Let SBn be the singular braid group generated by braid generators σi and singular braid generators τi, 1≤i≤n−1. Let STn denote the group that is the kernel of the homomorphism that maps, for each i, σi to the cyclic permutation (i,i+1) and τi to 1. In this paper we investigate the group ST3. We obtain a presentation for ST3. We prove that ST3 is isomorphic to the singular pure braid group SP3 on 3

We deal with the problem of finding sufficient and necessary conditions on a generalized ordered space X under which for every compact space Y and for every separately continuous function f:X×Y→R: (a) f is of the first Baire class, i.e. X is a Moran space; (b) the function f is a pointwise limit of a sequence of continuous functions which is uniformly convergent to f with respect to each variable at

We explore the special structure of the top-dimensional homology of any compact triangulable space X of dimension d. Since there are no (d+1)-dimensional cells, the top homology equals the top cycles and is thus a free abelian group. There is no obvious basis, but we show that there is a canonical embedding of the top homology into a canonical free abelian group which has a natural basis up to signs

A subset S of a group G invariably generates G if G is generated by {sg(s)|s∈S} for any choice of g(s)∈G,s∈S. A topological group G is said to be ℐ𝒢 if it is invariably generated by some subset S⊆G, and 𝒯ℐ𝒢 if it is topologically invariably generated by some subset S⊆G. In this paper, we study the problem of (topological) invariable generation for linear groups and for automorphism groups of trees

Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in 𝕊3. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using v-move and forbidden moves. In this paper, we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in

We introduce a generalization of oriented tangles, which are still called tangles, so that they are in one‐to‐one correspondence with the sutured manifolds. We define cobordisms between sutured manifolds (tangles) by generalizing cobordisms between oriented tangles. For every commutative algebra A over Z / 2 Z , we define A - Tangles to be the category consisting of A ‐tangles, which are balanced tangles

Given an integer n≥2 and a digit set 𝒟⊂{0,1,…,n−1}2, there is a self-similar set K⊂ℝ2 satisfying the set equation K=(K+𝒟)/n. This set K is called a fractal square. By studying the line segments contained in K, we give a lower estimate of the topological Hausdorff dimension of fractal squares. Moreover, we compute the topological Hausdorff dimension of fractal squares whose nontrivial connected components

A new extended cubic B-spline (ECBS) approximation is formulated, analyzed and applied to obtain the numerical solution of the time fractional Klein–Gordon equation. The temporal fractional derivative is estimated using Caputo’s discretization and the space derivative is discretized by ECBS basis functions. A combination of Caputo’s fractional derivative and the new approximation of ECBS together with

Since the directional permeability of fractured rock masses is significantly dependent on the geometric properties of fractures, in this work, a numerical study was performed to analyze the relationships between them, in which fracture length follows a fractal distribution. A method to estimate the representative elementary volume (REV) size and directional permeability (Kf) by extracting regular polygon

In this paper, we introduce differential exponential maps in Cartesian differential categories, which generalizes the exponential function \(e^x\) from classical differential calculus. A differential exponential map is an endomorphism which is compatible with the differential combinator in such a way that generalizations of \(e^0 = 1\), \(e^{x+y} = e^x e^y\), and \(\frac{\partial e^x}{\partial x} =

We introduce the notion of πF(Δ)–network and denote the combinatorial principles FELL(ΠF(Δ),ΠF(Δ)) and FELL⁎(ΠF(Δ),ΠF(Δ)), which will be applied to characterize the spaces X whose hyperspace, endowed with the Fell topology, satisfies the SSR condition and the SR condition. We use the selection principle SSΔ⁎(O,O) to characterize the SSR property for the spaces K(X), F(X) and [X]1, endowed with the

It is known that there exists a surjective map from the set of weak (1, 3) homotopy classes of knot projections to the set of positive knots [N. Ito and Y. Takimura, (1, 2) and weak (1, 3) homotopies on knot projections, J. Knot Theory Ramifications22 (2013) 1350085]. An interesting question whether this map is also injective, which question was formulated independently by S. Kamada and Y. Nakanishi

A finitely generated group G acting on a tree with infinite cyclic edge and vertex stabilizers is called a generalized Baumslag–Solitar group (GBS group). We prove that a one-knot group G is a GBS group if and only if G is a torus knot group, and describe all n-knot GBS groups for n≥3.

In [V. O. Manturov, An almost classification of free knots, Dokl. Math.88(2) (2013) 556–558.] the second author constructed an invariant which in some sense generalizes the quantum sl(3) link invariant of Kuperberg to the case of free links. In this paper, we generalize this construction to free graph-links. As a result, we obtain an invariant of free graph-links with values in linear combinations

In the 1950s Milnor defined a family of higher-order invariants generalizing the linking number. Even the first of these new invariants, the triple linking number, has received fruitful study since its inception. In the case that a link L has vanishing pairwise linking numbers, this triple linking number gives an integer-valued invariant. When the linking numbers fail to vanish, the triple linking

We study groups of some virtual knots with small number of crossings and prove that there is a virtual knot with long lower central series which, in particular, implies that there is a virtual knot with residually nilpotent group. This gives a possibility to construct invariants of virtual knots using quotients by terms of the lower central series of knot groups. Also, we study decomposition of virtual

In 2003, Ozsváth and Szabó defined the concordance invariant τ for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of τ for knots in S3 and a combinatorial proof that τ gives a lower bound for the slice genus of a knot. Recently, Harvey and O’Donnol defined a relatively bigraded combinatorial Heegaard Floer homology theory

Ng constructed an invariant of knots in ℝ3, a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in ℝ4 using marked graph diagrams.

A twisted torus knot is a torus knot with some consecutive strands twisted. More precisely, a twisted torus knot T(p,q,r,s) is a torus knot T(p,q) with r consecutive strands s times fully twisted. We determine which twisted torus knots T(p,q,p−kq,−1) are a torus knot.

For any classical knot k1, we can construct a ribbon 2-knot spun(k1) by spinning an arc removed a small segment from k1 about R2 in R4. A ribbon 2-knot is an embedded 2-sphere in R4. If k1 has an n-crossing presentation, by spinning this, we can naturally construct a ribbon presentation with n ribbon crossings for spun(k1). Thus, we can define naturally a notion on ribbon 2-knots corresponding to the

If a continuous multivariate function satisfies a Lipschitz condition on its domain, Box dimension of its graph equals to the number of its arguments. Furthermore, Box dimension of the graph of its Riemann–Liouville fractional integral also equals to the number of its arguments.

This paper is concerned with the growth rate of the maximal digits relative to the rate of approximation of the number by its convergents, as well as relative to the rate of the sum of digits for the Lüroth expansion of an irrational number. The Hausdorff dimension of the sets of points with a given relative growth rate is proved to be full.

A geometric triangulation of a Riemannian manifold is a triangulation where the interior of each simplex is totally geodesic. Bistellar moves are local changes to the triangulation which are higher dimensional versions of the flip operation of triangulations in a plane. We show that geometric triangulations of a compact hyperbolic, spherical or Euclidean manifold are connected by geometric bistellar

The unordered configuration space of n points on a graph Γ, denoted here by UCn(Γ), can be viewed as the space of all configurations of n unlabeled robots on a system of one-dimensional tracks, which is interpreted as a graph Γ. The topology of these spaces is related to the number of vertices of degree greater than 2; this number is denoted by m(Γ). We discuss a combinatorial approach to compute the

This paper deals with the extension of Pontryagin reflexivity of the topological groups to the class of convergence groups. First, we give an example of a non-topological convergence on a group which is compatible with the algebraic structure of the group. Then we introduce the notion of local quasi-convexity for the class of convergence groups and present the results obtained while investigating the

We present a series of axioms that describe the combinatorial core of the Mathias model (countable support iteration of ω2 Mathias reals over the Continuum Hypothesis). Our axioms are formulated in terms of games with Borel sets and functions and are strong enough to imply most of the propositions usually proved by means of the iterated Mathias forcing.

We show that the triple-crossing number of any knot is greater or equal to twice its (canonical) genus and we show an even stronger bound in the case of links. As an application we show that this bound is strong enough to obtain the triple-crossing numbers of all torus knots, and of many more knots and their connected sums.

We develop the theory of APD profiles introduced by J. Dydak for ∞-pseudometric spaces ([3]). We connect them with transfinite asymptotic dimension defined by T. Radul ([4]). We give a characterization of spaces with transfinite asymptotic dimension at most ω+n for n∈ω and a sufficient condition for a space to have transfinite asymptotic dimension at most m⋅ω+n for m,n∈ω, using the language of APD

Given a hereditarily meager ideal I on a countable set X we use Martin's axiom for countable posets to produce a zero-dimensional maximal topology τI on X such that τI∩I={∅} and, moreover, if I is p+ then τI is selectively separable (SS) and if I is q+, so is τI. In particular, we obtain regular maximal spaces satisfying all boolean combinations of the properties SS and q+.

We extend results of Positive Twist Knots and Thickenings [W. George, Positive twist knots and thickenings, J. Knot Theory Ramifications22 (2013) 1350046] to show that positive twist knots 𝒦m, for m≥3, satisfy the Uniform Thickness Property.

We prove a new inequality relating volume to length of closed geodesics on area minimizers for generic metrics on the complex projective plane. We exploit recent regularity results for area minimizers by Moore and White, and the Kronheimer–Mrowka proof of the Thom conjecture.

In this paper, we study on the numerical solution of fractional nonlinear system of equations representing the one-dimensional Cauchy problem arising in thermoelasticity. The proposed technique is graceful amalgamations of Laplace transform technique with q-homotopy analysis scheme and fractional derivative defined with Atangana–Baleanu (AB) operator. The fixed-point hypothesis is considered in order

As the Riemann–Liouville derivative is a derivative of a convolution of a function and the power law, the fractal–fractional derivative of a function is the fractal derivative of a convolution of that function with the power law or exponential decay. In order to further open new doors on ongoing investigations with field of partial differential equations with non-conventional differential operators

In this paper, we present a new numerical scheme for a model involving new mathematical concepts that are of great importance for interpreting and examining real world problems. Firstly, we handle a Labyrinth chaotic problem with fractional operators which include exponential decay, power-law and Mittag-Leffler kernel. Moreover, this problem is solved via Atangana-Seda numerical scheme which is based

In this paper, we investigate a nonlinear coupled system of fractional pantograph differential equations (FPDEs). The respective results address some adequate results for existence and uniqueness of solution to the problem under consideration. In light of fixed point theorems like Banach and Krasnoselskii’s, we establish the required results. Considering the tools of nonlinear analysis, we develop

Our motive in this scientific contribution is to deal with nonlinear reaction–diffusion equation having both space and time variable order. The fractional derivatives which are used are non-singular having exponential kernel. These derivatives are also known as Caputo–Fabrizio derivatives. In our model, time fractional derivative is Caputo type while spatial derivative is variable-order Riesz fractional

In this paper, we present a construction of new nonlinear recurrent hidden variable fractal interpolation curves. In order to get new fractal curves, we use Rakotch’s fixed point theorem. We construct recurrent hidden variable iterated function systems with function vertical scaling factors to generate more flexible fractal interpolation curves. We also give an illustrative example to demonstrate the

In this work, a novel fractal model for the laminar flow in roughened cylindrical microchannels is proposed. The average height of rough elements is derived using the fractal theory. The effects of relative roughness on the friction factor and the Poiseuille number are discussed. It is found that the Darcy friction factor and the Poiseuille number increase with the increase in the relative roughness

In this paper, we discuss a family of p.c.f. self-similar fractal networks which have reflection transformations. We obtain the average geodesic distance on the corresponding fractal in terms of finite pattern of integrals. With these results, we also obtain the asymptotic formula for average distances of the skeleton networks.

Surface finish is one of the most important issues that is discussed in machining of materials. In fact, reaching the required surface finish is the key scale in analysis of the quality of machined workpiece. Chip formation is an important factor that highly affects the surface finish of machined workpiece. In this research, we analyze the relationship between surface finish and chip formation by fractal

The coupling of heat transfer and water flow in rock fractures is a key issue to geothermal energy extraction. However, this coupling in a rough fracture has not been well studied so far. This paper will study this coupling in a rock fracture with different roughness. First, multi-scale and self-affine rough fracture are constructed through the Weierstrass–Mandelbrot function and embedded into a rock

Tool wear is one of the unwanted phenomena in machining operations where tool has direct contact with the workpiece. Tool wear is an important issue in milling operation that is caused due to different parameters such as machine vibration. Tool wear shows complex structure, and machine vibration is a chaotic signal that also is complex. In this research, we analyze the correlation between tool wear

Cardiac tissue is characterized by structural and cellular heterogeneities that play an important role in the cardiac conduction system. Under persistent atrial fibrillation (persAF), electrical and structural remodeling occur simultaneously. The classical mathematical models of cardiac electrophysiological showed remarkable progress during recent years. Among those models, it is of relevance the standard

We consider criteria for the differentiability of functions with continuous Laplacian on the Sierpiński Gasket and its higher-dimensional variants SGN, N>3, proving results that generalize those of Teplyaev [Gradients on fractals, J. Funct. Anal.174(1) (2000) 128–154]. When SGN is equipped with the standard Dirichlet form and measure μ, we show there is a full μ-measure set on which continuity of the

Magnetic resonance image (MRI) is an important tool to diagnose human diseases effectively. It is very important for research and clinical application to classify the normal and abnormal human brain MRI images automatically. In this paper, an accurate and efficient technique is proposed to extract features of MRIs and classify these images into normal and abnormal categories. We use two-dimensional

In this paper, we construct the Bäcklund transformations and the super-position formulas to the constant coefficients local fractional Riccati equation for the first time. Next, by means of the Bäcklund transformations and seed solutions which have been known in [X. J. Yang et al., Non-differentiable solutions for local fractional nonlinear Riccati differential equations, Fundam. Inform.151(1–4) (2017)

In this paper, by using the theory of local fractional calculus and some techniques of real analysis, the structural characteristics of Hilbert-type local fractional integral inequalities with abstract homogeneous kernel are studied. At the same time, the necessary and sufficient conditions for these inequalities to take the best constant factor are discussed. As an application, some best constant

Hydraulic properties of rock fractures are important issues for many geoengineering practices. Previous studies have revealed that natural rock fractures have variable apertures that could significantly influence the permeability of single rock fractures, yet the effect of aperture variation on the hydraulic properties of fractured rock masses has received little attention. The present study implemented

The understanding of the diffusion process and mechanisms of harmful species (e.g. chlorides) in porous cementitious materials is important to control and improve the material durability under harsh environments. In this paper, fractal analysis on the pore structure of porous cementitious materials was conducted and involved in a diffusion model. Macro material geometric parameters were considered

In this paper, we study the boundedness character and persistence, local and global behavior, and rate of convergence of positive solutions of following system of rational difference equations xn+1=α1+β1e−yna1+b1xn,yn+1=α2+β2e−zna2+b2yn,zn+1=α3+β3e−xna3+b3zn, wherein the parameters αi,βi,ai,bi for i∈{1,2,3} and the initial conditions x0,y0,z0 are positive real numbers. Some numerical examples are given

Let Kλ be the attractor of the following iterated function system(IFS): {f1(x)=λx, f2(x)=λx+1−λ},0<λ<1/2. Given α≥0, we say the line y=αx is visible through Kλ×Kλ if {(x,αx):x∈ℝ∖{0}}∩(Kλ×Kλ)=∅. Let V={α≥0:y=αx is visible through Kλ×Kλ}. In this paper, we give a complete description of V, containing its Hausdorff dimension and topological properties.

The q-fractional differential equation usually describes the physics process imposed on the time scale set Tq. In this paper, we first propose a difference formula for discretizing the fractional q-derivative cDqαx(t) on the time scale set Tq with order 0<α<1 and scale index 0

Using multifractal detrended cross-correlation analysis (MF-DCCA) and nonlinear Granger causality test, this paper examines the return predictability of margin-trading activities. Results show that the predictive power of margin-trading activities on subsequent stock returns varies with respect to the different aspects of margin trading. In line with previous studies, we find no significant correlation

A closed surface evolving under mean curvature flow becomes singular in finite time. Near the singularity, the surface resembles a self-shrinker, a surface that shrinks by dilations under mean curvature flow. If the singularity is modeled on a self-shrinker other than a round sphere or cylinder, then the singularity is unstable under perturbations of the flow. One can quantify this instability using

We study multi-moment maps on nearly Kähler six-manifolds with a two-torus symmetry. Critical points of these maps have non-trivial stabilisers. The configuration of fixed-points and one-dimensional orbits is worked out for generic six-manifolds equipped with an \(\mathrm {SU}(3)\)-structure admitting a two-torus symmetry. Projecting the subspaces obtained to the orbit space yields a trivalent graph

In this note we present corrected proofs of some results in [3] which are affected by the statement of the first part of Theorem 1.1. We also provide changes in the proofs of Theorems 1.4 and 1.5.

We introduce a notion of oriented dialgebra and develop a cohomology theory for oriented dialgebras by mixing the standard chain complexes computing group cohomology and associative dialgebra cohomology. We also introduce a formal deformation theory for oriented dialgebras and show that cohomology of oriented dialgebras controls such deformations.

Let A be a real projective line arrangement and M(A) its complement in CP2. We obtain an explicit expression in terms of Randell's generators of the meridians around the exceptional divisors in the blow-up X¯ of CP2 in the singular points of A. We use this to investigate the partial compactifications of M(A) contained in X¯ and give a counterexample to a statement suggested by A. Dimca and P. Eyssidieux

We directly connect topological changes that can occur in mathematical three-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena. This work widens the bridge between topology and natural sciences and creates a new platform for exploring geometrical physics.

We examine distortion of finitely generated normal subgroups. We show a connection between subgroup distortion and group divergence. We suggest a method computing the distortion of normal subgroups by decomposing the whole group into smaller subgroups. We apply our work to compute the distortion of normal subgroups of graph of groups and normal subgroups of right-angled Artin groups that induce infinite