# N EGATIVE C LIQUES IN S ETS OF

N/A
N/A
Protected

Share "N EGATIVE C LIQUES IN S ETS OF"

Copied!
47
0
0

Full text

(1)

### E QUIANGULAR L INES

Emily J. King

joint work with Matt Fickus, John Jasper, Dustin Mixon, and Xiaoxian Tang

University of Bremen

Tight Frames and Approximation February 20–23, 2018

(2)

QUIANGULAR

INES

OUNDS ON

EAL

QUIANGULAR

INES

IMPLICES

MBEDDED IN

S

(3)

QUIANGULAR

INES

OUNDS ON

EAL

QUIANGULAR

INES

IMPLICES

MBEDDED IN

S

3/35

(4)

### O RTHONORMAL B ASES

LetΦ={ϕj}be an orthonormal basis forFdforF=R,C.

What are the nice properties ofΦ? 1. Each vector isunit length.

(No vector is weighted more than the others.) 2. The inner products between the vectors are

2A. optimallysmalland 2B. equal.

(Vectors represent “different information.”) 3. For anyx∈Fd,x=jhx,ϕjiϕj.

(We can easily compute a change of basis.) How can we generalize these traits?

Equiangular tight frame =1,2A,2B,3(up to scaling) Equiangular lines =1,2B

(5)

### O RTHONORMAL B ASES

LetΦ={ϕj}be an orthonormal basis forFdforF=R,C.

What are the nice properties ofΦ?

1. Each vector isunit length.

(No vector is weighted more than the others.) 2. The inner products between the vectors are

2A. optimallysmalland 2B. equal.

(Vectors represent “different information.”) 3. For anyx∈Fd,x=jhx,ϕjiϕj.

(We can easily compute a change of basis.) How can we generalize these traits?

Equiangular tight frame =1,2A,2B,3(up to scaling) Equiangular lines =1,2B

4/35

(6)

### O RTHONORMAL B ASES

LetΦ={ϕj}be an orthonormal basis forFdforF=R,C.

What are the nice properties ofΦ?

1. Each vector isunit length.

(No vector is weighted more than the others.)

2. The inner products between the vectors are 2A. optimallysmalland

2B. equal.

(Vectors represent “different information.”) 3. For anyx∈Fd,x=jhx,ϕjiϕj.

(We can easily compute a change of basis.) How can we generalize these traits?

Equiangular tight frame =1,2A,2B,3(up to scaling) Equiangular lines =1,2B

(7)

### O RTHONORMAL B ASES

LetΦ={ϕj}be an orthonormal basis forFdforF=R,C.

What are the nice properties ofΦ?

1. Each vector isunit length.

(No vector is weighted more than the others.) 2. The inner products between the vectors are

2A. optimallysmalland 2B. equal.

(Vectors represent “different information.”)

3. For anyx∈Fd,x=jhx,ϕjiϕj.

(We can easily compute a change of basis.) How can we generalize these traits?

Equiangular tight frame =1,2A,2B,3(up to scaling) Equiangular lines =1,2B

4/35

(8)

### O RTHONORMAL B ASES

LetΦ={ϕj}be an orthonormal basis forFdforF=R,C.

What are the nice properties ofΦ?

1. Each vector isunit length.

(No vector is weighted more than the others.) 2. The inner products between the vectors are

2A. optimallysmalland 2B. equal.

(Vectors represent “different information.”) 3. For anyx∈Fd,x=jhx,ϕjiϕj.

(We can easily compute a change of basis.)

How can we generalize these traits?

Equiangular tight frame =1,2A,2B,3(up to scaling) Equiangular lines =1,2B

(9)

### O RTHONORMAL B ASES

LetΦ={ϕj}be an orthonormal basis forFdforF=R,C.

What are the nice properties ofΦ?

1. Each vector isunit length.

(No vector is weighted more than the others.) 2. The inner products between the vectors are

2A. optimallysmalland 2B. equal.

(Vectors represent “different information.”) 3. For anyx∈Fd,x=jhx,ϕjiϕj.

(We can easily compute a change of basis.) How can we generalize these traits?

Equiangular tight frame =1,2A,2B,3(up to scaling) Equiangular lines =1,2B

4/35

(10)

### O RTHONORMAL B ASES

LetΦ={ϕj}be an orthonormal basis forFdforF=R,C.

What are the nice properties ofΦ?

1. Each vector isunit length.

(No vector is weighted more than the others.) 2. The inner products between the vectors are

2A. optimallysmalland 2B. equal.

(Vectors represent “different information.”) 3. For anyx∈Fd,x=jhx,ϕjiϕj.

(We can easily compute a change of basis.) How can we generalize these traits?

Equiangular tight frame =1,2A,2B,3(up to scaling)

(11)

### E QUIANGULAR L INES D

EFINITION

LetF=CorR. LetΦ={ϕj}nj=1Fkwith ϕj

=1 for all j∈ {1, . . . ,n}. If there exists anαsuch that for allj6=`,

hϕj,ϕ`i=α, Φis a set ofequiangular lines.

If further for allx∈Fk

n j=1

|hx,ϕji|2= n

kkxk2 ⇔ x= n k

### ∑

j

hx,ϕjiϕj, thenΦis anequiangular tight frame (ETF).

By slight abuse of notation,

I Φ= ϕ1 ϕ2 . . . ϕn , and

I αis the “angle.”

### T

HEOREM

(Goyal, Kovaˇcevi´c, Kelner 2001; Strohmer, Heath 2003; Benedetto, Kolesar 2006) ETFs are optimally robust against erasures and noise.

5/35

(12)

### E XAMPLES IN R

2

I ONB, ETF

I ETF, maximal set of equiangular lines

I FUNTF, worst case coherence

I Equiangular lines which are not an ETF

(13)

### R ESEARCH Q UESTIONS

Q1 : Givend(and 0<α<1), what is themaximal sizes(d)(resp., sα(d)) of a set of equiangular lines (resp., with angleα) inRd?

Q2 : Given a specific ETF or class of ETFs, what is thestructure of linear dependenciesof the vectors?

7/35

(14)

### G RAM M ATRICES

Instead ofΦ, we will usually deal with theGram matrix G(Φ) =ΦΦ.

LetInbe then×nidentity andJnthen×nall-ones matrix (where we writeIandJwhen clear from context).

Basic linear alg: IfG= (a−b)In+bJn, thenG has a simple eigenvalueλ1=a+ (n−1)band an eigenvalueλ2=a−bwith multiplicityn−1.

a b . . . b b a . . . b ... ... . .. ...

b b . . . a

(15)

I

1 0

0 1

I

1 −1212

12 1 −12

1212 1

I

1 0 −1 0

0 1 0 −1

−1 0 1 0

0 −1 0 1

I 1

2

2 2

2 1

!

9/35

(16)

### S WITCHING E QUIVALENCE

Basic linear algebra:G(Φ) =G(Ψ) ⇔ Φ=for some unitaryU.

GivenΦ={ϕ1, . . . ,ϕn}, ˜Φ={ϕn, . . . ,−ϕ1}has the same geometric and linear algebraic properties.

### D

EFINITION

Two sets of unit vectorsΦandΨinFdareswitching equivalent, denoted byΦ∼=Ψ, if there exists a diagonal matrixBwith unit norm diagonal entries and a permutation matrixCsuch that

(BC)·G(Φ)·(BC)−1 = G(Ψ).

Φ∼=Ψ⇒there exists a unitaryU, diagonal(1,−1)-matrixBwith unit norm diagonal entries, and permutation matrixCsuch that

UΦ(BC)−1=Ψ.

=((Van Lindt & Seidel 1966+generalization toF) +permutations) = (Projective unitary equivalence+permutations)

(17)

### S WITCHING E QUIVALENCE

Basic linear algebra:G(Φ) =G(Ψ) ⇔ Φ=for some unitaryU.

GivenΦ={ϕ1, . . . ,ϕn}, ˜Φ={ϕn, . . . ,−ϕ1}has the same geometric and linear algebraic properties.

### D

EFINITION

Two sets of unit vectorsΦandΨinFdareswitching equivalent, denoted byΦ∼=Ψ, if there exists a diagonal matrixBwith unit norm diagonal entries and a permutation matrixCsuch that

(BC)·G(Φ)·(BC)−1 = G(Ψ).

Φ∼=Ψ⇒there exists a unitaryU, diagonal(1,−1)-matrixBwith unit norm diagonal entries, and permutation matrixCsuch that

UΦ(BC)−1=Ψ.

=((Van Lindt & Seidel 1966+generalization toF) +permutations) = (Projective unitary equivalence+permutations)

10/35

(18)

### S WITCHING E QUIVALENCE

Basic linear algebra:G(Φ) =G(Ψ) ⇔ Φ=for some unitaryU.

GivenΦ={ϕ1, . . . ,ϕn}, ˜Φ={ϕn, . . . ,−ϕ1}has the same geometric and linear algebraic properties.

### D

EFINITION

Two sets of unit vectorsΦandΨinFdareswitching equivalent, denoted byΦ∼=Ψ, if there exists a diagonal matrixBwith unit norm diagonal entries and a permutation matrixCsuch that

(BC)·G(Φ)·(BC)−1 = G(Ψ).

Φ∼=Ψ⇒there exists a unitaryU, diagonal(1,−1)-matrixBwith unit norm diagonal entries, and permutation matrixCsuch that

UΦ(BC)−1=Ψ.

=((Van Lindt & Seidel 1966+generalization toF) +permutations) =

(19)

### T

HEOREM

(Godsil and Royle 2001; Chien and Waldron 2016) LetΦ,Ψ⊂Fdwith

|Φ|=|Ψ|=n be equiangular. ThenΦ∼=Ψif and only if there exists a σ∈Snsuch that for all i6=j6=k6=i.

hϕi,ϕjihϕj,ϕkihϕk,ϕii=hψσ(i),ψσ(j)ihψσ(j),ψσ(k)ihψσ(k),ψσ(i)i. WhenF=Rand ignoring permutations, this gives precisely the structure of thetwo-graphwhich represents equivalence classes of switching equivalences. (With permutations, get isomorphisms of the two-graphs.)

11/35

(20)

### D

EFINITION

LetΦbe a set of equiangular lines with angleα. IfX⊂Φis such that X∼=YwithG(Y) = (1+α)I−αJ, then we callXanegative clique.

I (K, Tang 2016) LetXbe a maximal negative clique in a givenΦ.

Xis called aK-base.

I (Fickus, Jasper, K, Mixon 2017) IfXis a negative clique of size 1+ (1/α), we callXa1/α-regular simplex.

Negative cliques have size≤1+1/α.

When the bound is saturated, they form a tight frame for their span (Fickus, Jasper, K, Mixon 2017);

otherwise they are linearly independent (e.g., Lemmens & Seidel 1973).

(21)

### D

EFINITION

LetΦbe a set of equiangular lines with angleα. IfX⊂Φis such that X∼=YwithG(Y) = (1+α)I−αJ, then we callXanegative clique.

I (K, Tang 2016) LetXbe a maximal negative clique in a givenΦ.

Xis called aK-base.

I (Fickus, Jasper, K, Mixon 2017) IfXis a negative clique of size 1+ (1/α), we callXa1/α-regular simplex.

Negative cliques have size≤1+1/α.

When the bound is saturated, they form a tight frame for their span (Fickus, Jasper, K, Mixon 2017);

otherwise they are linearly independent (e.g., Lemmens & Seidel 1973).

12/35

(22)

QUIANGULAR

INES

OUNDS ON

EAL

QUIANGULAR

INES

IMPLICES

MBEDDED IN

S

(23)

### D

EFINITION

(Lemmens & Seidel 1973) LetΦ∈Rdbe equiangular withK-baseX.

LetΞdenote the subspace spanned byX. Elements ofΦwhich lie in the same coset ofΞare called pillars.

### P

ROPOSITION

(Lemmens & Seidel 1973; K, Tang 2016) Letϕ∈Φ\X. If any K-base is of size1+1/α, then the norm of PΞϕis equal toα. If any K-base is of size

<1+1/α, then the norm of PΞϕdepends on the number of negative inner productshϕ,xii, xi∈X.

14/35

(24)

### B ASIC P ROCEDURE OF K, T ANG 2016

I (Lemmens & Seidel 1973) We only need to compute upper bounds onsα(d)forαthe reciprocal of an odd integer between 5 and a√

2d+1. (3 solved.)

I For each possibleα, we consider all of the possible sizes of K-bases. (≥2,≤1+ (1/α)).

I For eachK-base size, we partitionΦ\Xinto equivalence classes based on thenumberof negative inner products withXand analyze these (using combinatorics, graph theory, and linear algebra).

I We further split the above equivalence classes into classes based onwith whichelements ofXthe elements have a negative inner product and analyze these. (This will sometimes involve a bound of a size of particular spherical two-distance sets.)

(25)

### B ASIC P ROCEDURE OF K, T ANG 2016

I (Lemmens & Seidel 1973) We only need to compute upper bounds onsα(d)forαthe reciprocal of an odd integer between 5 and a√

2d+1. (3 solved.)

I For each possibleα, we consider all of the possible sizes of K-bases. (≥2,≤1+ (1/α)).

I For eachK-base size, we partitionΦ\Xinto equivalence classes based on thenumberof negative inner products withXand analyze these (using combinatorics, graph theory, and linear algebra).

I We further split the above equivalence classes into classes based onwith whichelements ofXthe elements have a negative inner product and analyze these. (This will sometimes involve a bound of a size of particular spherical two-distance sets.)

15/35

(26)

### B ASIC P ROCEDURE OF K, T ANG 2016

I (Lemmens & Seidel 1973) We only need to compute upper bounds onsα(d)forαthe reciprocal of an odd integer between 5 and a√

2d+1. (3 solved.)

I For each possibleα, we consider all of the possible sizes of K-bases. (≥2,≤1+ (1/α)).

I For eachK-base size, we partitionΦ\Xinto equivalence classes based on thenumberof negative inner products withXand analyze these (using combinatorics, graph theory, and linear algebra).

I We further split the above equivalence classes into classes based onwith whichelements ofXthe elements have a negative inner product and analyze these. (This will sometimes involve a bound of a size of particular spherical two-distance sets.)

(27)

### B ASIC P ROCEDURE OF K, T ANG 2016

I (Lemmens & Seidel 1973) We only need to compute upper bounds onsα(d)forαthe reciprocal of an odd integer between 5 and a√

2d+1. (3 solved.)

I For each possibleα, we consider all of the possible sizes of K-bases. (≥2,≤1+ (1/α)).

I For eachK-base size, we partitionΦ\Xinto equivalence classes based on thenumberof negative inner products withXand analyze these (using combinatorics, graph theory, and linear algebra).

I We further split the above equivalence classes into classes based onwith whichelements ofXthe elements have a negative inner product and analyze these. (This will sometimes involve a bound of a size of particular spherical two-distance sets.)

15/35

(28)

### D

EFINITION

LetΦ={ϕ1, . . . ,ϕn} ⊂Rdbe a set of unit normed vectors and let α,βR. TheΦis aspherical two-distance setifhϕi,ϕji ∈ {α,β}for alli,j∈ {1, . . . ,n},i6=j. We denote bys(d,α,β)the largest size of a spherical two-distance set with the given parameters.

Note: An equiangular set of lines is a spherical two-distance set w.r.t α,α, and thussα(d) =s(d,α,α).

(29)

1/5

### T

HEOREM

(K, Tang 2016) LetΦ⊂Rdbe an equiangular set with angle1/5. If d>60, then

|Φ| ≤ 148+3·s(d, 1/13,−5/13) ≤ 148+648d(d+2) 47d+169 .

17/35

(30)

### A TASTE OF THE CASES T

HEOREM

(K, Tang 2016) For n=1, . . . ,bK/2cand for each equivalence class x⊂X(K,n), we have the following upper bounds on|x|.

(1) If n=1, then

|x| ≤

(r−K, 1≥K−(1/α)+12

1−α

l(K,1)−α, 1<K−(1/α)+12 . (2) If1<n<K−(1/α)+12 , then|x| ≤ r+1.

(3) If n=K−(1/α)+12 , then|x| ≤ r−K+b2αr−K1−αc. (4) If K−(1/α)+12 <n<bK2c, then

|x| ≤ s(r,β,γ), whereβ= α1−l(K,n)−l(K,n)andγ= 1−l(K,n)α−l(K,n).

(31)

d

HEOREM

ANG

### 2016)

Let m be the largest positive integer such that(2m+1)2≤d+2. Then

s(d) ≤













4d(m+1)(m+2)

(2m+3)2−d , d=44, 45, 46, 76, 77, 78, 117, 118, 166, 222, 286, 358

((2m+1)2−2)((2m+1)2−1)

2 , other k between44and400

.

Applied the SDP approach of [Bachoc, Valentin 2008; Barg, Yu 2014]

to bound the size of certain spherical two distance sets in the cases they arose.

19/35

(32)

### N EW B OUND VS . O LD

FIGURE:K, Tang 2016 andsdp(d,15,15)

— :d(d+1)2 ? ? ?:sdp(d,15,15) +++: K, Tang 2016

FIGURE:K, Tang 2016 andsdp(d,17,17)

— :k(k+1)2 ? ? ?:sdp(d,17,−17) +++: K, Tang 2016

(33)

### P OSITIVE C LIQUES

One may similarly define apositive clique.

For all 0≤α<1 there exist at leastdvectors In Fdwith pairwise inner productα

Geometrically, one may think of “pushing”

vectors in an onb together.

One may analyze projections onto orthogonal complements of positive cliques and use Ramsey theory to obtain asymptotic relative bounds. (Balla, Dr¨axler, Keevash, Sudakov 2018)

21/35

(34)

QUIANGULAR

INES

OUNDS ON

EAL

QUIANGULAR

INES

IMPLICES

MBEDDED IN

S

(35)

EFINITION

ICKUS

ASPER

IXON

### 2017)

LetΦbe an ETF. The set of subsets of vectors (or the corresponding incidence matrix) which are 1/α-regular simplices is thebinder.

These are the smallest sets of linearly dependent vectors in the ETF:

For a general set of unit vectorsΦ,

size of the smallest set of linearly dependent vectors inΦ ≥ 1+ 1 µ(Φ). (Gerschgorin circle theorem applied to the Gram matrix. Donoho, Elad 2003)

23/35

(36)

### B INDER F INDER

BinderFinderis a relatively short Matlab code that uses triple

products and some clever combinatorial tricks to compute the binder of a given ETF.

(Could also be used on sets of equiangular lines.)

http://www.math.uni-bremen.de/cda/

(37)

### B INDERS OF ETF S IN C

3×9

Perhaps the first investigation of linear dependencies in equiangular Gabor frames (SIC-POVMs) was presented in a talk by Hughston in 2007 (cited in Dang, Blanchfield, Bengtsson, Appleby 2013), where the linear dependencies of certain SIC-POVMs inC3were shown to be represented by the Hesse configuration.

I came to the question via the construction of ETFs in (Jasper, Mixon, Fickus 2013). This construction involves a tensor-like construction of an incidence matrix of a BIBD with a 1/α-regular simplex. (Bad algebraic spread, butgoodgeometric spread?!?!?)

25/35

(38)

### B INDERS OF ETF S IN C

3×9

Perhaps the first investigation of linear dependencies in equiangular Gabor frames (SIC-POVMs) was presented in a talk by Hughston in 2007 (cited in Dang, Blanchfield, Bengtsson, Appleby 2013), where the linear dependencies of certain SIC-POVMs inC3were shown to be represented by the Hesse configuration.

I came to the question via the construction of ETFs in (Jasper, Mixon, Fickus 2013). This construction involves a tensor-like construction of an incidence matrix of a BIBD with a 1/α-regular simplex. (Bad algebraic spread, butgoodgeometric spread?!?!?)

(39)

### H ESSE C ONFIGURATION

The Hesse configuration is the set of all lines inF23:

FIGURE: By David Eppstein - Own work, CC0,

https://commons.wikimedia.org/w/index.php?curid=18920067

26/35

(40)

### “N ORMAL ” C ONFIGURATION

Letζ=e2πi/3,θ∈[0, 2π/6]\{0, 2π/9}.

SIC-POVMs:(Hughston 2007; Dang, Blanchfield, Bengtsson, Appleby 2013)

0 0 0 −e −eζ −eζ2 1 1 1

1 1 1 0 0 0 −e −eζ −eζ2

−e −eζ −eζ2 1 1 1 0 0 0

Kirkman ETFs(Fickus, Jasper, Mixon 2013):

BIBD(3, 2, 1) D3

0 1 1 1 0 1 1 1 0

 &

1 1 1

1 ζ ζ2 1 ζ2 ζ

=:

 w0

w1

w2

0 0 0 w0 w1

w1 0 0 0 w0

w w 0 0 0

(41)

### “N ORMAL ” C ONFIG . B INDER & G RAM OF B INDER

Left: Binder of a “normal” SIC-POVM inC3, Right: The Gram matrix of the binder.

28/35

(42)

### O BTAINING THE H ESSE C OFIGURATION

Letζ=e2πi/3,θ∈ {0, 2π/9}.

SIC-POVMs:(Hughston 2007; Dang, Blanchfield, Bengtsson, Appleby 2013)

0 0 0 −e −eζ −eζ2 1 1 1

1 1 1 0 0 0 −e −eζ −eζ2

−e −eζ −eζ2 1 1 1 0 0 0

Polyphase BIBD ETFs(Fickus, Jasper, Mixon, Peterson, Watson 2017;

Fickus, Jasper, K, Mixon 2017):

BIBD(3, 2, 1) D3

0 1 1 1 0 1 1 1 0

 &

1 1 1

1 ζ ζ2 1 ζ2 ζ

=:

 w0 w1 w2

0 0 0 w0 −w1

−w1 0 0 0 w0

(43)

### H ESSE C ONFIGURATION AS A B INDER

Left: Hesse Configuration binder of a SIC-POVM inC3, Right: The Gram matrix of the binder.

30/35

(44)

### B INDER OF H OGGAR ’ S L INES

Left: Binder of Hoggar’s lines (non-Gabor SIC POVM inC8with 1008 simplices), Right: The Gram matrix of the binder.

(45)

### C ONCLUSION

I Fond memories (nightmares?) of using Sylowp-groups in the classification of finite simple groups.

I

32/35

(46)

(47)

## organizing/chauffeuring and to New Zealand for being so beautiful!

http://www.math.uni-bremen.de/cda/

“New Upper Bounds for Equiangular Lines by Pillar Decomposition”

on arXiv (the paper formerly known as “Computing Upper Bounds for Equiangular Lines in Euclidean Space”)

“Equiangular tight frames that contain regular simplices” on arXiv

34/35

References

Related documents

Figure 5.31 (A) shows the quadtree model of the environment together with distance transform associated with &#34;visit all&#34; path planning, and the start (S) and goal

or “big” it might be. This is why classroom teaching is not improved simply by adding the presence of technology to the picture – of course, but you can't say

(c) the class includes persons who are non-party consumers in relation to the contravening conduct or declared term; a court may, on the application of the regulator, make such

We have shown that the New Zealand Department of Education ordered an ICT 1201 computer in 1959, that it was in all probability operational by August 1960, and that its

We demonstrate this in specific cases, and illustrate the connection to higher order statistics by showing the sensitivity of local anisotropy to phase randomization, after which

Perhaps love in the context of professional relationships within the social work process is at the heart of a 21st century emancipation and liberation of Māori and other

Illustration of the range of wave frequencies obtained from linear Vlasov theory using typical solar wind parameters, emphasizing oblique and quasi-two-dimensional wave vectors

Use a CRUD model to produce a REST interface Doing the right thing should be easy. 15

In this thesis, we proved several new facts about the hierarchy pertaining to collapse. Firstly, we showed that above any totally α-c.a. Due to the manner in which that

We have observed all nine elements of the reduced power spectral tensor of MHD-scale fluctuations in fast solar wind using wavelet transforms of magnetic field observations by

• Considers the role that State government can play in facilitating and augmenting the growth of Sydney and, more particularly, of office development in Western Sydney.. As

The change for Clare in the frame story is from midwinter to midsummer, from cold to warmth, from death to new life, from the white of snow to the green of growth, and from the

Are there any differences in the reasons what women give for studying Computer Science compared to men?.

The fourth principle highlighted the important contribution of public sector decision- making to environmental, economic and social wellbeing in the Northern Territory. The

When a commercial streetscape is mostly intact as shown in Figure 16, an infill building should relate to the heritage buildings directly adjacent to it in form, scale, massing

To achieve this level of integration will require legislative and policy change to provide an overarching governance framework, which coordinates and integrates strategic

The policy framework supporting Enterprise Precincts and the aligned new zoning tool sit within the broader implementation of Plan Melbourne, Melbourne’s strategic planning

While high labour and capital inputs are used to build the business during the entry and growth stages; during the exit stage, farmers may give up control of

Benzene (ppb) change in annual max 1-hour (MDA1) ground level concentrations from Scenario 2 due to future industry (S3-S2) for a subset of the CAMx 1.33 km domain centred over

The processing of the 2007 claim for compensation under MRCA 8 The processing of a subsequent claim in 2010 under MRCA 8 The raising of debts in 2015 in relation to the

5.15 At the time of Mr C’s requests for access to the NDIS, the NDIA did not have any policy or guideline dealing specifically with incarcerated individuals and access to the NDIS.

When focussing on Marine Parks only (Table 8), the total is between 200,000 and 300,00 tonnes, equating to over 1 million tonnes of carbon dioxide equivalent units per year.

applied to the governing partial differential equation and boundary conditions in a given differential system; 2) the transformed differential system is then solved with the