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When does a projective morphism give an etale morphism (into affine space)? (Finite field) (normalization) - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T18:32:17Z http://mathoverflow.net/feeds/question/16331 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/16331/when-does-a-projective-morphism-give-an-etale-morphism-into-affine-space-fini When does a projective morphism give an etale morphism (into affine space)? (Finite field) (normalization) Ivan 2010-02-25T00:07:39Z 2010-02-26T00:29:37Z
Does anyone know of generalizations on what Mumford (Red Book) calls "uniformizing parameters"? For example, given a regular quasi-projective scheme over a finite field $\mathbb{F}$, is there an etale morphism into affine space over $\mathbb{F}$?.
 http://mathoverflow.net/questions/16331/when-does-a-projective-morphism-give-an-etale-morphism-into-affine-space-fini/16346#16346 Answer by S. Carnahan for When does a projective morphism give an etale morphism (into affine space)? (Finite field) (normalization) S. Carnahan 2010-02-25T02:08:22Z 2010-02-25T02:08:22Z
This paper by Kedlaya might be what you want, since it contains some rearrangement of the words you used, but I can't really tell from the question. If you want a proper F-scheme to have an etale map to affine space, it has to be a disjoint union of finite F-schemes, and the affine space has to be zero dimensional.
 http://mathoverflow.net/questions/16331/when-does-a-projective-morphism-give-an-etale-morphism-into-affine-space-fini/16397#16397 Answer by Qing Liu for When does a projective morphism give an etale morphism (into affine space)? (Finite field) (normalization) Qing Liu 2010-02-25T14:29:58Z 2010-02-26T00:29:37Z
[EDIT: I re-arrange the answer to make the statements clearer].

The answer to the question is no, for any base field $k$.

First, we can characterize smooth affine varieties $X$ which are étale over an affine space $A=\mathbb A^n_k$.

 A smooth affine variety $X$ over $k$ is étale over an affine space $A=\mathbb A^n_k$ if and only if the sheaf of differentials $\Omega_{X/k}$ is free of rank $n$, generated by $n$ closed differential forms $df_1, \ldots, df_n$.

Proof: (1) If there exists an étale morphism $f: X\to A={\rm Spec}k[t_1,\ldots, t_n]$, then we have an isomorphism $\Omega_{A/k}\otimes_{O_A} O_X=f^{\star}\Omega_{A/k}\simeq \Omega_{X/k}$. Then take $f_i$ equal to the image of $t_i$ in $O(X)$. (2) If $df_1,\ldots, df_n$ as above exist, we define a morphism $f: X\to A$ to be associated to the moprhism of $k$-algebras $k[t_1,\ldots, t_n]\to O(X)$, $t_i\mapsto f_i$. Then by hypothesis on the $df_i$'s, the canonical morphism $f^{\star}\Omega_{A/k}\to \Omega_{X/k}$ is an isomorphism. Therefore $f$ is étale.

Remark: it is not enough to suppose that $\Omega_{X/k}$ is free to insure that $X$ is étale over $A$. For example, if $X$ is an elliptic curve $E$ minus the origin, then $\Omega_{X/k}$ is free (because $\Omega_{E/k}$ is !), but at least in characteristic $0$, $X$ is not étale over $\mathbb A^1_k$ (if so, it would be finite and étale over $\mathbb A^1$ by considering the extension $E\to \mathbb P^1_k$, but $\mathbb A^1_k$ is simply connected in characteristic $0$). I don't known whether this can happen in positive characteristic. Note that this is already an example (in characteristic 0) of a smooth affine curve which is not étale over $\mathbb A^1$.

Now we construct in any characteristic a smooth affine curve $X$ such that $\Omega_{X/k}$ is not free. By the above, $X$ will not be étale over ${\mathbb A^1}$. Fix a projective smooth connected curve $C$ over $k$ of genus $g>1$. Suppose that for any affine open subset $X$ of $C$, $\Omega_{X/k}$ is free. We want to find a contradiction.

We first reduce to the case $k$ is algebraically closed (for simplicity, this is actually not necessary). I claim that over the algebraic closure $\bar{k}$ of $k$, $\Omega_{X'/\bar{k}}$ is free for any affine open subset $X'$ of $C_{\bar{k}}$. Indeed, $X'$ is defined over a finite extension $K/k$, and the projection $C_{K}\to C$ induces an étale morphism from $X'$ to an affine open subset $X$ of $C$, thus $\Omega_{X'/K}\simeq \Omega_{X/k}\otimes O_{X'}$ is free.

Now we suppose that $k$ is algebraically closed. For any closed point $x\in C$, $\Omega_{X/k}$, where $X=C \setminus {x}$, is free. So $\Omega_{C/k}|_X$ is trivial. This implies that the canonical divisor $K_C$ on $C$ is trivial on $X$, hence linearly equivalent to $(2g-2)[x]$. Fix a point $x_0\in C(k)$. Then the immersion $C\to {\rm Jac}(C)$, $x\mapsto [x-x_0]$, maps $C$ into the $(2g-2)$-torsion part of ${\rm Jac}(C)(k)$ which is finite. Contradicton.

Final remark: if you have a separated scheme $X$ which is étale (or more generally quasi-finite) over any affine scheme $A$, then it has to be quasi-affine. This is because by Zariski's Main Theorem, $X$ is an open subscheme of a scheme $\overline{X}$ which is finite over $A$, hence $\overline{X}$ is affine.


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Installation

Re: Problems with WEBWORK after upgr to 2.4

by Michael Gage -
Number of replies: 0
ok. The one other piece of standard advice (in addition to restarting
the server often while troubleshooting) is to keep an eye on the
apache server error log --it sometimes contains useful error messages which are not printed to the screen.

I think you need to go into the file at:

/opt/webwork/webwork2/lib/WeBWorK/DB/Driver/SQL.pmand try to check the the database user name is well defined.(The line where the call is made to the database is in theerror message above.)If you enter something likewarn "the database user name is ",\$params->{username};this message will be printed to the error log.If the username is wrong here, we'll need to chase the error upstream:One thing to double check is the contents of database.confin particular the section defining "problem_user"--Mike

format: plain


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# zbMATH — the first resource for mathematics

A general variable neighborhood search for solving the uncapacitated single allocation $$p$$-hub median problem. (English) Zbl 1188.90142
Summary: We present a new general variable neighborhood search approach for the uncapacitated single allocation $$p$$-hub median problem in networks. This NP hard problem is concerned with locating hub facilities in order to minimize the traffic between all origin-destination pairs. We use three neighborhoods and efficiently update data structures for calculating new total flow in the network. In addition to the usual sequential strategy, a new nested strategy is proposed in designing a deterministic variable neighborhood descent local search. Our experimentation shows that general variable neighborhood search based heuristics outperform the best-known heuristics in terms of solution quality and computational effort. Moreover, we improve the best-known objective values for some large Australia Post and PlanetLab instances. Results with the new nested variable neighborhood descent show the best performance in solving very large test instances.

##### MSC:
 90B80 Discrete location and assignment 90C59 Approximation methods and heuristics in mathematical programming
PlanetLab
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Outlook: VINTAGE ENERGY LTD is assigned short-term Ba1 & long-term Ba1 estimated rating.
Dominant Strategy : Wait until speculative trend diminishes
Time series to forecast n: 03 Mar 2023 for (n+4 weeks)
Methodology : Modular Neural Network (Emotional Trigger/Responses Analysis)

## Abstract

VINTAGE ENERGY LTD prediction model is evaluated with Modular Neural Network (Emotional Trigger/Responses Analysis) and Stepwise Regression1,2,3,4 and it is concluded that the VEN stock is predictable in the short/long term. According to price forecasts for (n+4 weeks) period, the dominant strategy among neural network is: Wait until speculative trend diminishes

## Key Points

2. Dominated Move
3. Why do we need predictive models?

## VEN Target Price Prediction Modeling Methodology

We consider VINTAGE ENERGY LTD Decision Process with Modular Neural Network (Emotional Trigger/Responses Analysis) where A is the set of discrete actions of VEN stock holders, F is the set of discrete states, P : S × F × S → R is the transition probability distribution, R : S × F → R is the reaction function, and γ ∈ [0, 1] is a move factor for expectation.1,2,3,4

F(Stepwise Regression)5,6,7= $\begin{array}{cccc}{p}_{a1}& {p}_{a2}& \dots & {p}_{1n}\\ & ⋮\\ {p}_{j1}& {p}_{j2}& \dots & {p}_{jn}\\ & ⋮\\ {p}_{k1}& {p}_{k2}& \dots & {p}_{kn}\\ & ⋮\\ {p}_{n1}& {p}_{n2}& \dots & {p}_{nn}\end{array}$ X R(Modular Neural Network (Emotional Trigger/Responses Analysis)) X S(n):→ (n+4 weeks) $∑ i = 1 n s i$

n:Time series to forecast

p:Price signals of VEN stock

j:Nash equilibria (Neural Network)

k:Dominated move

a:Best response for target price

For further technical information as per how our model work we invite you to visit the article below:

How do AC Investment Research machine learning (predictive) algorithms actually work?

## VEN Stock Forecast (Buy or Sell) for (n+4 weeks)

Sample Set: Neural Network
Stock/Index: VEN VINTAGE ENERGY LTD
Time series to forecast n: 03 Mar 2023 for (n+4 weeks)

According to price forecasts for (n+4 weeks) period, the dominant strategy among neural network is: Wait until speculative trend diminishes

X axis: *Likelihood% (The higher the percentage value, the more likely the event will occur.)

Y axis: *Potential Impact% (The higher the percentage value, the more likely the price will deviate.)

Z axis (Grey to Black): *Technical Analysis%

## IFRS Reconciliation Adjustments for VINTAGE ENERGY LTD

1. If there is a hedging relationship between a non-derivative monetary asset and a non-derivative monetary liability, changes in the foreign currency component of those financial instruments are presented in profit or loss.
2. An entity shall apply the impairment requirements in Section 5.5 retrospectively in accordance with IAS 8 subject to paragraphs 7.2.15 and 7.2.18–7.2.20.
3. However, the fact that a financial asset is non-recourse does not in itself necessarily preclude the financial asset from meeting the condition in paragraphs 4.1.2(b) and 4.1.2A(b). In such situations, the creditor is required to assess ('look through to') the particular underlying assets or cash flows to determine whether the contractual cash flows of the financial asset being classified are payments of principal and interest on the principal amount outstanding. If the terms of the financial asset give rise to any other cash flows or limit the cash flows in a manner inconsistent with payments representing principal and interest, the financial asset does not meet the condition in paragraphs 4.1.2(b) and 4.1.2A(b). Whether the underlying assets are financial assets or non-financial assets does not in itself affect this assessment.
4. For the purposes of applying the requirements in paragraphs 5.7.7 and 5.7.8, an accounting mismatch is not caused solely by the measurement method that an entity uses to determine the effects of changes in a liability's credit risk. An accounting mismatch in profit or loss would arise only when the effects of changes in the liability's credit risk (as defined in IFRS 7) are expected to be offset by changes in the fair value of another financial instrument. A mismatch that arises solely as a result of the measurement method (ie because an entity does not isolate changes in a liability's credit risk from some other changes in its fair value) does not affect the determination required by paragraphs 5.7.7 and 5.7.8. For example, an entity may not isolate changes in a liability's credit risk from changes in liquidity risk. If the entity presents the combined effect of both factors in other comprehensive income, a mismatch may occur because changes in liquidity risk may be included in the fair value measurement of the entity's financial assets and the entire fair value change of those assets is presented in profit or loss. However, such a mismatch is caused by measurement imprecision, not the offsetting relationship described in paragraph B5.7.6 and, therefore, does not affect the determination required by paragraphs 5.7.7 and 5.7.8.

*International Financial Reporting Standards (IFRS) adjustment process involves reviewing the company's financial statements and identifying any differences between the company's current accounting practices and the requirements of the IFRS. If there are any such differences, neural network makes adjustments to financial statements to bring them into compliance with the IFRS.

## Conclusions

VINTAGE ENERGY LTD is assigned short-term Ba1 & long-term Ba1 estimated rating. VINTAGE ENERGY LTD prediction model is evaluated with Modular Neural Network (Emotional Trigger/Responses Analysis) and Stepwise Regression1,2,3,4 and it is concluded that the VEN stock is predictable in the short/long term. According to price forecasts for (n+4 weeks) period, the dominant strategy among neural network is: Wait until speculative trend diminishes

### VEN VINTAGE ENERGY LTD Financial Analysis*

Rating Short-Term Long-Term Senior
Outlook*Ba1Ba1
Income StatementBaa2B3
Balance SheetBa3Baa2
Leverage RatiosB3Ba2
Cash FlowBa1Baa2
Rates of Return and ProfitabilityB3C

*Financial analysis is the process of evaluating a company's financial performance and position by neural network. It involves reviewing the company's financial statements, including the balance sheet, income statement, and cash flow statement, as well as other financial reports and documents.
How does neural network examine financial reports and understand financial state of the company?

### Prediction Confidence Score

Trust metric by Neural Network: 76 out of 100 with 508 signals.

## References

1. Arjovsky M, Bottou L. 2017. Towards principled methods for training generative adversarial networks. arXiv:1701.04862 [stat.ML]
2. Zou H, Hastie T. 2005. Regularization and variable selection via the elastic net. J. R. Stat. Soc. B 67:301–20
3. Abadie A, Diamond A, Hainmueller J. 2010. Synthetic control methods for comparative case studies: estimat- ing the effect of California's tobacco control program. J. Am. Stat. Assoc. 105:493–505
4. Zeileis A, Hothorn T, Hornik K. 2008. Model-based recursive partitioning. J. Comput. Graph. Stat. 17:492–514 Zhou Z, Athey S, Wager S. 2018. Offline multi-action policy learning: generalization and optimization. arXiv:1810.04778 [stat.ML]
5. G. J. Laurent, L. Matignon, and N. L. Fort-Piat. The world of independent learners is not Markovian. Int. J. Know.-Based Intell. Eng. Syst., 15(1):55–64, 2011
6. Bai J, Ng S. 2002. Determining the number of factors in approximate factor models. Econometrica 70:191–221
7. Zou H, Hastie T. 2005. Regularization and variable selection via the elastic net. J. R. Stat. Soc. B 67:301–20
Frequently Asked QuestionsQ: What is the prediction methodology for VEN stock?
A: VEN stock prediction methodology: We evaluate the prediction models Modular Neural Network (Emotional Trigger/Responses Analysis) and Stepwise Regression
Q: Is VEN stock a buy or sell?
A: The dominant strategy among neural network is to Wait until speculative trend diminishes VEN Stock.
Q: Is VINTAGE ENERGY LTD stock a good investment?
A: The consensus rating for VINTAGE ENERGY LTD is Wait until speculative trend diminishes and is assigned short-term Ba1 & long-term Ba1 estimated rating.
Q: What is the consensus rating of VEN stock?
A: The consensus rating for VEN is Wait until speculative trend diminishes.
Q: What is the prediction period for VEN stock?
A: The prediction period for VEN is (n+4 weeks)


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# Past Junior Topology and Group Theory Seminar

6 February 2013
16:00
Montserrat Casals-Ruiz
Abstract

The theory of equations over groups goes back to the very beginning of group theory and is linked to many deep problems in mathematics, such as the Diophantine problem over rationals. In this talk, we shall survey some of the key results on equations over groups, give an outline of the Makanin-Razborov process (an algorithm for solving equations over free groups) and its connections to other results in group theory and low-dimensional topology.

• Junior Topology and Group Theory Seminar
30 January 2013
16:00
David Hume; Robert Kropholler; Martin Palmer and Alessandro Sisto
Abstract

We will discuss (very) recent work by Hensel; Przytycki and Webb, who describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces.

• Junior Topology and Group Theory Seminar
30 January 2013
12:00
Lukasz Grabowski
Abstract

The eponymous result is due to Bridson and Vogtmann, and was proven in their paper "Automorphisms of Automorphism Groups of Free Groups" (Journal of Algebra 229). While I'll remind you all the basic definitions, it would be very helpful to be already somewhat familiar with the outer space.

• Junior Topology and Group Theory Seminar
23 January 2013
16:00
John Mackay
Abstract

I'll discuss work of Wise and Ollivier-Wise that gives cubulations of certain small cancellation and random groups, which in turn shows that they do not have property (T).

• Junior Topology and Group Theory Seminar
16 January 2013
16:00
Robert Kropholler
Abstract

I will be looking at some conjectures and theorems closely related to the h-cobordism theorem and will try to show some connections between them and some group theoretic conjectures.

• Junior Topology and Group Theory Seminar
3 December 2012
(All day)
Diana Davis
Abstract

We will start with the square torus, move on to all regular polygons, and then look at a large family of flat surfaces called Bouw-Möller surfaces, made by gluing together many polygons. On each surface, we will consider the action of a certain shearing action on geodesic paths on the surface, and a certain corresponding sequence.

• Junior Topology and Group Theory Seminar
28 November 2012
16:00
Will Cavendish
Abstract

A subgroup $H$ of a group $G$ is said to be engulfed if there is a
finite-index subgroup $K$ other than $G$ itself such that $H<K$, or
equivalently if $H$ is not dense in the profinite topology on $G$.  In
this talk I will present a variety of methods for showing that a
subgroup of a discrete group is engulfed, and demonstrate how these
methods can be used to study finite-sheeted covering spaces of
topological spaces.

• Junior Topology and Group Theory Seminar
21 November 2012
16:00
Andrew Sale
Abstract

Let F be a free group, and N a normal subgroup of F with derived subgroup N'. The Magnus embedding gives a way of seeing F/N' as a subgroup of a wreath product of a free abelian group over over F/N. The aim is to show that the Magnus embedding is a quasi-isometric embedding (hence "Q.I." in the title). For this I will use an alternative geometric definition of the embedding (hence "picture"), which I will show is equivalent to the definition which uses Fox calculus. Please note that we will assume no prior knowledge of calculus.

• Junior Topology and Group Theory Seminar
15 November 2012
16:30
Søren Fuglede Jørgensen
Abstract
In St John's College
In this part, I will redefine the quantum representations for $G = SU(2)$ making no mention of flat connections at all, instead appealing to a purely combinatorial construction using the knot theory of the Jones polynomial.

 Using these, I will discuss some of the properties of the representations, their strengths and their shortcomings. One of their main properties, conjectured by Vladimir Turaev and proved by Jørgen Ellegaard Andersen, is that the collection of the representations forms an infinite-dimensional faithful representation. As it is still an open question whether or not mapping class groups admit faithful finite-dimensional representations, it becomes natural to consider the kernels of the individual representations. Furthermore, I will hopefully discuss Andersen's proof that mapping class groups of closed surfaces do not have Kazhdan's Property (T), which makes essential use of quantum representations.

• Junior Topology and Group Theory Seminar
31 October 2012
16:00
Jason Semeraro
Abstract

Saturated fusion systems are a next generation approach to the theory of finite groups- one major motivation being the opportunity to borrow techniques from homotopy theory. Extending work of Broto, Levi and Oliver, we introduce a new object - a 'tree of fusion systems' and give conditions (in terms of the orbit graph) for the completion to be saturated. We also demonstrate that these conditions are 'best possible' by producing appropriate counterexamples. Finally, we explain why these constructions provide a powerful way of building infinite families of fusion systems which are exotic (i.e. not realisable as the fusion system of a finite group) and give some concrete examples.

• Junior Topology and Group Theory Seminar


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# Proof of $M$ Noetherian if and only if all submodules are finitely generated

Is my proof of proposition 6.2 on page 75 correct? (it's different from what they do in the book)

Proposition 6.2.: $M$ is a Noetherian $A$ module $\iff$ every submodule of $M$ is finitely generated.

My proof:

$\implies$ Assume $M$ has a submodule $N$ that is not finitely generated. Say, $N = \langle \bigcup_{i=1}^\infty \{ n_i \} \rangle$. Then the following is an increasing chain that is not stationary: $\langle n_1 \rangle \subset \langle n_1, n_2 \rangle \subset \langle n_1, n_2, n_3 \rangle \subset \dots$ Hence $M$ is not Noetherian.

$\Longleftarrow$ Assume $M$ is not Noetherian. Then there is a non-stationary increasing chain of submodules $N_1 \subset N_2 \subset \dots$. Then $N = \bigcup_{k=1}^\infty N_k$ is a submodule. If $N$ was finitely generated, $N_k$ would be stationary. Hence $N$ is a submodule that is not finitely generated.

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You might consider doing this. –  lentic catachresis Jul 20 '12 at 12:56
@BrunoStonek So I should post this as an answer? –  Rudy the Reindeer Jul 20 '12 at 12:58
@BrunoStonek Usually, people manage to write more than just "yes" to my questions. –  Rudy the Reindeer Jul 20 '12 at 12:59
I did not express any views on the subject, I just directed you to a meta question that is very related, and to the (only) answer, which was quite well received by the community. If you're worrying about preventing good answers by others by doing what is suggested in that post, then don't worry! There can be multiple answers to one question, no? If someone has something interesting to add, they can always do it, no matter if you posted your intended solution in the question or in an answer. –  lentic catachresis Jul 20 '12 at 13:01
@BrunoStonek Ok : ) –  Rudy the Reindeer Jul 20 '12 at 13:07

The first part is not quite correct: you have assumed $N$ is countably generated.

The second part looks good, though you should make sure you know why the fact that $N$ is finitely generated implies that the sequence is stationary.

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But if $N$ is generated by an uncountable set, couldn't you just pick out a countable set and make the exact same argument? I would think you would also have to make some statement about the non-redundancy of generators too, or minimality of the generating set. –  Derek Allums Jul 20 '12 at 13:40
I agree with Colin. If feels strange to me to write down a generating set when you are assuming $N$ is not finitely generated. After all, why not just write $N = \langle \cup_{n \in N} n \rangle$. You are right to be worried about redundancies, though that is an issue in your proof as well. Try this: choose elements $n_i \in N$ inductively so that $n_i \not\in \langle n_1, \ldots, n_{i-1}\rangle$... –  Morgan Sherman Jul 20 '12 at 14:28
I did indeed assume that it's countably generated, of course without intending to. Thank you! –  Rudy the Reindeer Jul 21 '12 at 10:30
But what do you think of unit3000-21's suggestion to fix my proof? –  Rudy the Reindeer Jul 21 '12 at 10:31
@ClarkKent Take a look at Benjamin's answer below this one. I think that ties up any loose ends. –  Derek Allums Jul 21 '12 at 21:44

Your first part is strange. I would rephrase the argument as follows: Suppose that $M$ is a Noetherian $A$ - module and there exists a submodule $N$ that is not finitely generated. Now choose $f_1 \in N$. Since $N$ is not finitely generated, $\langle f_1 \rangle \subsetneqq N$ and so we can choose $f_2$ such that $f_2 \in N - \langle f_1 \rangle$. By the same argument as before, we can choose $f_3 \in N - \langle f_1,f_2\rangle$ such that $\langle f_1,f_2,f_3 \rangle \subsetneqq N.$ We now have a chain of submodules of $N$ (and hence of $M$) such that

$$\langle f_1 \rangle \subsetneqq \langle f_1,f_2 \rangle \subsetneqq \ldots \subsetneqq \langle f_1,f_2,f_3 \rangle$$


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In Java, the dynamic array is implemented using an ArrayList. To allocate memory for real 2D array you need to use malloc(dim1 * dim2 * sizeof(int)). For declaring dynamic arrays you do not mention the array range. Dynamic arrays are the next logical extension of arrays. Actor) Class Types (e.g. Maybe the easiest way to create our color list array is the following: Listing1: Code with the color list array: With templates, the size could be known at compile-time, and with operator overloading the standard [] array syntax could be used instead of calling functions get and set, resulting in a cleaner API. Several cross-platform frameworks provide dynamic array implementations for C, including CFArray and CFMutableArray in Core Foundation, and GArray and GPtrArray in GLib. The dynamic array introduces some important overhead in both time and space. Additionally, they present a variant where growing and shrinking the buffer has not only amortized but worst-case constant time. This means that the first index in the list is 0, not 1. In a 1999 paper,[17] Brodnik et al. This extra space is what allows extra elements to be added to the array. Please try reloading this page Help Create Join Login. This is called an array. summary DWScript Dynamic Arrays Dynamic Arrays. In the worst case, this is 2n−n+12n - n + 12n−n+1. Register Start a Wiki. The elements of the dynamic array are stored contiguously at the start of the underlying array, and the remaining positions towards the end of the underlying array are reserved, or unused. describe a tiered dynamic array data structure, which wastes only n1/2 space for n elements at any point in time, and they prove a lower bound showing that any dynamic array must waste this much space if the operations are to remain amortized constant time. For example, .NET arrays are dynamic arrays, (resizeable via Array.Resize,) Java arrays are static only, and Delphi offers both static and dynamic arrays. 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Dynamic arrays are one of the most essential tools for any game-logic that you want to do, where the actions of the player, the in-game AI, and the rest of your team cannot be known in advance, but need to be tracked, organized, and facilitated via UE4 C++ code systems. So let's see how to create dynamic array to store our color list. Sign up, Existing user? Ada's Ada.Containers.Vectors generic package provides dynamic array implementation for a given subtype. So, the space used by a dynamic array is O(2n)O(2n)O(2n), for a wasted space factor of O(n)O(n)O(n). &\text{Add*/Delete at end - O(1),} &\text{Add/Delete at beginning - O(n),}\end{array}Add*/Delete at end - O(1),Get - O(1),Add/Delete at beginning - O(n),Set - O(1). It's important for the programmer to understand their program and whether or not it calls for an array or a dynamic array. Class) Dynamic arrays can be defined as a class member, or inside a function: When the array needs to double in size, the original array is first copied. Add … Dynamic arrays are supported as reference types, they are declared without bounds. Also, in a highly fragmented memory region, it may be expensive or impossible to find contiguous space for a large dynamic array, whereas linked lists do not require the whole data structure to be stored contiguously. The elements (items) in the array are numbered, starting from zero, and you use square brackets. Python's list datatype implementation is a dynamic array. Gap buffers are similar to dynamic arrays but allow efficient insertion and deletion operations clustered near the same arbitrary location. Static arrays, as opposed to dynamic arrays, have a fixed length that is specified as a constant value at compile time.. If some function expects pointer to 2D array, like foo(int * bar[5][6]) and you pass your x, weird things will happen. This disadvantage is mitigated by the gap buffer and tiered vector variants discussed under Variants below. Here, a dynamic array would be a good choice. Then values are added that force the dynamic array to grow. Elements can be removed from the end of a dynamic array in constant time, as no resizing is required. Dynamic arrays are a special composite type in UnrealScript that contains a variable number of elements in a well-defined order. In computer science, a dynamic array, growable array, resizable array, dynamic table, mutable array, or array list is a random access, variable-size list data structure that allows elements to be added or removed. Games Movies TV Video. Introduction to Dynamic Array in Java. Being a fundamental change to the way formulas work, we wanted to take some time to gather and incorporate user feedback. Below are some of … Dynamic Array Excel no longer needs this as it can output arrays natively. Dynamic arrays give the programmer more flexibility during the execution of a program. In the worst case asymptotically, inserting a new element takes O(n)O(n)O(n). In this case, accessing items in the array sequentially will actually involve accessing multiple non-contiguous areas of memory, so the many advantages of the cache-friendliness of this data structure are lost. The array type exists exactly for such purposes. The only limitation of arrays is that it is a fixed size. It takes initially array of size 4 and when it gets full, a new array is created with its double size and the data of first array get copied into second array, now the new item is inserted into new array. For example: It needs to be dynamic because you don't know how many different things you might eat during the day. This leads to compatibility issues. Dynamic arrays overcome a limit of static arrays, which have a fixed capacity that needs to be specified at allocation. A Dynamic array (vector in C++, ArrayList in Java) automatically grows when we try to make an insertion and there is no more space left for the new item.Usually the area doubles in size. It is supplied with standard libraries in many modern mainstream programming languages. Hashed array tree (HAT) is a dynamic array algorithm published by Sitarski in 1996. This is where C++ would really shine. Dynamic arrays are resizable versions of regular arrays, with built-in functions to allow easy management.. Example: Multidimensional dynamic array/fr. Dynamic arrays are a common example when teaching amortized analysis. /* The size of a single item in the array */, /* The capacity, eg how much room until resize is needed */, /* The actual number of items, this is always <= capacity */, /* The internal memory buffer that holds the items */, https://en.wikipedia.org/wiki/Dynamic_array, https://brilliant.org/wiki/dynamic-arrays/. Their lower bound is always zero. Example 1. A fixed-size array will suffice in applications where the maximum logical size is fixed (e.g. All three also have List data structures as part of their standard libraries. In addition to *Low*, *High*, *Length* and *Count*, they also … Typically resizing is expensive because it involves allocating a new underlying array and copying each element from the original array. A dynamic array might be preferred if: To avoid incurring the cost of resizing many times, dynamic arrays resize by a large amount, such as doubling in size, and use the reserved space for future expansion. Then the program finds a new, larger contiguous area of memory. Also the name of second array creates an alias of first so that it can be accessed by the same name as previous and the first array gets disposed Declaring Dynamic Arrays. The dynamic array is able to change its size during program execution. But, creating an array that is far bigger than you need is extremely wasteful of computer memory. In computer science, a dynamic array, growable array, resizable array, dynamic table, mutable array, or array list is a random access, variable-size list data structure that allows elements to be added or removed. It is supplied with standard libraries in many modern mainstream programming languages. 384,870 Pages. by specification), or can be calculated before the array is allocated. So you don't need to … Since the item size isn't known until runtime, it is necessary to pass in pointers to the values being read and written to the array. However, in languages like Python or Java that enforce reference semantics, the dynamic array generally will not store the actual data, but rather it will store references to the data that resides in other areas of memory. Unless you limit an array size, the default arrays are dynamic arrays. The types available are: Double; Int; String; Pointer (e.g. It is supplied with standard libraries in … A static array is a composite data type in UnrealScript that groups multiple values of the same type. Bagwell (2002)[18] presented the VList algorithm, which can be adapted to implement a dynamic array. Dynamic Arrays at Scratchpad, the place for everybody and everything! The actual length of the array must be set with the standard SetLength function, which will allocate the necessary memory for storing the array elements. There are more ways to create dynamic arrays in PHP. Smalltalk's OrderedCollection is a dynamic array with dynamic start and end-index, making the removal of the first element also O(1). Python's list built-in type is implemented as a dynamic array. This size is maintained until we've added enough elements to fill it. It is not possible to use an enumerative type or any other ordinal type as an index. The first element is alw… A dynamic array's definition will only allocate space for a pointer.During runtime, various routines will ensure convenient usage but, most importantly, the syntax of accessing an array's elements by placing indices in square brackets is supported by the compiler, implemented as automatic de-referencing of the pointer. Dynamic arrays are the next logical extension of arrays. Static arrays should be used when you do not want an array to hold more items that its predefined size. Dynamic arrays are growable arrays and have an advantage over static arrays. So, to set the value of index 1, we must have pushed two elements to the empty ArrayList. The algorithm has O(1) amortized performance when appending a series of objects to the end of a hashed array tree. An advantage is that it also allows addition at the end, until it reaches a certain capacity, in which case it reassigns a new capacity and reallocates itself. Original Author: Rama () Dear Community, Here is my introduction to UE4 C++ Dynamic Arrays! Student.Count) and a max (Student.Max). In case of a dynamic array type, the initial length of the array is zero. Elements can be added at the end of a dynamic array in constant timeby using the reserved space, until this space is completely consumed. For example, if you wanted to create a program that kept track of everything you ate during the day, you would need to use a dynamic data structure (one that can grow and shrink during program execution). Dynamic arrays have advantages and disadvantages when compared to arrays with a fixed number of elements (for these, see Variable Syntax ). [5] There have been various discussions on ideal growth factor values, including proposals for the golden ratio as well as the value 1.5. If the size is decreased, only the existing indices values will be preserved. You may recall the SINGLE function that was released last year. From Free Pascal wiki. The current array data can be preserved while changing the array element size. Teachers might have an array of student names, Student[0], Student[1] … Student[Student.Max]. The elements of the dynamic array are stored contiguously at the start of the underlying array, and the remaining positions towards the end of the underlying array are reserved, or unused. Oh no! Open Source Software. It represents an ordered collection of an object that can be indexed individually. Some styles failed to load. Compared to linked lists, dynamic arrays have faster indexing (constant time versus linear time) and typically faster iteration due to improved locality of reference; however, dynamic arrays require linear time to insert or delete at an arbitrary location, since all following elements must be moved, while linked lists can do this in constant time. In computer science, a dynamic array, growable array, resizable array, dynamic table, mutable array, or array list is a random access, variable-size list data structure that allows elements to be added or removed. With a normal array, if the array is completely filled during program execution, there's nothing that the programmer can do. Welcome to the Scratchpad Wiki! Already have an account? So instead of implicitly intersecting, DA Excel calls out where this could have happened in … If memory allocator uses a first-fit allocation algorithm, then growth factor values such as a=2 can cause dynamic array expansion to run out of memory even though a significant amount of memory may still be available. A simple dynamic array can be constructed by allocating an array of fixed-size, typically larger than the number of elements immediately required. A dynamic array is not the same thing as a dynamically allocated array, which is an array whose size is fixed when the array is allocated, although a dynamic array may use such a fixed-size array as a back end. A simple dynamic array can be constructed by allocating an array of fixed-size, typically larger than the number of elements immediately required. The number of elements used by the dynamic array contents is its logical size or size, while the size of the underlying array is called the dynamic array's capacity or physical size, which is the maximum possible size without relocating data.[2]. A dynamic array is initialized with a single value, 2. Let me try to explain the changes with a really basic example. Only the logical size is printed out to the user, so the process of doubling is hidden. Common Lisp provides a rudimentary support for resizable vectors by allowing to configure the built-in array type as adjustable and the location of insertion by the fill-pointer. [1], A simple dynamic array can be constructed by allocating an array of fixed-size, typically larger than the number of elements immediately required. # Available Types. _PRESERVE can preserve data when REDIM or $DYNAMIC is used to create dynamic arrays. They're awesome! With a dynamic array, one can keep pushing values into the array. Space - O(n)\begin{array}{c}&&\text{Space - O(n)} \end{array}Space - O(n). Jump to:navigation, search Increasing the size will preserve data inside of existing indices in the resized array. To create arrays dynamically in C#, use the ArrayList collection. Get - O(1),Set - O(1),Add*/Delete at end - O(1),Add/Delete at beginning - O(n),\begin{array}{c}&&\text{Get - O(1),} &\text{Set - O(1),} \\ This sample implementation uses realloc to resize the internal memory buffer when more space is needed. A dynamic array is an array with a big improvement: automatic resizing.. One limitation of arrays is that they're fixed size, meaning you need to specify the number of elements your array will hold ahead of time.. A dynamic array expands as you add more elements. Don’t go past Max or you will be out of bounds. Dynamic Arrays are not simply a new batch of functions added to the Excel Function Library. Compatibility Issue #1: @ and SINGLE. The array that this class implements can be added to. The total size of the array is often called the capacity, and the size of the actual values is often called the logical size. The following graphic shows this process. Some deque implementations use array deques, which allow amortized constant time insertion/removal at both ends, instead of just one end. Dynamic arrays are different from those in that they do not have a fixed length. This translates into a meaning that you can only specify the number of elements that your array can hold ahead of time. There will be huge amount of physical disk space that sit, unused, for the duration of your program. Forgot password? Essentially, a dynamic array is a structure with this info: A pointer to the first element; The length of the entire array When all space is consumed, and an additiona… Admittedly, you’ll find a few other differences between dynamic and static arrays, but the bottom line here is that, although dynamic arrays require a little more work to use because you must manage the memory yourself, they also provide added flexibility in working with data. In computer science, a dynamic array, growable array, resizable array, dynamic table, or array list is an array data structure that can be resized and allows elements to be added or removed. This property gives the dynamic array more power in programs where the programmer does not know how much data will enter the array at any given point. The operation of adding an element to the end might work as follows: As n elements are inserted, the capacities form a geometric progression. You might think that the programmer can just make an array with an extremely large size to get around the overhead of the dynamic array. When all space is consumed, and an additional element is to be added, then the underlying fixed-sized array needs to be increased in size. Arrays have a count (e.g. Dynamic arrays are allocated on the heap, which means they’re only limited by the size of memory. DA Excel: Excel that supports dynamic arrays, its functions and features; Legacy Excel: the “traditional” Excel that is still wrapped in the world of CTRL + SHIFT + ENTER and does not support dynamic arrays. The dynamic array is such a type of an array with a huge improvement for automatic resizing. So, if I wanted to make a dynamic array with the following elements, [1, 2], a dynamic array would be created with 4 spaces. This property gives the dynamic array more power in programs where the programmer does not know how much data will enter the array at any given point. Below are growth factors used by several popular implementations: The dynamic array has performance similar to an array, with the addition of new operations to add and remove elements: Dynamic arrays benefit from many of the advantages of arrays, including good locality of reference and data cache utilization, compactness (low memory use), and random access. "Dynamic array" has an actual meaning: an array whose size is not static. [3][4], The growth factor for the dynamic array depends on several factors including a space-time trade-off and algorithms used in the memory allocator itself. For growth factor a, the average time per insertion operation is about a/(a−1), while the number of wasted cells is bounded above by (a−1)n[citation needed]. Explore Wikis; Community Central; Start a Wiki; Search This wiki This wiki All wikis | Sign In Don't have an account? It then creates a new, doubly large array in that contiguous space, and saves the original array as the first half of the new array. A dynamic array is an array that can be lengthened or shortened to accomodate the number of elements in the array. If we tried to add a 5th element, the array would double in size (now with a capacity of 8) and add the 5th element. Wikis. A balanced tree can store a list while providing all operations of both dynamic arrays and linked lists reasonably efficiently, but both insertion at the end and iteration over the list are slower than for a dynamic array, in theory and in practice, due to non-contiguous storage and tree traversal/manipulation overhead. [16] Hashed array tree wastes order n1/2 amount of storage space, where n is the number of elements in the array. Log in. Overview. These elements all have the same type, called the "inner type" of the array. See also [StaticArrays static arrays]. So, accessing index 1 is actually indexing the second element in the list. Since announcing dynamic arrays, we’ve received an overwhelmingly positive response from the community. New user? Whenever you eat something, simply append the name of the food to the end of the dynamic data structure. Dynamic arrays are typically initialized with twice the number of initial array elements. Expanding the array by any constant proportion a ensures that inserting n elements takes O(n) time overall, meaning that each insertion takes amortized constant time. It is not a multidimensional array - it is array of pointers to int, or array of arrays. Any C++ type; Any UE4 C++ type, such as FLinearColor This excess space can be defined as capacity−logicalsizecapacity - logical sizecapacity−logicalsize. Excel’s calculation engine has been completely redesigned to make many functions work faster and with greater flexibility. 1. // resize a to twice its current capacity: // (copy the contents to the new memory location here), source code of java.util.ArrayList class from OpenJDK 6, "C++ STL vector: definition, growth factor, member functions", "Dissecting the C++ STL Vector: Part 3 - Capacity & Size", "Tiered Vectors: Efficient Dynamic Arrays for Rank-Based Sequences", Workshop on Algorithms and Data Structures, NIST Dictionary of Algorithms and Data Structures: Dynamic array, Open Data Structures - Chapter 2 - Array-Based Lists, https://en.wikipedia.org/w/index.php?title=Dynamic_array&oldid=994594699, Articles with unsourced statements from January 2018, Creative Commons Attribution-ShareAlike License, the maximum logical size is unknown, or difficult to calculate, before the array is allocated, it is considered that a maximum logical size given by a specification is likely to change, the amortized cost of resizing a dynamic array does not significantly affect performance or responsiveness, Getting or setting the value at a particular index (constant time), Iterating over the elements in order (linear time, good cache performance), Inserting or deleting an element in the middle of the array (linear time), Inserting or deleting an element at the end of the array (constant amortized time), This page was last edited on 16 December 2020, at 15:06. If for whatever reason realloc isn't available on your platform, then that line can be replaced with a call to malloc with the new size, a memcpy to copy the contents of the old buffer to the new one, and a free of the old buffer. Dynamic arrays' indices are always non-negative integers starting at zero for the first element. Many dynamic arrays also deallocate some of the underlying storage if its size drops below a certain threshold, such as 30% of the capacity. They usually have only a small fixed additional overhead for storing information about the size and capacity. As with static arrays, it allows O(1) random access. The dynamic array is able to change its size during program execution. A dynamic array is a resizable array data structure that can be dynamically resized as items are added to or removed from it. Elements can be added at the end of a dynamic array in constant time by using the reserved space, until this space is completely consumed. This makes dynamic arrays an attractive tool for building cache-friendly data structures. Goodrich[15] presented a dynamic array algorithm called tiered vectors that provides O(n1/k) performance for insertions and deletions from anywhere in the array, and O(k) get and set, where k ≥ 2 is a constant parameter. This worst case happens just after a growth of the dynamic array. Log in here. It is supplied with standard libraries in many modern mainstream programming languages. In the previous lesson, you learned about static arrays. However, it's important to look at the amortized analysis to see that the runtime is actually smaller; specifically, O(1)O(1)O(1). Delphi and D implement dynamic arrays at the language's core. This is because the size of an array is fixed. Dynamic arrays support is implemented in UnrealScript starting with UnrealEngine2. 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# Quantum tomography under prior information

Last week, I was at Texas A&M for David Larson’s 70th birthday conferenceBernhard Bodmann was also there, and we talked a little about phase retrieval.  He pointed me to this fantastic paper:

Quantum tomography under prior information

Teiko Heinosaari, Luca Mazzarella, Michael M. Wolf

It’s all about determining a positive semidefinite matrix $A$ from measurements of the form $\mathrm{Tr}[AS_n]$, where $S_n$ is a positive semidefinite matrix corresponding to some physical measurement process.   In the quantum setting, $A$ is called a mixed state, and its trace is $1$.  If a given collection of $S_n$‘s sum to the identity matrix, then the ensemble is known as a positive operator-valued measure (POVM); frame theorists might identify this as a Parseval “generalized” frame.  Furthermore, if $A$ and $S_n$ both have rank $1$, then we have the decompositions $A=xx^*$ and $S_n=f_nf_n^*$, and so the corresponding measurement has the form

$\mathrm{Tr}[AS_n]=\mathrm{Tr}[xx^*f_nf_n^*]=\mathrm{Tr}[f_n^*xx^*f_n^*]=|\langle x,f_n\rangle|^2.$

In this case, the rank-$1$ mixed state $A=xx^*$ is called as a pure state, and the unit-trace condition equivalently means $\|x\|^2=1$; recall that pure states were the subject of my previous post.  As such, mixed states form a natural generalization of pure states, and the trace-based measurement process corresponds nicely.

The main point of the paper is to find necessary and sufficient conditions for an ensemble of $S_n$’s to be informationally complete (IC) with respect to pure states.  For the reader’s sake, IC is just physics-speak for injective; $\{S_n\}$ is IC with respect to pure states if the measurements $\{\mathrm{Tr}[xx^*S_n]\}$ completely determine the pure state $xx^*$ (or $x$ up to a global phase factor).  As such, the results of this paper would be immediately applicable to the phase retrieval problem if it weren’t for two main differences:  the unknown mixed state necessarily has unit trace, and the $S_n$’s don’t necessarily have rank $1$.  But there are still implications for phase retrieval, and I detail some of them below.

The paper has two main sections (Sections 3 and 4).  Section 3 gives sufficient conditions for IC, while Section 4 gives necessary conditions.  As far as sufficient conditions are concerned, the authors start by using a construction from Cubitt, Montanaro and Winter to demonstrate that rank-$r$ mixed states can be distinguished with only $4r(d-r)-1$ measurements.  This corresponds nicely to a compressed sensing result of Gross, Liu, Flammia, Becker and Eisert, which gives a reconstruction procedure based on only $\mathcal{O}(rd\log^2d)$ measurements, meaning a mere log penalty is paid to achieve efficiency and stability in reconstruction.  Later, the authors completely characterize POVMs which are IC with respect to $2$– and $3$-dimensional pure states, although the characterization doesn’t appear to admit a useful generalization to larger dimensions.  Next, the authors use constructions from James and Milgram to produce ensembles of self adjoint matrices which are IC with respect to pure states.  In particular, the construction of Milgram manages to distinguish $d$-dimensional pure states with less than $4d-\alpha(d-1)$ measurements, where $\alpha(d-1)$ is the number of $1$’s in the binary representation of $d-1$.  The authors conclude the section by leveraging Minkowski dimension to show that almost every ensemble of $4d-3$ self adjoint matrices is IC with respect to $d$-dimensional pure states.

Unfortunately, the constructions in Section 3 are not immediately applicable to the phase retrieval problem, as the measurement matrices $S_n$ do not have rank $1$.  In fact, even the Minkowski-dimension-based argument fails to say anything about phase retrieval, as rank-$1$ matrices form a measure-zero subset of the set of self adjoint matrices; by comparison, Balan, Casazza and Edidin used an algebro-geometric approach to show that almost every ensemble of $4d-2$ vectors produce injective phaseless measurements.

Section 4, however, is immediately applicable to the phase retrieval problem.  In this section, the authors view the mapping $xx^*\mapsto\{\mathrm{Tr}[xx^*S_n]\}$ from the standpoint of topology.  First, they show that pure-state-IC measurements form a topological embedding of the set of pure states into euclidean space; this is proved using the facts that the mapping is linear and the set of pure states is compact.  However, this homeomorphic structure doesn’t appear to be strong enough to impose substantial necessary conditions for injectivity (see their Corollaries 2 and 3).  By contrast, diffeomorphic structure appears to be particularly telling.  The authors show that pure-state-IC measurements necessarily form a smooth embedding of the set of pure states into euclidean space, and then they use a well-known result about embedding complex projective space to conclude that $4d-2\alpha(d-1)-4$ measurements are necessary for injectivity.

Since injective phaseless measurements are necessarily IC with respect to pure states, this means that $4d-2\alpha(d-1)-4$ measurements are, in fact, necessary.  I alluded to this fact in a previous post, but now we have a paper we can actually cite.  Unbeknownst to the authors, this result actually disproves a conjecture of Ron Wright from 1978 (found in this paper).  The conjecture states that $d$-dimensional pure states are completely determined by their distributions with respect to three observables, implying that $3d$ measurements suffice to be pure-state-IC.  Interestingly, the main result they used about embedding complex projective space was first proved by Atiyah and Hirzebruch in 1959, almost 20 years before Ron Wright first posed his conjecture.

It remains to be determined how many phaseless measurements are necessary for injectivity.  Finkelstein’s work led some to believe that $3d-2$ was the minimum number, and now this paper shows that $4d-o(d)$ measurements are actually required.  In fact, we know that similar topological arguments cannot substantially improve this bound since they must also apply to $S_n$’s of arbitrary rank, with which $4d-\alpha(d-1)$ measurements suffice.  As such, an improvement to this lower bound must somehow account for how rank-$1$ measurements differ from general measurements.


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## tdawgg 3 years ago 1/4x^3=2 (x-2)^3=-64 1/8x^6-3=5

1. a132

1) $\frac14 x ^{3} =2\rightarrow x ^{3} = 2*4\rightarrow x ^{3} =8$ $x = \sqrt[3]{8} \rightarrow x =$

2. a132

oops sorry x = 2

3. a132

this is too frustrating to solve on OpenStudy's equation editor, so I'm going to solve it in GoogleDocs and then attach it. Be right back.


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Complex Conformal Group

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Hi all,\n\nFirst of all, some apologies.\n1) Sorry if this question falls below the academic level of this\nnewsgroup!\n2) Sorry for the length!\n\nI\'m having some difficulty reconciling an explicit isomorphism between\n\n1) the group of proper (orientation-preserving) conformal\ntransformations C(1,3) of compactified complex Minkowski space (CM#)\n2) the complex projective linear group PGL(4,C)\n\ngiven in some lecture notes I have been given.\n\n--------------------------------------------------------------------------------\nFirst, notation and preliminaries:\n\nLet z_1, z_2, w_1, w_2 be complex double null co-ordinates such that\nthe metric on CM is\n\nds^2 = 2(dz_1 dz_2 - dw_1 dw_2)\n\nand the orientation/volume 4-form is\nn = dw_1 ^ dw_2 ^ dz_1 ^ dz_2\n\n(We can then recover real Minkowski space, real Euclidean space, and\nother interesting spaces by imposing conditions relating these\ncoordinates to each other.)\n\nA (proper) conformal transformation is an orientation preserving map\ndenoted P from CM# -> CM# such that ds_new^2 = omega^2 ds_old^2, and\nn_new = omega^4 n_old.\n\nInfinitesmal conformal transformations are given by conformal Killing\nvectors, where the Lie derivative must be a multiple of the metric. The\nspace of such conformal Killing vectors is 15-dimensional. It can be\ndecomposed into 4 translation parameters, 6 rotation/Lorentz\nparameters, 1 dilation parameter, and 4 \'special\' conformal\ntransformation parameters.\n\n--------------------------------------------------------------------------------\nOK, here\'s the setup (phew!):\n\nWe can write a point of CM uniquely as the skew-symmetric degenerate\nmatrix:\n\n0 z_1 z_2 - w_1 w_2 -w_1 z_2\nz_1 z_2 - w_1 w_2 0 -z_1 w_2\nw_1 z_1 0 1\n-z_2 -w_2 -1 0\n\nwhich we label x^{a b}.\n\nFurther, unless x^{2 3} = 0, every skew-symmetric degenerate 4*4 matrix\nmust be of this form up to a scalar multiple.\n\nAlso, epsilon_abcd dx^{a b} dx^{c d} = lambda (dz_1 dz_2 - dw_1 dw_2)\nwhere epsilon_abcd is the 4-dimensional alternating symbol and lambda\nis some scalar.\n\n--------------------------------------------------------------------------------\nNow here are the claims I don\'t really understand.\n\nCLAIM 1: \'Hence\' if b denotes a general member of GL(4,C) then the\ntransformation x -> b x b^t induces a conformal transformation of CM#.\nFurthermore, every proper conformal transformation arises in this way.\n\nCLAIM 2: We can regard the six elements of x^{a b} as homogenous\ncoordinates on CP5. When we do so, we can identify CM# = {x^[a b x^c d]\n= 0} in CP5, the Klein quadric. Here the [] brackets denote\nantisymmetrization in the usual way.\n\nCLAIM 3: We can identify null geodesics in CM# with those projective\nlines in CP5 which lie in CM#.\n\nCan anyone help me shed light?\n\nThanks!\n\nPauca Sed Matura\n\n

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</jabberwocky>Hi all,

First of all, some apologies.
1) Sorry if this question falls below the academic level of this
newsgroup!
2) Sorry for the length!

I'm having some difficulty reconciling an explicit isomorphism between

1) the group of proper (orientation-preserving) conformal
transformations C(1,3) of compactified complex Minkowski space (CM#)
2) the complex projective linear group PGL(4,C)

given in some lecture notes I have been given.

--------------------------------------------------------------------------------
First, notation and preliminaries:

Let $z_1, z_2, w_1, w_2$ be complex double null co-ordinates such that
the metric on CM is

$$ds^2 = 2(dz_1 dz_2 - dw_1 dw_2)$$

and the orientation/volume 4-form is
$n = dw_1 ^ dw_2 ^ dz_1 ^ dz_2$

(We can then recover real Minkowski space, real Euclidean space, and
other interesting spaces by imposing conditions relating these
coordinates to each other.)

A (proper) conformal transformation is an orientation preserving map
denoted P from CM# -> CM# such that $ds_new^2 = \omega^2 ds_old^2,$ and
$n_{new} = \omega^4 n_{old}$.

Infinitesmal conformal transformations are given by conformal Killing
vectors, where the Lie derivative must be a multiple of the metric. The
space of such conformal Killing vectors is 15-dimensional. It can be
decomposed into 4 translation parameters, 6 rotation/Lorentz
parameters, 1 dilation parameter, and 4 'special' conformal
transformation parameters.

--------------------------------------------------------------------------------
OK, here's the setup (phew!):

We can write a point of CM uniquely as the skew-symmetric degenerate
matrix:

$$z_1 z_2 - w_1 w_2 -w_1 z_2z_1 z_2 - w_1 w_2-z_1 w_2w_1 z_1[/itex] 1 $-z_2 -w_2 -1$$ which we label [itex]x^{a b}$.

Further, unless $x^{2 3} = 0,$ every skew-symmetric degenerate 4*4 matrix
must be of this form up to a scalar multiple.

Also, $\epsilon_abcd dx^{a b} dx^{c d} = \lambda (dz_1 dz_2 - dw_1 dw_2)$
where $\epsilon_abcd$ is the 4-dimensional alternating symbol and $\lambda$
is some scalar.

--------------------------------------------------------------------------------
Now here are the claims I don't really understand.

CLAIM 1: 'Hence' if b denotes a general member of GL(4,C) then the
transformation $x -> b x b^t$ induces a conformal transformation of CM#.
Furthermore, every proper conformal transformation arises in this way.

CLAIM 2: We can regard the six elements of $x^{a b}$ as homogenous
coordinates on CP5. When we do so, we can identify CM# $= {x^[a b x^c d]= 0}$ in CP5, the Klein quadric. Here the [] brackets denote
antisymmetrization in the usual way.

CLAIM 3: We can identify null geodesics in CM# with those projective
lines in CP5 which lie in CM#.

Can anyone help me shed light?

Thanks!

Pauca Sed Matura

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text:
# The Universe of Discourse

Thu, 08 Nov 2018

I want to build an adjacency matrix for the vertices of a cube; this is a matrix that has m[a][b] = 1 exactly when vertices a and b share an edge. We can enumerate the vertices arbitrarily but a convenient way to do it is to assign them the numbers 0 through 7 and then say that vertices !!a!! and !!b!! are adjacent if, regarded as binary numerals, they differ in exactly one bit, so:

   import Data.Bits
a adj b = if (elem (xor a b) [1, 2, 4]) then 1 else 0

This compiles and GHC infers the type

   adj :: (Bits a, Num a, Num t) => a -> a -> t

Fine.

Now I want to build the adjacency matrix, which is completely straightforward:

    cube = [ [a adj b | b <- [0 .. 7] ] | a <- [0 .. 7] ]  where
a adj b = if (elem (xor a b) [1, 2, 4]) then 1 else 0

Ha ha, no it isn't; in Haskell nothing is straightforward. This produces 106 lines of type whining, followed by a failed compilation. Apparently this is because because 0 and 7 are overloaded, and could mean some weird values in some freakish instance of Num, and then 0 .. 7 might generate an infinite list of 1-graded torsion rings or something.

To fix this I have to say explicitly what I mean by 0. “Oh, yeah, by the way, that there zero is intended to denote the integer zero, and not the 1-graded torsion ring with no elements.”

        cube = [ [a adj b | b <- [0 :: Integer .. 7] ] | a <- [0 .. 7] ]  where
a adj b = if (elem (xor a b) [1, 2, 4]) then 1 else 0

Here's another way I could accomplish this:

        zero_i_really_mean_it = 0 :: Integer
cube = [ [a adj b | b <- [zero_i_really_mean_it .. 7] ] | a <- [0 .. 7] ] where
a adj b = if (elem (xor a b) [1, 2, 4]) then 1 else 0

        cube = [ [a adj b | b <- numbers_dammit [0 .. 7] ] | a <- [0 .. 7] ] where
p adj q = if (elem (xor p q) [1, 2, 4]) then 1 else 0