A saturation property of structures obtained by forcing with a compact family of random variables

A method how to construct Boolean-valued models of some fragments of arithmetic was developed in Krajicek (2011), with the intended applications in bounded arithmetic and proof complexity. Such a model is formed by a family of random variables defined on a pseudo-finite sample space. We show that under a fairly natural condition on the family (called compactness in K.(2011)) the resulting structure has a property that is naturally interpreted as saturation for existential types. We also give an example showing that this cannot be extended to universal types.Comment: preprint February 201

Random l-colourable structures with a pregeometry

We study finite $l$-colourable structures with an underlying pregeometry. The probability measure that is used corresponds to a process of generating such structures (with a given underlying pregeometry) by which colours are first randomly assigned to all 1-dimensional subspaces and then relationships are assigned in such a way that the colouring conditions are satisfied but apart from this in a random way. We can then ask what the probability is that the resulting structure, where we now forget the specific colouring of the generating process, has a given property. With this measure we get the following results: 1. A zero-one law. 2. The set of sentences with asymptotic probability 1 has an explicit axiomatisation which is presented. 3. There is a formula $\xi(x,y)$ (not directly speaking about colours) such that, with asymptotic probability 1, the relation "there is an $l$-colouring which assigns the same colour to $x$ and $y$" is defined by $\xi(x,y)$. 4. With asymptotic probability 1, an $l$-colourable structure has a unique $l$-colouring (up to permutation of the colours).Comment: 35 page

Prototyping Operational Autonomy for Space Traffic Management

Current state of the art in Space Traffic Management (STM) relies on a handful of providers for surveillance and collision prediction, and manual coordination between operators. Neither is scalable to support the expected 10x increase in spacecraft population in less than 10 years, nor does it support automated manuever planning. We present a software prototype of an STM architecture based on open Application Programming Interfaces (APIs), drawing on previous work by NASA to develop an architecture for low-altitude Unmanned Aerial System Traffic Management. The STM architecture is designed to provide structure to the interactions between spacecraft operators, various regulatory bodies, and service suppliers, while maintaining flexibility of these interactions and the ability for new market participants to enter easily. Autonomy is an indispensable part of the proposed architecture in enabling efficient data sharing, coordination between STM participants and safe flight operations. Examples of autonomy within STM include syncing multiple non-authoritative catalogs of resident space objects, or determining which spacecraft maneuvers when preventing impending conjunctions between multiple spacecraft. The STM prototype is based on modern micro-service architecture adhering to OpenAPI standards and deployed in industry standard Docker containers, facilitating easy communication between different participants or services. The system architecture is designed to facilitate adding and replacing services with minimal disruption. We have implemented some example participant services (e.g. a space situational awareness provider/SSA, a conjunction assessment supplier/CAS, an automated maneuver advisor/AMA) within the prototype. Different services, with creative algorithms folded into then, can fulfil similar functional roles within the STM architecture by flexibly connecting to it using pre-defined APIs and data models, thereby lowering the barrier to entry of new players in the STM marketplace. We demonstrate the STM prototype on a multiple conjunction scenario with multiple maneuverable spacecraft, where an example CAS and AMA can recommend optimal maneuvers to the spacecraft operators, based on a predefined reward function. Such tools can intelligently search the space of potential collision avoidance maneuvers with varying parameters like lead time and propellant usage, optimize a customized reward function, and be implemented as a scheduling service within the STM architecture. The case study shows an example of autonomous maneuver planning is possible using the API-based framework. As satellite populations and predicted conjunctions increase, an STM architecture can facilitate seamless information exchange related to collision prediction and mitigation among various service applications on different platforms and servers. The availability of such an STM network also opens up new research topics on satellite maneuver planning, scheduling and negotiation across disjoint entities

An Experimental Investigation of the Pressure Distribution on A 1/15-Scale Model of the Lockheed WS-117L Vehicle Plus Booster "B" at Mach Numbers from 0.70 to 1.45

Results obtained with two nose shapes tested at a Reynolds number per foot of 5 x 10(exp 6) at angles of attack from -4 deg to +10 deg at 0 deg angle of sideslip are presented in tabulated pressure coefficient form without analysis

On Recurrent Reachability for Continuous Linear Dynamical Systems

The continuous evolution of a wide variety of systems, including continuous-time Markov chains and linear hybrid automata, can be described in terms of linear differential equations. In this paper we study the decision problem of whether the solution $\boldsymbol{x}(t)$ of a system of linear differential equations $d\boldsymbol{x}/dt=A\boldsymbol{x}$ reaches a target halfspace infinitely often. This recurrent reachability problem can equivalently be formulated as the following Infinite Zeros Problem: does a real-valued function $f:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}$ satisfying a given linear differential equation have infinitely many zeros? Our main decidability result is that if the differential equation has order at most $7$, then the Infinite Zeros Problem is decidable. On the other hand, we show that a decision procedure for the Infinite Zeros Problem at order $9$ (and above) would entail a major breakthrough in Diophantine Approximation, specifically an algorithm for computing the Lagrange constants of arbitrary real algebraic numbers to arbitrary precision.Comment: Full version of paper at LICS'1

Splitting fields and general differential Galois theory

An algebraic technique is presented that does not use results of model theory and makes it possible to construct a general Galois theory of arbitrary nonlinear systems of partial differential equations. The algebraic technique is based on the search for prime differential ideals of special form in tensor products of differential rings. The main results demonstrating the work of the technique obtained are the theorem on the constructedness of the differential closure and the general theorem on the Galois correspondence for normal extensions..Comment: 33 pages, this version coincides with the published on

Tameness of holomorphic closure dimension in a semialgebraic set

Given a semianalytic set S in a complex space and a point p in S, there is a unique smallest complex-analytic germ at p which contains the germ of S, called the holomorphic closure of S at p. We show that if S is semialgebraic then its holomorphic closure is a Nash germ, for every p, and S admits a semialgebraic filtration by the holomorphic closure dimension. As a consequence, every semialgebraic subset of a complex vector space admits a semialgebraic stratification into CR manifolds satisfying a strong version of the condition of the frontier.Comment: Published versio

Completeness of dagger-categories and the complex numbers

The complex numbers are an important part of quantum theory, but are difficult to motivate from a theoretical perspective. We describe a simple formal framework for theories of physics, and show that if a theory of physics presented in this manner satisfies certain completeness properties, then it necessarily includes the complex numbers as a mathematical ingredient. Central to our approach are the techniques of category theory, and we introduce a new category-theoretical tool, called the dagger-limit, which governs the way in which systems can be combined to form larger systems. These dagger-limits can be used to characterize the dagger-functor on the category of finite-dimensional Hilbert spaces, and so can be used as an equivalent definition of the inner product. One of our main results is that in a nontrivial monoidal dagger-category with all finite dagger-limits and a simple tensor unit, the semiring of scalars embeds into an involutive field of characteristic 0 and orderable fixed field.Comment: 39 pages. Accepted for publication in the Journal of Mathematical Physic

An axiomatic approach to the non-linear theory of generalized functions and consistency of Laplace transforms

We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. Next, we identify an inconsistency in the conventional Laplace transform theory. As an application we offer a free of contradictions alternative in the framework of our algebra of generalized functions. The article is aimed at mathematicians, physicists and engineers who are interested in the non-linear theory of generalized functions, but who are not necessarily familiar with the original Colombeau theory. We assume, however, some basic familiarity with the Schwartz theory of distributions.Comment: 23 page