Two first-order logics of permutations

We consider two orthogonal points of view on finite permutations, seen as pairs of linear orders (corresponding to the usual one line representation of permutations as words) or seen as bijections (corresponding to the algebraic point of view). For each of them, we define a corresponding first-order logical theory, that we call $\mathsf{TOTO}$ (Theory Of Two Orders) and $\mathsf{TOOB}$ (Theory Of One Bijection) respectively. We consider various expressibility questions in these theories. Our main results go in three different direction. First, we prove that, for all $k \ge 1$, the set of $k$-stack sortable permutations in the sense of West is expressible in $\mathsf{TOTO}$, and that a logical sentence describing this set can be obtained automatically. Previously, descriptions of this set were only known for $k \le 3$. Next, we characterize permutation classes inside which it is possible to express in $\mathsf{TOTO}$ that some given points form a cycle. Lastly, we show that sets of permutations that can be described both in $\mathsf{TOOB}$ and $\mathsf{TOTO}$ are in some sense trivial. This gives a mathematical evidence that permutations-as-bijections and permutations-as-words are somewhat different objects.Comment: v2: minor changes, following a referee repor

Algebraic Relations Between Harmonic Sums and Associated Quantities

We derive the algebraic relations of alternating and non-alternating finite harmonic sums up to the sums of depth~6. All relations for the sums up to weight~6 are given in explicit form. These relations depend on the structure of the index sets of the harmonic sums only, but not on their value. They are therefore valid for all other mathematical objects which obey the same multiplication relation or can be obtained as a special case thereof, as the harmonic polylogarithms. We verify that the number of independent elements for a given index set can be determined by counting the Lyndon words which are associated to this set. The algebraic relations between the finite harmonic sums can be used to reduce the high complexity of the expressions for the Mellin moments of the Wilson coefficients and splitting functions significantly for massless field theories as QED and QCD up to three loop and higher orders in the coupling constant and are also of importance for processes depending on more scales. The ratio of the number of independent sums thus obtained to the number of all sums for a given index set is found to be $\leq 1/d$ with $d$ the depth of the sum independently of the weight. The corresponding counting relations are given in analytic form for all classes of harmonic sums to arbitrary depth and are tabulated up to depth $d=10$.Comment: 39 pages LATEX, 1 style fil

Теория модальностей А. С. Есенина-Вольпина

Yesenin-Volpin’s theory of modalities eliminates any beliefs from the foundations of mathematics and humanitarian knowledge. Some methods of moral and legal discussions are fuzzy and poorly formalized. The human activities on construction of theories and moral reasoning are submitted to the rules. Both cases include permissions, orders and forbidding different acts. The same act may be allowed in one situation and forbidden in the other situation. Yesenin-Volpin calls the set of orders correlated with suitable situations as “tactics”. He defines hierarchy of tactics which explains the principles of Kant’s autonomic ethics and legislative systems. All rational activities are described by modal words of different groups. Deontic modalities (to be bound, to be permitted, to be forbidden) have no similar axioms for alethic modalities (necessary, possible, impossible). The general meta-theory explains the relations between all kinds of modalities and grounds the construction of any strict mathematic or ethic theory.Теория модальностей А. С. Есенина-Вольпина  исключает  какие-либо  верования  из оснований  математики  и  гуманитарных  знаний. Некоторые методы моральных и правовых рассуждений  неотчетливы  и  не  формализованы  вследствие употребления деонтических модальностей по аналогии с алетическими. Построение научной теории и морального рассуждения осуществляется по правилам. Деонтическая логика формализует отношения между правилами любой природы. Одно и то же деяние может быть разрешено в одной ситуации и запрещено в другой. Есенин-Вольпин называет класс предписаний с соответствующимиситуациями «тактикой». Он определяет иерархию тактик, что, на наш взгляд, позволяет объяснить принципы автономной этики Канта и строение законодательных систем. Все рациональные действия описываются модальными словами разных групп. Правила связей деонтических модальностей («Обязательно», «Разрешено», «Запрещено») не всегда имеют аналогии с аксиомами для алетических модальностей («Необходимо», «Возможно», «Невозможно»). Обобщенная метатеория объясняет отношения между всеми видами модальностей, а также лежит в основании построениястрогих математических или этических теорий

N=1N=1 supersymmetry and the three loop anomalous dimension for the chiral superfield

We calculate the three loop anomalous dimension for a general $N=1$ supersymmetric gauge theory. The result is used to probe the possible existence of renormalisation invariant relationships between the Yukawa and gauge couplings.Comment: 18 pages. Uses Harvmac. Revised version includes discussion of the special case of the Wess-Zumino mode

Club guessing and the universal models

We survey the use of club guessing and other pcf constructs in the context of showing that a given partially ordered class of objects does not have a largest, or a universal element

Some limits of standard linguistic typology: the case of Cysouw's models for the frequencies of the six possible orderings of S, V and O

This article is a critical analysis of Michael Cysouw's comment "Linear order as a predictor of word order regularities".Peer ReviewedPostprint (author's final draft

Theories of practice and geography

Recent developments in theories of practice have seen place and space taken explicitly into account. In particular, THEODORE SCHATZKI’s ‘site ontology’ offers distinctive but as yet under-explored means of engaging with human geographies. By giving ontological priority to practices as constitutive of the social, this kind of practice theory provides an integrative conceptual framework that enables the analysis of diverse phenomena in relation to each other, over space and time, as they are constituted through practices. This article develops an outline agenda for bringing theories of practice, and particularly SCHATZKI’s ‘site ontology’, together with geographical inquiry. We elucidate this agenda through consideration of three contemporary preoccupations in human geography, comprising emotion, materiality and knowledge