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A simple directed graph

In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs.

Definition

In formal terms, a directed graph is an ordered pair G = (V, A) where[1]

• V is a set whose elements are called vertices, nodes, or points;
• A is a set of ordered pairs of vertices, called arcs, directed edges (sometimes simply edges with the corresponding set named E instead of A), arrows, or directed lines.

It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, which are usually called edges, links or lines.

The aforementioned definition does not allow a directed graph to have multiple arrows with the same source and target nodes, but some authors consider a broader definition that allows directed graphs to have such multiple arcs (namely, they allow the arc set to be a multiset). Sometimes these entities are called directed multigraphs (or multidigraphs).
On the other hand, the aforementioned definition allows a directed graph to have loops (that is, arcs that directly connect nodes with themselves), but some authors consider a narrower definition that does not allow directed graphs to have loops.[2] Directed graphs without loops may be called simple directed graphs, while directed graphs with loops may be called loop-digraphs (see section Types of directed graph).

Types of directed graphs

Subclasses

A simple directed acyclic graph
A tournament on 4 vertices
• Symmetric directed graphs are directed graphs where all edges appear twice, one in each direction (that is, for every arrow that belongs to the digraph, the corresponding inverse arrow also belongs to it). (Such an edge is sometimes called "bidirected" and such graphs are sometimes called "bidirected", but this conflicts with the meaning for bidirected graphs.)
• Simple directed graphs are directed graphs that have no loops (arrows that directly connect vertices to themselves) and no multiple arrows with same source and target nodes. As already introduced, in case of multiple arrows the entity is usually addressed as directed multigraph. Some authors describe digraphs with loops as loop-digraphs.[2]
• Complete directed graphs are simple directed graphs where each pair of vertices is joined by a symmetric pair of directed arcs (it is equivalent to an undirected complete graph with the edges replaced by pairs of inverse arcs). It follows that a complete digraph is symmetric.
• Semicomplete multipartite digraphs are simple digraphs in which the vertex set is partitioned into sets such that for every pair of vertices x and y in different sets, there is an arc between x and y. Note that there can be one arc between x and y or two arcs in opposite directions. [3]
• Semicomplete digraphs are simple digraphs where there is an arc between each pair of vertices. Every semicomplete digraph is a semicomplete multipartite digraph in a trivial way, with each vertex constituting a set of the partition. [4]
• Quasi-transitive digraphs are simple digraphs where for every triple x, y, z of distinct vertices with arcs from x to y and from y to z, there is an arc between x and z. Note that there can be just one arc between x and z or two arcs in opposite directions. A semicomplete digraph is a quasi-transitive digraph. There are extensions of quasi-transitive digraphs called k-quasi-transitive digraphs. [5]
• Oriented graphs are directed graphs having no opposite pairs of directed edges (i.e. at most one of (x, y) and (y, x) may be arrows of the graph). It follows that a directed graph is an oriented graph if and only if it has no 2-cycle.[6] (This is not the only meaning of "oriented graph"; see Orientation (graph theory).)
• Tournaments are oriented graphs obtained by choosing a direction for each edge in undirected complete graphs. Note that a tournament is a semicomplete digraph. [7]
• A directed graph is acyclic if it has no directed cycles. The usual name for such a digraph is directed acyclic graph (DAG).[8]
• Multitrees are DAGs in which there are no two distinct directed paths from the same starting vertex to the same ending vertex.
• Oriented trees or polytrees are DAGs formed by orienting the edges of trees (connected, acyclic undirected graphs).
• Rooted trees are oriented trees in which all edges of the underlying undirected tree are directed either away from or towards the root (they are called, respectively, arborescences or out-trees, and in-trees.

Digraphs with supplementary properties

This list is incomplete; you can help by adding missing items. (August 2016)
• Weighted directed graphs (also known as directed networks) are (simple) directed graphs with weights assigned to their arrows, similarly to weighted graphs (which are also known as undirected networks or weighted networks).[2]
• Flow networks are weighted directed graphs where two nodes are distinguished, a source and a sink.
• Rooted directed graphs (also known as flow graphs) are digraphs in which a vertex has been distinguished as the root.
• Control-flow graphs are rooted digraphs used in computer science as a representation of the paths that might be traversed through a program during its execution.
• Signal-flow graphs are directed graphs in which nodes represent system variables and branches (edges, arcs, or arrows) represent functional connections between pairs of nodes.
• Flow graphs are digraphs associated with a set of linear algebraic or differential equations.
• State diagrams are directed multigraphs that represent finite state machines.
• Commutative diagrams are digraphs used in category theory, where the vertices represent (mathematical) objects and the arrows represent morphisms, with the property that all directed paths with the same start and endpoints lead to the same result by composition.
• In the theory of Lie groups, a quiver Q is a directed graph serving as the domain of, and thus characterizing the shape of, a representation V defined as a functor, specifically an object of the functor category FinVctKF(Q) where F(Q) is the free category on Q consisting of paths in Q and FinVctK is the category of finite-dimensional vector spaces over a field K. Representations of a quiver label its vertices with vector spaces and its edges (and hence paths) compatibly with linear transformations between them, and transform via natural transformations.

Basic terminology

Oriented graph with corresponding incidence matrix

An arc (x, y) is considered to be directed from x to y; y is called the head and x is called the tail of the arc; y is said to be a direct successor of x and x is said to be a direct predecessor of y. If a path leads from x to y, then y is said to be a successor of x and reachable from x, and x is said to be a predecessor of y. The arc (y, x) is called the reversed arc of (x, y).

The adjacency matrix of a multidigraph with loops is the integer-valued matrix with rows and columns corresponding to the vertices, where a nondiagonal entry aij is the number of arcs from vertex i to vertex j, and the diagonal entry aii is the number of loops at vertex i. The adjacency matrix of a directed graph is a logical matrix, and is unique up to permutation of rows and columns.

Another matrix representation for a directed graph is its incidence matrix.

See direction for more definitions.

Indegree and outdegree

A directed graph with vertices labeled (indegree, outdegree)

For a vertex, the number of head ends adjacent to a vertex is called the indegree of the vertex and the number of tail ends adjacent to a vertex is its outdegree (called branching factor in trees).

Let G = (V, A) and vV. The indegree of v is denoted deg(v) and its outdegree is denoted deg+(v).

A vertex with deg(v) = 0 is called a source, as it is the origin of each of its outcoming arcs. Similarly, a vertex with deg+(v) = 0 is called a sink, since it is the end of each of its incoming arcs.

The degree sum formula states that, for a directed graph,

${\displaystyle \sum _{v\in V}\deg ^{-}(v)=\sum _{v\in V}\deg ^{+}(v)=|A|.}$

If for every vertex vV, deg+(v) = deg(v), the graph is called a balanced directed graph.[9]

Degree sequence

The degree sequence of a directed graph is the list of its indegree and outdegree pairs; for the above example we have degree sequence ((2, 0), (2, 2), (0, 2), (1, 1)). The degree sequence is a directed graph invariant so isomorphic directed graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a directed graph; in some cases, non-isomorphic digraphs have the same degree sequence.

The directed graph realization problem is the problem of finding a directed graph with the degree sequence a given sequence of positive integer pairs. (Trailing pairs of zeros may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the directed graph.) A sequence which is the degree sequence of some directed graph, i.e. for which the directed graph realization problem has a solution, is called a directed graphic or directed graphical sequence. This problem can either be solved by the Kleitman–Wang algorithm or by the Fulkerson–Chen–Anstee theorem.

Directed graph connectivity

 Main article: Connectivity (graph theory)

A directed graph is weakly connected (or just connected[10]) if the undirected underlying graph obtained by replacing all directed edges of the graph with undirected edges is a connected graph.

A directed graph is strongly connected or strong if it contains a directed path from x to y (and from y to x) for every pair of vertices (x, y). The strong components are the maximal strongly connected subgraphs.

A connected rooted graph (or flow graph) is one where there exists a directed path to every vertex from a distinguished root vertex.

Notes

1. ^ Bang-Jensen & Gutin (2000). Bang-Jensen & Gutin (2018), Chapter 1.Diestel (2005), Section 1.10. Bondy & Murty (1976), Section 10.
2. ^ a b c Chartrand, Gary (1977). Introductory Graph Theory. Courier Corporation. ISBN 9780486247755. Archived from the original on 2023-02-04. Retrieved 2020-10-02.
3. ^ Bang-Jensen & Gutin (2018), Chapter 7 by Yeo.
4. ^ Bang-Jensen & Gutin (2018), Chapter 2 by Bang-Jensen and Havet.
5. ^ Bang-Jensen & Gutin (2018), Chapter 8 by Galeana-Sanchez and Hernandez-Cruz.
6. ^ Diestel (2005), Section 1.10.
7. ^ Bang-Jensen & Gutin (2018), Chapter 2 by Bang-Jensen and Havet.
8. ^ Bang-Jensen & Gutin (2018), Chapter 3 by Gutin.
9. ^ Satyanarayana, Bhavanari; Prasad, Kuncham Syam, Discrete Mathematics and Graph Theory, PHI Learning Pvt. Ltd., p. 460, ISBN 978-81-203-3842-5; Brualdi, Richard A. (2006), Combinatorial Matrix Classes, Encyclopedia of Mathematics and Its Applications, vol. 108, Cambridge University Press, p. 51, ISBN 978-0-521-86565-4.
10. ^ Bang-Jensen & Gutin (2000) p. 19 in the 2007 edition; p. 20 in the 2nd edition (2009).


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# Tag Info

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format: plain


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# Rationalizing a Binomial Denominator with Radicals

There is an unspoken law in math that a radical cannot be left in the denominator. The process of eliminating the radical from the denominator is called rationalizing. When the denominator is a binomial (two terms) the conjugate of the denominator has to be used to rationalize.

Let's start be reviewing conjugate.

$3{+}\sqrt{2}$ is a binomial with a radical

$3{-}\sqrt{2}$ the conjugate (change the sign in the middle)

Example 1

• $\frac{4}{\sqrt{5}-3}$

= $\frac{4}{\left(\sqrt{5}-3\right)}.\frac{\left(\sqrt{5}+3\right)}{\left(\sqrt{5}+3\right)}$ multiply the numerator and denominator by the conjugate of the     denominator

= $\frac{4\sqrt{5}+12}{5+3\sqrt{5}-3\sqrt{5}-9}$  use the distributive property to simplify the top and bottom

= $\frac{4\sqrt{5}+12}{-4}$ combine like terms and notice that by multiplying by the conjugate      that radicals are eliminated in the denominator

= $\frac{4\sqrt{5}}{-4}+\frac{12}{-4}$ prepare to reduce fractions

= $-\sqrt{5}-3$ reduce fractions

Or

= answer written in equivalent a+bi form

Example 2

= multiply the numerator and denominator by the conjugate of the        denominator

=   use the distributive property to simplify the top and bottom

= $\frac{8+5\sqrt{2}}{7}$   combine like terms and notice that by multiplying by the conjugate       that radicals are eliminated in the denominator

Or

= $\frac{8}{7}+\frac{5\sqrt{2}}{7}$ answer written in equivalent a+bi form

To rationalize a radical expression, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial is obtained by changing the middle sign to its opposite.


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Algebraic Data Types: Kinds and cardinality

Let's get this out of the way before we go any further: This is a massive chapter. There is a so much content in here, and there is no way that I'll be able to summarize it all in one neatly packaged post. Instead, I'll be focusing on the things that struck me as particularly important: kinds and cardinality.

That means that in exchange for covering a few topics more thoroughly, I will not be devoting many words to /newtypes/, arity, what sum and product types are (see chapter 4 and chapter 9), or constructing and deconstructing data types (see chapter 7a).

## Kinds

This section has several references to the data type known as ~Maybe~. We have not covered this data type previously in this series, so a short introduction is in order if you're not already familiar with it.

~Maybe~ is used to model data that you may or may not have. It has two data constructors: ~Just~ and ~Nothing~, takes one type parameter, and is in some ways similar to how a lot of other languages have a ~null~ value, though more robust. It is similar to the ~Option~ type in other languages such as Rust, OCaml, and F#.

The data declaration is as follows:
data Maybe a = Just a | Nothing
deriving (Eq, Ord)
#+end_src

For more information, please see the Haskell wiki article.

Kinds are to types what types are to terms. Or, put differently, a kind can be seen as a type constructor's type or as a type one level up. It sounds convoluted, but bear with me.

Keep in mind that, as covered in chapter 4, a type constructor is one of two constructor types in Haskell (the other being data constructors). It is the name of the type. Type constructors are used only at the type level, in type signatures, and in typeclass declarations and instances.

We represent kinds with the * symbol. If you have a fully applied, concrete type constructor, such as Bool or [Int], then the kind is *. If you have a type constructor that is still waiting for a type parameter---such as Maybe---then that has the kind * -> *. A kind is not a type until it is fully applied. In other words, Maybe isn't a type, but Maybe String is.

Second, we only talk about kinds on the type level. That is, Bool is a type and has kind *. False is a term (or value) and does not have a kind. For data constructors that share names with their type constructors (e.g. []), this can be confusing, but remember that kinds only operate on type level constructors.

To make this clearer, let's turn to the ever faithful GHCi. To print the kind of a type constructor, we use the :k (or :kind) command followed by the constructor we want information on. Let's have a couple of examples and see what we can learn.

Prelude> :k Bool
Bool :: *

Bool is a fully applied type in and of itself. it takes no type parameters and as such has kind *.

Prelude> :k Maybe
Maybe :: * -> *

Maybe is our first higher-kinded type (more on that in a bit), which means it's a type constructor that requires another type before it's complete. In other words, you couldn't simply put Maybe in a type signature without saying what type of maybe it is.

Prelude> :k Maybe Integer
Maybe Integer :: *

Maybe Integer, on the other hand, is a concrete type. As opposed to just Maybe above, we also know what the type is (Integer). As such, it is fully applied and of kind *.

Prelude> :k (,)
(,) :: * -> * -> *

This one is quite interesting. This is the type constructor for a tuple. Without any type parameters, it's of kind * -> * -> * because it needs two types to be fully applied.

### Higher-kinded types

According to the book, there are only 'a few' kinds, and the default (and the only one covered in this chapter) is *. Kind signatures work the same as type signatures, i.e. using the same :: and -> syntax.

Higher-kinded types are types that require type parameters to be fully applied, to become real types---that is, their kind is at least * -> *. Lists, tuples, and Maybe are good examples of higher-kinded types.

## Cardinality

The cardinality of a type is the number of different values it defines. A value defined by a type is also known as an inhabitant of said type. Types' cardinalities can be anywhere from as small as 0 (empty types) and as large as infinite.

To demonstrate this concept we'll use these arbitrary sum types, modeled after the starters in the mainline Pokémon games (just pretend we're in 1999 for simplicity's sake):

data StarterType
= Grass
| Fire
| Water

data Starter
= GenI StarterType
| GenII StarterType

The cardinality of StarterType is 3 because it has only three possible inhabitants. The cardinality of Starter is 6, because it has two data constructors, each of which also require a StarterType. As such, we get 2 \cdot 3 (or 3 + 3), which is 6.

Above, the Starter type demonstrates why it's called a sum type: We can calculate the cardinality of the type by summing the number of possible values of each of it's data constructors.

If you've got a friend with a link cable, then maybe you've traded some pocket monsters and ended up with two starters---lucky you! In this case, you now have a tuple of starters, i.e. (Starter, Starter). A tuple is a product type, so that means we find the cardinality by multiplying the cardinality of each of its contained types. As such, the cardinality of (Starter, Starter) is 6 \cdot 6 = 36.

This is the basic concept of cardinality in types. It's pretty straightforward, and it's a good thing to keep in mind when designing systems.

### The function type is exponential

We've seen that sum types are the addition operator and that product types are the multiplication operator when it comes to calculating inhabitants of types. However, we haven't mentioned function types, which act as the exponent operator: given a function a -> b, the formula for the number of inhabitants is b^a. So if we have a function StarterType -> Bool, for instance, the number of possible implementations is 2^3 = 8. This relationship continues as the number of parameters increases, so for a -> b -> c it's equal to c^{ba} (or (c ^ b) ^ a).

If this is a bit confusing: don't worry, I'm right there with ya. But let's give it a go. Let's write out all possible permutations of the function StarterType -> Bool.

acceptStarter1 :: StarterType -> Bool
acceptStarter1 Grass = True
acceptStarter1 Fire = True
acceptStarter1 Water = True

acceptStarter2 :: StarterType -> Bool
acceptStarter2 Grass = True
acceptStarter2 Fire = True
acceptStarter2 Water = False

acceptStarter3 :: StarterType -> Bool
acceptStarter3 Grass = True
acceptStarter3 Fire = False
acceptStarter3 Water = True

acceptStarter4 :: StarterType -> Bool
acceptStarter4 Grass = False
acceptStarter4 Fire = True
acceptStarter4 Water = True

acceptStarter5 :: StarterType -> Bool
acceptStarter5 Grass = False
acceptStarter5 Fire = False
acceptStarter5 Water = True

acceptStarter6 :: StarterType -> Bool
acceptStarter6 Grass = False
acceptStarter6 Fire = True
acceptStarter6 Water = False

acceptStarter7 :: StarterType -> Bool
acceptStarter7 Grass = True
acceptStarter7 Fire = False
acceptStarter7 Water = False

acceptStarter8 :: StarterType -> Bool
acceptStarter8 Grass = False
acceptStarter8 Fire = False
acceptStarter8 Water = False

Whew. And that's with only two possible result values and three possible inputs. Yeah, this gets wild pretty quickly.

## Closing up

While the format is a bit different from previous posts in the series, I hope you found something useful here too. The rest of what the chapter covers is definitely interesting, but not eye-opening in the way that kinds and cardinality are. The next chapter is on handling errors and includes Maybe, Either, more higher-kindedness, and anamorphisms (if that sounds interesting, why not check out this post on corecursion and anamorphisms in the meantime?). Stay tuned, and it should land at some unspecified future date.

Until next time: stay safe.

Thomas Heartman is a developer, writer, speaker, and one of those odd people who enjoy lifting heavy things and putting them back down again. Preferably with others. Doing his best to gain and share as much knowledge as possible.


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# American Institute of Mathematical Sciences

November  2016, 21(9): 3191-3207. doi: 10.3934/dcdsb.2016093

## Takens theorem for random dynamical systems

 1 School of Mathematics, Peking University, Beijing 100871 2 Department of Mathematics, Brigham Young University, Provo, Utah 84602

Received  October 2015 Revised  January 2016 Published  October 2016

In this paper, we study random dynamical systems with partial hyperbolic fixed points and prove the smooth conjugacy theorems of Takens type based on their Lyapunov exponents.
Citation: Weigu Li, Kening Lu. Takens theorem for random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3191-3207. doi: 10.3934/dcdsb.2016093
##### References:
 [1] L. Arnold, Random Dynamical Systems,, Springer, (1998). Google Scholar [2] L. Arnold and K. Xu, Normal forms for random differential equations,, Journal of Differential Equations, 116 (1995), 484. doi: 10.1006/jdeq.1995.1045. Google Scholar [3] V. I. Arnold, Geometric Methods in the Theory of Ordinary Differential. Equations,, Springer-Verlag, (1983). Google Scholar [4] V. I. Arnold, Small denominators I. One the mapping of a circle into itself,, Izv. Akad. Nauk. Math., 25 (1961), 21. Google Scholar [5] A. Banyaga, R. de la Llave and C. E. Wayne, Cohomology equations near hyperbolic points and geometric versions of Sternberg linearization theorem,, J. Geom. Anal., 6 (1996), 613. doi: 10.1007/BF02921624. Google Scholar [6] G. R. Beliskii, Functional equations and the conjugacy of diffeomorphism of finite smoothness class,, Functional Anal. Appl., 7 (1973), 17. Google Scholar [7] G. R. Beliskii, Equivalence and normal forms of germs of smooth mappings,, Russian Math. Surveys, 33 (1978), 95. Google Scholar [8] A. D. Brjuno, Analytic form of differential equations I, II,, Trans. Mosc. Math. Soc., 25 (1971), 119. Google Scholar [9] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions,, LNM 580. Springer-Verlag, (1977). Google Scholar [10] K. T. Chen, Equivalence and decomposition of vector fields about an elementary critical point,, Amer.J. Math., 85 (1963), 693. doi: 10.2307/2373115. Google Scholar [11] S.-N. Chow, K. Lu and Y.-Q. Shen, Normal form and linearization for quasiperiodic systems,, Trans. Amer. Math. Soc., 331 (1992), 361. doi: 10.1090/S0002-9947-1992-1076612-1. Google Scholar [12] N. D. Cong, Topological Dynamics of Random Dynamical Systems,, Orford Mathematical Monographs, (1997). Google Scholar [13] M. ElBialy, Linearization of vector fields near resonant hyperbolic rest points,, J. Differential Equations, 118 (1995), 336. doi: 10.1006/jdeq.1995.1076. Google Scholar [14] D. M. Grobman, Topological classification of the neighborhood of a singular point in $n$-dimensional space,, Mat. Sb., 56 (1962), 77. Google Scholar [15] P. Guo and J. Shen, Smooth invariant manifolds for random dynamical systems,, Rocky Mountain Journal of Mathematics, (). Google Scholar [16] P. Hartman, On local homeomorphisms of Euclidean spaces,, Bol. Soc. Mat. Mexican, 5 (1960), 220. Google Scholar [17] P. Hartman, On the local linearization of differential equations,, Proc. Amer. Math. Soc., 14 (1963), 568. doi: 10.1090/S0002-9939-1963-0152718-3. Google Scholar [18] Yu. S. Ilyashenko and W. Li, Nonlocal Bifurcations. Mathematical Surveys and Monographs,, American Mathematical Society, (1999). Google Scholar [19] Yu. S. Ilyashenko and S. Yu. Yakovenko, Finitely smooth normal forms of local families of diffeomorphisms and vector fields, (Russian), Uspekhi Mat. Nauk, 46 (1991), 3. doi: 10.1070/RM1991v046n01ABEH002733. Google Scholar [20] Y. Kifer, Random Perturbations of Dynamical Systems,, Progress in Probability and Statistics, (1988). doi: 10.1007/978-1-4615-8181-9. Google Scholar [21] W. Li and K. Lu, Poincaré theorems for random dynamical systems,, Ergodic Theory Dynam. Systems, 25 (2005), 1221. doi: 10.1017/S014338570400094X. Google Scholar [22] W. Li and K. Lu, Sternberg theorems for random dynamical systems,, Comm. Pure Appl. Math., 58 (2005), 941. doi: 10.1002/cpa.20083. Google Scholar [23] W. Li and K. Lu, A Siegel theorem for random dynamical systems,, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 635. doi: 10.3934/dcdsb.2008.9.635. Google Scholar [24] P.-D. Liu, Random perturbations of Axiom A basic sets,, J. Statist. Phys., 90 (1998), 467. doi: 10.1023/A:1023280407906. Google Scholar [25] P.-D. Liu, Dynamics of Random perturbations: Smooth ergodic theory,, Ergod. Th. & Dynam. Sys., 21 (2001), 1279. doi: 10.1017/S0143385701001614. Google Scholar [26] P. Liu and M. Qian, Smooth Ergodic Theory of Random Dynamical Systems,, Lecture Notes in Mathematics, (1606). doi: 10.1007/BFb0094308. Google Scholar [27] K. R. Meyer, The implicit function theorem and analytic differential equations,, Lect. Notes in Math., 468 (1975), 191. Google Scholar [28] J. K. Moser, A rapidly covergent iteration method and nonlinear differential equations II,, Ann. Scuo. Norm. Sup. Pisa., 20 (1966), 499. Google Scholar [29] E. Nelson, Topics in Dynamics I. Flows,, Princeton University Press, (1969). Google Scholar [30] M. Nagumo and K. Isé, On the normal forms of differential equations in the neighborhood of an equilibrium point,, Osaka Math. J., 9 (1957), 221. Google Scholar [31] C. Pugh, On a theorem of P. Hartman,, Amer. J. Math., 91 (1969), 363. doi: 10.2307/2373513. Google Scholar [32] F. Takens, Partially hyperbolic fixed points,, Topology, 10 (1971), 133. doi: 10.1016/0040-9383(71)90035-8. Google Scholar [33] K. Palmer, Qualitative behavior of a system of ODE near an equilibrium point, A generalization of the Hartman- Grobman Theorem,, Technical Report, (1980). Google Scholar [34] R. Pérez-Marco, Linearization of holomorphic germs with resonant linear part,, preprint., (). Google Scholar [35] H. Poincaré, Sur le probléme des trois corps et les équations de la dynamique,, Acta Math., 13 (1890), 1. Google Scholar [36] D. Ruelle, Random Smooth Dynamical Systems,, Lecture Notes in Rutgers, (1996). Google Scholar [37] G. R. Sell, Smooth linearization near a fixed point,, Amer. J. Math., 107 (1985), 1035. doi: 10.2307/2374346. Google Scholar [38] G. R. Sell, Obstacles to linearization,, Diff. Urav., 20 (1984), 446. Google Scholar [39] C. L. Siegel, Iteration of analytic functions,, Ann. Math., 43 (1942), 607. doi: 10.2307/1968952. Google Scholar [40] C. L. Siegel, Ober die Normalform analytischer Differentialgleichungen in der Nahe einer Gleichgewichtslosung,, Nachr . Akad. Wiss. Gottingen, 1952 (1952), 21. Google Scholar [41] S. Sternberg, On the behavior of invariant curves near a hyberbolic point of a surface transformation,, Amer. J. Math., 77 (1955), 526. doi: 10.2307/2372639. Google Scholar [42] S. Sternberg, Local contractions and a theorem of Poincaré,, Amer. J. Math., 79 (1957), 809. doi: 10.2307/2372437. Google Scholar [43] T. Wanner, Linearization of random dynamical systems,, In C. Jones, 4 (1995), 203. Google Scholar [44] E. Zehnder, A simple proof of a generalization of a theorem by C. L. Siegel,, Lect. Notes in Math., 597 (1977), 855. Google Scholar [45] W. Zhang and W. Weinian, Sharpness for C1 linearization of planar hyperbolic diffeomorphisms,, J. Differential Equations, 257 (2014), 4470. doi: 10.1016/j.jde.2014.08.014. Google Scholar

show all references

##### References:
 [1] L. Arnold, Random Dynamical Systems,, Springer, (1998). Google Scholar [2] L. Arnold and K. Xu, Normal forms for random differential equations,, Journal of Differential Equations, 116 (1995), 484. doi: 10.1006/jdeq.1995.1045. Google Scholar [3] V. I. Arnold, Geometric Methods in the Theory of Ordinary Differential. Equations,, Springer-Verlag, (1983). Google Scholar [4] V. I. Arnold, Small denominators I. One the mapping of a circle into itself,, Izv. Akad. Nauk. Math., 25 (1961), 21. Google Scholar [5] A. Banyaga, R. de la Llave and C. E. Wayne, Cohomology equations near hyperbolic points and geometric versions of Sternberg linearization theorem,, J. Geom. Anal., 6 (1996), 613. doi: 10.1007/BF02921624. Google Scholar [6] G. R. Beliskii, Functional equations and the conjugacy of diffeomorphism of finite smoothness class,, Functional Anal. Appl., 7 (1973), 17. Google Scholar [7] G. R. Beliskii, Equivalence and normal forms of germs of smooth mappings,, Russian Math. Surveys, 33 (1978), 95. Google Scholar [8] A. D. Brjuno, Analytic form of differential equations I, II,, Trans. Mosc. Math. Soc., 25 (1971), 119. Google Scholar [9] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions,, LNM 580. Springer-Verlag, (1977). Google Scholar [10] K. T. Chen, Equivalence and decomposition of vector fields about an elementary critical point,, Amer.J. Math., 85 (1963), 693. doi: 10.2307/2373115. Google Scholar [11] S.-N. Chow, K. Lu and Y.-Q. Shen, Normal form and linearization for quasiperiodic systems,, Trans. Amer. Math. Soc., 331 (1992), 361. doi: 10.1090/S0002-9947-1992-1076612-1. Google Scholar [12] N. D. Cong, Topological Dynamics of Random Dynamical Systems,, Orford Mathematical Monographs, (1997). Google Scholar [13] M. ElBialy, Linearization of vector fields near resonant hyperbolic rest points,, J. Differential Equations, 118 (1995), 336. doi: 10.1006/jdeq.1995.1076. Google Scholar [14] D. M. Grobman, Topological classification of the neighborhood of a singular point in $n$-dimensional space,, Mat. Sb., 56 (1962), 77. Google Scholar [15] P. Guo and J. Shen, Smooth invariant manifolds for random dynamical systems,, Rocky Mountain Journal of Mathematics, (). Google Scholar [16] P. Hartman, On local homeomorphisms of Euclidean spaces,, Bol. Soc. Mat. Mexican, 5 (1960), 220. Google Scholar [17] P. Hartman, On the local linearization of differential equations,, Proc. Amer. Math. Soc., 14 (1963), 568. doi: 10.1090/S0002-9939-1963-0152718-3. Google Scholar [18] Yu. S. Ilyashenko and W. Li, Nonlocal Bifurcations. Mathematical Surveys and Monographs,, American Mathematical Society, (1999). Google Scholar [19] Yu. S. Ilyashenko and S. Yu. Yakovenko, Finitely smooth normal forms of local families of diffeomorphisms and vector fields, (Russian), Uspekhi Mat. Nauk, 46 (1991), 3. doi: 10.1070/RM1991v046n01ABEH002733. Google Scholar [20] Y. Kifer, Random Perturbations of Dynamical Systems,, Progress in Probability and Statistics, (1988). doi: 10.1007/978-1-4615-8181-9. Google Scholar [21] W. Li and K. Lu, Poincaré theorems for random dynamical systems,, Ergodic Theory Dynam. Systems, 25 (2005), 1221. doi: 10.1017/S014338570400094X. Google Scholar [22] W. Li and K. Lu, Sternberg theorems for random dynamical systems,, Comm. Pure Appl. Math., 58 (2005), 941. doi: 10.1002/cpa.20083. Google Scholar [23] W. Li and K. Lu, A Siegel theorem for random dynamical systems,, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 635. doi: 10.3934/dcdsb.2008.9.635. Google Scholar [24] P.-D. Liu, Random perturbations of Axiom A basic sets,, J. Statist. Phys., 90 (1998), 467. doi: 10.1023/A:1023280407906. Google Scholar [25] P.-D. Liu, Dynamics of Random perturbations: Smooth ergodic theory,, Ergod. Th. & Dynam. Sys., 21 (2001), 1279. doi: 10.1017/S0143385701001614. Google Scholar [26] P. Liu and M. Qian, Smooth Ergodic Theory of Random Dynamical Systems,, Lecture Notes in Mathematics, (1606). doi: 10.1007/BFb0094308. Google Scholar [27] K. R. Meyer, The implicit function theorem and analytic differential equations,, Lect. Notes in Math., 468 (1975), 191. Google Scholar [28] J. K. Moser, A rapidly covergent iteration method and nonlinear differential equations II,, Ann. Scuo. Norm. Sup. Pisa., 20 (1966), 499. Google Scholar [29] E. Nelson, Topics in Dynamics I. Flows,, Princeton University Press, (1969). Google Scholar [30] M. Nagumo and K. Isé, On the normal forms of differential equations in the neighborhood of an equilibrium point,, Osaka Math. J., 9 (1957), 221. Google Scholar [31] C. Pugh, On a theorem of P. Hartman,, Amer. J. Math., 91 (1969), 363. doi: 10.2307/2373513. Google Scholar [32] F. Takens, Partially hyperbolic fixed points,, Topology, 10 (1971), 133. doi: 10.1016/0040-9383(71)90035-8. Google Scholar [33] K. Palmer, Qualitative behavior of a system of ODE near an equilibrium point, A generalization of the Hartman- Grobman Theorem,, Technical Report, (1980). Google Scholar [34] R. Pérez-Marco, Linearization of holomorphic germs with resonant linear part,, preprint., (). Google Scholar [35] H. Poincaré, Sur le probléme des trois corps et les équations de la dynamique,, Acta Math., 13 (1890), 1. Google Scholar [36] D. Ruelle, Random Smooth Dynamical Systems,, Lecture Notes in Rutgers, (1996). Google Scholar [37] G. R. Sell, Smooth linearization near a fixed point,, Amer. J. Math., 107 (1985), 1035. doi: 10.2307/2374346. Google Scholar [38] G. R. Sell, Obstacles to linearization,, Diff. Urav., 20 (1984), 446. Google Scholar [39] C. L. Siegel, Iteration of analytic functions,, Ann. Math., 43 (1942), 607. doi: 10.2307/1968952. Google Scholar [40] C. L. Siegel, Ober die Normalform analytischer Differentialgleichungen in der Nahe einer Gleichgewichtslosung,, Nachr . Akad. Wiss. Gottingen, 1952 (1952), 21. Google Scholar [41] S. Sternberg, On the behavior of invariant curves near a hyberbolic point of a surface transformation,, Amer. J. Math., 77 (1955), 526. doi: 10.2307/2372639. Google Scholar [42] S. Sternberg, Local contractions and a theorem of Poincaré,, Amer. J. Math., 79 (1957), 809. doi: 10.2307/2372437. Google Scholar [43] T. Wanner, Linearization of random dynamical systems,, In C. Jones, 4 (1995), 203. Google Scholar [44] E. Zehnder, A simple proof of a generalization of a theorem by C. L. Siegel,, Lect. Notes in Math., 597 (1977), 855. Google Scholar [45] W. Zhang and W. Weinian, Sharpness for C1 linearization of planar hyperbolic diffeomorphisms,, J. Differential Equations, 257 (2014), 4470. doi: 10.1016/j.jde.2014.08.014. Google Scholar
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# Whats the absolute value of abs(4)?

$| 4 |$ is allready an absolute value, equal to $4$
So $| - 4 | = 4$ and so is $| 4 | = 4$.


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All Rights Reserved. However, a compass needle will not be steady in the magnetic meridian, because of the longitude from east to west being complete geodesic. It is called easterly (E) if the compass needle, aligned with the magnetic meridian, points eastward or to the right of true north, and westerly (W), if it points to the left What is the relation between geographic meridian and magnetic meridian? C) Every meridian is the same length and has the same beginning and end. The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. north pole (yes, north, not south - opposite poles attract each The magnetic meridian is an equivalent imaginary line connecting the magnetic south and north poles and can be taken as the horizontal component of magnetic force lines along the surface of the earth. Variation is the angular measurement between the magnetic meridian and the geographic meridian. A magnetic meridian lies parallel with the magnetic lines of force of the earth. True Meridian- sometimes known as the astronomic or geographic meridian. magnetic meridian is a line joining the two magnetic pole while geographic meridian is a line joining the two geographic poles What is a sample Christmas party welcome address? January 22, 2002 The National Scrapbooking and Ancestry Expo. The magnetic meridian is an equivalent imaginary line connecting the magnetic south and north poles and can be taken as the horizontal component of magnetic force lines along the surface of the earth. What are some samples of opening remarks for a Christmas party? Therefore, a compass needle will be parallel to the magnetic meridian. The physical quantities which determine the intensity of earth’s total magnetic field completely both in magnitude and direction are called magnetic elements. 1 – True Meridian. How many candles are on a Hanukkah menorah? A meridian connects points of equal longitude. A line having zero declination is known as Agonic line. What are the release dates for The Wonder Pets - 2006 Save the Ladybug? The Earth's geographic north pole and south pole are related Our technology know how to deliver accurately far infrared ray to target area stimulate Chinese meridian and acupuncture points is the most effective natural therapy which has been proved by thousands under user. A geographic meridian runs straight north and south, from the North pole to the South Pole. Where in California is San Jose located . A magnetic meridian is a curvy line, running from the north magnetic pole to the south magnetic pole. The true magnetic pole in the northern hemisphere presently lies in the Northwest Territories, Canada. Get answers by asking now. It is caused by the terrestrial magnetism. The angle between the magnetic and the true meridian is the magnetic declination, which is relevant for … magnetic meridian synonyms, magnetic meridian pronunciation, magnetic meridian translation, English dictionary definition of magnetic meridian. Therefore, a compass needle will be parallel to the magnetic meridian. True (Geographical) North is the direction to the North (Geographical) Pole. It is the direction of north pole indicated by magnetic needle. The old-fashioned magnetic photo albums with fading yellow pictures have been replaced by colorful, creative, informative photo albums. The 40th meridian, for example, has … The angular difference between the magnetic meridian and the true meridian is named as variation Variation is the angle between the geographic meridian and magnetic meridian . magnetic south pole, for example - don't match exactly. Angle of dip at a place is defined as the angle between the direction of intensity of earth’s magnetic field (B E) and the horizontal direction in magnetic meridian at that place. Why don't libraries smell like bookstores? A. true meridian and the axis of the compass card B. true meridian and the magnetic meridian C. magnetic meridian and the axis of the compass card D. axis of the compass card and the degaussing meridian 14 494 Ref: Compass, CE, Deviation, C to M D If the compass heading and the magnetic heading are the same then _____. And westerly or ( – ), when magnetic north is westward of geographic meridian. Define magnetic meridian. Study of Geomagnetism revealed that the Geomagnetic poles are situated slightly away from Geographic poles. This is any assume direction to a well defined object. Why do so many Europeans hate how North America considers themselves to be a separate continent when Europe is part of Asia? When did organ music become associated with baseball? points!). The magnetic field at the surface of the earth ranges from 3 × 10-5 T at the equator to 6 × 10-5 T at the poles. A geographic meridian runs straight north and south, from the North pole to the South Pole. Or else the error need to be subtracted from the 'magnetic bearing' to arrive at 'true bearing'. The angle between magnetic meridian and true meridian is known as magnetic declination. B) Every meridian is actually a circle rather than a curved line. The same Latin stem gives rise to the terms A.M. (Ante Meridian) and P.M. (Post Meridian) used to disambiguate hours of the day when using the 12-hour clock. How long will the footprints on the moon last? http://commons.wikimedia.org/wiki/File:IGRF_2000_m... Why is Canada always at the bottom in any US state rankings? On a more detailed level however, the magnetic lines are not straight, but bend and arc depen… The Earth's magnetic field is not uniform and also varies slowly over time. Magnetic declination at a place may be defined as the angle between its magnetic meridian and the earth geographic meridian at the place. A (geographic) meridian (or line of longitude) is the half of an imaginary great circle on the Earth's surface, terminated by the North Pole and the South Pole, connecting points of equal longitude, as measured in angular degrees east or west of the Prime Meridian. True and Magnetic Meridian MERIDIANS - a fixed reference line or direction which lies in a vertical plane passing through a fixed point of reference and through the observer’s position. The magnetic pole is not fixed in position but … The angle formed at any point between the magnetic meridian and the geographic meridian is called Magnetic Variation The horizontal component of the actual path … It is the generally adapted reference line in … Later it was observed that the North shown by compass is slightly different from True (Geographical) North. The position of a point along the meridian is given by its latitude. Normally it is expected that a 'magnetic compass' shows 'true' North. poles. … It is about 11.6° south of the geographic north pole, and about 104.3° west longitude. It is considered easterly or ( + ), when the magnetic north is eastward of geographic meridian. Know answer of objective question : The angle between the geographical meridian and magnetic meridian is called?. It is the reference direction of north pole of earth from a given station point. The earth acts very much like a bar magnet with a north magnetic pole located considerably south of the north pole defined by the earth’s rotational axis. Each meridian is perpendicular to all circles of latitude. [11] Therefore, a compass needle will be parallel to the magnetic meridian. The magnetic meridian is an equivalent imaginary line connecting the magnetic south and north poles and can be taken as the horizontal component of magnetic force lines along the surface of the earth. The magnetic equator passes through India, near Trivandrum, Kerala. Copyright © 2020 Multiply Media, LLC. for assessing the direction , true north or 0 degree is used as standard. Wet roads reduce traction but have no effect on braking distance.? Magnetic meridian is the line that is parallel to the direction taken by a freely moving magnetized needle. Assertion : The true geographic north direction is found by using a compass needle. Join Yahoo Answers and get 100 points today. The lines that have the same declination are known as Isogonic lines. Reason: The magnetic meridian of the earth is along the axis of rotation of the earth. There is lot of information on this and the 'error' (magnetic declination) maps on 'Mercator projection' of World in Wiki (-pedia & -media), including the link provided by the other answerer 'Morningf.. ' . Note that the magnetic field on the axis is twice at that on … Meridian, imaginary north–south line on the Earth’s surface that connects both geographic poles; it is used to indicate longitude. North pole of the earth’s magnet is in Antarctica while its south pole is in Northern Canada. Angle of Declination $(\phi)$: The angle between the magnetic meridian and geographic meridian at a place is called angle of declination. That is, they encircle the globe and converge at a common point in each hemisphere. Magnetic variation changes with a change in the vessel’s position. Answer this multiple choice objective question and get explanation and result.It is provided by OnlineTyari in English Scrapbooking is the fastest growing craft in America today. Magnetic Meridian : A vertical plane passing through the magnetic axis of a freely suspended magnet is called magnetic meridian. A meridian (or line of longitude) is an imaginary great circle on the Earth's surface connecting the North Pole and the South Pole. It has an error measured in the angle (difference between geographic meridian and magnetic meridian) to the east. The meridians run from north pole to south pole. This is the 3d animated lesson with explanation which is very interesting and easy to understand way of learning by Shiksha House. Following Pole Star gives accurate result when it is visible (in night time). See the link below for a map of magnetic meridians. It is also called geographic meridian. Meridian Energy Assessment (MEA) MEAS is a safe and easy way to provide fast, precise and non-invasive examination combining Traditional Chinese Medicine and technology to provide information on your physical energy, mental-emotional states, metabolism, musculoskeletal balance, autonomic nervous system, and your vital organs. The magnetic meridian is an equivalent imaginary line connecting the magnetic south and north poles and can be taken as the horizontal component of magnetic force lines along the surface of the earth. What does contingent mean in real estate? What is the relation between geographic meridian and magnetic meridian. to the rotation of the Earth - the axis passes through the poles. 2 – Magnetic Meridian. Still have questions? other, and "north" is where the north part of a compass needle D) Every meridian is distorted by magnetic declination. Its counterpart in the sky is 'Polaris' (Pole Star) which is the best guide. The magnetic meridian is an equivalent imaginary line connecting the magnetic south and north poles and can be taken as the magnetic force lines along the surface of the earth [1] . Magnetic compass needs to be relied on. A magnetic meridian is a curvy line, running from the north magnetic pole to the south magnetic pole. If the error is zero ('isogonal') it implies that the meridian lines coincide. Geographical meridian : A vertical plane passing through a place and geographic north and south axis is called geographic meridian. Who is the longest reigning WWE Champion of all time? The magnetic south pole, on the other hand, is near the geographic The magnetic axis of the earth is inclined at 20° to its geographical axis. Therefore, a compass needle will be parallel to the magnetic meridian. 3 – Arbitrary Meridian. Magnetic Variation or Declination is the horizontal angle formed between the geographic meridian and magnetic meridian. 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Can PCA work when the number of observations is smaller than the number of dimensions? [duplicate]

I understand how principal component analysis works. However, in a financial time series sense, I do not understand why the number of observations should be larger than the number of dimensions. I am having twenty securities (dimensions in this case) for ten time periods and I am told that PCA won't work here. Why wouldn't it work? Also, how does asymptotic principal component analysis help to solve this problem?

Can someone please explain this? If possible try to use less mathematical jargon (except for linear algebra basics such as eigenvalues/vectors). If it is not possible to explain it without using jargon, please go ahead and I will take my time and try to understand it.

• 1) Linear PCA does not require n>p because PCA is not afraid of singular data. This was answered on this site many times. 2) stats.stackexchange.com/a/43224/3277 is a question on Asymptotic PCA. This form of PCA seems to be a special way of use PCA in time series analysis. Jun 18 '14 at 7:23
• Closely related stats.stackexchange.com/questions/2772/…
– Momo
Jun 18 '14 at 11:47
• @ttnphns: Are you sure you gave the link you wanted to give? It seems unrelated to "asymptotic PCA", whatever it means. Jun 18 '14 at 22:08
• Yes, my link was erratum. The correct link, the one I really meant, is given by Momo. Thanks, @amoeba. Jun 19 '14 at 5:57
• Thanks for the thread. I have read that link and the answers there does not explain why should the number of observations be higher than the number of dimensions. I have searched this forum and I am not able to find solution to this. I'd be thankful if someone could explain this to me. Jun 19 '14 at 7:19


text:
## Kaggler 0.3.7 Released

Changes:

• Cython optimization for performance – boundscheck(False), wraparound(False), and cdivision(True) are used.
• Adaptive learning rate – instead of $\frac{1}{\sqrt{n_i} + 1}$, $\frac{1}{\sqrt{\sum{g_i^2}} + 1}$ is used where $g_i$ is the gradient of the associated weight.
• Type correction – change the type of index from double to int.

You can upgrade Kaggler either by using pip:
 $(sudo) pip install -U Kaggler  or from the source at github: $ git fetch origin $git rebase origin/master$ python setup.py build_ext --inplace $(sudo) python setup.py install  I haven’t had a chance to use it with real competition data yet – after the Avazu competition, I deleted whole build directory 🙁 – and I don’t have numbers for how much faster (or slower?!) it becomes after these changes yet. I will jump into another competition soon, and let you know how it works. 🙂 ## Kaggler – Python Package for Kaggler This article was originally posted on Kaggle’s Avazu competition forum and reposted here with a few edits. Here I’d like to share what I’ve put together for online learning as a Python package – named Kaggler. You can install it with pip as follows: $ pip install -U Kaggler

then, import algorithm classes as follows:

from kaggler.online_model import SGD, FTRL, FM, NN, NN_H2

Currently it supports 4 online learning algorithms – SGD, FTRL, FM, NN (1 or 2 ReLU hidden layers), and 1 batch learning algorithm – NN with L-BFGS AUC optimization.

It uses the liblinear style sparse input format – It is chosen so that the same input file can be used across other popular tools such as XGBoost, VWlibFM, SVMLight, etc.

Code and examples are available at https://github.com/jeongyoonlee/Kaggler, and package documentation is available at http://pythonhosted.org//Kaggler/.


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# Rotating a precession vector

Assume \hat x and \hat y are orthogonal ( at right angles ) vectors in the equatorial plane, and \hat z is an orthogonal vector perpendicular to the plane. These can be expressed as:

\hat x = [ x, 0, 0 ]

\hat y = [ 0, y, 0 ]

\hat z = [ 0, 0, z ]

Matrix multiplication works like this:

[ ax+by+cz,~ dx+ey+fz,~ gx+hy+yz ] = \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array} \right) \times [x, y, z]

The new vector elements are made of linear sums of the old vector elements.

Rotations can be expressed as multiplication matrices, which are multiplied by a vector to yield a new vector. If we rotate a point, the coordinates (expressed as vectors) are multiplied by the rotation matrix to give a rotationally transformed coordinate. Here is the rotation matrix for a rotation around the y axis by angle \phi :

\left( \begin{array}{ccc} \cos( \phi ) & 0 & -\sin( \phi ) \\ 0 & 1 & 0 \\ \sin( \phi ) & 0 & \cos( \phi ) \end{array} \right)

And here is a rotation by angle \beta around the z axis :

\left( \begin{array}{ccc} \cos( \beta ) & -\sin( \beta ) & 0 \\ \sin( \beta ) & \cos( \beta ) & 0 \\ 0 & 0 & 1 \end{array} \right)

These rotational transformations always preserve distance from the origin - the Pythagorean sum of the components sqrt( x^2 + y^2 + z^2 ) remains constant.

The equatorial plane can be represented as a unit vector in z (pointed towards the north pole):

Plane = [ 0, 0, 1 ]

The plane has an axial tilt \phi = 23.439281° . We can represent this as a matrix multiplication around the y axis:

Tilted = [ \sin( \phi ), 0, \cos( \phi ) ] = \left( \begin{array}{ccc} \cos( \phi ) & 0 & -\sin( \phi ) \\ 0 & 1 & 0 \\ \sin( \phi ) & 0 & \cos( \phi ) \end{array} \right) \times [ 0, 0, 1 ]

This is the way the north pole vector appears from the sun at the vernal (spring) equinox. If \beta is the time of year, that vector is rotated around the z axis:

Tilted~and~rotated = [ \cos( \beta ) \sin( \phi ), \sin( \beta ) \sin( \phi ), cos( \phi ) ] = \left( \begin{array}{ccc} \cos( \beta ) & -\sin( \beta ) & 0 \\ \sin( \beta ) & \cos( \beta ) & 0 \\ 0 & 0 & 1 \end{array} \right) \times [ \sin( \phi ), 0, \cos( \phi ) ]

Astronomers looking at the moon and other planets and stars are concerned about absolute orientation in space. For server sky, we are only concerned about the equator and sun angle relative to the equatorial plane. The sun angle can be computed by the " dot product " of our north pole vector with the sun, which is the cosine of the angle between the y axis and the north pole. The full dot product is more complicated, but in this simple case it is just the y term of the vector: \cos( Pole Angle ) = \sin( Equatorial Angle ) = \sin( \beta ) \sin( \phi )

\Large \sin( \theta_{eq} ) = \sin( \beta ) \sin( \phi )

In other words, the effect of precession is to tilt the equatorial plane by \theta_{eq} = \arcsin( \sin( \beta ) \sin( \phi ) ), where \beta is an angle representing the time of year, and \phi is the axial tilt. This can be approximated within 1.1 percent as \theta_{eq} \approx \phi \sin( \beta )

Precession (last edited 2018-04-03 20:44:14 by KeithLofstrom)


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# zbMATH — the first resource for mathematics

A multiplicity theorem for the Neumann problem. (English) Zbl 1134.35345
Summary: Here is a particular case of the main result of this paper: Let $$\Omega \subset{\mathbb{R}}^{n}$$ be a bounded domain, with a boundary of class $$C^{2}$$, and let $$f, g : {\mathbb{R}}\to{\mathbb{R}}$$ be two continuous functions, $$\alpha \in L^{\infty}(\Omega)$$, with $$\text{ess inf}_{\Omega}\alpha >0$$, $$\beta \in L^{p}(\Omega)$$, with $$p>n$$. If $\lim_{|\xi |\to +\infty}{\frac{f(\xi)}{{\xi}}}=0$ and if the set of all global minima of the function $$\xi \to{\frac{{\xi^{2}}}{{2}}}-\int_{0}^{\xi}f(t)\,dt$$ has at least $$k\geq 2$$ connected components, then, for each $$\lambda >0$$ small enough, the Neumann problem $\begin{cases} -\Delta u=\alpha(x)(f(u)-u) +\lambda \beta(x)g(u)&\text{ in } \Omega\\ \frac{\partial u}{\partial \nu}=0&\text{ on }\partial \Omega\end{cases}$ admits at least $$k+1$$ strong solutions in $$W^{2,p}(\Omega)$$.

##### MSC:
 35J60 Nonlinear elliptic equations 47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) 35J20 Variational methods for second-order elliptic equations 35J65 Nonlinear boundary value problems for linear elliptic equations
Full Text:
##### References:
 [1] A. Ambrosetti, A perturbation theorem for superlinear boundary value problems, M. R. C. Tech. Summ. Rep. (1974). [2] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. · Zbl 1042.35002 [3] Shujie Li and Zhaoli Liu, Perturbations from symmetric elliptic boundary value problems, J. Differential Equations 185 (2002), no. 1, 271 – 280. · Zbl 1032.35066 · doi:10.1006/jdeq.2001.4160 · doi.org [4] Zhaoli Liu and Jiabao Su, Solutions of some nonlinear elliptic problems with perturbation terms of arbitrary growth, Discrete Contin. Dyn. Syst. 10 (2004), no. 3, 617 – 634. · Zbl 1205.35118 · doi:10.3934/dcds.2004.10.617 · doi.org [5] Biagio Ricceri, Sublevel sets and global minima of coercive functionals and local minima of their perturbations, J. Nonlinear Convex Anal. 5 (2004), no. 2, 157 – 168. · Zbl 1083.49004 [6] C. G. Simader, Higher regularity of weak $$L_{q}$$-solutions of the Neumann problem for the Laplacian, Bayreuth. Math. Schr., to appear. [7] Guido Stampacchia, Contributi alla regolarizzazione delle soluzioni dei problemi al contorno per equazioni del secondo ordine ellitiche, Ann. Scuola Norm. Sup. Pisa (3) 12 (1958), 223 – 245 (Italian). · Zbl 0082.09701 [8] Eberhard Zeidler, Nonlinear functional analysis and its applications. III, Springer-Verlag, New York, 1985. Variational methods and optimization; Translated from the German by Leo F. Boron. · Zbl 0583.47051
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
My watch list
my.chemeurope.com

# Siphon

A siphon (also spelled syphon) is a continuous tube that allows liquid to drain from a reservoir through an intermediate point that is higher than the reservoir, the up-slope flow being driven only by hydrostatic pressure without any need for pumping. It is necessary that the final end of the tube be lower than the liquid surface in the reservoir.

## History

It is probable that Ctesibius was the discoverer of the principle of the siphon.[1] His student, Hero of Alexandria, wrote extensively about siphons in the treatise, Pneumatica.[2] Even earlier Egyptian reliefs from 1500 BC depict siphons used to extract liquids from large storage jars.[3].

The siphon was first used as a weapon by the Byzantine Navy, and the most common method of deployment was to emit Greek fire, a formula of burning oil, through a large bronze tube onto enemy ships. Usually the mixture would be stored in heated, pressurized barrels and projected through the tube by some sort of pump while the operators were sheltered behind large iron shields. It is not clear whether these were actual siphons or merely pumps that used air pressure to project the Greek fire. "Some apparatus called a 'siphon' (σιφων) was used". "The siphons were, apparently, flame-projectors, either hand-pumps or reservoirs worked by mechanical force-pumps".[4]

## Operation

### Theory

Among some physicists there is some dispute as to what causes the siphon to lift liquid from the upper reservoir to the crest of the siphon[5]. They argue that theoretically, internal molecular cohesion is sufficient to pull the liquid up the intake leg of the siphon to the crest. Furthermore, some (notably Encyclopedia Britannica[6]) argue that theoretically, "a siphon will work in a vacuum". In practice, atmospheric pressure is required, to maintain the cohesion of the liquid in the siphon. Liquids in vacuum are not in equilibrium and typically boil.

Once started, a siphon requires no additional energy to keep the liquid flowing up and out of the reservoir. The siphon works because the ultimate drain point is lower than the reservoir and the flow of liquid out the drain point creates a partial vacuum in the tube such that liquid is drawn up out of the reservoir.

The maximum height of the intermediate point (the crest) is limited by atmospheric pressure and the density of the liquid. At the high point of the siphon, gravity tends to draw the liquid down in both directions, creating a partial vacuum. Atmospheric pressure on the top surface of the higher reservoir is transmitted through the liquid in the reservoir and up the siphon tube and prevents a vacuum from forming. When the pressure exerted by the weight of the height of the column of liquid equals that of atmospheric pressure, a partial vacuum will form at the high point and the siphon effect is ended. For water at standard pressure, the maximum height is approximately 10 m (33 feet); for mercury it is 76 cm (30 inches).

### An analogy

An analogy to understand siphons is to imagine a long, frictionless train extending from a plain, up a hill and then down the hill into a valley below the plain. So long as the valley is below the plain, the part of the train on the valley side of the hill will be longer than the part on the plain side of the hill, so the portion of the train sliding into the valley can pull the rest of the train up the hill and into the valley. What is not obvious is what holds the train together when the train is a liquid in a tube. In this analogy, atmospheric pressure holds the train together. Once the force of gravity on the couplings between the cars of the train going up the hill exceeds that of atmospheric pressure, the coupling breaks and the train falls apart. The train analogy is demonstrated in a "siphon-chain model" [7] where a long chain on a pulley flows between two beakers.

## Practical requirements

A plain tube can be used as a siphon. An external pump has to be applied to start the liquid flowing and prime the siphon. This can be a human mouth and lungs. This is sometimes done with any leak-free hose to siphon gasoline from a motor vehicle's gasoline tank to an external tank. (Siphoning gasoline often results in the accidental swallowing of gasoline, which is quite poisonous.) If the tube is flooded with liquid before part of the tube is raised over the intermediate high point and care is taken to keep the tube flooded while it is being raised, no pump is required. Devices sold as siphons come with a siphon pump to start the siphon process. When applying a siphon to any application it is important that the piping be as closely sized to the requirement as possible. Using piping of too great a diameter and then throttling the flow using valves or constrictive piping appears to increase the effect of previously cited concerns over gases or vapor collecting in the crest which serve to break the vacuum. Once the vacuum is reduced the siphon effect is lost.

Reducing the size of pipe used closer to requirements appears to reduce this effect and creates a more functional siphon that does not require constant re-priming and restarting. In this respect, where the requirement is to match a flow into a container with a flow out of said container (to maintain a constant level in a pond fed by a stream, for example) it would be preferable to utilize two or three smaller separate parallel pipes that can be started as required rather than attempting to use a single large pipe and attempting to throttle it.

## Applications

Floodings: [8] Self-constructed siphons, made of pipes or tubes, can be used to evacuate water from cellars after floodings. Between the flooded cellar and a deeper place outside a connection is built, using a tube or some pipes. They are filled with water from the through an intake valve (at the highest end of the construction). When the ends are opened, the water flows through the pipe into the sewer or the river.

Large siphons may be used in municipal waterworks and industry. Their size requires control via valves at the intake, outlet and crest of the siphon. The siphon may be primed by closing the intake and outlets and filling the siphon at the crest. If intakes and outlets are submerged, a vacuum pump may be applied at the crest to prime the siphon. Alternatively the siphon may be primed by a pump at either the intake or outlet.

Gas in the liquid is a concern in large siphons[9]. The gas tends to accumulate at the crest and if enough accumulates to break the flow of liquid, the siphon stops working. The siphon itself will exacerbate the problem because as the liquid is raised through the siphon, the pressure drops, causing dissolved gases within the liquid to come out of solution. Higher temperature accelerates the release of gas from liquids so maintaining a constant, low temperature helps. The longer the liquid is in the siphon, the more gas is released, so a shorter siphon overall helps. Local high points will trap gas so the intake and outlet legs should have continuous slopes without intermediate high points. The flow of the liquid moves bubbles thus the intake leg can have a shallow slope as the flow will push the gas bubbles to the crest. Conversely, the outlet leg needs to have a steep slope to allow the bubbles to move against the liquid flow; though other designs call for a shallow slope in the outlet leg as well to allow the bubbles to be carried out of the siphon. At the crest the gas can be trapped in a chamber above the crest. The chamber needs to be occasionally primed again with liquid to remove the gas.

## Siphon terminology

Bowl siphon
Bowl siphons are part of flush toilets. Siphon action in the bowl siphon siphons out the contents of the toilet bowl and creates the characteristic toilet "sucking" sound.
Some toilets also use the siphon principle to obtain the actual flush from the cistern. The flush is triggered by a lever or handle that operates a simple diaphragm-like piston pump that lifts enough water to the crest of the siphon to start the flow of water which then completely empties the contents of the cistern into the toilet bowl. The advantage of this system was that no water would leak from the cistern excepting when flushed.
Early urinals incorporated a siphon in the cistern which would flush automatically on a regular cycle because there was a constant trickle of clean water being fed to the cistern by a slightly open valve.

Inverted siphon
An inverted siphon is not a siphon but a term applied to pipes that must dip below an obstruction to form a "U" shaped flow path. At no point does the siphon effect come into play; an inverted siphon will work fine in the absence of atmospheric pressure. Liquid flowing in one end simply forces liquid up and out the other end. Engineers must ensure that the flow rate in such a channel is fast enough to keep suspended solids from settling. Otherwise, the inverted siphon tends to act as a debris trap. This is especially important in sewage systems or culverts which must be routed under rivers or other deep obstructions where the better term is "depressed sewer".
Back siphonage
Back siphonage is a plumbing term applied to clean water pipes that connect directly into a reservoir without an air gap. As water is delivered to other areas of the plumbing system at a lower level, the siphon effect will tend to siphon water back out of the reservoir. This may result in contamination of the water in the pipes. Back siphonage is not to be confused with backflow. Back siphonage is a result of liquids at a lower level drawing water from a higher level. Backflow is driven entirely by pressure in the reservoir itself. Backflow cannot occur through an intermediate high-point. Back siphonage can flow through an intermediate high-point and is thus much more difficult to guard against.
Anti-siphon valve
Anti-siphon valves[10] are required in such designs. Building codes often contain specific sections on back siphonage and especially for external faucets. (See sample building code below.) The reason is that external faucets may be attached to hoses which may be immersed in an external body of water, such as a garden pond, swimming pool, aquarium or washing machine. Should the pressure within the water supply system fall, the external water may be siphoned back into the drinking water system through the faucet. Another possible contamination point is the water intake in the toilet tank. An anti-siphon valve is also required here to prevent pressure drops in the water supply line from siphoning water out of the toilet tank (which may contain additives such as "toilet blue") and contaminating the water system. Anti-siphon valves practically is a one-direction flow valve.
Anti-siphon valves are also used medically. Hydrocephalus, or excess fluid in the brain, maybe treated with a shunt which drains cerebrospinal fluid from the brain. All shunts have a valve to relieve excess pressure in the brain. The shunt may lead into the abdominal cavity such that the shunt outlet is significantly lower than the shunt intake when the patient is standing. Thus a siphon effect may take place and instead of simply relieving excess pressure, the shunt may act as a siphon, completely draining cerebrospinal fluid from the brain. The valve in the shunt may be designed to prevent this siphon action so that negative pressure on the drain of the shunt does not result in excess drainage. Only excess positive pressure from within the brain should result in drainage[11][12][13].
Note that the anti-siphon valve in medical shunts is preventing excess forward flow of liquid. In plumbing systems, the anti-siphon valve is preventing backflow.
Other anti-siphoning devices
Along with anti-siphon valves, anti-siphoning devices also exist. The two are unrelated in application. Siphoning can be used to remove fuel from tanks. With the cost of fuel increasing, it has been linked in several countries globally to the rise in fuel theft. Trucks, with their large fuel tanks, are most vulnerable. The anti-siphon device prevents thieves from inserting a tube into the fuel tank.
Siphon barometer
A siphon barometer is the term sometimes applied to the simplest of mercury barometers. A continuous U-shaped tube of the same diameter throughout is sealed on one end and filled with mercury. When placed into the upright position, mercury will flow away from the sealed end, forming a partial vacuum, until balanced by atmospheric pressure on the other end. The term "siphon" is used because the same principle of atmospheric pressure acting on a fluid is applied. The difference in height of the fluid between the two arms of the U-shaped tube is the same as the maximum intermediate height of a siphon. When used to measure pressures other than atmospheric pressure, a siphon barometer is sometimes called a siphon gauge and not to be confused with a siphon rain gauge. Siphon pressure gauges are rarely used today.
Siphon bottle

A siphon bottle (archaically called a siphoid [14]) is a pressurized bottle with a vent and a valve. Pressure within the bottle drives the liquid up and out a tube. It is a siphon in the sense that pressure drives the liquid through a tube. A special form was the gasogene.
Siphon cup
A siphon cup is the (hanging) reservoir of paint attached to a spray gun. This is to distinguish it from gravity-fed reservoirs. An archaic use of the term is a cup of oil in which the oil is siphoned out of the cup via a cotton wick or tube to a surface to be lubricated.
Siphon rain gauge
A siphon rain gauge is a rain gauge that can record rainfall over an extended period. A siphon is used to automatically empty the gauge. It is often simply called a "siphon gauge" and is not to be confused with a siphon pressure gauge.
Heron's siphon
Heron's siphon is a siphon that works on positive air pressure and at first glance appears to be a perpetual motion machine. In a slightly differently configuration, it is also known as Heron's fountain[15].

## Sample building code regulations regarding back siphonage

From Ontario's building code: [16]

7.6.2.3.Back Siphonage
1. Every potable water system that supplies a fixture or tank that is not subject to pressures above atmospheric shall be protected against back-siphonage by a backflow preventer.
2. Where a potable water supply is connected to a boiler, tank, cooling jacket, lawn sprinkler system or other device where a non-potable fluid may be under pressure that is above atmospheric or the water outlet may be submerged in the non-potable fluid, the water supply shall be protected against backflow by a backflow preventer.
3. Where a hose bibb is installed outside a building, inside a garage, or where there is an identifiable risk of contamination, the potable water system shall be protected against backflow by a backflow preventer.

## Self-siphons

The term self-siphon is used in a number of ways. Liquids that are composed of long polymers can "self-siphon"[17][18] and these liquids do not depend on atmospheric pressure. Self-siphoning polymer liquids work the same as the siphon-chain model where the lower part of the chain pulls the rest of the chain up and over the crest. This phenomenon is also called a tubeless siphon[19].

"Self-siphon" is also often used in sales literature by siphon manufacturers to describe portable siphons that contain a pump. With the pump, no external suction (e.g. from a person's mouth/lungs) is required to start the siphon and thus the product is inaccurately described as a "self-siphon".

If the upper reservoir is such that the liquid there can rise above the height of the siphon crest, the rising liquid in the reservoir can "self-prime" the siphon and the whole apparatus be described as a "self-siphon"[20]. Once primed, such a siphon will continue to operate until the level of the upper reservoir falls below the intake of the siphon. Such self-priming siphons are useful in some rain gauges and dams.

## Siphons in nature

The term "siphon" is used for a number of structures in human and animal anatomy, either because flowing liquids are involved or because the structure is shaped like a siphon, but in which no actual siphon effect is occurring: see Siphon (biology).

Biologists debate whether the siphon mechanism plays a role in blood circulation [21]. It is theorized that veins form a continuous loop with arteries such that blood flowing down veins help siphon blood up the arteries, especially in giraffes and snakes[22]. Some have concluded that the siphon mechanism aids blood circulation in giraffes [23]. Many others dispute this[24][25] and experiments show no siphon effects in human circulation[26]. Some cite negative pressure in the brain as supporting the role of the siphon effect in the brain[27].

## Explanation using Bernoulli's equation

Bernoulli's equation may be applied to a siphon to derive the flow rate and maximum height of the siphon.

Let the surface of the upper reservoir be the reference elevation.
Let point A be the start point of siphon, immersed within the higher reservoir and at a depth −d below the surface of the upper reservoir.
Let point B be the intermediate high point on the siphon tube at height +hB above the surface of the upper reservoir.
Let point C be the drain point of the siphon at height −hC below the surface of the upper reservoir.

Bernoulli's equation:

${v^2 \over 2}+gy+{P \over \rho}=\mathrm{constant}$
$v \;$ = fluid velocity along the streamline
$g \;$ = gravitational acceleration downwards
$y \;$ = elevation in gravity field
$P \;$ = pressure along the streamline
$\rho \;$ = fluid density

Apply Bernoulli's equation to the surface of the upper reservoir. The surface is technically falling as the upper reservoir is being drained. However, for this example we will assume the reservoir to be infinite and the velocity of the surface may be set to zero. Furthermore, the pressure at the surface is atmospheric pressure. Thus:

${0^2 \over 2}+g(0)+{P_\mathrm{atm} \over \rho}=\mathrm{constant}$ (Equation 1.)

Apply Bernoulli's equation to point A at the start of the siphon tube in the upper reservoir where P = PA, v = vA and y = −d

${v_A^2 \over 2}-gd+{P_A \over \rho}=\mathrm{constant}$ (Equation 2.)

Apply Bernoulli's equation to point B at the intermediate high point of the siphon tube where P = PB, v = vB and y = hB

${v_B^2 \over 2}+gh_B+{P_B \over \rho}=\mathrm{constant}$ (Equation 3.)

Apply Bernoulli's equation to point C where the siphon empties. Where v = vC and y = −hC. Furthermore, the pressure at the exit point is atmospheric pressure. Thus:

${v_C^2 \over 2}-gh_C+{P_\mathrm{atm} \over \rho}=\mathrm{constant}$ (Equation 4.)

### Velocity

As the siphon is a single system, the constant in all four equations are the same. Setting equations 1 and 4 equal to each other gives:

${0^2 \over 2}+g(0)+{P_\mathrm{atm} \over \rho}={v_C^2 \over 2}-gh_C+{P_\mathrm{atm} \over \rho}$

Solving for vC:

Velocity of siphon:
$v_C=\sqrt{2gh_C}$

The velocity of the siphon is thus driven solely by the height difference between the surface of the upper reservoir and the drain point. The height of the intermediate high point, hB, does not affect the velocity of the siphon. However, as the siphon is a single system, vB = vC and the intermediate high point does limit the maximum velocity. The drain point cannot be lowered indefinitely to increase the velocity. Equation 3 will limit the velocity to a positive pressure at the intermediate high point to prevent cavitation. The maximum velocity may be calculated by combining equations 1 and 3:

${0^2 \over 2}+g(0)+{P_\mathrm{atm} \over \rho}={v_B^2 \over 2}+gh_B+{P_B \over \rho}$

Setting PB = 0 and solving for vmax:

Maximum velocity of siphon:
$v_\mathrm{max}=\sqrt{2\left({P_\mathrm{atm} \over \rho}-gh_B\right)}$

The depth, −d, of the initial entry point of the siphon in the upper reservoir, does not affect the velocity of the siphon. No limit to the depth of the siphon start point is implied by Equation 2 as pressure PA increases with depth d. Both these facts imply the operator of the siphon may bottom skim or top skim the upper reservoir without impacting the siphon's performance.

Note that this equation for the velocity is the same as that of any object falling height hC. Note also that this equation assumes PC is atmospheric pressure. If the end of the siphon is below the surface, the height to the end of the siphon cannot be used; rather the height difference between the reservoirs should be used.

### Maximum height

Setting equations 1 and 3 equal to each other gives:

${0^2 \over 2}+g(0)+{P_\mathrm{atm} \over \rho}={v_B^2 \over 2}+gh_B+{P_B \over \rho}$

Maximum height of the intermediate high point occurs when it is so high that the pressure at the intermediate high point is zero. Setting PB = 0:

${P_\mathrm{atm} \over \rho}={v_B^2 \over 2}+gh_B$

Solving for hB:

General height of siphon:
$h_B={P_\mathrm{atm} \over \rho g} - {v_B^2 \over 2g}.$

This means that the height of the intermediate high point is limited by velocity of the siphon. Faster siphons result in lower heights. Height is maximized when the siphon is very slow and vB = 0:

Maximum height of siphon:
$h_B={P_\mathrm{atm} \over \rho g}$

This is the maximum height that a siphon will work. It is simply when the weight of the column of liquid to the intermediate high point equates to atmospheric pressure. Substituting values for water will give 10 metres for water and 0.76 metres for mercury.


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# Rhythmical Alternations

rhythmical

January 29, 2022

Last time, I looked at tidal.pegjs, which has an AST that is conceptually similar to rhythmical objects. In addition to sequential and parallel nodes, there are more ideas I haven't yet explored.

In this post, I want to take a look at alternating rhythms, which are enclosed with < and > in tidal mini notation.

Disclaimer: This post will be heavily repetitive and mathematical. As I am not a mathematician, I might not use conventional symbols for tree theory (if there are any).

## Example

C3 <Eb3 F3>

Playing back this pattern will have the following result:

C3 Eb3 C3 F3 C3 Eb3 C3 F3 C3 ...

In traditional western music notation, this could be notated like that:

In tidal.pegjs, the (unified) AST for C3 <Eb3 F3> looks like this:

not a browser

... which can be visualized with this tree:

## Event Querying

So far, rendering the events for a rhythmical tree involved traversing the tree and calculating time and duration for each leaf.

When having alternating nodes, we need to know at which run through of the tree we are, to know which of the possible choices to pick. In the example above, we have two different ways to go, so for every first run, we pick the first and for every second run, we pick the second.

This can be made clearer with a table:

 Q V B0 0 1 2 3 C3 Eb3 C3 F3 C3 Eb3 C3 F3 0 1 0 1
• Q = query count
• V = event value(s)
• B0 = index of selected value in branch
• the dash indicates where it repeats

In this case, we can say:

$B_0 = Q \% 2$

To know which value to pick, the C3 is completely irrelevant, so we can ignore any non alternating nodes before and after.

## Building Intuition

To further understand the behaviour of alternations, let's look at some more examples

### Variable Branching Factor

The branching factor is the number of children of a node. The pattern <A B C> has three children, so the branching factor is 3. The table looks like this:

 Q V B0 0 1 2 3 A B C A 0 1 2 0

Now, the index of the selected value is modulo 3:

$B_0 = Q \% 3$

More general, we can deduce:

$B_0 = Q \% F_0$

## Nested Alternations

It gets more complicated when we nest alternations into each other. Example: <<A B> C>

 Q V B0 B1 0 1 2 3 4 A C B C A 0 1 0 1 0 0 _ 1 _ 0
$B_1 = (Q / 2) \% 2 \newline F_0 = F_1 = 2 \newline Q_r = F_0 * F_1 = 4$

where Qr is the total length of the pattern until it repeats (dotted line).

### Nested + Variable Branching Factor

<<A B> C D>

 Q V B0 B1 0 1 2 3 4 5 6 A C D B C D A 0 1 2 0 1 2 0 0 _ _ 1 _ _ 0
$B_1 = (Q / 3) \% 2 \newline F_0 = 3, F_1 = 2 \newline Q_r = F_0 * F_1 = 6$

<<A B C> D>

 Q V B0 B1 0 1 2 3 4 5 6 A D B D C D A 0 1 0 1 0 1 0 0 _ 1 _ 2 _ 0
$B_1 = (Q / 2) \% 3 \newline F_0 = 2, F_1 = 3 \newline Q_r = F_0 * F_1 = 6$

From the two examples, we can deduce:

$B_1 = (Q / F_0) \% F_1$

### Nested with offset

<A <B C>>

 Q V B0 B1 0 1 2 3 4 A B A C A 0 1 0 1 0 _ 0 _ 1 _
$B_1 = ((Q - 1) / 2) \% 2$

<A B <C D>>

 Q V B0 B1 0 1 2 3 4 5 6 A B C A B D A 0 1 2 0 1 2 0 _ _ 0 _ _ 1 _
$B_1 = ((Q - 2) / 3) \% 2$

<A <B C D>>

 Q V B0 B1 0 1 2 3 4 5 6 A B A C A D A 0 1 0 1 0 1 0 _ 0 _ 1 _ 2 _
$B_1 = ((Q - 1) / 2) \% 3$

From the above, we can deduce:

$B_1 = ((Q - I_1) / F_0) \% F_1$

where I1 is the index of B1 inside B0.

### Nesting even more

<<<A B> C> D>

 Q V B0 B1 B2 0 1 2 3 4 5 6 7 8 A D C D B D C D A 0 1 0 1 0 1 0 1 0 0 _ 1 _ 0 _ 1 _ 0 0 _ _ _ 1 _ _ _ 0
$B_1 = (Q / 2) \% 2 \newline B_2 = (Q / 4) \% 2$

<<<A B> C> D E>

 Q V B0 B1 B2 0 1 2 3 4 5 6 7 8 9 10 11 12 A D E C D E B D E C D E A 0 1 2 0 1 2 0 1 2 0 1 2 0 0 _ _ 1 _ _ 0 _ _ 1 _ _ 0 0 _ _ _ _ _ 1 _ _ _ _ _ 0
$B_1 = (Q / 3) \% 2 \newline B_2 = (Q / 6) \% 2 \newline F_0 = 3, F_1 = 2, F_2 = 2$

From that, we can deduce:

$B_2 = (Q / (F_0*F_1)) \% F_2$

<A <B <C D>>>

 Q V B0 B1 B2 0 1 2 3 4 5 6 7 8 A B A C A B A D A 0 1 0 1 0 1 0 1 0 _ 0 _ 1 _ 0 _ 1 _ _ _ _ 0 _ _ _ 1 _
$B_2 = ((Q-3) / 4) \% 2$

<A <<B C> D>>

 Q V B0 B1 B2 0 1 2 3 4 5 6 7 8 A B A D A C A D A 0 1 0 1 0 1 0 1 0 _ 0 _ 1 _ 0 _ 1 _ _ 0 _ _ _ 1 _ _ _
$B_2 = ((Q-1) / 4) \% 2$

<A <B <C <D E>>>>

 Q V B0 B1 B2 B3 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A B A C A B A D A B A C A B A E A 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 _ 0 _ 1 _ 0 _ 1 _ 0 _ 1 _ 0 _ 1 _ _ _ _ 0 _ _ _ 1 _ _ _ 0 _ _ _ 1 _ _ _ _ _ _ _ _ 0 _ _ _ _ _ _ _ 1 _
$B_0 = Q \% 2 \newline B_1 = ((Q-1)/2) \% 2 \newline B_2 = ((Q-3)/4) \% 2 \newline B_3 = ((Q-7)/8) \% 2 \newline F_0 = F_1 = F_2 = F_3 = 2 \newline I_1 = I_2 = I_3 = 1$

## General Formula

The above example can be deduced to:

$B_0 = Q \% F_0 \newline B_1 = ((Q-I_1)/F_0) \% F_1 \newline B_2 = ((Q-(I_1*F_1+I_2))/F_0*F_1) \% F_2 \newline B_3 = ((Q-(I_1*F_1+I_2*F_2+F_3))/F_0*F_1*F_2) \% F_3$

Even more general:

$P_n = \prod_{\substack{0

In the example:

$P_0 = 1 \newline P_1 = F_0 = 2 \newline P_2 = P_1 * F_1 = 2 * 2 = 4 \newline P_3 = P_2 * F_2 = 4 * 2 = 8 \newline \newline O_0 = 0; \newline O_1 = I_1*P_0 = 1*1 = 1 \newline O_2 = O_1 + I_2*P_1 = 1 + 1*2 = 3 \newline O_3 = O_2 + I_3*P_2 = 3 + 1*4 = 7 \newline \newline B_0 = ((Q - 0)/1) \% 2 \newline B_1 = ((Q - 1)/2) \% 2 \newline B_2 = ((Q - 3)/4) \% 2 \newline B_3 = ((Q - 7)/8) \% 2$

## Conclusion

With the above general formula, we are able to know which branch will be selected at any query, without needing to calculate all preceeding query states. In a future post, I want to implement this + look at alternative ways to represent this. I think tidal cycles is doing this differently, using the slow function, which might be- simpler..

Felix Roos 2022