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The article shows that even a small number of iterations give a very large length of the Koch Curve. The overall purpose of this work is to show that even a small number of iterations can give a very complex fractal structure. In this case it is the very big length of Koch's curve. As examples, the distance from the Earth to the Moon and from the Earth to the Sun was compared to the length of the Koch curve. [ABSTRACT FROM AUTHOR]

The theory of iterated function systems (IFSs) has been an active area of research on fractals and various types of self-similarity in nature. The basic theoretical work on IFSs has been proposed by Hutchinson. In this paper, we introduce a new generalization of Hutchinson IFS, namely generalized q-contraction IFS, which is a finite collection of generalized q-contraction functions T1, . . ., TN from finite Cartesian product space X x ... x X into X, where (X, d) is a complete metric space. We prove the existence of attractor for this generalized IFS. We show that the Hutchinson operators for countable and multivalued q-contraction IFSs are Picard. Finally, when the map q is continuous, we show the relation between the code space and the attractor of q-contraction IFS. [ABSTRACT FROM AUTHOR]

The fractal manufacturing system (FrMS) is based on the concept of autonomously cooperating agents referred to as fractals. A fractal is a set of self-similar agents whose goal can be achieved through cooperation, coordination, and negotiation among the agents for themselves. A fractal has fractal-specific characteristics (e.g. self-similarity, self-organization, self-optimization, goal-orientation, and dynamics), and it also has the characteristics of an agent (e.g. autonomy, mobility, intelligence, cooperation, and adaptability) at the same time. In the FrMS, a goal can be regarded as the status which the system aspires to be in. The goal-formation process (GFP) in the FrMS is a process of generating goals and modifying them by coordination between agents. In the GFP, conflicts may occur between goals, which can drive a system to become inefficient. In this paper, a conflict resolution mechanism via agent-based negotiation is proposed for facilitating the GFP. The scheme deals with non-fixed goals. The mobile agent-based negotiation process (MANPro), in which a mobile agent is used for information-exchanging and problem-solving, is used for negotiations in this scheme. The proposed mechanism is illustrated with a goal formation scenario in an exemplary FrMS. [ABSTRACT FROM AUTHOR]

Summary: Geometric pore surface area is a significant parameter for the description of the irregular, somewhat random, porous structure of the catalyzed electrodes in polymer electrolyte membrane (PEM) fuel cells; however, its value is sensitive to the experimental methods employed, which necessitates the measurements via different methods. In this study, the geometric surface area of the porous electrode is determined by two different methods: the method of standard porosimetry (MSP) and Brunauer‐Emmett‐Teller (BET). The theory of fractal dimension is employed to analyze the data obtained from the MSP, and the fractal surface area calculated using nitrogen molecules as the scale is compared with the BET surface area. The results indicate that the geometric pore surface area is a property of the porous electrode that depends greatly on the "scale" size (ie, molecule size of the working fluid)—a smaller scale yields a larger value of the surface area. The surface area determined by the BET is found to be about one order of magnitude larger than that obtained by the MSP. Thus, the fractal dimension theory based on MSP demonstrates a useful tool to determine the accessible pore surface area at different length scales. [ABSTRACT FROM AUTHOR]

Abstract We study the similarities between the Fano varieties of lines on a cubic fourfold, a hyper-Kähler fourfold studied by Beauville and Donagi, and the hyper-Kähler fourfold constructed by Debarre and Voisin in [3]. We exhibit an analog of the notion of "triangle" for these varieties and prove that the 6-dimensional variety of "triangles" is a Lagrangian subvariety in the cube of the constructed hyper-Kähler fourfold. [ABSTRACT FROM AUTHOR]

Abstract Given a finite dimensional algebra A over an algebraically closed field, we consider the c -vectors such as defined by Fu in [18] and we give a new proof of its sign-coherence. Moreover, we characterise the modules whose dimension vectors are c -vectors as bricks respecting a functorially finiteness condition. [ABSTRACT FROM AUTHOR]

Abstract We illustrate an algorithm to classify nice nilpotent Lie algebras of dimension n up to a suitable notion of equivalence; applying the algorithm, we obtain complete listings for n ≤ 9. On every nilpotent Lie algebra of dimension ≤7, we determine the number of inequivalent nice bases, which can be 0, 1, or 2. We show that any nilpotent Lie algebra of dimension n has at most countably many inequivalent nice bases. [ABSTRACT FROM AUTHOR]

In this note we combine the "spin-argument" from Kitavtsev et al. (Proc R Soc Edinb Sect A Mater 147(5):1041-1089, 2017) and the n-dimensional incompatible, one-well rigidity result from Lauteri and Luckhaus (An energy estimate for dislocation configurations and the emergence of Cosserat-type structures in metal plasticity, 2016), in order to infer a new proof for the compactness of discrete multi-well energies associated with the modelling of surface energies in certain phase transitions. Mathematically, a main novelty here is the reduction of the problem to an incompatible one-well problem. The presented argument is very robust and applies to a number of different physically interesting models, including for instance phase transformations in shape-memory materials but also anti-ferromagnetic transformations or related transitions with an "internal" microstructure on smaller scales. [ABSTRACT FROM AUTHOR]

Abstract We combine aspects of the notions of finite decomposition complexity and asymptotic property C into a notion that we call finite APC-decomposition complexity. Any space with finite decomposition complexity has finite APC-decomposition complexity and any space with asymptotic property C has finite APC-decomposition complexity. Moreover, finite APC-decomposition complexity implies property A for metric spaces. We also show that finite APC-decomposition complexity is preserved by direct products of groups and spaces, amalgamated products of groups, and group extensions, among other constructions. [ABSTRACT FROM AUTHOR]

Abstract We prove a new sharpened version of the Strichartz inequality for radial solutions of the Schrödinger equation in two dimensions. We establish an improved upper bound for functions that nearly extremize the inequality, with a negative second term that measures the distance from the initial data to Gaussians. [ABSTRACT FROM AUTHOR]

We consider the self-assembly of fractals in one of the most well-studied models of tile based self-assembling systems known as the Two-Handed Tile Assembly Model (2HAM). In particular, we focus our attention on a class of fractals called discrete self-similar fractals (a class of fractals that includes the discrete Sierpiński carpet). We present a 2HAM system that finitely self-assembles the discrete Sierpiński carpet with scale factor 1. Moreover, the 2HAM system that we give lends itself to being generalized and we describe how this system can be modified to obtain a 2HAM system that finitely self-assembles one of any fractal from an infinite set of fractals which we call 4-sided fractals. The 2HAM systems we give in this paper are the first examples of systems that finitely self-assemble discrete self-similar fractals at scale factor 1 in a purely growth model of self-assembly. Finally, we show that there exists a 3-sided fractal (which is not a tree fractal) that cannot be finitely self-assembled by any 2HAM system. [ABSTRACT FROM AUTHOR]

We carry out a numerical study of the quantum walk search algorithm of Shenvi, Kempe and Whaley Shenvi et al. (2003) and the factors that affect its efficiency in finding an individual state from an unsorted set. Previous work has focused purely on the effects of the dimensionality of the dataset to be searched. In the current paper we consider the effects of interpolating between dimensions, the connectivity of the dataset and the possibility of disorder in the underlying substrate: all these factors affect the efficiency of the search algorithm. We show that in addition to the strong dependence on the spatial dimension of the structure to be searched, there are also secondary dependencies on the connectivity and symmetry of the lattice, with greater connectivity providing a more efficient algorithm. We also show that the algorithm can tolerate a non-trivial level of disorder in the underlying substrate. [ABSTRACT FROM AUTHOR]

A class of q-ary totally decomposed cumulative Γ(L,Gj)-codes with L={α∈:G(α)≠0} and Gj=Gj(x),1≤j≤q, where G(x) is a polynomial totally decomposed in GF(qm), are considered. The relation between different codes from this class is studied. Improved bounds of the minimum distance and dimension are obtained. [ABSTRACT FROM AUTHOR]

We determine the maximum size A2(8,6;4) of a binary subspace code of packet length v=8, minimum subspace distance d=6, and constant dimension k=4 to be 257. There are two isomorphism types of optimal codes. Both of them are extended LMRD codes. In finite geometry terms, the maximum number of solids in PG(7,2) mutually intersecting in at most a point is 257. The result was obtained by combining the classification of substructures with integer linear programming techniques. This result implies that the maximum size A2(8,6) of a binary mixed-dimension subspace code of packet length 8 and minimum subspace distance 6 is 257 as well. [ABSTRACT FROM AUTHOR]

We show that the configuration spaces of a product of parallelizable manifolds may be recovered from those of the factors as the Boardman- Vogt tensor product of right modules over the operads of little cubes of the appropriate dimension. We also discuss an analogue of this result for manifolds that are not necessarily parallelizable, which involves a new operad of skew little cubes. [ABSTRACT FROM AUTHOR]

We study the structure of the exteriors of gropes and Whitney towers in dimension 4, focusing on their fundamental groups. In particular we introduce a notion of unknottedness of gropes and Whitney towers in the 4-sphere. We prove that various modifications of gropes and Whitney towers preserve the unknottedness and do not enlarge the fundamental group. We exhibit handlebody structures of the exteriors of gropes and Whitney towers constructed by earlier methods of Cochran, Teichner, Horn, and the first author and use them to construct examples of unknotted gropes and Whitney towers. As an application, we introduce geometric bi-filtrations of knots which approximate the double sliceness in terms of unknotted gropes and Whitney towers. We prove that the bi-filtrations do not stabilize at any stage. [ABSTRACT FROM AUTHOR]

A generalization of Köthe rings is the family of rings whose ideals are direct sums of cyclic modules. These rings were previously studied in the commutative local case. This motivated us to study commutative rings with local dimension whose ideals are direct sums of cyclic modules. First, we obtain an structure theorem for rings of local dimension ω. Then we conclude that, for a ring of local dimension ω , every ideal of R is a direct sum of cyclic modules if and only if every indecomposable ideal of R is cyclic if and only if every maximal ideal of R / Soc (R) is cyclic if and only if every maximal ideal of R is cyclic if and only if R is a principal ideal ring. [ABSTRACT FROM AUTHOR]

We consider the problem Δ2u = V(x)up + ϵ in RN with u,Δu→0 as |x|→ + ∞, where p=N+4N−4, N ≥ 5, V is a positive continuous potential. Our aim is to construct high‐energy solutions for this equation by applying the finite‐dimensional reduction method and the penalization method. [ABSTRACT FROM AUTHOR]

We present novel nonlocal governing operators in 2D/3D for wave propagation and diffusion. The operators are inspired by peridynamics. They agree with the original peridynamics operator in the bulk of the domain and simultaneously enforce local boundary conditions (BC). The main ingredients are periodic, antiperiodic, and mixed extensions of separable kernel functions together with even and odd parts of bivariate functions on rectangular/box domains. The operators are bounded and self-adjoint. We present all possible 36 different types of BC in 2D which include pure and mixed combinations of Neumann, Dirichlet, periodic, and antiperiodic BC. Our construction is systematic and easy to follow. We provide numerical experiments that verify our theoretical findings. We also compare the solutions of the classical wave and heat equations to their nonlocal counterparts. [ABSTRACT FROM AUTHOR]

Highlights • An effective thermal conductivity model is derived based on fractal theory. • Predictions of proposed model show good consistent with experimental data. • The proposed theoretical model reveals the physical basis of heat transfer in porous media. Abstract Thermal conduction in porous media has received wide attention in science and engineering in the past decades. Previous models of the effective thermal conductivity of porous media contain empirical parameters typically with ambiguous or even no physical rationales. This study proposes a theoretical model of effective thermal conductivity in porous media with various liquid saturation based on the fractal geometry theory. This theoretical model considers geometrical parameters of porous media, including porosity, liquid saturation, fractal dimensions for both the granular matrix and liquid phases, and tortuosity fractal dimension of the liquid phase. Effects of these geometrical parameters on the effective thermal conductivity of porous media are also evaluated. This proposed fractal model has been validated using published experimental data, compared with previous models, and thus provides a physics-based theoretical model that can provide insight to geoscience and thermophysics studies on thermal conduction in porous media. [ABSTRACT FROM AUTHOR]

Optimal estimates on stabilized finite volume methods for the three dimensional Navier–Stokes model are investigated and developed in this paper. Based on the global existence theorem [23], we first prove the global bound for the velocity in the H1‐norm in time of a solution for suitably small data, and uniqueness of a suitably small solution by contradiction. Then, a full set of estimates is then obtained by some classical Galerkin techniques based on the relationship between finite element methods and finite volume methods approximated by the lower order finite elements for the three dimensional Navier–Stokes model. [ABSTRACT FROM AUTHOR]

Abstract We first give an example of a rigid structure of computable dimension 2 such that the unique isomorphism between two non-computably isomorphic computable copies has Turing degree strictly below 0 ″ , and not above 0 ′. This gives a first example of a computable structure with a degree of categoricity that does not belong to an interval of the form [ 0 (α) , 0 (α + 1) ] for any computable ordinal α. We then extend the technique to produce a rigid structure of computable dimension 3 such that if d 0 , d 1 , and d 2 are the degrees of isomorphisms between distinct representatives of the three computable equivalence classes, then each d i < d 0 ⊕ d 1 ⊕ d 2. The resulting structure is an example of a structure that has a degree of categoricity, but not strongly. [ABSTRACT FROM AUTHOR]

The financial world is typically seen as a miasma of numbers, which help to make it abstract and opaque. This essay argues that the forms and techniques of the financial world have a strong literary and linguistic dimension. This dimension can be seen in the form of the derivative, whose workings show how traders, central bankers, and analysts rely centrally on the written, spoken, and narrated word and its performative powers. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and Specs(M) denote the set all second submodules of an R-module M. In this paper, we investigate various properties of Specs(M) with respect to different topologies. We investigate the dual Zariski topology from the point of view of separation axioms, spectral spaces and combinatorial dimension. We establish conditions for Specs(M) to be a spectral space with respect to quasi-Zariski topology and second classical Zariski topology. We also present some conditions under which a module is cotop. [ABSTRACT FROM AUTHOR]

We endow a topological group (G,τ) with a coarse structure defined by the smallest group ideal Sτ on G containing all converging sequences with their limits and denote the obtained coarse group by (G,Sτ). If G is discrete then (G,Sτ) is a finitary coarse group studding in Geometric Group Theory. The main result: if a topological abelian group (G,τ) contains a non-trivial converging sequence then asdim (G,Sτ)=∞. We study metrizability, normality and functional boundedness of sequential coarse groups and put some open questions. [ABSTRACT FROM AUTHOR]

Abstract In 1961, Gerstenhaber proved the following theorem: if k is a field and X and Y are commuting d × d matrices with entries in k , then the unital k -algebra generated by these matrices has dimension at most d. The analog of this statement for four or more commuting matrices is false. The three matrix version remains open. We use commutative–algebraic techniques to prove that the three matrix analog of Gerstenhaber's theorem is true for some new classes of examples. In particular, we translate this three commuting matrix statement into an equivalent statement about certain maps between modules, and prove that this commutative–algebraic reformulation is true in special cases. We end with ideas for an inductive approach intended to handle the three matrix analog of Gerstenhaber's theorem more generally. [ABSTRACT FROM AUTHOR]

Abstract The recent emergence and use growth of cryptocurrencies based on Blockchain technology increased interest in the study of its economic dynamics and financial characteristics. Bitcoin is up to now the more widely known and disseminated cryptocurrency, with greater volume of transactions, market value and acceptance in exchange services. In order to contribute to the comprehension of the price behavior of the Bitcoin market, this study analyzes whether the historical series of prices of this currency, quoted every 12 h from September 14, 2011 to November 20, 2017 has multifractal behavior. The results of the research identified multifractal characteristics in the series and that both long-range correlations and fat tails distribution contribute to Bitcoin’s multifractal behavior. We compared the non-Gaussian properties and the multifractality degrees of Bitcoin series with the non-Gaussian properties and multifractality degrees of several stock market indices scattered around the world. In addition, we investigated the power of multifractal analysis in the study of volatility and forecast for this series, pointing to a possible use of multifractal parameters in Technical Analysis. Highlights • An investigation of the multifractality of the Bitcoin time series is addressed. • The multifractality was evidenced in the work through the MFDFA method. • The multifractality originates from both long-range correlations and fat tails. • The multifractality degrees (MD) of Bitcoin are higher than the MD of many indices. • There is a direct relation between multifractality degrees and volatility in the same time interval. [ABSTRACT FROM AUTHOR]

Considered herein is the Cauchy problem associated with the fifth-order -dimensional Kadomtsev-Petviashvili II equation. It is proved that, if a sufficiently smooth solution to this problem is supported compactly at two different instants of time, then it vanishes identically. [ABSTRACT FROM AUTHOR]

Abstract We study the spectral properties of positive absolutely minimum attaining operators defined on infinite dimensional complex Hilbert spaces and using this we derive a characterization theorem for such type of operators. We construct several examples and discuss some of the properties of this class. Also, we extend this characterization theorem for general absolutely minimum attaining operators by means of the polar decomposition theorem. [ABSTRACT FROM AUTHOR]

Abstract In the paper, we investigate a modified version of the shrinking target problem in beta dynamical systems which is induced by the β -transformation T β : x → β x (mod 1). The Hausdorff dimension of the lim sup set W (f , φ) = { x ∈ [ 0 , 1) : | T β n x − f (x) | < φ (n) for infinitely many n ∈ N } is determined, where f : [ 0 , 1 ] → [ 0 , 1 ] is a Lipschitz function and φ is a positive function defined on N. The set W (f , φ) can be viewed as a dynamical analogue of the inhomogeneous Diophantine approximation in which the inhomogeneous factor is allowed to vary. [ABSTRACT FROM AUTHOR]

Abstract We consider a class of the focusing nonlinear Schrödinger equation with inverse-square potential i ∂ t u + Δ u − c | x | − 2 u = − | u | α u , u (0) = u 0 ∈ H 1 , (t , x) ∈ R × R d , where d ≥ 3 , 4 d ≤ α ≤ 4 d − 2 and c ≠ 0 satisfies c > − λ (d) : = − (d − 2 2) 2. In the mass-critical case α = 4 d , we prove the global existence and blowup below ground states for the equation with d ≥ 3 and c > − λ (d). In the mass and energy intercritical case 4 d < α < 4 d − 2 , we prove the global existence and blowup below the ground state threshold for the equation. This extends similar results of [18] and [22] to any dimensions d ≥ 3 and a full range c > − λ (d). We finally prove the blowup below ground states for the equation in the energy-critical case α = 4 d − 2 with d ≥ 3 and c > − d 2 + 4 d (d + 2) 2 λ (d). [ABSTRACT FROM AUTHOR]

In this paper, a (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation is investigated, which can describe an inhomogeneous Heisenberg ferromagnetic spin chain with the octupole–dipole interaction or alpha helical protein with higher-order excitations and interactions under the continuum approximation. Bilinear forms, one- and two-soliton solutions are derived by virtue of the Hirota method and symbolic computation. Propagation and interaction properties of the solitons are discussed: Parabolic, cubic and periodic solitons are presented. Amplitudes of the solitons are only related to the wave numbers, while the velocities are related to both the wave numbers and variable coefficients. Interactions between the two parallel solitons are discussed. [ABSTRACT FROM AUTHOR]

A De La Vallée Poussin-type inequality is established for a certain class of partial differential equations posed in a bounded domain Ω in R N + 1 , N ≥ 1 , under Dirichlet boundary condition. Next, as applications of the obtained inequality, some sign-changing criteria are derived for the nontrivial solutions. [ABSTRACT FROM AUTHOR]