ac.in/article/fulltext/boms/042/06/0258

• # Keywords

Single crystal; scintillator; photodiode; gamma spectrometer.

• # Abstract

A compact gamma spectrometer was developed by employing an in-house grown single crystal of Gd$_3$Ga$_3$Al$_2$O$_{12}$:Ce,B scintillator optically coupled with a silicon photodiode. The performance of the detector was characterized in detail. The detector setup works with a low bias voltage of 9 V, drawn from a single battery. Power to the electronic components of the entire system is derived from a single universal serial bus port by employing required DC–DC converters. In addition to the low voltage operation, this developed spectrometer is very compact in size compared to the one developedby employing photo-multiplier tubes. The system offers excellent linearity over the gamma energy range of 344–1408 keV and an optimum energy resolution of about 13% at 662 keV.

• # Author Affiliations

1. Environmental Monitoring and Assessment Division, Bhabha Atomic Research Center, Mumbai 400085, India
2. Homi Bhabha National Institute, Mumbai 400094, India
3. Technical Physics Division, Bhabha Atomic Research Center, Mumbai 400085, India

• # Bulletin of Materials Science

Current Issue
Volume 42 | Issue 6
December 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019


text:
# Ellipsoid: Cutting Planes

## What is the Use Case?

The ellipsoid_cut example demonstrates using multiple cutting planes to constrain the distribution of particles on ellipsoids which are already aligned. This can be used in modeling scenarios where statistical modeling/analysis is needed for a region-of-interest on the anatomy/object-class at hand without having to affect the input data.

Here is an example of where the cutting planes intersect and ellipsoid:

## Grooming Steps

This use case starts with pre-aligned segmentations. The only grooming step is computing the signed distance transform for each segmentation.

## Optimization Parameters

The python code for the use case calls the optimize command of ShapeWorks which reads the project sheet with the shape filenames and optimization parameter values. See Project excel file for details regarding creating the project sheet. Below are the default optimization parameters for this use case.

{
"number_of_particles": 32,
"use_normals": 1,
"normals_strength": 15,
"checkpointing_interval": 200,
"keep_checkpoints": 0,
"iterations_per_split": 3000,
"optimization_iterations": 3000,
"starting_regularization": 100,
"ending_regularization": 10,
"recompute_regularization_interval": 2,
"domains_per_shape": 1,
"relative_weighting": 15,
"initial_relative_weighting": 0.05,
"verbosity": 0,
}

Here cutting_plane_counts is two for every ellipsoid and there are 15 ellipsoids so it is a list of twos of length 15. The cutting_planes parameter is a list of the two cutting plane points of length 15.

## Analyzing Shape Models

Below we can see the resulting shape model where correspondence points do not lie beyond the cutting planes and only a secton of the ellipsoids are modeled.