Thomas Pellard
email: thomas.pellard@cnrs.fr

The comparative study of the Japonic languages

Why do we compare?

(1) Goals and methods of science

• to advance our understanding of the world • answering "what?" (observation) is not sufficient, we also need to ask "how?" and "why?" (explanation) • the hallmark of the scientific method is the artificial experiment, but such experiments are often not possible or ethical in many fields, especially those who study the past and unique events: evolutionary biology, archaeology, but also epidemiology, ecology, sociology, etc. • one alternative to the controlled experiment is the "natural experiment", or "comparative method": manipulation is replaced by comparison with other attested cases • comparison has a long history and has been a standard method of enquiry in anatomy for centuries We consider that it is worth while to examine many animals of different species because what is often more condensed or more concealed in one species Nature displays more clearly and openly in another. (Regnier de Graaf 1663, Disputatio medica de natura et usu succi pancreatici, cited in Cole 1944: 12) (2) In linguistics too, comparison is an invaluable tool 1. typological comparison: search for the universal laws, tendencies, and possible variations of language (see the presentation by M. Shimoji) 2. historical comparison: explanation of languages by reference to their evolutionary changes, as revealed by comparison with related languages 1

The historical comparative method in linguistics

(3) Motivation and method • usually, we cannot observe ling. changes, only their results

• there are usually few historical records, so that we have to rely on comparison with related languages, which constitute as many pieces of the original puzzle • we infer (reconstruct) an earlier state (proto-language) from which all languages can be plausibly derived • regularity of sound change and arbitrariness of formmeaning associations allow to rule out coincidences (4) Alternatives to the traditional comparative method?

• mass comparison, lexical diffusion, allofams, typologybased classifications, etc., have not and will never produce any reliable etymology nor explain any irregular inflection • i.e. they do not solve interesting problems 1. "Comparative linguistics" has been used in the latter sense for more than 200 years. It would be very inappropriate to turn everything on its head by rebranding "linguistic typology" as "comparative linguistics", as awkwardly proposed by [START_REF] Haspelmath | What is the name of my subfield (or subcommunity): Language typology, linguistic typology, or comparative linguistics? Diversity linguistics comment: Language structures throughout the world[END_REF]. See Morpurgo [START_REF] Davies | Nineteenth-century linguistics (History of Linguistics 4)[END_REF] for a good overview of the early history of the field and [START_REF] Hock | Principles of historical linguistics[END_REF] for a good introduction to the concepts and methodology of historical linguistics.

(5) The goal of comparative linguistics is and has always been to explain languages and their evolution Another and not less important reason, which makes a critical comparison of the Sanskrit with its European sisters, worthy to be undertaken, is the light thrown thereby upon each of the languages compared, and the clearer view we thence obtain of the most ancient forms of each respectively, and probably some conception of the original and primitive signification of a great part of the grammatical inflections common to all. It is chiefly by comparison that we determine as far as our sensible and intellectual faculties reach, the nature of things. Frederic Schlegel justly expects, that comparative grammar will give us quite new explications of the genealogy of languages, in a similar way as comparative anatomy has thrown light on natural philosophy. (Bopp 1820: 2) It is usual to call comparative grammar the grammar that does not only describe but as far as possible explains linguistic forms, because it cannot as a rule limit itself to the examination of a single language. (Schleicher 1861: 2) Comparison is the only effective tool available to the linguist for writing the history of languages. We observe results of changes, not changes themselves. It is thus only with the help of combinations that we follow, and can follow, the development of languages. (Meillet 1925: 11) Even if we were, by some miracle, handed a complete grammar of Common Indo-European as spoken somewhere in, say, 4000 B.C. (the date is meaningless), the work of the Indo-Europeanist would scarcely be done. In fact, it would be barely begun. For his task would be, then as before, to relate the facts vouchsafed him to the facts of attested languages: to construct hypotheses, and to demonstrate precisely how it is possible, within a linguistic tradition or traditions, for a language to pass from one system at one point in time to another system at a later point. (Watkins 1973: 101) (6) Contrary to more or less common misunderstandings • proving relatedness is not much a goal as a result of the comparative method; nobody needed to prove that the Indo-European languages were related: it was selfevident since they shared irregularities that could only be explained through comparison • reconstructions are not speculative educated guesses but models that account for correspondences between related languages and that allow to derive attested forms • a phylogenetic tree is not a goal nor simply pigeonholing, it is a model of the historical diversification of languages (7) The most successful example of the comparative method is its application to the Indo-European languages • the verb 'be' is irregular in most languages, but it reconstructs as a regular verb alternating between full grade in singular forms and zero-grade in plural forms (*h₁és : *h₁s) (Chamberlain 1895: 6) Japanese etymologies, too, mostly remained at the guessing stage. As for the accidence, grammarians could enumerate forms, but were often unable to explain them; they could imagine theories, but could not build trustworthy theories. With Luchuan to refer to the case becomes different, for Luchuan stands to Japanese in about the same relationship as Italian does to French and Spanish. (Chamberlain 1897: 56) • still indebted to the breakthroughs achieved by Hattori Shirō, reprinted in [START_REF] Hattori | Nihongo no keitō[END_REF] and Hattori (2018) • the first attempt at a systematic reconstruction of proto-Ryukyuan is [START_REF] Thorpe | Ryūkyūan language history[END_REF], which also contains many comparisons with Japanese, though not all are convincing • Pellard (2019) presents an overview of the current state of the field of Japanese-Ryukyuan comparative linguistics • for comparison with Hachijō, see [START_REF] Kaneda | Hachijō hōgen dōshi no kiso kenkyū[END_REF], [START_REF] Hirako | Hachijōgo no furusa to atarashisa[END_REF], and references therein (12) Methodological problems • a distrust, if not contempt, for reconstruction and a bias towards philology: tendency to enshrine cherrypicked Old Japanese forms as proto-forms and to posit ad hoc adjustments rather than to resort to comparative reconstruction4 None of these scholars has attempted to reconstruct the phonology and morphology of the ancestral Proto-Ryukyuan language and to formulate the "sound laws" accounting for the linguistic situation found today in the islands. [… T]he comparative method […] has never been applied to these data. In Ryukyuan studies formal historical statements are generally confined to arrangements of variant forms in hypothetical evolutionary sequences, unsupported by comparative method evidence. (Thorpe 1983: 4-

midzo ̸ = mɨdzɨ çiɾu ̸ = ɸɨɾu Yonamine dʑúː ̸ = mìdʑíː pˀìɾúː ̸ = pʰìɾúː Hirara mdzu ̸ = midzɿ pɿː ̸ = piɿ Ishigaki ńdʑú ̸ = mídzɿ̀pɿːɾɿ̸̀ = píɴỲ onaguni ndú ̸ = míɴ tsˀuː ̸ = çìɾú 'horse' 'sea' 'mortar' 'medicine' pJ *uma ̸ = *omi *{u,o}su ̸ = *kusori oj uma = umi₁ usu = kusuri pR *uma ̸ = *omi *{u,o}su ̸ = *kusori Yamatohama mˀaː ̸ = ʔumi ʔusɨ ̸ = kˀusuɾi Shuri m̀ˀmà ̸ = ʔùmì ʔùːsì ̸ = kùsùì Hirara mma ̸ = im usɿ ̸ = fusuɿ Ishigaki ḿmá ̸ = íɴ úsɿ̸́ = ɸùɕíɾɿÝ onaguni mmà ̸ = ùnnáɡâ ùtɕî ̸ = tsˀùɾî
(15) Accent and tone (see also the presentation by Y. Igarashi)

• until recently, the Ryukyuan accent and tone systems were presented as simply deriving from Japanese • however, the Ryukyuan data requires to reconstruct more distinctions than in Japanese [START_REF] Hattori | Nihongo no keitō[END_REF](Hattori [1958]]; [START_REF] Hattori | Nihon sogo no saiken. With annotations by Uwano[END_REF]Hattori [1979]]), but the reconstruction of the phonetic content of those distinctions is problematic ( (26) Historical-comparative linguistics and typology • linguistic typology is interested in making cross-linguistic generalizations about the relative frequency or rarity of linguistic features • problem of sampling and genetic bias • is a feature found in many languages due to a universal tendency or to the fact that those languages inherited it? • in other words, did that feature arose only once or several times independently? • for instance, M. Shimoji argues that marked nominative alignment is cross-linguistically rare but frequent in Ryukyuan, but did that pattern arise several times independently, in which case an explanation is required, or did it arise only once and was simply inherited, in which case it will remain an isolated exception?

Conclusions

• much remains to be done within the field of Japonic historical-comparative linguistics • after a century of intense philological work and internal reconstruction, I expect most future breakthroughs to come from comparison with Ryukyuan, Hachijō and Japanese dialects • historical-comparative linguists need to pay more attention to descriptive work on endangered Japonic languages • descriptive linguists should not disregard historicalcomparative data nor reduce historical-comparative linguistics to grammaticalisation theory • it is widely acknowledged that descriptive grammars should be typologically and theoretically informed • I argue that they should also be historically and comparatively, or more concisely diachronically, informed

  It is proven that historical-comparative linguistics does work • Saussure (1879) made the bold hypothesis that some ie forms could be better explained if we assumed that some sounds had been lost, and it was later discovered (Kuryłowicz 1927) that Hittite preserved in some cases Saussure's reconstructed sounds: Comparison within Japonic • the first historical-comparative study of Ryukyuan and Japanese is Chamberlain's (1895) grammar of Shuri Okinawan • mainly of historiographic interest today, but Chamberlain did think of comparative grammar as explanatory It may be asked of what nature is the light which Luchuan [i.e. Ryukyuan] throws on Japanese. We reply that it is such as, in general, sister languages shed upon each other.

		Sanskrit Latin Gothic pIE
	'be.pres.3sg' ásti		est	ist	*h₁és-ti
	'be.pres.3pl' sánti	sunt	sind	*h₁s-énti
			V in Greek
		Greek	Sanskrit pIE
	nom.sg zéus	dyáus	*di̯ ḗu̯ -s
	gen.sg	di(w)ós divás	*diu̯ -és
	(8) Japonic comparative linguistics
	(9) Japonic (also Japanese-Ryukyuan, Japonesian)
	• Japanese, and its endangered dialects
	• Ryukyuan: ~5 endangered languages spoken in the Ryukyu
	Islands, a chain of around 50 inhabited islands stretching
	from the Southeast of Kyushu to the Northeast of Taiwan 3
	• Hachijō: an endangered language spoken 287 km south of
	Tokyo, in the Izu archipelago; its phylogenetic position has
	yet to be determined
	• the degree of diversity and the time depth of the family
	can be said to be not much different from that of the
	Romance or Slavic families (Pellard 2013; 2016a)
					Japonic	?
			Ryukyuan	Japanese Hachijō
		Southern		Northern
	Macro-Yaeyama	Miyako	Okinawa Amami
	Yonaguni Yaeyama
	(10) Comparisons have usually focused on Japanese and another
	language or language family (Altaic, Austronesian, etc.)
	• such hypotheses seldom try to explain idiosyncrasies of

• the name of the Greek god Zeus has an irregular declension which can be shown to go back to an earlier regular pattern where lengthened e-grade in strong cases alternates with zero-grade in weak cases (*di̯ ḗu̯ -: *diu̯ -), with a sound law *di̯ > z / 2 'protect' pIE *peh₂-s-> Hittite paḫš-, Latin pāstor, pāsco, Sanskrit pāsati • Bloomfield (1928) compared several Algonquian languages and reconstructed a cluster *çk not distinguished in any language, which he later discovered to be distinguished as htk in Swampee Cree Japanese (irregular verbs, tone and accent, apophony, etc.) • in absence of a convincing (i.e. explanatory) hypothesis of relationship with other languages, comparison within the Japonic family is the only alternative

[START_REF] Pellard | The historical position of the Ryukyuan languages[END_REF] 2016a) (11)

The now famous laryngeals, see[START_REF] Keiler | A phonological study of the Indo-European laryngeals[END_REF][START_REF] Lindeman | Introduction to the ' Laryngeal Theory[END_REF]

See Shimoji & Pellard (2010) and Heinrich et al. (2015) for a general introduction.

This is reminiscent of the situation in Romance linguistics[START_REF] Buchi | Des bienfaits de l'application de la méthode comparative à la matière romane: L'exemple de la reconstruction sémantique[END_REF]).

Abbreviations and sources

Amuro (Amami, Uke-jima: Shirata


Ginette Vagenheim

The «catalogue raisonné» of Pirro Ligorio's drawings My project started 3 years ago, is the publication of the first «catalogue raisonné» of Pirro Ligorio's drawings that are not preserved in his 40 volumes of Roman antiquities but in collections and museums around the world; the basis of my corpus is the checklist of around 300 drawings that David Coffin have increased in the course of his whole life and that was edited as an appendix in his posthumous biography on Ligorio published in 2004

At the beginning of my study, I immediately noticed that a great number of drawings of the checklist were not by Ligorio and I discovered that the first reason was because scholars and curators mostly based their expertise on the seminal article that John Gere published in 1971 in Master drawings, intitled «Some early drawings by Pirro Ligorio». The problem is that these 25 drawings have characteristics that are not of Ligorio's style but of Polidoro da Caravaggios style and as a result, also Coffin's checklist contain drawings that are "polidoresque" but not by Ligorio and in the other hand, because of that, his checklist is missing some drawings that are really by Ligorio.

How did that happen? As a keeper of the Department of Prints and Drawings, at the British Museum, and publisher of the catalogues of the Italian drawings, Gere was an amazing "connaisseur" and knew better than anyone the style of Ligorio's drawings, that he described as executed «mostly in his preferred medium of neat pen and golden-brown wash, peopled by his characteristic stocky rubbery, somewhat oriental-looking little figures with their tightly curled hair, prominent noses and large eyes, unduly large hands with flabby pointed fingers and clumsily articulated wrists." This recognizable style was, according to Gere, that of the Ligorio's full maturity but what he intended to present in his article was different; he intended to present drawings of Ligorio's first period, that is before his style was finally formed; and by doing so, Gere wanted to throw some light on a hitherto obscure aspect of Ligorio's artistic activity.

The proposal was of great interest but the problem was a methological one and that is the way Gere had identifyed drawings that are supposed to belong to a little known and little documented part of Ligorio's life? That is the obscure period before he became a famous antiquarian at the service of the Este family and of Pope Paul the fourth and Pius the fourth, that is before 1550.

Gere found Ligorio's early designs among a mass of drawings that he described as "Polidoresque in general character but not by Polidoro himself and which have tended -down the centuries-to accumulate under that name". The scholar idenfifyed in this mass, two coherent groups, one of them being attributable to Tadddeo Zuccaro and that he published as a monograph in ninetheen sixthy nine ([SHOW THE BOOK]1969). Among the second group of 25 drawings, five of them were traditionally given to Ligorio and even if Gere recognized that they were not typical of his style, he considered it as a sufficient element to attribute the whole group to Ligorio and therefore to assume the thesis that I try to qualify, that Ligorio's early style must be construed as completely dependent on that of Polidoro da Caravaggio. [[SHOW SLIDE 2]. Besides, Gere noticed the imprint of Polidoro in the attitudes and dispositions of Ligorio's figures represented in his only extant fresco that is the Dance of Salome, painted in the Oratorio of San Giovanni Decollato, in Rome and dated around 1540's .

Gere also correctly returned to Ligorio some drawings traditionally considered as by Polidoro, [[SHOW SLIDE 3] such as the Victory with Roman Trophies (in the Kupferstichkabinett in Berlin) and Roman Trophies with Prisoners in New York); it is true that these subjects were conform with the compositional patterns created by Polidoro in the 1520s, as a painter of facades, that he decorated with "trophies, friezes and stories illustrating the magnificenze romane", an activity that Ligorio, as well as Zuccaro, exercised in Polidoro's wake when Ligorio arrived in Rome around 1534 as a neapolitaner painter (pittore Napolitano). Consequently, some drawings were attributed to Ligorio mainly on the account that the subject represented was linked to Polidoro, as for example [[SHOW SLIDE 4] the drawing of Diana and Apollo Killing the Children of Niobe in the National Gallery, a subject that had been treated by Polidoro on the façade of the Palazzo Milesi in Rome . So from the similarity of subjects, we move to the similarity of style and that is the methodological mistake. From my part, I have tried to show, that the drawing was not by Ligorio and that all that we know is that it was engraved by Philipp Galle after Giulio Romano's invention [SHOW SLIDE 7].

One of the risk of Gere's method is the fact that some of the drawings now in Ligorio's corpus and in Coffin's may be by to Polidoro as supposed by Ursula Verena Fisher Pace [SHOW SLIDE 8]; she clearly makes the connection, in his catalogue, between the Copenhagen drawing attributed by Gere to Ligorio with a drawing of Polidoro kept in the Museum of Fine arts in San Francisco.

Another limit of Gere's method is that, in some cases, he only focused on one single stylistic feature, for example, Ligorio's big unfinished hands, with pointed fingers, as in the drawing in New York.

[[SHOW SLIDE 9] However, they sometimes merely indicate that the drawing is only a first draft or a study like the sheet of a woman with a child now in Edinburgh.

The same limits also appear in Coffin catalogue ( but I must now add not in his notes) with dubious results as the attribution to Ligorio, already made by Giorgio Vasari in his Libro dei disegni, of the drawing of Women and children at a fountain arrived in the national Gallery in two thousand and thirtheen 2013.

But after all, Gere's deep knowledge of Ligorio's characteristics of his late period and his complete description of Ligorio's style, at the beginning of his article, allowed me to show, through the precise dating of a number of drawings, that Ligorio's style clearly evolved over time, but also that certain characteristics of his late style were already present in his early drawings, and are, so to speak, irreducible; and thereby, they enable me to assess Ligorio's draftsmanship on more solid grounds.


CALL FOR CONTRIBUTIONS

 , 2017-2021). We are an international and multidisciplinary scholarly network some 170 people strong, funded by the COST Association in the framework of the Horizon 2020 Work Programme of the EU.

NEP4DISSENT is founded upon the conviction that resistance and dissent in former socialist Europe 1945-1989 constitutes a remarkable chapter of Europe's recent past, which not only informs in a decisive way the identities of post-socialist societies today, but has also reshaped the continent as a whole and still provides an important reference for contemporary social movements worldwide.

Building on the achievements of this field of study and the respective domain of cultural heritage, NEP4DISSENT will trigger the next phase in the discovery of that legacy through forging new, reflexive approaches and providing a platform for creating an international research network, designed to cross-fertilize and consolidate national research agendas and incubate novel collaborative initiatives, taking advantage of cutting-edge digital humanities tools and methods, as well as strengthening the capacity for dissemination through cooperation with art and cultural heritage curators.

If you are not part of our network yet, we wholeheartedly encourage you to join the activities of our Working Groups, which are described in detail here. Together with this survey, we are launching our first calls for Conference Grants and Short Term Scientific Missions, and our website, www.nep4dissent.eu will become active this Fall. If you want to get involved, please indicate so when submitting your response.

What is the aim?

The aim of this survey is to review the state-of-the-art in research on the legacy of resistance and dissent in former socialist countries. We are reaching out to you to help us identify ascending research trends and to scope possibilities of international collaboration by pooling together new research agendas, approaches and data, but also by putting into focus the uneven distribution and scope of scholarly outputs between local research fields and disciplines. Leveraging the collective intelligence of an international scholarly network is instrumental when it comes to identifying a diversity of scholarly approaches to the matter and describing a vast spectrum of actors, phenomena, events and processes under study.

The main conclusions of this survey will be collected in a Joint Review Report that will not only set priorities for the activities of the NEP4DISSENT research network in the coming years, but will also aim at achieving impact on wider scholarly community in this field of research. By exploring the emerging trends in this research area, we hope this state-of-the-art report will provide a helpful reference for scholars, evaluators, publishers, funders and policymakers, not least when it comes to requesting funding, evaluating such requests or making editorial decisions.

The report will provide an overview of the results of this survey, interpreted against research on the state-of-the-art carried out by NEP4DISSENT Working Groups. It will be edited by the Action's Core Group. In addition to the overview of the state-of-the-art, the report might also contain a supplement anthology of full responses to the survey, depending on their number, length and overall coherence of the submissions. Regardless whether your contribution will be referenced or quoted in the report, or published in full, you will be listed as a contributor to the report and to the project.

The report will be published as an open-access e-book and actively disseminated to the stakeholder groups mentioned above. English will be the language of the report, however if you'd like to choose another language for your contribution, you are free to do so.

What we want to know?

The questionnaire consists of six parts coterminous with and relevant to the activities of NEP4DISSENT Working Groups (outlined in more detail below): We are well aware that not all the parts may be relevant to your expertise, but we expect that you will be able to address at least one of them.

Each part contains a short introduction of the main theme together with some additional questions which may guide your response. In general, given the aim discussed above, we seek for the following information: What are key works and projects that took up the research in question? What approaches have been used? What are the key findings and results? Which historical actors, processes objects and phenomena has been well studied in this context, and which remain underrepresented? What aspects have been approached from a comparative or transnational perspective, and which are understood mostly as local developments? What is the institutional context of research and cultural heritage activities in the respective domain and which are the key governmental and/or non-governmental actors?

Towards a New Exploratory Phase in Research on East European Cultures of Dissent

  Alternative Cultures ........................................................................................................... 4. Cultural Memory of Dissent ............................................................................................... 5. Mediating Research through Technology .......................................................................... 6. Art and Cultural Heritage Curation .................................................................................... III. RESPONSE SHEET .......................................................................................................

	SURVEY OF THE STATE OF THE ART

I. COVER LETTER ............................................................................................................... Who is asking? ....................................................................................................................... What is the aim? ..................................................................................................................... What we want to know? ......................................................................................................... What is the format? ................................................................................................................ II. NEP4DISSENT QUESTIONNAIRE ................................................................................. 1. Culture under Surveillance ................................................................................................. 2. Culture in the Grey Zone ....................................................................................................

3.

SEND YOUR RESPONSES TO:

survey@nep4dissent.eu

I. COVER LETTER

Who is asking?

This survey will contribute to core activities of the COST Action 16213, New Exploratory Phase in Research on East European Cultures of Dissent (NEP4DISSENT

What is the format?

The responses should have a form of a research statement, app. 1,500 words long (however longer and shorter contributions are also welcome). Guided by the survey questions, you may wish to offer reflections on a single field or even a case study closely related to your own research interests. Or you may wish to offer a broader overview. We imagine it as something similar to a response to a survey circulated by a scholarly journal.

The responses, written in the text editor of your choosing, should be formatted according to the Response Sheet appended at the end of this document. Please send your responses to survey@nep4dissent.eu by 30 September 2018.

II. NEP4DISSENT QUESTIONNAIRE

The questionnaire is structured according to the interests of NEP4DISSENT Working Groups, defining their research priorities and proposing a set of guiding questions to which these translate. You are not required to refer to all of these questions in your response. You may wish to offer reflections on a single field or even a case study closely related to your own research interests. Or you may wish to offer a broader overview.

Culture under Surveillance

Scope: NEP4DISSENT will explore the effects of exposure of culture under socialism to state surveillance, ranging from monitoring, through censorship to political and judiciary interventions, as well as the counter-surveillance activities and practices of Western cultural diplomacy, transnational institutions, such as Radio Free Europe, and émigré cultural centers.

Guiding Questions:

• How does recent research address the question of the impact of censorship, the activities of the security apparatus and professional organizations (such as writers' unions, filmmakers' associations) on cultural life, on creators of culture, on institutions and events, and on cultural objects and repertoires of practices? • How has Western, transnational and émigré counter-surveillance effort been approached and assessed in recent scholarship? • How has surveillance and counter-surveillance been represented in artistic and cultural production, including literature, music, film and performance art? How have these representations evolved across the 1989 divide? How different approaches connect aesthetic and political dimension of these representations?

Culture in the Grey Zone

Scope: NEP4DISSENT will study concepts used in recent research on the resistance and dissent under state socialism to comprehend the grey zone phenomenon, understood as an ambiguous realm of negotiated space for expression situated on the borders between the domain of official and unofficial.

Guiding Questions:

• Which disciplines (history, anthropology, political science) employ conceptualizations of the grey zone as an analytical category in recent research?

• What concepts are used in recent research to address grey zone phenomena? Is there an agreed upon definition or a set of definitions of the grey zone? Is the category used in singular or in plural, as multiple and possibly mutually intertwined zones? • To what do recent conceptualizations refer, when speaking of the grey zone: distinct social or professional groups, sets of practices, mentalities and values? • Are these conceptualizations context-independent, or context-sensitive, including geographical, cultural, or habitual factors that affect the grey zone? Do they employ gender analysis? Do they take the special position of minority groups into account? Do they register conceptual change over time or local use variations?

Alternative Cultures

Scope: NEP4DISSENT will study alternative cultures in the former Eastern Bloc. By 'alternative culture' we understand broadly communities formed around cultural practices beyond the state-sanctioned mainstream of socialism, local as well as transnational circuits of unlicensed communication and their media, as well as marginal cultural spaces, either physical, such as home theatres, student clubs, or discotheques, or virtual spaces, such as periodicals.

Guiding Questions:

• What understandings of alternative culture, as defined above, are exhibited in recent research and what are the methodological and theoretical challenges faced by scholarship on alternative culture? • What kind of social groups, cultural practices and strategies have been associated with alternative culture in Eastern Europe under communist hegemony? • Has the relationship between state-licensed culture and alternative culture been defined in terms of coexistence, interdependence, conflict or still differently? How the former has shaped the latter? • What forms of expression have been designated as alternative media and what media strategies are considered instrumental for the alternative cultural identity? • Which physical and virtual spaces have been occupied by alternative culture and what kind of significance has been attached to these settings? • How have processes of cultural transfer and exchange, both among the former socialist countries and between these countries and the rest of the world, affected understanding of alternative culture on both sides of the divided continent?

Cultural Memory of Dissent

Scope: NEP4DISSENT will explore the cultures of remembrance of pre-1989 dissent across Europe, with the aim of examining the dissident legacy in both official and alternative narratives as well as the reception of such narratives; the contemporary uses of the dissident past; and the efforts aimed at integrating this legacy in the broader European culture of remembrance and their impact.

Guiding Questions:

• How have these stories legitimized or challenged the post-89 order? How and by whom has the language related to dissent and its aforementioned alternatives been appropriated since 1989, and how has this language been transformed as a result? How have the just mentioned questions interacted with the biographies and activities of former dissidents?

Mediating Research through Technology

Scope: NEP4DISSENT will examine digital research practices in the study of the legacy of resistance and dissent in former socialist countries, in order to map the needs of research community against the available digital infrastructure and to identify the potential for technical innovation in this regard.

Guiding Questions:

• How would you describe the the digital dimension of your research into history and culture (esp., but not limited to Eastern European cultures of dissent) including its limitations and challenges? This could encompass your sources, methods or your tools to capture evidence/data, structure, how you document, annotate, visualize/analyze and/or encode your evidence, or, even, your expectations from what a digital approach might bring to your research. • In terms of applying digital methods to your research, please describe any people, projects, groups, publications, or resources that you see as exemplary for your field, or would like to take as inspiration for your own approach/work.

Art and Cultural Heritage Curation

Scope: NEP4DISSENT will serve as a mutually-beneficial interface between the GLAM community (Galeries, Libraries, Archives, Museums) and scholars working on the legacy of dissent and resistance in former socialist countries, by both identifying best practices of their cooperation, and by exploring how exhibition practices of the period were themselves forms of resistance and dissent.

Guiding Questions:

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By providing your personal data you acknowledge that the colleagues from our COST Action who will write and edit the report, will prepare and present anonymised statistics about the respondents in the report. We take all necessary precautions to store the responses safely, according to local data protection laws. Your responses will not be disclosed to any third party. Access to your responses with your personal data will be granted only to you and the participants of the NEP4DISSENT research network, who will author and edit the report, and to COST Association officials, if required. We will not transfer the responses to any third party.

Without your expressed consent (see the above clauses), we will not cite or mention your name in the report. We will be happy to cite you anonymously, if you so desire. If you choose to provide us with your name and surname and give us permission to cite your response, your input will be quoted in the Report according to customary standards and your name will be feature on the list of contributors published in the Report.

Where to send? Please send your responses to survey@nep4dissent.eu


Namig J Guliyev
email: njguliyev@gmail.com

Vugar E Ismailov
email: vugaris@mail.ru

APPROXIMATION CAPABILITY OF TWO HIDDEN LAYER FEEDFORWARD NEURAL NETWORKS WITH FIXED WEIGHTS

Keywords: 2010 Mathematics Subject Classification. 41A30, 41A63, 65D15, 68T05, 92B20 multilayer feedforward neural network, hidden layer, sigmoidal function, activation function, weight, the Kolmogorov superposition theorem

We algorithmically construct a two hidden layer feedforward neural network (TLFN) model with the weights fixed as the unit coordinate vectors of the d-dimensional Euclidean space and having 3d + 2 number of hidden neurons in total, which can approximate any continuous d-variable function with an arbitrary precision. This result, in particular, shows an advantage of the TLFN model over the single hidden layer feedforward neural network (SLFN) model, since SLFNs with fixed weights do not have the capability of approximating multivariate functions.

Introduction

The topic of artificial neural networks is an important and vibrant area of research in modern science. This is due to a large number of application areas. Nowadays, neural networks are being successfully applied in areas as diverse as computer science, finance, medicine, geology, engineering, physics, etc. Perhaps the greatest advantage of neural networks is their ability to be used as an arbitrary function approximation mechanism. In this paper, we are interested in questions of density (or approximation with arbitrary accuracy) of the multilayer feedforward neural network (MLFN) model. Approximation capabilities of this model have been well studied for the past 30 years. Choosing various activation functions σ it was shown in a great number of papers that MLFNs can approximate any continuous function with an arbitrary precision. The most simple MLFN model is the single hidden layer feedforward neural network (SLFN) model. This model evaluates a multivariate function

k i=1 c i σ(w i • x -θ i ) (1.1)
of the variable x = (x 1 , . . . , x d ), d ≥ 1. Here the weights w i are vectors in R d , the thresholds θ i and the coefficients c i are real numbers, and the activation function σ is a univariate function. A multiple hidden layer network is defined by iterations of the SLFN model. For example, the output of the two hidden layer feedforward neural network (TLFN) model with k units in the first layer, m units in the second layer and the input x = (x 1 , . . . , x d ) is

m i=1 e i σ   k j=1 c ij σ(w ij • x -θ ij ) -ζ i   .
Here d i , c ij , θ ij and γ i are real numbers, w ij are vectors of R d , and σ is a fixed univariate function.

In many applications, it is convenient to take an activation function σ as a sigmoidal function, which is defined as lim t→-∞ σ(t) = 0 and lim t→+∞ σ(t) = 1.

The literature on neural networks abounds with the use of such functions and their superpositions.

The possibility of approximating a continuous function on a compact subset of R d , d ≥ 1, by SLFNs with a sigmoidal activation function has been tremendously studied in many papers. To the best of our knowledge, Gallant and White [START_REF] Gallant | There exists a neural network that does not make avoidable mistakes[END_REF] were the first to prove the universal approximation property for the SLFN model with a sigmoidal activation function. Their activation function, called the cosine squasher, has the ability to generate any trigonometric series. As such, this function has the density property. Carroll and Dickinson [START_REF] Carroll | Construction of neural nets using the Radon transform[END_REF] implemented the inverse Radon transformation to approximate L 2 functions, using any continuous sigmoidal function as an activation function. Cybenko [START_REF] Cybenko | Approximation by superpositions of a sigmoidal function[END_REF] proved that SLFNs with a continuous sigmoidal activation function can approximate any continuous function with arbitrary accuracy on compact subsets of R d . Funahashi [START_REF] Funahashi | On the approximate realization of continuous mapping by neural networks[END_REF], independently of Cybenko, proved the density property for a continuous monotone sigmoidal function. Hornik, Stinchcombe and White [START_REF] Hornik | Multilayer feedforward networks are universal approximators[END_REF] proved density of SLFNs with a discontinuous bounded sigmoidal function. Kůrková [START_REF] Kůrková | Kolmogorov's theorem and multilayer neural networks[END_REF] showed that staircase-like functions of any sigmoidal type has the capability of approximating continuous univariate functions on any compact subset of R within arbitrarily small tolerance. This result was substantially used in Kůrková's further results, which showed that a continuous multivariate function can be approximated arbitrarily well by TLFNs with a sigmoidal activation function (see [START_REF] Kůrková | Kolmogorov's theorem is relevant[END_REF][START_REF] Kůrková | Kolmogorov's theorem and multilayer neural networks[END_REF]). Chen, Chen and Liu [START_REF] Chen | A constructive proof of Cybenko's approximation theorem and its extensions[END_REF] generalized the result of Cybenko by proving that any continuous function on a compact subset of R d can be approximated by SLFNs with a bounded (not necessarily continuous) sigmoidal activation function. Almost the same result was independently obtained by Jones [START_REF] Jones | Constructive approximations for neural networks by sigmoidal functions[END_REF]. Costarelli and Spigler [START_REF] Costarelli | Constructive approximation by superposition of sigmoidal functions[END_REF] constructed special sums of the form (1.1), using a given function f ∈ C[a, b]. They then proved that these sums approximate f within any degree of accuracy. In their result, similar to [START_REF] Chen | A constructive proof of Cybenko's approximation theorem and its extensions[END_REF], σ is any bounded sigmoidal function. Chui and Li [START_REF] Chui | Approximation by ridge functions and neural networks with one hidden layer[END_REF] proved that SLFNs with a continuous sigmoidal activation function having integer weights and thresholds can approximate continuous univariate functions on any compact subset of the real line.

In a number of subsequent papers, which considered the density problem for the SLFN model, nonsigmoidal activation functions were allowed. Here we cite a few of them. The papers by Stinchcombe and White [START_REF] Stinchcombe | Approximating and learning unknown mappings using multilayer feedforward networks with bounded weights[END_REF], Cotter [START_REF] Cotter | The Stone-Weierstrass theorem and its application to neural networks[END_REF], Hornik [START_REF] Hornik | Approximation capabilities of multilayer feedforward networks[END_REF], Mhaskar and Micchelli [START_REF] Mhaskar | Approximation by superposition of a sigmoidal function and radial basis functions[END_REF] are among many others. It should be remarked that the more general result in this direction belongs to Leshno, Lin, Pinkus and Schocken [START_REF] Leshno | Multilayer feedforward networks with a non-polynomial activation function can approximate any function[END_REF]. They proved that the necessary and sufficient condition for any continuous activation function to have the density property is that it not be a polynomial. For more detailed discussion of the density problem, see the review paper by Pinkus [START_REF] Pinkus | Approximation theory of the MLP model in neural networks[END_REF].

The above results show that SLFNs with various activation functions enjoy the universal approximation property. In recent years, the theory of neural networks has been developed further in this direction. For example, from the point of view of practical applications, SLFNs with a restricted set of weights have gained a special interest (see, e.g., [START_REF] Draghici | On the capabilities of neural networks using limited precision weights[END_REF][START_REF] Ismailov | Approximation by neural networks with weights varying on a finite set of directions[END_REF][START_REF] Ismailov | Approximation by ridge functions and neural networks with a bounded number of neurons[END_REF][START_REF] Ismailov | Measure theoretic results for approximation by neural networks with limited weights[END_REF][START_REF] Jian | Neural networks with limited precision weights and its application in embedded systems[END_REF][START_REF] Liao | A neural network model with bounded-weights for pattern classification[END_REF]). It was proved that SLFNs with some restricted set of weights still possess the universal approximation property. For example, Stinchcombe and White [START_REF] Stinchcombe | Approximating and learning unknown mappings using multilayer feedforward networks with bounded weights[END_REF] showed that SLFNs with a polygonal, polynomial spline or analytic activation function and a bounded set of weights have the universal approximation property. Ito [START_REF] Ito | Representation of functions by superpositions of a step or sigmoid function and their applications to neural network theory[END_REF][START_REF] Ito | Approximation of continuous functions on R d by linear combinations of shifted rotations of a sigmoid function with and without scaling[END_REF] investigated this property of networks using monotone sigmoidal functions, with only weights located on the unit sphere. In [START_REF] Ismailov | Approximation by neural networks with weights varying on a finite set of directions[END_REF][START_REF] Ismailov | Approximation by ridge functions and neural networks with a bounded number of neurons[END_REF][START_REF] Ismailov | Measure theoretic results for approximation by neural networks with limited weights[END_REF], the second coauthor considered SLFNs with weights varying on a restricted set of directions, and gave several necessary and sufficient conditions for good approximation by such networks. For a set of weights consisting of two directions, he showed that there is a geometrically explicit solution to the problem. Hahm and Hong [START_REF] Hahm | An approximation by neural networks with a fixed weight[END_REF] went further in this direction, and showed that SLFNs with fixed weights can approximate arbitrarily well any continuous univariate function. Since fixed weights reduce the computational expense and training time, this result is of particular interest. In a mathematical formulation, the result says that for a bounded measurable sigmoidal function σ, networks of the form

k i=1 c i σ(αx -θ i ) are dense in C[a, b].
Cao and Xie [START_REF] Cao | The construction and approximation for feedforword neural networks with fixed weights[END_REF] strengthened this result by specifying the number of hidden neurons to realize ε-approximation to any continuous function. By implementing modulus of continuity, they established Jackson-type upper bound estimations for the approximation error.

Approximation capabilities of SLFNs with fixed weights were also analyzed in Lin, Guo, Cao and Xu [START_REF] Lin | Approximation by neural networks with scattered data[END_REF]. Taking the activation function σ as a continuous, even and 2π-periodic function, the authors of [START_REF] Lin | Approximation by neural networks with scattered data[END_REF] showed that neural networks of the form r i=1 c i σ(x -x i ) can approximate any continuous function on [-π, π] with an arbitrary precision ε. Note that all the weights are fixed equal to 1, and consequently do not depend on ε. To prove this, they first gave an integral representation for trigonometric polynomials, and constructed explicitly a network with the weight 1 that approximates this integral representation. Finally, the obtained result for trigonometric polynomials was used to prove a Jackson-type upper bound for the approximation error.

Note that SLFNs with a fixed number of weights cannot approximate d-variable functions if d > 1. That is, if in (1.1) we have n different weights w i (n is fixed), then there exist a compact set Q ⊂ R d and a function f ∈ C(Q), which cannot be approximated arbitrarily well by the networks formed as (1.1). This follows from a result of Lin and Pinkus on sums of n ridge functions (see [START_REF] Lin | Fundamentality of ridge functions[END_REF]Theorem 5.1]). For details, see our recent paper [START_REF] Guliyev | On the approximation by single hidden layer feedforward neural networks with fixed weights[END_REF]. Thus the above results of Hahm and Hong [START_REF] Hahm | An approximation by neural networks with a fixed weight[END_REF], Cao and Xie [START_REF] Cao | The construction and approximation for feedforword neural networks with fixed weights[END_REF], Lin, Guo, Cao and Xu [START_REF] Lin | Approximation by neural networks with scattered data[END_REF] cannot be generalized to the d-dimensional case if one allows only the SLFN model of neural networks.

It should be remarked that in all of the above-mentioned works the number of neurons k in the hidden layer is not fixed. As such to achieve a desired precision one may take an excessive number of hidden neurons. Unfortunately, practicality decreases with the increase of the number of neurons in the hidden layer. In other words, SLFNs are not always effective if the number of neurons in the hidden layer is prescribed. More precisely, they are effective if and only if we consider univariate functions. In [START_REF] Guliyev | A single hidden layer feedforward network with only one neuron in the hidden layer can approximate any univariate function[END_REF], we consider constructive approximation on any finite interval of R by SLFNs with a fixed number of hidden neurons. We construct algorithmically a smooth, sigmoidal, almost monotone activation function σ providing approximation to an arbitrary univariate continuous function within any degree of accuracy. Note that the result of [START_REF] Guliyev | A single hidden layer feedforward network with only one neuron in the hidden layer can approximate any univariate function[END_REF] is not applicable to multivariate functions.

The first crucial step in investigating approximation capabilities of MLFNs with a prescribed number of hidden neurons was made by Maiorov and Pinkus [START_REF] Maiorov | Lower bounds for approximation by MLP neural networks[END_REF]. Their remarkable result revealed that TLFNs with 3d units in the first layer and 6d + 3 units in the second layer can approximate an arbitrary continuous d-variable function. Using a different activation function than in [START_REF] Maiorov | Lower bounds for approximation by MLP neural networks[END_REF], the second coauthor [START_REF] Ismailov | On the approximation by neural networks with bounded number of neurons in hidden layers[END_REF] showed that the number of neurons in hidden layers can be reduced to d and 2d + 2 respectively. Note that the results of both papers carry a theoretical character, as they indicate only the existence of the corresponding TLFNs, their activation functions.

We see that in each result above at least one of the following general properties is violated.

(1) the number of hidden neurons is fixed;

(2) the weights are fixed;

(3) the activation function is computable;

(4) the network has the capability of approximating d-variable functions in the case d > 1. In this paper, we construct a special TLFN model that satisfies all of the properties ( 1)-( 4). In addition, we show that along with the number of hidden neurons and weights, it is also possible to fix some dilation coefficients of the constructed activation function.

The main result

In the sequel, we deal with an activation function, which is monotonic in the weak sense. Here by weak monotonicity we understand behavior of a function whose difference in absolute value from a monotonic function is a sufficiently small number. In this regard we say that a real function f defined on a set X ⊆ R is λ-increasing (respectively, λ-decreasing) if there exists an increasing (respectively, decreasing) function u : X → R such that |f (x)-u(x)| ≤ λ for all x ∈ X. Clearly, 0monotonicity coincides with the usual concept of monotonicity and a λ 1 -increasing function is

λ 2 -increasing if λ 1 ≤ λ 2 .
Our main result is the following theorem. Here the weights w q , q = 1, . . . , d, are fixed as follows:

w 1 = (1, 0, . . . , 0), w 2 = (0, 1, . . . , 0), . . . , w d = (0, 0, . . . , 1).

In addition, all the coefficients e p , except one, are equal.

Proof. We start with the algorithmic construction of σ mentioned in the theorem.

The algorithm consists of the following steps.

1. Consider the function

h(x) := 1 - min{1/2, λ} 1 + log(x -s + 1)
.

Obviously, this function is strictly increasing on the real line and satisfies the following properties:

(1) 0 < h(x) < 1 for all x ∈ [s, +∞);

(2) 1 -h(s) ≤ λ;

(3) h(x) → 1 as x → +∞. Our purpose is to construct σ satisfying the two-sided inequality

h(x) < σ(x) < 1 (2.1)
for x ∈ [s, +∞). Then our σ will approach 1 as x approaches +∞ and obey the inequality

|σ(x) -h(x)| ≤ λ,
that is, it will be a λ-increasing function.

2. In this step, we enumerate the monic polynomials with rational coefficients. Let q n be the Calkin-Wilf sequence (see [START_REF] Calkin | Recounting the rationals[END_REF]). We can enumerate all the rational numbers by setting r 0 := 0, r 2n := q n , r 2n-1 := -q n , n = 1, 2, . . . . Note that each monic polynomial with rational coefficients can uniquely be written as r k0 + r k1 x + . . . + r k l-1 x l-1 + x l , and each positive rational number determines a unique finite continued fraction [m 0 ; m 1 , . . . , m l ] := m 0 + 1

m 1 + 1 m 2 + 1 . . . + 1 m l
with m 0 ≥ 0, m 1 , . . . , m l-1 ≥ 1 and m l ≥ 2. We now construct a one-to-one mapping between the set of all monic polynomials with rational coefficients and the set of all positive rational numbers as follows. To the only zeroth-degree monic polynomial 1 we associate the rational number 1, to each first-degree monic polynomial of the form r k0 + x we associate the rational number k 0 + 2, to each second-degree monic polynomial of the form r k0 + r k1 x + x 2 we associate the rational number [k 0 ; k 1 + 2] = k 0 + 1/(k 1 + 2), and to each monic polynomial

r k0 + r k1 x + . . . + r k l-2 x l-2 + r k l-1 x l-1 + x l
of degree l ≥ 3 we associate the rational number [k 0 ;

k 1 + 1, . . . , k l-2 + 1, k l-1 + 2].
In other words, we define u 1 (x) := 1,

u n (x) := r qn-2 + x if q n ∈ Z, u n (x) := r m0 + r m1-2 x + x 2 if q n = [m 0 ; m 1 ],
and

u n (x) := r m0 + r m1-1 x + . . . + r m l-2 -1 x l-2 + r m l-1 -2 x l-1 + x l if q n = [m 0 ; m 1 , . . . , m l-2 , m l-1 ] with l ≥ 3.
Hence the first few elements of this sequence are defined as

1, x 2 , x, x 2 -x, x 2 -1, x 3 , x -1, x 2 + x, . . . .
The sequence of monic polynomials will be used in the sequel.

3. First we construct σ on the intervals [(2n -1)s, 2ns], n = 1, 2, . . .. For each monic polynomial u n (x) = ρ 0 + ρ 1 x + . . . + ρ l-1 x l-1 + x l with rational coefficients, set

B 1 := ρ 0 + ρ 1 -|ρ 1 | 2 + . . . + ρ l-1 -|ρ l-1 | 2 and B 2 := ρ 0 + ρ 1 + |ρ 1 | 2 + . . . + ρ l-1 + |ρ l-1 | 2 + 1.
Note that the numbers B 1 and B 2 depend on n, but for simplicity we will omit this in the notation.

Consider the sequence

M n := h((2n + 1)s), n = 1, 2, . . . .
Obviously, this sequence is strictly increasing and converges to 1. Now we define σ as the function

σ(x) := a n + b n u n x s -2n + 1 , x ∈ [(2n -1)s, 2ns]. (2.2)
Here

a 1 := 1 2 , b 1 := h(3s) 2 , (2.3)
and

a n := (1 + 2M n )B 2 -(2 + M n )B 1 3(B 2 -B 1 ) , b n := 1 -M n 3(B 2 -B 1 )
, n = 2, 3, . . . .

σ(x) = 1 + M 1 2 .
Thus, we obtain that Recall that the numbers σ(2ns) and σ((2n + 1)s) have already been defined in the previous step. Since both the numbers σ(2ns) and σ((2n + 1)s) belong to the interval (M n , 1), it follows that K n ∈ (M n , 1). First we extend σ smoothly to the interval [2ns, 2ns + s/2]. Take the number ε := (1 -M n )/6 and select δ ≤ s/2 such that

h(x) < M n < 1 + 2M n 3 ≤ σ(x) ≤ 2 + M n 3 < 1, ( 2
a n + b n u n x s -2n + 1 -(a n + b n u n (1)) ≤ ε, x ∈ [2ns, 2ns + δ]. (2.6)
One can select this δ as

δ := min εs b n C , s 2
,
where C > 0 is any number satisfying |u n (x)| ≤ C for x ∈ (1, 1.5). For example, if n = 1, then δ can be selected as s/2. Now define σ on the left-hand half of the interval [2ns, (2n + 1)s] as the function

σ(x) := K n -β 2ns,2ns+δ (x) × K n -a n -b n u n x s -2n + 1 , x ∈ 2ns, 2ns + s 2 . ( 2.7)
Let us prove that σ(x) satisfies the condition (2.1). Indeed, if 2ns + δ ≤ x ≤ 2ns + s/2, then there is nothing to prove, since σ(x) = K n ∈ (M n , 1). If 2ns ≤ x < 2ns + δ, then 0 < β 2ns,2ns+δ (x) ≤ 1 and hence from (2.7) we obtain that for each x ∈ [2ns, 2ns + δ), σ(x) is between the numbers K n and A n (x) := a n + b n u n x s -2n + 1 . On the other hand, from (2.6) it follows that

a n + b n u n (1) -ε ≤ A n (x) ≤ a n + b n u n (1) + ε.
The last inequality together with (2.2) and the inequalities (2.5) yields that

A n (x) ∈ 1+2Mn 3 -ε, 2+Mn 3 + ε for x ∈ [2ns, 2ns + δ). Since ε = (1 -M n )/6, the inclusion A n (x) ∈ (M n , 1) is valid. Now since both K n and A n (x) lie in the interval (M n , 1), we conclude that h(x) < M n < σ(x) < 1, for x ∈ 2ns, 2ns + s 2 .
We define σ on the right-hand half of the interval in a similar way:

σ(x) := K n -(1 -β (2n+1)s-δ,(2n+1)s (x)) × K n -a n+1 -b n+1 u n+1 x s -2n -1 , x ∈ 2ns + s 2 , (2n + 1)s ,
where

δ := min εs b n+1 C , s 2 , ε := 1 -M n+1 6 , C ≥ sup [-0.5,0] |u n+1 (x)|.
It is not difficult to verify, as above, that the constructed σ(x) satisfies the condition (2.1) on [2ns + s/2, 2ns + s] and

σ 2ns + s 2 = K n , σ (i) 2ns + s 2 = 0, i = 1, 2, . . . .
Steps 3 and 4 together construct σ on the interval [s, +∞). 5. On the remaining interval (-∞, s), we define σ as

σ(x) := 1 -β(s -x) 1 + M 1 2 , x ∈ (-∞, s).
Clearly, σ is a strictly increasing, smooth function on (-∞, s). In addition, σ(x) → σ(s) = (1 + M 1 )/2, as x tends to s from the left and σ (i) (s) = 0 for i = 1, 2, . . .. This final step completes the construction of σ on the whole real line. Note that the constructed σ is sigmoidal, infinitely differentiable on R, strictly increasing on (-∞, s) and λ-strictly increasing on [s, +∞).

It should be noted that the above algorithm allows one to compute σ at any point of the real axis instantly. The code of this algorithm is available at https: //sites.google.com/site/njguliyev/papers/tlfn. As a practical example, we give here the graph of σ (see Figure 2.1) and a numerical table (see Table 2.1) containing several computed values of this function on the interval [0, 50]. All computations were done in SageMath [START_REF] Stein | Sage Mathematics Software (Version 7.6), The Sage Developers[END_REF]. Figure 2.2 shows how the graph of the λ-increasing function σ changes on the interval [0, 100] as the parameter λ decreases. Now we show that in addition to its nice properties such as computability, smoothness and weak monotonicity, our σ enjoys an important property of approximating each continuous d-variable function as an activation function for TLFNs with a fixed number of hidden neurons.

It follows from (2.2) that

σ(sx + (2n -1)s) = a n + b n u n (x), x ∈ [0, 1] (2.8)
for n = 1, 2, . . .. Here a n and b n are computed by (2.3) and (2.4) for n = 1 and n > 1 respectively. From (2.8) we obtain that each monic polynomial u n , n = 1, 2, . . ., can be represented in the form

u n (x) = 1 b n σ(sx + (2n -1)s) - a n b n .
(2.9) [START_REF] Kolmogorov | On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition (Russian)[END_REF] in the form given by Lorentz [START_REF] Lorentz | Metric entropy, widths, and superpositions of functions[END_REF] and Sprecher [START_REF] Sprecher | On the structure of continuous functions of several variables[END_REF], there exist constants λ q > 0, q = 1, . . ., d with (2.10) for some g ∈ C[0, 1] depending on f . By the density of polynomials with the rational coefficients in the space of continuous functions over any compact subset of R, for the exterior continuous univariate function g in (2.10) and any ε > 0 there exists a polynomial p(x) of the mentioned form such that |g(x) -p(x)| < ε 2(2d + 1) This together with (2.9) means that

|g(x) -(α 0 σ(sx -β 0 ) -γ 0 )| < ε 2(2d + 1) (2.11)
for some α 0 , β 0 , γ 0 ∈ R and all x ∈ [0, 1]. Namely,

α 0 = p 0 b n , β 0 = s -2ns, γ 0 = p 0 a n b n .
(2.12) Substituting (2.11) in (2.10) we obtain that

f (x 1 , . . . , x d ) - 2d+1 p=1 α 0 σ s d q=1 λ q φ p (x q ) -β 0 -γ 0 < ε 2 (2.13) for all (x 1 , . . . , x d ) ∈ [0, 1] d .
For each p = 1, . . ., 2d + 1, the function φ p in (2.10) is defined on [a, b]. For this function, using the linear transformation x = (t -a)/s from [a, b] to [0, 1] and the same procedure for the function g above, we can obtain the inequality

|φ p (t) -(α p σ(t -β p ) -γ p )| < δ, (2.14)
for all t ∈ [a, b].
Here δ is any positive real number, and the parameters α p , β p and γ p depend on δ. Note that these parameters can be computed similarly as in (2.12). Since λ q > 0 for q = 1, . . ., d, and

d q=1 λ q = 1, it follows from (2.14) that d q=1 λ q φ p (x q ) - d q=1 λ q α p σ(x q -β p ) -γ p < δ, (2.15)
for all p = 1, . . ., 2d + 1, and (x 1 , . . . , x d ) ∈ [0, 1] d . Now since the function α 0 σ(sx -β 0 ) is uniformly continuous on every closed interval of the real line, we can choose δ as small as necessary and obtain from (2.15) that

2d+1 p=1 α 0 σ s d q=1 λ q φ p (x q ) -β 0 - 2d+1 p=1 α 0 σ s d q=1 λ q α p σ(x q -β p ) -γ p -β 0 < ε 2 .
This inequality may be rewritten in the form

2d+1 p=1 α 0 σ s d q=1 λ q φ p (x q ) -β 0 - 2d+1 p=1 α 0 σ d q=1 c pq σ(w q • x -β p ) -ζ p < ε 2 ,
(2.16) where c pq = sλ q α p , ζ p = sγ p + β 0 , and w q is the q-th coordinate vector. From (2.13) and (2.16) it follows that

f (x) - 2d+1 p=1 α 0 σ d q=1 c pq σ(w q • x -β p ) -ζ p -(2d + 1)γ 0 < ε, (2.17)
Clearly, the constant (2d + 1)γ 0 can be written in the form

(2d + 1)γ 0 = ασ d q=1 c q σ(w q • x -θ q ) -ζ , (2.18)
for c q = 0, q = 1, . . ., d, and suitable coefficients α and ζ. Considering (2.18) in (2.17 Remark 2.2. Theorem 2.1, in particular, shows that TLFNs are more powerful than SLFNs, since SLFNs with a fixed number of hidden neurons and/or weights have not the capability of approximating multivariate functions (see Introduction). We refer the reader to [START_REF] Lin | Limitations of shallow nets approximation[END_REF] for interesting results and discussions around the comparison of performances between MLFNs and SLFNs.

Remark 2.3. In [START_REF] Gripenberg | Approximation by neural networks with a bounded number of nodes at each level[END_REF], Gripenberg showed that the general approximation property of feedforward multilayer perceptron networks can be achieved in networks where the number of neurons in each layer is bounded, but the number of layers grows to infinity. This is the case provided the activation function is twice continuously differentiable and not linear. Taking an exceedingly large number of layers is an indispensable part of Gripenberg's method. Can one develop a different method which enables to use only a preliminarily prescribed number of layers for all approximated functions? To answer this question, we started with SLFNs. It turned out that in this case the answer is "yes" if approximated functions are univariate (see [START_REF] Guliyev | A single hidden layer feedforward network with only one neuron in the hidden layer can approximate any univariate function[END_REF]). Moreover, one can fix the weights of constructed SLFNs. But SLFNs with fixed weights or bounded number of neurons are proved not capable of approximating multivariate functions (see [START_REF] Guliyev | On the approximation by single hidden layer feedforward neural networks with fixed weights[END_REF]). Then how many hidden layers with bounded number of neurons are needed to approximate multivariate functions with arbitrary precision? First of all, one may want to know if any such constrained approximation is possible in practice. Theorem 2.1 shows that even two hidden layers and a specifically constructed activation function are sufficient to solve this problem affirmatively.

Theorem 2 . 1 .c

 21 Assume a closed interval [a, b] ⊂ R is given, s = b -a, and λ is any sufficiently small positive real number. Then one can algorithmically construct a computable, infinitely differentiable, sigmoidal activation function σ : R → R which is strictly increasing on (-∞, s), λ-strictly increasing on [s, +∞) and satisfies the following property: For any continuous function f on the d-dimensional box [a, b] d and ε > 0, there exist constants e p , c pq , θ pq and ζ p such that the inequality f (x)pq σ(w q • x -θ pq ) -ζ p < ε holds for all x = (x 1 , . . . , x d ) ∈ [a, b] d .

(2. 4 )

 4 It is not difficult to see that for n > 2 the numbers a n , b n are the coefficients of the linear function y = a n + b n x mapping the closed interval [B 1 , B 2 ] onto the closed interval [(1 + 2M n )/3, (2 + M n )/3]. In addition, for n = 1, i.e. on the interval [s, 2s],

. 5 )

 5 for all x ∈ [(2n -1)s, 2ns], n = 1, 2, . . .. 4. In this step, we construct σ on the intervals [2ns, (2n + 1)s], n = 1, 2, . . .. To this end we use the smooth transition function β a,b (x) := β(b -x) β(b -x) + β(x -a) , where β(x) := e -1/x , x > 0, 0, x ≤ 0. Clearly, β a,b (x) = 1 for x ≤ a, β a,b (x) = 0 for x ≥ b, and 0 < β a,b (x) < 1 for a < x < b. Consider the sequence K n := σ(2ns) + σ((2n + 1)s) 2 , n = 1, 2, . . . .

Figure 2 . 1 .

 21 Figure 2.1. The graph of σ on [0, 50] (s = 3, λ = 1/2)

Figure 2 .

 2 3 displays variations in the graph of σ with respect to the length s of a closed interval [a, b].

Figure 2 . 2 .

 22 Figure 2.2. Changes in the graph of σ with respect to λ (s = 1)

dλ

  q=1 λ q = 1 and nondecreasing continuous functionsφ p : [a, b] → [0, 1], p = 1, . . ., 2d+ 1 such that every continuous function f : [a, b] d → R admits the representation f (x 1 , . . . , x d ) = q φ p (x q )

Figure 2 . 3 .

 23 Figure 2.3. Changes in the graph of σ with respect to s (λ = 0.75)

c

  pq σ(w q • x -θ pq ) -ζ p < ε, where e 1 = e 2 = . . . = e 2d+1 . The last inequality completes the proof of the theorem. Remark 2.1. Obviously, a compact subset Q of the space R d can be embedded into a box [-l, l] d , and by the Tietze extension theorem (see [41, Theorem 15.8]), any continuous function f on Q can be extended to [-l, l] d . Hence Theorem 2.1 is valid not only for boxes of the form [a, b] d but for any compact set Q ⊂ R d , with the proviso that s = 2l.

Table 2 .

 2 [START_REF] Calkin | Recounting the rationals[END_REF]. Some computed values of σ (s = 3, λ = 1/2)

	t	σ	t	σ	t	σ	t	σ	t	σ
	0 0.25941 10 0.91169 20 0.94932 30 0.96241 40 0.94166
	1 0.36008 11 0.92728 21 0.94074 31 0.94506 41 0.95333
	2 0.57848 12 0.95325 22 0.93635 32 0.94003 42 0.96499
	3 0.91514 13 0.93437 23 0.93635 33 0.92771 43 0.95602
	4 0.91514 14 0.92551 24 0.94074 34 0.92905 44 0.94295
	5 0.91514 15 0.91549 25 0.93278 35 0.93842 45 0.93186
	6 0.91514 16 0.92958 26 0.93177 36 0.96385 46 0.93943
	7 0.91198 17 0.94366 27 0.92482 37 0.94692 47 0.95079
	8 0.91105 18 0.95775 28 0.92900 38 0.93923 48 0.96593
	9 0.90650 19 0.95532 29 0.94153 39 0.92999 49 0.95800

Acknowledgements

The research of the second author was supported by the Azerbaijan National Academy of Sciences under the program "Approximation by neural networks and some problems of frames".