Vihara architecture: defining the existential foothold of VIII century Buddhist monastery "Somapura Mahavihara" of ancien Bengal

Lost in history, the ruin of ‘Somapura Mahavihara’ was not recognized separated from its birthplace, i.e. nature, for more than 700 years. Yet, within its silent presence, the monument dominated the name of the region: ‘Paharpur’ (land of hillock), according to its appearance surrounds by its flat land topo¬graphy. Discovered in 1919, the single largest Buddhist Vihara (monas¬tery) of ancient Bengal came into light, pronouncing the flou¬rishing minute of Buddhist architecture, once dominant religious force of the subcontinent. The earliest historical monumental architecture of greater Asia, had long been deriving itself from the Buddhist monastic architecture as early as VI century BC. In line of history, the discovery of ‘Somapura Mahavihara’ contributed attesting the sensitivities of a highly sophisticated architectonic typology of Vihara Architecture in the land of ancient Bengal. The recovery of ‘Somapura Mahavihara’ was not only from its cradle of nature, but also from its remarkable existence imprinted in the reign of Pala dynasty (750 - 1155 AD) announcing the existential foothold of man in his nature. The existential foothold of ‘Somapura Mahavihara’ comprises the factors, responsible in shaping the anchorage of the mo¬nument since the birth of Vihara architecture, as early as 530 BC. These factors not only denote the building technology in response to its environment but also the amalgamation of be¬lief, upon which the dwellers transformed the site as a place announcing their existence on earth. This research paper aims at exploring the existential foothold of ‘Somapura Mahavihara’, in terms of its territorial, functional, structural, social, cultural, religious sym¬bolic hierarchies of human achievement while clarifying the architectonic typology that shaped ‘Somapura Mahavihara’ through evolution process of ‘Vihara Architecture’. This understanding intends to combine the archaeological knowledge with comparative architectural analysis of contem¬porary Viharas of ancient Bengal, to define the singularity of ‘Somapura Mahavihara’. In consequence, the glorious past of ‘Somapura Mahavihara’ is intended to portray through iden¬tifying the relation of religious and functional rationalism with the connotation of art, architecture and belief moulded within natural forces, as one complete entity; RESUMO: Vihara Arquitetura: Definindo a posição existencial do século VIII Budista mosteiro “Somapura Mahavihara” de Bengala antiga. Perdidas na História, as ruínas de ‘Somapura Mahavihara’ foram confundidas com uma montanha durante mais de setecentos anos. Contudo, no seu silêncio presente, o monumento marcou a toponímia da região; ‘Paharpur’ significa ‘a terra do outeiro’, evidenciando a singularidade deste monumento numa região dominada por uma extensa planície. Em 1919, foi descoberto o maior mosteiro budista da antiga região de Bengal, demonstrando a prosperidade da arquitectura budista. Tem¬poralmente, a descoberta de ‘Somapura Mahavihara’ contribuiu para atestar a evolução e a sofisticação da tipologia arquitectónica denominada ‘Arquitectura Vihara’, existente na antiga região de Bengal. A noção de pegada existencial de ‘Somapura Mahavihara’ compreende os factores responsáveis por moldar a ancoragem do monumento ao lugar em que se insere desde o início da arquitectura Vihara, que remonta a 530 a.C. Estes factores evidenciam a tecnologia construtiva empregue para responder ao ambiente envolvente mas também a evolução da religião, factores estes que os monges construtores consideraram ao transformar o lugar e anunciar a sua existência na Terra. Esta investigação tem por objectivo explorar a noção de pegada existencial de ‘Somapura Mahavihara’, nas suas dimensões territoriais, funcionais, estruturais, sociais, culturais e nas hierarquias simbólicas das realizações humanas para clarificar a tipologia arquitectónica que deu forma a ‘Somapura Mahavihara’ durante a evolução da arquitectura Vihara. Este entendimento pretende combinar/cruzar o conhecimen¬to arqueológico com estudos arquitectónicos comparativos de Viharas na antiga região de Bengal, com o objectivo de definir a singularidade de ‘Somapura Mahavihara’. Neste estudo estudar-se-á também o confronto entre a dimensão religiosa e a artística (divino vs. humano), integrados na arquitectura de ‘Somapura Mahavihara’ em perfeita harmonia.

A pointwise constrained version of the Liapunov convexity theorem for vectorial linear first-order control systems

We generalize the Liapunov convexity theorem's version for vectorial control systems driven by linear ODEs of first-order p = 1 , in any dimension d ∈ N , by including a pointwise state-constraint. More precisely, given a x ‾ ( ⋅ ) ∈ W p , 1 ( [ a , b ] , R d ) solving the convexified p-th order differential inclusion L p x ‾ ( t ) ∈ co { u 0 ( t ) , u 1 ( t ) , … , u m ( t ) } a.e., consider the general problem consisting in finding bang-bang solutions (i.e. L p x ˆ ( t ) ∈ { u 0 ( t ) , u 1 ( t ) , … , u m ( t ) } a.e.) under the same boundary-data, x ˆ ( k ) ( a ) = x ‾ ( k ) ( a ) & x ˆ ( k ) ( b ) = x ‾ ( k ) ( b ) ( k = 0 , 1 , … , p − 1 ); but restricted, moreover, by a pointwise state constraint of the type 〈 x ˆ ( t ) , ω 〉 ≤ 〈 x ‾ ( t ) , ω 〉 ∀ t ∈ [ a , b ] (e.g. ω = ( 1 , 0 , … , 0 ) yielding x ˆ 1 ( t ) ≤ x ‾ 1 ( t ) ). Previous results in the scalar d = 1 case were the pioneering Amar & Cellina paper (dealing with L p x ( ⋅ ) = x ′ ( ⋅ ) ), followed by Cerf & Mariconda results, who solved the general case of linear differential operators L p of order p ≥ 2 with C 0 ( [ a , b ] ) -coefficients. This paper is dedicated to: focus on the missing case p = 1 , i.e. using L p x ( ⋅ ) = x ′ ( ⋅ ) + A ( ⋅ ) x ( ⋅ ) ; generalize the dimension of x ( ⋅ ) , from the scalar case d = 1 to the vectorial d ∈ N case; weaken the coefficients, from continuous to integrable, so that A ( ⋅ ) now becomes a d × d -integrable matrix; and allow the directional vector ω to become a moving AC function ω ( ⋅ ) . Previous vectorial results had constant ω, no matrix (i.e. A ( ⋅ ) ≡ 0 ) and considered: constant control-vertices (Amar & Mariconda) and, more recently, integrable control-vertices (ourselves).