Mehdi Hage

M Hage-Hassan

New simple applications of symmetry and division algebra

published or not. The documents may come

New simple applications of symmetry and division algebra

II-Division algebras and religions

1-The definition of God in the Quran: "Who has never begotten and has not been begotten, and that no one is able to match!" a-So God is not composed of matter and he is Pure Imaginary "As time in relativity " b-The imaginary in division algebras are: Complex, Quaternion and Octonion. This is reflected in the Christian religion by: the father, the son and saint-spirit (3 in1).

2-Christ was a man so it is formed of matter and this means following the definition (2)

He is only a prophet.

3-In popular culture we thank God in the case of success but: a-In an interview a writer said that his son missed his exam. Because "God does not help him?" b-Einstein said that "God does not play with dice" to refute the probabilistic interpretation of quantum mechanics. "Undisputable and irrefutable proof! "

III-Symmetry in Arabic poetry

In Arabic language a lot of words have symmetry, for example: door ب( ا ,)ب bed ت( خ ,)ت etc. The generalizations of this symmetry to phrases and poems have been made since long time by several authors and poets.

Example:

مودته تدوم ل كل هول وهل كل مودته تدوم One of the pioneers of the coffee showed me a lovely poem that has a wide spread on social networking sites. This poem is composed of four lines and can be read horizontally and vertically. This poem is:

وهذا صديقي الوم محال يقال كالم احبه صديقي الجمال بليغ كالم هذا و خيال الجمال يقال محال

We write the representation matrix M4 of this poem by: M4 =

We therefore find a symmetric matrix, M4, with axis of symmetry the second diagonal.

By analogy with the Octonion we must find the representation matrix M8, "Work left to poets".

Annex: what is an atom?

One of the workers asked me what an atom is. I said, it was a man named proton and his wife named neutron and children their name Electron. At night, proton and neutron slept in a room called the nucleus and all the boys slept in a separate room around the nucleus, this is the atom. In short, the atom is made up of a nucleus consisting of set of protons and neutrons and around electrons.


Yoann Verger

First steps for a Sraan ecological economics

, as suggested by Vivian Walsh (2003).

Introduction

The idea of Sraan ecological economics advanced in Verger (2017b) combines Piero [START_REF] Sraa | Production of Commodities by Means of Commodities[END_REF] interpretation of classical political economy with ecological economics goals and principles. In a recent critique of this conception, [START_REF] Martins | The classical circular economy, sraan ecological economics and the capabilities approach[END_REF], although noticing that the idea of Sraan ecological economics is essentially based on the criticism of neoclassical position and neoclassical imperialism, focuses on the concept of classical subsistence wage and Standard commodity to show that a constructive stance can be build on Sraa's theory.

The nal outcome of Martins' (2018) comments is not so much a discussion of the idea of Sraan ecological economics, but rather a more specic thing, that is, the advancement of the connection between the idea of classical circular approach and the capability approach as originally proposed by Amartya [START_REF] Sen | Equality of what?[END_REF] and Martha [START_REF] Nussbaum | Women and Human Development: The Capabilities Approach[END_REF], following a line of research subsequently advanced by Vivian [START_REF] Walsh | Smith after sen[END_REF]2003), and that he has begun to develop in [START_REF] Martins | Sustainability economics, ontology and the capability approach[END_REF]2013;2016).

Here I shall elaborate further on the idea of Sraan ecological economics and its articulation with the capability approach, while providing a reply to [START_REF] Martins | The classical circular economy, sraan ecological economics and the capabilities approach[END_REF] critique of the use of Sraa's theory when advancing the idea of Sraan ecological economics in Verger (2017b). To do so, I rstly discuss two ideas that [START_REF] Martins | Ecosystems, strong sustainability and the classical circular economy[END_REF][START_REF] Martins | The classical circular economy, sraan ecological economics and the capabilities approach[END_REF] presents as constructive: Sraa's interpretation of the classical meaning of wages, and the Standard commodity as a common unit in terms of which biophysical processes can be compared in theory, while at the same time using it as a possible way to achieve eciency in the production of commodities [START_REF] Martins | The classical circular economy, sraan ecological economics and the capabilities approach[END_REF].

While I agree that the classical idea of a subsistence wage can be used in a constructive way, and especially in connection with the capability theory, I deny the fact that the Standard commodity could be used in the way Martins has suggested. The Standard commodity is an analytical tool that is part Sraa's theory of value, itself inspired by the classical theory of value of Ricardo and Marx. The goal of this theory of value is mainly critical: it is to show that the neoclassical distribution theory is not coherent, especially when it asserts that factor revenue should be linked with marginal productivity.

The discussion on these ideas also facilitates the clarication of the role that Sraan ecological economics can play within ecological economics. So besides replying to [START_REF] Martins | The classical circular economy, sraan ecological economics and the capabilities approach[END_REF], this is a good opportunity to provide some further elucidations on what can be a research program that would seek to use Sraa's theory to develop ecological economics. Therefore, after explaining that Sraa's ideology and epistemology should be claried before adapting them to ecological economics, I explain within which areas of economic and environmental analysis and management Sraan ecological economics should be developed. Finally, I present the case of hyper-subsystem, a concept rst introduced by [START_REF] Pasinetti | Growing subsystems, vertically hyperintegrated sectors and the labour theory of value[END_REF][START_REF] Pasinetti | Growing subsystems and vertically hyper-integrated sectors: a note of clarication[END_REF], as an analytical tool that can be used to integrate the capability approach into Sraan ecological economics, while addressing the environmental impacts of production.

2 On [START_REF] Martins | The classical circular economy, sraan ecological economics and the capabilities approach[END_REF] 2.1 On what I agree I shall refer, in this subsection, only at the points of agreement with [START_REF] Martins | The classical circular economy, sraan ecological economics and the capabilities approach[END_REF], leaving the points of disagreement to the next subsection. My points of agreement mainly are on two notions: the importance of the classical notion of subsistence in Sraa, and Martins' personal view on the development of the capability approach.

First, Martins rightly highlights the fact that Sraa conceives the more appropriate notion of wage as a wage composed by two separate parts, the subsistence wage: as consisting of specied necessaries determined by physiological or social conditions which are independent of prices or the rate of prots (Sraa, 1960, 44); and the surplus wage: besides the ever-present element of subsistence, [wages] may include a share of the surplus product (Sraa, 1960, 8).

Furthermore, Sraa asserts that, if the whole wage must be treated as a variable, as he does in his analysis, we could introduce a limit below which the wage cannot fall; a limit which would itself fall with any improvement in the method of production of necessaries, carrying with it a rise in the rate of prots and a change in the prices of other products (Sraa, 1960, 8).

Thus I agree with Martins that Sraa's revival of classical political economy presupposes an objective analysis of the physiological and social conditions that determine a certain standard of living [START_REF] Martins | The classical circular economy, sraan ecological economics and the capabilities approach[END_REF]. This objective analysis can help to dene, for each system of production, what should be the subsistence wage, leaving aside the question of what should be the level of the surplus wage. And to dene the subsistence wage, Martins advances that the capability approach is a promising pathway.

In this perspective, the subsistence wage should be the one that allows the purchase of a set of goods and services that are considered necessary to guarantee access to all the basic capabilities. The list of the basic capabilities that each one should have access to should be decided through a democratic process of decision informed both by universal principles about the basic capabilities that should be accessible to all humans, and by local cultural and technical features, as well as local preferences.

Martins indicates that the construction of universal principles should be informed by an ontology of Aristotelian potentials or causal powers enabled by underlying physiological and social structure [START_REF] Martins | The classical circular economy, sraan ecological economics and the capabilities approach[END_REF]. I agree with him that this appears to be a promising strategy, as well as I agree with his defense of [START_REF] Nussbaum | Tragedy and human capabilities: a response to vivian walsh[END_REF] when she asserts, against Sen, that it is possible to construct a universal list of basic capabilities that would be independent from local preferences (whereas Sen focuses more on the importance of liberty, suggesting that each community of agents should be left free to decide its own list of basic capabilities -see for instance Sen, 2013).

On what I do not agree

My main points of disagreement with [START_REF] Martins | The classical circular economy, sraan ecological economics and the capabilities approach[END_REF] refer to his use of the Standard commodity and to his apparent interpretation of Sraa's epistemic values.

Standard commodity and waste

The Standard commodity is a mixed commodity whose commodities are in the same proportion as the commodities in the surplus of the Standard system (Sraa, 1960, Chapter IV). The Standard system is an imaginary construction: from the real system, the processes are reduced or increased in size, with constant returns to scale, so that the commodities used as inputs are in the same proportions as the commodities used as outputs (we can call these proportions the Standard proportions).

Martins asserts that the Standard commodity and the Standard system could be used as analytical tools to investigate the environmental eciency of the system, and more precisely to diminish waste production:

the Standard commodity is suggested in [START_REF] Martins | Ecosystems, strong sustainability and the classical circular economy[END_REF] as a common unit in terms of which biophysical processes can be compared in theory, while at the same time using it as a possible way to achieve eciency in the production of commodities. This is so because in Sraa's Standard system, the proportions in which commodities are used as inputs are the same proportions in which they are used as outputs. If inputs are being produced exactly in the same quantities and proportions as they are used, it is possible to achieve a situation in which no waste is generated in the system, if the total quantity of commodities produced is equal to the total quantity of commodities used, since inputs are being exactly replaced by the outputs produced. This contributes to sustainability, even if it is not a sucient condition for it to exist.

...

The point emphasised in [START_REF] Martins | Ecosystems, strong sustainability and the classical circular economy[END_REF] when resorting to the idea of a Standard system and the Standard commodity is the need of achieving no waste in the production of reproducible commodities, that is, reproducing whatever inputs are used in the same proportions and quantities as they are used, which leads to no waste when the total quantities of outputs and inputs are equal.

...

In short, the Standard system is used to address the problem of waste, reducing the latter through the use of balanced proportions [START_REF] Martins | The classical circular economy, sraan ecological economics and the capabilities approach[END_REF].

Hence, Martins seems to say that if a system only produces what it is using (simple reproduction without surplus), the system will be in the Standard proportions and no waste will be produced. What I want to answer is that, while it is true that a system in a self-replacing state is a Standard system, this does not imply that no waste are produced, unless the system under consideration is the Earth.

First, a commodity is not a waste: when Martins says that the quantities of outputs will be exactly the same as the quantities of inputs, so that there is no surplus and no waste, he is talking (as Sraa does) about commodities. But during each process of production and consumption, wastes are produced (heat, gaseous emissions, tiny part of materials, scraped xed material, etc.) and they are not represented in the system of production of commodities. A clarication of the distinction between commodities and waste is needed.

1 A commodity is something that is sold to someone and, hence, receives a price. More precisely, a commodity is dened by Sraa as something produced by an industry and exchanged on a market for other commodities. A waste is something produced by an industry, but not exchanged on the market. Hence, a waste does not usually receive a price. Only if it is recycled, it can have a price, and can thus be represented in Sraa's system:

if the scrap (metal, timber, etc.) is interchangeable in use with some other material already accounted for, it simply assumes the price of the latter without need of an additional process; if it is not completely interchangeable (e.g. scrap iron as compared with pig iron), then there will be room for two processes, producing the same commodity (e.g. steel), but diering in the proportions in which they use the two types of material (Sraa, 1960, 74).

Otherwise, a waste does not receive a price and does not appear in Sraa's system. 2

Second, I understand that Martins wants to achieve a situation where all processes of production manage their wastes in such a way that no waste is released in the environment. But all processes of production and consumption, even if they are optimized to recycle the most of the waste produced, will generate some waste that will be lost in the environment. Hence, the only possibility to have no material waste in a system of production is to consider the whole Earth as the system of production. In this case, we would indeed obtain a system in standard proportions, as the Earth system is in a steady-state position from the material point of view (this could even be disputed, as some matter are transformed into energy in nuclear reactions, [START_REF] Georgescu-Roegen | Energy analysis and economic valuation[END_REF]. But in this case, although the theory is exact, the application of the theory would be meaningless, as it would mean that we are already in a sustainable system with no waste.

Epistemic values

This brings me to the second point of disagreement with Martins, his interpretation of Sraa's epistemic values. On this subject, Martins says that Sraa is measuring value at a theoretical level, in order to achieve an exact theory, and not at an empirical level. ... no economic model provides a complete description of the conditions for sustainability, not least because it is not possible to measure with mathematical exactness all the biophysical processes taking place at an empirical level. Hence, it seems to me that Martins suggests that to consider the whole Earth as a system of production would be valid from a theoretical point of view, even though it is not possible to apply the theory at the empirical level. The suggestion implies that a theory could be accepted as long as it is exact, although it cannot be applied at the empirical level. As a reply, 2 It should be added that the price of the service of waste removal should not be confused with the price of the waste (the removal of a waste can be costly, and thus the service of the waste removal can receive a price, if an industry sells this kind of service -see Verger, 2016).

I advance that this suggestion would be a bad interpretation of the epistemic values defended by Sraa. 3

Sraa has never explicitly written on the matter of epistemic values. He has nevertheless expressed some thoughts on the dierence between the theoretical level and the empirical level, and on the usefulness of theory, during a conversation with John Hicks at a conference in Corfu (1958, cf. Hague andLutz, 1961 for the proceedings):

Mr. Sraa thought one should emphasize the distinction between two types of measurement. First, there was the one in which the statisticians were mainly interested. Second there was measurement in theory. The statisticians' measures were only approximate and provided a suitable eld for work in solving index number problems. The theoretical measures required absolute precision. Any imperfections in these theoretical measures were not merely upsetting, but knocked down the whole theoretical basis. One could measure capital in pounds or dollars and introduce this into a production function. The denition in this case must be absolutely water-tight, for with a given quantity of capital one had a certain rate of interest so that the quantity of capital was an essential part of the mechanism. One therefore had to keep the denition of capital separate from the needs of statistical measurement, which were quite dierent. The work of J. B. Clark, Bohm-Bawerk and others was intended to produce pure denitions of capital, as required by their theories, not as a guide to actual measurement. If we found contradictions, then these pointed to defects in the theory, and an inability to dene measures of capital accurately. It was on this -the chief failing of capital theory -that we should concentrate, rather than on problems of measurement.

Professor Hicks was not quite clear about this. Did Mr. Sraa mean to equate models with theories? He could see that in a particular model one could only make that model water-tight by introducing drastic simplications. Only thus, for example, could one have a clear and precise denition of capital stock. But some simplications were so drastic that he himself was simply not interested in any theory based on them.

Mr. Sraa replied that Wicksell's might be a simple model in that he worked out a simple and general theory for future development.

Surely the usefulness of any theory lay in its explanatory value. Was one only interested in a theory if one could t actual gures into it;

or was one interested independently of that?

Professor Hicks argued that if a theory was to explain the working of the social mechanism, it ought to be capable of having measurable concepts tted into it.

Mr. Sraa took the view that if one could not get the measures required by the theorists' denitions, this was a criticism of theory, which the theorists could not escape by saying that they hoped their theory would not often fail. If a theory failed to explain a situation, it was unsatisfactory. (Hague and Lutz, 1961, pp. 305 -306).

It could be that the discussion between Hicks and Sraa was not recorded ex- Going back to my discussion on [START_REF] Martins | The classical circular economy, sraan ecological economics and the capabilities approach[END_REF], I argue that if someone, in order to have a theory of economic-environment interactions that is exact, says that we should implement in it all the existing natural and economic processes, then the theory will be exact but not useful, as no-one will be able to use it for empirical purposes. It is true that it is not easy to apply Sraa's theory at the empirical level, because the number of commodities and the number of processes required to produce these commodities are important. But at least each commodity sold is usually recorded in an accounting book, and most of the countries have reliable national accounting of commodities produced and sold. This is not the case for all wastes produced in the industrial system (although some physical accounts are starting to be implemented, taking explicitly into account waste -see Fischer-Kowalski and Haberl, 2015 for an overview) and more importantly, this is not the case for all the natural processes, in the whole world.

I must nally say that these points of disagreement are minor, and do not overcome the fact that Sraan ecological economics should be articulated with the capability approach as developed by [START_REF] Martins | Sustainability economics, ontology and the capability approach[END_REF]2013;2016;[START_REF] Martins | The classical circular economy, sraan ecological economics and the capabilities approach[END_REF]. But rst Sraan ecological economics must be more precisely dened, so I turn now to the work that must be done in this direction.

Sraan ecological economics

In this section I will describe the relevant research pathway that will help to First results can be estimated from the published and unpublished writings of Sraa and its conferences talks. objectivist approach is also a fact that supports the articulation between Sraan ecological economics and the capability approach.

The second line of research aims at constructing an epistemological, ethical, and ontological proposal that could be the basis for the development of Sraan ecological economics. The hypothesis underlining this line of research is that heterodox economics should base their criticism of the dominant theory, not only referring to logical and rational arguments, but also referring to concurrent ethical and political projects. This means that a concurrent ideology should be constructed in order to revive heterodox economics [START_REF] Combemale | L'hétérodoxie encore : continuer le combat, mais lequel ?[END_REF]. In this respect, the goal will be to analyze and try to reconcile in a relevant way the results of the rst line of research with the current attempts to reconstruct a coherent set of epistemological, ethical, and ontological positions for the ecological economics eld [START_REF] Martins | Sustainability economics, ontology and the capability approach[END_REF][START_REF] Spash | New foundations for ecological economics[END_REF]. I believe a rst direction in this line of research would be to investigate the political neo-realist approach developed by [START_REF] Amable | A neorealist approach to institutional change and the diversity of capitalism[END_REF], which is underpinned by the philosophical and political analyses of [START_REF] Gramsci | Quaderni del Carcere[END_REF][START_REF] Gramsci | Quaderni del Carcere[END_REF] and [START_REF] Castoriadis | L'institution Imaginaire de la Société[END_REF]. Unfortunately I do not have the possibility to develop further in this article this point of view, and I must now turn to the tools that could be integrated and developed within Sraan ecological economics.

Relevant areas of analysis

The third line of research needs to show the relevant research areas for a Sraan ecological economics. Ideas will emerge from the two rst lines of the research, but three directions could already be pointed at.

First a reection on the property regimes that should be attached to natural resources and wastes should be attempted, in relation to the question of their valuation and the rents that can be earned through their possession. The rst attempts to use Sraa in order to solve environmental questions usually concluded that natural resources and wastes should be included in the market of commodities to be managed in an ecient way (see Verger, 2016Verger, , 2017b)). My hypothesis is that a solution for their sustainable management which would be more in accordance with Sraa's ethical, ontological and political standpoint should avoid giving an exchange value to nature. For instance, the natural resource may be managed under the regime of common property, with rules of sustainable use that do not imply exchange of money between the users.

A second direction is pointed at by some authors in ecological economics [START_REF] Ballet | A note on sustainability economics and the capability approach[END_REF][START_REF] Ballet | Environment, justice and the capability approach[END_REF][START_REF] Martins | The place of capability approach within sustainability economics[END_REF] and concerns the capability theory. My hypothesis is that the capability approach would t well in Sraan ecological economics, although an important work has to be made to make the approach operational, especially if we want to include environmental issues. The main question is: how can we control the economy so that it would produce the required set of basic capabilities, while not endanger the environment? Hence, this direction of research should be concerned with the construction of tools and indicators that will help to solve this question.

The third direction concerns ecological macroeconomics. Macro-ecological economics is a little eld, with few models, and only one which explicitly refers to Sraa (Kemp-Benedict, 2014) 5 : all of them use either neoclassical or post- Keynesian models when they represent the production of goods and the distribution of surplus. It is to be noticed that the most promising path for developing ecological macroeconomics is described by [START_REF] Hardt | Ecological macroeconomic models: assessing current developments[END_REF] in this way:

Although current models are still at an early stage of development, the combination of environmentally-extended input-output analysis and stock-ow consistent modelling stands out as a promising avenue for integrating concerns about ecological impacts and nancial stability.

They also point to the most important questions to be answered by such models:

One very important question, as highlighted in the introduction to this study, is how to achieve nancial stability while decreasing consumption. Another important question is how a shift to lowerproductivity sectors would aect incomes and inequality, or how the necessary investment for a post-growth transition can be realised without restarting the engine of economic growth itself [START_REF] Hardt | Ecological macroeconomic models: assessing current developments[END_REF].

I believe that Sraa's theory can be used to solve these questions. The interest of Sraa's theory is that it is a theory of value based on input-output analysis that allows the construction of rigorous macro-economic accounting tools. But two problems needs to be handled rst. To begin with, Sraa's analysis is static, so there is no possible discussion on the coordination of agents' decision and possible equilibrium. To render its static analysis into a dynamic one, one could start with the work of [START_REF] Pasinetti | Growing subsystems, vertically hyperintegrated sectors and the labour theory of value[END_REF][START_REF] Pasinetti | Growing subsystems and vertically hyper-integrated sectors: a note of clarication[END_REF][START_REF] Pasinetti | Structural economic dynamics: a theory of the economic consequences of human learning[END_REF]. An example of a development following Pasinetti's approach is given in the following section.

Second, money is not considered in Sraa's system. As Sraa is only looking at one point of time, there is no need for money, or rather money would be only neutral in his system (it is obtained by selling commodities, and it is only used to buy commodities; there is no buying or savings of money taken into consideration). Of course, if dynamics are introduced, money and a nancial system should be introduced next.

Once the research questions are dened, appropriate tools for theoretical and empirical analysis can be designed. In the next section, a tool is presented that 5 [START_REF] Kemp-Benedict | The inverted pyramid: A neo-ricardian view on the economy-environment relationship[END_REF] tries to express absolute rents on natural resources and to develop physical indicators of the eciency of a system of production, by using a vertical integration analysis [START_REF] Pasinetti | The notion of vertical integration in economic analysis[END_REF]. He attempts to express the part of GDP which is based on the use of environmental resource, as he tries to reveal Daly's inverted pyramid (natural resources extraction accounts for only 5% of the global GDP but actually all production is directly or indirectly related to this initial extraction, [START_REF] Daly | Consumption and welfare: Two views of value added[END_REF]. However, my point of view is that he is not revealing the part of GDP related to the extraction of natural resources, because extracting natural resources is a necessary condition, in his system, to produce economic values. If there would not be any extraction, then GDP would be equal to zero. What he is actually revealing in his analysis is by how much the value of GDP increases when rents on natural resources are added.

can be used to link ecological macroeconomics with the capability approach:

the concept of hyper-subsystem.

4 An example of a Sraan ecological economics tool of analysis Here I will now present an example of how Sraan ecological economics could be developed, by introducing Pasinetti's concept of hyper-subsystem and linking this concept with the management of waste, the management of natural resources, and the capability approach (the latter connection -between Pasinetti's and the capability approach -has been suggested by [START_REF] Walsh | Sen after putnam[END_REF]).

Pasinetti's concept of hyper-subsystem [START_REF] Pasinetti | Growing subsystems, vertically hyperintegrated sectors and the labour theory of value[END_REF][START_REF] Pasinetti | Growing subsystems and vertically hyper-integrated sectors: a note of clarication[END_REF][START_REF] Pasinetti | From sraa: backwards, to a better understanding of marx's values/prices "transformation"; and forward, to a novel view of growing economic systems[END_REF][START_REF] Garbellini | Productivity accounting in vertically (hyper-)integrated terms: Bridging the gap between theory and empirics[END_REF]) is a theoretical renement of Pasinetti's previous concept of vertically integrated sector [START_REF] Pasinetti | The notion of vertical integration in economic analysis[END_REF].

This older approach was inspired by the notion of sub-system presented by Sraa (Sraa, 1960, Appendix A). Pasinetti showed that a system of production could be decomposed into vertically integrated sectors. Each sector produces only one commodity -either a capital good or a nal consumption good -and is composed of the requirements in terms of labor and commodities for the production of this commodity at a given period (direct requirements), and for its reproduction during the next period (indirect requirements).

The new hyper-subsystem approach is used to investigate the direct, indirect, and what Pasinetti calls hyper-indirect requirements, for the production of one consumption good at a given period (direct requirements), for its reproduction during the next period (indirect requirements), and for the change in the rate of consumption of this good at the next period (hyper-indirect requirements). The main dierence between the two approaches is that in the latter, each hyper-subsystem produces only nal consumption goods, so that they are totally independent from one another. 6 Their dynamics depend only on the growth rate of the consumption of the good they produce. On the contrary, vertically integrated sectors are not independent, as an evolution in the rate of consumption of a good i will impact all sectors producing capital goods that are required for the production of this good i [START_REF] Garbellini | Productivity accounting in vertically (hyper-)integrated terms: Bridging the gap between theory and empirics[END_REF].

The previous approach of vertically integrated sectors has already been used in ecological economics. After reviewing this literature, I will show how the use of the more recent approach of hyper-subsystems improves these analyses.

4.1 Review of literature [START_REF] Alcántara | Economía y contaminación atmosférica: Hacia un nuevo enfoque desde el análisis input-output[END_REF] seems to be the rst to use the vertically integrated approach for the analysis of pollution. His approach is simple: vertically integrated coecients of pollution can be constructed just as vertically integrated coecients of labor, once we know the pollution produced per unit of output by each industry.

Sánchez-Chóliz and Duarte ( 2003) use this approach to analyze the pollution associated with the demand of one sector of production (which produces either consumption or capital goods). [START_REF] Alcántara | Input-output subsystems and pollution: An application to the service sector and co2 emissions in spain[END_REF] use the same approach to study the CO 2 emissions of the service sector in Spain and Butnar and Llop (2011) pursue their analysis with a study of the determinants of a change in CO 2 emissions, structural decomposition. Structural decomposition to understand the causes of change of atmospheric pollution from industry in Spain has been also investigated by [START_REF] Roca | Income growth and atmospheric pollution in spain: An input-output approach[END_REF], using the NAMEA input-output accounting.

In Sanchez-Choliz and Duarte ( 2005), a slightly aggregated model is presented, each sector being composed of several dierent industries with dierent processes. They investigate changes in productivity and nal demand, guring how these changes impact the vertically embodied pollution of each sector. Furthermore, assuming constant returns to scale, they try to optimize the system depending on value added, pollution, and employment. They try to dierentiate consumer responsibility from producer responsibility (this distinction is also implemented in [START_REF] Cadarso | Co2 emissions of international freight transport and oshoring: Measurement and allocation[END_REF]. Finally Cadarso et al. ( 2012) study the impact of international trade on the responsibility of pollution. They decide to account for the responsibility of the pollution depending on the share of net product value -i.e. added value -that goes to each country.

We should note that one of the problem highlighted by Sanchez-Choliz and Duarte ( 2005) is that investments are mixed with nal consumption in their account of nal demand. We actually nd this problem in all the literature.

Anticipation of changes in the structure of nal consumption is not considered, whereas the investments required by this changing structure may actually explain a large part of the residual that appears in structural decomposition analyses made by [START_REF] Roca | Income growth and atmospheric pollution in spain: An input-output approach[END_REF][START_REF] Roca | Income growth and atmospheric pollution in spain: An input-output approach[END_REF][START_REF] Butnar | Structural decomposition analysis and input-output subsystems: Changes in co2 emissions of spanish service sectors (2000-2005)[END_REF][START_REF] Butnar | Structural decomposition analysis and input-output subsystems: Changes in co2 emissions of spanish service sectors (2000-2005)[END_REF]. Hence, we can say that the dierent models presented in the literature are not really dynamic. They compare dierent static states, without properly taking into account how to manage the transition between those states.

If we want to take into account for each subsystem the evolution of nal consumption and thus the investments required by its evolution, we should adopt the hyper-subsystem approach developed by [START_REF] Pasinetti | Growing subsystems, vertically hyperintegrated sectors and the labour theory of value[END_REF][START_REF] Pasinetti | Growing subsystems and vertically hyper-integrated sectors: a note of clarication[END_REF], where dierent rates of growth are attributed to each hyper-subsystem. In this way, the evolution of the pollution will be more precisely attributed to each component of nal demand, at each period of production. Secondly, it will also prove useful for the integration of waste productions and natural resource needs, and for the study of the evolution of these productions and needs at each periods, relating them precisely with each component of nal demand. In this way, we will move from static comparison to dynamic analysis.

Hyper-subsystems

Here I will follow the clear formulation given by [START_REF] Pasinetti | Growing subsystems, vertically hyperintegrated sectors and the labour theory of value[END_REF]. Consider the rate of growth of the nal consumption of one commodity i as:

g i (t) = c i (t + 1) -c i (t) c i (t) (1)
With c i (t) the level of nal consumption of commodity i at period t.

Our system of production of n commodities by means of commodities and labor organized within n industries can be described like this:7

A (t) x (t) + c (t) + A (t) g i (t) x (i) (t) = B (t) x (t) (2)
a l (t) x (t) = L (t)

(3)

A (t) x (t) = s (t) (4) x (i) (t) = [B (t) -(1 + g i (t)) A (t)] -1 c (i) (t) (5)
s (i) (t) = A (t) [B (t) -(1 + g i (t)) A (t)] -1 c (i) (t) (6) L i (t) = a l (t) [B (t) -(1 + g i (t)) A (t)] -1 c (i) (t) (7)
c (i) (t) = c (t) ; x (i) (t) = x (t) ; s (i) (t) = s (t) ; L i (t) = L (t) (8)
Where c (i) (t) is a column vector the n components of which are all zeros except the i th one8 . A and B are respectively the input and output coecient inter-industry matrices, a l is the input labor coecient row vector, x (t) is the industry intensity level column vector (i.e. the intensities at which the respective activities are carried on), c (t) the nal consumption column vector, s (t) the means of production column vector, L (t) the quantity of labor. The physical units are chosen so as to make the main diagonal of the output coecient matrix B a list of ones.

A unit of vertically hyper-integrated productive capacity for commodity i is dened as the i th column m i (t) of matrix M (i) (t), this latter matrix being dened in this way:

M (i) (t) = A (t) [B (t) -(1 + g i (t)) A (t)] -1 (9)
The unit of vertically hyper-integrated productive capacity represents the commodities needed to maintain and expand the productive capacity of the nal consumption commodity i, i.e. the commodities needed to produce one unit of commodity i at period t the investments needed to produce 1 + g i (t) units at period t + 1. We can note that m i (t) = s (i) (t)when c i (t) = 1.

Similarly, we can dene a vertically hyper-integrated labor coecient for commodity i as the i th component l i (t) of row vector l (i) (t), this latter vector being dened in this way:

l (i) (t) = a l (t) [B (t) -(1 + g i (t)) A (t)] -1 (10)
As for the price system, if we assume a uniform rate of prots π (t), we have:

p (t) A (t) + p (t) A (t) π (t) + a l (t) w (t) = p (t) B (t) (11)
With p (t) the price row vector and w (t) the wage rate. If we assume nonuniform rates of prots, with π i (t) the rate of prots of industry i, we have:

p (t) A (t) + p (t) A (t) Π (t) + a l (t) w (t) = p (t) B (t) (12)
Where Π (t) is the diagonal matrix with the dierent rates of prots in the main diagonal.

Use for managing pollution

A matrix of pollution per industry, with one row for each dierent kind of pollution, can be easily added, so that the pollution related to the nal consumption of one good and the investments required by the evolution of its nal consumption, can be precisely described.

Consider z ij (t) as a pollution coecient describing the amount of pollution i emitted during the process of production of one unit of commodity j, and a pollution matrix Z (t) with the same number of columns n than matrices A and B (it may have a dierent number of rows). Then the pollution directly, indirectly and hyper-indirectly caused by the production of the quantity c i of commodity i can be represented in this way:

z (i) (t) = Z (t) [B (t) -(1 + g i (t)) A (t)] -1 c (i) (t) (13)
We can thus dene a unit of vertically hyper-integrated pollution z i (t) as the vector describing the amounts of pollution emitted during the process of production of one unit of consumption good i and the investments required by its consumption growth rate. We dene it in this way: z i (t) = z (i) (t) when c i (t) = 1.

On the value side, if a tax on the emission of a pollution is implemented and that u (t) is a vector summarizing all the taxes per emission, then the value system will be (if the rate of prots is uniform and if the taxes are paid at the end of the period of production):

p (t) A (t) + p (t) A (t) π (t) + a l (t) w (t) + u (t) Z (t) = p (t) B (t) (14)
And if the rates of prots π i (t) are not uniform, the value system can be presented in this way:

p (t) A (t) + p (t) A (t) Π (t) + a l (t) w (t) + u (t) Z (t) = p (t) B (t) (15)
4.4 Use for managing natural resource extraction

In the same way, a matrix of natural resource extraction per industry, with one row for each dierent kind of natural resource, can be added very easily.

Consider e ij (t) as a natural resource extraction coecient describing the amount of natural resource i that has to be extracted for the production of one unit of commodity j, and a natural resource extraction matrix E (t) with the same number of columns n than matrices A and B (it may have a dierent number of rows). Then the natural resources directly, indirectly and hyperindirectly required by the production of the quantity c i of commodity i can be represented like this:

e (i) (t) = E (t) [B (t) -(1 + g i (t)) A (t)] -1 c (i) (t) (16)
We can thus dene a unit of vertically hyper-integrated natural resources extraction e i (t) as the vector describing the quantities of natural resources extracted to produce one unit of consumption good i and the investments required by its consumption growth rate. We can dene it in this way: e i (t) = e (i) (t) when c i (t) = 1.

On the value side, if a rent on the extraction of a natural resource is implemented and that r (t) is a vector summarizing all the rents per natural resource extraction, then the value system will be (if the rate of prots is uniform and if the rents are paid at the end of the period of production):

p (t) A (t) + p (t) A (t) π (t) + a l (t) w (t) + r (t) E (t) = p (t) B (t) (17)
And if the rates of prots π i (t) are not uniform, the value system can be represented in this way:

p (t) A (t) + p (t) A (t) Π (t) + a l (t) w (t) + r (t) E (t) = p (t) B (t) (18)

Use for capability

Finally, following the suggestion made by [START_REF] Martins | Ecosystems, strong sustainability and the classical circular economy[END_REF][START_REF] Martins | The classical circular economy, sraan ecological economics and the capabilities approach[END_REF], we could link Sraan ecological economics and the capability approach by dening a vector of subsistence commodities d (t), that would provide each member of a population N (t) the access to a specic standard of living, which is socially dened as the minimum standard. This vector will connect the classical concept of subsistence wage and Nussbaum's concrete view on capabilities, where the focus is on objective functionings to be achieved by means of a nite quantity of commodities.

From our formal system, we can know the value of the subsistence wage:

r min (t) = p (t) d (t) (19)
We can also compute the part of the subsistence wage in the national revenue:

C basic (t) = N (t) p (t) d (t) (20)
And nally we can dene the minimum wage rate (or wage per hour) below which no one will be able to earn the subsistence wage at the end of the year:

w basic (t) = N (t) p (t) d (t) L (t) (21)
To sum up, from the technologies of production, the industry intensity levels, the rates of prots in the dierent industries, and the way the national wage is shared in the dierent industries, we can know the prices and the wage rate.

Hence, if we can dene a set of subsistence commodities, we will nd their prices at the current level of distribution. We will be also able to dene the minimum wage per capita and per year, at this level of distribution, as well as the minimum wage rate. With this, we can know if enough subsistence commodities are produced to guarantee the access of each capability for all, and, knowing the wage earned by each individual, we can know the number of individuals that cannot aord all the subsistence commodities, and must make dramatic choices [START_REF] Walsh | Sen after putnam[END_REF].

If we want to avoid dramatic choices, several options are possible: we can change the distribution of the national revenue so that each individual receives at least the subsistence wage, or we can increase production so that enough subsistence commodities are produced for everybody, or we can decrease the cost of production of the subsistence commodities by technological change. In order to decide between these options, it may be interesting to know the emissions linked with the production of the subsistence commodities and the evolution of their consumption, as well as the extraction of natural resources needed to produce this set and to increase or decrease its production. They can be calculated in the following way:

e basic (t) = N (t)

n i=1 E (t) [B (t) -(1 + g i (t)) A (t)] -1 d (i) (t) (22)
z basic (t) = N (t) The hyper-subsystem approach is thus a tool showing that Sraan ecological economics can have a constructive side. It furthermore reveals that the capability approach can be included in the development of this constructive side.

n i=1 Z (t) [B (t) -(1 + g i (t)) A (t)] -1 d (i) (t) (23
Sraan ecological economics, if it proves itself useful as an heterodox school, should also develop its critical side. But the hegemony of neoclassical theory will not be challenged by criticism only. The construction of an epistemological, ethical, and ontological proposal, linked with the development of concepts and tools useful for theoretical and empirical studies is thus a really important part of a program that would seek to develop Sraan ecological economics.

Martins ( 2018) is thus right when he criticizes Verger (2017b) by saying that the focus in this article was too much on the critical side of Sraa's theory, overlooking Sraa's objective approach that can lead to interesting constructive ideas. However, his use of the concepts of Standard commodity and Standard system does not seem to me to be appropriate, as well as his suggestion that some developments may only be of theoretical interest, while not being of any practical use. I believe that the Standard system and the Standard commodity are currently more useful on the critical side of Sraan ecological economics and I hope to have shown that Sraa believed that a good theory was a one that could be used on empirical level.

It is important to note that this article provides only the rst steps for a long research project. I do believe that the journey must be done, even though I do not know where it will lead us.

  construct a coherent and useful description of what should be Sraan ecological economics. The research on this development should be conducted along three lines. Following what Spash (2012) tried to do for ecological economics, it is important, rst, to do an historical work on the epistemological, ethical, and ontological positions of Sraa that underline and shed light on his theory of value and, second, to engage in the development of an epistemological, ethical, and ontological position that would be the basis for Sraan ecological economics.By doing this, the understanding of the relevant areas of environmental and economic analysis on which the development of Sraan ecological economics can shed a new light will be improved. Once this is done, it will be possible to develop a coherent corpus of environmental policy tools for Sraan ecological economics, and I will give an example of one of these tools in the next section.3.1 Epistemology, ethics and ontology of Sraa and Srafan ecological economicsThe rst line of the work should be directed at understanding the meaning of Sraa's equation from an epistemological, ethical, and ontological point of view. What was Sraa aiming at? Was he only looking at correcting logical mistakes in the development of economic theory, or was his theory grounded on a completely dierent standpoint than the one of the neoclassical theory? The goal is to have a better understanding of his theory, especially on the question of the connection between theory and empirical facts (e.g. the empirical relevance of the uniform rate of prots in the equations, the meaning of the photograph metaphorVerger (2017a) or the importance of natural resources and their possible depletion in his model).

  of hyper-subsystems, we are more precisely allocating pollution and natural resource extraction to each component of nal demand, by taking explicitly into account the investments required by the evolution of nal demand. But this does not mean that we are taking all environmental problems into consideration. In particular, pollution during the process of consumption is not considered (private house consumption of energy and water, or private car atmospheric pollution, etc.). Only the responsibility of the production side is considered. Nevertheless, this investigation can facilitate the construction of scenarios on the evolution of the production process or in the composition of demand in order to increase the universal access of basic capabilities, while decreasing our impact on the environment.

  actly, but what I understand from the record is that Sraa defends two epistemic values for a theory: precision and explanatory value. If the theory cannot precisely dene a concept, for instance in that case capital (how to measure capital, on which unit), it means that the theory is wrong. If the statistician cannot precisely measure it, but only roughly, this should not be a reason to discard the theory. However, if the statistician cannot measure the concept at

all, i.e. if he cannot apply the theory, then it is because the theoretician is wrong, even though his theory is coherent and precise. The theory in this case is not useful. So there is two cases where we can say that we have a good theory: if it is precise (or exact), and if it is useful. Both are necessary to have a good theory.

  If this is true, then we can fairly assume that Sraa would have accepted the ontological assumptions that form the basis of the capability theory. And as highlighted by Martins (2016; 2018), Sraa's 4 Sraa's unpublished papers, including his lectures, are kept at the Wren Li- brary, Trinity College, University of Cambridge (UK) and they are for the most part available online.The online eort is directed by Giancarlo de Vivo and

4 We have just seen that about epistemic values, he is considering exactness and explanatory value as important.

[START_REF] Martins | Ecosystems, strong sustainability and the classical circular economy[END_REF]

also highlights Sraa's objectivist approach. On ethics, Sraa observes that Marx's defense of the right of the worker to get the whole national product is based on ethical positions. He criticizes the following circular reasoning, which was used to dismiss Marx's labor theory of value: Distribution determines values, & values justify that distribution

(Sraa's unpublished papers, August 1946 -1948, D3/12/44: 7)

. For Sraa, it is ethics and politics that determine the distribution of the national product, and this distribution that consequently determines the values. Now, if it is accepted that Sraa wanted to revive Marx (a thesis which is highly debated both in Sraan and Marxist literature -see for instance

[START_REF] Kurz | Don't treat too ill my piero! interpreting sraa's papers[END_REF][START_REF] De Vivo | Some notes on marx's role in the development of sraa's thought[END_REF][START_REF] Sinha | A revolution in economic theory[END_REF]

, we could assume that he was in favor of the ethics of Marx, and also of Marx's ontological presuppositions, which, according to

[START_REF] Nussbaum | Women and Human Development: The Capabilities Approach[END_REF]

, are Aristotelian.

This refers to the distinction I made inVerger (2017b) between the system of physical exchanges of matter and energy and the system of value equations, where in the latter only commodities are represented. Flows of matter and energy can be used to characterize the system of production of commodities in a physical way, but the commodities are dened at the social level, not at the physical level.

I may add that it is possible that this suggestion was not intended by Martins: however, reading his article, I think this interpretation can naturally arise, so it seems to me important to clarify this point.

https://janus.lib.cam.ac.uk/db/node.xsp?id=EAD%2FGBR%2F0016%2FSRAFFA

This also means that the number of hyper-subsystems, which is equal to the number of nal consumption goods, is smaller than the total number of industries as soon as one industry only produces capital goods.

Bold capital letters refer to matrices, bold lower case letters to vectors, italic letters to scalars.

Even the i th component is equal to zero if industry i only produces capital goods.


Alexander Shen

Making randomness tests more robust

published or not. The documents may come

Making randomness tests more robust

1 What is a randomness test?

Imagine that we want to check some "null hypothesis", e.g., the hypothesis of a fair coin (all bits are independent with probabilities 1/2 for head and tail). The standard way is to choose some test, a subset 𝐹 ⊂ {0, 1} 𝑛 of the set of all possible outcomes that has small probability, and agree that an appearance of a sequence from 𝐹 will be interpreted as a failure (the hypothesis is rejected). [START_REF] Marsaglia | DIEHARD: a battery of tests of randomness[END_REF] The probability of 𝐹, usually called the significance level, is the probability of rejecting a sequence that is produced according to the null hypothesis. Its choice depends on the level of certainty we need when rejecting the null hypothesis. For example, a simple test could consists of all strings of some length 𝑁 where the frequency of ones exceeds some threshold 𝑡 > 1/2. The significance level is then the fraction of 𝑁-bit strings that contain more than 𝑡𝑁 ones, and this fraction should be small to make the rejection meaningful.

More general, we may consider any test statictic, i.e., a random variable 𝜉 that is a real-valued function defined on the outcomes (𝑛-bit strings in our example), and for every outcome 𝑐 consider the p-value that corresponds to this outcome, i.e., the probability of the event {𝑥 | 𝜉(𝑥) ≥ 𝜉(𝑐)}. In this way we get a family of tests: for every ε > 0 we get a test 𝑇 ε that consist of all outcomes that have 𝑝-value at most ε. The probability of 𝑇 ε is at most ε (more precisely, it is equal to the maximal p-value that does not exceed ε). So different observers may look at the p-value for the tested string, compare this p-value with their favorite significance level and decide whether to reject the null hypothesis based on the test result (this happens if 𝑝-value is below the significance level) or not. Popular collections of randomness tests (like diehard [START_REF] Marsaglia | DIEHARD: a battery of tests of randomness[END_REF] and more extensive dieharder [START_REF] Brown | Dieharder: A Random Number Test Suite[END_REF]) use this approach: they do not say just "rejected" or "accepted" but provide the 𝑝-value for many different tests (random variables 𝜉 in our notation).

Secondary tests

An important part of mentioned test collections [START_REF] Marsaglia | DIEHARD: a battery of tests of randomness[END_REF][START_REF] Brown | Dieharder: A Random Number Test Suite[END_REF] are "secondary" tests based on the following observation. Let 𝜉 be an arbitrary random variable; consider the new random variable 𝜂 = 𝑝(𝜉), where 𝑝 is the p-value function constructed for the variable 𝜉. The main observation: the distribution of this random variable is close to the uniform distribution. It would be exactly the uniform distribution if 𝜉 were a continuous random variable such that every individual value has probability 0. In general, the cumulative distribution function Pr[𝜉 ≤ 𝑐] equals the maximal p-value that does not exceed 𝑐. The distance between the consecutive p-values is the probability of an individual outcome for 𝜉, so for almost all tests the distribution is very close to the uniform one.

So we can apply the test to non-overlapping (and therefore independent) segments of the input sequence, and then test the sequence of p-values obtained for them against the uniform distribution. The test collections we mentioned [START_REF] Marsaglia | DIEHARD: a battery of tests of randomness[END_REF][START_REF] Brown | Dieharder: A Random Number Test Suite[END_REF] use the Kolmogorov-Smirnov test for this comparison. And this is important: in many cases the p-values obtained by the primary tests are not very close to zero; the significant discrepancy appears only at the second stage.

Problems with secondary tests

The first problem is already mentioned: the distribution of p-values is not exactly uniform. But the difference usually is negligible, since the distance between empirical and theoretical cumulative distributions is much bigger anyway. More dangerous are the assumptions that are made in the tests. The birthday spacing test from the original collection uses the overlapping sequences of bits for several tests, so there are no reasons to claim that the p-values will have uniform distribution. [START_REF] Brown | Dieharder: A Random Number Test Suite[END_REF] The same problem appears with 6 × 8 matrix rank test from diehard (Kolmogorov-Smirnov test is applied to dependent distrubutions).

The bigger problem arises because for most of these tests the distribution is not really computed exactly; instead, some heuristic assumptions are used (based on asymptotic convergence, or some experimental measurements3). So we do not have proofs that the approximate p-values coincide with the correct ones. The problem becomes even more serious when the secondary tests are used: in a primary test we may believe for a good reason that the approximation provides some upper bound for a correct p-value. However, for the secondary tests bounds are not enough, and we have no reason to believe in the correctness of the tests.

"True randomness" calibration

The natural idea would be to check first that the tests are valid by applying them to sequences that are believed to be random. If some test systematically rejects sequences that (we believe) are produces by a reliable source of random bits, then the test should be discarded. The dieharder documentation says about this problem: Many dieharder tests, despite our best efforts, are numerically unstable or have only approximately known target statistics or are 2"The first test uses bits 1-24 (counting from the left) from integers in the specified file. Then the file is closed and reopened, then bits 2-25 of the same integers are used to provide birthdays, and so on to bits 9-32. Each set of bits provides a p-value, and the nine p-values provide a sample for a KSTEST" (cdbday.c from diehard).

3The comments for one of the tests (OQSO) in diehard say "The mean is based on theory; sigma comes from extensive sumulation". straight up asymptotic results, and will eventually return a failing result even for a gold-standard generator (such as AES), or for the hypercautious the XOR generator with AES, threefish, kiss, all loaded at once and xor'd together. ⟨…⟩ Failure with numbers of psamples within an order of magnitude of the AES thresholds should probably be considered possible test failures, not generator failures. Failures at levels significantly less than the known gold standard generator failure thresholds are, of course, probably failures of the generator. So some tests in [START_REF] Brown | Dieharder: A Random Number Test Suite[END_REF] are labeled as suspect tests that consistently fail "good" generators. It would be desirable to use some more reliable approach, and there is an obvious possibility. Instead of using Kolmogorov-Smirnov test to compare the p-values distribution with the hypothetical uniform distribution, we may use two-sample Kolmogorov-Smirnov test (or any other test for comparing two distrubutions) to check whether the p-values obtained for the generator under testing are indistinguishable (up to some significance level) from the p-values obtained for the "gold standard generator" (assume for now that we have one).

The additional advantage of this approach is that now we do not need to compute the p-values, since only the ordering of them matters, so we can directly work with the values of the underlying test variable and do not need any information about its distribution. The price we pay is that we need more data. Indeed, we need the p-values both for generator we test and for the etalon generator. Also we need more samples for the same precision, since we now compare two empirical distributions and each has some sampling error.

What if do not have "true randomness"?

The trick mentioned in the preceding section assumes that we have some generator that we trust to be random. This assumption is crucial: If we use some faulty generator as an etalon for comparison, we may easily reject the good generator or accept a bad one. The second case may happen anyway for every test, but the first situation is hardly acceptable.

However, there is an easy way to get around this problem. Instead of using 𝑁 test runs (strings 𝑥 1 , … , 𝑥 𝑁 ) of our generator and comparing then with 𝑁 test runs (𝑒 1 , … , 𝑒 𝑁 ) of an "etalon" (presumably good) generator, we may take 2𝑁 runs 𝑥 1 , … , 𝑥 2𝑁 of our generator and 𝑁 test runs 𝑒 1 , … , 𝑒 𝑁 of the etalon generator, and then compare the distribution of test values for 𝑥 1 , … , 𝑥 𝑁 and for 𝑥 𝑁+1 ⊕ 𝑒 1 , … , 𝑥 2𝑁 ⊕ 𝑒 𝑁 (here ⊕ stands for bitwise xor). This is a valid test for every 𝑒 1 , … , 𝑒 𝑁 (assuming they are independent from 𝑥 1 , … , 𝑥 2𝑛 ; we may use some fixed values since this guarantees independence). On the other hand, if we use true random generator to obtain 𝑒 1 , … , 𝑒 𝑁 , then the second half will contain true random variables, so the test is as powerful as before. The only disadvantage is that we need twice more bits for testing.

* LIRMM CNRS & University of Montpellier. Supported by RaCAF ANR grant. 1There are many philosophical and practical questions related to this procedure. Normally we should choose a test before the experiment, but what can be done if the experiment already happened? What are the "reasonable" tests that can be meaningful in such a situation? Algorithmic information theory considers "simple" tests and identifies "randomness" with incompressibility, but how can we test incompressibility? Does it make any sense to test algorithms that claim that they produce a "pseudorandom" sequence -without a seed, like the binary expansion of 𝜋, or with a seed? (This latter setting corresponds to the Yao-Blum-Micali definition of pseudo-random number generators.) We do not discuss these questions here.