A Topos Perspective on State-Vector Reduction

A preliminary investigation is made of possible applications in quantum theory of the topos formed by the collection of all $M$-sets, where $M$ is a monoid. Earlier results on topos aspects of quantum theory can be rederived in this way. However, the formalism also suggests a new way of constructing a `neo-realist' interpretation of quantum theory in which the truth values of propositions are determined by the actions of the monoid of strings of finite projection operators. By these means, a novel topos perspective is gained on the concept of state-vector reduction

Paper Session II-C - On-Orbit Outfitted Shuttle External Tanks Applied to Lunar Exploration and Industrialization

Exploration of the Moon and development of lunar resources will require delivering habitat facilities to a number of lunar surface locations and include grouping habitats to form colonies plus emplacement of storage facilities for lunar-derived fluids. Liquid Oxygen Tank-lntertank Structure subassemblies derived from the National Space Transportation System (NSTS) External Tank (Figure 1) can be outfitted as Lunar Tank Assemblies (LTA) at the International Space Station Alpha (ISSA) to provide such support facilities. The ISSA would require the addition of an auxiliary truss and a second Space Station Remote Manipulator System (SSRMS); these additions would not compromise the ISSA functionality. On-orbit outfitting would not degrade the structural integrity of the external tanks. These LTA concepts would be capable of autonomous unmanned flight and soft landing at a preselected lunar site. These concepts include habitats and industrial configurations for liquification/storage of oxygen or hydrogen. Previous evaluations of on-orbit outfitting, utilizing the U. S. Space Station Freedom, were performed in support of the NASA Langley Research Center (References 1 and 2). As a result of independent efforts, Bionetics obtained patents for the habitat configuration shown in Figure 2 and for outfitting an external tank as a cryogenic transporter for the lunar-derived isotope Helium-3 (not shown). The current study describes concepts for LTA outfitting at the ISSA with specific features for habitats and industrial storage configurations. Assessments of transit and lunar surface operations led to the conclusion that on-orbit outfitting of LTA\u27s would be an asset to any lunar program

Topos-Theoretic Extension of a Modal Interpretation of Quantum Mechanics

This paper deals with topos-theoretic truth-value valuations of quantum propositions. Concretely, a mathematical framework of a specific type of modal approach is extended to the topos theory, and further, structures of the obtained truth-value valuations are investigated. What is taken up is the modal approach based on a determinate lattice \Dcal(e,R), which is a sublattice of the lattice \Lcal of all quantum propositions and is determined by a quantum state $e$ and a preferred determinate observable $R$. Topos-theoretic extension is made in the functor category \Sets^{\CcalR} of which base category \CcalR is determined by $R$. Each true atom, which determines truth values, true or false, of all propositions in \Dcal(e,R), generates also a multi-valued valuation function of which domain and range are \Lcal and a Heyting algebra given by the subobject classifier in \Sets^{\CcalR}, respectively. All true propositions in \Dcal(e,R) are assigned the top element of the Heyting algebra by the valuation function. False propositions including the null proposition are, however, assigned values larger than the bottom element. This defect can be removed by use of a subobject semi-classifier. Furthermore, in order to treat all possible determinate observables in a unified framework, another valuations are constructed in the functor category \Sets^{\Ccal}. Here, the base category \Ccal includes all \CcalR's as subcategories. Although \Sets^{\Ccal} has a structure apparently different from \Sets^{\CcalR}, a subobject semi-classifier of \Sets^{\Ccal} gives valuations completely equivalent to those in \Sets^{\CcalR}'s.Comment: LaTeX2

Revolutionaries and spies: Spy-good and spy-bad graphs

We study a game on a graph $G$ played by $r$ {\it revolutionaries} and $s$ {\it spies}. Initially, revolutionaries and then spies occupy vertices. In each subsequent round, each revolutionary may move to a neighboring vertex or not move, and then each spy has the same option. The revolutionaries win if $m$ of them meet at some vertex having no spy (at the end of a round); the spies win if they can avoid this forever. Let $\sigma(G,m,r)$ denote the minimum number of spies needed to win. To avoid degenerate cases, assume |V(G)|\ge r-m+1\ge\floor{r/m}\ge 1. The easy bounds are then \floor{r/m}\le \sigma(G,m,r)\le r-m+1. We prove that the lower bound is sharp when $G$ has a rooted spanning tree $T$ such that every edge of $G$ not in $T$ joins two vertices having the same parent in $T$. As a consequence, \sigma(G,m,r)\le\gamma(G)\floor{r/m}, where $\gamma(G)$ is the domination number; this bound is nearly sharp when $\gamma(G)\le m$. For the random graph with constant edge-probability $p$, we obtain constants $c$ and $c'$ (depending on $m$ and $p$) such that $\sigma(G,m,r)$ is near the trivial upper bound when $r<c\ln n$ and at most $c'$ times the trivial lower bound when $r>c'\ln n$. For the hypercube $Q_d$ with $d\ge r$, we have $\sigma(G,m,r)=r-m+1$ when $m=2$, and for $m\ge 3$ at least $r-39m$ spies are needed. For complete $k$-partite graphs with partite sets of size at least $2r$, the leading term in $\sigma(G,m,r)$ is approximately $\frac{k}{k-1}\frac{r}{m}$ when $k\ge m$. For $k=2$, we have \sigma(G,2,r)=\bigl\lceil{\frac{\floor{7r/2}-3}5}\bigr\rceil and \sigma(G,3,r)=\floor{r/2}, and in general $\frac{3r}{2m}-3\le \sigma(G,m,r)\le\frac{(1+1/\sqrt3)r}{m}$.Comment: 34 pages, 2 figures. The most important changes in this revision are improvements of the results on hypercubes and random graphs. The proof of the previous hypercube result has been deleted, but the statement remains because it is stronger for m<52. In the random graph section we added a spy-strategy resul

An advanced technology space station for the year 2025, study and concepts

A survey was made of potential space station missions that might exist in the 2020 to 2030 time period. Also, a brief study of the current state-of-the-art of the major subsystems was undertaken, and trends in technologies that could impact the subsystems were reviewed. The results of the survey and study were then used to arrive at a conceptual design of a space station for the year 2025. Factors addressed in the conceptual design included requirements for artificial gravity, synergies between subsystems, and the use of robotics. Suggestions are made relative to more in-depth studies concerning the conceptual design and alternative configurations

Paper Session III-B - A Lunar Orbiting Node in Support of Missions to Mars

Future Mars missions may use lunar-derived oxygen as a propellant for interplanetary transit. A man-tended platform as a Node in low lunar orbit offers a site for storage and transfer of lunar oxygen to the transport vehicles as well as rendezvous and transfer for lunar-bound cargo and crews. In addition, it could provide an emergency safe-haven for a crew awaiting rescue. A conceptual design study yielded an approximate size for the platform needed to support typical oxygen transfer rates which were based upon NASA studies of Mars missions. The Node consists of a gravity gradient stabilized lunar orbiting tank-farm with a storage capacity of 100,000 kg of lunar oxygen, 3,300 kg of lunar cargo and 9,300 kg of Earth supplied hydrogen. An emergency habitat configuration accomodates 14 persons on-board for 110 days. The Node supports an annual lunar oxygen Ereduction of 106 kg with 220,000 kg of oxygen delivered to Earth orbit for an expenditure of 109,000 g of Earth supplied hydrogen

Analysis of a rotating advanced-technology space station for the year 2025

An analysis is made of several aspects of an advanced-technology rotating space station configuration generated under a previous study. The analysis includes examination of several modifications of the configuration, interface with proposed launch systems, effects of low-gravity environment on human subjects, and the space station assembly sequence. Consideration was given also to some aspects of space station rotational dynamics, surface charging, and the possible application of tethers

Some operational aspects of a rotating advanced-technology space station for the year 2025

The study of an Advanced Technology Space Station which would utilize the capabilities of subsystems projected for the time frame of the years 2000 to 2025 is discussed. The study includes tradeoffs of nuclear versus solar dynamic power systems that produce power outputs of 2.5 megawatts and analyses of the dynamics of the spacecraft of which portions are rotated for artificial gravity. The design considerations for the support of a manned Mars mission from low Earth orbit are addressed. The studies extend to on-board manufacturing, internal gas composition effects, and locomotion and material transfer under artificial gravity forces. The report concludes with an assessment of technology requirements for the Advanced Technology Space Station

Concepts for manned lunar habitats

The design philosophy that will guide the design of early lunar habitats will be based on a compromise between the desired capabilities of the base and the economics of its development and implantation. Preferred design will be simple, make use of existing technologies, require the least amount of lunar surface preparation, and minimize crew activity. Three concepts for an initial habitat supporting a crew of four for 28 to 30 days are proposed. Two of these are based on using Space Station Freedom structural elements modified for use in a lunar-gravity environment. A third concept is proposed that is based on an earlier technology based on expandable modules. The expandable modules offer significant advantages in launch mass and packaged volume reductions. It appears feasible to design a transport spacecraft lander that, once landed, can serve as a habitat and a stand-off for supporting a regolith environmental shield. A permanent lunar base habitat supporting a crew of twelve for an indefinite period can be evolved by using multiple initial habitats. There appears to be no compelling need for an entirely different structure of larger volume and increased complexity of implantation

A topos for algebraic quantum theory

The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr's idea that the empirical content of quantum physics is accessible only through classical physics, we show how a C*-algebra of observables A induces a topos T(A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum S(A) in T(A), which in our approach plays the role of a quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on S(A), and self-adjoint elements of A define continuous functions (more precisely, locale maps) from S(A) to Scott's interval domain. Noting that open subsets of S(A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos T(A).Comment: 52 pages, final version, to appear in Communications in Mathematical Physic