Superelliptic curve

A multi-input, multi-output system represented as a noncommutative matrix signal-flow graph.

In automata theory and control theory, branches of mathematics, theoretical computer science and systems engineering, a noncommutative signal-flow graph is a tool for modelingTemplate:Sfn interconnected systems and state machines by mapping the edges of a directed graph to a ring or semiring.

A single edge weight might represent an array of impulse responses of a complex system (see figure to the right),Template:Sfn or a character from an alphabet picked off the input tape of a finite automaton,Template:Sfn while the graph might represent the flow of information or state transitions.

As diverse as these applications are, they share much of the same underlying theory.Template:SfnTemplate:Sfn

Definition

Signal-flow graph fragment.

Consider n equations involving n+1 variables {x₀, x₁,...,x_n}.

$x_{i}=\sum _{j=0}^{n}a_{ij}x_{j},\;\;\;1\leq i\leq n,$

with a_ij elements in a ring or semiring R. The free variable x₀ corresponds to a source vertex v₀, thus having no defining equation. Each equation corresponds to a fragment of a directed graph G=(V,E) as show in the figure.

The edge weights define a function f from E to R. Finally fix an output vertex v_m. A signal-flow graph is the collection of this data S = (G=(V,E), v₀,v_m $\in$ V, f : E → R). The equations may not have a solution, but when they do,

$x_{m}=Tx_{0},$

with T an element of R called the gain.

Return Loop Method

There exist severalTemplate:Sfn noncommutative generalizations of Mason's rule. The most common is the return loop method (sometimes called the forward return loop method (FRL), having a dual backward return loop method (BRL)). The first rigorous proof is attributed to Riegle,Template:Sfn so it is sometimes called Riegle's rule.Template:Sfn

As with Mason's rule, these gain expressions combine terms in a graph-theoretic manner (loop-gains, path products, etc). They are known to hold over an arbitrary noncommutative ring and over the semiring of regular expressions.Template:Sfn

Formal Description

The method starts by enumerating all paths from input to output, indexed by j $\in$ J. We use the following definitions:

• The j-th path product is (by abuse of notation) a tuple of k_j edge weights along it:

$p_{j}=(w_{k_{j}}^{(j)},\ldots ,w_{2}^{(j)},w_{1}^{(j)}).$

• To split a vertex v is to replace it with a source and sink respecting the original incidence and weights (this is the inverse of the graph morphism taking source and sink to v).
• The loop gain of a vertex v w.r.t. a subgraph H is the gain from source to sink of the signal-flow graph split at v after removing all vertices not in H.
• Each path defines an ordering of vertices along it. The along path j, the i-th FRL (BRL) node factor is (1-S_i^(j))⁻¹ where S_i^(j) is the loop gain of the i-th vertex along the j-th w.r.t. the subgraph obtained by removing v₀ and all vertices ahead of (behind) it.

The contribution of the j-th path to the gain is the product along the path, alternating between the path product weights and the node factors:

$T_{j}=\prod _{i=k_{j}}^{1}(1-S_{i}^{(j)})^{-1}w_{i}^{(j)},$

so the total gain is

$T=\sum _{j\in J}T_{j}.$

An Example

A noncommutative signal-flow graph from x to z

Consider the signal-flow graph shown. From x to z, there are two path products: (d) and (e,a). Along (d), the FRL and BRL contributions coincide as both share same loop gain (whose split reappears in the upper right of the table below):

$f+e(1-b)^{-1}c,$

Multiplying its node factor and path weight, its gain contribution is

$T_{d}=\left[1-f-e(1-b)^{-1}c\right]^{-1}d.$

Along path (e,a), FRL and BRL differ slightly, each having distinct splits of vertices y and z as shown in the following table.

Adding to T_d, the alternating product of node factors and path weights, we obtain two gain expressions:

$T^{(FRL)}=\left[1-f-e(1-b)^{-1}c\right]^{-1}d+\left[1-f-e(1-b)^{-1}c\right]^{-1}e(1-b)^{-1}a,$

and

$T^{(BRL)}=\left[1-f-e(1-b)^{-1}c\right]^{-1}d+(1-f)^{-1}e\left[1-b-c(1-f)^{e}\right]^{-1}a,$

These values are easily seen to be the same using identities (ab)⁻¹ = b⁻¹a⁻¹ and a(1-ba)⁻¹=(1-ab)⁻¹a.

Applications

Matrix Signal-Flow Graphs

Consider equations

$y_{i}=\sum _{j=1}^{2}a_{ij}x_{j}+\sum _{j=1}^{2}b_{ij}y_{j}$

and

$z_{i}=\sum _{j=1}^{2}c_{ij}y_{j},$

This system could be modeled as scalar signal-flow graph with multiple inputs and outputs. But, the variables naturally fall into layers, which can be collected into vectors x=(x₁,x₂)^t y=(y₁,y₂)^t and z=(z₁,z₂)^t. This results in much simpler matrix signal-flow graph as shown in the figure at the top of the article.

Applying the forward return loop method is trivial as there's a single path product (C,A) with a single loop-gain B at y. Thus as a matrix, this system has a very compact representation of its input-output map

$T=C(1-B)^{-1}A.$

Finite Automata

Representation of a finite automaton as a (noncommutative) signal flow graph over a semiring.

An important kind of noncommutative signal-flow graph is a finite state automaton over an alphabet $\Sigma$.Template:SfnTemplate:Sfn

Serial connections correspond to the concatenation of words, which can be extended to subsets of the free monoid $\Sigma ^{*}$. For A, B $\subseteq \Sigma ^{*}$

$A\cdot B=\{ab\mid a\in A,b\in B\}.$

Parallel connections correspond to set union, which in this context is often written A+B.

Finally, self-loops naturally correspond to the Kleene closure

$A^{*}=\{\lambda \}+A+AA+AAA+\cdots ,$

where $\lambda$ is the empty word. The similarity to the infinite geometric series

$(1-x)^{-1}=1+x+x^{2}+x^{3}\cdots ,$

is more than superficial, as expressions of this form serve as 'inversion' in this semiring.Template:Sfn

In this way, the subsets of $\Sigma ^{*}$ built of from finitely many of these three operations can be identified with the semiring of regular expressions. Similarly, finite graphs whose edges are weighted by subsets of $\Sigma ^{*}$ can be identified with finite automata, though generally that theory starts with singleton sets as in the figure.

This automaton is deterministic so we can unambiguously enumerate paths via words. Using the return loop method, path contributions are:

• path ab, has node factors (c^*, $\lambda$), yielding gain contribution

$ac^{*}b,$

• path ada, has node factors (c^*, c^*, $\lambda$), yielding gain contribution

$ac^{*}dc^{*}a,$

• path ba, has node factors (c^*, $\lambda$), yielding gain contribution

$bc^{*}a.$

Thus the language accepted by this automaton (the gain of its signal-flow graph) is the sum of these terms

$L=ac^{*}b+ac^{*}dc^{*}a+bc^{*}a.$

See also

• Signal-flow graph
• Mason's rule
• Finite automata
• Regular expressions

Notes

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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
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  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang