Integrability for Nonlinear Time-Delay Systems

In this note, the notion of integrability is defined for 1-forms defined in the time-delay context. While in the delay-free case, a set of 1-forms defines a vector space, it is shown that 1-forms computed for time-delay systems have to be viewed as elements of a module over a certain non-commutative polynomial ring. Two notions of integrability are defined, strong and weak integrability, which coincide in the delay-free case. Necessary and sufficient conditions are given to check if a set of 1-forms is strongly or weakly integrable. To show the importance of the topic, integrability of 1-forms is used to characterize the accessibility property for nonlinear time-delay systems. The possibility of transforming a system into a certain normal form is also considered

Eigenvalue of a semi-infinite elastic strip

A semi-infinite elastic strip, subjected to traction free boundary conditions, is studied in the context of in-plane stationary vibrations. By using normal (Rayleigh–Lamb) mode expansion the problem of existence of the strip eigenmode is reformulated in terms of the linear dependence within infinite system of normal modes. The concept of Gram's determinant is used to introduce a generalized criterion of linear dependence, which is valid for infinite systems of modes and complex frequencies. Using this criterion, it is demonstrated numerically that in addition to the edge resonance for the Poisson ratio ν=0, there exists another value of ν≈0.22475 associated with an undamped resonance. This resonance is best explained physically by the orthogonality between the edge mode and the first Lamé mode. A semi-analytical proof for the existence of the edge resonance is then presented for both described cases with the help of the augmented scattering matrix formalism

Patterns in rational base number systems

Number systems with a rational number $a/b > 1$ as base have gained interest in recent years. In particular, relations to Mahler's 3/2-problem as well as the Josephus problem have been established. In the present paper we show that the patterns of digits in the representations of positive integers in such a number system are uniformly distributed. We study the sum-of-digits function of number systems with rational base $a/b$ and use representations w.r.t. this base to construct normal numbers in base $a$ in the spirit of Champernowne. The main challenge in our proofs comes from the fact that the language of the representations of integers in these number systems is not context-free. The intricacy of this language makes it impossible to prove our results along classical lines. In particular, we use self-affine tiles that are defined in certain subrings of the ad\'ele ring $\mathbb{A}_\mathbb{Q}$ and Fourier analysis in $\mathbb{A}_\mathbb{Q}$. With help of these tools we are able to reformulate our results as estimation problems for character sums

Grammars over the Lambek Calculus with Permutation: Recognizing Power and Connection to Branching Vector Addition Systems with States

In [Van Benthem, 1991] it is proved that all permutation closures of context-free languages can be generated by grammars over the Lambek calculus with the permutation rule (LP-grammars); however, to our best knowledge, it is not established whether converse holds or not. In this paper, we show that LP-grammars are equivalent to linearly-restricted branching vector addition systems with states and with additional memory (shortly, lBVASSAM), which are modified branching vector addition systems with states. Then an example of such an lBVASSAM is presented, which generates a non-semilinear set of vectors; this yields that LP-grammars generate more than permutation closures of context-free languages. Moreover, equivalence of LP-grammars and lBVASSAM allows us to present a normal form for LP-grammars and, as a consequence, prove that LP-grammars are equivalent to LP-grammars without product

Cooperation Evolution in Random Multiplicative Environments

Most real life systems have a random component: the multitude of endogenous and exogenous factors influencing them result in stochastic fluctuations of the parameters determining their dynamics. These empirical systems are in many cases subject to noise of multiplicative nature. The special properties of multiplicative noise as opposed to additive noise have been noticed for a long while. Even though apparently and formally the difference between free additive vs. multiplicative random walks consists in just a move from normal to log-normal distributions, in practice the implications are much more far reaching. While in an additive context the emergence and survival of cooperation requires special conditions (especially some level of reward, punishment, reciprocity), we find that in the multiplicative random context the emergence of cooperation is much more natural and effective. We study the various implications of this observation and its applications in various contexts.Comment: 20 pages 7 figur

Naked Object File System (NOFS): A Framework to Expose an Object-Oriented Domain Model as a File System

We present Naked Objects File System (NOFS), a novel framework that allows a developer to expose a domain model as a file system by leveraging the Naked Objects design principle. NOFS allows a developer to construct a file system without having to understand or implement all details related to normal file systems development. In this paper we explore file systems frameworks and object-oriented frameworks in a historical context and present an example domain model using the framework. This paper is based on a fully-functional implementation that is distributed as free/open source software, including virtual machine images to demonstrate and study the referenced example file systems

Dual-context multicategories as models for implicit computational complexity

In this thesis we study dual-context type systems and their models. A dual-context type system is one in which the context of a term is split into two parts, a normal part and a safe part. This separation allows one to put different kinds of usage restrictions on the two types of variable. For example, in implicit computational complexity, such a separation is used to implement resource-free characterisations of complexity classes. A similar separation between two kinds of variable occurs in type systems for linear logic, where different structural operations are permitted in the normal and safe parts of the context.We start by defining two basic dual-context calculi II(Σ) and IL(Σ) of typed terms, built using two kinds of free variables and dual-typed function symbols. In the IL(Σ) calculus we use safe variables in a 'linear' fashion, forbidding any weakening and contraction, while, in the II(Σ) calculus, contraction and weakening are allowed for both safe and normal variables.We then consider models for II(Σ) and IL(Σ) with basic equational judgements. Rather than following the traditional approach of encoding dual-contexts through additional structure on a category, we consider dual-context multicategories in which dual-contexts are built into the definition. The main advantage this approach is that it covers a wider class of models, including some particularly natural models from the field of implicit computational complexity.Next we enrich our basic calculi with different type constructions such as products and sums and provide their multicategorical interpretation. We consider a dual-context list type constructor together with a type-theoretic analogue of the safe recursion of Bellantoni and Cook's system ß, which characterises polynomial time computability on natural numbers. We define an interpretation for such dual-context lists using safe list objects. We show that the polytime computable functions are exactly the functions definable in every dual-context multicategory with safe binary list object.Finally, motivated by the standard approach to the categorical interpretation of primitive recursion using the notion of initial algebra, we develop a notion of safe initial algebra, which generalises the safe recursion scheme and provides us with insights about the choice of initial functions in system ß

The field theory of symmetrical layered electrolytic systems and the thermal Casimir effect

We present a general extension of a field-theoretic approach developed in earlier papers to the calculation of the free energy of symmetrically layered electrolytic systems which is based on the Sine-Gordon field theory for the Coulomb gas. The method is to construct the partition function in terms of the Feynman evolution kernel in the Euclidean time variable associated with the coordinate normal to the surfaces defining the layered structure. The theory is applicable to cylindrical systems and its development is motivated by the possibility that a static van der Waals or thermal Casimir force could provide an attractive force stabilising a dielectric tube formed from a lipid bilayer, an example of which are t-tubules occurring in certain muscle cells. In this context, we apply the theory to the calculation of the thermal Casimir effect for a dielectric tube of radius $R$ and thickness $\delta$ formed from such a membrane in water. In a grand canonical approach we find that the leading contribution to the Casimir energy behaves like $-k_BTL\kappa_C/R$ which gives rise to an attractive force which tends to contract the tube radius. We find that $\kappa_C \sim 0.3$ for the case of typical lipid membrane t-tubules. We conclude that except in the case of a very soft membrane this force is insufficient to stabilise such tubes against the bending stress which tend to increase the radius. We briefly discuss the role of lipid membrane reservoir implicit in the approach and whether its nature in biological systems may possibly lead to a stabilising mechanism for such lipid tubes.Comment: 28 pages, 2 figures, LaTe