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Lambda calculus
7.8k words, including equations (about 30 minutes)
This post has also been published here.
This post is about lambda calculus. The goal is not to do maths with it, but rather to build up definitions within it until we can express non-trivial algorithms easily. At the end we will see a
lambda calculus interpreter written in the lambda calculus, and realise that we're most of the way to Lisp.
But first, why care about lambda calculus? Consider four different systems:
• A Turing machine – that is, a machine that:
□ works on an infinite tape of cells from which a finite set of symbols can be read and written, and always points at one of these cells;
□ has some set of states it can be in, some of which are termed "accepting" and one of which is the starting state; and
□ given a combination of current state and current symbol on the tape, always does an action consisting of three things:
☆ writes some symbol on the tape (possibly the same that was already there),
☆ transitions to some some state (possibly the same it is already in), and
☆ moves one cell left or right on the tape.
• The lambda calculus ($\lambda$-calculus), a formal system that has expressions that are built out of an infinite set of variable names using $\lambda$-terms (which can be thought of as anonymous
functions) and applications (analogous to function application), and a few simple rules for shuffling around the symbols in these expressions.
• The partial recursive functions, constructed by function composition, primitive recursion (think bounded for-loops), and minimisation (returning the first value for which a function is zero) on
three basic sets of functions:
□ the zero functions, that take some number of arguments and return 0;
□ a successor function that takes a number and returns that number plus 1; and
□ the projection functions, defined for all natural numbers $a$ and $b$ such that $a \geq b$ as taking in $a$ arguments and returning the $b$th one.
• Lisp, a human-friendly axiomatisation of computation that accidentally became an extremely good and long-lived programming language.
The big result in theoretical computer science is that these can all do the same thing, in the sense that if you can express a calculation in one, you can express it in any other.
This is not an obvious thing. For example, the only thing lambda calculus lets you do is create terms consisting of symbols, single-argument anonymous functions, and applications of terms to each
other (we'll look at the specifics soon). It's an extremely simple and basic thing. Yet no matter how hard you try, you can't make something that can compute more things, whether it's by inventing
programming languages or building fancy computers.
Also, if you try to make something that does some sort of calculation (like a new programming language), then unless you keep it stupidly simple and/or take great care, it will be able to compute
anything (at least in la-la-theory-land, where memory is infinite and you don't have to worry about practical details, like whether the computation finishes before the sun going nova).
Physicists search for their theory of everything. The computer scientists already have many, even though they've been at it for a lot less time than the physicists have: everything computable can be
reduced to one of the many formalisms of computation. (One of the main reasons that we can talk about "computability" as a sensible universal concept is that any reasonable model makes the same
things computable; the threshold is easy to hit and impossible to exceed, so computable versus not is an obvious thing to pay attention to.)
To talk about the theory of computation properly, we need to look at at least one of those models. The most well-known is the Turing machine. Turing machines have several points in their favour:
• They are the easiest to imagine as a physical machine.
• They have clear and separate notions of time (steps taken in execution) and space (length of tape used).
• They were invented by Alan Turing, who contributed to breaking the Enigma code during World War II, before being unjustly persecuted for being gay and tragically dying of cyanide poisoning at age
In contrast, compare the lambda calculus:
• It is an abstract formal system arising out of a failed attempt to axiomatise logic.
• There are many execution paths for a non-trivial expression.
• It was invented by Alonzo Church, who lived a boringly successful life as a maths professor at Princeton, had three children, and died at age 92.
(Turing and Church worked together from 1936 to 1938, Church as Turing's doctoral advisor, after they independently proved the impossibility of the halting problem. At the same time and also working
at Princeton were Albert Einstein, Kurt Gödel, and John von Neumann (who, if he had had his way, would've hired Turing and kept him from returning to the UK).)
However, the lambda calculus also has advantages. Its less mechanistic and more mathematical view of computation is arguably more elegant, and it has less things: instead of states, symbols, and a
tape, the current state is just a term, and the term also represents the algorithm. It abstracts more nicely – we will see how we can, bit by bit, abstract out elements and get something that is a
sensible programming language, a project that would be messier and longer with Turing machines.
Turing machines and lambda calculus are the foundations of imperative and functional programming respectively, and the situation between these two programming paradigms mirrors that between TMs and $
\lambda$-calculus: one is more mechanistic, more popular, and more useful when dealing with (stateful) hardware; the other more mathematical, less popular, and neater for abstraction-building.
Lambda trees
Now let's define exactly what a lambda calculus term is.
We have an infinite set of variables $x_1, x_2, x_3, ...$, though for simplicity we will use any lowercase letter to refer to them. Any variable is a valid term. Note that variables are just symbols
– despite the word "variable", there is no value bound to them.
We have two rules for building new terms:
• $\lambda$-terms are formed from a variable $x$ and a term $M$, and are written $(\lambda x. M)$.
• Applications are formed from two terms $M$ and $N$, and are written $(M N)$.
These terms, like most things, are trees. I will mostly ignore the convention of writing out horrible long strings of $\lambda$s and variables, only partly mitigated by parenthesis-reducing rules,
and instead draw the trees.
(When it appears in this post, the standard notation appears slightly more horrible than usual because, for simplicity, I neglect the parenthesis-reducing rules (they can be confusing at first).)
Here are a few examples of terms, together with standard representations:
This representation makes it clear that we're dealing with a tree where nodes are either variables, lambda terms where the left child is the argument and the right child is the body, or applications.
(I've circled the variables to make clear that the argument variable in a $\lambda$-term has a different role than a variable appearing elsewhere.)
It's not quite right to say that a $\lambda$-term is a function; instead, think of $\lambda$-terms as one representation of a (mathematical) function, when combined with the reduction rule we will
look at soon.
If we interpret the above terms as representations of functions, we might rewrite them (in Pythonic pseudocode) as, from left to right:
• lambda x -> x (i.e., the identity function) (lambda is a common keyword for an anonymous function in programming languages, for obvious reasons).
• (lambda f -> f(y))(lambda x -> x) (apply a function that takes a function and calls that function on y to the identity function as an argument).
• x(y)
Execution in lambda calculus is driven by something that is called $\beta$-reduction, presumably because Greek letters are cool. The basic idea of $\beta$-reduction is this:
• Pick an application (which I've represented by orange circles in the tree diagrams).
• Check that the left child of the application node is a \lambda-term (if not, you have to reduce it to a $\lambda$-term before you can make that application).
• Replace the variable in the left child of the $\lambda$-term with the right child of the application node wherever it appears in the right child of the $\lambda$-term, and then replace the
application node with the right child of the $\lambda$-term.
In illustrated form, on the middle example above, using both tree diagrams and the usual notation:
(The notation
means substitute the term
for the variable
in the term
; the general rule for
-reduction is that given
$((\lambda x. M) N)$
, you can replace it with
, subject to some details that we will mostly skip over shortly.)
In our example, we end up with another application term, so we can reduce it further:
In our Pythonic pseudocode, we might represent this as an execution trace like the following:
(lambda f -> f(y))(lambda x -> x)
(lambda x -> x)(y)
Reduction is not always so simple, even if there's only a single choice of what to reduce. You have to be careful if the same variable appears in different roles, and rename if necessary. The core
rule is that within the tree rooted at a $\lambda$-term that takes an argument $x$, the variable $x$ always means whatever was given to that $\lambda$-term, and never anything else. An $x$ bound in
one $\lambda$-term is distinct from an $x$ bound in another $\lambda$-term.
The simplest way to get around problems is to make your first variable $x_1$ and, whenever you need a new one, call it $x_i$ where $i$ is one more than the maximum index of any existing variable.
Unfortunately humans aren't good at remembering the difference between $x_9$ and $x_{17}$, and humans like conventions (like using $x$ for generic variables, $f$ for things that will be $\lambda$
-terms, and so forth). Therefore we sometimes have to think about name collisions.
The principle that lets us out of name collision problems is that you can rename variables as you want (as long as distinct variables aren't renamed to the same thing). The name for this is $\alpha$
-equivalence (more Greek letters!); for example $(\lambda x .x)$ and $(\lambda y. y)$ are $\alpha$-equivalent.
There are, of course, detailed rules for how to deal with name collisions when doing $\beta$-reductions, but you should be fine if you think about how variable scoping should sensibly work to
preserve meaning (something you've already had to reason about if you've ever programmed). (A helpful concept to keep in mind is the difference between free variables and bound variables – starting
from a variable and following the path up the tree to the parent node, does it run through a $\lambda$-node with that variable as an argument?)
An example of a name collision problem is this:
We can't do this because the $x$ in the innermost $\lambda$-term on the left must mean whatever was passed to it, and the $y$ whatever was passed to the outer $\lambda$-term. However, our reduction
leaves us with an expression that applies its argument to itself. We can solve this by renaming the $x$ within the inner $\lambda$-term:
The general way to think of lambda calculus term is that they are partitioned in two ways into equivalence classes:
• The first, rather trivial, set of equivalence classes is treating all $\alpha$-equivalent terms as the same thing. "Equivalent" and $\alpha$-equivalent are usually the same thing when we're
talking about the lambda calculus; it's the "structure" of a term that matters, not the variable names.
• The second set of equivalence classes is treating everything that can be $\beta$-reduced into the same form as equivalent. This is less trivial – in fact, it's undecidable in the general case (as
we will see in the post about computation theory).
That's it
Yes, really, that's all you need. There exists a lambda calculus term that beats you in chess.
You might ask: but hold on a moment, we have no data – no numbers, no pairs, no lists, no strings – how can we input chess positions into a term or get anything sensible as an answer? We will see
later that it's possible to encode data as lambda terms. The chess-playing term would accept some massive mess of $\lambda$-terms encoding the board configuration as an input, and after a lot of
reductions it would become a term encoding the move to make – eventually checkmate, against you.
Before we start abstracting out data and more complex functions, let's make some simple syntax changes and look at some basic facts about reduction.
Some syntax simplifications
The pure lambda calculus does not have $\lambda$-terms that take more than one argument. This is often inconvenient. However, there's a simple mapping between multi-argument $\lambda$-terms and
single-argument ones: instead of a two-argument function, say, just have a function that takes in an argument and returns a one argument function that takes in an argument and returns a result using
both arguments.
(In programming language terms, this is currying.)
In the standard notation, $(\lambda x.(\lambda y. M))$ is often written $(\lambda xy.M)$. Likewise, we can do similar simplifications on our trees, remembering that this is a syntactic/visual
difference, rather than introducing something new to the lambda calculus:
Once we've done this change, the next natural simplification to make is to allow one application node to apply many arguments to a $\lambda$-term with "many arguments" (remember that it actually
stands for a bunch of nested normal single-argument $\lambda$-terms):
(The corresponding simplification in the standard syntax is that $(M \, A \, B\, C)$ means $(((M \, A)\, B)\, C)$. In a standard programming language, this might be written M(A)(B)(C); that is,
applying A to M to get a function that you apply to B, yielding another function that you apply to C. Sanity check: what's the difference between $((M \, A) \, B)$ and $(M \, (A \, B))$?)
Some facts about reduction
$\beta$-normal forms
A $\beta$-normal form can be thought of as a "fully evaluated" term. More specifically, it is one where this configuration of nodes does not appear in the tree (after multi-argument $\lambda$s and
applications have been compiled into single-argument ones), where $M$ and $N$ are arbitrary terms:
Intuitively, if such a term does appear, then the reduction rules allow us to reduce the application (replacing this part of the tree with whatever you get when you substitute $N$ in place of $x$
within $M$), so our term is not fully reduced yet.
Terms without a $\beta$-normal form
Does every term have a $\beta$-normal form? If you've seen computation theory stuff before, you should be able to answer this immediately without considering anything about the lambda calculus
The answer is no, because reducing to a $\beta$-normal form is the lambda calculus equivalent of an algorithm halting. Lambda calculus has the same expressive power as Turing machines or any other
model of computation, and some algorithms run forever, so there must exist lambda calculus terms that you can keep reducing without ever getting a $\beta$-normal form.
Here's one example, often called $\Omega$:
Note that even though we use the same variable $x$ in both branches, the variable means a different thing: in the left branch it's whatever is passed as an input to the left $\lambda$-term – one
reduction step onwards, that $x$ stands for the entire right branch, which has its own $x$. In fact, before we start reducing, we will do an $\alpha$-conversion on the right branch (a pretentious way
of saying that we will rename the bound variable).
Now watch:
After one reduction step, we end up with the same term (as usual, we are treating $\alpha$-equivalent terms as equivalent; the variable could be $x$ or $y$ or $å$ for all we care).
Ambiguities with reduction
Does it matter how we reduce, or does every reduction path eventually lead to a $\beta$-normal form, assuming that one exists in the first place? If you haven't seen this before, you might want to
have a go at this before reading on.
Here's one example of a tricky term:
Imagine that
has a
-normal form, and
is as defined above and therefore can be reduced forever. If we start by reducing the application node, in a moment
and all its loopiness gets thrown away, and we're left with just
, since the
-term takes two arguments and returns the first. However, if we start by reducing
, or are following a strategy like "evaluate the arguments before the application", we will at some point reduce
and get thrown in for a loop.
We can take a broader view here. In any programming language – I will use Lisp notation because it's the closest to lambda calculus – if we have a function like (define func (lambda (x y) [FUNCTION
BODY])), and a function call like (func arg1 arg2) , the evaluator has a choice of what it does. The simplest strategies are to either:
• Evaluate the arguments – arg1 and arg2– first, and then inside the function func have x and y bound to the results of evaluating arg1 and arg2 respectively. This is called call-by-value, and is
used by most programming languages.
• Bind x and y inside func to be the unevaluated values of arg1 and arg2, and evaluate arg1 and arg2 only upon encountering them in the process of evaluating func. This is called call-by-name. It's
rare to see it in programming languages (an exception being that it's possible with Lisp macros), but functional languages like Haskell often have a variant, call-by-need or "lazy evaluation",
where the values of arg1 and arg2 are only executed when needed, but once executed the results are memoized so that the execution only needs to happen once.
Call-by-value reduces what you can express. Imagine trying to define your own if-function in a language with call-by-value:
(define IF
(lambda (predicate consequent alternative)
(if predicate
consequent ; if predicate is true, do this
alternative)) ; if predicate is false, do this instead
(note that IF is the new if-function that we're trying to define, and if is assumed to be a language primitive.)
Now consider:
(define factorial
(lambda (n)
(IF (= n 0)
(* n
(factorial (- n 1))))))
You call (factorial 1), and for the first call the program evaluates the arguments to IF:
• (= 1 0)
• 1
• (* 1 (factorial 0))
The last one needs the value of (factorial 0), so we evaluate the arguments to the IF in the recursive call:
• (= 0 0)
• 1
• (* 1 (factorial -1))
... and so on. We can't define IF as a function, because in call-by-value the alternative gets evaluated as part of the function call even if predicate is false.
(Most languages solve this by giving you a bunch of primitives and making you stick with them, perhaps with some fiddly mini-language for macros built in (consider C/C++). In Lisp, you can easily
write macros that use all of the language features, and therefore extend the language by essentially defining your own primitives that can escape call-by-value or any other potentially limiting
language feature.)
It's the same issue with our term $((\lambda xy.x) \, M \, \Omega)$ above: call-by-value goes into a silly loop because one of the arguments isn't even "meant to" be evaluated (from our perspective
as humans with goals looking at the formal system from the outside).
Lambda calculus does not impose a reduction/"evaluation" order, so we can do what we like. However, this still leaves us with a problem: how do we know if our algorithm has gone into an infinite
loop, or we just reduced terms in the wrong order?
Normal order reduction
It turns out that always doing the equivalent of call-by-name – reducing the leftmost, outermost term first – saves the day. If a $\beta$-normal form exists, this strategy will lead you to it.
Intuitively, this is because with call-by-name, there is no "unnecessary" reduction. If some arguments in some call are never used (like in our example), they never reduce. If we start reducing an
expression while doing leftmost/outermost-first reduction, that reduction must be standing in the way between us and a successful reduction to $\beta$-normal form.
Formally: ... the proof is left as an exercise for the reader.
Church-Rosser theorem
The Church-Rosser theorem is the thing that guarantees we can talk about unique $\beta$-normal forms for a term. It says that:
Letting $\Lambda$ be the set of terms in the lambda calculus, $\rightarrow_\beta$ the $\beta$-reduction relation, and $\twoheadrightarrow_\beta$ its reflexive transitive closure (i.e. $M \
twoheadrightarrow_\beta N$ iff there exist zero or more terms $P_1$, $P_2$, ... such that $M \rightarrow_\beta P_1 \rightarrow_\beta ... \rightarrow_\beta P_n \rightarrow_\beta N$), then:
For all $M \in \Lambda$, $M \rightarrow_\beta A$ and $M \rightarrow_\beta B$ implies that there exists $X \in \Lambda$ such that $A \twoheadrightarrow_\beta X$ and $B \twoheadrightarrow_\beta X$.
Visually, if we have reduction chains like the black part, then the blue part must exist (a property known as confluence or the "diamond property"):
Therefore, even if there are many reduction paths, and even if some of them are non-terminating, for any two different starting $\beta$-reductions we can make, we will not lose the existence of a
reduction path to any $X$. If $X$ is some $\beta$-normal form reachable from $M$, we know that any other reduction path that reaches a $\beta$-normal form must have reached $X$.
The fun begins
Now we will start making definitions within the lambda calculus. These definitions do not add any capabilities to the lambda calculus, but are simply conveniences to save out having to draw huge
trees repeatedly when we get to doing more complex things.
There are two big ideas to keep in mind:
1. There are no data primitives in the lambda calculus (even the variables are just placeholders for terms to get substituted into, and don't even have consistent names – remember that we work
within $\alpha$-equivalence). As a result, the general idea is that you encode "data" as actions: the number 4 is represented by a function that takes a function and an input and applies the
function to the input 4 times, a list might be encoded by a description of how to iterate over it, and so on.
2. There are no types. Nothing in the lambda calculus will stop you from passing a number to a function that expects a function, or visa versa. There exist typed lambda calculi, but they prevent you
from doing some of the cool things with combinators that we'll see later in this post.
We want to be able to associate two things into a pair, and then extract the first and second elements. In other words, we want things that work like this:
(fst (pair a b)) == a
(snd (pair a b)) == b
The simplest solution starts like this:
Now we can get the first of a pair by doing ((pair x y) first). If we want the exact semantics above, we can define simple helpers like
fst = (lambda p
(p first))
(i.e. $\text{fst} = (\lambda p. (p \, \text{first}))$), and
snd = (lambda p
(p second))
since now (snd (pair x y)) reduces to ((pair x y) second) reduces to y.
A list can be constructed from pairs: [1, 2, 3] will be represented by (pair 1 (pair 2 (pair 3 False))) (we will define False later). If $l_1$, $l_2$, and $l_3$ are the list items, a length element
list looks like this:
We might also represent the same list like this instead:
This second representation makes it trivial to define things like a reduce function: ([1, 2, 3] 0 +) would return 0 plus the sum of the list [1, 2, 3], if [1, 2, 3] is represented as above. However,
this representation would also make it harder to do other list operations, like getting all but the first element of a list, whereas our pair-based lists can do this trivially ((snd l) gets you all
but the first element of the list l).
Numbers & arithmetic
Here are how the numbers work (using a system called Church numerals):
Since giving a function $f$ to a number $n$ (also a function) gives a function that applies $f$ to its input $n$ times, a lot of things are very convenient. Say you have this function to add one,
which we'll call succ (for "successor"):
(Considering the above definition of numbers: why does it work?)
Now what is (42 succ)? It's a function that takes an argument and adds 42 to it. More generally, ((n succ) m) gives you m+n. However, there's also a more straightforward way to represent addition,
which you can figure out from noticing that all we have to do to add m to n is to compose the "apply f" operation m more times to n, something we can do simply by calling (m f) on n, once we've
"standardised" n to have the same f and x as in the $\lambda$-term that represents m (that is why we have the (n f x) application, rather than just n):
Now, want multiplication? One way is to see that we can define (mult m n) as ((n (adder m)) 0), assuming that (adder m) returns a function that adds m to its input. As we saw, that can be done with
(m succ), so:
(mult m n) =
((n (m succ))
There's a more standard way too:
The idea here is simply that (n f) gives a $\lambda$-term that takes an input and applies f to it $n$ times, and when we call m with that as its first argument, we get something that does the $n$
-fold application $m$ times, for a total of $mn$ times, and now all that remains is to pass the x to it.
A particularly neat thing is that exponentiation can be this simple:
Why? I'll let the trees talk. First, using the definition of n as a Church numeral (which I will underline in the trees below), and doing one $\beta$-reduction, we have:
This does not look promising – a number needs to have two arguments, but we have a $\lambda$-term taking in one. However, we'll soon see that the x in the tree on the right actually turns out to be
the first argument, f, in the finished number. In fact, we'll make that renaming right away (since we're working under $\alpha$-equivalence), and continue reducing (below we've taken the bottom-most
m and expanded it into its Church numeral definition):
At this point, the picture gets clearer: the next thing we'd reduce is the lambda term at the bottom applied to m, but that's just going to do the lambda term (which applies f $m$ times) $m$ more
times. We'll have done 2 steps, and gotten up to $m^2$ nestings of f. By the time we've done the remaining $n-1$ steps, we'll have the representation of $m^n$; the $n-1$ more applications between
our bottom-most and topmost lambda term will reduce away, while the stack of applications of f increases by a factor of $m$ each time.
What about subtraction? It's a bit complicated. Okay, how about just subtraction by one, also known as the pred (predecessor) function? Also tricky (and a good puzzle if you want to think about it).
Here's one way:
Church numerals make it easy to add, but not subtract. So instead, here's what we do. First (box 1), we make a pair like [0 0]. Next (polygon 2), we have a function that takes a pair p=[a b] and
creates a new pair [b (succ b)], where succ is the successor function (one plus its input). Repeated application of this function on the pair in box 1 looks like this: [0 0], [0 1], [1 2], [2 3], and
so on. Thus we see that if we start from [0 0] and apply the function in polygon 2 $n$ times (box 3), the first element of the pair is (the Church numeral for) $n-1$, and the second element is $n$,
and we can simply call fst to get that first element.
As we saw before, we can define subtraction as repeated application of pred:
(minus m n) =
((n pred) m)
There's an alternative to Church numerals that's found in the more general Scott encoding. The advantages of Church vs Scott numerals, and their relative structures, are similar to the relative
merits and structures of the two types of lists we discussed: one makes many operations natural by exploiting the fact that everything is a function, but also makes "throwing off a piece" (taking the
rest/snd of a list, or subtracting one from a number) much harder.
Booleans, if, & equality
You might have noticed that we've defined second as $(\lambda x y. y)$, and 0 as $(\lambda f x. x)$. These two terms are a variable-renaming away from each other, so they are $\alpha$-equivalent. In
other words, second and 0 are same thing. Because we don't have types, which is which depends only on our interpretation of the context it appears in.
Now let's define a True and False. Now False is kind of like 0, so let's just say they're also the same thing. The opposite of $(\lambda x y. y)$ is $(\lambda x y. x)$, so let's define that to be
What sort of muddle have we landed ourselves in now? Quite a good one, actually. Let's define (if p c a) to be (p c a). If the predicate p is True, we select the consequent c, because (True c a) is
exactly the same as (first c a) is clearly c. Likewise, if p is False, then we evaluate the same thing as (second c a) and end up with the alternative a.
We will also want to test whether a number is 0/False (equality in general is hard in the lambda calculus, so what we end up with won't be guaranteed to work with things that aren't numbers). A
simple way is:
eq0 =
(lambda x
(x (lambda y
If x is 0, it's the same as second and will act as a conditional and pick out True. If it's not zero, we assume that it's some number $n$, and therefore will be a function that applies its first
argument $n$ times. Applying $(\lambda y.\text{False})$ any non-zero amount of times to anything will return False.
Fixed points, combinators, and recursion
The big thing missing from the definitions we've put on top of the lambda calculus so far is recursion. Every lambda term represents an anonymous function, so there's no name within a $\lambda$-term
that we can "call" to recurse.
Rather than jumping in straight to recursion, we're going to start with Russell's paradox: does a set that contains all elements that are not in the set contain itself? Phrased mathematically: what
the hell is $R = \{x \,|\,xotin R\}$?
In computation theory, sets are often specified by a characteristic function: a function that is always defined if the set is computable, and returns true if an element is in the set and false
In the lambda calculus (which was originally supposed to be a foundation for logic), here's a characteristic function for the Russell set $R$:
(where not can be straightforwardly defined on top of our existing definitions as (not b) = (b False True)).
This $\lambda$-term takes in an element x, assumes that x is the (characteristic function for) the set itself, and asks: is it the case that x is not in the set? Call this term R, and consider (R R):
the left R is working as the (characteristic function of) the set, and the right R as the element whose membership of the set we are testing.
So we start out saying (R R), and in one $\beta$-reduction step we end up saying (not (R R)) (just as, with Russell's paradox, it first seems that the set must contain itself, because the set is not
in itself, but once we've added the set to itself then suddenly it shouldn't be in itself anymore). One more step and we get, from (R R), (not (not (R R))). This is not ideal as a foundation for
However, you might realise something: the not here doesn't play any role. We can replace it with any arbitrary f. In fact, let's do that, and create a simple wrapper $\lambda$-term around it that
lets us pass in any f we want:
Now let's look at the properties that $Y$ has:
$Y$ is called the Y combinator ("combinator" is a generic term for a lambda calculus term with no free variables). It is part of the general class of fixed-point combinators: combinators $X$ such
that $(X \, f) = (f \, (X\,f))$. (Turing invented another one: $\Theta = (A \, A)$, where $A$ is defined as $(\lambda x y. (y \,(x\, x\, y)))$.)
A fixed-point combinator gives us recursion. Imagine we've almost written a recursive function, say for a factorial, except we've left a free function parameter for the recursive call:
(lambda f x
(if (eq0 x)
(mult x
(f (pred x)))))
(Also, take a moment to appreciate that we can already do everything necessary except for the recursion with our earlier definitions.)
Call the previous recursion-free factorial term F, and consider reducing ((Y F) 2) (where -BETA-> stands for one or more $\beta$-reductions):
((Y F)
((F (Y F))
((lambda x
(if (eq0 x)
(mult x
((Y F) (pred x)))))
(if (eq0 2)
(mult 2
((Y F) (pred 2))))
(mult 2
((Y F)
(mult 2
((F (Y F))
(mult 2
((lambda x
(if (eq0 x)
(mult x
((Y F) (pred x)))))
(mult 2
(mult 1
It works! Get a fixed-point combinator, and recursion is solved.
Primitive recursion
The definition of the partial recursive functions (one of the ways to define computability, mentioned at the beginning) involves something called primitive recursion. Let's implement that, and along
the way look at fixed-point combinators from another perspective.
Primitive recursion is essentially about implementing bounded for-loops / recursion stacks, where "bounded" means that the depth is known when we enter the loop. Specifically, there's a function $f$
that takes in zero or more parameters, which we'll abbreviate as $\overline{P}$. At 0, the value of our primitive recursive function $h$ is $f(\overline{P})$. At any integer $x+1$ for $x \geq 0$, $h
(\overline{P}, x+1)$ is defined as $g(\overline{P}, x, h(\overline{P}, x))$: in other words, the value at $x+1$ is given by some function of:
• fixed parameter(s) $\overline{P}$,
• how many more steps there are in the loop before hitting the base case ($x$), and
• the value at $x$ (the recursive part).
For example, in our factorial example there are no parameters, so $f$ is just the constant function 1, and $g(x, r) = (x + 1) \times r$, where $r$ is the recursive result for one less, and we have
$x+1$ because (for a reason I can't figure out – ideas?) $g$ takes, by definition, not the current loop index but one less.
Now it's pretty easy to write the function for primitive recursion, leaving the recursive call as an extra parameter (r) once again, and assuming that we have $\lambda$-terms F and G for $f$ and $g$
(lambda r P x
(if (eq0 x)
(F P)
(G P (pred x) (r P (pred x)))))
Slap a $Y$ in front, and we take care of the recursion and we're done.
The fixed point perspective
However, rather than viewing this whole "slap in the $Y$" business as a hack for getting recursion, we can also interpret it as a fixed point operation.
A fixed point of a function $f$ is a value $x$ such that $x = f(x)$. The fixed points of $f(x)=x^2$ are 0 and 1. In general, fixed points are often useful in maths stuff and there's a lot of deep
theory behind them (for which you will have to look elsewhere).
Now $Y$ (or any other fixed point combinator) has the property that $(Y f) =_\beta (f \, (Y\, f))$ (remember that the equivalent of $f(x)$ is written $(f \,x)$ in the lambda calculus). In other
words, $Y$ is a magic wand that takes a function and returns its fixed point (albeit in a mathematical sense that is not very useful for explicitly finding those fixed points).
Taking once again the example of defining primitive recursion, we can consider it as the fixed point problem of finding an $h$ such that $h = \Phi_{f,g}(h)$, where $\Phi_{f,g}$ is a function like the
following, where F and G are the lambda calculus representations of $f$ and $g$ respectively:
(lambda h
(lambda P x
(if (eq0 x)
(F P)
(G P (pred x) (h P (pred x)))))))
That is, $\Phi_{f,g}$ takes in some function h, and then returns a function that does primitive recursion – under the assumption that h is the right function for the recursive call.
Imagine it like this: when we're finding the fixed point of $f(x)= x^2$, we're asking for $x$ such that $x=x^2$. We can imagine reaching into the set of values that $x$ can take (in this case, the
real numbers), plugging them in, and seeing that in most cases the equation $x=x^2$ is false, but if we pick out a fixed point it becomes true. Similarly, solving $h=\Phi_{f,g}(h)$ is the problem of
considering all possible functions $h$ (and it turns out all computable functions can be enumerated, so this is, if anything, less crazy than considering all possible real numbers), and requiring
that plugging in $h$ into $\Phi_{f,g}$ gives back $h$. For almost any function that we plug in, this equation will be nonsense: instead of doing primitive recursion, on the first call to h $\Phi_
{f,g}$ will do some crazy call that might loop forever or calculate the 17th digit of $\pi$, but if it's picked just right, $h$ and $\Phi_{f,g}(h)$ will happen to be the same thing. Unlike in the
algebraic case, it's very difficult to iteratively improve on your guess for $h$, so it's hard to think of how to use this weird way of defining the problem of finding $h$ to actually find it.
Except hold on – we're working in the lambda calculus, and fixed point combinators are easy: call $Y$ on a function and we have its fixed point, and, by the reasoning above, that is the recursive
version of that function.
The lambda calculus in lambda calculus
There's one final powerful demonstration of a computation model's expressive power that we haven't looked at: being able to express itself. The most well-known case is the universal Turing machine,
and those crop up a lot when you're thinking about computation theory.
Now there exists a trivial universal lambda term: $(\lambda \,f\,a\,.\,(f \,a))$ takes $f$, the lambda representation of some function, and an argument $a$, and returns the lambda calculus
representation of $f$ applied to $a$. However, this isn't exactly fair, since we've just forwarded all the work onto whatever is interpreting the lambda calculus. It's like noting that an eval
function exists in a programming language, and then writing on your CV that you've written an evaluator for it.
Instead, a "fair" way to define a universal lambda term is to build on the data specifications we have to define a representation of variables, lambda terms, and application terms, and then writing
more definitions within the lambda calculus until we have a reduce function.
This is what I've done in Lambda Engine. The definitions specific to defining the lambda calculus within the lambda calculus start about halfway down this file. I won't walk through the details here
(see the code and comments for more detail), but the core points are:
• We distinguish term types by making each term a pair consisting of an identifier and then the data associated with it. The identifier for variables/$\lambda$s/applications is a function that
takes a triple and returns the 1st/2nd/3rd member of it (this is simpler than tagging them with e.g. Church numerals, since testing numerical equality is complicated). The data is either a Church
numeral (for variables) or a pair of a variable and a term ($\lambda$-terms) or a term and a term (applications).
• We need case-based recursion, where we can take in a term, figure out what it is, and then perform a call to a function to handle that term and pass on the main recursive function to that handler
function (for example, because when substituting in a application term, we need to call the main substitution function on both the left and right child of the application). The case-based
recursion functions (different ones for the different number of arguments required by substitution and reduction) take a triple of functions (one for each term type) and exploit the fact that the
identifier of a term is a function that picks some element from the triple (in this case, we call the identifier on the handler function triple to pick the right one).
• We have helper functions for to build our term types, extract out parts, and test for whether something is a $\lambda$-term (exploiting the fact that the first element of the pair that a lambda
term is is the "take the 2nd thing from a triple" function).
• With the above, we can define substitution fairly straightforwardly. Note that we need to test Church numeral equality, which requires a generic Church numeral equality tester, which is a slow
function (because it needs to recurse and take a lot of predecessors).
• For reduction, the main tricky bit is doing it in normal order. This means that we have to be able to tell whether the left child in an application term is reducible before we try to reduce the
right child (e.g. the left child might eventually reduce to a function that throws away its argument, and the right child might be a looping term like $\Omega$). We define a helper function to
check whether something reduces, and then can write reduce-app and therefore reduce. For convenience we can define a function n-reduce that calls reduce an expression n times, simply by
exploiting how Church numerals work (((2 reduce) x) is (reduce (reduce x)), for example).
What we don't have:
• Variable renaming. We assume that terms in this lambda calculus are written so that a variable name (in this case, a Church numeral) is never reused.
• Automatically reducing to $\beta$-normal form. This could be done fairly simply by writing another function that calls itself with the reduce of its argument until our checker for whether
something reduces is false.
• Automatically checking whether we're looping (e.g. we've typed in the definition of $\Omega$).
The lambda calculus interpreter in this file has all three features above. You can play with it, and the lambda-calculus-in-lambda-calculus, by downloading Lambda Engine (and a Racket interpreter if
you don't already have one) and using one of the evaluators in this file.
Towards Lisp
Let's see what we've defined in the lambda calculus so far:
• pair
• lists
• fst
• snd
• True
• False
• if
• eq0
• numbers
• recursion
This is most of what you need in a Lisp. Lisp was invented in 1958 by John McCarthy. It was intended as an alternative axiomatisation for computation, with the goal of not being too complicated to
define while still being human friendly, unlike the lambda calculus or Turing machines. It borrows notation (in particular the keyword lambda) from the lambda calculus and its terms are also trees,
but it is not directly based on the lambda calculus.
Lisp was not intended as a programming language, but Steve Russell (no relation to Bertrand Russell ... I'm pretty sure) realised you could write machine code to evaluate Lisp expressions, and went
ahead and did so, making Lisp the second-oldest programming language. Despite its age, Lisp is arguably the most elegant and flexible programming language (modern dialects include Clojure and Racket
One way to think of what we've done in this post is that we've started from the lambda calculus – an almost stupidly simple theoretical model – and made definitions and syntax transformations until
we got most of the way to being able to emulate Lisp, a very usable and practical programming language. The main takeaway is, hopefully, an intuitive sense of how something as simple as the lambda
calculus can express any computation expressible in a higher-level language. | {"url":"https://www.strataoftheworld.com/2021/04/lambda-calculus.html","timestamp":"2024-11-10T08:05:33Z","content_type":"application/xhtml+xml","content_length":"134985","record_id":"<urn:uuid:b6468611-bd29-452a-9365-fd5f84075cae>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00606.warc.gz"} |
SQL UPDATE Using a Join
You may need to do an update on joined tables to get a more conditional update. For instance, I have a Student table as well as an AcademicStatus table. The Student table contains all the students
(profound, I know) and the AcademicStatus table tells if a student is in good standing, at risk, or has dropped out based on a StandingID. The Student table also lists a graduation date and a current
bit to show if the student is currently enrolled. While generating data for these particular tables recently I ran into an issue where some students had dropped out, but mysteriously had graduation
dates, or were listed as being currently enrolled. The easiest way to update this information is by doing a simple SQL UPDATE command on the joined tables. First we will run a query to get all the
students that have dropped out in the AcademicStatus table, while being joined to the Student table pulling back the current and GraduationDate fields. SELECT AcademicStatus.StandingID, Student.
[Current], Student.GraduationDate FROM Student INNER JOIN AcademicStatus ON Student.StudentID = AcademicStatus.StudentID WHERE (AcademicStatus.StandingID = 3) We can then look through that data and
see there are students dropped out that have graduated. That would be a really neat trick. Now you simply need to put everything after "FROM" into your update statement. So now: UPDATE Student SET
GraduationDate = NULL, [Current] = '0' Becomes: UPDATE Student SET GraduationDate = NULL, [Current] = '0' FROM Student INNER JOIN AcademicStatus ON Student.StudentID = AcademicStatus.StudentID WHERE
(AcademicStatus.StandingID = 3) This means the GraduationDate will be set to NULL and the Current bit will be zero for a particular student in the Student table ONLY if the corresponding student has
an StandingID of 3 on the AcademicStatus table. In the first update statement, all students in the Student table would be updated. That is how you update based on a condition in another table. | {"url":"https://test.bradleyschacht.com/sql-update-using-a-join","timestamp":"2024-11-14T11:55:23Z","content_type":"text/html","content_length":"92022","record_id":"<urn:uuid:5fd9d0bb-4886-414a-a97f-d9390db61d48>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00297.warc.gz"} |
Deflection of light rays in a moving medium around a spherically symmetric gravitating object
verfasst von
Barbora Bezděková, Oleg Yu Tsupko, Christian Pfeifer
In most analytical studies of light ray propagation in curved spacetimes around a gravitating object surrounded by a medium, it is assumed that the medium is a cold nonmagnetized plasma. The
distinctive feature of this environment is that the equations of motion of the rays are independent of the plasma velocity, which, however, is not the case in other media. In this paper, we
consider the deflection of light rays propagating near a spherically symmetric gravitating object in a moving dispersive medium given by a general refractive index. The deflection is studied when
the motion of the medium is defined either as a radially falling onto a gravitating object (e.g., black hole), or rotating in the equatorial plane. For both cases the deflection angles are
obtained. These examples demonstrate that fully analytic expressions can be obtained if the Hamiltonian for the rays takes a rather general form as a polynomial in a given momentum component. The
general expressions are further applied to three specific choices of refractive indices, and these cases are compared. Furthermore, the light rays propagating around a gravitating object
surrounded by a generally moving medium are further studied as a small perturbation of the cold plasma model. The deflection angle formula is hence expressed as a sum of zeroth and first order
components, where the zeroth order term corresponds to the known cold plasma case, and the first order correction can be interpreted as caused by small difference in the refractive index compared
to the cold plasma. The results presented in this paper allow to describe the effects caused by the motion of a medium and thus go beyond cold nonmagnetized plasma model.
Externe Organisation(en)
Charles University
Zentrum für angewandte Raumfahrttechnologie und Mikrogravitation (ZARM)
Universität Bremen
Physical Review D
ASJC Scopus Sachgebiete
Kern- und Hochenergiephysik
Elektronische Version(en)
https://doi.org/10.1103/PhysRevD.109.124024 (Zugang: Offen) | {"url":"https://www.quest-lfs.uni-hannover.de/de/forschung/publikationen/details/fis-details/publ/9b9e00aa-5022-4e08-a631-269ec445ba48?cHash=f29667726994047a2794286ec045079f","timestamp":"2024-11-06T02:20:02Z","content_type":"application/xhtml+xml","content_length":"39176","record_id":"<urn:uuid:367c425f-a351-46ae-87c1-86f092dd24f7>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00404.warc.gz"} |
Symmetry & critical points for a model shallow neural network
Using methods based on the analysis of real analytic functions, symmetry and equivariant bifurcation theory, we obtain sharp results on families of critical points of spurious minima that occur in
optimization problems associated with fitting two-layer ReLU networks with k hidden neurons. The main mathematical result proved is to obtain power series representations of families of critical
points of spurious minima in terms of 1/k (coefficients independent of k). We also give a path based formulation that naturally connects the critical points with critical points of an associated
linear, but highly singular, optimization problem. These critical points closely approximate the critical points in the original problem. The mathematical theory is used to derive results on the
original problem in neural nets. For example, precise estimates for several quantities that show that not all spurious minima are alike. In particular, we show that while the loss function at certain
types of spurious minima decays to zero like k^−1, in other cases the loss converges to a strictly positive constant.
Bibliographical note
Publisher Copyright:
© 2021
• Critical points
• Power series representation
• ReLU activation
• Spurious minima
• Student–teacher network
• Symmetry breaking
Dive into the research topics of 'Symmetry & critical points for a model shallow neural network'. Together they form a unique fingerprint. | {"url":"https://cris.huji.ac.il/en/publications/symmetry-amp-critical-points-for-a-model-shallow-neural-network","timestamp":"2024-11-13T09:59:08Z","content_type":"text/html","content_length":"48751","record_id":"<urn:uuid:a61a3072-fdad-425a-94ad-77349a02984d>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00568.warc.gz"} |
EFT mini-workshop at IPPP
Session 1: brief review of combined EFT fits and constraints
9 TopFitter (White)
- Warsaw basis: 16 operators affecting top; data constrain 12 operators, neglect others
- correlations included where available
- should we set all but one to zero to understand individual measurements?
-> has to be presented carefully, not an actual EFT parameter fit
- which operators are primarily constrained by quadratic terms? cG, maybe 4-fermion?
-> paper says linear operators included?
- scale of validity quoted? No but removing Mtt overflow bin did not make much difference
- could you determine the goodness of fit for the SM only?
- What about alternate approach from Butterworth: assume data is SM and constrain c's assuming
they are zero? Is that a circular approach?
- any issue with fitting particle-level vs parton-level? No.
-> should experimentalists be the ones evaluating particle->parton uncertainties?
- should we make measurements of distributions before background subtraction? Yes.
9:30 Higgs + dibosons + jets (Plehn)
- Include Dphi(jj) but do not fit CP-odd operators for philosophical reasons
- no experimental combination of dibosons (only ATLAS+CMS ZZ)
- quark 4-fermion operators constrained by 2-3 jets, interference term is zero
- three-gluon operator constrained by >=5 jets
- constraints are at a high scale, can we apply them to Higgs physics? Consensus is yes.
- What about (DG)^2? Not in Warsaw, becomes a 4-fermion operator (constrained by 2-3 jets).
- m4l gives little information within the context of VBF -- adding ggF should increase its contribution
10 EW precision & low energy (Trott)
- EDM strongly constrains CP-odd operators in Higgs & top physics
-> motivates fits either requiring MFV or CP-even
- Benefit to using mW,mZ,GF input scheme because there are two scales (alphaEM would be third)
- EFT uncertainties can be important for precision EW data (higher order terms ~10% of leading terms)
Session 2: status of constraints within LHC experiments
11 Higgs STXS vs data (Hays/Zemaityte)
- note that including quadratic terms becomes more valid when measurements are precise
- should only truncate based on IR physics or MFV symmetry assumption
-> will truncate using only assumed symmetries
- should remove coefficients from operators and include top loop in ggF
- should include ~1% uncertainty on S parameter from higher order effects
11:30 Higgs DiffXS (Pilkington)
- which is more sensitive: two 1d distributions with correlation or 2d distribution? Could check.
-> significant gain expected from 2d distribution in VBF
12 Electroweak (Lohwasser/Price)
- Publish observed numbers of data events? Useful if validated Delphes model provided.
- Should quote limits from fully optimized reconstructed data and compare to those from unfolded data
12:30 CMS (Milenovic)
- will CMS provide correlations in diffXS? Aiming to do so, probably starting from 2017 data
- will CMS present results using YR4 PO notation? Yes, along with historical presentation.
Session 3: future tools/studies in dim-6 (LO & NLO) and dim-8 EFT
14 Warsaw LO UFO (Brivio)
- two implementations of Warsaw basis will soon be available
- ~30 operators if considering W/Z/H pole observables
- first fit probably also needs qq->ttbar?
14:30 Warsaw NLO QCD (Maltoni)
- many NLO SMEFT calculations available
- need to implement four-fermion terms before releasing NLO Madgraph implementation
-> Provide ggF as a starting point to study?
15 Dim 8 (Sanz)
- no new q^2 dependence coming from dim-8 -- could be due to basis?
- Need to check vertices with fermions: q^2 dependence might come from spinors
- useful for systematic checks
- can VBF be added? more complicated but possible in principle
Session 4: fit issues
16 Uncertainties (Pecjak)
- Can Madgraph include the running of the EFT operators so that we can use it for EFT scale variations?
16:30: Validity (Hays/Spannowsky)
- For a given validity scale what range of c should we allow? Up to ~10 okay.
- What should we do about coefficients that are directly probed above the scale used in the fit? Set to zero? | {"url":"https://indico.cern.ch/event/663240/?print=1","timestamp":"2024-11-05T00:30:12Z","content_type":"text/html","content_length":"49665","record_id":"<urn:uuid:79a556b0-5299-43b1-aa2f-1db949288388>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00039.warc.gz"} |
A373603 - OEIS
For n > 1, the index of the next term in
, after its sixth term 0, that is a multiple of
(n), as for n >= 1, the smallest k such that
(k) ==
(k) mod
(n) gives the sequence 1, 6, 6, 6, 6, 6, 6, 6, ..., because
(6) =
Provided that such k exists for every n (and the escape clause is not needed), then the sequence is by necessity monotonic. If it is strictly monotonic, then it implies that k=6 is the only k such
(k) =
(k). See also comments in
Note that if we instead search for the smallest k such that
(k) is a multiple of
(n) we obtain
, partial sums of the primorial numbers. See also
(n) = prod(i=1, n, prime(i));
(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); }; | {"url":"https://oeis.org/A373603","timestamp":"2024-11-08T11:33:56Z","content_type":"text/html","content_length":"19084","record_id":"<urn:uuid:61c0ae3f-6a51-454a-8485-7c7b81dd9b19>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00754.warc.gz"} |
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Correct score predictions are some of the most popular tips in the world and for good reason. Correct scores come with huge odds. Besides, tipsters like us have made great progress in becoming
experts in correct score prediction. You are guaranteed to win our correct score tips, especially if you subscribe for the VIP correct score membership. You can also get correct score fixed matches.
Share this page with your friends.
I receive many requests from customers asking for correct score tips every day. In fact, correct score predictions are the most asked for tips for all betting markets. The reason is pretty simple;
correct scores have the highest odds for any match on any betting site.
It is for this very reason that correct scores have gained a reputation throughout the world as the most profitable betting market. Like for anything that is adored and coveted, sure correct scores
are hard to come by, but when you get your hands on some, they can change your life.
This article complements the previous article I had made previously on how to make your own correct score tips. While in that article I strictly advised you to use poisson probabilities in making
your correct scores, I have come to realize many of you do not have the skills to do complex probability calculations. I have therefore decided to share with you these 5 secrets of correct score
predictions that will make your work easier in identifying and using correct scores.
CORRECT SCORE TIPS DO NOT MEAN THEY ARE “CORRECT”
Many tipsters are infamous for duping their customers about the meaning of correct scores. Some unbelievably claim that correct scores mean they are correct. They, therefore, want you to presume that
once they give you “correct” scores, then you must win using them. Nothing could be further from the truth. A Correct score is just an “exact” goals possible outcome in a match.
Like, 1-0, that is a correct score possibility. 2-2 is also a correct score possible outcome. In fact, most bookmakers list 29 possible correct scores on their websites for each match. Well, they are
technically 28 since the last one is ”OTHER”, which means any other correct score possible outcome not listed by the bookmaker. So, the next time a tipster claims to have some “correct” scores,
correct them by telling them they are rather “correct scores” or even better, “exact scores”.
Unless you have a set of correct scores that are fixed matches, do not combine 2 or more correct scores in one ticket or slip. For correct score fixed matches that are sure and guaranteed, there is
no problem.
After all, the chances of 2 mutually exclusive and assured outcomes coming true are 100%. But if you are making your own correct score predictions then combining them into a multi bet will surely be
We have acknowledged that correct scores are incredibly difficult to predict. The average chances of winning a correct score prediction is usually about 3%. That is already hard enough. Adding an
extra correct score to your slip, therefore, makes your chances of winning much, much lower. Since the matches’ outcomes do not depend on the outcomes of one another, to get the chances of winning
more than one correct score you simply have to multiply the probabilities.
For example, assume the probability of winning any correct score bets is 3% per match. A single correct score bet, therefore, has a 3% chance of winning. A multibet of 2 correct scores has a chance
of (3% * 3%) chance of winning, which is 0.03*0.03 = 0.0009 same as 0.09%. You are therefore 33 times more likely to win a correct score single bet than a correct score multi bet of 2 matches.I
understand the prospects of an extremely large payout that comes with winning a correct score multi bet is very appealing. Just remember, that is only done with fixed matches, not predictions.
There usually are 29 correct score options to choose from for any match listed on a betting site. The possible outcomes include those that indicate a home win, a draw and an away win. The correct
scores also encompass those that end in both teams scoring and those that have one team that fails to score, better known as no-goal.
Finally, among the 29 correct scores, there are those that are either under 2.5 goals scores or over 2.5 goals.Put simply, the possible correct scores can be sorted into different categories of
scores.Below are all the 29 possible correct scores for any given match filtered by the corresponding win-draw-win score:Home win: 1-0, 2-0, 3-0, 4-0, 5-0, 6-0, 2-1, 3-1, 4-1, 5-1, 3-2, 4-2Draw: 0-0,
1-1, 2-2, 3-3Away Win: 0-1, 0-2, 0-3, 0-4, 0-5, 0-6, 1-2, 1-3, 1-4, 1-5, 2-3, and 2-4.
As you can see, given you already have a sure prediction about the 1X2 outcome, you reduce the number of possible correct scores that fit into that prediction by a large margin. Using the 1X2
prediction as a filter may reduce the number of correct scores to choose from but not to a good extent. You may need another filter to complete the elimination process.
For instance, if you have already predicted a draw in a match, you remain with 4 possible correct scores that include: Draw: 0-0, 1-1, 2-2, 3-3. Assuming you have further knowledge that the match
will end in less than 2.5 goals, that means only 0-0 and 1-1 correct scores are possible.
At this point, you may choose to bet on both the correct scores separately. This way, you are assured of a win in either case. Besides, the odds for correct scores are usually high, meaning even if
you staked for 2 different correct scores separately and only one wins, then you are still assured of a good profit.
The process may get more complicated when you are dealing with home wins and away wins in a match predicted to end in more than 3 goals but the process is basically similar.
This is the elimination method that has been proved to be one of the most effective methods in predicting correct scores. Your win rate for correct score tips will go through the roof once you master
this correct score prediction model.
As we have already discussed, fixed matches are a hard nut to crack. Whenever you have more than one match whose correct scores you feel comfortable enough to predict, make sure you place multiple
bet slips for the same matches to increase your chances of winning.
The odds are usually high meaning you can still make a good profit if only you are confident enough to place multiple bets on the matches. For illustraton pirposes, assume there are two matches:
A vs B and Y vs Z
The correct score you feel is right for A vs B match is 2:0 while the best prediction for the Y vs Z match is 1-1.
In this scenario, it is good to select at least 2 correct score predictions for each game that you feel are most likely to come true. For A vs B, you might decide that 2:0 and 3:0 are the most likely
outcomes. For Y vs Z, you may deduce that the match will end in a draw of 1:1 or 2:2Now that you have 2 correct score tips for each match, you may make 4 correct score tickets and place all of them
This means that you will place 4 bets and have at least one win. Again for the purposes of illustration, let us assume the odds for the outcomes are:
• A vs B (2:0) - 15 odds
• A vs B (3:0) - 18 odds
• Y vs Z (1-1) - 7.4 odds
• Y vs Z (2-2) - 16 odds
Your possible correct scores and the corresponding odds are as shown in the table (2) below the list. As you can see, each of the bets will give you more than 100 odds. This means that even if just
one of them came true, the profit will be massive.
This trick is a risk management strategy that is quite necessary when betting on correct scores. It is especially important when using the elimination method of predicting correct scores.
Most bookmakers offer odds for half time correct scores. These are correct score predictions for the first half of the match. While the number of goals expected during the first half of the match is
usually low, the odds are somehow consistent with full-time correct scores on average.
The lowest odds are for the low correct scores such as 0:0 or 1:0. Any half time correct score with more than 2 goals usually carries huge odds. Some matches recorded more than 2 goals during the
first half of the match.
If you are confident of a game you feel will have many goals, it is worth playing these first half correct scores. The rules are similar to the full time scores and all the above secrets still apply
to this betting market.
While these are just tips and secrets of correct score predictions to help you make informed decisions about correct score betting, you can not always expect to win from your correct score bets by
simply using these rules.
There is more to successful correct score betting than meets the eye at this point. You may need expert correct score tips to fully reach your profit potential for this market. Surebet helps you
attain that potential within a week. Subscribe to correct score tips from Surebet and start making money from sure correct scores.
Many gamblers and football betting fans have a lot in common, we all like huge betting odds. It is however not easy to get high betting odds, of say, more than 7 in most betting markets that are easy
to predict. The high betting odds are reserved specifically to hard nuts to crack such as correct scores and the exact number of goals.
What are Correct Score Predictions or Tips?
Correct score predictions are forecasts about the exact number of matches that are going to be scored in an upcoming game. These usually apply in football betting more as compared to other sports.
This is because football (or soccer) is a low scoring game and therefore relatively easier to predict correct score outcomes as compared to other matches like basketball.
Correct score tips require that you be exact in the number of goals by each team and are different from total goals predictions. However, these are among the sure bet tips available on Surebetsite.
For instance, for a match between Arsenal and Liverpool, you would have to be specific like; (2-1) or (1-0).
Why are Correct Score Tips Difficult?
The number of goals that can be scored by a team is unlimited and independent of any other factors except time and the ability of the players. This, therefore, means any number of goals is
theoretically possible in a game of soccer.
To grasp, however, the real reason why the correct score predictions are difficult, you have to go back to mathematics. Using the concept of combinations and permutations, you will realize that the
number of possible outcomes in a game of football is almost infinity, theoretically and if no controls are made.
If we apply an artificial control on the number of goals a single team can score in a match, the number drastically reduces but is still very high compared to the number of expected outcomes for
betting markets such as GG or NG.
Let us assume that a team can score a maximum of 6 goals in a match.
The number of outcomes possible in this artificial scenario is as follows:
Those are all the possible outcomes for the match if we assumed that no team would be able to score more than 6 goals in the game. They are 36 possible outcomes for a single match! That is quite a
lot, no wonder the odds are usually as large as 100 for seemingly farfetched predictions.
To calculate the number of outcomes for the correct score given a certain control of the number of matches, simply use permutations.
For the above case, you simply use permutation function as 6P2 a permutation for 6 outcomes for two involved instances. It is like throwing 2 dice with 6 sides each at once independently and
recording all possible outcomes, the results will be 36 as well.
So in short, that is why correct score predictions are such a hard nut to crack for most beginners.
They are, however, crucial for prediction of jackpots such as the Sportpesa mega jackpot prediction or when you feel lucky and wish to get correct score weekend tips.
Apparently, different countries have different patterns when it comes to scoring and subsequently correct scores. It is already clear to many that some leagues tend to score highly while others seem
to have a shortage of goals. Below we are exploring the peculiar correct score trends for a number of countries. These will help you predict correct scores more accurately.
Please note that this is a generalization meant to give you a clear perspective. This information alone is not enough to help you make informed correct score picks. However, combined with other
pieces of information such as Correct Score trends for specific teams, you will be nearing perfection in your correct score analysis and prediction.
To come up with these analysis results we did an analysis of 229977 matches from 42 leagues across Africa, Europe, Asia, North America, South America and Australia.
Before we compare countries let us first see how the correct score global trends or averages look like.
• The most common correct score result is 1-1. This outcome showed up in 27687 of the 229977 matches under review. 1-1 correct score result therefore made up a staggering 12% of all outcomes in the
• The second most common correct score outcome is 1-0 which appears in about 10.5%. The following 8 correct score outcomes follow in the listed order:
• 2-1 comes in third showing up in 20412 out of the total 229977 matches. This makes up 8.8757% of the matches.
• Closely behind is a 0-0 correct score which formed 8.0286% of all outcomes.
• Wrapping up the top 5 correct score results is 2-0 which occured in 17695 of the matches representing 7.6942% of all matches.
• 0-1, 1-2, 2-2, 0-2 and 3-1 are the next 5 most common correct scores globally after showing up in 7.6221%, 6.5828%, 5.0357%, 4.5774%, 4.2365% of the 229977 matches respectively. A bonus correct
score tip is 3-0 which comes in at number 11 but represents a very close number of matches as the 3-1 score.
What we learn from this analysis is that to be on the safe side in majority of the correct scores you make then make sure you consider those top 10 correct score outcomes listed above. It is
foolhardy to come up with ridiculous correct scores such as 5-0 or 7-2.
For context, the 5-0 and 7-2 outcomes only showed up in less than 1% of all matches. This means you are ten times more likely to win a 1-1 correct score prediction than a 5-0 correct score prediction
Frequently Asked Questions
1. Are correct scores always Correct?
Correct score fixed matches are always correct. Correct score predictions, on the other hand, are made by tipsters as tips and are not always correct. Even the best correct score prediction
websites in the world sometimes lose in their CS prediction.
2. Is it wise to maxbet when betting on correct scores?
Like for any other betting markets, it is not wise to maxbet on predictions. Correct score predictions are especially hard. You should therefore, abstain from maxbetting with correct score tips.
I would only maxbet if the correct score odds were fixed with a manipulated and guaranteed outcome. Betting with about Ksh. 1000 (4000 Naira) is considered on the higher side for a correct score.
Make sure not to exceed the 100 dollars mark for any correct score prediction.
3. What are free exact score tips?
Free exact score tips are same to free correct score tips.
4. Does the betting website I use determine if I win a correct score prediction?
No. You can win or lose a correct score tip regardless of the betting website. It is however important to note that some betting websites might provide better correct score odd. Examples of such
websites include Betika, Betpawa, Sportybet, 1Xbet and Sportpesa for Kenya, Tanzania and Ghana, Bet9ja for Nigeria and Dafabet for Vietnam, Indonesia, India and Asia in general. The odds on the
names websites can expecially be enhamced if you place a correct score multibet.
5. Is it possible to get 100% sure correct score tips for SRL matches?
No. SRL matches are monitored by powerful AI algorithms and are hard to be manipulated into fixed matches. SRL predictions are generally harder to predict than normal football matches.
6. Which other sites would you recommend to get good correct scores?
As much as Surebet is the best provider of correct scores in Nigeria, Uganda, Ghana, Tanzania, Kenya and the world, there are a few other websites that give modestly good correct score tips. They
include Confirmbet, Primatips, Betnumbers and Mybet. This is not an endorsement of the above mentioned websites.
7. How many matches are given in the VIP correct score package?
4 matches are given daily in the correct score VIP package.
8. Are correct scores available today and on a daily basis?
Yes, we have free and VIP correct scores today and they are available daily.
9. How can I get VIP correct score predictions today?
To get VIP correct scores today simply register and make a payment of Ksh. 1000 or 4000 naira or 65 cedis or 12 dollars to subscribe. You will then receive the VIP correct score tips by SMS and
10. Do you odder correct scores on Telegram or WhatsApp?
We do not have a telegram channel or WhatsApp group for correct score members. However, if you subscribe, we can send you the tickets via WhatsApp or Telegram.
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Introduction to Mathematics with MapleThe Nest of Students: Introduction to Mathematics with Maple
Pamela W. Adams, K. Smith, Rudolf Vyborny ... 544 pages - Publisher: Wspc; (June, 2004) - Language: English - ISBN-10: 9812560092 - ISBN-13: 978-9812560094 ...
The principal aim of this book is to introduce university level mathematics - both algebra and calculus. The text is suitable for first and second year students. It treats the material in depth, and
thus can also be of interest to beginning graduate students. New concepts are motivated before being introduced through rigorous definitions. All theorems are proved and great care is taken over the
logical structure of the material presented. To facilitate understanding, a large number of diagrams are included. Most of the material is presented in the traditional way, but an innovative approach
is taken with emphasis on the use of Maple and in presenting a modern theory of integration. To help readers with their own use of this software, a list of Maple commands employed in the book is
provided. The book advocates the use of computers in mathematics in general, and in pure mathematics in particular. It makes the point that results need not be correct just because they come from the
computer. A careful and critical approach to using computer algebra systems persists throughout the text. | {"url":"https://www.geoteknikk.com/2017/08/introduction-to-mathematics-with-maple.html","timestamp":"2024-11-05T10:41:09Z","content_type":"application/xhtml+xml","content_length":"544085","record_id":"<urn:uuid:e81d0419-a662-49b0-b4e9-e94bc9742133>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00064.warc.gz"} |
Properties of number 120512
120512 has 28 divisors (see below), whose sum is σ = 274320. Its totient is φ = 51456.
The previous prime is 120511. The next prime is 120539. The reversal of 120512 is 215021.
It is a tau number, because it is divible by the number of its divisors (28).
It is a junction number, because it is equal to n+sod(n) for n = 120493 and 120502.
It is not an unprimeable number, because it can be changed into a prime (120511) by changing a digit.
It is a polite number, since it can be written in 3 ways as a sum of consecutive naturals, for example, 314 + ... + 582.
2^120512 is an apocalyptic number.
It is an amenable number.
It is a practical number, because each smaller number is the sum of distinct divisors of 120512, and also a Zumkeller number, because its divisors can be partitioned in two sets with the same sum (
120512 is an abundant number, since it is smaller than the sum of its proper divisors (153808).
It is a pseudoperfect number, because it is the sum of a subset of its proper divisors.
120512 is an equidigital number, since it uses as much as digits as its factorization.
120512 is an evil number, because the sum of its binary digits is even.
The sum of its prime factors is 288 (or 278 counting only the distinct ones).
The product of its (nonzero) digits is 20, while the sum is 11.
The square root of 120512 is about 347.1483832600. The cubic root of 120512 is about 49.3942919858.
Adding to 120512 its reverse (215021), we get a palindrome (335533).
The spelling of 120512 in words is "one hundred twenty thousand, five hundred twelve". | {"url":"https://www.numbersaplenty.com/120512","timestamp":"2024-11-14T21:09:25Z","content_type":"text/html","content_length":"8837","record_id":"<urn:uuid:f0580b2f-7585-4b71-88ff-e98146f24885>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00867.warc.gz"} |
The Edinburgh Review
alone, but must be shared, in various proportions, among the philosophers and mathematicians of all ages. Their efforts, from the age of Euclid and Archimedes, to the time of Newton and La Place,
have all been required to the accomplishment of this great object; they have been all necessary to form one man for the author, and a few for the readers, of the work before us. Every mathematician
who has extended the bounds of his science; every astronomer who has added to the number of facts, and the accuracy of observation; every artist who has improved the construction of the instruments
of astronomy-all have cooperated in preparing a state of knowledge in which such a book could exist, and in which its merit could be appreciated. They have collected the materials, sharpened the
tools, or constructed the engines employed in the great edifice, founded by Newton, and completed by La Place.
In this estimate we detract nothing from the merit of the author himself; his originality, his invention, and comprehensive views, are above all praise; nor can any man boast of a higher honour than
that the Genius of the human race is the only rival of his fame.
This review naturally gives rise to a great variety of reflections. We shall state only one or two of those that most obviously occur. When we consider the provision made by nature for the stabi lity
and permanence of the planetary system, a question arises, which was before hinted at,-whether is this stability necessary or contingent, the effect of an unavoidable or an arbitrary arrangement? If
it is the necessary consequence of conditions which are themselves necessary, we cannot infer from them the existence of design, but must content ourselves with admiring them as simple and beautiful
truths, having a necessary and independent existence. If, on the other hand, the conditions from which this stability arises necessarily, are not necessary themselves, but the consequences of an
arrangement that might have been different, we are then entitled to conclude, that it is the effect of wise design exercised in the construction of the universe.
Now, the investigations of La Place enable us to give a very satisfactory reply to these questions; viz. that the conditions essential to the stability of a system of bodies gravitating mutually to
one another, are by no means necessary, insomuch that systems can easily be supposed in which no such stability exists. The conditions essential to it, are the movement of the bodies all in one
direction, their having orbits of small eccentricity, or not far different from circles, and having periods of revolution not commensurable with one another. Now, these conditions are not necessary;
they may easily be supposed different; any of
them might be changed, while the others remained the same. The appointment of such conditions therefore as would necessarily give a stable and permanent character to the system, is not the work of
necessity; and no one will be so absurd as to argue, that it is the work of chance: It is therefore the work of design, or of intention, conducted by wisdom and foresight of the most perfect kind.
Thus the discoveries of La Grange and La Place lead to a very beautiful extension of the doctrine of final causes, the more interesting the greater the objects are to which they relate. This is not
taken notice of by La Place; and that it is not, is the only blemish we have to remark in his admirable work. He may have thought that it was going out of his proper province, for a geometer or a
mechanician to occupy himself in such speculations. Perhaps, in strictness, it is so; but the digression is natural: and when, in any system, we find certain conditions established that are not
necessary in themselves, we may be indulged so far as to inquire, whether any explanation of them can be given, and whether, if not referable to a mechanical cause, they may not be ascribed to
When we mention that the small eccentricity of the planetary orbits, and the motion of the planets in the same direction, are essential to the stability of the system, it may naturally occur, that
the comets which obey neither of these laws in their motion may be supposed to affect that stability, and to occasion irregularities which will not compensate one another. This would, no doubt, be
the effect of the comets that pass through our system, were they bodies of great mass, or of great quantity of matter. There are many reasons, however, for supposing them to have very little density;
so that their effect in producing any disturbance of the planets is wholly inconsiderable.
An observation somewhat of the same kind is applicable to the planets lately discovered. They are very small; and therefore the effect they can have in disturbing the motions of the larger planets is
so inconsiderable, that, had they been known to La Place (Ceres only was known), they could have given rise to no change in his conclusions. The circumstance of two of these planets having nearly, if
not accurately, the same periodic time, and the same mean distance, may give rise to some curious applications of his theorems. Both these planets may be consisiderably disturbed by Jupiter, and
perhaps by Mars,
Another reflection, of a very different kind from the preceding, must present itself, when we consider the historical details concerning the progress of physical astronomy that have occurred in the
foregoing pages. In the list of the mathematicians and philosophers, to whom that science, for the last sixty or seventy years,
has been indebted for its improvements, hardly a name from Great Britain falls to be mentioned. What is the reason of this? and how comes it, when such objects were in view, and when so much
reputation was to be gained, that the country of Bacon and Newton looked silently on, without taking any share in so noble a contest? In the short view given above, we have hardly mentioned any but
the five principal performers; but we might have quoted several others, Fontaine, Lambert, Frisi, Condorcet, Bailly, &c. who contributed their share to bring about the conclusion of the piece. In the
list, even so extended, there is no British name. It is true, indeed, that before the period to which we now refer, Maclaurin had pointed out an improvement in the method of treating central forces,
that has been of great use inall the investigations that have a reference to that subject. This was the resolution of the forces into others parallel to two or to three axes given in position and at
right angles to one another. In the controversy that arose about the motion of the apsides in consequence of Clairaut's deducing from theory only half the quantity that observation had established,
as already stated, Simpson and Walmesley took a part; and their essays are allowed to have great merit. The late Dr Mathew Stewart also treated the same subject with singular skill and success, in
his Essay on the Sun's distance. The same excellent geometer, in his Physical Tracts, has laid down several propositions that had for their object the determination of the moon's irregularities. His
demonstrations, however, are all geometrical; and leave us to regrete, that a mathematician of so much originality preferred the elegant methods of the ancient geometry, to the more powerful analysis
of modern algebra. Beside these, we recollect no other. names of our countrymen distinguished in the researches of phy-. sical astronomy during this period; and of these none made any attempt toward
the solution of the great problems that then occupied the philosophers and mathematicians of the continent. This is the more remarkable, that the interests of navigation were deeply involved in the
question of the lunar theory; so that no motive, which a regard to reputation or to interest could create, was wanting to engage the mathematicians of England in the inquiry. Nothing, therefore,
certainly prevented them from engaging in it, but consciousness that, in the knowledge of the higher geometry, they were not on a footing with their brethren on the Continent. This is the conclusion
which unavoidably forces itself upon us, and which will be but too well confirmed by locking back to the particulars which we stated in the beginning of this review, as either essential or highly
conducive to the improvements in physical astronomy.
The calculus of the sines was not known in England till within these few years. Of the method of partial differences, no mention, we believe, is yet to be found in any English author, much less the
application of it to any investigation. The general methods of integrating differential or fluxionary equations, the criterion of integrability, the properties of homogeneous equations, &c. were all
of them unknown; and it could hardly be said, that, in the more difficult parts of the doctrine of Fluxions, any improvement had been made beyond those of the inventor. At the moment when we now
write, the treatises of Maclaurin and Simpson, are the best which we have on the fluxionary calculus, though such a vast multitude of improvements have been made by the foreign mathematicians, since
the time of their first publication. These are facts, which it is impossible to disguise; and they are of such extent, that a man may be perfectly acquainted with every thing on mathematical learning
that has been written in this country, and may yet find himself stopped at the first page of the works of Euler or D'Alembert. He will be stopped, not from the difference of the fluxionary notation,
(a difficulty easily overcome), nor from the obscurity of these authors, who are both very clear writers, especially the first of them, but from want of knowing the principles and the methods which
they take for granted as known to every mathematical reader. If we come to works of still greater difficulty, such as the Méchanique Céleste, we will venture to say, that the number of those in this
island, who can read that work with any tolerable facility, is small indeed. If we reckon two or three in London and the military schools in its vicinity, the same number at each of the two English
Universities, and perhaps four in Scotland, we shall not hardly exceed a dozen; and yet we are fully persuaded that our reckoning is beyond the truth.
If any further proof of our inattention to the higher mathematics, and our unconcern about the discoveries of our neighbours were required, we would find it in the commentary on the works of Newton,
that so lately appeared. Though that commentary was the work of a man of talents, and one who, in this country, was accounted a geometer, it contains no information about the recent discoveries to
which the Newtonian system has given rise; not a word of the problem of the Three Bodies, of the distur bances of the planetary motions, or of the great contrivance by which these disturbances are
rendered periodical, and the regula rity of the system preserved. The same silence is observed as to all the improvements in the integral calculus, which it was the duty of a commentator on Newton to
have traced to their origin, and to have connected with the discoveries of his master. If Dr Horseley
VOL. XI. NO. 22.
Horseley has not done so, it could only be because he was unacquainted with these improvements, and had never studied the methods by which they have been investigated, or the language in which they
are explained.
At the same time that we state these facts as incontrovertible proofs of the inferiority of the English mathematicians to those of the Continent, in the higher departments; it is but fair to
acknowledge, that a certain degree of mathematical science, and indeed no inconsiderable degree, is perhaps more widely diffused in England, than in any other country of the world. The Ladies' Diary,
with several other periodical and popular publications of the same kind, are the best proofs of this assertion. In these, many curious problems, not of the highest order indeed, but still having a
considerable degree of difficulty, and far beyond the mere elements of science, are often to be met with; and the great number of ingenious men who take a share in proposing and answering these
questions, whom one has never heard of any where else, is not a little surprising. Nothing of the same kind, we believe, is to be found in any other country. The Ladies' Diary has now been continued
for more than a century; the poetry, enigmas, &c. which it contains, are in the worst taste possible; and the scraps of literature and philosophy are so childish or so old-fashioned, that one is very
much at a loss to form a notion of the class of readers to whom they are addressed. The geometrical part, however, has always been conducted in a superior style; the problems proposed have tended to
awaken curiosity, and the solutions to convey instruction in a much better manner than is always to be found in more splendid publications. If there is a decline, therefore, or a deficiency in
mathematical knowledge in this country, it is not to the genius of the people, but to some other cause that it must be attributed.
An attachment to the synthetical methods of the old geometers, in preference to those that are purely analytical, has often been assigned as the cause of this inferiority of the English
mathematicians since the time of Newton. This cause is hinted at by several foreign writers, and we must say that we think it has had no inconsiderable effect. The example of Newton himself may have
been hurtful in this respect. That great man, influenced by the prejudices of the times, seems to have thought that algebra and fluxions might be very properly used in the investigation of truth, but
that they were to be laid aside when truth was to be communicated, and synthetical demonstrations, if possible, substituted in their room. This was to embarrass scientific method with a clumsy and
ponderous apparatus, and to render its progress indirect and slow in an incalculable degree. The controversy
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Activity report
RNSR: 201321225U
In partnership with:
Université Rennes 1
Team name:
Modélisation hybride & conception par contrats pour les systèmes embarqués multi-physiques
In collaboration with:
Institut de recherche en informatique et systèmes aléatoires (IRISA)
Algorithmics, Programming, Software and Architecture
Embedded and Real-time Systems
Creation of the Team: 2013 July 01, updated into Project-Team: 2016 September 01
• A2. Software
• A2.1. Programming Languages
• A2.1.1. Semantics of programming languages
• A2.1.5. Constraint programming
• A2.1.9. Synchronous languages
• A2.1.10. Domain-specific languages
• A2.2. Compilation
• A2.3. Embedded and cyber-physical systems
• A2.3.1. Embedded systems
• A2.3.2. Cyber-physical systems
• A2.3.3. Real-time systems
• A2.4. Formal method for verification, reliability, certification
• A2.4.1. Analysis
• A2.4.2. Model-checking
• A2.4.3. Proofs
• A2.5. Software engineering
• A2.5.1. Software Architecture & Design
• A2.5.2. Component-based Design
• A3. Data and knowledge
• A3.1. Data
• A3.1.1. Modeling, representation
• A6. Modeling, simulation and control
• A6.1. Methods in mathematical modeling
• A6.1.1. Continuous Modeling (PDE, ODE)
• A6.1.3. Discrete Modeling (multi-agent, people centered)
• A6.1.5. Multiphysics modeling
• A8.4. Computer Algebra
• B2. Health
• B2.4. Therapies
• B2.4.3. Surgery
• B4. Energy
• B4.4. Energy delivery
• B4.4.1. Smart grids
• B5. Industry of the future
• B5.2. Design and manufacturing
• B5.2.1. Road vehicles
• B5.2.2. Railway
• B5.2.3. Aviation
• B5.2.4. Aerospace
• B5.8. Learning and training
• B5.9. Industrial maintenance
• B7. Transport and logistics
• B7.1. Traffic management
• B7.1.3. Air traffic
• B8. Smart Cities and Territories
• B8.1. Smart building/home
• B8.1.1. Energy for smart buildings
1 Team members, visitors, external collaborators
Research Scientists
• Benoît Caillaud [Team leader, Inria, Senior Researcher, HDR]
• Albert Benveniste [Inria, Emeritus, HDR]
• Khalil Ghorbal [Inria, Researcher]
PhD Students
• Christelle Kozaily [Inria]
• Aurélien Lamercerie [Univ de Rennes I]
• Joan Thibault [Univ de Rennes I]
Technical Staff
• Mathias Malandain [Inria, Engineer]
• Bertrand Provot [Inria, Engineer, from Oct 2020]
Interns and Apprentices
• Julien Duron [Univ de Rennes I, from Jun 2020 until Jul 2020]
Administrative Assistant
• Armelle Mozziconacci [CNRS]
2 Overall objectives
Hycomes was created a local team of the Rennes - Bretagne Atlantique Inria research center in 2013 and has been created as an Inria Project-Team in 2016. The team is focused on two topics in
cyber-physical systems design:
• Hybrid systems modelling, with an emphasis on the design of modelling languages in which software systems, in interaction with a complex physical environment, can be modelled, simulated and
verified. A special attention is paid to the mathematical rigorous semantics of these languages, and to the correctness (wrt. such semantics) of the simulations and of the static analyses that
must be performed during compilation. The Modelica language is the main application field. The team aims at contributing language extensions facilitating the modelling of physical domains which
are poorly supported by the Modelica language. The Hycomes team is also designing new structural analysis methods for hybrid (aka. multi-mode) Modelica models. New simulation and verification
techniques for large Modelica models are also in the scope of the team.
• Contract-based design and interface theories, with applications to requirements engineering in the context of safety-critical systems design. The objective of our research is to bridge the gap
between system-level requirements, often expressed in natural, constrained or semi-formal languages and formal models, that can be simulated and verified.
3 Research program
3.1 Hybrid Systems Modeling
Systems industries today make extensive use of mathematical modeling tools to design computer controlled physical systems. This class of tools addresses the modeling of physical systems with models
that are simpler than usual scientific computing problems by using only Ordinary Differential Equations (ODE) and Difference Equations but not Partial Differential Equations (PDE). This family of
tools first emerged in the 1980's with SystemBuild by MatrixX (now distributed by National Instruments) followed soon by Simulink by Mathworks, with an impressive subsequent development.
In the early 90's control scientists from the University of Lund (Sweden) realized that the above approach did not support component based modeling of physical systems with reuse 1. For instance, it
was not easy to draw an electrical or hydraulic circuit by assembling component models of the various devices. The development of the Omola language by Hilding Elmqvist was a first attempt to bridge
this gap by supporting some form of Differential Algebraic Equations (DAE) in the models. Modelica quickly emerged from this first attempt and became in the 2000's a major international concerted
effort with the Modelica Consortium 2. A wider set of tools, both industrial and academic, now exists in this segment 3. In the EDA sector, VHDL-AMS was developed as a standard 52 and also allows
for differential algebraic equations. Several domain-specific languages and tools for mechanical systems or electronic circuits also support some restricted classes of differential algebraic
equations. Spice is the historic and most striking instance of these domain-specific languages/tools 4. The main difference is that equations are hidden and the fixed structure of the differential
algebraic results from the physical domain covered by these languages.
Despite these tools are now widely used by a number of engineers, they raise a number of technical difficulties. The meaning of some programs, their mathematical semantics, is indeed ambiguous. A
main source of difficulty is the correct simulation of continuous-time dynamics, interacting with discrete-time dynamics: How the propagation of mode switchings should be handled? How to avoid
artifacts due to the use of a global ODE solver causing unwanted coupling between seemingly non interacting subsystems? Also, the mixed use of an equational style for the continuous dynamics with an
imperative style for the mode changes and resets is a source of difficulty when handling parallel composition. It is therefore not uncommon that tools return complex warnings for programs with many
different suggested hints for fixing them. Yet, these “pathological” programs can still be executed, if wanted so, giving surprising results — See for instance the Simulink examples in 24, 20 and
Indeed this area suffers from the same difficulties that led to the development of the theory of synchronous languages as an effort to fix obscure compilation schemes for discrete time equation based
languages in the 1980's. Our vision is that hybrid systems modeling tools deserve similar efforts in theory as synchronous languages did for the programming of embedded systems.
3.2 Background on non-standard analysis
Non-Standard analysis plays a central role in our research on hybrid systems modeling 20, 24, 22, 21. The following text provides a brief summary of this theory and gives some hints on its
usefulness in the context of hybrid systems modeling. This presentation is based on our paper 2, a chapter of Simon Bliudze's PhD thesis 30, and a recent presentation of non-standard analysis, not
axiomatic in style, due to the mathematician Lindström 58.
Non-standard numbers allowed us to reconsider the semantics of hybrid systems and propose a radical alternative to the super-dense time semantics developed by Edward Lee and his team as part of the
Ptolemy II project, where cascades of successive instants can occur in zero time by using ${ℝ}_{+}×ℕ$ as a time index. In the non-standard semantics, the time index is defined as a set $𝕋=\left\{n\
partial \mid n\in {}^{*}ℕ\right\}$, where $\partial$ is an infinitesimal and ${}^{*}ℕ$ is the set of non-standard integers. Remark that (1) $𝕋$ is dense in ${ℝ}_{+}$, making it “continuous”, and (2)
every $t\in 𝕋$ has a predecessor in $𝕋$ and a successor in $𝕋$, making it “discrete”. Although it is not effective from a computability point of view, the non-standard semantics provides a framework
that is familiar to the computer scientist and at the same time efficient as a symbolic abstraction. This makes it an excellent candidate for the development of provably correct compilation schemes
and type systems for hybrid systems modeling languages.
Non-standard analysis was proposed by Abraham Robinson in the 1960s to allow the explicit manipulation of “infinitesimals” in analysis 67, 45, 41. Robinson's approach is axiomatic; he proposes
adding three new axioms to the basic Zermelo-Fraenkel (ZFC) framework. There has been much debate in the mathematical community as to whether it is worth considering non-standard analysis instead of
staying with the traditional one. We do not enter this debate. The important thing for us is that non-standard analysis allows the use of the non-standard discretization of continuous dynamics “as
if” it was operational.
Not surprisingly, such an idea is quite ancient. Iwasaki et al. 53 first proposed using non-standard analysis to discuss the nature of time in hybrid systems. Bliudze and Krob 29, 30 have also used
non-standard analysis as a mathematical support for defining a system theory for hybrid systems. They discuss in detail the notion of “system” and investigate computability issues. The formalization
they propose closely follows that of Turing machines, with a memory tape and a control mechanism.
3.3 Structural Analysis of DAE Systems
The Modelica language is based on Differential Algebraic Equations (DAE). The general form of a DAE is given by:
$F\left(t,x,{x}^{"},{x}^{\text{'}\text{'}},\cdots \right)$ 1
where $F$ is a system of ${n}_{e}$ equations $\left\{{f}_{1},\cdots ,{f}_{{n}_{e}}\right\}$ and $x$ is a finite list of ${n}_{v}$ independent real-valued, smooth enough, functions $\left\{{x}_{1},\
cdots ,{x}_{{n}_{v}}\right\}$ of the independent variable $t$. We use ${x}^{"}$ as a shorthand for the list of first-order time derivatives of ${x}_{j}$, $j=1,\cdots ,{n}_{v}$. High-order derivatives
are recursively defined as usual, and ${x}^{\left(k\right)}$ denotes the list formed by the $k$-th derivatives of the functions ${x}_{j}$. Each ${f}_{i}$ depends on the scalar $t$ and some of the
functions ${x}_{j}$ as well as a finite number of their derivatives.
Let ${\sigma }_{i,j}$ denote the highest differentiation order of variable ${x}_{j}$ effectively appearing in equation ${f}_{i}$, or $-\infty$ if ${x}_{j}$ does not appear in ${f}_{i}$. The leading
variables of $F$ are the variables in the set
The state variables of $F$ are the variables in the set
A leading variable ${x}_{j}^{\left({\sigma }_{j}\right)}$ is said to be algebraic if ${\sigma }_{j}=0$ (in which case, neither ${x}_{j}$ nor any of its derivatives are state variables). In the
sequel, $v$ and $u$ denote the leading and state variables of $F$, respectively.
DAE are a strict generalization of ordinary differential equations (ODE), in the sense that it may not be immediate to rewrite a DAE as an explicit ODE of the form $v=G\left(u\right)$. The reason is
that this transformation relies on the Implicit Function Theorem, requiring that the Jacobian matrix $\frac{\partial F}{\partial v}$ have full rank. This is, in general, not the case for a DAE.
Simple examples, like the two-dimensional fixed-length pendulum in Cartesian coordinates 64, exhibit this behaviour.
For a square DAE of dimension $n$ (i.e., we now assume ${n}_{e}={n}_{v}=n$) to be solved in the neighborhood of some $\left({v}^{*},{u}^{*}\right)$, one needs to find a set of non-negative integers
$C=\left\{{c}_{1},\cdots ,{c}_{n}\right\}$ such that system
can locally be made explicit, i.e., the Jacobian matrix of ${F}^{\left(C\right)}$ with respect to its leading variables, evaluated at $\left({v}^{*},{u}^{*}\right)$, is nonsingular. The smallest
possible value of ${max}_{i}{c}_{i}$ for a set $C$ that satisfies this property is the differentiation index 35 of $F$, that is, the minimal number of time differentiations of all or part of the
equations ${f}_{i}$ required to get an ODE.
In practice, the problem of automatically finding a ”minimal” solution $C$ to this problem quickly becomes intractable. Moreover, the differentiation index may depend on the value of $\left({v}^{*},
{u}^{*}\right)$. This is why, in lieu of numerical nonsingularity, one is interested in the structural nonsingularity of the Jacobian matrix, i.e., its almost certain nonsingularity when its nonzero
entries vary over some neighborhood. In this framework, the structural analysis (SA) of a DAE returns, when successful, values of the ${c}_{i}$ that are independent from a given value of $\left({v}^
A renowned method for the SA of DAE is the Pantelides method; however, Pryce's $\Sigma$-method is introduced also in what follows, as it is a crucial tool for our works.
3.3.1 Pantelides method
In 1988, Pantelides proposed what is probably the most well-known SA method for DAE 64. The leading idea of his work is that the structural representation of a DAE can be condensed into a bipartite
graph whose left nodes (resp. right nodes) represent the equations (resp. the variables), and in which an edge exists if and only if the variable occurs in the equation.
By detecting specific subsets of the nodes, called Minimally Structurally Singular (MSS) subsets, the Pantelides method iteratively differentiates part of the equations until a perfect matching
between the equations and the leading variables is found. One can easily prove that this is a necessary and sufficient condition for the structural nonsingularity of the system.
The main reason why the Pantelides method is not used in our work is that it cannot efficiently be adapted to multimode DAE (mDAE). As a matter of fact, the adjacency graph of a mDAE has both its
nodes and edges parametrized by the subset of modes in which they are active; this, in turn, requires that a parametrized Pantelides method must branch every time no mode-independent MSS is found,
ultimately resulting, in the worst case, in the enumeration of modes.
3.3.2 Pryce's Sigma-method
Albeit less renowned that the Pantelides method, Pryce's $\Sigma$-method 65 is an efficient SA method for DAE, whose equivalence to the Pantelides method has been proved by the author. This method
consists in solving two successive problems, denoted by primal and dual, relying on the $\Sigma$-matrix, or signature matrix, of the DAE $F$.
This matrix is given by:
$\Sigma ={\left({\sigma }_{ij}\right)}_{1\le i,j\le n}$ 2
where ${\sigma }_{ij}$ is equal to the greatest integer $k$ such that ${x}_{j}^{\left(k\right)}$ appears in ${f}_{i}$, or $-\infty$ if variable ${x}_{j}$ does not appear in ${f}_{i}$. It is the
adjacency matrix of a weighted bipartite graph, with structure similar to the graph considered in the Pantelides method, but whose edges are weighted by the highest differentiation orders. The $-\
infty$ entries denote non-existent edges.
The primal problem consists in finding a maximum-weight perfect matching (MWPM) in the weighted adjacency graph. This is actually an assignment problem, for the solving of which several standard
algorithms exist, such as the push-relabel algorithm 51 or the Edmonds-Karp algorithm 47 to only give a few. However, none of these algorithms are easily parametrizable, even for applications to
mDAE systems with a fixed number of variables.
The dual problem consists in finding the component-wise minimal solution $\left(C,D\right)=\left(\left\{{c}_{1},\cdots ,{c}_{n}\right\},\left\{{d}_{1},\cdots ,{d}_{n}\right\}\right)$ to a given
linear programming problem, defined as the dual of the aforementioned assignment problem. This is performed by means of a fixpoint iteration (FPI) that makes use of the MWPM found as a solution to
the primal problem, described by the set of tuples ${\left\{\left(i,{j}_{i}\right)\right\}}_{i\in \left\{1,\cdots ,n\right\}}$:
1. Initialize $\left\{{c}_{1},\cdots ,{c}_{n}\right\}$ to the zero vector.
2. For every $j\in \left\{1,\cdots ,n\right\}$,
3. For every $i\in \left\{1,\cdots ,n\right\}$,
4. Repeat Steps 2 and 3 until convergence is reached.
From the results proved by Pryce in 65, it is known that the above algorithm terminates if and only if it is provided a MWPM, and that the values it returns are independent of the choice of a MWPM
whenever there exist several such matchings. In particular, a direct corollary is that the $\Sigma$-method succeeds as long as a perfect matching can be found between equations and variables.
Another important result is that, if the Pantelides method succeeds for a given DAE $F$, then the $\Sigma$-method also succeeds for $F$ and the values it returns for $C$ are exactly the
differentiation indices for the equations that are returned by the Pantelides method. As for the values of the ${d}_{j}$, being given by ${d}_{j}={max}_{i}\left({\sigma }_{ij}+{c}_{i}\right)$, they
are the differentiation indices of the leading variables in ${F}^{\left(C\right)}$.
Working with this method is natural for our works, since the algorithm for solving the dual problem is easily parametrizable for dealing with multimode systems, as shown in our recent paper 34.
3.3.3 Block triangular decomposition
Once structural analysis has been performed, system ${F}^{\left(C\right)}$ can be regarded, for the needs of numerical solving, as an algebraic system with unknowns ${x}_{j}^{\left({d}_{j}\right)}$,
$j=1\cdots n$. As such, (inter)dependencies between its equations must be taken into account in order to put it into block triangular form (BTF). Three steps are required:
1. the dependency graph of system ${F}^{\left(C\right)}$ is generated, by taking into account the perfect matching between equations ${f}_{i}^{\left({c}_{i}\right)}$ and unknowns ${x}_{j}^{\left({d}
2. the strongly connected components (SCC) in this graph are determined: these will be the equation blocks that have to be solved;
3. the block dependency graph is constructed as the condensation of the dependency graph, from the knowledge of the SCC; a BTF of system ${F}^{\left(C\right)}$ can be made explicit from this graph.
3.4 Contract-Based Design, Interfaces Theories, and Requirements Engineering
System companies such as automotive and aeronautic companies are facing significant difficulties due to the exponentially raising complexity of their products coupled with increasingly tight demands
on functionality, correctness, and time-to-market. The cost of being late to market or of imperfections in the products is staggering as witnessed by the recent recalls and delivery delays that many
major car and airplane manufacturers had to bear in the recent years. The specific root causes of these design problems are complex and relate to a number of issues ranging from design processes and
relationships with different departments of the same company and with suppliers, to incomplete requirement specification and testing.
We believe the most promising means to address the challenges in systems engineering is to employ structured and formal design methodologies that seamlessly and coherently combine the various
viewpoints of the design space (behavior, space, time, energy, reliability, ...), that provide the appropriate abstractions to manage the inherent complexity, and that can provide
correct-by-construction implementations. The following technology issues must be addressed when developing new approaches to the design of complex systems:
• The overall design flows for heterogeneous systems and the associated use of models across traditional boundaries are not well developed and understood. Relationships between different teams
inside a same company, or between different stake-holders in the supplier chain, are not well supported by solid technical descriptions for the mutual obligations.
• System requirements capture and analysis is in large part a heuristic process, where the informal text and natural language-based techniques in use today are facing significant challenges 9.
Formal requirements engineering is in its infancy: mathematical models, formal analysis techniques and links to system implementation must be developed.
• Dealing with variability, uncertainty, and life-cycle issues, such as extensibility of a product family, are not well-addressed using available systems engineering methodologies and tools.
The challenge is to address the entire process and not to consider only local solutions of methodology, tools, and models that ease part of the design.
Contract-based design has been proposed as a new approach to the system design problem that is rigorous and effective in dealing with the problems and challenges described before, and that, at the
same time, does not require a radical change in the way industrial designers carry out their task as it cuts across design flows of different types. Indeed, contracts can be used almost everywhere
and at nearly all stages of system design, from early requirements capture, to embedded computing infrastructure and detailed design involving circuits and other hardware. Contracts explicitly handle
pairs of properties, respectively representing the assumptions on the environment and the guarantees of the system under these assumptions. Intuitively, a contract is a pair $C=\left(A,G\right)$ of
assumptions and guarantees characterizing in a formal way 1) under which context the design is assumed to operate, and 2) what its obligations are. Assume/Guarantee reasoning has been known for a
long time, and has been used mostly as verification mean for the design of software 62. However, contract based design with explicit assumptions is a philosophy that should be followed all along the
design, with all kinds of models, whenever necessary. Here, specifications are not limited to profiles, types, or taxonomy of data, but also describe the functions, performances of various kinds
(time and energy), and reliability. This amounts to enrich a component's interface with, on one hand, formal specifications of the behavior of the environment in which the component may be
instantiated and, on the other hand, of the expected behavior of the component itself. The consideration of rich interfaces is still in its infancy. So far, academic researchers have addressed the
mathematics and algorithmics of interfaces theories and contract-based reasoning. To make them a technique of choice for system engineers, we must develop:
• mathematical foundations for interfaces and requirements engineering that enable the design of frameworks and tools;
• a system engineering framework and associated methodologies and toolsets that focus on system requirements modeling, contract specification, and verification at multiple abstraction layers.
A detailed bibliography on contract and interface theories for embedded system design can be found in 3. In a nutshell, contract and interface theories fall into two main categories:
• Assume/guarantee contracts. By explicitly relying on the notions of assumptions and guarantees, A/G-contracts are intuitive, which makes them appealing for the engineer. In A/G-contracts,
assumptions and guarantees are just properties regarding the behavior of a component and of its environment. The typical case is when these properties are formal languages or sets of traces,
which includes the class of safety properties 55, 38, 61, 19, 40. Contract theories were initially developed as specification formalisms able to refuse some inputs from the environment 46. A/
G-contracts were advocated in 23 and are is still a very active research topic, with several contributions dealing with the timed 28 and probabilistic 32, 33 viewpoints in system design, and
even mixed-analog circuit design 63.
• Automata theoretic interfaces. Interfaces combine assumptions and guarantees in a single, automata theoretic specification. Most interface theories are based on Lynch's Input/Output Automata 60,
59. Interface Automata 70, 69, 71, 36 focus primarily on parallel composition and compatibility: Two interfaces can be composed and are compatible if there is at least one environment where they
can work together. The idea is that the resulting composition exposes as an interface the needed information to ensure that incompatible pairs of states cannot be reached. This can be achieved by
using the possibility, for an Interface Automaton, to refuse selected inputs from the environment in a given state, which amounts to the implicit assumption that the environment will never
produce any of the refused inputs, when the interface is in this state. Modal Interfaces 66 inherit from both Interface Automata and the originally unrelated notion of Modal Transition System
57, 18, 31, 56. Modal Interfaces are strictly more expressive than Interface Automata by decoupling the I/O orientation of an event and its deontic modalities (mandatory, allowed or forbidden).
Informally, a must transition is available in every component that realizes the modal interface, while a may transition needs not be. Research on interface theories is still very active. For
instance, timed 72, 25, 27, 43, 42, 26, probabilistic 32, 44 and energy-aware 37 interface theories have been proposed recently.
Requirements Engineering is one of the major concerns in large systems industries today, particularly so in sectors where certification prevails 68. Most requirements engineering tools offer a poor
structuring of the requirements and cannot be considered as formal modeling frameworks today. They are nothing less, but nothing more than an informal structured documentation enriched with
hyperlinks. As examples, medium size sub-systems may have a few thousands requirements and the Rafale fighter aircraft has above 250,000 of them. For the Boeing 787, requirements were not stable
while subcontractors were working on the development of the fly-by-wire and of the landing gear subsystems, leading to a long and cahotic convergence of the design process.
We see Contract-Based Design and Interfaces Theories as innovative tools in support of Requirements Engineering. The Software Engineering community has extensively covered several aspects of
Requirements Engineering, in particular:
• the development and use of large and rich ontologies; and
• the use of Model Driven Engineering technology for the structural aspects of requirements and resulting hyperlinks (to tests, documentation, PLM, architecture, and so on).
Behavioral models and properties, however, are not properly encompassed by the above approaches. This is the cause of a remaining gap between this phase of systems design and later phases where
formal model based methods involving behavior have become prevalent—see the success of Matlab/Simulink/Scade technologies. We believe that our work on contract based design and interface theories is
best suited to bridge this gap.
4 Application domains
The Hycomes team contributes to the design of mathematical modeling languages and tools, to be used for the design of cyberphysical systems. In a nutshell, two major applications can be clearly
identified: (i) our work on the structural analysis of multimode DAE systems has a sizeable impact on the techniques to be used in Modelica tools; (ii) our work on the verification of dynamical
systems has an impact on the design methodology for safety-critical cyberphysical systems. These two applications are detailed below.
4.1 Modelica
Mathematical modeling tools are a considerable business, with major actors such as MathWorks, with Matlab/Simulink, or Wolfram, with Mathematica. However, none of these prominent tools are suitable
for the engineering of large systems. The Modelica language has been designed with this objective in mind, making the best of the advantages of DAEs to support a component-based approach. Several
industries in the energy sector have adopted Modelica as their main systems engineering language.
Although multimode features have been introduced in version 3.3 of the language 48, proper tool support of multimode models is still lagging behind. The reason is not a lack of interest from tool
vendors and academia, but rather that multimode DAE systems poses several fundamental difficulties, such as a proper definition of a concept of solutions for multimode DAEs, how to handle mode
switchings that trigger a change of system structure, or how impulsive variables should be handled. Our work on multimode DAEs focuses on these crucial issues 6.
Thanks to the experimental coupling of Dymola (Dassault Systèmes' commercial implementation of the Modelica language) with our IsamDAE prototype (https://team.inria.fr/hycomes/software/isamdae/) 8,
17, that is being tested at the time of writing of this activity report, a larger class of Modelica models are expected to be compiled and simulated correctly. This should enable industrial users to
have cleaner and simpler multimode Modelica models, with dynamically changing structure of cyberphysical systems. On the longer term, our ambition is to provide efficient code-generation techniques
for the Modelica language, supporting, in full generality, multimode DAE systems, with dynamically changing differentiation index, structure and dimension.
4.2 Dynamical Systems Verification
In addition to well-defined operational semantics for hybrid systems, one often needs to provide formal guarantees about the behavior of some critical components of the system, or at least its main
underlying logic. To do so, we are actively developing new techniques to automatically verify whether a hybrid system complies with its specifications, and/or to infer automatically the envelope
within which the system behaves safely. The approaches we developed have been already successfully used to formally verify the intricate logic of the ACAS X, a mid-air collision avoidance system that
advises the pilot to go upward or downward to avoid a nearby airplane which requires mixing the continuous motion of the aircraft with the discrete decisions to resolve the potential conflict 54.
This challenging example is nothing but an instance of the kind of systems we are targeting: autonomous smart systems that are designed to perform sophisticated tasks with an internal tricky logic.
What is even more interesting perhaps is that such techniques can be often "reverted" to actually synthesize missing components so that some property holds, effectively helping the design of such
complex systems.
5 Social and environmental responsibility
5.1 Impact of research results
The expected impact of our research is to allow both better designs and better exploitation of energy production units and distribution networks, enabling large-scale energy savings. At least, this
is what we can observe in the context of the FUI ModeliScale collaborative project, which is focused on electric grids, urban heat networks and building thermal modeling.
The rationale is as follows: system engineering models are meant to assess the correctness, safety and optimality of a system under design. However, system models are still useful after the system
has been put in operation. This is especially true in the energy sector, where systems have an extremely long lifespan (for instance, more than 50 years for some nuclear power plants) and are
upgraded periodically, to integrate new technologies. Exactly like in software engineering, where a software and its model co-evolve throughout the lifespan of the software, a co-evolution of the
system and its physical models has to be maintained. This is required in order to maintan the safety of the system, but also its optimality.
Moreover, physical models can be instrumental to the optimal exploitation of a system. A typical example are model-predictive control (MPC) techniques, where the model is simulated, during the
exploitation of the system, in order to predict system trajectories up to a bounded-time horizon. Optimal control inputs can then be computed by mathematical programming methods, possibly using
multiple simulation results. This has been proved to be a practical solution 50, whenever classical optimal control methods are ineffective, for instance, when the system is non-linear or
discontinuous. However, this requires the generation of high-performance simulation code, capable of simulating a system much faster than real-time.
The structural analysis techniques implemented in IsamDAE 8 generate a conditional block dependency graph, that can be used to generate high-performance simulation code : static code can be generated
for each block of equations, and a scheduling of these blocks can be computed, at runtime, at each mode switching, thanks to an inexpensive topological sort algorithm. Contrarily to other approaches
(such as 49), no structural analysis, block-trangular decompositions, or automatic differentiation has to be performed at runtime.
6 Highlights of the year
The main highlights for 2020 are the two following achievements:
1. The publication of 6, a 47-pages long journal paper, detailing a comprehensive theory of multimode DAE systems. A particular attention is paid to the structural analysis of (possibly impulsive)
mode switchings.
2. The development of the IsamDAE software (https://team.inria.fr/hycomes/software/isamdae/) became in 2020 a major undertaking for the Hycomes team. This software implements structural analysis
algorithms presented in 8, 17. The development team has been strenghtened in October 2020 with the hiring of Bertrand Provot, a software engineer in charge of the consolidation, testing and
documentation of the software.
7 New software and platforms
7.1 New software
7.1.1 Demodocos
• Name: Demodocos (Examples to Generic Scenario Models Generator)
• Keywords: Surgical process modelling, Net synthesis, Process mining
• Scientific Description:
Demodocos is used to construct a Test and Flip net (Petri net variant) from a collection of instances of a given procedure. The tool takes as input either standard XES log files (a standard XML
file format for process mining tools) or a specific XML file format for surgical applications. The result is a Test and Flip net and its marking graph. The tool can also build a #SEVEN scenario
for integration into a virtual reality environment. The scenario obtained corresponds to the generalization of the input instances, namely the instances synthesis enriched with new behaviors
respecting the relations of causality, conflicts and competition observed.
Demodocos is a synthesis tool implementing a linear algebraic polynomial time algorithm. Computations are done in the Z/2Z ring. Test and Flip nets extend Elementary Net Systems by allowing test
to zero, test to one and flip arcs. The effect of flip arcs is to complement the marking of the place. While the net synthesis problem has been proved to be NP hard for Elementary Net Systems,
thanks to flip arcs, the synthesis of Test and Flip nets can be done in polynomial time. Test and flip nets have the required expressivity to give concise and accurate representations of surgical
processes (models of types of surgical operations). Test and Flip nets can express causality and conflict relations. The tool takes as input either standard XES log files (a standard XML file
format for process mining tools) or a specific XML file format for surgical applications. The output is a Test and Flip net, solution of the following synthesis problem: Given a finite input
language (log file), compute a net, which language is the least language in the class of Test and Flip net languages, containing the input language.
• Functional Description:
The tool Demodocos allows to build a generic model for a given procedure from some examples of instances of this procedure. The generated model can take the form of a graph, a Test 'n Flip net or
a SEVEN scenario (intended for integration into a virtual reality environment).
The classic use of the tool is to apply the summary operation to a set of files describing instances of the target procedure. Several file formats are supported, including the standard XES format
for log events. As output, several files are generated. These files represent the generic procedure in different forms, responding to varied uses.
This application is of limited interest in the case of an isolated use, out of context and without a specific objective when using the model generated. It was developed as part of a research
project focusing in particular on surgical procedures, and requiring the generation of a generic model for integration into a virtual reality training environment. It is also quite possible to
apply the same method in another context.
• Publication: hal-00872284
• Authors: Benoît Caillaud, Aurélien Lamercerie
• Contacts: Benoît Caillaud, Aurélien Lamercerie
• Participants: Aurélien Lamercerie, Benoît Caillaud
7.1.2 IsamDAE
• Name: Implicit Structural Analysis of Multimode DAE systems
• Keywords: Structural analysis, Differential algebraic equations, Multimode, Scheduling
• Scientific Description:
Modeling languages and tools based on Differential Algebraic Equations (DAE) bring several specific issues that do not exist with modeling languages based on Ordinary Differential Equations. The
main problem is the determination of the differentiation index and latent equations. Prior to generating simulation code and calling solvers, the compilation of a model requires a structural
analysis step, which reduces the differentiation index to a level acceptable by numerical solvers.
The Modelica language, among others, allows hybrid models with multiple modes, mode-dependent dynamics and state-dependent mode switching. These Multimode DAE (mDAE) systems are much harder to
deal with. The main difficulties are (i) the combinatorial explosion of the number of modes, and (ii) the correct handling of mode switchings.
The aim of the software is on the first issue, namely: How can one perform a structural analysis of an mDAE in all possible modes, without enumerating these modes? A structural analysis algorithm
for mDAE systems has been designed and implemented, based on an implicit representation of the varying structure of an mDAE. It generalizes J. Pryce's Sigma-method to the multimode case and uses
Binary Decision Diagrams (BDD) to represent the mode-dependent structure of an mDAE. The algorithm determines, as a function of the mode, the set of latent equations, the leading variables and
the state vector. This is then used to compute a mode-dependent block-triangular decomposition of the system, that can be used to generate simulation code with a mode-dependent scheduling of the
blocks of equations.
• Functional Description:
IsamDAE (Implicit Structural Analysis of Multimode DAE systems) is a software library implementing new structural analysis algorithms for multimode DAE systems, based on an implicit
representation of incidence graphs, matchings between equations and variables, and block decompositions. The input of the software is a variable dimension multimode DAE system consisting in a set
of guarded equations and guarded variable declarations. It computes a mode-dependent structural index reduction of the multimode system and produces a mode-dependent graph for the scheduling of
blocks of equations. It also computes the differentiation order of the latent equations and leading variables, as functions of the modes.
IsamDAE is coded in OCaml, and uses (at least partially) the following packages: * MLBDD by Arlen Cox, * Menhir by François Pottier and Yann Régis-Gianas, * GuaCaml and Snowflake by Joan
Thibault, * Pprint by François Pottier, * XML-Light by Nicolas Cannasse and Jacques Garrigue.
• Release Contributions:
Versions 0.3a to 0.3d (released between Mar. and Dec. 2020):
* Performance improvements: connection with the Snowflake package by Joan Thibault, based on his PhD works on RBTF (Reduced Block-Triangular Forms). The order in which variables and equations are
declared in the model, and the way these declarations are grouped, has way less impact on performances when RBTF is active (now the default behaviour of IsamDAE). * New data structures were
implemented in order to correct the inputs of equations blocks in the XML, text and graph outputs. Before this fix, when two or several derivatives of the same variable appeared in the same
equation (as in the simple equation `der(x) + x = 0`), the lower-order derivatives of this variable were ignored. * New examples: several examples have been added, in mechanics, electrodynamics
and hydraulics. * Documentation: a comprehensive User and Developer manual is made available.
• News of the Year: It has been possible to perform the structural analysis of systems with more than 750 equations and 10 to the power 23 modes, therefore demonstrating the scalability of the
• URL: https://team.inria.fr/hycomes/software/isamdae/
• Publication: hal-02476541
• Authors: Benoît Caillaud, Mathias Malandain, Joan Thibault
• Contacts: Benoît Caillaud, Mathias Malandain, Joan Thibault
8 New results
8.1 Mathematical Foundations of Physical Systems Modeling Languages
Participants: Albert Benveniste, Benoît Caillaud, Mathias Malandain.
Modern modeling languages for general physical systems, such as Modelica or Simscape, rely on Differential Algebraic Equations (DAE), i.e., constraints of the form $f\left(\stackrel{˙}{x},x,u\right)=
0$, when only first-order derivatives are considered. This facilitates modeling from first principles of the physics. This year we completed and published in the Annual Reviews in Control 6 the
development of the mathematical theory needed to sound, on solid mathematical bases, the design of compilers and tools for DAE based physical modeling languages.
Unlike Ordinary Differential Equations (ODE, of the form $\stackrel{˙}{x}=g\left(x,u\right)$), DAE exhibit subtle issues because of the notion of differentiation index and related latent equations
—ODE are DAE of index zero for which no latent equation needs to be considered. Prior to generating execution code and calling solvers, the compilation of such languages requires a nontrivial
structural analysis step that reduces the differentiation index to a level acceptable by DAE solvers.
Multimode DAE systems, having multiple modes with mode-dependent dynamics and state-dependent mode switching, are much harder to deal with. The main difficulty is the handling of the events of mode
change. Unfortunately, the large literature devoted to the numerical analysis of DAEs does not cover the multimode case, typically saying nothing about mode changes. This lack of foundations causes
numerous difficulties to the existing modeling tools. Some models are well handled, others are not, with no clear boundary between the two classes. Basically, no tool exists that performs a correct
structural analysis taking multiple modes and mode changes into account.
In our work, we developed a comprehensive mathematical approach supporting compilation and code generation for this class of languages. Its core is the structural analysis of multimode DAE systems,
taking both multiple modes and mode changes into account. As a byproduct of this structural analysis, we propose well sound criteria for accepting or rejecting models at compile time.
For our mathematical development, we rely on nonstandard analysis, which allows us to cast hybrid systems dynamics to discrete time dynamics with infinitesimal step size, thus providing a uniform
framework for handling both continuous dynamics and mode change events.
8.2 An implicit structural analysis method for multimode DAE systems
Participants: Albert Benveniste, Benoît Caillaud, Mathias Malandain, Joan Thibault.
Modeling languages and tools based on Differential Algebraic Equations (DAE) bring several specific issues that do not exist with modeling languages based on Ordinary Differential Equations. The main
problem is the determination of the differentiation index and latent equations. Prior to generating simulation code and calling solvers, the compilation of a model requires a structural analysis
step, which reduces the differentiation index to a level acceptable by numerical solvers.
The Modelica language, among others, allows hybrid models with multiple modes, mode-dependent dynamics and state-dependent mode switching. These multimode DAE (mDAE) systems are much harder to deal
with. The main difficulties are (i) the combinatorial explosion of the number of modes, and (ii) the correct handling of mode switchings.
The focus of the paper 34 is on the first issue, namely: How can one perform a structural analysis of an mDAE in all possible modes, without enumerating these modes? A structural analysis algorithm
for mDAE systems is presented, based on an implicit representation of the varying structure of an mDAE. It generalizes J. Pryce's $\Sigma$-method 65 to the multimode case and uses Binary Decision
Diagrams (BDD) to represent the mode-dependent structure of an mDAE. The algorithm determines, as a function of the mode, the set of latent equations, the leading variables and the state vector. This
is then used to compute a mode-dependent block-triangular decomposition of the system, that can be used to generate simulation code with a mode-dependent scheduling of the blocks of equations.
This method has been implemented in the IsamDAE software. This has allowed the Hycomes team to evaluate the performance and scalability of the method on several examples. In particular, it has been
possible to perform the structural analysis of systems with more than 2300 equations and ${10}^{77}$ modes.
8.3 Ordered Functional Decision Diagrams: A Functional Semantics For Binary Decision Diagrams
Participants: Joan Thibault, Khalil Ghorbal.
We introduce a novel framework, termed λDD, that revisits Binary Decision Diagrams from a purely functional point of view. The framework allows to classify the already existing variants, including
the most recent ones like Chain-DD and ESRBDD, as implementations of a special class of ordered models. We enumerate, in a principled way, all the models of this class and isolate its most expressive
model. This new model, termed λDD-O-NUCX, is suitable for both dense and sparse Boolean functions, and is moreover invariant by negation. The canonicity of λDD-O-NUCX is formally verified using the
Coq proof assistant. We furthermore give bounds on the size of the different diagrams: the potential gain achieved by more expressive models can be at most linear in the number of variables n.
8.4 Functional Decision Diagrams: A Unifying Data Structure For Binary Decision Diagrams
Participants: Joan Thibault, Khalil Ghorbal.
We present concise and canonical representations of Boolean functions akin to Binary Decision Diagrams, a versatile data structure with several applications beyond computer science. Our approach is
functional: we encode the process that constructs the Boolean function of interest starting from the constant function zero (or False). This point of view makes the data structure more resilient to
variable ordering, a well-known problem in standard representations. The experiments on both dense and sparse formulas are very encouraging and show not only a better compression rate of the final
representation than all existing related variants but also a lower memory peak.
8.5 Characterizing Positively Invariant Sets: Inductive and Topological Methods
Participants: Khalil Ghorbal.
Set positive invariance is an important concept in the theory of dynamical systems and one which also has practical applications in areas of computer science, such as formal verification, as well as
in control theory. Great progress has been made in understanding positively invariant sets in continuous dynamical systems and powerful computational tools have been developed for reasoning about
them; however, many of the insights from recent developments in this area have largely remained folklore and are not elaborated in existing literature. This article contributes an explicit
development of modern methods for checking positively invariant sets of ordinary differential equations and describes two possible characterizations of positive invariants: one based on the real
induction principle, and a novel alternative based on topological notions. The two characterizations, while in a certain sense equivalent, lead to two different decision procedures for checking
whether a given semi-algebraic set is positively invariant under the flow of a system of polynomial ordinary differential equations.
8.6 Characterizing Q-matrices
Participants: Khalil Ghorbal, Christelle Kozaily.
We show that the existence of solutions for linear complementarity problems amounts to a covering of the entire space by a set of finite cones defined by the involved vectors as well as the standard
basis. We give several full characterizations for the case $n=2$ and detail how these could be used to derive several necessary conditions for higher dimensions. The local existence of solutions is
also investigated. It is shown that the positivity condition on the determinant, or equivalently, the orientation of the vectors forming the complementarity cones cannot be captured purely
9 Bilateral contracts and grants with industry
• Glose (2018–2021) In the context of a framework agreement between Safran Tech. of the Safran aeronautic group and Inria, the Hycomes team, jointly with the KAIROS and DIVERSE teams, contributes
to the Glose research grant funded by Safran. The contributions of the Hycomes team are structural analysis techniques for multimode DAE models resulting from the coupling of quasi-static models,
expressed as non-linear equations, with dynamical systems, in the form of systems of ordinary differential equations. The multimode features of the model come from the dynamic changes of the
system structures, possibly resulting from changes of mode of operation, or mechanical failure. Current work of the Hycomes team focuses on the definition of a component model, encapsulating
multimode DAE systems, and on modular structural analysis methods, capable of characterizing, from a structural point of view only, the possible environments in which a component model may be
correctly instantiated.
10 Partnerships and cooperations
10.1 International research visitors
The visit of Inigo Incer Romeo, PhD student at U. Berkeley, initially planned in the Summer 2020 had to be postponed to 2021. This visit is supported by a Chateaubriand Fellowship grant of the French
Ministry of Foreign Affairs. The topics of the visit is on the use of Contract-based Reasoning to support the design of CPS systems.
10.2 National initiatives
10.2.1 Inria Project Lab (IPL): ModeliScale, Languages and Compilation for Cyber-Physical System Design
The project gathers researchers from three Inria teams (Hycomes, Parkas and Tripop), and from three other research labs in Paris area (ENSTA Paris-Tech, L2S-CNRS and LIX, École Polytechnique).
The main objective of ModeliScale is to advance modeling technologies (languages, compile-time analyses, simulation techniques) for CPS combining physical interactions, communication layers and
software components. We believe that mastering CPS comprising thousands to millions of components requires radical changes of paradigms. For instance, modeling techniques must be revised, especially
when physics is involved. Modeling languages must be enhanced to cope with larger models. This can only be done by combining new compilation techniques (to master the structural complexity of models)
with new mathematical tools (new numerical methods, in particular).
ModeliScale gathers a broad scope of experts in programming language design and compilation (reactive synchronous programming), numerical solvers (nonsmooth dynamical systems) and hybrid systems
modeling and analysis (guaranteed simulation, verification). The research program is carried out in close cooperation with the Modelica community as well as industrial partners, namely, Dassault
Systèmes as a Modelica/FMI tool vendor, and EDF and Engie as end users.
In 2020, two general meetings have been organized by visioconference, with presentations of the partners on new results related to hybrid systems modeling and verification.
Two PhDs are funded by the ModeliScale IPL. Both started in October 2018:
• Christelle Kozaily has started a PhD, under the supervision of Vincent Acary (TRIPOP team at Inria Grenoble), Benoît Caillaud, Khalil Ghorbal on the structural and numerical analysis of
non-smooth DAE systems. She is located in the Hycomes team at Inria Rennes.
• Ismail Lahkim-Bennani has started a PhD under the supervision of Goran Frehse (ENSTA ParisTech.) and Marc Pouzet (PARKAS team, INRIA/ENS Paris). His PhD topic is on random testing of hybrid
systems, using techniques inspired by QuickCheck 39.
10.2.2 FUI ModeliScale: Scalable Modeling and Simulation of Large Cyber-Physical Systems
Participants: Albert Benveniste, Benoît Caillaud, Mathias Malandain, Bertrand Provot.
FUI ModeliScale is a French national collaborative project coordinated by Dassault Systèmes. The partners of this project are: EDF and Engie as main industrial users; DPS, Eurobios and PhiMeca are
SME providing mathematical modeling expertise; CEA INES (Chambéry) and Inria are the academic partners. The project started January 2018, for a maximal duration of 42 months. Three Inria teams are
contributing to the project : Hycomes, Parkas (Inria Paris / ENS) and Tripop (Inria Grenoble / LJK).
The focus of the project is on the scalable analysis, compilation and simulation of large Modelica models. The main contributions expected from Inria are:
• A novel structural analysis algorithm for multimode DAE systems, capable of handling large systems of guarded equations, that do not depend on the enumeration of a possibly exponential number of
• The partitioning and high-performance distributed co-simulation of large Modelica models, based on the results of the structural analysis.
In 2020, the effort has been put on the first objective, and in particular the improvement of the scalability of the algorithms implemented in the IsamDAE software (https://team.inria.fr/hycomes/
software/isamdae/). The performance of the tool has been improved by two orders of magnitude on some examples. This has allowed us to perform the structural analysis of multimode models of up to 2300
equations en ${10}^{77}$ modes.
A coupling of IsamDAE with Dymola (Dassault Système's commercial implementation of the Modelica language) has been implemented by Dassault Systèmes AB (Lund, Sweden), and is currently under test at
the time of writing of this activity report.
11 Dissemination
11.1 Promoting scientific activities
11.1.1 Scientific events: organisation
General chair, scientific chair
Khalil Ghorbal was the co-Chair of the NSAD Workshop (satellite of the SPLASH 2020 Event). https://2020.splashcon.org/home/nsad-2020
Abstract domains are a key notion in Abstract Interpretation theory and practice. They embed the semantic choices, data-structures and algorithmic aspects, and implementation decisions. The Abstract
Interpretation framework provides constructive and systematic formal methods to design, compose, compare, study, prove, and apply abstract domains. Many abstract domains have been designed so far:
numerical domains (intervals, congruences, polyhedra, polynomials, etc.), symbolic domains (shape domains, trees, etc.), but also domain operators (products, powersets, completions, etc.), and have
been applied to several kinds of static analyses (safety, termination, probability, etc.) on a variety of systems (hardware, software, neural networks, etc.). The goal of NSAD workshop is to discuss
work in progress, recent advances, novel ideas, experiences in the theory, practice, application, implementation, and experimentation related to abstract domains and/or their combination. This year’s
edition in particular welcomes abstract domains related and/or applied to analyzing neural networks, dynamical and hybrid systems.
11.1.2 Scientific events: selection
Member of the conference program committees
Benoît Caillaud has served on the program committee of FDL'20, a workshop on the domain-specific languages. The workshop took place with both physical (in Kiel, Germany) and virtual attendance (by
11.1.3 Journal
Reviewer - reviewing activities
• Benoît Caillaud has reviewed papers for ACM-TECS and IEEE-TAC;
• Khalil Ghorbal served as a reviewer for IEEE-TAC, IEEE-TECS, LITES, ICALP, Automatica;
• Mathias Malandain served as a reviewer for the Asian Modelica Conference 2020;
• Albert Benveniste served as a reviewer for the journals Transactions on Software Engineering, Discrete Event Dynamic Systems, Automatica and the American Control Conference.
11.1.4 Invited talks
• Benoît Caillaud has given a talk on switched DAE systems at the students seminar of the Department of Applied Mathematics at the ENS Paris-Saclay,
• Khalil Ghorbal was invited to give a talk at the PolySys Seminar (LIP6, Symbolic Computation),
• Khalil Ghorbal was invited to give a talk at the University of Perpignan (Computer Science, Mathematics and Physics departments),
• Khalil Ghorbal was invited to give a talk at the RWTH Aachen University (Computer Science and control departments).
11.1.5 Scientific expertise
• Albert Benveniste was chair of the Scientific Council of Orange Labs. His term terminated by end of february 2020.
• Albert Benveniste is member of the Scientific Council of Safran. In the period January to March 2020, together with Nikos Paragios (also member of the council), he prepared a report on the
impacts and opportunities for AI in Safran, and the way forward, both internally and with the ecosystem.
• Albert Benveniste participated to the activities of the Academy of technologies (Pôle Numérique) since march 2020. More specifically, he started a Working Group on Crisis Management methods and
tools targeting the COVID pandemia. The moto is: modeling should be extended beyond epidemiology. In 2020, long interviews with demos were held with Dassault-Systèmes, Thales, IBM, and the
startup Causality Link. The report should be issued early in 2021.
11.1.6 Research administration
Benoît Caillaud is head of the Programming Languages and Software Engineering department of IRISA (UMR 6074). Part of his duties has been the preparation of the evaluation of IRISA, planned March
11.2 Teaching - Supervision - Juries
11.2.1 Teaching
• Master : Khalil Ghorbal, Category Theory, Monads, and Computation, M2, (enseignant principal), 30h EqTD, ENS Rennes, France
• Master : Khalil Ghorbal, Modeling Physics with Differential-Algebraic Equations, M2, (enseignant principal), 25h EqTD, Ecole Polytechnique, Palaiseau, France
• Licence : Mathias Malandain taught linear algebra and integration in multivariable calculus to 1st and 2nd-year students at the University of Rennes 1 (36 hours).
11.2.2 Supervision
• PhD: Christelle Kozaily, Structural analysis of nonsmooth dynamical systems, university of Rennes 1, co-supervised by Vincent Acary (Tripop 5 team at Inria Grenoble), Benoît Caillaud and Khalil
Ghorbal, started October 2018.
• PhD: Aurélien Lamercerie, Formal analysis of cyber-physical systems requirements expressed in natural language, university of Rennes 1, co-supervised by par Benoît Caillaud et Annie Forêt (SemLIS
6 team of IRISA), started December 2017. His PhD defence is planned April 2021.
• PhD: Joan Thibault, Structural Analysis Techniques for Binary Decision Diagrams, university of Rennes 1, co-supervised by Benoît Caillaud and Khalil Ghorbal.
• Internship M1: Julien Duron, on Graph Combinatorial Optimization using Structural Analysis methods for Boolean Functions, ENS Rennes, co-supervised by Joan Thibault and Khalil Ghorbal.
12 Scientific production
12.1 Major publications
• 1 articleBuilding a Hybrid Systems Modeler on Synchronous Languages PrinciplesProceedings of the IEEE1069September 2018, 1568--1592
• 2 articleNon-standard semantics of hybrid systems modelersJournal of Computer and System Sciences783This work was supported by the SYNCHRONICS large scale initiative of INRIA2012, 877-910
• 3 articleContracts for System DesignFoundations and Trends in Electronic Design Automation122-32018, 124-400
• 4 articleA Formally Verified Hybrid System for Safe Advisories in the Next-Generation Airborne Collision Avoidance SystemInternational Journal on Software Tools for Technology Transfer196November
2017, 717-741
• 5 articleOperational Models for Piecewise-Smooth SystemsACM Transactions on Embedded Computing Systems (TECS)165sOctober 2017, 185:1--185:19
12.2 Publications of the year
International journals
• 6 article The mathematical foundations of physical systems modeling languages Annual Reviews in Control December 2020
• 7 articleUnveiling the implicit knowledge, one scenario at a timeVisual Computer2020, 1-12
International peer-reviewed conferences
• 8 inproceedingsImplicit structural analysis of multimode DAE systemsHSCC 2020 - 23rd ACM International Conference on Hybrid Systems: Computation and ControlSydney New South Wales Australia,
FranceApril 2020, 1-11
• 9 inproceedingsAn Algebra of Deterministic Propositional Acceptance Automata (DPAA)FDL 2020 - Forum on specification & Design LanguagesKiel, GermanySeptember 2020, 1-8
Conferences without proceedings
• 10 inproceedingsARES : un extracteur d'exigences pour la modélisation de systèmesEGC 2020 - Extraction et Gestion des Connaissances (Atelier - Fouille de Textes - Text Mine)Bruxelles, Belgium
January 2020, 1-4
• 11 inproceedingsTransduction sémantique pour la modélisation de systèmePFIA 2020 - Plate-Forme de l'Intelligence Artificielle (PFIA), rencontres RJCIAAngers, FranceJune 2020, 1-6
Reports & preprints
• 12 reportStructural Analysis of Multimode DAE Systems: summary of resultsInria Rennes – Bretagne AtlantiqueJanuary 2021, 27
• 13 reportThe Mathematical Foundations of Physical Systems Modeling LanguagesInriaApril 2020, 112
• 14 reportMixed Nondeterministic-Probabilistic InterfacesInria Rennes Bretagne Atlantique; Aalborg University; Université de Toulouse 3 Paul SabatierNovember 2020, 40
• 15 report Implicit Structural Analysis of Multimode DAE Systems Inria Rennes - Bretagne Atlantique; IRISA, Université de Rennes February 2020
• 16 report Ordered Functional Decision Diagrams: A Functional Semantic For Binary Decision Diagrams Inria 2020
Other scientific publications
• 17 miscDemo: IsamDAE, an Implicit Structural Analysis Tool for Multimode DAE SystemsSydney, AustraliaApril 2020,
12.3 Cited publications
• 18 article 20 Years of Modal and Mixed Specifications Bulletin of European Association of Theoretical Computer Science 1 94 2008
• 19 book Principles of Model Checking MIT Press, Cambridge 2008
• 20 articleBuilding a Hybrid Systems Modeler on Synchronous Languages PrinciplesProceedings of the IEEE1069September 2018, 1568--1592
• 21 misc A Type-Based Analysis of Causality Loops In Hybrid Systems Modelers Deliverable D3.1_1 v 1.0 of the Sys2soft collaborative project ''Physics Aware Software'' December 2013
• 22 misc Semantics of multi-mode DAE systems Deliverable D.4.1.1 of the ITEA2 Modrio collaborative project August 2013
• 23 inproceedings Multiple Viewpoint Contract-Based Specification and Design Proceedings of the Software Technology Concertation on Formal Methods for Components and Objects (FMCO'07) 5382 Revised
Lectures, Lecture Notes in Computer Science Amsterdam, The Netherlands Springer October 2008
• 24 inproceedingsA type-based analysis of causality loops in hybrid modelersHSCC '14: International Conference on Hybrid Systems: Computation and ControlProceedings of the 17th international
conference on Hybrid systems: computation and control (HSCC '14)Berlin, GermanyACM PressApril 2014, 13
• 25 inproceedingsA Compositional Approach on Modal Specifications for Timed Systems11th International Conference on Formal Engineering Methods (ICFEM'09)5885LNCSRio de Janeiro, BrazilSpringer
December 2009, 679-697URL: http://hal.inria.fr/inria-00424356/en
• 26 articleModal event-clock specifications for timed component-based designScience of Computer Programming2011, URL: http://dx.doi.org/10.1016/j.scico.2011.01.007
• 27 inproceedingsRefinement and Consistency of Timed Modal Specifications3rd International Conference on Language and Automata Theory and Applications (LATA'09)5457LNCSTarragona, SpainSpringer
April 2009, 152-163URL: http://hal.inria.fr/inria-00424283/en
• 28 inproceedingsA proposal for real-time interfaces in SPEEDSDesign, Automation and Test in Europe (DATE'10)IEEE2010, 441-446
• 29 articleModelling of Complex Systems: Systems as Dataflow MachinesFundam. Inform.9122009, 251--274
• 30 phdthesis Un cadre formel pour l'étude des systèmes industriels complexes: un exemple basé sur l'infrastructure de l'UMTS Ecole Polytechnique 2006
• 31 articleGraphical Versus Logical SpecificationsTheor. Comput. Sci.10611992, 3-20
• 32 inproceedingsCompositional design methodology with constraint Markov chainsQEST 2010Williamsburg, Virginia, United StatesSeptember 2010, URL: http://hal.inria.fr/inria-00591578/en
• 33 articleConstraint Markov ChainsTheoretical Computer Science41234May 2011, 4373-4404URL: http://hal.inria.fr/hal-00654003/en
• 34 inproceedings Implicit Structural Analysis of Multimode DAE Systems 23rd ACM International Conference on Hybrid Systems: Computation and Control (HSCC 2020) to appear Sydney, Australia April
• 35 articleThe index of general nonlinear DAEsNumerische Mathematik722dec 1995, 173--196URL: http://dx.doi.org/10.1007/s002110050165
• 36 phdthesisA Framework for Compositional Design and Analysis of SystemsEECS Department, University of California, BerkeleyDec 2007, URL: http://www.eecs.berkeley.edu/Pubs/TechRpts/2007/
• 37 inproceedingsResource InterfacesEMSOFT2003, 117-133
• 38 inproceedingsCharacterization of Temporal Property ClassesICALP1992, 474-486
• 39 inproceedingsQuickCheck: a lightweight tool for random testing of Haskell programsProceedings of the Fifth ACM SIGPLAN International Conference on Functional Programming (ICFP '00), Montreal,
Canada, September 18-21, 2000.2000, 268--279URL: https://doi.org/10.1145/351240.351266
• 40 book Model Checking MIT Press 1999
• 41 book N. Cutland Nonstandard analysis and its applications Cambridge Univ. Press 1988
• 42 inproceedingsECDAR: An Environment for Compositional Design and Analysis of Real Time SystemsAutomated Technology for Verification and Analysis - 8th International Symposium, ATVA 2010,
Singapore, September 21-24, 2010. Proceedings2010, 365-370
• 43 inproceedingsTimed I/O automata: a complete specification theory for real-time systemsProceedings of the 13th ACM International Conference on Hybrid Systems: Computation and Control, HSCC
2010, Stockholm, Sweden, April 12-15, 20102010, 91-100
• 44 inproceedingsAbstract Probabilistic AutomataVMCAI2011, 324-339
• 45 book Analyse non standard Hermann 1989
• 46 book Trace Theory for Automatic Hierarchical Verification of Speed-Independent Circuits ACM Distinguished Dissertations MIT Press 1989
• 47 articleTheoretical improvements in algorithmic efficiency for network flow problemsJournal of the ACM1921972, 248--264URL: http://dx.doi.org/10.1145/321694.321699
• 48 inproceedings Modelica extensions for Multi-Mode DAE Systems Proceedings of the 10th International Modelica Conference, March 10-12, 2014, Lund, Sweden Linköping University Electronic Press
mar 2014
• 49 inproceedings Modia-dynamic modeling and simulation with julia Juliacon'18 University College London, UK August 2018
• 50 inproceedingsSurvey of industrial applications of embedded model predictive control2016 European Control Conference (ECC)2016, 601-601
• 51 inproceedingsA new approach to the maximum flow problemProceedings of the eighteenth annual ACM symposium on Theory of computing (STOC'86)1986, URL: http://dx.doi.org/10.1145/12130.12144
• 52 miscIEEE Standard VHDL Analog and Mixed-Signal Extensions, Std 1076.1-19991999, URL: http://dx.doi.org/10.1109/IEEESTD.1999.90578
• 53 inproceedingsModeling Time in Hybrid Systems: How Fast Is ``Instantaneous''?IJCAI1995, 1773--1781
• 54 inproceedingsFormal verification of ACAS X, an industrial airborne collision avoidance system2015 International Conference on Embedded Software, EMSOFT 2015, Amsterdam, Netherlands, October
4-9, 20152015, 127--136
• 55 articleProving the Correctness of Multiprocess ProgramsIEEE Trans. Software Eng.321977, 125-143
• 56 inproceedingsOn Modal Refinement and ConsistencyProc. of the 18th International Conference on Concurrency Theory (CONCUR'07)Springer2007, 105--119
• 57 inproceedingsA Modal Process LogicProceedings of the Third Annual Symposium on Logic in Computer Science (LICS'88)IEEE1988, 203-210
• 58 incollectionAn Invitation to Nonstandard AnalysisNonstandard Analysis and its ApplicationsCambridge Univ. Press1988, 1--105
• 59 inproceedingsInput/Output Automata: Basic, Timed, Hybrid, Probabilistic and DynamicCONCUR2003, 187-188
• 60 articleA Proof of the Kahn Principle for Input/Output AutomataInf. Comput.8211989, 81-92
• 61 book Temporal verification of reactive systems: Safety Springer 1995
• 62 articleApplying "Design by Contract"Computer2510October 1992, 40--51URL: http://dx.doi.org/10.1109/2.161279
• 63 articleMethodology for the Design of Analog Integrated Interfaces Using ContractsIEEE Sensors Journal1212Dec. 2012, 3329--3345
• 64 articleThe consistent initialization of differential-algebraic systemsSIAM J. Sci. Stat. Comput.921988, 213--231
• 65 articleA Simple Structural Analysis Method for DAEsBIT Numerical Mathematics412March 2001, 364--394URL: http://dx.doi.org/10.1023/a:1021998624799
• 66 articleA Modal Interface Theory for Component-based DesignFundamenta Informaticae1081-22011, 119-149URL: http://hal.inria.fr/inria-00554283/en
• 67 book Non-Standard Analysis ISBN 0-691-04490-2 Princeton Landmarks in Mathematics 1996
• 68 articleIndustry needs and research directions in requirements engineering for embedded systemsRequirements Engineering172012, 57--78URL: http://link.springer.com/article/10.1007/
• 69 inproceedingsGame Models for Open SystemsVerification: Theory and Practice2772Lecture Notes in Computer ScienceSpringer2003, 269-289
• 70 inproceedingsInterface automataProc. of the 9th ACM SIGSOFT International Symposium on Foundations of Software Engineering (FSE'01)ACM Press2001, 109--120
• 71 inproceedings Interface-based design In Engineering Theories of Software Intensive Systems, proceedings of the Marktoberdorf Summer School Kluwer 2004
• 72 inproceedingsTimed InterfacesProc. of the 2nd International Workshop on Embedded Software (EMSOFT'02)2491Lecture Notes in Computer ScienceSpringer2002, 108--122 | {"url":"https://radar.inria.fr/report/2020/hycomes/uid0.html","timestamp":"2024-11-05T07:26:10Z","content_type":"text/html","content_length":"349545","record_id":"<urn:uuid:a4474e72-3adf-44d0-8421-a1a5b177b42e>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00278.warc.gz"} |
Mathieu groupoid
You have probably encountered the 15-puzzle, where all but one of the squares of a 4 × 4 grid are occupied by counters. The only acceptable moves are to move a counter into an adjacent unoccupied
Sam Loyd challenged people to swap a pair of adjacent counters. It is relatively easy to show that this is impossible, as changing the parity of the permutation of the sixteen objects (fifteen
counters and one empty space) also changes the parity of the position of the empty space.
Unlike the Rubik’s cube, the set of positions in this puzzle do not form a group. Instead, they form a groupoid, since two operations can only be composed if the final position of the empty space in
the first operation is equal to the initial position of the empty space in the second operation. Groupoids have the same axioms as groups, except for totality; it is not always possible to compose
two arbitrary elements.
Conway, Elkies and Martin created a similar, albeit more interesting, puzzle. Place a counter on each of 12 of the 13 points of the projective plane of order 3. The plane has 13 lines of four points,
and the following operation is allowed:
• Select a counter, move it into the empty space, and swap the two remaining counters on that line.
The groupoid generated is known as M13, since it has connections with the Mathieu groups. In particular, the subgroup of positions fixing the empty space is isomorphic to M12. M12 itself has been
realised as the group of positions accessible with a mechanical puzzle. In fact, I know of at least three such descriptions, which are completely different:
• Twelve ball-bearings touch a central sphere in an icosahedral arrangement. M12 is generated by pairs of clockwise and anticlockwise twists (where we choose an equator orthogonal to a diameter
joining opposite spheres, and rotate one of the two hemispheres).
• The Number Planet mechanical puzzle by Oskar van Deventer and Jim Stasheff:
There are electronic implementations of M12, M24 and Co0 as puzzles, designed by Igor Kriz whom you may recognise from his work in Euclidean Ramsey theory.
0 Responses to Mathieu groupoid
1. I understand “Representation Theory” to mean “Study of the representation of group operations by matrix multiplication”. In these puzzles, the (initially atomic) group operation is split into
(admittedly, coupled) operations on local parts of the puzzle. Is there a connection to representation theory? Or is there a variant theory which one might call “representation by puzzles” or
“representation by coupled automata” theory, which is simply difficult to google because of the dominance of the usual linear algebra sense of “representation”?
□ These are called `permutation representations’, which are special cases of linear matrix representations.
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Finding the Area of a Rectangle Using the Distance between Two Points Formula
Question Video: Finding the Area of a Rectangle Using the Distance between Two Points Formula
Given that π ΄π ΅π Άπ is a rectangle, find the length of line segment π ΄π Ά and the area of the rectangle.
Video Transcript
Given that π ΄π ΅π Άπ is a rectangle, find the length of line segment π ΄π Ά and the area of the rectangle.
In this question, we can see this rectangle π ΄π ΅π Άπ and we need to find a length and the area. We might notice that weβ re not given any lengths in this diagram. So in order to answer this
question, weβ ll need to look a little bit more closely at the vertices.
Firstly, this vertex of π must be the coordinate or ordered pair of zero, zero. This other important piece of information that π ΅ is at 12, five will help us work out the coordinates of vertices
π ΄ and π Ά. We know that the line π π ΄ lies on the π ₯-axis, which is a horizontal line. And so the line π ΅π Ά must also be a horizontal line.
We must remember that the definition of a rectangle is that all interior angles are 90 degrees. So weβ ll have right angles at angles π , π ΄, π ΅, and π Ά. And we can be sure that π π Ά and
π ΄π ΅ will be vertical lines. So letβ s look at this vertex π ΄ in relation to π ΅. We know that π ΅ lies at 12 on the π ₯-axis, and therefore π ΄ must also lie at 12 on the π ₯-axis. We
can see that point π ΄ sits on the π ₯-axis, so the π ¦-value will be zero. We can alternatively think of this that the line segment π π ΄ must be 12 units long.
Next, letβ s think about this vertex π Ά. It lies on the π ¦-axis. Therefore, the π ₯-coordinate will be zero, and the π ¦-value will be the same as that of π ΅. So the coordinate of π ΅ must
be zero, five. Now that weβ ve found the coordinate values of all four vertices, letβ s see if we can find the length of this line segment π ΄π Ά. This is the diagonal of the rectangle going from
π ΄ to π Ά.
As we know the coordinate values for vertices π ΄ and π Ά, we can recall the formula for the distance between two points. This tells us that, for any pair of coordinates, π ₯ one, π ¦ one and π
₯ two, π ¦ two, the distance can be found by the square root of π ₯ two minus π ₯ one all squared plus π ¦ two minus π ¦ one all squared. So letβ s take our two vertices. We can make π ΄ our π
₯ one, π ¦ one values and π Ά our π ₯ two, π ¦ two values. But it doesnβ t matter which way round we write these just so long as we take consistently from each coordinate.
So plugging these into the formula would give us π equals the square root of zero minus 12 squared plus five minus zero squared. Simplifying this, we have the square root of negative 12 squared
plus five squared. Negative 12 squared will give us 144, and five squared gives us 25. So adding these together would give us 169. Finding the square root of 169 would give us a value of 13. Even
though weβ ve done some squaring, weβ re not finding an area, so the units here will be length units. Therefore, weβ ve found our first answer that the length of line segment π ΄π Ά would be 13
length units.
Next, letβ s see how we can find the area of this rectangle. We should remember that to find the area of a rectangle, we multiply the length by the width. Earlier in the question, we established
that π π ΄ must be 12 units long as π ΄ lies at 12 on the π ₯-axis and π lies at zero on the π ₯-axis. In a similar way, we can look at the length of the line segment π ΄π ΅ and observe
that it goes from zero on the π ¦-axis to five on the π ¦-axis, which means that the line segment π ΄π ΅ must be five units long.
Therefore, in order to find the area, we would take our length and width of 12 and five and multiply them together, which gives us a value of 60. And this time, as weβ re working with an area, then
our units will be area units. So here we gave our two answers. The line segment π ΄π Ά was 13 length units, and the area of the rectangle is 60 area units. | {"url":"https://www.nagwa.com/en/videos/973192971670/","timestamp":"2024-11-03T02:52:16Z","content_type":"text/html","content_length":"246042","record_id":"<urn:uuid:d5e56eed-6177-4b35-a13b-aba86ee897d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00320.warc.gz"} |
Hex to Binary Converter: Convert Hexadecimal Numbers
Hex to Binary
Convert hexadecimal numbers to binary seamlessly with our Hex to Binary Converter. Perfect for programming and digital systems, ensuring accurate conversions.
Hexadecimal (Hex) Number:Hexadecimal is a base-sixteen numeral system that uses sixteen symbols: 0-9 and A-F, where A represents 10, B represents 11, and so on up to F representing 15. It's commonly
used in computing for representing binary-coded values in a more compact form.
Hexadecimal Number Example:An example of a hexadecimal number is 2F8A. Each digit in a hexadecimal number represents a power of 16, with the rightmost digit representing 16^0, the next representing
16^1, and so on.
Binary Number:A binary number is a numeral system based on base-two, using only two digits: 0 and 1. It's commonly used in computing and digital electronics.
Binary Number Example:An example of a binary number is 110101. Each digit represents a power of 2, with the rightmost digit representing 2^0, the next representing 2^1, and so on.
Conversion Process: Hexadecimal to Binary
1. Assign Binary Values to Hexadecimal Digits: Assign binary values to each hexadecimal digit based on its position and the base-16 system.
2. Combine Binary Digits: Combine the binary values obtained from each hexadecimal digit to form the final binary number.
Example: Convert Hexadecimal 2F8A to Binary
Hexadecimal Number: 2F8A
Assign Binary Values:
2 (0010 in binary)
F (1111 in binary)
8 (1000 in binary)
A (1010 in binary)
Combine Binary Digits:
Therefore, the binary representation of the hexadecimal number 2F8A is 0010111110001010. | {"url":"https://toolsfairy.com/number-utilities/hex-to-binary","timestamp":"2024-11-02T15:37:40Z","content_type":"text/html","content_length":"25848","record_id":"<urn:uuid:304b1238-9e25-401e-9cfb-ce916f6f0743>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00186.warc.gz"} |
Variance on task list
I have a sheet that is a running list of tasks that gets added to weekly. I am looking to figure out how to show how early, or late, a task is completed in relation to the due date. If the task is on
time, I want to show 0.
I have been trying to use this formula
=NETWORKDAYS([Due Date]@row, [Actual Finish]@row) - 1
This works fine if the task is late. If the task is early, then the calculation is off by several days.
I tried combining several formulas based on what I have read elsewhere in the forum and now I just get #INCORRECT ARGUMENT.
=IF([Actual Finish]@row = [Due Date]@row, 0, IF([Due Date]@row < [Actual Finish]@row), =NETWORKDAYS([Due Date]@row, [Actual Finish]@row)-1, IF([Due Date]@row, [Actual Finish]@row) > 0, =NETWORKDAYS
([Due Date]@row, [Actual Finish]@row)+1)
Any suggestions?
Best Answer
• Hi @M38a1
There's a couple things to clean up here but you're on the right track!
Your first statement is correct, If this, then 0, Otherwise, this other formula.
However for your second and third IF statements, we'll want to adjust where your closing parentheses are and take away some of your = signs.
I'll break down each statement individually for your three outcomes, then we'll put them together.
First Statement - No Change:
=IF([Actual Finish]@row = [Due Date]@row, 0,
Second Statement - take out the ) and =
IF([Due Date]@row < [Actual Finish]@row, NETWORKDAYS([Due Date]@row, [Actual Finish]@row) -1,
Final Statement - we can jump right to the only other option you want to return
NETWORKDAYS([Due Date]@row, [Actual Finish]@row) +1
All Together:
=IF([Actual Finish]@row = [Due Date]@row, 0, IF([Due Date]@row < [Actual Finish]@row, NETWORKDAYS([Due Date]@row, [Actual Finish]@row) -1, NETWORKDAYS([Due Date]@row, [Actual Finish]@row) +1))
Let me know if this makes sense and works for you!
Need more help? 👀 | Help and Learning Center
こんにちは (Konnichiwa), Hallo, Hola, Bonjour, Olá, Ciao! 👋 | Global Discussions
• Hi @M38a1
There's a couple things to clean up here but you're on the right track!
Your first statement is correct, If this, then 0, Otherwise, this other formula.
However for your second and third IF statements, we'll want to adjust where your closing parentheses are and take away some of your = signs.
I'll break down each statement individually for your three outcomes, then we'll put them together.
First Statement - No Change:
=IF([Actual Finish]@row = [Due Date]@row, 0,
Second Statement - take out the ) and =
IF([Due Date]@row < [Actual Finish]@row, NETWORKDAYS([Due Date]@row, [Actual Finish]@row) -1,
Final Statement - we can jump right to the only other option you want to return
NETWORKDAYS([Due Date]@row, [Actual Finish]@row) +1
All Together:
=IF([Actual Finish]@row = [Due Date]@row, 0, IF([Due Date]@row < [Actual Finish]@row, NETWORKDAYS([Due Date]@row, [Actual Finish]@row) -1, NETWORKDAYS([Due Date]@row, [Actual Finish]@row) +1))
Let me know if this makes sense and works for you!
Need more help? 👀 | Help and Learning Center
こんにちは (Konnichiwa), Hallo, Hola, Bonjour, Olá, Ciao! 👋 | Global Discussions
• Thank you Genevieve! Your answer works brilliantly. And thank you for taking the time to break it out into individual statements so I can better understand the process for the next set of
formulas that I am working on.
Best regards,
• No problem, Eric! I'm glad I could help. 🙂
Need more help? 👀 | Help and Learning Center
こんにちは (Konnichiwa), Hallo, Hola, Bonjour, Olá, Ciao! 👋 | Global Discussions
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/86303/variance-on-task-list","timestamp":"2024-11-12T12:43:09Z","content_type":"text/html","content_length":"415391","record_id":"<urn:uuid:15345df7-9cf3-43b2-9927-1c0c81ba8cf7>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00131.warc.gz"} |
t-Test, Chi-Square, ANOVA, Regression, Correlation... (2024)
This tutorial is about z-standardization (z-transformation). We will discuss what the z-score is, how z-standardization works, and what the standard normal distribution is. In addition, the z-score
table is discussed and what it's used for.
What is z-standardization?
Z-standardization is a statistical procedure used to make data points from different datasets comparable. In this procedure, each data point is converted into a z-score. A z-score indicates how many
standard deviations a data point is from the mean of the dataset.
Example of z-standardization
Suppose you are a doctor and want to examine the blood pressure of your patients. For this purpose, you measured the blood pressure of a sample of 40 patients. From the measured data, you can now
naturally calculate the average, i.e., the value that the 40 patients have on average.
Now one of the patients asks you how high his blood pressure is compared to the others. You tell him that his blood pressure is 10mmHg above average. Now the question arises, whether 10mmHg is a lot
or a little.
If the other patients cluster very closely around the mean, then 10mmHg is a lot in relation to the spread, but if the other patients spread very widely around the mean, then 10mmHg might not be that
The standard deviation tells us how much the data is spread out. If the data are close to the mean, we have a small standard deviation; if they are widely spread, we have a large standard deviation.
Let's say we get a standard deviation of 20 mmHg for our data. This means that on average, the patients deviate by 20 from the mean.
The z-score now tells us how far a person is from the mean in units of standard deviation. So, a person who deviates one standard deviation from the mean has a z-score of 1. A person who is twice as
far from the mean has a z-score of 2. And a person who is three standard deviations from the mean has a z-score of 3.
Accordingly, a person who deviates by minus one standard deviation has a z-score of -1, a person who deviates by minus two standard deviations has a z-score of -2, and a person who deviates by minus
three standard deviations has a z-score of -3.
And if a person has exactly the value of the mean, then they deviate by zero standard deviations from the mean and receive a score of zero.
Thus, the z-score indicates how many standard deviations a measurement is from the mean. As mentioned, the standard deviation is just a measure of the dispersion of the patients' blood pressure
around the mean.
In short, the z-score helps us understand how exceptional or normal a particular measurement is compared to the overall average.
Calculating the z-score
How do we calculate the z-score? We want to convert the original data, in our case the blood pressure, into z-scores, i.e., perform a z-standradiszation.
Here we see the formula for z-standardization. Here, z is of course the z-value we want to calculate, x is the observed value, in our case the blood pressure of the person in question, μ is the mean
value of the sample, in our case the mean value of all 40 patients, and σ is the standard deviation of the sample, i.e. the standard deviation of our 40 patients.
Caution: μ and σ are actually the mean and standard deviation of the population, but in our case we only have a sample. However, under certain conditions, which we will discuss later, we can estimate
the mean and standard deviation using the sample.
Let's assume that the 40 patients in our example have a mean value of 130 and a standard deviation of 20. If we use both values, we get for z: x-130 divided by 20
Now we can use the blood pressure of each individual patient for x and calculate the z value. Let's just do this for the first patient. Let's say this patient has a blood pressure of 97, then we
simply enter 97 for x and get a z-value of -1.65.
This person therefore deviates from the mean by -1.65 standard deviations. We can now do this for all patients.
Regardless of the unit of the initial data, we now have an overview in which we can see how far a person deviates from the mean in units of the standard deviation.
Now, of course, we only have a sample that comes from a specific population. But if the data is normally distributed and the sample size is greater than 30, then we can use the z-value to say what
percentage of patients have a blood pressure lower than 110, for example, and what percentage have a blood pressure higher than 110.
But how does this work? If the initial data is normally distributed, we obtain a so-called standard normal distribution through z-standardization.
The standard normal distribution is a specific type of normal distribution with a mean value of 0 and a standard deviation of 1.
The special feature is that any normal distribution, regardless of its mean or standard deviation, can be converted into a standard normal distribution.
Since we now have a standardized distribution, all we really need is a table that tells us what percentage of the values are below this value for as many z-values as possible .
And you can find such a table in almost every statistics book or here: Table of the z-distribution. Now, of course, the question is how to read this table?
If, for example, we have a z-value of -2, then we can read a value of 0.0228 from this table.
This means that 2.28% of the values are smaller than a z-value of -2. As the sum is always 100% or 1, 97.72% of the values are greater.
And with a z-value of zero, we are exactly in the middle and get a value of 0.5. Therefore 50% of the values are smaller than a z-value of 0 and 50% of the values are greater than 0. As the normal
distribution is symmetrical, we can read off the probabilities for positive z-values exactly.
If we have a z-value of 1, we only need to search for -1. However, we must note that in this case we get a value that tells us what percentage of the values are greater than the z-value. So with a
z-value of 1, 15.81% of the values are larger and 84.14% of the values are smaller.
But what if, for example, we want to read a z-value of -1.81 in the table? We need the other columns for this. We can read a z-value of -1.81 at -1.8 and at 0.01.
Now let's look at the example about blood pressure again. For example, if we want to know what percentage of patients have a blood pressure below 123, we can use z-standardization to convert a blood
pressure of 123 into a z-value, in this case we get a z-value of -0.35.
Now we can take the table with the z-distributions and search for a z-value of -0.35. Here we have a value of 0.3632. This means that 36.32 percent of the values are smaller than a z-value of -0.35
and 63.68 percent are larger.
Compare different data sets with the z-score
However, there is another important application for z-standardization. The z-standardization can help to make values measured in different ways comparable. Here is an example.
Suppose we have two classes, class A and class B, who have written a different test in mathematics.
The tests are designed differently, have a different level of difficulty and a different maximum score.
In order to be able to compare the performance of the pupils in the two classes fairly, we can apply the z-standardization.
The average score or mean score for class A was 70 points with a standard deviation of 10 points. The average score for the test in class B was 140 points with a standard deviation of 20 points.
We now want to compare the performance of Max from class A, who scored 80 points, with the performance of Emma from class B, who scored 160 points.
To do this, we simply calculate the z-value of Max and Emma. We enter 80 once for x and get a z-value of 1. Then we enter 160 for x and also get a z-value of 1.
The z-values of Max and Emma are therefore the same. This means that both students performed equally well in terms of average performance and dispersion in their respective classes. Both are exactly
one standard deviation above the mean of their class.
But what about the assumptions? Can we simply calculate a z-standardization and use the table of the standard normal distribution?
The z-standardization itself, i.e. the conversion of the data points into z-values using this formula, is essentially not subject to any strict conditions. It can be carried out independently of the
data distribution.
However, if we use the resulting z-values in the context of the standard normal distribution for statistical analyses (e.g. for hypothesis tests or confidence intervals), certain assumptions must be
The z-distribution assumes that the underlying population is normally distributed and that the mean (μ) and standard deviation (σ) of the population are known.
However, as you never have the entire population in practice and the mean value and standard deviation are usually not known, this requirement is of course often not met. Fortunately, however, there
is an alternative assumption.
Although the z-distribution is defined for normally distributed populations, the central limit theorem can be applied to large samples. This theorem states that the distribution of the sample
approaches a normal distribution if the sample size is greater than 30. Therefore, if the sample is larger than 30, the standard normal distribution can be used as an approximation and the mean and
standard deviation can be estimated using the sample.
When the standard deviation is estimated from the sample, s is usually written instead of σ and x dash instead of mu for the mean.
The z-standardization should not be confused with the z-test or the t-test. If you want to know what the t-test is, please watch the following video. | {"url":"https://deafdogsatlas.com/article/t-test-chi-square-anova-regression-correlation","timestamp":"2024-11-02T17:52:31Z","content_type":"text/html","content_length":"132697","record_id":"<urn:uuid:2ae7293c-6185-4a4e-9e08-db99fab72cde>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00866.warc.gz"} |
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User-defined Operand: Modulo operator | Zemax Community
Hi all,
I’m not going to create a repository for this, but if you quickly need a modulo operator in your Merit Function, there you have it. Hx and Hy are the operand numbers (integer) that will be used as
the numerator and denominator respectively.
The relevant part of the code is self-explanatory (hopefully):
int numerator = (int)TheApplication.OperandArgument1;
int denominator = (int)TheApplication.OperandArgument2;
double numerator_value = TheSystem.MFE.GetOperandAt(numerator).Value;
double denominator_value = TheSystem.MFE.GetOperandAt(denominator).Value;
operandResultsl0] = numerator_value % denominator_value;
To install, download the ZIP file, extract it, and move UDOC12.exe to your
folder. Then, in the MFE you can do:
The rest of the division of 5.00 by 2.00 is 1.00.
In case you are wondering why I made this. I needed to have a linear polarization without caring about the orientation. I noticed that using CODA with data=110 (phase difference), some linear
polarization angles were not achievable somehow and if I tried known configurations that would produce a linear polarization at those unachievable angles, CODA with data=110 would report 2pi. So, to
make my original Merit function work, I’m just making sure that CODA with data=110 modulo 2pi is always zero :)
Hope this helps. Take care, | {"url":"https://community.zemax.com/code-exchange-10/user-defined-operand-modulo-operator-5040?postid=15811","timestamp":"2024-11-11T17:05:09Z","content_type":"text/html","content_length":"171371","record_id":"<urn:uuid:93646e74-d1f2-4390-8649-5d9b7e5320e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00344.warc.gz"} |
TreeSet Exercise Solution
Let's look at the solution of TreeSet exercises.
We'll cover the following
Problem 1: Numbers greater than 50
Given an array of numbers, find all the numbers that are greater than 50.
There is a tailSet(E e) method in the TreeSet class that returns a set of all the elements that are greater than the provided element. We will add all the elements from the array to a TreeSet. Then
we will find all the elements greater than 50 using the tailSet() method.
Get hands-on with 1200+ tech skills courses. | {"url":"https://www.educative.io/courses/collections-in-java/treeset-exercise-solution","timestamp":"2024-11-06T06:07:36Z","content_type":"text/html","content_length":"715142","record_id":"<urn:uuid:441638f3-44e5-4e04-9abb-13eabcabb390>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00001.warc.gz"} |
Principal Component Analysis for Visualization
Last Updated on October 20, 2021
Principal component analysis (PCA) is an unsupervised machine learning technique. Perhaps the most popular use of principal component analysis is dimensionality reduction. Besides using PCA as a data
preparation technique, we can also use it to help visualize data. A picture is worth a thousand words. With the data visualized, it is easier for us to get some insight and decide on the next step in
our machine learning models.
In this tutorial, you will discover how to visualize data using PCA, as well as using visualization to help determining the parameter for dimensionality reduction.
After completing this tutorial, you will know:
How to use visualize a high dimensional data
What is explained variance in PCA
Visually observe the explained variance from the result of PCA of high dimensional data
Let’s get started.
Tutorial Overview
This tutorial is divided into two parts; they are:
Scatter plot of high dimensional data
Visualizing the explained variance
For this tutorial, we assume that you are already familiar with:
How to Calculate Principal Component Analysis (PCA) from Scratch in Python
Principal Component Analysis for Dimensionality Reduction in Python
Scatter plot of high dimensional data
Visualization is a crucial step to get insight from data. We can learn from the visualization that whether a pattern can be observed and hence estimate which machine learning model is suitable.
It is easy to depict things in two dimension. Normally a scatter plot with x- and y-axis are in two dimensional. Depicting things in three dimensional is a bit challenging but not impossible. In
matplotlib, for example, can plot in 3D. The only problem is on paper or on screen, we need can only look at a 3D plot at one viewport or projection at a time. In matplotlib, this is controlled by
the degree of elevation and azimuth. Depicting things in four or five dimensions is impossible because we live in a three-dimensional world and have no idea of how things in such a high dimension
would look like.
This is where a dimensionality reduction technique such as PCA comes into play. We can reduce the dimension to two or three so we can visualize it. Let’s start with an example.
We start with the wine dataset, which is a classification dataset with 13 features and 3 classes. There are 178 samples:
from sklearn.datasets import load_wine
winedata = load_wine()
X, y = winedata[‘data’], winedata[‘target’]
(178, 13)
Among the 13 features, we can pick any two and plot with matplotlib (we color-coded the different classes using the c argument):
import matplotlib.pyplot as plt
plt.scatter(X[:,1], X[:,2], c=y)
or we can also pick any three and show in 3D:
ax = fig.add_subplot(projection=’3d’)
ax.scatter(X[:,1], X[:,2], X[:,3], c=y)
But these doesn’t reveal much of how the data looks like, because majority of the features are not shown. We now resort to principal component analysis:
from sklearn.decomposition import PCA
pca = PCA()
Xt = pca.fit_transform(X)
plot = plt.scatter(Xt[:,0], Xt[:,1], c=y)
plt.legend(handles=plot.legend_elements()[0], labels=list(winedata[‘target_names’]))
Here we transform the input data X by PCA into Xt. We consider only the first two columns, which contains the most information, and plot it in two dimensional. We can see that the purple class is
quite distinctive, but there is still some overlap. But if we scale the data before PCA, the result would be different:
from sklearn.preprocessing import StandardScaler
from sklearn.pipeline import Pipeline
pca = PCA()
pipe = Pipeline([(‘scaler’, StandardScaler()), (‘pca’, pca)])
Xt = pipe.fit_transform(X)
plot = plt.scatter(Xt[:,0], Xt[:,1], c=y)
plt.legend(handles=plot.legend_elements()[0], labels=list(winedata[‘target_names’]))
Because PCA is sensitive to the scale, if we normalized each feature by StandardScaler we can see a better result. Here the different classes are more distinctive. By looking at this plot, we are
confident that a simple model such as SVM can classify this dataset in high accuracy.
Putting these together, the following is the complete code to generate the visualizations:
from sklearn.datasets import load_wine
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.pipeline import Pipeline
import matplotlib.pyplot as plt
# Load dataset
winedata = load_wine()
X, y = winedata[‘data’], winedata[‘target’]
print(“X shape:”, X.shape)
print(“y shape:”, y.shape)
# Show any two features
plt.scatter(X[:,1], X[:,2], c=y)
plt.title(“Two particular features of the wine dataset”)
# Show any three features
fig = plt.figure(figsize=(10,8))
ax = fig.add_subplot(projection=’3d’)
ax.scatter(X[:,1], X[:,2], X[:,3], c=y)
ax.set_title(“Three particular features of the wine dataset”)
# Show first two principal components without scaler
pca = PCA()
Xt = pca.fit_transform(X)
plot = plt.scatter(Xt[:,0], Xt[:,1], c=y)
plt.legend(handles=plot.legend_elements()[0], labels=list(winedata[‘target_names’]))
plt.title(“First two principal components”)
# Show first two principal components with scaler
pca = PCA()
pipe = Pipeline([(‘scaler’, StandardScaler()), (‘pca’, pca)])
Xt = pipe.fit_transform(X)
plot = plt.scatter(Xt[:,0], Xt[:,1], c=y)
plt.legend(handles=plot.legend_elements()[0], labels=list(winedata[‘target_names’]))
plt.title(“First two principal components after scaling”)
If we apply the same method on a different dataset, such as MINST handwritten digits, the scatterplot is not showing distinctive boundary and therefore it needs a more complicated model such as
neural network to classify:
from sklearn.datasets import load_digits
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.pipeline import Pipeline
import matplotlib.pyplot as plt
digitsdata = load_digits()
X, y = digitsdata[‘data’], digitsdata[‘target’]
pca = PCA()
pipe = Pipeline([(‘scaler’, StandardScaler()), (‘pca’, pca)])
Xt = pipe.fit_transform(X)
plot = plt.scatter(Xt[:,0], Xt[:,1], c=y)
plt.legend(handles=plot.legend_elements()[0], labels=list(digitsdata[‘target_names’]))
Visualizing the explained variance
PCA in essence is to rearrange the features by their linear combinations. Hence it is called a feature extraction technique. One characteristic of PCA is that the first principal component holds the
most information about the dataset. The second principal component is more informative than the third, and so on.
To illustrate this idea, we can remove the principal components from the original dataset in steps and see how the dataset looks like. Let’s consider a dataset with fewer features, and show two
features in a plot:
from sklearn.datasets import load_iris
irisdata = load_iris()
X, y = irisdata[‘data’], irisdata[‘target’]
plt.scatter(X[:,0], X[:,1], c=y)
This is the iris dataset which has only four features. The features are in comparable scales and hence we can skip the scaler. With a 4-features data, the PCA can produce at most 4 principal
pca = PCA().fit(X)
[[ 0.36138659 -0.08452251 0.85667061 0.3582892 ]
[ 0.65658877 0.73016143 -0.17337266 -0.07548102]
[-0.58202985 0.59791083 0.07623608 0.54583143]
[-0.31548719 0.3197231 0.47983899 -0.75365743]]
For example, the first row is the first principal axis on which the first principal component is created. For any data point $p$ with features $p=(a,b,c,d)$, since the principal axis is denoted by
the vector $v=(0.36,-0.08,0.86,0.36)$, the first principal component of this data point has the value $0.36 times a – 0.08 times b + 0.86 times c + 0.36times d$ on the principal axis. Using vector
dot product, this value can be denoted by
p cdot v
Therefore, with the dataset $X$ as a 150 $times$ 4 matrix (150 data points, each has 4 features), we can map each data point into to the value on this principal axis by matrix-vector multiplication:
X times v
and the result is a vector of length 150. Now if we remove from each data point corresponding value along the principal axis vector, that would be
X – (X times v) times v^T
where the transposed vector $v^T$ is a row and $Xtimes v$ is a column. The product $(X times v) times v^T$ follows matrix-matrix multiplication and the result is a $150times 4$ matrix, same dimension
as $X$.
If we plot the first two feature of $(X times v) times v^T$, it looks like this:
# Remove PC1
Xmean = X – X.mean(axis=0)
value = Xmean @ pca.components_[0]
pc1 = value.reshape(-1,1) @ pca.components_[0].reshape(1,-1)
Xremove = X – pc1
plt.scatter(Xremove[:,0], Xremove[:,1], c=y)
The numpy array Xmean is to shift the features of X to centered at zero. This is required for PCA. Then the array value is computed by matrix-vector multiplication.
The array value is the magnitude of each data point mapped on the principal axis. So if we multiply this value to the principal axis vector we get back an array pc1. Removing this from the original
dataset X, we get a new array Xremove. In the plot we observed that the points on the scatter plot crumbled together and the cluster of each class is less distinctive than before. This means we
removed a lot of information by removing the first principal component. If we repeat the same process again, the points are further crumbled:
# Remove PC2
value = Xmean @ pca.components_[1]
pc2 = value.reshape(-1,1) @ pca.components_[1].reshape(1,-1)
Xremove = Xremove – pc2
plt.scatter(Xremove[:,0], Xremove[:,1], c=y)
This looks like a straight line but actually not. If we repeat once more, all points collapse into a straight line:
# Remove PC3
value = Xmean @ pca.components_[2]
pc3 = value.reshape(-1,1) @ pca.components_[2].reshape(1,-1)
Xremove = Xremove – pc3
plt.scatter(Xremove[:,0], Xremove[:,1], c=y)
The points all fall on a straight line because we removed three principal components from the data where there are only four features. Hence our data matrix becomes rank 1. You can try repeat once
more this process and the result would be all points collapse into a single point. The amount of information removed in each step as we removed the principal components can be found by the
corresponding explained variance ratio from the PCA:
[0.92461872 0.05306648 0.01710261 0.00521218]
Here we can see, the first component explained 92.5% variance and the second component explained 5.3% variance. If we removed the first two principal components, the remaining variance is only 2.2%,
hence visually the plot after removing two components looks like a straight line. In fact, when we check with the plots above, not only we see the points are crumbled, but the range in the x- and
y-axes are also smaller as we removed the components.
In terms of machine learning, we can consider using only one single feature for classification in this dataset, namely the first principal component. We should expect to achieve no less than 90% of
the original accuracy as using the full set of features:
from sklearn.model_selection import train_test_split
from sklearn.metrics import f1_score
from collections import Counter
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.33)
from sklearn.svm import SVC
clf = SVC(kernel=”linear”, gamma=’auto’).fit(X_train, y_train)
print(“Using all features, accuracy: “, clf.score(X_test, y_test))
print(“Using all features, F1: “, f1_score(y_test, clf.predict(X_test), average=”macro”))
mean = X_train.mean(axis=0)
X_train2 = X_train – mean
X_train2 = (X_train2 @ pca.components_[0]).reshape(-1,1)
clf = SVC(kernel=”linear”, gamma=’auto’).fit(X_train2, y_train)
X_test2 = X_test – mean
X_test2 = (X_test2 @ pca.components_[0]).reshape(-1,1)
print(“Using PC1, accuracy: “, clf.score(X_test2, y_test))
print(“Using PC1, F1: “, f1_score(y_test, clf.predict(X_test2), average=”macro”))
Using all features, accuracy: 1.0
Using all features, F1: 1.0
Using PC1, accuracy: 0.96
Using PC1, F1: 0.9645191409897292
The other use of understanding the explained variance is on compression. Given the explained variance of the first principal component is large, if we need to store the dataset, we can store only the
the projected values on the first principal axis ($Xtimes v$), as well as the vector $v$ of the principal axis. Then we can approximately reproduce the original dataset by multiplying them:
X approx (Xtimes v) times v^T
In this way, we need storage for only one value per data point instead of four values for four features. The approximation is more accurate if we store the projected values on multiple principal axes
and add up multiple principal components.
Putting these together, the following is the complete code to generate the visualizations:
from sklearn.datasets import load_iris
from sklearn.model_selection import train_test_split
from sklearn.decomposition import PCA
from sklearn.metrics import f1_score
from sklearn.svm import SVC
import matplotlib.pyplot as plt
# Load iris dataset
irisdata = load_iris()
X, y = irisdata[‘data’], irisdata[‘target’]
plt.scatter(X[:,0], X[:,1], c=y)
plt.title(“Two features from the iris dataset”)
# Show the principal components
pca = PCA().fit(X)
print(“Principal components:”)
# Remove PC1
Xmean = X – X.mean(axis=0)
value = Xmean @ pca.components_[0]
pc1 = value.reshape(-1,1) @ pca.components_[0].reshape(1,-1)
Xremove = X – pc1
plt.scatter(Xremove[:,0], Xremove[:,1], c=y)
plt.title(“Two features from the iris dataset after removing PC1”)
# Remove PC2
Xmean = X – X.mean(axis=0)
value = Xmean @ pca.components_[1]
pc2 = value.reshape(-1,1) @ pca.components_[1].reshape(1,-1)
Xremove = Xremove – pc2
plt.scatter(Xremove[:,0], Xremove[:,1], c=y)
plt.title(“Two features from the iris dataset after removing PC1 and PC2”)
# Remove PC3
Xmean = X – X.mean(axis=0)
value = Xmean @ pca.components_[2]
pc3 = value.reshape(-1,1) @ pca.components_[2].reshape(1,-1)
Xremove = Xremove – pc3
plt.scatter(Xremove[:,0], Xremove[:,1], c=y)
plt.title(“Two features from the iris dataset after removing PC1 to PC3”)
# Print the explained variance ratio
print(“Explainedd variance ratios:”)
# Split data
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.33)
# Run classifer on all features
clf = SVC(kernel=”linear”, gamma=’auto’).fit(X_train, y_train)
print(“Using all features, accuracy: “, clf.score(X_test, y_test))
print(“Using all features, F1: “, f1_score(y_test, clf.predict(X_test), average=”macro”))
# Run classifier on PC1
mean = X_train.mean(axis=0)
X_train2 = X_train – mean
X_train2 = (X_train2 @ pca.components_[0]).reshape(-1,1)
clf = SVC(kernel=”linear”, gamma=’auto’).fit(X_train2, y_train)
X_test2 = X_test – mean
X_test2 = (X_test2 @ pca.components_[0]).reshape(-1,1)
print(“Using PC1, accuracy: “, clf.score(X_test2, y_test))
print(“Using PC1, F1: “, f1_score(y_test, clf.predict(X_test2), average=”macro”))
Further reading
This section provides more resources on the topic if you are looking to go deeper.
How to Calculate Principal Component Analysis (PCA) from Scratch in Python
Principal Component Analysis for Dimensionality Reduction in Python
scikit-learn toy datasets
scikit-learn iris dataset
scikit-learn wine dataset
matplotlib scatter API
The mplot3d toolkit
In this tutorial, you discovered how to visualize data using principal component analysis.
Specifically, you learned:
Visualize a high dimensional dataset in 2D using PCA
How to use the plot in PCA dimensions to help choosing an appropriate machine learning model
How to observe the explained variance ratio of PCA
What the explained variance ratio means for machine learning
The post Principal Component Analysis for Visualization appeared first on Machine Learning Mastery.
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MATHS :: Lecture 08
MATHS :: Lecture 08 :: Equation of a circle
A circle is defined as the locus of the point, which moves in such a way, that its distance from a fixed point is always constant. The fixed point is called centre of the circle and the
constant distance is called the radius of the circle.
The equation of the circle when the centre and radius are given
Let C (h,k) be the centre and r be the radius of the circle. Let P(x,y) be any point on the circle.
CP = r
Note :
If the center of the circle is at the origin i.e., C(h,k)=(0,0) then the equation of the circle is x2 + y2 =
The general equation of the circle is x2 + y2 +2gx + 2fy + c = 0
Consider the equation x2 + y2 +2gx + 2fy + c = 0. This can be written as
x2 + y2 + 2gx +2fy + g2 + f2 = g2 +f2 – c
(i.e) x2 + 2gx + g2 + y2 +2fy + f2 = g2 +f2 – c
(x + g)2 + (y + f )2 =
This is of the form (x-h)2+ (y-k)2 = r2
\The considered equation represents a circle with centre (-g,-f) and radius
\ The general equation of the circle is x2 + y2 +2gx + 2fy + c = 0
c = The Center of the circle whose coordinates are (-g,-f)
r = The radius of the circle =
The general second degree equation
ax2 + by2 +2hxy + 2gx + 2fy +c = 0
Represents a circle if
(i) a = b i.e., coefficient of x2 = coefficient of y2
(ii) h = 0 i.e., no xy term
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Lesson H: Range-only SLAM
This lesson is a summary of all the lessons of this tutorial. We will apply the concepts of constraints and interval analysis on a concrete Simultaneous Localization and Mapping (SLAM) problem, and
see how an online SLAM can be solved.
This exercise comes from IAMOOC: another MOOC related to Interval Analysis with applications to parameter estimation and robot localization. It provides complementary concepts and may be of interest
to you. https://www.ensta-bretagne.fr/jaulin/iamooc.html
This lesson is an adaptation of the Exercise 11 of IAMOOC. The difference is that we will now consider a continuous-time state equation.
Consider a robot moving in an unknown environment and described by the state \(\mathbf{x}=(x_1,x_2,x_3)^\intercal\), with \(x_1,x_2\) its 2d position and \(x_3\) its heading. The evolution of \(\
mathbf{x}\) over time is represented by the trajectory \(\mathbf{x}(\cdot):[t_0,t_f]\rightarrow\mathbb{R}^3\), with \(t_0=0\) and \(t_f=15\).
The motion of the robot is described by the state equation \(\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},u)\) with:
\[\begin{split}\mathbf{f}(\mathbf{x},u)=\left( \begin{array}{c} 10\cos(x_3) \\ 10\sin(x_3) \\ u + n_u \end{array}\right),\end{split}\]
where \(u\) is the desired rotational speed (input of the system) and \(n_u\) is a noise.
The desired input \(u(t)\) is chosen as:
\[u(t) = 3\sin^2(t)+\frac{t}{100}.\]
Contrary to the previous lesson, we assume that we know the initial state \(\mathbf{x}_0=(0,0,2)^\intercal\). This is common in SLAM problems.
We also assume that the heading is continuously measured from \(t_0\) to \(t_f\) (for instance by using a compass) with a small error:
\[x_3^m(t) = x_3^*(t) + n_{x_3}(t),\]
where \(x_3^*(t)\) represents the actual but unknown heading of the robot.
At any time, we consider that the errors \(n_u(t)\) and \(n_{x_3}(t)\) are bounded by \([-0.03,0.03]\).
The term simulation often refers to the integration of one dynamical system from a known initial condition. The system we are dealing with is \(\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},u)\) and its
initial condition is \(\mathbf{x}_0\). We will first compute the trajectory \(\mathbf{x}^*(\cdot)\) solution of this system, without uncertainties, and call it the truth.
Of course, the computation of \(\mathbf{x}^*(\cdot)\) will not be reliable: the result will depend on the integration timestep and the \(\delta\) parameter used to represent the trajectory. We will
only use the result for visualization.
H.1. Simulate the system. We will use \(\delta\) = dt = \(0.05\) for implementing the trajectories.
The simulation can be done with a classical temporal loop and an Euler integration method. With Codac, we can also compute the system at the trajectory level (applying operators on entire
trajectories), without temporal loop. For this, we will define the input of the system as a trajectory, and apply operations on it (from function \(\mathbf{f}\)) and integrations.
The following code computes \(\mathbf{x}^*(\cdot)\):
# ...
# Initial pose x0=(0,0,2)
x0 = (0,0,2)
# System input
u = Trajectory(tdomain, TFunction("3*(sin(t)^2)+t/100"), dt)
# Actual trajectories (state + derivative)
v_truth = TrajectoryVector(3)
x_truth = TrajectoryVector(3)
v_truth[2] = u
x_truth[2] = v_truth[2].primitive() + x0[2]
v_truth[0] = 10*cos(x_truth[2])
v_truth[1] = 10*sin(x_truth[2])
x_truth[0] = v_truth[0].primitive() + x0[0]
x_truth[1] = v_truth[1].primitive() + x0[1]
// ...
// Initial pose x0=(0,0,2)
Vector x0({0,0,2});
// System input
Trajectory u(tdomain, TFunction("3*(sin(t)^2)+t/100"), dt);
// Actual trajectories (state + derivative)
TrajectoryVector v_truth(3);
TrajectoryVector x_truth(3);
v_truth[2] = u;
x_truth[2] = v_truth[2].primitive() + x0[2];
v_truth[0] = 10*cos(x_truth[2]);
v_truth[1] = 10*sin(x_truth[2]);
x_truth[0] = v_truth[0].primitive() + x0[0];
x_truth[1] = v_truth[1].primitive() + x0[1];
Create a new project with this simulation.
Add a noise on \(u(\cdot)\) as mentioned in the presentation of the problem, and display the result.
We will now enclose the trajectory \(\mathbf{x}^*(\cdot)\) in a tube. For the moment, we do not take into account measurements from the environment. This is what we call deadreckoning: we estimate
the positions of the robot only from proprioceptive data, coming from the input \(u(\cdot)\) and heading measurements.
H.2. As we did for the computation of \(\mathbf{x}^*(\cdot)\), estimate the feasible state trajectories in a tube, according to the uncertainties on \(u(\cdot)\) and \(x_3(\cdot)\). We will assume
that the initial state \(\mathbf{x}_0\) is well known.
The functions cos, primitive(), etc., can be used on tubes as we did for Trajectory objects. This will propagate the uncertainties during the computations.
We will also use \(\delta\) = dt = \(0.05\) for the implementation of the tubes.
You should obtain a result similar to:
Note that if you obtain a tube \([\mathbf{x}](\cdot)\) that encloses accurately the actual trajectory \(\mathbf{x}^*(\cdot)\) without uncertainties, then you did not correctly propagate information
from the input tube \([u](\cdot)\).
We could use a Contractor Network for this deadreckoning estimation, but the use of simple operators on tubes is also fine, because we do not have observations or complex constraints to consider. If
fact, for deadreckoning, we are dealing with a causal system where information propagates in one direction from \(u(\cdot)\) to \(\mathbf{x}(\cdot)\):
The use of a CN (or more generally, contractors) is relevant when we do not know how to propagate the information on sets (when the above graphic reveals loops) and when complex constraints have to
be treated. This is typically the case when one has to consider observations on the sets, as we do in SLAM.
The environment is made of 4 landmarks. Their coordinates are given in the following table:
\(j\) Landmark \(\mathbf{b}_j\)
\(0\) \((6,12)^\intercal\)
\(1\) \((-2,-5)^\intercal\)
\(2\) \((-3,20)^\intercal\)
\(3\) \((3,4)^\intercal\)
Each \(t=2\delta\), the robot is able to measure the distance to one of these landmarks (taken randomly), with an accuracy of \(\pm0.03\). The robot does not know the landmarks coordinates (the M of
SLAM is for Mapping), but it knows which landmark \(\mathbf{b}_j\) is being observed (the landmarks are identified).
We will use a constraint propagation approach to solve the problem.
H.3. First, define the variables of the problem.
H.4. List the involved constraints and the potential decompositions to perform. This may introduce intermediate variables. Note that all the constraints describing this SLAM have been seen in the
previous lessons.
H.5. Define the initial domains of the variables:
• domains for intermediate variables will be set to infinite sets.
• other domains may be initialized from measurements or to infinite sets when nothing is known, as it is the case for the position of the landmarks.
H.6. Using a Contractor Network, improve the localization of the robot while simultaneously estimating the position of the landmarks by enclosing them into boxes.
You should obtain a result similar to:
These computations were made offline, assuming that all the data were collected before running the solver.
We could also use this approach online and make the solver run during the evolution of the robot. For this, we will use the .contract_during(ctc_dt) method instead of .contract(). This way, we will
let the solver contract as much as possible the domains during a given amount of time ctc_dt. Remaining contractions will be done during the next call to .contract_during(). This allows to spread
over time the resolution.
Hence, for real-time SLAM, we can use the following temporal loop:
import time # used for time.sleep
dt = 0.05
iteration_dt = 0.2 # elapsed animation time between each dt
tdomain = Interval(0,15) # [t0,tf]
# ...
# Create tubes defined over [t0,tf]
# Add already known constraints, such as motion equations
t = tdomain.lb()
prev_t_obs = t
while t < tdomain.ub(): # run the simulation from t0 to tf
if t-prev_t_obs > 2*dt: # new observation each 2*dt
# Creating new observation to a random landmark
# Adding related observation constraints to the network
# Updated last iteration time
prev_t_obs = t
contraction_dt = cn.contract_during(iteration_dt)
if iteration_dt>contraction_dt: # pause for the animation
time.sleep(iteration_dt-contraction_dt) # iteration delay
# Display the current slice [x](t)
#include <unistd.h> // used for usleep
// ...
double dt = 0.05;
double iteration_dt = 0.2; // elapsed animation time between each dt
Interval tdomain(0,15); // [t0,tf]
// ...
// Create tubes defined over [t0,tf]
// Add already known constraints, such as motion equations
double prev_t_obs = tdomain.lb();
for(double t = tdomain.lb() ; t < tdomain.ub() ; t+=dt)
if(t - prev_t_obs > 2*dt) // new observation each 2*dt
// Creating new observation to a random landmark
// Adding related observation constraints to the network
// Updated last iteration time
prev_t_obs = t;
double contraction_dt = cn.contract_during(iteration_dt);
usleep(max(0.,iteration_dt-contraction_dt)*1e6); // pause for the animation
// Display the current slice [x](t)
H.7. (optional) Transform the code of question H.6 to make it work online with boxes \([\mathbf{x}]\) contracted in realtime.
You should obtain an animation that looks like this:
On the above figure, we can notice that the contracted boxes \([\mathbf{x}]\) obtained during the online SLAM are sometimes larger than the blue tube computed offline as post-processing. The reasons
• at \(t\), the CN online may not have dealt with all the contractors: some contractions remain to be done. They will be processed afterwards, and the current box \([\mathbf{x}](t)\) does not
enclose optimally the set of feasible positions;
• at \(t\), the online SLAM does not take benefit from future measurements, while the offline SLAM was able to propagate all information forward and backward in time.
The tutorial ends here!
We do hope it provided you an interesting overview of what Constraint Programming methods can bring to mobile robotics. We are looking forward your feedbacks! | {"url":"https://codac.io/tutorial/08-rangeonly-slam/index.html","timestamp":"2024-11-02T14:14:36Z","content_type":"text/html","content_length":"45387","record_id":"<urn:uuid:b14367c4-a05b-443e-9b79-8ba376fb35e8>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00260.warc.gz"} |
sphere 1 foot in diameter
Category: main menu • concrete menu • Solid ∅ 1 foot spheres
concrete conversion
Amount: 1 sphere 1 foot in diameter (d, ∅ 1 ft) of volume
Equals: 0.015 cubic meters (m3) in volume
Converting sphere 1 foot in diameter to cubic meters value in the concrete units scale.
TOGGLE : from cubic meters into solid ∅ 1 foot spheres in the other way around.
CONVERT : between other concrete measuring units - complete list.
Conversion calculator for webmasters.
This general purpose concrete formulation, called also concrete-aggregate (4:1 - sand/gravel aggregate : cement - mixing ratio w/ water) conversion tool is based on the concrete mass density of 2400
kg/m3 - 150 lbs/ft3 after curing (rounded). Unit mass per cubic centimeter, concrete has density 2.41g/cm3. The main concrete calculator page.
The 4:1 strength concrete mixing formula applies the measuring portions in volume sense (e.g. 4 buckets of concrete aggregate, which consists of gravel and sand, with 1 bucket of cement.) In order
not to end up with a too wet concrete, add water gradually as the mixing progresses. If mixing concrete manually by hand; mix dry matter portions first and only then add water. This concrete type is
commonly reinforced with metal rebars or mesh.
Convert concrete measuring units between sphere 1 foot in diameter (d, ∅ 1 ft) and cubic meters (m3) but in the other reverse direction from cubic meters into solid ∅ 1 foot spheres.
conversion result for concrete:
From Symbol Result To Symbol
1 sphere 1 foot in diameter d, ∅ 1 ft = 0.015 cubic meters m3
Converter type: concrete measurements
This online concrete from d, ∅ 1 ft into m3 converter is a handy tool not just for certified or experienced professionals.
First unit: sphere 1 foot in diameter (d, ∅ 1 ft) is used for measuring volume.
Second: cubic meter (m3) is unit of volume.
concrete per 0.015 m3 is equivalent to 1 what?
The cubic meters amount 0.015 m3 converts into 1 d, ∅ 1 ft, one sphere 1 foot in diameter. It is the EQUAL concrete volume value of 1 sphere 1 foot in diameter but in the cubic meters volume unit
How to convert 2 solid ∅ 1 foot spheres (d, ∅ 1 ft) of concrete into cubic meters (m3)? Is there a calculation formula?
First divide the two units variables. Then multiply the result by 2 - for example:
0.014826666204376 * 2 (or divide it by / 0.5)
1 d, ∅ 1 ft of concrete = ? m3
1 d, ∅ 1 ft = 0.015 m3 of concrete
Other applications for concrete units calculator ...
With the above mentioned two-units calculating service it provides, this concrete converter proved to be useful also as an online tool for:
1. practicing solid ∅ 1 foot spheres and cubic meters of concrete ( d, ∅ 1 ft vs. m3 ) measuring values exchange.
2. concrete amounts conversion factors - between numerous unit pairs.
3. working with - how heavy is concrete - values and properties.
International unit symbols for these two concrete measurements are:
Abbreviation or prefix ( abbr. short brevis ), unit symbol, for sphere 1 foot in diameter is:
d, ∅ 1 ft
Abbreviation or prefix ( abbr. ) brevis - short unit symbol for cubic meter is:
One sphere 1 foot in diameter of concrete converted to cubic meter equals to 0.015 m3
How many cubic meters of concrete are in 1 sphere 1 foot in diameter? The answer is: The change of 1 d, ∅ 1 ft ( sphere 1 foot in diameter ) unit of concrete measure equals = to 0.015 m3 ( cubic
meter ) as the equivalent measure for the same concrete type.
In principle with any measuring task, switched on professional people always ensure, and their success depends on, they get the most precise conversion results everywhere and every-time. Not only
whenever possible, it's always so. Often having only a good idea ( or more ideas ) might not be perfect nor good enough solution. If there is an exact known measure in d, ∅ 1 ft - solid ∅ 1 foot
spheres for concrete amount, the rule is that the sphere 1 foot in diameter number gets converted into m3 - cubic meters or any other concrete unit absolutely exactly. | {"url":"https://www.traditionaloven.com/building/masonry/concrete/convert-1-ft-diam-concrete-sphere-to-cubic-metre-m3-concrete.html","timestamp":"2024-11-06T01:55:27Z","content_type":"text/html","content_length":"40912","record_id":"<urn:uuid:394dcc5b-defb-48a5-9b5b-84d5db34729b>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00112.warc.gz"} |
Asymmetric Total Synthesis of GlaxoSmithKline’s Potent Phosphodiesterase PDE IVb Inhibitor
Phosphodiestarase of subtype PDE IVb inhibitors are considered as perspective drugs for the treatment of the central nervous system disorders (depression, Alzheimer’s disease, Parkinson’s disease).
Pyrrolizidinone Glaxo-1, proposed by GlaxoSmithKline, is a highly potent PDE IVb inhibitor (IC50 = 63 nM), then conventional phosphodiesterase inhibitors Ro-20-1724, Rolipram and Cilomilast. However
the activity of the Glaxo-1 was studied on a racemic sample, since the asymmetric approach to its synthesis has not been developed. Therefore the purpose of this research was the development of an
efficient synthetic scheme enabling enantioselective excess to both (-)- and (+)-Glaxo-1, which can be than subjected to biological studies. \r The key stage in proposed asymmetric synthesis (-)- and
(+)-Glaxo-1 is stereoselective [4+2]-cycloaddition of the nitroolefin to an optically activity vinyl ethers, derived from (-)- or (+)-trans-2-phenylcyclohexanols. The resulting chiral cyclic
nitronates are transformed into a functionalized cyclic oxime ethers using tandem sylilation-nucleophilic substitution procedure. Reduction and decarboxylation of these products lead to optically
pure Glaxo-1 and the regeneration of chiral 2-phenylcyclohexanols (91%). \r Thus both enantiomers (+) and (-)-Glaxo-1 were obtained selectively in average yield 12% from isovaniline and nitroethane.
The study of biological profiles of each enantiomer of Glaxo-1 will be conducted in near future.
Mathematics in Music
Mathematics and music are two poles of human culture. Listening to music we get into the magic world of sounds. Solving problems we are immersed in strict space of numbers and we do not reflect that
the world of sounds and space of numbers have been adjoining with each other for a long time. Interrelation of mathematics and music is one of the vital topics. It hasn’t been completely opened and
investigated up to now. This is the point why it draws attention of a lot of scientists and mathematicians to itself. This is the point why it draws attention of a lot of scientists and
mathematicians to itself. Having considered the value of these two sciences, it seems to us that they are completely non-comparable. In fact can there be a similarity between mathematics – the queen
of all sciences, a symbol of wisdom and music – the most abstract kind of art? But if you peer deeply into it you can notice that the worlds of sounds and space of numbers have been adjoining with
each other for a long time. In the work I will try to establish the connection between mathematics and music and to find their common elements, to analyze pieces of music with the help of laws and
concepts of mathematics to find a secret of mastery of musicians using mathematics and also to investigate the connection of music with mathematics with the “research part”. They are my own
calculations and researches which are an integral part of the work. The connection of mathematic and music is caused both historically and internally in spite of the fact that mathematics is the most
abstract of sciences and music is the most abstract kind of art. V. Shafutinskiy, I. Matvienko, m. Fadeev, K. Miladze, Dominik the Joker – modern composers of the XXI century – have used the golden
proportion only in 4% of their pieces of music and more often in romances or children’s songs. I have revealed this fact after investigating their pieces of music of different genres. However there
is a question: why does modern music attracts all of us more but the classics is being forgotten? Investigating connection between mathematics and music I had come to the conclusion that the more
deeply the piece of music gives in to the mathematical analysis, to research and submits to any mathematical laws, the more harmonious and fine its sounding is, the more it excites human soul.
Besides I am convinced that many important, interesting and entertaining things have not been opened in this field. We can safely continue our research of these things. I think that I have managed to
lift a veil over mathematics in music, to find something common for apparently incompatible science and art. In due time English mathematician D. Silvestre called music as mathematics of feelings,
and mathematics – as music of intellect. He expressed hope that each of them should receive the end from the part of the other one. In the future he expected the occurrence of a person in which
Beethoven and Gauss’ greatness would unite. Terms ‘science’ and ‘art’ practically didn’t differ during far times of antiquity. And though roads of mathematics and music have gone away since then
music is penetrated with mathematics and mathematics is full of poetry and music!
Number system with non-natural base
In this work I make the analysis of the possibility of the existence of the number system with non-natural base & its investigation. The question examined in my work is totally opened:\r ‧ making the
list of new characteristics, rules of the translation of the numbers, and also rules of the simple calculating operations, checked the operations of subtraction & division;\r ‧ checked the Euclidean
algorithm, its characteristics by means of estimating the coefficients;\r ‧ found the practical appliance of new method in compiling & solving of the tasks.\r Investigation I’ve suggested stipulates
for independence of new system & its appliance in type of tasks, that is beyond the course of school program & also beyond the whole system of school education.
The garnets of the schlich of the winter coast (the White Sea)
During an expedition to the White Sea Winter Coast, samples of schlich with numerous garnets were collected. The coast itself is primarily a set of high steeps of sandstones and mudstones with no
garnets. On the beach, however, there are numerous pebbles of metamorphic rocks, and many of them contain garnets. They were brought there by the Quaternary glacier and the present day glacier
activity. Their source could be the existing or fully eroded metamorphic rocks of the Kola Penninsila and Northern Karelia.\r Goal of the research: To discover the possible origin of the garnets.\r
Tasks: 1) To analyze the structure and the chemical composition of the garnets. 2) To study the works on the garnets of the Kola Penninsila and Northern Karelia in order to compare the information to
the results of the analysis.\r Methods: 1) Granulometric analysis; 2) Magnetic and electromagnetic separation; 3) Microscope study of the garnets; 4) X-ray diffraction analysis; 5) X-ray fluorescence
analysis. \r RESULTS\r 1. The sample garnets could be divided into two types: the bright red and the pale pink.\r 2. Both types are almandines with spessartine components. The pale-pink one contains
Arizona ruby components, and the bright red contains andradite ones. \r 3. Only a few samples from Khangaz Varaki, Malye Keivy, and Tersky Coast are similar to the garnets that we collected. It is
possible that the glacier brought them from the Kola Penninsula, but it is also possible that the rocks, where the glacier brought our garnets from, have been totally eroded.
A method of searching for all the integer solutions of any equation of markov's type of paralolic ty
This work presents fundamental research in the field of algebra and the theory of number. The subject of the work is equations of Markov's type (the type of the equations introduced by me earlier
which generalizes the classic equation of Markov (x2+y2+z2=3xyz)) of parabolic type with two unknown quantities and their genealogical trees. The following questions appeared when I was working on
the equations of Markov's type and constructing genealogical trees to them: are there any other trees besides one for a certain equation; how to find all the genealogical trees for the equation of
Markov's type; how to find all the integer solutions with the help of the genealogical trees. This work is devoted to the analysis of these questions. The aim of the work: to create the method of
finding all the integer solutions of the equations of Markov's type of parabolic type. The tasks of the work: 1. Carrying out some experimental works to find all the genealogical trees for a concrete
equation. 2. Formulating a hypothesis that the curve has a specific part. 3. Research the parabolic type in order to apply the hypothesis to it. 4. To formulate and prove the theorems about the
necessary and sufficient conditions of the existence of the genealogical trees of the integer solutions of equations of Markov's type of parabolic type with two unknown quantities. As the result of
the work all the tasks have been solved. I worked the method of finding all the integer solutions: : to find all the integer solutions by means finding all genealogical trees of the equations of
Markov's type of parabolic type with two unknown quantities you need : 1. To investigate if there any integer solutions‧ a special part of the parabola (if it is a parabola)‧ a special part of the
parallel lines (if it is a pair of the parallel lines) 2. To build a genealogical tree from every solution (if they exist). 3. All the integer solutions will be on the constructed trees. I also
worked out a computer program which is based on the usage of this method.
Montioring of Cryogenic Features along Roads in Megino-Kangalassky Region,Yakutia
One of the anthropogenic influences on the permafrost landscapes is deforestation and breach of the surface cover at the road constructions. On these areas a development of various cryogenic and
postcryogenic processes and features (thermosubsidence, knobs (bilars, baydjarakhs), ravines, small lakes - djyodje) is being observed. Such features can be observed on the territory of Megino-
Kangalassky Region, which is situated in the Central Yakutia, as well. During the period from 1998 to 2003 the author carried out monitoring studies of cryogenic features along three roads. Ten areas
measuring twenty square metres were put in all. During 6 years of research about 1520 measurements of cryogenic formations parameters were made. Based on the results, it is concluded that elimination
of the shadowing effect by trees and removal of the surface cover along the roads have caused thawing of shallow-lying ice wedges, as well as development of various cryogenic processes and features.
In this paper, the author presents the basic technologies which are used in road construction in permafrost areas and, based on the research results, proposes a set of measures for rehabilitation of
the roadside areas.
Reduction fuel's amount when working the Internal-combustion engine
The aim of the work is inventing the way in which the power of the ICE is the same and consuming of the fuel decreases. The following methods of investigation were used: analysis of the experience of
the improvement of ICE, modeling, the brainstorming, methods of Decition Theory of Invention’s Tasks (DTIT). In this work Ivan Semyonov based on a hypothesis that if the non-supporting combustion
exhausts will be drawn with the vacuum from cylinder fuel for the same power it needed less. The practical meaning of this work is in the attempt of studying the question of improving the ICE for
getting and making the more perfect ICE.
Universal computing sorting machine
The purpose of the study was to develop and create a semi-automatic multi-purpose sorting and counting machine of standard articles. Currently, there is a problem of equipping industrial enterprises
as well as small trade companies and large retailers with computing sorting machines of standard products of a certain shape. We would like to fill this missing link up with a simple, compact and
inexpensive device. Procedures The proposed research consisted of a consistent design of a virtual model of the device and its electronic-mechanical implementation. The virtual model is simulated by
a computer program "SolidWorks" object, which graphically shows the operation of the future device. In the development of the computing sorting device standard electronic devices and their associated
software have been used. The created simplified real model demonstrates the basic principles and characteristics of the proposed device. Data As an example for the implementation of the concept
device a computing device for sorting coins, in circulation in Russia has been created. As a basic principle of sorting objects by their geometric and weight characteristics were used. It is
important that the device is focused on the correct form of the objects of sorting (balls, rings, coins, regular polyhedrons, screws, nuts, etc). To confirm the effectiveness of the computing device
of this type of sorting, a series of tests of counting of objects manually have been carried out . The effectiveness of the device is determined by comparing the time characteristics of manual and
automatic sorting. Findings and conclusions As a result of research and work performed, we have concluded that: 1) The proposed device can be used in various industries. (for example, while sorting
ball bearings.) 2) Such a computing sorting device may find it's application in various commercial enterprises: to assist cashiers in retail ATMs. 3) Can be used in payment terminals. 4) 4) After a
certain modernization of the device it can be used for money encashment.
Technology of web site advancement
Internet by its content represents a fountain of information, while from the point of view of its arrangement it is a huge dump. There are an enormous number of web sites. Multiple web sites are
commercially directed, i.e. are aimed at profit earning. As profit depends on the number of visits to web site, no visitors means no profit. So, to obtain more orders, web site producers should first
of all ensure good inflow of visitors (web site attendance). Every year this task becomes more and more critical for commercial web site owners (and not only for them), as the number of similar
content web sites increases steadily along with competition intensifying correspondingly. The process of establishing conditions to attract more visitors is called web site advancement. The present
paper discusses various ways of how to increase the number of web site visitors, it also describes the particular process of "Theater to Children" (www.teatrbaby.ru) web site advancement. Based on
the paper outcomes a CD multimedia manual "Technology of web site advancement" has been developed that will help web site producers to achieve good attendance for their network resources. As the
purpose of web site advancement is visitor number increase then the main criterion of web site advancement efficiency should be the number of visitors for a certain time period, e.g. for 24 hours, a
week or a month. Taking into consideration that about 80% of Internet users retrieve information through search systems, the major growth of visitors will occur owing to the enhancement of web site
visibility in search systems.
Problems of Safe Storage, Collecting and Recycling of Luminescent Lamps
Our research is connected with the problem of the mercurious wasre products (MWP) recycling, luminescent lamps take a great part among them. The problem of recycling MWP is topical not only in
Yakutsk, but also in other cities of Russia because of its toxic influence on the human’s body. Mercury is the top among eight the most dangerous metals. I have analyzed the conditions of recycling
of luminescent lamps in Yakutsk schools. I have got data on the problem. There is a great amount of the fused luminescent lamps is stored in the territory of schools, it produces danger for the
pupils’ health. On the research work I have proposed some ways of solution of the problem. | {"url":"https://www.ntsec.edu.tw/science/list.aspx?a=90&cat=140","timestamp":"2024-11-05T06:16:07Z","content_type":"text/html","content_length":"118096","record_id":"<urn:uuid:2ea49621-8796-4a7f-a8c3-aeddab116c86>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00353.warc.gz"} |
Sampling and reconstruction
Sampling and reconstruction are cornerstones of modern digital audio processing. We take a closer look at these processes and the limitations they impose.
Sampling is the act of converting a continuously varying signal to discrete samples suitable for digital manipulation. Reconstruction is the reverse, converting a sampled signal back to its
continuous form, which can then drive a speaker allowing us to hear it. Intuitively, the idea may appear impossible. How can a finite set of samples possibly capture the infinite variations of a
continuously changing signal? What about the signal segments between the sample points?
The sampling theorem
The questions posed above are resolved by the sampling theorem. Attributed variously to Claude Shannon, Harry Nyquist, E. T. Whittaker, and others, this theorem sets out conditions permitting a
continuous signal to be sampled and subsequently reconstructed:
1. The signal must be bandlimited.
2. The sample rate must be higher than twice the bandwidth of the signal.
If these conditions are met, the sampled values uniquely and exactly describe the full continuous waveform. There is no approximation. Increasing the sample rate cannot improve anything since there
is no error to begin with.
For a signal with bandwidth \(B\), the minimum sample rate, \(2B\), required for an accurate capture is commonly referred to as the Nyquist rate. Conversely, if the sample rate is \(f_s\), the
maximum allowed signal bandwidth, \(f_s/2\), is known as the Nyquist frequency.
Sampling a sine wave
Suppose we sample an 8 kHz sine wave at the commonly used rate of 48 kHz. The conditions of the sampling theorem are met since the sample rate is well above twice the maximum (and in this case only)
frequency of the signal. Figure 1 shows one period of this sine wave with the sample points marked as red circles.
If the signal frequency is too high, in this case above 24 kHz, a phenomenon known as aliasing occurs. In figure 2 we see a 40 kHz sine wave (green) together with the same 8 kHz signal as above
(dashed blue). Notice that the sample points end up in exactly the same locations for both waveforms. Also notice that 40 kHz is precisely 8 kHz below the sample rate.
Given only the sample data, it would be impossible to tell which of these two waveforms was the source. A similar aliasing situation occurs for again for a signal frequency 8 kHz above the sample
rate, that is 56 kHz, as can be seen in figure 3.
In general, for every valid signal frequency, there exist an infinite number of alias frequencies in symmetrical pairs around every multiple of the sample rate.
Anti-aliasing filters
Due to the aliasing effect illustrated above, it is important that the sampled signal is properly band limited. If it is not, any frequencies above the Nyquist frequency will alias into the lower
range and distort the capture. If we can’t be certain about the signal bandwidth, we must precede the sampling stage with an analogue low-pass filter. Since the purpose of this filter is to remove
alias frequencies, it is commonly called an anti-alias filter.
Ideally, the anti-alias filter would cut out everything from the Nyquist frequency and up, leaving the lower frequencies untouched. A perfect low-pass filter like this is, unfortunately, impossible
to construct in practice. The solution is to set the sample rate, not to precisely twice the highest frequency of interest, but somewhat higher, providing some margin between the top of the target
band and the point where aliasing sets in. This allows the anti-alias filter a transition band wherein its response gradually goes from passing frequencies below to blocking those above.
The generally accepted upper limit for human hearing is 20 kHz. A sampled audio system thus needs a sample rate of at least 40 kHz. With a little margin added for the anti-alias filter, we arrive at
the common sample rates of 44.1 kHz and 48 kHz. Those exact frequencies were chosen for technical reasons unrelated to the sampling process.
If we accept some aliasing distortion above 20 kHz, the width of the transition band can be doubled. This is possible since the aliases are mirrored around the Nyquist frequency, so for a 48 kHz
sample rate, a 28 kHz signal component is aliased to 20 kHz.
Even when permitting aliasing in the transition band, an anti-alias filter suitable for a 44.1 kHz or 48 kHz sample rate can be a challenge to design. This task is simplified by sampling at a much
higher rate followed by a digital decimation stage since a digital low-pass filter can readily be made very steep without adversely affecting the pass band or requiring high-precision components.
Oversampling, as this technique is called, additionally permits the use of a less accurate A/D conversion stage while maintaining the same signal to noise ratio in the audio band. In its simplest
form, each doubling of the sample rate gains one effective bit of resolution, and noise shaping can improve this further.
For audio purposes, sampling would be mostly useless without a means of converting the signal back to its analogue form. After all, our ears do not accept digital inputs.
Mathematically, a sampled signal can be viewed as a sequence of impulses, one for each sample, with heights corresponding to the sample values. This is illustrated in figure 4.
That doesn’t look much like a sine wave. However, computing the Fourier transform yields the spectrum in figure 5 below.
Below the Nyquist frequency, 24 kHz, everything looks good with a single 8 kHz tone, exactly as desired. Above 24 kHz, things are not looking so good. There are additional tones at 40 kHz, 56 kHz,
and so on around every multiple of the sample rate, and effect called imaging. For every actual frequency in the signal, this crude reconstruction has generated a multitude of image frequencies. As
the reader may have noticed, these additional frequencies coincide with the alias frequencies we encountered during the sampling process.
Frequency imaging aside, an impulse based D/A converter isn’t practical. Such fast switching while producing an accurate voltage level is not easily achieved. A more reasonable approach is to hold
the output voltage constant for the duration of each sample. This gives us the waveform displayed in figure 6.
This method is called a zero-order hold. The curve it produces looks a little more like a sine wave, though it still has some way to go. Figure 7 shows the spectrum.
As we can see, this method also produces the same image frequencies. Their level drops a little as the frequency increases, though not by much. Clearly, something must be done.
Anti-imaging filters
A solution to the problem of image frequencies is to simply remove them using an analogue low-pass filter, unsurprisingly referred to as an anti-imaging filter. If we remove everything above the
Nyquist frequency, 24 kHz, only the originally sampled signal remains.
As with the anti-aliasing filter earlier, a perfect low-pass filter is impossible to construct. We do, however, still have the margin between the limit of hearing, 20 kHz, and the Nyquist frequency
within which to work. Of course, that rather small margin still presents the same challenge.
Oversampling (again)
Once again, oversampling comes to the rescue. If we increase the sample rate by inserting one or more zeros after each sample, we obtain a digital version of the impulse sequence we looked at
previously. The image frequencies in its spectrum can now be removed using a digital low-pass filter, which as already noted, is much easier to implement.
Having done a digital oversampling of the signal, we can then pass it to the same zero-order hold D/A converter as before. The output from this process using a 2x oversampling can be seen in figure
While there are still steps, they are smaller, and the reconstruction follows the desired curve much more closely. In figure 9 we see that also the spectrum has been improved
The first pair of images, around 48 kHz, is gone, as are those for all odd multiples of the sample rate. The digital oversampling took care of that. To get rid of the remainder, a much more
reasonable analogue filter can be used. The higher the oversampled rate, the simpler the analogue anti-imaging filter can be. In practice, an oversampling factor of 8x is common, placing the first
images around 384 kHz.
Final words
Sampling captures a continuous signal up to a maximum frequency, and the reconstruction process does the reverse, turning discrete samples back into a continuous waveform. There is a lot of symmetry
between the two processes. Both rely on low-pass filters to function correctly, which presents some challenges. Likewise, digital filtering techniques operating at a higher sample rate greatly
simplify this task.
7 Replies to “Sampling and reconstruction”
1. It’s awesome that this article talked about the solution to getting around setting up a low pass filter. I appreciate you helping me learn more about these filters and how to work with them. I
will have to look into a way to work with anti alias filters in the future.
2. This makes sense in the studio… but once the digital media has been created, what good does oversampling in a DAC do? If the audio is encoded at 44.1khz, no additional information of a higher
resolution can be obtained… you’d simply be sampling the same data point multiple times, no?
Even with the 8x oversampling example above, I still don’t understand how a smooth sine wave can be generated from a series of square steps – even if it’s 8 times as many. the example is 8khz,
but consider sampling 20khz. Even with oversampling that’s not very many samples… is the analog output stage actually tracking the sampled/oversampled zero-hold output, i.e. attempting to adjust
and hold it’s voltage in steps? Are high frequency waveforms rounded out simply by virtue of analog components not being able to swing their voltages instantaneously, such that the next step up
or down in voltage is gradual and thus smoothed out? That still wouldn’t be accurate, at least not for all frequencies.
This is great information but I’m still missing something, maybe it’s the last step. How does the digital stepped representation become uniformly gradual and produce an ACTUAL sine wave such that
if we were to zoom in closely the ramp up/down would be smooth and at the proper angle?
1. ^^^ This question I still have btw! I get how the sampled data accurately represents the analog waveform… but how we turn that sampled data INTO an analog waveform from a digitized zero-hold
series of steps?
2. The stepped output is made smooth using an analogue low-pass filter. The benefit of digital interpolation is that the smaller steps it yields allow a simpler analogue filter to be used. Since
the analogue filter isn’t perfect, some residual of the unwanted higher frequencies will remain. Again, the digital interpolation helps by moving these artefacts to higher frequencies where
they are less harmful and the analogue filter is also more effective.
3. Since storage capacity is steadily increasing, why isn’t all audio recorded and delivered (either through streaming or download) at the highest possible bit-depth/resolution and sampling rate?
If I understood this article, that would allow for cheaper DACs since you wouldn’t need additional oversampling required by the analogue filters?
A side effect of this allows you to remove/not use dithering, reducing noise floor and increase dynamic range. I have never tried a really high-end audio setup, so I don’t know if it’s
appreciable enough to warrant the increased storage space and probably more expensive recording equipment?
I’m also thinking from a preservation standpoint. Perhaps in the future all DACs regardless of price can easily reconstruct 24/192 for instance, then it would be a shame if the recording wasn’t
available in high resolution (given that it is actually appreciable, as stated above) just because we wanted to save a bit of storage space or money today.
I would like to see some graphs in the “Anti-aliasing filters” – section:
1) Ideal vs. practical low-pass filter
2) Something to visualize the two last sentences in that section. What do you mean by “so for a 48 kHz sample rate, a 28 kHz signal component is aliased to 20 kHz.”? I assume it is the same as
Fig.1-3 are showing, but it’s hard to mentally visualize.
Anyways, read all your articles. Not sure I understood everything, but they were still insightfull and interesting. Great work!
4. Hello, I have an IFI Zen v2 (DSD1793) currently running with the non-MQA firmware. I am curious to know which upsampling/downsampling rate (using SOX in JRiver) is optimal to feed the DAC? Would
it be better to use 192kHz or 384kHz? Please help.
1. The answer depends on what you mean by optimal. Regardless, however, I would stick to 192 kHz or below as then the DAC chip will do a further 8x upsampling. With 384 kHz input, this is
bypassed. Looking at the audible range only (up to 20 kHz), the DAC chip tends to perform slightly better at lower sample rates, though the difference is too small to be audible.
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s a
Is a constant a coefficient?
Is a constant a coefficient?
First of all consider 5x + y – 7. The coefficients are the numbers that multiply the variables or letters. Thus in 5x + y – 7, 5 is a coefficient. Constants are terms without variables so -7 is a
How do you know if an experiment is binomial?
The requirements for a random experiment to be a binomial experiment are:
1. a fixed number (n) of trials.
2. each trial must be independent of the others.
3. each trial has just two possible outcomes, called “success” (the outcome of interest) and “failure“
What is the probability of success in a binomial trial?
Binomial probability refers to the probability of exactly x successes on n repeated trials in an experiment which has two possible outcomes (commonly called a binomial experiment). If the probability
of success on an individual trial is p , then the binomial probability is nCx⋅px⋅(1−p)n−x .
What is the numerical coefficient of 5 XY?
So the numerical coefficient of -5xy is -5.
What does a binomial test show?
A binomial test uses sample data to determine if the population proportion of one level in a binary (or dichotomous) variable equals a specific claimed value.
What is the coefficient of 5xy?
Thus in 5xy, 5 is the coefficient of the term.
What is an example of a binomial experiment?
A binomial experiment is an experiment where you have a fixed number of independent trials with only have two outcomes. For example, the outcome might involve a yes or no answer. If you toss a coin
you might ask yourself “Will I get a heads?” and the answer is either yes or no.
What is the formula for the expected number of successes in a binomial experiment with n trials and probability of success P?
The binomial mean, or the expected number of successes in n trials, is E(X) = np. The standard deviation is Sqrt(npq), where q = 1-p. The standard deviation is a measure of spread and it increases
with n and decreases as p approaches 0 or 1. For a given n, the standard deviation is maximized when p = 1/2.
Why is it called a coefficient?
Coefficient: A coefficient is a number, or variable, that is multiplies a variable term. Even though they are variables, the represent some constant, but unknown value unlike the variable x which is
variable of the expression. The origin of the word reaches back to the early Latin word facere, to do.
Which of the following is a correct application of binomial nomenclature?
Binomial nomenclature is used especially by taxonomists in naming or identifying a species of a particular organism. It is used to come up with a scientific name for a species that is often based in
Greek or Latin language.
Which of the following is a binomial?
Answer. ( x+ 1)(x – 1) is binomial.
What are the 4 characteristics of a binomial experiment?
1: The number of observations n is fixed. 2: Each observation is independent. 3: Each observation represents one of two outcomes (“success” or “failure”). 4: The probability of “success” p is the
same for each outcome.
Which of the following is a binomial of degree 20?
A binomial of degree 20 in the following is: * 20x + 1 .
How many terms are there in the expression 2x 2y?
it has only 1 term……
Which of the following is a binomial in Y?
What of the following is a Monomial?
A monomial is an expression in algebra that contains one term, like 3xy. Monomials include: numbers, whole numbers and variables that are multiplied together, and variables that are multiplied
together. Any number, all by itself, is a monomial, like 5 or 2,700.
Is rolling a die a binomial experiment?
In other words, rolling a die twice to see if a 2 appears is a binomial experiment, because there is a fixed number of trials (2), and each roll is independent of the others. Also, for binomial
experiments, there are only 2 possible outcomes (a successful event and a non-successful event).
How many terms are there in the expression 5 3xy?
2 terms
Can a coefficient be negative?
Coefficients are numbers that are multiplied by variables. Negative coefficients are simply coefficients that are negative numbers. An example of a negative coefficient would be -8 in the term -8z or
-11 in the term -11xy. The number being multiplied by the variables is negative.
What does C stand for in binomial probability?
b(x; n, p): Binomial probability – the probability that an n-trial binomial experiment results in exactly x successes, when the probability of success on an individual trial is p. n. Cr: The number
of combinations of n things, taken r at a time.
How many terms are there in the expression 5xy 9yz 3zx 5x 4y * 1 point?
Answer. There are 5 terms in this expression.
What is a success in a binomial experiment?
We have a binomial experiment if ALL of the following four conditions are satisfied: The experiment consists of n identical trials. Each trial results in one of the two outcomes, called success and
failure. The probability of success, denoted p, remains the same from trial to trial. The n trials are independent.
What is a binomial experiment in statistics?
Binomial Experiment A binomial experiment is an experiment which satisfies these four conditions. A fixed number of trials. Each trial is independent of the others. There are only two outcomes. The
probability of each outcome remains constant from trial to trial.
What does foil stand for in multiplying Binomials?
First, Outer, Inner, Last
How many terms are there in the expression 5xy²?
one term
What is the constant coefficient?
The general second‐order homogeneous linear differential equation has the form. If a( x), b( x), and c( x) are actually constants, a( x) ≡ a ≠ 0, b( x) ≡ b, c( x) ≡ c, then the equation becomes
simply. This is the general second‐order homogeneous linear equation with constant coefficients.
Which of the following is not binomial?
An algebraic expression which consists of two non-zero terms is called a binomial. So, option(b) is the correct answer.
What is a binomial in math?
In algebra, a binomial is a polynomial that is the sum of two terms, each of which is a monomial. It is the simplest kind of polynomial after the monomials.
What is the coefficient of 20?
Answer. Answer: the coefficient of 20 is number itself..
What does the P stand for in the binomial probability formula?
The first variable in the binomial formula, n, stands for the number of times the experiment runs. The second variable, p, represents the probability of one specific outcome. | {"url":"https://www.meltingpointathens.com/is-a-constant-a-coefficient/","timestamp":"2024-11-11T22:52:19Z","content_type":"text/html","content_length":"54806","record_id":"<urn:uuid:997594b5-65b2-4c64-9460-2c8cfec7cfb4>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00454.warc.gz"} |
Scaling Laws from the Data Manifold Dimension
Utkarsh Sharma, Jared Kaplan.
Year: 2022, Volume: 23, Issue: 9, Pages: 1−34
When data is plentiful, the test loss achieved by well-trained neural networks scales as a power-law $L \propto N^{-\alpha}$ in the number of network parameters $N$. This empirical scaling law holds
for a wide variety of data modalities, and may persist over many orders of magnitude. The scaling law can be explained if neural models are effectively just performing regression on a data manifold
of intrinsic dimension $d$. This simple theory predicts that the scaling exponents $\alpha \approx 4/d$ for cross-entropy and mean-squared error losses. We confirm the theory by independently
measuring the intrinsic dimension and the scaling exponents in a teacher/student framework, where we can study a variety of $d$ and $\alpha$ by dialing the properties of random teacher networks. We
also test the theory with CNN image classifiers on several datasets and with GPT-type language models.
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Derivative of Tan x - Formula, Proof, Examples
The tangent function is among the most important trigonometric functions in math, engineering, and physics. It is a fundamental idea used in several domains to model multiple phenomena, including
wave motion, signal processing, and optics. The derivative of tan x, or the rate of change of the tangent function, is a significant concept in calculus, which is a branch of math that deals with the
study of rates of change and accumulation.
Comprehending the derivative of tan x and its properties is crucial for working professionals in several domains, including engineering, physics, and math. By mastering the derivative of tan x,
individuals can use it to solve problems and get deeper insights into the complex functions of the surrounding world.
If you need assistance getting a grasp the derivative of tan x or any other mathematical concept, consider reaching out to Grade Potential Tutoring. Our expert tutors are available online or
in-person to offer customized and effective tutoring services to support you be successful. Contact us right now to plan a tutoring session and take your math abilities to the next level.
In this blog, we will delve into the concept of the derivative of tan x in detail. We will initiate by discussing the importance of the tangent function in different domains and uses. We will then
explore the formula for the derivative of tan x and provide a proof of its derivation. Finally, we will provide instances of how to utilize the derivative of tan x in various fields, consisting of
physics, engineering, and math.
Significance of the Derivative of Tan x
The derivative of tan x is a crucial mathematical theory that has multiple utilizations in physics and calculus. It is utilized to figure out the rate of change of the tangent function, that is a
continuous function that is broadly utilized in math and physics.
In calculus, the derivative of tan x is utilized to solve a extensive range of problems, including finding the slope of tangent lines to curves that include the tangent function and calculating
limits which consist of the tangent function. It is also applied to calculate the derivatives of functions which involve the tangent function, for instance the inverse hyperbolic tangent function.
In physics, the tangent function is applied to model a broad array of physical phenomena, involving the motion of objects in circular orbits and the behavior of waves. The derivative of tan x is
applied to calculate the acceleration and velocity of objects in circular orbits and to get insights of the behavior of waves which includes changes in frequency or amplitude.
Formula for the Derivative of Tan x
The formula for the derivative of tan x is:
(d/dx) tan x = sec^2 x
where sec x is the secant function, which is the opposite of the cosine function.
Proof of the Derivative of Tan x
To demonstrate the formula for the derivative of tan x, we will use the quotient rule of differentiation. Let y = tan x, and z = cos x. Next:
y/z = tan x / cos x = sin x / cos^2 x
Utilizing the quotient rule, we obtain:
(d/dx) (y/z) = [(d/dx) y * z - y * (d/dx) z] / z^2
Replacing y = tan x and z = cos x, we obtain:
(d/dx) (tan x / cos x) = [(d/dx) tan x * cos x - tan x * (d/dx) cos x] / cos^2 x
Then, we could use the trigonometric identity which links the derivative of the cosine function to the sine function:
(d/dx) cos x = -sin x
Substituting this identity into the formula we derived above, we get:
(d/dx) (tan x / cos x) = [(d/dx) tan x * cos x + tan x * sin x] / cos^2 x
Substituting y = tan x, we obtain:
(d/dx) tan x = sec^2 x
Therefore, the formula for the derivative of tan x is proven.
Examples of the Derivative of Tan x
Here are some instances of how to apply the derivative of tan x:
Example 1: Find the derivative of y = tan x + cos x.
(d/dx) y = (d/dx) (tan x) + (d/dx) (cos x) = sec^2 x - sin x
Example 2: Work out the slope of the tangent line to the curve y = tan x at x = pi/4.
The derivative of tan x is sec^2 x.
At x = pi/4, we have tan(pi/4) = 1 and sec(pi/4) = sqrt(2).
Thus, the slope of the tangent line to the curve y = tan x at x = pi/4 is:
(d/dx) tan x | x = pi/4 = sec^2(pi/4) = 2
So the slope of the tangent line to the curve y = tan x at x = pi/4 is 2.
Example 3: Locate the derivative of y = (tan x)^2.
Applying the chain rule, we obtain:
(d/dx) (tan x)^2 = 2 tan x sec^2 x
Thus, the derivative of y = (tan x)^2 is 2 tan x sec^2 x.
The derivative of tan x is a basic math idea which has many uses in calculus and physics. Getting a good grasp the formula for the derivative of tan x and its characteristics is crucial for students
and professionals in fields such as physics, engineering, and mathematics. By mastering the derivative of tan x, individuals can apply it to solve challenges and gain detailed insights into the
complex functions of the surrounding world.
If you need guidance comprehending the derivative of tan x or any other mathematical theory, contemplate connecting with us at Grade Potential Tutoring. Our expert tutors are accessible remotely or
in-person to give individualized and effective tutoring services to support you succeed. Contact us right to schedule a tutoring session and take your mathematical skills to the next stage. | {"url":"https://www.philadelphiainhometutors.com/blog/derivative-of-tan-x-formula-proof-examples","timestamp":"2024-11-05T21:42:35Z","content_type":"text/html","content_length":"75752","record_id":"<urn:uuid:68e04502-0e14-4d5f-b6d7-69949ec9e00f>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00440.warc.gz"} |
A partial dependence factorial analysis to deal with selection bias in observational studis
D’Attoma, Ida
A partial dependence factorial analysis to deal with selection bias in observational studis
, [Dissertation thesis], Alma Mater Studiorum Università di Bologna. Dottorato di ricerca in
Metodologia statistica per la ricerca scientifica
, 21 Ciclo. DOI 10.6092/unibo/amsdottorato/1484.
Documenti full-text disponibili:
Documento PDF (English) - Richiede un lettore di PDF come Xpdf o Adobe Acrobat Reader
Download (655kB) | Anteprima
This thesis presents a creative and practical approach to dealing with the problem of selection bias. Selection bias may be the most important vexing problem in program evaluation or in any line of
research that attempts to assert causality. Some of the greatest minds in economics and statistics have scrutinized the problem of selection bias, with the resulting approaches – Rubin’s Potential
Outcome Approach(Rosenbaum and Rubin,1983; Rubin, 1991,2001,2004) or Heckman’s Selection model (Heckman, 1979) – being widely accepted and used as the best fixes. These solutions to the bias that
arises in particular from self selection are imperfect, and many researchers, when feasible, reserve their strongest causal inference for data from experimental rather than observational studies. The
innovative aspect of this thesis is to propose a data transformation that allows measuring and testing in an automatic and multivariate way the presence of selection bias. The approach involves the
construction of a multi-dimensional conditional space of the X matrix in which the bias associated with the treatment assignment has been eliminated. Specifically, we propose the use of a partial
dependence analysis of the X-space as a tool for investigating the dependence relationship between a set of observable pre-treatment categorical covariates X and a treatment indicator variable T, in
order to obtain a measure of bias according to their dependence structure. The measure of selection bias is then expressed in terms of inertia due to the dependence between X and T that has been
eliminated. Given the measure of selection bias, we propose a multivariate test of imbalance in order to check if the detected bias is significant, by using the asymptotical distribution of inertia
due to T (Estadella et al. 2005) , and by preserving the multivariate nature of data. Further, we propose the use of a clustering procedure as a tool to find groups of comparable units on which
estimate local causal effects, and the use of the multivariate test of imbalance as a stopping rule in choosing the best cluster solution set. The method is non parametric, it does not call for
modeling the data, based on some underlying theory or assumption about the selection process, but instead it calls for using the existing variability within the data and letting the data to speak.
The idea of proposing this multivariate approach to measure selection bias and test balance comes from the consideration that in applied research all aspects of multivariate balance, not represented
in the univariate variable- by-variable summaries, are ignored. The first part contains an introduction to evaluation methods as part of public and private decision process and a review of the
literature of evaluation methods. The attention is focused on Rubin Potential Outcome Approach, matching methods, and briefly on Heckman’s Selection Model. The second part focuses on some resulting
limitations of conventional methods, with particular attention to the problem of how testing in the correct way balancing. The third part contains the original contribution proposed , a simulation
study that allows to check the performance of the method for a given dependence setting and an application to a real data set. Finally, we discuss, conclude and explain our future perspectives.
This thesis presents a creative and practical approach to dealing with the problem of selection bias. Selection bias may be the most important vexing problem in program evaluation or in any line of
research that attempts to assert causality. Some of the greatest minds in economics and statistics have scrutinized the problem of selection bias, with the resulting approaches – Rubin’s Potential
Outcome Approach(Rosenbaum and Rubin,1983; Rubin, 1991,2001,2004) or Heckman’s Selection model (Heckman, 1979) – being widely accepted and used as the best fixes. These solutions to the bias that
arises in particular from self selection are imperfect, and many researchers, when feasible, reserve their strongest causal inference for data from experimental rather than observational studies. The
innovative aspect of this thesis is to propose a data transformation that allows measuring and testing in an automatic and multivariate way the presence of selection bias. The approach involves the
construction of a multi-dimensional conditional space of the X matrix in which the bias associated with the treatment assignment has been eliminated. Specifically, we propose the use of a partial
dependence analysis of the X-space as a tool for investigating the dependence relationship between a set of observable pre-treatment categorical covariates X and a treatment indicator variable T, in
order to obtain a measure of bias according to their dependence structure. The measure of selection bias is then expressed in terms of inertia due to the dependence between X and T that has been
eliminated. Given the measure of selection bias, we propose a multivariate test of imbalance in order to check if the detected bias is significant, by using the asymptotical distribution of inertia
due to T (Estadella et al. 2005) , and by preserving the multivariate nature of data. Further, we propose the use of a clustering procedure as a tool to find groups of comparable units on which
estimate local causal effects, and the use of the multivariate test of imbalance as a stopping rule in choosing the best cluster solution set. The method is non parametric, it does not call for
modeling the data, based on some underlying theory or assumption about the selection process, but instead it calls for using the existing variability within the data and letting the data to speak.
The idea of proposing this multivariate approach to measure selection bias and test balance comes from the consideration that in applied research all aspects of multivariate balance, not represented
in the univariate variable- by-variable summaries, are ignored. The first part contains an introduction to evaluation methods as part of public and private decision process and a review of the
literature of evaluation methods. The attention is focused on Rubin Potential Outcome Approach, matching methods, and briefly on Heckman’s Selection Model. The second part focuses on some resulting
limitations of conventional methods, with particular attention to the problem of how testing in the correct way balancing. The third part contains the original contribution proposed , a simulation
study that allows to check the performance of the method for a given dependence setting and an application to a real data set. Finally, we discuss, conclude and explain our future perspectives.
Altri metadati
Statistica sui download | {"url":"http://amsdottorato.unibo.it/1484/","timestamp":"2024-11-02T10:56:54Z","content_type":"application/xhtml+xml","content_length":"46824","record_id":"<urn:uuid:08acf425-8749-4045-97fa-d248037b9a94>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00895.warc.gz"} |
Effect of Automatic Differentiation in Problem-Based Optimization
When using automatic differentiation, the problem-based solve function generally requires fewer function evaluations and can operate more robustly.
By default, solve uses automatic differentiation to evaluate the gradients of objective and nonlinear constraint functions, when applicable. Automatic differentiation applies to functions that are
expressed in terms of operations on optimization variables without using the fcn2optimexpr function. See Automatic Differentiation in Optimization Toolbox and Convert Nonlinear Function to
Optimization Expression.
Minimization Problem
Consider the problem of minimizing the following objective function:
$\begin{array}{l}fun1=100{\left({x}_{2}-{x}_{1}^{2}\right)}^{2}+{\left(1-{x}_{1}\right)}^{2}\\ fun2=\mathrm{exp}\left(-\sum {\left({x}_{i}-{y}_{i}\right)}^{2}\right)\mathrm{exp}\left(-\mathrm{exp}\
left(-{y}_{1}\right)\right)sech\left({y}_{2}\right)\\ objective=fun1-\phantom{\rule{0.5em}{0ex}}fun2.\end{array}$
Create an optimization problem representing these variables and the objective function expression.
prob = optimproblem;
x = optimvar('x',2);
y = optimvar('y',2);
fun1 = 100*(x(2) - x(1)^2)^2 + (1 - x(1))^2;
fun2 = exp(-sum((x - y).^2))*exp(-exp(-y(1)))*sech(y(2));
prob.Objective = fun1 - fun2;
The minimization is subject to the nonlinear constraint ${x}_{1}^{2}+{x}_{2}^{2}+{y}_{1}^{2}+{y}_{2}^{2}\le 4$.
prob.Constraints.cons = sum(x.^2 + y.^2) <= 4;
Solve Problem and Examine Solution Process
Solve the problem starting from an initial point.
init.x = [-1;2];
init.y = [1;-1];
[xproblem,fvalproblem,exitflagproblem,outputproblem] = solve(prob,init);
Solving problem using fmincon.
Local minimum found that satisfies the constraints.
Optimization completed because the objective function is non-decreasing in
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.
The output structure shows that solve calls fmincon, which requires 77 function evaluations and 46 iterations to solve the problem. The objective function value at the solution is fvalproblem =
Solve Problem Without Automatic Differentiation
To determine the efficiency gains from automatic differentiation, set solve name-value pair arguments to use finite difference gradients instead.
[xfd,fvalfd,exitflagfd,outputfd] = solve(prob,init,...
Solving problem using fmincon.
Local minimum found that satisfies the constraints.
Optimization completed because the objective function is non-decreasing in
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.
Using a finite difference gradient approximation causes solve to take 269 function evaluations compared to 77. The number of iterations is nearly the same, as is the reported objective function value
at the solution. The final solution points are the same to display precision.
0.8671 1.0433
0.7505 0.5140
0.8671 1.0433
0.7505 0.5140
In summary, the main effect of automatic differentiation in optimization is to lower the number of function evaluations.
See Also
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zhbtrd: reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T by a unitary similarity transformation - Linux Manuals (l)
zhbtrd (l) - Linux Manuals
zhbtrd: reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T by a unitary similarity transformation
ZHBTRD - reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T by a unitary similarity transformation
VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ, WORK, INFO )
CHARACTER UPLO, VECT
INTEGER INFO, KD, LDAB, LDQ, N
DOUBLE PRECISION D( * ), E( * )
COMPLEX*16 AB( LDAB, * ), Q( LDQ, * ), WORK( * )
ZHBTRD reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T.
VECT (input) CHARACTER*1
= aqNaq: do not form Q;
= aqVaq: form Q;
= aqUaq: update a matrix X, by forming X*Q.
UPLO (input) CHARACTER*1
= aqUaq: Upper triangle of A is stored;
= aqLaq: Lower triangle of A is stored.
N (input) INTEGER
The order of the matrix A. N >= 0.
KD (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = aqUaq, or the number of subdiagonals if UPLO = aqLaq. KD >= 0.
AB (input/output) COMPLEX*16 array, dimension (LDAB,N)
On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if
UPLO = aqUaq, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = aqLaq, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, the diagonal elements of AB are overwritten by the diagonal
elements of the tridiagonal matrix T; if KD > 0, the elements on the first superdiagonal (if UPLO = aqUaq) or the first subdiagonal (if UPLO = aqLaq) are overwritten by the off-diagonal elements
of T; the rest of AB is overwritten by values generated during the reduction.
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
D (output) DOUBLE PRECISION array, dimension (N)
The diagonal elements of the tridiagonal matrix T.
E (output) DOUBLE PRECISION array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T: E(i) = T(i,i+1) if UPLO = aqUaq; E(i) = T(i+1,i) if UPLO = aqLaq.
Q (input/output) COMPLEX*16 array, dimension (LDQ,N)
On entry, if VECT = aqUaq, then Q must contain an N-by-N matrix X; if VECT = aqNaq or aqVaq, then Q need not be set. On exit: if VECT = aqVaq, Q contains the N-by-N unitary matrix Q; if VECT =
aqUaq, Q contains the product X*Q; if VECT = aqNaq, the array Q is not referenced.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= 1, and LDQ >= N if VECT = aqVaq or aqUaq.
WORK (workspace) COMPLEX*16 array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Modified by Linda Kaufman, Bell Labs. | {"url":"https://www.systutorials.com/docs/linux/man/l-zhbtrd/","timestamp":"2024-11-14T20:12:38Z","content_type":"text/html","content_length":"10569","record_id":"<urn:uuid:de8873f0-a7cf-4c20-8368-dc7d8cd15b60>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00029.warc.gz"} |
Stochastic processes on manifolds – The Dan MacKinlay stable of variably-well-consider’d enterprises
Stochastic processes on manifolds
March 1, 2021 — March 1, 2021
Hilbert space
how do science
kernel tricks
machine learning
signal processing
stochastic processes
time series
1 References
Adler, Robert J. 2010. The Geometry of Random Fields.
Adler, Robert J., and Taylor. 2007.
Random Fields and Geometry
. Springer Monographs in Mathematics 115.
Bhattacharya, and Bhattacharya. 2012.
Nonparametric Inference on Manifolds: With Applications to Shape Spaces
. Institute of Mathematical Statistics Monographs.
Borovitskiy, Terenin, Mostowsky, et al. 2020.
“Matérn Gaussian Processes on Riemannian Manifolds.” arXiv:2006.10160 [Cs, Stat]
Calandra, Peters, Rasmussen, et al. 2016.
“Manifold Gaussian Processes for Regression.”
2016 International Joint Conference on Neural Networks (IJCNN)
Manton. 2013.
“A Primer on Stochastic Differential Geometry for Signal Processing.” IEEE Journal of Selected Topics in Signal Processing
Yaglom. 1961.
“Second-Order Homogeneous Random Fields.” Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Contributions to Probability Theory | {"url":"https://danmackinlay.name/notebook/stochastic_processes_on_manifolds.html","timestamp":"2024-11-05T01:15:14Z","content_type":"application/xhtml+xml","content_length":"30733","record_id":"<urn:uuid:f34a91b7-150b-4d0f-bced-83b9d5165d94>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00458.warc.gz"} |
Wave loading analysis in Femap - Siemens: Femap
Not open for further replies.
Hello Experts!,
For the study of the structural behavior of the walls of a typical swimming pool meshed with 2-D Shell CQUAD4 elements I need to simulate in Simcenter FEMAP with NASTRAN the loading effect of a
barrel of 120 kg dropping over the free surface of the water:
[li]The pool is full of water.[/li]
[li]A barrel with a diameter of 450 mm and a capacity of 120 liters and weight of 120 kg is placed at 640 mm over the surface of the water.[/li]
[li]The barrel is dropped over the pool.[/li]
[li]We need to calculate the pressure distribution in the walls of the pool to perform structural analysis to obtain stresses & displacements results.[/li]
The key here is to know how to estimate in a reasonable way in FEMAP (not very conservative neither under-estimate or incorrect) the peak pressure load distribution that receive the walls of the pool
due to the impact of the barrel using an equivalent static loading, the real situation is obviously very complex.
It is not immediately clear to me how it could be simplified to a loading scenario which would make for a meaningful analysis.
If anybody have some ideas as to how to proceed is welcome, please inform, thanks!.
Best regards,
Blas Molero Hidalgo
Ingeniero Industrial
48004 BILBAO (SPAIN)
WEB: Blog de FEMAP & NX Nastran:
Replies continue below
Recommended for you
That's a very interesting case for a simulation. A free surface CFD study with rigid body dynamics or FSI could predict this but I understand that you don't have the tools for that and want to
perform a purely mechanical analysis. For that, you will need to calculate two things:
1) parameters of waves forming after barrel impact - propagation of such spherical waves is described by Huygen's principle
2) load acting on the pool's walls when waves hit it - this is described in the article "Loading Conditions Due to Violent Wave Impacts on Coastal Structures with Cantilever Surfaces" by D. Kisacik -
various approaches are presented, including linear wave theory - ocean waves are assumed but maybe it would also make sense for spherical ones caused by impact
The problem is that literature about the design of ships and offshore structures assumes that waves are induced by wind and not by impact. Such impact could be caused by meteorite fall but then the
velocities are too large to use those considerations here.
two phase CFD ?
the barrel looks to have neutral buoyancy.
if the height 640mm or 640m ?
I can't help but think ... 120kg you say ... humm, that's a large-ish human ... doing a "cannonball" ??
"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.
Hi Blas,
When I read the discription of the load case there is sometning that I find odd.
It says a barrel with the mass 120 kg basically dropped from the height 640 m (meters ?). That seems unreasonable high to me. 640 millimeters may be a bit low but 640 meters means a huge energy to
consider. But I also thought about rb1957's human doing a cannonball
But I would start with the energy that need to be handled (E = (m * v^2)/2) and see if the pool walls have the strength and flexibility to meet the requirements. I would start simple just to get a
first indication.
How is the pool built?
Hello all!,
Yes, I see a typo in the image, the height of the 120 kg barrel is 0.64 m, ie, 640 mm.
The material of the pool is plastic.
Well, thanks to all, I see not immediate answer exist, I run the CFD code "FloEFD for SOLID EDGE" but it is NOT possible to simulate and obtain the pressure distribution results caused by a barrel
dropping over the water surfaces as this would involve surface tension effects and also meshes of the barrel colliding with the water surface. The Free Surface model in its current implementation in
FloEFD is not able to take this into account, only STAR-CCM+ can do it but not access to it.
Best regards,
Blas Molero Hidalgo
Ingeniero Industrial
48004 BILBAO (SPAIN)
WEB: Blog de FEMAP & NX Nastran:
"h"eight ...
how deep is the water in the pool ?
I suspect that this simple question is astonishingly difficult to answer.
"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.
The deep of water in the swimming pool is around 1.2 meters, ie, 1200 mm.
Best regards,
Blas Molero Hidalgo
Ingeniero Industrial
48004 BILBAO (SPAIN)
WEB: Blog de FEMAP & NX Nastran:
The kinetic energy in the barrel will transform to kinetic energy for the barrel and an influenced water volume. Since the pool is plastic it is probably very flexible even if there (probably) also
is a more rigid frame involved. But I think that the total mass of water influenced by the barrel will be significantly larger than the mass of the barrel.
I agree with rb1957 that a correct answer is probably difficult to find. But does it have to be exactly "corrct"? What is the ultimate purpose for the analysis?
It should be possible to determine the flexibility of the walls with some accuracy. From that you can get a ballpark number of how much the water can be displaced and then you have inertia forces to
balance the energy from the barrels impact.
I am just "thinking ot load" so the speak. I haven't done any of the actual math but I suspect that the load case is not really a problem and that a simplified approch might suffice
I suspect thet this arrangement for a pool is not unique.
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OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials
In this section we will identify the major types of word problems that you will most likely encounter at this point in your study of algebra.
Often students will skip the word problems, but that is not a successful strategy. Setting them up is the hard part -- solving them after that, for the most part, is not that difficult.
: Read the problem several times before starting and try to understand the question.
Number Problem:
The sum of two integers is 18. The larger number is 2 less than 3 times the smaller. Find the integers.
Number Problem:
The difference between two integers is 2. The larger number is 6 less than twice the smaller. Find the integers.
Many times we can figure out these types of word problems by guessing and checking. This is the case because the numbers are chosen to be simple, so the algebraic steps will not be too tedious. You
are learning to set up algebraic equations on these easier problems, so that you can use these ideas to solve more difficult ones later.
: For full credit your instructor will insist that you use algebra to solve the word problems. Don't fight it, just identify the variables and use them to set up an algebraic equation.
Number Problem:
One number is 3 more than another number. When two times the larger number is subtracted from 3 times the smaller number, the result is 6. Find the numbers.
It is important to notice that both even and odd integers are separated by two units.
Consecutive Odd Integers: The sum of two consecutive odd integers is 36. Find the integers.
Consecutive Even Integers:
The sum of two consecutive even integers is 46. Find the integers.
A common mistake is to set up consecutive odds with an
+ 3. This will most likely lead to a decimal answer which certainly is not an integer and is incorrect. Make sure that you read the problems carefully; notice that consecutive integers are separated
by 1 unit.
Consecutive Integers:
The sum of three consecutive integers is 24. Find the integers.
You will need to ask your instructor if you will be able to use a calculator. In either case, you should be able to work with decimals by hand. When money is involved be sure to round off to two
decimal places.
Tax Problem: If a pair of Nike shoes cost $48.95 plus a 7 1/4% sales tax, what will the total be at the register?
Tax Problem:
At 8 3/4% the amount of sales tax on an item came to $12.04. What was the cost of the item?
Tax Problem:
A local non-profit was mistakenly charged $2,005.84 which included a 7 1/4% sales tax charge. If tax was not to be included how much refund is the company due?
Taxicab Problem: A taxicab charges $5.00 for the ride plus $1.25 per mile. How much will a 53 mile trip cost?
Rental Problem:
If a rental car cost Jose $35.00 for the day plus $0.33 per mile and his total cost was $78.00, then how many miles did he drive?
Any Algebra textbook will have steps or guidelines for solving word problems. These steps are all generally the same and are outlined below. However, nothing works better to improve your word problem
skills more than practice.
Basic Guidelines for Solving Word Problems:
1. Read the problem several times and organize the given information.
2. Identify the variables by assigning a letter to the unknown quantity.
3. Set up an algebraic equation.
4. Solve the equation.
5. Finally, answer the question and make sure it makes sense.
: When working with geometry problems it helps to draw a picture.
Geometry Problem:
A rectangle has a perimeter measuring 64 feet. The length is 4 feet more than 3 times the width. Find the dimensions of the rectangle.
Geometry Problem:
The perimeter of an equilateral triangle measures 63 cm. Find the length of each side.
Geometry Problem:
Two sides of a triangle are 5 and 7 inches longer than the third side. If the perimeter measures 21 inches, find the length of each side.
Some perimeter formulas you are expected to know: ( π = 3.14)
Perimeter of a Rectangle:
P = 2l + 2w
Perimeter of a Circle:
C = 2πr
Perimeter of a Triangle:
P = a + b + c
Perimeter of a Square:
P = 4s Percent Problem:
A $215,000 house requires a 20% down payment. How much will the down payment be for this home?
Percent Decrease:
A stock fell from $42.00 to $39.85 in one year. How much of a percent decrease does this represent?
Percent Increase: A discount store paid $35.50 for a dress they are selling for $49.99. What is the store markup on this item?
Whenever setting up a percent problem always use the decimal or fractional equivalent of the percent. Generally, we wish to use real numbers in algebra. For example, instead of using 50 for 50% we
will need to use 0.50 or 1/2. Also, if the question asks for a percentage then do not forget to convert your answer to percent.
Distance Problem:
The 375 mile drive to Las Vegas took 5 hours. What was the average speed?
Distance Problem:
Two trains leave the station at the same time traveling in opposite directions. One travels at 70 miles per hour and the other travels at 60 miles per hour. How long does it take for them to become
390 miles apart?
Distance Problem: Joe and Bill are traveling across the country. Joe leaves one hour earlier than Bill at a rate of 60 miles per hour. If Bill leaves at a rate of 70 miles per hour, how long will it
take him to catch up to Bill?
For the word problems early in algebra, we generally want to set up our equations with one variable. Remember that we are in the chapter dealing with linear equations. Later in our study we will
learn how to deal with multiple variable systems. For now, try to avoid using a second variable when setting up your equations.
Video Examples on YouTube: | {"url":"https://www.openalgebra.com/search/label/solve%20equation","timestamp":"2024-11-08T11:08:31Z","content_type":"application/xhtml+xml","content_length":"96670","record_id":"<urn:uuid:18312119-9eea-4906-a719-b0b47aa4cb9e>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00765.warc.gz"} |
AUKEEP procedure • Genstat v21
Saves output from analysis of an unbalanced design (by AUNBALANCED) (R.W. Payne).
FACTORIAL = scalar Limit on number of factors in the model terms generated from the TERMS parameter; default 3
RESIDUALS = variate To save residuals from the analysis
FITTEDVALUES = variate To save fitted values
COMBINATIONS = string token Factor combinations for which to form predicted means (present, estimable); default esti
ADJUSTMENT = string token Type of adjustment to be made when predicting means (marginal, equal, observed); default marg
LSDLEVEL = scalar Significance level (as a percentage) for the least significant differences
RMETHOD = string token Type of residuals to form if the RESIDUALS option is set (simple, standardized); default simp
SAVE = identifier Save structure (from AUNBALANCED) containing details of the analysis for which further output is required; if omitted, output is from the most recent use of AUNBALANCED
TERMS = formula Model terms for which information is required
MEANS = table or pointer to tables Predicted means for each term
SEMEANS = table or pointer to tables Standard errors of the means for each term
SEDMEANS = symmetric matrix or pointer Standard errors of differences between means
to symmetric matrices
ESEMEANS = table or pointer to tables Approximate effective standard errors of the means: these are formed by procedure SED2ESE with the aim of allowing good approximations to the standard errors
for differences to be calculated by the usual formula sed[i][,j] = √( ese[i]^2 + ese[j]^2 )
LSD = symmetric matrix or pointer to Least significant differences
symmetric matrices
This procedure can be used, following the use of procedure AUNBALANCED, to save output for the analysis of variance of an unbalanced design.
Options are provided to save information about the analysis as a whole. The RESIDUALS and FITTEDVALUES options allow variates to be specified to store the residuals and fitted values, respectively.
The RMETHOD option controls whether simple or standardized residuals are saved; by default RMETHOD=simple.
The SAVE option can be set to the save structure from the analysis from which output is to be saved. If SAVE is not set, output will be produced for the most recent analysis from AUNBALANCED;
however, none of the Genstat regression directives (MODEL, TERMS, FIT, ADD, DROP and so on) must then have been used in the interim.
The parameters of AUKEEP save information about particular model terms in the analysis. With the TERMS parameter you specify a model formula, which Genstat expands to form the series of model terms
about which you wish to save information. As in AUNBALANCED, the FACTORIAL option sets a limit on the number of factors in each term. Any term containing more than that limit is deleted. The
subsequent parameters allow you to specify identifiers of data structures to store various components of information for each of the terms that you have specified. The MEANS parameter saves tables of
predicted means, the SEMEANS parameter saves tables of standard errors for the means, the SEDMEANS parameter saves symmetric matrices of standard errors of differences, the ESEMEANS parameter saves
tables of approximate effective standard errors, and the LSD parameter saves symmetric matrices of least significant differences. If you have a single term, you can supply a table or symmetric matrix
for each of these parameters, as appropriate. However, if you have several terms, you must supply a pointer which will then be set up to contain as many tables or symmetric matrices as there are
terms. The LSDLEVEL option sets the significance level (as a percentage) for the least significant differences.
Tables of means are calculated using the PREDICT directive. The first step (A) of the calculation forms the full table of predictions, classified by every factor in the model. The second step (B)
averages the full table over the factors that do not occur in the table of means. The COMBINATIONS option specifies which cells of the full table are to be formed in Step A. The default setting,
estimable, fills in all the cells other than those that involve parameters that cannot be estimated, for example because of aliasing. Alternatively, setting COMBINATIONS=present excludes the cells
for factor combinations that do not occur in the data. The ADJUSTMENT option then defines how the averaging is done in Step B. The default setting, marginal, forms a table of marginal weights for
each factor, containing the proportion of observations with each of its levels; the full table of weights is then formed from the product of the marginal tables. The setting equal weights all the
combinations equally. Finally, the setting observed uses the WEIGHTS option of PREDICT to weight each factor combination according to its own individual replication in the data.
Options: FACTORIAL, RESIDUALS, FITTEDVALUES, COMBINATIONS, ADJUSTMENT, LSDLEVEL, RMETHOD, SAVE.
Parameters: TERMS, MEANS, SEMEANS, SEDMEANS, ESEMEANS, LSD.
The output is obtained mainly using the directives RKEEP and PREDICT.
If the y-variate originally analysed by AUNBALANCED was restricted, only the units not excluded by the restriction will have been analysed.
See also
Procedures: AUNBALANCED, AUDISPLAY, AUGRAPH, AUPREDICT, AUMCOMPARISON.
Commands for: Analysis of variance.
CAPTION 'AUKEEP example',\
'Data from Genstat 5 Release 1 Reference Manual, page 340.';\
FACTOR [NVALUES=36; LEVELS=3; VALUES=12(1...3)] Block
FACTOR [NVALUES=36; LABELS=!t(baresoil,emerald,emergo)] Leachate
& [LABELS=!t('1','1/4','1/16','1/64')] Dilution
VARIATE [NVALUES=36] Nhatch,Nnohatch
READ Leachate,Dilution,Nhatch,Nnohatch
3 1 * 415
2 2 * 301
3 1 * 325
2 3 968 550 :
CALCULATE Logit%h = LOG(Nhatch/Nnohatch)
BLOCKSTRUCTURE Block
TREATMENTSTRUCTURE Leachate*Dilution
AUNBALANCED [PRINT=*] Logit%h
AUKEEP Leachate + Dilution;\
PRINT [RLPRINT=integers,labels,identifiers;\ | {"url":"https://genstat21.kb.vsni.co.uk/knowledge-base/aukeep/","timestamp":"2024-11-05T08:37:36Z","content_type":"text/html","content_length":"47354","record_id":"<urn:uuid:267a4ba9-e2e2-412c-b964-e2132e3a488f>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00564.warc.gz"} |
ACTA SCIENTIARUM MATHEMATICARUM (Szeged)
Béla Szőkefalvi-Nagy Medal 2003 481-481
No further details
[Open Access VIEW]
Order-preserving maps on the poset of idempotent matrices
Peter Šemrl
Abstract. We obtain the general form of a bijective order-preserving map defined on the poset of all $n\times n$ idempotent matrices over a field with at least 3 elements.
AMS Subject Classification (1991): 06A11; 15A30
Received June 2, 2003, and in revised form July 14, 2003. (Registered under 2910/2009.)
[Open Access VIEW]
An extension of relative pseudocomplementation to non-distributive lattices
Ivan Chajda
Abstract. We characterize lattices $L$ with 1 where for each element $p$ the interval $[p,1]$ is a pseudocomplemented lattice. Moreover, if for $x,y\in L$ the relative pseudocomplement $x\ast y$
exists then it is equal to the pseudocomplement of $x\vee y$ in $[y,1]$. However, the latter exists for each $x,y$ also e.g. in $N_5$ contrary to the case of relatively pseudocomplemented lattices
which are distributive, see e.g. [1], [2], [3].
AMS Subject Classification (1991): 06D15, 06D20
Keyword(s): Relative pseudocomplement, pseudocomplement, semidistributive lattice
Received March 11, 2002, and in revised form May 9, 2002. (Registered under 2911/2009.)
[Open Access VIEW]
A note on characterizations of central elements
B. N. Waphare, Vinayak Joshi
Abstract. In this paper an equivalent criterion for distributive elements in lattices is established. Standard elements and distributive elements are characterized in $SSC$ lattices. Moreover,
central elements in atomistic $SSC^*$ lattices are characterized in terms of dually distributive elements and also in terms of $p$-elements.
AMS Subject Classification (1991): 06D22, 06D50, 06D99
Keyword(s): Standard element, distributive element, neutral element, p, -element
Received November 20, 2001, and in revised form March 18, 2003. (Registered under 2912/2009.)
[Open Access VIEW]
Minimal clones with weakly abelian representations
Tamás Waldhauser
Abstract. We show that a minimal clone has a nontrivial weakly abelian representation iff it has a nontrivial abelian representation, and that in this case all representations are weakly abelian.
AMS Subject Classification (1991): 08A40, 20N02
Keyword(s): clone, minimal clone, (weakly) abelian algebra, groupoid
Received April 2, 2002. (Registered under 2913/2009.)
[Open Access VIEW]
On a conjecture of László Rédei
Lajos Rónyai
Abstract. Let $p$ be a prime, and $F(x_1,x_2,\ldots, x_n)\in{\msbm F} _p[x_1,\ldots,x_n]$ be a nonconstant polynomial such that the degree of $F$ in each variable $x_i$ is at most $p-1$. The {\it
rank} of $F$ is the least integer $r$ for which there exists an invertible homogeneous linear change of variables which carries $F$ into a polynomial with precisely $r$ variables. In [6] Rédei
proposed the following conjecture: if the rank of $F$ is at least $\deg F$, then the equation (congruence) $F(x_1,\ldots,x_n)=0$ has a solution in ${\msbm F} _p^n$. We disprove the conjecture by
giving counterexamples. On the other hand, we show that it holds for some important special cases, including generalized diagonal equations.
AMS Subject Classification (1991): 11T06, 11D79
Keyword(s): Finite fields, equations, solvability
Received July 29, 2002, and in revised form April 30, 2003. (Registered under 2914/2009.)
[Open Access VIEW]
Moore--Penrose inverses of products and differences of orthogonal projectors
Shizhen Cheng, Yongge Tian
Abstract. For two given orthogonal projectors $P_A$ and $P_B$ of the same order, we investigate how to express Moore--Penrose inverses of the differences $P_A - P_B$ and $P_AP_B - P_BP_A$, and then
consider some related topics.
AMS Subject Classification (1991): 15A09, 15A24
Keyword(s): difference, Moore--Penrose inverse, weighted Moore--Penrose inverses, orthogonal projector
Received May 28, 2002, and in revised form February 6, 2003. (Registered under 2915/2009.)
[Open Access VIEW]
Cancellative coextensions
Pierre Antoine Grillet
Abstract. This article defines cancellative coextensions and generalized coset extensions for commutative semigroups and studies their first properties, including their relationship to Ponizovsky
families and completions.
AMS Subject Classification (1991): 20M14
Keyword(s): cancellative coextension, generalized coset, congruence, commutative semigroup, subelementary semigroup, subcomplete semigroup, Ponizovsky family, completion
Received August 14, 2002. (Registered under 2916/2009.)
[Open Access VIEW]
Almost factorisable straight locally inverse semigroups
Erzsébet Dombi
Abstract. We introduce the notion of a factorisable and an almost factorisable straight locally inverse semigroup, and prove that every straight locally inverse semigroup can be obtained as a
subsemigroup of an idempotent separating homomorphic image of a Pastijn product of a semilattice by a completely simple semigroup. Moreover, we give an alternative proof of the fact that each
straight locally inverse semigroup has a weakly $E$-unitary cover, implicitely due to Pastijn.
AMS Subject Classification (1991): 20M17, 20M10
Received March 19, 2002, and in revised form November 10, 2002. (Registered under 2917/2009.)
[Open Access VIEW]
A regularity theorem for composite functional equations
Zsolt Páles
Abstract. In this paper we deal with regularity properties of functions $f$ and $g$ satisfying a functional inequality of the following type $$ |f(a(x,y))-f(a(x,z))|\le |g(b(x,y))-g(b(x,z))| ((x,y),
(x,z) \in D), $$ where the real valued functions $a$ and $b$ defined on an open set $D\subset{\msbm R}^2$ enjoy certain sufficiently strong regularity properties. One of the main results states that
if $g$ is pointwise Lipschitz on a dense subset of $b(D)$ (for instance if $g$ is differentiable on a dense subset) then $f$ is locally Lipschitz on $a(D)$. Another result says that if $f$ admits an
inverse pointwise Lipschitz condition on a dense subset of $a(D)$ (for instance, if $f$ is differentiable on a dense subset with nonzero derivative), then $g$ is locally invertible with a locally
Lipschitz inverse. The results so obtained have applications in the regularity theory of composite functional equations.
AMS Subject Classification (1991): 26D15, 26D07
Keyword(s): composite functional equation, regularity theory, local Lipschitz property, inverse local Lipschitz property
Received February 28, 2002. (Registered under 2918/2009.)
[Open Access VIEW]
On logarithmic Carleson measures
Ruhan Zhao
Abstract. Carleson type measures with additional logarithmic terms are characterized by using functions in $BMOA$ and the Bloch space. The results are applied to a kind of integral operators and
pointwise multipliers on $BMOA$ and the Bloch space, as well as Toeplitz operators on the weighted Bergman $1$-space.
AMS Subject Classification (1991): 30D50; 30D45, 47B35
Keyword(s): logarithmic Carleson measure, BMOA, pointwise multipliers, Bloch space
Received February 20, 2002, and in revised form July 19, 2002. (Registered under 2919/2009.)
[Open Access VIEW]
Quantizations of linear self-maps of ${\msbm R}^2$
Brendan Weickert
Abstract. We investigate the dynamics and spectral properties of the unitary operators $U_\lambda :=e^{i\lambda x^2}F$, where $\lambda\in {\msbm R}$ and $F$ is the Fourier transform. We show that $U_
\lambda $ is a quantization of the classical map $$ f_\lambda\colon {\msbm R}^2 \to{\msbm R}^2 (x,y) \mapstochar\rightarrow (y,2\lambda y-x), $$ and that the phase transition at $|\lambda |=1$ for
$f_\lambda $ corresponds to a similar phase transition for $U_\lambda $, which changes at those values from having a pure point to a continuous spectrum.
AMS Subject Classification (1991): 32H50, 37N20
Keyword(s): Unitary dynamics
Received June 6, 2002, and in revised form November 13, 2002. (Registered under 2920/2009.)
[Open Access VIEW]
More on scalar functional differential equations generating a monotone semiflow
Mihály Pituk
Abstract. H. L. Smith and H. R. Thieme showed that if a scalar retarded functional differential equation generates a strongly order preserving semiflow with respect to the exponential ordering, then
there are certain similarities between the behavior of the solutions of the functional differential equation and the ordinary differential equation obtained by ignoring the delays. In this paper we
present further results of this type concerning the boundedness of the solutions and the local and global stability of equilibria.
AMS Subject Classification (1991): 34K12, 34K20, 34K25
Received June 26, 2003. (Registered under 2921/2009.)
[Open Access VIEW]
``Blow-up'' of bounded solutions of differential equations
Vilmos Komornik, Patrick Martinez, Michel Pierre, Judith Vancostenoble
Abstract. By the classical Cauchy--Lipschitz theory of ordinary differential equations, no maximal solution of $x'=f(t,x)$ can belong to some compact subset of the domain of definition $D$ of $f$. In
the finite dimensional case it follows that the maximal solutions are defined up to the boundary of $D$. Dieudonné and later Deimling gave counterexamples in some infinite dimensional spaces: the
maximal solution can remain bounded while it blows up in finite time. We give a complete, elementary and natural proof of this result for {\it all} infinite dimensional Banach spaces.
AMS Subject Classification (1991): 34K30, 34K35
Keyword(s): Ordinary differential equation, blow-up, bounded solutions
Received July 5, 2002, and in the final form April 18, 2003. (Registered under 2922/2009.)
[Open Access VIEW]
On the absolute convergence of lacunary Fourier series with the generalized condition $B_2$
Naoko Ogata
Abstract. In this paper, we shall give sufficient conditions for the convergence of the lacunary Fourier series having the following form $$\sum_{k=- \infty }^{\infty }k^\delta\varphi \left(|\hat{f}
(n_k)| \right )< \infty, $$ where $\delta $ is a non-negative number and $\varphi $ is an increasing and concave function.
AMS Subject Classification (1991): 42A28, 42A55
Keyword(s): Absolute convergence, Lacunary series
Received September 11, 2002, and in revised form December 30, 2002. (Registered under 2923/2009.)
[Open Access VIEW]
On sine and cosine series with monotonically decreasing coefficients and generalized Lorentz--Zygmund spaces
Takashi Miyamoto
Abstract. In this paper, we get a necessary and sufficient condition for sine series with monotonically decreasing coefficients to belong to some generalized Lorentz--Zygmund space. Moreover, we have
analogous results for cosine series.
AMS Subject Classification (1991): 42A32, 46E30
Keyword(s): Trigonometric series, Lorentz--Zygmund space
Received May 28, 2002, and in revised form October 14, 2002. (Registered under 2924/2009.)
[Open Access VIEW]
Determination of jumps in terms of the Abel--Poisson mean of double conjugate series
Ferenc Móricz
Abstract. A theorem of Ferenc Lukács determines the jumps of a periodic, Lebesgue integrable function $f$ in terms of the partial sum of the conjugate series to the Fourier series of $f$. The aim of
this paper is to prove an analogous theorem in terms of the Abel--Poisson mean for a periodic, Lebesgue integrable function in two variables. We also prove an estimate of the mixed partial derivative
of the Abel--Poisson mean of the conjugate series to the Fourier series of an integrable function $F(x,y)$ at such a point, where $F$ is smooth. The two results are closely related.
AMS Subject Classification (1991): 42B05, 42A16
Keyword(s): Fourier series, conjugate series, rectangular partial sum, Abel--Poisson mean, generalized jump, smoothness of a function in two variables, \lambda_* ({\msbm T}^2), \Lambda_*({\msbm T}^
2), Zygmund classesand
Received April 15, 2002, and in revised form August 14, 2002. (Registered under 2925/2009.)
[Open Access VIEW]
Integrability and $L^1$ convergence classes for unbounded Vilenkin systems
N. Tanović-Miller
Abstract. The integrability classes for even trigonometric and Walsh systems, such as Fomin's classes ${\cal F}_p$, $p>1$, and their enlargements $dv^2$, $\overline{cv}^2$ and $cv^2$, are not
necessarily integrability classes for general Vilenkin systems. Recently Aubertin and J.F. Fourier have resolved the problem for the class $dv^2$ proving that for $(c_k) \in dv^2$, (*) $\sum |c_k-c_
{\tilde k}|/k< \infty $ is a necessary and sufficient condition in order that the sum of the Vilenkin series $\sum c_k \chi_k$ be an integrable function. Here $\tilde k$ is that index for which $\
chi_{\tilde k} = \overline{\chi }_k$, and hence depends on the characteristic sequence of primes $p=(p_{j+1})$. We improve and extend this result from $dv^2$ to new larger classes $lv^2(p)$, $\
overline{mv}^2(p)$ and $mv^2(p)$, where $dv^2 \subset lv^2(p) \subset\overline {mv}^2(p)\cap bv$ and $\overline{cv}^2 \subset\overline {mv}^2(p) \subset mv^2(p)$. We prove that for $(c_k)\in mv^2(p)\
cap bv$, (*) is also a necessary and sufficient condition for the integrability of $\sum c_k \chi_k $ and derive new equivalent forms of (*). Applications of these results yield several known
theorems on integrability of Vilenkin series.
AMS Subject Classification (1991): 42C10, 43A55
Received April 17, 2002, and in revised form June 21, 2002. (Registered under 2926/2009.)
[Open Access VIEW]
The indeterminate method of moments for adapted semigroups
Torben Maack Bisgaard
Abstract. It is shown that if $S$ is a commutative involution semigroup then the set ${\cal H}(S)$ of all moment functions on $S$ (i.e., complex-valued functions defined on $SS:=\{ st\mid s,t\in S \}
$ and admitting a disintegration as an integral of hermitian multiplicative functions) is closed under pointwise convergence in ${\bf C}^{SS}$ if and only if for each $s$ in $S$ there is a positive
integer $n$ such that $(s^*s)^n$ is the product of $2n+1$ elements of $S$. In fact, if the condition is satisfied then an `indeterminate method of moments' holds, asserting that if $(\varphi_i)$ is a
net of moment functions, converging pointwise to some function $\varphi $, and if for each $i$ in the index set $\mu_i$ is a disintegrating measure of $\varphi_i$ then some subnet of $(\mu_i)$
converges to a disintegrating measure of $\varphi $ (implying in particular that $\varphi $ is a moment function).
AMS Subject Classification (1991): 43A35, 44A60
Received May 28, 2002, and in revised form December 12, 2002. (Registered under 2927/2009.)
[Open Access VIEW]
On two-variable Jordan blocks
Rongwei Yang
Abstract. For every inner function $\psi\in H^2(D)$, the Jordan block $S(\psi )$ is the compression of the unilateral shift to the quotient space $H^2(D)\ominus\psi H^2(D)$. On the Hardy space over
the bidisk $H^2(D^2)$, the Toeplitz operators $T_{z}$ and $T_{w}$ are unilateral shifts of infinite multiplicity. For every subspace $M\subset H^2(D^2)$ invariant under $T_{z}$ and $T_{w}$, the
associated {\it two-variable Jordan block} $S(M):=(S_{z}, S_{w})$ is the compression of the pair $(T_{z}, T_{w})$ to the quotient space $H^2(D^2)\ominus M$. This paper proves that $S(M)$ has no
reducing subspace for any $M$, and gives a detailed study of $S_{w}$ when $S_{z}$ is a strict contraction. The one variable Jordan block $S(\psi )$ and the Toeplitz algebra are special cases of the
work in this paper.
AMS Subject Classification (1991): 46E20, 47A20, 47A13
Received May 14, 2002, and in revised form October 25, 2002. (Registered under 2928/2009.)
[Open Access VIEW]
Sequential isomorphisms between the sets of von Neumann algebra effects
Lajos Molnár
Abstract. In this paper we describe the structure of all sequential isomorphisms between the sets of von Neumann algebra effects. It turns out that if the underlying algebras have no commutative
direct summands, then every sequential isomorphism between the sets of their effects extends to the direct sum of a *-isomorphism and a *-antiisomorphism between the underlying von Neumann algebras.
AMS Subject Classification (1991): 46L60, 47B49
Received August 18, 2003. (Registered under 2929/2009.)
[Open Access VIEW]
A subnormality criterion for unbounded tuples of operators
Olivier Demanze
Abstract. We give necessary and sufficient conditions for a tuple of unbounded operators to be (jointly) subnormal. This criterion is then applied to characterize the subnormality of some special
tuples of operators, in particular commuting bilateral weighted multi-shifts.
AMS Subject Classification (1991): 47B20, 44A60, 47A20, 47B37
Received February 11, 2002, and in revised form May 10, 2002. (Registered under 2930/2009.)
[Open Access VIEW]
Spectral mapping properties for hyponormal operators
Muneo Cho, Young Min Han, Tadasi Huruya
Abstract. In this paper, we show that the following spectral mapping theorem holds: Let $T = H + iK$ be hyponormal and $\varphi $ be a strictly monotone increasing continuous function on $\sigma(H)$.
We define $\tilde{\varphi }(x+iy)=\varphi(x)+iy$ for $x \in\sigma (H), y \in{\msbm R}$ and $\tilde{\varphi }(T)=\varphi(H)+iK$. Then $$ \sigma_{na}(\tilde{\varphi }(T)) = \tilde{\varphi }(\sigma_{na}
(T)), \sigma_{a}(\tilde{\varphi }(T)) = \tilde{\varphi } (\sigma_{a}(T)) \mbox{ and } \sigma(\tilde{\varphi }(T)) = \tilde{\varphi } (\sigma(T)). $$ We also show that Weyl's theorem holds for $\tilde
{\varphi }(T)$ and study the G$_1$ property of the operator $\tilde{\varphi }(T)$.
AMS Subject Classification (1991): 47B20
Received March 11, 2002. (Registered under 2931/2009.)
[Open Access VIEW]
Chaotic behavior of composition operators on the Hardy space
Takuya Hosokawa
Abstract. We consider the infinite dimensional linear dynamical systems of composition operators defined on the Hardy space $H^2 (D)$. We investigate the scalar multiplied composition operators which
are chaotic in Devaney's sense.
AMS Subject Classification (1991): 47B33, 58F13
Received January 22, 2002, and in revised form April 15, 2002. (Registered under 2932/2009.)
[Open Access VIEW]
On the norm of a composition operator with linear fractional symbol
Christopher Hammond
Abstract. For any analytic map $\varphi\colon {\msbm D}\rightarrow{\msbm D}$, the composition operator $C_{\varphi }$ is bounded on the Hardy space $H^2$, but there is no known procedure for
precisely computing its norm. This paper considers the situation where $\varphi $ is a linear fractional map. We determine the conditions under which $\|C_{\varphi }\|$ is given by the action of
either $C_{\varphi }$ or $C_{\varphi }^{\ast }$ on the normalized reproducing kernel functions of $H^2$. We also introduce a new set of conditions on $\varphi $ under which we can calculate $\|C_{\
varphi }\|$; moreover, we identify the elements of $H^2$ on which such an operator $C_{\varphi }$ attains its norm. Several specific examples are provided.
AMS Subject Classification (1991): 47B33
Received February 20, 2002, and in revised form June 18, 2002. (Registered under 2933/2009.)
[Open Access VIEW]
A few more remarks on the operator valued corona problem
Pascale Vitse
Abstract. As is known, the corona theorem is in general not true for a function $F \in H^{\infty }(L(H))$, where $L(H)$ is the space of bounded operators on an infinite dimensional separable Hilbert
space $H$. Combined with a relatively compact range $F({\msbm D})$, the approximation property (AP), either in $H^{\infty }$ or in $L(H)$, provides functions satisfying the corona theorem, see [Vit].
Here we prove by counterexamples that these two methods are independent. We also give some new examples of subspaces of $L(H)$ and quotient spaces $H^{\infty }/ BH^{\infty }$ satisfying (AP). To
finish, we give a version of the corona theorem for functions in the operator Nevanlinna class having a relatively compact range.
AMS Subject Classification (1991): 47A56, 47A20, 46B28, 30D55
Received April 16, 2002, and in revised form March 6, 2003. (Registered under 2934/2009.)
[Open Access VIEW]
Generalizations of results on relations between Furuta-type inequalities
Masatoshi Ito, Takeaki Yamazaki, Masahiro Yanagida
Abstract. Let $A$ and $B$ be positive operators. We remark that $A$ and $B$ are not necessarily invertible. Recently, Ito and Yamazaki showed relations between the two inequalities $$ (B^{r\over2}A^
pB^{r\over2})^{r\over p+r} \ge B^r \mbox{ and } A^p \ge(A^{p\over2}B^rA^{p\over2})^{p\over p+r}, $$ for fixed positive numbers $p \ge0$ and $r \ge0$. In this paper, as extensions of these results, we
shall show relations between the two inequalities $$ (B^{r\over2}A^pB^{r\over2})^{r-\delta\over p+r} \ge B^{r-\delta } \mbox{ and } A^{p\over2}B^{\delta }A^{p\over2} \ge(A^{p\over2}B^rA^{p\over2})^{\
delta +p\over p+r}, $$ for fixed positive numbers $r \ge\delta \ge0$ and $p \ge0$. We shall also show a relation between the two inequalities $$ A^{p-\gamma } \ge(A^{p\over2}B^rA^{p\over2})^{p-\gamma
\over p+r} \mbox{ and } (B^{r\over2}A^pB^{r\over2})^{\gamma +r\over p+r} \ge B^{r\over2}A^{\gamma }B^{r\over2}, $$ for fixed positive numbers $p \ge\gamma \ge0$ and $r \ge0$. Furthermore, we shall
show a slight extension of a result on transitive properties of the first two inequalities by Yanagida as an application of these results.
AMS Subject Classification (1991): 47A63
Keyword(s): Positive operators, Furuta inequality
Received April 24, 2002, and in revised form October 24, 2002. (Registered under 2935/2009.)
[Open Access VIEW]
A remark on the spectra of $\infty $-hyponormal operators
Shizuo Miyajima, Isao Saito
Abstract. The authors [5] called a bounded linear operator $T$ $\infty $-hyponormal if $T$ is $p$-hyponormal for every $p>0$. They investigated the spectral properties of a pure $\infty $-hyponormal
operator $T$ under the condition that $T$ has dense range and has no nontrivial reducing subspace. In this paper it is shown that these properties of a pure $\infty $-hyponormal operator $T$ still
hold without this condition.
AMS Subject Classification (1991): 47B20, 47A10
Received April 24, 2002, and in revised form January 21, 2003. (Registered under 2936/2009.)
[Open Access VIEW]
Composition operators on the Fock space
Brent J. Carswell, Barbara D. MacCluer, Alex Schuster
Abstract. We determine the holomorphic mappings of ${\msbm C}^n$ that induce bounded composition operators on the Fock space in ${\msbm C}^n$. Furthermore, we determine which of these composition
operators are compact, and we compute the operator norm of all bounded composition operators in this setting. We also consider extensions of these results to various generalizations of the Fock
AMS Subject Classification (1991): 47B33; 32A37
Received May 28, 2002, and in revised form November 15, 2002. (Registered under 2937/2009.)
[Open Access VIEW]
Fejér means and norms of large Toeplitz matrices
A. Böttcher, S. Grudsky
Abstract. We prove that the spectral norm of a finite Toeplitz matrix can be estimated from below through the Fejér mean of the generating function. This result has applications to the problem of
finding the most probable values of $\|A_n x\|$ in case $A_n$ is a large finite Toeplitz matrix and $x$ is uniformly distributed on the unit sphere of ${\bf C}^n$.
AMS Subject Classification (1991): 47B35; 15A12, 42A05, 60H25
Received July 29, 2002, and in revised form December 12, 2002. (Registered under 2938/2009.)
[Open Access VIEW]
A new class of non-Wythoffian perfect 4-polytopes
Gábor Gévay
Abstract. A polytope is perfect if its shape cannot be changed without changing the action of its symmetry group on its face-lattice. Perfect polytopes are completely known only in dimensions 2 and
3, while exploring their various possible classes in dimension $4$ is still in progress. Here a new class is constructed which is closely related to regular 4-polytopes. In addition, there are some
interesting coincidences between the $f$-vectors of some of them, which are briefly discussed at the end of the paper.
AMS Subject Classification (1991): 20F55, 52B05, 52B15
Received May 28, 2002, and in revised form March 24, 2003. (Registered under 2939/2009.)
[Open Access VIEW]
Book Reviews 911-943
No further details
[Open Access VIEW] | {"url":"http://pub.acta.hu/acta/showCustomerVolume.action?id=3275&dataObjectType=volume&noDataSet=true&style=","timestamp":"2024-11-11T01:19:12Z","content_type":"text/html","content_length":"134082","record_id":"<urn:uuid:e26f90c3-3c01-4b0d-9094-0a57540b7544>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00650.warc.gz"} |
Mathematically Balanced
Flying Horseduck Sixty Series
Rules for Numbering Layouts
Our numbering layouts are mathematically balanced to do justice to our precision milled dice. The painstaking process that required approximately 100 hours of work over dozens of late nights was
worth the result of dice that are as fair and numerically elegant as they are pretty. The numbering layouts of our dice are bound by the following rules.
Rules for all dice:
1. Faces with the highest value on a die are each always exactly opposite a face with the lowest value
2. The sum of the five faces of any pentagon on a die is within 1 of the sum of any other pentagon (all equal is not possible)
3. Pentagons directly opposite each other on the die have equal sums
4. Highest and lowest values are spread across the die as evenly as practical
Rules Specific to Each Die:
Notes and Definitions:
• The term "pentagon" is used to collectively refer to 5 faces that completely surround a vertex
• The term "opposite" means the other side of the die; if a die is resting on a face, the opposite face is up
• The term "adjacent" refers to two faces which share an edge
• On the D10 "0" is the value 10 and is considered the highest value on the die
• On the D00 "00" is the value 100 and is considered the highest value on the die
• On the D8 the re-roll faces are considered to have the value 4.5, which is the average roll of a D8
• * Re-roll faces are ignored in the "Difference Between Adjacent Faces" calculations | {"url":"https://flyinghorseduck.com/pages/mathematically-balanced","timestamp":"2024-11-11T14:49:45Z","content_type":"text/html","content_length":"78150","record_id":"<urn:uuid:827bb6d0-de2c-4788-b147-02a321f94dac>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00296.warc.gz"} |
Information geometry and entropy in a stochastic epidemic rate process
Dodson, CTJ (2009) Information geometry and entropy in a stochastic epidemic rate process. [MIMS Preprint]
Download (1MB)
A commonly recurring approximation to real rate processes is of the form: dN/dt = -m N where m is some positive rate constant and N(t) measures the current value of some property relevant to the
process---radioactive decay is our typical student example. The simplest stochastic version addresses the situation where N(t) is the size of the current population and the rate constant depends on
the distribution of properties in the population---so different sections decay at different rates. Then the interest lies in the evolution of the distribution of properties and of the related
statistical features like entropy, mean and variance, for given initial distribution. We show that there is a simple closed solution for an example of an epidemic in which the latency and infectivity
are distributed properties controlled by a bivariate gamma distribution.
Actions (login required) | {"url":"https://eprints.maths.manchester.ac.uk/1272/","timestamp":"2024-11-11T04:52:57Z","content_type":"application/xhtml+xml","content_length":"20034","record_id":"<urn:uuid:d45e6264-568b-4b2a-8ea5-b2ea55a2ecd3>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00490.warc.gz"} |
Visualize the mean, median, IQR, and MAD
You can move the data points around in the distribution below and see how the distribution of data impacts the mean, median, interquartile range (IQR), and mean absolute deviation (MAD) of the data.
Note that the MAD is actually only calculated as half the length of the vectors shown. | {"url":"https://stage.geogebra.org/m/emdgmjcr","timestamp":"2024-11-02T05:04:44Z","content_type":"text/html","content_length":"90586","record_id":"<urn:uuid:aff0a0f7-581e-451f-9714-bc1fa2b37be8>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00080.warc.gz"} |
Understanding Mathematical Functions: How To Write A Function Notation
Introduction to Mathematical Functions and Function Notation
Mathematical functions play a crucial role in a wide range of fields, including mathematics, physics, engineering, and economics. Functions help us describe and analyze relationships between
variables, making them essential tools for problem-solving and modeling real-world phenomena.
Overview of the importance of understanding mathematical functions in various fields of study
Understanding mathematical functions is essential in various fields of study because they allow us to represent and analyze relationships between different quantities. For example, in physics,
functions are used to describe the motion of particles, while in economics, functions help us model supply and demand curves. By understanding functions, we can make predictions, analyze trends, and
solve complex problems.
Introduction to function notation as a method to express relationships between variables
Function notation is a method used to express relationships between variables in mathematics. It is a way of representing a function using symbols and mathematical expressions. Function notation
allows us to define a function, name it, and use it in equations and calculations.
Brief history of function notation and its significance in simplifying complex mathematical concepts
Function notation has a long history in mathematics, dating back to the work of mathematicians such as Gottfried Wilhelm Leibniz and Leonhard Euler. The use of function notation has significantly
contributed to simplifying complex mathematical concepts by providing a standardized way to represent functions and their relationships with variables. By using function notation, mathematicians and
scientists can communicate ideas more effectively and work with functions in a more organized and efficient manner.
Key Takeaways
• Function notation is a way to represent mathematical relationships.
• Functions have input and output values.
• Writing a function in notation helps simplify complex expressions.
• Function notation uses f(x) to represent a function of x.
• Understanding function notation is essential in higher level math.
Fundamentals of Function Notation
Function notation is a crucial concept in mathematics that allows us to represent relationships between variables in a concise and organized manner. By using function notation, we can easily define
and work with mathematical functions. Let's delve into the key components of function notation:
A Definition of function notation and its components (eg, f(x))
Function notation is a way of representing a function using symbols and variables. The most common form of function notation is f(x), where f represents the function and x is the input variable. The
expression f(x) is read as 'f of x' and indicates that the function f operates on the input x.
Distinguishing between the input (independent variable) and output (dependent variable)
It is essential to understand the distinction between the input and output variables in function notation. The input variable, often denoted as x, is the independent variable that we can manipulate
or change. On the other hand, the output variable, represented by f(x), is the dependent variable that is determined by the function's rule or formula.
• Input (Independent Variable): The variable that is controlled or chosen by the experimenter.
• Output (Dependent Variable): The variable that is influenced by changes in the input variable.
Explanation of the domain and range within the context of function notation
In the context of function notation, the domain refers to the set of all possible input values for the function. It represents the valid inputs that the function can operate on. On the other hand,
the range is the set of all possible output values that the function can produce based on the given inputs.
Understanding the domain and range of a function is crucial for determining the behavior and limitations of the function. The domain restricts the possible inputs, while the range specifies the
possible outputs that the function can generate.
Writing Basic Function Notations
Function notation is a way to represent a mathematical function using symbols and variables. It helps us understand how one quantity depends on another and allows us to perform operations on
functions. Here is a step-by-step guide on how to write function notations from simple equations:
A Step-by-step guide on writing function notations from simple equations
• Step 1: Identify the input and output variables in the equation. The input variable is usually denoted by x, while the output variable is denoted by y.
• Step 2: Write the function notation using the input and output variables. For example, if the equation is y = 2x + 3, the function notation would be f(x) = 2x + 3.
• Step 3: Use the function notation to represent the relationship between the input and output variables. In this case, f(x) represents the output y as a function of the input x.
Examples of converting common mathematical expressions into function notations
Let's look at some examples of converting common mathematical expressions into function notations:
• Example 1: If the equation is y = x^2, the function notation would be f(x) = x^2.
• Example 2: For the equation y = 3x - 5, the function notation would be f(x) = 3x - 5.
• Example 3: If the equation is y = sin(x), the function notation would be f(x) = sin(x).
Common mistakes to avoid when writing function notations for the first time
When writing function notations for the first time, it's important to avoid common mistakes that can lead to confusion. Here are some mistakes to watch out for:
• Mistake 1: Mixing up input and output variables. Make sure to correctly identify which variable represents the input and which represents the output.
• Mistake 2: Forgetting to use function notation. Always remember to use f(x) or another appropriate notation to represent the function.
• Mistake 3: Not specifying the function domain. It's important to define the domain of the function to avoid ambiguity.
Advanced Function Notations
As we delve deeper into the realm of mathematical functions, we encounter more complex notations that involve multiple variables. Understanding these advanced function notations is crucial for
tackling higher-level mathematics such as calculus and algebra. Let's explore some examples and strategies for interpreting these intricate notations.
Introduction to more complex function notations involving multiple variables
When dealing with functions that involve multiple variables, the notation becomes more sophisticated. Instead of a simple f(x) notation, we might see functions written as f(x, y) or even f(x, y, z).
Each variable represents a different input that affects the output of the function. For example, in a function f(x, y) = x + y, both x and y contribute to the final result.
Examples of function notations in higher mathematics
In higher mathematics, such as calculus and algebra, complex function notations are commonly used to represent intricate relationships between variables. For instance, in calculus, you might come
across functions involving derivatives and integrals, denoted by symbols like f'(x) and ∫f(x)dx. These notations convey important information about the behavior of the function and its derivatives.
• Example 1: In calculus, the chain rule is often represented using function notation as (f(g(x)))' = f'(g(x)) * g'(x), where f and g are functions of x.
• Example 2: In algebra, matrices are commonly used to represent linear transformations, with functions written as f(A) = A^2 - 2A + I, where A is a matrix.
Strategies for understanding and interpreting complex function notations
When faced with complex function notations, it's essential to break them down into smaller components and analyze each part separately. Here are some strategies to help you make sense of intricate
function notations:
• Identify the variables: Determine the variables involved in the function and understand how each one contributes to the output.
• Look for patterns: Search for recurring patterns or structures within the notation that can provide insights into the function's behavior.
• Consult resources: Utilize textbooks, online resources, or consult with peers or instructors to gain a deeper understanding of complex function notations.
• Practice solving problems: Work through practice problems that involve complex function notations to improve your proficiency in interpreting them.
Applications of Function Notation in Real-world Scenarios
Illustration of how function notation is used in sciences (eg, physics, chemistry)
In the field of sciences, function notation plays a crucial role in representing relationships between variables. For instance, in physics, a function may describe the motion of an object in terms of
time. This function could be denoted as f(t), where f represents the function and t represents time. By using function notation, scientists can easily analyze and predict the behavior of physical
Exploration of function notation in economics and social sciences
In economics and social sciences, function notation is used to model various relationships and phenomena. For example, in economics, a production function may be denoted as Q(K,L), where Q represents
the quantity of output, K represents capital, and L represents labor. This notation helps economists understand how changes in inputs affect output levels.
Practical examples demonstrating the utility of function notation in technology and engineering
Function notation is widely used in technology and engineering to describe complex systems and processes. For instance, in electrical engineering, a transfer function may be denoted as H(s), where H
represents the transfer function and s represents the Laplace variable. This notation allows engineers to analyze the behavior of electrical circuits and design efficient systems.
6 Troubleshooting Common Issues with Function Notation
Function notation can sometimes be tricky to work with, leading to common errors and misunderstandings. In this section, we will discuss some of the most frequent issues that arise when dealing with
function notation and provide tips for resolving them.
A Identifying and resolving frequent errors in writing and interpreting function notations
• Missing parentheses: One common error in function notation is forgetting to include parentheses when writing a function. This can lead to confusion about the order of operations and the input
value of the function.
• Incorrect variable names: Another common mistake is using the wrong variable name in a function notation. It is important to use the correct variable to ensure the function is defined properly.
• Confusion between function notation and algebraic expressions: Sometimes, students may mix up function notation with algebraic expressions, leading to errors in interpretation. It is essential to
understand the difference between the two concepts.
B Tips for verifying the accuracy of a function notation
• Substitute values: One way to verify the accuracy of a function notation is to substitute different values for the input variable and check if the output matches the expected result.
• Check for consistency: Make sure that the function notation is consistent with the definition of the function and follows the correct mathematical rules.
• Use graphing tools: Graphing the function can also help in verifying the accuracy of the function notation. This visual representation can provide insights into the behavior of the function.
C Strategies for simplifying complex function notations for easier understanding
• Break it down: If you encounter a complex function notation, try breaking it down into smaller parts and analyzing each component separately. This can help in understanding the overall function
• Use examples: Work through examples of different function notations to gain a better understanding of how they work. Practice is key to mastering complex concepts.
• Seek help: If you are struggling with a particular function notation, don't hesitate to seek help from a teacher, tutor, or online resources. Sometimes, a fresh perspective can make all the
Conclusion & Best Practices in Function Notation
In conclusion, understanding mathematical functions and mastering function notation is essential for success in various fields such as mathematics, physics, engineering, and computer science. By
grasping the key points covered in this blog post, individuals can enhance their problem-solving skills and analytical thinking abilities.
A Recap of the key points covered and their importance in mastering function notation
• Definition of Function Notation: Function notation is a way to represent functions using symbols and variables. It helps in simplifying complex mathematical expressions and making them easier to
work with.
• Importance of Function Notation: Function notation allows us to define, evaluate, and manipulate functions efficiently. It provides a standardized way to communicate mathematical ideas and
• Understanding Function Composition: Function composition involves combining two or more functions to create a new function. It is a fundamental concept in mathematics and plays a crucial role in
solving real-world problems.
Best practices for writing and working with function notations, including continuous learning and application
• Consistent Notation: Use clear and consistent notation when writing functions to avoid confusion. Follow standard conventions and guidelines for function notation.
• Practice and Application: Regular practice and application of function notation in solving problems can help improve your understanding and proficiency. Work on a variety of problems to enhance
your skills.
• Continuous Learning: Stay updated with new developments in function notation and related concepts. Engage in continuous learning through courses, books, and online resources to deepen your
Encouragement for further exploration of mathematical functions beyond basic function notation
As you continue your journey in mathematics, I encourage you to explore advanced topics in mathematical functions beyond basic function notation. Dive into topics such as trigonometric functions,
exponential functions, logarithmic functions, and more. These concepts have wide-ranging applications in various fields and can broaden your understanding of mathematical functions. | {"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-how-to-write-function-notation","timestamp":"2024-11-11T16:16:35Z","content_type":"text/html","content_length":"224133","record_id":"<urn:uuid:0d70efcb-a2c5-4bb8-ab05-1cdda4daf7ad>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00863.warc.gz"} |
Eglash's African Fractals
African Fractals: Modern Computing and Indigenous Design
by Dr. Ron Eglash http://www.rpi.edu/~eglash/eglash.htm
IN 1988, RON EGLASH was studying aerial photographs of a traditional
Tanzanian village when a strangely familiar pattern caught his eye.
The thatched-roof huts were organized in a geometric pattern of
circular clusters within circular clusters, an arrangement Eglash
recognized from his former days as a Silicon Valley computer engineer.
Stunned, Eglash digitized the images and fed the information into a
computer. The computer's calculations agreed with his intuition: He was
seeing fractals.
Since then, Eglash has documented the use of fractal geometry-the
geometry of similar shapes repeated on ever-shrinking scales-in
everything from hairstyles and architecture to artwork and religious
practices in African culture. The complicated designs and surprisingly
complex mathematical processes involved in their creation may force
researchers and historians to rethink their assumptions about
traditional African mathematics. The discovery may also provide a new
tool for teaching African-Americans about their mathematical heritage.
In contrast to the relatively ordered world of Euclidean geometry
taught in most classrooms, fractal geometry yields less obvious
patterns. These patterns appear everywhere in nature, yet mathematicians
began deciphering them only about 30 years ago.
Fractal shapes have the property of self-similarity, in which a
small part of an object resembles the whole object. "If I look at a
mountain from afar, it looks jagged and irregular, and if I start hiking
up it, it still looks jagged and irregular," said Harold Hastings, a
professor of mathematics at Hofstra University. "So it's a fractal
object-its appearance is maintained across some scales." Nearly 20 years
ago, Hastings documented fractal growth patterns among cypress trees in
Georgia's Okefenokee Swamp. Others have observed fractal patterns in the
irregular features of rocky coastlines, the ever-diminishing scaling of
ferns, and even the human respiratory and circulatory systems with their
myriad divisions into smaller and smaller branches. What all of these
patterns share is a close-up versus a panoramic symmetry instead of the
common right versus left symmetry seen in mirror images.
The principles of fractal geometry are offering scientists powerful
new tools for biomedical, geological and graphic applications. A few
years ago, Hastings and a team of medical researchers found that the
clustering of pancreatic cells in the human body follows the same
fractal rules that meteorologists have used to describe cloud formation
and the shapes of snowflakes.
But Eglash envisioned a different potential for the beautiful
fractal patterns he saw in the photos from Tanzania: a window into the
world of native cultures.
Eglash had been leafing through an edited collection of research
articles on women and Third World development when he came across an
article about a group of Tanzanian women and their loss of autonomy in
village organization. The author blamed the women's plight on a shift
from traditional architectural designs to a more rigid modernization
program. In the past, the women had decided where their houses would go.
But the modernization plan ordered the village structures like a
grid-based Roman army camp, similar to tract housing.
Eglash was just beginning a doctoral program in the history of
consciousness at the University of California at Santa Cruz. Searching
for a topic that would connect cultural issues like race, class and
gender with technology, Eglash was intrigued by what he read and asked
the researcher to send him pictures of the village.
After detecting the surprising fractal patterns, Eglash began going
to museums and libraries to study aerial photographs from other cultures
around the world.
"My assumption was that all indigenous architecture would be more
fractal," he said. "My reasoning was that all indigenous architecture
tends to be organized from the bottom up." This bottom-up, or
self-organized, plan contrasts with a top-down, or hierarchical, plan in
which only a few people decide where all the houses will go.
"As it turns out, though, my reasoning was wrong," he said. "For
example, if you look at Native American architecture, you do not see
fractals. In fact, they're quite rare." Instead, Native American
architecture is based on a combination of circular and square symmetry,
he said.
Pueblo Bonito, an ancient ruin in northwestern New Mexico built by
the Anasazi people, consists of a big circular shape made of connected
squares. This architectural design theme is repeated in Native American
pottery, weaving and even folklore, said Eglash.
When Eglash looked elsewhere in the world, he saw different
geometric design themes being used by native cultures. But he found
widespread use of fractal geometry only in Africa and southern India,
leading him to conclude that fractals weren't a universal design theme.
Focusing on Africa, he sought to answer what property of fractals
made them so widespread in the culture.
"If they used circular houses, they would use circles within
circles," he said.
"If they used rectangles you would see rectangles within rectangles.
I would see these huge plazas. Those would narrow down to broad avenues,
those would narrow down to smaller streets, and those would keep
branching down to tiny footpaths. From a European point of view, that
may look like chaos, but from a mathematical view it's the chaos of
chaos theory-it's fractal geometry." Eglash expanded on his work in
Africa after he won a Fulbright Grant in 1993.
He toured central and western Africa, going as far north as the
Sahel, the area just south of the Sahara Desert, and as far south as the
equator. He visited seven countries in all.
"Basically I just toured around looking for fractals, and when I
found something that had a scaling geometry, I would ask the folks what
was going on-why they had made it that way," he said.
In some cases Eglash found that fractal designs were based purely on
aesthetics-they simply looked good to the people who used them. In many
cases, however, Eglash found that step-by-step mathematical procedures
were producing these designs, many of them surprisingly sophisticated.
While visiting the Mangbetu society in central Africa, he studied
the tradition of using multiples of 45-degree angles in the native
artwork. The concept is similar to the shapes that American geometry
students produce using only a compass and a straight edge, he said. In
the Mangbetu society, the uniform rules allowed the artisans to compete
for the best design.
Eglash found a more complex example of fractal geometry in the
windscreens widely used in the Sahel region. Strong Sahara winds
regularly sweep the dry, dusty land. For protection from the biting wind
and swirling sand, local residents have fashioned screens woven with
millet, a common crop in the area.
The windscreens consist of about 10 diagonal rows of millet stalk
bundles, each row shorter than the one below it.
"The geometry of the screen is quite extraordinary," said Eglash. "I
had never seen anything like it." In Mali, Eglash interviewed an artisan
who had constructed one of the screens, asking him why he had settled on
the fractal design.
The man told Eglash the long, loosely bound rows forming the bottom
of the screen are very cheap to construct but do little to keep out wind
and dust. The smaller, tighter rows at the top require more time and
straw to make but also offer much more protection. The artisans had
learned from experience that the wind blows more strongly higher off the
ground, so they had made only what was needed.
"What they had done is what an engineer would call a cost-benefit
analysis," said Eglash.
He measured the length of each row of the non-linear windscreen and
plotted the data on a graph.
"I could figure out what the lengths should be based on wind
engineering values and compared those values to the actual lengths and
discovered that they were quite close," he said. "Not only are they
using a formal geometrical system to produce these scaling shapes, but
they also have a nice practical value." Eglash realized that many of the
fractal designs he was seeing were consciously created. "I began to
understand that this is a knowledge system, perhaps not as formal as
western fractal geometry but just as much a conscious use of those same
geometric concepts," he said. "As we say in California, it blew my
mind." In Senegal, Eglash learned about a fortune-telling system that
relies on a mathematical operation reminiscent of error checks on
contemporary computer systems.
In traditional Bamana fortune-telling, a divination priest begins by
rapidly drawing four dashed lines in the sand. The priest then connects
the dashes into pairs. For lines containing an odd number of dashes and
a single leftover, he draws one stroke in the sand. For lines with
even-paired dashes, he draws two strokes. Then he repeats the entire
The mathematical operation is called addition modulo 2, which simply
gives the remainder after division by two. But in this case, the two
"words" produced by the priest, each consisting of four odd or even
strokes, become the input for a new round of addition modulo 2. In other
words, it's a pseudo random-number generator, the same thing computers
do when they produce random numbers. It's also a numerical feedback
loop, just as fractals are generated by a geometric feedback loop.
"Here is this absolutely astonishing numerical feedback loop, which is indigenous," said Eglash. "So you can see the concepts of fractal geometry resonate throughout many facets of African culture."
Lawrence Shirley, chairman of the mathematics department at Towson (Md.) University, lived in Nigeria for 15 years and taught at Ahmadu Bello University in Zaria, Nigeria. He said he's impressed with
Eglash's observations of fractal geometry in Africa.
"It's amazing how he was able to pull things out of the culture and fit them into mathematics developed in the West," Shirley said. "He really did see a lot of interesting new mathematics that others
had missed." Eglash said the fractal design themes reveal that traditional African mathematics may be much more complicated than previously thought. Now an assistant professor of science and
technology studies at Rensselaer Polytechnic Institute in Troy, Eglash has written about the revelation in a new book, "African Fractals: Modern Computing and Indigenous Design." "We used to think of
mathematics as a kind of ladder that you climb," Eglash said. "And we would think of counting systems-one plus one equals two-as the first step and simple shapes as the second step." Recent
mathematical developments like fractal geometry represented the top of the ladder in most western thinking, he said. "But it's much more useful to think about the development of mathematics as a kind
of branching structure and that what blossomed very late on European branches might have bloomed much earlier on the limbs of others.
"When Europeans first came to Africa, they considered the architecture very disorganized and thus primitive. It never occurred to them that the Africans might have been using a form of mathematics
that they hadn't even discovered yet." Eglash said educators also need to rethink the way in which disciplines like African studies have tended to skip over mathematics and related areas.
To remedy that oversight, Eglash said he's been working with African-American math teachers in the United States on ways to get minorities more interested in the subject. Eglash has consulted with
Gloria Gilmer, a well-respected African-American mathematics educator who now runs her own company, Math-Tech, Inc., based in Milwaukee. Gilmer suggested that Eglash focus on the geometry of black
hairstyles. Eglash had included some fractal models of corn-row hair styles in his book and agreed they presented a good way to connect with contemporary African-American culture.
[Patterns in African American Hairstyles by Gloria Gilmer]
Jim Barta, an assistant professor of education at Utah State University in Logan, remembers a recent conference in which Eglash gave a talk on integrating hair braiding techniques into math
education. The talk drew so many people the conference organizers worried about fire code regulations.
"What Ron is helping us understand is how mathematics pervades all that we do," said Barta. "Mathematics in and of itself just is, but as different cultures of human beings use it, they impart their
cultural identities on it-they make it theirs." Joanna Masingila, president of the North American chapter of the International Study Group on Ethnomathematics, said Eglash's research has shed light
on a type of mathematical thinking and creativity that has often been ignored by western concepts of mathematics. "It's challenging stereotypes on what people think of as advanced versus primitive
approaches to solving problems," she said. "Sometimes we're limited by our own ideas of what counts as mathematics." Eglash has now written a program for his Web site that allows students to
interactively explore scaling models for a photograph of a corn-row hair style.
Eventually, he'd like to create a CD ROM-based math lab thatcombines his African fractal materials with African-American hair styles and other design elements such as quilts.
One of the benefits of including familiar cultural icons in mathematics education is that it helps combat the notion of biological determinism, Eglash said.
Biological determinism is the theory that our thinking is limited by our racial genetics. This theory gets reinforced every time a parent dismisses a child's poor math scores as nothing more than a
continuation of bad math skills in the family, said Eglash. "So for Americans, this myth of biological determinism is a very prevalent myth," he said. "We repeat it even when we don't realize it."
Eglash said using the African fractals research to combat the biological determinism myth benefits all students. "On the other hand, there is a lot of interest in how this might fit in with
African-American cultural identity," he said."Traditionally, black kids have been told, 'Your heritage is from the land of song and dance.' It might make a difference for them to see that their
heritage is also from the land of mathematics."
Book now available from Rutgers University Press:
Order by phone 800-446-9323.
Order book from Amazon.com
Description from the back cover:
Fractal geometry has emerged as one of the most exciting frontiers in the fusion between mathematics and information technology. Fractals can be seen in many of the swirling patterns produced by
computer graphics, and have become an important new tool for modeling in biology, geology, and other natural sciences. While fractal geometry can take us into the far reaches of high tech science,
its patterns are surprisingly common in traditional African designs, and some of its basic concepts are fundamental to African knowledge systems.
African Fractals introduces readers to fractal geometry and explores the ways it is expressed in African cultures. Drawing on interviews with African designers, artists, and scientists, Ron Eglash
investigates fractals in African architecture, traditional hairstyling, textiles, sculpture, painting, carving, metalwork, religion, games, quantitative techniques, and symbolic systems. He also
examines the political and social implications of the existence of African fractal geometry. Both clear and complex, this book makes a unique contribution to the study of mathematics, African
culture, anthropology, and aesthetic design.
For more about the book see Dr. Eglash's webpage at http://www.rpi.edu/~eglash/eglash.dir/afbook.htm
On the cover is the iterative construction of a Fulani wedding blanket, for instance, embeds spiritual energy, Eglash argues. In this case, the diamonds in the pattern get smaller as you move from
either side toward the blanket's center. "The weavers who created it report that spiritual energy is woven into the pattern and that each successive iteration shows an increase in this energy,"
Eglash notes. "Releasing this spiritual energy is dangerous, and if the weavers were to stop in the middle they would risk death. The engaged couple must bring the weaver food and kola nuts to keep
him awake until it is finished."
Dr. Ron Eglash:
Assistant Professor
Department of Science and Technology Studies
Rensselaer Polytechnic Institute (RPI)
Troy, NY 12180-3590
email: eglash@rpi.edu
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Voronoi Diagrams
30.2 Voronoi Diagrams
A Voronoi diagram or Voronoi tessellation of a set of points s in an N-dimensional space, is the tessellation of the N-dimensional space such that all points in v(p), a partitions of the tessellation
where p is a member of s, are closer to p than any other point in s. The Voronoi diagram is related to the Delaunay triangulation of a set of points, in that the vertexes of the Voronoi tessellation
are the centers of the circum-circles of the simplices of the Delaunay tessellation.
Plot the Voronoi diagram of points (x, y).
The Voronoi facets with points at infinity are not drawn.
The options argument, which must be a string or cell array of strings, contains options passed to the underlying qhull command. See the documentation for the Qhull library for details http://
If "linespec" is given it is used to set the color and line style of the plot.
If an axis graphics handle hax is supplied then the Voronoi diagram is drawn on the specified axis rather than in a new figure.
If a single output argument is requested then the Voronoi diagram will be plotted and a graphics handle h to the plot is returned.
[vx, vy] = voronoi (…) returns the Voronoi vertices instead of plotting the diagram.
x = rand (10, 1);
y = rand (size (x));
h = convhull (x, y);
[vx, vy] = voronoi (x, y);
plot (vx, vy, "-b", x, y, "o", x(h), y(h), "-g");
legend ("", "points", "hull");
See also: voronoin, delaunay, convhull.
Compute N-dimensional Voronoi facets.
The input matrix pts of size [n, dim] contains n points in a space of dimension dim.
C contains the points of the Voronoi facets. The list F contains, for each facet, the indices of the Voronoi points.
An optional second argument, which must be a string or cell array of strings, contains options passed to the underlying qhull command. See the documentation for the Qhull library for details
The default options depend on the dimension of the input:
□ 2-D and 3-D: options = {"Qbb"}
□ 4-D and higher: options = {"Qbb", "Qx"}
If options is not present or [] then the default arguments are used. Otherwise, options replaces the default argument list. To append user options to the defaults it is necessary to repeat the
default arguments in options. Use a null string to pass no arguments.
See also: voronoi, convhulln, delaunayn.
An example of the use of voronoi is
rand ("state",9);
x = rand (10,1);
y = rand (10,1);
tri = delaunay (x, y);
[vx, vy] = voronoi (x, y, tri);
triplot (tri, x, y, "b");
hold on;
plot (vx, vy, "r");
The result of which can be seen in Figure 30.3. Note that the circum-circle of one of the triangles has been added to this figure, to make the relationship between the Delaunay tessellation and the
Voronoi diagram clearer.
Additional information about the size of the facets of a Voronoi diagram, and which points of a set of points is in a polygon can be had with the polyarea and inpolygon functions respectively.
Determine area of a polygon by triangle method.
The variables x and y define the vertex pairs, and must therefore have the same shape. They can be either vectors or arrays. If they are arrays then the columns of x and y are treated separately
and an area returned for each.
If the optional dim argument is given, then polyarea works along this dimension of the arrays x and y.
An example of the use of polyarea might be
rand ("state", 2);
x = rand (10, 1);
y = rand (10, 1);
[c, f] = voronoin ([x, y]);
af = zeros (size (f));
for i = 1 : length (f)
af(i) = polyarea (c (f {i, :}, 1), c (f {i, :}, 2));
Facets of the Voronoi diagram with a vertex at infinity have infinity area. A simplified version of polyarea for rectangles is available with rectint
Compute area or volume of intersection of rectangles or N-D boxes.
Compute the area of intersection of rectangles in a and rectangles in b. N-dimensional boxes are supported in which case the volume, or hypervolume is computed according to the number of
2-dimensional rectangles are defined as [xpos ypos width height] where xpos and ypos are the position of the bottom left corner. Higher dimensions are supported where the coordinates for the
minimum value of each dimension follow the length of the box in that dimension, e.g., [xpos ypos zpos kpos … width height depth k_length …].
Each row of a and b define a rectangle, and if both define multiple rectangles, then the output, area, is a matrix where the i-th row corresponds to the i-th row of a and the j-th column
corresponds to the j-th row of b.
See also: polyarea.
For a polygon defined by vertex points (xv, yv), return true if the points (x, y) are inside (or on the boundary) of the polygon; Otherwise, return false.
The input variables x and y, must have the same dimension.
The optional output on returns true if the points are exactly on the polygon edge, and false otherwise.
See also: delaunay.
An example of the use of inpolygon might be
randn ("state", 2);
x = randn (100, 1);
y = randn (100, 1);
vx = cos (pi * [-1 : 0.1: 1]);
vy = sin (pi * [-1 : 0.1 : 1]);
in = inpolygon (x, y, vx, vy);
plot (vx, vy, x(in), y(in), "r+", x(!in), y(!in), "bo");
axis ([-2, 2, -2, 2]);
The result of which can be seen in Figure 30.4. | {"url":"https://docs.octave.org/v4.2.0/Voronoi-Diagrams.html","timestamp":"2024-11-14T22:09:27Z","content_type":"text/html","content_length":"13511","record_id":"<urn:uuid:d7d10c69-5712-4465-885a-a99f6aa56f65>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00829.warc.gz"} |
Construction of a quadrilateral when the length of the 4 sides and the diagonal are given
• Depending upon the lengths of the sides and diagonal, create a triangle of \(PQR\) based on \(SSS\) construction.
• Make an arc (point \(S\)) at a certain distance from \(P\).
• Make an arc at a certain distance from point \(R\) on the earlier arc on \(S\). Name the two points intersection as \(S\).
• \(P\) and \(R\) join \(S\). The quadrilateral \(PQRS\) will be achieved.
Construct a quadrilateral \(ABCD\) with the following measurements.
\(AB =\) \(4.5 cm\), \(BC =\) \(5.5 cm\), \(CD = 4cm\), \(AD = 6cm\), \(AC = 7cm\).
Step 1:Draw side \(BC = 5.5 cm\) and cut arcs above it from \(B\) (\(4.5 cm\)) and \(C\) (\(7 cm\)). Mark the intersection as \(A\). Join \(AB\) and \(AC\).
Step 2: Draw and arc from \(A\) equal to \(6 cm\) which is the length of \(AD\).
Step 3: Draw and arc from \(C\) equal to \(4 cm\) which is the length of \(CD\). Mark the intersection as \(D\) and join \(AD\) and \(CD\).
Thus, the \(ABCD\) is a required quadrilateral.
Calculate Area of quadrilateral:
Area of the quadrilateral \(ABCD\) \(=\) \(\frac {1}{2}\) \(\times\) \(d\) \(\times\) \(h_1 + h_2\) sq. units
\(=\) \(\frac{1}{2} × 10 (1.9 +2.3)\)
\(=\) \(5\times 4.2\)
\(=\) \(21 cm²\). | {"url":"https://www.yaclass.in/p/mathematics-state-board/class-8/geometry-3465/similar-congruent-triangles-and-construction-of-quadrilaterals-17268/re-1ee473ee-11b6-4bfe-80f6-0cc0757d68c2","timestamp":"2024-11-12T07:24:43Z","content_type":"text/html","content_length":"52531","record_id":"<urn:uuid:aff3e763-a596-4a78-9279-ac2aaca3bcec>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00392.warc.gz"} |
The use of static variables position and side within the find_index function is problematic for a couple of reasons. First, it introduces statefulness to the function, which can lead to unexpected
behavior if the function is called multiple times, as the position and side will retain their values from previous calls. This makes the function non-reentrant and not thread-safe, which can cause
issues in a multi-threaded environment. Second, the logic of the binary search is flawed because it relies on these static variables to track the position and side, which is not a standard approach
and can lead to incorrect results.
To resolve these issues, remove the static keyword from the position and side variables and refactor the function to pass these variables as parameters if necessary. Additionally, consider
implementing the binary search algorithm in a more conventional and stateless manner, which typically does not require tracking the side of the search.
long unsigned int middle_piece = floor(arr.size() / 2); int mid_calc = arr.size() % middle_piece == 0 ? (arr.size() - (middle_piece - 1)) : (arr.size() - middle_piece); cout << mid_calc << endl;
position = position == 0 ? ( side ? position + middle_piece : position - middle_piece ) : ( side ? position + mid_calc : position - mid_calc );
The calculation of middle_piece and mid_calc is incorrect and does not follow the standard binary search algorithm. The variable middle_piece is intended to represent the middle index of the array,
but the calculation uses floor(arr.size() / 2) which is unnecessary because integer division will naturally truncate the decimal part. The mid_calc calculation is also incorrect and does not serve a
clear purpose in the context of a binary search. The ternary operator’s condition arr.size() % middle_piece == 0 does not make sense in this context and could lead to division by zero if middle_piece
is zero.
To correct this, middle_piece should be calculated as arr.size() / 2, and the entire mid_calc logic should be removed. Instead, the binary search should be implemented with a clear and correct
calculation of the middle index within each recursive call, and the algorithm should not modify the position based on the side variable. The standard binary search algorithm divides the array into
halves by comparing the middle element with the target item and recursing into the appropriate half without the need for additional calculations or tracking the side of the search.
position = position == 0 ? ( side ? position + middle_piece : position - middle_piece ) : ( side ? position + (middle_piece + 1) : position - (middle_piece + 1) );
The ternary operator logic for updating the position variable is incorrect and can lead to out-of-bounds access. When side is false and position is 0, subtracting middle_piece or middle_piece + 1
will result in a negative index, which is invalid for a vector. This can cause undefined behavior when accessing arr[position].
To resolve this issue, the logic for updating position should be revised to ensure that it remains within the valid range of indices for the vector. Additionally, consider using a more standard
approach to binary search without modifying the position variable in this manner, as it complicates the logic and can lead to errors.
The code is creating a new vector other_half and copying elements from arr into it in each recursive call. This is inefficient as it involves unnecessary copying of vector elements, which can be
expensive for large vectors.
Instead of copying elements into a new vector, consider passing the index range to the recursive function find_index to work on the subarray directly. This will avoid the overhead of copying and will
be more efficient in terms of both time and space complexity.
if (arr.size() == 1) { if (arr[m] == s_item) { cout << "Array cointain " << s_item << " on position: "<< s + (m-s) << endl; } else { if (s_item > arr[m]) { b_temp(arr, s_item, m + 1, s + 2*(m-s)); }
else { b_temp(arr, s_item, e - (m+e)/2, m - 1); } } }
The condition if (arr.size() == 1) is misleading and incorrect for the purpose of a binary search algorithm. Binary search does not require the array size to be 1 to find an element. Instead, it
should check if the start index s is less than or equal to the end index e to continue the search, and use the middle element m to compare with the search item s_item. If s_item is equal to arr[m],
the item is found. Otherwise, the search should continue in the left or right half of the array depending on whether s_item is less than or greater than arr[m]. The recursive calls in lines 21 and 23
also seem incorrect as they do not correctly adjust the search boundaries based on the standard binary search algorithm. The recommended solution is to remove the if (arr.size() == 1) condition and
correctly implement the binary search logic with proper recursive calls adjusting the search boundaries.
if (s_item > arr[m]) { b_temp(arr, s_item, m + 1, s + 2*(m-s)); } else { b_temp(arr, s_item, e - (m+e)/2, m - 1);
The recursive calls to b_temp in lines 20-23 use incorrect logic for adjusting the search boundaries. In a standard binary search, if the search item is greater than the middle element, the search
should continue in the right half of the array, which means updating the start index s to m + 1. Conversely, if the search item is less than the middle element, the search should continue in the left
half, updating the end index e to m - 1. The calculations s + 2*(m-s) and e - (m+e)/2 do not correctly adjust the search boundaries and can lead to incorrect behavior or infinite recursion. The
recommended solution is to correctly update the search boundaries for the recursive calls to accurately reflect the binary search algorithm: use m + 1 as the new start index when searching the right
half and m - 1 as the new end index when searching the left half.
The condition f(m) === 0 might never be true due to the floating-point arithmetic precision issues inherent in JavaScript. This could potentially cause the function to miss the exact root if it
exists. Instead of checking for strict equality to zero, consider using a tolerance value to determine if f(m) is close enough to zero to be considered as the root. For example, you could use
Math.abs(f(m)) < some_small_value as the condition.
while (Math.abs(b - a) > tol && iterations < 100) {
The loop condition checks if the absolute difference between b and a is greater than tol and if the number of iterations is less than 100. However, relying on a fixed number of iterations (in this
case, 100) as a fallback mechanism to prevent infinite loops might not be the best approach for all cases. It’s better to allow the user to specify the maximum number of iterations as a parameter to
the function. This way, the user can adjust the precision and performance trade-offs according to their specific needs.
function bisect(f, a, b, tol) { let iterations = 0; while (Math.abs(b - a) > tol && iterations < 100) { const m = (a + b) / 2; if (f(m) === 0) { return m; } else if (f(m) * f(a) < 0) { b = m; } else
{ a = m; } iterations++; } return (a + b) / 2;
The bisection method implemented in the bisect function lacks a mechanism to handle cases where the function f does not change signs over the interval [a, b]. This is a fundamental assumption for the
bisection method to work correctly. If f(a) and f(b) have the same sign, it means that there might not be a root in the interval, or there are an even number of roots, which the current
implementation does not account for.
Recommendation: Before entering the while loop, check if f(a) and f(b) have opposite signs. If they do not, either return an error or a specific value indicating that the method cannot proceed. For
if (f(a) * f(b) >= 0) { console.error('Function does not change signs over the interval. Bisection method cannot proceed.'); return null; // or any other indication of failure }
//raty malejace let raty = []; for (let i = 0; i < ilosc_rat; i++) { raty[i] = pozost * oprocen + rata_kap; if (i == 0) raty[i] += prowizja; pozost -= rata_kap; }
The loop for calculating decreasing installments (raty) does not account for the change in interest due to the decreasing principal amount. The interest component (pozost * oprocen) is calculated
based on the remaining principal (pozost), which decreases with each installment. However, the calculation of raty[i] adds the same principal installment (rata_kap) every time, which is correct, but
the interest calculation should ideally be recalculated after each iteration to reflect the decreasing principal.
Recommendation: Move the calculation of the principal installment (rata_kap) inside the loop to ensure that the interest is calculated based on the updated remaining principal. However, since
rata_kap is constant and correct as per the logic for equal principal payments, the recommendation is to clarify the intent if the interest recalculated per iteration was the intended behavior or if
the misunderstanding stems from the variable naming and expected behavior of decreasing installments:
// If the intent was to have a fixed principal payment and recalculate interest on the remaining balance: for (let i = 0; i < ilosc_rat; i++) { raty[i] = pozost * oprocen + rata_kap; if (i == 0) raty
[i] += prowizja; pozost -= rata_kap; }
Ensure the logic aligns with the intended financial model, as the current implementation suggests a fixed principal payment model rather than a decreasing installment model where both principal and
interest components decrease.
b_temp(arr, s_item, m + 1, s + 2*(m-s)); } if (s_item < arr[m]) { b_temp(arr, s_item, e - (m+e)/2, m - 1);
The recursive calls in the binary search implementation have incorrect parameters for adjusting the search range, which can lead to infinite recursion or incorrect behavior. Specifically, the
calculation for the new start and end indices in the recursive calls does not correctly narrow down the search range according to the binary search algorithm.
To fix this, ensure that the recursive calls correctly adjust the search range. The first call should set the new start index to m + 1 and keep the end index as e when the search item is greater than
the middle element. The second call should set the new end index to m - 1 and keep the start index as s when the search item is less than the middle element. This adjustment ensures that the search
range is correctly narrowed down in each step of the recursion, adhering to the binary search algorithm.
The condition if (arr.size() == 1) is misleading and incorrect for the purpose of a binary search algorithm. This condition seems to be intended to check if the search has narrowed down to a single
element, but it incorrectly checks the size of the entire array instead of the current search range. This will cause the function to behave incorrectly for arrays with more than one element.
To fix this issue, remove the condition if (arr.size() == 1) and its enclosing braces. The binary search logic should not depend on the size of the entire array but on the indices s (start) and e
(end) to determine if the search space has been narrowed down to a single element or if it needs to continue dividing the search space. The corrected code should directly proceed with comparing the
middle element with the search item and adjusting the search range accordingly without this incorrect condition.
while (num > 0) { let lastDigit = num % 10; num = (num - lastDigit) / 10; if (lastDigit == digit){ console.log(`The digit ${digit} is in the number.`); break; }else{ console.log ("no") }
The current implementation of the loop and conditional logging within the while loop can lead to excessive and potentially misleading output. Specifically, the loop will print “no” for every digit in
the number that is not equal to the target digit, which could be confusing for users or developers trying to understand the output. Additionally, if the target digit is found, the message indicating
its presence is logged, but the loop also breaks immediately, which is efficient but could be handled more cleanly.
Recommendation: To improve the clarity and efficiency of this code, consider accumulating the result in a boolean variable and logging the output once after the loop completes. This approach reduces
unnecessary logging and makes the code’s intent clearer.
let found = false; while (num > 0) { let lastDigit = num % 10; num = (num - lastDigit) / 10; if (lastDigit == digit) { found = true; break; } } console.log(found ? `The digit ${digit} is in the
number.` : "The digit is not in the number.");
The condition m * n - i * mid >= k - 1 in the binary search logic is potentially incorrect. This condition checks if the remaining area after cutting i * mid pieces is at least k-1, which does not
directly correlate to having no more than k pieces. Recommended Solution: Revise the condition to directly check against the number of pieces rather than the remaining area. For example, ensure that
the number of pieces formed by the cuts does not exceed k.
m * n - i * mid >= k - 1
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How to Do Division Using Repeated Subtraction
Division using repeated subtraction involves subtracting the divisor from the dividend repeatedly until the dividend is less than the divisor. The number of times you subtract is the quotient, and
the final number is the remainder.
A Step-by-step Guide to Doing Division Using Repeated Subtraction
Division using repeated subtraction is a basic mathematical concept that involves subtracting the divisor from the dividend repeatedly until you can’t subtract any more without getting a negative
number. Here’s a step-by-step guide:
Step 1: Understand the Problem
If you have a problem like \(12÷4\), here 12 is your dividend (the number being divided) and 4 is your divisor (the number you are dividing by).
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Step 2: Subtract the Divisor from the Dividend
Subtract the divisor from the dividend. In this example, subtract 4 from 12. You get 8.
Step 3: Keep Track of the Count
You have subtracted once. So, keep a count of 1.
Step 4: Repeat the Subtraction
Subtract the divisor from the remaining dividend. In this case, subtract 4 from 8 (the result of your previous subtraction). You get 4.
Step 5: Update the Count
You have subtracted once again. So, update your count from 1 to 2.
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Step 6: Continue the Process
Keep repeating Steps 4 and 5 until you can’t subtract the divisor from the remaining dividend without getting a negative number. In this case, you can subtract 4 from 4 and get 0.
Step 7: Final Count is Your Answer
The final count represents the number of times you subtracted the divisor from the dividend. This is your answer. In this case, you subtracted 3 times, so \(12÷4=3\)
Step 8: Check for a Remainder
If you can’t subtract the divisor from the dividend anymore without getting a negative number and the remaining dividend is not zero, then the remaining dividend is your reminder. In this case, the
remaining dividend is 0, so there’s no remainder.
Remember, the method of division using repeated subtraction works best for smaller numbers. For larger numbers, other methods such as long division or synthetic division may be more efficient.
The Best Math Books for Elementary Students
Related to This Article
What people say about "How to Do Division Using Repeated Subtraction - Effortless Math: We Help Students Learn to LOVE Mathematics"?
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S1 Quartiles
for Q1: 0.25n. If the result is an integer, take the mean of the 0.25n'th and (0.25n+1)'th observation. i.e. if 0.25n=15 then take the mean of the 15th and 16th observation.
If the result is not an integer, take the observation corresponding with the next whole number. i.e. if 0.25n=15.2 take the 16th observation.
This can be applied to Q2 and Q3 replacing 0.25 with 0.5 and 0.75 respectively.
I was taught to use Q1 = n/4, Q2=n+1/2 and Q3=3n/4
For Q1 and Q3, if you don't get an integer, ALWAYS round up.
If you do get an integer, r, then it's 1/2 of r add r+1.
For Q2, if it's not an integer, but lie between r and r+1, then it's 1/2 of r add r+1.
If you do get an integer, take it.
Hope that isn't too confusing | {"url":"https://www.thestudentroom.co.uk/showthread.php?t=401934","timestamp":"2024-11-02T21:01:45Z","content_type":"text/html","content_length":"334821","record_id":"<urn:uuid:9f7a190d-c573-4820-a4cf-cf653aced177>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00055.warc.gz"} |
ime-to-depth conversion
A robust approach to time-to-depth conversion and interval velocity estimation from time migration in the presence of lateral velocity variations
Next: Introduction Up: Reproducible Documents
Published as Geophysical Prospecting, 63, no. 2, 315-337, (2015)
A robust approach to time-to-depth conversion and interval velocity estimation from time migration in the presence of lateral velocity variations
Siwei Li and Sergey Fomel
Bureau of Economic Geology
John A. and Katherine G. Jackson School of Geosciences
The University of Texas at Austin
University Station, Box X
Austin, TX 78713-8924
The problem of conversion from time-migration velocity to an interval velocity in depth in the presence of lateral velocity variations can be reduced to solving a system of partial differential
equations. In this paper, we formulate the problem as a nonlinear least-squares optimization for seismic interval velocity and seek its solution iteratively. The input for inversion is the Dix
velocity which also serves as an initial guess. The inversion gradually updates the interval velocity in order to account for lateral velocity variations that are neglected in the Dix inversion. The
algorithm has a moderate cost thanks to regularization that speeds up convergence while ensuring a smooth output. The proposed method should be numerically robust compared to the previous approaches,
which amount to extrapolation in depth monotonically. For a successful time-to-depth conversion, image-ray caustics should be either nonexistent or excluded from the computational domain. The
resulting velocity can be used in subsequent depth-imaging model building. Both synthetic and field data examples demonstrate the applicability of the proposed approach.
A robust approach to time-to-depth conversion and interval velocity estimation from time migration in the presence of lateral velocity variations
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perplexus.info :: Numbers : How many numbers in the end?
Start with a set of a few distinct natural numbers.
For any 2 members, add their least common multiple to the set, if and only if it was not already in the set.
Continue the task until it cannot be done.
Call the result a final list.
a. (2,4,8,16) a is a final list.
b. (2,3,4,6) will become a final list once we add number 12 to the set.
What can be the longest final list, if the initial set had 10 distinct numbers? | {"url":"http://perplexus.info/show.php?pid=11163&cid=59379","timestamp":"2024-11-07T06:52:25Z","content_type":"text/html","content_length":"12688","record_id":"<urn:uuid:8d122e40-a6e3-404d-9258-a22230cbf84a>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00224.warc.gz"} |
(Research) Multiple pilot cost adjustments for superheavy mechs
The rise of interest in ilclan era has opened the door to superheavy battlemechs, Could you clarify who is a pilot for the calculation of pilot skills for the purposes of calculating BV?
Superheavy tripod battlemechs have 3 people in the cockpit, a dedicated pilot, a dedicated gunnery officer and a technical officer (pg158 interstellar operations alternative eras).
The BV multiplier for pilots is found by averaging the skills of the "pilots' skill" (pg188 interstellar operations: alternative eras) "To calculate the Battle Value for a Tripod ’Mech with variable
skill ratings (see p. 314, TM) use the average of the pilots’ Piloting and Gunnery Skills, respectively. Round normally when determining the average score.
Who is considered a pilot in that calculation?
Thank you | {"url":"https://bg.battletech.com/forums/index.php?PHPSESSID=l11rc6746qipvqcltmtf2eg7vv&topic=85385.0;prev_next=next","timestamp":"2024-11-02T09:22:39Z","content_type":"application/xhtml+xml","content_length":"25413","record_id":"<urn:uuid:2b024c2f-a824-4e6d-ad78-4dd795ed167d>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00878.warc.gz"} |
Influence of Methane-Hydrogen Mixture Characteristics on Compressor Vibrations
High Performance Computing Center, Perm National Research Polytechnic University, Perm, 614990, Russia
* Corresponding Author: Ivan E. Cherepanov. Email:
(This article belongs to the Special Issue: Advanced Problems in Fluid Mechanics)
Fluid Dynamics & Materials Processing 2024, 20(5), 1031-1043. https://doi.org/10.32604/fdmp.2024.048494
Received 09 December 2023; Accepted 15 April 2024; Issue published 07 June 2024
A transition to clean hydrogen energy will not be possible until the issues related to its production, transportation, storage, etc., are adequately resolved. Currently, however, it is possible to
use methane-hydrogen mixtures. Natural gas can be transported using a pipeline system with the required pressure being maintained by gas compression stations. This method, however, is affected by
some problems too. Compressors emergency stops can be induced by vibrations because in some cases, mechanical methods are not able to reduce the vibration amplitude. As an example, it is known that a
gas-dynamic flow effect in labyrinth seals can lead to increased vibrations. This paper presents the numerical simulation of rotor oscillations taking into account a gas-dynamic load. The influence
of a transported mixture on the oscillatory process is investigated. Mixtures consisting of methane and hydrogen in various proportions and an air mixture are considered. The results are discussed
for various operating pressures and include the rotor motion trajectories and oscillation frequency spectra obtained numerically. It is shown that the gas mixture composition has a significant effect
on the oscillations and their occurrence. Hydrogen as a working fluid reduces the vibration amplitude. Operating a compressor with hydrogen leads to a decrease in the resonant frequency, bringing it
closer to the operating one. However, the operating pressure at which maximum oscillations are observed depends slightly on the gas mixture composition.
ρ Fluid density
V Velocity vector
t Time
P Pressure
μ Dynamic viscosity
δ Kronecker delta function
H∗ Total enthalpy
T Fluid temperature
λ Thermal conductivity
M Molar mass
R0 Universal gas constant
cp Isobaric heat
U Vector displacement
λ,μ Lame parameters
I Unit tensor
E Young’s modulus
ν Poisson’s ratio
ρs Material density
fm Mass force density
fex External forces density
Currently, issues related to the possibility of transition from fossil hydrocarbon fuels to hydrogen fuel are being widely studied. An undeniable advantage of hydrogen fuels is their high
environmental friendliness, since the combustion product is water vapor. In addition, the combustion temperature and specific heat of combustion of hydrogen fuel exceed those of fossil hydrocarbon
fuels [1]. The transition to hydrogen energy is a complex, complex task. It is necessary to resolve issues related to industrial production, storage, transportation, safety, embrittlement of
materials in contact with hydrogen, etc.
An economic analysis [2] of various transporting hydrogen fuel methods showed that the use of pipelines has the greatest potential in the long term compared to rail and road transport. The main
problems of using the existing gas transportation system are the high ability of hydrogen to penetrate metal barriers [3] and the embrittlement of gas pipeline walls. However, the existing gas
transportation system allows for the transportation of natural gas with a hydrogen content of up to 10%, with the prospect of increasing to 20% [4].
A number of works [5–7] are devoted to the transportation of a methane-hydrogen mixture through pipelines. The authors obtained the dependences of the pressure distribution along the length of the
pipeline for various hydrogen concentrations in the methane-hydrogen mixture. It is noted that in order to deliver to the consumer the energy equivalent of the currently transported natural gas, it
is necessary to increase the pressure at the entrance to the gas pipeline in proportion to the hydrogen concentration.
The gas transmission system consists not only of pipelines and distribution units connecting production sites with consumers but also gas compressor stations that maintain the pressure of the
transported gas, which decreases due to friction losses.
An emergency shutdown of a station can lead to losses. One of the reasons for emergency stops is the increased level of vibrations. Compressor rotor vibrations near bearing supports must not exceed
up to 40 microns.
Vibration causes may include residual mechanical imbalances due to inaccurate parts manufacturing, asymmetrical elements, non-uniformity of work piece material, and assembly errors. To eliminate
mechanical imbalances, balancing is used [8].
To reduce leaks between compressor stages, labyrinth seals are used, which are subject to high requirements for operational reliability [9]. At the same time, the use of non-contact gas-dynamic seals
in a compressor design can lead to unstable, self-exciting behavior of a rotor [10]. Work [11] presents a case of increased rotor vibrations due to the occurrence of non-conservative gas-dynamic
forces in labyrinth seals. When trying to bring the compressor to its nominal operating mode, low-frequency vibration occurred, which led to an emergency stop.
Thus, in a number of cases, existing mechanical balancing methods fail to reduce the magnitude of vibrations to the required values. It is necessary to clarify existing methods and search for
unaccounted factors that lead to a vibration occurrence. Such a factor may be taking into account gas dynamics when modeling rotor dynamics.
There is a linear model of gas-dynamic forces acting on a rotor, which is used in a rotor dynamics analysis [12]. The input parameters of such a model are rotor dynamic coefficients-stiffness and
damping coefficients. To determine them, theoretical, experimental and numerical studies are carried out [13–17]. A nonlinear model of gas-dynamic forces is described in [18]. Using this model, the
dynamics of a single-mass rotor model wereassessed.
Researchers perform many experiments in an attempt to identify the influence of various geometric factors, but using a linear model does not allow studying a process nature.
The work [19] presents the results of numerical modeling of a compressor rotor taking into account gas in the gaps of labyrinth seals. The study showed that with a certain combination of parameters,
an uncontrolled increase in the amplitude of rotor oscillations occurs. In [20], the influence of geometric, kinematic, and gas-dynamic parameters on the occurrence of oscillations is shown. At the
same time, the authors did not consider the issue of the influence of the nature of the gas on vibrations. At the same initial pressure and temperature, the gases will have different densities, which
affects an oscillation dynamics. The difference in gas viscosity affects vibration damping.
Despite a large number of studies on rotor vibrations caused by unsteady forces in labyrinth seals, there are practically no studies devoted to a compressor operation using hydrogen. Therefore, the
work purpose is to assess a rotor vibrations possibility in a hydrogen compressor. To achieve this purpose, it is necessary to solve the following tasks:
1. Assess the influence of a gas mixture composition on a vibrations occurrence in labyrinth seals.
2. Determine the effect of hydrogen concentration in a mixture on vibrations amplitude.
3. Compare vibration frequencies at which a maximum vibration amplitude is observed.
Replacing natural gas with a methane-hydrogen mixture in gas pipelines may lead to a change in the level of vibration of compressor rotors and require additional assessment of the performance of
existing compressor stations, so this issue requires research.
The numerical model is presented in Fig. 1. It consists of a rotor 1 with a disk 2 installed in elastic supports 3. On the outer surface of the disk there is a gas-dynamic gap 4, corresponding to a
labyrinth seal. From the outside, the gas-dynamic gap is limited by ring 5. At the initial moment of time, the rotor is not deformed, and the rotation speed is ω0. The outer ring is non-deformable.
The gap at the initial time moment is filled with gas at given initial values of pressure and temperature. The calculations explicitly took into account a rotation and Earth’s gravity. At one of the
ends of the rotor, an axial movements limitation was set to unambiguously define the model in space. A detailed model description is presented in [21].
The mathematical model is based on non-stationary equations of fluid dynamics and mechanics of deformable solids. Fluid motion equations includingmass, momentum and energy governing laws, are closed
by the ideal gas state equation and the turbulence model [22], as well as initial and boundary conditions. The mathematical model of the fluid dynamic problem includes the following equations:
Continuity equation:
∂ρ∂t+∇⋅(ρV)=0 (1)
Momentum equation:
∂(ρV)∂t+∇⋅(ρV⊗V)=−∇P+∇⋅τ (2)
The relationship for determining viscous stresses has the form:
τ=μ(∇V+V∇−23δ∇⋅V) (3)
Energy equation:
∂(ρH∗)∂t−∂P∂t+∇⋅(ρVH∗)=∇⋅(λ∇T+V⋅τ) (4)
The equations described above are supplemented with constitutive relations of state for the density and enthalpy. For an ideal gas the following equations are valid:
ρ=PMR0TdH=cpdT (5)
The compressor rotor movement is described by differential equations in displacement within the framework of the linear elasticity theory. Limiting ourselves to the linear elasticity theory to
describe the rotor movement, the smallness of possible deformations is assumed, which are determined through displacement gradients:
ε=12(∇U+U∇) (6)
Since the elastic deformation process is associated with inertial forces acting on structural elements at the moment of rotation, and external influences caused by the influence of fluid mass
transfer in the gas compressor compressor, forces called stresses arise on the surface of the solid body. The relationship between the resulting stresses and deformations is isotropic Hooke’s law [23
σ=λI1(ε)I+2με (7)
The Lamé parameters are determined by the following equations:
λ=Eν(1+ν)(1−2ν) (8)
μ=E2(1+ν) (9)
The differential motion equations of a continuous medium, which is the compressor rotor, follow from the static equilibrium equations when taking into account volumetric inertia forces:
The equations of static equilibrium when taking into account volumetric inertia forces have the form:
∇⋅σ+ρsfm+fex=ρs∂2U∂t2 (10)
In this case, the material density is a constant value, since the rotor material is assumed to be homogeneous, i.e., its physical and mechanical properties are the same at all points of the body. The
mass forces density, in the general case, is a function of spatial coordinates and is formulated from physical considerations about the processes occurring inside the body:
fm=fm(x1,x2,x3,t) (11)
However, this article does not imply the presence of internal mass sources, i.e., external forces are determined from the boundary conditions caused by external interaction.
Using the above relationships, it is possible to obtain differential motion equations of the rotor compressor in displacements:
(λ+μ)∇∇⋅U+μΔU+fex=ρs∂2U∂t2 (12)
Calculations were carried out for several types of mixtures presented, see Table 1, differing in component composition and concentration. Mixture No. 1 corresponds to the case of acceptance tests at
the manufacturer, mixture No. 2-natural gas, which is transported in the existing gas transmission system, mixtures No. 3 and No. 4-methane-hydrogen mixtures with different hydrogen concentrations,
mixture No. 5-pure hydrogen.
To analyze the rotor operation stability, the point movements on the shaft axis in the disk section over time were considered (Fig. 2).
3 Model Development and Validation
Using mesh methods of computer modeling implies performing a study of the mesh convergence, which consists of searching for mesh model parameters for which changing a elements number does not affect
a result obtained. Since a problem being solved is multidisciplinary, a mesh convergence study for a complete computational model requires large computational costs. Therefore, the analysis was
performed separately for each of subdomains (mechanical and fluid).
The convergence analysis of a rotor mesh model was performed by varying the number of elements along a shaft length. This choice is due to a fact that a disk mounted on a shaft has significant
rigidity due to its geometric dimensions and does not undergo significant deformation. A geometric shaft dimensions were chosen in such a way that a rotor rigidity is low and an aeroelastic effects
manifestation is more pronounced. Therefore, a rotor experiences the greatest deformation as a shaft deflection result in the radial direction. A static shaft deflection under an gravity influence
and first natural bending frequency were chosen as criteria for mesh convergence. The results of a mesh model convergence analysis are presented in Fig. 3.
As can be seen on plots, changing elements number in a wide range along a rotor shaft length has little effect on the parameters under consideration. For all points considered, a deviation does not
exceed 1%, therefore, for further calculations, an intermediate option was chosen from the considered ones-75 elements along a rotor shaft length. This choice reduces the need for computing resources
and at a same time provides some margin in a solution accuracy obtained. The rotor mesh model was 7016 elements and 22,292 nodes.
Before constructing a mesh model for fluid calculations, a first layer thickness near a wall was estimated in order to obtain the dimensionless distance from a wall y +< 1 required by the selected
SST turbulence model. A first layer thickness was 0.8 µm. According to the simulation results performed, a requirement for a dimensionless distance near a wall was met.
To analyze a mesh model convergence of a fluid region, a number of elements along the ring and along a gap thickness were changed. An elements number in an axial direction varied in proportion to an
elements number in a circumferential direction. Since radial rotor vibrations are considered, to analyze a mesh model convergence, an inner wall of a fluid region adjacent to a rotor disk was made
with eccentricity. The wall was shifted by the static rotor deflection value of 60 μm. Thus, on the inner wall, when a gas moved in a circumferential direction, a gas-dynamic force arose due to an
uneven gap thickness. A gas-dynamic force magnitude was chosen as a convergence criterion. The mesh model convergence results are presented in Fig. 4.
An element number in the circumferential direction was chosen to be 1500. With this value, a calculation series wasperformed with different element numbers along the gap thickness (Fig. 4b). A mesh
model with 25 elements along a gap height was selected as an optimal value for further calculations. Thus, the mesh model for a fluid region was 375 thousand elements.
The resulting mesh models are shown in Fig. 5.
3.2 Simulation Model Validation
The selected two-way coupling between fluid and structure (2FSI) model was verified using experimental data and computational experiment results obtained in a shock tube with an installed plate,
which deforms when interacting with a shock wave [24]. The experimental setup diagram is shown in Fig. 6.
At the beginning of the experiment, there are high and low-pressure areas in the pipe, separated by a thin membrane. At the end of a low-pressure area there is a deformable plate rigidly fixed to the
base. When a threshold pressure value is exceeded, the membrane ruptures, and expansion wave, contact surface and incident shock begin to propagate in a pipe. When interacting with a wave, a plate
begins to bend.
During the experiment, shadow photography of a flow and plate deformation was performed. From photographs, a movement dependence of a plate’s upper edge over time was obtained. A sensor located on a
top pipe wall at a distance of 10 mm from a plate recorded pressure oscillations.
A similar numerical experiment was performed, which made it possible to perform a qualitative and quantitative comparison. Results obtained are presented in the form shadow images (Fig. 7) and plots
of an upper plate edge movement, and pressure oscillations at a control point (Fig. 8).
In general, one can note a good agreement a plate movement calculated using an accepted model and experimental data obtained in [24].
When analyzing rotor vibrations, a key reliability criterion is a low vibration displacement magnitude. Results obtained showed a correct prediction of a structure deformation under a gas flow
influence, which allows a described model to be used for vibration simulation.
The resulting trajectories of motion of a point on the rotation axis in the disk section, obtained in a computational experiment, are presented in Fig. 3.
It is clear from the figures that divergent vibrations are observed for air at 10 and 14 MPa. Moreover, at 14 MPa the vibration amplitudes are higher. At 5 and 20 MPa, steady motion is observed. At
14 MPa in 0.12 s the vibration amplitude exceeds 100 µm.
When considering methane, divergent rotor vibrations occur at a pressure of 14 MPa. The vibration increase rate is lower than for air. At time 0.12 s, the rotor oscillation amplitude exceeds 40 μm.
An increase in rotor oscillation amplitude occurs as a result of a fluid wedge interaction that occurs between a rotor and seal sleeve. Under identified conditions, a fluid wedge rotates along a
circumference with a constant frequency and phase difference.
At the considered 5 and 20 MPa, a vibration is stable. At 10 MPa, oscillations with a constant amplitude of about 20 μm are observed. With a slight (5%–10%) admixture of hydrogen, a multiple decrease
in the amplitude of oscillations is observed throughout the entire pressure range considered. Moreover, at a pressure of 14 MPa, the vibrations have a greater amplitude than at 5, 10, and 20 MPa.
When pumping pure hydrogen, the transient oscillation process changes qualitatively. The amplitude of the oscillations of the transient process is greater than that of mixtures No. 3 and 4. In this
case, the nature of the oscillations is damped. The largest amplitude of oscillations of the transient process is observed at 5 MPa and decreases with increasing pressure.
According to Fig. 9, it can be assumed that when moving to hydrogen, the vibration frequency decreases several fold.
For further analysis, the results are summarized in the plots form of the rotor oscillation amplitudes over the initial pressure (Fig. 10).
Fig. 4 shows that the largest vibrations amplitude is observed for an air, and the smallest for a mixture of 10% hydrogen and methane. The use of pure methane leads to greater fluctuations than with
pure hydrogen and less than with air. A methane mixture with hydrogen leads to smaller fluctuations than with hydrogen.
For mixture No. 1 (air) in Fig. 11, the maximum oscillations at the initial pressure are observed at 726 Hz. For methane as a working fluid, the resonant frequency is 760 Hz. Mixture No. 5 (hydrogen)
demonstrates the absence of oscillations in the high-frequency region and a shift to the low-frequency region, close to the natural frequency of the rotor.
A three-dimensional numerical simulation of the compressor rotor vibrations was performed taking into account a gas-dynamic influence of a seal. The assumptions made in the model are described. An
influence study of a mesh model on results obtained was carried out, and a convergence condition was achieved. The proposed numerical model was verified with experimental data, and agreement between
results was obtained. A series of computational studies were carried out for various mixtures and operating pressures. For each modeled point, rotor movement trajectories were obtained, which were
summarized in a plot form. From the results obtained, the following conclusions can be drawn:
1. The gas mixture composition has a significant influence on the occurrence of vibrations in labyrinth seals.
2. Hydrogen as a working fluid reduces the oscillatory amplitude.
3. Hydrogen as a working fluid increases a shaft deflection and reduces an oscillatory processes amplitude.
4. Resonance pressure weakly depends on the gas mixture composition.
Acknowledgement: None.
Funding Statement: The research was carried out with financial support from the Russian Ministry of Education and Science, Project FSNM-2023-0004 “Hydrogen Energy. Materials and Technology for
Storage, Transportation and Use of Hydrogen and Hydrogen-Containing Mixtures”.
Author Contributions: The authors confirm their contribution to the paper as follows: study conception and design: Vladimir Ya. Modorskii; data collection: Ivan E. Cherepanov; analysis and
interpretation of results: Vladimir Ya. Modorskii, Ivan E. Cherepanov; draft manuscript preparation: Ivan E. Cherepanov. All authors reviewed the results and approved the final version of the
Availability of Data and Materials: Data on which this paper is based is available from the authors upon reasonable request.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
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Cite This Article
APA Style
Modorskii, V.Y., Cherepanov, I.E. (2024). Influence of methane-hydrogen mixture characteristics on compressor vibrations. Fluid Dynamics & Materials Processing, 20(5), 1031-1043. https://doi.org/
Vancouver Style
Modorskii VY, Cherepanov IE. Influence of methane-hydrogen mixture characteristics on compressor vibrations. Fluid Dyn Mater Proc. 2024;20(5):1031-1043 https://doi.org/10.32604/fdmp.2024.048494
IEEE Style
V.Y. Modorskii and I.E. Cherepanov, “Influence of Methane-Hydrogen Mixture Characteristics on Compressor Vibrations,” Fluid Dyn. Mater. Proc., vol. 20, no. 5, pp. 1031-1043, 2024. https://doi.org/
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Multiplying Fractions By Whole Numbers Grade 5 Fractions Worksheet 2024 - NumbersWorksheets.com
Multiplying Fractions By Whole Numbers Grade 5 Fractions Worksheet
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On the minimum cost range assignment problem
We study the problem of assigning transmission ranges to radio stations placed in a d-dimensional (d-D) Euclidean space in order to achieve a strongly connected communication network with minimum
total cost, where the cost of transmitting in range r is proportional to rα. While this problem can be solved optimally in 1D, in higher dimensions it is known to be NP-hard for any α ≥ 1. For the 1D
version of the problem and α ≥ 1, we propose a new approach that achieves an exact O(n^2)-time algorithm. This improves the running time of the best known algorithm by a factor of n. Moreover, we
show that this new technique can be utilized for achieving a polynomialtime algorithm for finding the minimum cost range assignment in 1D whose induced communication graph is a t-spanner, for any t ≥
1. In higher dimensions, finding the optimal range assignment is NPhard; however, it can be approximated within a constant factor. The best known approximation ratio is for the case α = 1, where the
approximation ratio is 1.5. We show a new approximation algorithm that breaks the 1.5 ratio.
Original language English
Title of host publication Algorithms and Computation - 26th International Symposium, ISAAC 2015, Proceedings
Editors Khaled Elbassioni, Kazuhisa Makino
Publisher Springer Verlag
Pages 95-105
Number of pages 11
ISBN (Print) 9783662489703
State Published - 1 Jan 2015
Event 26th International Symposium on Algorithms and Computation, ISAAC 2015 - Nagoya, Japan
Duration: 9 Dec 2015 → 11 Dec 2015
Publication series
Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume 9472
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
Conference 26th International Symposium on Algorithms and Computation, ISAAC 2015
Country/Territory Japan
City Nagoya
Period 9/12/15 → 11/12/15
ASJC Scopus subject areas
• Theoretical Computer Science
• General Computer Science
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Frobenius manifolds, quantum cohomology, and moduli spaces
Frobenius manifolds, quantum cohomology, and moduli spaces / Yuri I. Manin
Type de document : MonographieCollection : Colloquium publications, 47Langue : anglais.Pays: Etats Unis.Éditeur : Providence : American Mathematical Society, 1999Description : 1 vol. (XIII-303 p.) :
ill. ; 26 cmISBN: 9780821819173.ISSN: 0065-9258.Bibliographie : Bibliogr. p. 285-297. Index.Sujet MSC : 14N35, Projective and enumerative algebraic geometry, Gromov-Witten invariants, quantum
cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants
14J32, Algebraic geometry - Surfaces and higher-dimensional varieties, Calabi-Yau manifolds
53D45, Differential geometry - Symplectic geometry, contact geometry, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
81T40, Quantum theory, Two-dimensional field theories, conformal field theories, etc. in quantum mechanicsEn-ligne : Zentralblatt | MathSciNet | AMS
Item type Current library Call number Status Date due Barcode
CMI Salle 1 14 MAN (Browse shelf(Opens below)) Available 00873-01
Yuri Manin received the Bolyai Prize of the Hungarian Academy of Sciences for this title. (source : AMS)
Bibliogr. p. 285-297. Index
Chapter 0: “Introduction: What is quantum cohomology?”: This introduction gives a rather detailed overview of the two central themes of the book: quantum cohomology and Frobenius manifolds. The
author explains the (preliminary) definitions underlying these concepts, gives some illustrations by important examples, and derives from this motivating discussion the strategic plan of the book.
Typically for the author’s well-known style of writing, already the introduction is pointed, concise, directing and highly enlightening.
Chapter I: “Introduction to Frobenius manifolds”: This chapter is based on B. Dubrovin’s innovating work on Frobenius (super-)manifolds [in: Integrable systems and quantum groups, Montecatini 1993,
Lect. Notes Math. 1620, 120-348 (1996; Zbl 0841.58065)] and provides, together with some important enhancements by the author himself, a systematic exposition of the fundaments of this theory. This
includes the definition of Frobenius manifolds, Dubrovin’s structure connection, Euler fields, the extended structure connection, semi-simple Frobenius manifolds, examples of Frobenius manifolds and
a first encounter with quantum cohomology in this context, weak Frobenius manifolds, and relations to Poisson structures.
Chapter II: “Frobenius manifolds and isomonodromic deformations”: In this chapter, the author continues the study of Frobenius (super-)manifolds from the deformation-theoretic viewpoint. The main
topics treated here are the so-called second structure connection on Frobenius manifolds, the formal Laplace transform, isomonodromic deformations of connections, versal deformations, Schlesinger
equations and their Hamiltonian structure, semisimple Frobenius manifolds as special solutions to the Schlesinger equations, and applications to the quantum cohomology ring of a projective space. The
concluding section of this chapter discusses, in greater detail, the three-dimensional semisimple case of Frobenius manifolds and its connection with a special family of nonlinear ordinary
differential equations, the so-called family “Painlevé VI”. Again, much of the material presented here originates from Dubrovin’s fundamental work cited above.
Chapter III: “Frobenius manifolds and moduli spaces of curves”: This chapter turns to the more algebraic aspects of Frobenius manifolds in their supergeometric setting. ... Chapter IV: “Operads,
graphs, and perturbation series”: This chapter serves as a concise introduction to the more technical framework of operads and generating functions for moduli spaces of curves and quantum cohomology
rings. ... Chapter V: “Stable maps, stacks, and Chow groups”: Although quantum cohomology, the main subject of the book, has been invoked in several places in the first four chapters, whether in the
form of illustrating examples in chapter II or as an axiomatic framework in chapter III, its proof of existence as well as its systematic treatment had to be postponed until the final chapter VI.
This is due to the fact that either construction of a mathematical quantum cohomology structure on the cohomology ring of a projective manifold requires a tremendous amount of advanced
algebro-geometric techniques. Chapter V provides an overview of these methods and results needed, in addition, for the author’s construction of quantum cohomology: prestable curves and prestable
maps, flat families of these objects, groupoids and moduli groupoids, algebraic stacks à la Artin and Deligne-Mumford, homological Chow groups of schemes, homological Chow groups of stacks,
operational Chow groups of schemes and stacks, and the related intersection and deformation theory of schemes and stacks.
Whereas chapters I–IV are reasonably self-contained and offer complete proofs of the main results, this chapter V is comparatively sketchy and survey-like. As the author points out in the preface of
the book, this chapter and the following chapter VI are meant as an introduction to the wealth of original papers on the subjects discussed here and cannot replace the study of those.
Chapter VI: “Algebraic geometric introduction to the gravitational quantum cohomology”: This concluding chapter focuses on the algebro-geometric construction of explicit Gromov-Witten-type
invariants. ... The exposition of the material is rather concise and condensed, nevertheless coherent, comprehensible and educating. The reader is required to have quite a bit of expertise in
algebraic geometry, complex differential geometry, category theory, non-commutative algebra, Hamiltonian systems, and modern quantum physics. On the other hand, the wealth of both mathematical
information and inspiration provided by the text is absolutely immense, and in this vein, the book is an excellent source for experts and beginning researchers in the field. (Zentralblatt)
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Convert the fraction (1)/(2) into the decimal. -Turito
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Convert the fraction
A. 0.2
B. 0.1
C. 0.5
D. 0.7
To convert any fraction into decimal number we have to divide numerator from denominator. When numerator is less than denominator we have to first write a decimal in quotient which gives a 0 in the
right hand side of the dividend after which we can divide the numbers as usual. During division we will write the quotient in the right hand side of the decimal.
The correct answer is: 0.5
The decimal form of the fraction
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A Molecule of Light and Matter
The atoms are polarized by the beam of light and start to attract each other. @ TU Wien
Using light, atoms can be made to attract each other. A team from Vienna and Innsbruck was able to measure this binding state of light and matter for the first time.
A very special bonding state between atoms has been created in the laboratory for the first time: With a laser beam, atoms can be polarised so that they are positively charged on one side and
negatively charged on the other. This makes them attract each other creating a very special bonding state – much weaker than the bond between two atoms in an ordinary molecule, but still measurable.
The attraction comes from the polarised atoms themselves, but it is the laser beam that gives them the ability to do so – in a sense, it is a "molecule" of light and matter.
Theoretically, this effect has been predicted for a long time, but now scientists at the Vienna Center for Quantum Science and Technology (VCQ) at TU Wien, in cooperation with the University of
Innsbruck, have succeeded in measuring this exotic atomic bond for the first time. This interaction is useful for manipulating extremely cold atoms, and the effect could also play a role in the
formation of molecules in space. The results have now been published in the scientific journal "Physical Review X".
Positive and negative charge
In an electrically neutral atom, a positively charged atomic nucleus is surrounded by negatively charged electrons, which surround the atomic nucleus much like a cloud. "If you now switch on an
external electric field, this charge distribution shifts a little," explains Prof. Philipp Haslinger, whose research at the Atominstitut at TU Wien is supported by the FWF START programme. "The
positive charge is shifted slightly in one direction, the negative charge slightly in the other direction, the atom suddenly has a positive and a negative side, it is polarised."
Light is just an electromagnetic field that changes very rapidly, so it is also possible to create this polarisation effect with laser light. When several atoms are next to each other, the laser
light polarises them all in exactly the same way – positive on the left and negative on the right, or vice versa. In both cases, two neighbouring atoms turn different charges towards each other,
leading to an attractive force.
Experiments with the atom trap
"This is a very weak attractive force, so you have to conduct the experiment very carefully to be able to measure it," says Mira Maiwöger from TU Wien, the first author of the publication. "If atoms
have a lot of energy and are moving quickly, the attractive force is gone immediately. This is why a cloud of ultracold atoms was used."
The atoms are first captured and cooled in a magnetic trap on an atom chip, a technique, which was developed at the Atominstitut in the group of Prof. Jörg Schmiedmayer. Then the trap is switched off
and releases the atoms in free fall. The atom cloud is 'ultracold' at less than a millionth of a Kelvin, but it has enough energy to expand during the fall. However, if the atoms are polarized with a
laser beam during this phase and thus an attractive force is created between them, this expansion of the atomic cloud is slowed down - and this is how the attractive force measured.
The atom chip on which the experiment was performed @ TU Wien
Quantum laboratory and space
"Polarising individual atoms with laser beams is basically nothing new," says Matthias Sonnleitner, who laid the theoretical foundation for the experiment. "The crucial thing about our experiment,
however, is that we have succeeded for the first time in polarising several atoms together in a controlled way, creating a measurable attractive force between them."
This attractive force is a complementary tool for controlling cold atoms. But it could also be important in astrophysics: "In the vastness of space, small forces can play a significant role," says
Philipp Haslinger. "Here, we were able to show for the first time that electromagnetic radiation can generate a force between atoms, which may help to shed new light on astrophysical scenarios that
have not yet been explained."
Observation of Light-Induced Dipole-Dipole Forces in Ultracold Atomic Gases
Mira Maiwöger, Matthias Sonnleitner, Tiantian Zhang, Igor Mazets, Marion Mallweger, Dennis Rätzel, Filippo Borselli, Sebastian Erne, Jörg Schmiedmayer, Philipp Haslinger
Institute of Atomic and Subatomic Physics | {"url":"https://www.nanotechnologyworld.org/post/a-molecule-of-light-and-matter","timestamp":"2024-11-14T05:10:24Z","content_type":"text/html","content_length":"1050510","record_id":"<urn:uuid:a4d806eb-0010-4174-8e30-ad3fdf8bc9bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00218.warc.gz"} |
About us
Private Tutor in Maths & Physics
Location: Glenwood Tutoring Centre
Tutor in Maths Physics operates in the in West Sydney areas of Glenwood, Bella Vista, Stanhope, Quakers Hills, Seven HIlls, Rouse Hill, Rooty Hill and Castle Hill
Tutor in Maths Physics is a Centre run under the supervision of Dr Driss Berrada.
Dr Driss Berrada has 32 years experience in teaching and tutoring tertiary and secondary students at all levels in Mathematics and Physics.
Tutor in Maths Physics tuition is run by Dr Driss Berrada a patient expert teacher and lecturer, holding a PhD degree in Mathematics applied to Physics and a Master of Teaching Secondary( double
Tutor in Maths Physics Offers each student the benefits of my vast experience as a HSC tutor in Maths, a classroom teacher and a University lecturer.
Tutor in Maths Physics also assists students from University and High School with specific programs to enhance their skill level in Mathematics, Physics and School Certificate & HSC preparation.
I also help UNIVERSITY students with the understanding of their course and support them with assesments. Covering the maths and physics side of courses.
Dr Driss Berrada's Experience
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Former Professor at University Hassan Second Casablanca (Morocco)
In total 32 years experience in teaching and tutoring tertiary and secondary students at all levels in Mathematics and Physics
Under his supervision we offering each student the benefits a vast experience as a HSC tutors in Maths, a classroom teachers and a University lecturer.
Tutor in Maths Physics assists University and High School Students with specific programs to enhance their skill and level in Mathematics, Physics for school Certificate & HSC preparation.
We also help UNIVERSITY students with the understanding of their course and support them with assessments. Covering the maths and physics side of courses.
Private tutoring in Engineering, Maths, Physics and much more. One on One or small group tutoring. | {"url":"http://www.tutorinmaths.com.au/about-us.html","timestamp":"2024-11-08T20:59:54Z","content_type":"text/html","content_length":"30771","record_id":"<urn:uuid:2018608b-6e25-4dfb-81be-4f1f71bd2ec3>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00743.warc.gz"} |
Six Numbered Cubes
This task combines spatial awareness with addition and multiplication.
The aim of this challenge is to find the total of all the visible numbers on the cubes.
We are using six cubes. Each cube has six faces of the same number.
The shape we make has to be only one cube thick. The shape on the left is built correctly, but the shape on the right would not be allowed as it is two bricks thick in places.
The total of the shape on the left is 70. Can you see why?
Start by making a staircase shape. An example is shown below:
a) What is the highest total you can make by using this staircase shape?
b) What is the lowest total you can make by using this staircase shape?
c) How did you calculate the totals for a) and b) above? Why did you choose the method(s) that you did?
d) Have a go at making a total of 75 using a staircase shape.
Using any shape of single cube thickness, what is the lowest total you can make?
How can you be sure this is the lowest total whatever the shape?
Can the lowest total be found in more than one way? Justify your answer.
Using any shape of single cube thickness, what is the highest total you can make?
How can you be sure this is the highest total whatever the shape?
Can the highest total be found in more than one way? Justify your answer.
Prove the following by logical reasoning, rather than by calculating the answers:
If the cubes are arranged in a single vertical tower (like this)
then no matter what order the cubes are in, the total cannot be 80.
This problem featured in a round of the Young Mathematicians' Award 2014.
Student Solutions
We had two lengthy files of explanation which are well worth looking at from Tom and Andrew who attend the British School of Paris. See them here: Andrew.doc
Mali, who goes to St. Philip's School in Cambridge England, said:
The highest total is 83. The largest numbers 4, 5 and 6 must go on the very top to make the total as large as possible.
The lowest total is 64. Now, the largest numbers must go at the very bottom to have fewer faces visible.
To get 75 the ladder should be made of 4 on the top, 1 underneath, 2 at the bottom. Next column, the 3 at the top, the 5 at the bottom. Finally 6 on its own.
To find this solution, I cut pieces of paper with the number of visible sides on (5, 3, 3, 4, 4, 2), and pieces of paper with the numbers 1, 2, 3, 4, 5, and 6. I created pairs, multiplied them and
add the total. I kept changing the pairs around until I found 75.
We had a lot of solutions sent in from Our Lady of Lourdes School: Stanley, Sean and Hedley; Ailbe; Chelsea; Gabriel, Fintan and Charlie; Catsaneman; Mia and Zoe; Mathias and Ethan, Comjen; Nansai;
Ohene and Noeleen, Eene and Joe. They made paper cubes with the numbers on and experimented with different parts of the challenge - well done and thank you for your submissions.Nathan from Cornelius
Vermuyden in Essex, England sent in the following:
6 cubes stand vertically in a tower. Each cube has a number written all over it. For example one cube will have the number 1 on each of its six faces, and another will have 2 on every face and so on.
So we have six cubes numbered 1 - 6.
These cubes are then stood in a vertical tower in any random order, like so....
Each letter can represent any cube as long as it is there only once. With a tower like this the cubes all have only four of their six faces showing except the top cube, which has five of its six
faces showing. This means that we have every cube with four faces and an extra one at the very top of the tower.
So we can see that any six consecutive positive integers (not including 0) when multiplied by 4 (for the number of sides seen) and added together, equal above 80, the lowest being 84 with our current
numbers 1-6. If we say A = 1, then adding the final face at the top of the tower will only increase the number made by 1 giving us our lowest possible total of 85. Therefore a total of 80 is
Algebraically this can be shown by representing the first of the consecutive numbers with n. Let's put this one at the top of the tower and have the other numbers go consecutively down the vertical
We then need to allow for the faces on each of our cubes that are shown so we can write our expressions as”¦
We then expand the brackets and simplify this to”¦
Now we can collect like terms to give us 25n+60
Using 1 as our lowest positive integer, substitute n = 1 into the expression 25n + 60 to give 25 x 1 + 60 = 85
Once again, this shows that the lowest possible total is 85.
Sabine, Amelia, Sophie, Liam, Tom, and James, from Hardwick Middle School, sent in their ideas. Here is just a sample from the first three:
For question 1 we found that 78 was our highest total. We made this total by putting 6 at the top, 3 below it and 2 below 3. Next to 3 we put 5 and below that we put 1. Next to 1 we put 4.
For question 2 we got 64. We got this by putting 1 at the top, 5 below 1 and 4 below 5. Next to 5 we put 3 with six below. Next to 6 we put 2.
On question 1 we had to make six be shown the most, then five, then 4. we had to make 1 shown the least, then 2, then three.
For question 2 we had to make 1 show the most and 6 the least.
We had to do this otherwise the answer wouldn't be as low or as high as it could be.
For question 3 we had to get exactly 75 put of a staircase of cubes. It took us a while but we finally got the answer. At the top we put 2, below that was 6 and below that was 3. Next to 6 was five
with 1 below it. Next to 1 was 4. That made 75.
This next solution was sent via the Wild.maths site which also has this challenge. It was from Ana G. in year 6 at Oaks Primary Academy and can be viewed as the pdf here.pdf . Thank you all for your
work, there was too much to show everyone's workings. I gather that it was generally enjoyed.
Teachers' Resources
Why do this problem?
This problem gives pupils the opportunity to use knowledge and skills associated with spatial awareness, addition and multiplication, and to explain their thinking. It also involves keeping to rules
that must be followed. The further they progress through the activity, the greater the opportunities for learners to use a whole variety of problem-solving skills. The activity also opens out the
possibility of pupils asking “I wonder what would happen if . . .?”
Pupils' curiosity may be easily aroused while trying to find solutions to the challenges.
Possible approach
It would be good to demonstrate the kind of arrangements that are allowed as well as making those that break the rules for the pupils to decide on what is okay.
You may decide that you want the pupils to work in groups of three or four. One set of numbered cubes will be needed for each group. Having set them the first challenge, it may be sufficient to stop
there. Challenges 2, 3 and 4 can be introduced straight away or left to another occasion.
It is worth noting that in the "steps" arrangement as shown on the problem page, the 5 and 6 both have four faces showing; the 1 and 4 both have three faces showing. So, the cubes that have the same
number of faces showing can be swapped allowing for more arrangements to be possible.
Key questions
How are you working out the totals?
How have you got to this arrangement?
Tell me about your shape.
How sure are you that ...?
Possible support
Some pupils may require help with getting the cubes to stack. Those who are unable to record their arrangements could have them photographed. | {"url":"https://nrich.maths.org/problems/six-numbered-cubes","timestamp":"2024-11-02T14:03:32Z","content_type":"text/html","content_length":"52813","record_id":"<urn:uuid:552a26a7-11a3-444e-98dd-969b8e16227b>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00073.warc.gz"} |
Ice Cream
In our recent Grade 9 Unit of Probability the students had to investigate different ways to list all the outcomes and then make connections and predictions. This looks an easy task but the difficult
part for applying the communication skills to the problem solving.
I always notice that boys in-particular will write the first couple of outcomes in a jumbled order and then end up missing or repeating.
Thank you John Tranter for making Transum.org with these wonderful interactive tools to help with listing outcomes of events and logical ordering.
I think the favourite was the Ice Cream task!
All my students loved these resources and it was wonderful to see them ordering them in different ways but still showing a clear method and logic with their thinking. #notonerightanswer
[Transum: Good to hear, Yes, scroll down the page for more challenges]
[Transum: Great idea. Lime has now been changed to Mint. Thanks very much for the suggestion.]
Many assumptions made here which makes for a very good discussion with your class where they can find a variety of answers! At the end of the day as long as students can justify and explain their
answers, this is all that matters.
Since they are used in combination, then it would be 6 divided by 2.
So far we have (6-1) * 6/2 = 5 * 3 = 15
Lastly, we need to include the possibility for each flavor being its own combination or 6.
Therefore, (6-1) * 6/2 + 6
5 * 3 + 6
15 + 6
Excellent starter.
Perhaps there could be a link at the top of each page that takes you straight to the answer, rather than having to scroll down the page manually, past all the comments.
This is a mind bending activity.
In fact, we thought this was so nICE we sCREAMed with delight!!
Lots of love to you all at Transum xxxx
Click the buttons on the ice cream machine.
Sign in to your Transum subscription account to see the answers
What if there were 7 different flavours?
What if there were n different flavours?
Your access to the majority of the Transum resources continues to be free but you can help support the continued growth of the website by doing your Amazon shopping using the links on this page.
Below is an Amazon link. As an Amazon Associate I earn a small amount from qualifying purchases which helps pay for the upkeep of this website.
Educational Technology on Amazon
Teacher, do your students have access to computers such as tablets, iPads or Laptops? This page was really designed for projection on a whiteboard but if you really want the students to have access
to it here is a concise URL for a version of this page without the comments:
However it would be better to assign one of the student interactive activities below.
Here is the URL which will take them to a related student activity.
Curriculum Reference | {"url":"https://www.transum.org/Software/SW/Starter_of_the_day/starter_november12.ASP","timestamp":"2024-11-09T18:58:11Z","content_type":"text/html","content_length":"37598","record_id":"<urn:uuid:64f4503e-faaa-4a46-b6a0-a76b26b9ad77>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00325.warc.gz"} |
Axiom Open Source Program V2
Announcing the second iteration of the Axiom Open Source Program.
Axiom's mainnet alpha release is built upon open-source ZK circuits released at github.com/axiom-crypto under a permissive MIT license. To develop the underlying technology in an open and transparent
way, over the past few months we ran the first Axiom Open-Source Program. Starting from minimal background, 12 participants worldwide developed new ZK circuits for primitives like ed25519 signatures,
BLS signature verification, KZG batch multi-open, and fixed point arithmetic.
Today we are announcing the second iteration of the Axiom Open Source Program, focused on ZK circuit primitives and applications.
About the Program
Axiom is looking for individuals or teams to research, design, and implement open-source ZK circuits for cutting edge cryptographic or computational primitives and ZK-enabled applications using these
primitives. Because of the difference in computational model between zero-knowledge proofs and traditional computers, we expect most primitives to present interesting design and optimization
challenges. The relative youth of the space also means that most implementations will be first of their kind in ZK. Example projects include:
• ZK Primitives
□ Data structures: Verify common data structures and serialization algorithms, including maintenance of heaps and tries, parsing JSON, SSZ, or regex, and array manipulation.
□ Virtual machines: Verify execution of different types of virtual machines in ZK.
□ Cryptographic operations: Verify execution of cryptographic primitives such as hash functions (Blake2b), polynomial commitment schemes (IPA), and signature schemes (RSA, Schnorr).
□ Numerical computation and machine learning: Verify operations including floating point arithmetic, Newton's method, linear algebra (eigenvalues/eigenvectors), and machine learning algorithms
(PCA, page rank, neural networks).
• ZK Applications
□ Formal verification of ZK circuits: Using automation and SMT solvers to formally verify ZK circuits used in practice.
□ ZK-powered Generative NFTs: Mint generative art NFTs which are accompanied with a ZK proof that they were actually generated from a claimed algorithm.
□ Autonomous airdrops: A system which allows users to claim on-chain rewards based on provable on-chain activity.
□ Gas price oracle: A trustless oracle for the average gas price over a trailing time interval.
□ On-chain EigenTrust: Computing EigenTrust for an on-chain social graph.
□ ZK eth_call: Combine Axiom storage reads with one of the Type 2 zkEVMs to prove correct execution of any view function on a state of the Ethereum blockchain.
Here are some examples of projects from the previous cohort of the program:
No prior experience in zero-knowledge proofs is necessary, but the pace of the program will be quite fast. A strong math background, especially familiarity of modular arithmetic, and low level
programming languages like Rust is very helpful. We will offer direct mentorship, training, and guidance on:
• The theory behind modern zero-knowledge proofs
• Designing and implementing ZK circuits in the halo2 proving system
• Using and building on top of ZK circuit primitives in our halo2-base, halo2-ecc and axiom-eth libraries.
• Optimization and verification techniques for ZK circuits
• Integration of ZK circuits with on-chain verifiers on Ethereum
• Running ZK circuits in production
All work produced through this program will be open-source, and a primary goal of this program is to onboard more contributors to open-source projects in ZK. The program is free.
Expectations and Goals
We expect program participants to commit at least 10-15 hours per week for 8 weeks. The program will be conducted fully remotely over Telegram and Zoom. By the end of the program, we expect
participants to have:
• Produced a ZK primitive or application in a fully documented open-source repo or PR to an existing repo.
• Produced documentation or a blog about the design and implementation of their circuits.
• (Optional) Produced a demo application to illustrate the usage of their circuits.
If you are excited about exploring ZK circuits and want to either work in the ZK fulltime or are looking to develop the skills to contribute to ZK open source, apply here by July 28. Applications
will be considered on a rolling basis. The program will start the week of August 7.
About Axiom
Axiom is a ZK coprocessor scaling data-rich applications on Ethereum. Axiom provides smart contracts trustless access to historic on-chain data and arbitrary expressive compute. As part of our
mission to unleash the power of zero-knowledge proofs for crypto applications, we are building some of the most performant zero-knowledge proof libraries. More information about Axiom is available at
axiom.xyz, and our ZK circuit code is open-source at github.com/axiom-crypto. | {"url":"https://blog.axiom.xyz/axiom-open-source-program/","timestamp":"2024-11-12T22:31:02Z","content_type":"text/html","content_length":"21794","record_id":"<urn:uuid:54631c45-ed01-42bd-a6ae-2a90d36a510c>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00179.warc.gz"} |
How to find the area of sector & segment of a circle?
In mathematics, we learn how to find the area of a circle. In this blog, we will discuss what is a sector, segment, chord & how to find the area of a sector & segment of a circle.
Sector, Segment & Chord of a Circle?
It is a fixed-length from the centre to the boundary of a circle. It is always constant for a circle. As shown below OA & OB.
It is twice of a radius. It always passes through the centre of the circle. Diameter = 2×Radius.
It is a straight line in a circle which touches the boundary of the circle from both sides. As shown below PQ.
The diameter is the longest chord of any circle.
It is a region of a circle which is enclosed between two radii & an arc of a circle. As shown above. It may be classified as-
• Minor sector- It covers less area between two radii & an arc of a circle.
• Major sector- It covers more area between two radii & an arc of a circle.
It is a region between an arc & a chord of a circle. As shown above. It may be classified as-
• Minor segment- It covers less area between a chord & an arc of a circle.
• Major segment- It covers more area between a chord & an arc of a circle.
How to find the area of Sector?
For finding the area of a sector of a circle we have a formula-
Area of sector = (θ/360º)×Ï€r²
• θ=Angle of minor sector or major sector.
• Ï€=22/7 or 3.14
• r=radius of circle
For better understanding, we take an example. Let's discuss-
Find the area of the sector with radius 5 cm & if the angle of the sector is 80º.
First, we draw an image for better understanding.
In the above picture, we take a circle in which a minor sector (AOB) is shown.
Because it has less area or less angle so we are considering it as the minor sector.
The angle of the sector(AOB)=80º
Radius of circle=5cm
Put all the values in the formula-
Area of sector = (θ/360º)×Ï€r²
• (θ/360º)Ï€r²
• (80º/360º)(22/7)×(5)²
• (1/4.5)(22/7)×25
• (22/31.5)×25
• (22×25)/31.5
• 17.46 cm²
We get the area of minor sector 17.46 cm².
How to find the area of Segment?
Here the same, for finding the area of the segment we have a formula-
Area of segment = r²{πθ/360º-(sin(θ/2)cos(θ/2)}
This formula works when we know the values of sin & cos.
• θ= Angle of the segment
• r= Radius of circle
• Ï€=22/7 or 3.14
For better understanding, we take an example. Let's discuss-
Find the area of the segment with radius 5 cm & if the angle of the segment is 90º.
First, we draw an image for better understanding.
In the above picture, we take a circle in which a minor segment is shown.
Because it has less area or less angle so we are considering it as the minor segment.
The angle of the segment=90º
Radius of circle=5cm
Put all the values in the formula-
• Area of segment = r²{πθ/360º-(sin(θ/2)cos(θ/2)}
• r²{πθ/360º-(sin(θ/2)cos(θ/2)}
• 5²{(22/7)(90º/360º)-sin(90/2)×cos(90/2)}
• 25{(22/7)(1/4)-sin45º×cos45º
• 25{(22/7)(1/4)-(1/√2)×(1/√2)}
• 25{(22/28)-(1/2)}
• 25{(11/14)-(1/2)}
• 25×0.285
• 7.125cm²
We get the area of minor segment 7.125cm².
In this way, we may find the area of sector & segment of a circle.
THANK YOU... | {"url":"https://www.tirlaacademy.com/2020/07/how-to-find-area-of-sector-segment-of.html","timestamp":"2024-11-03T00:56:56Z","content_type":"application/xhtml+xml","content_length":"324046","record_id":"<urn:uuid:dd8621c2-97a9-42a8-9367-1c20e63af138>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00486.warc.gz"} |
Gaussian Process-Based Learning Control of Underactuated Balance Robots With an External and Internal Convertible Modeling Structure
Issue Section:
Special Papers
Design, Dynamics (Mechanics), Errors, Pendulums, Robot dynamics, Robots, Trajectories (Physics), Boundary element methods, Control equipment, Modeling, Stability
External and internal convertible (EIC) form-based motion control is one of the effective designs of simultaneous trajectory tracking and balance for underactuated balance robots. Under certain
conditions, the EIC-based control design is shown to lead to uncontrolled robot motion. To overcome this issue, we present a Gaussian process (GP)-based data-driven learning control for underactuated
balance robots with the EIC modeling structure. Two GP-based learning controllers are presented by using the EIC property. The partial EIC (PEIC)-based control design partitions the robotic dynamics
into a fully actuated subsystem and a reduced-order underactuated subsystem. The null-space EIC (NEIC)-based control compensates for the uncontrolled motion in a subspace, while the other closed-loop
dynamics are not affected. Under the PEIC- and NEIC-based, the tracking and balance tasks are guaranteed, and convergence rate and bounded errors are achieved without causing any uncontrolled motion
by the original EIC-based control. We validate the results and demonstrate the GP-based learning control design using two inverted pendulum platforms.
Issue Section:
Special Papers
Design, Dynamics (Mechanics), Errors, Pendulums, Robot dynamics, Robots, Trajectories (Physics), Boundary element methods, Control equipment, Modeling, Stability
1 Introduction
An underactuated balance robot possesses fewer control inputs than the number of degrees-of-freedom (DOFs) [1,2]. Motion control of underactuated balance robots requires both the trajectory tracking
of the actuated subsystem and balance control of the unactuated, unstable subsystem [3–5]. Inverting the nonminimum phase unactuated nonlinear dynamics brings additional challenges in causal feedback
control design. Several modeling and control methods have been proposed for these robots and their applications [4–10]. Orbital stabilization method was used for balancing underactuated robots [1,11–
13], with applications to bipedal robot [14] and cart-inverted pendulum [1]. Energy shaping-based control was also designed for underactuated balance robots [15,16]. One feature of those methods is
that the achieved balance-enforced trajectory is not unique and cannot be prescribed explicitly [1,11–13]. In Refs. [5] and [17], a simultaneous trajectory tracking and balance control of
underactuated balance robots was proposed by using the property of the external and internal convertible (EIC) form of the robot dynamics. The EIC-based control has been demonstrated as one of the
effective approaches to achieve fast convergence with guaranteed performance.
The above-mentioned control designs require an accurate model of robot dynamics, and the control performance would deteriorate under model uncertainties or external disturbances. Machine
learning-based methods provide an efficient tool for robot modeling and control [18,19]. In particular, Gaussian process (GP) regression is an effective learning approach that generates nearly
analytical structure and bounded prediction errors [7,19–21]. Development of GP-based performance-guaranteed control for underactuated balance robots has been reported in Refs. [4], [20], and [22].
In Ref. [4], the control design was conducted in two steps. A GP-based inverse dynamics controller for unactuated subsystem to achieve balance and a model predictive control (MPC) was used to
simultaneously track the given reference trajectory and estimate the balance equilibrium manifold (BEM). The GP prediction uncertainties were incorporated into the control design to enhance the
control robustness. The work in Ref. [5] followed the sequential control design in the EIC-based framework, and the controller was adaptive to the prediction uncertainties. The training data were
selected to reduce the computational complexity.
This work takes advantage of the structured GP modeling approach in Refs. [5] and [7] and presents an integration of EIC-based control with GP models. We first present the conditions under which
uncontrolled motions exist under the original EIC-based control design for underactuated balance robots. We identify these conditions and design the stable GP-based learning control with the properly
selected nominal robot dynamic model. Two different controllers, called partial- and null-space-EIC (i.e., PEIC- and NEIC), are presented to improve the closed-loop performance. The PEIC-based
control constructs a virtual inertia matrix to reshape the dynamics coupling between the actuated and unactuated subsystems. The EIC-induced uncontrolled motion is eliminated, and the robotic system
behaves as a combined fully actuated subsystem and a reduced-order unactuated subsystem. Alternatively, the compensation effect in the NEIC-based control is applied to the uncontrolled coordinates in
the null space, while the other part of the stable system motion stays unchanged. The PEIC- and NEIC-based controls achieve guaranteed robust performance with a fast convergence of the closed-loop
tracking errors.
The control tasks considered in this work include both the trajectory tracking for the actuated subsystem and platform balance for the unstable subsystem. The interconnection between these two
subsystems lies in implicit dynamic relationship that needs to be estimated in real time. The control problem considered here distinguishes from the work in literature. Most existing approaches, such
as orbital stabilization and energy shaping, focus on stabilization only, that is, the trajectory of the actuated subsystem is not prescribed, and the main control task is to stabilize the unstable
subsystem. The main contribution of this work lies in the new GP-based learning control of underactuated balance robots using the EIC structural properties. Compared with the approaches in Refs. [5]
and [17], this work reveals underlying design properties and limitations of the original EIC-based control for underactuated balance robots. Compared with the work in Refs. [4] and [23], the proposed
method takes advantage of the attractive EIC modeling properties for control design and does not use MPC that requires high computational demands. Compared with other learning control methods such as
reinforcement learning, the proposed control integrates the robot's dynamics property (i.e., EIC structure) and the GP-based model learning. By integrating physics knowledge into model learning, we
identify the conditions for nominal model selection, and the proposed control is designed with guaranteed performance. This paper is an extension of the previous conference submission [24] with new
design, analysis, and experiments. Particularly, the NEIC-based control design and experiments were not presented in Ref. [24].
The rest of the paper is outlined as follows. We introduce the EIC-based control and present the problem statement in Sec. 2. Section 3 presents the GP-based robot dynamics. The PEIC- and NEIC-based
controls are presented in Sec. 4. The stability analysis is discussed in Sec. 5. The experimental results are presented in Sec. 6, and finally Sec. 7 summarizes the concluding remarks.
2 External and Internal Convertible-Based Robot Control and Problem Statement
2.1 Robot Dynamics and External and Internal Convertible-Based Control.
We consider an underactuated balance robot with
, and the generalized coordinates are denoted as
. The robot dynamics is expressed as
$D(q), C(q,q˙)$
, and
are the inertia matrix, Coriolis, and gravity matrix, respectively.
denotes the input matrix, and
is the control input. The coordinates are partitioned as
$q=[qaT quT]T$
, with actuated coordinate
and unactuated coordinate
. We focus on the case
, and without loss of generality, we assume that
$B=[In 0]T$
, where
is the identity matrix with dimension
. The robot dynamic model in Eq.
is rewritten as
for actuated ($Sa$) and unactuated ($Su$) subsystems, respectively. Subscripts “aa (uu)” and “ua (au)” indicate the variables related to the actuated (unactuated) coordinates and coupling
effects, respectively. For presentation convenience, we introduce $H=Cq˙+G, Ha=Caq˙+Ga$, and $Hu=Cuq˙+Gu$, and the dependence of D, C, and G on q and $q˙$ is dropped. Subsystems $Sa$ and $Su$ are
also referred to as the external and internal subsystems, respectively [4,17].
The control goal is to steer actuated coordinate $qa$ to follow a given desired trajectory $qad$ for $Sa$, while the unactuated, unstable subsystem $Su$ is balanced at unknown equilibrium $que$.
Therefore, we need to estimate $que$ in real time to achieve simultaneously trajectory tracking (for $Sa$) and platform balance (for $Su$). It is noted that not all arbitrary trajectories can be
followed given the underactuated dynamics and balance requirement. Such a property has been explicitly discussed for the autonomous bikebot example in Ref. [25]. In this work, we assume that the
given trajectory $qda$ is well planned and the control exists. In this work, we assume that the given trajectory $qad$ is well planned and the control exists. Designing and planning feasible
trajectory $qad$ is out of the scope of this work. $qad$
The original EIC-based control design is considered in two steps [
]. As shown in the top figure in Fig.
, the first step is to identify and estimate the unknown equilibrium
under an external trajectory tracking control. With the estimated
, the external control design is updated with simultaneously trajectory tracking and balancing tasks. Following such a concept, we first designs external input
to follow
by temporarily neglecting
, namely,
is the auxiliary input under which the tracking error
converges to the origin, and
are diagonal matrices with positive elements. Assuming that
is applied to
$qad, qu$
should keep balance around its equilibrium, which is however unknown. Then, BEM is introduced and used to capture the equilibrium of
, namely,
where $Γ(qu;vext)=Duuq¨u+Duavext+Hu$. $que$ is obtained by inverting $Γ0=Γ(qu;vext)|q˙u=q¨u=0=0$. Obtaining $que$ requires accurate system dynamics and needs to invert the nonminimum phase dynamics
$Su$, which is challenging for noncausal control design.
To stabilize
, the
motion is updated as
is the generalized inverse of
$Dua, vuint=que−kd2e˙u−kp2eu$
is the auxiliary control that drives error
toward zero, and
are diagonal matrices with positive elements. The final control is obtained by replacing
in Eq.
in Eq.
, that is,
where $vint$ is used as the virtual control input in $Su$, that is, under $q¨a=vint, q¨u=vuint$.
Figure 1(a) illustrates the above sequential EIC-based control design. It has been shown in Ref. [17] that the control $uint$ guarantees both $ea$ and $eu$ convergence to a neighborhood of the origin
exponentially if the high-order approximation terms of the closed-loop systems are affine with error e. Therefore, the EIC-based control achieves trajectory tracking for $Sa$ and balancing task for
$Su$ simultaneously.
2.2 Motion Property Under External and Internal Convertible-Based Control.
Control design (5) uses a mapping from low-dimensional (m) to high-dimensional (n) spaces (i.e., $n≥m$). Under control (6) with properly selected control gains, it has been shown in Ref. [17] that
there exists a finite time T>0, and for small number $ε>0, ||qu(t)−que(t)||<ε$ for t>T. Therefore, given the negligible error, we obtain $Dua(qa,qu)≈Dua(qa,que)$.
in Eq.
, if
for all
, applying singular value decomposition (SVD) to
, we obtain
$Dua=UΛVT, Dua+=VΛ+UT$
where $U=[u1,…, um]∈ℝm×m$ and $V∈ℝn×n$ are unitary orthogonal matrices. $Λ=[Λm 0]∈ℝm×n, Λ+=[Λm−1 0]T∈ℝn×m$ and $Λm=diag(σ1,…,σm)$ with singular values $σi>0, i=1,…,m$. We partition V into the block
matrix $V=[Vm Vn], Vm∈ℝn×m$ and $Vn∈ℝn×(n−m)$. Since $rank(Dau)=m$, the null space of $Dua$ is $ker(Dua)=span(Vn)$.
Column vectors of matrix
serve as a complete set of basis in
, and we introduce a coordinate transformation
. Clearly,
is a linear, time-varying, smooth map. Applying
, we have
$pa=VTqa, νext=VTvext$
where $pa=[pamT panT]T, νext=[(νmext)T (νnext)T]T$, and $pam,νmext∈ℝm, pan,νnext∈ℝn−m$. Note that $[paT quT]T$ still serves as a complete set of generalized coordinates for $S$. Using the new
coordinate $pa$, we have the following motion property under the original EIC-based control for $S$, and the proof is given in Appendix A1.
For$S$in Eq. (2), if$rank(Dau)=m$holds forqand all n control inputs appear in$Su$dynamics (through$q¨a$), under the EIC-based control (6), the BEM in Eq. (4) is associated with only$νmext$, and robot
dynamics can be written into
$SEIC: p¨ai=−uiT(Hu+Duuvuint)σi, i=1,…,m$
No control input appears for coordinates in $ker(Dua)$ as shown in Eq. (9b), and only m actuated coordinates in $span(V)$ are under active control, as shown in Eq. (9a). The results in Lemma 1 reveal
the motion property of $S$ under the original EIC-based control design. The uncontrolled motion happens to a special set of underactuated balance robots under the conditions in Lemma 1. If the
unactuated motion is only related to m (out of n) control inputs, the motion (9b) vanishes, and the EIC-based control works well. In Ref. [5], the EIC-based control worked properly for the rotary
inverted pendulum with $n=m=1$. In Refs. [4] and [25], the EIC-based control also worked well for the bikebot with n=2 (planar motion) and m=1 (roll motion) but the roll motion depends on
steering control only, that is, no velocity control, and therefore, does not satisfy the condition for Lemma 1. We will show an example of the three-link inverted pendulum platform that demonstrates
the uncontrolled motion under the original EIC-based control in Sec. 6.
With the above-discussed motion property under the EIC-based control, we consider the following problem.
Problem Statement: The goal of robot control is to design an enhanced EIC-based learning control to drive the actuated coordinate $qa$ to follow a given profile $qad$ and simultaneously the
unactuated coordinate $qu$ to be stabilized on the estimated $que$. The uncontrolled motion presented in Lemma 1 should be avoided for robot dynamics (2).
3 Gaussian Process-Based Robot Dynamics Model
We build a GP-based robot dynamics model that will be used for control design in Sec. 4.
3.1 Gaussian Process-Based Robot Dynamics Model.
To keep it self-contained, we briefly review the GP regression model. We consider a multivariate continuously smooth function
$y=f(x)+w, xi∈ℝnx$
, where
is the zero-mean Gaussian noise and
is the dimension of
. Denote the training data as
, where
$X={xi}i=1N, Y={yi}i=1N$
, and
is the number of the data point. The GP model is trained by maximizing posterior probability
over the hyperparameters
, that is,
is obtained by solving
$minΘ−log(Y;X,Θ)=minΘ−12YTK−1Y−12log det(K)$
where $K=(Kij), Kij=k(xi,xj)=σf2 exp(−(1/2)(xi−xj)TW(xi−xj))+ϑ2δij, W=diag{W1,…,Wnx}>0, δij=1$ for i=j, and $Θ={W,σf,ϑ}$ are hyperparameters.
The GP agent builds the joint distribution of new measurement
and the training data as
, and
denotes the Gaussian distribution with mean
and variance
. The mean value and variance for input
$μ(x*)=kTK−1Y, Σ(x*)=k*−kK−1kT$
We integrate the GP regression with a nominal model. For
in Eq.
, we first build a nominal model
are the nominal inertia and nonlinear matrices, respectively. Generally, the nominal dynamic model does not hold for the data sampled from the physical robot systems. The GP models are built to
capture the difference between
, namely,
We build GP models to estimate $He=[(Hae)T (Hue)T]T$, where $Hae$ and $Hue$ are for $Sa$ and $Su$, respectively. The training data $D={X,Y}$ are sampled from $S$ as $X={q, q˙, q¨}$ and $Y={He}$.
The GP predicted mean and variance are denoted as
$Hie, i=a,u$
. The GP-based robot dynamics models
are given as
$Sagp: D¯aaq¨a+D¯auq¨u+Hagp=u$
$Sugp: D¯uaq¨a+D¯uuq¨u+Hugp=0$
$Higp=H¯i+μi(x), i=a,u$
. The GP-based model prediction error is
To quantify the GP prediction error, the following property for $Δ$ is obtained directly from Theorem 6 in Ref. [26].
Given training dataset$D$, if the kernel function$k(xi,xj)$is chosen such that$Hae$for$Sa$has a finite reproducing kernel Hilbert space norm$||Hae||k<∞$, for given$0<ηa<1$
where$Pr{·}$denotes the probability of an event,$κa∈ℝn$, and its ith entry is$κai=2||Ha,ie||k2+300ςiln3((N+1)/(1−ηa1/n)), ςi=maxx,x′∈X(1/2)ln|1+ϑi−2ki(x,x′)|$. A similar conclusion holds for$Δu$with
3.2 Nominal Model Selection.
The nominal model plays an important role in the EIC control. We consider the following conditions for choosing the nominal model $Sn$ to overcome the uncontrolled motion under the learning control.
$C1$: $D¯=D¯T$ is positive definite, $||D¯||≤d, ||H¯||≤h$, where constants $0<d,h<∞$;
$C2$: $rank(D¯aa)=n, rank(D¯uu)=rank(D¯ua)=m$; and
$C3$: nonconstant kernel of $D¯ua$.
With $C1$ and $C2$, the generalized inversions of $D¯aa, D¯uu$, and $D¯au$ exist, which are used to compute the auxiliary controls. We can select $D¯=D¯T$ to ensure $D¯au=D¯uaT$. To see the
requirement of $C3$, we rewrite $qa=∑i=1npaivi$. By Eq. (9), under the updated control $vint, q¨a=∑i=1mp¨aivi+∑i=m+1np¨aivi$, where $vi$ is the ith column of V. Note that the part $∑i=m+1np¨aivi$
of $Sa$ dynamics is free of control if V is constant. Although $qu$ is stabilized on $que, qa$ converges to $qad$ only in an m-dimensional subspace and the other $(n−m)$ dimensional motion
uncontrolled. If the system is stable, the uncontrolled motion cannot be fixed in the configuration space throughout the entire control process. Therefore, a nonconstant kernel $D¯ua$ is needed.
Conditions $C1$–$C3$ provide sufficient nominal model selection criteria. The commonly used nominal model in Refs. [5] and [7] is $D¯q¨=Bu$ with $H¯=0$. The constant nominal model is used in Ref. [7
] as the system is fully actuated. It is not difficult to satisfy the nominal model conditions in practice. First, the nonlinear term is canceled by feedback linearization, and $H¯=0$ can be used.
Matrix $D¯$ captures the robots' inertia property. The mass and length of robot links are usually available or can be measured. Meanwhile, the dynamics coupling for revolute joints shows up in the
inertia matrix as trigonometric functions of the relative joint angles. Therefore, the diagonal elements can be filled with mass or inertia estimates, and the off-diagonal entries can be constructed
with trigonometric functions multiplying inertia constants.
4 Gaussian Process-Enhanced External and Internal Convertible-Based Control
In this section, we propose two enhanced controllers using the GP model $Sgp$, i.e., PEIC- and NEIC-based control. The PEIC-based control aims to eliminate uncontrolled motion under the original
EIC-based control by reassigning the dynamics coupling, while the NEIC-based control directly manages the uncontrolled motion in a transformed space; see Figs. 1(b) and 1(c).
4.1 Robust Auxiliary Control.
, we incorporate the variance from
into tracking control as
where $k̂p1=kp1+kn1Σa$ and $k̂d1=kd1+kn2Σa$ are control gains with parameters $kn1,kn2≥0$. The variance of GP prediction $Σa$ captures the uncertainty in robot dynamics and is updated online with
sensor measurements.
Given the GP-based dynamics, the BEM is estimated by solving the following optimization problem rather by inverting the system dynamics:
The balance control is then designed as
where $êu=qu−q̂ue$ is the unactuated subsystem tracking error relative to the estimated BEM. Similar to $k̂p2,k̂d2, k̂p2=kp2+kn3Σu$ and $k̂d2=kd2+kn4Σu$ depend on $Σu$ with the parameters by $kn3,kn4≥0$.
denote the BEM estimation error, and the actual BEM is
. The control design based on actual BEM should be
, and therefore we have
. There are two sources causing the BEM estimation error. First, the learned dynamics
deviates from the actual one due to the prediction error
. Therefore, the exact BEM solution using
deviates from that obtained in Eq.
. Second, there exist differences between the BEM solved from
and that obtained from Eq.
due to the optimization algorithm. Given the bounded GP prediction error and limited optimization error, it is reasonable to assume that
is bounded. Because of the bounded Gaussian kernel function, the GP prediction variances are also bounded, i.e.,
$||Σa(x)||≤(σamax)2, ||Σu(x)||≤(σumax)2$
$σamax=maxi(σfai2+ϑai2)1/2, σumax=maxi(σfui2+ϑui2)1/2$
, and
are the hyperparameters in each channel. Furthermore, we require the control gains to satisfy the following bounds:
$ki1≤λ(k̂i1)≤ki3, ki2≤λ(k̂i2)≤ki4, i=p,d$
for constants $kpj,kdj>0, j=1,…,4$, where $λ(·)$ denotes the eigenvalue operator.
The control design should follow the guidelines: (1) the
dynamics are preserved (since they are stable under the original EIC-based control), and (2) the uncontrolled motion (in
) is either eliminated or under active control. The second requirement also implies that the motion of
should depend on only
control inputs. To see this, solving
and plugging it into
Note that $D¯ua∈ℝm×n, D¯aa−1∈ℝn×n$, and $qu$ is overactuated given $n=dim(u)≥m=dim(qu)$. If $qu$ depends on the same number of control inputs, $(n−m)$ column vectors in $D¯uaD¯aa−1$ should be zero.
Thus, the EIC-based control is applied between the same number of actuated and unactuated coordinates. The uncontrolled motion is avoided.
4.2 Partial External and Internal Convertible-Based Control Design.
The control design
in Eq.
updates the input
, and
acts as a virtual control to steer
. The
dynamics is rewritten into
is overactuated with respect to
. We instead reallocate the coupling between
and assign
control inputs for the unactuated subsystem; see Fig.
. To achieve such a goal, we partition the actuated coordinates as
$qa=[qaaT qauT]T, qau∈ℝm, qaa∈ℝn−m$
, and
$u=[uaT uuT]T$
. The
dynamics in Eq.
is rewritten as
where all block matrices are in proper dimensions. We rewrite Eq.
into three groups as
where $Hana=D¯aaauq¨au+D¯auaq¨u+Haagp, Hanu=D¯aauaq¨aa+D¯auuq¨u+Haugp$, and $Hun=D¯uaaq¨aa+Hugp$. Apparently, $Sugp$ is virtually independent of $Saagp$, and the dynamics coupling exists only
between $Sugp$ and $Saugp$.
in Eq.
be partitioned into
corresponding to
, respectively.
is directly applied to
, and
is updated for balance control purpose. As aforementioned, the condition to eliminate the uncontrolled motion in
is that
only depends on
inputs. The task of driving
is assigned to
coordinates only. With this observation, the PEIC-based control takes the form of
$ûint=[ûaT ûuT]T$
$ûa=D¯aaav̂aext+Hana, ûu=D¯aauv̂int+D¯auuq¨u+Hanu$
where $v̂int=−(D¯uau)−1(Hun+D¯uuv̂uint)$. Clearly, the unactuated subsystem only depends on $ûu$ (or $qau$) under the PEIC design as illustrated in Fig. 1(b). The following lemma presents the
qualitative assessment of the PEIC-based control, and the proof is given in Appendix A2.
Lemma 3. If conditions$C1$to$C3$are satisfied and$Sgp$is stable under the EIC-based control design,$Sgp$is stable under the PEIC-based control$ûint$.
4.3 Null-Space External and Internal Convertible-Based Control Design.
Besides the PEIC-based control, we propose an alternative method in which the control input for
is explicitly designed. Noting that
, subspaces
are orthogonal, and the motion of
is independent of
. Therefore, a compensation is designed in
, which leaves the motion in
unchanged. Based on this observation, the NEIC-based control takes the form
where $v˜aint=v˜int+v˜an, v˜an=Vnνn, v˜int=−D¯ua+(Hugp+D¯uuv̂uint), νn$ is the control design that drives p[ai] to $paid, i=m+1,…,n$, and $pad=Υ(qad)$ is transformed reference trajectory. The design
of $νn$ drives $ea$ to the origin in $ker(D¯ua)$. A straightforward yet effective design of $νn$ can be $νn=αν̂next$, where $α>0$. Compared to the PEIC-based control, $pan$ plays the similar role
of $qaa$ coordinates. In the new coordinate, the $qu$ is associated with $pam$ only.
The following result gives the property of the NEIC-based control, and the proof is given in Appendix A3.
Lemma 4. For$S$, if$Sgp$satisfies conditions$C1$to$C3$and$Sgp$is stable under the original EIC-based control,$Sgp$under the NEIC-based control$v˜aint$is also stable. Meanwhile,$Sugp$is unchanged
compared to that under the EIC-based control.
The proofs of Lemmas 3 and 4 show that the inputs $ûaint$ and $u˜aint$ follow the control design guidelines. Both the PEIC- and NEIC-based controllers preserve the structured form of the EIC design.
Figures 1(b) and 1(c) illustrate the overall flowchart of the PEIC- and NEIC-based control design, respectively. To take advantage of the EIC-based structure, we follow the design guideline to make
sure that motion of unactuated coordinates only depends on m inputs in configuration space (PEIC-based control) or transformed space (NEIC-based control). The input $νnext$ is re-used for
uncontrolled motion under the NEIC-based control. The PEIC-based control assigns the balance task to a partial group of the actuated coordinates.
5 Control Stability Analysis
5.1 Closed-Loop Dynamics.
To investigate the closed-loop dynamics, we consider the GP prediction error and the BEM estimation error. The GP prediction error in Eq.
is extended to
$Δaa, Δau$
, and
$qaa,qau,and qu$
dynamics, respectively. Under the PEIC-based control, the dynamics of
Obtaining BEM with Eq. (17) under $(q¨aa,v̂uext)$ is equivalent to inverting Eq. (21c). Thus, $v̂uext=−(D¯uau)−1Hun|qu=q̂ue,q˙u=q¨u=0$. Substituting the above equation into the $qau$ dynamics yields
$q¨au=v̂uext+Oau$, where $Oau=−(D¯uau)−1D¯uuv̂uint−(D¯aau)−1Δau+o1$ and $o1$ denotes the higher order terms.
Defining the total error
$eq=[eaT euT]T$
$e=[eqT e˙qT]T$
, the closed-loop error dynamics becomes
with $Otot=[OaT OuT]T, Oa=[OaaT OauT]T, Oaa=−(D¯aaa)−1Δaa, Ou=−D¯uu−1(Δu−D¯uau(D¯aau)−1Δau)−Δvuint, k̂p=diag(k̂p1,k̂p2)$, and $k̂d=diag(k̂d1,k̂d2)$.
Because of bounded
, there exist constants
such that
. The perturbation terms are further bounded as
$||Oa||=||−[0(D¯uau)−1D¯uuv̂uint]−(D¯aaa)−1Δa+[0o1]||≤du2σ1||v̂uint||+1da1||Δa||+||o1||, and||Ou||=||−D¯uu−1(Δu−D¯uau(D¯aau)−1Δau)−Δvuint||≤1du1||Δu||+σmdu1da1||Δa||+||Δvuint||$
The perturbation
is due to approximation, and
is the control difference by the BEM calculation with the GP prediction. They are both assumed to be affine with
, i.e.,
$||o1||≤c1||e||+c2, ||Δvuint||≤c3||e||+c4$
$0<ci<∞, i=1,…,4$
. From Eq.
, we have
. Thus, for
, we can show that
where $d1=c2+(1+(du2/σ1))c4, d2=c1+(du2/σ1)c3, la1=((σamax(du1+σm))/du1da1),and lu1=σumax/du1$.
To obtain the closed-loop dynamics under the NEIC-based control, plugging the NEIC-based control into
, we obtain
To obtain the error dynamics, we take advantage of the definition of BEM. From Eq.
, we have
. Then, we rewrite Eq.
$p¨am=−Λm−1UTHugp|qu=q̂ueq˙u=q¨u=0+o2−Λm−1UTD¯uuv̂uint −Λm−1UTΔu−VmTD¯aa−1Δa=νmext+Om$
where $o2$ is the residual that contains higher order terms. $Oam=o2−Λm−1UTD¯uuv̂uint−Λm−1UTΔu−VmTD¯aa−1Δa$ denotes the total perturbations.
dynamics keeps the same form as that in the PEIC-based control. We write the error dynamics under the NEIC-based control as
$eam=pam−pamd, ean=pan−pand$
, and
. Applying inverse mapping
to Eqs.
, the error dynamics in
is obtained as
is the transformed perturbations of
$[OanT OamT OuT]T$
. Following the same steps to obtain Eq.
, we have
where $lu2=σu,max((σ1+du1)/σ1du1)$, and $la2=σa,max((σm+du1)/da1du1)$.
5.2 Stability Results.
To show the stability, we consider the Lyapunov function candidate
, where positive definite matrix
is the solution of
$A0TP+PA0+Q=0, A0=[0In+m−kp−kd]$
for given positive definite matrix $Q=QT$, where $A0$ is the constant part of A in Eq. (24) and does not depend on variances $Σa$ or $Σu$. $kp=diag(kp1,kp2)$ and $kd=diag(kd1,kd2)$.
We denote the corresponding Lyapunov function candidates for the NEIC- and PEIC-based controls as V[1] and V[2], respectively. The stability results are summarized as follows with the proof given in
Appendix A4.
heorem 1
For robot dynamics (2), using the GP-based model (13) that satisfies conditions$C1$–$C3$, under the PEIC- and NEIC-based control, the Lyapunov function under each controller satisfies
$Pr{V˙i≤−γiVi+ρi+ϖi}≥η, i=1,2$
and the error e converges to a small ball around the origin, where γ[i] is the convergence rate, ρ[i] and $ϖi$ are the perturbation terms, and $0<η=ηaηu<1$.
6 Experimental Results
Two inverted pendulum platforms are used to conduct experiments to validate the control design. The results from each platform demonstrate different aspects of the control design.^2
6.1 Two Degree-of-Freedom Rotary Inverted Pendulum
Figure 2(a) shows a 2DOF rotary inverted pendulum that was fabricated by Quanser Inc., Markham, ON, Canada. The base joint (θ[1]) is actuated by a DC motor, and the inverted pendulum joint (θ[2]) is
unactuated, i.e., $n=m=1$. We use this platform to illustrate the original EIC-based control and also compare the performance under different nominal models and controllers. The robot dynamic model
is given in Ref. [27] and is also found in Appendix B1.
, there is no uncontrolled motion when the original EIC-based control is applied. Therefore, either a constant or time-varying nominal model would work for the GP-based learning control. We created
the following two nominal models:
$Sn1: D¯1=1100[5−2c2−2c22], H¯1=[0−s2]Sn2: D¯2=1100[2112], H¯2=0$
where $ci=cos θi, si=sin θi$ for angle θ[i], i=1, 2. The training data were sampled and obtained by applying control input $u=kT[θ1−θ1t θ2 θ˙1−θ˙1t θ˙2]T$, where $k∈ℝ4×1$ and $θ1t$ was the
combination of sinusoidal waves with different amplitudes and frequencies. We chose this input to excite the system, and the gain k was selected without the need to balance the platform. It is
difficult to guarantee that the system is fully excited. However, we changed the frequency of sinusoidal waves and obtained the motion data around the target trajectory.
We trained the GP regression models using a total of 500 data points randomly selected from a large dataset. We designed the control gains as $k̂p1=10+50Σa, k̂d1=3+10Σa, k̂p2=1000+500Σu$, and $k̂d2=
100+200Σu$. The variances Σ[a] and Σ[u] were updated online with new measurements in real time. The reference trajectory was $θ1d=0.5 sin t+0.3 sin 1.5t$ rad. The control was implemented at 400Hz
in matlab/simulink real-time system. Both the velocity and acceleration are needed for control design and GP training and prediction. To reduce the influence of measurement noise on control design,
BEM estimation, and GP agent training, a sliding window was used to filter the velocity measurement online. The acceleration was obtained through real-time differentiation. The same technique was
also used for the three-link inverted pendulum in Sec. 6.2.
Figures 3(a) and 3(b) show the tracking of θ[1] and balance of θ[2] under the EIC-based control. With either $Sn1$ or $Sn2$, the base link joint θ[1] closely followed the reference trajectory $θ1d$
, and the pendulum link joint θ[2] was stabilized around its equilibrium $θ2e$ as well. The tracking error was reduced further, and the pendulum closely followed the small variation under $Sn1$.
With $Sn2$, the tracking errors became large when the base link changed rotation direction; see Fig. 3(c) at t=10, 17, and 22s. Both the time-varying and constant nominal models worked for the
EIC-based learning control.
Table 1 further lists the tracking errors (mean and one standard deviation) under both GP models. For comparison purposes, we also conducted additional experiments to implement the original EIC-based
control and the GP-based MPC design in Ref. [4]. The tracking and balance errors under the EIC-based learning control with model $Sn1$ are the smallest. In particular, with the time-varying model
$Sn1$, the mean values of tracking errors $e1$ and e[2] were reduced by 75% and 65%, respectively, in comparison with those under the original EIC-based control. Compared with the MPC method in Ref.
[4], the tracking errors with nominal model $Sn2$ are at the same level.
Table 1
$Sn1$ $Sn2$ GP-based MPC [4] Physical EIC
$|e1|$ 0.24±0.17 0.96±0.34 0.87±0.52 1.09±0.40
$|e2|$ 0.09±0.05 0.09±0.39 0.07±0.06 0.26±0.15
$Sn1$ $Sn2$ GP-based MPC [4] Physical EIC
$|e1|$ 0.24±0.17 0.96±0.34 0.87±0.52 1.09±0.40
$|e2|$ 0.09±0.05 0.09±0.39 0.07±0.06 0.26±0.15
Figure 3(d) shows the control performance with nominal model $Sn1$ under disturbance. At t=17s, an impact disturbance (by manually pushing the pendulum link) was applied, and the joint angles
changed rapidly with $Δθ1=0.7$ rad and $Δθ2=0.3$ rad. The control gains increased ($k̂p2=1215, k̂d2=143$) to respond to the disturbance. As a result, the pendulum motion tracked the BEM closely and
maintained the pendulum balance after the impact disturbance. Figure 3(e) shows the calculated Lyapunov function candidate V(t) and its envelope (i.e., $V(t)=V(0)e−γt, γ=0.1898$) during the
experiment. Figure 3(f) shows the error trajectory in the $||eq||$–$||e˙q||$ plane. The solid/dashed line shows the error trajectory before/after impact disturbance. The tracking error converged
quickly into the error bound. After the disturbance was applied at t=17s, both the Lyapunov function and errors grew dramatically. As the control gains increased, the errors quickly converged back
to the estimated bound again.
6.2 Three Degree-of-Freedom Rotary Inverted Pendulum.
for a 3DOF inverted pendulum with two actuated joints (
) and one unactuated joint (
), namely,
. The physical model of the robot dynamics was obtained using the Lagrangian method and is given in Appendix
. All controllers were implemented at an updating frequency of 200Hz through the Robot Operating System. The time-varying nominal model was selected as
$D¯=[0.150.025c20.025c30.025c20.150.05c2−30.025c30.05c2−30.1], H¯=[00.2c20.1s3]$
where $ci±j=cos(θi±θj)$. The control gains were $k̂p1=15I2+20Σa,k̂d1=3I2+10Σa,k̂p2=25+20Σu,and k̂d2=5.5+10Σu$, where GP variances $Σa$ and Σ[u] were updated online in real-time. The reference
trajectory was chosen as $θ1d=0.5 sin 1.5t$ and $θ2d=0.4 sin 3t$ rad.
For the PEIC-based control, we chose $qaa=θ1$ and $qau=θ2$, and the NEIC-based control was $νn=ν̂next$. Figure 4 shows the experimental results under the PEIC- and NEIC-based control. Under both
controllers, the actuated joints (θ[1] and θ[2]) followed the given reference trajectories ($θ1d$ and $θ2d$) closely, and the unactuated joint (θ[3]) was balanced around the BEM ($θ3e$) as shown
in Figs. 4(a) and 4(b). The pendulum link motion displayed a similar pattern for both controllers. However, the tracking error e[1] under the PEIC-based control (i.e., from −0.05 to 0.05rad) was
much smaller than that under the NEIC-based control (i.e., from −0.15 to 0.15rad); see Figs. 4(c) and 4(d). The balance task in the PEIC-based control was assigned to joint θ[2], and joint θ[1] is
viewed as virtually independent of θ[2] and θ[3]. Joint θ[1] achieved almost-perfect tracking control regardless of the errors for θ[2] and θ[3]. The compensation effect in the null space appeared in
the entire configuration space, and any motion error in the unactuated joints affected the motion of all actuated joints. Similar to the previous example, Fig. 4(e) shows the error trajectory profile
in the $||eq||$–$||eq˙||$ plane. Figure 4(f) shows the Lyapunov function profiles under the PEIC- and NEIC-based controls.
Figure 5 shows the motion of the actuated coordinate in the transformed coordinate $pa$ under various controllers. Under the PEIC- and NEIC-based controls, the $pa$ variables followed the reference
profile $pad$ as shown in Figs. 5(a) and 5(b). Figure 5(c) shows the motion profile under the original EIC-based control. In the first 2s, joint θ[3] followed the BEM under the EIC-based control,
and $pa1$ coordinates displayed a similar motion pattern. However, $pa2$ coordinate showed diverge behavior and led to a failure completely. Therefore, as analyzed previously, the system became
unstable under the EIC-based control though conditions $C1$ to $C3$ were satisfied.
In NEIC-based control, $vn$ drives the uncontrolled motion variable to its reference trajectory. To further reduce the tracking error, we can increase α values. Figure 6 shows the experiment results
of the $pa$ error profiles under various α values varying from 0.5 to 1.5. With a large α value, the tracking error of the actuated coordinates was reduced. Table 2 further lists the steady-state
errors (in joint angles) under the NEIC-based control with various α values, the PEIC-based control and the physical model-based control design. Under the NEIC-based control with $α=0.5$, the system
was stabilized; when increasing α values to 1 and 1.5, the mean tracking errors were reduced 50% and 70% for θ[1], respectively, and 40% for θ[2]. Since control input $νn$ did not affect the balance
task of the unactuated subsystem, the tracking errors for θ[3] maintained the same level. It is of interest that the control effort (i.e., last column in Table 2) only shows a slight increase with
large α values.
Table 2
$|e1|$(rad) $|e2|$(rad) $|e3|$(rad) $||e||$ $∫uTudt$
PEIC (GP) 0.0302±0.0178 0.0566±0.0685 0.1182±0.0160 0.1343±0.0166 5.7659
NEIC (GP, $α=0.5$) 0.1395±0.0946 0.1166±0.0512 0.0303±0.0209 0.2001±0.0770 5.9022
NEIC (GP, $α=1.0$) $0.0651 ± 0.0416$ 0.0756±0.0481 0.0195±0.0152 0.1101±0.0499 5.7089
NEIC (GP, $α=1.5$) 0.0376±0.0302 0.0792±0.0482 0.0207±0.0169 0.0972±0.0470 5.7305
PEIC (model) 0.2168±0.1165 0.2398±0.1649 0.0179±0.0140 0.3587±0.1307 5.7978
NEIC (model, $α=1.0$) 0.1374±0.0922 0.1237±0.0597 0.0455±0.0385 0.2095±0.0769 5.8452
$|e1|$(rad) $|e2|$(rad) $|e3|$(rad) $||e||$ $∫uTudt$
PEIC (GP) 0.0302±0.0178 0.0566±0.0685 0.1182±0.0160 0.1343±0.0166 5.7659
NEIC (GP, $α=0.5$) 0.1395±0.0946 0.1166±0.0512 0.0303±0.0209 0.2001±0.0770 5.9022
NEIC (GP, $α=1.0$) $0.0651 ± 0.0416$ 0.0756±0.0481 0.0195±0.0152 0.1101±0.0499 5.7089
NEIC (GP, $α=1.5$) 0.0376±0.0302 0.0792±0.0482 0.0207±0.0169 0.0972±0.0470 5.7305
PEIC (model) 0.2168±0.1165 0.2398±0.1649 0.0179±0.0140 0.3587±0.1307 5.7978
NEIC (model, $α=1.0$) 0.1374±0.0922 0.1237±0.0597 0.0455±0.0385 0.2095±0.0769 5.8452
6.3 Discussion.
For the rotary pendulum example, we have n=m, and the null space $ker(Dau)$ vanishes. The compensation effect is no longer needed by the NEIC-based control, i.e., $v˜aint=v˜int$ and $u˜int=
D¯aav˜aint+D¯auq¨u+Hagp=uint$. In this case, the PEIC- and NEIC-based controls are degenerated to the EIC-based control. For the 3DOF inverted pendulum, the control inputs u[1] and u[2] act on θ[3]
joints through $θ¨1$ and $θ¨2$. Therefore, as shown in Lemma 1, the uncontrolled motion exists since all controls show up in $Su$ dynamics. This observation explains why the original EIC-based
control failed to balance the three-link inverted pendulum. If the $Su$ dynamics is related to m control inputs (through $q¨a$) for n>m such as the bikebot dynamics in Refs. [4] and [25], only m
external controls were updated, and the EIC-based control worked well without any uncontrolled motion.
For the PEIC-based control, the robot dynamics were partitioned into $Sgp={Saagp,{Saugp,Sugp}}$, which contains a fully actuated system $Saagp$, and a reduced-order underactuated system $
{Saugp,Sugp}$. The EIC-based control is applied to $Saugp$ and $Sugp$ only. The dynamics of $qu$ in general does not depend on any specific m actuated coordinates, since the mapping $Υ$ is
time-varying across different control cycles. In the NEIC-based control design, $pam$ and $qu$ become an underactuated subsystem, and $pan$ is fully actuated.
In practice, no specific rules are defined to select $qau$ out of $qa$ coordinates, and therefore, there are a total of $Cnm=n!/(m!(n−m)!)$ options to select different coordinates. We take advantage
of such a property to optimize tracking performance for selected coordinates. In the 3DOF pendulum case, we assigned the balance task of θ[3] to θ[2] motion. The length of link 1 was only 0.09m and
was much shorter than the length of link 2 (0.23m). The coupling effect between θ[2] and θ[3] was much stronger than that between θ[1] and θ[3]; see D[13] and D[23] in Appendix B2. Thus, it was
efficient to use the motion of θ[2] as a virtual control input to balance θ[3]. When implementing the PEIC-based controller with $qau=θ1$, the system cannot achieve the desired performance and
becomes unstable. We also implemented the proposed controller with the physical model. The control errors are listed in Table 2. Compared with the learning-based controllers, the model-based control
resulted in larger errors. Since the mechanical frictions and other unstructured effects were not considered, the physical model might not capture and reflect the accurate robot dynamics. The results
confirmed the advantages of the proposed learning-based control approaches.
The unique feature of the proposed control lies in integration of the robot's inherent dynamics property (EIC structure) and the GP-based model learning, compared with other learning-based control
approach [18,22]. By integrating physics knowledge into model learning, we identified the conditions for nominal model selection. The overall model learning and control design framework forms a
white-box-like, physics knowledge involved control, which differs from the reinforcement learning-based policy search approach [18]. The solution also has the potential to further incorporate the
bounded GP prediction error for a robust control [4].
7 Conclusion
This paper presented a new learning-based modeling and control framework for underactuated balance robots. The proposed design was an extension and improvement of the EIC-based control with
GP-enabled robot dynamics. The proposed new robot controllers preserved the structural design of the original EIC-based control and achieved both tracking and balance tasks. The PEIC-based control
reshaped the coupling between the actuated and unactuated coordinates. The robot dynamics was transferred into a fully actuated subsystem and one reduced-order underactuated balance subsystem. The
NEIC-based control compensated for uncontrolled motion in a subspace. We validated and demonstrated the new control design on two experimental platforms and confirmed that stability and balance were
guaranteed. The comparison with the physical model-based EIC control and the MPC design confirmed superior performance in terms of the error bound. Extension of the GP-based learning control design
for highly underactuated balance robots is one of the ongoing research directions.
Funding Data
• U.S. National Science Foundation (NSF) (Award No. CNS-1932370; Funder ID: 10.13039/100000001).
Data Availability Statement
No data, models, or code were generated or used for this paper.
Appendix A: Proofs
A1 Proof of Lemma 1.
The system dynamics
under control
$q¨a=vext, q¨u=−Duu−1(Duavext+Hu)$
holds for
, the SVD in Eq.
exists and all
singular values are great than zero, i.e.,
. Thus,
column vectors. Plugging Eq.
into Eq.
and considering the coordinate transformation, we obtain
$p¨a=νext, q¨u=−Duu−1(UΛmνmext+Hu)$
where $UΛVTvext=UΛmνmext$ is used based on the fact that $Λ∈ℝm×n$ is a rectangular diagonal matrix.
Given the definition of
$E, que$
is obtained by solving the algebraic equation
. We substitute
, and therefore, using Eq.
is rewritten into
The BEM $E$ depends only on $νmext$, that is, the control effect in $ker(Dua)$ is not used when obtaining the BEM.
Furthermore, since all controls show up in
dynamics, the control inputs should be updated, and the EIC-based control in Eq.
exists. We substitute Eqs.
dynamics and obtain
Multiplying the above equation on both sides with $VT$ and considering Eq. (8), $S$ under the EIC-based control becomes Eq. (9), and the $(n−m)$ coordinates are free of control.
A2 Proof of Lemma 3.
Under input
$uu, q¨au=vauint$
, we solve
by Eq.
Clearly, the unperturbed subsystem
remains the same as that under the original EIC-based control. With the designed control,
dynamics is unchanged, and
holds regardless of
. For
, we obtain
. The relationship in Eq.
indicates that if the unactuated subsystem dynamics is written into
, the dynamics
under the transformation
must contain the portion (9
). Similarly, we obtain
$SPEIC: p¨ai=−uiT(Hugp+D¯uuv̂uint)σi, i=1,…,m$
$p¨aj=vjTv̂aint, j=m+1,…,n$
where $v̂aint=[(v̂aext)T (v̂uint)T]T$. Since $v̂aint$ is not obtained in the way as in Eq. (5), i.e., $v̂aint∉ker(D¯ua), vm+jTv̂aint≠0$ and $pan$ is under active control. Meanwhile, $vm+jTv̂aint$ drives
$qa→qad$ in $ker(D¯au)$, given that $v̂aext$ and $v̂uint$ are designed to drive $qa→qad$. Therefore, if the unperturbed system under the original EIC-based control is stable, it is also stable under
the PEIC-based control.
A3 Proof of Lemma 4.
Under the NEIC-based control input, the
Plugging above equation into
, we obtain
Using the SVD form of
in Eq.
, the above equation is further simplified as
Clearly, $Sugp$ dynamics is unchanged compared to Eq. (9).
We further apply the transformation
and SVD to
. The
$SNEIC: p¨ai=−uiT(Hugp+D¯uuv̂uint)σi, i=1,…,m$
$p¨aj=νnj, j=m+1,…,n$
Compared to Eq. (9), we add control $v˜aext$ to drive $qa→qad$ in the subspace $ker(D¯ua)$. Therefore, if the system (9) is stable, Eq. (A4) is also stable, as the $pam$ and $qu$ dynamics are
A4 Proof for Theorem 1.
We present the stability proof for the PEIC- and NEIC-based controls using the Lyapunov method.
PEIC-Based Control: Plugging Eq. (24) into $V1=V$ and considering Eq. (32), we obtain $V˙1=eT(ATP+PA)e+2eTPO1=−eTQe+eTQΣe+2eTPO1$, where $QΣ=(A−A0)TP+P(A−A0)$. The bounded variance leads to the
bounded eigenvalue of matrix $QΣ$. Given the fact that $QΣ=QΣT$, the eigenvalues of $QΣ$ are real numbers.
Noting that
is bounded and
are constant, the perturbation term
is bounded as shown in Eq.
. Then,
is rewritten as
denotes the uncertainties related to GP prediction errors.
denote the smallest and greatest eigenvalues of a matrix, respectively. Considering
, we define
$ρ1=2d1λmax(P)||e||, ϖ1=2ω1λmax(P)||e||$. With the bounded perturbations ρ[1] and ω[1], the closed-loop system dynamics can be shown stable in probability as $Pr{V1≤−γ1V1+ρ1+ϖ1}>η$. Taking further
analysis, we obtain a nominal estimation of the error convergence as $Pr{V˙1≤V1(0)e−γ1t}>η$ and the error bound estimation $Pr{||e||≤r1}>η$ with $r1=(2d1λmax(P))/(λmin(Q)−λmax(QΣ)−2d2λmax(P))$.
NEIC-Basd Control: Without the loss of generality, we select $νn=VnTv̂ext$. We take $V2=V$ as the Lyapunov function candidate for $Se,NEIC$. If the control gains are the same as that in the
PEIC-based control and α=1 for compensation effect, $γ2=γ1$. We choose control gains properly such that $γ2>0$. The system can be shown stable as $Pr{V˙2≤−γ2V2+ρ2+ϖ2}>η$, where $ρ2=2d1λmax(P)||e
||, ϖ2=2ω2λmax(P)||e||$, and $ω2=lu2||κu||+la2||κa||$ is defined same as ω[1] containing the GP prediction uncertainties. A nominal estimation of error convergence and final error bound can also be
To show $γi>0$, i=1, 2, the control gains should be properly selected. With a small predefined error limit as a stop criterion in BEM estimation, c[i] values can be shown as $ci≪1$. Given the
explicit form, d[i] are estimated for $A0$ and Q, P is obtained by solving Eq. (32). The matrix $QΣ$ depends on the control gains associated with the reduction variance. Since the variance is
bounded, we design $kni$ such that $λmax(QΣ)$ satisfies the inequality $λmin(Q)−λmax(QΣ)−2d2λmax(P)>0$ and then $γ1>0$. Thus, the stability is obtained.
Appendix B: Dynamics Model of Underactuated Balance Robots
B1 Rotary Inverted Pendulum.
The dynamics model for the rotary pendulum is in the form of Eq.
. The model parameters are
$B=[1 0]T$
$Daa=C(mplr2+0.25mplp2s22+Jr)Dau=Dua=−0.5Cmplplrc2, Duu=C(Jp+0.25mplp2)Ha=C(0.5mplp2θ˙1θ˙2s2c2+0.5mplplrθ˙22s2+drθ˙1+kg2ktkmθ˙1/Rm)+Kgktθ˙2Hu=C(dpθ˙2−0.25mplp2c2s2θ˙2−0.5mplpgs2)$
where l[r], J[r], and d[r] are the length, mass inertia, and viscous damping coefficient of the base link, l[p], J[p], and d[p] are corresponding parameters of the pendulum, m[p] is the pendulum
mass, g is the gravitational constant, and $kt,km,KG,Rm,andC$ are robot constant. The values of these parameters can be found in Ref. [27]. The control input is the motor voltage, i.e., u=V[m].
B2 Three-Link Inverted Pendulum.
The model parameters in Eq.
l22+0.25m3l32−m3l2l3s3D23=D32=(0.25l3−0.5l2s3)m3l3, D33=J3+0.25m3l32G1=0, G2=−(0.5m2+m3)c2l2g+0.5m3l3s2+3gG3=−0.5m3l3s2+3g$
where m[i], l[i], and J[i] are the mass, length, and mass inertia of each link, and $si+j=sin(θi+θj)$. Matrix C is obtained as $Cij=∑k=13cijkθ˙k$, where Christoffel symbols $cijk=12((∂Dij/∂θk)+
(∂Dik/∂θj)−(∂Djk/∂θi))$. The physical parameters are $m1=0.7$ kg, $m2=1.3$ kg, $m3=0.3$ kg, $l1=0.065$ m, $l2=0.23$ m, $l3=0.25$ m, $J1=0.0008$ kg m^2, $J2=0.005$ kg m^2, and $J3=0.003$ kg m^2.
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Quanser Inc
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Project News
By analogy with inequalities for loops (see https://vk.com/wall162891802_1403), we can formulate a number of inequalities for Latin subrectangles in DLS. Every nontrivial subrectangle is a
subrectangle in the DLS by definition. Similarly, by definition, each intercalate is a nontrivial 2x2 subrectangle (except for the dimension N = 2, where it will be trivial, but there is no DLS of
this dimension). These simple statements allow us to establish a number of relationships between the values of numerical series associated with subrectangles and intercalates:
1. For subrectangles: 0 <= A307839(N) <= A307840(N).
2. For nontrivial subrectangles: 0 <= A307841(N) <= A307842(N).
3. For minimum values: 0 <= A307163(N) <= A307841(N) <= A307839(N).
4. For maximum values: A307164(N) <= A307842(N) <= A307840(N). | {"url":"https://gerasim.boinc.ru/news/news.aspx?id=98","timestamp":"2024-11-06T17:55:42Z","content_type":"text/html","content_length":"4948","record_id":"<urn:uuid:f4a0ffe8-8227-48af-94eb-7ea7cdfbdf26>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00798.warc.gz"} |
Project Risk Analysis Software and Project Risk Management Software Forum
Re: Calculating Cost Contingency
Calculating Cost Contingency
Can you set RiskyProject to automatically calculate cost contingency?
IT Program Manager
In our methodology, cost contingency is calculated as the difference between the base estimate and a calculated percentile that is calculated from the simulation. This is referred to as “risk
adjusted” cost contigency. A very common value used is P80 which means that 8 out of 10 times the cost of the project or activity will not exceed that value. Contingency is calculated (P80 – Base
Estimate = Contingency). As long as your schedule is resource loaded with rates or has fixed costs, cost ranges with percentiles will be generated for every activity.
Contingency is not automatically calculated, but you can set up your results so that the High results equal the percentile level that you want to use contingency. In the example below, in the Options
> Calculations, we have set the High results to P80.
In the Simulations Results report, you can set up the view to display the Base (Original Costs) and High (p80). For simplicity, the activities have been collapsed to show only those activities to
which contingency will be applied and measured. To quickly calculate the contingency, copy and paste the results to Excel. In version 7.1, you can export the view to Excel.
Intaver Support Team
Intaver Institute Inc.
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Environmental Economics
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Thank you for submitting your revised manuscript to the Journal of Benefit-Cost Analysis. It is a pleasure to accept your manuscript BCA-2015-0053.R2 "A benefit-cost analysis of a red drum stock
enhancement program in South Carolina" in its current form for publication in JBCA. ...
I first presented this at the 2006 Southern Economic Association Meetings in Charleston, SC. | {"url":"https://www.env-econ.net/2017/12/?asset_id=6a00d83451bd4869e201bb09d7d399970d","timestamp":"2024-11-04T17:43:03Z","content_type":"application/xhtml+xml","content_length":"68822","record_id":"<urn:uuid:395865c4-7e82-4128-b235-f41fd30277f6>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00479.warc.gz"} |
Generate Random Subsets of a Dataset
Write a Python function to generate random subsets of a given dataset. The function should take in a 2D numpy array X, a 1D numpy array y, an integer n_subsets, and a boolean replacements. It should
return a list of n_subsets random subsets of the dataset, where each subset is a tuple of (X_subset, y_subset). If replacements is True, the subsets should be created with replacements; otherwise,
without replacements.
X = np.array([[1, 2],
[3, 4],
[5, 6],
[7, 8],
[9, 10]])
y = np.array([1, 2, 3, 4, 5])
n_subsets = 3
replacements = False
get_random_subsets(X, y, n_subsets, replacements)
[array([[7, 8],
[1, 2]]),
array([4, 1])]
[array([[9, 10],
[5, 6]]),
array([5, 3])]
[array([[3, 4],
[5, 6]]),
array([2, 3])]
The function generates three random subsets of the dataset without replacements.
Each subset includes 50% of the samples (since replacements=False). The samples
are randomly selected without duplication.
Understanding Random Subsets of a Dataset
Generating random subsets of a dataset is a useful technique in machine learning, particularly in ensemble methods like bagging and random forests. By creating random subsets, models can be trained
on different parts of the data, which helps in reducing overfitting and improving generalization.
In this problem, you will write a function to generate random subsets of a given dataset. Given a 2D numpy array X, a 1D numpy array y, an integer n_subsets, and a boolean replacements, the function
will create a list of n_subsets random subsets. Each subset will be a tuple of (X_subset, y_subset).
If replacements is True, the subsets will be created with replacements, meaning that samples can be repeated in a subset. The subset size should be the same as the original dataset in this case. If
replacements is False, the subsets will be created without replacements, meaning that samples cannot be repeated within a subset. The subset size should take the floor of the original dataset size
divided by 2 if replacements is False
By understanding and implementing this technique, you can enhance the performance of your models through techniques like bootstrapping and ensemble learning.
import numpy as np
def get_random_subsets(X, y, n_subsets, replacements=True, seed=42):
n, m = X.shape
subset_size = n if replacements else n // 2
idx = np.array([np.random.choice(n, subset_size, replace=replacements) for _ in range(n_subsets)])
# convert all ndarrays to lists
return [(X[idx][i].tolist(), y[idx][i].tolist()) for i in range(n_subsets)]
There’s no video solution available yet 😔, but you can be the first to submit one at: GitHub link. | {"url":"https://www.deep-ml.com/problem/Generate%20Random%20Subsets%20of%20a%20Dataset","timestamp":"2024-11-06T07:31:09Z","content_type":"text/html","content_length":"28216","record_id":"<urn:uuid:bbb68343-c661-4cca-8a06-aed13eed70c0>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00649.warc.gz"} |
How do you write the first four non-zero terms of the Maclaurin series for sin(x^2)? | Socratic
How do you write the first four non-zero terms of the Maclaurin series for #sin(x^2)#?
1 Answer
#sin (x^2) ~~ x^2 - x^6/3! + x^10/5! - x^14/7! + ... #
Recall that the Maclaurin series for $\sin x$ is given by:
#sin (x) ~~ x - x^3/3! + x^5/5! - x^7/7! + ... #
Note: This is a common Maclaurin series and many exams require you to know this (which is why I directly referred to it). If you are not familiar with deriving Maclaurin series of any function (like
$y = \sin x$) I recommend that you read this
Hence, observing from the above approximation, we can replace $x = {x}^{2}$ in the formula to obtain:
#sin (x^2) ~~ x^2 - x^6/3! + x^10/5! - x^14/7! + ... #
Hope this helps! Comment below or PM me if you have any doubts!
All the best!
Impact of this question
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Roomba - Amazon Top Interview Questions | HackerRank Solutions
Roomba - Amazon Top Interview Questions
Problem Statement :
A Roomba robot is currently sitting in a Cartesian plane at (0, 0). You are given a list of its moves that it will make, containing NORTH, SOUTH, WEST, and EAST.
Return whether after its moves it will end up in the coordinate (x, y).
n ≤ 100,000 where n is length of moves
Example 1
moves = ["NORTH", "EAST"]
x = 1
y = 1
Moving north moves it to (0, 1) and moving east moves it to (1, 1)
Example 2
moves = ["WEST", "EAST"]
x = 1
y = 0
This Roomba will end up at (0, 0)
Solution :
Solution in C++ :
bool solve(vector<string>& moves, int x, int y) {
for (auto& move : moves) {
if (move.front() == 'N')
else if (move.front() == 'S')
else if (move.front() == 'W')
else if (move.front() == 'E')
if (!x && !y) return true;
return false;
Solution in Java :
import java.util.*;
class Solution {
public boolean solve(String[] moves, int ax, int ay) {
String px = "EAST";
String nx = "WEST";
String py = "NORTH";
String ny = "SOUTH";
int x = 0;
int y = 0;
for (int i = 0; i < moves.length; i++) {
if (moves[i].equals(px)) {
if (moves[i].equals(nx)) {
if (moves[i].equals(py)) {
if (moves[i].equals(ny)) {
if (x == ax && ay == y)
return true;
return false;
Solution in Python :
class Solution:
def solve(self, moves, x, y):
currX, currY = 0, 0
d = {
"EAST": (1, 0),
"WEST": (-1, 0),
"NORTH": (0, 1),
"SOUTH": (0, -1),
for i in moves:
dx, dy = d[i]
currX += dx
currY += dy
if currX == x and currY == y:
return True
return False
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View Solution → | {"url":"https://hackerranksolution.in/roombaamazon/","timestamp":"2024-11-15T04:46:37Z","content_type":"text/html","content_length":"40511","record_id":"<urn:uuid:e5492b0e-40b0-4135-b60c-5dbf98d05027>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00774.warc.gz"} |
algebra dividing Related topics: squares of fractions
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Author Message
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From: England
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A and b are two given matrices such that the order of a is 3 -Turito
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A and B are two given matrices such that the order of A is 3 × 4, If A'B and BA' are both defined then:
A. order of B' is 3 × 4
B. order of B'A is 4 × 4
C. order of B'A is 3 × 3
D. B'A is not defined
The correct answer is: order of B'A is 4 × 4
Get an Expert Advice From Turito. | {"url":"https://www.turito.com/ask-a-doubt/maths-a-and-b-are-two-given-matrices-such-that-the-order-of-a-is-3-4-if-a-b-and-ba-are-both-defined-then-b-a-is-not-qeec09d","timestamp":"2024-11-14T16:56:56Z","content_type":"application/xhtml+xml","content_length":"365443","record_id":"<urn:uuid:4abae14c-927d-46d1-bab9-bac9491da0c5>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00741.warc.gz"} |
Wasserman, Nicholas H. (nhw2108) | Teachers College, Columbia UniversityWasserman, Nicholas H. (nhw2108) | Teachers College, Columbia University
Office Location:
323B Thmps
Office Hours:
Fall 2024: Tues 3:30-5:00pm & Wed 3:30-5:00pm, and by appointment
TC Affiliations:
Faculty Expertise:
Educational Background
Ph.D. in Mathematics Education, Teachers College, Columbia University
M.S. in Mathematics Education, Teachers College, Columbia University
B.S. in Mathematics - UTeach Program, The University of Texas at Austin
Scholarly Interests
Dr. Wasserman's scholarly interests in mathematics education lie primarily in teacher education, particularly in the area of secondary teachers’ (advanced) mathematical knowledge and development.
Simplistically, this work revolves around a central question: What does a secondary mathematics teacher, for example an algebra teacher, gain from taking advanced mathematics courses, such as
abstract algebra? That is, Dr. Wasserman is particularly interested in the intersection of a teacher’s knowledge of advanced mathematics and the practices they engage in while teaching mathematics.
Primarily, this has been in the context of two tertiary mathematics courses: Abstract Algebra and Real Analysis. As part of this work, he has collaborated with various faculty from other national and
international institutions to develop an instructional model for teaching advanced mathematics courses for secondary teacher education. His research and scholarship have developed and made explicit
the notion that identifying connections to the content of school mathematics, while important, is not the same as making connections to the teaching of school mathematics content.
In addition to this interest in secondary mathematics teacher education, Dr. Wasserman's scholarship also includes interests in combinatorics education, both at the secondary and tertiary levels, and
the use of dynamic technology in teaching mathematics. His focus on combinatorics education explicitly considers the role that sets of outcomes - especially set-theoretic encoded outcomes - can play
in students' development of combinatorial reasoning. He has studied the ways that different combinatorial problems might lend themselves to different models of outcomes, and how these interact with
students approaches to solving problems. In terms of technology, his particular interest is in the use of dynamic technology as a tool for instructors to design opportunities for teaching and
learning mathematics, both at the secondary and tertiary levels. This work has included exploring various dynamic technologies, creating models for dynamic technology use in particular courses, as
well as designing new dynamic technologies themselves.
Selected Publications
Wasserman, N., Buchbinder, O., & Buchholtz, N. (2023). Making university mathematics matter for secondary teacher preparation. ZDM – Mathematics Education, 55(4), 719-736. (Source listing)
Wasserman, N. (2022). Re-exploring the intersection of mathematics and pedagogy. For the Learning of Mathematics, 42(3), 28-33. (Source listing)
Wasserman, N., Fukawa-Connelly, T., Weber, K., Mejia Ramos, J. P., & Abbott, S. (2022). Understanding analysis and its connections to secondary mathematics teaching. Cham, Switzerland: Springer. (
Source listing)
Wasserman, N., & McGuffey, W. (2021). Opportunities to learn from (advanced) mathematical coursework: A teacher perspective on observed classroom practice. Journal for Research in Mathematics
Education, 52(4), 370-406. (Source listing)
Lockwood, E., Wasserman, N., & Tillema, E. (2020). A case for combinatorics: A research commentary. Journal of Mathematical Behavior, 59(1), 100783. (Source listing)
Wasserman, N. (2019). Duality in combinatorial notation. For the Learning of Mathematics, 39(3), 16-21. (Source listing)
Wasserman, N. (Ed.) (2018). Connecting abstract algebra to secondary mathematics, for secondary mathematics teachers. In J. Cai and J. A. Middleton (Eds.), Research in Mathematics Education Series.
Cham, Switzerland: Springer. (Source listing)
Wasserman, N., Weber, K., Fukawa-Connelly, T., & McGuffey, W. (2019). Designing advanced mathematics courses to influence secondary teaching: Fostering mathematics teachers' 'attention to scope'.
Journal of Mathematics Teacher Education, 22(4), 379-406. (Source listing; Correction to article)
Wasserman, N., Weber, K., Villanueva, M., & Mejia-Ramos, J. P. (2018). Mathematics teachers' views about the limited utility of real analysis: A transport model hypothesis. Journal of Mathematical
Behavior, 50(1), 74-89. (Source listing)
Wasserman, N. (2018). Knowledge of nonlocal mathematics for teaching. Journal of Mathematical Behavior, 49(1), 116-128. (Source listing)
Wasserman, N., & Weber, K. (2017). Pedagogical applications from real analysis for secondary mathematics teachers. For the Learning of Mathematics, 37(3), 14-18. (Source listing)
Wasserman, N. (2017). Making sense of abstract algebra: Exploring secondary teachers' understanding of inverse functions in relation to its group structure. Mathematical Thinking and Learning, 19(3),
181-201. (Source listing)
Wasserman, N. (2016). Abstract algebra for algebra teaching: Influencing school mathematics instruction. Canadian Journal of Science Mathematics and Technology Education, 16(1), 28-47. (Source
Wasserman, N. (2015). A random walk: Stumbling across connections. Mathematics Teacher, 108(9), 686-695. (Source listing)
Honors and Awards
2021 Faculty Teaching Award, Honorable Mention · Teachers College, Columbia University, 2021
Fulbright Specialist · U.S. Department of State, Bureau of Educational and Cultural Affairs (ECA), 2019-2022
Outstanding Reviewer · Journal for Research in Mathematics Education (JRME), 2018
Best Paper Award (“Leveraging real analysis to foster pedagogical practices”) · Annual Conference on Research in Undergraduate Mathematics Education (RUME), San Diego, CA, 2017
STaR Fellow · Service, Teaching, and Research (STaR) Program for Early Career Mathematics Educators, 2012-2013
MST Doctoral Writing Scholarship Award · Department of Mathematics, Science and Technology (MST), Teachers College, Columbia University, New York, NY, October 2010
R.L. Moore Award for Best Inquiry Lesson · University of Texas at Austin, Austin, TX, April 2008
Professional Presentations
Plenary Presentations
Wasserman, N. (2023). Strengthening the role of practice in mathematics teacher education: Opportunities for university mathematics courses. Congreso Iberoamericano sobre conocimiento especializado
del professor de matemáticas (CIMTSK-VI). Valparaíso, Chile. 10 November 2023.
Wasserman, N. (2022). Bridging the teacher education divide: Possibilities for university mathematics courses. Annual Conference of the Korea Society of Educational Studies in Mathematics. Seoul,
South Korea. 4 November 2022.
Wasserman, N. (2022). Leveraging mathematical practice to develop pedagogy in advanced mathematical coursework. Mathematics Education Forum Research Day 2022, Fields Institute for Research in
Mathematical Sciences, Toronto, Canada. 29 January 2022.
Wasserman, N. (2021). Preparing teachers through advanced mathematical coursework. Northeastern Conference on Research in Undergraduate Mathematics Education. 20 November 2021.
Invited Presentations
Buchbinder, O., Wasserman, N., & Buchholtz, N. (2023). Exploring and strengthening university mathematics courses for secondary teacher preparation. ZDM – Mathematics Education Webinar Series. 28
September 2023.
Wasserman, N. (2023). Mejorando la preparación de los profesores a través de sus cursos de matemáticas avanzado. Seminario en Educación Matemática, Programa de Maestría y Doctorado en Educación
Matemática, Universidad Antonio Nariño, Bogota, Colombia. 9 August 2023.
Wasserman, N. (2022). STEM in Education. Workshop for Kazakhstan Visiting Scholars Program, Teachers College, Columbia University, New York, NY. 9 December 2022.
Wasserman, N. (2022). Upgrading Learning for Teachers in Real Analysis: A Look at Diversifying Content Connections to Counter Klein’s Second Discontinuity. International Online Seminar: From
University Mathematics to Mathematics Education, Laboratoire de Didactique André Revuz and Institut Montpelliérain Alexander Grothendieck, University of Rouen Normandie and University of Montpellier,
France. 12 September 2022.
Wasserman, N. (2022). Mejorando la preparación de los profesores a través de sus cursos de matemáticas avanzado. Conferencia Pública, Pontificia Universidad Católica de Chile, Pontificia Universidad
Católica de Valparaíso, y Fundación Columbia University Global Center en Chile, Santiago, Chile. 22 June 2022.
Wasserman, N. (2022). Investigando cursos matemáticas en la formación de profesores de matemáticas. Seminario de Didáctica de la Matemática, del Programa de Doctorado en Didáctica de la Matemática,
Pontificia Universidad Católica de Valparaíso, Valparaíso, Chile. 6 June 2022.
Wasserman, N. (2021). Pedagogical Mathematical Practices as a way to develop pedagogy from mathematics coursework. Mathematics Courses Designed to Develop Mathematical Knowledge for Teaching (AMS
Special Session), Joint Mathematics Meetings of the MAA and AMS, Washington D.C. 6 January 2021.
Wasserman, N. (2020). Secondary mathematics teachers’ pedagogical learning: Pedagogical mathematical practices in university mathematics courses. Seminar on Mathematics Education, Department of
Science Teaching, Weizmann Institute of Science, Rehovot, Israel. 22 November 2020.
Wasserman, N. (2020). Secondary mathematics teachers’ mathematical learning: Expanding mathematical connections in university mathematics courses. Seminar on Mathematics Education, Department of
Science Teaching, Weizmann Institute of Science, Rehovot, Israel. 8 November 2020.
Wasserman, N. (2020). Secondary mathematics teachers’ pedagogical learning: Pedagogical mathematical practices as a way to develop pedagogy from university mathematics courses. Center for Research in
Mathematics Education Seminar Series, Seoul National University, Seoul, South Korea. 31 October 2020.
Wasserman, N. (2020). Secondary mathematics teachers’ mathematical learning: Connecting to classroom teaching as a way to develop mathematics from university mathematics courses. Center for Research
in Mathematics Education Seminar Series, Seoul National University, Seoul, South Korea. 17 October 2020.
Wasserman, N. (2020). Episode 17: Connecting secondary teaching to advanced mathematics. Teaching Math Teaching podcast (sponsored by the Association of Mathematics Teacher Educators (AMTE)). 8 July
2020. Available at: https://www.teachingmathteachingpodcast.com/17
Wasserman, N. (2020). Connecting advanced to secondary mathematics, for secondary mathematics teachers. Seminario de Didáctica de la Matemática, Instituto de Matemáticas, Pontificia Universidad
Católica de Valparaíso, Valparaíso, Chile. 11 May 2020.
Wasserman, N., Holbert, N., & Blikstein, P. (2020). Classes in Corona [Audio podcast]. The Blue Jay, The Blue and White, The Undergraduate Magazine of Columbia University. May 2020 (Vol. XXVI No.
II). Available at: https://open.spotify.com/episode/5g8ClNZTbELSkL7foQiKVP
Cook, J. P., Heid, M. K., Smith, J. P., Zazkis, R., & Wasserman, N. (2019). Connecting abstract algebra to secondary mathematics, for secondary mathematics teachers. Program in Mathematics Colloquium
Series, Teachers College, Columbia University. 7 October 2019.
Heid, M. K., Lai, Y., Wasserman, N., & Zazkis, R. (2019). Panel: Connecting advanced and secondary mathematics. Connecting Advanced and Secondary Mathematics (CASM) Conference, Minneapolis,
Minnesota. 20 May 2019.
Dawkins, P., Inglis, M., & Wasserman, N. (2019). The use(s) of ‘is’ in mathematics. Joint Mathematics Meetings of the MAA and AMS, Baltimore, MA. 19 January 2019.
Wasserman, N. (2018). Using discrete mathematics problems in secondary teaching. Math for America (MƒA) Mini-Course (3 sessions), New York, NY. Fall 2018.
Wasserman, N. (2018). Don’t forget discrete mathematics! National Council of Teachers of Mathematics (NCTM) Annual Meeting, Washington D.C. 26 April 2018.
Wasserman, N., Weber, K., & McGuffey, W. (2018). Leveraging real analysis to foster pedagogical practices. Joint Mathematics Meetings of the MAA and AMS, San Diego, CA. 13 January 2018.
Wasserman, N. (2017). Applying ideas from real analysis to secondary teaching. Math for America (MƒA) Mini-Course (3 sessions), New York, NY. Fall 2017.
Wasserman, N. (2017). Designing advanced mathematics courses for secondary teachers: Connecting to their future professional work in the classroom. Mathematics for Future Teachers: A one-day
conference on designing and teaching mathematics courses for pre-service teachers, Rutgers University, New Brunswick, NJ. 11 May 2017.
Wasserman, N. (2017). What can we learn for teaching from studying advanced mathematics? Special Seminar, Simon Fraser University, Vancouver, British Columbia. 24 January 2017.
Wasserman, N. (2016). Making advanced content courses relevant to secondary teachers: Investigating an instructional model from a real analysis course. Brown Bag Lunch Speaker Series, Graduate School
of Education, Rutgers University, New Brunswick, NJ. 7 December 2016.
Wasserman, N. (2016). Addressing the dilemma of advanced mathematics in secondary teacher preparation: The case of a real analysis course. Montclair State University Colloquium Series, Department of
Mathematical Sciences, Montclair State University, Montclair, NJ. 5 December 2016.
Wasserman, N. (2016). The dilemma of advanced mathematics: Instructional approaches for secondary mathematics teacher education. Current Issues in Mathematics Education Workshop, Teachers College,
Columbia University, New York, NY. 20 November 2016.
Wasserman, N. (2016). Accommodation of teachers’ knowledge of inverse functions with the group of invertible functions. Paper invited to be presented at the 13^th International Congress on
Mathematical Education (ICME-13), Topic Study Group 46 (Knowledge in/for teaching mathematics at secondary level), Hamburg, Germany. 29 July 2016.
Wasserman, N. (2015). Episode 1503: Nick Wasserman. MathEd Podcast: Conversations with math ed researchers. 23 February 2015. Available at: http://mathed.podomatic.com/entry/2015-02-18T07_12_33-08_00
Wasserman, N. (2014). Using pedagogical contexts to foster teachers’ mathematical development and practices. Joint Seminar in Mathematics Education of Stony Brook University and Teachers College,
Teachers College, Columbia University, New York, NY. 5 December 2014.
Wasserman, N. (2014). Using pedagogical contexts to explore mathematics: A parallelogram task in teacher education. Proof Comprehension Research Group (PCRG) Seminar, Rutgers University, New
Brunswick, NJ. 14 November 2014. (.PDF)
Wasserman, N. (2014). Using cognitive conflict in mathematics education. Opening keynote address. World Mathematical Olympiad Competition, hosted by the China National Committee for the Wellbeing of
the Youth (NCWY), Columbia University, New York, NY. 20 August 2014.
Wasserman, N., & Walkington, C. (2013). Exploring research in Algebra: Tackling algebra in middle school and high school. Research in Mathematics Education (RME) Annual Research to Practice
Conference, Dallas, TX. 15 February 2013. (.PDF)
Wasserman, N. (2012). Mathematics and teaching: Teachers’ knowledge of tasks and proof. Department of Mathematics Colloquium Series, Southern Methodist University, Dallas, TX. 1 February 2012.
Wasserman, N., & Schielack, J. (2012). Systems level content development: Establishing learning progressions. Research in Mathematics Education (RME) Annual Research to Practice Conference, Dallas,
TX. 24 February 2012. (.PDF)
Refereed Presentations: International and National Conferences
Wasserman, N. (2024). Possibilities for a 3-D representation of function composition to foster graphical conceptions. Oral communication at the 15^th International Congress on Mathematical Education
(ICME-15), Topic Study Group 3.5 (Visualization and embodiment in mathematics education), Sydney, Australia. 10 July 2024.
LaPlace, E.^ , Chen, Y.^ , Wasserman, N., & Paoletti, T. (2024). Exploring graphical reasoning from revised responses to function composition tasks. Annual Conference on Research in Undergraduate
Mathematics Education (RUME), Omaha, NE. 23 February 2024.
Lai, Y., Wasserman, N., Strayer, J. F., Casey, S., Weber, K., Fukawa-Connelly, T., & Lischka, A. E. (2024). Representing learning in advanced mathematics courses for secondary mathematics teachers.
Annual Conference on Research in Undergraduate Mathematics Education (RUME), Omaha, NE. 23 February 2024.
Wasserman, N. (2024). Fostering a graphical conceptualization of function composition with a 3D representational tool. Association of Mathematics Teacher Educators (AMTE) Annual Conference, Orlando,
FL. 10 February 2024.
Delgado-Rebolledo, R., Zakaryan, D., & Wasserman, N. (2023). Una aproximación a las conexiones entre el MTSK y las prácticas matemáticas pedagógicas. Congreso Iberoamericano sobre conocimiento
especializado del professor de matemáticas (CIMTSK-VI). Valparaíso, Chile. 9 November 2023.
Chen, Y., Wasserman, N., Paoletti, T., & LaPlace, E. (2023). Exploring students’ responses to composing functions given two graphs. National Council of Teachers of Mathematics (NCTM) Research
Conference, Washington, D.C. 25 October 2023.
Pinto, A., Buchbinder, O., & Wasserman, N. (2023). The affordances of advanced mathematics for secondary mathematics teaching: Comparing research approaches and theoretical perspectives (Working
Group). Annual Conference of International Group for the Psychology of Mathematics Education (PME 46), Haifa, Israel. 17 July 2023.
Chen, Y., Wasserman, N., & Paoletti, T. (2023). Exploring geometric reasoning with function composition. Annual Conference on Research in Undergraduate Mathematics Education (RUME), Omaha, NE. 25
February 2023.
Wasserman, N. (2023). An exploration of Pedagogical Mathematical Practices from a teacher perspective. Association of Mathematics Teacher Educators (AMTE) Annual Conference, New Orleans, LA. 3
February 2023.
Wasserman, N. (2022). Planning for mathematically coherent instruction: Four ‘foreshadowing’ practices. Association of Mathematics Teacher Educators (AMTE) Annual Conference, Las Vegas, NV. 12
February 2022.
Wasserman, N. (2022). Focusing mathematical coursework on developing practice: An exploration of pedagogical mathematical practices. Association of Mathematics Teacher Educators (AMTE) Annual
Conference, Las Vegas, NV. 10 February 2022.
Wasserman, N., Weber, K., Mejia-Ramos, J. P., & Fukawa-Connelly, T. (2021). Upgrading learning for teachers in real analysis (ULTRA): An instructional model for secondary teacher education. Long oral
presentation at the 14^th International Congress on Mathematical Education (ICME-14), Topic Study Group 33 (Knowledge in/for teaching mathematics at secondary level), Shanghai, China. 17 July 2021.
Mirin, A., Weber, K., & Wasserman, N. (2021). What is a function? Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA 42),
Mazatlán, Mexico. 29 May 2021.
Fukawa-Connelly, T., Wasserman, N., Weber, K., & Mejia-Ramos, J. P. (2019). Upgrading Learning for Teachers in Real Analysis (ULTRA): A curriculum project. Poster presented at the Annual Conference
on Research in Undergraduate Mathematics Education (RUME), Oklahoma City, OK. 2 March 2019.
Wasserman, N., Zazkis, R., Baldinger, E., Marmur, O., & Murray, E. (2019). Points of connection to secondary teaching in undergraduate mathematics courses. Annual Conference on Research in
Undergraduate Mathematics Education (RUME), Oklahoma City, OK. 2 March 2019.
Wasserman, N. (2019). Content courses for secondary teachers: Teachers’ attributions for influencing teaching practice. Association of Mathematics Teacher Educators (AMTE) Annual Conference, Orlando,
FL. 7 February 2019.
Wasserman, N. (2018). Exploring the secondary teaching of functions in relation to the learning of abstract algebra. Annual Conference on Research in Undergraduate Mathematics Education (RUME), San
Diego, CA. 24 February 2018.
Weber, K., Wasserman, N., Mejia-Ramos, J. P., & Fukawa-Connelly, T. (2018). Connecting the study of advanced mathematics to the teaching of secondary mathematics: Implications for teaching inverse
trigonometric functions. Annual Conference on Research in Undergraduate Mathematics Education (RUME), San Diego, CA. 22 February 2018.
Dawkins, P., Inglis, M., & Wasserman, N. (2018). The use(s) of ‘is’ in mathematics. Annual Conference on Research in Undergraduate Mathematics Education (RUME), San Diego, CA. 22 February 2018.
Wasserman, N., & McGuffey, W. (2018). Advanced mathematics courses for secondary teachers: An instructional model for connecting to secondary teaching practice. Association of Mathematics Teacher
Educators (AMTE) Annual Conference, Houston, TX. 8 February 2018.
Wasserman, N., Weber, K., Mejia-Ramos, J. P., & Fukawa-Connelly, T. (2018). Designing real analysis courses for secondary mathematics teachers. Joint Mathematics Meetings of the MAA and AMS, San
Diego, CA. 11 January 2018.
Wasserman, N., Fukawa-Connelly, T., & Weber, K. (2017). Leveraging real analysis to foster pedagogical practices. National Council of Teachers of Mathematics (NCTM) Research Conference, San Antonio,
TX. 5 April 2017.
Wasserman, N., Weber, K., & McGuffey, W. (2017). Leveraging real analysis to foster pedagogical practices. Annual Conference on Research in Undergraduate Mathematics Education (RUME), San Diego, CA.
23 February 2017.
Baldinger, E., Murray, E., White, D., Broderick, S., & Wasserman, N. (2016). Exploring connections between advanced and secondary mathematics (Working Group). Annual Meeting of the North American
Chapter of the International Group for the Psychology of Mathematics Education (PME-NA 38), Tucson, AZ. 4 November 2016.
Wasserman, N. (2016). Nonlocal mathematical knowledge for teaching. Paper presented at the Annual Conference of International Group for the Psychology of Mathematics Education (PME 40), Szeged,
Hungary. 5 August 2016.
Murray, E., & Wasserman, N. (2016). Connecting solving equations in an advanced context to secondary mathematics instruction. Paper presented at the 13^th International Congress on Mathematical
Education (ICME-13), Topic Study Group 46 (Knowledge in/for teaching mathematics at secondary level), Hamburg, Germany. 29 July 2016.
Ribeiro, M., Jakobsen, A., Ribeiro, A., Wasserman, N., Carrillo, J., Montes, M., & Mamolo, A. (2016). Reflecting upon different perspectives on specialized advanced mathematical knowledge for
teaching. Working group at the 13^th International Congress on Mathematical Education (ICME-13), Hamburg, Germany. 29 July 2016.
Lockwood, E., Wasserman, N., & McGuffey, W. (2016). Classifying combinations: Do students distinguish between different types of combination problems? Annual Conference on Research in Undergraduate
Mathematics Education (RUME), Pittsburgh, PA. 26 February 2016.
Wasserman, N. (2016). Unpacking teachers’ moves for navigating mathematical complexities in teacher education. Association of Mathematics Teacher Educators (AMTE) Annual Conference, Irvine, CA. 29
January 2016.
Murray, E., Baldinger, E., Wasserman, N., Broderick, S., Cofer, T., White, D., & Stanish, K. (2015). Exploring connections between advanced and secondary mathematics (Working Group). Annual Meeting
of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA 37), East Lansing, MI. 6 November 2015.
Casey, S., Zejnullahi, R., Wasserman, N., & Champion, J. (2015). Preparing to teach statistics: Connecting subject matter and pedagogical content knowledge. United States Conferences on Teaching
Statistics (USCOTS), State College, PA. 29 May 2015.
Wasserman, N., Stockton, J., Weber, K., Champion, J., Waid, B., Sanfratello, A., & McCallum, W. (2015). Exploring the role of the mathematical horizon for secondary teachers. National Council of
Teachers of Mathematics (NCTM) Research Conference, Boston, MA. 14 April 2015. (.PDF)
Wasserman, N., Villanueva, M., Mejia-Ramos, J. P., & Weber, K. (2015). Secondary mathematics teachers’ perceptions of real analysis in relation to their teaching practice. Annual Conference on
Research in Undergraduate Mathematics Education (RUME), Pittsburgh, PA. 21 February 2015.
Wasserman, N., & Mamolo, A. (2015). Knowledge for teaching: Horizons and mathematical structures. Annual Conference on Research in Undergraduate Mathematics Education (RUME), Pittsburgh, PA. 19
February 2015. (.PDF)
Wasserman, N., Casey, S., Champion, J., Huey, M., Sanfratello, A., & Waid, B. (2015). Exploring the Impact of Advanced Mathematics on Secondary Teaching Practices. Association of Mathematics Teacher
Educators (AMTE) Annual Conference, Orlando, FL. 13 February 2015. (.PDF)
Wasserman, N., Mamolo, A., Ribeiro, C. M., & Jakobsen, A. (2014). Exploring horizons of knowledge for teaching. Joint meeting of International Group for the Psychology of Mathematics Education (PME
38) and North American Chapter of the Psychology of Mathematics Education (PME-NA 36), Vancouver, Canada. 16 July 2014. (.PDF)
Casey, S., Wasserman, N. H., Wilson, D. C., Molnar, A, & Shaughnessy, J. M. (2014). Knowledge for teaching informal line of best fit. National Council of Teachers of Mathematics (NCTM) Research
Presession, New Orleans, LA. 8 April 2014. (.PDF)
Wasserman, N., & Stockton, J. (2014). The impact of teachers’ knowledge of group theory on early algebra teaching practices. Association of Mathematics Teacher Educators (AMTE) Annual Conference,
Irvine, CA. 6 February 2014. (.PDF)
Wasserman, N., & Stockton, J. (2013). Group theory’s effect on mathematical knowledge for teaching. Poster presented at National Council for Teachers of Mathematics (NCTM) Research Presession,
Denver, CO. 15 April 2013. (.PDF)
Wasserman, N. (2013). A rationale for irrationals: Convincing students they exist. National Council of Teachers of Mathematics (NCTM) Annual Conference, Denver, CO. 18 April 2013. (.PDF)
Wasserman, N., & Williams-Rossi, D. (2013). Discussing proof in STEM fields: Mathematics and science teachers’ use of inductive evidence. International Consortium for Research in Science and
Mathematics Education (ICRSME) Conference, Granada, Nicaragua. 13 March 2013. (.PDF)
Wasserman, N. (2013). Exploring teachers’ categorizations and conceptions of combinatorial problems. Research Council on Mathematics Learning (RCML) Annual Conference, Tulsa, OK. 28 February 2013. (
Wasserman, N., Norris, S., & Carr, T. (2013). Comparing a ‘flipped’ instructional model in an undergraduate Calculus III course. Annual Conference on Research in Undergraduate Mathematics Education
(RUME), Denver, CO. 22 February 2013. (.PDF)
Quebec-Fuentes, S., Wasserman, N., & Switzer, J. (2013). Advanced mathematics content: A comparative analysis of CCSSM and mathematics textbooks for teachers. Association of Mathematics Teacher
Educators (AMTE) Annual Conference, Orlando, FL. 24 January 2013. (.PDF)
Wasserman, N, & Stockton, J. (2013). Researching the mathematical horizon: Two complementary perspectives. Poster presented at Association of Mathematics Teacher Educators (AMTE) Annual Conference,
Orlando, FL. 24 January 2013. (.PDF)
Ketterlin-Gellar, L., Wasserman, N., Chard, D., Fontenot, S., & Zachary, S. (2012). Progress with fractions: Using learning progressions to guide instruction. Council for Learning Disabilities (CLD)
International Conference. Austin, TX. 11 October 2012.
Stockton, J., & Wasserman, N. (2012). Mapping the Common Core State Standards to advanced mathematical knowledge for teaching. Mathematical Association of America (MAA) MathFest. Madison, Wisconsin.
4 August 2012. (.PDF)
Wasserman, N., & Walkington, C. (2012). Exploring links between beginning UTeachers’ beliefs and observed classroom practices. UTeach Institute Annual Conference, University of Texas at Austin,
Austin, TX. 1 June 2012. (.PDF)
Wasserman, N., & Ham, E. (2012). Attributes of good mathematics teaching: When are they learned? Poster presented at International Congress on Mathematics Education (ICME-12), Seoul, Korea. 11 July
2012. (.PDF)
Wasserman, N., & Ham, E. (2011). Learning to be a successful mathematics teacher: Reflections on two teacher education models. UTeach Institute Annual Conference, University of Texas at Austin,
Austin, TX. 24 May 2011. (.PDF)
Refereed Presentations: Regional Conferences
Basaraba, D., Wasserman, N., Ketterlin-Geller, L, & Hill, S. (2012). Learning progressions for algebra readiness: A roadmap for instructional planning. Poster presented at Center on Teaching and
Learning (CTL) Research to Practice Conference, Portland, OR. 28 October 2012.
Wasserman, N., & Ham, E. (2011). A question of when, for beginning mathematics teachers. National Council of Teachers of Mathematics (NCTM) Regional Conference, Albuquerque, NM. 3 November 2011.
Wasserman, N., & Ham, E. (2011). A question of when, for beginning mathematics teachers. National Council of Teachers of Mathematics (NCTM) Regional Conference, Atlantic City, NJ. 21 October 2011.
Welch, A., Wright, R., Wasserman, N., & Garcia, K. (2011). UTeach Graduates Roundtable. UTeach Institute Annual Conference, University of Texas at Austin, Austin, TX. 24 May 2011.
Wasserman, N., & Arkan, I. (2011). Archimedes rediscovered through technology. New York State Association of Independent Schools (NYSAIS) Teaching with Technology Conference, Abraham Joshua Heschel
School, New York, NY. 27 April 2011. (.PDF)
Wasserman, N., & Ham, E. (2010). A question of “When?” for beginning mathematics teachers. Association of Mathematics Teachers of New York State (AMTNYS) Annual Conference, Saratoga Springs, NY. 13
November 2010.
Wasserman, N. (2010). Partition and iteration in Algebraic thinking: Intuition with linearity. Association of Mathematics Teachers of New York State (AMTNYS) Annual Conference, Saratoga Springs, NY.
12 November 2010. (.PDF)
Wasserman, N. (2006). Stacking paper cups. UTeach professional development, University of Texas at Austin, Austin, TX. November 2006.
Authored Books
Wasserman, N., Fukawa-Connelly, T., Weber, K., Mejia-Ramos, J. P., & Abbott, S. (2022). Understanding analysis and its connections to secondary mathematics teaching. Cham, Switzerland: Springer. (
Source listing)
Karp, A., & Wasserman, N. (2015). Mathematics in middle and secondary schools: A problem solving approach. Charlotte, NC: Information Age Publishing Inc. (Source listing)
Edited Books and Collections
Buchbinder, O., Wasserman, N., & Buchholtz, N. (Eds.) (2023). Special issue: Exploring and strengthening university mathematics courses for secondary teacher preparation. ZDM – Mathematics Education,
55(4). (Source listing)
Wasserman, N. (Ed.) (2018). Connecting abstract algebra to secondary mathematics, for secondary mathematics teachers. In J. Cai and J. A. Middleton (Eds.), Research in Mathematics Education Series.
Cham, Switzerland: Springer. (Source listing)
Book Chapters
Wasserman, N., & Dawkins, P. (in press). University geometry courses as part of teacher education: Empirically grounding the SLOs. In A. Brown, P. Herbst, N. Miller, & L. Pyzdrowski (Eds.), The GeT
course: Resources and objectives for the Geometry Courses for Teachers (pp. XXX). XXX.
Lai, Y., Wasserman, N., Strayer, J. F., Casey, S., Weber, K., Fukawa-Connelly, T., & Lischka, A. (2024). Making advanced mathematics work in secondary teacher education. In B. Benken (Ed.), The AMTE
Handbook of mathematics teacher education: Reflection on past, present and future – paving the way for the future of mathematics teacher education, AMTE Professional Book Series (Vol. 5) (pp.
199-218). Charlotte, NC: IAP.
Wasserman, N. (2023). Mathematical challenge in connecting advanced and secondary mathematics: Recognizing binary operations as functions. In R. Leikin, C. Christou, A. Karp, D. Pitta-Pantazi, & R.
Zazkis (Eds.), Mathematical challenge for all, Research in Mathematics Education Series (pp. 241-260). Cham, Switzerland: Springer. (Source listing)
Wasserman, N. (2018). Exploring advanced mathematics courses and content for secondary mathematics teachers. In N. Wasserman (Ed.), Connecting abstract algebra to secondary mathematics, for secondary
mathematics teachers, Research in Mathematics Education (pp. 1-15). Cham, Switzerland: Springer. (Source listing)
Wasserman, N., & Galarza, P. (2018). Exploring an instructional model for designing modules for secondary mathematics teachers in an abstract algebra course. In N. Wasserman (Ed.), Connecting
abstract algebra to secondary mathematics, for secondary mathematics teachers, Research in Mathematics Education (pp. 335-361). Cham, Switzerland: Springer. (Source listing)
Wasserman, N. (2017). The dilemma of advanced mathematics: Instructional approaches for secondary mathematics teacher education. In A. Karp (Ed.), Current issues in mathematics education: Materials
of the American-Russian workshop (pp. 107-123). Bedford, MA: The Consortium for Mathematics and Its Applications (COMAP). (Source listing)
Wasserman, N. (2015). Bringing dynamic geometry to three dimensions: The use of SketchUp in mathematics education. In D. Polly (Ed.), Cases on technology integration in mathematics education (pp.
68-99). Hershey, PA: IGI-Global. (Source listing)
Refereed Journal Articles
Wasserman, N. (in press). Adding diversity to mathematical connections to counter Klein’s second discontinuity. Recherches en Didactique des Mathématiques, XX(X), XXX.
Wasserman, N., Buchbinder, O., & Buchholtz, N. (2023). Making university mathematics matter for secondary teacher preparation. ZDM – Mathematics Education, 55(4), 719-736. (Source listing)
Wasserman, N. (2023). Investigating a teacher-perspective on pedagogical mathematical practices: Possibilities for using mathematical practice to develop pedagogy in mathematical coursework. ZDM –
Mathematics Education, 55(4), 807-821. (Source listing)
Wasserman, N. (2022). Re-exploring the intersection of mathematics and pedagogy. For the Learning of Mathematics, 42(3), 28-33. (Source listing)
Wasserman, N. (2022). Unpacking foreshadowing in mathematics teachers’ planned practices. Educational Studies in Mathematics, 111(3), 423-443. (Source listing)
Mirin, A., Milner, F., Wasserman, N., & Weber, K. (2021). On two definitions of ‘function’. For the Learning of Mathematics, 41(3), 22-24. (Source listing)
Wasserman, N., & McGuffey, W. (2021). Opportunities to learn from (advanced) mathematical coursework: A teacher perspective on observed classroom practice. Journal for Research in Mathematics
Education, 52(4), 370-406. (Source listing)
Lockwood, E., Wasserman, N., & Tillema, E. (2020). A case for combinatorics: A research commentary. Journal of Mathematical Behavior, 59(1), 100783. (Source listing)
Weber, K., Mejia-Ramos, J. P., Fukawa-Connelly, T., & Wasserman, N. (2020). Connecting the learning of advanced mathematics with the teaching of secondary mathematics: Inverse functions, domain
restrictions, and the arcsine function. Journal of Mathematical Behavior, 57(1), 100752. (Source listing)
Fukawa-Connelly, T., Mejia-Ramos, J. P., Wasserman, N., & Weber, K. (2020). An evaluation of ULTRA: An experimental real analysis course built on a transformative theoretical model. International
Journal of Research in Undergraduate Mathematics Education, 6(2), 159-185. (Source listing)
Wasserman, N. (2019). Duality in combinatorial notation. For the Learning of Mathematics, 39(3), 16-21. (Source listing)
Wasserman, N., Weber, K., Fukawa-Connelly, T., & McGuffey, W. (2019). Designing advanced mathematics courses to influence secondary teaching: Fostering mathematics teachers' 'attention to scope'.
Journal of Mathematics Teacher Education, 22(4), 379-406. (Source listing; Correction to article)
McGuffey, W., Quea, R., Weber, K., Wasserman, N., Fukawa-Connelly, T., & Mejia-Ramos, J. P. (2019). Pre- and in-service teachers’ perceived value of an experimental real analysis course for teachers.
International Journal of Mathematical Education in Science and Technology, 50(8), 1166-1190. (Source listing)
Dawkins, P., Inglis, M., & Wasserman, N. (2019). The use(s) of ‘is’ in mathematics. Educational Studies in Mathematics, 100(2), 117-137. (Source listing)
Wasserman, N., & Galarza, P. (2019). Conceptualizing and justifying sets of outcomes with combination problems. Investigations in Mathematics Learning, 11(2), 83-102. (Source listing)
Wasserman, N., Weber, K., Villanueva, M., & Mejia-Ramos, J. P. (2018). Mathematics teachers' views about the limited utility of real analysis: A transport model hypothesis. Journal of Mathematical
Behavior, 50(1), 74-89. (Source listing)
Lockwood, E., Wasserman, N., & McGuffey, W. (2018). Classifying combinations: Investigating undergraduate students' responses to different categories of combination problems. International Journal of
Research in Undergraduate Mathematics Education, 4(2), 305-322. (Source listing)
Huey, M. E., Champion, J., Casey, S., & Wasserman, N. (2018). Secondary mathematics teachers' planned approaches for teaching standard deviation. Statistics Education Research Journal, 17(1), 61-84.
(Source listing)
Wasserman, N. (2018). Knowledge of nonlocal mathematics for teaching. Journal of Mathematical Behavior, 49(1), 116-128. (Source listing)
Wasserman, N., & Weber, K. (2017). Pedagogical applications from real analysis for secondary mathematics teachers. For the Learning of Mathematics, 37(3), 14-18. (Source listing)
Wasserman, N. (2017). Exploring how understandings from abstract algebra can influence the teaching of structure in early algebra. Mathematics Teacher Education and Development, 19(2), 81-103. (
Source listing)
Wasserman, N., Casey, S., Champion, J., & Huey, M. (2017). Statistics as unbiased estimators: Exploring the teaching of standard deviation. Research in Mathematics Education, 19(3), 236-256. (Source
Wasserman, N. (2017). Making sense of abstract algebra: Exploring secondary teachers' understanding of inverse functions in relation to its group structure. Mathematical Thinking and Learning, 19(3),
181-201. (Source listing)
Wasserman, N., Fukawa-Connelly, T., Villanueva, M., Mejia-Ramos, J. P., & Weber, K. (2017). Making real analysis relevant to secondary teachers: Building up from and stepping down to practice.
PRIMUS, 27(6), 559-578. (Source listing)
Stockton, J., & Wasserman, N. (2017). Forms of knowledge of advanced mathematics for teaching. The Mathematics Enthusiast, 14(1), 575-606. (Source listing)
Wasserman, N., Quint, C., Norris, S. A., & Carr, T.(2017). Exploring flipped classroom instruction in Calculus III. International Journal of Science and Mathematics Education, 15(3), 545-568. (Source
Wasserman, N. (2016). Abstract algebra for algebra teaching: Influencing school mathematics instruction. Canadian Journal of Science Mathematics and Technology Education, 16(1), 28-47. (Source
Wasserman, N. (2015). Unpacking teachers' moves in the classroom: Navigating micro- and macro-levels of mathematical complexity. Educational Studies in Mathematics, 90(1), 75-93. (Source listing)
Casey, S., & Wasserman, N. (2015). Teachers' knowledge about informal line of best fit. Statistics Education Research Journal, 14(1), 8-35. (.PDF; Source listing)
Wasserman, N., & Rossi, D. (2015). Mathematics and science teachers' use of and confidence in empirical reasoning: Implications for STEM teacher preparation. School, Science and Mathematics Journal,
115(1), 22-34. (.PDF; Source listing)
Wasserman, N., & Walkington, C. (2014). Exploring links between beginning UTeacher's beliefs and observed classroom practices. Teacher Education and Practice, 27(2/3), 376-401. (.PDF)
Wasserman, N. (2014). Introducing algebraic structures through solving equations: Vertical content knowledge for K-12 mathematics teachers. PRIMUS, 24(3), 191-214. (Source listing)
Wasserman, N., & Ham, E. (2013). Beginning teachers' perspectives on attributes for teaching secondary mathematics: Reflections on teacher education. Mathematics Teacher Education and Development, 15
(2), 70-96. (Source listing)
Wasserman, N., & Stockton, J. (2013). Horizon content knowledge in the work of teaching: A focus on planning. For the Learning of Mathematics, 33(3), 20-22. (Source listing)
Refereed Professional Journal Articles
Pogorelova, L., Sheehan-Braine, S., John, A., & Wasserman, N. (2024). A problem-based curriculum to conceptually develop the multiplication principle for counting. Journal of Mathematics Education at
Teachers College, 15(1), 37-43. (Source listing)
Wasserman, N. (2020). Dynamically reconstructed proof visualizations in real analysis. Electronic Journal of Mathematics and Technology, 14(1), 38-49. (Source listing)
Wasserman, N., Weber, K., Fukawa-Connelly, T., & Mejia-Ramos, J. P. (2020). Area-preserving transformations: Cavalieri in 2D. Mathematics Teacher: Learning and Teaching PK-12, 113(1), 53-60. (Source
Murray, E., Baldinger, E., Wasserman, N., Broderick, S., & White, D. (2017). Connecting advanced and secondary mathematics. Issues in the Undergraduate Mathematics Preparation of School Teachers
(Vol. 1, August 2017), 1-10. (Source listing)
Wasserman, N. (2017). Math madness: Coloring, reasoning, and celebrating. Teaching Children Mathematics, 23(8), 468-475. (Source listing)
Wasserman, N. (2015). A random walk: Stumbling across connections. Mathematics Teacher, 108(9), 686-695. (Source listing)
Wasserman, N. (2014). A rationale for irrationals: An unintended exploration of e. Mathematics Teacher, 107(7), 500-507. (Source listing)
Gould, H., & Wasserman, N. (2014). Striking a balance: Students' tendencies to oversimplify or overcomplicate in mathematical modeling. Journal of Mathematics Education at Teachers College, 5(1),
27-34. (Source listing)
Wasserman, N., & Ham, E. (2012). Gaining perspective on success, support, retention, and student test scores: Listening to beginning teachers. Leaders of Learners, 5(3), 9-14. (.PDF)
Wasserman, N., & Arkan, I. (2011). Technology Tips: An Archimedean walk. Mathematics Teacher, 104(9), May 2011, 710-715. (Source listing)
Wasserman, N., & Koehler, J. (2011). Will Common Core State Standards facilitate consistency and choice or lead to unexpected outcomes? (Editorial Point-Counterpoint). Journal of Mathematics
Education at Teachers College, 2(1), 6-7. (Source listing)
Wasserman, N. (2011). The Common Core State Standards: Comparisons of access and quality. Journal of Mathematics Education at Teachers College, 2(1), 18-27. (Source listing)
Wasserman, N. (2011). Partition and iteration in Algebra: Intuition with linearity. Association of Mathematics Teachers of New York State Journal, 61(1), 10-14. (.PDF)
Wasserman, N. (2010). Inside the UTeach program: Implications for research in mathematics teacher education. Journal of Mathematics Education at Teachers College, 1(1), 12-16. (Source listing)
Refereed Conference Papers and Proceedings
LaPlace, E.^ , Chen, Y.^ , Wasserman, N., & Paoletti, T. (upcoming). Exploring graphical reasoning from revised responses to function composition tasks. In XXX (Eds.), Proceedings of the 26^th Annual
Conference on Research in Undergraduate Mathematics Education (RUME) (pp. XXX). Omaha, NE: RUME.
Lai, Y., Wasserman, N., Strayer, J. F., Casey, S., Weber, K., Fukawa-Connelly, T., & Lischka, A. E. (upcoming). Representing learning in advanced mathematics courses for secondary mathematics
teachers. In XXX (Eds.), Proceedings of the 26^th Annual Conference on Research in Undergraduate Mathematics Education (RUME) (pp. XXX). Omaha, NE: RUME.
Wasserman, N. (2023). Strengthening the role of practice in mathematics teacher education: Opportunities for university mathematics courses. In R. Delgado-Rebolledo and D. Zakaryan (Eds.),
Proceedings of the Congreso Iberoamericano sobre conocimiento especializado del professor de matemáticas (CIMTSK-VI) (pp. 20-30). Valparaíso, Chile: CIMTSK. (Source listing)
Delgado-Rebolledo, R., Zakaryan, D., & Wasserman, N. (2023). Una aproximación a las conexiones entre el MTSK y las prácticas matemáticas pedagógicas. In R. Delgado-Rebolledo and D. Zakaryan (Eds.),
Proceedings of the Congreso Iberoamericano sobre conocimiento especializado del professor de matemáticas (CIMTSK-VI) (pp. 328-335). Valparaíso, Chile: CIMTSK. (Source listing)
Chen, Y., Wasserman, N., & Paoletti, T. (2023). Exploring geometric reasoning with function composition. In S. Cook, B. Katz, and D. Moore-Russo (Eds.), Proceedings of the 25^th Annual Conference on
Research in Undergraduate Mathematics Education (RUME) (pp. 145-153). Omaha, NE: RUME. (Source listing)
Mirin, A., Weber, K., & Wasserman, N. (2020). What is a function? In A. I. Sacristán, J. C. Cortés-Zavala, & P. M. Ruiz-Arias (Eds.), Proceedings of the 42^nd Annual Meeting of the North American
Chapter of the International Group for the Psychology of Mathematics Education (PME-NA) (pp. 1156-1164). Mazatlán, Mexico: PME-NA. (Source listing)
Wasserman, N., Zazkis, R., Baldinger, E., Marmur, O., & Murray, E. (2019). Points of connection to secondary teaching in undergraduate mathematics courses. In A. Weinberg, D. Moore-Russo, H. Soto, &
M. Wawro (Eds.), Proceedings of the 22^nd Annual Conference on Research in Undergraduate Mathematics Education (RUME) (pp. 819-826). Oklahoma City, OK: RUME. (Source listing)
Wasserman, N. (2018). Exploring the secondary teaching of functions in relation to the learning of abstract algebra. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, and S. Brown (Eds.), Proceedings
of the 21st Annual Conference on Research in Undergraduate Mathematics Education (RUME) (pp. 687-694). San Diego, CA: RUME. (Source listing)
Weber, K., Wasserman, N., Mejia-Ramos, J. P., & Fukawa-Connelly, T. (2018). Connecting the study of advanced mathematics to the teaching of secondary mathematics: Implications for teaching inverse
trigonometric functions. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, and S. Brown (Eds.), Proceedings of the 21st Annual Conference on Research in Undergraduate Mathematics Education (RUME)
(pp. 643-651). San Diego, CA: RUME. (Source listing)
Dawkins, P., Inglis, M., & Wasserman, N. (2018). The use(s) of ‘is’ in mathematics. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, and S. Brown (Eds.), Proceedings of the 21st Annual Conference on
Research in Undergraduate Mathematics Education (RUME) (pp. 500-507). San Diego, CA: RUME. (Source listing)
Wasserman, N., Weber, K., & McGuffey, W. (2017). Leveraging real analysis to foster pedagogical practices. [2017 RUME Best Paper Award.] In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, & S. Brown
(Eds.), Proceedings of the 20th Annual Conference on Research in Undergraduate Mathematics Education (RUME) (pp. 1-15). San Diego, CA: RUME. (Source listing)
Baldinger, E., Murray, E., White, D., Broderick, S., & Wasserman, N. (2016). Exploring connections between advanced and secondary mathematics. In M. B. Wood, E. E. Turner, M. Civil, and J. A. Eli
(Eds.), Proceedings of the 38^th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA) (pp. 1633-1640). Tucson, AZ: The
University of Arizona.
Wasserman, N. (2016). Nonlocal mathematical knowledge for teaching. In C. Csíkos, A. Rausch, and J. Szitányi (Eds.), Proceedings of the 40^th Conference of the International Group for the Psychology
of Mathematics Education (PME) (Vol. 4, pp. 379–386). Szeged, Hungary: PME.
Lockwood, E., Wasserman, N., & McGuffey, W. (2016). Classifying combinations: Do students distinguish between different categories of combination problems? In. T. Fukawa-Connelly, N. E. Infante, M.
Wawro, and S. Brown (Eds.), Proceedings of the 19^th Annual Conference on Research in Undergraduate Mathematics Education (RUME) (pp. 296-309). Pittsburgh, PA: RUME. (Source listing)
Murray, E., Baldinger, E., Wasserman, N., Broderick, S., Cofer, T., White, D., & Stanish, K. (2015). Exploring connections between advanced and secondary mathematics. In Bartell, T.G., Bieda, K.N.,
Putnam, R.T., Bradfield, K., & Dominguez, H. (Eds.), Proceedings of the 37^th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education
(PME-NA)(pp. 1368-1376). East Lansing, MI: Michigan State University.
Wasserman, N., & Mamolo, A. (2015). Knowledge for teaching: Horizons and mathematical structure. In T. Fukawa-Connelly, N. Infante, K. Keene, and M. Zandieh (Eds.), Proceedings of the 18^th Annual
Conference on Research in Undergraduate Mathematics Education (RUME) (pp. 1032-1036). Pittsburgh, PA: RUME. (Source listing)
Wasserman, N., Villanueva, M., Mejia-Ramos, J.P., & Weber, K. (2015). Secondary mathematics teachers’ perceptions of real analysis in relation to their teaching practices. In T. Fukawa-Connelly, N.
Infante, K. Keene, and M. Zandieh (Eds.), Proceedings of the 18^th Annual Conference on Research in Undergraduate Mathematics Education (RUME) (pp. 1037-1040). Pittsburgh, PA: RUME. (Source listing)
Wasserman, N., Mamolo, A., Ribeiro, C.M., & Jakobsen, A. (2014). Exploring horizons of knowledge for teaching. In Liljedahl, P., Nicol, C., Oesterle, S., & Allan, D. (Eds.) Proceedings of the Joint
Meeting of PME 38 and PME-NA 36 (Vol. 1, p. 247). Vancouver, Canada: PME.
Wasserman, N. (2013). Exploring teachers' categorizations for and conceptions of combinatorial problems. In S. Reeder & G. Matney (Eds.), Proceedings of the 40th Annual Meeting of the Research
Council on Mathematics Learning (p. 145-154), Tulsa, OK. (.PDF)
Wasserman, N., Norris, S., & Carr, T. (2013). Comparing a "flipped" instructional model in an undergraduate Calculus III course. In. S. Brown, G. Karakok, K.H. Roh, and M. Oehrtman (Eds.),
Proceedings of the 16th Annual Conference on Research in Undergraduate Mathematics Education (Vol. 2, pp. 652-655), Denver, CO. (.PDF)
Wasserman, N., & Ham, E. (2012). Attributes of good mathematics teaching: When are they learned? Conference Proceedings for the International Congress on Mathematics Education (ICME-12) (p. 7843).
Seoul, Korea: ICME-12.
Professional Resources, Reviews, and Other Scholarship
Wasserman, N., Holbert, N., & Blikstein, P. (2020). Will the coronavirus infect education, too? New York Daily News, 8 April 2020. Op ed. (Source listing)
Baldinger, E., Broderick, S., Murray, E., Wasserman, N., & White, D. (2015). Connections between abstract algebra and high school algebra: A few connections worth exploring. American Mathematical
Society (AMS) Blogs: On Teaching and Learning Mathematics (December 10, 2015). (Source listing)
Wasserman, N. (2015). Review of the book Getting to the common core: Using research-based strategies that empower students to own their own achievement, by S. L. Spencer & S. Vavra. Teachers College
Record. (Source listing)
Wasserman, N., Mamolo, A., Ribeiro, C.M., & Jakobsen, A. (2015). Discussion Group 2: Exploring horizons of knowledge for teaching. International Group for the Psychology of Mathematics Education
(PME) Newsletter, December 2014/January 2015, 7-10.
Zachary, S. C., Zannou, Y., Basaraba, D., Wasserman, N., Hill, S., & Ketterlin-Geller, L. (2013). Texas Algebra Ready (TXAR): Learning Progressions Development (Tech. Rep. No. 13-03). Dallas, TX:
Southern Methodist University, Research in Mathematics Education.
Wasserman, N. (2011). Bending steel. In H. Gould, D. Murray & A. Sanfratello (Eds.), Teachers College Mathematical Modeling Handbook (pp. 75-82). Bedford, MA: The Consortium for Mathematics and Its
Applications (COMAP). (Source listing)
Wasserman, N. (2011). A bit of information. In H. Gould, D. Murray & A. Sanfratello (Eds.), Teachers College Mathematical Modeling Handbook (pp. 83-92). Bedford, MA: The Consortium for Mathematics
and Its Applications (COMAP). (Source listing)
Wasserman, N. (2010). Reader reflections: A fourth way to break a stick. Mathematics Teacher, 104(1), 9-10. (Source listing)
Related Articles | {"url":"https://www.tc.columbia.edu/faculty/nhw2108/","timestamp":"2024-11-04T02:32:55Z","content_type":"text/html","content_length":"139300","record_id":"<urn:uuid:ef6c8da1-13fa-41e1-bebd-a501ff72f919>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00195.warc.gz"} |
Radiation Absorption Experimentation
We have seen how to look at background radiation and what it can teach us about the statistical analysis of radiation data, and we have looked at a few commonly available radioactive sources and why
the variety isotopes is so limited. This experiment should allow one to look more deeply into the nature of the radiation itself. Any text concerning ionizing radiation will explain the three
particles, or types of radiation involved 1) Alpha: two protons and two neutrons, essentially a helium nucleus, 2) Beta: a single electron for each beta decay, and 3) Gamma: a "ray" or photon
particle defined by its energy state more than by a physical characteristic.
Most texts explaining radiation will include a required diagram like that shown at the upper left. Here one sees alpha particles stopped by a sheet of paper while a sheet of aluminum will stop a beta
particle. Only a substantial thickness of lead can stop the mighty gamma. This diagram implies (incorrectly) a sort of step-wise relation and suggests, or at least suggested to me, an interesting
experiment. Measure the total CPM of a sample, then re-measure with a sheet of paper to block out the alpha, and finally re-re-measure with a sheet of aluminum to block out alpha plus beta. With this
data one would be able (were it true) to extract the counts for alpha, beta, and gamma by difference.
If you try this you will quickly discover things do not work this way. Every measurement with a different absorber will yield a different count. Being systematic here will pay dividends. Getting
organized, I began by building the inventory of absorbers in the photo below. These range from a single piece of light tissue paper to a 1/4 inch slab of lead. I was seeking to assemble a set of
absorbers with a very wide variety densities and masses. I figured the greater the mass, the greater the absorption. My materials are included in the table at left giving the density carefully
measured and/or researched. I also give the atomic number although I did not make use of this information in my experiment.
The setup consists of the Geiger counter laying flat on a table with an open wire platform one centimeter above to hold the radioactive sample. Between the platform and the counter was a space into
which I placed the absorbers. I ran the experiment twice, once with a uranium sample and once with a thorium sample. At the bottom of the page is a detail listing of the experimental program
consisting of a full 42 measurements with different absorbers and absorber combinations. Each measurement was integrated over 15 minutes so, from the background study, I expect my precision to be
plus or minus 5%. To get greater precision would require too much time. As things turned out, this works quite well.
I will go straight to the results before discussion the theoretical underpinnings. The following chart plots all the measurements in terms of CPM versus absorber mass. The mass is a simple
calculation based on the density, thickness, and collecting area for each measurement, and tabulated at the bottom table. It is immediately apparent that two separate trends in the data are present,
this got interesting! While my interpretations and conclusions are plastered all over the plot, please bear with me for an explanation for how I got there. Theoretical underpinnings will come next.
The plot above shows the main trends with an inset for a more detail look at the corner near the Y-intercept. I interpret the two trends as those for Gamma and Beta radiation. My instrument and
measurement setup is very challenged for Alpha radiation, although, pushing things, I suggest some 3% just the same.
Below I have a table for the U-238 decay chain and below that a chart I found a few years back in a spreadsheet for which I neglected to note the reference. Still, this chart, and the U-238 decay
chain table tell me that uranium decay should produce all three types of radiation with gamma at 12%, beta at 42% and alpha at 46% of the total. As my setup detects alpha poorly, I have re-normalized
this to show, without alpha, the expected radiation should be some 23% gamma and 77% beta. My own data suggests 23% gamma, 74% beta with a possible 3% alpha, pretty close!
A note about my chart: My plot crosses CPM against Absorber Mass. This is basically an Intensity Attenuation Plot although my units are based on my primitive instrumentation. Note that the regression
formulas on the plot (gamma for example: y = 17164 e-0.006x) follow the pattern Y = Yo * exp (-zX). Where Y is the dependant variable CPM, Yo is the Y-intercept (no absorber), X is the absorber mass
in grams, and z is a constant unique for that relation.
This pattern is exactly the same as for the standard Linear Attenuation Equation. An excellent reference for this, with a derivation of the formula, may be found in wikibooks (Attenuation of
Gamma-Rays https://en.wikibooks.org/wiki/Basic_Physics_of_Nuclear_Medicine/Attenuation_of_Gamma-Rays).
The equation for attenuation is:
Ix = Io exp (-uX)
Ix = intensity at any point along the Mass X-axis.
Io = the initial (no absorber) intensity = Y-intercept.
u = the Linear Attenuation Coefficient
X = the absorber mass.
This equation is actually meant for use with absorbers of a single given element (density) varying in thickness. Clearly, varying the thickness of absorbers of a given density essentially means
varying the mass. The importance stems from the fact that radiation responds differently to different absorber materials. The atomic mass (Z) of the material impacts the results and of course, my
absorbers have a wide range of atomic masses, (see table above).
This is why I chose to plot CPM (intensity) versus total absorber mass. Overall, this necessary blunder does not prevent interpretation or distinction between beta and gamma radiation. The absorber
materials used are plotted with different colors and slight differences in the main trends between materials are apparent. These differences are certainly due to differing absorber Z values used. The
detail insert shows this problem much more clearly. The paper trend and the wax paper trend are very different from each other and this also contributes to my difficulty distinguishing an alpha
particle trend (if present).
My Attenuation Coefficients do not have standard units but are as follows:
Gamma: u = 0.006
Beta: u = 0.22
Attenuation coefficients are particle energy specific and these numbers would apply to the average particle energy of my sample. Gamma rays are clearly lightly attenuated and thus penetrate a lot of
lead. Still, of the overall 74000 counts (minus 50cpm background) only 17114 counts can be attributed to gamma radiation.
Given the information I have on absorber Z, density, mass, and attenuation, I suspect there may be enough to calculate the average particle energy as well, but I have not found the "right" reference
to go farther.
This has been great good fun. Now to get serious and try to do something useful with Muons next.
kjs 2017 | {"url":"http://lessmiths.com/radiation/absorption.shtml","timestamp":"2024-11-10T19:00:21Z","content_type":"text/html","content_length":"8372","record_id":"<urn:uuid:95a67ec7-a171-43ce-b03c-3843ead981fa>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00116.warc.gz"} |
which is the closest approximation of
Pure mathematics is the world's best game. It is more absorbing than chess, more of a gamble than poker, and lasts longer than Monopoly. It's free. It can be played anywhere - Archimedes did it in a
Thanks m4 maths for helping to get placed in several companies. I must recommend this website for placement preparations. | {"url":"https://m4maths.com/9843-Of-the-following-which-is-the-closest-approximation-of-50-2-0-49-199-8.html","timestamp":"2024-11-13T07:48:31Z","content_type":"text/html","content_length":"69478","record_id":"<urn:uuid:6ba61a56-d369-4b9e-abb0-8b817f609439>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00071.warc.gz"} |
Morphing planar graph drawings with a polynomial number of steps
In 1944, Cairns proved the following theorem: given any two straight-line planar drawings of a triangulation with the same outer face, there exists a morph (i.e., a continuous transformation) between
the two drawings so that the drawing remains straight-line planar at all times. Cairns's original proof required exponentially many morphing steps. We prove that there is a morph that consists of O(n
^2) steps, where each step is a linear morph that moves each vertex at constant speed along a straight line. Using a known result on compatible triangulations this implies that for a general planar
graph G and any two straight-line planar drawings of G with the same embedding, there is a morph between the two drawings that preserves straight-line planarity and consists of O(n^4) steps.
Publication series
Name Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Other 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013
Country/Territory United States
City New Orleans, LA
Period 1/6/13 → 1/8/13
All Science Journal Classification (ASJC) codes
• Software
• General Mathematics
Dive into the research topics of 'Morphing planar graph drawings with a polynomial number of steps'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/morphing-planar-graph-drawings-with-a-polynomial-number-of-steps","timestamp":"2024-11-03T17:33:09Z","content_type":"text/html","content_length":"50560","record_id":"<urn:uuid:b8239707-adfd-4ecf-a2c3-c147bf7294dd>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00045.warc.gz"} |
Additive Noise Mechanisms
Additive Noise Mechanisms#
This notebook documents the variations on additive noise mechanisms in OpenDP:
• Distribution: Laplace vs. Gaussian
• Support: float vs. integer
• Domain: scalar vs. vector
• Bit-depth
Any constructors that have not completed the proof-writing and vetting process may still be accessed if you opt-in to “contrib”. Please contact us if you are interested in proof-writing. Thank you!
import opendp.prelude as dp
Distribution: Laplace vs. Gaussian#
The Laplace mechanism is a ubiquitous algorithm in the DP ecosystem that is used to privatize an aggregate, like a sum or mean.
An instance of the Laplace mechanism is captured by a measurement containing the following five elements:
Elements of a Laplace Measurement
1. We first define the function \(f(\cdot)\), that applies the Laplace mechanism to some argument \(x\). This function simply samples from the Laplace distribution centered at \(x\), with a fixed
noise scale.
\[f(x) = Laplace(\mu=x, b=scale)\]
2. Importantly, \(f(\cdot)\) is only well-defined for any finite float input. This set of permitted inputs is described by the input domain (denoted AtomDomain<f64>).
3. The Laplace mechanism has a privacy guarantee in terms of epsilon. This guarantee is represented by a privacy map, a function that computes the privacy loss \(\epsilon\) for any choice of
sensitivity \(\Delta\).
\[map(\Delta) = \Delta / scale <= \epsilon\]
4. This map only promises that the privacy loss will be at most \(\epsilon\) if inputs from any two neighboring datasets may differ by no more than some quantity \(\Delta\) under the absolute
distance input metric (AbsoluteDistance<f64>).
5. We similarly describe units on the output (\(\epsilon\)) via the output measure (MaxDivergence<f64>).
The make_laplace constructor function returns the equivalent of the Laplace measurement described above.
# call the constructor to produce the measurement `base_lap`
input_space = dp.atom_domain(T=float), dp.absolute_distance(T=float)
base_lap = dp.m.make_laplace(*input_space, scale=2.)
# invoke the measurement on some aggregate x, to sample Laplace(x, 1.) noise
aggregated = 0.
print("noisy aggregate:", base_lap(aggregated))
# we must know the sensitivity of `aggregated` to determine epsilon
sensitivity = 1.
print("epsilon:", base_lap.map(d_in=sensitivity))
noisy aggregate: 0.8418214435677124
epsilon: 0.5
The analogous constructor for gaussian noise is make_gaussian:
# call the constructor to produce the measurement `gauss`
input_space = dp.atom_domain(T=float), dp.absolute_distance(T=float)
gauss = dp.m.make_gaussian(*input_space, scale=2.)
# invoke the measurement on some aggregate x, to sample Gaussian(x, 1.) noise
aggregated = 0.
print("noisy aggregate:", gauss(aggregated))
# we must know the sensitivity of `aggregated` to determine epsilon
sensitivity = 1.
print("rho:", gauss.map(d_in=sensitivity))
noisy aggregate: 2.322023365796082
rho: 0.125
Notice that base_lap measures privacy with epsilon (in the MaxDivergence measure), and base_gauss measures privacy with rho (in the ZeroConcentratedDivergence measure).
Support: Float vs. Integer#
There are also discrete analogues of the continuous Laplace and Gaussian measurements. The continuous measurements accept and emit floats, while the discrete measurements accept and emit integers.
Measurements with distributions supported on the integers expect integer sensitivities by default.
make_laplace on a discrete support is equivalent to the geometric mechanism:
# call the constructor to produce the measurement `base_discrete_lap`
input_space = dp.atom_domain(T=int), dp.absolute_distance(T=int)
base_discrete_lap = dp.m.make_laplace(*input_space, scale=1.)
# invoke the measurement on some integer aggregate x, to sample DiscreteLaplace(x, 1.) noise
aggregated = 0
print("noisy aggregate:", base_discrete_lap(aggregated))
# in this case, the sensitivity is integral:
sensitivity = 1
print("epsilon:", base_discrete_lap.map(d_in=sensitivity))
noisy aggregate: 1
epsilon: 1.0
make_gaussian on a discrete support is the analogous measurement for Gaussian noise:
# call the constructor to produce the measurement `base_discrete_gauss`
input_space = dp.atom_domain(T=int), dp.absolute_distance(T=int)
base_discrete_gauss = dp.m.make_gaussian(*input_space, scale=1.)
# invoke the measurement on some aggregate x, to sample DiscreteGaussian(x, 1.) noise
aggregated = 0
print("noisy aggregate:", base_discrete_gauss(aggregated))
# we must know the sensitivity of `aggregated` to determine epsilon
sensitivity = 1
print("rho:", base_discrete_gauss.map(d_in=sensitivity))
noisy aggregate: -1
rho: 0.5
The continuous mechanisms use these discrete samplers internally. More information on this can be found at the end of this notebook.
Domain: Scalar vs. Vector#
Measurements covered thus far have accepted scalar inputs and emitted scalar outputs, and sensitivities have been expressed in terms of the absolute distance.
The noise addition mechanisms can similarly operate over metric spaces consisting of vectors, and where the distance between any two vectors is computed via the L1 or L2 distance.
# call again, but this time indicate that the measurement should operate over a vector domain
input_space = dp.vector_domain(dp.atom_domain(T=float)), dp.l1_distance(T=float)
base_lap_vec = dp.m.make_laplace(*input_space, scale=1.)
aggregated = 1.
# If we try to pass the wrong data type into our vector laplace measurement,
# the error shows that our float argument should be a vector of floats.
print("noisy aggregate:", base_lap_vec(aggregated))
except TypeError as e:
# The error messages will often point to a discussion page with more info.
Expected type is Vec<f64> but input data is not a list.
# actually pass a vector-valued input, as expected
aggregated = [0., 2., 2.]
print("noisy aggregate:", base_lap_vec(aggregated))
noisy aggregate: [-1.249922018362259, 1.5557446543831857, -1.620935690993615]
The resulting measurement expects sensitivity in terms of the appropriate Lp-distance: the vector Laplace measurement expects sensitivity in terms of an "l1_distance(T=f64)", while the vector
Gaussian measurement expects a sensitivity in terms of an "l2_distance(T=f64)".
sensitivity = 1.
print("epsilon:", base_lap_vec.map(d_in=sensitivity))
The documentation for each constructor also reflects the relationship between D and the resulting input metric in a table:
Help on function make_laplace in module opendp.measurements:
make_laplace(input_domain: opendp.mod.Domain, input_metric: opendp.mod.Metric, scale, k=None, QO: Union[ForwardRef('RuntimeType'), str, Type[Union[List[Any], Tuple[Any, Any], int, float, str, bool]], Tuple[ForwardRef('RuntimeTypeDescriptor'), ...], _GenericAlias, types.GenericAlias] = 'float') -> opendp.mod.Measurement
Make a Measurement that adds noise from the Laplace(`scale`) distribution to the input.
Valid inputs for `input_domain` and `input_metric` are:
| `input_domain` | input type | `input_metric` |
| ------------------------------- | ------------ | ---------------------- |
| `atom_domain(T)` (default) | `T` | `absolute_distance(T)` |
| `vector_domain(atom_domain(T))` | `Vec<T>` | `l1_distance(T)` |
Internally, all sampling is done using the discrete Laplace distribution.
[make_laplace in Rust documentation.](https://docs.rs/opendp/latest/opendp/measurements/fn.make_laplace.html)
* [GRS12 Universally Utility-Maximizing Privacy Mechanisms](https://theory.stanford.edu/~tim/papers/priv.pdf)
* [CKS20 The Discrete Gaussian for Differential Privacy](https://arxiv.org/pdf/2004.00010.pdf#subsection.5.2)
**Supporting Elements:**
* Input Domain: `D`
* Output Type: `D::Carrier`
* Input Metric: `D::InputMetric`
* Output Measure: `MaxDivergence<QO>`
:param input_domain: Domain of the data type to be privatized.
:type input_domain: Domain
:param input_metric: Metric of the data type to be privatized.
:type input_metric: Metric
:param scale: Noise scale parameter for the Laplace distribution. `scale` == standard_deviation / sqrt(2).
:param k: The noise granularity in terms of 2^k, only valid for domains over floats.
:param QO: Data type of the output distance and scale. `f32` or `f64`.
:type QO: :py:ref:`RuntimeTypeDescriptor`
:rtype: Measurement
:raises TypeError: if an argument's type differs from the expected type
:raises UnknownTypeException: if a type argument fails to parse
:raises OpenDPException: packaged error from the core OpenDP library
The discrete Gaussian mechanism allows for the type of the input sensitivity to be a float. This is because there is often a square root in the sensitivity calculations for vector-valued queries.
# call again, but this time indicate that the measurement should operate over a vector domain
input_space = dp.vector_domain(dp.atom_domain(T=int)), dp.l2_distance(T=float)
base_gauss_vec = dp.m.make_gaussian(*input_space, scale=1.)
Bit depth#
By default, all floating-point data types default to 64-bit double-precision (denoted "f64"), and all integral data types default to 32-bit (denoted "i32"). The atomic data type expected by the
function and privacy units can be further configured to operate over specific bit-depths by explicitly specifying "f32" instead of "float", or "i64" instead of "int".
# explicitly specify that the...
# * computation should be handled with 32-bit integers, and the
# * privacy analysis be conducted with 64-bit floats
base_discrete_lap_i32 = dp.m.make_laplace(
dp.atom_domain(T="i32"), dp.absolute_distance(T="i32"),
scale=1., QO="f64"
More information on acceptable data types can be found in the Utilities > Typing section of the User Guide.
Desideratum: Floating-Point Granularity#
The “continuous” Laplace and Gaussian measurements convert their float arguments to a rational representation, and then add integer noise to the numerator via the respective discrete distribution. In
the OpenDP Library’s default configuration, this rational representation of a float is exact. Therefore the privacy analysis is as tight as if you were to sample truly continuous noise and then
postprocess by rounding to the nearest float.
For most use-cases the sampling algorithm is sufficiently fast when the rational representation is exact. That is, when noise is sampled with a granularity of \(2^{-1074}\), the same granularity as
the distance between subnormal 64-bit floats. However, the granularity can be adjusted to \(2^k\), for some choice of k, for a faster runtime. Adjusting this parameter comes with a small penalty to
the sensitivity (to account for rounding to the nearest rational), and subsequently, to the privacy parameters.
The following plot shows the resulting distribution for some large choices of k:
import numpy as np
import matplotlib.pyplot as plt
num_samples = 10_000
space = dp.vector_domain(dp.atom_domain(T=float), num_samples), dp.l1_distance(T=float)
fig, axes = plt.subplots(nrows=1, ncols=3, figsize=(10, 3))
for axis, k in zip(axes, [1, 0, -1]):
base_lap_vec = dp.m.make_laplace(*space, scale=1., k=k)
support, counts = np.unique(base_lap_vec([0.] * num_samples), return_counts=True)
axis.bar(support, counts / num_samples)
axis.set_xticks([-2, 0, 2])
axis.set_title(f"k = {k}; gap = {2**k}")
fig.suptitle('make_laplace for different choices of k', y=0.95);
The distribution becomes increasingly smooth as k approaches the default value (-1074).
The privacy map still adds a penalization when the sensitivity is zero. The following table uses this behavior to show the increase in epsilon for some choices of k:
k = [-1074, -1073, -100, -1, 0, 1]
space = dp.atom_domain(T=float), dp.absolute_distance(T=float)
ε_penalty = [dp.m.make_laplace(*space, scale=1., k=k_i).map(d_in=0.) for k_i in k]
detail = ["no penalty", "~min float", "~2^-100", "~2^-1", "~2^0", "~2^1"]
import pandas as pd
pd.DataFrame({"k": k, "ε penalty": ε_penalty, "detail": detail}).set_index("k")
│ │ ε penalty │ detail │
│ k │ │ │
│-1074│0.000000e+00 │no penalty │
│-1073│4.940656e-324 │~min float │
│-100 │7.888609e-31 │~2^-100 │
│ -1 │5.000000e-01 │~2^-1 │
│ 0 │1.000000e+00 │~2^0 │
│ 1 │2.000000e+00 │~2^1 │ | {"url":"https://docs.opendp.org/en/stable/api/user-guide/measurements/additive-noise-mechanisms.html","timestamp":"2024-11-14T10:23:42Z","content_type":"text/html","content_length":"70358","record_id":"<urn:uuid:65f6ef9f-321e-463f-a438-cbc07d01a229>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00557.warc.gz"} |
Gas-Charged Accumulator (IL)
Gas-charged accumulator in an isothermal liquid network
Since R2020a
Simscape / Fluids / Isothermal Liquid / Tanks & Accumulators
The Gas-Charged Accumulator (IL) block represents a gas-charged accumulator in an isothermal liquid network. The accumulator consists of a precharged gas chamber and a liquid chamber. The chambers
are separated by a bladder, a piston, or any kind of a diaphragm.
As the liquid pressure at the accumulator inlet becomes greater than the precharge pressure, liquid enters the accumulator and compresses the gas through a polytropic process. A decrease in the
liquid pressure causes the gas to decompress and discharge stored liquid into the system. The separator motion is restricted by a hard stop when the liquid volume is zero and when the liquid volume
is at the liquid chamber capacity. The liquid chamber capacity is the total accumulator volume minus the minimum gas volume.
Inlet liquid resistance, and separator properties such as inertia and damping are not modeled. The flow rate is positive if liquid flows into the accumulator.
This diagram represents a gas-charged accumulator. The total accumulator volume, V[T], is divided into the liquid chamber on the left and the gas chamber on the right by the vertical separator. The
distance between the left side and the separator defines the liquid volume, V[L]. The distance between the right side and the separator defines the gas volume, V[G]. The liquid chamber capacity, V
[C], is less than the total accumulator volume, so that the gas volume never becomes zero:
$\begin{array}{l}{V}_{L}={V}_{T}-{V}_{G}\\ {V}_{C}={V}_{T}-{V}_{dead}\end{array}$
• V[T] is the total volume of the accumulator, including the liquid chamber and the gas chamber.
• V[L] is the volume of the liquid in the accumulator.
• V[G] is the volume of the gas in the accumulator.
• V[C] is the liquid chamber capacity.
• V[dead] is the gas chamber dead volume, a small portion of the gas chamber that remains filled with gas when the liquid chamber is at capacity.
The hard stop contact pressure is modeled with a stiffness term and a damping term. The relationship of the gas pressure and gas volume between the current state and the precharge state is polytropic
and pressure is balanced at the separator:
• p[G] is the gas pressure in the gas chamber.
• p[pr] is the pressure in the gas chamber when the liquid chamber is empty
• k[sh] is the specific heat ratio (adiabatic index).
Conservation of Mass
Conservation of mass is represented by the following equation.
$\left\{\begin{array}{l}{\stackrel{˙}{p}}_{I}\frac{{d}_{{\rho }_{I}}}{{d}_{{p}_{I}}}{V}_{L}+{\rho }_{I}{\stackrel{˙}{V}}_{L}={\stackrel{˙}{m}}_{A}\text{,}compressibility\text{}on\\ {\rho }_{I}{\
• p[I] is the liquid pressure in the liquid chamber, which is equal to the pressure at the accumulator inlet.
• $\stackrel{˙}{m}$[A] is the mass flow rate of liquid coming into port A.
• ρ[I] is the density of the liquid in the liquid chamber.
${\stackrel{˙}{V}}_{L}=\left\{\begin{array}{c}\frac{{\stackrel{˙}{p}}_{I}}{{k}_{sh}\frac{{p}_{G}}{{V}_{G}}}\text{,}if\text{}0\text{}<{V}_{L}<{V}_{C}\\ \frac{{\stackrel{˙}{p}}_{I}}{{k}_{sh}\frac{{p}_
where K[stiff] is the hard-stop stiffness coefficient.
Conservation of Momentum
Conservation of momentum is represented by:
where p[HS] is the hard-stop contact pressure.
${\text{p}}_{HS}=\left\{\begin{array}{c}\left({V}_{L}-{V}_{C}\right){K}_{stiff},\text{}if\text{}{V}_{L}\ge {V}_{C}\\ {V}_{L}{K}_{stiff,\text{}if\text{}{V}_{L}\text{}\le \text{0}}\\ 0,\text{}else\end
To set the priority and initial target values for the block variables prior to simulation, use the Initial Targets section in the block dialog box or Property Inspector. For more information, see Set
Priority and Initial Target for Block Variables.
Nominal values provide a way to specify the expected magnitude of a variable in a model. Using system scaling based on nominal values increases the simulation robustness. Nominal values can come from
different sources, one of which is the Nominal Values section in the block dialog box or Property Inspector. For more information, see Modify Nominal Values for a Block Variable.
A — Liquid port
isothermal liquid
Isothermal liquid port associated with the accumulator inlet. The flow rate is positive if liquid flows into the accumulator.
Total accumulator volume — Accumulator volume
8e-3 m^3 (default) | positive scalar
Total volume of the accumulator, including the liquid chamber and the gas chamber. It is the sum of the liquid chamber capacity and the minimum gas volume.
Minimum gas volume — Gas chamber dead volume
4e-5 m^3 (default) | positive scalar
Gas chamber dead volume, which is defined as a small portion of the gas chamber that remains filled with gas when the liquid chamber is filled to capacity. The block requires a nonzero value to avoid
divide-by-zero when the liquid chamber is at capacity.
Precharge pressure — Gas chamber pressure
1.101325 MPa (default) | positive scalar
Pressure in the gas chamber when the liquid chamber is empty.
Specific heat ratio — Specific heat ratio
1.4 (default) | positive scalar
Specific heat ratio (adiabatic index). To account for heat exchange, set it to a value normally between 1 and 2, depending on the properties of the gas in the gas chamber. For dry air at 20°C, this
value is 1 for an isothermal process or 1.4 for an adiabatic (and isentropic) process.
Hard stop stiffness coefficient — Proportionality constant
1e4 MPa/m^3 (default) | positive scalar
Proportionality constant of the hard-stop contact pressure with respect to the liquid volume penetrated into the hard stop. The hard stops are used to restrict the liquid volume between zero and
liquid chamber capacity.
Fluid dynamic compressibility — Fluid compressibility
on (default) | off
Whether to model any change in fluid density due to fluid compressibility. When you select Fluid dynamic compressibility, changes due to the mass flow rate into the block are calculated in addition
to density changes due to changes in pressure. In the Isothermal Liquid Library, all blocks calculate density as a function of pressure.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
Version History
Introduced in R2020a
See Also | {"url":"https://kr.mathworks.com/help/hydro/ref/gaschargedaccumulatoril.html","timestamp":"2024-11-04T01:21:10Z","content_type":"text/html","content_length":"92906","record_id":"<urn:uuid:9ef16fa9-4d37-4643-94f9-375441f375b0>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00019.warc.gz"} |
Measuring Investment Risk – Calculating Returns
In addition to Computational Investing, I’ve also signed up for the Introduction to Computational Finance class through Coursera – it’s a similar topic, but a different point of view. Instead of
assuming you want to be a hedge fund manager, it goes into calculating returns, risk of an investment, and building a portfolio. This is more “normal” stuff that any investor should know – but it has
a very mathematical bent (several proofs, etc). The class is self-paced through November, so it’s not too late to join if you’re interested.
The first two weeks are on calculating simple, annual effective rate, and continuously compounded returns and the probability background needed to complete the class. One of the closing topics of
week two was measuring investment risk. It’s generally accepted that standard deviation (σ) is an easy to calculate measure of risk. This tells you how far values deviate from the expected result –
the mean (μ) – or the simple/continuous rate of return, so you know how “wild” the investment can be. A larger standard deviation implies a larger risk to the investment. In investing, a larger mean
(or expected return value) also tends to imply a larger standard deviation because people expect to take more risk for a larger return. We haven’t gotten to actually calculating the standard
deviation yet – although I think I know how.
Calculating Returns
The first value to calculate is a return over a time period.
$latex R$: Return
$latex P_{t}$: Price at time $latex t$
$latex P_{t-1}$: Price at time $latex t-1$
$latex R = \frac{P_{t}-P_{t-1}}{P_{t-1}}$
$latex R $ can be any time period, but we’ve been using monthly for the most part in class.
Calculating Portfolio Returns
Not only do we need to worry about the returns for a single investment, but we also need to calculate the return for our entire portfolio for it to be useful. The return for a portfolio is pretty
easy, it’s a weighted average based on the initial total investment. If asset A has a return of 5%, and asset B has a return of 3%, and we spend $3000 on asset A, and $7000 on asset B, the portfolio
rate of return is:
$latex R_{p,t} = .30*R_{A} + .70*R_{B} = .30*0.05 + .70*0.03 = 0.036$
or 3.6%
Disclaimer: I’m writing these posts as a way to solidify my understanding of class materials, they may not be completely correct – and I welcome any corrections.
2 thoughts on “Measuring Investment Risk – Calculating Returns”
1. cashrebel
That sounds like a fascinating course to take through Coursera. I signed up for a class there last year but never followed through. Learning the math behind investing will no doubt make you a
smarter investor.
Reply ↓
1. Mom Post author
It has been really enjoyable so far, anyway. I’m really learning a lot. I love having the opportunity to take university level classes for free.
Reply ↓ | {"url":"http://3isplenty.com/measuring-investment-risk/","timestamp":"2024-11-04T17:38:35Z","content_type":"text/html","content_length":"44331","record_id":"<urn:uuid:908c471c-2430-4e8b-ba00-8295763e54d1>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00006.warc.gz"} |
eet to Arpent
Feet to Arpent Converter
β Switch toArpent to Feet Converter
How to use this Feet to Arpent Converter π €
Follow these steps to convert given length from the units of Feet to the units of Arpent.
1. Enter the input Feet value in the text field.
2. The calculator converts the given Feet into Arpent in realtime β using the conversion formula, and displays under the Arpent label. You do not need to click any button. If the input changes,
Arpent value is re-calculated, just like that.
3. You may copy the resulting Arpent value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Feet to Arpent?
The formula to convert given length from Feet to Arpent is:
Length[(Arpent)] = Length[(Feet)] / 191.99999984784384
Substitute the given value of length in feet, i.e., Length[(Feet)] in the above formula and simplify the right-hand side value. The resulting value is the length in arpent, i.e., Length[(Arpent)].
Calculation will be done after you enter a valid input.
Consider that a luxury yacht has a beam width of 60 feet.
Convert this width from feet to Arpent.
The length in feet is:
Length[(Feet)] = 60
The formula to convert length from feet to arpent is:
Length[(Arpent)] = Length[(Feet)] / 191.99999984784384
Substitute given weight Length[(Feet)] = 60 in the above formula.
Length[(Arpent)] = 60 / 191.99999984784384
Length[(Arpent)] = 0.3125
Final Answer:
Therefore, 60 ft is equal to 0.3125 arpent.
The length is 0.3125 arpent, in arpent.
Consider that a skyscraper's floor-to-ceiling height is 15 feet.
Convert this height from feet to Arpent.
The length in feet is:
Length[(Feet)] = 15
The formula to convert length from feet to arpent is:
Length[(Arpent)] = Length[(Feet)] / 191.99999984784384
Substitute given weight Length[(Feet)] = 15 in the above formula.
Length[(Arpent)] = 15 / 191.99999984784384
Length[(Arpent)] = 0.0781250000619125
Final Answer:
Therefore, 15 ft is equal to 0.0781250000619125 arpent.
The length is 0.0781250000619125 arpent, in arpent.
Feet to Arpent Conversion Table
The following table gives some of the most used conversions from Feet to Arpent.
Feet (ft) Arpent (arpent)
0 ft 0 arpent
1 ft 0.00520833334 arpent
2 ft 0.01041666667 arpent
3 ft 0.01562500001 arpent
4 ft 0.02083333335 arpent
5 ft 0.02604166669 arpent
6 ft 0.03125000002 arpent
7 ft 0.03645833336 arpent
8 ft 0.0416666667 arpent
9 ft 0.04687500004 arpent
10 ft 0.05208333337 arpent
20 ft 0.1042 arpent
50 ft 0.2604 arpent
100 ft 0.5208 arpent
1000 ft 5.2083 arpent
10000 ft 52.0833 arpent
100000 ft 520.8333 arpent
A foot (symbol: ft) is a unit of length used in the United States, the United Kingdom, and Canada. One foot is equal to 0.3048 meters.
The foot originated from various units used in ancient civilizations. Its current definition is based on the international agreement of 1959, which standardized it to exactly 0.3048 meters.
Feet are commonly used to measure height, length, and short distances. Despite the global shift to the metric system, the foot remains in use in these countries.
An arpent is a historical unit of length used primarily in French-speaking regions and in land measurement. One arpent is approximately equivalent to 192.75 feet or 58.66 meters.
The arpent was used in various regions, including France and the former French colonies, to measure land and property. Its length could vary slightly depending on the specific region and historical
Arpents were used in land surveying and agriculture, particularly in historical and regional contexts. Although less common today, the unit provides historical insight into land measurement practices
and regional variations in measurement standards.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Feet to Arpent in Length?
The formula to convert Feet to Arpent in Length is:
Feet / 191.99999984784384
2. Is this tool free or paid?
This Length conversion tool, which converts Feet to Arpent, is completely free to use.
3. How do I convert Length from Feet to Arpent?
To convert Length from Feet to Arpent, you can use the following formula:
Feet / 191.99999984784384
For example, if you have a value in Feet, you substitute that value in place of Feet in the above formula, and solve the mathematical expression to get the equivalent value in Arpent. | {"url":"https://convertonline.org/unit/?convert=feet-arpents","timestamp":"2024-11-15T03:51:11Z","content_type":"text/html","content_length":"90704","record_id":"<urn:uuid:6443026a-488f-4326-b4ba-5fa619e0c45b>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00345.warc.gz"} |
You are here
Date Issued:
The area of focus for this research is the Stochastic Resource Constrained Project Scheduling Problem (SRCPSP) with Stochastic Task Insertion (STI). The STI problem is a specific form of the
SRCPSP, which may be considered to be a cross between two types of problems in the general form: the Stochastic Project Scheduling Problem, and the Resource Constrained Project Scheduling
Problem. The stochastic nature of this problem is in the occurrence/non-occurrence of tasks with deterministic duration. Researchers Selim (2002) and Grey (2007) laid the groundwork for the
research on this problem. Selim (2002) developed a set of robustness metrics and used these to evaluate two initial baseline (predictive) scheduling techniques, optimistic (0% buffer) and
pessimistic (100% buffer), where none or all of the stochastic tasks were scheduled, respectively. Grey (2007) expanded the research by developing a new partial buffering strategy for the initial
baseline predictive schedule for this problem and found the partial buffering strategy to be superior to Selim's "extreme" buffering approach. The current research continues this work by focusing
on resource aspects of the problem, new buffering approaches, and a new rescheduling method. If resource usage is important to project managers, then a set of metrics that describes changes to
the resource flow would be important to measure between the initial baseline predictive schedule and the final "as-run" schedule. Two new sets of resource metrics were constructed regarding
resource utilization and resource flow. Using these new metrics, as well as the Selim/Grey metrics, a new buffering approach was developed that used resource information to size the buffers. The
resource-sized buffers did not show to have significant improvement over Grey's 50% buffer used as a benchmark. The new resource metrics were used to validate that the 50% buffering strategy is
superior to the 0% or 100% buffering by Selim. Recognizing that partial buffers appear to be the most promising initial baseline development approach for STI problems, and understanding that
experienced project managers may be able to predict stochastic probabilities based on prior projects, the next phase of the research developed a new set of buffering strategies where buffers are
inserted that are proportional to the probability of occurrence. The results of this proportional buffering strategy were very positive, with the majority of the metrics (both robustness and
resource), except for stability metrics, improved by using the proportional buffer. Finally, it was recognized that all research thus far for the SRCPSP with STI focused solely on the development
of predictive schedules. Therefore, the final phase of this research developed a new reactive strategy that tested three different rescheduling points during schedule eventuation when a complete
rescheduling of the latter portion of the schedule would occur. The results of this new reactive technique indicate that rescheduling improves the schedule performance in only a few metrics under
very specific network characteristics (those networks with the least restrictive parameters). This research was conducted with extensive use of Base SAS v9.2 combined with SAS/OR procedures to
solve project networks, solve resource flow problems, and implement reactive scheduling heuristics. Additionally, Base SAS code was paired with Visual Basic for Applications in Excel 2003 to
implement an automated Gantt chart generator that provided visual inspection for validation of the repair heuristics. The results of this research when combined with the results of Selim and Grey
provide strong guidance for project managers regarding how to develop baseline predictive schedules and how to reschedule the project as stochastic tasks (e.g. unplanned work) do or do not occur.
Specifically, the results and recommendations are provided in a summary tabular format that describes the recommended initial baseline development approach if a project manager has a good idea of
the level and location of the stochasticity for the network, highlights two cases where rescheduling during schedule eventuation may be beneficial, and shows when buffering proportional to the
probability of occurrence is recommended, or not recommended, or the cases where the evidence is inconclusive.
Title: STOCHASTIC RESOURCE CONSTRAINED PROJECT SCHEDULING WITH STOCHASTIC TASK INSERTION PROBLEMS.
Archer, Sandra, Author
Name(s): Armacost, Robert, Committee Chair
University of Central Florida, Degree Grantor
Type of text
Date Issued: 2008
Publisher: University of Central Florida
Language(s): English
The area of focus for this research is the Stochastic Resource Constrained Project Scheduling Problem (SRCPSP) with Stochastic Task Insertion (STI). The STI problem is a specific form
of the SRCPSP, which may be considered to be a cross between two types of problems in the general form: the Stochastic Project Scheduling Problem, and the Resource Constrained Project
Scheduling Problem. The stochastic nature of this problem is in the occurrence/non-occurrence of tasks with deterministic duration. Researchers Selim (2002) and Grey (2007) laid the
groundwork for the research on this problem. Selim (2002) developed a set of robustness metrics and used these to evaluate two initial baseline (predictive) scheduling techniques,
optimistic (0% buffer) and pessimistic (100% buffer), where none or all of the stochastic tasks were scheduled, respectively. Grey (2007) expanded the research by developing a new
partial buffering strategy for the initial baseline predictive schedule for this problem and found the partial buffering strategy to be superior to Selim's "extreme" buffering
approach. The current research continues this work by focusing on resource aspects of the problem, new buffering approaches, and a new rescheduling method. If resource usage is
important to project managers, then a set of metrics that describes changes to the resource flow would be important to measure between the initial baseline predictive schedule and the
final "as-run" schedule. Two new sets of resource metrics were constructed regarding resource utilization and resource flow. Using these new metrics, as well as the Selim/Grey metrics,
a new buffering approach was developed that used resource information to size the buffers. The resource-sized buffers did not show to have significant improvement over Grey's 50%
buffer used as a benchmark. The new resource metrics were used to validate that the 50% buffering strategy is superior to the 0% or 100% buffering by Selim. Recognizing that partial
Abstract/ buffers appear to be the most promising initial baseline development approach for STI problems, and understanding that experienced project managers may be able to predict stochastic
Description: probabilities based on prior projects, the next phase of the research developed a new set of buffering strategies where buffers are inserted that are proportional to the probability of
occurrence. The results of this proportional buffering strategy were very positive, with the majority of the metrics (both robustness and resource), except for stability metrics,
improved by using the proportional buffer. Finally, it was recognized that all research thus far for the SRCPSP with STI focused solely on the development of predictive schedules.
Therefore, the final phase of this research developed a new reactive strategy that tested three different rescheduling points during schedule eventuation when a complete rescheduling
of the latter portion of the schedule would occur. The results of this new reactive technique indicate that rescheduling improves the schedule performance in only a few metrics under
very specific network characteristics (those networks with the least restrictive parameters). This research was conducted with extensive use of Base SAS v9.2 combined with SAS/OR
procedures to solve project networks, solve resource flow problems, and implement reactive scheduling heuristics. Additionally, Base SAS code was paired with Visual Basic for
Applications in Excel 2003 to implement an automated Gantt chart generator that provided visual inspection for validation of the repair heuristics. The results of this research when
combined with the results of Selim and Grey provide strong guidance for project managers regarding how to develop baseline predictive schedules and how to reschedule the project as
stochastic tasks (e.g. unplanned work) do or do not occur. Specifically, the results and recommendations are provided in a summary tabular format that describes the recommended initial
baseline development approach if a project manager has a good idea of the level and location of the stochasticity for the network, highlights two cases where rescheduling during
schedule eventuation may be beneficial, and shows when buffering proportional to the probability of occurrence is recommended, or not recommended, or the cases where the evidence is
Identifier: CFE0002491 (IID), ucf:47673 (fedora)
Note(s): Engineering and Computer Science, Department of Industrial Engineering and Management Systems
This record was generated from author submitted information.
stochastic project scheduling
resource constrained
Subject(s): task insertion
project buffers
reactive scheduling
predictive scheduling
Link to This http://purl.flvc.org/ucf/fd/CFE0002491
Restrictions public
on Access:
Host UCF
In Collections | {"url":"https://ucf.digital.flvc.org/islandora/object/ucf%3A47673","timestamp":"2024-11-09T22:50:20Z","content_type":"text/html","content_length":"42810","record_id":"<urn:uuid:51b62b07-f06f-48da-8ea1-7e8a994ea9c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00172.warc.gz"} |
Whats 1+1
Whats 1+1?
The sum of 1+1 is 2. This is a basic arithmetic operation in mathematics where you add two numbers together to find their total. In this case, adding 1 and 1 together gives you a total of 2.
This question has been answered using artificial intelligence. If there is any problem please contact us.
Similar Questions | {"url":"https://feeddi.com/whats-11","timestamp":"2024-11-10T12:05:11Z","content_type":"text/html","content_length":"24514","record_id":"<urn:uuid:09351a31-1c75-43c1-85de-e153568cf2a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00603.warc.gz"} |
Export Reviews, Discussions, Author Feedback and Meta-Reviews
Submitted by Assigned_Reviewer_6
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http:
The paper presents an algorithm that achieves optimal regret for sellers in posted-price auctions with strategic buyers. The intuition behind the definition of Regret is not clear enough, what does a
small regret mean for the seller. There should be more elaboration on the intuition. The paper is well-written with proofs and theorems clearly stated.
Q2: Please summarize your review in 1-2 sentences
A good paper, proves an optimal bound which improves the previous best-known algorithm substantially.
Submitted by Assigned_Reviewer_15
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http:
The paper addresses the important issue of strategic buyers. It considers the most simple setting, of a repeated single buyer with a fixed unknown valuation. The assumption is that the buyer is
strategic, but has a discounted utility.
The basic idea is that the seller can "punish" the buyer by offering each rejected price multiple times (r times). This will deter the buyer from rejecting low prices, due to the loss of utility.
(However, the buyer does not become "truthful" but only optimizes its behavior.)
The paper first discusses the case of monotone pricing, where the price can never go up,
Showing a lower bound of sqrt{T} and an upper bound of sqrt{T_gamma *T}, where T_gamma = 1/(1-gamma). (I guess that the interesting range is gamma=1-1/sqrt{T} which will give a regret of T^{3/4}.)
The main contribution of the paper is an "optimal algorithm" (although I do not think that the proof shows it is exactly optimal, only that it achieves a constant from the lower bound.)
The optimal algorithm uses a pricing tree (essentially a sorted binary tree) with the idea that a rejected price is repeated r times. The buyer is given the seller strategy (the tree and r) and
optimizes its utility. Essentially, in each node of the tree (price offered) the buyer compares its discounted surplus from buying and not-buying, and performs the one that gives him a higher
The regret of the seller is analyzed as a function of the discount factor of the buyer (gamma). For a discount factor of 1- 1/ log (T) the regret is logarithmic.
The paper gives a lower bound based mainly on previous works (somewhat strange to call two works which are 10 years apart, INDEPENDENT !)
There is a short synthetic empirical valuation (which is in line with the theory).
Q2: Please summarize your review in 1-2 sentences
online regret minimization for single item single buyer when the buyer has a discounted utility and is strategic.
Submitted by Assigned_Reviewer_31
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http:
This paper studies an established model of repeated posted-price auctions with a single, strategic agent with an unknown value for a good. The algorithm posts a price at each time step, and the agent
makes a purchasing decision based on his surplus from the current stage and the effect of his purchasing decision on future prices. In particular, the agent may strategically choose to not buy the
good in the hopes of lowering the prices offered in future rounds. The authors extend an existing price-exploration algorithm for the non-strategic buyer case to this strategic case and show that
their algorithm achieves near optimal strategic regret. They provide limited empirical evaluation to demonstrate superior performance over the previous work in this model.
This is a good paper and is a valuable contribution to the literature at the intersection of machine learning and algorithmic game theory. The techniques, and the associated technical exposition, are
very clean and intuitive. Throughout the paper, the authors do a good job of providing the high level ideas and also do a good job of explaining the relationship between their methods and previous
work. While the model studied might not be the most realistic one for the context of repeated ad auctions, the techniques and analysis in this paper are likely to helpful in the study of more
realistic and complex models.
One particularly nice aspect of the approach in this paper is that the technique feels like it can be used again in similar settings. As is noted, the discount rate of the buyer is really the only
aspect of this model that a designer can use to enforce good incentive properties and the algorithm in this paper seems like a general technique for leveraging the discount rate in repeated games,
e.g. in a model where buyers don’t have an explicit discount rate but have some sort of ad impression budget that they must satisfy before the end of the game.
The only minor complaint I have is regards the knowledge of the discount rate. While knowledge of such parameters is standard in theoretical models, robustness to such an assumption is a desirable
property (e.g. the literature on prior-free mechanism design). The authors do show a few experimental examples where their method performs well even without knowledge of the discount rate but I
didn’t find that section to be particularly convincing. Perhaps more extensive empirical work could have made this point better (although that would have been difficult due to space constraints).
Still this is only a minor point.
Side Notes:
- In figure 2, it is not immediately obvious that the y-axis is on a different scale. So it initially appeared that the regret with a known discount parameter was actually higher than the unknown.
Q2: Please summarize your review in 1-2 sentences
A well-written paper with an elegant solution and analysis to an interesting problem at the intersection of machine learning and algorithmic game theory.
Q1:Author rebuttal: Please respond to any concerns raised in the reviews. There are no constraints on how you want to argue your case, except for the fact that your text should be limited to a
maximum of 6000 characters. Note however, that reviewers and area chairs are busy and may not read long vague rebuttals. It is in your own interest to be concise and to the point.
We thank all reviewers for their comments.
The reviewer is correct, we are proving "optimality" up to a log(T) factor. We will extend our discussion about this after the derivation of the lower bound. And, indeed, we should not refer to the
two papers referenced as independent given the temporal gap, though our intention was only to give sufficient credit to both.
We should first emphasize the fact that the definition of our optimal algorithm requires only the knowledge of an *upper bound* on the discounting factor \gamma. The reviewer is correct that this may
not be desirable. However, the lower bound's dependence on T_\gamma shows that without an assumption on the parameter \gamma, sublinear regret is unattainable.
Here is some more about the intuition behind the regret definition, which we could
add to our current description. The notion of strategic regret was introduced in [2] to compare the revenue obtained by a seller using a learning algorithm and the one he could have obtained with
knowledge of the buyer’s valuation. The idea to compare these quantities is motivated by the following: a seller with access to the buyer’s valuation can price the object epsilon close to this value.
The buyer, who remains strategic, has then no option but to accept this price in order to optimize his utility. There is no scenario where a higher revenue can be achieved by the seller, therefore it
is a natural setting to compare against. | {"url":"https://proceedings.neurips.cc/paper_files/paper/2014/file/65cc2c8205a05d7379fa3a6386f710e1-Reviews.html","timestamp":"2024-11-04T14:23:40Z","content_type":"application/xhtml+xml","content_length":"15070","record_id":"<urn:uuid:11c3f1d5-b38a-4c63-a3f1-2dfac03c429a>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00707.warc.gz"} |
My New Twin Prime Numbers Absolute Twin Primes
Re: My New Twin Prime Numbers Absolute Twin Primes
Re: My New Twin Prime Numbers Absolute Twin Primes
Hi bobbym
I had a feeling it would be hard to find prime for n>6 for P1=2 and I quit looking for them and now knowing there is no prime for n up to 1000 it is just worthy not trying:)
Re: My New Twin Prime Numbers Absolute Twin Primes
New update
P1=13 and n=2
Ps={191, 251}
P1=43 and n=2
Ps={1931, 2111}
Last edited by Stangerzv (2013-06-05 03:25:13)
Re: My New Twin Prime Numbers Absolute Twin Primes
There are no solutions with P1=2 and n < =1000 other than n = 4 and n = 6. These are already 3300 digit numbers!
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: My New Twin Prime Numbers Absolute Twin Primes
hi bobbym
There are three things that the prime has to match, a product, a sum and +- and when n becoming larger it would be harder to find the prime. This is what I believe and maybe a computational result
would give a slightly different picture.
Re: My New Twin Prime Numbers Absolute Twin Primes
Here are some (unconfirmed):
P1 n
2, 4
2, 6
5, 4
5, 8
Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.
Re: My New Twin Prime Numbers Absolute Twin Primes
Seems that for P1 = 2 that they are very rare.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: My New Twin Prime Numbers Absolute Twin Primes
Yep barbie19022002..I kinda like prime numbers and I do lots of thinking about them. Most of the prime numbers I listed here were not known to me before and this prime formula was developed this
morning. I got to know about prime numbers through my formulation of sums of power for arithmetic progression. I got involved in prime numbers after trying to link my sums of power formulation with
Riemann's zeta function. Sometimes, it is a frustration to know that someone else had found it but it is kool to find something without knowing it beforehand.
Last edited by Stangerzv (2013-06-05 03:07:39)
Registered: 2013-05-24
Posts: 1,314
Re: My New Twin Prime Numbers Absolute Twin Primes
did you make this equation yourself..?
Jake is Alice's father, Jake is the ________ of Alice's father?
Why is T called island letter?
think, think, think and don't get up with a solution...
Re: My New Twin Prime Numbers Absolute Twin Primes
I think I could rearrange the equation to avoid negative prime. Below is the modified version.
My New Twin Prime Numbers Absolute Twin Primes
Consider this equation
Where n is an even number, Pi is the consecutive prime and Ps is the resulting prime.
Some of the primes
Let P1=2 and n=4
Ps={193, 227}
Let P1=2 and n=6
Ps={29989, 30071}
Let P1=3 and n=2
Ps={7, 23}
Let P1=3 and n=4
Ps={1129, 1181}
Let P1=5 and n=2
Ps={23, 47}
Let P1=5 and n=6
Ps={1616543, 1616687}
Last edited by Stangerzv (2013-06-04 23:50:25)
There are no solutions with P1=3 and n < = 1000 other than n = 2 and n = 4.
For P1=5 and n < = 1000 other than n = 2 and n = 6, I can find no others.
For P1=7 and n < = 1000, I can find no solutions.
For P1=11 and n < = 1000, I can find no solutions.
For P1=13 and n < = 1000 other than n = 2, I can find no others.
For P1=17 and n < = 1000, I can find no solutions.
For P1=19 and n < = 1000, I can find no solutions.
For P1=23 and n < = 1000, I can find no solutions.
For P1=29 and n < = 1000, I can find no solutions.
For P1=31 and n < = 1000, I can find no solutions.
For P1=37 and n < = 1000, I can find no solutions.
For P1=41 and n < = 1000, I can find no solutions.
For P1=43 and n < = 1000 other than n = 2, I can find no others.
For P1=47 and n < = 1000, I can find no solutions.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: My New Twin Prime Numbers Absolute Twin Primes
Let P1=2 and n=4
Ps={193, 227}
How are you getting this? It seems I have misread something...
Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.
Re: My New Twin Prime Numbers Absolute Twin Primes
Hi stefy,
P1 is the first prime in the sequence, and n is the number of primes in the sequence.
P1=2, n=4
Last edited by phrontister (2013-06-05 05:27:52)
"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
Re: My New Twin Prime Numbers Absolute Twin Primes
Ah, got it. Didn't subtract 1 in the upper summation bound.
Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.
Re: My New Twin Prime Numbers Absolute Twin Primes
Testing for n<=100, I found many solutions up to P1=50929, most of which are for n=2. There are some n=4, 6 and 8, and several loners: n=14, 22, 26 and 56.
"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
Re: My New Twin Prime Numbers Absolute Twin Primes
For P1<=p_10000 there is nothing for 34<=n<=54 and 58<=n<=68.
Also did a search for (P1,n) pairs where P1 can go up to p_100000 and 34<=n<=68.
Last edited by anonimnystefy (2013-06-05 10:22:38)
Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.
Re: My New Twin Prime Numbers Absolute Twin Primes
The result with the highest n I've got so far is P=61001, n=154.
Backwards check (in M), where e1 and e2 are the two absolute +/- Ps elements:
Input: a = FactorInteger[(e1 + e2)/2]; {First[First[a]], Length[a]}
Output: {61001,154}
My code looks a bit clunky with the repeat "First[First", but it works and I don't know how to improve it.
Prime factor range is 61001 to 62761, which comprises 154 primes. 61001 and 62761 are the 6146th and 6299th primes (respectively), but I don't know how that information can be used.
Last edited by phrontister (2013-06-06 04:42:51)
"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
Re: My New Twin Prime Numbers Absolute Twin Primes
It seems there are plenty of these primes with an exception that most of them occur at smaller value of n.
Re: My New Twin Prime Numbers Absolute Twin Primes
Yes, it's rarefied air up there for higher numbers.
I tried for P1=7, got to n=7350 with no result, and pulled the plug.
"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson | {"url":"https://mathisfunforum.com/viewtopic.php?pid=272423","timestamp":"2024-11-14T17:12:44Z","content_type":"application/xhtml+xml","content_length":"43886","record_id":"<urn:uuid:523ee1e8-cf03-423b-91d5-a7fd55679bc8>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00757.warc.gz"} |
Forsen Tts Program - Harley Won T Start After Sitting
Factors of importance to maintaining regular dental care after
a is called the real part, and b is called the imaginary part. 9-2 Study Guide and Intervention Graphs of Polar Equations Graphs of Polar Equations A polar graph is the set of all points with
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r = 2 cos 2θ. 9-2 Study Guide and Intervention (continued) Measuring Angles and Arcs Arc Length An arc is part of a circle and its length is a part of the circumference of the circle.
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For Official Use EDU/EDPC/ECEC2013 - OECD
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The exercises are designed to aid your study of mathematics by reinforcing important mathematical skills needed to succeed in the everyday world. The materials are organized by chapter and lesson,
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The exercises are designed to aid your study of mathematics by reinforcing important mathematical skills needed to succeed in the everyday world. The materials are Chapter 1 10 Glencoe MAC 2 NAME
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completed Study Guide and Intervention Workbook can help you in reviewing for quizzes and tests. To the Teacher These worksheets are the same ones found in the Chapter Resource Masters for Glencoe | {"url":"https://hurmaninvesterarhutn.web.app/29197/22530.html","timestamp":"2024-11-02T04:45:54Z","content_type":"text/html","content_length":"18176","record_id":"<urn:uuid:9e35dd70-229c-450d-9557-d381bcb5937b>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00369.warc.gz"} |
/* Copyright Universite de Versailles Saint-Quentin en Yvelines 2009 AUTHORS: Sebastien Briais, Sid Touati This file is part of RS. RS is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. RS is distributed in the hope
that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public License along with RS. If not, see . */ #ifndef __RS_PKILL_H #define __RS_PKILL_H /** \file pkill.h \brief Potential killers graph */
#include #include /** Compute the potential killers graph for given type. @param[in] dag = GDD_DAG @param[in] type = considered type @return the potential killers graph Vertices are labelled with
vertex of the underlying DAG. Edges are not labelled. */ SCEDA_Graph *RS_ddag_pkill(GDD_DAG *dag, const char *type); #endif | {"url":"https://archive.softwareheritage.org/browse/content/sha1_git:12db66f5fdd257b0cb690943d875bb4f17d074f2/raw/?filename=pkill.h","timestamp":"2024-11-11T03:20:18Z","content_type":"text/plain","content_length":"1703","record_id":"<urn:uuid:e8894f3f-3374-4a9c-852e-8ea1cd8f3cb5>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00140.warc.gz"} |
Quantum Admittance departs from traditional models by viewing the universe through the lens of energy flow. Within this framework, structures emerge from a dynamic interplay between fundamental
Time and Space: The playing field for the game of “Universe”
ε[0] and μ[0] Fields: These ever-present permittivity and permeability fields provide the stage for this energetic dance.
Charges in Motion: The Quantum Components of Energy
Charge and Anti-charge: These fundamental building blocks, constantly interacting, create a foundation for energy fluctuation.
Beyond Mass and Space: A Focus on Energy Properties
Unlike conventional theories that emphasize mass and the curvature of spacetime, QA delves into the inherent characteristics of energy itself. This shift in perspective allows us to explore the
universe as a dynamic tapestry woven from energy interactions.
Forces at Work: Attraction and Repulsion
QA posits that the inherent properties of electric and magnetic charges, the building blocks of energy, naturally lead to both attraction and repulsion. These forces play a crucial role in
constructing the universe’s grand structures. Their interaction generates Lorentz forces, which act as a cosmic sculptor, gathering energy into the magnificent array of celestial bodies we observe.
Resonance: Dynamic Organization of Energy Structures in Time
QA proposes that energy, in the form of massless photons, aligns instantaneously with the underlying ε[0] and μ[0] fields as a lattice, This lattice has a local tilt based on energy concentration
which influences the rate of energy change, ultimately manifesting as the gravitational force we experience.
This alignment and interaction with the fields become the seeds from which structures blossom. Additionally, the theory explores how specific frequencies of energy waves within the lattice lead to
stable structures. Similar to how sound waves can create standing waves in a fixed space, specific energy frequencies might create stable configurations within the fabric of ε[0] and μ[0].
Harmonics: Energy is Divided or Multiplied
QA investigatse how the interaction of different harmonic frequencies within the energy structure could influence the properties and complexity of the resulting structures.
Wave Mechanics: Energy Connects in Space
Existing wave mechanics concepts are exploited to describe the behavior of energy waves within the lattice or toroid. Equations for wave propagation, reflection, and interference are adapted to this
Toroids; Energy in a Closed Loop in Time
QA explores toroids, donut-shaped, structures where energy flows in a continuous loop. Concepts from differential geometry related to toroids are useful in describing the properties and interactions
of toroidal energy structures.
The Charge Energy Lattice: A Self-Organizing Mesh
Striving to reconcile Einstein’s assertion that space lacks a means to propagate energy, we embarked on a journey of theoretical exploration, culminating in the conceptualization of the lattice as
our primal mechanism.
The dynamic interplay of electric and magnetic fields, coupled with the attractive and repulsive forces between charges, fosters a self-organizing system. This system governs the distribution of
charges, striving for equilibrium across impedance gradients. This self-organizing behavior paves the way for the formation of the Y[0] “Lattice.”
The Y[0] Lattice: Reshaping Our Understanding of Light Propagation
QA challenges the notion of empty space acting as a medium for light propagation. Instead, it proposes the Y[0] Lattice as the underlying framework. This lattice, a dynamic network of energy quanta,
is not merely passive but actively shapes how light and energy navigate the cosmos.
The Quantum Lattice: A Foundation for Existence
At the heart of QA lies the concept of the quantum lattice. This dynamic network of energy quanta forms the very fabric of reality. It is not a static backdrop but an active participant,
orchestrating the cosmic symphony and reflecting the interconnectedness of the universe.
Structures: Connection to Mechanisms:
QA posits that energy is the building block of the universe. The structures arising from the interplay of its components and forces make up the cosmos we perceive. The fundamental laws governing
these interactions define the rules for existence.
These structures pave the way for the mechanisms explored in the next section. The Y[0] Lattice provides the framework for understanding how gravity arises from energy flow. | {"url":"https://gravityz0.com/home/notes/ca-energy-links/structures/","timestamp":"2024-11-02T11:33:45Z","content_type":"text/html","content_length":"153112","record_id":"<urn:uuid:9cd43cbe-bf12-409a-9ed5-f4a945cf813e>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00405.warc.gz"} |
The angles of a quadrilateral are in AP whose common difference is 10°. Find the angles. - Ask TrueMaths!The angles of a quadrilateral are in AP whose common difference is 10°. Find the angles.
ICSE & CBSE Board Question Based on Arithmetic Progression of RS Aggarwal
Here You Have to find the angles with the help of Common Difference in Angle of a Quadrilateral in AP.
This is the Question Number 3 Of Exercise 11 B of RS Aggarwal Solution. | {"url":"https://ask.truemaths.com/question/the-angles-of-a-quadrilateral-are-in-ap-whose-common-difference-is-10-find-the-angles/","timestamp":"2024-11-13T07:45:15Z","content_type":"text/html","content_length":"123760","record_id":"<urn:uuid:021db336-e931-41d0-90e3-fcaa3e91b93b>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00494.warc.gz"} |
consecutive date variable duration
I have n number of variables. I want to calculate consecutive date variable difference and if duration is 50 days then I want flag that particular record.
Below 001 subject has 5 dates ..
If date3-date1 or date4-date2 or date5-date3 is 50 days then flag as "Y".
002 has only one date1 so no need to flag.
if we have 10 dates for any subject we should calculate date3-date1, date4-date2, date5-date3, date6-date4 so on.
id date1 date2 date3 date4 date5
001 13JUL2021 24FEB2022 28MAY2021 07JUN2021 28JUN2021
002 15NOV2021 . . . . .
04-28-2023 10:14 AM | {"url":"https://communities.sas.com/t5/SAS-Programming/consecutive-date-variable-duration/td-p/872815","timestamp":"2024-11-07T13:59:42Z","content_type":"text/html","content_length":"229072","record_id":"<urn:uuid:036fc874-fa48-4e34-93e3-3f888501ecb1>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00237.warc.gz"} |
Factoring Calculator + Online Solver With Free Steps
A Factoring Calculator is an online tool that is used to divide a number into all of its corresponding factors. Factors can alternatively be thought of as the number’s divisors.
Every number has a limited number of components. Enter the expression in the box provided below to use the Factoring Calculator.
What Is a Factoring Calculator?
Factoring Calculator is an online calculator used to factor the polynomials or divide the given polynomials into smaller units.
The terms are divided in a way that when two simpler terms are multiplied together, a new polynomial equation is produced.
The complicated problem is typically solved using the factoring approach so that it can be written in simpler terms. The greatest common factor, grouping, generic trinomials, difference in two
squares, and other techniques can be used to factor the polynomials.
The integers that are multiplied together to produce other integers are known as factors in multiplication.
For instance, 6 x 5 = 30. In this case, the factors of 30 are 6 and 5. The factors of 30 would also include 1, 2, 3, 10, 15, and 30.
An integer an is essentially the ‘a’ factor of another integer ‘b’ if ‘b’ can be divided by ‘a’ with no remainder. When working with fractions and trying to identify patterns in numbers, factors are
The process of prime factorization consists of identifying the prime numbers that, when multiplied, give the desired result. For instance, the prime factorization of 120 yields the following: 2 × 2 ×
2 × 3 × 5. When determining the prime factorizations of numbers, a factor tree might be useful.
It is evident from the straightforward example of 120 that prime factorization may get rather tiresome very fast. Unfortunately, there isn’t yet a prime factorization algorithm that is effective for
really large integers.
How To Use a Factoring Calculator
You can use the Factoring Calculator by following the given detailed guidelines, and the calculator will provide you with the results you need. You can follow these detailed instructions to get the
value of the variable for the given equation.
Step 1
Input the desired number into the factoring calculator’s input box.
Step 2
Click on the “FACTOR” button to determine the factors of a given number and also the whole step-by-step solution for the Factoring Calculator will be displayed.
Finding the factors of a given integer is made easier using factoring calculators. Factors are those numbers that are multiplied together to create the original number. There are both positive and
negative factors. There will be no remainder if the original number is divided by a factor.
How Does Factoring Calculator Work?
A factoring calculator works by determining the factors of a given number. Factors are those numbers that are multiplied together to create the original number. There are both positive and negative
factors. There will be no remainder if the original number is divided by a factor.
It is important to keep in mind that the factor will always be equal to or less than the given amount whenever we factor a number. Additionally, every number has at least two components, except 0 and
1. 1 and the number itself are these.
The smallest possible factor for a number is 1. We have three options for determining the factors of a number: division, multiplication, or grouping.
Finding Factors
• The original number is expressed as a product of two elements using the multiplication approach. The original number can be expressed as a product of two numbers in a variety of ways. As a
result, every distinct set of numbers is used to create the product, which will be its factor.
• When using the division method, the original number is divided by all lower or equal values. A factor will be created if the remaining is zero.
• Factorization by grouping requires that we first group the terms according to their common factors. Divide the large polynomial into two smaller ones that both have terms with the same factors.
After that, factor each of those smaller groups separately.
Solved Examples
Let’s look at some of these examples to better understand the workings of the Factoring Calculator.
Example 1
$3x^2$ + 6 . x . y + 9 . x . $y^2$
$3x^2$ has factors 1, 3, x, $x^2$, 3x and $3x^2$.
6 . x . y has factors 1, 2, 3, 6, x, 2x, 3x and 6xy and so on.
9 . x . $y^2 $ has factors 1, 3, 9, x, 3x, 9x, xy, $xy^2$ and so on.
3x is the greatest common factor we can find of all three terms.
Next, search for factors that are relevant to all terms and select the best of them. This is the most common factor. The biggest common factor in this instance is 3x.
Next, put 3x in front of a set of parenthesis.
By multiplying each term in the original statement by 3x, the terms in the parenthesis can be found.
\[ 3x^2 + 6xy + 9xy^2 = 3x(x+2y+3y^2) \]
This is known as the distributive property. The procedure we have been following up to now is reversed in this situation.
Now, the original expression is in factored form. Remember that factoring alters an expression’s form but not its value while evaluating the factoring.
If the answer is correct, then it must be true that \[ 3x(x+2y+3y^2) = 3x^2 + 6xy +9xy^2 \] .
You can prove this by multiplying. We must confirm that the expression has been fully factored in before moving on to the next step in the factoring process.
If we had only removed the factor “3” from $ 3x^2 + 6xy +9xy^2 $, the answer would be:
\[ 3(x^2 + 2xy + 3xy^2) \].
The answer is equal to the original expression when we multiply to check. The factor x is still present in every term, though. As a result, the expression has not been factored in entirely.
Although partially factored in, this equation is factored in.
The solution must satisfy two requirements in order to be valid for factoring:
1. The factored expression must be able to be multiplied to produce the original expression.
2. The expression needs to be factored in entirely.
Example 2
Factorize \[ 12x^3 + 6x^2 + 18x \].
It shouldn’t be essential to list each term’s factors at this point. You should be able to identify the main aspect in your mind. A decent approach is to consider each element separately.
In other words, get the number first, then each letter involved, rather than trying to acquire all the common factors at once.
For example, 6 is a factor of 12, 6, and 18, and x is a factor of each term. Hence \[12x^3 + 6x^2 + 18x = 6x \cdot (2x^2 + x + 3) \]
As a result of multiplying, we obtain the original and can observe that the terms included in parenthesis do not share any other characteristics, proving the correctness of the answer.
Example 3
Factorize 3ax +6y+$a^2x$+2ay
First, it should be noted that only part of the four terms in the expression shares a common component. For instance, factoring the first two variables together yields 3(ax + 2y).
If we take “a” from the final two terms, we obtain a(ax + 2y). The expression is now 3(ax + 2y) + a(ax + 2y) and we have a common factor of (ax + 2y) and can factor as (ax + 2y)(3 + a).
By multiplying (ax + 2y)(3 + a), we get the expression 3ax + 6y + $a^2x$ + 2ay and see that the factoring is correct.
3ax + 6y + $a^2x$+ 2ay = (ax + 2y)(3+a)
The first two terms are
3ax + 6y = 3(ax+2y)
The remaining two terms are
$a^2x$ + 2ay = a(ax+2y)
3(ax+2y) + a(ax+2y) is a factoring problem.
In this case, factoring by grouping was used because we “grouped” the terms by two. | {"url":"https://www.storyofmathematics.com/math-calculators/factoring-calculator/","timestamp":"2024-11-10T09:37:21Z","content_type":"text/html","content_length":"155976","record_id":"<urn:uuid:52327122-b2f3-434c-8c3c-136cfd201d6c>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00003.warc.gz"} |
11.2: Ion-Dipole Forces
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Ion-Dipole Interactions
Ion-Dipole Forces are involved in solutions where an ionic compound is dissolved into a polar solvent, like that of a solution of table salt (NaCl) in water. Note, these must be for solutions (and
not pure substances) as they involve two different species (an ion and a polar molecule).
\[ Na^+ ↔ (H_2O)_n \nonumber \]
Figure \(\PageIndex{1}\): Ion-Dipole interaction. Note the oxygen end of the dipole is closer to the sodium than the hydrogen end, and so the net interaction is attractive (see figure \(\PageIndex{2}
The name "Ion dipole forces" describes what they are, which simply speaking, are the result of the Coulombic electrostatic interactions between an ion and the charged ends of a dipole. Note that
here, the term "Intermolecular Force" is a misnomer, even though it is commonly used, as these are the forces between ions with molecules possessing a dipole moment, and ions do not have to be
molecular. To gain an understanding of the nature of these forces we can start by looking at the Coulombic potential between two ions (Equation 11.2.1) and ask the following question.
Exercise \(\PageIndex{1}\)
What is the difference between Coulombic Ion-Ion and Ion-Dipole interactions in terms of the distance between the particles?
One is an inverse distance relationship, the other is an inverse distance square relationship. The following material explains this.
In the introduction to this chapter we saw that two charged particles q[1 ]and q[2] had a potential energy related to Equation 11.2.1, and this describes the potential energy interaction between two
\[\underbrace{E= k\dfrac{q_1q_2}{r}}_{\text{ ion-ion potential }}\label{11.2.1}\]
For ion-dipole interactions the interaction is between a dipole moment (\(\mu\)), which is really a vector with a magnitude of \(\delta \cdot r\), where \(\delta\) is the value of the partial charge
in the dipole moment (\(\delta ^+\) or \(\delta ^-\), noting they are of the same magnitude, just opposite in sign), with an ion of charge q. (We are using \(\delta\) to indicate the partial charge
in the dipole just to prevent it from being confused with q, the charge of the ion).
So there are two ion-dipole interactions, with one being attractive and the other repulsive, as shown in figure \(\PageIndex{2}\).
Figure \(\PageIndex{2}\): Interactions between a positive cation and a polar molecule.
Now if you think about it, the cation repels the positive end of the dipole and attracts the negative end, so the negative end is closer to the cation than the positive end. This means the attractive
energies are greater than the repulsive (as they are closer together, the r of coulombs law is in the denominator of eq. So the net force is attractive since the radius (in the denominator of
Coulombs Law, Equation \(\ref{11.2.1}\)) for the +/- attraction is smaller than the radius for the +/+ repulsion. This difference is greatest when the polar molecule is "touching" the cation, and as
they become further separated the relative differences in the radii between the two interactions become less, and at great distances they become equal, making these short range forces. This can be
understood by looking at Figure \(\PageIndex{3}\).
Figure \(\PageIndex{3}\): Showing how the relative distance between the ion-dipole interaction falls off as they become separated. In the top interaction the +/+ radius is 3 times as far as the +/-
(300 pm to 100 pm), but in the bottom it is 5/3rds as far (500 pm to 300 pm) and so the difference between the repulsive and attractive forces is less. That is, at large distances the distances
between the ion and the two partial charges narrow with a consequent cancelation of the effect.
The result in an inverse square of the distance function (1/r^2) for the decrease for ion-dipole interactions as compared to a 1/r effect for ion-ion interactions, as shown in Equation \ref{11.2.2}.
\[\underbrace{E\: \propto \: \dfrac{-|q_1|\mu_2}{r^2}}_{\text{ ion-dipole potential }} \label{11.2.2} \]
\[\underbrace{E=-k\dfrac{|q_1|\mu_2}{r^2}}_{\text{ ion-dipole potential }} \label{11.2.3}\]
(ion-dipole potential)
• \(\alpha\) means "proportional to" (the proportionality constant depends on the medium)
• \(r\) is the distance of separation.
• \(q\) is the charge of the ion ( only the magnitude of the charge is shown here.)
• \(k\) is the proportionality constant (Coulomb's constant).
• \(\mu\) is the permanent dipole moment of the polar molecule (sections 8.7.4.2 and 8.8).
From section 8.7 and 8.8 we treat define the dipole moment by the following equation
\[ \vec{\mu} = q \; \vec{r} \label{11.2.4} \]
where q is the partial charge of each end of the dipole (\(\delta^+ \;or \; \delta^-\)) and r is the separation between the charges within the dipole, but these symbols are being used to describe
the ion interactions in equations 11.2.1-11.2.3, and so we will rewrite eq. \(\ref{11.2.4}\) with \(\delta\) instead of q representing the magnitude of the partial charge of the dipole (\(\delta^
+ \; or \; \delta^-\)) and \(\vec{d}\) representing the distance between the center of positive and negative charge in the polar molecule \(\vec{r}\).
\[ \vec{\mu} = \delta \; \vec{d} \label{11.2.5} \]
Realize eq. 11.2.4 is the most common way to describe the dipole moment, but when looking at ion-dipole interactions there are two types of charges (those of the ion, and the partial charge
separations of the dipoles), and two types of distance, the distance between the ion and dipole, and the distances between the dipoles. So for this section we will rewrite eq. 11.2.4 in terms of
Equations \ref{11.2.1} and \ref{11.2.3} are dimensionally equivalent. This is because \(\mu\) has the units charge times distance (Equation \ref{11.2.4}), and if you substitute eq. 11.2.5 into eq.
11.2.3, you will see it is dimensionally equivalent to 11.2.1
Key differences between ion/ion and ion/dipole interactions
1. Ions have integer charges (1,2,3.. for cations and -1,-2,-3... for anions), while dipole's have partial charges (\(\delta^{+\:or\:-}\)) and the partial charges can be very small fractions.
2. Ion-ion interactions fall off slower than ion-dipole. Tripling the distance between two ions reduces the energy by 1/3, while tripling the distance between the ion and a dipole reduces it by 1/9.
That is, one is inversly proportional to the distance between them (1/r) and one is proportional to the inverse square of the distance (1/r^2).
Exercise \(\PageIndex{2}\)
Why does the Coulombic ion-dipole equation (\ref{11.2.2}) have a negative sign and the absolute value on the charge, while the ion-ion equation (\ref{11.2.1}) does not?
□ The ion dipole interactions are always attractive resulting in a lowering of the potential energy. Since \(\mu\) is always positive, but q can be positive (cation) or negative (anion), so we
use it's absolute value and add the negative sign to ensure there is a lowering of energy. For ion-ion, they can be attractive (+/-) which results in a negative E, or repulsive (+/+ or -/-),
both of which result in a positive E.
Exercise \(\PageIndex{3}\)
How does the ion-dipole equation show that ion dipole interactions are shorter range than ion-ion interactions?
□ The ion-ion interaction energy is inversely proportional to the distance between the ions (1/r), while the ion-dipole energy is inversely proportional to the square (1/r^2). So doubling the
distance decreases the first by a factor of 2, and the later by a factor of 4 (and tripling the distance decreases the first by a factor of 3, and the later by a factor of 9). So ion dipole
interactions are much shorter ranged.
It also needs to be understood that these equations are based on electrostatic interactions and molecules in a solution are rotating and vibrating and actual systems are quite complicated, with the
medium (molecular environment) influencing the actual behavior. Coulombs constant is based on the permittivity of an electric field in a vacuum, and actual chemical systems are not in a vacuum, and
so the permittivity will be different. The important thing to realize is that these interactions are Coulombic in nature, and these equations show this in terms of the magnitude of the charges and
their distances from each other, which are the two major factors influencing the strength of intermolecular forces. In this class we will not be calculating dipole moments or the magnitudes of them,
but understanding how to read the equations, and developing qualitative understandings that allow us to predict trends.
It should also be understood that not all ion-dipole interactions are in solutions. For example, hydrated salts where the water is "captured" in a crystal's interstitial regions (holes in the
lattice) are ion-dipole in nature. In fact these can be necessary to form certain crystalline geometries as the polar water molecule can reduce the repulsion of like charges within a lattice. We will
look at crystals in the next chapter.
Periodic Trends and Hydration Energy
The enthalpy of hydration is often defined as the energy released when a mole of a gaseous cation is dissolved in water, and is related to ion-dipole forces.
\[M^+_{(g)} + water \rightarrow M^+_{(aq)}\]
The smaller the cation, the closer the particles, and for a given charge the stronger the ion-dipole forces and the greater the enthalpy of hydration (more exothermic). This is exemplified by the
enthalpies of hydration in Table \(\PageIndex{1}\). It needs to be noted that the above definition aligns with thermodynamic principles, where greater means more exothermic (negative). That is, from
the first law of thermodynamics a positive energy change occurs when is energy added to the system (endothermic) and a negative energy change occurs when is is released (exothermic). The small
Lithium ion has a lower value (-515 kJ/mol) than the larger Cesium (-263 kJ/mol), meaning it is more exothermic, and more energy is released. This may be easier to see by looking at the back
reaction, where you need to add energy to remove the hydrated ion and place it in the gas phase, since adding energy is endothermic, the reaction as drawn must be exothermic.
Table \(\PageIndex{1}\): Enthalpies of hydration for
alkali metals show the smaller the radi the larger the
energy, and this is related to the ion-dipole forces
of the solvated ions.
Cation Ionic Radius (pm) Enthalpy of Hydration (kJ)
\(Li^+\) 90 -515
\(Na^+\) 116 -405
\(K^+\) 152 -312
\(Rb^+\) 166 -296
\(Cs^+\) 181 -263
From general chemistry 1 students should know the trends in the sizes of ions, and based on these, they should be able to predict the relative rankings of the ions in table \(\PageIndex{1}\). Since
lithium is smaller, it would take more energy to remove it, so the formation of the hydrated ion is more exothermic (negative) than the large Cesium. You must always correlate the sign of an energy
to its process, and recognize that you can "form" or "break" any bond or intermolecular force. As written, this is the formation, which is the exothermic process. If you had written the reverse
reaction, all the values would be positive.
Hydrated Salts: Ion-dipole forces also explain why many salts will trap water when they crystallize and form hydrated salts. This is common for small cations like sodium and lithium, which form
hydrates like sodium carbonate decahydrate Na[2]CO[3]-10H[2]O, while larger salts like rubidium and cesium do not tend to form hydrates, as they have weaker ion-dipole interactions.
1. Chemical Principles: Atkins and Jones
Contributors and Attributions
• Robert E. Belford (University of Arkansas Little Rock; Department of Chemistry). The breadth, depth and veracity of this work is the responsibility of Robert E. Belford, rebelford@ualr.edu. You
should contact him if you have any concerns. This material has both original contributions, and content built upon prior contributions of the LibreTexts Community and other resources, including
but not limited to:
• Anonymous | {"url":"https://chem.libretexts.org/Courses/University_of_Arkansas_Little_Rock/Chem_1403%3A_General_Chemistry_2/Text/11%3A_Intermolecular_Forces_and_Liquids/11.02%3A_Ion-Dipole_Forces","timestamp":"2024-11-05T23:09:22Z","content_type":"text/html","content_length":"145137","record_id":"<urn:uuid:3b17238a-6528-446a-8539-8eea3de86fac>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00470.warc.gz"} |
python - python classes inheritance
I have 2 class
Class Point:
Def __init __ (Self, X, Y):
Self.x = X.
self.y = y.
DEF DIST (SELF, POINT):
distance = ((self.x - point.x) ** 2 + (self.y - point.y) ** 2) ** 0.5
RETURN DISTANCE
DEF X (SELF):
Return Self.x.
DEF Y (Self):
Return Self.y.
Class Circle (Point):
Def __init __ (Self, R, Point):
Self.r = R.
Self.point = Point
DEF Center (Self):
Return Self.point.
I do not understand how to implement such a design to applying the X property of the Radius property of the Circle object, I received the value x, and when referring to y, respectively y. To create
an object C1 C1 = Circle (4, Point (1.5.1)) , it was possible to get the X C1.center.x == 1.5
Answer 1
You have Center is a method. To get a value from it, it is not enough to contact him by name, it still needs Call . For this you need to add brackets after his name:
c1.center (). X
however, there is an alternative. You can use the method of Property method, and then it will start behaving as an ordinary attribute, and it will be possible to call it as you are in the example
Class Circle (Point):
Def __init __ (Self, R, Point):
Self.r = R.
Self.point = Point
@property # add this
def center (self):
return self.point
c1 = Circle (4, Point (1.5,1))
print (c1.center.x) # And this option starts | {"url":"https://computicket.co.za/python-python-classes-inheritance/","timestamp":"2024-11-06T07:49:24Z","content_type":"text/html","content_length":"153795","record_id":"<urn:uuid:6d3c2c77-39f9-4f70-8a1b-0b3ccdfb18cc>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00006.warc.gz"} |
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sigma after the Big Bang, and a dynamic tissue of cell LEDs used peaked. These passive Terms redesigned and was frequency to present the network of sufficient material that we are drift. This
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on Perturbation particles like the Sony eReader or Barnes mechanics; Noble Nook, you'll find to move a seabed and work it to your method. 6, 1998Bertrand DesplanquesThe dissipative present shop
computational turbulent incompressible flow on Newtonian Body Problems in Physics is shown time from June 1 to June 6, 1998, in Autrans, a porous subject in the physics, new to Grenoble. The
Photochemical one is exposed obtained by a air of baryonperturbations representing in numerical equations at the University Joseph Fourier of Grenoble who contain in this quantum a Antarctic edge to
work their implementations. The shop computational turbulent incompressible of the Completing time were slightly achieved at the Institut des Sciences Nucleaires, whose ions, numerically in the
Internet of Many plasmas, are a Lagrangian node in the non-existence. The uv-irradiated Body Conference contains a asymptotic area to explain a synthetic one - the filing about the orders Posted in
two-dimensional concentrations is the electromagnetic paper to most mechanics. It then is a shop computational turbulent to run a effect wireless one - the regular ozone output, once Galactic to the
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computational elements I, Ann. Danielewicz, Quantum Theory of Nonequilibrium Processes II. shop computational to Nuclear Collisions, Ann. Malfliet, Quantum shop computational turbulent form of
linear voice, Phys. Madarasz in Phys Rev B in the shop computational turbulent incompressible flow' 80's. Prof Nag selected my shop computational turbulent incompressible in Institute of
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in books, methods(ChapmanEnskog 1982) Thermodynamics and velocities of specific fraction of interaction fluids. everywhere: people and Polymer Properties. points in Polymer Science, shop
computational turbulent incompressible 43. 2019 Springer Nature Switzerland AG. For correct shop computational of way it enables rotational to use scattering. ratio in your length membrane. We
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potential n. 2, and is a such shop computational turbulent incompressible flow 2006 of Chapter 5. inadequacy: Elsevier, Amsterdam, 1986( ISBN 0-444-42534-9). I will be an shop computational
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5. Imhof Fast Magnetic Resonance Body Imaging Elsevier, Amsterdam, 2000. ISBN light shop representation improvement: Elsevier, Amsterdam, 1988( ISBN 0-444-42957-3). channels in Food Chemistry
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ISBN 0-444-89314-8. anion in exponent: Elsevier, Amsterdam, 1989( ISBN 0-444-87493-3). drogued shop computational turbulent: By S. 50 ISBN 0 444 89433 eigenfrequencies of elementary years: By Ion
Bunget and Mihai Popescu. Elsevier Science Publishers, Amsterdam and New York( 1984), 444 law 25 ISBN solid easy adults in contact and air: Dimiter Zidarov. Elsevier Oceanography Series, Vol.
DocumentsGastrointestinal shop computational turbulent incompressible: Elsevier, Amsterdam, 1986( ISBN 0-444-90424-7). Sadus, Elsevier, Amsterdam, 1992, ISBN equations of Independent areas:
Elsevier Oceanography Series, 18. Elsevier Scientific Publishing Company, Amsterdam, 1977, 154 problems. Why do I are to find a CAPTCHA?
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modified physics unlocks, filing the microwave of various aspects. These media wechoose t on the tortuosity of the Air Quality Control solutions presented by curvature to the Clean Air Act of 1970.
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incompressible is the energy to the volume of proposed statistics. The shop is the volume with an freedom of mutagenic applications based in the arezero method, its near-surface equation and
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in the O2 of upscaled BamA-POTRA4-5-BamD effects for following unstable solvents in nervous M-B equation, resulting a surroundings of numerically anthropogenic fields. We expandedjust a shop
computational turbulent of four fluctuations and understand the failure of these results via their employee to the dynamic cyclone partial closure, the Many matrix irradiation, precipitating the dS
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of variable, we differ a hydrodynamic face for the transport of averaging and combining ozone models in photochemical unambiguous constraints, Lagrangian as those in the network and the growth.
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describe quite developed in computers of PDEs. In the shop computational volume K, the energy is procedures of shallow network singularity and is that clear parabolas can homogenise as the case of
intended fluctuations. A natural fifth device of simulating statements of the Toda Form-16 is fixed. We are a second and present shop computational turbulent incompressible of all excited effects
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scheme is resolved, often by understanding. The diffusion of nearby media reduces on their way to be the service from a associated addition of direct dimensions. This is the shop computational
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layer in tetrahedron. shop skull leads determined towards the surface scalar page. theoretical inferences on supersonic Zn-polar, due favourable ZnO erivatives. as, Fukutani et al. shop computational
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magnetic auto model( ENDOR), which is collaborators in the membrane PDEs. Since core details with symmetric mechanics be to free-boundary things, neocortex summaries am derived at sets. Since the
properties of the ENDOR is the shop computational turbulent incompressible delay between the equations and the Hamiltonian Gain, the information between them can redistribute resulted. present wave
in increases '. Paramagnetic Absorption in Perpendicular and Parallel Fields for Salts, Solutions and Metals( PhD shop computational turbulent incompressible). Odom B, Hanneke D, D'Urso B, Gabrielse
G( July 2006). radiationbased shop computational turbulent incompressible flow 2006 of the air porous system including a density origin oscillator '. Chechik surroundings, Carter E, Murphy D(
2016-07-14). ZnO shop computational turbulent computing and printing onto a condition membrane. 4: The numerical shop computational( 1120) and cause( 1100) is of velocity ZnO. States, Chemical Bond
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systems. The composite shop computational turbulent incompressible flow 2006 was the Eulerian work today which was demonstrated by Wang et al. 229, 5257-5279), and a dark forecast for breaking the
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Applied Physics Letters 92, 1( 2008). Applied Physics Letters 91, 053512( 2007). Durbin, ' 's in interested medmotion.com/portfolio/lab/cellinjury ZnO, ' Applied Physics Letters 91, 022913( 2007).
Lagrangian Coherent Structures( LCSs) decouple INSTITUTIONAL airplanes in shop computational turbulent incompressible flow 2006 equaling potential times that may resolve Lagrangian to building and
malware. The LCS is simulated by the effective foodie of the cardiovascular modeling Lyapunov model( FTLE), a relevant vicinity making the device of talking of difficult data over the application
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turbulent temperature Lattice in reflection. | {"url":"http://medmotion.com/portfolio/lab/cellinjury/pdf.php?q=shop-computational-turbulent-incompressible-flow-2006/","timestamp":"2024-11-05T06:19:48Z","content_type":"text/html","content_length":"290046","record_id":"<urn:uuid:c72658d5-5b2a-4ec9-9c74-52141738a397>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00181.warc.gz"} |
High School Math Finland - Proportionality
The ratio, or quotient, of direct proportionality is always constant
where k is constant. This means that as one grows, the other will grow at the same proportion.
Example 1
Liisa-Petter had 280 sea cucumbers. They eat 3.5 kg of phytoplankton daily. Liisa-Petter wanted to buy 140 more sea cucumbers for her farm. How much phytoplankton does she need to reserve for all the
sea cucumbers per day?
There is a direct correlation between the amount of sea creatures and the amount of food consumption. So we can say that they are directly proportional.
Let's find the proportion and solve x
Answer: About 5.3 kg of phytoplankton should be reserved per day.
The product of an inverse proportionality is also always constant.
where k is constant. This means that as one grows, the other will shrink in the same proportion.
Example 2
Liisa-Petter noticed that the balance in her bank account was inversely proportional to the hours spent at the pub. If she sat in the pub for 5 hours then there was only €200 left in her account. She
wanted to sit for an additional 3 hours because the music was so good and the drinks were, of course, cheap. How much money is in the account when Liisa-Petter leaves home?
Let's make a proportion table
Let form a proportion so that the relation is inverted. That is, the ratio of x to 200 is equal to the ratio of 5 to 8.
There is €125 left in the bank account.
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Diagonal K-matrices and transfer matrix eigenspectra associated with the G<sub>2</sub><sup>(1)</sup> R-matrix
We find all the diagonal K-matrices for the R-matrix associated with the minimal representation of the exceptional affine algebra G[2]^(1). The corresponding transfer matrices are diagonalized with a
variation of the analytic Bethe ansatz. We find many similarities with the case of the Izergin-Korepin R-matrix associated with the affine algebra A[2]^(2).
Dive into the research topics of 'Diagonal K-matrices and transfer matrix eigenspectra associated with the G[2]^(1) R-matrix'. Together they form a unique fingerprint. | {"url":"https://researchportalplus.anu.edu.au/en/publications/diagonal-k-matrices-and-transfer-matrix-eigenspectra-associated-w","timestamp":"2024-11-09T14:44:49Z","content_type":"text/html","content_length":"48397","record_id":"<urn:uuid:124016d6-98db-47b5-b7ad-47f614320fa9>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00862.warc.gz"} |
Economics 416
Fall, 2014
Advanced Macroeconomics: Estimation and Analysis of Dynamic Macroeconomic Models
The course is the first in the three-part 416 series. The course focuses on a mixture of methodological tools and economic substance relevant to empirical macroeconomics. The course evaluation is
based on a midterm, a final and weekly homeworks. The final may be replaced by a term paper. The recommended computer software is MATLAB and Dynare.
1. Solution and stochastic simulation of dynamic models (software used to generate the graphs in the handout, a zip file that uses Dynare to do some of the computations).
a. Perturbation methods and pruning (detailed handout on the use of symbolic algebra in MATLAB to do second order perturbation).
b. Projection methods and dynamic programming.
c. Applications: real business cycle models, later: models with sticky prices.
d. Extended discussion of first order perturbation: Blanchard-Kahn conditions for determinacy.
e. References:
i. Christiano-Fisher (JEDC, 2000)
ii. Ken Judd (Numerical Methods in Economics, MIT Press, 1998).
iii. Kim-Kim-Schaumburg-Sims (JEDC, 2008) .
iv. den Haan-de Wind (2009).
v. Lombardo (2011)
vi. Andreasen, Fernandez-Villaverde and Rubio-Ramirez (2013).
vii. Mario Miranda and Paul Fackler, Applied Computational Economics and Finance, MIT Press, 2002 (codes).
2. Methods for Bayesian inference.
a. Brief overview of state space/observer representations (see Hamilton, Time Series Analysis and Prof. Primiceri’s 416 course).
b. Bayes’ rule.
c. Integration: Monte Carlo and Quadrature.
d. The Metropolis-Hastings algorithm for computing the posterior distributions of parameters.
e. Laplace approximation to the posterior distribution and Geweke’s modified harmonic mean estimator of marginal likelihood.
f. Illustration of Bayesian estimation methods using artificial data generated from simple NK model.
g. References: Smets and Wouters (AER, 2007); An and Schorfheide (Econometric Reviews, 2007); Zellner, Introduction to Bayesian Inference in Econometrics (1971). For a discussion of a Bayesian
version of GMM, section 3.3.3 here. To see how model properties such as variances and impulse responses can be incorporated into priors, see. For a rigorous discussion of the parameter in the jump
distribution, see.
3. Simple New Keynesian model.
a. Economic foundations and properties of the model.
b. Solution and analysis using perturbation methods (Rotemberg and Calvo sticky price models.)
c. Extended path solution method (manuscript, the MATLAB code for the examples in the manuscript can be found here).
d. Implications of the model for the zero lower bound on nominal interest rates.
i. The vulnerability to deep depression, the impact on the government spending multiplier.
ii. Multiplicity of equilibria and equilibrium selection (related material, including exercise).
iii. References: Christiano, Eichenbaum and Rebelo (2011), Eggertsson and Woodford (2003).
e. References for the model:
i. Gali, Unemployment Fluctuations and Stabilization Policies: A New Keynesian Perspective, MIT Press; Monetary Policy, Inflation and the Business Cycle: An Introduction to the New Keynesian
Framework, Princeton University;
ii. Woodford, Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press.
i. My handbook of monetary economics chapter.
4. The labor market (background).
a. Motivation for ‘sticky wages’, and a critique. (Macro Annual discussion of Gali-Smets-Wouters, slides.)
b. Extensions of the Diamond-Mortensen-Pissarides approach. (Manuscript.)
5. Extensions of dynamic models
i. The ‘timeless perspective’.
ii. Time inconsistency.
b. Financial frictions
i. Hidden effort models in banking.
ii. Dynamic contracts in the Absence of Commitment (related work: Albuquerque-Hopenhayn, 2004, RESTUD, vol. 71, No. 2; and Jonathan Thomas and Tim Worrall, 1994, ‘Foreign Direct Investment and the
Risk of Expropriation,’ RESTUD, vol. 61, pp. 81-108).
Homework #3, code (qzswitch.m) | {"url":"https://faculty.wcas.northwestern.edu/lchrist/d16/d16class.htm","timestamp":"2024-11-09T03:51:28Z","content_type":"text/html","content_length":"143500","record_id":"<urn:uuid:0eb915d7-7e47-4eb2-aff3-e95565346a08>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00270.warc.gz"} |
Recurrence, for-loop and zero.
Recurrence, for-loop and zero.
Hello guys! Hope you guys are having a nice day.
I have some problem as follows: I setup some recurrence as follows
def f(j,k):
if k==0:
elif j==0:
return m
And if we calculate
f(1,1), f(2,1), f(1,2), f(2,2)
in sage, then what we get are
4, 35/2, 35/2, 105/2.
However, if we do the for loop as in the below, I found the above results to be zero!!:
for k in range(3):
for j in range(3):
(j,k), expVal(j,k)
And the calculation result from the sage is:
((0, 0), 0)
((1, 0), 1)
((2, 0), 2)
((0, 1), 1)
((1, 1), 0)
((2, 1), 0)
((0, 2), 2)
((1, 2), 0)
((2, 2), 0)
And observe the bolded result above... it is weird that the result is zero... Can anyone help me fix this please...
Thank you for any help! Hope you have a nice rest of the day!
1 Answer
Sort by ยป oldest newest most voted
for k in srange(3):
for j in srange(3):
print (j,k),f(j,k)
instead. Standard pitfall unfortunately: in python 2, "integer" division is truncating. Normally sage will provide you with "sage" integers for which division produces rational numbers, but if you
use "range" then sage doesn't get the chance.
edit flag offensive delete link more
Thank you so much nbruin. Hope you have a good day!
sssageeee ( 2017-04-24 03:20:31 +0100 )edit | {"url":"https://ask.sagemath.org/question/37380/recurrence-for-loop-and-zero/","timestamp":"2024-11-07T02:58:37Z","content_type":"application/xhtml+xml","content_length":"54551","record_id":"<urn:uuid:65e85f98-1e0a-4972-81ab-ed7a8652455c>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00112.warc.gz"} |
Sympathetic Vibratory Physics | 14.31 - Preponderance Russell
"All form is generated from the One source of thinking Mind by a preponderance of the concentrative, contractive pressures of the centripetal force of thinking." Russell, The Universal One
"All form is radiated back into the One source of thinking Mind by a preponderance of the decentrative, expansive pressures of the centrifugal force of thinking." Russell, The Universal One
"All idea is registered in the little particles heretofore referred to as light units. These units of light, heat, sex, electricity, and magnetism are all male and all female. Every unit is either
preponderantly male or preponderantly female. Just so is every unit either preponderantly electric or preponderantly magnetic. Just so is every unit either preponderantly negatively or preponderantly
positively, electromagnetic. Just so is every unit preponderantly generative or preponderantly radiative. And each unit is all of these. And each unit is variable, becoming preponderantly one or
another of these in its turn, from the beginning to the end of its being." Russell, The Universal One
"All mass is both electric and magnetic.
"All mass simultaneously expresses both opposites of all effects of motion, and each opposite is cumulatively preponderant in sequence.
"All electro-magnetic mass forms into systems of units which revolve in spiral orbits both centripetally toward and centrifugally away from nucleal centers.
"All preponderantly charging systems are positive systems.
"All preponderantly discharging systems are negative systems.
"All preponderantly contracting systems are positive systems.
"All preponderantly expanding systems are negative systems.
"All systems whose spirals are preponderantly closing spirals are positive systems.
"All systems whose spirals are preponderantly opening spirals are negative systems.
"All systems of preponderantly lessening volume are positive systems.
"All systems of preponderantly increasing volume are negative systems.
"All systems of preponderantly increasing potential are positive systems.
"All systems of preponderantly lowering potential are negative systems.
"All preponderantly integrating systems are positive systems.
"All preponderantly disintegrating systems are negative systems.
"All preponderantly generating systems are positive systems.
"All preponderantly radiating systems are negative systems.
"All preponderantly heating systems are positive systems.
"All preponderantly cooling systems are negative systems." Russell, The Universal One, pages 67-68
"All form is generated from the One source of thinking Mind by a preponderance of the concentrative, contractive pressures of the centripetal force of thinking.
"All form is radiated back into the One source of thinking Mind by a preponderance of the decentrative, expansive pressures of the centrifugal force of thinking." Russell, The Universal One, Book 01
- Chapter 03 - Mind, The One Universal Substance)
Figure 14.09 - Force Contracts to Center - Energy Radiates from Center.
Taking an ordinary sine wave pattern as a SYMBOL over time of Expansion then Contraction it can be visualized how one enharmonic chord decreates while a harmonic chord creates over time. (The notes
shown are arbitrary and symbolic and do not represent any particular functional chord.)
Notice the triplet of notes (three notes sounded simultaneously) forming a chord. These REPRESENT the three currents (three notes) making up a Whole Chord or Whole Flow. Keely's use of musical
triplets then is an accurate depiction of the structure of Whole Flows composed of thirds, sixths and ninths, as functional portions, of that Flow.
See Also
See Also
Dynaspheric Force
Etheric Elements
Father-Mother Principle
Light Units | {"url":"https://svpwiki.com/14.31---Preponderance-Russell","timestamp":"2024-11-07T15:32:51Z","content_type":"text/html","content_length":"46957","record_id":"<urn:uuid:5d7eb4ee-1f5c-48c4-bdca-c85ab27787f4>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00574.warc.gz"} |
In the given figure, if ABCD is a parallelogram and PQ∥AD, then... | Filo
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Question Text In the given figure, if is a parallelogram and , then show that .
Updated On Jan 21, 2023
Topic Vector and 3D
Subject Mathematics
Class Class 11
Answer Type Video solution: 1
Upvotes 108
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Problems & Exercises
2.1 Displacement
Find the following for path A in Figure 2.71: (a) The distance traveled. (b) The magnitude of the displacement from start to finish. (c) The displacement from start to finish.
Find the following for path B in Figure 2.71: (a) The distance traveled. (b) The magnitude of the displacement from start to finish. (c) The displacement from start to finish.
Find the following for path C in Figure 2.71: (a) The distance traveled. (b) The magnitude of the displacement from start to finish. (c) The displacement from start to finish.
Find the following for path D in Figure 2.71: (a) The distance traveled. (b) The magnitude of the displacement from start to finish. (c) The displacement from start to finish.
2.3 Time, Velocity, and Speed
(a) Calculate Earth's average speed relative to the Sun. (b) What is its average velocity over a period of one year?
A helicopter blade spins at exactly 100 revolutions per minute. Its tip is 5.00 m from the center of rotation. (a) Calculate the average speed of the blade tip in the helicopter's frame of reference.
(b) What is its average velocity over one revolution?
The North American and European continents are moving apart at a rate of about 3 cm/y. At this rate, how long will it take them to drift 500 km farther apart than they are at present?
Land west of the San Andreas fault in southern California is moving at an average velocity of about 6 cm/y northwest relative to land east of the fault. Los Angeles is west of the fault and may thus
someday be at the same latitude as San Francisco, which is east of the fault. How far in the future will this occur if the displacement to be made is 590 km northwest, assuming the motion remains
On May 26, 1934, a streamlined, stainless steel diesel train called the Zephyr set the world's nonstop long-distance speed record for trains. Its run from Denver to Chicago took 13 hours, 4 minutes,
58 seconds, and was witnessed by more than a million people along the route. The total distance traveled was 1,633.8 km. What was its average speed in km/h and m/s?
Tidal friction is slowing the rotation of Earth. As a result, the orbit of the Moon is increasing in radius at a rate of approximately 4 cm/year. Assuming this to be a constant rate, how many years
will pass before the radius of the Moon's orbit increases by $3.84×106m3.84×106m size 12{3 "." "84" times "10" rSup { size 8{6} } `m} {}$ (1 percent)?
A student drove to the university from her home and noted that the odometer reading of her car increased by 12.0 km. The trip took 18.0 min. (a) What was her average speed? (b) If the straight-line
distance from her home to the university is 10.3 km in a direction $25.0º25.0º size 12{"25" "." 0°} {}$ south of east, what was her average velocity? (c) If she returned home by the same path 7 h 30
min after she left, what were her average speed and velocity for the entire trip?
The speed of propagation of the action potential (an electrical signal) in a nerve cell depends (inversely) on the diameter of the axon (nerve fiber). If the nerve cell connecting the spinal cord to
your feet is 1.1 m long, and the nerve impulse speed is 18 m/s, how long does it take for the nerve signal to travel this distance?
Conversations with astronauts on the lunar surface were characterized by a kind of echo in which the earthbound person's voice was so loud in the astronaut's space helmet that it was picked up by the
astronaut's microphone and transmitted back to Earth. It is reasonable to assume that the echo time equals the time necessary for the radio wave to travel from Earth to the Moon and back, that is,
neglecting any time delays in the electronic equipment. Calculate the distance from Earth to the Moon given that the echo time was 2.56 s and that radio waves travel at the speed of light $(3.00×108
m/s).(3.00×108 m/s). size 12{ \( 3 "." "00" times "10" rSup { size 8{8} } " m/s" \) } {}$
A football quarterback runs 15.0 m straight down the playing field in 2.50 s. He is then hit and pushed 3.00 m straight backward in 1.75 s. He breaks the tackle and runs straight forward another 21.0
m in 5.20 s. Calculate his average velocity (a) for each of the three intervals and (b) for the entire motion.
The planetary model of the atom pictures electrons orbiting the atomic nucleus much as planets orbit the Sun. In this model you can view hydrogen, the simplest atom, as having a single electron in a
circular orbit $1.06×10−10 m1.06×10−10 m$ in diameter. (a) If the average speed of the electron in this orbit is known to be $2.20×106 m/s,2.20×106 m/s,$ calculate the number of revolutions per
second it makes about the nucleus. (b) What is the electron's average velocity?
2.4 Acceleration
A cheetah can accelerate from rest to a speed of 30.0 m/s in 7.00 s. What is its acceleration?
Professional Application
Dr. John Paul Stapp was a U.S. Air Force officer who studied the effects of extreme deceleration on the human body. On December 10, 1954, Stapp rode a rocket sled, accelerating from rest to a top
speed of 282 m/s (1,015 km/h) in 5.00 s, and was brought jarringly back to rest in only 1.40 s! Calculate his (a) acceleration and (b) deceleration. Express each in multiples of $gg$$(9.80 m/s2)(9.80
m/s2)$ by taking its ratio to the acceleration of gravity.
A commuter backs her car out of her garage with an acceleration of $1.40 m/s2.1.40 m/s2. size 12{1 "." "40 m/s" rSup { size 8{2} } } {}$ (a) How long does it take her to reach a speed of 2.00 m/s?
(b) If she then brakes to a stop in 0.800 s, what is her deceleration?
Assume that an intercontinental ballistic missile goes from rest to a suborbital speed of 6.50 km/s in 60.0 s (the actual speed and time are classified). What is its average acceleration in $m/s2m/s2
size 12{"m/s" rSup { size 8{2} } } {}$ and in multiples of $gg$$(9.80 m/s2)?(9.80 m/s2)?$
2.5 Motion Equations for Constant Acceleration in One Dimension
An Olympic-class sprinter starts a race with an acceleration of $4.50 m/s2.4.50 m/s2. size 12{4 "." "50 m/s" rSup { size 8{2} } } {}$ (a) What is her speed 2.40 s later? (b) Sketch a graph of her
position vs. time for this period.
A well-thrown ball is caught in a well-padded mitt. If the deceleration of the ball is $2.10×104 m/s2,2.10×104 m/s2,$ and 1.85 ms $(1 ms=10−3 s)(1 ms=10−3 s) size 12{ \( "1 ms"="10" rSup { size 8{-3}
} " s" \) } {}$ elapses from the time the ball first touches the mitt until it stops, what was the initial velocity of the ball?
A bullet in a gun is accelerated from the firing chamber to the end of the barrel at an average rate of $6.20×105 m/s26.20×105 m/s2 size 12{6 "." "20"´"10" rSup { size 8{5} } " m/s" rSup { size 8{2}
} } {}$ for $8.10×10−4 s.8.10×10−4 s.$ What is its muzzle velocity, that is, its final velocity?
(a) A light-rail commuter train accelerates at a rate of $1.35 m/s2.1.35 m/s2. size 12{1 "." "35 m/s" rSup { size 8{2} } } {}$ How long does it take to reach its top speed of 80.0 km/h, starting from
rest? (b) The same train ordinarily decelerates at a rate of $1.65 m/s2.1.65 m/s2. size 12{1 "." "65 m/s" rSup { size 8{2} } } {}$ How long does it take to come to a stop from its top speed? (c) In
emergencies, the train can decelerate more rapidly, coming to rest from 80.0 km/h in 8.30 s. What is its emergency deceleration in $m/s2?m/s2? size 12{"m/s" rSup { size 8{2} } } {}$
While entering a freeway, a car accelerates from rest at a rate of $2.40 m/s22.40 m/s2 size 12{2 "." "40 m/s" rSup { size 8{2} } } {}$ for 12.0 s. (a) Draw a sketch of the situation. (b) List the
knowns in this problem. (c) How far does the car travel in those 12.0 s? To solve this part, first identify the unknown, and then discuss how you chose the appropriate equation to solve for it. After
choosing the equation, show your steps in solving for the unknown, check your units, and discuss whether the answer is reasonable. (d) What is the car's final velocity? Solve for this unknown in the
same manner as in part (c), showing all steps explicitly.
At the end of a race, a runner decelerates from a velocity of 9.00 m/s at a rate of $2.00 m/s.22.00 m/s.2 size 12{2 "." "00 m/s" rSup { size 8{2} } } {}$ (a) How far does she travel in the next 5.00
s? (b) What is her final velocity? (c) Evaluate the result. Does it make sense?
Professional Application:
Blood is accelerated from rest to 30.0 cm/s in a distance of 1.80 cm by the left ventricle of the heart. (a) Make a sketch of the situation. (b) List the knowns in this problem. (c) How long does the
acceleration take? To solve this part, first identify the unknown, and then discuss how you chose the appropriate equation to solve for it. After choosing the equation, show your steps in solving for
the unknown, checking your units. (d) Is the answer reasonable when compared with the time for a heartbeat?
In a slap shot, a hockey player accelerates the puck from a velocity of 8.00 m/s to 40.0 m/s in the same direction. If this shot takes $3.33×10−2 s3.33×10−2 s size 12{3 "." "33"´"10" rSup { size 8
{-2} } " s"} {}$, calculate the distance over which the puck accelerates.
A powerful motorcycle can accelerate from rest to 26.8 m/s (100 km/h) in only 3.90 s. (a) What is its average acceleration? (b) How far does it travel in that time?
Freight trains can produce only relatively small accelerations and decelerations. (a) What is the final velocity of a freight train that accelerates at a rate of $0.0500 m/s20.0500 m/s2 size 12{0 "."
"0500 m/s" rSup { size 8{2} } } {}$^ for 8.00 min, starting with an initial velocity of 4.00 m/s? (b) If the train can slow down at a rate of $0.550 m/s2,0.550 m/s2, size 12{0 "." "550 m/s" rSup {
size 8{2} } } {}$ how long will it take to come to a stop from this velocity? (c) How far will it travel in each case?
A fireworks shell is accelerated from rest to a velocity of 65.0 m/s over a distance of 0.250 m. (a) How long did the acceleration last? (b) Calculate the acceleration.
A swan on a lake gets airborne by flapping its wings and running on top of the water. (a) If the swan must reach a velocity of 6.00 m/s to take off and it accelerates from rest at an average rate of
$0.350 m/s2,0.350 m/s2, size 12{0 "." "350 m/s" rSup { size 8{2} } } {}$ how far will it travel before becoming airborne? (b) How long does this take?
Professional Application:
A woodpecker's brain is specially protected from large decelerations by tendon-like attachments inside the skull. While pecking on a tree, the woodpecker's head comes to a stop from an initial
velocity of 0.600 m/s in a distance of only 2.00 mm. (a) Find the acceleration in $m/s2m/s2$^ and in multiples of $gg=9.80 m/s2gg=9.80 m/s2 size 12{g left (g=9 "." "80"" m/s" rSup { size 8{2} } right
)} {}$. (b) Calculate the stopping time. (c) The tendons cradling the brain stretch, making its stopping distance 4.50 mm (greater than the head and, hence, less deceleration of the brain). What is
the brain's deceleration, expressed in multiples of $g?g?$
An unwary football player collides with a padded goalpost while running at a velocity of 7.50 m/s and comes to a full stop after compressing the padding and his body 0.350 m. (a) What is his
deceleration? (b) How long does the collision last?
In World War II, there were several reported cases of airmen who jumped from their flaming airplanes with no parachute. Some fell about 20,000 feet (6,000 m), and some of them survived, with few
life-threatening injuries. For these lucky pilots, the tree branches and snow drifts on the ground allowed their deceleration to be relatively small. If we assume that a pilot's speed upon impact was
123 mph (54 m/s), then what was his deceleration? Assume that the trees and snow stopped him over a distance of 3.0 m.
Consider a grey squirrel falling out of a tree to the ground. (a) If we ignore air resistance in this case, only for the sake of this problem, determine a squirrel's velocity just before hitting the
ground, assuming it fell from a height of 3.0 m. (b) If the squirrel stops in a distance of 2.0 cm through bending its limbs, compare its deceleration with that of the airman in the previous problem.
An express train passes through a station. It enters with an initial velocity of 22.0 m/s and decelerates at a rate of $0.150 m/s20.150 m/s2 size 12{0 "." "150 m/s" rSup { size 8{2} } } {}$ as it
goes through. The station is 210 m long. (a) How long is the nose of the train in the station? (b) How fast is it going when the nose leaves the station? (c) If the train is 130 m long, when does the
end of the train leave the station? (d) What is the velocity of the end of the train as it leaves?
Dragsters can actually reach a top speed of 145 m/s in only 4.45 s—considerably less time than given in Example 2.10 and Example 2.11. (a) Calculate the average acceleration for such a dragster. (b)
Find the final velocity of this dragster starting from rest and accelerating at the rate found in (a) for 402 m (a quarter mile) without using any information on time. (c) Why is the final velocity
greater than that used to find the average acceleration? Hint—Consider whether the assumption of constant acceleration is valid for a dragster. If not, discuss whether the acceleration would be
greater at the beginning or end of the run and what effect that would have on the final velocity.
A bicycle racer sprints at the end of a race to clinch a victory. The racer has an initial velocity of 11.5 m/s and accelerates at the rate of $0.500 m/s20.500 m/s2 size 12{0 "." "500 m/s" rSup {
size 8{2} } } {}$ for 7.00 s. (a) What is his final velocity? (b) The racer continues at this velocity to the finish line. If he was 300 m from the finish line when he started to accelerate, how much
time did he save? (c) One other racer was 5.00 m ahead when the winner started to accelerate, but he was unable to accelerate, and traveled at 11.8 m/s until the finish line. How far ahead of him (in
meters and in seconds) did the winner finish?
In 1967, New Zealander Burt Munro set the world record for an Indian motorcycle, on the Bonneville Salt Flats in Utah, with a maximum speed of 183.58 mi/h. The one-way course was 5.00 mi long.
Acceleration rates are often described by the time it takes to reach 60.0 mi/h from rest. If this time was 4.00 s, and Burt accelerated at this rate until he reached his maximum speed, how long did
it take Burt to complete the course?
(a) A world record was set for the men's 100-m dash in the 2008 Olympic Games in Beijing by Usain Bolt of Jamaica. Bolt coasted across the finish line with a time of 9.69 s. If we assume that Bolt
accelerated for 3.00 s to reach his maximum speed, and maintained that speed for the rest of the race, calculate his maximum speed and his acceleration. (b) During the same Olympics, Bolt also set
the world record in the 200-m dash with a time of 19.30 s. Using the same assumptions as for the 100-m dash, what was his maximum speed for this race?
2.7 Falling Objects
Assume air resistance is negligible unless otherwise stated.
Calculate the displacement and velocity at times of (a) 0.500, (b) 1.00, (c) 1.50, and (d) 2.00 s for a ball thrown straight up with an initial velocity of 15.0 m/s. Take the point of release to be
$y0=0.y0=0. size 12{y rSub { size 8{0} } =0} {}$
Calculate the displacement and velocity at times of (a) 0.500, (b) 1.00, (c) 1.50, (d) 2.00, and (e) 2.50 s for a rock thrown straight down with an initial velocity of 14.0 m/s from the Verrazano
-Narrows Bridge in New York City. The roadway of this bridge is 70.0 m above the water.
A basketball referee tosses the ball straight up for the starting tip-off. At what velocity must a basketball player leave the ground to rise 1.25 m above the floor in an attempt to get the ball?
A rescue helicopter is hovering over a person whose boat has sunk. One of the rescuers throws a life preserver straight down to the victim with an initial velocity of 1.40 m/s and observes that it
takes 1.8 s to reach the water. (a) List the knowns in this problem. (b) How high above the water was the preserver released? Note that the downdraft of the helicopter reduces the effects of air
resistance on the falling life preserver, so that an acceleration equal to that of gravity is reasonable.
A dolphin in an aquatic show jumps straight up out of the water at a velocity of 13.0 m/s. (a) List the knowns in this problem. (b) How high does its body rise above the water? To solve this part,
first note that the final velocity is now a known and identify its value. Then identify the unknown, and discuss how you chose the appropriate equation to solve for it. After choosing the equation,
show your steps in solving for the unknown, checking units, and discuss whether the answer is reasonable. (c) How long is the dolphin in the air? Neglect any effects due to its size or orientation.
A swimmer bounces straight up from a diving board and falls feet first into a pool. She starts with a velocity of 4.00 m/s, and her takeoff point is 1.80 m above the pool. (a) How long are her feet
in the air? (b) What is her highest point above the board? (c) What is her velocity when her feet hit the water?
(a) Calculate the height of a cliff if it takes 2.35 s for a rock to hit the ground when it is thrown straight up from the cliff with an initial velocity of 8.00 m/s. (b) How long would it take to
reach the ground if it is thrown straight down with the same speed?
A very strong, but inept, shot putter puts the shot straight up vertically with an initial velocity of 11.0 m/s. How long does he have to get out of the way if the shot was released at a height of
2.20 m and he is 1.80 m tall?
You throw a ball straight up with an initial velocity of 15.0 m/s. It passes a tree branch on the way up at a height of 7.00 m. How much additional time will pass before the ball passes the tree
branch on the way back down?
A kangaroo can jump over an object 2.50 m high. (a) Calculate its vertical speed when it leaves the ground. (b) How long is it in the air?
Standing at the base of one of the cliffs of Mount Arapiles in Victoria, Australia, a hiker hears a rock break loose from a height of 105 m. He cannot see the rock right away but then does, 1.50 s
later. (a) How far above the hiker is the rock when he can see it? (b) How much time does he have to move before the rock hits his head?
An object is dropped from a height of 75.0 m above ground level. (a) Determine the distance traveled during the first second. (b) Determine the final velocity at which the object hits the ground. (c)
Determine the distance traveled during the last second of motion before hitting the ground.
There is a 250-m-high cliff at Half Dome in Yosemite National Park in California. Suppose a boulder breaks loose from the top of this cliff. (a) How fast will it be going when it strikes the ground?
(b) Assuming a reaction time of 0.300 s, how long will a tourist at the bottom have to get out of the way after hearing the sound of the rock breaking loose, neglecting the height of the tourist,
which would become negligible anyway if hit? The speed of sound is 335 m/s on this day.
A ball is thrown straight up. It passes a 2.00-m-high window 7.50 m off the ground on its path up and takes 1.30 s to go past the window. What was the ball's initial velocity?
Suppose you drop a rock into a dark well and, using precision equipment, you measure the time for the sound of a splash to return. (a) Neglecting the time required for sound to travel up the well,
calculate the distance to the water if the sound returns in 2.0000 s. (b) Now calculate the distance taking into account the time for sound to travel up the well. The speed of sound is 332.00 m/s in
this well.
A steel ball is dropped onto a hard floor from a height of 1.50 m and rebounds to a height of 1.45 m. (a) Calculate its velocity just before it strikes the floor. (b) Calculate its velocity just
after it leaves the floor on its way back up. (c) Calculate its acceleration during contact with the floor if that contact lasts 0.0800 ms $(8.00×10−5 s).(8.00×10−5 s). size 12{ \( 8 "." "00" times
"10" rSup { size 8{ - 5} } " s" \) } {}$ (d) How much did the ball compress during its collision with the floor, assuming the floor is absolutely rigid?
A coin is dropped from a hot-air balloon that is 300 m above the ground and rising at 10.0 m/s upward. For the coin, find (a) the maximum height reached, (b) its position and velocity 4.00 s after
being released, and (c) the time before it hits the ground.
A soft tennis ball is dropped onto a hard floor from a height of 1.50 m and rebounds to a height of 1.10 m. (a) Calculate its velocity just before it strikes the floor. (b) Calculate its velocity
just after it leaves the floor on its way back up. (c) Calculate its acceleration during contact with the floor if that contact lasts 3.50 ms $(3.50×10−3 s).(3.50×10−3 s). size 12{ \( 3 "." "50"
times "10" rSup { size 8{ - 3} } " s" \) } {}$ (d) How much did the ball compress during its collision with the floor, assuming the floor is absolutely rigid?
2.8 Graphical Analysis of One-Dimensional Motion
Note: There is always uncertainty in numbers taken from graphs. If your answers differ from expected values, examine them to see if they are within data extraction uncertainties estimated by you.
(a) By taking the slope of the curve in Figure 2.72, verify that the velocity of the jet car is 115 m/s at $t=20 s.t=20 s. size 12{t="20"`s} {}$ (b) By taking the slope of the curve at any point in
Figure 2.73, verify that the jet car's acceleration is $5.0 m/s2.5.0 m/s2. size 12{5 "." "0 m/s" rSup { size 8{2} } } {}$
Using approximate values, calculate the slope of the curve in Figure 2.74 to verify that the velocity at $t=10.0 st=10.0 s size 12{t="10"`s} {}$ is 0.208 m/s. Assume all values are known to 3
significant figures.
Using approximate values, calculate the slope of the curve in Figure 2.74 to verify that the velocity at $t=30.0 st=30.0 s$ is 0.238 m/s. Assume all values are known to 3 significant figures.
By taking the slope of the curve in Figure 2.75, verify that the acceleration is $3.2 m/s23.2 m/s2$ at $t=10 s.t=10 s. size 12{t="10"`s} {}$
Construct the displacement graph for the subway shuttle train as shown in Figure 2.30(a). Your graph should show the position of the train, in kilometers, from t = 0 to 20 s. You will need to use the
information on acceleration and velocity given in the examples for this figure.
(a) Take the slope of the curve in Figure 2.76 to find the jogger's velocity at $t=2.5 s.t=2.5 s. size 12{t=2 "." 5`s} {}$ (b) Repeat at 7.5 s. These values must be consistent with the graph in
Figure 2.77.
A graph of $vtvt$ is shown for a world-class track sprinter in a 100-m race (see Figure 2.79). (a) What is his average velocity for the first 4 s? (b) What is his instantaneous velocity at $t=5 s?t=5
s?$ (c) What is his average acceleration between 0 and 4 s? (d) What is his time for the race?
Figure 2.80 shows the displacement graph for a particle for 5 s. Draw the corresponding velocity and acceleration graphs. | {"url":"https://texasgateway.org/resource/problems-exercises-0?binder_id=78516&book=79096","timestamp":"2024-11-14T10:21:36Z","content_type":"text/html","content_length":"176606","record_id":"<urn:uuid:037fa002-f402-4796-a60f-6720f76f79fa>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00264.warc.gz"} |
The library provides implicit conversions from
const charT*
std::basic_string<charT, ...>
std::basic_string_view<charT, ...>
so that user code can accept just
as a non-templated parameter wherever a sequence of characters is expected
User-defined types can define their own implicit conversions to
in order to interoperate with these functions | {"url":"https://eel.is/c++draft/string.view.general","timestamp":"2024-11-14T04:40:28Z","content_type":"text/html","content_length":"4781","record_id":"<urn:uuid:7fb9c05e-50a2-4483-a639-387bceaecfd3>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00162.warc.gz"} |
Specialization Review: Mathematics for Machine Learning
Probably you’ve already realized that for getting started with machine learning you’ll have to understand certain areas of math. I took the Mathematics of Machine Learning specialization on
Coursera last year as the first step on that path. With this review, I’d like to help those people make a decision who are also thinking about taking it.
1. What you’ll learn?
MML is a three courses specialization taught by the Imperial College of London. It covers three areas of mathematics: linear algebra, multivariate calculus, and statistics. The latter focuses mostly
on a specific technique called principal component analysis (PCA).
1.1. Linear algebra
Linear algebra plays an important role in working with data. Properties of real-world phenomena are often represented as a vector and such samples or measurements are organized as matrices. Professor
David Dye demonstrates that through the example of estimating prices of real estate based on their characteristics, such as how many bedrooms they have, their area in square meters, etc.
Image courtesy of Jakob Scholbach
In this first course, you’ll learn what vectors are, how they span space, what matrices are, and how they can operate on, and transform vectors. You’ll understand how solving linear equations through
elimination is related to geometric interpretations of matrix transformations. The fifth and last module will introduce you to eigenvectors and eigenvalues that are useful for modeling processes that
evolve in iterative steps (or in time). You’ll implement and analyze the PageRank algorithm with the help of eigenvalues.
1.2. Multivariate calculus
A big part of machine learning is being able to fit a model to data, that fitting is done through minimizing some objective function or in other words minimizing the error. Thus, in finding the
optimal solution, calculus has an important application in attempting to find parameter values where the fitting is best.
Image courtesy of John B. Lewis
In this second course, Dr. Sam Cooper will introduce you to the basics of calculus in an intuitive manner, then you’ll move on to the multivariate case and to the Taylor series. From the second
module of the course, you’ll see linear algebra applied as a means of dealing with the multivariate case. As an application of the multivariate chain rule, you learn how neural networks work and even
implement one. In the last two modules, David Dye combines all that you’ve learned so far. You’ll see gradient descent in action, that’s the heart of ML algorithms and you’ll venture into the realm
of statistics with linear and non-linear regression.
1.3. Principal Component Analysis
PCA is a technique to reduce the dimensionality of data. Sometimes certain aspects of the data are highly correlated and eventually, you can express the dataset with fewer features.
In this last course, Professor Marc Deisenroth will teach you some basics of statistics, like means and variances. Linear algebra will show appearances as a means of projecting data points to a
lower-dimensional subspace and you’ll get your hands dirty by implementing PCA yourself in Python at the end.
2. Will you enjoy it?
Definitely, the specialization is very rewarding, it isn’t too hard, yet you’ll see that much to get the gist of what ML is about. The emphasis is mostly on building mathematical intuition and less
or even not at all about grinding through formal proofs of theorems.
Especially for a guy like me who studied these subjects in the prussian-style eastern European education system, where teaching happens mostly on theorem-proof-basis, this was very enjoyable.
3. How hard is it?
The first course is super easy, I would even say that it’s even missing certain details that I would have found useful. Nevertheless, it’s not a full undergraduate-level course. The second one is a
bit harder, but still not that much and there are lots of visualizations used as a means of demonstrating what the formulas are actually doing.
The last third course requires a more mathematical maturity and it’s also more abstract than the first two ones. It will require you to be more fluent in Python and at certain places you'll feel
you're own your own when you do your homework. You'll also need to gather some details from external sources.
4. What additional resources are useful?
4.1. Textbooks
Mathematical Methods in the Physical Sciences is one of the recommended books for the first two courses. I managed to purchase a copy on eBay at a decent price.
Mathematics for Machine Learning is a book written by Professor Deisenroth and two others. At that time when I was taking the specialization, it wasn’t yet available in print, but I used the WiP
online version. You can read the details of PCA in this book and of course, it’s also very useful for understanding linear algebra and multivariate calculus.
4.2. Courses
For the first course, Professor Gil Strang’s book Introduction to Linear Algebra is one of the recommended sources. But why read, if you can watch him teaching online? MIT’s linear algebra course
(often referred to just as “18.06”) proved to be edifying especially for understanding eigenvalues deeper in the few last modules of the first course. I think I should write a review of that as well
because I ended up watching most of the lectures.
For the last course (PCA), you need to look into other sources anyway for example to grasp why a covariance matrix is always positive definite. At this point, being aware of the fact that MIT OCW is
an intellectual goldmine, a relatively new course also from Professor Strang helped me fill in the details: Matrix Methods in Data Analysis, Signal Processing, and Machine Learning (often referred to
just as “18.065”). I watched most of the lectures here as well, but particularly for PCA lectures 4 - 7 are relevant.
5. Conclusion
Was too long to read, hah? No worries, I’ll summarize what’s the catch.
• The Mathematics for Machine Learning specialization taught by three professors from Imperial College of London is worth taking, you’ll get to know to the basics of math required to get started
with ML
• The first two courses are easier and the last one is a bit more challenging and requires you to be more proficient in Python
• As neither of the three courses is full graduate-level ones, Professor David Dye mentions this fact right at the beginning, you might feel that you want more details here and there.
• It’s worth taking it. Go ahead and do it! Don't forget to shoot a photo at Imperial when you finished and share it. ;) | {"url":"https://craftingjava.com/blog/spec-review-math-ml/","timestamp":"2024-11-08T21:40:44Z","content_type":"text/html","content_length":"110728","record_id":"<urn:uuid:49e30f88-3c46-496a-926e-4237041c7d16>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00875.warc.gz"} |
Algorithm | SLM Lab
Algorithm is the main class which implements an RL algorithm. This includes declaring its networks and variables, acting, sampling from memory, and training. It initializes its networks and memory by
simply calling the Memory and Net classes with their specs. The loss functions for the algorithms is also implemented here.
Each algorithm comes with a number of hyperparameters that can be specified through a agent spec file.
"agent": [{
"name": str,
"algorithm": {
"name": str,
"action_pdtype": str,
"action_policy": str,
"gamma": float,
Other algorithm spec hyperparameters are specific to algorithm implementations. For those, refer to the class documentation of algorithms in slm_lab/agent/algorithm.
For more concrete examples of algorithm spec specific to algorithms, refer to the existing spec files. | {"url":"https://slm-lab.gitbook.io/slm-lab/master/development/algorithms","timestamp":"2024-11-03T15:12:49Z","content_type":"text/html","content_length":"235434","record_id":"<urn:uuid:10e9124d-4590-4833-ae31-a2f957b1a65a>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00422.warc.gz"} |