Quantum Set Theory Extending the Standard Probabilistic Interpretation of Quantum Theory

The notion of equality between two observables will play many important roles in foundations of quantum theory. However, the standard probabilistic interpretation based on the conventional Born formula does not give the probability of equality between two arbitrary observables, since the Born formula gives the probability distribution only for a commuting family of observables. In this paper, quantum set theory developed by Takeuti and the present author is used to systematically extend the standard probabilistic interpretation of quantum theory to define the probability of equality between two arbitrary observables in an arbitrary state. We apply this new interpretation to quantum measurement theory, and establish a logical basis for the difference between simultaneous measurability and simultaneous determinateness.     

Some Bulk-Viscous Solutions in a First-Order Theory

We first motivate the study of viscosity in cosmology. Whilst most studies assume that the universe is filled with a perfect fluid, viscosity is expected to play a role, at least during some stages of the evolution of the Universe. There are several theories of viscosity. Eckart’s first-order theory was found to permit superluminal signals, and equilibrium states were found to be unstable. To solve these problems, the Israel-Stewart second-order theory was proposed. More recently, a relatively new first-order theory has appeared, which is claimed to also solve these problems.We briefly reviewthis first-order theory and present the basic field equations. Then we attempt to find homogeneous and isotropic solutions in the theory. It is noted that there do not exist stiff matter (pressure = energy density) solutions in the theory, in contrast to other theories. We then find power-law solutions without a cosmological term. Surprisingly, there do not exist simple exponential solutions, again in contrast to other theories. Finally, we present a solution with a cosmological term and make some concluding remarks.     

From Cognitive Load Theory to Collaborative Cognitive Load Theory

Cognitive load theory has traditionally been associated with individual learning. Based on evolutionary educational psychology and our knowledge of human cognition, particularly the relations between working memory and long-term memory, the theory has been used to generate a variety of instructional effects. Though these instructional effects also influence the efficiency and effectiveness of collaborative learning, be it computer supported or face-to-face, they are often not considered either when designing collaborative learning situations/environments or researching collaborative learning. One reason for this omission is that cognitive load theory has only sporadically concerned itself with certain particulars of collaborative learning such as the concept of a collective working memory when collaborating along with issues associated with transactive activities and their concomitant costs which are inherent to collaboration. We illustrate how and why cognitive load theory, by adding these concepts, can throw light on collaborative learning and generate principles specific to the design and study of collaborative learning.     

Soft Linear Set Theory

A formulation of naïve set theory is given in Lafont’s Soft Linear Logic, a logic with polynomial time cut-elimination. We demonstrate that the provably total functions of this set theory are precisely the PTIME functions. A novelty of this approach is the representation of the unary/binary natural numbers by two distinct sets (the safe naturals and the soft naturals).     

Signature Theory in Holographic Algorithms

In the theory of holographic algorithms proposed by Valiant, computation is expressed and processed in terms of signatures. We substantially develop the signature theory in holographic algorithms. This theory is developed in terms of d-realizability and d-admissibility. For the class of 2-realizable signatures we prove a Birkhoff-type theorem which determines this class. It gives a complete structural understanding of the relationship between 2-realizability and 2-admissibility. This is followed by characterization theorems for 1-realizability and 1-admissibility. Finally, using this theory of general (i.e., unsymmetric) signatures we give additional counting problems solvable in polynomial time by holographic algorithms.     

控制理论中的频率定理 (Kalman-Yakubovich引理)*

频率定理（Ｋａｌｍａｎ－Ｙａｋｕｂｏｖｉｃｈ）是控制理论中的重要结果之一，它给出了Ｌｕｒ’ｅ方程和Ｒｉｃｃａｔｉ方程可解的一个充分必要条件，本文简要叙述了频率定理建立和发展的历史，描述了它在非线性系统绝对稳定性理论，线性二次型最优控制和自适应控制理论中的部分应用。     

关键字驱动自动化测试的原理和实现

由GUI自动化功能测试存在的问题，引出了一种以关键字驱动思想为指导的自动化测试模型。首先解释了关键字驱动思想的原理，进而根据这个原理介绍了一种自动化功能测试模型，详细讨论了这个模型的优越性，以及如何应用它来进行测试开发的流程。     

现代控制理论中的某些问题(Ⅱ)

六、分布参数系统的控制有时,要控制的系统不是几个参数的演化,而是一个场的演化,例如温度场、弹性振动、人造地球卫星中液体燃料的晃动、化工系统中浓度与温度等等的控制.场的演化一般表达成偏微分方程,或函数空间中的微分方程.这种系统叫做分布参数系统.这种系统比集中参数系统(即一、二节中所论的那种由常数微分方程描述的系统)复杂得多.许多新问题被提出来.     

证据论与约集论

约集论用于处理模糊性与不确定性.在某些方面,它同Dempster-Shafer的证据论相重迭,但约集论使用分划来描述约集、下界近似及上界近似,从而获取知识表示中的不确定性,而证据论使用证据函数来实现同一目的.该文针对两个理论表述上的差异,揭示出其内在的关系,以有助于人们对两者的理解,从而为开拓它们的应用铺平道路.此外,在证据论中,组合证据的基本操作是正交和,而在约集论中,基本操作是分划的交,因而存在“证据组合是否对应于分划的交”的问题.通过一个例子来说明回答是否定的.     

Large Deviations in Quantum Information Theory

We obtain asymptotic estimates for the probabilities of events of special types, which are useful in quantum information theory, especially for identification in noisy channels.     

Entropy in Thermodynamics and in Information Theory

Problems of Information Transmission - We discuss the relation between the concepts of entropy in thermodynamics and in information theory.     

Some Issues in Quantum Information Theory

Quantum information theory is a new interdisciplinary research field related to quantum mechanics, computer science, information theory, and applied mathematics. It provides completely new paradigms to do information processing tasks by employing the principles of quantum mechanics. In this review, we first survey some of the significant advances in quantum information theory in the last twenty years. We then focus mainly on two special subjects： discrimination of quantum objects and transformations between entanglements. More specifically, we first discuss discrimination of quantum states and quantum apparatus in both global and local settings. Secondly, we present systematical characterizations and equivalence relations of several interesting entanglement transformation phenomena, namely entanglement catalysis, multiple-copy entanglement transformation, and partial entanglement recovery.     

Progress in Computational Complexity Theory

We briefly survey a number of important recent achievements in Theoretical Computer Science (TCS), especially Computational Complexity Theory. We will discuss the PCP Theorem, its implications to inapproximability on combinatorial optimization problems; space bounded computations, especially deterministic logspace algorithm for undirected graph connectivity problem; deterministic polynomial-time primality test; lattice complexity, worst-case to average-case reductions; pseudorandomness and extractor constructions; and Valiant's new theory of holographic algorithms and reductions.     

A Postmodern Theory of Cyberterrorism: Game Theory

ABSTRACT

This article analyzes how the battle between computer security experts and cyberterrorists can be explained through game theory. This article is important because it not only applies game theory to the study of cyberterrorism, which has been rarely done so far, but it also breaks new ground by intersecting the game theoretical model with another theory, social network theory. An important thesis of this analysis is that under the principles of game theory, each player is assumed to be rational; all players wish the outcome to be as positive or rewarding as possible. Another key argument is that game theory is a postmodern theory; against opponents who wage attacks in a postmodern fashion, conventional strategies lead nowhere. The cyberterrorist and the cyber forensics expert not only engage in real-time game play but also use tactics that are not conceivable in conventional conflict.     

Productivity of Edalat-Potts Exact Arithmetic in Constructive Type Theory

In this work we focus on a formalisation of the algorithms of lazy exact arithmetic a la Edalat-Potts in type theory. We choose the constructive type theory extended with coinductive types as our formal verification tool. We show examples of how infinite objects such as streams and expression trees can be formalised as coinductive types. We study the type theoretic notion of productivity which ensures the infiniteness of the outcome of the algorithms on infinite objects. Syntactical methods are not always strong enough to ensure the productivity. However, if some information about the complexity of a function is provided, one may be able to show the productivity of that function. In the case of the normalisation algorithm we show that such information can be obtained from the choice of real number representation that is used to represent the input and the output. Furthermore, we show how to employ this semantic information to formalise a restricted case of normalisation algorithm in our type theory.     

Computational Reconstruction of Cognitive Music Theory

In order to obtain a computer-tractable model of music, we first discuss what conditions the music theory should satisfy from the various viewpoints of artificial intelligence and/or other computational notions. Then, we look back on the history of cognitive theory of music, i.e., various attempts to represent our mental understandings and to show music structures. Among which, we especially pay attention to the Generative Theory of Tonal Music (GTTM) by Lehrdahl and Jackendoff, as the most promising candidate of cognitive/computational theory of music. We briefly overview the theory as well as its inherent problems, including the ambiguity of its preference rules. By our recent efforts, we have solved this ambiguity problem by assigning parametrized weights, and thus we could implement an automatic tree analyzer. After we introduce the system architecture, we show our application systems.     

Theory Change via View Application in Instructionless Learning

IE is a model of learning by experimentation in the domain of complex devices. In this paper we describe view application,the principal component of that model. This mechanism combines abstract knowledge structures into the learner's theory of the device. View application organizes complex changes in the learner's theory, thus ensuring that the space of theories is searched rapidly and that only coherent theories are tried. We evaluate the mechanism along three dimensions – its psychological validity, its generality, and its ability to constrain search. We also compare view application to other knowledge-rich learning techniques.     

Explaining Environmental Sustainability in Supply Chains Using Graph Theory

The need for theory building in environmental supply chains has been at the centre of many discussions in recent years. Existing research, however, does not typically consider methods that aim at theory generation. Current methods such as econometric modelling or structural equation modelling face challenges related to how causality is established due to potential issues regarding cross-sectional data sets. To address this gap, this paper suggests a total interpretive structural modelling based approach. We use graph theory logic to synthesize expert interpretations in the form of a theoretical supply chain model. This method may prove to be an alternative method to econometric based modelling or structural equation modelling. We provide an application of the method in exploring the drivers of low carbon supply chain and their relationships. Limitations and future research opportunities are also provided.     

Mathematical Theory of the Duality Computer in the Density Matrix Formalism

We give the mathematical theory of duality computer in the density matrix formalism. This result complements the mathematical theory of duality computer of Gudder in the pure state formalism.      

On the Theory of Matchgate Computations

Valiant has proposed a new theory of algorithmic computation based on perfect matchings and Pfaffians. We study the properties of matchgates—the basic building blocks in this new theory. We give a set of algebraic identities which completely characterizes these objects for arbitrary numbers of inputs and outputs. These identities are derived from Grassmann-Plücker identities. The 4 by 4 matchgate character matrices are of particular interest. These were used in Valiant’s classical simulation of a fragment of quantum computations. For these 4 by 4 matchgates, we use Jacobi’s theorem on compound matrices to prove that the invertible matchgate matrices form a multiplicative group. Our results can also be expressed in the theory of Holographic Algorithms in terms of realizable standard signatures. These results are useful in establishing limitations on the ultimate capabilities of Valiant’s theory of matchgate computations and Holographic Algorithms. J.-Y. Cai supported by NSF CCR-0208013 and CCR-0511679. A preliminary version of this paper will appear in Proceedings of IEEE Conference on Computational Complexity 2007, pp. 305–318. V. Choudhary supported by NSF CCR-0208013. P. Lu supported by NSF CCR-0511679 and by the National Natural Science Foundation of China Grant 60553001 and the National Basic Research Program of China Grant 2007CB807900, 2007CB807901.