For an element in BS(1,n)=〈t,a|tat−1=an〉 written in the normal form t−uavtw with u,w≥0 and v∈ℤ, we exhibit a geodesic word representing the element and give a formula for its word length with respect to the generating set {t,a}. Using this word length formula, we prove that there are sets of elements of positive density of positive, negative and zero conjugation curvature, as defined by Bar Natan,

We study the nonlinear embeddability of Banach spaces and the equi-embeddability of the family of Kalton’s interlaced graphs ([ℕ]k,d𝕂)k into dual spaces. Notably, we define and study a modification of Kalton’s property 𝒬 that we call property 𝒬p (with p∈(1,+∞]). We show that if ([ℕ]k,d𝕂)k equi-coarse Lipschitzly embeds into X∗, then the Szlenk index of X is greater than ω, and that this is optimal

We show that the Torelli group of a closed surface of genus ≥3 acts nontrivially on the rational cohomology of the space of 3-element subsets of that surface. This implies that for a Riemann surface of genus ≥3, the mixed Hodge structure on the space of its positive, reduced divisors of degree 3 does in general not split over ℚ.

We generalize the notions of asymptotic dimension and coarse embeddings from metric spaces to quantum metric spaces in the sense of Kuperberg and Weaver [A von Neumann algebra approach to quantum metrics, Mem. Am. Math. Soc.215(1010) (2012) 1–80]. We show that quantum asymptotic dimension behaves well with respect to metric quotients and direct sums, and is preserved under quantum coarse embeddings

We define and explore rack objects internal to categories with products. In demonstration, we classify the group-racks, and use homotopy to prove both existence and exclusion theorems for path-connected topological racks.

In this paper, we recall the geometric definition of the braid group by Emil Artin and we give a complete, elementary geometric/topological proof of the standard presentation of the braid group on n strings.

The aim of this paper is to analyze the chaotic dynamics of a novel 2D fractional order discrete Ushiki map using Caputo-like delta fractional difference operator. The dynamical nature of the proposed discrete fractional Ushiki map is examined with evolution of time states and bifurcation diagrams. In addition, control law aimed at stabilizing the proposed map and the synchronization of the discrete

Abstract In this paper, we are applied a new approach to prove that the solution of the linear Fredholm operator equation of the third kind given on a segment and having a finite number of multipoint singularities is equivalent to the solution of the linear Fredholm operator equation of the second kind with additional conditions. We are showed an example of solving the system of linear integral Fredholm

Abstract An initial-boundary value problem for a time-fractional subdiffusion equation with an arbitrary order elliptic differential operator is considered. Uniqueness and existence of the classical solution of the posed problem are proved by the classical Fourier method. Sufficient conditions for the initial function and for the right-hand side of the equation are indicated, under which the corresponding

Abstract In this paper, certain spectral properties related with the first order linear differential expression in the weighted Hilbert space at finite interval have been examined. Firstly, the minimal and maximal operators which are generated by the first order linear differential expression in the weighted Hilbert space have been determined. Then, the deficiency indices of the minimal operator have

Abstract A linear boundary value problem for essentially loaded differential equations is investigated. Using the properties of essentially loaded differential equation and assuming the invertibility of the matrix compiled through the coefficients at the values of the derivative of the desired function at load points, we reduce the considering problem to a two-point boundary value problem for loaded

Abstract Two-point boundary value problem for fourth order partial integro-differential equation is considered. By new unknown function the problem is reduced to an equivalent family of two-point boundary value problems for the Volterra integro-differential equations of second order. By the method of introduction functional parameters are constructed the algorithms for finding of solution to family

Abstract In this paper, certain spectral properties related with the first order linear differential-operator expression with involution in the Hilbert space of vector-functions at finite interval have been examined. Firstly, the minimal and maximal operators which are generated by the first order linear differential-operator expression with involution in the Hilbert spaces of vector-functions has

Abstract We consider a one-dimensional two-particle quantum system interacted by two identical point interactions situated symmetrically with respect to the origin at the points \(\pm x_{0}\). The corresponding Schrödinger operator (energy operator) is constructed as a self-adjoint extension of the symmetric Laplace operator. An essential spectrum is described and the condition for the existence of

Abstract The Dirichlet and the Keldysh problems for a three-dimensional elliptic equation with three singular coefficients are investigated in a semi-infinite parallelepiped. The uniqueness and the existence theorems are proved using the spectral analysis method. For the problem posed, using the Fourier method, one dimensional spectral problems are obtained. Based on the completeness property of systems

Abstract In this paper we consider of a multidimensional loaded mixed type equation of the second order with some seminonlocal conditions on the coefficients. The existence and uniqueness of solution of the seminonlocal boundary value problem of second kind is proved in the Sobolev space \(W\ _{2}^{3}(Q)\). In proof of the theorems are used the methods of ‘‘\(\varepsilon\)-regularizations’’, a priori

Abstract In this work for a class of second order model and non-model partial integro-differential equations with singularity in the kernel by the aid of arbitrary functions obtained integral representation for solution manifold. In this given article it is entered a new class of functions that at point (a, b) is converted to zero with some asymptotic behavior and the solution of given equation is

Abstract In this article in a rectangular domain inverse boundary value problems with local and non-local boundary conditions for a fourth order partial differential equation were formulated and investigated. The solutions of the inverse problems were obtained in the form of Fourier series. It was proved the theorem on uniqueness and existence of the solution of the inverse problems under consideration

Abstract A two-parameter family of algorithms for finding an approximate solution to a linear two-point boundary value problem for a system of ordinary differential equations is offered. The convergence conditions for the algorithms are obtained. The necessary and sufficient coefficient conditions for the well-posedness of considered problem are established.

Abstract We considered the mixed parabolic-hyperbolic type equation with discontinuous coefficient in a rectangular domain.We established a criterion for the unique solvability of this problem,using this criterion constructed solution as the sum of Fourier series. Finally, the stability of the solution with respect to initial function is proved.

Abstract The nonlinear two-point boundary value problem with parameter for systems of ordinary differential equations is investigated by the parametrization method. The method consists in partition of the interval apart, the introduction of additional parameters and reducing the original problem to the equivalent multi-point boundary value problem with parameters. The algorithms for finding a solution

Abstract An inverse problem for determining the order of time-fractional derivative in a nonhomogeneous subdiffusion equation with an arbitrary elliptic differential operator with constant coefficients in \(N\)-dimensional torus is considered. Using the classical Fourier method it is proved, that the value of the solution at a fixed time instant as the observation data recovers uniquely the order of

Abstract Problems of optimal control of linear and nonlinear stochastic systems with quadratic criterion qualities are studied. For such problems the existence of optimal control in feedback control form is proved by method of dynamic programming. The work consists of two parts. The first part deals with the linear problem. In the first part, the existence of a solution to the Cauchy problem for the

Abstract The article investigates the Robin boundary-value problem for a linear inhomogeneous ordinary differential equation of the second order on a segment with a small parameter at the highest derivative of the unknown function. The aim of the study is to construct a uniform asymptotic expansion of the solution of the Roben problem with an arbitrary degree of accuracy on the considered segment,

Abstract In this article, we consider the scattering problem for the Sturm–Liouville operator on the positive half-line with boundary condition depending quadratically of spectral parameter.The scattering data is defined. The resolvent operator is constructed and the expansion formula is obtained.

Abstract In this paper, we consider an inverse boundary value problem for a mixed type partial differential equation with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. The differential equation depends from another positive parameter in mixed derivatives. With respect to first variable this equation

Abstract In this article, a boundary value problem for a loaded partial differential equations of fourth order was considered on a rectangle domain. The solution of the problem is obtained in the form of Fourier series. Also, theorems about uniqueness and existence of the solution of the problem under consideration were proved by spectral method.

We prove a law of large numbers for the volumes of families of random hyperbolic mapping tori and Heegaard splittings providing a sharp answer to a conjecture of Dunfield and Thurston.

Weak sobriety is a topological property that gives rise to a categorical equivalence between topological spaces and distributive lattices without a least element. In this paper, we give some characterizations of weak sobriety and obtain some characterizations of local compactness, core-compactness and coherence of the special topological space by its upper space, where the upper space of a space is

In this paper, applying the new concept of aperiodic functions, we characterize disjoint hypercyclic and disjoint topologically mixing weighted translation operators on general Banach lattices. The obtained results in addition to previous modes of Lebesgue and Orlicz spaces, also cover new classes such as some closed subspaces of Morrey spaces.

In an earlier paper the second author made a study of the knotted periodic orbits in a strange attractor for a set of differential equations in a paper by Clark Robinson. The attractor is modeled by a Lorenz-like template. It was shown that the knots and links are positive but need not be positive braids. Here we show that they are fibered, have positive signature, and that each knot-type appears infinitely

As a generalization of the classical knots, virtual knots deal with the problem of circles embedded in a thickened orientable surface with arbitrary genus. In the study of virtual knots, the classification of virtual knots remains an important issue. In recent years, many polynomial invariants have appeared, such as affine index polynomial, a new invariant RD[t](h) of virtual knots, etc. In this paper

This work is aimed at establishing a unified fractional thermoelastic model for piezoelectric structures, and shedding light on the influence of different definitions on the transient responses. Theoretically, based upon Cattaneo-type equation, a unified form of fractional heat conduction law is proposed by adopting Caputo Fabrizio fractional derivative, Atangana Baleanu fractional derivative and Tempered

Coal failure behavior under dynamic loading is significant for dealing with the failure issues encountered in underground coal mines. In this study, the crack propagation and internal fracture process of coal under dynamic loading were investigated by a split Hopkinson pressure bar (SHPB) experimental system and numerical approach, respectively. The experimental and numerical results show that the

In this work, we analyze various stability (from stability to global uniform asymptotic stability) of Caputo fractional-order nonautonomous systems (CFONSs). With f(t,x) being only continuous, it is proven that the Caputo fractional derivative (CFD) of a general Lyapunov function V(t,x(t)) along solutions is continuous and moreover, t0CDtαV(t,x(t))≤∂V(t,x)∂t(t,x(t))t0CDtαt+∂V(t,x)∂x⋅f(t,x)(t,x(t))

We obtain the Assouad dimensions of two Moran classes with zero infimum contraction under suitable conditions.

We introduce the notion of weighted limit in an arbitrary quasi-category, suitably generalizing ordinary limits in a quasi-category, and classical weighted limits in an ordinary category. This is accomplished by generalizing Joyal’s approach: we identify a meaningful construction for the quasi-category of weighted cones over a diagram in a quasi-category, whose terminal object is the weighted limit

In this paper, a novel direct scheme to solve a set of time-delay fractional optimal control problems is introduced. The method firstly uses a set of Dickson polynomials as basis functions to approximate the states and control variables of the system. Next, the context of these basis functions and the use of a collocation method allow to transform the problem into a system of nonlinear algebraic equations

This study experimentally estimated the fractal pore size distribution of granites, which is then linked to the hydro-mechanical properties of rocks treated by temperatures that range from 25∘C to 900∘C. The mercury intrusion experiment was carried out to characterize the pore size distributions and the MTS815.02 triaxial testing system was used to investigate hydro-mechanical properties of rocks.

We identify a countable infinity of parameter regimes with strange nonchaotic attractors (SNAs). At the edge of each arc parameter area, there is an uncountable infinity of SNAs with torus intermittency. The mechanism for the creation of SNAs in different regime is induced by an n-frequency quasiperiodic orbit through a quasiperiodic analog of saddle-node bifurcation (Type-I intermittent route). We

We investigate a class of self-similar networks modeled on the three-dimensional Vicsek fractal, and obtain the asymptotic formula of node-weighted average geodesic distances, which is an integral of geodesic distance in terms of the self-similar measure.

In this work, we initially study the strange nonchaotic dynamics of a two-degree-of-freedom quasiperiodically forced vibro-impact system. It is shown that strange nonchaotic attractors (SNAs) occur between two chaotic regions, but not between the quasiperiodic region and the chaotic one. Subsequently, we mainly focus on the abundant multistability in the system, especially the coexistence of SNAs and

When an oscillator vibrates in a porous medium or along an unsmooth surface, a fractal model can be adopted using the two-scale fractal derivative. This paper elucidates a simple but effective method for fast and exclusive insight into the asymptotically periodic property at the initial stage and the extremely low frequency property when time tends to infinity. Some examples are illustrated to show

In this paper, we investigate how many real numbers can be well approximated by their convergents in the Engel expansions. Furthermore, the relative growth rate of convergence speed of convergents in the Engel expansion of an irrational number is studied to the rate of growth of its digits. The Hausdorff dimension of exceptional sets of points with a given relative growth rate is established.

During the development of deep-water oil and gas resources, severe engineering problems or even marine geological hazards might be induced by the complex geological characteristics of deep-water weakly consolidated (DWC) formation. To solve these problems, the power-law cementing fluid was used to solidify the DWC formation during the well-cementing operation. However, the transportation process of

We examine N-star compactifications of non-compact regular continuous frames and provide conditions under which these kinds of compactifications are perfect. Some properties and characterisations of such compactifications are established for the cases where N∈{1,2}. Connectedness of the remainder of a non-compact regular continuous frame in these compactifications is also investigated.

In this paper, we consider δ-ergodic pseudo-orbits. If any δ-pseudo-orbit can be shadowed by a point along a syndetic (aperiodic) set, then we try to connect this type of partial shadowing to the standard one. Also, a few relations on partial shadowing, not necessarily relevant, are obtained.

We present a 3-manifold automorphism f:M→M with rankFix(f⁎)=2rankπ1(M). Related facts are discussed.

In this paper, we derive a sharp condition on the equivalence of topological transitivity among an interval autonomous dynamical system, its induced set-valued system and induced normal fuzzified system. We also prove that their sensitivity (resp., total transitivity) are equivalent. For a general non-autonomous dynamical system, we show the equivalence of topological mixing (resp., mild mixing, cofinite

Extending upon our previous work, we verify the Jones Unknot Conjecture for all knots up to 24 crossings. We describe the method of our approach and analyze the growth of the computational complexity of its different components.

In this paper, we show that every spinal open book decomposition of a closed oriented 3-manifold M spinal open book embeds into a certain spinal open book decomposition of #kS2×̃S3, the connected sum of k copies of the twisted S3-bundle over S2, where k depends on the spinal open book decomposition of M. We also discuss spinal open book embeddings of a huge class of spinal open books of closed oriented

Assuming the existence of c incomparable selective ultrafilters, we classify the non-torsion Abelian groups of cardinality c that admit a countably compact group topology. We show that for each κ∈[c,2c] each of these groups has a countably compact group topology of weight κ without non-trivial convergent sequences and another that has convergent sequences. Assuming the existence of 2c selective ultrafilters

In 2002 Á. Császár introduced the notion of generalized topology, which differs from the notion of topology by the lack of the intersection property. Many kinds of generalized continuity may be considered as a continuity in a generalized topology, for example quasicontinuity, precontinuity, porouscontinuity, qualitative continuity, Denjoy property. In the paper we give full characterization of set

We investigate how various linear and nonlinear filters affect the scaling properties of long-range power-law multivariate synthetic series quantified by multivariate multifractal detrended fluctuation analysis (MV-MFDFA). We consider four types of transforms which are often encountered in physical and physiological processes: linear, nonlinear polynomial, logarithmic and power-law filters. The effect

Based on the theories of multivariate multiscale dispersion entropy, we propose an improved new model — dual-embedded dimensional multivariate multiscale dispersion entropy (mvMDE), and generalize the new model to fractional order (GmvMDE). The mvMDE and GmvMDE simultaneously consider the cross-correlation between multiple channels, and they provide a dynamic complexity measure for measuring the multivariable

In this paper, we study the solution theory of the nonlinear q-fractional differential equation of Caputo type cDqαy(t)=f(t,y(t)) with given initial values Dqky(0+)=dk,k=0,1,…,n−1 where α>0 is the order, n=[α] and 0

In this paper, a new notion of super coalescence hidden-variable fractal interpolation function (SCHFIF) is introduced. The construction of SCHFIF involves choosing an IFS from a pool of several non-diagonal IFS at each level of iteration. Further, the integral of a SCHFIF is studied and shown to be a SCHFIF passing through a different set of interpolation data.

We find all subsets of N which occur as the set of possible cardinalities of preimages of a continuous function. We also study and answer this question for various subclasses of continuous functions.

For a compact, orientable, irreducible, ∂-irreducible, and an-annular 3-manifold, it is shown there are only finitely many boundary slopes for incompressible and ∂-incompressible surfaces of a bounded Euler characteristic. We use normal surface theory and the inverse relationship of crushing a triangulation along a normal surface [8] and that of inflating an ideal triangulation [12] to introduce and

We prove that if X is a metric space of Assouad dimension s∈(0,∞), then for every α∈[0,s] there is a countable subset of X of Assouad dimension α. We reduce this to the special case of bounded X due to Wang and Wen in 2016 by using the inversion invariance of Assouad dimension.