Narrow Results

Quantum-dot Cellular Automata (QCA) presents a new model at Nano-scale for possible substitution of conventional Complementary Metal–Oxide–Semiconductor (CMOS) technology. On the other hand, an Arithmetic Logic Unit (ALU) is a digital electronic circuit which performs arithmetic and bitwise logical operations on integer binary numbers. Therefore, QCA-based ALU is an important part of the processor in order to develop a full capability processor. Although the QCA has become very important, there is not any comprehensive and systematic work on studying and analyzing its important techniques in the field of ALU design. This paper provides the comprehensive, systematic and detailed study and survey of the state-of-the-art techniques and mechanisms in the field of QCA-based ALU designing. There are three categories in which QCA plays a role: ALU, logic unit (LU) and arithmetic unit (AU). Each category presents the important studies. In addition, this paper reviews the major developments in these three categories and it plans the new challenges. Furthermore, it provides the identification of open issues and guidelines for future research. Also, a Systematic Literature Review (SLR) on QCA-based ALU, LU and AU is discussed in this paper. We identified 1,960 papers, which are reduced to 26 primary studies through our paper selection process. According to the obtained results from 2001 to 2015, the number of published articles are very high in 2014 and low in 2005 and 2009. This survey paper also provides a discussion of considered mechanisms in terms of ALU, LU and AU attribute as well as directions for future research.

Many constructs in mathematical physics entail notational complexities, deriving from the manipulation of various types of index sets which often can be reduced to labelling by various multisets of integers. In this work, we develop systematically the "Dirichlet Hopf algebra of arithmetics" by dualizing the addition and multiplication maps. Then we study the additive and multiplicative antipodal convolutions which fail to give rise to Hopf algebra structures, but form only a weaker Hopf gebra obeying a weakened homomorphism axiom. A careful identification of the algebraic structures involved is done featuring subtraction, division and derivations derived from coproducts and chochains using branching operators. The consequences of the weakened structure of a Hopf gebra on cohomology are explored, showing this has major impact on number theory. This features multiplicativity versus complete multiplicativity of number theoretic arithmetic functions. The deficiency of not being a Hopf algebra is then cured by introducing an "unrenormalized" coproduct and an "unrenormalized" pairing. It is then argued that exactly the failure of the homomorphism property (complete multiplicativity) for non-coprime integers is a blueprint for the problems in quantum field theory (QFT) leading to the need for renormalization. Renormalization turns out to be the morphism from the algebraically sound Hopf algebra to the physical and number theoretically meaningful Hopf gebra (literally: antipodal convolution). This can be modelled alternatively by employing Rota–Baxter operators. We stress the need for a characteristic-free development where possible, to have a sound starting point for generalizations of the algebraic structures. The last section provides three key applications: symmetric function theory, quantum (matrix) mechanics, and the combinatorics of renormalization in QFT which can be discerned as functorially inherited from the development at the number-theoretic level as outlined here. Hence the occurrence of number theoretic functions in QFT becomes natural.

Let $K$ be the attractor of the following iterated function system (IFS)

As a consequence, we prove that the following conditions are equivalent:

(1) For any $u\in [0,1]$, there are some $x,y\in K$ such that $u=x\cdot y$.

(2) For any $u\in [0,1]$, there are some $x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_{10}\in K$ such that

(3) $c\ge {(1-\lambda )}^2$.

Let $({ℳ},c_k,n_k)$ be a class of Moran sets. We assume that the convex hull of any $E\in ({ℳ},c_k,n_k)$ is $[0,1]$. Let $A,B$ be two nonempty sets in $ℝ$. Suppose that $f$ is a continuous function defined on an open set $U\subset {ℝ}^2$. Denote the continuous image of $f$ by

We present a decision procedure for the ∀∃ theory of the lattice of Σ₁ sentences of Peano Arithmetic.

Cognitive psychologists have relied on dual-task interference experiments to understand the low-capacity and serial nature of conscious mental operations. Two widely studied paradigms, the Attentional Blink (AB) and the Psychological Refractory Period (PRP) have demonstrated a first-come first-served policy; processing a stimulus either impedes conscious access (AB) or postpones treatment (PRP) of a concurrent stimulus. Here we explored the transition from dual-task paradigms to multi-step human cognition. We studied the relative weight of individual addends in a sequential arithmetic task, where number notation (symbolic/non-symbolic) and presentation speed were independently manipulated. For slow presentation and symbolic notation, the decision relied almost equally on all addends, whereas for fast or non-symbolic notation, the decision relied almost exclusively on the last item reflecting a last-come first-served policy. We suggest that streams of stimuli may be chunked in events in which the last stimuli may override previous items from sensory buffers.

The simulation model of four stroke gasoline engine is established first according to the four stroke gasoline engine's operation principle in this paper. Then the simulation result and the result contrasting to the experiment are given out and the gasoline engine speed-governing system model is established based on the valve control, and the simulation result is also given out. Guided by the simulation model and simulation result, the design of software and hardware for electronic Speed-governing System is carried out, and the result of the experiment is given out.

This paper presents an analysis of arithmetic based on the view that natural numbers are properties.

This work considers Chaos aspects in a setting of arithmetic provided by Observer's Mathematics (see http://www.mathrelativity.com). We prove that the physical speed is a random variable, cannot exceed some constant, and this constant does not depend on an inertial coordinate system. Certain results and communications pertaining to these theorems are provided.