text:
# Kinematics with Product-of-Exponentials terminology: not forward, not inverse

I'm working on a kinematic estimation algorithm, and am trying to come up with the right terminology.

I'm using the Product-of-Exponentials formulation for kinematic chains (and ignoring position for now):

$$R = e^{\omega_1 \theta_1}e^{\omega_2 \theta_2}...e^{\omega_n \theta_n}R_0$$

where:
$$R \in SO(3)$$: orientation of end effector
$$\theta_{1..n}$$: joint coordinates
$$\omega_{1..n} \in so(3)$$: joint screw axes

Forward Kinematics: Solve for $$R$$ given $$\omega_{1..n}, \theta_{1..n}$$
Inverse Kinematics: Solve for $$\theta_{1..n}$$ given $$R, \omega_{1..n}$$
?___?: Solve for $$\omega_{1..n}$$ given $$R, \theta_{1..n}$$

My question is, what phrase fills that blank? I've been using the PoE formula for so long that I am starting to forget the DH paradigm but I'd imagine there's an equivalent with DH params too.

The literature I'm comparing against calls this "system identification" but I think that's because they're treating it like a dynamical system...I am treating it like a standard robot kinematics problem.

Is there an agreed-upon term for this process?

• \omega is the so(3) matrix and \theta is the joint coordinate? Shouldn't there be an SO(3) matrix on the end, the orientation when all \theta are zero? – Peter Corke Apr 27 at 22:14
• @PeterCorke correct, I had omitted for brevity, just edited for clarity ;) – Eric Peltola Apr 28 at 2:26


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# To find the net charge of distribution

## Homework Statement

You're 1.5 m from a charge distribution whose size is much less than 1 m.
You measure an electric field strength of 282 N/C. You move to a distance of
2.0 m, and the field strength becomes 119 N/C.

Hint: Don't try to calculate the charge. Instead determine how the field
decreases with distance and from that, infer the charge.

## The Attempt at a Solution

All I was taught by the prof was that I need to break charge distribution
into elements of dq and then integrate over the distance. However, the
question does not necessarily state the distance of the distribution, which is
why I think the question gives two instances of field measurements, i'm not
sure how to approach the question from here. Hope that I can get a good
lead on how to solve this question. Thanks.

Related Advanced Physics Homework Help News on Phys.org
diazona
Homework Helper
What does the question ask you to figure out?

Hi, the question asks to find the net charge of the distribution

gabbagabbahey
Homework Helper
Gold Member
Hint: Since the charge distribution is much less than 1m in size, and the field is measured at points further away than one meter, you are effectively far away from the charge distribution....what terms in the field's multipole expansion do you expect to dominate when you are far away from the charge distribution?

I think what you're looking for has something to do with the inverse cube of the distance between the charge distribution and the point where you are measuring. As to the net distribution...I'm still unclear how to infer that unless it's infinite ( ie. 1/0^3 when you're measuring at the charge distribution) but it doesn't make sense physically I think.

gabbagabbahey
Homework Helper
Gold Member
I think what you're looking for has something to do with the inverse cube of the distance between the charge distribution and the point where you are measuring. As to the net distribution...I'm still unclear how to infer that unless it's infinite ( ie. 1/0^3 when you're measuring at the charge distribution) but it doesn't make sense physically I think.
This thread is almost a year-old, I doubt the OP is still working on the problem.

Anyways, the point is that the electric field of any charge distribution can be expanded in a so-called multipole expansion... there will be terms that are proportional to $\frac{1}{r^2}$ (monopole term) $\frac{1}{r^3}$ (dipole term) and so on. When you are "far" from the distribution, only the first 2 or 3 terms will be significant, so in this case, you assume only a monopole and dipole terms are present and use the given information to find the coefficients of each....the coefficient of the monopole moment is directly proportional to the total charge of the distribution.

format: plain


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Article Contents
Article Contents

# Anisotropically diffused and damped Navier-Stokes equations

• The incompressible Navier-Stokes equations with anisotropic diffusion and anisotropic damping is considered in this work. For the associated initial-boundary value problem, we prove the existence of weak solutions and we establish an energy inequality satisfied by these solutions. We prove also under what conditions the solutions of this problem extinct in a finite time.
Mathematics Subject Classification: Primary: 35Q35, 35D30, 35B40; Secondary: 76D05, 76D03.

 Citation:

•  [1] S.N. Antontsev, J.I. Díaz and S.I. Shmarev., Energy methods for free boundary problems. Progr. Nonlinear Differential Equations Appl. 48, Birkhäuser, 2002. [2] S.N. Antontsev and H.B. de Oliveira., Analysis of the existence for the steady Navier-Stokes equations with anisotropic diffusion. Adv. Differential Equations 19 (2014) no. 5-6, 441-472. [3] S.N. Antontsev and H.B. de Oliveira., Evolution problems of Navier-Stokes type with anisotropic diffusion. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 1-26, First online: 17 December 2015. [4] I. Fragalà, F. Gazzola and B. Kawohl., Existence and nonexistence results for anisotropic quasilinear elliptic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), no. 5, 715-734. [5] J.K. Djoko, and P.A. Razafimandimby., Analysis of the Brinkman-Forchheimer equations with slip boundary conditions. Appl. Anal. 93 (2014), no. 7, 1477-1494. [6] J. Frehse, J. Málek and M. Steinhauer., On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method. SIAM J. Math. Anal. 34 (2003), no. 5, 1064-1083. [7] J. Haškovec and C. Schmeiser., A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems. Monatsh. Math. 158 (2009), no. 1, 71-79. [8] V. Kalantarov and S. Zelik., Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities. Commun. Pure Appl. Anal. 11 (2012), no. 5, 2037-2054. [9] J.-L. Lions., Quelques mèthodes de résolution des problèmes aux limites non liniaires. Dunod, Paris, 1969. [10] J. Málek, J. Nečas, M. Rokyta and M.R$\dot u$žička., Weak and measure-valued solutions to evolutionary PDEs. Chapman & Hall, London, 1996. [11] H.B. de Oliveira., On the influence of an absorption term in incompressible fluid flows. Adv. Math. Fluid Mech. Springer-Verlag (2010), 409-424. [12] H.B. de Oliveira., Existence of weak solutions for the generalized Navier-Stokes equations with damping. NoDEA Nonlinear Differential Equations Appl. 20 (2013) no. 3, 797-824. [13] J. Rákosník., Some remarks to anisotropic Sobolev spaces. II. Beiträge Anal. No. 15 (1980), 127-140 (1981). [14] M. Troisi., Teoremi di inclusione per spazi di Sobolev non isotropi. Ricerche Mat. 18 (1969), 3-24.
Open Access Under a Creative Commons license


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## Profitability Ratios

The two ratios selected to do profitability analysis are covered in the below mentioned sections.

### ROE

It stands for return on equity and is calculated by dividing net income with total equity. The analysis clearly shows that ROE is in increasing trend for UPS and is in decreasing trend for FDX due to increasing and decreasing income for both companies respectively as interpreted by our financial accounting homework help online interpreter.

### ROA

It stands for return on assets and is calculated by dividing net income with total assets. The analysis clearly shows that ROA is in decreasing trend for both companies .However, the value is higher for UPS compare to FDX .

## Liquidity Ratios

The two ratios selected to do Liquidity analysis are covered in the below mentioned sections.

### Current Ratio

It is calculated by dividing Current Assets with current liabilities. The analysis clearly shows that it remains same for UPS and is in increasing trend for FDX due to increasing current assets in case of FDX as given below.

### Quick Ratios

It is calculated by dividing Current Assets subtracted with inventory with current liabilities. The analysis clearly shows that it is in decreasing trends for UPS and is in increasing trend for FDX due to decreasing and increasing current assets respectively as given below.

## Capital Structure Ratios

The two ratios selected to do Capital structure analysis are covered in the below mentioned sections.

### Debt ratio

It is calculated by dividing total debt with total assets. The analysis clearly shows that it is in decreasing trends for UPS and is in increasing trend for FDX due to decreasing and increasing debts respectively as given below.

### Equity ratio

It is calculated by dividing total Equity with total assets. The analysis clearly shows that it is in decreasing trends for both companies due to decreasing equity respectively as given below.

## Efficiency Ratios

The two ratios selected to do efficiency ratios analysis are covered in the below mentioned sections.

### Payables turnover ratio

It is calculated by dividing cost of goods sold with total payables as per our corporate finance assignment help experts. The analysis clearly shows that it is in decreasing trends for both companies due to increasing payables respectively as given below.

Receivables turnover ratio
It is calculated by dividing revenue with total receivables. The analysis clearly shows that it is in increasing trends for UPS and is in decreasing trend for FDX due to increasing and decreasing receivables respectively as given below.

## Market Performance Ratios

The two ratios selected to do Market Performance ratios analysis are covered in the below mentioned sections.

### P/E Ratios

It is calculated by dividing Market price per share with Earning per share. The analysis clearly shows that it is in increasing trends for both companies due to increasing market price over the years.


text:
# How do you rewrite 1/8^6 with negative exponents?

${8}^{- 6}$
In general, ${x}^{- n} = \frac{1}{x} ^ n$


text:
# Congruence of invertible skew symmetric matrices

I am asking for some hints to solve this excercise. Given an invertible skew symmetric matrix $A$, then show that there are invertible matrices $R, R^T$ such that $R^T A R = \begin{pmatrix} 0 & Id \\ -Id & 0 \end{pmatrix}$, meaning that this is a block matrix that has the identity matrix in two of the four blocks and the lower one with a negative sign.

I am completely stuck!

The key fact is that every real and normal matrix is orthogonally similar to its real Jordan form. However, as real Jordan forms are seldom taught in universities, we may need to start from scratch here.

1. As $A$ is skew symmetric and invertible, its eigenvalues are conjugate pairs of nonzero pure imaginary numbers $\pm b_1i,\,\pm b_2i,\,\ldots,\pm b_ni$ (where the size of $A$ is $2n\times2n$).
2. Furthermore, as $A$ is skew symmetric, it is a normal matrix. Hence it possesses an orthonormal eigenbasis. (In some lecture notes, this fact is sometimes stated as "if $A$ is normal, $A=UDU^\ast$ for some unitary matrix $U$ and diagonal matrix $D$".) Let $u_1+iv_1,\,u_2+iv_2,\,\ldots,\,u_n+iv_n$ be $n$ orthogonal eigenvectors corresponding the eigenvalues $b_1i,\,b_2i,\,\ldots,\,b_ni$.
3. Note that $\left\{\frac{u_1}{\|u_1\|},\frac{u_2}{\|u_2\|},\ldots,\frac{u_n}{\|u_n\|},\frac{v_1}{\|v_1\|},\frac{v_2}{\|v_2\|},\ldots,\frac{v_n}{\|v_n\|}\right\}$ is an orthonormal basis of $\mathbb{R}^{2n}$ (why?). So, these $2n$ vectors form a real orthogonal matrix $Q$ and $A=Q\pmatrix{0&B\\ -B&0}Q^T$ (again, why?), where $B=\operatorname{diag}(b_1,b_2,\ldots,b_n)$. Now the rest should be straightforward.

Hint: a skew-symmetric matrix commutes with its transpose, and so it is diagonalizable. Your block matrix on your right hand side is also skew-symmetric, and so it is also diagonalizable.


text:
In the previous modules, we introduced and made occasional reference to pattern matching. Now that we have developed some familiarity with the language, it is time to take a proper, deeper look. We will kick-start the discussion with a condensed description, which we will expand upon throughout the chapter:

In pattern matching, we attempt to match values against patterns and, if so desired, bind variables to successful matches.

## Analysing pattern matching

Pattern matching is virtually everywhere. For example, consider this definition of `map`:

```map _ []     = []
map f (x:xs) = f x : map f xs
```

At surface level, there are four different patterns involved, two per equation.

• `f` is a pattern which matches anything at all, and binds the `f` variable to whatever is matched.
• `(x:xs)` is a pattern that matches a non-empty list which is formed by something (which gets bound to the `x` variable) which was cons'd (by the `(:)` function) onto something else (which gets bound to `xs`).
• `[]` is a pattern that matches the empty list. It doesn't bind any variables.
• `_` is the pattern which matches anything without binding (wildcard, "don't care" pattern).

In the `(x:xs)` pattern, `x` and `xs` can be seen as sub-patterns used to match the parts of the list. Just like `f`, they match anything - though it is evident that if there is a successful match and `x` has type `a`, `xs` will have type `[a]`. Finally, these considerations imply that `xs` will also match an empty list, and so a one-element list matches `(x:xs)`.

From the above dissection, we can say pattern matching gives us a way to:

• recognize values. For instance, when `map` is called and the second argument matches `[]` the first equation for `map` is used instead of the second one.
• bind variables to the recognized values. In this case, the variables `f`, `x`, and `xs` are assigned to the values passed as arguments to `map` when the second equation is used, and so we can use these values through the variables in the right-hand side of `=`. As `_` and `[]` show, binding is not an essential part of pattern matching, but just a side effect of using variable names as patterns.
• break down values into parts, as the `(x:xs)` pattern does by binding two variables to parts (head and tail) of a matched argument (the non-empty list).

## The connection with constructors

Despite the detailed analysis above, it may seem a little too magical how we break down a list as if we were undoing the effects of the `(:)` operator. Be careful: this process will not work with any arbitrary operator. For example, one might think of defining a function which uses `(++)` to chop off the first three elements of a list:

```dropThree ([x,y,z] ++ xs) = xs
```

But that will not work. The function `(++)` is not allowed in patterns. In fact, most other functions that act on lists are similarly prohibited from pattern matching. Which functions, then, are allowed?

In one word, constructors – the functions used to build values of algebraic data types. Let us consider a random example:

```data Foo = Bar | Baz Int
```

Here `Bar` and `Baz` are constructors for the type `Foo`. You can use them for pattern matching `Foo` values and bind variables to the `Int` value contained in a `Foo` constructed with `Baz`:

```f :: Foo -> Int
f Bar     = 1
f (Baz x) = x - 1
```

This is exactly like `showAnniversary` and `showDate` in the Type declarations module. For instance:

```data Date = Date Int Int Int   -- Year, Month, Day
showDate :: Date -> String
showDate (Date y m d) = show y ++ "-" ++ show m ++ "-" ++ show d
```

The `(Date y m d)` pattern in the left-hand side of the `showDate` definition matches a `Date` (built with the `Date` constructor) and binds the variables `y`, `m` and `d` to the contents of the `Date` value.

### Why does it work with lists?

As for lists, they are no different from `data`-defined algebraic data types as far as pattern matching is concerned. It works as if lists were defined with this `data` declaration (note that the following isn't actually valid syntax: lists are actually too deeply ingrained into Haskell to be defined like this):

```data [a] = [] | a : [a]
```

So the empty list, `[]` and the `(:)` function are constructors of the list datatype, and so you can pattern match with them. `[]` takes no arguments, and therefore no variables can be bound when it is used for pattern matching. `(:)` takes two arguments, the list head and tail, which may then have variables bound to them when the pattern is recognized.

```Prelude> :t []
[] :: [a]
Prelude> :t (:)
(:) :: a -> [a] -> [a]
```

Furthermore, since `[x, y, z]` is just syntactic sugar for `x:y:z:[]`, we can achieve something like `dropThree` using pattern matching alone:

```dropThree :: [a] -> [a]
dropThree (_:_:_:xs) = xs
dropThree _          = []
```

The first pattern will match any list with at least three elements. The catch-all second definition provides a reasonable default[1] when lists fail to match the main pattern, and thus prevents runtime crashes due to pattern match failure.

Note

From the fact that we could write a `dropThree` function with bare pattern matching it doesn't follow that we should do so! Even though the solution is simple, it is still a waste of effort to code something this specific when we could just use Prelude and settle it with `drop 3 xs` instead. Mirroring what was said before about baking bare recursive functions, we might say: don't get too excited about pattern matching either...

### Tuple constructors

Analogous considerations are valid for tuples. Our access to their components via pattern matching...

```fstPlusSnd :: (Num a) => (a, a) -> a
fstPlusSnd (x, y) = x + y

norm3D :: (Floating a) => (a, a, a) -> a
norm3D (x, y, z) = sqrt (x^2 + y^2 + z^2)
```

... is granted by the existence of tuple constructors. For pairs, the constructor is the comma operator, `(,)`; for larger tuples there are `(,,)`; `(,,,)` and so on. These operators are slightly unusual in that we can't use them infix in the regular way; so `5 , 3` is not a valid way to write `(5, 3)`. All of them, however, can be used prefix, which is occasionally useful.

```Prelude> (,) 5 3
(5,3)
Prelude> (,,,) "George" "John" "Paul" "Ringo"
("George","John","Paul","Ringo")
```

## Matching literal values

As discussed earlier in the book, a simple piece-wise function definition like this one

```f :: Int -> Int
f 0 = 1
f 1 = 5
f 2 = 2
f _ = -1
```

is performing pattern matching as well, matching the argument of `f` with the `Int` literals 0, 1 and 2, and finally with `_` . In general, numeric and character literals can be used in pattern matching on their own[2] as well as together with constructor patterns. For instance, this function

```g :: [Int] -> Bool
g (0:[]) = False
g (0:xs) = True
g _ = False
```

will evaluate to False for the [0] list, to True if the list has 0 as first element and a non-empty tail and to False in all other cases. Also, lists with literal elements like [1,2,3], or even "abc" (which is equivalent to ['a','b','c']) can be used for pattern matching as well, since these forms are only syntactic sugar for the (:) constructor.

The above considerations are only valid for literal values, so the following will not work:

```k = 1
--again, this won't work as expected
h :: Int -> Bool
h k = True
h _ = False
```
Exercises
1. Test the flawed `h` function above in GHCi, with arguments equal to and different from 1. Then, explain what goes wrong.
2. In this section about pattern matching with literal values, we made no mention of the boolean values `True` and `False`, but we can do pattern matching with them as well, as demonstrated in the Next steps chapter. Can you guess why we omitted them? (Hint: is there anything distinctive about the way we write boolean values?)

## Syntax tricks

### As-patterns

Sometimes, when matching a pattern with a value, it may be useful to bind a name to the whole value being matched. As-patterns allow exactly this: they are of the form `var@pattern` and have the additional effect to bind the name `var` to the whole value being matched by `pattern`. For instance, here is a toy variation on the map theme:

```contrivedMap :: ([a] -> a -> b) -> [a] -> [b]
contrivedMap f [] = []
contrivedMap f list@(x:xs) = f list x : contrivedMap f xs
```

`contrivedMap` passes to the parameter function `f` not only `x` but also the undivided list used as argument of each recursive call. Writing it without as-patterns would have been a bit clunky because we would have to either use `head` or needlessly reconstruct the original value of `list`, i.e. actually evaluate `x:xs` on the right side:

```contrivedMap :: ([a] -> a -> b) -> [a] -> [b]
contrivedMap f [] = []
contrivedMap f (x:xs) = f (x:xs) x : contrivedMap f xs
```
Exercises
Implement `scanr`, as in the exercise in Lists III, but this time using an as-pattern.

### Introduction to records

For constructors with many elements, records provide a way of naming values in a datatype using the following syntax:

```data Foo2 = Bar2 | Baz2 {bazNumber::Int, bazName::String}
```

Using records allows doing matching and binding only for the variables relevant to the function we're writing, making code much clearer:

```h :: Foo2 -> Int
h Baz2 {bazName=name} = length name
h Bar2 {} = 0

x = Baz2 1 "Haskell"     -- construct by declaration order, try ":t Baz2" in GHCi
y = Baz2 {bazName = "Curry", bazNumber = 2} -- construct by name

h x -- 7
h y -- 5
```

Also, the `{}` pattern can be used for matching a constructor regardless of the datatype elements even if you don't use records in the `data` declaration:

```data Foo = Bar | Baz Int
g :: Foo -> Bool
g Bar {} = True
g Baz {} = False
```

The function `g` does not have to be changed if we modify the number or the type of elements of the constructors `Bar` or `Baz`.

There are further advantages to using record syntax which we will cover in more details in the Named fields section of the More on datatypes chapter.

## Where we can use pattern matching

The short answer is that wherever you can bind variables, you can pattern match. Let us have a glance at such places we have seen before; a few more will be introduced in the following chapters.

### Equations

The most obvious use case is the left-hand side of function definition equations, which were the subject of our examples so far.

```map _ []     = []
map f (x:xs) = f x : map f xs
```

In the `map` definition we're doing pattern matching on the left hand side of both equations, and also binding variables on the second one.

#### `let` expressions and `where` clauses

Both `let` and `where` are ways of doing local variable bindings. As such, you can also use pattern matching in them. A simple example:

```y = let (x:_) = map (*2) [1,2,3]
in x + 5
```

Or, equivalently,

```y = x + 5
where
(x:_) = map (*2) [1,2,3]
```

Here, `x` will be bound to the first element of `map ((*) 2) [1,2,3]`. `y`, therefore, will evaluate to ${\displaystyle 2+5=7}$.

### Lambda abstractions

Pattern matching can be used directly in lambda abstractions:

```swap = \(x,y) -> (y,x)
```

It is clear, however, that this syntax permits only one pattern (or one for each argument in the case of a multi-argument lambda abstraction).

### List comprehensions

After the `|` in list comprehensions you can pattern match. This is actually extremely useful, and adds a lot to the expressiveness of comprehensions. Let's see how that works with a slightly more sophisticated example. Prelude provides a `Maybe` type which has the following constructors:

```data Maybe a = Nothing | Just a
```

It is typically used to hold values resulting from an operation which may or may not succeed; if the operation succeeds, the `Just` constructor is used and the value is passed to it; otherwise `Nothing` is used.[3] The utility function `catMaybes` (which is available from Data.Maybe library module) takes a list of Maybes (which may contain both "Just" and "Nothing" Maybes), and retrieves the contained values by filtering out the `Nothing` values and getting rid of the `Just` wrappers of the `Just x`. Writing it with list comprehensions is very straightforward:

```catMaybes :: [Maybe a] -> [a]
catMaybes ms = [ x | Just x <- ms ]
```

Another nice thing about using a list comprehension for this task is that if the pattern match fails (that is, it meets a Nothing) it just moves on to the next element in `ms`, thus avoiding the need of explicitly handling constructors we are not interested in with alternate function definitions.[4]

### `do` blocks

Within a `do` block like the ones we used in the Simple input and output chapter, we can pattern match with the left-hand side of the left arrow variable bindings:

```putFirstChar = do
(c:_) <- getLine
putStrLn [c]
```

Furthermore, the `let` bindings in `do` blocks are, as far as pattern matching is concerned, just the same as the "real" let expressions.

## Notes

1. Reasonable for this particular task, and only because it makes sense to expect that `dropThree` will give `[]` when applied to a list of, say, two elements. With a different problem, it might not be reasonable to return any list if the first match failed. In a later chapter, we will consider one simple way of dealing with such cases.
2. As perhaps could be expected, this kind of matching with literals is not constructor-based. Rather, there is an equality comparison behind the scenes
3. The canonical example of such an operation is looking up values in a dictionary - which might just be a `[(a, b)]` list with the tuples being key-value pairs, or a more sophisticated implementation. In any case, if we, given an arbitrary key, try to retrieve a value there is no guarantee we will actually find a value associated to the key.
4. The reason why it works this way instead of crashing out on a pattern matching failure has to do with the real nature of list comprehensions: They are actually wrappers for the list monad. We will eventually explain what that means when we discuss monads.
 Pattern matching Solutions to exercises Elementary Haskell edit this chapter Haskell edit book structure


text:
## Get DLLs Loaded in Current Session

### Description

This function provides a way to get a list of all the DLLs (see dyn.load) that are currently loaded in the R session.

### Usage

getLoadedDLLs()

### Details

This queries the internal table that manages the DLLs.

### Value

An object of class "DLLInfoList" which is a list with an element corresponding to each DLL that is currently loaded in the session. Each element is an object of class "DLLInfo" which has the following entries.

 name the abbreviated name. path the fully qualified name of the loaded DLL. dynamicLookup a logical value indicating whether R uses only the registration information to resolve symbols or whether it searches the entire symbol table of the DLL. handle a reference to the C-level data structure that provides access to the contents of the DLL. This is an object of class "DLLHandle".

Note that the class DLLInfo has a method for \$ which can be used to resolve native symbols within that DLL. Therefore, one must access the R-level elements described above using [[, e.g. x[["name"]] or x[["handle"]].

### Note

We are starting to use the handle elements in the DLL object to resolve symbols more directly in R.

### Author(s)

Duncan Temple Lang duncan@wald.ucdavis.edu.

getDLLRegisteredRoutines, getNativeSymbolInfo
getLoadedDLLs()


text:
American Institute of Mathematical Sciences

2011, 1(1): 1-14. doi: 10.3934/naco.2011.1.1

On methods for solving nonlinear semidefinite optimization problems

 1 School of Business, National University of Singapore, 119245, Singapore

Received  November 2010 Revised  November 2010 Published  February 2011

The nonlinear semidefinite optimization problem arises from applications in system control, structural design, financial management, and other fields. However, much work is yet to be done to effectively solve this problem. We introduce some new theoretical and algorithmic development in this field. In particular, we discuss first and second-order algorithms that appear to be promising, which include the alternating direction method, the augmented Lagrangian method, and the smoothing Newton method. Convergence theorems are presented and preliminary numerical results are reported.
Citation: Jie Sun. On methods for solving nonlinear semidefinite optimization problems. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 1-14. doi: 10.3934/naco.2011.1.1
References:
 [1] A. Ben-Tal, F. Jarre, M. Kocvara, A. Nemirovski and J. Zowe, Optimal design of trusses under a nonconvex global buckling constraint,, Optim. and Eng., 1 (2000), 189. doi: 10.1023/A:1010091831812. Google Scholar [2] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, "Linear Matrix Inequalities in System and Control Theory,", SIAM Studies in Applied Mathematics, (1994). Google Scholar [3] X. Chen, H. D. Qi and P. Tseng, Analysis of nonsmooth symmetric-matrix functions with applications to semidefinite complementarity problems,, SIAM J. Optim., 13 (2003), 960. doi: 10.1137/S1052623400380584. Google Scholar [4] X. Chen, D. Sun and J. Sun, Complementarity functions and numerical experiments on some smoothing Newton methods for second-order-cone complementarity problems, , Comput. Optim. Appl., 25 (2003), 39. doi: 10.1023/A:1022996819381. Google Scholar [5] X. Chen and P. Tseng, Non-interior continuation methods for solving semidefinite complementarity problems,, Math. Program., 95 (2003), 431. doi: 10.1007/s10107-002-0306-1. Google Scholar [6] F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley and Sons, (1983). Google Scholar [7] R. Correa and C. H. Ramirez, A global algorithm for nonlinear semidefinite programming,, SIAM J. Optim., 15 (2004), 303. doi: 10.1137/S1052623402417298. Google Scholar [8] M. Diehl, F. Jarre and C. H. Vogelbusch, Loss of superlinear convergence for an SQP-type method with conic constraints,, SIAM J. Optim., 16 (2006), 1201. doi: 10.1137/050625977. Google Scholar [9] M. Doljansky, An interior proximal algorithm and the exponential multiplier method for semidefinite programming,, SIAM J. Optim., 9 (1999), 1. doi: 10.1137/S1052623496309405. Google Scholar [10] A. Forsgren, Optimality conditions for nonconvex semidefinite programming,, Math. Program., 88 (2000), 105. doi: 10.1007/PL00011370. Google Scholar [11] J. Eckstein, "Splitting Methods for Monotone Operators with Applications to Parallel Optimization,", PhD thesis, (1989). Google Scholar [12] B. Fares, D. Noll and P. Apkarian, Robust control via sequential semidefinite programming,, SIAM J. Contr. and Optim., 40 (2002), 1791. doi: 10.1137/S0363012900373483. Google Scholar [13] M. L. Flegel and C. Kanzow, Equivalence of two nondegeneracy conditions for semidefinite programs,, J. Optim. Theory Appl., 135 (2007), 381. doi: 10.1007/s10957-007-9270-5. Google Scholar [14] M. Fukushima, Application of the alternating directions method of multipliers to separable convex programming problems,, Comput. Optim. Appl., 1 (1992), 93. doi: 10.1007/BF00247655. Google Scholar [15] M. Fukushima, Z. Q. Luo and P. Tseng, Smoothing functions for second-order cone complementarity problems,, SIAM J. Optim., 12 (2002), 436. doi: 10.1137/S1052623400380365. Google Scholar [16] Y. Gao and D. Sun, Calibrating least squares covariance matrix problems with equality and inequality constraints,, SIAM J. Matrix Anal. Appl. 31 (2009), 31 (2009), 1432. doi: 10.1137/080727075. Google Scholar [17] B. S. He, L. Z. Liao, D. R. Han and H. Yang, A new inexact alternating directions method for monotone variational inequalities,, Math. Program., 92 (2002), 103. Google Scholar [18] B. S. He, L. Z. Liao and M. J. Qian, Alternating projection based prediction-correction methods for structured variational inequalities,, J. Compu. Math., 24 (2002), 693. Google Scholar [19] M. R. Hestenes, Multiplier and gradient methods,, J. Optim. Theory Appl., 4 (1969), 303. Google Scholar [20] N. J. Higham, Computing a nearest symmetric positive semidefinite matrix,, Linear Algebra Appl., 103 (1998), 103. doi: 10.1016/0024-3795(88)90223-6. Google Scholar [21] F. Jarre, An interior method for nonconvex semidefinite programs,, Optim. and Eng., 1 (2000), 347. doi: 10.1023/A:1011562523132. Google Scholar [22] C. Kanzow, I. Ferenczi and M. Fukushima, On the local convergence of semismooth newton methods for linear and nonlinear second-order cone programs without strict complementarity,, SIAM J. Optim., 20 (2009), 297. doi: 10.1137/060657662. Google Scholar [23] C. Kanzow and C. Nagel, Semidefinite programs: New search directions, smoothing-type methods, and numerical results,, SIAM J. Optim., 13 (2002), 1. doi: 10.1137/S1052623401390525. Google Scholar [24] C. Kanzow and C. Nagel, Some structural properties of a Newton-type method for semidefinite programs,, J. Optim. Theory Appl., 122 (2004), 219. doi: 10.1023/B:JOTA.0000041737.19689.4c. Google Scholar [25] C. Kanzow and C. Nagel, Quadratic convergence of a nonsmooth newton-type method for semidefinite programs without strict complementarity,, SIAM J. Optim., 15 (2005), 654. Google Scholar [26] C. Kanzow, C. Nagel, H. Kato and M. Fukushima, Successive linearization methods for nonlinear semidefinite programs,, Comput. Optim. Appl., 31 (2005), 251. doi: 10.1007/s10589-005-3231-4. Google Scholar [27] D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,", Academic Press, (1980). Google Scholar [28] M. Kocvara and M. Stingl, PENNON - A Generalized augmented Lagrangian method for semidefinite programming,, In, (2003), 297. Google Scholar [29] M. Kocvara and M. Stingl, PENNON: a code for convex nonlinear and semidefinite programming,, Optim. Meth. Soft., 18 (2003), 317. doi: 10.1080/1055678031000098773. Google Scholar [30] M. Kocvara and M. Stingl, Solving nonconvex SDP problems of structural optimization with stability control,, Optim. Meth. Soft., 19 (2004), 595. doi: 10.1080/10556780410001682844. Google Scholar [31] L. Kong, J. Sun and N. Xiu, A regularized smoothing Newton method for symmetric cone complementarity problems,, SIAM J. Optim., 19 (2008), 1028. doi: 10.1137/060676775. Google Scholar [32] F. Leibfritz, COMP $l_{ e}$ ib 1.1: Constraint matrix-optimization problem library - a collection of test examples for nonlinear semidefinite programs, control system design and related problems,, Technical Report, (2005). Google Scholar [33] F. Leibfritz and M. E. Mostafa, An interior point constrained trust region method for a special class of nonlinear semidefinite programming problems,, SIAM J. Optim., 12 (2002), 1048. doi: 10.1137/S1052623400375865. Google Scholar [34] C. Li and W. Sun, On filter-successive linearization methods for nonlinear semidefinite programming,, China Sci. Ser. A, 52 (2009), 2341. doi: 10.1007/s11425-009-0168-6. Google Scholar [35] D. Noll and P. Apkarian, Spectral bundle methods for non-convex maximum eigenvalue functions: first-order methods,, Math. Program., 104 (2005), 701. doi: 10.1007/s10107-005-0634-z. Google Scholar [36] D. Noll and P. Apkarian, Spectral bundle methods for non-convex maximum eigenvalue functions: second-order methods,, Math. Program., 104 (2005), 729. doi: 10.1007/s10107-005-0635-y. Google Scholar [37] J. S. Pang, D. Sun and J. Sun, Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems,, Math. Oper. Res., 28 (2003), 39. doi: 10.1287/moor.28.1.39.14258. Google Scholar [38] T. Pennanen, Local convergence of the proximal point algorithm and multiplier methods without monotonicity,, Math. Oper. Res., 27 (2002), 170. doi: 10.1287/moor.27.1.170.331. Google Scholar [39] M. J. D. Powell, A method for nonlinear constraints in minimization problems,, In, (1972), 283. Google Scholar [40] H. Qi and D. Sun, An augmented Lagrangian dual approach for the H-weighted nearest correlation matrix problem,, IMA J. Numer. Anal., (2011). Google Scholar [41] L. Qi and J. Sun, A nonsmooth version of Newton's method,, Math. Program., 58 (1993), 353. doi: 10.1007/BF01581275. Google Scholar [42] R. T. Rockafellar, Augmented Lagrange multiplier functions and duality in nonconvex programming,, SIAM J. Control, 12 (1974), 268. doi: 10.1137/0312021. Google Scholar [43] R. T. Rockafellar, Monotone operators and the proximal point algorithm,, SIAM J. Control Optim., 14 (1976), 877. doi: 10.1137/0314056. Google Scholar [44] R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming,, Math. Oper. Res., 1 (1976), 97. doi: 10.1287/moor.1.2.97. Google Scholar [45] R. T. Rockafellar, Lagrange multipliers and optimality,, SIAM Review, 35 (1993), 183. Google Scholar [46] A. Shapiro, First and second order analysis of nonlinear semidefinite programs,, Math. Program., 77 (1997), 301. doi: 10.1007/BF02614439. Google Scholar [47] A. Shapiro and J. Sun, Some properties of the augmented Lagrangian in cone constrained optimization,, Math. Oper. Res., 29 (2004), 479. doi: 10.1287/moor.1040.0103. Google Scholar [48] D. Sun, The strong second-order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications,, Math. Oper. Res., 31 (2006), 761. doi: 10.1287/moor.1060.0195. Google Scholar [49] D. Sun and J. Sun, Semismooth matrix valued functions,, Math. Oper. Res., 27 (2002), 150. doi: 10.1287/moor.27.1.150.342. Google Scholar [50] D. Sun and J. Sun, Strong semismoothness of the Fischer-Burmeister SDC and SOC complementarity functions,, Math. Program., 103 (2005), 575. doi: 10.1007/s10107-005-0577-4. Google Scholar [51] D. Sun and J. Sun, Löwner's operator and spectral functions in Euclidean Jordan algebras,, Math. Oper. Res., 33 (2008), 421. doi: 10.1287/moor.1070.0300. Google Scholar [52] D. Sun, J. Sun and L. Zhang, Rates of convergence of the augmented Lagrangian method for nonlinear semidefinite programming,, Math. Program., 114 (2008), 349. doi: 10.1007/s10107-007-0105-9. Google Scholar [53] J. Sun, D. Sun and L. Qi, A squared smoothing Newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems,, SIAM J. Optim., 14 (2004), 783. doi: 10.1137/S1052623400379620. Google Scholar [54] J. Sun, L. Zhang and Y. Wu, Properties of the augmented Lagrangian in nonlinear semidefinite optimization,, J. Optim. Theory Appl., 129 (2006), 437. doi: 10.1007/s10957-006-9078-8. Google Scholar [55] J. Sun and S. Zhang, A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs,, Eur. J. Oper. Res., 207 (2010), 1210. doi: 10.1016/j.ejor.2010.07.020. Google Scholar [56] J. Sun, G. Zhao and J. Zhu, A predictor-corrector algorithm for a class of nonlinear saddle point problems,, SIAM J. Contr. Optim., 35 (1997), 532. doi: 10.1137/S0363012994276111. Google Scholar [57] N. K. Tsing, M. K. H. Fan and E. I. Verriest, On analyticity of functions involving eigenvalues,, Linear Algebra Appl., 207 (1994), 159. doi: 10.1016/0024-3795(94)90009-4. Google Scholar [58] P. Tseng, Merit functions for semidefinite complementarity problems,, Math. Program., 83 (1998), 159. doi: 10.1007/BF02680556. Google Scholar [59] C. Wang, D. Sun and K. C. Toh, Solving log-determinant optimization problems by a Newton-CG proximal point algorithm,, SIAM J. Optim., 20 (2010), 2994. doi: 10.1137/090772514. Google Scholar [60] Z. Wen, D. Goldfarb and W. Yin, Alternating direction augmented Lagrangian methods for semidefinite programming,, Optimization Online, (2009). Google Scholar [61] H. Yamashita, H. Yabe and K. Harada, A primal-dual interior point method for nonlinear semidefinite programming,, Technical report, (2007). Google Scholar [62] Z. S. Yu, Solving semidefinite programming problems via alternating direction methods,, J. Compu. Appl. Math., 193 (2006), 437. doi: 10.1016/j.cam.2005.07.002. Google Scholar [63] S. Zhang, J. Ang and J. Sun, An alternating direction method for solving convex nonlinear semidefinite programming problems,, Technical Report, (2010). Google Scholar [64] X. Zhao, D. Sun and K. Toh, A Newton-CG augmented Lagrangian method for semidefinite programming,, SIAM J. Optim., 20 (2010), 1737. doi: 10.1137/080718206. Google Scholar

show all references

References:
 [1] A. Ben-Tal, F. Jarre, M. Kocvara, A. Nemirovski and J. Zowe, Optimal design of trusses under a nonconvex global buckling constraint,, Optim. and Eng., 1 (2000), 189. doi: 10.1023/A:1010091831812. Google Scholar [2] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, "Linear Matrix Inequalities in System and Control Theory,", SIAM Studies in Applied Mathematics, (1994). Google Scholar [3] X. Chen, H. D. Qi and P. Tseng, Analysis of nonsmooth symmetric-matrix functions with applications to semidefinite complementarity problems,, SIAM J. Optim., 13 (2003), 960. doi: 10.1137/S1052623400380584. Google Scholar [4] X. Chen, D. Sun and J. Sun, Complementarity functions and numerical experiments on some smoothing Newton methods for second-order-cone complementarity problems, , Comput. Optim. Appl., 25 (2003), 39. doi: 10.1023/A:1022996819381. Google Scholar [5] X. Chen and P. Tseng, Non-interior continuation methods for solving semidefinite complementarity problems,, Math. Program., 95 (2003), 431. doi: 10.1007/s10107-002-0306-1. Google Scholar [6] F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley and Sons, (1983). Google Scholar [7] R. Correa and C. H. Ramirez, A global algorithm for nonlinear semidefinite programming,, SIAM J. Optim., 15 (2004), 303. doi: 10.1137/S1052623402417298. Google Scholar [8] M. Diehl, F. Jarre and C. H. Vogelbusch, Loss of superlinear convergence for an SQP-type method with conic constraints,, SIAM J. Optim., 16 (2006), 1201. doi: 10.1137/050625977. Google Scholar [9] M. Doljansky, An interior proximal algorithm and the exponential multiplier method for semidefinite programming,, SIAM J. Optim., 9 (1999), 1. doi: 10.1137/S1052623496309405. Google Scholar [10] A. Forsgren, Optimality conditions for nonconvex semidefinite programming,, Math. Program., 88 (2000), 105. doi: 10.1007/PL00011370. Google Scholar [11] J. Eckstein, "Splitting Methods for Monotone Operators with Applications to Parallel Optimization,", PhD thesis, (1989). Google Scholar [12] B. Fares, D. Noll and P. Apkarian, Robust control via sequential semidefinite programming,, SIAM J. Contr. and Optim., 40 (2002), 1791. doi: 10.1137/S0363012900373483. Google Scholar [13] M. L. Flegel and C. Kanzow, Equivalence of two nondegeneracy conditions for semidefinite programs,, J. Optim. Theory Appl., 135 (2007), 381. doi: 10.1007/s10957-007-9270-5. Google Scholar [14] M. Fukushima, Application of the alternating directions method of multipliers to separable convex programming problems,, Comput. Optim. Appl., 1 (1992), 93. doi: 10.1007/BF00247655. Google Scholar [15] M. Fukushima, Z. Q. Luo and P. Tseng, Smoothing functions for second-order cone complementarity problems,, SIAM J. Optim., 12 (2002), 436. doi: 10.1137/S1052623400380365. Google Scholar [16] Y. Gao and D. Sun, Calibrating least squares covariance matrix problems with equality and inequality constraints,, SIAM J. Matrix Anal. Appl. 31 (2009), 31 (2009), 1432. doi: 10.1137/080727075. Google Scholar [17] B. S. He, L. Z. Liao, D. R. Han and H. Yang, A new inexact alternating directions method for monotone variational inequalities,, Math. Program., 92 (2002), 103. Google Scholar [18] B. S. He, L. Z. Liao and M. J. Qian, Alternating projection based prediction-correction methods for structured variational inequalities,, J. Compu. Math., 24 (2002), 693. Google Scholar [19] M. R. Hestenes, Multiplier and gradient methods,, J. Optim. Theory Appl., 4 (1969), 303. Google Scholar [20] N. J. Higham, Computing a nearest symmetric positive semidefinite matrix,, Linear Algebra Appl., 103 (1998), 103. doi: 10.1016/0024-3795(88)90223-6. Google Scholar [21] F. Jarre, An interior method for nonconvex semidefinite programs,, Optim. and Eng., 1 (2000), 347. doi: 10.1023/A:1011562523132. Google Scholar [22] C. Kanzow, I. Ferenczi and M. Fukushima, On the local convergence of semismooth newton methods for linear and nonlinear second-order cone programs without strict complementarity,, SIAM J. Optim., 20 (2009), 297. doi: 10.1137/060657662. Google Scholar [23] C. Kanzow and C. Nagel, Semidefinite programs: New search directions, smoothing-type methods, and numerical results,, SIAM J. Optim., 13 (2002), 1. doi: 10.1137/S1052623401390525. Google Scholar [24] C. Kanzow and C. Nagel, Some structural properties of a Newton-type method for semidefinite programs,, J. Optim. Theory Appl., 122 (2004), 219. doi: 10.1023/B:JOTA.0000041737.19689.4c. Google Scholar [25] C. Kanzow and C. Nagel, Quadratic convergence of a nonsmooth newton-type method for semidefinite programs without strict complementarity,, SIAM J. Optim., 15 (2005), 654. Google Scholar [26] C. Kanzow, C. Nagel, H. Kato and M. Fukushima, Successive linearization methods for nonlinear semidefinite programs,, Comput. Optim. Appl., 31 (2005), 251. doi: 10.1007/s10589-005-3231-4. Google Scholar [27] D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,", Academic Press, (1980). Google Scholar [28] M. Kocvara and M. Stingl, PENNON - A Generalized augmented Lagrangian method for semidefinite programming,, In, (2003), 297. Google Scholar [29] M. Kocvara and M. Stingl, PENNON: a code for convex nonlinear and semidefinite programming,, Optim. Meth. Soft., 18 (2003), 317. doi: 10.1080/1055678031000098773. Google Scholar [30] M. Kocvara and M. Stingl, Solving nonconvex SDP problems of structural optimization with stability control,, Optim. Meth. Soft., 19 (2004), 595. doi: 10.1080/10556780410001682844. Google Scholar [31] L. Kong, J. Sun and N. Xiu, A regularized smoothing Newton method for symmetric cone complementarity problems,, SIAM J. Optim., 19 (2008), 1028. doi: 10.1137/060676775. Google Scholar [32] F. Leibfritz, COMP $l_{ e}$ ib 1.1: Constraint matrix-optimization problem library - a collection of test examples for nonlinear semidefinite programs, control system design and related problems,, Technical Report, (2005). Google Scholar [33] F. Leibfritz and M. E. Mostafa, An interior point constrained trust region method for a special class of nonlinear semidefinite programming problems,, SIAM J. Optim., 12 (2002), 1048. doi: 10.1137/S1052623400375865. Google Scholar [34] C. Li and W. Sun, On filter-successive linearization methods for nonlinear semidefinite programming,, China Sci. Ser. A, 52 (2009), 2341. doi: 10.1007/s11425-009-0168-6. Google Scholar [35] D. Noll and P. Apkarian, Spectral bundle methods for non-convex maximum eigenvalue functions: first-order methods,, Math. Program., 104 (2005), 701. doi: 10.1007/s10107-005-0634-z. Google Scholar [36] D. Noll and P. Apkarian, Spectral bundle methods for non-convex maximum eigenvalue functions: second-order methods,, Math. Program., 104 (2005), 729. doi: 10.1007/s10107-005-0635-y. Google Scholar [37] J. S. Pang, D. Sun and J. Sun, Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems,, Math. Oper. Res., 28 (2003), 39. doi: 10.1287/moor.28.1.39.14258. Google Scholar [38] T. Pennanen, Local convergence of the proximal point algorithm and multiplier methods without monotonicity,, Math. Oper. Res., 27 (2002), 170. doi: 10.1287/moor.27.1.170.331. Google Scholar [39] M. J. D. Powell, A method for nonlinear constraints in minimization problems,, In, (1972), 283. Google Scholar [40] H. Qi and D. Sun, An augmented Lagrangian dual approach for the H-weighted nearest correlation matrix problem,, IMA J. Numer. Anal., (2011). Google Scholar [41] L. Qi and J. Sun, A nonsmooth version of Newton's method,, Math. Program., 58 (1993), 353. doi: 10.1007/BF01581275. Google Scholar [42] R. T. Rockafellar, Augmented Lagrange multiplier functions and duality in nonconvex programming,, SIAM J. Control, 12 (1974), 268. doi: 10.1137/0312021. Google Scholar [43] R. T. Rockafellar, Monotone operators and the proximal point algorithm,, SIAM J. Control Optim., 14 (1976), 877. doi: 10.1137/0314056. Google Scholar [44] R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming,, Math. Oper. Res., 1 (1976), 97. doi: 10.1287/moor.1.2.97. Google Scholar [45] R. T. Rockafellar, Lagrange multipliers and optimality,, SIAM Review, 35 (1993), 183. Google Scholar [46] A. Shapiro, First and second order analysis of nonlinear semidefinite programs,, Math. Program., 77 (1997), 301. doi: 10.1007/BF02614439. Google Scholar [47] A. Shapiro and J. Sun, Some properties of the augmented Lagrangian in cone constrained optimization,, Math. Oper. Res., 29 (2004), 479. doi: 10.1287/moor.1040.0103. Google Scholar [48] D. Sun, The strong second-order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications,, Math. Oper. Res., 31 (2006), 761. doi: 10.1287/moor.1060.0195. Google Scholar [49] D. Sun and J. Sun, Semismooth matrix valued functions,, Math. Oper. Res., 27 (2002), 150. doi: 10.1287/moor.27.1.150.342. Google Scholar [50] D. Sun and J. Sun, Strong semismoothness of the Fischer-Burmeister SDC and SOC complementarity functions,, Math. Program., 103 (2005), 575. doi: 10.1007/s10107-005-0577-4. Google Scholar [51] D. Sun and J. Sun, Löwner's operator and spectral functions in Euclidean Jordan algebras,, Math. Oper. Res., 33 (2008), 421. doi: 10.1287/moor.1070.0300. Google Scholar [52] D. Sun, J. Sun and L. Zhang, Rates of convergence of the augmented Lagrangian method for nonlinear semidefinite programming,, Math. Program., 114 (2008), 349. doi: 10.1007/s10107-007-0105-9. Google Scholar [53] J. Sun, D. Sun and L. Qi, A squared smoothing Newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems,, SIAM J. Optim., 14 (2004), 783. doi: 10.1137/S1052623400379620. Google Scholar [54] J. Sun, L. Zhang and Y. Wu, Properties of the augmented Lagrangian in nonlinear semidefinite optimization,, J. Optim. Theory Appl., 129 (2006), 437. doi: 10.1007/s10957-006-9078-8. Google Scholar [55] J. Sun and S. Zhang, A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs,, Eur. J. Oper. Res., 207 (2010), 1210. doi: 10.1016/j.ejor.2010.07.020. Google Scholar [56] J. Sun, G. Zhao and J. Zhu, A predictor-corrector algorithm for a class of nonlinear saddle point problems,, SIAM J. Contr. Optim., 35 (1997), 532. doi: 10.1137/S0363012994276111. Google Scholar [57] N. K. Tsing, M. K. H. Fan and E. I. Verriest, On analyticity of functions involving eigenvalues,, Linear Algebra Appl., 207 (1994), 159. doi: 10.1016/0024-3795(94)90009-4. Google Scholar [58] P. Tseng, Merit functions for semidefinite complementarity problems,, Math. Program., 83 (1998), 159. doi: 10.1007/BF02680556. Google Scholar [59] C. Wang, D. Sun and K. C. Toh, Solving log-determinant optimization problems by a Newton-CG proximal point algorithm,, SIAM J. Optim., 20 (2010), 2994. doi: 10.1137/090772514. Google Scholar [60] Z. Wen, D. Goldfarb and W. Yin, Alternating direction augmented Lagrangian methods for semidefinite programming,, Optimization Online, (2009). Google Scholar [61] H. Yamashita, H. Yabe and K. Harada, A primal-dual interior point method for nonlinear semidefinite programming,, Technical report, (2007). Google Scholar [62] Z. S. Yu, Solving semidefinite programming problems via alternating direction methods,, J. Compu. Appl. Math., 193 (2006), 437. doi: 10.1016/j.cam.2005.07.002. Google Scholar [63] S. Zhang, J. Ang and J. Sun, An alternating direction method for solving convex nonlinear semidefinite programming problems,, Technical Report, (2010). Google Scholar [64] X. Zhao, D. Sun and K. Toh, A Newton-CG augmented Lagrangian method for semidefinite programming,, SIAM J. Optim., 20 (2010), 1737. doi: 10.1137/080718206. Google Scholar
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text:
## Notes on Transformation Orderings

• An important property and also a crucial point ofa term rewriting system is its termination. Transformation or-derings, developed by Bellegarde & Lescanne strongly based on awork of Bachmair & Dershowitz, represent a general technique forextending orderings. The main characteristics of this method aretwo rewriting relations, one for transforming terms and the otherfor ensuring the well-foundedness of the ordering. The centralproblem of this approach concerns the choice of the two relationssuch that the termination of a given term rewriting system can beproved. In this communication, we present a heuristic-based al-gorithm that partially solves this problem. Furthermore, we showhow to simulate well-known orderings on strings by transformationorderings.

$Rev: 13581$


text:
WOfS Imputation ¶

• Compatibility: Notebook currently compatible with both the NCI and DEA Sandbox environments.

• Products used: ga_ls7e_ard_3

Description¶

There are plenty of missing data in our time series of Australian waterbodies, not least due to Landsat 7’s scan line corrector failing in May 2003 (see below for an image of missing data in WOfS). In this notebook we impute missing data on a scene-by-scene basis in a few different ways.

We will run WOfS ourself since we do not want the terrain mask — we know where the waterbody is and we can ignore the terrain masking for this analysis.

Getting started¶

To run this analysis, work through this notebook starting with the “Load packages” cell.

Import Python packages that are used for the analysis.

Analysis parameters¶

This section lets you set parameters for the analysis.

Choose a waterbody¶

Specify the geohash of a waterbody here:

[3]:

geohash = "r282x882w"  # Lagoon of Islands

Estimate WOfS for the full time stack¶

Load all data for this waterbody and estimate WOfS.

Fetch the waterbody polygon extents:

[4]:

wb = get_waterbody(geohash)

[5]:

xlim = wb.total_bounds[::2]
ylim = wb.total_bounds[1::2]

Set up the datacube:

[6]:

# Set up the datacube to get DEA data.
dc = datacube.Datacube(app="WOfSImputation")

# Some query parameters.
dask_chunks = {"x": 3000, "y": 3000, "time": 1}

[7]:

# To load in native resolution and projection we need to find the native CRS.
output_crs = mostcommon_crs(
dc, product="ga_ls7e_ard_3", query=dict(x=xlim, y=ylim, crs="EPSG:3577")
)

[8]:

query = dict(
x=xlim,
y=ylim,
crs="EPSG:3577",
output_crs=output_crs,  # Native CRS
align=(
15,
15,
),  # Required for loading native resolution and CRS Landsat Collection 3
resolution=(-30, 30),  # Native resolution
group_by="solar_day",
measurements=[
"nbart_blue",
"nbart_green",
"nbart_red",
"nbart_nir",
"nbart_swir_1",
"nbart_swir_2",
"oa_nbart_contiguity",
],
)

[9]:

ls7 = dc.load("ga_ls7e_ard_3", **query)

Check that everything will fit into memory before we load it:

[10]:

n_px = ls7.sizes["time"] * ls7.sizes["x"] * ls7.sizes["y"]
est = (6 * n_px * 16 + 2 * n_px * 8) // 8 / 1e9
print("Estimated:", est, "GB")

assert est < 40

Estimated: 0.187979792 GB

[11]:

ls7.load()

[11]:

<xarray.Dataset>
Dimensions:              (time: 776, x: 143, y: 121)
Coordinates:
* time                 (time) datetime64[ns] 1999-08-10T23:45:11.304012 ......
* y                    (y) float64 -4.659e+06 -4.659e+06 ... -4.662e+06
* x                    (x) float64 4.921e+05 4.921e+05 ... 4.963e+05 4.963e+05
spatial_ref          int32 32655
Data variables:
nbart_blue           (time, y, x) int16 53 53 52 1 101 ... 147 192 194 173
nbart_green          (time, y, x) int16 170 358 288 218 ... 311 312 289 318
nbart_red            (time, y, x) int16 401 347 338 374 ... 385 407 345 394
nbart_nir            (time, y, x) int16 1727 1796 1679 ... 1496 1412 1475
nbart_swir_1         (time, y, x) int16 1114 1114 1019 ... 1286 1325 1227
nbart_swir_2         (time, y, x) int16 579 578 492 470 ... 672 761 709 778
oa_fmask             (time, y, x) uint8 1 1 1 1 1 1 1 1 ... 1 1 1 1 1 1 1 1
oa_nbart_contiguity  (time, y, x) uint8 1 1 1 1 1 1 1 1 ... 1 1 1 1 1 1 1 1
Attributes:
crs:           epsg:32655
grid_mapping:  spatial_ref

We can then combine some masks. We will mask with a combination of fmask, contiguity, valid data, and within the waterbody maximum extent:

[12]:

wb_raster = xr_rasterize(wb, ls7)

[13]:

mask = (ls7.oa_fmask == 1) | (ls7.oa_fmask == 4) | (ls7.oa_fmask == 5)

Convert the cube into a format that WOfS will accept.

[14]:

ls7_cube = ls7[
[
"nbart_blue",
"nbart_green",
"nbart_red",
"nbart_nir",
"nbart_swir_1",
"nbart_swir_2",
]
].to_array(dim="band")

Finally we can go ahead and estimate WOfS.

[15]:

wofs = fuzzy_wofs.wofs.predict(ls7_cube)

We can reproduce the WOfS all-time summary by taking the mean, which will let us confirm that everything worked OK:

[16]:

plt.imshow(np.nanmean(np.where(mask, wofs, np.nan), axis=0), cmap="jet_r")

/env/lib/python3.6/site-packages/ipykernel_launcher.py:1: RuntimeWarning: Mean of empty slice
"""Entry point for launching an IPython kernel.

[16]:

<matplotlib.image.AxesImage at 0x7ff672120c88>

That looks like it matches the all-time summary (see here on DEA Maps), and the islands in the middle of the lagoon are clearly visible.

Baseline comparison: Time series without imputing WOfS¶

First we will try matching DEA Waterbodies. Throw out all time steps with <90% pixels:

[17]:

wb_area_px2 = wb_raster.sum()

[18]:

ok = mask.sum(axis=(1, 2)) >= wb_area_px2 * 0.9

Then just sum up the pixel counts.

[19]:

ts_baseline = pd.Series(
np.nansum(np.where(mask, wofs, np.nan), axis=(1, 2)), index=ls7.time.values
)

[20]:

n_nodata = pd.Series((~mask & (wb_raster == 1)).sum(axis=(1, 2)), index=ls7.time.values)
pc_nodata = n_nodata / int(wb_area_px2)

We can plot this with linear interpolation to join the water surface area time series:

[21]:

fig, ax = plt.subplots()
ts_baseline.plot(ax=ax, linewidth=0, marker="x", c="C0")
ts_baseline.interpolate(method="time").plot(ax=ax, linewidth=1, c="C0")
ax.set_xlabel("Time")
ax.set_ylabel("Area (px$^2$)");

Different thresholds for “good data” can change this substantially. Let’s compare the 90% threshold above to a 75% threshold:

[22]:

ok_low = mask.sum(axis=(1, 2)) >= wb_area_px2 * 0.75

[23]:

ts_baseline_low = pd.Series(
np.nansum(np.where(mask, wofs, np.nan), axis=(1, 2)), index=ls7.time.values
)

[24]:

fig, ax = plt.subplots()
ts_baseline.plot(ax=ax, linewidth=0, marker="x", c="C0", label="_nolegend_")
h1 = ts_baseline.interpolate(method="time").plot(
ax=ax, linewidth=1, c="C0", label="90% OK"
)
ts_baseline_low.plot(ax=ax, linewidth=0, marker="x", c="C1", label="_nolegend_")
h2 = ts_baseline_low.interpolate(method="time").plot(
ax=ax, linewidth=1, c="C1", label="75% OK"
)
plt.legend()
ax.set_xlabel("Time")
ax.set_ylabel("Area (px$^2$)");

It’s useful here to estimate how far out we might be, because this is really a lower bound on how many wet pixels there are — we’re essentially assuming all missing pixels are dry, so why not assume all missing pixels are wet? That would instead give us an upper bound.

[25]:

ts_baseline_upper = pd.Series(
np.where(ok, wofs.sum(axis=(1, 2)) + n_nodata, np.nan), index=ls7.time.values
)
ts_baseline_low_upper = pd.Series(
np.where(ok_low, wofs.sum(axis=(1, 2)) + n_nodata, np.nan), index=ls7.time.values
)

We can then plot the lower and upper bounds:

[26]:

fig, ax = plt.subplots()

as_bars = False

if as_bars:
ax.errorbar(
ts_baseline_low.index,
ts_baseline_low,
yerr=np.stack(
[np.zeros(len(ts_baseline_low)), ts_baseline_low_upper - ts_baseline_low]
),
label="75% OK data",
c="C1",
)
ax.errorbar(
ts_baseline.index,
ts_baseline,
yerr=np.stack([np.zeros(len(ts_baseline)), ts_baseline_upper - ts_baseline]),
label="90% OK data",
c="C0",
)
else:
is_nan = pd.isnull(ts_baseline_low)
ax.fill_between(
ts_baseline_low.index[~is_nan],
ts_baseline_low[~is_nan],
ts_baseline_low_upper[~is_nan],
label="75% OK data",
color="C1",
zorder=1,
alpha=0.5,
)
is_nan = pd.isnull(ts_baseline)
ax.fill_between(
ts_baseline.index[~is_nan],
ts_baseline[~is_nan],
ts_baseline_upper[~is_nan],
label="90% OK data",
color="C0",
alpha=0.5,
zorder=2,
)
plt.legend()
ax.set_xlabel("Time")
ax.set_ylabel("Area (px$^2$)");

Nearest observation imputation¶

We can assume that if we have no observed WOfS data for a pixel on some date, then we can fill that pixel using the last available WOfS value for that pixel. We can then compute the time series as before, but no pixels will be missing as they will already have been imputed.

[27]:

# Convert into a dataframe so we can use pandas.
df = pd.DataFrame(wofs.reshape(wofs.shape[0], -1), index=ls7.time.values)

[28]:

# Mask the masked values using pandas.

[29]:

# Forward-fill masked values.

[30]:

# Calculate the surface area time series.
ts_ffill = df_ffilled.sum(axis=1)

[31]:

fig, ax = plt.subplots()
ax.errorbar(
ts_baseline_low.index,
ts_baseline_low,
yerr=np.stack(
[np.zeros(len(ts_baseline_low)), ts_baseline_low_upper - ts_baseline_low]
),
label="75% OK data",
c="C1",
)
ax.errorbar(
ts_baseline.index,
ts_baseline,
yerr=np.stack([np.zeros(len(ts_baseline)), ts_baseline_upper - ts_baseline]),
label="90% OK data",
c="C0",
)
ts_ffill.plot(ax=ax, c="C2", label="ffill")
plt.legend()
ax.set_xlabel("Time")
ax.set_ylabel("Area (px$^2$)");

An equally meaningful approach would be to take the next observation instead.

[32]:

df_bfilled = df_masked.bfill(axis=0)

[33]:

ts_bfill = df_bfilled.sum(axis=1)

[34]:

fig, ax = plt.subplots()
ts_ffill.plot(ax=ax, c="C2", label="ffill")
ts_bfill.plot(ax=ax, c="purple", label="bfill", linestyle="--")
plt.legend()
ax.set_xlabel("Time")
ax.set_ylabel("Area (px$^2$)");

We can also average these (on a per-pixel level!), which means we’re taking the average of the previous and next WOfS values:

[35]:

ts_avgfill = (df_ffilled + df_bfilled).sum(axis=1) / 2

[36]:

fig, ax = plt.subplots()
ts_baseline.interpolate(method="time").plot(ax=ax, c="C0", label="90% OK")
ts_baseline_low.interpolate(method="time").plot(ax=ax, c="C1", label="75% OK")
ts_avgfill.plot(ax=ax, c="C3", label="avgfill")
plt.legend()
ax.set_xlabel("Time")
ax.set_ylabel("Area (px$^2$)");

Finally we could also linearly interpolate each pixel. This will largely be very similar to the average fill, except if there is a long series of unobserved times.

[37]:

df_linfilled = df_masked.interpolate(method="time", axis=0)

[38]:

ts_linfilled = df_linfilled.sum(axis=1)

[39]:

fig, ax = plt.subplots()
ts_baseline.interpolate(method="time").plot(ax=ax, c="C0", label="90% OK")
ts_baseline_low.interpolate(method="time").plot(ax=ax, c="C1", label="75% OK")
ts_avgfill.plot(ax=ax, c="C3", label="avgfill")
ts_linfilled.plot(ax=ax, c="C4", label="linfill")
plt.legend()

[39]:

<matplotlib.legend.Legend at 0x7ff6bc1c8208>

Mean observation imputation¶

Take the mean over all-time as the value for each missing pixel. This is a fairly standard method for imputing values in prediction tasks, but it’s very rough.

[40]:

mean_values = df_masked.mean(axis=0)

[41]:

df_meanfill = df_masked.fillna(value=mean_values)

[42]:

ts_meanfill = df_meanfill.sum(axis=1)

[43]:

fig, ax = plt.subplots()
ts_baseline.interpolate(method="time").plot(ax=ax, c="C0", label="90% OK")
ts_baseline_low.interpolate(method="time").plot(ax=ax, c="C1", label="75% OK")
ts_avgfill.plot(ax=ax, c="C3", label="avgfill")
ts_meanfill.plot(ax=ax, c="magenta", label="meanfill")
plt.legend()

[43]:

<matplotlib.legend.Legend at 0x7ff671ca3be0>

A very clear pattern can be seen where the mean water level is preferentially chosen, which will correspond to days with very few observations. Where there are not many observations, filling with the mean water level will cause the waterbody to appear as if it spends most of the time at the mean water level.

Conclusion¶

Imputing WOfS by setting missing values to the average of their previous and next valid observations, on a per-pixel basis, provides a simple and efficient way to impute WOfS across missing data, while seeming to add less bias toward dry values. Forward- or backward-filling would also be OK, but we might as well be time-symmetric. In the imputed time series seasonal patterns of filling and emptying are seen, whereas these patterns are less clear or missing in the non-imputed series.

Contact: If you need assistance, please post a question on the Open Data Cube Slack channel or on the GIS Stack Exchange using the open-data-cube tag (you can view previously asked questions here). If you would like to report an issue with this notebook, you can file one on Github.

Compatible datacube version:

[61]:

datacube.__version__

[61]:

'1.8.3'

Tags¶

Tags: DEA Waterbodies, WOfS, water


text:
# Beyond Einstein’s Velocity Addition Law, by Abraham A. Ungar. Fundamental Theories of Physics 117. Kluwer Academic, 2001, xlii + 413 pp.

Scott A. Walter University of Nantes, François-Viète Center (EA 1161)
Foundations of Physics 32(2), 2002, 327–330.

By most standards, the basic four-dimensional spacetime formalism proposed by Hermann Minkowski in 1908 for use in special relativity has performed passably well. Spacetime diagrams in particular are ubiquitous to teaching and research alike. However, from the inception of the spacetime approach, and afterwards at an interval of roughly twenty years, rival formalisms have arisen with the intent either to complement the dominant technique or to supplant it entirely. The most recent entry in the lists is Abraham Ungar’s gyrovector formalism, which after its discovery in 1988 underwent significant development by its author, culminating in the work under review. As with most new formal methods, it has little to recommend its study in the way of new physical results or insights, but may be compared with the spacetime approach in terms of the elegance of respective proofs of selected theorems. One might also compare Ungar’s method to those of his immediate forerunners. On both counts, the gyroformalism proves to be worthy of physicists’ attention.

When H. Minkowski began pondering the structure of the Lorentz group in 1907, one of the first things he noticed was that geometrical relations between velocity vectors measured in inertial frames of reference are not Euclidean (as in classical kinematics), but hyperbolic. The spacetime formalism he went on to develop for the physics of relativity, however, did not exploit this insight, rather, its reliance on an imaginary temporal coordinate tended to obscure the non-Euclidean nature of the Minkowski metric. Nonetheless, his system, developed by A. Sommerfeld and others into a full-blown vectorial analysis, and doted with a visually intuitive model in the form of a spacetime diagram, rose rapidly to dominate theoretical work in relativity. Mathematicians like V. Varičak and E. Borel then saw that by employing a real temporal coordinate, they could exploit hyperbolic trigonometry, and went on inaugurate a new, non-Euclidean style of relativity. This alternative style was largely neglected by contemporary physicists, who also ignored Borel’s subsequent discovery of a kinematic phenomenon later known as Thomas precession. Even so, the non-Euclidean style found steady employment in relativity textbooks, where it was used to present velocity composition.

Over the years, there have been a handful of attempts to promote the non-Euclidean style for use in problem solving in relativity and electrodynamics, the failure of which to attract any substantial following, compounded by the absence of any positive results must give pause to anyone considering a similar undertaking. Until recently, no one was in a position to offer an improvement on the tools available since 1912. In his new book, Ungar furnishes the crucial missing element from the panoply of the non-Euclidean style: an elegant nonassociative algebraic formalism that fully exploits the structure of Einstein’s law of velocity composition. The formalism relies on what the author calls the “missing link” between Einstein’s velocity addition formula and ordinary vector addition: Thomas precession, or the angular difference in relative velocity that results when one changes the order of frame velocities in the velocity addition formula, the magnitude of this relative velocity being invariant with respect to frame order.

Ungar lays out for the reader a sort of vector algebra in hyperbolic space, based on the notion of a gyrovector. A gyrovector space differs in general from a vector space in virtue of inclusion of Thomas precession, and exclusion of the vector distributive law. As a result, when expressed in terms of gyrovectors, Einstein’s (non-commutative) velocity addition law becomes “gyrocommutative”, in that the two possible expressions for relative velocity for two inertial frames in non-parallel motion are related by a Thomas precession. One advantage of this approach is that hyperbolic geometry segues into Euclidean geometry, with notions such as group, vector, and line passing over to their hyperbolic gyro-counterparts (gyrogroup, etc.). Ungar’s book is devoted to the presentation of the theory of gyrogroups and gyrovector spaces derived from the formal wedding of velocity addition to Thomas precession.

One might suppose that there is a price to pay in mathematical regularity when replacing ordinary vector addition with Einstein’s addition, but Ungar shows that the latter supports gyrocommutative and gyroassociative binary operations, in full analogy to the former. Likewise, some gyrocommutative and gyroassociative binary operations support scalar multiplication, giving rise to gyrovector spaces, which provide the setting for various models of hyperbolic geometry, just as vector spaces form the setting for the common model of Euclidean geometry. In particular, Einstein gyrovector spaces provide the setting for the Beltrami ball model of hyperbolic geometry, while Möbius gyrovector spaces provide the setting for the Poincaré ball model of hyperbolic geometry.

The book begins with a potted history of Thomas precession and its abstract counterpart, Thomas “gyration” (thus explaining the plethora of “gyro” prefixes). This chapter will be of most interest to physicists, as it presents the basic method of gyrovectors and their motivation in a nutshell. Subsequent chapters are more formal, laying out in definition-lemma-theorem style the theory of gyrogroups and gyrovector spaces, with reference to the Beltrami and Poincaré models of hyperbolic geometry, including a nice discussion of common and hyperbolic parallel transport, studied with the methods of nonassociative algebra rather than those of differential geometry. In the book’s final chapters, one finds a parametrization of the abstract Lorentz boost by gyrovectors, and two further parametrizations of nonlinear, pseudo-Lorentz transformations.

A trip “Beyond the Einstein Addition Law” will require a grasp of the basics of group theory, as well as access to a computer algebra program. For the perplexed, the author points to a wealth of references on both elementary and advanced concepts. Further assistance is provided by over eighty figures, most of which employ either Beltrami or Poincaré discs. The figures do yeoman service in aiding the reader to follow the geometric arguments, the assimilation of which may be tested by working out the exercises that close each chapter. Unfortunately, the text suffers from inadequate editing, with unnecessary repetition between chapters. A self-congratulatory tone, and an irrelevant recapitulation of an ongoing priority dispute are also to be regretted. Perhaps in relation to the latter, the author refers again and again to his first publication on K-loops, when he could have saved space and time by simply including the paper in an appendix.

Minor flaws in presentation aside, what can be said of the epistemic value of Ungar’s formalism? There are three areas susceptible to application of the gyrogroup method: physics, non-Euclidean geometry and abstract algebra. All three areas appear ready to benefit from the method, which simplifies certain calculations and exploits Euclidean space intuitions. But will these advantages bring about new insights, and new results? For abstract algebra and non-Euclidean geometry, the author summarizes several recent results, for example, a new model of hyperbolic geometry (the “Ungar model”), and the discovery of cohyperbolic geometry (in which the angle sum of a triangle is $\pi$, but triangle medians are not concurrent).

For physics, however, one still wonders if a compact, Euclidean vector notation accompanied by Beltrami or Poincaré models will ever outperform tensor calculus aligned with spacetime diagrams. To argue the case for gyrogroup methods, Ungar revives the age-old question of the reality of the Lorentz-FitzGerald contraction (LFC). He applies his gyrogroup-theoretic method to the problem of the apparent form of a sphere in uniform motion, and finds that it is measured as a flattened ellipsoid by an observer at rest. According to Ungar, this result vindicates Einstein’s stance on the appearance of a sphere in uniform motion against the well-known argument of R. Penrose (1959), which holds that such flattening is not photographically observable. Penrose, however, considers light propagating with finite velocity from the sphere to an ideal photographic plate at rest, where Ungar assumes infinite signal velocity, a difference which clearly moots the comparison. For the case considered by the author, a more fruitful analogy might be drawn with the three-dimensional, mental “snapshots” described by J. Synge in his 1955 textbook on special relativity. Given the form of an object in motion as judged by a comoving inertial observer, Synge’s method determines the object form as judged by an observer at rest, via iterative Lorentz transformation of surface points, where Ungar’s algebra recovers the entire form in one fell swoop.

Although apodictic proof of the gyrogroup method’s creative power of discovery in physics is still forthcoming, physicists will appreciate the new geometric view of Einstein’s velocity addition law and Thomas precession. Announced as a likely companion text to both undergraduate and graduate courses in physics, non-Euclidean geometry and abstract algebra, the book stands to be of most profit to advanced students and researchers. It will be a useful addition to all research libraries.


text:
# Testing White Dwarf Crystallization Theory with Asteroseismology of the Massive Pulsating DA Star BPM 37093

@article{Metcalfe2004TestingWD,
title={Testing White Dwarf Crystallization Theory with Asteroseismology of the Massive Pulsating DA Star BPM 37093},
author={Travis S. Metcalfe and Michael H. Montgomery and Antonio Kanaan},
journal={The Astrophysical Journal Letters},
year={2004},
volume={605},
pages={L133 - L136}
}
• Published 2 February 2004
• Physics, Geology
• The Astrophysical Journal Letters
It was predicted more than 40 years ago that the cores of the coolest white dwarf stars should eventually crystallize. This effect is one of the largest sources of uncertainty in white dwarf cooling models, which are now routinely used to estimate the ages of stellar populations in both the Galactic disk and the Galactic halo. We are attempting to minimize this source of uncertainty by calibrating the models, using observations of pulsating white dwarfs. In a typical mass white dwarf model…
• Physics, Geology
• 2013
We present the first asteroseismological study for 42 massive ZZ Ceti stars based on a large set of fully evolutionary carbon–oxygen core DA white dwarf models characterized by a detailed and
White dwarfs are the most common stellar evolutionary end-point. Moreover, they can be considered as reliable cosmic clocks to infer the age of a wide variety of stellar populations, including
• Physics
• 2005
In view of recent claims that asteroseismology could supply invaluable insight into the crystallization process oc- curring in the interiors of massive white dwarf stars, we present in this work new
• Physics, Geology
The Astronomy and Astrophysics Review
• 2019
Stars are extremely important astronomical objects that constitute the pillars on which the Universe is built, and as such, their study has gained increasing interest over the years. White dwarf
• Physics, Geology
Astronomy & Astrophysics
• 2019
Context. Ultra-massive hydrogen-rich white dwarf stars are expected to harbor oxygen/neon cores resulting from the progenitor evolution through the super-asymptotic giant branch phase. As evolution
• Physics
• 2009
We explore the physics of crystallization in the deep interiors of white dwarf (WD) stars using the color–magnitude diagram and luminosity function constructed from proper-motion cleaned Hubble Space
• Physics, Geology
• 2010
We compute new chemical profiles for the core and envelope of white dwarfs appropriate for pulsational studies of ZZ Ceti stars. These profiles are extracted from the complete evolution of progenitor
• Physics, Geology
• 2005
BPM 37093 is the only hydrogen-atmosphere white dwarf currently known which has sufficient mass (~1.1 $M_\odot$) to theoretically crystallize while still inside the ZZ Ceti instability strip (\$T_{\rm
• Physics
• 2013
We announce the discovery of the most massive pulsating hydrogen-atmosphere white dwarf (WD) ever discovered, GD 518. Model atmosphere fits to the optical spectrum of this star show it is a 12, 030 ±
• Physics, Geology
• 2018
The thermally pulsing phase on the asymptotic giant branch (TP-AGB) is the last nuclear burning phase experienced by most of low and intermediate mass stars. During this phase, the outer chemical

## References

SHOWING 1-10 OF 24 REFERENCES

• Physics
• 1999
We consider the pulsational properties of white dwarf star models with temperatures appropriate for the ZZ Ceti instability strip and with masses large enough that they should be substantially
Because white dwarfs are the most common end state of stellar evolution, determining their internal structure will yield many clues about the final stages of stellar evolution and the physics of
• Physics, Geology
• 1997
The white dwarfs are promising laboratories for the study of cosmochronology and stellar evolution. Through observations of the pulsating white dwarfs, we can measure their internal structures and
• Physics
• 2000
A detailed photometric and spectroscopic analysis of cool (Teff ≲ 12,000 K) white dwarf stars is presented. The sample has been drawn from the Yale Parallax Catalog and from a proper motion survey in
The white dwarf disk luminosity function is explored using observational results of Liebert et al. (1988, 1989) as a template for comparison, and the cooling curves of Wood (1990, 1991) as the input
• Physics, Geology
• 1999
We present new determinations of effective temperature, surface gravity, and masses for a sample of 46 hot DA white dwarfs selected from the Extreme Ultraviolet Explorer (EUVE) and ROSAT Wide Field
• Physics
• 2003
▪ Abstract Old, cool white dwarfs convey valuable information about the early history of our Galaxy. They have been used to determine the age of the galactic disk, several open clusters, and a
• Physics
• 1975
We describe the first results of a quantitative exploration of white dwarf evolution for models incorporating an accurate dense plasma equation of state and a full treatment of partial ionization and
• Physics, Geology
• 2003
We investigate the global pulsation properties of DBV white dwarf models that include the double-layered envelope structure expected from time-dependent diffusion calculations, as well as a
• Physics
• 2004
We report the discovery of two new ZZ Ceti pulsators, LP 133-144 and HE 1258+0123, selected on the basis of model atmosphere fits to optical spectroscopic data. The atmospheric parameters for LP


text:
# Find the Value of K for Which the System Kx + 2y = 5 3x + Y = 1 Has (I) a Unique Solution, and (Ii) No Solution. - Mathematics

Find the value of k for which the system
kx + 2y = 5
3x + y = 1
has (i) a unique solution, and (ii) no solution.

#### Solution

The given system of equation may be written as

kx + 2y - 5 = 0

3x + y - 1 = 0

It is of the form

a_1x + b_1y + c_1 = 0

a_2x + b_2y + c_2 = 0

Where a_1 = k, b_1 = 2, c_1 = -5

And a_2 = 3, b_2 = 1, c_2 = -1

Where, a_1 = k, b_1 = 2, c_1 = -5

And a_2 = 3, b_2 = 1, c_2 = -1

1) The given system will have a unique solution, if

a_1/a_2 != b_1/b_2

=> k/3 != 2/1

=> k != 6

So, the given system of equations will have a unique solution, if k != 6

2) The given system will have no solution, if

a_1/a_2 - b_1/b_2 != c_1/c_2

we have

b_1/b_2 = 2/1 and c_1/c_2 = (-5)/(-1) = 5/1

Clearly b_1/b_2 != c_1/c_2

So, the given system of equations will have no solution, if

a_1/a_2 = b_1/b_2

=> k/3 = 2/1

=> k = 6

Hence, the given system of equations will have no solution if k = 6

Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 10 Maths
Chapter 3 Pair of Linear Equations in Two Variables
Exercise 3.5 | Q 5 | Page 73


text:
# Define 和 Identify 有什麼分別？

### 3 個解答

• SC147
Lv 6
2 月前
最愛解答

Define --下定義, 界定

Identify -- 認出, 識別, 確定

eg.

a) We DEFINE “acidic” as a solution having a pH value of less than 7

b) We IDENTIFY whether a solution is "acidic" by using a blue litmus paper, which will turn red if the solution is acidic.

eg.

a) We DEFINE “a square” as a quadrilateral with 4 equal sides and 4 right angles.

b) We IDENTIFY a square by :

i) measuring its sides and angles, and so finding that it's got 4 equal sides and 4 right angles;

ii) proving that a given quadrilateral has got 4 equal sides & 4 right angles.

✰✰ 從例子可見, DEFINE 只是把某事物給出一個「定義」, 而 IDENTIFY 就是用實際行動/証明 去找出這事物是否符合給出的定義

• 2 月前

define是定義，identify是辯認，兩回事

• 匿名
2 月前

Define

動詞  解釋，給……下定義

Identify

動詞  認出，識別


text:
# Definition:Set/Definition by Predicate

## Definition

An object can be specified by means of a predicate, that is, in terms of a property (or properties) that it possesses.

Whether an object $x$ possesses a particular property $P$ is either true or false (in Aristotelian logic) and so can be the subject of a propositional function $\map P x$.

Hence a set can be specified by means of such a propositional function:

$S = \set {x: \map P x}$

which means:

$S$ is the set of all objects which have the property $P$

or, more formally:

$S$ is the set of all $x$ such that $\map P x$ is true.

In this context, we see that the symbol $:$ is interpreted as such that.

## Also known as

This construction is sometimes known as the set-builder notation or as set comprehension.

This is also sometimes rendered as set builder notation.

An alternative notation for this is $S = \set {x \mid \map P x}$, but it can be argued that the use of $\mid$ for such that can cause ambiguity and confusion, as $\mid$ has several other meanings in mathematics.

On the other hand, if the expression defining the predicate is thick with $:$ characters, it may improve clarity to use $\mid$ for such that after all.

Some authors, mindful of such confusion, use the notation $S = \set {x; \map P x}$ as the semicolon is relatively rare in mathematical notation.

Sometimes it is convenient to abbreviate the notation by simply writing $S = \set {\map P x}$ or even just $S = \set P$.

For example, to describe the set $\set {x \in \R: \map f x \le \map g x}$ (for appropriate functions $f, g$), one could simply use $\set {f \le g}$.

Some sources simply identify $x$ as a variable, and then refer to $A = \set x$ as the set of all the values that $x$ can take.

Some sources use the notation $\boldsymbol [x: \map P x \boldsymbol ]$ for $\set {\map P x}$.

## Axiomatic Set Theory

In the context of axiomatic set theory, a more strictly rigorous presentation of this concept is:

$S = \set {x \in A: \map P x}$

which means:

$S$ is the set of all objects in $A$ which have the property $P$

or, more formally:

$S$ is the set of all $x$ in $A$ such that $\map P x$ is true.

This presupposes that all the objects under consideration for inclusion in $S$ already belong to some previously-defined set $A$.

Thus any set $S$ can be expressed as:

$S = \set {s: s \in S}$

See the Axiom of Specification.

## Examples

### University Professors

An example in natural language of a set definition by predicate is:

$S := \text {the set of all university professors}$

### Musical Mathematicians

Let $M$ denote the set of all the mathematicians in the world.

Let $I$ denote the set of all people who can play a musical instrument.

Let $S$ denote the set of all mathematicians who can play a musical instrument.

Then we can define $S$ as:

$S := \set {x: x \in M \text { and } x \in I}$

or as:

$S := \set {x \in M: x \in I}$

### Set of Integers $x$ such that $2 \le x$

Let $S$ be the set defined as:

$S := \set {x \in \Z: 2 \le x}$

Then $S$ is the set of all integers greater than or equal to $2$:

$S = \set {2, 3, 4, \ldots}$

### Set of Integers $x$ such that $x \le 5$

Let $S$ be the set defined as:

$S := \set {x \in \Z: x \le 5}$

Then $S$ is the set of all integers less than or equal to $5$:

$S = \set {\ldots, 2, 3, 4, 5}$

### Set Indexed by Natural Numbers between $1$ and $100$

Let $V$ be the set defined as:

$V := \set {v_i: 1 \le i \le 100, i \in \N}$

Then $V$ is the set of the $100$ elements:

$V = \set {v_1, v_2, \ldots, v_{100} }$

and can also be written:

$V := \set {v_i: i = 1, 2, \ldots, 100}$

or even:

$V := \set {v_i: 1 \le i \le 100}$

as it is understood that the domain of $i$ is the set of natural numbers.

### Set Indexed by Natural Numbers between $1$ and $10$

Let $U$ be the set defined as:

$U := \set {u_i: 1 < i < 10, i \in \N}$

Then $U$ has exactly $8$ elements:

$U = \set {u_2, u_3, u_4, u_5, u_6, u_7, u_8, u_9}$


text:
The Korean Physical Society 06130 22, Teheran-ro 7-gil, Gangnam-gu, Seoul, Republic of Korea 610 Representation : Tae Won NOH TEL: 02-556-4737 FAX: 02-554-1643 E-mail : webmaster@kps.or.kr Copyright(C) KPS, All rights reserved.

# The XXX International Conference on Neutrino Physics and Astrophysics (Neutrino 2022)

30 May 2022 to 4 June 2022
Virtual Seoul
Asia/Seoul timezone

## GUNDAM: A next-generation fitting tool for future T2K analyses

Not scheduled
5m
Virtual Seoul

#### Virtual Seoul

Poster Accelerator neutrinos

### Description

The phase II of T2K will permit a measurement of the CP violating phase in the leptonic sector ($\delta_{CP}$) with unprecedented level of accuracy. This will be achieved by the increase of statistics made possible by the upgrades of the neutrino beam production at the JPARC facility. In order to get the best precision on $\delta_{CP}$, a better understanding on the involved systematic uncertainties is essential. The T2K experiment takes advantage of a detector placed near the production site (called ND280) which measure the flux and interaction properties of neutrinos before they oscillate.

In practice, the ND280 data is analysed prior to the oscillation analysis to provide correction on the predictions (i.e. best fit values) as well as additional constraints on each error. This task requires a substantial effort, since it involves a fit of several neutrino samples with hundreds of fit parameters, and will soon become substantially more complicated as upgrades allow the collection of more data that can be analysed in increasingly multi-faceted ways. To meet the challenges of analyses using this upgraded hardware it is necessary to create similarly upgraded analysis tools. In this poster such a tool is presented: GUNDAM (Generic fitter for Upgraded Near Detector Analysis Methods) is a next-generation fitting tool capable of analysing large amounts of of near-detector data in the context of both oscillation and cross-section measurements.

In this poster we will showcase the main features of GUNDAM, demonstrating its flexible and fast set of libraries which can be setup with a set of user-friendly yaml configuration files, before showing various examples of its applications to T2K analyses. These include the potential to perform complete near detector fits in a fraction of the time they have previously taken, and to provide diagnostic tools to identify pathologies in parameter configurations that can impact MINUIT-based fitters.


text:
# Calculus Examples

Find the Derivative Using Chain Rule - d/dx
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Power Rule which states that is where .
Replace all occurrences of with .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
The derivative of with respect to is .
Replace all occurrences of with .
Differentiate.
Since is constant with respect to , the derivative of with respect to is .
Move to the left of .
Differentiate using the Power Rule which states that is where .
Multiply by .
Simplify.
Multiply by .
Reorder the factors of .


text:
In the late 1980s I was contracted to write some software for a company that produced video-based educational systems. They had video cassette machines that had been modified to interface with a PC, which could send instructions to the VCRs such as play, stop, fast forward, and rewind. The educational programs consisted of short recorded segments which typically ended with a question, and based on the answer the user provided they wanted to continue play at different portions of the tape. The software they wanted me to write was meant to be able to be given a current position (i.e. play time) and advance to a new position. For example I might be told that they are at minute 8 currently and want to resume play at minute 12.

Without going into all the detail, I can tell you this was a Very Hard Problem. There was no way to tell where you were on the tape without actually playing, and if you are familiar with VCRs (which are rapidly going the way of the dinosaur) you may be aware that engaging the play head takes a few seconds. There were variations in motor speed, tape stretching, and all sorts of other variables to take into account. The approach I used was to try to calculate the “optimal” fast forward time, shorten that a bit, and then narrow down to the exact position with a modified binary search (or, to put it more crudely, “trial and error”).

We got the system to work although it was a pretty crude affair but before anything shipped new VCRs came out that recorded “time code” on the tape and had much more precise electronics essentially supporting just this functionality we had duct-taped together, and so it was all for naught (and not long after CD-ROMs came and made it all redundant anyway).

I don’t remember all the math that I worked out for this system, but it is easy enough to recall the basic parameters. There was the diameter of the empty plastic spools, the diameter of a full spool, and the playing time of the tape (typically 120 minutes), which was a linear function of the length (assuming an unstretched tape :-))

While playback happened at a constant speed, as the tape ran through an independent roller mechanism unaffected by the spool state, fast-forwarding and rewinding happened at a variable rate, as it was controlled by the motors driving the spool, and the spool diameters changed.

All in all, a nasty business, and we are better off without tape! But there is still a common household item with some similar characteristics, namely the toilet roll. So, here are some problems surrounding toilet rolls to think about. To save you having to measure your toilet rolls here are some typical measurements from mine (Charmin 2 ply from Costco) - spool diameter 45mm, roll diameter 120mm, 250 x 101mm sheets = 25250mm length.

• how thick is each sheet of paper? (and thus, how many times does it go around the spool?)
• what would the diameter of a roll be that went all the way around the earth (assuming the earth’s diameter is 12,756.32km)?
• what if it went around the earth but 1m higher than in the example before? How long would the roll be? (you may recognize this as a familiar puzzle involving rope but I thought I’d sneak it in here!)

(BTW no less a figure than Donald Knuth has considered toilet paper worthy of investigation, as evidenced here).

If we call the paper thickness $t$, the radius of the spool $r_s$ and the radius of the roll (spool plus paper) $r_r$, then it should be obvious that the number of winds $n$ is:

$$n = \frac{r_r - r_s}{t}$$

If we approximate the roll as a number of concentric circles rather than a spiral, then the innermost roll has length $2\pi r_s$, and the next roll has length $2\pi (r_s+t)$, and so on; each successive roll is $2\pi t$ longer than the prior one, and the final roll has length $2\pi r_r$.

Because each successive roll is longer by a constant amount, we have an arithmetic sequence. Recall that the sum of an arithmetic sequence is given by $\frac{n}{2}(a_1 + a_n)$, where $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the last term. So the length of the entire roll is given by:

$$l = \frac{n}{2} ( 2\pi r_s + 2 \pi r_r) = \pi \frac{r_r^2 - r_s^2}{t}$$

Thus my toilet paper is about 0.38 mm thick - quite luxurious!

For the second problem it is reasonable to assume $r_s = 0$, to give us the simpler formula:

$$l \approx \pi \frac{r_r^2}{t}$$

or:

$$r_r \approx \sqrt{\frac{l t}{\pi}}$$

or:

$$r_r \approx \sqrt{d t}$$

where $d$ is the earth’s diameter. So for my paper this would be about 70 meters. If I had a cheaper brand of one ply paper this might be considerably less.

I’ll leave the last question as an exercise but it should be fairly clear that the difference in $l$ in this case is quite small and so the extra radius needed for the toilet roll is very small.