text:
A Computational Algebraic Geometry

# Lattice Landau Gauge and Algebraic Geometry

## Abstract

Finding the global minimum of a multivariate function efficiently is a fundamental yet difficult problem in many branches of theoretical physics and chemistry. However, we observe that there are many physical systems for which the extremizing equations have polynomial-like non-linearity. This allows the use of Algebraic Geometry techniques to solve these equations completely. The global minimum can then straightforwardly be found by the second derivative test. As a warm-up example, here we study lattice Landau gauge for compact U(1) and propose two methods to solve the corresponding gauge-fixing equations. In a first step, we obtain all Gribov copies on one and two dimensional lattices. For simple systems their number can already be of the order of thousands. We anticipate that the computational and numerical algebraic geometry methods employed have far-reaching implications beyond the simple but illustrating examples discussed here.

\ShortTitle

Lattice Landau Gauge and Algebraic Geometry \FullConferenceInternational Workshop on QCD Green’s Functions,
Confinement, and Phenomenology - QCD-TNT09
September 07 - 11 2009
ECT Trento, Italy

## 1 Introduction

Finding the global minimum of a multivariate function is one of the most important tasks in statistical mechanics, condensed matter theory, lattice gauge theories and theoretical chemistry. Most of the usual methods to minimize a multivariate function are based on the Newton-Raphson method, where a start solution is guessed and then is refined by successive iterations. There are several efficient refinements of this minimization procedure, e.g., simulated annealing. However, they are often not successful as they can get trapped in a local minimum instead of the global one. This problem introduces an error of an unknown order.

A typical example is the Landau gauge on the lattice: in the continuum the corresponding Landau gauge fixing condition is solved to fix the gauge, whereas on the lattice, usually the so-called standard lattice Landau gauge-fixing (LLG) functional (whose first derivatives with respect to the gauge parameters are the corresponding gauge-fixing conditions on the lattice) is numerically minimized [10, 11, 12, 13]. We take the example of the LLG functional for compact U(), which is given by

 F=∑i,μ(1−cos(ϕi,μ+θi+^μ−θi)), (1)

where the variables sit on the th lattice site of a -dimensional lattice grid, is the unit vector in direction, and the variables sit on the links between the th and th lattice sites. Here the -variables are related to the gauge potential in the corresponding continuum gauge theory and the -variables are the gauge transformations with respect to which is minimized. The special case when all are zero is called the trivial orbit. We will also frequently use random choices for the link angles which we then refer to as random orbits. The Hessian matrix of this functional is the Faddeev-Popov operator and its determinant is known as the Faddeev-Popov determinant [18] on the lattice.

In order to compare the results from lattice Landau gauge to those from functional methods in the continuum such as studies of Dyson-Schwinger equations [8], one should in principle find all stationary points of under variation with respect to the gauge variables [3, 4, 6, 2], that is all solutions to the corresponding LLG equations on the lattice,

 ∂F∂θi=0. (2)

The solutions to these equations are the lattice analogues of Gribov copies [7].1 The sum over all those Gribov copies weighted by the sign of the Faddeev-Popov determinant evaluated at each copy would then be independent of the gauge orbit by the Poincaré-Hopf theorem and this could thus be used to define the measure of Landau gauge on the lattice analogous to standard Faddeev-Popov theory with Becchi-Rouet-Stora-Tyutin (BRST) symmetry.

The reason why this procedure fails is that the signs of all Gribov copies exactly cancel one another, always yielding an exact zero for their sum which computes the vanishing Euler characteristic of the lattice gauge group manifold in the standard implementation of the Landau gauge. This zero sum is the origin for the famous Neuberger problem of standard BRST formulations on the lattice [16, 17], which is the statement that the expectation value of any gauge-invariant observable is of indefinite form in such a formulation.

The problem can be overcome by stereographically projecting the lattice gauge group onto a manifold whose Euler characteristic is non-zero [6]. This provides a well-defined lattice BRST formulation on the projected manifold and it can serve as a non-perturbative definition of BRST symmetry in the continuum limit.

The Gribov copies of compact U() are of course lattice artifacts. However, the gauge group is a direct product of odd-dimensional and compact manifolds (circles in this case), as it is for every SU() gauge theory on the lattice also. This is sufficient to conclude that it also shares the same problem with SU() theories. Moreover, it turns out that the Neuberger problem in SU() is avoided when that of its maximal Abelian subgroup is [9], because the coset manifold consist of even dimensional spheres whose Euler characteristic is 2. It is therefore obviously important to understand the Gribov copy and Neuberger problems in compact U() gauge models.

At the same time, for the trivial orbit the functional in Eq. (1) also represents the Hamiltonian of an XY model which stands alongside the Ising and Heisenberg models as one of the most intensely studied systems in statistical mechanics and condensed matter theory. For a random orbit, it is that of a random phase XY model (RPXYM). These have been widely used as simple models of classical superconductivity, to study aspects of high- superconductors, to describe the XY magnet with random Dzyaloshinski-Moriya interactions or a positional disordered Josephson junction array, to name a few. In either case, the calculations (related to Domain Wall Renormalization Group studies) boil down to obtaining the global minimum of the Hamiltonian with respect to the -variables for a given set of ’s (see, e.g., [25]). In Ref. [26], it was shown in lower dimensional models that one can first perform a duality transformation then followed by a numerical minimization so that a minimization algorithm has to search only within the space of local minima. Recently, Akino and Kosterlitz [25] have used this method to obtain the global minimum of a RPXYM Hamiltonian. This method, while giving more confidence in the final results with less numerical effort, still relies on numerical algorithms which are known to fail for larger systems.

The knowledge of all stationary points is important in recent studies of the potential energy landscape of the classical statistical mechanical systems such as the RPXYM and its relation to phase transitions [21, 22, 23]. An effort to numerically classify all Gribov copies of compact U(1) in 2 dimensions was done in [27].

Here, we take up the general problem of finding all solutions to the standard LLG equations for compact U() or the stationary points of the corresponding XY model. The corresponding equations are non-linear which makes them difficult to deal with analytically or numerically. We observe, however, that this non-linearity is polynomial-like and the equations can be transformed into systems of polynomial equations. Then we can use Algebraic Geometry techniques to solve these systems and obtain all stationary points of the corresponding function and in particular the global minimum accurately. This procedure is applicable to a wide range of interesting physical problems such as obtaining all stationary points of the classical Hamiltonians of Ising and Heisenberg models; solving classical field equations for pure lattice gauge field theories, Abelian Higgs model, etc.; finding a generic form of SU() matrices; obtaining the vacua of Supersymmetric potentials; and many more (see Ref. [2]). The corresponding gauge-fixing equations for the modified LLG, proposed in Refs. [3, 4], for the compact U() case can also be shown to have polynomial-like non-linearity but their structure is more complicated [2]. Here, we restrict the discussion to the standard LLG.

In the one-dimensional lattice case, the LLG equations have been exactly solved for both periodic and anti-periodic boundary conditions elsewhere [2, 3, 4] where the corresponding equations are treated as systems of linear equations. However, for higher dimensional lattices the corresponding equations are genuinely non-linear and the same method can not be used. We will show that the systems of these non-linear equations can be viewed as systems of multivariate polynomial equations and therefore can be described by the language of Algebraic Geometry. We will then discuss two promising approaches to solve the polynomial systems: the numerical polynomial homotopy continuation (NPHC) method and the computational Algebraic Geometry approach (in the Appendix) and present our results. Though our goal is to solve the gauge fixing conditions for higher dimensional lattices, we will explain the methods using the one-dimensional case because: (1) we already know the exact solutions for this case and so the results from the new approaches will have a precise comparison, (2) the one-dimensional case, interpreted as an Algebraic Geometry problem, provides all the essence of the higher dimensional generalizations.

## 2 The Extremizing Equations as Polynomial Equations

Here, we show that the problem of solving the extremizing equations in terms of the -variables can be transformed into that of solving a system of multivariate polynomial equations. We first note that, for a one-dimensional lattice having lattice sites, the directional index is irrelevant and is just . Thus, the corresponding extremizing equations are

 fi(ϕθ)=sin(ϕθi)−sin(ϕθi−1)=0, (3)

where , for all . These are the gauge-fixing equations (2) on the lattice and reproduce the continuum Landau gauge () in the naive continuum limit. These equations are also the steady state equations of a variant of the famous Kuramoto model to study synchronization in mathematical biology [24]. It can be shown that by using anti-periodic boundary conditions on both - and -variables the global gauge freedom can be completely fixed [3, 4, 2]. While dealing with periodic boundary conditions one can get rid of the residual gauge freedom by taking, say, to be zero and removing the equation from the system [2]. For simplicity, we take the trivial orbit case (i.e., all ). Using the trigonometric identity, and writing and , we get

 fi(c,s)=ci(si+1+si−1)−si(ci+1+ci−1)=0, (4)

for all . This is merely a change of notation. However, we add additional equations in the system for each site , namely,

 gi(c,s)=s2i+c2i−1=0, (5)

for all . Now, the combined system of all and is not just a change of notation but all the and are algebraic variables and the equations are multivariate polynomial equations, i.e., the fact that and are originally and is taken care of by the constraint equations (5). In general, for the one-dimensional lattice with lattice sites, we have in total polynomial equations and variables.

## 3 Numerical Polynomial Homotopy Continuation Method

In the Appendix, a method to exactly solve systems of multivariate equations, called the Groebner basis technique, and its current status are briefly explained. Here we discuss a numerical method to solve a system of multivariate equations, called the numerical polynomial homotopy continuation (NPHC) method. To explain the NPHC method let us consider a system of multivariate polynomial equations, say , where and , that is known to have isolated solutions, e.g., the above mentioned LLG equations after eliminating the global gauge freedom. Now, there is a classical result, called the Classical Bezout Theorem, that asserts that for a system of polynomial equations in variables, for generic values of coefficients, the maximum number of solutions in is , where is the degree of the th polynomial. This bound, called the classical Bezout bound (CBB), is exact for generic values (i.e., roughly speaking, non-zero random values) of coefficients, e.g., for the one-dimensional LLG equations with number of lattice sites and with periodic boundary conditions, this number is (because there are polynomials each of which is a degree polynomial). The genericity is well-defined and the interested reader is referred to Ref. [28, 29] for details.

Using the so-called Bernstein-Khovanskii-Kushnirenko theorem, a tighter bound, which takes the sparsity of the system into account, can be computed. This bound is called the BKK root count or the mixed volume (or stable mixed volume (SMV) [29, 33, 34] in general). The reader is referred to the above references to get a precise definition of the mixed volume and its computation.

Based on either of these bounds on the number of complex solutions, a homotopy can be constructed as

 H(x,t)=γ(1−t)Q(x)+tP(x), (6)

where is a random complex number. is a system of polynomial equations with the following properties: (1) the solutions of are known or can be easily obtained. is called the start system and the solutions are called the start solutions, (2) the number of solutions of is equal to the CBB or SMV for , (3) the solution set of for consists of a finite number of smooth paths, called homotopy paths, each parameterized by , and (4) every isolated solution of can be reached by some path originating at a solution of . One can then track all the paths corresponding to each solution of from to and reach . Here, a randomly chosen complex ensures the paths are well-behaved. By implementing an efficient path tracker algorithm, all isolated solutions of a system of multivariate polynomials system can be obtained.

To illustrate how the method works, we take a univariate polynomial from Ref. [28] to make this discussion self-consistent, say , pretending that we do not know its solutions, i.e., . We first define a family of problems as

 H(z,t)=(1−t)(z2−1)+t(z2−5)=z2−(1+4t)=0, (7)

where is a parameter. For , we have and at we recover our original problem. The problem of getting all solutions of the original problem now reduces to tracking solutions of from where we know the solutions, i.e., , to . The choice of the piece in Eq. (7), called the start system, should be clear now: this system has the same number of solutions as the CBB of the original problem and is easy to solve. Now, one of the ways to track the paths is to solve the differential equation that is satisfied along all solution paths, say for the th solution path:

 dH(z∗i(t),t)dt=∂H(z∗i(t),t)∂zdz∗i(t)dt+∂H(z∗i(t),t)∂t=0. (8)

This equation is called the Davidenko differential equation. Inserting (7) into this equation, we have

 dz∗i(t)dt=−2z∗i(t). (9)

This initial value problem can be solved numerically (again, pretending that an exact solution is not known) with the initial conditions as and . The other approach is to use Euler’s predictor and Newton’s corrector methods. We do not intend to discuss the actual path tracker algorithm used in practice, but it is important to mention that in the path tracker algorithms used in practice, almost all apparent difficulties have been resolved, such as tracking singular solutions, multiple roots, solutions at infinity, etc. For the sake of completeness, we should also mention here that in the actual path tracker algorithms the homotopy is randomly complexified to avoid singularities by , called the gamma trick, i.e., taking

 H(z,t)=γ(1−t)(z2−1)+t(z2−5)=0. (10)

There are several sophisticated computational packages such as PHCpack [30], PHoM [32], Bertini [28] and HOM4PS2 [31] which can be used to solve systems of multivariate polynomial equations. They are all available as freewares from the respective research groups. In the Appendix, we compare the Groebner basis technique and the NPHC method with technical remarks.

### 3.1 Results

We present the results by classifying the obtained solutions in terms of the number of positive and negative eigenvalues of the corresponding Hessian matrix, or the Faddeev-Popov operator, because then we can use the Neuberger zero as a necessary condition for having all the solutions.

#### Anti-periodic Boundary Conditions

We now explore the simplest non-trivial case in higher dimensional lattices which is the standard LLG functional on a lattice with the trivial orbit and anti-periodic boundary conditions (i.e., the classical XY model) and also a random orbit case (i.e., randomly chosen ). For both cases, the CBB was and the SMV was . For the trivial orbit case, the total number of real and complex solutions was out of which there were real solutions, i.e., Gribov copies. Similarly, for the random orbit case, the total number of real and complex solutions was out of which there were real solutions2. It is important at this stage to mention a few specifics about these solutions. We give details for the trivial orbit case, which are similar to the other cases, below.

1. A solution means that a set of values of ’s and ’s satisfies each of the equations with tolerances . All the solutions come with real and imaginary parts. A solution is a real solution if the imaginary part of each of the variables (i.e., all and ) is less than or equal to the tolerance (below which the number of real solutions does not change, i.e., it is robust for all the cases we consider in this discussion). The original trigonometric equations are satisfied with tolerance after and are transformed back to . All these solutions can be further refined with an arbitrary precision. This is a remarkable success of the method because then these solutions are close to the exact solutions.

2. The sum of the signs of the Faddeev-Popov determinants for all real solutions for the trivial and random orbit cases is , which is numerically zero, yielding the expected Neuberger zero as mentioned in the Introduction.

3. For the trivial orbit case, there are exactly real solutions which have zero Faddeev-Popov determinant with tolerance . These solutions constitute the set of singular loci or the Gribov horizons. They can be further classified in terms of the number of zero eigenvalues of the Faddeev-Popov operator at each of the solutions. This amounts to classifying the singular solutions of the polynomial system in terms of their multiplicities using the so-called deflation singularities technique[37]. For both the random orbit cases, there is no Gribov horizon, i.e., all the solutions are non-singular.

4. For the trivial orbit case, the remaining real solutions have nonsingular Faddeev-Popov determinant, i.e., nonsingular solutions. All the nonsingular solutions of both the trivial and the random orbit cases can be classified by the number of negative eigenvalues (Table 2), which shows the expected two-fold symmetry giving rise to the Neuberger zero.

See Table 1 for the summary.

#### Periodic Boundary Conditions

After eliminating the constant zero mode for the lattice case with periodic boundary conditions (by taking, say, two of the ’s at the corners of the lattice to be , without losing generality. See Ref. [2] for more details), we have equations and variables. The number of solutions is out of which are non-singular and are singular. The Faddeev-Popov operator is now a dimensional matrix. The sum of signs of the Faddeev-Popov determinants is zero. Similarly, for a random orbit the number of real solutions is and again the sum of signs of the Faddeev-Popov determinants is . The results are summarized in Table 2.

## 4 Summary

By taking the standard lattice Landau gauge-fixing (LLG) functional (the classical Hamiltonian of the random phase XY model (RPXYM)) as an example, we showed that the non-linearity of the corresponding gauge-fixing equations (extremizing equations) is polynomial-like. With additional constraint equations, the combined system of equations can be treated as a system of polynomial equations with all variables defined over . Though we are interested in only real solutions of these polynomial systems, we used Complex Algebraic Geometry concepts over Real Algebraic Geometry due to the stronger results available in the former. To solve the corresponding polynomial systems exactly, one can use an elegant algorithm, called the Buchberger algorithm (see Appendix A). Though the algorithm works well for the one-dimensional lattice, for the lattice, it is beyond the capabilities of a standard desktop machine. However, the method can be useful in finding a parametrization of the SU() matrices [39] and hence can then be used, for example, to extend the method of restricting a path integral of the SU() Georgi-Glashow model to the ’t Hooft-Polyakov sectors by using the twisted C-periodic boundary conditions [5] for odd and to precisely define the modified lattice Landau gauge for the SU() case.

For the two-dimensional case, we used the Numerical Polynomial Homotopy Continuation (NPHC) method, which gives all solutions of a polynomial system numerically. This method does not suffer from the technical difficulties of the Buchberger algorithm or its variants, and in principle this method can find the solutions for those systems which may be intractable for the former method. By computing the eigenvalues of the Hessian matrix at these solutions, one can obtain the global minimum, up to machine precision, of a multivariate function whose extremizing equations can be translated to polynomial equations.

The anti-periodic boundary conditions system for the trivial and a random orbit can be solved using the NPHC method and that there is an exact cancelation of the signs of the Faddeev-Popov determinants in this case, i.e., the Neuberger zero. We classified all the solutions in terms of the number of negative and positive eigenvalues of the corresponding Faddeev-Popov operator. Apparently, the number of Gribov copies is orbit dependent, though the Neuberger zero is orbit independent. It would be important to compute the number of solutions for different random orbits for the lattice with the modified LLG using the NPHC method. Efforts to solving the corresponding gauge-fixing equations for the modified LLG for compact U() and also the linear covariant gauge-fixing equations for compact U(), in the spirit of [41, 42], are in progress. It would also be very interesting to solve the Langevin dynamics equations for the XY model and other models  [40] using algebraic geometry methods.

## 5 Acknowledgement

DM would like to thank the developers of the PHCpack, HOM4PS2, Bertini, PHoM and Singular, and to Hirokazu Anai, Guillaume Moroz, Antonio Montes, Akira Suzuki, Vladimir Gerdt, Daniel Robertz, Anton Ilderton, Yang-Hui He, JM Kosterlitz, Jon-Ivar Skullerud, Teresa Mendes, Attilio Cucchieri and Paul Watts for their help and critical remarks on this work. DM was supported by the Science Foundation of Ireland and by the British Council Researcher Exchange Programme to visit the Imperial College London. Support by the Australian Research Council is also acknowledged.

## Appendix A Computational Algebraic Geometry

Since many important and useful results in algebraic geometry are mainly developed for the complex field , we treat our polynomial equations as having complex variables, and in the end we use only real solutions and throw the remaining solutions away (see, e.g.,  [19]). One can then use an important result from computational algebraic geometry: after specifying an ordering of the monomials of the system (denoted as for the ordering in which is placed prior to and is placed prior to ), one can transform a given system of multivariate polynomial equations, referred to as an ideal, to another one which has the same solutions but is easier to solve. This new system is called a Groebner basis for the given monomial ordering.

There is a well defined procedure to find a Groebner Basis for any given ideal and monomial order, called the Buchberger algorithm. It should be noted that the Buchberger algorithm reduces to Gaussian elimination in the case of linear equations, i.e., it is a generalization of the latter. Similarly it is a generalization of the Euclidean algorithm for the computation of the Greatest Common Divisors of a univariate polynomial. Recently, more efficient variants of the Buchberger algorithm have been developed to obtain a Groebner basis, e.g., F4, F5 and Involution Algorithms [35]. Symbolic computation packages such as Mathematica, Maple, Reduce, etc., have built-in commands to calculate a Groebner basis for a given monomial. Singular, COCOA and McCauley2 are specialized packages for Groebner basis and Computational Algebraic Geometry, available as freeware. MAGMA is also such a specialized package available as a non-free package. Rather than going into the details of the specifics and technicalities of this algorithm, we dive into the practical applications of the Groebner basis technique relevant to our problem and refer the reader to the above mentioned references for further details.

### a.1 The Groebner Basis Technique At Work

Here, we provide a practical example of how the Groebner basis technique can be used for our systems. Firstly, for the polynomials for the one-dimensional lattice with three lattice sites, anti-periodic boundary conditions and the trivial orbit case, the corresponding ideal (denoted as the polynomials put between ’’ and ’’) is

 I3 = <−c2s1−c3s1+c1s2−c1s3,c2s1−c1s2−c3s2+c2s3,−c3s1+c3s2−c1s3−c2s3, (11) c21+s21−1,c22+s22−1,c23+s23−1>.

For the ideal in Eq. (11), a Groebner basis for the so-called lexicographic ordering is

 G3 = <−s3+s33,c3s3,−1+c23+s23,−s2+s2s23,c3s2,s22−s23,c2s3,c2s2, (12) −1+c22+s23,−s1+s1s23,c3s1,c2s1,s21−s23,c1s3,c1s2,c1s1,−1+c21+s23>.

As noted earlier, the solutions of this system are the same as the original system. Here, the first equation in is a univariate polynomial in variable and solving it is simple because it can be factorized as giving and . Using this and back-substitution, all the solutions can be obtained and are as follows:

 (c1,c2,c3,s1,s2,s3)={(0,0,0,±1,±1,±1),(±1,±1,±1,0,0,0)}. (13)

Thus, the solution space of the system , called the variety of and denoted as , is the above mentioned set of isolated points in a -dimensional affine space. This is the known result from Refs. [3, 4, 2] in terms of ’s, i.e.,

 (θ1,θ2,θ3)={all 23 permutations of 0/π% }∪ {all 23 permutations of −π2/π2}. (14)

All of these solutions are correctly reproduced in Eq. (13). Similar computation can be performed for the periodic boundary conditions case [2].

For an ideal which is known to have only isolated solutions (called a zero-dimensional ideal) with a lexicographic ordering one can always find a Groebner basis in an upper diagonal form, analogous to the Gaussian Elimination method, such that at least one polynomial is univariate with the others having an increasing number of variables.

Interpreting the gauge fixing equations in terms of Algebraic Geometry allowed us to deal with the actual non-linearity of the equations, rather than treating them as linear equations as done in Refs. [2, 3, 4]. Specifically, in this interpretation the corresponding method does not make any distinction between the equations arising from a one-dimensional lattice or those arising from a higher dimensional lattice. In theory, as long as one can obtain a Groebner basis, the equations can be exactly solved. For the one-dimensional case, it was easy to go up to a fairly big number of lattice sites on a single machine. However, in general, obtaining a Groebner basis is very difficult due to an algorithmic complexity, known as Exponential Space complexity, which roughly means that the RAM required by the computation blows up exponentially. In particular, on a regular single desktop machine with GB RAM, we could not obtain a Groebner basis for the trivial orbit case (i.e., the classical XY model) and with anti-periodic boundary conditions for a lattice, using Singular3.2. The corresponding system is made of equations, each of degree , in variables. However, a more powerful machine should be able to obtain a Groebner basis for these systems.

Recently, V. Gerdt and Daniel Robertz were able to compute a Groebner Basis for this system over using a Linux machine with AMD Opteron (TM) processor MHz, cores, GB memory and with Magma (V2.14.14) in less than hours with degree reverse lexicographic ordering [36]. This is a very important step towards solving the system. However, since the computation is over not all the solutions over can be obtained.

### a.2 Comparison between the Groebner basis technique and the NPHC Method

The NPHC method is strikingly different from the Groebner basis technique in that the algorithm for the former suffers from no known major complexities. Moreover, the path tracking is embarrassingly parallelizable, because all the start solutions can be tracked completely independently of each other. This feature along with the rapid progress towards the improvements of the algorithms makes the NPHC well suited for many physical problems arising in condensed matter theory, lattice QCD, etc.

For most systems of polynomial equations in practice, we do not know the actual number of solutions from the beginning. So even after obtaining all solutions through the homotopy continuation method, we would still prefer to have some kind of verification of the solutions. In the standard LLG fixing case, we used the Neuberger zero as a necessary condition. However, this is certainly not a sufficient condition and further checks may be required for bigger systems. This is called certifying the solutions, which has been recently developed using the so-called Tropical Algebraic Geometry [38]. We anticipate that our results mentioned above will serve as ideal test systems for all such new ideas.

To solve the corresponding equations for the periodic boundary conditions case, another method is to leave the constant zero modes in the system and use the Numerical Algebraic Geometry (NAG) [43], which is the generalization of the NPHC method to positive-dimensional ideals.

Recently, a computational Algebraic Geometry approach to find vacuum configurations in string phenomenology (where mathematically the problem reduces to minimization of the so-called superpotentials which exhibit polynomial-like non-linearity) is proposed in Refs. [19, 20]. A very efficient Mathematica package, called STRINGVACUA, has been developed with an interface with the computer algebra system Singular. This package relies on the Groebner basis technique in solving the extremizing polynomial equations. It is anticipated that incorporating the NPHC method can extend the applicability of the package and the approach beyond its present status.

It should be noted that there are several promising methods such as the Discriminant Variety [44] and finding real solutions out of complex curves [45] from computational real algebraic geometry which are emerging as alternatives to the above mentioned methods in some of the important problems mentioned in the Introduction.

### Footnotes

2. Note that we do not intend to compare the efficiency of the available packages. However, to give an idea how computationally intensive such a calculation is, we note that the HOM4PS2 package took around and minutes, respectively, to run these systems on a Linux single-processor desktop machine.

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## Thinking Mathematically (6th Edition)

Given $x^{2}$+ 4; x = 5 Now put the value of x = 5 in the given expression. = $5^{2}$ + 4 = 25 + 4 = 29

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# Minimum of Poissons

Let $X_i\sim\text{Pois}(\lambda_i)$ for $i=1,2,\ldots,n$ and $Y = \min X_i$. Can we show that, for example $\mathbb{E}[Y] \leq f(\lambda,n)\min\lambda_i$ for some $f : (\mathbb{R}^n,\mathbb{N}) \to [0,1]$ and lower bound the variance $\mathbb{V}[Y]$ by anything meaningful?

Jensen's inequality tells us $$\mathbb{E}[Y] = \mathbb{E}[\min_i X_i] \leq \min_i\mathbb{E}[X_i] = \min_i\lambda_i$$ already but I'd like something more concrete.

Note that this question asks a similar question, but is concerned with the entire distribution over $Y$ while I only need bounds on 2 moments. Accordingly, I'd expect a more closed-form answer available.

If it helps, in my case $X_i\sim\text{Pois}(\lambda)$ for $i=1,\ldots,n-1$ and $X_n \sim \text{Pois}(\lambda + \gamma)$ for $\lambda,\gamma>0$ so for $\gamma$ and $n$ large enough we may effectively assume $X_i\sim\text{Pois}(\lambda)$ are i.i.d (with high probability) for the purposes of estimating $Y$.

• What do you mean by "assume $X_i$" in the last sentence? Are some words missing?
– whuber
Jan 20, 2018 at 17:41
• @whuber ah thanks. I meant that $X_i$ are effectively i.i.d because for large enough $\gamma$, the probability that $Y = X_n$ is low. Jan 20, 2018 at 18:02
• are you sure you can use Jensen's inequality here? Jan 21, 2018 at 5:55
• @Taylor I’m quite sure: the minimum of a collection of linear functionals over $\mathbb{R}^n$ is concave (draw a picture to check). Denote $X=(X_1,\ldots,X_n)\in\mathbb{R}^n$. Then $\min X_i = \min \langle X,e_i\rangle$ is a minimum of linear functionals and hence concave, so Jensen applies. Jan 21, 2018 at 6:41
• You do not even need Jensen: $\min X_i\le X_i$ for all $i$'s, hence $\mathbb{E}[\min_i X_i] \leq\mathbb{E}[X_i]$ for all $i$'s. Jan 21, 2018 at 10:01

This only answers half of my question. Lower bounding the variance of $Y$ is still open.
Without loss of generality assume $\lambda_{\min} = \lambda_1\leq \lambda_2\leq \dots \leq \lambda_n = \lambda_{\max}$. Note that $$\mathbb{P}(Y > 0) = \prod_{i=1}^n (1 - e^{-\lambda_i}).$$ Moreover, $\mathbb{E}[Y | Y>0] \leq \mathbb{E}[X_1 | X_1>0]$. Bringing these together,
\begin{align} \mathbb{E}[Y] &= \mathbb{E}[Y | Y>0]\mathbb{P}(Y > 0)\\ &\leq \mathbb{E}[X_1 | X_1>0]\mathbb{P}(Y>0)\\ &= \lambda_{\min} \frac{\prod_{i=1}^n(1 - e^{-\lambda_{i}})}{1 - e^{-\lambda_{\min}}}\\ &= \lambda_{\min} \prod_{i=2}^n (1 - e^{-\lambda_i})\tag{$*$}\\ &\leq \lambda_{\min} (1 - e^{-\lambda_{\max}})^{n-1} \end{align} as desired. Note that this is sharp for $n=1$, and depending on the situation $(*)$ might be more helpful than the final bound.


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Open access peer-reviewed chapter

# Design of Urban Air Quality Monitoring Network: Fuzzy Based Multi-Criteria Decision Making Approach

Written By

Abdullah Mofarrah, Tahir Husain and Badr H. Alharbi

Submitted: October 14th, 2010 Reviewed: April 28th, 2011 Published: July 8th, 2011

DOI: 10.5772/16716

From the Edited Volume

## Air Quality Monitoring, Assessment and Management

Edited by Nicolás A. Mazzeo

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## 1. Introduction

The air quality monitoring network (AQMN) is the essential part for air quality management, strategies planning, and performance assessment (Mofarrah and Husain, 2010). Existing methods of establishing ambient air quality monitoring networks typically evaluate the parameters related to air pollutant concentrations, emission source characteristics, atmospheric transport and dispersion, secondary reactions, deposition characteristics, and local topography (Harrison and Deacon, 1998; Bladauf et al., 2002). In most of the cases, AQMN is designed to measure the pollutants of concern such as particulate matter (PM10), carbon monoxide (CO), sulfur dioxide (SO2), ozone (O3), nitrogen oxides (NOx), and total hydrocarbons (Chang and Tseng, 1999). Most of the reported AQMN design methods applied to a specific situation wherein one or two specific objectives are considered (Harrison and Deacon, 1998; Mofarrah and Husain, 2010). However, design of AQMN considering the multiple-criteria including multiple pollutants is complicated because air pollution phenomena are complex and dynamic in nature, depends on the meteorological and topographical conditions and involves not only irregularity of atmospheric movement but also uncertainty of human activities. The objective of this study is to develop a systematic approach for designing urban AQMN considering multi-criteria including multiple air pollutants in the system. The optimization is approached based on the utility scores gained from the fuzzy analytical hierarchy process associated with a candidate station, which is estimated over the representative zone (RZ) of the potential station.

## 2. Fuzzy Analytical Hierarchy Process

Fuzzy Analytical Hierarchy Process (AHP) is the extension of analytical hierarchy process (Saaty, 1980) is used for structuring the problem. AHP is an efficient method in which hierarchical structure is developed by a pair-wise comparison between any two criteria. The levels of the pair-wise comparisons range from 1 to 9, where ‘1’ represents that two criteria are equally important, while the other extreme ‘9’ represents that one criterion is absolutely more important than the other (Saaty, 1980). The AHP uses objective mathematics to process the subjective and personal preferences of an individual or a group of decision maker (Saaty, 1980). Generally, decision-making processes are subject to insufficiency of data and lack of knowledge (Tesfamariam and Sadiq, 2006). In fact, even if the data are available, criteria often contain linguistic definitions involving human judgment and subjectivity, which introduce uncertainties in the decision making process. In application to actual system traditional AHP is not so effective in capturing uncertainty and subjective judgments of different experts. The fuzzy AHP developed Zadeh (1965) is modified version of AHP, and can be used to handle the fuzziness of the data. It is easier to understand and can effectively handle both qualitative and quantitative data in the multiple-criteria problems. In this paper triangular fuzzy numbers (TFNs) are used to judge the qualitative information related to AQMN design. The TFN is defined by three real numbers, expressed as (l, m, n) with a membership function between 0 and 1 (Fig. 1). The parameters l, m, and n, respectively, indicate the smallest possible value, the most promising value, and the largest possible value that describe a fuzzy event (Zadeh, 1965). The mathematical definition of a TFN can be described as (Kaufmann and Gupta, 1988):

μ(x/M˜)={0,(xl)/(ml),(nx)/(nm),0,   x<l,lxm,mxn,x>nE1

The TFNs for this study are developed in such a way that the most likely value has a membership grade of unity, considering the fact that the lower and upper bonds have a membership value of zero in that fuzzy set. The arithmetic of fuzzy set is little different than regular arithmetic. For example the fuzzy algebraic operations of two TFNs, namely A(l1,m1,n1)andB(l2,m2,n2)are as follows (Kaufmann and Gupta, 1988):

A+B=(l1+l2,m1+m2,n1+n2)

AB=(l1n2,m1m2,n1l2)

A.B

0(l2,n2); and if A/B=A.B1=min(l1/l2,l1/n2,n1/l2,n1/n2),mostlikely(m1/m2),andmax(l1/l2,l1/n2,n1/l2,n1/n2)

AQIj=j=1mCij*pijCsj

## 3. Methodology used in this work

A systematic methodology for designing urban AQMN is developed by using multiple criteria, which covered environmental (e.g., air quality), social (i.e., location sensitivity (LS), population density (PD), population sensitivity (PS), and cost parameter (CP). The aim of this AQMN design is to study air pollutants’ characteristics, human health, social sensitivity, and cost objective. Thus, the proposed design will protect public health, sensitive locations/receptors from exposures to ambient air pollutants, and it will measure the maximum pollutants’ concentration for the study area. The framework of this methodology is shown in Fig. 2. This technique will help the decision maker to optimize the AQMN with limited financial and human resources.

At the beginning, the study area is divided into a continuous grid system in which each grid represents a potential candidate location for monitoring station. Fuzzy synthetic optimization technique is used to identify the potential monitoring sites. The key functions of the methodology are described in the following sections.

### 3.1. Air quality exceedance index

An air quality index (AQI) function is generated comparing the local air quality with the national standards. AQIprovides overall information about the local air quality. It hints how clean or polluted the local air comparing with the national standards. If the ratio of measured data (Cij) and standard (Csj) grater then one, it means the local air quality is violating the national air quality standards. For this study, concentrations of the major air pollutants such as CO, NO2, SO2, and PM10 are monitored and subsequently converted into an AQIby assigning a probability of occurrence factor to each air pollutants based on their occurrence of exceedance during study periods as eq. 2.

CIi=Minimize[CiCi]E2

Cij = jthpollutant concentration (i.e., CO, NO2, SO2, and PM10) in ithgrid; Csj= national air quality standard of jthpollutant, pijis the probability of occurrence of jthpollutant in the ithlocation over the measurement periods.

 Criteria Assigned scores Location sensitivity (LS)/ available amenities /grid LSi No-basic facility (1,2,4) Facilities of low value (e.g., storage facilities) (1,3,5) Factories and industry (2,4,6) Residential, parks (3,5,7) Schools, churches, heritage places (4,6,8) Hospitals, sensitive locations (5,7,9) Population density (PD)/ number of people /grid PDi <50 (1,2,4) 51- 250 (1,3,5) 251-450 (2,4,6) 451-650 (3,5,7) 650-850 (4,6,8) "/>850 (5,7,9) Population sensitivity (PS)/ sensitive population /grid PSi <10 (1,2,4) 11-20 (1,3,5) 21-30 (2,4,6) 31-40 (3,5,7) 41-50 (4,6,8) "/>50 (5,7,9) Cost criteria Ci Installation cost "/>$7000 (1,2,4)$7000-$6000 (1,3,5)$5900-$5000 (2,4,6)$4900-$4000 (3,5,7)$3900-$3000 (4,6,8) Installation cost <$3000 (5,7,9)

### Table 1.

Definitions for criteria scores

### 3.2. Cost objective

An important objective of any AQMN is to minimization of its cost. This objective can also be interpreted as a budgetary constraint. Cost criteria consideration in this evaluation is installation cost. Generally, the installation cost is varied depending on the site location, local labour force and communication facilities. To compare the installation cost within the area of interest a cost index (CI) is introduced as:

SCIi=Maximized[LSiLSi+PDiPDi+PSiPSi]E3

where, CIi= cost index for the grid i, Ci= cost score at the ithlocation over the study area. Considering the local market and communication facility the installation cost score (Ci) of ithgrid was assigned as shown in Table 1.

### 3.3. Definitions of social criteria

In each grid, the social criteria such as LS, PS, and PS are consequently converted to a score with the help of pre assigned scale shown in Table 1. Finally, the scores are normalized to get social criteria index (SCI) at ithlocation as shown in eq. 4.

[AQIiJCIiLSiPDiPSi]SCi=[A1A2A3.An][AQIA1CIA1LSA1PDA1PSA1AQIA2CIA2LSA2PDA2PSA2AQIA3CIA3LSA3PDA3PSA3.....AQInCIAnLSAnPDAnPSAn]×[wjwcwLSwPDwPS]E4

### 3.4. Determining of screening scores

The screening score is the composition components of an environmental parameters (eq. 2), cost objective (eq. 3), and social criteria (eq. 4). It is defined by a dimension less function, called screening score (SC). The SCcombines, under a mathematical approach, and is defined by eq. 5.

SCAiE5

where, A1,A2,……,Anare the possible monitoring station location, and wj, wc, wLS, wPd,and wPSare weighting factors for AQI, CI, LS, PD,and PSrespectively.

The SCfrom eq. 5 is also fuzzy value. For decision purpose the comparisons of fuzzy data is not straightforward. To obtained crisp value of SC, centroidal method (Yager, 1980) is used. The centroid index of the fuzzy number represents the crisp score of an alternative Ai. If the fuzzy SCfor a grid Aiis SCx(Ai)=(β1α1)(α1+2/3(β1α1))+(λ1β1)(β1+1/3(λ1β1))(β1α1)+(λ1β1)1, β1, γ1), then the crisp score of that location can be computed as follows:

SCAiE6

where, SCx(Ai) is the crisp score of grid Aiand α1, β1, and γ1 are the lowest, most likely and maximum values ofDr={j=14R2,if(RRc)00,Otherwise. Depending on the SCx(Ai) values the grids location are screened, and potential locations were ranked for second step analysis.

### 3.5. Determination of representative zone (RZ)

After the identification and quantification of the objectives of the monitoring network, the second step is to determine the degree of representativeness (Dr) and the representative zone (RZ) associated with each candidate monitoring location. The RZfor a monitoring station is established on the basis of the concept of a sphere of influence area surrounding the potential station, for which the pollutants measurements can either be regarded as representative or can be extrapolated with known confidence. Sphere of influence (SOI) is defined as the zone over which the metrological (MET) data for a given monitoring location can be considered representative (Mofarrah and Husain, 2010). A grid cell (i) will belong to the RZof a monitoring station at grid cell (k), if the Drof cell (i) is greater than zero (eq. 7).

The Dris dictated by a predetermine cutoff value (Rc) in the spatial correlation coefficient (R) between the pollutant’s concentration at the monitoring locations identified and the neighboring locations surrounding it. The spatial correlation coefficient (R) gives an indication of the relationship among locations to be selected in the monitoring network (Elkamel et al., 2008). The Rlies between -1 and +1 (Liu et al., 1986). The computation of Ris carried out in all radial directions surrounding each potential location until the Rfalls below the predetermined cut-off value (Rc). In this study RZis considered the area surrounding it in which the Rof this location with the nearby locations is higher than the cut-off value (Rc). This means the pollutants concentration measured at this location are representatively correlated with a certain degree of confidence to any location in the network within the area. Fig. 3 illustrates the general concept of RZ. By this concept it is assumed that, when a station is installed in a grid square (i.e A5), the nearby grid squares, as are marked on Fig. 3, are not allowed to be installed with the same class of station if their Dris higher than zero. When searching for the next station location, the marked grid squares will be skipped for enhancing the solution efficiency.

R=i=1p(x1ix¯1)(x2ix¯2)i=1p(x1ix¯1)2i=1p(x2ix¯2)2E7

where, R isspatial correlation coefficient of the concentrations between two adjacent monitoring locations x1 = (x11, x12,….x1p) and x2 = (x21, x22,….x2p) with a sample size p can be expressed (Elkamel et al., 2008) as:

x¯1=1pi=1px1iE8

where, x¯2=1pi=1px2iand U(g)are the average concentrations at location 1 and 2, respectively. Once Drof the each pathetical location is calculated, the ranking of the potential location can be quantified in terms of grid utility scores. A grid score for gth candidate location is defined as the sum of SCxof all grids correlated with gth location as:

U(g)=SC(xg)+i=1nDr×SCxiE9

where, nis the number of grids correlated with gth location. Highest U(g)means better location for air quality monitoring station.The RZof the sphere can be defined as the number of square grids place inside it.

## 4. Application of the methodology

Riyadh, the capital city of Saudi Arabia was considered to demonstrate the proposed methodology. Riyadh is one of the major industrial cities in the Kingdom of Saudi Arabia; it has multiple types of heavy and light industries such as oil refinery, power plant, cement industry etc. The population of the city is above four million with very high growth rates. Therefore, to maintain the air quality standard the city authorities have planned to re-assess the current air quality monitoring network. There are six existing air quality monitoring stations in Riyadh city owned and operated by different organizations. Most of these existing air AQMNs are not working properly or not serving at satisfactory levels. The main objective of this study is to design the air quality monitoring network for Riyadh city and to identify the optimal station locations to satisfy the future air quality monitoring demands. To design the AQMN three major emission sources such as point sources, area sources and line sources were considered. The detailed emission inventory can be found elsewhere (Mofarrah and Husain, 2010). The major point sources in Riyadh city are power plants, refinery and cement industries. The old and new industrial cities under development are considered as the area source. The automobile sources for the selected major roads based on traffic counts, composition of traffic, and model years were considered as the line sources in this study. The database for emission inventory was developed based on production rate, fuel consumption and the emission factors as suggested by USEPA.

At the beginning, the study area (40km x 60km) was conceptually divided into 441 square grids as subsystems. Each grid component includes environmental parameters (i.e., sulfur dioxide (SO2), nitrogen dioxides (NO2), carbon monoxide (CO) and fine particulate matters (PM10)), social objectives (i.e., LS, PD, PS) and cost criteria (CI). The concentration level of each air pollutant was simulated on; hourly, 8-hourly, and 24-hourly basis using Industrial Source Complex (ISC3) air quality software programs. The concentration distribution of selected air pollutants over the study area is shown in Fig. 4. If we compare the pollutants distribution (Fig. 4) with the Saudi Arabian national air quality standard (Table 2), it is clear that some regions within the study are experiencing high level of air pollution threats. For this study, the social and cost objective data of each grid were also modeled as fuzzy variables according to the fuzzy scale mentioned in Table 1.

 Pollutant Measurement period Limit SO2 30 day period, one hour average 730 µg/m3 12 month period, 24 hour average 365 µg/m3 12month period, annual average 80(µg/m3) Inhalable Particulates (fpm) 12-month period, the 24-hour maximum 340(µg/m3) 12-month period, the annual average 80(µg/m3) Nitrogen Oxides Defined as Nitrogen Dioxide (NO2) 30 day period, the one-hour average 660(µg/m3) 12-month period, the annual average 100(µg/m3) Carbon Monoxide (CO) 30-day period, the one-hour average 40 (mg/m3) 30-day period, the 8-hour average 10(mg/m3)

### Table 2.

Ambient air quality standards for Saudi Arabia (Source Presidency of Meteorology and Environment, Saudi Arabia)

### 4.1. Criteria weight computation

Based on the importance of each criterion on AQMN design, a fuzzy pair-wise comparisons matrix (PCM) A˜Fis formed considering C1, C2, C3,C4and C5are respectively, location sensitive (LS), population sensitive (PS), cost, air quality (AQ)and population density (PD). The preference scale as shown in Table 3 was used in this case.

 How important is A relative to B? Preference index(Saaty 1988) Fuzzy value( l, m, u; Jie et al. 2006) Equally important 1 (1, 1, 1) Moderately more important 3 (1,3,5) Strongly more important 5 (3,5,7) Very strongly more important 7 (5,7,9) Overwhelmingly more important 9 (7,9,11) Intermediate values (Need to judge two) 2 (1,2,4) 4 (2,4,6) 6 (4,6,8) 8 (6,8,10)

### Table 3.

Criteria preference scale

 ÃF= C1 C2 C3 C4 C5 C1 1, 1, 1 0.33,0.5,1.0 0.25,0.33,1 0.33,0.5,1 2,3,4 C2 1, 2,3 1, 1, 1 0.33,0.5,1 0.17,0.2,0.25 0.5,1,1 C3 1,3,4 1,2,3 1, 1, 1 0.2,0.25,0.33 0.33,0.5,1 C4 1,2,3 4,5,6 3,4,5 1, 1, 1 0.17,0.2,0.25 C5 0.25,0.33,0.50 1,1,2 1,2,3 4,5,6 1, 1, 1

After constructing, relative weights of each criterion is calculated by using fuzzy extent analysis (Lee et al., 2006) as follows:

 Row Left Middle right The first row sum 3.92 5.34 8.00 The 2nd row sum 3.00 4.70 6.25 The 3rd row sum 3.53 6.75 9.33 The 4th row sum 9.17 12.20 15.25 The 5th row sum 7.25 9.33 12.50 Total 26.87 38.32 51.34

 Criteria Left Middle right LS (C1) 3.92/51.34= 0.0763 5.34/38.32 = 0.1393 8.0/26.87 = 0.2978 PS (C2) 3.00/51.34= 0.0584 7.40/38.32 = 0.1227 6.25/26.87 = 0.2326 Cost (C3) 3.53/51.34= 0.0688 6.75/38.32 = 0.1762 9.33/26.87 = 0.3474 AQ (C4) 9.17/51.34= 0.1786 12.20/38.32 = 0.3184 15.25/26.87 = 0.5676 PD (C5) 7.25/51.34= 0.1412 9.33/38.32 = 0.2436 12.5/26.87 = 0.4653

The weights of AQwere re-distributed to the CO, SO2,PM10and NO2on the basis of their impotence (i.e., considering the local environment and human health point of view). The complete set of criteria weights are shown in Table 4.

 Criteria Weights Sub-criteria CO SO2 PM10 NO2 LS (C1) w1=(0.0763,0.1393,0.2978) - - - - PS (C2) w2=(0.0584,0.1227,0.2326) - - - - Cost (C3) w3=(0.0688,0.1762,0.3474) - - - - AQ (C4) w4=(0.1786,0.3184, .5676) C41 = 20% of C4 C42 = 25% of C4 C43 = 35% of C4 C44 = 20% of C4 PD (C5) w5=(0.1412,0.2436,0.4653) - - -

### Table 4.

Weights of each criteria and sub-criteria

## 5. Results and discussions

The concentration level of each pollutant was compared with the Saudi National Air quality standards (Table 2) to calculate the air quality index (AQI). Social and cost objectives data of each grid were also predicted and converted into fuzzy scores according to Table 1. The step by step calculations of potential location identification is described in the following section by considering few grids. Table 5 shows the AQIsand assigned scores of different parameters. The criteria scores (Table 5) are multiplied with the weighting factors (Table 4) to form fuzzy screening scores matrix as shown in the Table 6.

 Grid no AQI (SO2) AQI (NO2) AQI (CO) AQI (PM10) Assigned scores(LS) Assigned scores(PS) Assigned scores(CI) Assigned scores(PD) A1 22.7798 38.8779 58.3944 27.4490 (1,2,4) (1,2,4) (3,5,7) (5,7,9) A2 22.4193 38.9320 65.7578 24.7020 (1,2,4) (1,2,4) (3,5,7) (5,7,9) A3 31.5411 51.0081 81.7773 32.3196 (1,3,5) (1,3,5) (3,5,7) (4.6.8) A4 45.5648 69.3000 97.5538 45.8194 (2,4,6) (2,4,6) (3,5,7) (5,7,9) A5 66.8583 96.5431 107.2574 58.2306 (2,4,6) (1,2,4) (3,5,7) (4,6,8) A6 46.5463 78.2129 109.7475 45.0471 (1,2,4) (1,2,4) (3,5,7) (5,7,9) .. .. .. .. .. .. .. .. A436 2.486 14.478 19.775 24.351 (1,2,4) (1,2,4) (3,5,7) (5,7,9) A437 3.204 17.229 21.650 21.579 (1,2,4) (1,2,4) (3,5,7) (5,7,9) A438 5.808 21.943 25.775 17.283 (1,3,5) (1,3,5) (3,5,7) (4.6.8) A439 13.402 69.518 73.369 17.328 (2,4,6) (2,4,6) (3,5,7) (5,7,9) A440 31.328 346.130 343.296 21.558 (1,2,4) (1,2,4) (3,5,7) (5,7,9) A441 5.118 21.599 23.418 16.571 (1,2,4) (1,2,4) (3,5,7) (5,7,9)

### Table 5.

AQIs and assigned scores for different criteria

 Grid no AQI (SO2) AQI (NO2) AQI (CO) AQI (PM10) (LS) (PS) (CI) (PD) A1 1.017,1.813,3.232 1.389,2.476,4.413 2.086,3.719,6.629 1.716,3.059,5.453 0.076,0.279,1.191 0.058,0.245,0.93 0.206,0.881,2.432 0.706,1.705,4.188 A2 1.001,1.785,3.181 1.391,2.479,4.42 2.349,4.187,7.465 1.544,2.753,4.907 0.076,0.279,1.191 0.058,0.245,0.93 0.206,0.881,2.432 0.706,1.705,4.188 A3 1.408,2.511,4.476 1.822,3.248,5.79 2.921,5.208,9.283 2.02,3.602,6.421 0.076,0.418,1.489 0.058,0.368,1.163 0.206,0.881,2.432 0.565,1.462,3.722 A4 2.034,3.627,6.466 2.475,4.413,7.867 3.485,6.212,11.074 2.864,5.106,9.102 0.153,0.557,1.787 0.117,0.491,1.396 0.206,0.881,2.432 0.706,1.705,4.188 A5 2.985,5.322,9.487 3.449,6.148,10.96 3.831,6.83,12.176 3.64,6.489,11.568 0.153,0.557,1.787 0.117,0.491,1.396 0.206,0.881,2.432 0.565,1.462,3.722 A6 2.078,3.705,6.605 2.794,4.981,8.879 3.92,6.989,12.459 2.816,5.02,8.949 0.076,0.279,1.191 0.058,0.245,0.93 0.206,0.881,2.432 0.706,1.705,4.188 .. .. .. .. .. .. .. .. .. A436 0.646,1.152,2.054 0.706,1.259,2.245 0.87,1.551,2.764 1.056,1.883,3.357 0.076,0.279,1.191 0.058,0.245,0.93 0.206,0.881,2.432 0.706,1.705,4.188 A437 0.769,1.371,2.445 0.773,1.379,2.458 0.771,1.374,2.45 1.362,2.427,4.327 0.076,0.418,1.489 0.058,0.368,1.163 0.206,0.881,2.432 0.565,1.462,3.722 A438 0.98,1.747,3.114 0.921,1.641,2.926 0.617,1.101,1.962 2.468,4.4,7.843 0.153,0.557,1.787 0.117,0.491,1.396 0.206,0.881,2.432 0.706,1.705,4.188 A439 3.104,5.534,9.865 2.621,4.672,8.329 0.619,1.103,1.967 5.695,10.153,18.098 0.153,0.557,1.787 0.117,0.491,1.396 0.206,0.881,2.432 0.565,1.462,3.722 A440 15.455,27.552,49.116 12.263,21.861,38.971 0.77,1.373,2.447 13.312,23.732,42.307 0.076,0.279,1.191 0.058,0.245,0.93 0.206,0.881,2.432 0.706,1.705,4.188 A441 0.964,1.719,3.065 0.836,1.491,2.658 0.592,1.055,1.881 2.175,3.877,6.912 0.382,0.975,2.68 0.117,0.491,1.396 0.138,0.705,2.084 0.706,1.705,4.188

### Table 6.

Weighted screening scores matrix

The crisp values of weighted fuzzy screening scores (SC) of each grid were estimated by applying eq. 6 and reported in Table 7. Based on the top SCx scores, 50 potential locations were indentified for second step analysis.

 Grid no Grid screening scores (SC) Crisp values of (SCx) A1 0.065,0.133,0.278 0.1586 A2 0.067,0.137,0.285 0.1628 A3 0.076,0.16,0.326 0.1871 A4 0.099,0.198,0.392 0.2298 A5 0.112,0.223,0.436 0.2569 A6 0.099,0.194,0.388 0.2273 .. .. .. A436 0.045,0.098,0.217 0.1202 A437 0.043,0.103,0.223 0.1231 A438 0.055,0.12,0.253 0.1428 A439 0.083,0.171,0.344 0.1990 A440 0.231,0.429,0.806 0.4887 A441 0.06,0.125,0.261 0.1483

### Table 7.

Grid screening scores

To determine the degree of representativeness (Dr) and the representative zone (RZ) associated with each candidate monitoring location, three different cutoff values (Rc), 0.45, 0.60 and 0.75 were used separately and compared with the coefficient in the spatial correlation (R) between the pollutant concentration of the potential monitoring station and the neighboring locations surrounding it. Ten optimal locations and corresponding number of grids coverage were indentified for different cutoff value as shown in Fig. 5. The results show that cutoff value has significant effect on the representative zone (RZ) assessment. However, considering the geometry of the study area and analysis of the different cutoff values it was found that the 0.6 cutoff value is the best suited for the Riyadh city. Hence, the grid scores U(g) and top ten optimal locations distribution with cutoff value 0.6 is evaluated as shown in Fig. 6. Due to heavy industries and high populations’ exposure, 1-2 optimal locations were found near a major point source such as power plant at grids A397 to A441 (Fig. 6). However, this part is outside of the present study area, but respecting the study results, we suggested putting at least one air monitoring station at grid A419 or A398.

For a specific situation the agency can choose either a high or a low value of Rc. A high Rc based network may not necessarily cover more area, but the covered region is well represented. On the other hand a low Rc based network, would offer more coverage of the region, but the covered region may not be satisfactorily represented (Mofarrah and Husain, 2010; Elkamel et al., 2008). The final decision in such a case is of course dependent on the respective agency. It should be noted that the design of an air quality network with higher cutoff values (Rc) is the addition of some more monitoring stations in the network, as compared to that of a network designed with a lower Rcvalues. The selection of the Rcvaries case by case, based on budget, type of air monitoring station, meteorological condition, and the purpose of the monitoring network.

## 6. Conclusions

The AQMN represents an essential tool to monitor and control atmospheric pollution. The use of some specific criteria in conjunction with the mathematical models provides a general approach to determine the optimal number of monitoring stations. In this study, fuzzy multiple-criteria approach in conjunction with the degree of representativeness technique was used to develop optimal AQMN design. The triangular fuzzy numbers (TFNs) were used to capture the uncertainty associated from human judgement (e.g., assigning weights, scoring). The coverage area of the monitoring station is an essential part of an AQMN which was determined on the basis of representative zone. The effect of the correlation coefficient as well as the cutoff values on coverage of the network was also studied by changing the cutoff values. This methodology provides a systematic approach, which allows multiple-criteria and multiple pollutants in AQMN design. However, the design of an AQMN depends on many site-specific issues and good upfront planning is therefore crucial in properly assessing the problem and designing an optimal AQMN.

## Acknowledgments

Financial support provided by the Natural Science and Engineering Research Council of Canada (NSERC) is highly appreciated.

## References

1. 1. BaldaufR. W.LaneD. D.MarotzG. A.BarkmanH. W.PierceT.2002Application of a risk assessment based approach to designing ambient air quality monitoring networks for revaluating non-cancer health impacts,Environmental Monitoring and Assessment,78213227,1573-2959
2. 2. ChangN. B.TsengC. C.1999Optimal Design of Multi-pollutant Air Quality Monitoring Network in a Metropolitan Region using Kaohsiung, Taiwan as an Example,Environmental Monitoring and Assessment,572121148,1573-2959
3. 3. ElkamelA.FatehifarE.TaheriM.Al-RashidiM. S.LohiA.2008A heuristic optimization approach for Air Quality Monitoring Network design with the simultaneous consideration of multiple pollutants,Environmental Management,88507516(May 2007),1432-1009
4. 4. Harrison, R.M. and Deacon A.R.1998Spatial correlation of automatic air quality monitoring at urban background sites: implications for network design,Environmental Technology,19121132,0147-9487X
5. 5. KaufmannA.GuptaM. M.1988Fuzzy mathematical models in engineering and management science,Elsevier Science Publishers,0-44470-501-5York, U.S.A
6. 6. LeeH. J.MengM. C.CheongC. W.2006Web Based Fuzzy Multicriteria Decision Making Tool,International Journal of the Computer, the Internet and Management,14,2May-August, 2006),114,0858-7027
7. 7. LiuM. K.AvrinJ.PollackR. I.BeharJ. V.Mc EloryJ. L.1986Methodology for designing air quality monitoring networksEnvironmental Monitoring and Assessment,6111,1573-2959
8. 8. MofarrahA.TahirHusain.2010A Holistic Approach for optimal design of Air Quality Monitoring Network Expansion in an Urban Area,Atmospheric Environment,443January 2010),432440,1352-2310
9. 9. SaatyT. L.1980The analytic hierarchy process: planning, priority setting, resource allocation,McGraw-Hill,0-07054-371-2York
10. 10. TesfamariamS.SadiqR.2006Risk-based environmental decision-making using fuzzy analytic hierarchy process (F-AHP).Stoch Environ Res Risk Assess,213550(March 2006),1436-3259
11. 11. YagerR. R.1980On a general class of fuzzy connectives.Fuzzy Set Syst,4235242,0165-0114
12. 12. ZadehL. A.1965Fuzzy sets,Information and control,83June 1965),338353,0890-5401

Written By

Abdullah Mofarrah, Tahir Husain and Badr H. Alharbi

Submitted: October 14th, 2010 Reviewed: April 28th, 2011 Published: July 8th, 2011


text:
# Example of a function $f:\mathbb{R}^2\to\mathbb{R}$ not differentiable at $(0,0)$, but has a directional derivative at $(0,0)$ in all directions

Give an example of a function $f:\mathbb{R}^2\to\mathbb{R}$ such that $f'_u(0,0)$ exists in all directions $\|u\| = 1$, but $f$ is not differentiable at $(0,0)$. You have to show that your example satisfy the above requirement.

-

$$f(x,y) = \left\{\begin{array}{ll}0 & (x,y)=(0,0)\\0&y\neq x^2\\1 & y=x^2, (x,y) \neq (0,0) \end{array} \right.$$

In every direction the line spanned by $u$ can only intersect the parabola twice, so for $(x,y)$ close to the origin we have $y\neq x^2$ and $f(x,y)=0$, so $f'_u(0,0) = 0$. However, taking the path $f(x,x^2)$ we have that $f(x,x^2) \to 1$ as $x \to 0$. It follows that $f$ is not continuous at $(0,0)$ so it cannot be differentiable.

As a bonus, it is actually possible to construct a continuous function which has this property, but it is a little more complicated and not as easy to prove, so I leave thinking about that to you.

-

Define $f:\mathbb R^2\to\mathbb R$, by $$f(x,y)= \frac{x^2y}{x^4+y^2}, \text{when} (x,y)\neq (0,0)$$ and $$f(0,0)= 0$$ This function is not continuous at $(0,0)$. Hence not differentiable. But direction derivative exits for all $v$. This is not continuous because, if you approach to $(0,0)$ via line $y= mx$, we have different limit of $f$.

For $v= (v_1,v_2)$, $\|v\|=1$, we have direction derivative of $f$ in the direction $v$ at $(0,0)$ is $$D_vf(0,0)=\lim_{t\to 0}\frac{f[(0,0)+t(v_1,v_2)]-f(0,0)}{t}=0 \text{ if } v_2=0$$ and above limit equal to $\frac{v_1^2}{v_2}$ if $v_2\neq 0$. Hence limit exits for all $v$. Hence direction derivative exists for all $v$. But not continuous.

-

Define $f$ arbitrarily on the semicircle $x^2+y^2=1, x>0$. For example make it wildly discontinuous, or even non-Borel-measurable. Then define $f$ to be linear on each line through the origin.

-

Not an answer: See comment below:

$$f(x,y)=\frac{y}{|y|}\sqrt{x^2+y^2}\text{ if } y\neq 0$$ and $f(x,y)=0$ if $y=0$. It is continuous at $(0,0)$ but not differentiable. Though direction derivative exists for all $v$. Check it.

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This function is not defined along an entire line in $\mathbb{R}^2$, and it can't be continuously extended over the line $y=0$ either. So I don't think this would count. I also think it's misleading to say that it's continuous at $(0,0)$. Maybe with respect to its true domain, but not if $\mathbb{R}^2$ is the input space. –  alex.jordan May 6 '12 at 17:00
Yes you are right... Thanks for pointing out.. Let me write this in answer itself... –  Junu May 6 '12 at 17:09
This is still an example worth thinking about. I did not downvote. –  alex.jordan May 6 '12 at 18:02
@alex.jordan this example works!! –  Neeraj Bhauryal Aug 28 '13 at 15:05
@Neeraj Making me think about a post that's over a year old, eh? This function is not even defined when $y=0$ (along the $x$-axis), so how can it have a directional derivative at $(0,0)$ in the direction of the $x$-axis? –  alex.jordan Aug 29 '13 at 0:59


text:
Overall Objectives
Application Domains
Partnerships and Cooperations
Bibliography
Inria | Raweb 2014 | Exploratory Action
 PDF e-Pub

## Section: Research Program

### From proof-checking to Interoperability

A new turn with Deduction modulo was taken when the idea of reasoning modulo an arbitrary equivalence relation was applied to typed $\lambda$-calculi with dependent types, that permits to express proofs as algorithms, using the Brouwer-Heyting-Kolmogorov interpretation and the Curry-de Bruijn-Howard correspondence [32] . It was shown in 2007, that extending the simplest $\lambda$-calculus with dependent types, the $\lambda \Pi$-calculus, with an equivalence relation, led to a calculus we called the $\lambda \Pi$-calculus modulo, that permitted to simulate many other $\lambda$-calculi, such as the Calculus of Constructions, designed to express proofs in specific theories.

This led to the development of a general proof-checker based on the $\lambda \Pi$-calculus modulo [3] , that could be used to verify proofs coming from different proof systems, such as Coq [30] , HOL [39] , etc. To emphasize this versatility of our proof-system, we called it Dedukti —“to deduce” in Esperanto. This system is currently developed together with companion systems, Coqine, Holide, Focalide, and Zenonide, that permits to translate proofs from Coq, HOL, Focalize, and Zenon, to Dedukti. Other tools, such as Zenon Modulo, directly output proofs that can be checked by Dedukti.

Dedukti proofs can also be exported to other systems, in particular to the MMT format [47] .

A thesis, which is at the root of our research effort, and which was already formulated by the team of the Logical Framework [38] is that proof-checkers should be theory independent. This is for instance expressed in the title of our invited talk at Icalp 2012: A theory independent Curry-De Bruijn-Howard correspondence.

Using a single prover to check proofs coming from different provers naturally led to investigate how these proofs could interact one with another. This issue is of prime importance because developments in proof systems are getting bigger and, unlike other communities in computer science, the proof-checking community has given little effort in the direction of standardization and interoperability. On a longer term we believe that, for each proof, we should be able to identify the systems in which it can be expressed.


text:
# fontawesome package deforms orginal font style

Why do I notice a shift and deformation in the fonts when I use the fontawesome package. The firs image doesnt use the fontawesome package and the font looks thicker and neat, when the package is included its thinner. MWE 1: without fontawesome pack:

    \documentclass[10pt]{article}
\RequirePackage[T1]{fontenc}
\usepackage{times}
\usepackage{calc}

\reversemarginpar
%\usepackage[paper=letterpaper,
%%includefoot, % Uncomment to put page number above margin
%marginparwidth=1.2in,     % Length of section titles
%marginparsep=.05in,       % Space between titles and text
%margin=1in,               % 1 inch margins
%includemp]{geometry}
%\setlength{\parindent}{0in}
%%%%%%%%%%%%%%%%%%%%%%%% PDTOOLTIP & FONTAWESOME DECLARE %%%%%%%
\usepackage{pdfcomment}% http://ctan.org/pkg/pdfcomment
\usepackage{amsmath}% http://ctan.org/pkg/amsmath
\usepackage{mathtools}% http://ctan.org/pkg/mathtools
\usepackage{xcolor}
%\usepackage{fontawesome} % add skype, FB and Gtalk symbol. Has a problem.
%%%%%%%%%%%%%%%%%%%%%%%% End DECLARE %%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%% Begin CV Document %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Research Interests}
\textbf{Complex adaptive systems in control systems engineering and
behavioral science:} distributed algorithms, computational agent-based
modeling, hybrid dynamic systems, decentralized decision making,
emergence and self organization, amorphous computing, autonomous
systems, control, communications, verification, cooperation,
optimization, game theory, resource allocation, parallel computation,
robotics, energy systems, sustainability in the built environment,
behavioral ecology, engineering education, bio-mimicry and
bio-inspiration
\end{document}

MWE 2: With fontawesome pack.

\documentclass[10pt]{article}
\RequirePackage[T1]{fontenc}
\usepackage{times}
\usepackage{calc}

\reversemarginpar
%\usepackage[paper=letterpaper,
%%includefoot, % Uncomment to put page number above margin
%marginparwidth=1.2in,     % Length of section titles
%marginparsep=.05in,       % Space between titles and text
%margin=1in,               % 1 inch margins
%includemp]{geometry}
%\setlength{\parindent}{0in}
%%%%%%%%%%%%%%%%%%%%%%%% PDTOOLTIP & FONTAWESOME DECLARE %%%%%%%
\usepackage{pdfcomment}% http://ctan.org/pkg/pdfcomment
\usepackage{amsmath}% http://ctan.org/pkg/amsmath
\usepackage{mathtools}% http://ctan.org/pkg/mathtools
\usepackage{xcolor}
\usepackage{fontawesome} % add skype, FB and Gtalk symbol. Has a problem.
%%%%%%%%%%%%%%%%%%%%%%%% End DECLARE %%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%% Begin CV Document %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Research Interests}
\textbf{Complex adaptive systems in control systems engineering and
behavioral science:} distributed algorithms, computational agent-based
modeling, hybrid dynamic systems, decentralized decision making,
emergence and self organization, amorphous computing, autonomous
systems, control, communications, verification, cooperation,
optimization, game theory, resource allocation, parallel computation,
robotics, energy systems, sustainability in the built environment,
behavioral ecology, engineering education, bio-mimicry and
bio-inspiration
\end{document}

-
Please show us your code! All questions should include a Minimal Working Example. The code used to generate that image, for example, as a small, complete document. Or any similar minimal document illustrating the problem. – cfr Apr 12 '14 at 2:18
See my answer. I get pretty much the opposite effect from the one you claim to get. Either you have switched things in your question or something else is going on. Impossible to say, really, without the code... – cfr Apr 12 '14 at 2:26
Have added the MWE with the corresponding outputs. – abhilash sukumari Apr 12 '14 at 2:36
OK. fontawesome requires fontspec which requires Xe/LuaLaTeX. Yet you are loading fontenc and times. times is deprecated anyway and ought not be be used. But neither should its replacement if you are using fontspec. Moreover, you almost certainly don't want fontenc unless you know exactly what you are doing. Summary: I am not surprised you are getting odd output. You are doing all kinds of strange and mutually contradictory things with fonts in your preamble. – cfr Apr 12 '14 at 2:40
yes but the idea is to get the kind of output I have shown in MWE1. What package can I use to get that and still have fontawesome running successfully ? – abhilash sukumari Apr 12 '14 at 2:43

fontawesome loads fontspec and fontspec changes the default font from Computer Modern to Latin Modern.

Computer Modern:

\documentclass{article}
\usepackage{kantlipsum}

\begin{document}
\kant[1]
\end{document}

Latin Modern with fontspec via fontawesome:

\documentclass{article}
\usepackage{fontawesome}
\usepackage{kantlipsum}

\begin{document}
\kant[1]
\end{document}

Latin Modern via fontspec without fontawesome:

\documentclass{article}
\usepackage{fontspec}
\usepackage{kantlipsum}

\begin{document}
\kant[1]
\end{document}

In the context of your code, what this means is that when you load fontawesome, you are getting Latin Modern:

\documentclass[10pt]{article}
\usepackage{calc}

\usepackage{fontawesome} % add skype, FB and Gtalk symbol. Has a problem.

\begin{document}
\section{Research Interests}
\textbf{Complex adaptive systems in control systems engineering and
behavioral science:} distributed algorithms, computational agent-based
modeling, hybrid dynamic systems, decentralized decision making,
emergence and self organization, amorphous computing, autonomous
systems, control, communications, verification, cooperation,
optimization, game theory, resource allocation, parallel computation,
robotics, energy systems, sustainability in the built environment,
behavioral ecology, engineering education, bio-mimicry and
bio-inspiration
\end{document}

To obtain a Times-like font, try something like the following:

\documentclass[10pt]{article}
\usepackage{calc}

\usepackage{fontawesome} % add skype, FB and Gtalk symbol. Has a problem.
\defaultfontfeatures{Ligatures=TeX}
\setmainfont
[ UprightFont = *-Regular ,
BoldFont = *-Bold ,
ItalicFont = *-Italic ,
BoldItalicFont = *-BoldItalic ,
]
{TeXGyreTermes}

\begin{document}
\section{Research Interests}
\textbf{Complex adaptive systems in control systems engineering and
behavioral science:} distributed algorithms, computational agent-based
modeling, hybrid dynamic systems, decentralized decision making,
emergence and self organization, amorphous computing, autonomous
systems, control, communications, verification, cooperation,
optimization, game theory, resource allocation, parallel computation,
robotics, energy systems, sustainability in the built environment,
behavioral ecology, engineering education, bio-mimicry and
bio-inspiration
\end{document}

If you have Times or Times New Roman installed for your system, you could use those instead. Or else choose one of the other Times-alike TeX fonts which is available in opentype.

-
Is it possible to capitalize the section heading using \scshape for TeXGyreTermes font ? – abhilash sukumari Apr 12 '14 at 3:42
Yes. That should work just as it would do otherwise. The fonts support small caps (smcp feature) and fontspec makes that available. Just try \textsc{whatever} to confirm it. (I just did this.) No additional configuration is required. – cfr Apr 12 '14 at 3:46


text:
ΔU Equal 0

$\Delta U=q+w$

Abigail Menchaca_1H
Posts: 104
Joined: Sat Sep 07, 2019 12:19 am

ΔU Equal 0

If ΔU is zero why does the gas expand

Posts: 50
Joined: Mon Jun 17, 2019 7:24 am

Re: ΔU Equal 0

There is another similar question about change in internal energy for isothermal processes. Hopefully this gives some insight: Internal energy, or U, is a state function dependent on temperature. This means that for a process that is isothermal (delta T is 0), even if it is an expansion of a gas, the delta U will equal 0 because the internal energy does not change. Because delta U = q + w, even if work is done, q will be the exact opposite to ensure that delta U is 0. To clarify: delta U does not refer to the same quantity as work.

Posts: 53
Joined: Fri Aug 02, 2019 12:16 am

Re: ΔU Equal 0

If you are referring to an isothermal process, a gas can expand (-w) if it is heated (+q) for the same amount of energy. The energy lost by expanding is supplied by heat, and the change in internal energy is 0.

Sofia Barker 2C
Posts: 101
Joined: Wed Sep 18, 2019 12:21 am

Re: ΔU Equal 0

Delta U can be zero despite expansion because delta U represents a change in energy, so for there to be no change in energy, the energy gained / lost due to work would be made up for in a gain or loss of heat.

Tauhid Islam- 1H
Posts: 64
Joined: Fri Aug 02, 2019 12:15 am

Re: ΔU Equal 0

Delta U is the change in the internal energy if the system and that is equal to Q (heat added to the system) - W (work done by the system). If delta U is zero, the heat added to the system is the same as the work done by the system. The amount of heat that is added to the system has the same amount of energy as the amount of expansionary work done by the system to the surrounding. The work energy lost to the surrounding by the expansion of a gas is basically replaced by heat energy into the system, resulting in a delta U value of 0.


text:
# Math Help - anyone know the answers to these?

1. ## anyone know the answers to these?

1. A test was carried out on a electrical circuit and the following resaults obtained:

4t1+t2=6
-3t1+5t2=0

the 1s and 2s are lower case dont know how to do it sorry!

solve analytically the above simultaneous equatios and check your answer by plotting their graphs on the same set of axes

2. the area of a rectangle is 25.2m2 (squared) and the width is 3.2 metres shorter than the length. calculate the dimentions of the rectangle.

3. solve by factorising the following equations:
x(squared)-6x+9=0
6x(squared)+x-15=0

if anyone can work these out or help me i would be much app cheers!

2. Originally Posted by jenko
1. A test was carried out on a electrical circuit and the following resaults obtained:

4t1+t2=6
-3t1+5t2=0
by plotting their graphs on the same set of axes
Let $t_{1} = x$ and $t_{2} = y$ and then draw a graph using it as the $y = mx + c$ formula

Originally Posted by jenko

2. the area of a rectangle is 25.2m2 (squared) and the width is 3.2 metres shorter than the length. calculate the dimentions of the rectangle.
$L \times W = A$

$A = 25,2 m^2$

$L = x \ metres$

$W = (x - 3,2) \ metres$

So we know that $(x)(x - 3,2) = 25,2$

Originally Posted by jenko

3. solve by factorising the following equations:
x(squared)-6x+9=0
6x(squared)+x-15=0
$x^2 - 6x + 9 = 0$

$(x - 3)(x - 3) = 0$

$(x - 3)^2 = 0$

---

$6x^2 + x - 15 = 0$

$(3x + 5)(2x - 3) = 0$

You can solve them.

3. are these the answers or formulas?

4. Originally Posted by jenko
are these the answers or formulas?
These are basically answers. You just have to finish them off.

5. what are the final answers ive worked them out i need to check them now

6. Originally Posted by jenko
what are the final answers ive worked them out i need to check them now
What are your answers?

7. i found the length and width of the square but i did not to do it the way of the question but heres what i got

Area=LxW
4.675x5.390= 25.19 is this the correct answer ? but how do i work it out as the question asked?

8. Originally Posted by jenko
i found the length and width of the square but i did not to do it the way of the question but heres what i got

Area=LxW
4.675x5.390= 25.19 is this the correct answer ? but how do i work it out as the question asked?
You have to expand L x W and you have to take the Area over to the left side, to get the right side equal to 0. Then solve.

9. sorry completely confused can you give me an example using different numbers? tha i can work off? for the questions above? cheers

10. Originally Posted by janvdl
$L \times W = A$

$A = 25,2 m^2$

$L = x \ metres$

$W = (x - 3,2) \ metres$

So we know that $(x)(x - 3,2) = 25,2$
I am quite certain you know how to use the quadratic formula.

$x^2 - 3,2x = 25,2$

$x^2 - 3,2x - 25,2 = 0$

Now use the quadratic formula to solve for $x$. It's that simple.

11. no i dont this has just completly confused me ! i realy havent a clue for these ive looked at my notes and i couldnt find anything :S i need each question broken dwn induvadually so do these i cannot find any notes on these in my books from previous lessons these re only practice questions ready for my summer exam it jus the fact that im trying to get these into my head with out using notes as my exams are not open book (allowed to look at notes whilst in exam)

12. Originally Posted by jenko
no i dont this has just completly confused me ! i realy havent a clue for these ive looked at my notes and i couldnt find anything :S i need each question broken dwn induvadually so do these i cannot find any notes on these in my books from previous lessons these re only practice questions ready for my summer exam it jus the fact that im trying to get these into my head with out using notes as my exams are not open book (allowed to look at notes whilst in exam)
Well then it wont help to just give you the answers.

The quadratic formula is the following:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Where a, b and c are co-efficients of:

$ax^2 + bx + c = 0$

13. where shalll we start?