Narrow Results

Let $R$ be a finite commutative ring. In this paper, we consider the probability that the product of randomly chosen three elements of $R$ is zero, which we denote by $z{p}_{{}_3}(R)$. First we obtain bounds for $z{p}_{{}_3}(R)$ for various classes of rings and then show that for any ring $R$ with identity $z{p}_{{}_3}(R)\le \frac{7}{8}$. Finally, we characterize all commutative rings $R$ with identity with certain range of $z{p}_{{}_3}(R)$.

Probability, be it inherent or explicitly introduced, has become an important issue in the verification of programs. In this paper we study a formalism which allows reasoning about programs which can act probabilistically. To describe probabilistic programs, a basic programming language with an operator for probabilistic choice is introduced and a denotational semantics is given for this language. To specify propertics of probabilistic programs, standard first order logic predicates are insufficient, so a notion of probabilistic predicates is introduced. A Hoare-style proof system to check properties of probabilistic programs is given. The proof system for a sublanguage is shown to be sound and complete; the properties that can be derived are exactly the valid properties. Finally some typical examples illustrate the use of the probabilistic predicates and the proof system.

We consider heap-ordered trees in this paper. The main theorems in this paper show that the average and the variance of the altitude (or level) of the nodes in a random n-node heap-ordered tree are and , respectively where γ=0.5772156649… is the Euler’s constant.

A theoretical measure of the ionicity based on different mathematical concepts is presented in this research. Considering the distribution of valence electrons on each atom in a bond, we assume that the chemical properties of the atom can be expressed by means of probability. Using the introduced probability, different probabilistic models such as classical, fuzzy and information theoretical models have been employed to introduce new descriptors of bond ionicity. The ionicities were calculated for 12 heterodiatomic molecules and the bonds were classified in terms of covalency vs. ionicity. It was found that our proposed ionicity descriptors correlate well with the partial ionic character of the bonds.

The Nagel–Schreckenberg (NS) cellular automata (CA) model for describing the vehicular traffic flow in a street with open boundaries is studied. To control the traffic flow, a traffic signalization light operating for a fixed-time scheme is placed in the middle of the street. Extensive Monte Carlo simulations are carried out to calculate various model characteristics. Essentially, we investigate the formation of the cars queue behind traffic light dependence on the duration of green light T_g, injecting and extracting probabilities α and β, respectively. Two phases of average training queues were found. Besides, the dependence of car accident probability per site and per time step on T_g, α and β is computed.

After a brief review of classical probability theory (measure theory), we present an observation (due to Sorkin) concerning an aspect of probability in quantum mechanics. Following Sorkin, we introduce a generalized measure theory based on a hierarchy of "sum-rules". The first sum-rule yields classical probability theory, and the second yields a generalized probability theory that includes quantum mechanics as a special case. We present some algebraic relations involving these sum-rules. This may be useful for the study of the higher-order sum-rules and possible generalizations of quantum mechanics. We conclude with some open questions and suggestions for further work.

I give a brief introduction to many worlds or "no wave function collapse" quantum mechanics, suitable for non-specialists. I then discuss the origin of probability in such formulations, distinguishing between objective and subjective notions of probability.

The reverse quantum speed limit (RQSL) gives an upper limit to the time for evolution between initial and final quantum states. We show that, in conjunction with the existence of a minimum time scale, the RQSL implies a lower limit to the norm of the change in a quantum state, and confirm that this limit is satisfied in two-state and ideal-measurement models. Such a lower limit is of relevance for interpretational issues in probability and for understanding the meaning of probability in Everett quantum theory.

The fact that certain “extraordinary” probabilistic phenomena — in particular, macroscopic violations of the second law of thermodynamics — have never been observed to occur can be accounted for by taking hard preclusion as a basic physical law, i.e. precluding from existence events corresponding to very small but nonzero values of quantum-mechanical weight. This approach is not consistent with the usual ontology of the Everett interpretation, in which outcomes correspond to branches of the state vector, but can be successfully implemented using a Heisenberg-picture-based ontology in which outcomes are encoded in transformations of operators. Hard preclusion can provide an explanation for biological evolution, which can in turn explain our subjective experiences of, and reactions to, “ordinary” probabilistic phenomena, and the compatibility of those experiences and reactions with what we conventionally take to be objective probabilities arising from physical laws.

The afterpulse probability has been measured using the photon statistical method on the base of theoretical analysis for single-photon detection. It is found that the measured afterpulses obey Gaussian distribution, and the Gauss curves are left shifted with increasing incident photon intensity. We have analyzed the physical process of the phenomenon and obtained the afterpulse probability of our single-photon detector. Compared with the common start–stop method, this robust photon statistical method has the advantages of direct measurement and not requiring additional equipment.

We present the modification of optical transition probability in the superlattices (SLs) due to the presence of structural defect and the variation of layer thickness. The results show that optical transition probability between two bound states is mainly determined by the thicknesses of layer in defect region and its nearest constituent layers for a symmetric structure. When the well layer thickness varies, the variation of transition probabilities between two bound states depend on the parity of the index of minigap n, while those from the bound states to the delocalized scattering is irrelevant to n. It is believed that applying appropriate structure and layer thickness may enhance the optical transition probability in designing optical device.

For the limitation that current node influence ranking algorithms can only be applied in a single type of network and the results are inaccurate, an algorithm based on similarity is proposed. When a node is similar to many nodes in the network, it is representative and can be treated as an influential node. Firstly, probability walking model is used to simulate the initiative visit between nodes in different types of networks. Secondly, superposed probabilistic transfer similarity is defined based on the model considering nodes’ inbound and outbound information. Finally, node ranking algorithm is set up using the new similarity measuring method. Experiments show that the algorithm can evaluate different kinds of networks with high accuracy, whether the network is directed or undirected, weighted or unweighted.

The first equation in a hierarchy of equations due to Vlasov is studied. The equation can be described as a Schrödinger equation for the probabilistic description of a system. It has applications in other areas as well. A physical meaning can be assigned to the phase of the wavefunction. The approach allows construction of solutions of the Schrödinger equation from known solutions and conversely much like a Bäcklund transformation. Finally, the process of introducing a potential into the Schrödinger equation is generalized to the case of Bohmian mechanics.

Traffic congestion is now nearly ubiquitous in many urban areas and frequently occurs during rush hour periods. Rush hour avoidance is an effective way to ease traffic congestion. It is significant to calculate the rush hour for alleviating traffic congestion. This paper provides a method to calculate the fuzzy peak hour of the urban traffic network considering the flow, speed and occupancy. The process of calculation is based on betweenness centrality of network theory, optimal separation method, time period weighting, probability–possibility transformations and trapezoidal approximations of fuzzy numbers. The fuzzy peak hour of the urban road traffic network (URTN) is a trapezoidal fuzzy number [m₁, m₂, m₃, m₄]. It helps us (i) to confirm a more detailed traffic condition at each moment, (ii) to distinguish the five traffic states of the traffic network in one day, (iii) to analyze the characteristic of appearance and disappearance processes of the each traffic state and (iv) to find out the time pattern of residents travel in one city.

In this work, the crystal structures and phase transitions of compounds with perovskite structure were investigated. The classification of structural phase transitions in perovskites was carried out, the most common crystal structures and structural phase transitions were shown. A mathematical model was constructed, a theorem was given and proved for the probability of a possible transition. The formulas $E({\tau }_1)=(\frac{{(\lambda \beta )}^2}{8}+\frac{\lambda \beta }{2}+1)E({\xi }_1)$ and $D({\tau }_1)=(\frac{{(\lambda \beta )}^4}{96}+\frac{{(\lambda \beta )}^3}{12}-\frac{3{(\lambda \beta )}^2}{8}-\frac{\lambda \beta }{2}){(E{\xi }_1)}^2+(\frac{{(\lambda \beta )}^2}{8}+\frac{\lambda \beta }{2}+1)D({\xi }_1)$ are given for the mathematical expectation and variance of random variable ${\tau }_1$, which is the moment when the stochastic process $X(t)$ deviation from the boundary [0, $\beta$] interval for the first time. According to the mathematical model, one of the trajectories of random processes corresponding to the phase transitions that occur in perovskites is constructed.

The role of dissipation with respect to a microscopic superposition of quantum states is investigated by means of master equations. This has implications for the study of the emergence of classicality from the quantum level. In particular, it illustrates why it is difficult to observe a macroscopic quantum state. The role of the environment is assumed by the measuring apparatus. A pure state is reduced to a mixture in the pointer basis of the system by means of the interaction with the apparatus. It is the intention that this type of analysis will have applications to experiments which are designed to better understand the environmental-assisted invariance formulation of quantum mechanics.

In this paper, the reasons for the shortcoming of Zadeh's fuzzy set theory — it cannot correctly reflect different kinds of fuzzy phenomenon in the natural world — are discussed. In addition, the proof of the error of Zadeh's fuzzy set theory — it incorrectly defined the set complement that cannot exist in Zadeh's fuzzy set theory — is proposed. This error of Zadeh's fuzzy set theory causes confusion in thinking, logic and conception. It causes the seriously mistaken belief that logics of fuzzy sets necessarily go against classical and normal thinking, logic and conception. Two new fuzzy set theories, C-fuzzy set theory and probability-like fuzzy set theory, the N-fuzzy set theory, are proposed. The two are equivalent, and both overcome the error and shortcoming of Zadeh's fuzzy set theory, and they are consistent with normal, natural and classical thinking, logic and concepts. The similarities of N-fuzzy set theory with probability theory are also examined.

In the case of expert systems, when premises from two different rules argue for the same conclusion, there exist a number of methods for combining the certainties in the conclusion, obtained from each premise individually, to yield a total certainty in the conclusion. When these certainties are interpreted as probability values, it is necessary in all these methods to make assumptions of independence concerning the premises and conclusion. It is proven here that there does not exist a procedure which yields, for all probability functions, meaningful bounds for the probability of the conclusion based on two premises in terms of the probabilities of the conclusion based on each premise individually. Additionally, it is proven that there does not even exist a procedure which yields, for all probability functions, meaningful bounds for the probability of the conclusion based on two premises in terms of the probabilities of all one- and two-member combinations of the premises and the conclusion.

Finally, suggestions are made as to when to make assumptions of independence and when to take other alternatives.

We show that the probability of generating an iterated wreath product of non-abelian finite simple groups converges to 1 as the order of the first simple group tends to infinity provided the wreath products are constructed with transitive and faithful actions. This has the consequence that the profinite group which is the inverse limit of these iterated wreath products is positively finitely generated.

Belief revision characterizes the process of revising an agent's beliefs when receiving new evidence. In the field of artificial intelligence, revision strategies have been extensively studied in the context of logic-based formalisms and probability kinematics. However, so far there is not much literature on this topic in evidence theory. In contrast, combination rules proposed so far in the theory of evidence, especially Dempster rule, are symmetric. They rely on a basic assumption, that is, pieces of evidence being combined are considered to be on a par, i.e. play the same role. When one source of evidence is less reliable than another, it is possible to discount it and then a symmetric combination operation is still used. In the case of revision, the idea is to let prior knowledge of an agent be altered by some input information. The change problem is thus intrinsically asymmetric. Assuming the input information is reliable, it should be retained whilst the prior information should be changed minimally to that effect. To deal with this issue, this paper defines the notion of revision for the theory of evidence in such a way as to bring together probabilistic and logical views. Several revision rules previously proposed are reviewed and we advocate one of them as better corresponding to the idea of revision. It is extended to cope with inconsistency between prior and input information. It reduces to Dempster rule of combination, just like revision in the sense of Alchourrón, Gärdenfors, and Makinson (AGM) reduces to expansion, when the input is strongly consistent with the prior belief function. Properties of this revision rule are also investigated and it is shown to generalize Jeffrey's rule of updating, Dempster rule of conditioning and a form of AGM revision.