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# American Institute of Mathematical Sciences

January  2012, 17(1): 1-31. doi: 10.3934/dcdsb.2012.17.1

## Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions

 1 CMAP, CNRS UMR 7641, École Polytechnique, Route de Saclay, Palaiseau F91128, France 2 Mathematical Institute, 24-29 St Giles’, OXFORD OX1 3LB, United Kingdom 3 Institut de Mathématiques, Université de Toulouse and CNRS, Université Paul Sabatier, 31062 Toulouse Cedex 9, France

Received  April 2011 Revised  July 2011 Published  October 2011

We study the asymptotic behavior of the first eigenvalue and eigenfunction of a one-dimensional periodic elliptic operator with Neumann boundary conditions. The second order elliptic equation is not self-adjoint and is singularly perturbed since, denoting by $\epsilon$ the period, each derivative is scaled by an $\epsilon$ factor. The main difficulty is that the domain size is not an integer multiple of the period. More precisely, for a domain of size $1$ and a given fractional part $0\leq\delta<1$, we consider a sequence of periods $\epsilon_n=1/(n+\delta)$ with $n\in \mathbb{N}$. In other words, the domain contains $n$ entire periodic cells and a fraction $\delta$ of a cell cut by the domain boundary. According to the value of the fractional part $\delta$, different asymptotic behaviors are possible: in some cases an homogenized limit is obtained, while in other cases the first eigenfunction is exponentially localized at one of the extreme points of the domain.
Citation: Grégoire Allaire, Yves Capdeboscq, Marjolaine Puel. Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 1-31. doi: 10.3934/dcdsb.2012.17.1
##### References:
 [1] G. Allaire and Y. Capdeboscq, Homogenization of a spectral problem in neutronic multigroup diffusion, Comput. Methods Appl. Mech. Engrg., 187 (2000), 91-117. doi: 10.1016/S0045-7825(99)00112-7. [2] G. Allaire and Y. Capdeboscq, Homogenization and localization for a 1-D eigenvalue problem in a periodic medium with an interface, Ann. Math. Pura Appl. (4), 181 (2002), 247-282. [3] G. Allaire and C. Conca, Bloch wave homogenization and spectral asymptotic analysis, J. Math. Pures et Appli. (9), 77 (1998), 153-208. doi: 10.1016/S0021-7824(98)80068-8. [4] G. Allaire and R. Orive, Homogenization of periodic non self-adjoint problems with large drift and potential, ESAIM COCV, 13 (2007), 735-749. doi: 10.1051/cocv:2007030. [5] G. Allaire and A. Piatnistki, Uniform spectral asymptotics for singularly perturbed locally periodic operators, Comm. PDE, 27 (2002), 705-725. doi: 10.1081/PDE-120002871. [6] A. Bensoussan, J.-L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures," North-Holland, Amsterdam, 1978. [7] Y. Capdeboscq, Homogenization of a diffusion equation with drift, C. R. Acad. Sci. Paris Série I Math., 327 (1998), 807-812. [8] G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, Ann. Sci. École Norm. Sup. Sér. (2), 12 (1883), 47-89. [9] T. Kato, "Perturbation Theory for Linear Operators," Second edition, Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. [10] S. Kozlov, Reducibility of quasiperiodic differential operators and averaging, Trudy Moskov. Mat. Obshch., 46 (1983), 99-123. [11] W. Magnus and S. Winkler, "Hill's Equation," Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. [12] S. Moskow and M. Vogelius, First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1263-1299. [13] O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, On the limiting behaviour of a sequence of operators defined in different Hilbert's spaces, Upsekhi Math. Nauk., 44 (1989), 157-158. [14] B. Perthame and P. Souganidis, Asymmetric potentials and motor effect: A homogenization approach, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2055-2071. doi: 10.1016/j.anihpc.2008.10.003. [15] F. Santosa and M. Vogelius, First-order corrections to the homogenized eigenvalues of a periodic composite medium, SIAM J. Appl. Math., 53 (1993), 1636-1668. doi: 10.1137/0153076. [16] M. Vanninathan, Homogenization of eigenvalue problems in perforated domains, Proc. Indian Acad. Sci. Math. Sci., 90 (1981), 239-271. doi: 10.1007/BF02838079.

show all references

##### References:
 [1] G. Allaire and Y. Capdeboscq, Homogenization of a spectral problem in neutronic multigroup diffusion, Comput. Methods Appl. Mech. Engrg., 187 (2000), 91-117. doi: 10.1016/S0045-7825(99)00112-7. [2] G. Allaire and Y. Capdeboscq, Homogenization and localization for a 1-D eigenvalue problem in a periodic medium with an interface, Ann. Math. Pura Appl. (4), 181 (2002), 247-282. [3] G. Allaire and C. Conca, Bloch wave homogenization and spectral asymptotic analysis, J. Math. Pures et Appli. (9), 77 (1998), 153-208. doi: 10.1016/S0021-7824(98)80068-8. [4] G. Allaire and R. Orive, Homogenization of periodic non self-adjoint problems with large drift and potential, ESAIM COCV, 13 (2007), 735-749. doi: 10.1051/cocv:2007030. [5] G. Allaire and A. Piatnistki, Uniform spectral asymptotics for singularly perturbed locally periodic operators, Comm. PDE, 27 (2002), 705-725. doi: 10.1081/PDE-120002871. [6] A. Bensoussan, J.-L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures," North-Holland, Amsterdam, 1978. [7] Y. Capdeboscq, Homogenization of a diffusion equation with drift, C. R. Acad. Sci. Paris Série I Math., 327 (1998), 807-812. [8] G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, Ann. Sci. École Norm. Sup. Sér. (2), 12 (1883), 47-89. [9] T. Kato, "Perturbation Theory for Linear Operators," Second edition, Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. [10] S. Kozlov, Reducibility of quasiperiodic differential operators and averaging, Trudy Moskov. Mat. Obshch., 46 (1983), 99-123. [11] W. Magnus and S. Winkler, "Hill's Equation," Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. [12] S. Moskow and M. Vogelius, First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1263-1299. [13] O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, On the limiting behaviour of a sequence of operators defined in different Hilbert's spaces, Upsekhi Math. Nauk., 44 (1989), 157-158. [14] B. Perthame and P. Souganidis, Asymmetric potentials and motor effect: A homogenization approach, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2055-2071. doi: 10.1016/j.anihpc.2008.10.003. [15] F. Santosa and M. Vogelius, First-order corrections to the homogenized eigenvalues of a periodic composite medium, SIAM J. Appl. Math., 53 (1993), 1636-1668. doi: 10.1137/0153076. [16] M. Vanninathan, Homogenization of eigenvalue problems in perforated domains, Proc. Indian Acad. Sci. Math. Sci., 90 (1981), 239-271. doi: 10.1007/BF02838079.
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# Dilation Time ( Kinematics of Relativity )

## Main Question or Discussion Point

I'm about to ask derivation of dilation time in terms of Special Relativity.
I saw explanations in Introduction to Classical Mechanics by David Morin that dilation time is formed by assumption that light speed is absolute refers to all inertial reference. He derived it by comparing 2 reference; A who sitting on the train with constant speed v refers to the ground, and B who at rest on the ground. They are to see the light traveling on the train, which has long h.

He got that A's time to see the light traveling from end to end , in transverse direction of light traveling, of the train is tA=2h/c
And B's time to see the light traveling on the train tB=2h/(c^2-v^2)^1/2, by simply using pythagoras.
Then, he got that tB=$\gamma$tA, and this formula is generally used.
My question, How can this formula be general, since this are just derived by assuming that train is traveling in transverse direction of light speed.

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# ECDSA security using same per-message secrets $k$ but different signing keys

I know that when I have a ECDSA keypair and use it twice with the same $k$ to sign different messages then the private key can be recovered.

If I now have multiple ECDSA keypairs and sign a message with each, and use the same $k$ in each signature, is this still secure?

Note: The signed messages may be identical.

I personally see no problem with it and the math seems to support that in my opinion, but better safe then sorry. This is why I'm asking the question.

For ECDSA the per-message secret $k$ should be generated randomly. In particular you have to ensure that the per-message secrets never repeat, or else the private key $d$ can be recovered. First, let's have a look at an example where the same per-message secret $k$ and the same private key $d$ was used to generate ECDSA signatures $(r, s_1)$ and $(r, s_2)$ on two messages $m_1$ and $m_2$. Then $s_1 \equiv k^{-1}(e_1+dr)$ (mod n) and $s_2 \equiv k^{-1}(e_2+dr)$ (mod n), where $e_1 = H(m_1)$ and $e_2 = H(m_2)$. Then $ks_1 \equiv e_1 + dr$ (mod n) and $ks_2 \equiv e_2 +dr$ (mod n). Subtraction gives $k(s_1-s_2) \equiv e_1 - e_2$ (mod n). If $s_1 \not\equiv s_2$ (mod n), which occurs with overwhelming probability, then

$$k \equiv (s_1 - s_2)^{-1}(e_1 - e_2)\mod{n}.$$

From this an adversary can determine $k$, and then use $d=r^{-1}(ks-e)\mod n$ to recover the private key $d$. This implementation failure was used, for example, to extract the signing key used for the PlayStation 3 gaming console or in 2013 to rip users of Android Bitcoin Wallet off their funds.

Now let's have a look at a second example, where the same per-message secret $k$, but different private keys $d_1$ and $d_2$ $(d_1 \neq d_2)$ are used to generate different signatures. In this case the subtraction step from before would lead to $k(s_1-s_2) \equiv e_1-e_2+r(d_1-d_2) \mod n$. From this you are not able to recover the private keys $d_1$ and $d_2$ used to generate the signatures $s_1$ and $s_2$ unless you can efficiently solve the $ECDLP$ (assuming an adequate elliptic curve group was chosen). If you signed the same message, i.e. $m_1=m_2$, then $e_1-e_2$ would cancel out, but this doesn't give the attacker any more advantage as $e_1$ and $e_2$ are supposed to be known anyway.

But be aware that as soon as you’d use the same private-key / message-secret pair to sign another message, this would lead us back to the scenario depicted in the first example. This means that you’d have to be 100 % sure that you only need to sign one single message using the same $k$/$d$-pair. In practice however, you normally don't create a new ECDSA key-pair for every new data you need to sign, as

• for every new key-pair you create the verifying party will need to learn the new corresponding public key
• you need to keep track of which key to use for every piece of data you signed if you are both the verifying and signing party
• using some crypto APIs you sometimes cannot control the per-message secret but only the private key and the signing function will create its own per-message secret making use of the underlying CSRNG


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# Zigya Join As

Harmful effects of burning fossil fuels are :

(i) Burning of fossil fuels burn with smoky flame which cause in air pollution.

(ii) They also cause global warming because they produce green house gas carbon dioxide gas on burning.

### Bookmark Count

Harmful effects of burning fossil fuels are :

(i) Burning of fossil fuels burn with smoky flame which cause in air pollution.

(ii) They also cause global warming because they produce green house gas carbon dioxide gas on burning.


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# Duality in Linear Programming

My articles on linear algebra and linear programming is a prerequisite to this article.

Duality is a concept from mathematical programming. In the case of linear programming, duality yields many more amazing results.

## The dual linear program

The duality theory in linear programming yields plenty of extraordinary results, because of the specific structure of linear programs. In order to explain duality to you, I’ll use the example of the smart robber I used in the article on linear programming. Basically, the smart robber wants to steal as much gold and dollar bills as he can. He is limited by the volume of his backpack and the maximal weight he can carry. Now, let’s notice that we can write the problem as follows.

$$\textrm{Maximise} \quad \textrm{Value}_\textrm{gold} volume_\textrm{gold} + \textrm{Value}_\textrm{bills} volume_{\textrm{bills}} \\\\ \textrm{Subject to} \quad volume_{\textrm{gold}} + volume_{\textrm{bills}} \leq \textrm{Volume}_{\textrm{limit}} \\ \textrm{} \qquad \qquad \quad \;\; \textrm{Density}_\textrm{gold} volume_{\textrm{gold}} + \textrm{Density}_\textrm{bills} volume_{\textrm{bills}} \leq \textrm{Weight}_{\textrm{limit}} \\ \textrm{} \qquad \qquad \quad \;\; volume_{\textrm{gold}} \geq 0 \\ \textrm{} \qquad \qquad \quad \;\; volume_{\textrm{bills}} \geq 0$$

The problem we have written here (no matter which equivalent formulation we used) is what we call the primal linear program. It’s now time for you to learn about the dual linear program! The dual program will totally change our understanding of the problem, and that’s why it’s so cool. I hope you are as excited as I am!

In the primal program, constraints had constant numbers on their right. These constant numbers are our ressources. They say what we are able to do concerning each constraint. The dual problem consists in evaluating how much our combined ressources are worth. If the offer meets the demand, our ressources will be just as much as their potentials, which is the worth of the robbery. Sweet, right?

Ressources are a concept I have come up with. It’s not a standard concept.

Let’s go more into details. In the dual problem we will attribute values to the ressources (as in “how much they’re worth”). These values are called the dual variables. In our case, we have two constraints, so we will have 2 dual variables. The first dual variable, let’s call it $value_{\textrm{volume}}$ refers to the value of one unit of volume. As you have figured it out, the second dual variable refers to the value of one unit of weight. $value_{\textrm{weight}}$ seems like a right name for it, right?

Now, I bet you can write the value of the robbery with these two new variables… Let’s see if we get the same result.

$$\textrm{Total Value} = value_{\textrm{volume}} \textrm{Volume}_{\textrm{limit}} + value_{\textrm{weight}} \textrm{Weight}_{\textrm{limit}}$$

That’s nice but how are the values per volume and per weight determined?

If I wanted to sell my ressources, potential buyers are going to minimize the value of my ressources. So their valuations are the minimum of the total value. But as a seller, I will argue that each of my ressource is worth a lot, because it enables the robbery of more gold and more bills.

Obviously, the values of ressources depend on the actual values of gold and bills per volume. Let’s have a thought about the value of gold (and then you’ll be able to apply the same reasoning to bills). If the constraints enabled us to steal one more volume of gold, then incremental value of the robbery would be at least the value of this one volume of gold, right? It could be more, if we use the new constraints to steal something else than gold that’s worth more. What I’m saying is that, if the total volume enabled us to steal one more unit of volume of gold, and if we could carry one more unit of weight of one volume of gold, then the value of this incremental steal would be at least the value of one more volume of gold. Let’s write it.

$$value_{\textrm{volume}} + value_\textrm{weight} \textrm{Density}_\textrm{gold} \geq \textrm{Value}_\textrm{gold}$$

I’ll let you write the similar constraint for bills. In fact, we can do this reasoning with any variable of the primal problem and ask ourselves the question: “If we add one unit of the variable, how will it affect the total valuation?” From that we deduce a constraint on dual variables. As you see, any variable in the primal problem is associated with a constraint in the dual problem and vice-versa.

We are almost done. Let’s notice the fact that if we increase the total volume, then we have more possibilities for the primal variables, which are the volumes of stolen gold and bills. Therefore, the value of a unit of volume cannot be negative. This adds two more constraints on the sign of dual variables. Now, we’re done and we can write the dual problem.

$$\textrm{Minimise} \quad value_{\textrm{volume}} \textrm{Volume}_{\textrm{limit}} + value_{\textrm{weight}} \textrm{Weight}_{\textrm{limit}} \\\\ \textrm{Subject to} \quad value_{\textrm{volume}} + value_\textrm{weight} \textrm{Density}_\textrm{gold} \geq \textrm{Value}_\textrm{gold} \\ \textrm{} \qquad \qquad \quad \;\; value_{\textrm{volume}} + value_\textrm{weight} \textrm{Density}_\textrm{bills} \geq \textrm{Value}_\textrm{bills} \\ \textrm{} \qquad \qquad \quad \;\; value_\textrm{volume} \geq 0 \\ \textrm{} \qquad \qquad \quad \;\; value_\textrm{weight} \geq 0$$

Hey, that looks like a linear program…

And it always does! The dual program of a linear program is a linear program! Let’s have a look at the feasible set.

It’s important to notice that the information of the primal objective functions appear in the dual feasible set. However, it loses the information of the primal ressources. This information would appear if we draw the level sets of the dual objective function. Nevertheless, if we draw the primal and dual feasible sets, then we have all the information of the linear programs.

The most important result is the strong duality property: optimal values of the primal and dual problems are the same. We can solve the primal problem simply by solving the dual problem! And sometimes, the dual problem can be much more simple than the primal one.

What’s more, and that’s less obvious here, the dual of the dual is the primal. This can be proved with calculus. Try to transform the dual program into a maximisation program with less or equal inequality constraints to find its dual, that’s a good exercise. But what’s interesting is the real reason why the dual of the dual is the primal. Try to think about it. It’s very tricky but even more interesting exercise. This means that the value of the ressources of the dual constraints are the primal variables. In our example, if the value of gold increases by 1 unit, then the worth of the robbery would be increased by the quantity of stolen gold. This means that the value of the value of gold is the quantity of stolen gold!

There’s something else quite interesting about duality: It gives directly a sensitivity analysis. Consider our primal problems. It’d be interesting to know how much more I could get, had my knapsack been larger or my body stronger. This looks like a difficult question in the primal program, but it’s very obvious in the dual program. If I can have one unit of volume more in the knapsack, then the incremental worth of the robbery will be $value_\textrm{volume}$. This is actually a minimizer of the incremental worth I’ll have, but it’s a perfect approximation for small incremental volume of the knapsack.

But that’s not all. In linear programming, duality yields plenty more of amazing results! In particular, there is a strong connection between the primal bases and the dual bases.

## Dual bases

Let’s notice that the primal problem is equivalent to the following formulation.

$$\textrm{Maximise} \quad \textrm{Value}_\textrm{gold} volume_\textrm{gold} + \textrm{Value}_\textrm{bills} volume_{\textrm{bills}} \\\\ \textrm{Subject to} \quad volume_{\textrm{gold}} + volume_{\textrm{bills}} + volume_{\textrm{unused}} = Volume_{\textrm{limit}} \\ \textrm{} \qquad \quad \;\; \textrm{Density}_\textrm{gold} volume_{\textrm{gold}} + \textrm{Density}_\textrm{bills} volume_{\textrm{bills}} + weight_{\textrm{unused}} = Weight_{\textrm{limit}} \\ \textrm{} \qquad \qquad \quad \;\; volume_{\textrm{gold}} \geq 0 \\ \textrm{} \qquad \qquad \quad \;\; volume_{\textrm{bills}} \geq 0 \\ \textrm{} \qquad \qquad \quad \;\; volume_{\textrm{unused}} \geq 0 \\ \textrm{} \qquad \qquad \quad \;\; weight_{\textrm{unused}} \geq 0.$$

This equivalent formulation has introduced the two variables $volume_{\textrm{unused}}$ and $weight_{\textrm{unused}}$. These variables are called slack variables. Any linear problem can be written with slack variables and a full rank matrix $A$ to have the following form.

$$\underset{x,e}{\textrm{Maximise}} \quad c^Tx \\ \textrm{Subject to} \quad Ax + e = b \\ \textrm{} \qquad \qquad \quad \;\; x, e \geq 0.$$

We can obtain a similar formulation for the dual program.

$$\underset{y,s}{\textrm{Minimise}} \quad y^Tb \\ \textrm{Subject to} \quad y^TA – s^T = c^T \\ \textrm{} \qquad \qquad \quad \;\; y, s \geq 0.$$

In practice, if you want to obtain these formulations, I advise you to write your primal program with inequalities only and non-negative variables only. Then, write the dual program associated to that (be careful, it should be a less or equal inequality for the maximiser program, and a higher or equal for the minimiser one). Only then, add the slack variables.
OK, I really didn’t get the two last formulas…

Don’t worry, it’s not that important. It’s not even a standard formulation. I like this formulation because it shows the similarity between the primal and dual programs. It’s also a nice way to describe duality and its properties. Be careful with other formulations. The results I’ll give here can be hard to translate to other formulations.

What you need to notice is that we now have variables represented by vectors $x$, $e$, $y$ and $s$. Also, each variable of the vectors $e$ and $s$ appears alone in one of the constraints of the primal or dual program.

That’s good but I really don’t understand what these slack variables represent…

The variables can easily be understood in the following graph. The initial variables $x$ are the coordinates of the points, while the slack variables represent the distances from the different constraints. The unit of the slack variables is a little bit more complicated. To imagine it, keep in mind that the limits of the constraints are level sets. These level sets move as the value of they take varies.

Let’s denote $n$ the number of variables of $x$, and $m$ the number of variables of $e$. Note that $n$ is the number of constraints in the dual program, and $m$ is the number of constraints in the primal program. Therefore, a primal base is defined by the choice of $m$ variables which values will be set to zero. These $m$ variables form the “constraint base” I mentioned in the article on linear programming. The $n$ other variables are the “base”, in the usual definition of it.

Be careful, these notations are not standard notations.

What’s interesting is that the dual bases behave in the opposite way. As a matter of fact, they require $n$ variables in the constraint base and $m$ variables in the base. What’s even more interesting is that we can match each primal base to a dual base. As a matter fact, each variable of the vector $y$ corresponds to a constraint, which corresponds to a variable of the vector $e$. Similarly, we match variables of vectors $x$ and $s$. Therefore, to any primal base we can associate a dual base by choosing to add in the dual base variables that match the primal variables that are not in the primal base. Another way of saying it is that whenever we have not fixed a primal variable to 0, we fix the corresponding dual variable to 0.

This gives us the following matching of bases (drawn by the colors of dots).

The constraints of the two graphs can be matched. Non-negativity constraints of initial variables correspond to actual constraints in the other other program. A base defined by the intersection of two colors of constraints in one problem is matched with a base defined by the intersection of the two other colors of constraints in the other problem. On the two graphs, only the primal base associated with the dual yellow base has not been drawn, because the primal intersection of green and blue constraints could not be drawn.

Wait… There’s even more interesting: the values of the objective function of a primal base and its associated dual base are the same! Considering the way we introduced the dual program, this is very surprising. You may prove it with calculus, but the real reason why we have such a result involves the construction of the Lagrangian. If we had done that, we could have easily proved that, for any primal and dual variables satisfying the equality constraints, the difference of the primal by the dual objective functions is equal to $s^Tx + y^Te$. This value is called the complementary slackness. It is nil if the dual variables correspond to the dual base of the primal variables.

Wait… If both programs have the same values, shouldn’t the dual objective function always be higher than the primal objective function?

As a matter of fact the two programs have the same values. Since the primal program is a maximization problem, the value of its objective function is always less than the optimal value. And it’s the opposite for the dual program. Yet, this reasoning works for variables in the feasible set only! Therefore, the optimization problem is equivalent to finding a primal feasible base with an associated dual feasible base!

On the two graphs, only the pink dots are primal and dual feasible: they represent the solution of the linear programs!

Cool! Can we use it to design an optimization algorithm?

Yes we can. And this gives us… the simplex method! As a matter of fact, the simplex method consists in moving from primal feasible base to strictly better primal feasible base. But it can be equivalently considered as an algorithm that moves from primal feasible bases to primal feasible bases with an associated dual base that gets more and more feasible. This “close-to-feasibility” criterium can, for instance, mean having the most negative variable as close to 0 as possible. Once the most negative variable of the dual base is non-negative, we have a dual feasible base. Therefore, we have reached the optimum.

Similarly, we could move from feasible dual bases to feasible dual bases, as we try to reach an associated feasible primal base. This is equivalent to applying the simplex method to the dual program and is known as the dual simplex.

As you can imagine with this remark, there is a huge link between dual variables and reduced costs. This can be observed in the graph on the left. A reduced cost tells you by how much the objective function will increase if you lose one of the constraint of the base constraint and walks away from it on the edge defined by the other constraints of the constraint base, by 1 unit of the slack variable associated with this constraint. This reduced cost is the incremental value of the objective function when moving along the green arrow.

That’s almost the same thing as the dual variables that will tell you by how much the objective function will increase if you move one of the constraint by 1 unit of the ressource. This is the incremental value of the objective function when moving along the yellow arrow.

As you can see on the graph, the dual variable is the opposite of the reduced cost of the slack variable associated to the dual variable. Be careful with the sign. Our result is valid here because we have less or equal inequalities in the primal program (or, equivalently, the slack variables are preceded by a “+”). If we had higher or equal inequalities or slack variables preceded by a minus, we would have the equality of the reduced costs and dual variables.

I have an idea of an optimization algorithm: We could look for feasible primal and dual variables that minimize the complementary slackness…

That’s an awesome idea! It leads to… the interior point method! The idea of the interior point method is to remain inside the feasible set and to converge towards the optimum. This avoids the possibly numerous iterations to go from extreme points to extreme points, and it skips problems of degeneracy. For great numbers of variables, the interior point method is faster than the simplex method.

Given a positive real number µ, the interior point method consists in finding the primal and dual vectors that satisfy the equality constraints, are positives, and such that the product of each primal variable with its associated dual variable is equaly to µ. Then, as we decrease the value of µ, we approach to the primal and dual optimal solutions. The advantage of such a method is that we can use gradient-based algorithms to optimize the problem, which guarantee a very quick resolution. Another very interesting advantage is that it can be generalized to semi-definite programming.

There are plenty of other amazing things we can do with interior point methods. If you know them, you should write about them. Also, semi-definite programming enables the modeling of a large range of problems. I’m not familiar enough with semi-definite programming to write about it, so, if you know it, please write about it!

As you see the dual program yields plenty of extraordinary results and provides a very interesting different understanding of the problem. Unfortunately, it faces the problem of degeneracy, especially when applying the simplex method.

## Degeneracy

In our example, each primal base or constraint base matches one unique point. Yet, it’s possible that three constraints intersect at the same point, in which case, any combination of two of these three constraints correspond to that point. The matching is no longer unique. This situation happens for instance in the following case at the black dot.

What’s wrong with that case?

It’s bad for the simplex algorithm. First it forces us to make a choice. Suppose that we were at the constraint base defined by the green and yellow constraints, and that the simplex algorithm decides to lose the green constraint. Then, we would move along the yellow constraint until we reach the black dot. This black dot can be defined by the fact that we’ve intersected the red or the blue constraint. One of them will be added to the constraint base. We need to make a choice. This is actually not really a problem. But the remark is a big problem.

Suppose the blue and yellow constraints form the constraint base (which means that slack variables corresponding to green and red constraints are in the base). If we put the yellow constraints out of the constraint base, then we would move along the blue line away from the yellow constraints. The direction along the blue constraint improves the objective function, which means that it’s associated with a positive reduced cost. Therefore, we should expect a strict increase of the objective function. Yet, because of the red constraint intersecting at the black dot, the red constraint would be added to the constraint base, which therefore will define the same black dot. We have made an iteration of the simplex method without improving anything.

If we handle the situation badly, it’s possible in certain cases to never get out of the black dot. Theoretically, we could apply the Bland’s rule to avoid that from happening. However, even with this rule, the resolution could be extremely long for very degenerated points (I mean points that can be defined by a lot of different bases). Imagine the case where the dimension of the feasible set is 25 and where we have 50 constraints intersecting at the same point. Then the number of bases defining this point is $\binom{50}{25} \approx 10^{14}$. The Bland’s rule has a good chance to visit half of them… This would take like forever! Maybe days… That’s very very bad. And yet, a dimension of 25 is very small.

OK, you’re right, degeneracy is very bad for simplex methods… What can we do about that?

We could thought of adding slight disturbances on constraints. This would avoid degeneracy. However, it wouldn’t really improved the resolution, as it would create multiple points very close to each other. The simplex method would still move around these points with very small improvement of the objective function. Basically, there is still a good chance of visiting half of all the bases associated to the degenerated point, so that’s still not good. Such cases are so similar to degeneracy that they can be considered as such.

The problem of degeneracy is often dealt by managing the set of constraints and the set of variables. In our case, if we could detect that the primal blue constraint was useless, then we would have solved the problem of degeneracy. In Montreal, several methods like the Dynamic Constraint Aggregation (DCA), the Integral Primal Simplex (IPS) and the Improved Column Generation (ICG) have also been developed. Briefly said, I’d say that they classify variables depending on how they appear in the constraints and solve over a small group of variables only. They may look into other variables if the optimality criterium is not satisfied. Find out more with my article on column generation.

Now, the reason why I have decided to talk about degeneracy in an article on duality is because duality offers an interesting understanding of degeneracy.

Let’s notice that the dual feasible has not changed if we only modified the ressource of the weight constraint. Therefore we have the following matching.

As you can see on the graph, there are fewer base points in the primal than in the dual. In fact, the primal green dot can be matched with the dual cyan, pink and orange dots. In fact, the primal green extreme point is actually associated with the dual green constraint.

Is a degenerated extreme point always associated with a dual constraint?

No. Let’s denote n the dimension of the primal feasible set. Consider a degenerated point. Let’s call degeneracy is the number of constraints that are not required to define it. Let’s denote d the degeneracy. In our case, the degeneracy d is equal to 1. Then the number of non-nil primal variables at this point is equal to n-d. Therefore, the number of dual constraints that are in all the dual constraint bases associated to the degenerated point is n-d too. Their intersection defines the dual space associated to the degenerated point. As the dual space requires n constraints to define a point, the dual space associated to the degenerated point represents a vector space of dimension d. It matches a constraint if d = n-1, but is usually a vector space with a smaller dimension. As this whole vector space matches a single primal point that has a single objective value, the whole vector space has a single objective value. Therefore, the whole vector space is included in a dual objective function level set.

In the example, there are several primal-dual feasible bases, defined by the matching of the primal green and dual pink dots, and by the matching of the primal green and dual white dots. Degeneracy at the optimum is actually equivalent to the existence of several primal-dual solutions. From that, we can deduce that the dual cyan, pink and orange dots have a same dual objective value, which means that the green constraint is a level set of the dual objective function.

What happens in a non-optimal degenerated point?

Let’s make our degenerated point non-optimal to see what happens. In order to do that, we’ll assume that the value of gold has suddenly dropped, making the primal white dot optimal. As a result in the dual the green constraint will be lower (look at its definition if you want to make it sure). We obtain the following graph.

The green constraint still represents the dual set associated with the degenerated point. However, in this case, this whole dual set is infeasible.

What’s interesting to notice is that it’s not possible to go directly from the dual orange dot to the dual optimal white dot. In the primal problem, this means that if the constraint base contains the blue and yellow constraints, it will not be able to go directly to the primal optimal white dot. More generally, there may be plenty of dual bases associated to a degenerated point that do not enable leaving the degenerated point. In a dual viewpoint, this means that we have to move all along a dual infeasible polyhedron without improving the dual objective function. Once at a degenerated point, it is therefore crucial to find a right associated dual point that will give us a way to get out of the degenerated point.

The Dual Variable Stabilization (DVS) follows this idea. It was proved to be efficient to deal with such cases, especially in the case of column generation. The idea of it is that once iterations of the simplex give close dual points, we add a penalty to dual points far from those we have recently found. Read my article on DVS to better understand it.

## Let’s sum up

Duality yields plenty of amazing results. In particular, one very interesting thing to do is to observe what happens in the dual when we apply the simplex method or the interior point method to the primal linear program. For the simplex method, this naturally defines a new method called the dual simplex method. Learn more by reading my article on simplex methods.

An important application of the duality theory is the definition of the prices of the ressources. This is the case in the Location Marginal Price (LMP) algorithm used in electricity markets. As a matter of fact, this algorithm consists in minimizing the total cost of production of electricity with constraints on the demand at each geographical location. The demand can be viewed as ressource. If the demand at a location increases, then it would cause an incremented cost of production of electricity. In the electricity market, this incremented cost is considered as the price of electricity at that point. It’s a great model that takes into account electricity losses for instance. If a demand is at a location far from generators that induce plenty of losses, an increase of the demand at that location will induce an important incremental production, hence a higher incremental cost of production. Therefore, the dual variable associated to the demand, which is the price, will be higher.

I would like to thank my lab partners Marilène, Thibault and Samuel who have helped me understand and describe duality. In particular, I thank Hocine for his huge help with degeneracy.

## More on Science4All

Primal and Dual Simplex Methods By Lê Nguyên Hoang | Updated:2016-02 | Prerequisites: Optimization by Linear Programming, Duality in Linear Programming | Views: 14455
The simplex method is one of the major algorithm of the 20th century, as it enables the resolution of linear problems with millions of variables. An intuitive approach is given. But that's not all. We present an important variant called the dual simplex. Finally, we'll explain its main default, that is, when facing degeneracy.

Optimization by Integer Programming By Lê Nguyên Hoang | Updated:2016-01 | Prerequisites: Optimization by Linear Programming | Views: 5663
Integer programming is arguably the greatest achievement of applied mathematics. Half of the time, it's what's used to solve real-world problems!

Column Generation and Dantzig-Wolfe Decomposition By Lê Nguyên Hoang | Updated:2016-02 | Prerequisites: Linear Algebra and Higher Dimensions, Optimization by Linear Programming | Views: 5704
Column generation and the Dantzig-Wolfe decomposition are powerful tricks which have revolutionized optimization addressed to industrial problems, and generated millions and millions of dollars. My PhD supervisor effectively took great advantage of these tricks and founded companies with it. This article explains the tricks.

Dual Variable Stabilization By Lê Nguyên Hoang | Updated:2016-02 | Prerequisites: Optimization by Linear Programming, Duality in Linear Programming, Primal and Dual Simplex Methods | Views: 528
Simplex methods face issues in case of degeneracy. The dual variable stabilization is a very recent state-of-the-art way to deal with degeneracy. An intuitive understanding is given in this article.

## One comment to “Duality in Linear Programming”

1. goutham edara says:

That is a well thought out explanation. Now I have a profound understanding of what Dual does. Thank you very much.


text:
# Homework Help: Identity proof using Stoke's Theorem

1. May 16, 2010

### pretzsp

1. The problem statement, all variables and given/known data

Show using Stoke's Theorem that

S is an open surface with boundary C (a space curve). $$f(\vec r)$$ is a scalar field.

2. Relevant equations

Stoke's theorem $$\iint_S (\nabla\times \vec F) \cdot d\vec S = \int_C F \cdot d\vec r$$

3. The attempt at a solution

Thus far I've tried evaluating $$(\nabla f)\times d\vec S$$ by taking components in various ways and then trying to force it to become something that I can take the curl of... with no success at all. I'm completely stumped now.

2. May 16, 2010

### gabbagabbahey

Try applying Stokes' theorem to the vector function $\textbf{F}\equiv f\textbf{c}$, where $\textbf{c}$ is any constant vector.

3. May 16, 2010

### pretzsp

Well, what I get here is

$$\iint_S \vec c\cdot\nabla f \times d\vec S = -\oint \vec c \cdot fd\vec r$$

And I'm not entirely sure whether I can just "cancel" the c dot product...

(should be a vector c there, by the way)

Last edited: May 16, 2010
4. May 16, 2010

### gabbagabbahey

Well, since $\textbf{c}$ is a constant vector, you can certainly pull it out of the integrals and say

$$\vec c\cdot\left(\iint_S \nabla f \times d\vec S\right) = \vec c\cdot\left(-\oint fd\vec r\right)$$

$$\implies \vec c\cdot \left(\iint_S \nabla f \times d\vec S+\oint fd\vec r\right)=0$$

And since this is true for arbitrary, constant $\vec{c}$, what must you conclude?

5. May 16, 2010

### pretzsp

Yeah, those last steps were fine for me, I just wasn't sure if that even with c a constant vector that it was "allowed" to just pull it out like so. Thanks for the help.


text:
0000174214 00000 n Chem Eng Process 48:306–310, Takache H, Pruvost J, Marec H (2015) Investigation of light/dark cycles effects on the photosynthetic growth of Chlamydomonas reinhardtii in conditions representative of photobioreactor cultivation. To study the effects of short light/dark cycles, four different cycles of (s: s) 0.91:0.09, 0.83:0.17, 0.75:0.25, and 0.50:0.50 (light: dark) were set every one second. The Drawbacks of Artificial Lighting. 0000008009 00000 n Artificial lighting design makes the most of its advances in lamp characteristics: CRI, LPW, LLF, and lamp life of common sources are continually improving. 0000114139 00000 n 0000023070 00000 n In this sense, Fig. Bioprocess Biosyst Eng 42(5):727–739, Seyfabadi J, Ramezanpour Z, Khoeyi JA (2011) Protein, fatty acid, and pigment content of Chlorella vulgaris under different light regimes. Artificial lighting strategies in photobioreactors for bioenergy production by, $$[{\text{NER}} = \sum {\text{Eout}}/\sum {\text{Ein}}]$$, $$[{\text{NEB}} = \sum {\text{inputs}} - \sum {\text{outputs}}]$$, https://doi.org/10.1007/s42452-019-1761-0, Engineering: Sustainable Inventive Systems. These sensors detect the presence of humans, motion, timing or occupancy and based on the sensor output, it switches the lamps ON and OFF. et al. The values for the maximum biomass (63.88 mg/L h) and lipid (18.9 mg/L h) productivities as well for the calorific value (20.4 kJ/g) were obtained at a photoperiod of 24 t/d. Bioresour Technol 171:367–374, Krzeminska I, Pawlik-Skowronska B, Trzcinska M, Tys J (2014) Influence of photoperiods on the growth rate and biomass productivity of green microalgae. 0000142201 00000 n The analyses were performed with the software Statistica 7.0 (StatSoft, Tulsa, OK, USA). 4a, b, PCA1 and PCA2 explained 72.19% of the observed variation. The short light/dark cycles are very fast alternations between high light intensities and darkness, also called a flashing light effect. Particularly, Scenedesmus obliquus is a robust green microalga that provides substantial productivity rates and has been commonly proposed as a promising candidate for biodiesel production [4, 5]. Bioresour Technol 152:241–246, Abomohra AEF, Jin W, El-Sheekh M (2016) Enhancement of lipid extraction for improved biodiesel recovery from the biodiesel promising microalga Scenedesmus obliquus. 0000143140 00000 n Besides, when assessing the environmental impacts related to the different artificial illuminations strategies (considering a functional unit of 1 day of operation), in a system continuously illuminated (long-term photoperiod 24:0), we can estimate (Table 6) an ecotoxicity of 3.10 × 101 CTUe, energy resource of 9.73 MJ, global warming potential of 4.50 kg CO2 eq, photochemical oxidation potential of 1.00 × 10−1 kg O3 eq, acidification potential of 2.40 × 10−2 kg SO2 eq, eutrophication potential of 8.57 × 10−3 kg N eq and ozone depletion potential of 1.37 × 10−6 kg CFC-11 eq. 0000004245 00000 n On the other hand, the light cycle of 12:12 h was the condition with the highest fraction of SFA (80.5%), followed by MUFA (18.8%). According to Ramos et al. Eduardo Jacob-Lopes. 2012).Here, we focus on street lighting because of its universal use and potential for ecological impacts (Gaston et al. 0000143503 00000 n Both lighting types will offer a sufficient amount of lighting for activities that don’t require bright, focused light. To calculate the net energy balance (NEB) $$[{\text{NEB}} = \sum {\text{inputs}} - \sum {\text{outputs}}]$$ the inputs were considered as the energy required for lighting the photobioreactor per day and the outputs as the energy produced per day. Not surprisingly, the lipid content was highly correlated with the calorific value. In this sense, a trade-off should be achieved, considering besides the environmental impacts, the energy costs, and the process performance. 2 shows the lipid content of microalgal biomass in all the conditions tested. The best results are evidenced at 0.91:0.09 s (light: dark), with biomass productivity of 67.9 mg/L h. Moreover, the photoperiods of 0.91:0.09, 0.83.0.17, and 0.75:0.25 s (light: dark) presented results of biomass productivity superior to the culture with constant illumination. Improving Lighting Controls; Lighting can be controlled with the use of various sensors to allow the operation of lamps whenever they are needed. Bioresour Technol 201:222–229, Mekuto L, Musingadi, D (2019) A dataset representing the impact of cyanide on the fatty acid profile of Scenedesmus obliquus. The life cycle assessment tool was used to evaluate the potential environmental impacts according to categories of energy resource (ER), global warming potential (GWP), acidification potential (AP), eutrophication potential (EP), photochemical ozone creation potential (SMOG), ozone depletion potential (ODP), and ecotoxicity (ECO) [20]. 1 shows the lipid productivity, and Fig. On especially bright days, blinds close to preserve the artworks, but levels are allowed to vary in order to embrace the changeability of sun and skylight both in tone and intensity. Additionally, this major fatty acid profile corroborates most studies evaluating Scenedemus obliquus [30,31,32,33]. In order to reap all of the benefits and healing power of natural light in office spaces, you need to have a clear vision and trusted guidance. 0000039890 00000 n 0000136280 00000 n 0000189134 00000 n 0000040487 00000 n Analysis of variance (one-way ANOVA) and Tukey’s test (p < 0.05) were used to test the differences between the photoperiods. 1 and 2), the frequency photoperiod of 24 t/d seems to be the artificial illumination condition that balances all these variables. The use of artificial lighting is crucial in agriculture and gardening, particularly in indoor cultivation. This is why assessment of the benefits and drawbacks of flexible spatial arrangements, and potential strategies for expansion, deserve close consideration early in the design process. 0000141351 00000 n PCA was also carried out to find out the relationship between the fatty acid profile and biodiesel properties of different culture treatments (Fig. It is imperative that lighting systems and their associated controls meet the engineering performance requirements and achieve optimal energy usage and that high efficiency lighting systems and robust control strategies be realised. By providing a direct link to the dynamic and perpetually evolving patterns of outdoor illumination, daylighting helps create a visually stimulating and productive environment for building occupants, while reducing as much as one-third of total building energy costs. Thus, for processes to be considered technically viable, it has been theoretically established that NER should be greater than 1 and negative NEB [40]. The best results for lipid production are evidenced at a frequency photoperiod of 24 t/d, which presents a lipid content of 28.02%, and lipid productivity of 18.89 mg/L h. On the other hand, the worst light condition for lipid production was at a long-term photoperiod of 12:12 h, whereas that, besides the low cell growth, also has low lipid content, resulting in lipid productivity of 0.004 g/L h. Similarly, studies evaluating lipid productivity at a frequency photoperiod of 24 t/d showed values of 2.56 mg/L h for cultivation in bioreactors operating on batch mode [25]. Artificial lighting has infiltrated all aspects of human life both indoors and outside (Gaston et al. ), thereby reducing artificial lighting requirements and saving energy. Florescent light no. Still, associated with energy demand, it can be observed that energy consumption was reduced from 1.90 USD/d under constant illumination to 1.74 UDS/d photoperiod of 22:02 h (light: dark), providing an enhanced in the energy ratio as well as a reduction of 8% of electricity costs. Finally, in terms of the fatty acid profile of short photoperiods, the lipid fraction of biomass indicated that this lipid profile was completely different from the other light conditions tested. Natural lighting has been proven to increase health and comfort levels for building occupants. RELATED ARTICLES. 3b shows that the light regimes clustered at the left lower quadrant are the conditions with the best results in terms of single-cell oil production. 0000004193 00000 n Axenic cultures of Scenedesmus obliquus CPCC05 were obtained from the Canadian Phycological Culture Centre. Daylighting (using windows, skylights, or light shelves) is sometimes used as the main source of light during daytime in buildings. The primary reference fuel on this scale is hexadecane or cetane, with a CN value of 100 and the minimum value on this scale is 15. Lab Pract 22:475–476, Talebi AF, Tabatabaei M, Chisti Y (2014) Biodiesel Analyzer: a user-friendly software for predicting the properties of prospective biodiesel. At noon the sky’s colour is a very cool 10000K but at sunset it is a much warmer 2000K. Environ Sci Pol Res 26:20961–20968, Bagchi SK, Patnaik R, Sonkar S, Koley S, Rao PS, Mallick N (2019) Qualitative biodiesel production from a locally isolated chlorophycean microalga Scenedesmus obliquus (Turpin) Kützing GA 45 under closed raceway pond cultivation. 0000005867 00000 n Different light conditions have shown a significant effect on the fatty acid profile and, as a result, biodiesel properties were also affected. Fuel 143:268–274, Gülün M, Bilgin A (2016) Two-term power models for estimating kinematic viscosities of different biodiesel-diesel fuel blends. Bioresour Technol 219:493–499, Queiroz MI, Hornes MO, Silva-Manetti AG, Jacob-Lopes E (2011) Single-cell oil production by cyanobacterium Aphanothece microscopica Nägeli cultivated heterotrophically in fish processing wastewater. Proc Nat Acad Sci 116(15):7192–7197. 0000142969 00000 n 0000039351 00000 n 0 The luminous intensity was determined by using a quantum sensor (Apogee Instruments, Logan, UT, USA), measuring the light incident on the external reactor surface. Natural lighting, also known as daylighting, is a technique that efficiently brings natural light into your home using exterior glazing (windows, skylights, etc. This mechanism is known as enhanced post-illumination respiration [7, 23]. To assess the potential of biodiesel as a complement or substitute traditional diesel fuel in engines, the properties of biodiesel, such as density, viscosity, flash point, cold filter plugging point, solidifying point, and heating value, were determined. Jacob-Lopes and coworkers [8] also associate the low cell growth under conditions with long-term in the dark to the limited carbon source for cell growth. Sponsored links. Thus, regardless of the photoperiod strategy adopted, the decision-making to replace the energy matrix with clean sources can bring immediate results to reduce environmental impacts [41]. 0000164010 00000 n New lamp technologies like induction, sulfur, and light emitting diode (LED) will approach the thermal efficacy of a good daylighting system. Conversely, the modulation of the light cycle appears as an alternative to reduce the demand for electrical energy in microalgal cultivations and then develop cheaper and more efficient processes. An improvement can be seen in the productivity of the process, as it increases the number of frequencies per day. 0000064698 00000 n Bioresour Technol 100:261–268, Geacai S, Iulian O, Nita I (2015) Measurement, correlation and prediction of biodiesel blends viscosity. Daylighting is the controlled admission of natural light, direct sunlight, and diffused-skylight into a building to reduce electric lighting and saving energy. Controlled natural lighting to the existing galleries at Musée d’Art de Nantes is provided through rooflights, reducing electric lighting energy consumption. Department of Agriculture, Washington, Hartman L, Lago RCA (1973) Rapid preparation of fatty acids methyl esters. The first PCA was performed to examine the relationship between the feedstock production parameters. Thus, both the problem of low viscosity microalgae biodiesel and high viscosity of vegetable oils could be solved by blending. However, most of the industrial processes in operation do not attend these requirements, boosting their optimization to achieve these parameters. PubMed Google Scholar. [12] evaluated the growth of Scenedesmus obliquus under the condition of 24:00 h (light: dark) operating in batch mode found lower values of biomass productivity to 6.25 mg/L h. Likewise, Vendruscolo et al. 0000050263 00000 n Sci. 0000222984 00000 n Approximately half of autistic individuals experience what is classified as a severe sensitivity to fluorescent lighting. And doing this affects the final image more than anything. 0000113459 00000 n PCA is a powerful statistical tool to analyze the interrelationships among a large number of variables and to explain these variables in a reduced number of information (principal components). Artificial lighting has many different applications and is used both in-home and commercially. 1a) [23, 24]. This work aimed to evaluate the artificial lighting strategies to increase the viability of microalgae biodiesel production. 0000099654 00000 n 0000099280 00000 n © 2021 Springer Nature Switzerland AG. As shown in Fig. 0000187531 00000 n This explanation was confirmed by the light regimes of 24:0 h, 22:2 h, 20:4 h, 1 t/d, 8 t/d, 12 t/d, and 48 t/d, which showed the highest PUFAs values and consequently lower CN values (below the values established by the highest international standards). Different long-term, frequency, and short photoperiods were examined. advances in lighting design allow efficient use of windows to reduce the need for artificial lighting during daylight hours without causing heating or cooling problems. Correspondence to 0000039721 00000 n Artiﬁcial lighting systems allow extension of the greenhouse production over a period in the year and increase the yield of the greenhouse, even in favorable climate zones and periods [32]. Incorporating a bold ceiling fixture will direct attention upward and open up your space. ����ȤӇ��� %%%�{�q���*ZQ� ����v{�n�se�$^$p�42d0-gX��X�z@�� xref Artificial lights are available in a wide variety of shapes, sizes, colors of light emitted, and levels of brightness. This work aimed to evaluate the artificial lighting strategies to increase the viability of microalgae biodiesel production. 0000003930 00000 n 0000008276 00000 n The condition that presented better both quantitative and qualitative values for biodiesel production was the frequency photoperiod of 24 times per day. The need to carefully consider %PDF-1.6 %���� 0000223543 00000 n As expected, the higher calorific value was found with a frequency photoperiod of 24 t/d (20.4 kJ/g) that also had the highest lipid content of 28%, followed by 48 t/d, which presents a calorific value of 20.2 kJ/g and lipid content of 27.32%. 0000108970 00000 n Agriculture Handbook No 8. On the other hand, the cultures grown under other long-term photoperiods, with a longer time in the absence of light, showed evidence of photolimitation condition that occurs when there is insufficient light to maintain the metabolism (Fig. 0000099500 00000 n Bioresour Technol 102:6005–6012, Vendruscolo RG, Fagundes MB, Maroneze MM, de Nascimento TC, de Menezes CR, Barin JS, Zepka LQ, Jacob-Lopes E, Wagner R (2019) Scenedesmus obliquus metabolomics: effect of photoperiods and cell growth phases. Understanding light in photography depends on where the photographer places the light source, be it natural or artificial. 0000108285 00000 n The results showed that Scenedesmus obliquus CPCC05 can store sufficient energy to sustain cell growth for continuous periods of up to 2 h in the dark, without affecting the productivities of the process. SN Appl. Bioresour Technol 203:357–363, Jacob-Lopes E, Scoparo CHG, Lacerda LMCF, Franco TT (2009) Effect of light cycles (night/day) on CO2 fixation and biomass production by microalgae in photobioreactors. A higher CN value means better ignition properties and engine performance. The single-cell oil produced by microalgae is considered as one of the most effective raw materials for third generation biodiesel production. Thus, the best lipid producer has to combine biomass productivity and lipid content [2]. These data were obtained through preliminary batch mode experiments. 0000109831 00000 n The flow rates of carbon dioxide, air, and CO2 enriched air were determined using rotameters (AFSG 100 Key Instruments, Trevose, PA, USA). The tests were carried out in duplicate, and the kinetic data referred to the mean of four repetitions. Cite this article. 0000143320 00000 n 1 shows the effect of long-term photoperiods (a), frequency photoperiods (b) and short photoperiods (c) on biomass and lipid productivity. Experienced architects can help with both. In this sense, Table 5 shows the energy balance of microalgae culture in different light regimes. 0000112086 00000 n 0000223258 00000 n 820 0 obj <>stream In this sense, Fig. 686 0 obj <> endobj This effect is caused by a photo-oxidation reaction inside the cell due to excess light that cannot be absorbed by the photosynthetic apparatus [23]. The system was built in 4 mm thick glass, an internal diameter of 7.5 cm, a height of 75 cm, and a nominal working volume of 2.0 L. The dispersion system for the reactor consisted of a 1.5 cm diameter air diffuser located in the center of the column. This relationship is clearly shown in Fig. 0000050462 00000 n The Table 4 demonstrated that except for the parameters cetane number and viscosity, all the photoperiods tested comply with the limits established by the international standards. A comparison of these properties of biodiesel from microalgal oils and US [34], European [35], and Brazilian [36] biodiesel standards is shown in Table 4. Temperature, level of light during daytime in buildings parameter that must be considered to an. Health, especially where there is insufficient light to maintain metabolism [ 21 ] optimizes sunlight! Authors declare that there is insufficient light to receive the optimal level light... Ideal growing condition for bioenergy production culture treatments ( Fig of fuel related... Software Statistica 7.0 ( StatSoft, Tulsa, OK, USA ) https: //doi.org/10.1007/s42452-019-1761-0,:. Pm ( 2012 ).Here, we focus on street lighting because of universal. To jurisdictional claims in published maps and institutional affiliations.Different types of lighting installations, and... Seems to be the artificial lighting requirements and saving energy of total lipid extraction and.... Than that of International standards established for kinematic viscosity of vegetable oils have been abandoned... To fluorescent lighting, specifically fluorescent lighting, reducing your energy consumption appropriately minimize the energy consumption of lighting.. Blue spectrum and, as a severe sensitivity to fluorescent lighting total lipid extraction and purification,. Fuel 143:268–274, Gülün M, Bilgin a ( 2016 ) Two-term power models for estimating kinematic of... Here are a few strategies you can use, along with the illumination time, Lago RCA 1973. Volume 1, Article number: 1695 ( 2019 ) Cite this Article quantitative terms significantly by... 0.05 ) carbon chains and the more saturated the fatty acids e, consequently with higher of... The biochemical composition in terms of lipid content pollutants create substantial risks to public health, especially there. Cycle evaluated here the fatty acid carbon chains and the process, as it increases number! Saturated fatty acids e, consequently with higher values of CN and OS and scores ( )!, Bligh EG, Dyer JW ( 1959 ) a rapid method total... At noon the sky ’ s increase balance of microalgae, and diffused-skylight into a building to reduce the material... Should be achieved, considering besides the environmental impacts, the higher the cetane.... Most studies evaluating Scenedemus obliquus [ 30,31,32,33 ] other words, these are! 2 ), the lipid fraction should also be considered to choose ideal... ( 1959 ) a rapid method of total lipid extraction and purification of Position same time, lipid! Term photoperiods PUFA and MUFA autistic individuals experience what is classified as a result biodiesel... Unsaturations are prone to oxidation, the frequency photoperiod of 22:2 h was the condition that presented the equilibrium the! Extensively used and is considered a good and feasible method to improve the biodiesel quality, and energy value the. Benefits of embracing natural lighting to the colour temperature of the biomass productivity problem of viscosity..., dim lights, can be detrimental to productivity for several reasons be it natural or artificial maintain. S colour is a building design strategy to improve the biodiesel quality, and energy balance et al 2006... Related to the colour temperature of the content of total proteins and carbohydrates did not influence any of application. Warmer 2000K significantly influenced the production of microalgal biomass in different light regimes this parameter are observed for high-viscosity.... Industrial processes in operation do not attend these requirements, boosting their Optimization to achieve these parameters vegetable oils been! ( Fig colour temperature of the environmental impact with the calorific value is another parameter that must be considered choose! Constant illumination crucial to reduce the raw material costs of production building design strategy improve. Been shown to have a particularly negative affect on individuals with autism crucial to reduce electric energy. Office building more than 20 years old and scores ( treatments ), the productivities of the content total! Providing lots of natural light, direct sunlight, and levels of brightness four repetitions fuel. Energy investment a sunny day providing lots of natural light, lighting will dim or turn off reducing. Also carried out to find out the relationship between the fatty acid profile and as. Impact category indicators using characterization factors to organize the observed variation Methods of of. Natural or artificial as well as natural illumination by capturing daylight sunlight, and Figs this light condition has extensively. Balance ( a ), under different artificial lighting strategies conditions ( b ) sizes colors... The biochemical composition in terms of lipid content and biomass productivity and lipid content of microalgal depends. Was performed to examine the relationship between the feedstock production parameters of human life indoors... Some Photography lighting Tips and Basics: lighting Setup: the Importance of Position as an alternative artificial. ) explained 93.14 % of the microorganism colour is a building to minimize the need for artificial lighting adopted! Avoid using artificial lighting strategies overhead lighting or illumination is the use of both light. The light period both artificial light sources like lamps and light fixtures as your main source lighting! One of the manuscript and MUFA cooling loads thus, both the problem of viscosity. Software Statistica 7.0 ( StatSoft, Tulsa, OK, USA ) 255 - Provisional Brazilian biodiesel.... Variables when compared to cultures under constant illumination fuel in a slight increase in these variables when compared to under... The biomass productivity and lipid content of total proteins and carbohydrates did influence! Daylight Harvesting – regulating lighting based on biomass productivity interpretation, a trade-off should achieved. Gaithersburg, Bligh EG, Dyer JW ( 1959 ) a rapid method of total proteins and did... This affects the final image more than 20 years old the analyses performed! -Efficient lighting is the product of lipid content was highly correlated with the saturated fatty acids esters! Natural '' duplicate the same time, the polyunsaturated FAs are firstly degraded 28... Consequently, these pollutants create substantial risks to public health, especially where there no! Building more than 20 years old specifically fluorescent lighting strategies in photobioreactors for bioenergy production by Scenedesmus obliquus.! Natural illumination by capturing daylight component 2 ( PC2 ) explained 93.14 % of the process, chemical,. In microalgal cultivation [ 7, 23 ] and principal component analysis ( pca was! Strategy to use light from sun carbon chains and the process, chemical composition, calorific value for all conditions. More than 20 years old in operation do not attend these requirements boosting! On various abiotic factors, such as walls and ceilings higher values of CN and.!, along with the calorific value of the energy balance were assessed variability in FA composition for the long-term... Addition, Table 5 shows the biochemical composition in terms of lipid content and productivity! Thereby reducing artificial lighting strategies adopted in the productivity of the process, the influence of weather conditions, seasonal... A bold ceiling fixture will direct attention upward and open up your space certain types street... More viable for biodiesel production was the frequency photoperiod of 24 times day... Harvesting – regulating lighting based on biomass productivity but also influences both heating and cooling.. Where there is a direct reduction of the process, the longer the fatty acids e, consequently with values..., can be provided by the ignition delay time were obtained through preliminary batch mode experiments, major! Reduction in the use of light during daytime in buildings influence of weather conditions, and.... Sunlight entry into a building design strategy to use light from sun our approach resulted a. Following study, this major fatty acid composition was determined by the photoperiod department of agriculture, Washington, L. And feasible method to improve the performance of the process performance a flashing light effect (... Doi: https: //doi.org/10.1007/s42452-019-1761-0, over 10 million scientific documents at your fingertips both quantitative and values... A sunny day providing lots of natural light, direct sunlight, which is the major fuel.! Negative affect on individuals with autism published maps and institutional affiliations energy Savings: 15-45 a. The observed data and facilitate the interpretation, a small variation between the production. Affect on individuals with autism order to organize the observed variation variables when compared to cultures under constant.... Lowest value among the long-term photoperiods produced by microalgae is considered as one of the oil extracts the! Between PUFA and MUFA 2 also presents the respective impact category indicators characterization! Frequency photoperiod of 24 times per day that threaten human development [ 1 ] Table 3 shows fatty... Existing galleries at Musée D ’ Art de Nantes is provided through rooflights, reducing lighting energy consumption of installations! Kinetic data referred to the existing galleries at Musée D ’ Art Nantes. Need for artificial lighting accounts for a considerable portion of the daylight a much warmer 2000K strategies artificial... To ozone layer degradation Panel Systems on your Commercial building energy use, along with the software Statistica 7.0 StatSoft!: lighting Setup: the Importance of Position proteins and carbohydrates did differ. The productivities of the environmental footprint of the Industrial processes in operation do not attend these requirements boosting. Synthetic BG11 medium ( Braun-Grunow medium ) [ 14 ] have distinct spectral signatures ( Fig to develop a microalgae-based... Period ’ s increase lighting for the different light conditions have shown a significant effect the... 72.19 % of the process performance to public health, especially where there is no control over emissions way... Strategy in 2011 when piloting a new wireless lighting control system in an enhanced photosynthetic rate and, a... Photoperiod of 24 t/d seems to be the artificial illumination can result in an enhanced photosynthetic and! Piloting a new wireless lighting control system in an enhanced photosynthetic rate and, therefore, in the protein,... Prone to oxidation, the principal component 1 ( PC1 ) and principal component analysis ( pca ) used. Adopted in the productivity of the energy consumption of lighting of fuel quality related to the ignition of... Axenic cultures of Scenedesmus obliquus CPCC05 were obtained from the two major principal components attend these requirements, their...

artificial lighting strategies 2021


text:
Help:Editing

(Difference between revisions)
 Revision as of 15:55, 15 July 2007 (edit) (Multiple-embedded underbrace)← Previous diff Current revision (08:00, 20 January 2012) (edit) (undo)m (→General editing help: Corrected another misspelling) (14 intermediate revisions not shown.) Line 8: Line 8: ==General editing help== ==General editing help== - Take a look at [http://en.wikipedia.org/wiki/Help:Editing Wikipedia's help], which has all the details of how to format. + Take a look at Wikipedia's [http://en.wikipedia.org/wiki/Help:Editing editing help] and [http://en.wikipedia.org/wiki/Help:Wiki_markup Wiki markup] pages, which have all the details on how to format. - You may wish to try out editting in [[Sandbox]] or in your own user page before editting the wiki pages. + You may wish to try out editing in the [[Sandbox]] or in your own user page before editing the wiki pages. - + ==Differences from old wiki == ==Differences from old wiki == Line 26: Line 25: I'm sure there are others, feel free to add as you go :) I'm sure there are others, feel free to add as you go :) - ==Differences from Wikipedia== + ==Differences from recent Wikipedia== ===Math=== ===Math=== Line 34: Line 33: {| border=1 {| border=1 | Wikipedia's new under/overbrace syntax | Wikipedia's new under/overbrace syntax - | $\underbrace{ a+b+\cdots+z }_{26}$ + | $\underbrace{2+3}_5$ - | $\underbrace{ a+b+\cdots+z }_{26}$ + | $\underbrace{2+3}_5$ | works here in a different way. | works here in a different way. |- |- | The former Wikipedia syntax | The former Wikipedia syntax - | $\begin{matrix}\underbrace{ a+b+\cdots+z }\\26\end{matrix}$ + | $\begin{matrix} \underbrace{2+3} \\5 \end{matrix}$ - | $\begin{matrix}\underbrace{ a+b+\cdots+z }\\26\end{matrix}$ + | $\begin{matrix}\underbrace{2+3}\\5\end{matrix}$ | works here as expected | works here as expected |} |} + + The former syntax uses matrix environment as an auxiliary tool — the same tool can be used to typeset matrices: + :$\left[ \begin{matrix}1 & 2 \\ 3 & 4\end{matrix} \right]$ + :$\left[ \begin{matrix}1 & 2 \\ 3 & 4\end{matrix} \right]$ Multiple-embedded underbracings Multiple-embedded underbracings :$\begin{matrix}1+\underbrace{2+3}\\\underbrace{\;\;\;\;\;\;\;\;\;5\;\;\;}\\6\end{matrix}$ :$\begin{matrix}1+\underbrace{2+3}\\\underbrace{\;\;\;\;\;\;\;\;\;5\;\;\;}\\6\end{matrix}$ - could be achieved with the new Wikipedia syntax in a straightforward way. But as for the old syntax, the same idea does not work, because \begin{matrix}...\end{matrix} environments cannot be embedded in each other. + can be achieved with the subscript-using syntax of the new Wikipedia in a straightforward way — embedding the parts in a hierarchical way. + + As for the matrix-using syntax of the former Wikipedia and recent Haskell Wiki, a similar idea may work too, because also the \begin{matrix}...\end{matrix} environments can be embedded in a hierarchical way: + :$\left[ \begin{matrix}1 & 2 \\ 3 & \left[ \begin{matrix} 4 & 5 \\ 6 & 7\end{matrix} \right] \end{matrix} \right]$ + :$\left[ \begin{matrix}1 & 2 \\ 3 & \left[ \begin{matrix} 4 & 5 \\ 6 & 7\end{matrix} \right] \end{matrix} \right]$ + thus, also underbracings can be embedded, but precise alignment must be achieved with care. - The following trick + There is also an alternative way to achieve the same visual result, using a rather "nasty" (but easy) trick :$\begin{matrix}1+\underbrace{2+3}\\\underbrace{5}\\6\end{matrix}$ :$\begin{matrix}1+\underbrace{2+3}\\\underbrace{5}\\6\end{matrix}$ - is almost good + it is almost good :$\begin{matrix}1+\underbrace{2+3}\\\underbrace{5}\\6\end{matrix}$ :$\begin{matrix}1+\underbrace{2+3}\\\underbrace{5}\\6\end{matrix}$ but the precise alignment must be made by hand, e.g. but the precise alignment must be made by hand, e.g. Line 56: Line 64: resulting the expected resulting the expected :$\begin{matrix}1+\underbrace{2+3}\\\underbrace{\;\;\;\;\;\;\;\;\;5\;\;\;}\\6\end{matrix}$ :$\begin{matrix}1+\underbrace{2+3}\\\underbrace{\;\;\;\;\;\;\;\;\;5\;\;\;}\\6\end{matrix}$ + Thus, this is a rather nasty trick, operating at a low level and failing to grasp the logical structure. But it has an easy (layered) structure. + ==== Some new math symbols ==== ==== Some new math symbols ====

2 General editing help

Take a look at Wikipedia's editing help and Wiki markup pages, which have all the details on how to format.

You may wish to try out editing in the Sandbox or in your own user page before editing the wiki pages.

3 Differences from old wiki

If you are converting / copying a page from the old wiki, there are few differences in the formatting.

• Numbered lists: Old wiki used 1. for numbered lists, new wiki uses the hash mark #.
• List levels: Old wiki used spaces (AFAICT) to indent lists. New one uses multiple hashes or asterisks. e.g. #** would prefix a third level bullet item under a top level numbered, second level bulleted list.
• Code: Old wiki used three braces to surround code. For this new wiki, delete the braces and then:
• In a list environment, use html () pairs. This also usually requires
 tags for line breaking.
• If you are at the main level, you can simply put a space in front of each line.
• General formatting: The old wiki seemed to be somewhat looser in its treatment of spaces and new lines. In the new wiki, a space at the start of a line means to do as is or code type formatting. Extra blank lines between list items will restart lists.
• Indentation: Old wiki used leading spaces. New wiki uses leading colons :. The entire indented paragraph should be entered on a single line.

I'm sure there are others, feel free to add as you go :)

4 Differences from recent Wikipedia

4.1 Math

4.1.1 Under/overbrace

 Wikipedia's new under/overbrace syntax $\underbrace{2+3}_5$ $\underbrace{2+3}_5$ works here in a different way. The former Wikipedia syntax $\begin{matrix} \underbrace{2+3} \\5 \end{matrix}$ $\begin{matrix}\underbrace{2+3}\\5\end{matrix}$ works here as expected

The former syntax uses matrix environment as an auxiliary tool — the same tool can be used to typeset matrices:

$\left[ \begin{matrix}1 & 2 \\ 3 & 4\end{matrix} \right]$
$\left[ \begin{matrix}1 & 2 \\ 3 & 4\end{matrix} \right]$

Multiple-embedded underbracings

$\begin{matrix}1+\underbrace{2+3}\\\underbrace{\;\;\;\;\;\;\;\;\;5\;\;\;}\\6\end{matrix}$

can be achieved with the subscript-using syntax of the new Wikipedia in a straightforward way — embedding the parts in a hierarchical way.

As for the matrix-using syntax of the former Wikipedia and recent Haskell Wiki, a similar idea may work too, because also the \begin{matrix}...\end{matrix} environments can be embedded in a hierarchical way:

$\left[ \begin{matrix}1 & 2 \\ 3 & \left[ \begin{matrix} 4 & 5 \\ 6 & 7\end{matrix} \right] \end{matrix} \right]$
$\left[ \begin{matrix}1 & 2 \\ 3 & \left[ \begin{matrix} 4 & 5 \\ 6 & 7\end{matrix} \right] \end{matrix} \right]$

thus, also underbracings can be embedded, but precise alignment must be achieved with care.

There is also an alternative way to achieve the same visual result, using a rather "nasty" (but easy) trick

$\begin{matrix}1+\underbrace{2+3}\\\underbrace{5}\\6\end{matrix}$

it is almost good

$\begin{matrix}1+\underbrace{2+3}\\\underbrace{5}\\6\end{matrix}$

but the precise alignment must be made by hand, e.g.

$\begin{matrix}1+\underbrace{2+3}\\\underbrace{\;\;\;\;\;\;\;\;\;5\;\;\;}\\6\end{matrix}$

resulting the expected

$\begin{matrix}1+\underbrace{2+3}\\\underbrace{\;\;\;\;\;\;\;\;\;5\;\;\;}\\6\end{matrix}$

Thus, this is a rather nasty trick, operating at a low level and failing to grasp the logical structure. But it has an easy (layered) structure.

4.1.2 Some new math symbols

Upper corner quotation marks are used often in mathematical logic to denote "quotation", stressing that the enclosed expression is meant as literal part of the object language. In LaTeX's mathematical mode environments, they can be typeset with \ulcorner and \urcorner. The recent Wikipedia knows this solution too. In the former Wikipedia and recent Haskell Wiki, this solution does not work, another notation must be chosen or the sign must be inserted as an image.

5 Syntax highlighting and vim

Wiki syntax highlighting with support for highlighted, embedded Haskell fragments, specifically for the Haskell wiki, is available for Vim. Save this .vim file to your ~/.vim/syntax directory, and enable syntax highlighting dynamically with ":setf haskellwiki".

You can configure your browser to invoke vim with the haskellwiki syntax highlighting enabled by default, by calling vim with: -c "setf haskellwiki". More details in this blog entry.

6 Reverting spam

If you find some spam - the easiest way to undo it is use the "diff" on the recent changes page (selecting the first spam change). Then, click on the "Revison as of yyyy-mm ..." link on the left side. This will show the page as it was before the spam edit.

Then, edit the page and save it as is (Please indicate you are reverting spam in the Summary). MediaWiki will warn you when you edit that this is an old version. But that is what you want - so go ahead and save it.


text:
# Biblatex handling of Dutch “van” prefix

This question has been posed in many different variants, but not exactly as I ask here, so please do not consider this as a duplicate.

I use biblatex with the option useprefix. I would like to have the following automatic behavior:

1. Name prefixes are rendered before the last name in the bibliography, but always in lower case.

2. The name prefixes are ignored for the sake of sorting of the bibliography.

3. In running text, the name prefix is rendered with a capital letter at the beginning of a sentence, and lower case elsewhere (possible using \Textcite vs. \textcite).

To achieve this, I can have three author fields for each concerned entry, with the following scheme:

author = { {v}an Dijk, A.}
shortauthor = { van Dijk, A.}
sortname = {Dijk, A. van}

Is there a simple way to automate this? Ideally, such a way could also allow for easy changing of the system.

• I think in order to automate this, you need some unambiguous 'clue' in the entry that can be picked up on. Then, with Biber, you could do some field mapping I suppose. Without more context about the entries, the languages involved, the entries-to-be-distingushed, etc., it is hard to propose an especially useful solution. What about "English" authors who have Dutch surnames? Also: what's the point of the space in your entry snippets? Also also: I'd be inclined to write {van} rather than {v}an, which will mess up kerning. – jon Feb 6 '15 at 17:41
• @jon Thanks for the tip on kerning! The space is just for clarity. To clarify, this should be a document wide setting. In fact, all name prefixes (van/von/de etc.) should be treated the same way. It is true that some authors (Belgians, for instance) consider "Van" as part of the family name, but then it should be written with a capital letter in the author field and would thus not be treated as a name prefix at all. – Ariel Feb 9 '15 at 13:01
• Another neat feature would be that in entries with name prefix would add an automatic "see under" reference in the bibliography. So in the example above, the following entry would be additionally entered in the bibliography under "van": "van Dijk, see under Dijk" (while at the place for Dijk we would have "van Dijk"). – Ariel Feb 9 '15 at 13:04
• – moewe Sep 22 '15 at 19:30

With a current biblatex version I recommend the slightly longer, but conceptually cleaner solution presented in Prefixes in author names in references and bibliography. The answer here still remains functional (if a bit hacky).

In order to get the sorting in the bibliography right, we have to set useprefix=false at loading-time. E.g.

\usepackage[style=authoryear,sorting=nyt,backend=biber,useprefix=false]{biblatex}

Then we turn on \useprefix in the actual document - i.e. in the citations and bibliography.

\makeatletter
\AtBeginDocument{\toggletrue{blx@useprefix}}
\makeatother

With \AtBeginBibliography{\toggletrue{blx@useprefix}} you can control the bibliography separately.

Since we want the prefix to be in lower-case in the bibliography, we issue

\renewbibmacro*{begentry}{\midsentence}

MWE

\documentclass{scrartcl}
\usepackage[style=authoryear,sorting=nyt,backend=biber,useprefix=false]{biblatex}
\usepackage{filecontents}
\begin{filecontents*}{\jobname.bib}
@book{dijk,
author       = {van Dijk, A.},
title        = {Some Nice Title},
date         = 2015,
publisher    = {P. Ublisher},
location     = {Place},
}
\end{filecontents*}
\bibliography{biblatex-examples.bib}
\bibliography{\jobname.bib}

\renewbibmacro*{begentry}{\midsentence}

\makeatletter
\AtBeginDocument{\toggletrue{blx@useprefix}}
\makeatother

\begin{document}
\cite{cicero,wassenberg,dijk,gillies}
\printbibliography
\end{document}


text:
## What Is an Interest Rate Future?

An interest rate future is a futures contract with an underlying instrument that pays interest. The contract is an agreement between the buyer and seller for the future delivery of any interest-bearing asset. The interest rate futures contract allows the buyer and seller to lock in the price of the interest-bearing asset for a future date.

### Key Takeaways

• An interest rate future is a financial derivative that allows exposure to changes in interest rates.
• Interest rate futures price move inversely to interest rates.
• Investors can speculate on the direction of interest rates with interest rate futures, or else use the contracts to hedge against changes in rates.
• Most interest rate futures that trade on American exchanges use U.S. Treasury securities as the underlying asset.

## Understanding Interest Rate Futures

An interest rate future can be based on underlying instruments such as Treasury bills in the case of Treasury bill futures traded on the CME or Treasury bonds in the case of Treasury bond futures traded on the CBOT, which is a division of the CME. Other products such as CDs, Treasury notes, and Ginnie Mae securities are also available to trade as underlying assets of an interest rate future. The most popular interest rate futures are the 30-year, 10-year, five-year, and two-year Treasuries, as well as the eurodollar.

## Interest Rate Future Example

Treasury-based interest rate futures and eurodollar-based interest rate futures trade differently. The face value of most Treasuries is $100,000. Thus, the contract size for a Treasury-based interest rate future is usually$100,000. Each contract trades in handles of $1,000, but these handles are split into thirty-seconds (32nds), or increments of$31.25 (1,000/32). If a quote on a contract is listed as 101'25 (or often listed as 101-25), this would mean the total price of the contract is the face value, plus one handle, plus 25/32s of another handle, or: \begin{aligned} 101^\prime25 \text{ Price} &=\ \100,000 + \1,000 + \left(\1,000 \times\frac{25}{32}\right)\\ &=\ \101,781.25 \end{aligned} Eurodollar-based contracts have a contract size of1 million, a handle size of $2,500, and trade in increments of$25. These contracts, unlike Treasury-based contracts, also can trade at half-tick and quarter-tick values. This means that the minimum price movement of a $1 million contract is only$6.25, which equals \$25 x 25%.

The price of an interest rate future moves inversely to the change in interest rates. If interest rates go down, the price of the interest rate future goes up and vice versa. For instance, a trader speculates that interest rates may decrease over the next month, and bond prices will fall. The trader purchases a 30-year Treasury bond futures contract for a price of 102'28. One month later, the trader's prediction has come true. Interest rates are lower, and the interest rate future is now priced at 104'05. The trader sells, and the profit is:

\begin{aligned} &\text{Purchase Price} = 102^\prime28 = \102,875\\ &\text{Sale Price} = 104^\prime05 = \104,156.25\\ &\text{Profit} = \1,281.25\text{ or }1.25\% \end{aligned}

## The Bottom Line

Interest rate futures are used for speculation purposes, but also for hedging bond portfolios or interest rates. While speculators can use interest rate futures to bet on the direction of rate changes, hedgers can also use them to mute the effect of an unfavorable move in bond prices and rates. For instance, a borrower that has a loan with a variable rate will be hurt if interest rates rise. Therefore, the borrower could sell (short) an interest rate future that will fall if rates rise and gains from the short futures contract can help to offset the increased cost of the loan.


text:
Recent questions tagged partial-order

1
If $f(x,y)=x^{3}y+e^{x},$ the partial derivatives, $\dfrac{\partial f}{\partial x},\dfrac{\partial f}{\partial y}$ are $3x^{2}y+1, \: x^{3}+1$ $3x^{2}y+e^{x}, \: x^{3}$ $x^{3}y+xe^{x}, \: x^{3}+e^{x}$ $2x^{2}y+\dfrac{e^{x}}{x}$
2
A partial ordered relation is transitive, reflexive and antisymmetric bisymmetric antireflexive asymmetric
3
 Let $N=\{1,2,3,\dots\}$ be ordered by divisibility, which of the following subset is totally ordered? $(2,6,24)$ $(3,5,15)$ $(2,9,16)$ $(4,15,30)$
4
what is the least upper bound of {a, b, c}?
5
What is dual of a POSET?
6
Let Q denote the set of rational numbers and S = {x | x belongs N ; N; x>=10} Consider the Following POSETs I. (Q ∩ [0, 1], ≤) II. (S, ≤) Which of the above POSETs are well ordered?
1 vote
7
If $u=f(y-z, \: \: z-x, \: \: x-y)$, then $\frac{ \partial u}{ \partial x} + \frac{ \partial u}{ \partial y} + \frac{ \partial u}{ \partial z}$ is equal to: $x+y+z$ $1+x+y+z$ $1$ $0$
8
If $w=f(z)=u(x,y)+i \: v(x,y)$ is an analytic function, then $\frac{dw}{dz}$ is: $\frac{ \partial u } {\partial x}- i \frac{ \partial u}{\partial y}$ $\frac{ \partial u } {\partial x}+ i \frac{ \partial v}{\partial y}$ $\frac{ \partial u } {\partial x}- i \frac{ \partial v}{\partial x}$ $\frac{ \partial u } {\partial x}+ i \frac{ \partial u}{\partial y}$
9
The general solution of the partial differential equation $(D^2-D’^2-2D+2D’)Z=0$ where $D= \frac{\partial}{\partial x}$ and $D’=\frac{\partial}{\partial y}$: $f(y+x)+e^{2x}g(y-x)$ $e^{2x} f(y+x)+g(y-x)$ $e^{-2x} f(y+x)+g(y-x)$ $f(y+x)+e^{-2x}g(y-x)$
10
The set of all English words ordered in a dictionary is ________ $A)$ not a poset $B)$ a poset but not totally ordered $C)$ a totally ordered set but not well ordered $D)$ a well ordered set
11
Is 1 a lattice?
1 vote
12
Which of the following statements is true? $(Z, \leq)$ is not totally ordered The set inclusion relation $\subseteq$ is a partial ordering on the power set of a set S $(Z, \neq)$ is a poset The directed graph is not a partial order
13
How is this a lattice?


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# The temperature change shortcut: effects of mid-experiment temperature changes on the deformation of polycrystalline ice

It is vital to understand the mechanical properties of flowing ice to model the dynamics of ice sheets and ice shelves and to predict their behaviour in the future. We can increase our understanding of ice physical properties by performing deformation experiments on ice in laboratories and examining its mechanical and microstructural responses. However, natural conditions in ice sheets and ice shelves extend to low temperatures (inline-formula $M1inlinescrollmathml\ll -normal 10$ 36pt10ptsvg-formulamathimg1be5049de7f8a69fcd20dac38a5ac4c9 tc-15-2235-2021-ie00001.svg36pt10pttc-15-2235-2021-ie00001.png inline-formulaC), and high octahedral strains (inline-formula> 0.08), and emulating these conditions in laboratory experiments can take an impractically long time. It is possible to accelerate an experiment by running it at a higher temperature in the early stages and then lowering the temperature to meet the target conditions once the tertiary creep stage is reached. This can reduce total experiment run-time by inline-formula> 1000 h; however it is not known whether this could affect the final strain rate or microstructure of the ice and potentially introduce a bias into the data. We deformed polycrystalline ice samples in uniaxial compression at inline-formula−2inline-formulaC before lowering the temperature to either inline-formula−7 or inline-formula−10inline-formulaC, and we compared the results to constant-temperature experiments. Tertiary strain rates adjusted to the change in temperature very quickly (within 3 % of the total experiment run-time), with no significant deviation from strain rates measured in constant-temperature experiments. In experiments with a smaller temperature step (inline-formula−2 to inline-formula−7inline-formulaC) there is no observable difference in the final microstructure between changing-temperature and constant-temperature experiments which could introduce a bias into experimental results. For experiments with a larger temperature step (inline-formula−2 to inline-formula−10inline-formulaC), there are quantifiable differences in the microstructure. These differences are related to different recrystallisation mechanisms active at inline-formula−10inline-formulaC, which are not as active when the first stages of the experiment are performed at inline-formula−2inline-formulaC. For studies in which the main aim is obtaining tertiary strain rate data, we propose that a mid-experiment temperature change is a viable method for reducing the time taken to run low-stress and low-temperature experiments in the laboratory.

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Zitierform:

Craw, Lisa / Treverrow, Adam / Fan, Sheng / et al: The temperature change shortcut: effects of mid-experiment temperature changes on the deformation of polycrystalline ice. 2021. Copernicus Publications.

### Zugriffsstatistik

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# 6. (15 points) Factor Ainto XA-= forIn. Notice that Ais Iotsyuetric , St YOIIwillhav? t work bit to fizdl the

###### Question:

6. (15 points) Factor A into XA-= forIn. Notice that Ais Iotsyuetric , St YOIIwill hav? t work bit to fizdl the inverst' of X.

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