Zeta functions of discrete self-similar sets

In this paper we study a class of countable and discrete subsets of a Euclidean space that are “self-similar” with respect to a finite set of (affine) similarities. Any such set can be interpreted as having a fractal structure. We introduce a zeta function for these sets, and derive basic analytic properties of this “fractal” zeta function. Motivating examples that come from combinatorial geometry and arithmetic are given particular attention.     

Dimensions of fractals in the large

Fractals in the large can be generated as the invariant set of an expansive, iterated function system. A number of dimensions have been introduced and studied for such fractals. In this note we show that these dimensions coincide for large fractals generated by functions with arithmetic expansion factors, and that this common dimension is equal to the dimension of the (small) fractal generated by the inverse functions.     

On the arithmetic of fractal dimension using hyperhelices

A hyperhelix is a fractal curve generated by coiling a helix around a rect line, then another helix around the first one, a third around the second… an infinite number of times. A way to generate hyperhelices with any desired fractal dimension is presented, leading to the result that they have embedded an algebraic structure that allows making arithmetic with fractal dimensions and to the idea of an infinitesimal of fractal dimension.     

Fractal structure and fractal dimension determination at nanometer scale

Three-dimensional fractures of different fractal dimensions have been constructed with successive random addition algorithm, the applicability of various dimension determination methods at nanometer scale has been studied. As to the metallic fractures, owing to the limited number of slit islands in a slit plane or limited datum number at nanometer scale, it is difficult to use the area-perimeter method or power spectrum method to determine the fractal dimension. Simulation indicates that box-counting method can be used to determine the fractal dimension at nanometer scale. The dimensions of fractures of valve steel 5Cr21Mn9Ni4N have been determined with STM. Results confirmed that fractal dimension varies with direction at nanometer scale. Our study revealed that, as to theoretical profiles, the dependence of frsctal dimension with direction is simply owing to the limited data set number, i.e. the effect of boundaries. However, the dependence of fractal dimension with direction at nanometer scale in real metallic fractures is correlated to the intrinsic characteristics of the materials in addition to the effect of boundaries. The relationship of fractal dimensions with the mechanical properties of materials at macrometer scale also exists at nanometer scale. Project supported by the National Natural Science Foundation of China (Grant Nos. 59771050 and 59872004) and the Foundation Fund of Ministry of Metallurgical Industry.     

Randomized proofs in arithmetic

A randomized proof system for arithmetic is introduced. A proof of an arithmetical formula is defined as its derivation from the axioms of arithmetic by the standard rules of inference of arithmetic and also one more rule which we call the random substitution rule. Such proofs can be regarded as a special kind of interactive proof and, more exactly, as a special kind of the Arthur-Merlin proofs. The main result of the paper shows that a proof in arithmetic with the random substitution rule can be considerably shorter than an arithmetical proof of the same formula. Namely, there exists a set of formulas such that (i) these formulas are provable in arithmetic but, unless PSPACE=NP, do not have polynomially long proofs; (ii) these proofs have polynomially long proofs in arithmetic with random substitution (whatever random numbers appear) and the probability of error of these proofs is exponentially small. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 220, 1995, pp. 49–71. Translated by E. Ya. Dantsin.     

Fractal Approximation

In the present article every complex square integrable function defined in a real bounded interval is approached by means of a complex fractal function. The approximation depends on a partition of the interval and a vectorial parameter of the iterated function system providing the fractal attractor. The original may be discontinuous or undefined in a set of zero measure. The fractal elements can modify the features of the originals, for instance their character of smooth or non-smooth. The properties of the operator mapping every function into its fractal analogue are studied in the context of the uniform and least square norms. In particular, the transformation provides a decomposition of the set of square integrable maps. An orthogonal system of fractal functions is constructed explicitly for this space. Sufficient conditions for the uniform convergence of the fractal series expansion corresponding to this basis are also deduced. The fractal approximation of real functions is obtained as a particular case.     

Fourier transforms on Cantor sets: A study in non-Diophantine arithmetic and calculus

Fractals equipped with intrinsic arithmetic lead to a natural definition of differentiation, integration, and complex structure. Applying the formalism to the problem of a Fourier transform on fractals we show that the resulting transform has all the required basic properties. As an example we discuss a sawtooth signal on the ternary middle-third Cantor set. The formalism works also for fractals that are not self-similar.     

Logic and probabilistic systems

Following some ideas of Roberto Magari, we propose trial and error probabilistic functions, i.e. probability measures on the sentences of arithmetic that evolve in time by trial and error. The set of the sentences that get limit probability 1 is a theory, in fact can be a complete set. We prove incompleteness results for this setting, by showing for instance that for every there are true sentences that get limit probability less than . No set as above can contain the set of all true sentences, although we exhibit some containing all the true sentences. We also consider an approach based on the notions of inner probability and outer probability, and we compare this approach with the one based on trial and error probabilistic functions. Although the two approaches are shown to be different, we single out an important case in which they are equivalent. Received March 20, 1995     

Estimation of fractal dimension and fractal curvatures from digital images

Most of the known methods for estimating the fractal dimension of fractal sets are based on the evaluation of a single geometric characteristic, e.g. the volume of its parallel sets. We propose a method involving the evaluation of several geometric characteristics, namely all the intrinsic volumes (i.e. volume, surface area, Euler characteristic, etc.) of the parallel sets of a fractal. Motivated by recent results on their limiting behavior, we use these functionals to estimate the fractal dimension of sets from digital images. Simultaneously, we also obtain estimates of the fractal curvatures of these sets, some fractal counterpart of intrinsic volumes, allowing a finer classification of fractal sets than by means of fractal dimension only. We show the consistency of our estimators and test them on some digital images of self-similar sets.     

Fractal Polynomials and Maps in Approximation of Continuous Functions

The primary goal of this article is to establish some approximation properties of fractal functions. More specifically, we establish that a monotone continuous real-valued function can be uniformly approximated with a monotone fractal polynomial, which in addition agrees with the function on an arbitrarily given finite set of points. Furthermore, the simultaneous approximation and \mboxinterpolation which is norm-preserving property of fractal polynomials is established. In the final part of the article, we establish differentiability of a more general class of fractal functions. It is shown that these smooth fractal functions and their derivatives are good approximants for the original function and its \mboxderivatives.     

机械臂运动路径设计

采用D-H方法推导出了机械臂角度姿态与指尖空间位置的关系,以距离偏差为目标,建立了给定位置求机械臂参数的优化模型,并提出了二分步长接近式搜索算法求解该模型.进一步从实际工程精度要求出发,提出了确定精度求指令的方法.针对按照确定曲线运行的问题,采取了离散曲线而且相邻离散点之间一条指令可达的方法,并用改进的二分步长接近式搜索算法来求解.提出了解决避障问题的启发式算法和位姿空间变换法.还对机械臂结构进行了理论分析及论证,根据分析结果,针对旋转角度范围、角度量化、杆长选择等方面提出了一些建议.     

Euclidean and Lorentzian quantum gravity—lessons from two dimensions

No theory of four-dimensional quantum gravity exists as yet. In this situation the two-dimensional theory, which can be analyzed by conventional field-theoretical methods, can serve as a toy model for studying some aspects of quantum gravity. It represents one of the rare settings in a quantum-gravitational context where one can calculate quantities truly independent of any background geometry.We review recent progress in our understanding of 2d quantum gravity, and in particular the relation between the Euclidean and Lorentzian sectors of the quantum theory. We show that conventional 2d Euclidean quantum gravity can be obtained from Lorentzian quantum gravity by an analytic continuation only if we allow for spatial topology changes in the latter. Once this is done, one obtains a theory of quantum gravity where space-time is fractal: the intrinsic Hausdorff dimension of usual 2d Euclidean quantum gravity is four, and not two. However, certain aspects of quantum space-time remain two-dimensional, exemplified by the fact that its so-called spectral dimension is equal to two.     

Fractal properties of refined box dimension on functional graph

Many fractals, which have theoretical and practical significance, take the form of functional graphs. For continuous functions whose graphs are fractal sets, their fractal characteristics are studied, and the relations between refined box dimensions of functional graphs before and after four arithmetic operations are discussed.     

A Characterization of Self-similar Symbolic Spaces

In this paper we use fractal structures to study self-similar sets and self-similar symbolic spaces. We show that these spaces have a natural fractal structure, justifying the name of fractal structure, and we characterize self-similar symbolic spaces in terms of fractal structures. We prove that self-similar symbolic spaces can be characterized in a similar way, in the form, to the definition of classical self-similar sets by means of iterated function systems. We also study when a self-similar symbolic space is a self-similar set. Finally, we study relations between fractal structures with “pieces” homeomorphic to the space and different concepts of self-homeomorphic spaces. Along the paper, we propose several methods in order to construct self-similar sets and self-similar symbolic spaces from a geometrical approach. This allows to construct these kind of spaces in a very easy way.     

Geometry of the common dynamics of Pisot substitutions with the same incidence matrix

Any Pisot substitution can be associated with a bounded set with interesting properties, called the Rauzy fractal. This set is obtained by projection of the broken line associated with an infinite fixed point. Two substitutions having the same incidence matrix can have different Rauzy fractals. We show that under weak conditions, the intersection of these two fractals has strictly positive measure, and can also be generated by a substitution.     

Open set condition for graph directed self-similar structure

Graph directed self-similar structure generalizes the concept of self-similar set and contains some important instants of fractal sets. We characterize the open set condition (OSC), which is fundamental in the study of self-similar set, for graph directed self-similar structure in terms of the post critical set. Using this characterization, we establish the relations between OSC and other separation conditions including post-critically finite, finitely ramified condition and finite preimage property. It turns out that whether the intrinsic metric is doubling makes difference. In particular, finitely ramified condition implies OSC in case of doubling metric but does not in case of non-doubling metric.     

The pλn fractal decomposition: Nontrivial partitions of conserved physical quantities

A mathematical method for constructing fractal curves and surfaces, termed the pλn fractal decomposition, is presented. It allows any function to be split into a finite set of fractal discontinuous functions whose sum is equal everywhere to the original function. Thus, the method is specially suited for constructing families of fractal objects arising from a conserved physical quantity, the decomposition yielding an exact partition of the quantity in question. Most prominent classes of examples are provided by Hamiltonians and partition functions of statistical ensembles: By using this method, any such function can be decomposed in the ordinary sum of a specified number of terms (generally fractal functions), the decomposition being both exact and valid everywhere on the domain of the function.     

Geometry of plane sections of the infinite regular skew polyhedron {4, 6 | 4}

The asymptotic behavior of open plane sections of triply periodic surfaces is dictated, for an open dense set of plane directions, by an integer second homology class of the three-torus. The dependence of this homology class on the direction can have a rather rich structure, leading in special cases to a fractal. In this paper we present in detail the results for the skew polyhedron {4, 6 | 4} and in particular we show that in this case a fractal arises and that such a fractal can be generated through an elementary algorithm, which in turn allows us to verify for this case a conjecture of Novikov that such fractals have zero measure.      

The catenary degrees of elements in numerical monoids generated by arithmetic sequences

We compute the catenary degree of elements contained in numerical monoids generated by arithmetic sequences. We find that this can be done by describing each element in terms of the cardinality of its length set and of its set of factorizations. As a corollary, we find for such monoids that the catenary degree becomes fixed on large elements. This allows us to define and compute the dissonance number- the largest element with a catenary degree different from the fixed value. We determine the dissonance number in terms of the arithmetic sequence’s starting point and its number of generators.