Duals of Feynman integrals. Part I. Differential equations

We elucidate the vector space (twisted relative cohomology) that is Poincaré dual to the vector space of Feynman integrals (twisted cohomology) in general spacetime dimension. The pairing between these spaces — an algebraic invariant called the intersection number — extracts integral coefficients for a minimal basis, bypassing the generation of integration-by-parts identities. Dual forms turn out to be much simpler than their Feynman counterparts: they are supported on maximal cuts of various sub-topologies (boundaries). Thus, they provide a systematic approach to generalized unitarity, the reconstruction of amplitudes from on-shell data. In this paper, we introduce the idea of dual forms and study their mathematical structures. As an application, we derive compact differential equations satisfied by arbitrary one-loop integrals in non-integer spacetime dimension. A second paper of this series will detail intersection pairings and their use to extract integral coefficients.

Crystal morphology of spherical viruses

The article discusses modern views on the structure of spherical virus capsids, which have the shape of icosahedrons (icosahedral viruses). Each face of icosahedron is composed of a single-layer closest packing of protein globules, which can have different orientation relative to the edges of icosahedron. If the lines of globules are parallel to the edges of icosahedron, then the capsid has a point symmetry group Ih (with symmetry planes), if they are not parallel – the symmetry group I (without planes). From a mathematical point of view, in both symmetry groups there are series that unite equally (up to similarity) arranged capsids. They are connected pairwise by transitions to dual forms (homologous series). A hypothesis is formulated that the largest spherical viruses can have even more diverse and complex capsid structures. Along with icosahedron, their basic forms can be any simple shapes, allowed in Ih and I symmetry groups (8 in total). A suggestion is made that transitions within similarity series and between homologous series have a phylogenetic significance. There are known spherical viruses of both symmetry groups. For example, the SARS-CoV-2 coronavirus has a symmetry group Ih and belongs to a well-known series. The crystallographic approach allows to construct a strict morphological classification of spherical viruses. This is important for their early recognition and separate examination. The article demonstrates practical application of crystal morphology in the study of viral systems – an urgent problem of geoecology and life protection.

The dual form of education is an embodied reality

The organizational principles and results of the implementation of the pilot project of dual form of education for applicants for the second (master's) level of higher education in the specialty 151 "Automation and computer-integrated technologies" are considered. A brief retrospective look at the history of dual education is given. A comparative analysis of the institutional, individual and dual forms of education according to the most popular professional competencies in the modern labor market is given. Recommendations of the Working Group of the Ministry of Education and Science of Ukraine on the organization of dual education, the procedure for enrollment of students in dual education, the peculiarities of the implementation of this form of education. The enterprises that took part in the pilot project on the basis of the memorandum of cooperation are identified. The results of a pilot project on the organization and implementation of the dual form of education at the Kharkiv National University of Construction and Architecture are presented. For the practical implementation of the dual form of education, an integrated model was chosen – a model of a divided week, which provides for training two days a week in an educational institution, three days – in the workplace at the enterprise. The step-by-step algorithm of introduction of dual education is offered, practically worked out during realization of the pilot project, components of efficiency of such form of receiving education are defined. The connection of the pilot project with the Licensing conditions for the educational activities of educational institutions in terms of recognition of the qualifications of research and teaching staff in the relevant specialty is shown. It is proposed to use the experience of the pilot project for other technical specialties.

Robust Optimization Model with Shared Uncertain Parameters in Multi-Stage Logistics Production and Inventory Process

In this paper, we focus on a class of robust optimization problems whose objectives and constraints share the same uncertain parameters. The existing approaches separately address the worst cases of each objective and each constraint, and then reformulate the model by their respective dual forms in their worst cases. These approaches may result in that the value of uncertain parameters in the optimal solution may not be the same one as in the worst case of each constraint, since it is highly improbable to reach their worst cases simultaneously. In terms of being too conservative for this kind of robust model, we propose a new robust optimization model with shared uncertain parameters involving only the worst case of objectives. The proposed model is evaluated for the multi-stage logistics production and inventory process problem. The numerical experiment shows that the proposed robust optimization model can give a valid and reasonable decision in practice.

The Category of Dual in the Two Slavic Grammar Books: Adam Bohorič and Meletius Smotrytsky

In the Slavic grammar books of the 16th-17th centuries, the category of dual was represented in the fi rst Slovenian grammar book «Arcticae horulae succisivae» (1584, Wit-tenberg) by the Protestant grammarian A. Bohorič (1520 –1588) and in the grammar book of the Church Slavonic lan-guage «Slavonic Grammar with Correct Syntax» (1619, Jevje) by M. Smotrytsky (1578–1633 (?)).It was assumed earlier that the Slovenian grammar had had a possible impact on Smotrytsky’s Grammar. However, the analysis of the dual forms in Bohorič’s G r a m m a r and Smotrytsky’s Grammar showed that such an impact was most likely impossible. When creating their gram-mar books, the authors were guided by different aims: if A. Bohorič was following the practice of using the forms of a living language, then Smotrytsky was orienting towards the fi xed stable forms of dead written language.

Seeing doubles: On two of a kind

This article expands upon a previously published study in this journal that examined a literary strategy which employs linguistic devices of doubling in biblical narratives that reference twins. It demonstrated that Israelite authors employed several devices, including dual forms, gemination, doubled vocabulary, polysemy, and paronomasia on the number ‘two’, in order to match form to content. Here, we add studies of four biblical passages (Judges 5, Proverbs 20, 30, 31) as well as a brief excursus on a similar strategy in the Epic of Gilgamesh, and we argue that the strategy of doubling also appears in narratives that feature pairs or the doubling of amounts (i.e. not necessarily texts that employ the term ‘twin’).

Liczba podwójna w archaicznej poezji rzymskiej

The archaic epitaph of Lucius Cornelius Scipio (son of Barbatus), consul of 259 BC, contains the Old Latin form Tempestātēbus (in the dative), referring probably to two goddesses of weather (CIL I2 9). This theonym proves the existence in the Old Latin language of the dual formation *Tempestātē ‘two deities of weather’. The word aide, attested in the epitaph of Scipio, son of Barbatus, can be easily interpreted as a possible form of the dual number, namely *aideē (nom.-acc. du. f.) ‘a bipartite (two-sided) temple; a sacral building with two sanctuaries’. Livius Andronicus used some forms of the dual number in his Latin translation of Homer’s Odyssey, among others genū (nom.-acc. du. n.) ‘two knees’ (fr. 14), inque manū (acc. du. f.) ‘and to both hands’ (fr. 5). It is not known, however, whether the use of the dual number in Andronicus’ Odussia was consistent or optional. Naevius introduced some dual forms into the epic work of Bellum Punicum, e.g. hōc Samnītē ‘these two Samnites’ (fr. 36). It seems, however, that the poet did not do it consistently. The metrical and linguistic analysis of the preserved fragments does not give a definite answer as to whether the use of forms of the dual number was obligatory in Naevius’ work, or was the result of the free choice of the Roman poet. In some passages passed down through indirect tradition, it was easy to arrive at the secondary plurality of the old dual forms.