Modeling of ALFA Programs Using PVS Theorem Prover

In Safety critical software failure can have a high price. Such software should be free of errors before it is put into operation. Application of formal methods in the Software Development Life Cycle helps to ensure that the software for safety critical missions are ultra reliable. PVS theorem prover, a formal method tool, can be used for the formal verification of software in ADA Language for Flight Software Application (ALFA.). This paper describes the modeling of ALFA programs for PVS theorem prover. An ALFA2PVS translator is developed which automatically converts the software in ALFA to PVS specification. By this approach the software can be verified formally with respect to underflow/overflow errors and divide by zero conditions without the actual execution of the code.

Modeling of ALFA Programs Using PVS Theorem Prover

In Safety critical software failure can have a high price. Such software should be free of errors before it is put into operation. Application of formal methods in the Software Development Life Cycle helps to ensure that the software for safety critical missions are ultra reliable. PVS theorem prover, a formal method tool, can be used for the formal verification of software in ADA Language for Flight Software Application (ALFA.). This paper describes the modeling of ALFA programs for PVS theorem prover. An ALFA2PVS translator is developed which automatically converts the software in ALFA to PVS specification. By this approach the software can be verified formally with respect to underflow/overflow errors and divide by zero conditions without the actual execution of the code

Robot free will

We introduce the perspex machine which unifies projective geometry and Turing computation and results in a supra-Turing machine. We show two ways in which the perspex machine unifies symbolic and non-symbolic AI. Firstly, we describe concrete geometrical models that map perspexes onto neural networks, some of which perform only symbolic operations. Secondly, we describe an abstract continuum of perspex logics that includes both symbolic logics and a new class of continuous logics. We argue that an axiom in symbolic logic can be the conclusion of a perspex theorem. That is, the atoms of symbolic logic can be the conclusions of sub-atomic theorems. We argue that perspex space can be mapped onto the spacetime of the universe we inhabit. This allows us to discuss how a robot might be conscious, feel, and have free will in a deterministic, or semi-deterministic, universe. We ground the reality of our universe in existence. On a theistic point, we argue that preordination and free will are compatible. On a theological point, we argue that it is not heretical for us to give robots free will. Finally, we give a pragmatic warning as to the double-edged risks of creating robots that do, or alternatively do not, have free will.

Solution-free sets for sums of binary forms

In this paper, we obtain quantitative estimates for the asymptotic density of subsets of the integer lattice Z2 that contain only trivial solutions to an additive equation involving binary forms. In the process we develop an analogue of Vinogradov’s mean value theorem applicable to binary forms.

On a Gromoll-Meyer type theorem in globally hyperbolic stationary spacetimes

Following the lines of the celebrated Riemannian result of Gromoll and Meyer, we use infinite dimensional equivariant Morse theory to establish the existence of infinitely many geometrically distinct closed geodesics in a class of globally hyperbolic stationary Lorentzian manifolds.

The Borsuk-Ulam theorem for homotopy spherical space forms

In this work, we show for which odd-dimensional homotopy spherical space forms the Borsuk-Ulam theorem holds. These spaces are the quotient of a homotopy odd-dimensional sphere by a free action of a finite group. Also, the types of these spaces which admit a free involution are characterized. The case of even-dimensional homotopy spherical space forms is basically known.

BASS CYCLIC UNITS AS FACTORS IN A FREE GROUP IN INTEGRAL GROUP RING UNITS

Marciniak and Sehgal showed that if u is a non-trivial bicyclic unit of an integral group ring then there is a bicyclic unit v such that u and v generate a non-abelian free group. A similar result does not hold for Bass cyclic units of infinite order based on non-central elements as some of them have finite order modulo the center. We prove a theorem that suggests that this is the only limitation to obtain a non-abelian free group from a given Bass cyclic unit. More precisely, we prove that if u is a Bass cyclic unit of an integral group ring ZG of a solvable and finite group G, such that u has infinite order modulo the center of U(ZG) and it is based on an element of prime order, then there is a non-abelian free group generated by a power of u and a power of a unit in ZG which is either a Bass cyclic unit or a bicyclic unit.

Central limit theorem for asymmetric kernel functionals

Asymmetric kernels are quite useful for the estimation of density functions with bounded support. Gamma kernels are designed to handle density functions whose supports are bounded from one end only, whereas beta kernels are particularly convenient for the estimation of density functions with compact support. These asymmetric kernels are nonnegative and free of boundary bias. Moreover, their shape varies according to the location of the data point, thus also changing the amount of smoothing. This paper applies the central limit theorem for degenerate U-statistics to compute the limiting distribution of a class of asymmetric kernel functionals.

Arbitrage-Free Conditions and Hedging Strategies for Markets with Penalty Costs on Short Positions

We consider a discrete-time financial model in a general sample space with penalty costs on short positions. We consider a friction market closely related to the standard one except that withdrawals from the portfolio value proportional to short positions are made. We provide necessary and sufficient conditions for the nonexistence of arbitrages in this situation and for a self-financing strategy to replicate a contingent claim. For the finite-sample space case, this result leads to an explicit and constructive procedure for obtaining perfect hedging strategies.

A syntactic realization theorem for justification logics

Justification logics are refinements of modal logics where modalities are replaced by justification terms. They are connected to modal logics via so-called realization theorems. We present a syntactic proof of a single realization theorem that uniformly connects all the normal modal logics formed from the axioms \$mathsfd\$, \$mathsft\$, \$mathsfb\$, \$mathsf4\$, and \$mathsf5\$ with their justification counterparts. The proof employs cut-free nested sequent systems together with Fitting's realization merging technique. We further strengthen the realization theorem for \$mathsfKB5\$ and \$mathsfS5\$ by showing that the positive introspection operator is superfluous.

The Automorphism Group of the Free Algebra of Rank Two

The theorem of Czerniakiewicz and Makar-Limanov, that all the automorphisms of a free algebra of rank two are tame is proved here by showing that the group of these automorphisms is the free product of two groups (amalgamating their intersection), the group of all affine automorphisms and the group of all triangular automorphisms. The method consists in finding a bipolar structure. As a consequence every finite subgroup of automorphisms (in characteristic zero) is shown to be conjugate to a group of linear automorphisms.

Invariants of Unipotent Transformations Acting on Noetherian Relatively Free Algebras

2000 Mathematics Subject Classification: 16R10, 16R30.

Path Optimization in Free Energy Calculations

Free energy calculations are a computational method for determining thermodynamic quantities, such as free energies of binding, via simulation.

Currently, due to computational and algorithmic limitations, free energy calculations are limited in scope.

In this work, we propose two methods for improving the efficiency of free energy calculations.

First, we expand the state space of alchemical intermediates, and show that this expansion enables us to calculate free energies along lower variance paths.

We use Q-learning, a reinforcement learning technique, to discover and optimize paths at low computational cost.

Second, we reduce the cost of sampling along a given path by using sequential Monte Carlo samplers.

We develop a new free energy estimator, pCrooks (pairwise Crooks), a variant on the Crooks fluctuation theorem (CFT), which enables decomposition of the variance of the free energy estimate for discrete paths, while retaining beneficial characteristics of CFT.

Combining these two advancements, we show that for some test models, optimal expanded-space paths have a nearly 80% reduction in variance relative to the standard path.

Additionally, our free energy estimator converges at a more consistent rate and on average 1.8 times faster when we enable path searching, even when the cost of path discovery and refinement is considered.

Integration of cad/ tool path data for 5-axis step-nc machining of free form / irrregular contoured surfaces

This research paper presents the work on feature recognition, tool path data generation and integration with STEP-NC (AP-238 format) for features having Free form / Irregular Contoured Surface(s) (FICS). Initially, the FICS features are modelled / imported in UG CAD package and a closeness index is generated. This is done by comparing the FICS features with basic B-Splines / Bezier curves / surfaces. Then blending functions are caculated by adopting convolution theorem. Based on the blending functions, contour offsett tool paths are generated and simulated for 5 axis milling environment. Finally, the tool path (CL) data is integrated with STEP-NC (AP-238) format. The tool path algorithm and STEP- NC data is tested with various industrial parts through an automated UFUNC plugin.

Free Resolutions from Involutive Bases

We show that the theory of involutive bases can be combined with discrete algebraic Morse Theory. For a graded k[x0 ...,xn]-module M, this yields a free resolution G, which in general is not minimal. We see that G is isomorphic to the resolution induced by an involutive basis. It is possible to identify involutive bases inside the resolution G. The shape of G is given by a concrete description. Regarding the differential dG, several rules are established for its computation, which are based on the fact that in the computation of dG certain patterns appear at several positions. In particular, it is possible to compute the constants independent of the remainder of the differential. This allows us, starting from G, to determine the Betti numbers of M without computing a minimal free resolution: Thus we obtain a new algorithm to compute Betti numbers. This algorithm has been implemented in CoCoALib by Mario Albert. This way, in comparison to some other computer algebra system, Betti numbers can be computed faster in most of the examples we have considered. For Veronese subrings S(d), we have found a Pommaret basis, which yields new proofs for some known properties of these rings. Via the theoretical statements found for G, we can identify some generators of modules in G where no constants appear. As a direct consequence, some non-vanishing Betti numbers of S(d) can be given. Finally, we give a proof of the Hyperplane Restriction Theorem with the help of Pommaret bases. This part is largely independent of the other parts of this work.