text:
Showing Associativity and Commutativity of a binary operation given by a Cayley table. Example 13.1.4. Here, the output result relies on the operation executed on the input or proposition values and the value can be either true or false. A binary operation, , is defined on the set {1, 2, 3, 4}. Situation 2: Sometimes, a binary operation on a finite set (a set with a limited number of elements) is displayed in a table which shows how the operation is to be performed. More explicitly, let S S S be a set and â * â be a binary operation on S. S. S. Then 1. can be generalised as a binary operation is performed on two elements (say a and b) from set X. Apart from these differences, operations such as addition, subtraction, multiplication, and division are all computed following the same rules as the decimal system. This table shows the operation * (âstarâ). In order to do the binary calculations yourself most would prefer using a table for smaller numbers and a calculator for larger ones. Oracle Database Lite SQL also supports set operators. Examples of binary operation on from to include addition (), subtraction (), multiplication) and division (). 11.2 Multiplication tables For small sets, we may record a binary operation using a table, called the multiplication table (whether or not the binary operation is multiplication). As is the case for other functions, there are several ways of specifying a binary operation. If the set is small, we sometimes specify the binary operation by a table. The result of a not operation is â¦ Deï¬nition 3.2 Binary Operation. The result of AND operation in Hex Ascii Result:.. Binary Operation. There are many properties of the binary operations which are as follows: 1. 2.10 Examples. This is a binary operation. The result of AND operation in Octal Decimal Result:.. Just as we get a number when two numbers are either added or subtracted or multiplied or are divided. Pandas include a couple useful twists, however: for unary operations like negation and trigonometric functions, these ufuncs will preserve index and column labels in the output, and for binary operations such as addition and multiplication, Pandas will automatically align indices when passing the objects to the ufunc. Bases: sage.structure.sage_object.SageObject An object that represents a binary operation as a table. Almost all modern technology and computers use the binary system due to its ease of implementation in digital circuitry using logic gates. Hot Network Questions How did musicians acquire samples for tracker music (MOD, S3M, XM and the like)? The empty in the jth row and the kth column represent the elements a j *a k.. A binary operation on S is a correspondence that associates with each ordered pair (a, b) of elements of S a uniquely ... â A free PowerPoint PPT presentation (displayed as a Flash slide show) on PowerShow.com - id: 47753e-NTE0N Properties of Binary Operations. How many elements of this operation have an inverse?. A binary operation in a finite set can completely be described by means of a table. Given an operation table with n rows and n columns, and each entry being an element of A = {a 1, a 2... a n}, a binary operation *: A × A â A can be defined where a i * a j is the entry in the i th row and the j th column of the operation table. While performing binary operations, it is important to know the convention being used in order to perform the operation following the applicable rules. For this quiz and worksheet combo, you are reviewing binary operation and structure in abstract algebra. Given below is the table corresponding to some binary operation a * b on a set {0,1,2,3,4,5}. The usual addition + is a binary operation on the set R, and also on the sets Z, Q, Z+, and C. 2. Not operation is defined for Aâ or NOT A if A = 1, then Aâ = 0 or else Aâ = 1. class sage.matrix.operation_table.OperationTable (S, operation, names = 'letters', elements = None) ¶. A binary operation on a nite set is commutative the table is symmetric about the diagonal running from upper left to lower right. B. Binary operations Deï¬nition (2.1) A binary operation â on a set S is a function mapping S ×S into S. For each (a,b) â S ×S, we denote the element â((a,b)) of S by a âb. Binary Operations Let S be any given set. The binary operations include two variables for input values. The result of AND operation in Decimal Hex Result:.. The result of the operation on a and b is another element from the same set X. Binary Logic Operations. In â¦ D. 4. Operation Tables¶. 6. Unlike the situation of ordinary numbers, the values of the variables in binary logic can be only two in number. Example 1. Binary calculator,Hex calculator: add,sub,mult,div,xor,or,and,not,shift. The resultant of the two are in the same set.Binary operations on a set are calculations that combine two elements of the set (called operands) to produce another element of the same set. The usual division / is not a binary operation on R since / Learn how to make aâ¦ Definition: Binary operation. 0. A binary operation given by a table. (vii) Let S = N, with de ned by a b = ab (e.g., 2 3 = 23 = 8). * a b a a b b b a In studying binary operations on sets, we tend to be interested in those operations that have certain properties which we discuss next. Binary Result:.. Is an associative binary operation with trivial squares necessarily commutative? The binary operations associate any two elements of a set. This module implements general operation tables, which are very matrix-like. As you will discover in this lesson, binary operations need not be applied only to numbers. There are two general classes of operators: unary and binary. Binary logic presupposes two distinguishing characteristics : two-valued variables, and appropriate logical operations. Because of the many interesting examples of binary operations â¦ Represent operation * as a table on A. Closure Property: Consider a non-empty set A and a binary operation * on A. The number of binary operations * : A X A â A is equal to [n(A)] n(A X A). The operation Î¦ is not associative for real numbers. An identity element in a set is an element that is special with respect to a binary operation on the set: when an identity element is paired with any element via the operation, it returns that element. For the operation on , every element has an inverse, namely .. For the operation on , the only element that has an inverse is ; is its own inverse.. For the operation on , the only invertible elements are and .Both of these elements are equal to their own inverses. For example, the following is the multiplication table of a binary operation â : {a,b}×{a,b} ââ {a,b}. In the video, the commutativity of a given binary operation is verified, and identity element as well as inverse of some elements are found. The levels of precedence among the Oracle Database Lite SQL operators from high to low are listed in Table 2-1. To perform this operation we need a minimum of 1 input variable that can take the values of binary numbers i.e., 0 or 1 to get an output with one binary value (0/1). Chapter 4: Binary Operations and Relations 4.1: Binary Operations DEFINITION 1. Solution: The table of the operation is shown in fig: The result of AND operation in Ascii Input Base:.. The composition table helps us to verify most of the properties satisfied by the binary operations. This table can be formed as follows: In other words, $$\star$$ is a rule for any two elements in the set $$S$$. Using our tool in binary calculator mode you can perform the four basic arithmetic operations on binary numbers: addition, subtraction, multiplication and division. Let $$S$$ be a non-empty set, and $$\star$$ said to be a binary operation on $$S$$, if $$a \star b$$ is defined for all $$a,b \in S$$. A. 1. is defined for every pair of elements in , and . The result of AND operation in Binary Octal Result:.. 2. uniquely associates each pair of elements in to some element of .. (Note that it would be very hard to decide if a binary operation on a nite set is associative just by looking at the table.) A binary operation on a nonempty set Ais a function from A Ato A. Addition, subtraction, multiplication are binary operations on Z. A binary operation on a finite set (a set with a limited number of elements) is often displayed in a table that demonstrates how the operation is performed. Are divided from upper left to lower right Decimal Hex result: associates each pair of elements,... 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Worksheet combo, you are reviewing binary operation is shown in fig: examples! Numbers, the values of the many interesting examples of binary operations need not be applied only to.!, \ ( \star\ ) is a rule for any two elements ( say a b. It is important to know the convention being used in order to do the operations! Elements = None ) ¶ an inverse? ) is a rule for any elements. Is not associative for real numbers rule for any two elements ( say a and a calculator for larger.. Logic presupposes two distinguishing characteristics: two-valued variables, and is equal to its of! Solutions for Class 12 Maths Chapter 3 â binary operations implementation in circuitry... 4.1: binary operations which are as follows: 1 larger ones due to its inverse! Calculator: add, sub, mult, div, XOR, or and...
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# Automating model selection criteria production

I want to perform model comparison according to several criteria using R. My dataframe's name is df

head(df)

Y  X1  X2   X3   X4
1 18 307 130 3504 12.0
2 15 350 165 3693 11.5
3 18 318 150 3436 11.0
4 16 304 150 3433 12.0
5 17 302 140 3449 10.5
6 15 429 198 4341 10.0

I want to compare all possible combinations with Y as the dependent variable. I'll give you what I've done so far to get the R.squared for all possible models. I would like your comments for better coding, especially on getting all those model formulas

# First get all possible combinations of the 4 inependent variables
comb_list <- lapply(1:4,function(i) combn(4,i))

# now create the formulas
comb_list_forms <- lapply(unlist(
sapply(1:length(comb_list), function(i)
sapply(1:dim(comb_list[[i]])[2], function(x) {nam <- names(df[1+comb_list[[i]][,x]])
formul <- "Y~"
sapply(1:length(nam), function(y) formul <<- paste(formul,nam[y],sep="+"))
formul <- sub("~+", "~", formul,fixed = T)
}))),as.formula)

# finally get r.squared
attach(df)
models.r.sq <- sapply(comb_list_forms, function(i) summary(lm(i))$r.squared)  Thank you ## 4 Answers I think package leaps will help you. It is designed specifically to select a "best" subset of variables for linear regression. As for the formula creation, you can use foreach package like this:  #Create the list where each list element contains one combination xcomb <- foreach(i=1:4, .combine=c) %do% { combn(paste("X", 1:4, sep=""), i, simplify=FALSE) } #Convert list elements to formula formlist <- lapply(xcomb, function(l) formula(paste("Y", paste(l, collapse="+"), sep="~")))  Ok, here is another way (which is certainly not as elegant or concise than the other ones, but it has the merit of not requiring Emacs to check parenthesis matches :-) Let's say we have a vector of predictors of interest like this > Xs <- paste("X", 1:4, sep="")  Then, we can just use > allXs <- lapply(seq(along=Xs), function(x) combn(Xs, x, simplify=FALSE, FUN=paste, collapse="+"))  where > unlist(allXs)  gives you all 15 combinations of the X's. Another option is to just change the right-hand side of a formula, say  fm <- as.formula(paste("Y ~ ", paste(Xs, collapse= "+")))  so as to reflect the different combinations that you enumerate in your comb_list object. This can be done using the Formula package: > fm <- as.formula(paste("Y ~ ", paste(Xs, collapse= "|"))) > fm2 <- Formula(fm) > foo <- function(x) formula(fm2, rhs=x, collapse=TRUE) > foo(1:2) Y ~ X1 + X2 <environment: 0x102593040> > foo(c(1,3,4)) Y ~ X1 + X3 + X4 <environment: 0x1021d02a0>  • +1. Did not know about this Formula feature. Thanks – mpiktas Dec 16 '10 at 15:16 This is another way to get the formulas comb_list_forms <- unlist(lapply(1:length(comb_list), function(i) lapply(1:dim(comb_list[[i]])[2], function(x) as.formula(paste("Y ~ ", paste(names(df[1+comb_list[[i]][,x]]), collapse= "+"))))))  You could consider using step on the full model. This uses AIC rather than$R^2\$ as the criterion for selection as this is considered to be superior.

In any case, the functions add1 and drop1 that step uses may be of use.

format: plain


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# Physical meaning of Phase matching

Can anyone help me understand what exactly is meant by phase matching? I want to know whether momentum can be conserved only under phase matching condition or phase matching condition ensures momentum conservation. How does momentum conservation ensure efficient transfer of energy between waves? How is phase matching related to various matrix elements of susceptibility? I have been struggling with these fundamental questions for a long time.

• uh, you linked a google search? – anon01 Apr 4 '16 at 16:42
• momentum conservation is true whether or not a phase matching condition is met. You may want to ask a single, specific question about phase matching to get a good answer. – anon01 Apr 4 '16 at 16:44
• "Momentum is conserved, as is necessary for phase-matching" (en.wikipedia.org/wiki/Quasi-phase-matching) Please explain this. Thanks – Aswin Alex Apr 4 '16 at 17:53
• You'll get a better answer if you tell us what the context is, what you know, and what confuses you. In nonlinear optics, you generally want your field to build up. For that, the two waves have to have the same phase. If not, there will be alternating growth and reduction of the field, resulting in almost no output. – garyp Apr 4 '16 at 19:43
• I want to understand the interaction between the various waves inside a non-linear crystal with a space-time picture. Is phase matching a condition which will sustain an efficient transfer of energy between waves throughout the non-linear crystal? Should momentum be conserved at each instant of time everywhere inside the crystal? If yes, is this the reason why phase matching requirement is imposed for efficient transfer of energy? – Aswin Alex Apr 5 '16 at 4:56

For sum frequency generation ($\omega_1+\omega_2=\omega_3$) $\Delta\phi=\phi_3-\phi_2-\phi_1$. If $\Delta\phi=-\pi/2$ the energy transfer will occur from $\omega_3$ to $\omega_2$ and $\omega_1$ making photon split. If $\Delta\phi=+\pi/2$ the energy transfer will occur from $\omega_1$ and $\omega_2$ to $\omega_3$ making photon merger.
Between the two cases the efficiency of one of the two process will decrease. If the phase matching relation tells you that $\phi_3=\pi/2+\phi_1+\phi_2$. The phase $\phi_1$ and $\phi_2$ are given by your optical sources at $\omega_1$ and $\omega_2$. But if you add a signal at $\omega_3$ to stimulate the nonlinear process it will deteriorate the efficiency if $\phi_3$ is not exactly the phase given by the phase matching condition.


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# Manning's Roughness Coefficient

Written by Jerry Ratzlaff on . Posted in Fluid Dynamics

Manning's roughness coefficient, abbreviated as n, also called Manning's coefficient, roughness coefficient, a dimensionless number, measures the roughness or frictional resistance exerted by a flow against the ground surface in an open channel, culvert or pipe.  The Manning roughness coefficient is one of the most important ways to describe the water flow over the ground surface.

The roughness coefficient is used to calculate the friction factor using a Moody Diagram and the Reynolds number.

## Manning's roughness coefficient formula

 $$\large{ n = \frac{1.49 \; r_h^{\frac{2}{3} } \; S^{\frac{1}{2} } }{ v } }$$

### Where:

 Units English Metric $$\large{ n }$$ = Manning's roughness coefficient $$\large{ dimensionless }$$ $$\large{ S }$$ = channel slope or energy slope line $$\large{ ft }$$ $$\large{ m }$$ $$\large{ v }$$ = flow velocity in a channel, culvert, or pipe $$\large{ \frac{ft}{sec} }$$ $$\large{ \frac{m}{s} }$$ $$\large{ r_h }$$ = hydraulic radius $$\large{ ft }$$ $$\large{ m }$$

### Solve For:

 $$\large{ v = \frac{1.49 \; r_h^{\frac{2}{3} } \; S^{\frac{1}{2} } }{ n } }$$ $$\large{ r_h = \left( \frac{ v \; n }{ 1.49 \; S^{\frac{1}{2}} } \right)^{ \frac{1}{0.66} } }$$ $$\large{ S = \left( \frac{ v \; n }{ 1.49 \; r^{\frac{2}{3}} } \right)^2 }$$ $$\large{ Q = \frac{ 1.49 }{ n } \; A_c \; r_h^{\frac{2}{3} } \; S^{\frac{1}{2}} }$$

## Related Manning's Roughness Coefficient formula

 $$\large{ n = \frac{ 0.56 }{ Q } \; m_c^{\frac{5}{3} } \; m_l^{\frac{1}{2} } \; Q^{\frac{8}{3} } }$$ (gutter flow rate)

### Where:

$$\large{ n }$$ = Manning's roughness coefficient

$$\large{ r_h }$$ = hydraulic radius

$$\large{ Q }$$ = flow rate in a channel, culvert, or pipe

$$\large{ m_c }$$ = cross slope of pavement

$$\large{ m_l }$$ = longitudinal slope of pavement


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# Juniper

Juniper (Jump Non linear Integer Program solver) is a solver for MixedIntegerNonLinearPrograms (MINLPs) written in Julia. Juniper solves these kind of problems using a NLP solver and then branch and bound. If the NLP solver isn't global optimal then Juniper is a heuristic. You need the global optimum? Check out Alpine.jl

# Why?

You have a non linear problem with discrete variables (MINLP) and want some more control over the branch and bound part? => Use Juniper

You have a really good solver for the relaxation and just want to solve problems with discrete variables as well? Just combine your solver with Juniper.

# Basic usage

] add Juniper as ] is used to get to interact with the package manager.

using Juniper

You also have to import your NLP solver i.e.

using Ipopt

as well as JuMP

Define Juniper as the optimizer:

optimizer = Juniper.Optimizer
nl_solver = optimizer_with_attributes(Ipopt.Optimizer, "print_level" => 0)

And give it a go:

using LinearAlgebra # for the dot product
m = Model(optimizer_with_attributes(optimizer, "nl_solver"=>nl_solver))

v = [10,20,12,23,42]
w = [12,45,12,22,21]
@variable(m, x[1:5], Bin)

@objective(m, Max, dot(v,x))

@NLconstraint(m, sum(w[i]*x[i]^2 for i=1:5) <= 45)

optimize!(m)

# retrieve the objective value, corresponding x values and the status
println(JuMP.value.(x))
println(JuMP.objective_value(m))
println(JuMP.termination_status(m))

This solver is a NLP solver therefore you should have at least one NLconstraint or NLobjective.

Juniper is specialized for non convex problems which get solved locally optimal. It will solve convex problems as well but specialized solvers for convex problems should be preferred then.

# Performance

You can find detailed performance measurements in our paper or technical report.

The most recent stats can be found on the website BnBVisual.


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# Minimal graphs with a prescribed number of spanning trees

 Importance: Medium ✭✭
 Author(s): Azarija, Jernej Skrekovski, Riste
 Subject: Graph Theory
 Keywords: number of spanning trees, asymptotics
 Posted by: azi on: April 22nd, 2012

\begin{conjecture} Let $n \geq 3$ be an integer and let $\alpha(n)$ denote the least integer $k$ such that there exists a simple graph on $k$ vertices having precisely $n$ spanning trees. Then $\alpha(n) = o(\log{n}).$ \end{conjecture}

% You may use many features of TeX, such as % arbitrary math (between $...$ and $$...$$) % \begin{theorem}...\end{theorem} environment, also works for question, problem, conjecture, ... % % Our special features: % Links to wikipedia: \Def {mathematics} or \Def[coloring]{Graph_coloring} % General web links: \href [The On-Line Encyclopedia of Integer Sequences]{http://www.research.att.com/~njas/sequences/} Observe that $\alpha(n)$ is well defined for $n \geq 3$ since $C_n$ has $n$ spanning trees.

The function was introduced by Sedlacek \cite{S} who has shown that for large enough $n$ $\alpha(n) \leq \frac{n+6}{3} \mbox{if } n \equiv 0 \pmod{3}$ and $\alpha(n) \leq \frac{n+4}{3} \mbox{if } n \equiv 2 \pmod{3}.$

Using the fact that almost all positive integers $n$ are expressible as $n = ab+ac+bc$ for integers $0 < a < b < c$ it can be shown \cite{A} that for large enough $n$

$\alpha(n) \leq \frac{n+4}{3} \mbox{if } n \equiv 2 \pmod{3}$ and $\alpha(n) \leq \frac{n+9}{4}$ otherwise.

Moreover, the only fixed points of $\alpha$ are 3, 4, 5, 6, 7, 10, 13 and 22.

The conjecture is motivated by the following graph (ploted for a very small sample of vertices)

\Image{graf_0.png}

The conjecture \cite{C} is justifiable for highly composite numbers $n$ since in this case one can construct the graph obtained after taking cycles $C_{p_1}, \ldots,C_{p_k}$ for every odd prime factor $p_i$ of $n$.

## Bibliography

% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries) [S] J. Sedlacek, On the minimal graph with a given number of spanning trees, Canad. Math. Bull. 13 (1970) 515-517.

[A] J. Azarija, R. Skrekovski, Euler's idoneal numbers and an inequality concerning minimal graphs with a prescribed number of spanning trees, IMFM preprints 49 (2011) \href[Link to paper]{http://www.imfm.si/preprinti/PDF/01157.pdf}

* [C] \href[Minimal graphs with a prescribed number of spanning trees]{http://mathoverflow.net/questions/93656/minimal-graphs-with-a-prescribed-number-of-spanning-trees}

* indicates original appearance(s) of problem.


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# Poppet Valve (TL)

Flow control valve actuated by longitudinal motion of ball element

• Library:
• Simscape / Fluids / Thermal Liquid / Valves & Orifices / Flow Control Valves

## Description

The Poppet Valve (TL) block models the flow control with a ball valve in a thermal liquid network. You can specify the seat geometry as either sharp-edged or conical. The ball displacement is set by a physical signal at port S.

Ball Valve Seat Types

### Opening Area With a Sharp-Edged Seat

When is set to Sharp-edged, the valve opening area is based on the geometrical expression:

$A=\pi {r}_{o}\left(1-{\left(\frac{{r}_{b}}{{d}_{OB}}\right)}^{2}\right){d}_{OB}\left(h\right)+{A}_{leak},$

where:

• ro is the valve orifice radius.

• rb is the valve ball radius.

• dOB(h) is the distance between the center of the ball and the edge of the orifice. This distance is a function of the valve lift (h).

• Aleak is the leakage area.

The maximum displacement, hmax, is:

${h}_{\mathrm{max}}=\sqrt{\frac{2{r}_{b}^{2}-{r}_{o}^{2}+{r}_{o}\sqrt{{r}_{o}^{2}+4{r}_{b}^{2}}}{2}}-\sqrt{{r}_{b}^{2}-{r}_{o}^{2}}.$

Seat Schematic

### Opening Area With a Sharp-Edged Seat

When is set to Conical, the valve opening area is based on the geometrical expression:

$A=\pi {r}_{b}\mathrm{sin}\left(\theta \right)h+\frac{\pi }{2}\mathrm{sin}\left(\frac{\theta }{2}\right)\mathrm{sin}\left(\theta \right){h}^{2},$

The maximum displacement, hmax, is:

${h}_{\mathrm{max}}=\frac{\sqrt{{r}_{b}^{2}+\frac{{r}_{o}^{2}}{\mathrm{cos}\left(\frac{\theta }{2}\right)}}-{r}_{b}}{\mathrm{sin}\left(\frac{\theta }{2}\right)},$

where θ is the Cone angle.

Seat Schematic

The

### Numerically-Smoothed Displacement

The block calculates the poppet displacement, h, such that

$h=\left\{\begin{array}{ll}0,\hfill & \left(S-{S}_{\mathrm{min}}\right)\le 0\hfill \\ {h}_{Max},\hfill & \left(S-{S}_{\mathrm{min}}\right)\ge {h}_{Max}\hfill \\ \left(S-{S}_{\mathrm{min}}\right),\hfill & \text{Else}\hfill \end{array}$

where:

• S is the physical signal input.

• Smin is the Poppet position when in the seat parameter.

• hMax is the maximum displacement.

At the extremes of the orifice opening range, you can maintain numerical robustness in your simulation by adjusting the block . When the smoothing factor is nonzero, a smoothing function is applied to every calculated displacement, but primarily influences the simulation at the extremes of this range.

The normalized orifice opening is:

$\stackrel{^}{h}=\frac{h}{{h}_{\mathrm{max}}}.$

The Smoothing factor, s, is applied to the normalized opening:

${\stackrel{^}{h}}_{smoothed}=\frac{1}{2}+\frac{1}{2}\sqrt{{\stackrel{^}{h}}_{}^{2}+{\left(\frac{s}{4}\right)}^{2}}-\frac{1}{2}\sqrt{{\left(\stackrel{^}{h}-1\right)}^{2}+{\left(\frac{s}{4}\right)}^{2}}.$

The smoothed opening is:

${h}_{smoothed}={\stackrel{^}{h}}_{smoothed}{h}_{\mathrm{max}}.$

This smoothed opening is used in the valve opening area, either Aopen,conical or Aopen,sharp-edged.

### Momentum Balance

The pressure differential over the valve is:

${p}_{A}-{p}_{B}=\frac{\stackrel{˙}{m}\sqrt{{\stackrel{˙}{m}}^{2}+{\stackrel{˙}{m}}_{cr}^{2}}}{2{\rho }_{Avg}{C}_{d}^{2}{S}^{2}}\left[1-{\left(\frac{{S}_{R}}{S}\right)}^{2}\right]P{R}_{Loss},$

where:

• pA is the pressure at port A.

• pB is the pressure at port B.

• $\stackrel{˙}{m}$ is the mass flow rate.

• ρAvg is the average liquid density.

• Cd is the Discharge coefficient.

• ${\stackrel{˙}{m}}_{cr}$ is the critical mass flow rate:

${\stackrel{˙}{m}}_{cr}={\mathrm{Re}}_{cr}{\mu }_{Avg}\sqrt{\frac{\pi }{4}{S}_{R}}.$

where:

• Recr is the Critical Reynolds number.

• μAvg is the average fluid dynamic viscosity.

• S is the Cross-sectional area at port A and B.

• PRLoss is the pressure ratio:

$P{R}_{Loss}=\frac{\sqrt{1-{\left({S}_{R}/S\right)}^{2}\left(1-{C}_{d}^{2}\right)}-{C}_{d}\left({S}_{R}/S\right)}{\sqrt{1-{\left({S}_{R}/S\right)}^{2}\left(1-{C}_{d}^{2}\right)}+{C}_{d}\left({S}_{R}/S\right)}.$

### Mass Balance

The mass conservation equation in the valve is

${\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}=0,$

where:

• ${\stackrel{˙}{m}}_{A}$ is the mass flow rate into the valve through port A.

• ${\stackrel{˙}{m}}_{B}$ is the mass flow rate into the valve through port B.

### Energy Balance

The energy conservation equation in the valve is

${\varphi }_{A}+{\varphi }_{B}=0,$

where:

• ϕA is the energy flow rate into the valve through port A.

• ϕB is the energy flow rate into the valve through port B.

## Ports

• A — Thermal liquid conserving port representing valve inlet A

• B — Thermal liquid conserving port representing valve inlet B

• S — Physical signal input port for the control member displacement

## Parameters

### Parameters Tab

Valve seat specification

Choice of valve seat geometry. Options include Sharp-edged and Conical. The default setting is Sharp-edged.

Cone angle

Angle formed by the sides of the conical seat. This parameter is active only when the Valve seat specification parameter is active. The default value is 120 deg.

Ball diameter

Diameter of the spherical control member. The default value is 0.01 m.

Orifice diameter

Diameter of the valve opening. The default value is 7e-3 m.

Poppet position when in the seat

Poppet offset when the valve is shut. A positive, nonzero value indicates a partially open valve. A negative, nonzero value indicates an overlapped valve that remains shut for an initial displacement set by the physical signal at port S. The default is 0 m.

Leakage area

Area through which fluid can flow in the fully closed valve position. This area accounts for leakage between the valve inlets. The default value is 1e-12 m^2.

Smoothing factor

Portion of the opening-area curve to smooth expressed as a fraction. Smoothing eliminates discontinuities at the minimum and maximum flow valve positions. The smoothing factor must be between 0 and 1. Enter a value of 0 for zero smoothing. Enter a value of 1 for full-curve smoothing. The default value is 0.01.

Cross-sectional area at ports A and B

Area normal to the direction of flow at the valve inlets. This area is assumed the same for all the inlets. The default value is 0.01 m^2.

Discharge coefficient

Ratio of the actual mass flow rate through the valve to its ideal, or theoretical, value. The discharge coefficient accounts for the effects of valve geometry. The value must be between 0 and 1.

Critical Reynolds number

Reynolds number at which flow transitions between laminar and turbulent regimes. Flow is laminar below this number and turbulent above it. The default value is 12.

### Variables Tab

Mass flow rate into port A

Mass flow rate into the component through port A at the start of simulation. The default value is 1 kg/s.

## Version History

Introduced in R2016a


text:
# Part 5.6Transferring a file with PHP

For a website which allows downloads of specific files for each user there need to be restrictions. For instance, there might be an application only John Smith is permitted to download and another that only Joe Bloggs is allowed to open. It may be good to implement a system like the one on this site where registered and accepted users are permitted to download software.

## Reading the bytes from the file

The first step is to read the file in to the script. It should not go into a variable:

PHP
<?php
$filename = "myFile.txt"; readfile($filename);
?>

## Generating a file

The word generating is used somewhat loosely here as the file is technically not generated. The file is provided as a byte stream. There is more about byte steams on the Wikipedia page.

PHP runs on the server, so the server is in control of what files the PHP page can access. Below is a sample of the header of a page that is prepared to download:

PHP
<?php
?>

The header of the page is sent as raw HTTP data that the user's system can interpret and the server can interpret. By placing this at the top, the page tricks the browser into thinking that it is downloading an x-ole-storage file. But if this is run, nothing will download. The combination of the header and the readfile parts are what make it work. With the same sample myFile.txt the Content-Type header would be declared using the text MIME (Multipurpose Internet Mail Extensions) type (text/plain) and then read from the file.

PHP
<?php
header("Content-Disposition: attachment; filename=".basename($file)); header("Content-Type: text/plain"); ob_clean(); flush(); readfile("myFile.txt"); exit; ?>  The flush and ob_clean commands clean (or flush) the output buffer. And that is how to emulate a file download in PHP. ## .htaccess (Apache only) One of the best things about the way that web servers work is the way that they can be controlled. Creating a simple file that specifies a few lines in can change the way part of a server works. .htaccess is beyond the scope of this tutorial, but it will cover just a bit about it. .htaccess can be used to restrict access from everything except the server and FTP (File Transfer Protocol). This means that only users who are permitted to download files if they are logged in can do so. PHP <?php$loggedIn = ""
if (isset($_SESSION["loggedIn")) {$loggedIn = $_SESSION["loggedIn"]; } if ($loggedIn == true)	{

ob_clean();
flush();

exit;
}
?>

And now the .htaccess file that resides in the directory that is not to be accessible.

.htaccess
Order Deny,Allow
Deny from all

Placing that in the root of the directory structure will prevent access to this directory and any sub directories, yet PHP can still get around this.

Code preview
Feedback 👍
Comments are sent via email to me.


text:
Written by J. H. “Red” Hayes/Jack Rhodes
Released by Bob Dylan on Saved (1980), and played live once in 1999
Tabbed by Eyolf Østrem

## Album version

This is played very loosely. The chords used are all in the 7th position, so either use a capo on the 7th fret with the chords E, A/e and B11, or play without a capo with the barre chords B, E/b and F#11. Since the outcome is the same, I thought I might as well use the capoed chords in the tab.
The A/e could be an Esus4 as well (022200). When the bass comes in towards the end of the first verse, it playes E and F# (i.e. A and B relative to the capo), which fits well with Esus4 also.
The second line is played without the first A/e in verses 2 and 3.

Chords (capo 7th fret):

A/e     0xx220
B11     x2222x

E

E              A/e           E
How many times have you heard someone say,
A/e     E         A/e            E
"If I had his money I'd do things my way."
E      A/e  E
Mmm, but little they know
A/e        E
Mmmmm, that it's so hard to find
B11       E
One rich man in ten with a satisfied mind.

E      A/e E
Mmmmm, mmmmm

## Live version (Philadelphia, Nov 9, 1999)

Add a [C Csus4 C] or a [C Csus4 C Csus2 C] flourish at the end of each line and a Gsus4 or C/g for the Gs at midline.
Note that the first line in the first verse is sung completely different from the rest of the lines. Probably to fool the audience into thinking that he was going to sing Blowin’ in the wind…

Csus4   x33010
C       x32010
Csus2   x30010
Gsus4   320013
C/g     302013

C Csus4 C (repeat ad lib)

C        F     G                          C
How many times have     you heard someone say,
C         G                 F            C
"If I had his money     I could do things my way."
C           G                 F          C
But little they know    that it's so hard to find
C           G              F         C
One rich man in ten     with a satisfied mind.

C          G               F           C
Once I was winning      in fortune and fame;
C            G                   F              C
Everything that I dreamed of to get a start in lifes game.
C           F   G                  C
But suddenly it happened, I lost every dime,
C         G            F         C
But I'm richer by far   with a satisfied mind.

C           G                F                 C
C                  G             F                 C
Or a friend when you're lonesome or a love that's grown cold;
C          G             F         C
The wealthiest person   is a pauper at times
C            G              F         C
Compared to the man     with a satisfied mind.

C            G             F            C
When life is over,      my time has run out,
C              G               F                 C
My friends and my loved ones I'll leave, there's no doubt.
C             F  G                      C
But one thing for certain, when it comes my time,
C              G            F         C
I'll leave this old world with a satisfied mind.

C        F     G                          C
How many times have     you heard someone say.


text:
# Thread: Determine if each of the following sets is a basis for the indicated subspace

1. ## Determine if each of the following sets is a basis for the indicated subspace

{[1,2,3]T,[4,5,6]T,[7,8,9]T} in R3

So, I've learned that for a set of vectors to be a basis it has to satisfy 2 of 3 of the following rules (if it meets two, the third is given):
1.) Must be linearly independent
2.) Must be a spanning set
3.) the set has dimension(V) elements

I've shown that the set is not linearly independent so it doesn't meet #1.
The set does meet #3.

I am running in to some confusion on how to test if it is a spanning set. Would I set up an augmented matrix and solve to test for consistency?
Also, I had previously thought that if a set was LI, then it could not be a basis...

2. ## Re: Determine if each of the following sets is a basis for the indicated subspace

$\mathbb{R}^3 \text{ has 3 dimensions so we must have 3 linearly independent vectors of 3 elements each in order to span it.}$

$\text{Since the 3 vectors are not linearly independent they cannot be a spanning set of }\mathbb{R}^3$

A more interesting problem is when you have greater than 3, 3 dimensional vectors.

Clearly they aren't all linearly independent but do they span $\mathbb{R}^3?$

They do if a subset of 3 of them are linearly independent.

3. ## Re: Determine if each of the following sets is a basis for the indicated subspace

Ok, then being that they are not linearly independent, and not a spanning set then it is not a basis?

Also, I noticed that the middle vector is the average of the first two vectors.. does this mean I can remove that and get a basis?

4. ## Re: Determine if each of the following sets is a basis for the indicated subspace

Originally Posted by MrJank
Ok, then being that they are not linearly independent, and not a spanning set then it is not a basis?

Also, I noticed that the middle vector is the average of the first two vectors.. does this mean I can remove that and get a basis?
Yes, a basis both spans it's space and it's elements are linearly independent.

yes but only for a 2 dimensional subspace of $\mathbb{R}^3$, i.e. a plane


text:
This is the first in a category of retreads where I look again at past work which could do with some loving care. This might be as they are broken, could do with some updating or be usefully enhanced

One of my first shiny projects was using the wikipediatrend package to monitor daily page views on Wikipedia. Plotted in ggvis - which at the time seemed to be the future of interactive charts - it was also set up so that specific days could be clicked and links to articles in the Guardian newspaper were made available via the GuardianR package

Since it was written, Wikipedia have released an API and Oliver Keyes has created the pageviews package

Data is only available from mid 2015 but the response is way faster than before and there are additional features available which I will be coming back to at a later date

Initially, I have broadly recreated the original app as a flexdashboard where you can obtain daily page view data for any of the more than 5 million articles on the English wikipedia.

Here, I use a fixed input and some snipped images. The actual dashboard is an interactive shiny app

First we load the required packages (some are not essential for the code in the blog)

library(flexdashboard)
library(pageviews)
library(DT)
library(plotly)
library(stringr)
library(GuardianR)
library(httr)
library(rvest)
library(XML)
library(selectr)
library(feather)
library(tidyverse)

### Nothing in his life became him like the leaving it

2016 was a bad year for music fans with several of the biggest names dying

# Prince has to be distinguished as shown in Wikipedia page
comps <- c("George Michael","Glenn Frey","Prince (musician)","David Bowie","Leonard Cohen")

## set current date in format required for function
today <- paste0(str_replace_all(Sys.Date(),"-",""),"00")

# collect daily data from when records are available
comp_pageviews <- article_pageviews(article = comps,
start = "2015010100", end = today)

glimpse(comp_pageviews)
## Observations: 4,320
## Variables: 8
## $project <chr> "wikipedia", "wikipedia", "wikipedia", "wikipedia"... ##$ language    <chr> "en", "en", "en", "en", "en", "en", "en", "en", "e...
## $article <chr> "George_Michael", "George_Michael", "George_Michae... ##$ access      <chr> "all-access", "all-access", "all-access", "all-acc...
## $agent <chr> "all-agents", "all-agents", "all-agents", "all-age... ##$ granularity <chr> "daily", "daily", "daily", "daily", "daily", "dail...
## $date <dttm> 2015-07-01, 2015-07-02, 2015-07-03, 2015-07-04, 2... ##$ views       <dbl> 2640, 2730, 2601, 2772, 2967, 2936, 3948, 3588, 28...

The almost 3000 rows (at time of article) are retrieved almost instantaneously

Lets first summarize the data into a tabular form

comp_pageviews %>%
group_by(article) %>%
summarise(tot=sum(views),max=max(views),min=min(views),median=round(median(views),0),toppc=round(100*max/tot),maxdate=as.Date(date[which(views==max(views))]))%>%
arrange(desc(median)) %>%
DT::datatable(width=700,class='compact stripe hover row-border order-column',rownames=FALSE,options= list(paging = FALSE, searching = FALSE,info=FALSE))

So David Bowie and Prince stand apart with George Michael also separating himself in terms of interest from the other two stars. The peak day ranged between a 20-40% of total for individual artist (this will obviously decline over time)

Lets look at a chart to see if anything else pops out. As the range in pageviews is immense, I have implemented it using a log scale. Hover the chart(on the actual site) for individual daily views

comp_pageviews %>%
plot_ly(x=~date, y=~log10(views), color=~article) %>%
)


text:
# I Wigner-Eckart theorem

#### Silviu

Hello! I am a bit confused about the dimensionality of the vectors in Wigner-Eckart theorem. Here it is how it gets presented in my book. Given a vector space V and a symmetry group on it G, with the representation U(G) we have the irreducible tensors $${O_i^\mu,i=1,...,n_\mu}$$ (where $n_\mu$ is the dimension of the $D^\mu$ irreducible representation (irrep) of G) having the property that $$U(g)O_i^\mu U(g)^{-1}=O_i^\mu D^\mu(g)^j_i$$ for all $g\in G$. Now for a set of irreducible tensors $O_i^\mu$ and a set of orthonormal vectors $e_j^\nu$, vectors which span an invariant subspace of V, we have: $$O_i^\mu |e_j^\nu>=O_k^\mu |e_l^\nu>D^\mu(g)^k_i D^\nu(g)^l_j$$ which shows that the vectors $O_i^\mu |e_j^\nu>$ transform under the direct product representation $D^{\mu \times \nu}$. Then using Clebsch-Gordan coefficients, we can diagonalize $D^{\mu \times \nu}$ and implicitly decompose the vector space it acts on into invariant subspaces. Thus we get: $$O_i^\mu |e_j^\nu>=\sum_{\alpha,\lambda,l}|w^\lambda_{\alpha l}><\alpha,\lambda,l(\mu \nu)i,j>$$ where $<\alpha,\lambda,l(\mu \nu)i,j>$ are the CG coefficients and $|w^\lambda_{\alpha l}>$ are the basis vectors corresponding to the irrep $D^{\lambda}$ in the decomposition of $D^{\mu \times \nu}$. Lastly, to get the Wigner-Eckart theorem they calculate this product: $$<e^l_\lambda|O_i^\mu |e_j^\nu>$$ and here it is where I get confused. The vector (actually I think it becomes a one form, but anyway) $<e^l_\lambda|$ is in the vector space V (it has lot's of zeros as it is invariant under a certain subspaces, but still in V). However $O_i^\mu |e_j^\nu>$ is in the vector space associated with $D^{\mu \times \nu}$. Now the first one would have the dimension the same as V while the second one will have a dimension $n_\mu \times n_\nu$ which don't have to be equal. How can you take the dot product of 2 vectors with different dimensionalities? What am I missing here? For reference the book I am using is Group Theory in Physics by Wu-Ki Tung and this is presented towards the end of chapter 4. Thank you!

Related Linear and Abstract Algebra News on Phys.org

#### Silviu

"Wigner-Eckart theorem"

### Physics Forums Values

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text:
##### Gesamte Ressourcen (13) function elasticsubmit() { $('#type').val($('#type-clone').val()); document.getElementById("elastic-search").submit(); }

Students take measurements, determine a pattern, make a prediction and then check it in a competition or activity.

Verlag: Texas Instruments Inc.

Fach:  Mathematics

Schlagwörter: Distance, Linear, Measures, Predictions, Time, Velocity

Students will learn and use foundational concepts for robotic vehicle control along with math concepts to write a program to drive their Rover around the base of the Olympus Mons volcano on…

Verlag: Texas Instruments Inc.

Fach:  Computer Science, STEM

Students take measurements, determine a pattern, make a prediction and then check it in a competition or activity.

Verlag: Texas Instruments Inc.

Fach:  Mathematics

Schlagwörter: Distance, Linear Functions, Measures, Predictions, TI-Innovator Rover, Time, Velocity

Students will learn and use basic concepts for robotic vehicle control along with math concepts to write a program to drive their Rover around the base of the Olympus Mons volcano on Mars.

Verlag: Texas Instruments Inc.

Students take measurements, determine a pattern, make a prediction and then check it in a competition or activity.

Verlag: Texas Instruments Inc.

Fach:  Mathematics

Schlagwörter: Distance, Linear Functions, Measures, Predictions, TI-Innovator Rover, Time, Velocity

Students will learn and use basic concepts for robotic vehicle control along with math concepts to write a program to drive their Rover around the base of the Olympus Mons volcano on Mars.

Verlag: Texas Instruments Inc.

Students take measurements, determine a pattern, make a prediction and then check it in a competition or activity.

Verlag: Texas Instruments Inc.

Fach:  Mathematics

Schlagwörter: Distance, Linear, Measures, Predictions, TI-Innovator Rover, Time, Velocity

Students take measurements, determine a pattern, make a prediction and then check it in a competition or activity.

Verlag: Texas Instruments Inc.

Fach:  Mathematics

Schlagwörter: Distance, Linear Functions, Measures, Predictions, TI-Innovator Rover, Time, Velocity

Använd matematiska verktyg för att analysera data från ett vetenskapligt experiment.

Fach:  Mathematics, Physics

Students will learn and use basic concepts for robotic vehicle control along with math concepts to write a program to drive their Rover around the base of the Olympus Mons volcano on Mars.

Autor: Texas Instruments Inc.

Students will learn and use basic concepts for robotic vehicle control along with math concepts to write a program to drive their Rover around the base of the Olympus Mons volcano on Mars.

Autor: Texas Instruments Inc.

Expérimentations de 3 projets avec des élèves de 15 ans sur le thème de la domotique pour induire la structure du raisonnement des démonstrations en mathématiques.

Autor: Sébastien Verspecht

Fach:  Mathematics, STEM


text:
97-406 Hunziker, W., Sigal, I.M.
Time-Dependent Scattering Theory of N-Body Quantum Systems (511K, PS-Adobe-2.0) Jul 16, 97
Abstract , Paper (src), View paper (auto. generated ps), Index of related papers

Abstract. We give a full and selfcontained account of the basic results in $N$--body scattering theory which emerged over the last ten years: The existence and completeness of scattering states for potentials decreasing like $r^{-\mu}$, $\mu>\sqrt{3}-1$ . Our approach is a synthesis of earlier work and of new ideas. Global conditions on the potentials are imposed only to define the dynamics. Asymptotic completeness is derived from the fact that the mean square diameter of the system diverges like\ \ $t^2$\ \ as\ \ $t\to \pm\infty$\ \ for any orbit $\psi_t$ which is separated in energy from thresholds and eigenvalues (a generalized version of Mourre's theorem involving only the tails of the potentials at large distances). We introduce new propagation observables which considerably simplify the phase--space analysis. As a topic of general interest we describe a method of commutator expansions.

Files: 97-406.ps


text:
# Find area with partial unit squares

Count unit squares and partial unit squares to find the area of shapes.

### Problem

Each square in the grid is a unit square with an area of 1 square meter.
What is the area of the figure?
square meters


text:
# zbMATH — the first resource for mathematics

Mean values of Dedekind sums. (English) Zbl 0851.11028
For a positive integer $$k$$ and an arbitrary integer $$h$$, the Dedekind sum $$s(h,k)$$ is defined by $s(h,k) = \sum^k_{a = 1} \left( \left( {a \over k} \right) \right) \left( \left( {ah \over k} \right) \right)$ where $$((x)) = x - [x] - 1/2$$ if $$x \neq 0$$ and 0 if $$x = 0$$. The mean values of the $$2m$$th moment of $$s(h,k)$$ is investigated via the circle method of Hardy and Littlewood in this paper. Suppose that $$k$$ is a large prime number; then it is proved that $\sum^{k - 1}_{h = 1} s(h,k)^{2m} = 2 \cdot {\zeta (2m)^2 \over \zeta (4m)} \left( {k \over 12} \right)^{2m} + O \bigl( (k^{9/5} + k^{2m - 1 + 1/(m + 1)}) \log^3k \bigr).$ When $$k$$ is a large integer, a similar estimate can also be obtained.

##### MSC:
 11F20 Dedekind eta function, Dedekind sums 11M41 Other Dirichlet series and zeta functions 11P55 Applications of the Hardy-Littlewood method
##### Keywords:
Dedekind sum; mean values; circle method
Full Text:


text:
# Continuous Multiclass Labeling Approaches and Algorithms

We study convex relaxations of the image labeling problem on a continuous domain with regularizers based on metric interaction potentials. The generic framework ensures existence of minimizers and covers a wide range of relaxations of the originally combinatorial problem. We focus on two specific relaxations that differ in flexibility and simplicity -- one can be used to tightly relax any metric interaction potential, while the other one only covers Euclidean metrics but requires less computational effort. For solving the nonsmooth discretized problem, we propose a globally convergent Douglas-Rachford scheme, and show that a sequence of dual iterates can be recovered in order to provide a posteriori optimality bounds. In a quantitative comparison to two other first-order methods, the approach shows competitive performance on synthetical and real-world images. By combining the method with an improved binarization technique for nonstandard potentials, we were able to routinely recover discrete solutions within 1 the combinatorial image labeling problem.

## Authors

• 14 publications
• 23 publications
• ### MAP Estimation of Semi-Metric MRFs via Hierarchical Graph Cuts

We consider the task of obtaining the maximum a posteriori estimate of d...
05/09/2012 ∙ by M. Pawan Kumar, et al. ∙ 0

• ### Bottleneck potentials in Markov Random Fields

We consider general discrete Markov Random Fields(MRFs) with additional ...
04/17/2019 ∙ by Ahmed Abbas, et al. ∙ 0

• ### Parsimonious Labeling

We propose a new family of discrete energy minimization problems, which ...
07/05/2015 ∙ by Puneet K. Dokania, et al. ∙ 0

• ### A Global Approach for Solving Edge-Matching Puzzles

We consider apictorial edge-matching puzzles, in which the goal is to ar...
09/21/2014 ∙ by Shahar Z. Kovalsky, et al. ∙ 0

• ### Structured prediction models for RNN based sequence labeling in clinical text

Sequence labeling is a widely used method for named entity recognition a...
08/01/2016 ∙ by Abhyuday Jagannatha, et al. ∙ 0

• ### Semantic 3D Reconstruction with Continuous Regularization and Ray Potentials Using a Visibility Consistency Constraint

We propose an approach for dense semantic 3D reconstruction which uses a...
04/11/2016 ∙ by Nikolay Savinov, et al. ∙ 0

• ### A constant-ratio approximation algorithm for a class of hub-and-spoke network design problems and metric labeling problems: star metric case

Transportation networks frequently employ hub-and-spoke network architec...
03/16/2018 ∙ by Yuko Kuroki, et al. ∙ 0

##### This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

## 1 Problem Formulation

The multi-class image labeling problem consists in finding, for each pixel in the image domain , a label which assigns one of class labels to so that the labeling function adheres to some local data fidelity as well as nonlocal spatial coherency constraints.

This problem class occurs in many applications, such as segmentation, multiview reconstruction, stitching, and inpainting [PCF06]. We consider the variational formulation

 infℓ:Ω→{1,…,l}f(ℓ),f(ℓ):=∫Ωs(x,ℓ(x))dxdata term+J(ℓ).% regularizer (1)

The data term assigns to each possible label a local cost , while the regularizer enforces the desired spatial coherency. We will in particular be interested in regularizers that penalize the weighted length of interfaces between regions of constant labeling. Minimizing is an inherently combinatorial problem, as there is a discrete decision to be made for each point in the image.

In the fully discrete setting, the problem can be expressed in terms of a Markov Random Field [Win06] with a discrete state space, where the data and regularization terms can be thought of as unary and binary potentials, respectively. For graph-based discretizations of , the resulting objective only contains terms that depend on the labels at one or two points, and the problem can be approached with fast graph cut-based methods. Unfortunately, this scheme introduces an anisotropy [Boy03] and thus does not represent isotropic regularizers well. Using ternary or higher-order terms, the metrication error can be reduced, but graph-based methods then cannot be directly applied.

However, it can be shown that even in the graph-based representation the problem is NP-hard for relatively simple [BVZ01], so we cannot expect to easily derive fast solvers for this problem. This is in part caused by the discrete nature of the feasible set. In the following, we will relax this set. This allows to solve the problem in a globally optimal way using convex optimization methods. On the downside, we cannot expect the relaxation to be exact for any problem instance, i.e. we might get non-discrete (or discrete but suboptimal) solutions.

There are several choices for the relaxation method, of which in our opinion the following is the most transparent (Fig. 1): Identify label with the -th unit vector , set , and solve

 infu:Ω→Ef(u),f(u):=∫Ω⟨u(x),s(x)⟩dx+J(u). (2)

The data term is now linear in and fully described by the vector

 s(x):=(s1(x),…,sl(x))⊤:=(s(x,1),…,s(x,l))⊤. (3)

Due to the linearization, the local costs may be arbitrarily complicated, possibly derived from a probabilistic model, without affecting the overall problem class. In this form, we relax the label set by allowing to take “intermediate” values in the convex hull of the original label set. This is just the standard simplex ,

 Δl := conv{e1,…,el}={a∈Rl|a⩾0,l∑i=1ai=1}. (4)

The problem is then considered on the relaxed feasible set ,

 C:={u∈BV(Ω)l|u(x)∈Δl for a.e. x∈Ω}. (5)

The space of functions of bounded variation guarantees a minimal regularity of the discontinuities of , see Sect. 2.2. Assuming we can extend the regularizer to the whole relaxed set , we get the relaxed problem

 infu∈Cf(u),f(u):=∫Ω⟨u(x),s(x)⟩dx+J(u). (6)

If can be made convex, the overall problem is convex as well, and it may likely be computationally tractable. In addition, should ideally have a closed-form expression, or at least lead to a computationally tractable problem.

Whether these points are satisfied depends on the way a given regularizer is extended to the relaxed set. The prototypical example for such a regularizer is the total variation,

 TV(u)=∫Ω∥Du∥, (7)

where denotes the Frobenius norm, in this case on . Note that may be discontinuous, so the gradient has to be understood in a distributional sense (Sect. 2.2). Much of the popularity of stems from the fact that it allows to include boundary-length terms: The total variation of the indicator function of a set ,

 Per(S):=TV(χS). (8)

called the perimeter of , is just the classical length of the boundary .

In this paper, we will in more generality consider ways to construct regularizers which penalize interfaces between two adjacent regions with labels according to the perimeter (i.e. length or area) of the interface weighted by an interaction potential depending on the labels (in a slight abuse of notation the interaction potential is also denoted by , since there is rarely any ambiguity with respect to the ambient space dimension). The simplest case is the Potts model with the uniform metric iff and otherwise . In this case, the regularizer penalizes the total interface length, as seen above for the total variation.

As a prime motivation for our work, consider the two-class case with . As here the second component of is given by the first via , we may pose the relaxed problem in the form

 minu′∈BV(Ω),u′(x)∈[0,1] for a.e. x∈Ω∫Ωu′(x)(s1(x)−s2(x))dx+2TV(u′), (9)

where is a scalar. This formulation is also referred to as continuous cut in analogy to graph cut methods. It can be shown [CEN06] that while there may be non-discrete solutions of the relaxed problem, a discrete – i.e. – global optimal solution can be recovered from any solution of the relaxed problem. We can thus reduce the combinatorial problem to a convex problem. While there are reasons to believe that this procedure cannot be extended to the multi-class case, we may still hope for “nearly” discrete solutions.

### 1.1 Related Work

The difficulty of the labeling problem varies greatly with its precise definition. Formulations of the labeling problem can be categorized based on

1. whether they tackle the binary (two-class) or the much harder multiclass problem, and

2. whether they rely on a graph representation or are formulated in a spatially continuous framework.

An early analysis of a variant of the binary continuous cut problem and the associated dual maximal flow problem was done by Strang [Str83]. Chan et al. [CEN06] pointed out that by thresholding a nonbinary result of the relaxed, convex problem at almost any threshold one can generate binary solutions of the original, combinatorial problem (this can be carried over to any threshold and to slightly more general regularizers [Ber09, ZNF09]. The proof heavily relies on the coarea formula [FR60], which unfortunately does not transfer to the multiclass case. The binary continuous case is also closely related to the thoroughly analyzed Mumford-Shah [MS89] and Rudin-Osher-Fatemi (ROF) [ROF92] problems, where a quadratic data term is used.

For the graph-based discretization, the binary case can be formulated as a minimum-cut problem on a grid graph, which allows to solve the problem exactly and efficiently for a large class of metrics using graph cuts [KB05, BK04]. Graph cut algorithms have become a working horse in many applications as they are very fast for medium sized problems. Unfortunately they offer hardly any potential for parallelization. As mentioned in the introduction, the graph representation invariably introduces a grid bias  [Boy03] (Fig. 2). While it is possible to reduce the resulting artifacts by using larger neighborhoods, or by moving higher-order potentials through factor graphs, this greatly increases graph size as well as processing time.

Prominent methods to handle the graph-based multiclass case rely on finding a local minimum by solving a sequence of binary graph cuts [BVZ01] (see [KT07] for a recent generalization). These methods have recently been extended to the continuous formulation [TPCB08] with similar theoretical performance [Ols09]. Our results can be seen as a continuous analogon to [Ish03], where it was shown that potentials of the form can be exactly formulated as a cut on a multi-layered graph. An early analysis can be found in [KT99]

, where the authors also derive suboptimality bounds of a linear programming relaxation for metric distances. All these methods rely on the graph representation with pairwise potentials.

In this paper, we will focus on the continuous multiclass setting with higher order potentials in the discretization. Closely related to our approach is [CCP08]. In contrast to approaches that rely on a linear ordering of the labels [Ish03, BT09], the authors represent labels in a higher-dimensional space. In a certain sense, [LLT06] can be seen as a predecessor of this approach: in this work, the authors represent the label assignment using a piecewise constant real-valued function, but parametrize this function using a set of polynomial basis functions, which enables them to employ the Potts regularizer.

The approach of [CCP08] allows to formulate interaction potentials of the form with nondecreasing, positive, concave . The authors provide a thorough analysis of the continuous model and propose a relaxation based on the convex envelope, which gives almost discrete solutions in many cases. We will extend this approach to the more general class of regularizers where is an arbitrary metric. The same authors proposed a “Fast Primal-Dual” algorithm with proven convergence to solve the associated saddle point problem [PCBC09a]. By lifting the objective to a higher dimensional space, it turns out that the same method can be used to solve many problems also with nonlinear data term [PCBC09b].

Our approach is a generalization of [ZGFN08, LKY09], where a similar linearization is used with the regularizer restricted to the Potts distance, and with less strong convergence results. These methods have also been extended to segmentation on manifolds [DFPH09].

Regarding optimization, several authors proposed smoothing of the primal or dual objective together with gradient descent [Ber09, BYT10]. In contrast, our approach does not require any a priori smoothing. Using Nesterov’s method [Nes04] for the labeling problem was proposed in [LKY09]. An earlier analysis of the method in the context of -norm and TV minimization can be found in [WABF07]. In [BBC09] the method is applied to a class of general regularized problems. In [GBO09] a predecessor of the proposed Douglas-Rachford approach was presented in the Split Bregman framework [Set09a] and restricted to the two-class case. We provide an extension to the multi-class case, with proof of convergence and a sound stopping criterion.

### 1.2 Contribution

The paper is organized as follows:

1. We formulate natural requirements on the regularizer and show their implications on the choice of the interaction potential (Sect. 3). In particular, must necessarily be a metric under these requirements (Prop. 1).

2. Given such a metric, we study two possible approaches to extend regularizers on the relaxed set (Sect. 4):

• The “envelope” approach, which is a generalization of the method recently suggested by Chambolle et al. (Sect. 4.3). While there is no simple closed form expression, we show that it can be used to construct a true extension for any metric (Prop. 5).

• The “Euclidean distance” approach (Sect. 4.4), which yields exact extensions for Euclidean metrics only but has a closed form expression. We review some methods for the approximation of non-Euclidean metrics.

We provide a unified continuous framework and show existence of a minimizer.

3. Both approaches lead to non-smooth convex problems, which can be studied in a general saddle point formulation (Sect. 5). Within this framework, we propose an improved binarization technique for nonstandard potentials to recover approximate solutions for the non-relaxed problem (Sect. 5.6).

4. We provide and analyze two different methods that are capable of minimizing the specific class of saddle point problems (Sect. 6):

• A specialization of a method for nonsmooth optimization as suggested by Nesterov (Sect. 6.2). The method is virtually parameter-free and provides explicit a priori and a posteriori optimality bounds.

• A Douglas-Rachford splitting approach (Sect. 6.3). We show that the approach allows to compute a sequence of dual iterates that provide an optimality bound and stopping criterion in form of the primal-dual gap.

Both methods are highly parallelizable and are shown to converge. For reference, we also summarize the primal-dual technique from [PCBC09a] (Sect. 6.1).

5. Finally, we illustrate and compare the above methods under varying conditions and demonstrate the applicability of the proposed approaches on real-world problems (Sect. 7).

In contrast to existing graph-based methods, we provide a continuous and isotropic formulation, while in comparison with existing continuous approaches, we provide a unified framework for arbitrary, non-uniform metrics . The Euclidean metric method and Nesterov optimization have been announced in less generality in [LKY09, LBS09].

## 2 Mathematical Preliminaries

In the following sections we provide a reference of the notation used, and a brief introduction to the concept of functions of bounded variation and corresponding functionals. We aim to provide the reader with the basic ideas. For more detailed expositions we refer to [AFP00, Zie89].

### 2.1 Notation

In the following, superscripts denote a collection of vectors or matrix columns, while subscripts denote vector components, i.e. we denote, for ,

 A = (a1|…|al)=(Aij),Aij=(aj)i=aji,1⩽i⩽d,1⩽j⩽l. (10)

An additional bracket indicates an element of a sequence . We will frequently make use of the Kronecker product [Gra81]

 ⊗:Rn1×m1×Rn2×m2→R(n1n2)×(m1m2) (11)

in order to formulate all results for arbitrary dimensions. The standard simplex in is denoted by , where .

is the identity matrix in

and the usual Euclidean norm for vectors resp. the Frobenius norm for matrices. Analogously, the standard inner product extends to pairs of matrices as the sum over their elementwise product. denotes the ball of radius at , and the set of with of a set is defined as iff and otherwise. By iff and otherwise we denote the corresponding indicator function. For a convex set , is the support function from convex analysis. denotes the classical Jacobian of .

is the space of -times continuously differentiable functions on with compact support, and the completion of under the supremum norm. As usual, denotes the -dimensional Lebesgue measure, while denotes the -dimensional Hausdorff measure. For some measure and set , denotes the restriction of to , i.e. .

### 2.2 Total Variation and BV

The total variation will be our main tool to construct the regularizer . For a differentiable scalar-valued function , the total variation is simply the integral over the norm of its gradient:

 TV(u) = ∫Ω∥∇u∥dx. (12)

As is the designated labeling function, which ideally should be piecewise constant, the differentiability and continuity assumptions have to be dropped. In the following we will shortly review the general definition of the total variation and its properties.

We require the image domain to be a bounded open domain with compact Lipschitz boundary, that is the boundary can locally be represented as the graph of a Lipschitz-continuous function. For simplicity, we will assume in the following that .

We consider general vector-valued functions which are locally absolutely integrable, i.e. . As is bounded this is equivalent to being absolutely integrable, i.e. . For any such function , its total variation is defined in a dual way [AFP00, (3.4)] as

 TV(u) := supv∈DTVl∑j=1∫Ωujdivvjdx=supv∈DTV∫Ω⟨u,Divv⟩dx, (13) DTV := {v∈C∞c(Ω)d×l|∥v(x)∥⩽1∀x∈Ω}, (14) Divv := (divv1,…,divvl)⊤.

This definition can be derived for continuously differentiable by extending (12) to vector-valued ,

 TV(u) = ∫Ω∥J(u)∥dx, (15)

replacing the norm by its dual formulation and partial integration. If has finite total variation, i.e. , is said to be of bounded variation. The vector space of all such functions is denoted by :

 BV(Ω)l = {u∈(L1(Ω))l|TV(u)<∞}. (16)

Equivalently, iff and its distributional derivative corresponds to a finite Radon measure; i.e. and there exist -valued measures on the Borel subsets of such that [AFP00, p.118]

 l∑j=1∫Ωujdivvjdx=−l∑j=1d∑i=1∫ΩvjidDiuj,∀v∈(C∞c(Ω))d×l. (17)

These measures form the distributional gradient , which is again a measure that vanishes on any -negligible set. If then , where is the total variation of the measure in the measure-theoretic sense [AFP00, 3.6]. The total variation of characteristic functions has an intuitive geometrical interpretation: For a Lebesgue-measurable subset , its perimeter is defined as the total variation of its characteristic function,

 Per(S) := TV(χS). (18)

Assuming the boundary is sufficiently regular, is just the classical length () or area () of the boundary.

### 2.3 Properties of TV and Compactness

We review the most important ingredients for proving existence of minimizers for variational problems on involving involving .

Convexity. As is the pointwise supremum of a family of linear functions, it is convex and positively homogeneous, i.e. for .

Lower Semicontinuity. A functional is said to be lower semicontinuous with respect to some topology, if for any sequence converging to ,

 liminfk→∞J(u(k)) ⩾ J(u). (19)

It can be shown that for fixed , the total variation is well-defined on and lower semicontinuous in w.r.t. the topology [AFP00, 3.5,3.6]; hence also in due to the boundedness of .

Compactness. For , instead of the norm topology induced by

 ∥u∥BV := ∫Ω∥u∥dx+TV(u), (20)

which makes a Banach space but is often too strong, one frequently uses the weak* convergence: Define weakly* iff

1. ,

2. in and

3. weakly* in measure, i.e.

 ∀v∈C0(Ω):limk→∞∫ΩvdDu(k)=∫ΩvdDu. (21)

For this is equivalent to in , and being uniformly bounded in , i.e. there exists a constant such that [AFP00, 3.13]. For the weak* topology in , a compactness result holds [AFP00, 3.23]: If and is uniformly bounded in , then contains a subsequence weakly*-converging to some .

### 2.4 General Functionals on BV

We will now review how general functionals depending on the distributional gradient can be defined. Recall that for any the distributional gradient is a measure. Moreover, it can be uniquely decomposed into three mutually singular measures

 Du = Dau+Dju+Dcu, (22)

that is: An absolutely continuous part ,  the jump part , and the Cantor part . Mutual singularity refers to the fact that can be partitioned into three subsets, such that each of the measures is concentrated on exactly one of the sets. We will give a short intuitive explanation, see [AFP00, 3.91] for the full definitions.

The part is absolutely continuous with respect to the -dimensional Lebesgue measure , i.e. it vanishes on any -negligible set. It captures the “smooth” variations of : in any neighborhood where has a (possibly weak) Jacobian , the jump and Cantor parts vanish and

 Du=Dau=J(u)Ld. (23)

The jump part is concentrated on the set of points where locally jumps between two values and along a -dimensional hypersurface with normal (unique up to a change of sign). In fact, there exists a jump set of discontinuities of and Borel functions and such that [AFP00, 3.78, 3.90]

 Dju=Du└Ju=νu(u+−u−)⊤Hd−1└Ju, (24)

where denotes the restriction of the -dimensional Hausdorff measure on the jump set , i.e. for measurable sets . The Cantor part captures anything that is left.

As an important consequence of the mutual singularity, the total variation decomposes into . Using this idea, one can define functionals depending on the distributional gradient [AFP00, Prop. 2.34]. For and some convex, lower semi-continuous , define

 J(u) := ∫ΩΨ(Du):=∫ΩΨ(J(u)(x))dx+… (25) ∫JuΨ∞(νu(x)(u+(x)−u−(x))⊤)dHd−1+… ∫ΩΨ∞(Dcu|Dcu|)d|Dcu|.

Here is the recession function of , and denotes the polar decomposition of , which is the density of with respect to its total variation measure . If is positively homogeneous, holds, and

 J(u) = ∫ΩΨ(Du|Du|)d|Du|. (26)

From (25) it becomes clear that the meaning of acting on the Jacobian of is extended to the jump set as acting on the difference of the left and right side limits of at the discontinuity. This is a key point: by switching to the measure formulation, one can handle noncontinuous functions as well.

## 3 Necessary Properties of the Interaction Potential

Before applying the methods above to the labeling problem, we start with some basic considerations about the regularizer and the interaction potential . We begin by formalizing the requirements on the regularizer of the relaxed problem as mentioned in the introduction. Let us assume we are given a general interaction potential . Intuitively, assigns a weight to switching between label and label . We require

 d(i,j) > 0,i≠j, (27)

but no other metric properties (i.e. symmetry or triangle inequality) for now. Within this work, we postulate that the regularizer should satisfy

1. is convex and positively homogeneous on .

2. for any constant , i.e. there is no penalty for constant labelings.

3. For any partition of into two sets with , and any ,

 J(eiχS+ejχS∁) = d(i,j)Per(S), (28)

i.e. a change from label to label gets penalized proportional to as well as the perimeter of the interface. Note that this implies that is isotropic (i.e. rotationally invariant).

We require convexity in (P1) in order to render global optimization tractable. Indeed, if is convex, the overall objective function (6) will be convex as well due to the linearization of the data term. Positive homogeneity is included as it allows to be represented as a support function (i.e. its convex conjugate is an indicator function and for some closed convex set ), which will be exploited by our optimization methods.

Requirements (P3) and (P2) formalize the principle that the multilabeling problem should reduce to the classical continuous cut (9) in the two-class case. This allows to include boundary length-based terms in the regularizer that can additionally be weighted depending on the labels of the adjoining region (Fig. 4).

Together, these requirements pose a natural restriction on :

###### Proposition 1.

Let satisfy (P1) – (P3). Then must necessarily be a metric, i.e. for all ,

1. ,

2. ,

###### Proof.

1. follows from (P2) and (P3) by choosing and with . Symmetry in 2. is obtained from (P3) by replacing with , as . To show 3., first note that for any constant and all , since . Fix any set with perimeter

 c:=Per(S)>0. (29)

Then, using the above mentioned fact and the positive homogeneity of ,

 cd(i,k) = J(eiχS+ekχS∁) (30) = 2J(12(eiχS+ekχS∁)+12ej) (31) = 2J(12(eiχS+ejχS∁)+12(ejχS+ekχS∁)) (32) ⩽ J(eiχS+ejχS∁)+J(ejχS+ekχS∁) (33) = c(d(i,j)+d(j,k)). (34)

Note that if the requirement (27) is dropped, it is easy to show that if for some , then for any . In this case the classes and can be collapsed into a single class as far as the regularizer is concerned. The decision between label and is then completely local, i.e. depends only on the data term and can be postponed to a post-processing step by modifying the data term to

 s′i(x):=s′j(x) := min{si(x),sj(x)}. (35)

Thus (27) is not a real limitation and can be always assured. As a side note, it can be shown that, under some assumptions and in the space of piecewise constant functions, the subadditivity of already follows if is required to be lower semicontinuous [Bra02, p.88].

Proposition 1 implies that for non-metric , we generally cannot expect to find a regularizer satisfying (P1)–(P3). Note that here is not required to be of any particular form. In the following sections, we will show that on the other hand, if is metric as in Proposition 1, such a regularizer can always be constructed. This implies that the interaction potentials allowing for a regularizer that satisfies (P1)–(P3) are exactly the metrics.

## 4 Constructing Regularizers from the Interaction Potential

We study next how to construct regularizers on satisfying (P1)–(P3). As in (25) we set

 J(u):=∫ΩΨ(Du). (36)

We additionally require to be a support function, i.e. for some closed, convex ,

 Ψ(z)=σDloc(z)=supv(x)∈Dloc⟨z,v(x)⟩. (37)

As a support function, coincides with its recession function , thus

 J(u) = ∫ΩΨ(J(u)(x))dx+… (38) ∫JuΨ(νu(x)(u+(x)−u−(x))⊤)dHd−1+… ∫ΩΨ(Dcu|Dcu|)d|Dcu|.

Also, we have an equivalent dual formulation in analogy to (13),

 J(u)=sup{∫Ω ⟨u,Divv⟩dx|v∈C∞c(Ω)d×l,v(x)∈Dloc∀x∈Ω}. (39)

For simplicity we will also assume that is rotationally invariant along the image dimensions, i.e. for any rotation matrix ,

 v=(v1,…,vl)∈Dloc ⇔ (Rv1,…,Rvl)∈Dloc. (40)

This is equivalent to being isotropic.

We will now show under which circumstances a minimizer exists in , and then see how the approach can be used to construct regularizers for specific interaction potentials.

### 4.1 Existence of Minimizers

The complete problem considered here is of the form (cf. (6) and (38))

 infu∈Cf(u),f(u):=∫Ω⟨u,s⟩dx+J(u) (41)

where as in (36),

 Ψ(z)=supv(x)∈Dloc⟨z,v(x)⟩ (42)

for some closed convex , and

 C={u∈BV(Ω)l|u(x)∈ΔlLd-a.e. x∈Ω}. (43)

Note that is convex, as it is the pointwise supremum of affine functions (39). Again for simplicity we set . Then we have the following

###### Proposition 2.

Let be closed convex, , , and

 f(u)=∫Ω⟨u,s⟩dx+∫ΩΨ(Du). (44)

Additionally assume that for some . Then is lower semicontinuous in with respect to convergence.

###### Proof.

As the data term is continuous, it suffices to show that the regularizer is lower semicontinuous. This is an application of [AFP00, Thm. 5.47]. In fact, the theorem shows that is the relaxation of ,

 ~f(u):=∫Ω⟨u,s⟩dx+∫ΩΨ(Ju(x))dx, (45)

on w.r.t. (thus here ) convergence and thus lower semicontinuous in . To apply the theorem, we have to show that is quasiconvex in the sense of [AFP00, 5.25], which holds as it is convex by construction. The other precondition is (at most) linear growth of , which holds with

###### Proposition 3.

Let as in Prop. 2 and additionally assume that

 Bρl(0)∩{(v1,…,vl)|∑ivi=0)}⊆Dloc⊆Bρu(0) (46)

for some . Then the problem

 minu∈Cf(u) (47)

has at least one minimizer.

###### Proof.

From the inner and outer bounds it holds that for any with . Moreover, the constraint implies that satisfies . Combining this with the positive homogeneity of , it follows from (26) that

 0⩽ρlTV(u)⩽J(u) ⩽ ρuTV(u). (48)

From

 ∫Ω⟨u,s⟩dx⩾−∫Ω∥u(x)∥∞∥s(x)∥1dx, (49)

the fact that , and boundedness of and , it follows that the data term is bounded from below on .

We now show coercivity of with respect to the norm: Let with . As the data term is bounded from below, and using the fact that , it follows that . Thus is coercive.

Equations (48) and (49) also show that is bounded from below. Thus we can choose a minimizing sequence . Due to the coercivity, the sequence must then be bounded from above. From this and [AFP00, Thm. 3.23] we conclude that there is a subsequence of weakly*- (and thus -) converging to some . With the lower semicontinuity from Prop. 2 and closedness of with respect to convergence, existence of a minimizer follows. ∎

### 4.2 Relation to the Interaction Potential

To relate such to the labeling problem in view of (P3), we have the following

###### Proposition 4.

Let as in Prop. 2. For some and vectors , let . Then for any vector with ,

 J(u) = (50)

In particular, if for some with , then fulfills (P3).

###### Proof.

To show the first equality, (26) implies

 J(u) = ∫ΩΨ(Du|Du|)|Du| (51) = ∫ΩΨ(D(a+u′(b−a))|D(a+u′(b−a))|)|D(a+u′(b−a))| (52) = ∫ΩΨ((Du′)(b−a)⊤|(Du′)(b−a)⊤|)|(Du′)(b−a)⊤| (53)

We make use of the property , which is a direct consequence of the definition of the total variation measure and the fact that for any vector (note that are also vectors). Therefore

 J(u) = ∫ΩΨ((Du′)(b−a)⊤|Du′|∥b−a∥)|Du′|∥b−a∥, (54)

which by positive homogeneity of implies

 J(u) = ∫ΩΨ(Du′|Du′|(b−a)⊤)|Du′|. (55)

Since the density function assumes values in , there exists, for a.e. , a rotation matrix mapping to . Together with the rotational invariance of from (40) this implies

 J(u) = ∫ΩΨ(y(b−a)⊤)|Du′|=Ψ(y(b−a)⊤)TV(u′), (56)

which proves the first equality in (50). The second equality can be seen as follows:

 r := supv∈Dloc∥v(b−a)∥ (57) = supv∈Dlocsupz∈Rd,∥z∥⩽1⟨z,v(b−a)⟩ (58) = supz∈Rd,∥z∥⩽1supv∈Dloc⟨z,v(b−a)⟩. (59)

Denote by a rotation matrix mapping to , i.e. , then

 r = supz∈Rd,∥z∥⩽1supv∈Dloc⟨Rzz,Rzv(b−a)⟩ (60) = supz∈Rd,∥z∥⩽1supv′∈RzDloc⟨y,v′(b−a)⟩. (61)

The rotational invariance of provides , therefore

 r = supz∈Rd,∥z∥⩽1supv∈Dloc⟨y,v(b−a)⟩ (62) = supv∈Dloc⟨y,v(b−a)⟩=Ψ(y(b−a)⊤). (63)

As a consequence, if the relaxed multiclass formulation is restricted to two classes by parametrizing for , it essentially reduces to the scalar continuous cut problem: Solving

 minu′∈BV(Ω),u′(x)∈[0,1],Ld-a.e. x∈Ω∫Ω⟨(1−u′)a+u′b,s⟩dx+J(u) (64)

is equivalent to solving

 minu′∈BV(Ω),u′(x)∈[0,1],Ld-a.e. x∈Ω∫Ωu′(b−a)dx+Ψ(y(b−a)⊤)TV(u′) (65)

which is just the classical binary continuous cut approach with data and regularizer weight , where is some arbitrary unit vector. For the multiclass case, assume that

 u=uP = e1χP1+…+elχPl (66)

for some partition with . Then the absolutely continuous and Cantor parts vanish [AFP00, Thm. 3.59, Thm. 3.84, Rem. 4.22], and only the jump part remains:

 J(uP) = ∫SuPΨ(νuP(uP+−uP−)⊤)dHd−1, (67)

where


text:
#### Software installation

gabriele somma

Hi Simon, yesterday I helped a friend to install the software on windows 10 pro, the installation is successful but when I open the software from this error, I am attaching a video. What do you recommend me to do? Thanks

Simon Brown

From: main@SDR-Radio.groups.io <main@SDR-Radio.groups.io> on behalf of gabriele somma via groups.io <gabrielesomma@...>
Sent: 04 July 2022 08:46
To: main@SDR-Radio.groups.io <main@SDR-Radio.groups.io>
Subject: [SDR-Radio] Software installation #rtl

Hi Simon, yesterday I helped a friend to install the software on windows 10 pro, the installation is successful but when I open the software from this error, I am attaching a video. What do you recommend me to do? Thanks

--
- + - + -
Please use https://forum.sdr-radio.com:4499/ when posting questions or problems.

Gisle Vanem

On Mon, Jul 4, 2022 at 09:46 AM, gabriele somma wrote:

What do you recommend me to do? Thanks

The installer and the redistributable stuff have bug issues:

• these MFC140u.dll and VCOMP140.dll files are not always copied to a directory on PATH.

• One sometimes has to do it manually.

• Not to speak about the uninstall.exe program that deleted much more than what was installed.

Simon Brown

Hi,

(If you have problems after reading this let me know.)

1. Very probably the DLLs you need have been removed by another install / uninstall.
2. Uninstalling SDR Console never removes C++ redistributable files.
3. Uninstalling SDR Console only removes files in the SDR-Radio.com (V3) folder.
4. Never ever copy system DLLs. System DLLs live in the system folders.
To reinstall, select C++ 2019 Redistributable from the SDR-Radio.com (V3) Tools entry in the Start Menu:

Uninstall, then Install. This ensures the redistributable is correctly installed. The latest can be downloaded here: https://aka.ms/vs/17/release/vc_redist.x64.exe .

If you copy files into the PATH then you will always have problems, never do this.

From: main@SDR-Radio.groups.io <main@SDR-Radio.groups.io> on behalf of Gisle Vanem via groups.io <gvanem@...>
Sent: 04 July 2022 09:50
To: main@SDR-Radio.groups.io <main@SDR-Radio.groups.io>
Subject: Re: [SDR-Radio] Software installation #rtl

On Mon, Jul 4, 2022 at 09:46 AM, gabriele somma wrote:

What do you recommend me to do? Thanks

The installer and the redistributable stuff have bug issues:

• these MFC140u.dll and VCOMP140.dll files are not always copied to a directory on PATH.

• One sometimes has to do it manually.

• Not to speak about the uninstall.exe program that deleted much more than what was installed.

--
- + - + -
Please use https://forum.sdr-radio.com:4499/ when posting questions or problems.

gabriele somma

Simon thanks for your clarification, I solved thanks.

Gisle Vanem

On Mon, Jul 4, 2022 at 12:27 PM, Simon Brown wrote:

1. Uninstalling SDR Console only removes files in the SDR-Radio.com (V3) folder.

Try this:
* Install version 3.x under c:\foo.
* Install version 3.y under c:\foo\bar.
* Run the c:\foo\uninstall.exe.
* Result; it removes everything under c:\foo\bar too!

Simon Brown

Yes it will, that's the way it's designed. Don't install in that nested tree structure. Give each installation its own root.

Simon Brown

From: main@SDR-Radio.groups.io <main@SDR-Radio.groups.io> on behalf of Gisle Vanem via groups.io <gvanem@...>
Sent: 04 July 2022 18:18
To: main@SDR-Radio.groups.io <main@SDR-Radio.groups.io>
Subject: Re: [SDR-Radio] Software installation #rtl

On Mon, Jul 4, 2022 at 12:27 PM, Simon Brown wrote:

1. Uninstalling SDR Console only removes files in the SDR-Radio.com (V3) folder.

Try this:
* Install version 3.x under c:\foo.
* Install version 3.y under c:\foo\bar.
* Run the c:\foo\uninstall.exe.
* Result; it removes everything under c:\foo\bar too!

--
- + - + -
Please use https://forum.sdr-radio.com:4499/ when posting questions or problems.

Gisle Vanem

On Mon, Jul 4, 2022 at 08:56 PM, Simon Brown wrote:

Yes it will, that's the way it's designed.

Pretty lame. Other professional installers / uninstaller use a manifest of what's installed. And only uninstall that.

Simon Brown

True, but this is crappy freeware.

Simon Brown

From: main@SDR-Radio.groups.io <main@SDR-Radio.groups.io> on behalf of Gisle Vanem via groups.io <gvanem@...>
Sent: 04 July 2022 20:27
To: main@SDR-Radio.groups.io <main@SDR-Radio.groups.io>
Subject: Re: [SDR-Radio] Software installation #rtl

On Mon, Jul 4, 2022 at 08:56 PM, Simon Brown wrote:

Yes it will, that's the way it's designed.

Pretty lame. Other professional installers / uninstaller use a manifest of what's installed. And only uninstall that.

--
- + - + -
Please use https://forum.sdr-radio.com:4499/ when posting questions or problems.

jdow

That is not even effective as all the registry and appdata settings will be shared.

{o.o}

On 20220704 11:56:14, Simon Brown wrote:
Yes it will, that's the way it's designed. Don't install in that nested tree structure. Give each installation its own root.

Simon Brown

From: main@SDR-Radio.groups.io <main@SDR-Radio.groups.io> on behalf of Gisle Vanem via groups.io <gvanem@...>
Sent: 04 July 2022 18:18
To: main@SDR-Radio.groups.io <main@SDR-Radio.groups.io>
Subject: Re: [SDR-Radio] Software installation #rtl

On Mon, Jul 4, 2022 at 12:27 PM, Simon Brown wrote:

1. Uninstalling SDR Console only removes files in the SDR-Radio.com (V3) folder.

Try this:
* Install version 3.x under c:\foo.
* Install version 3.y under c:\foo\bar.
* Run the c:\foo\uninstall.exe.
* Result; it removes everything under c:\foo\bar too!

--
- + - + -
Please use https://forum.sdr-radio.com:4499/ when posting questions or problems.

jdow

Bat pucky.

{o.o}

On 20220704 12:27:43, Gisle Vanem wrote:

On Mon, Jul 4, 2022 at 08:56 PM, Simon Brown wrote:

Yes it will, that's the way it's designed.

Pretty lame. Other professional installers / uninstaller use a manifest of what's installed. And only uninstall that.

Gord Seifert

Lame?   What seems lame to me is installing one version under another. If all versions were installed in seperate directories on the same level under foo (foo\3.x, foo\3.y, ect.) your problem would not exist and the hard drive would still not look like a mess.

jdow

SDRC is not designed to support side-by-side installations. A proper such design would not use the same registry and %apdata% folders for storing settings, thus making them truly independent of each other. SDRSharp does this if that is what you want.

With this in mind a settings snapshot feature that archives all settings before the user gets into "experimental" mode might be a good thing.

{^_^}

On 20220704 23:25:13, Gord Seifert wrote:

Lame?   What seems lame to me is installing one version under another. If all versions were installed in seperate directories on the same level under foo (foo\3.x, foo\3.y, ect.) your problem would not exist and the hard drive would still not look like a mess.

 1 - 13 of 13


text:
# zbMATH — the first resource for mathematics

Riesz and Fredholm theory in Banach algebras. (English) Zbl 0534.46034
Research Notes in Mathematics, 67. Boston-London-Melbourne: Pitman Advanced Publishing Program. VI, 123 p. £7.95 (1982).
In this monograph the authors show how the classical theory of Fredholm and Riesz operators on Banach and Hilbert spaces can be extended to the setting of Banach and $$C^*$$-algebras.
There are six chapters. The first of these summarises results from Fredholm and Riesz operator theory, most of which are familiar. There follows a chapter on Fredholm theory in Banach algebras. The theory is initially developed in a unital primitive Banach algebra. A Fredholm element in such an algebra is defined as one which is invertible modulo the socle of the algebra. The concept is motivated by Atkinson’s characterization of a Fredholm operator on a Banach space as one which is invertible modulo the ideal of finite rank operators. The nullity, defect, and index of a Fredholm element can be defined, and a representation connects these quantities with the nullity, defect, and index of an associated Fredholm operator. This representation enables many results of Fredholm theory in primitive Banach algebras to be derived from their counterparts in operator theory. An extension to general unital Banach algebras is then possible by factoring out the primitive ideals. The numerical valued nullity, defect, and index in the primitive case are replaced by functions (with finite support) on the structure space of the general algebra. The classical spectral theory of Fredholm operators carries over in this general algebraic setting.
A development of Riesz theory in Banach algebras is the subject of the next chapter. A Riesz operator on a Banach space has been characterized by Ruston as one whose canonical image in the Calkin algebra has spectral radius zero. This characterization motivates the concept of a Riesz element x in a Banach algebra A, with respect to a closed two-sided ideal K, as one whose canonical image in the quotient algebra A/K has spectral radius zero. If K is an inessential ideal then, by building on the Fredholm theory developed in the previous chapter, a spectral theory for Riesz elements, analogous to that for Riesz operators, emerges for unital algebras. This is later extended to general Banach algebras. A theory of Riesz algebras is developed, and numerous examples of such algebras are given.
In the fourth chapter, results concerning Hilbert space operators are extended to a $$C^*$$-algebra setting. In particular the decomposition theorems of West and Stampfli are extended. It is shown that a $$C^*$$- algebra is a Riesz algebra if, and only if, the spectrum of every Hermitean element has no non-zero accumulation points.
A chapter is devoted to applications. Here the general theory is applied to the study of semi-normal elements in a $$C^*$$-algebra, to certain algebras of Banach space operators, as well as to measures on compact groups. A final chapter is intended to be used as a reference to that part of the general theory of Banach algebras which underlies the material in the text.
The subject matter is well presented. Each chapter concludes with a section of notes and comments which contain the historical development of the ideas discussed, and refer to an extensive bibliography. There are many new results, as well as old ones in a more general setting. The book will provide a welcome addition to the literature on Banach algebras and spectral theory.
Reviewer: H.P.Rogosinski

##### MSC:
 46H05 General theory of topological algebras 47A53 (Semi-) Fredholm operators; index theories 47-02 Research exposition (monographs, survey articles) pertaining to operator theory 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis 46L05 General theory of $$C^*$$-algebras 46J05 General theory of commutative topological algebras 47B06 Riesz operators; eigenvalue distributions; approximation numbers, $$s$$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators 47B40 Spectral operators, decomposable operators, well-bounded operators, etc.