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This paper presents a numerical method to solve the non-linear vibration problem of composite structures made of many unidirectional fibers/matrix layers. To take into account damping properties of structures, fractional derivative Zener model for complex moduli is used in each layer. The frequency dependence of these models leads to a non-linear vibration problem. To solve it, a High Order Newton (HON) solver based on homotopy and perturbation techniques is proposed. Thus, the initial non-linear problem turns into a set of linear systems, easier to solve. Numerical results obtained by the HON solver are compared with experimental and numerical results of literature. [ABSTRACT FROM AUTHOR]

A classical theorem of braid theory, dating back to Artin's work, says that two closed braids in a solid torus are ambient isotopic if and only if they represent the same conjugacy class of the braid group. This theorem can be reformulated in the framework of link theory without referring to the group structure. A link in a surface bundle over the circle is transversal whenever it covers the circle. In this terminology, Artin's theorem states that in a solid torus trivially fibered over the circle transversal links are ambient isotopic if and only if they are isotopic in the class of transversal links. We generalize this result by proving that (in the piecewise linear category) transversal links in an arbitrary compact orientable -manifold fibered over the circle with a compact fiber are ambient isotopic if and only if they are isotopic in the class of transversal links. [ABSTRACT FROM AUTHOR]

Partial differential equations are widely used to describe complex phenomena in various branches of science, including: physics, mechanics, etc. Therefore, obtaining high-precision target solutions and, if possible, finding exact analytical solutions of these equations play an important role in these sciences. In this paper, we first examine Lie symmetry and invariant transformations. Next, we obtain the equation solutions with the help of symmetries. We compute invariants and similarity solutions. Spectral method was used to numerically solve the equation. Then, we introduce two approaches, the Homotopy method and the direct method, which are used to construct conservation laws. Then we construct three conservation laws for the Degasperis-Procesi equation by using the direct method. Numerical solution and analysis of the Degasperis-Procesi equation by the methods proposed in this research have not been done in previous studies. Also, in Iran, this is the first time that this research has been done, which is a combination of mathematics, physics and computer science, and is considered interdisciplinary and applied. [ABSTRACT FROM AUTHOR]

A graphoid is a mixed multigraph with multiple directed and/or undirected edges, loops, and semiedges. A covering projection of graphoids is an onto mapping between two graphoids such that at each vertex, the mapping restricts to a local bijection on incoming edges and outgoing edges. Naturally, as it appears, this definition displays unusual behaviour since the projection of the corresponding underlying graphs is not necessarily a graph covering. Yet, it is still possible to grasp such coverings algebraically in terms of the action of the fundamental monoid and combinatorially in terms of voltage assignments on arcs. In the present paper, the existence theorem is formulated and proved in terms of the action of the fundamental monoid. A more conventional formulation in terms of the weak fundamental group is possible because the action of the fundamental monoid is permutational. The standard formulation in terms of the fundamental group holds for a restricted class of coverings, called homogeneous. Further, the existence of the universal covering and the problems related to decomposing regular coverings via regular coverings are studied in detail. It is shown that with mild adjustments in the formulation, all the analogous theorems that hold in the context of graphs are still valid in this wider setting. [ABSTRACT FROM AUTHOR]

The current paper proposes and tests algorithms for finding the diameter of a compact convex set and the farthest point in the set to another point. For these two nonconvex problems, I construct Frank–Wolfe and projected gradient ascent algorithms. Although these algorithms are guaranteed to go uphill, they can become trapped by local maxima. To avoid this defect, I investigate a homotopy method that gradually deforms a ball into the target set. Motivated by the Frank–Wolfe algorithm, I also find the support function of the intersection of a convex cone and a ball centered at the origin and elaborate a known bisection algorithm for calculating the support function of a convex sublevel set. The Frank–Wolfe and projected gradient algorithms are tested on five compact convex sets: (a) the box whose coordinates range between −1 and 1, (b) the intersection of the unit ball and the non-negative orthant, (c) the probability simplex, (d) the Manhattan-norm unit ball, and (e) a sublevel set of the elastic net penalty. Frank–Wolfe and projected gradient ascent are about equally fast on these test problems. Ignoring homotopy, the Frank–Wolfe algorithm is more reliable. However, homotopy allows projected gradient ascent to recover from its failures. [ABSTRACT FROM AUTHOR]

We study the structure of self-homeomorphism groups of fibered manifolds. A fibered topological space is a Birman–Hilden space whenever in each isotopic pair of its fiber-preserving (taking each fiber to a fiber) self-homeomorphisms the homeomorphisms are also fiber-isotopic (isotopic through fiber-preserving homeomorphisms). We prove in particular that the Birman–Hilden class contains all compact connected locally trivial surface bundles over the circle, including nonorientable ones and those with nonempty boundary, as well as all closed orientable Haken 3-manifold bundles over the circle, including nonorientable ones. [ABSTRACT FROM AUTHOR]

Pseudo-arclength continuation is a scheme used to generate a set of solutions for nonlinear equations and has been applied in numerous fields. In continuation schemes, solutions are obtained by tracking a parameter associated with the system as it varies, thereby creating a solution path. Pseudo-arclength continuation employs the nullspace vector, which is tangent to the path, as an approximation of the arclength in order to determine the direction of variation at each point on the path. However, the nullspace vector has both positive and negative solutions that are equally valid, leading to uncertainty in the direction of travel along the path at each location. Selecting the wrong nullspace vector can result in returning to a previously located solution instead of continuing the mapping of the solution path. Techniques to ensure consistent path direction have been developed; however, their efficacy does not extend to all scenarios. This paper addresses this gap by introducing a "flipping condition" to track previously determined path directions and ensure continuous following of the same path direction. A clear description of the process used to incorporate the flipping condition into the pseudo-arclength continuation scheme, practical implementation of the scheme, and examples demonstrating its application to the nonlinear circular restricted three-body problem (CR3BP), is provided. The inclusion of the flipping condition ensures a consistent direction of travel along the solution path and allows for families of solutions to be developed. • Pseudo-arclength continuation develops solutions to nonlinear equations along a path. • Uncertainty in path direction comes from equally valid signs of the nullspace. • Previously-located solutions are repeatedly found without a consistent path direction. • Implementation process for pseudo-arclength continuation to track path direction. • Application to periodic orbits families in the circular restricted three-body problem. [ABSTRACT FROM AUTHOR]

The degree of a based graph is the number of essential non-basepoint vertices after generic perturbation. Hatcher-Vogtmann's degree theorem states that the subcomplex of Auter Space of graphs of degree at most d is (d - 1)-connected. We extend the definition of degree to the simplicial closure of Auter Space and prove a version of Hatcher-Vogtmann's result in this context. [ABSTRACT FROM AUTHOR]

A topological fibered space is a Birman–Hilden space whenever in each isotopic pair of its fiber-preserving (taking each fiber to a fiber) self-homeomorphisms the homeomorphisms are also fiber-isotopic (isotopic through fiber-preserving homeomorphisms). We present a series of sufficient conditions for a fiber bundle over the circle to be a Birman–Hilden space. [ABSTRACT FROM AUTHOR]

Let $X$ , $Y$ be nonsingular real algebraic sets. A map $\varphi \colon X \to Y$ is said to be $k$ -regulous, where $k$ is a nonnegative integer, if it is of class $\mathcal {C}^k$ and the restriction of $\varphi$ to some Zariski open dense subset of $X$ is a regular map. Assuming that $Y$ is uniformly rational, and $k \geq 1$ , we prove that a $\mathcal {C}^{\infty }$ map $f \colon X \to Y$ can be approximated by $k$ -regulous maps in the $\mathcal {C}^k$ topology if and only if $f$ is homotopic to a $k$ -regulous map. The class of uniformly rational real algebraic varieties includes spheres, Grassmannians and rational nonsingular surfaces, and is stable under blowing up nonsingular centers. Furthermore, taking $Y=\mathbb {S}^p$ (the unit $p$ -dimensional sphere), we obtain several new results on approximation of $\mathcal {C}^{\infty }$ maps from $X$ into $\mathbb {S}^p$ by $k$ -regulous maps in the $\mathcal {C}^k$ topology, for $k \geq 0$. [ABSTRACT FROM AUTHOR]

The relation between the holonomy along a loop with the curvature form is a well-known fact, where the small square loop approximation of aholonomy H γ , O is proportional to R σ . In an attempt to generalize the relation for arbitrary loops, we encounter the following ambiguity. For a given loop γ embedded in a manifold M , H γ , O is an element of a Lie group G ; the curvature R σ ∈ g is an element of the Lie algebra of G. However, it turns out that the curvature form R σ obtained from the small loop approximation is ambiguous, as the information of γ and H γ , O are insufficient for determining a specific plane σ responsible for R σ . To resolve this ambiguity, it is necessary to specify the surface S enclosed by the loop γ ; hence, σ is defined as the limit of S when γ shrinks to a point. In this article, we try to understand this problem more clearly. As a result, we obtain an exact relation between the holonomy along a loop with the integral of the curvature form over a surface that it encloses. The derivation of this result can be viewed as an alternative proof of the non-Abelian Stokes theorem in two dimensions with some generalizations. [ABSTRACT FROM AUTHOR]

The aim of the present research article is to investigate the existence and uniqueness of a solution to the integral equation and homotopy result. To achieve our objective, we introduce the notion of ( α , η , ψ )-contraction in the framework of F -bipolar metric space and prove some fixed point results for covariant and contravariant mappings. Some coupled fixed point results in F -bipolar metric space are derived as outcomes of our principal theorems. A non-trivial example is also provided to validate the authenticity of the established results. [ABSTRACT FROM AUTHOR]

This paper proposes a homotopy-based approach to solve the power flow problem (PFP) in islanded microgrid networks with droop-controlled distributed generation (DG) units. The technique is based on modifying an "easy" problem solution that evolves with the computation of intermediate results to the PFP solution of interest. These intermediate results require the solution of nonlinear equations through Newton–Raphson (NR) method. In favor of convergence, the intermediate solutions are close to each other, strengthening the convergence qualities of the technique for the solution of interest. The DG units are modeled with operational power limits and three types of droop-control strategies, while the loads are both magnitude voltage- and frequency-dependent. To evaluate the method performance, simulations are performed considering the proposed and classical NR methods, both departing from a flat start estimation. Tests are carried out in three test systems. Different load and DG unit scenarios are implemented for a 6-, 38-, and 69-bus test system. A base case is studied for all systems, while for the two larger models, a loading factor is used to simulate the load augmenting up to the maximum value. The results demonstrated that for the largest-size model system, only the homotopy-based approach could solve the PFP for stringent requirements such as the diversification of the load profile and hard loading operation point. [ABSTRACT FROM AUTHOR]

We consider the space ΛM:= H¹(S¹,M) of loops of Sobolev class H1 of a compact smooth manifold M, the so-called free loop space of M. We take quotients ΛM/G where G is a finite subgroup of O(2) acting by linear reparametrization of S¹. We use the existence of transfer maps tr: H∗(ΛM/G) → H∗(ΛM) to define a homology product on ΛM/G via the Chas-Sullivan loop product. We call this product PG the transfer product. The involution ϑ: ΛM → ΛM which reverses orientation, ϑ􀀀 γ(t):= γ(1 - t), is of particular interest to us. We compute H∗(ΛSn/ϑ;Q), n > 2, and the product Pϑ: Hi(ΛSn/ϑ;Q) × Hj(ΛSn/ϑ;Q) → Hi+j-n(ΛSn/ϑ;Q) associated to orientation reversal. Rationally Pϑ can be realized "geometrically" using the concatenation of equivalence classes of loops. There is a qualitative difference between the homology of ΛSn/ϑ and the homology of ΛSn/G when G ⊂ S¹ ⊂ O(2) does not "contain" the orientation reversal. This might be interesting with respect to possible differences in the number of closed geodesics between non-reversible and reversible Finsler metrics on Sn, the latter might always be infinite. [ABSTRACT FROM AUTHOR]

Suppose that p and q are projections in a unital C ∗ -algebra A such that ‖ p (1 - q) ‖ < 1 . It is shown that there exists a unitary u in A which is homotopic to the unit of A , and satisfies p u p = p u ∗ p , u (p q p) u ∗ = q p q and ‖ 1 - u ‖ ≤ 2 ‖ (q p) † ‖ 1 + ‖ (q p) † ‖ · ‖ p (1 - q) ‖ , where (q p) † denotes the Moore–Penrose inverse of qp. Under the same restriction of ‖ p (1 - q) ‖ < 1 , it is proved that ‖ p - q ‖ < 1 if and only if there exists a unitary u in A such that pup is normal and q = u p u ∗ . An example is constructed to show that there exist certain Hilbert space H and projections p and q on H such that ‖ p - q ‖ = 1 and q = u p u ∗ for some unitary operator u on H. [ABSTRACT FROM AUTHOR]

Given a smooth map f_V\colon V \to K with f^*_V (\nu _K) = \nu _V, a general question arises: under which conditions there exists a smooth extension f\colon M \to N of f_V such that f is transverse to K and f^{-1}(K) = V, where M, N are smooth closed manifolds of dimension m and n, V, K are closed submanifolds of M and N, respectively, of same codimension and \nu _K, \nu _V are the normal bundles of K in N and V in M, respectively. In this paper, we give conditions to the existence of extensions, by using bordism intersection product. Moreover, we present an interesting and non-trivial example illustrating the systematic construction of such extensions, skeletonwise. [ABSTRACT FROM AUTHOR]

This paper is an introduction to combinatorial knot theory via state summation models for the Jones polynomial and its generalizations. It is also a story about the developments that ensued in relation to the discovery of the Jones polynomial and a remembrance of Vaughan Jones and his mathematics. [ABSTRACT FROM AUTHOR]

The aim of this research article is to obtain fixed point results in the context of F -bipolar metric spaces. The obtained results extend some fixed point theorems in the existing literature. We also provide a non-trivial example to validate our claims. The existence and uniqueness of the solution of the integral equation are proved as applications of our leading results. Furthermore, the existence of the unique solution in homotopy theory is also investigated. [ABSTRACT FROM AUTHOR]

We study the extent of countable saturation for coronas of abelian \mathrm {C}^{*}-algebras. In particular, we show that the corona algebra of C_0(\mathbb {R}^n) is countably saturated if and only if n=1. [ABSTRACT FROM AUTHOR]

Necessary conditions for asymptotic stability and stabilizability of subsets for dynamical and control systems are obtained. The main necessary condition is homotopical and is in turn used to obtain a homological one. A certain extension is ruled out. Questions are posed. [ABSTRACT FROM AUTHOR]

Topological spaces can be induced by various algebraic ordering relations such as, linear, partial and the inclusion-ordering of open sets forming chains and chain complexes. In general, the classifications of covering spaces are made by using fundamental groups and lifting. However, the Riesz ordered n-spaces and Urysohn interpretations of real-valued continuous functions as ordered chains provide new perspectives. This paper proposes the formulation of covering spaces of n-space charts of a foliated n-manifold containing linearly ordered chains, where the chains do not form topologically separated components within a covering section. The chained subspaces within covering spaces are subjected to algebraic split–join operations under a bijective function within chain-subspaces to form simply directed chains and twisted chains. The resulting sets of chains form simply directed chain-paths and oriented chain-paths under the homotopy path-products involving the bijective function. It is shown that the resulting embedding of any chain in a leaf of foliated n-manifold is homogeneous and unique. The finite measures of topological subspaces containing homotopies of chain-paths in covering spaces generate multiplicative and cyclic group varieties of different orders depending upon the types of measures. As a distinction, the proposed homotopies of chain-paths in covering spaces and the homogeneous chain embedding in a foliated n-manifold do not consider the formation of circular nerves and the Nachbin topological preordering, thereby avoiding symmetry/asymmetry conditions. [ABSTRACT FROM AUTHOR]

Current hygrothermal behaviour prediction models neglect the hysteresis phenomenon. This leads to a discrepancy between numerical and experimental results, and a miscalculation of buildings' durability. In this paper, a new mathematical model of hysteresis is proposed and implemented in a hygrothermal model to reduce this discrepancy. The model is based on a symmetry property between sorption curves and uses also a homotopic transformation relative to a parameter s ∈ [ 0 , 1 ]. The advantage of this model lies in its ease of use and implementation since it could be applied with the knowledge of only one main sorption curve by considering s = 0 , in other words, we only use the axisymmetric property here. In the case where the other main sorption curve is known, we use this curve to incorporate the homotopy property in order to calibrate the parameter s.The full version of the proposed model is called Axisymmetric + Homotopic. Furthermore, it was compared not only with the experimental sorption curves of different types of materials but also with a model that is well known in the literature (CARMELIET's model). This comparison shows that the Axisymmetric + Homotopic model reliably predicts hysteresis loops of various types of materials even with the knowledge of only one of the main sorption curves. However, the full version of Axisymmetric + Homotopic model is more reliable and covers a large range of materials. The proposed model was incorporated into the mass transfer model. The simulation results strongly match the experimental ones. [ABSTRACT FROM AUTHOR]

Taylor-Couette flow between two concentric cylinders has received much attention due to its use in various applications, including biomedical devices, micro electro-mechanical systems, polymer pumping and electric motor cooling. Due to the complex interaction of the viscosity and the involved geometry within the confined space, different flow regimes are dominant under different conditions, affecting the fluid dynamics and heat transfer. In analyzing the mentioned flow, besides the experimental and computational studies, analytical models have been developed with varying levels of complication. In the present study, using the homotopy of both the flow and the domain geometry between the concentric and eccentric cylinders, a practical formula for flow between eccentric cylinders is developed. In doing so, an appropriate transformation function for the geometry is developed and embedded into the velocity equation for the concentric cylinders. The resultant equation is tested against flow simulation results. A validity margin analysis is performed based on the variation of the mass flow rate between the cylinders. It is seen that the proposed model for eccentric cylinders is applicable for all gap distances, unlike the previous models that are restricted to narrow gaps. Finally, a separate formula to quantify the error in the estimates of the present method is also derived, which involves the ratio of the cylinders and the eccentricity. [ABSTRACT FROM AUTHOR]

This study deals with the entropy investigation of the peristalsis of a water–copper nanofluid through an asymmetric inclined channel. The asymmetric channel is anticipated to be filled with a uniform permeable medium, with a constant magnetic field impinging on the wall of the channel. The physical effects, such as Hall current, mixed convection, Ohmic heating, and heat generation/annihilation, are also considered. Mathematical modeling from the given physical description is formulated while employing the "long wavelength, low Reynolds number" approximations. Analytical and numerical procedures allow for the determination of flow behavior in the resulting system, the results of which are presented in the form of tables and graphs, in order to facilitate the physical analysis. The results indicate that the growth of nanoparticle volume fraction corresponds to a reduction in temperature, entropy generation, velocity, and pressure gradient. The enhanced Hall and Brinkman parameters reduce the entropy generation and temperature for such flows, whereas the enhanced permeability parameter reduces the velocity and pressure gradient considerably. Furthermore, a comparison of the heat transfer rates for the two results, for different physical parameters, indicates that these results agree well. The significance of the underlying study lies in the fact that it analyzes the peristalsis of a non-Newtonian nanofluid, where the rheological characteristics of the fluid are predicted using the Carreau-Yasuda model and by considering the various physical effects. [ABSTRACT FROM AUTHOR]

We present a short proof of the adek–Král–Matouek–Vokřínek–Wagner result from the title (in the following form due to Filakovský–Wagner–Zhechev). For any fixed evenlthere is no algorithm recognizing the extendability of the identity map of S l to a PL map X → S l of given 2l-dimensional simplicial complexXcontaining a subdivision of S l as a given subcomplex. We also exhibit a gap in the Filakovský–Wagner–Zhechev proof that embeddability of complexes is undecidable in codimension > 1 . [ABSTRACT FROM AUTHOR]

This paper generalizes a result of Lynn on the “degree” of an equivariant cohomology ring H* G(X). The degree of a graded module is a certain coefficient of its Poincar´e series, and is closely related to multiplicity. In the present paper, we study these commutative algebraic invariants for equivariant cohomology rings. The main theorem is an additivity formula for degree: deg(H* G(X)) = Σ [A,c]∈Q′max(G,X) 1/ |WG(A, c)| deg(H* CG(A,c) (c)). We also show how this formula relates to the additivity formula from commutative algebra, demonstrating both the algebraic and geometric character of the degree invariant [ABSTRACT FROM AUTHOR]

This paper introduces proximal path cycles, which lead to the main results in this paper, namely, extensions of the Mitsuishi-Yamaguchi Good Coverning Theorem with different forms of Tanaka good cover of an Alexandrov space equipped with a proximity relation as well as extension of the Jordan curve theorem. In this work, a path cycle is a sequence of maps h1,...,hi,...,hn-1 mod n in which hi : [ 0,1 ] → X and hi(1) = hi+1(0) provide the structure of a path-connected cycle that has no end path. An application of these results is also given for the persistence of proximal video frame shapes that appear in path cycles. [ABSTRACT FROM AUTHOR]

In this article, we study the existence of a weak solutions for the nonlinear fractional elliptic systems with Dirichlet boundary conditions in three cases. We use the Leray–Schauder degree and some sufficient conditions for the solvability of a resonance and non‐resonance systems with respect to the spectrum of the fractional Laplacian. [ABSTRACT FROM AUTHOR]

We show that weak monoidal Quillen equivalences induce equivalences of symmetric monoidal \infty-categories with respect to the Dwyer-Kan localization of the symmetric monoidal model categories. The result will induce a Dold-Kan correspondence of coalgebras in \infty-categories. Moreover it shows that Shipley's zig-zag of Quillen equivalences provides an explicit symmetric monoidal equivalence of \infty-categories for the stable Dold-Kan correspondence. We study homotopy coherent coalgebras associated to a monoidal model category and we show examples when these coalgebras cannot be rigidified. That is, their \infty-categories are not equivalent to the Dwyer-Kan localizations of strict coalgebras in the usual monoidal model categories of spectra and of connective discrete R-modules. [ABSTRACT FROM AUTHOR]

This study employed the global optimization method called Modified Trusted Region Method (MTRM) to resolve the portfolio selection risk problem. An objective of unconstrained optimization problem was formulated with four sets of fund data. The relationship between the level of acceptable risk and the weighting factor was analyzed numerically. The return of portfolio increased along with the level of acceptable risk since a high return was always accompanied by higher risk. By contrast, the risk of portfolio decreased as the weighting factor increased. The MTRM could resolve the portfolio optimization problem. [ABSTRACT FROM AUTHOR]

The linear convex log-homotopy has been used in the derivation of particle flow filters. One natural question is whether it is beneficial to consider other forms of homotopy. We revisit this question by considering a general linear form of log-homotopy for which we derive particle flow filters, validate the distribution of flows, and obtain conditions for the stability of particle flows. We then formulate the problem of stiffness mitigation as an optimal control problem by minimizing the condition number of the Hessian matrix of the posterior density function. The optimal homotopy can be efficiently obtained by solving a 1-D second-order two-point boundary value problem. Compared with traditional matrix analysis based approaches to improving condition numbers, this approach explicitly exploits the special structure of the stochastic differential equations in particle flow filters. The effectiveness of the proposed approach is demonstrated by numerical examples. [ABSTRACT FROM AUTHOR]

This study aims to investigate L B -valued GFA from algebraic and topological perspectives, where L stands for residuated lattice and B is a set of propositions about the general fuzzy automata, in which its underlying structure is a complete infinitely distributive lattice. Further, the concepts of L B -valued general fuzzy automata (or simply L B -valued GFA) contractible spaces, L B -valued GFA path homotopy, L B -valued GFA retraction, L B -valued GFA deformation retraction, L B -valued GFA path connected space and L B -valued GFA homotopy equivalent space are introduced and explicated. In addition, L B -valued GFA fundamental groups are proposed and studied. Regarding these issues, some properties are also established and explained. [ABSTRACT FROM AUTHOR]

A simplicial set is non-singular if the representing map of each non-degenerate simplex is degreewise injective. The simplicial mapping set XK has n-simplices given by the simplicial For a fixed N, we analyze the space of all sequences z = (z¹, . . ., zN), approximating a continuous function on the circle, with a given persistence diagram P, and show that the typical components of this space are homotopy equivalent to S¹. We also consider the space of functions on Y -shaped (resp., starshaped) trees with a 2-point persistence diagram, and show that this space is homotopy equivalent to S¹ (resp., to a bouquet of circles). [ABSTRACT FROM AUTHOR]

The main purpose of this study is to compare a semi-analytical method and numerical method namely the homotopy perturbation method (HPM) and finite difference method (FDM) respectively. These methods were employed for solving the nonlinear problem of the magnetohydrodynamic (MHD) couette flow of third-grade fluid between the two parallel plates. The comparison was made between a solution of HPM and FDM against a solution obtained from regular perturbation and the results are tabulated. From a computational viewpoint, it is revealed that the HPM is more reliable and efficient than FDM. Also, the results show that the FDM requires slightly more computational effort than the HPM, although the HPM yields more accurate results than the FDM. [ABSTRACT FROM AUTHOR]

This paper deals with surfaces covering a set of spheres, whose centers form polyhedra. We propose novel methods of skinning based on homotopic deformation for the considered case. A method starts with a regular surface with a simple construction which can be deformed in a many ways. We demonstrate some of them in a few examples. The method is compared to the existing solutions by the new approach implementation and the visualization of the obtained results. [Display omitted] • Locally scalable surface skinning a configuration of spheres with centers forming a polyhedron or its part. • Homotopy-based construction of a skinning surface for the spheres created from a technically simple initial surface. • Visual examples and comparison of the proposed method with the existing skinning approaches. [ABSTRACT FROM AUTHOR]

The Cauchy problem for a nonlinear system of differential equations with a Stieltjes integral (loads) of the desired solution is considered. The equation contains bifurcation parameters where the system has a trivial solution for any values. The necessary and sufficient conditions are derived for those parameter values (bifurcation points) in the neighborhood of which the Cauchy problem has a non-trivial real solution. The constructive method is proposed for the solution of real solutions in the neighborhood of those points. The method uses successive approximations and builds asymptotics of the solution. The theoretical results are illustrated by example. The Cauchy problem with loads and bifurcation parameters has not been studied before. [ABSTRACT FROM AUTHOR]

Previous work (Pradines, C.R. Acad. Sci. Paris 263 (1966) 907, Aof and Brown, Topology Appl. 47 (1992) 97) has given a setting for a holonomy Lie groupoid of a locally Lie groupoid. Here we develop analogous 2-dimensional notions starting from a locally Lie crossed module of groupoids. This involves replacing the Ehresmann notion of a local smooth coadmissible section of a groupoid by a local smooth coadmissible homotopy (or free derivation) for the crossed module case. The development also has to use corresponding notions for certain types of double groupoids. This leads to a holonomy Lie groupoid rather than double groupoid, but one which involves the 2 2 -dimensional information. [ABSTRACT FROM AUTHOR]

In the present paper, we aim to extend and unify the results obtained for the multivalued F -contractions, which have been frequently studied recently, in a different way from the results in the literature without using the Pompeiu-Hausdorff metric. Hence, we first introduce a new class of multivalued mappings that includes multivalued F -contractions. Then, we obtain some best proximity point results for new kind of F -contraction mappings. Thus, we unify and improve many results in the literature. To see this fact, we give some nontrivial and interesting examples. Also, considering the strong relationship between homotopy theory and various branches of mathematics, we obtain an application to homotopy theory of our main result. [ABSTRACT FROM AUTHOR]

The goal of the present study is to characterize solutions under a travelling wave formulation to a degenerate Fisher-KPP problem. With the degenerate problem, we refer to the following: a heterogeneous diffusion that is formulated with a high order operator; a non-linear advection and non-Lipstchitz spatially heterogeneous reaction. The paper examines the existence of solutions, uniqueness and travelling wave oscillatory properties (also called instabilities). Such oscillatory behaviour may lead to negative solutions in the proximity of zero. A numerical exploration is provided with the following main finding to declare: the solutions keeps oscillating in the proximity of the null stationary solution due to the high order operator, except if the reaction term is quasi-Lipschitz, in which it is possible to define a region where solutions are positive locally in time. [ABSTRACT FROM AUTHOR]

Let X be a real algebraic variety (resp. nonsingular real algebraic variety) and let Y be a homogeneous space for some linear real algebraic group. We prove that a continuous (resp. C ∞) map f : X → Y can be approximated by regular maps in the C 0 (resp. C ∞) topology if and only if it is homotopic to a regular map. Taking Y = S p , the unit p -dimensional sphere, we obtain solutions of several problems that have been open since the 1980's and which concern approximation of maps with values in the unit spheres. This has several consequences for approximation of maps between unit spheres. For example, we prove that for every positive integer n every C ∞ map from S n into S n can be approximated by regular maps in the C ∞ topology. Up to now such a result has only been known for five special values of n , namely, n = 1 , 2 , 3 , 4 or 7. [ABSTRACT FROM AUTHOR]

A homotopy is a continuous one-parameter family of continuous functions. This enquiry sought to find out how the various forms ranging from paths, inverses, reflexivity, symmetry and transitivity and other instances could be given in descriptive survey. [ABSTRACT FROM AUTHOR]

We prove a Künneth theorem for the Vietoris–Rips homology and cohomology of a semi-uniform space. We then interpret this result for graphs, where we show that the Künneth theorem holds for graphs with respect to the strong graph product. We finish by computing the Vietoris–Rips cohomology of the torus endowed with diferent semi-uniform structures. [ABSTRACT FROM AUTHOR]

This paper examines winding number of a closed planar curve through various aspects. It is related to a range on an arc in the complex plane and a point not in the range. Functions of such natures are considered to be continuous real-valued functions. Conclusion was drawn by naming a number of areas of applications. [ABSTRACT FROM AUTHOR]

Human beings exist in a biological and social system from a micro to a macro level, by means of "collectivity", a dynamic collaboration that they have established together with the elements in that system in a way to complement each other and realize a common goal. Many neuroscientific concepts used today to explain neuronal processes from which mental functions originate are far from searching answers to traditional philosophical questions. However, the brain — as the generator of highly abstract concepts — is so complex that it cannot be explained by minimalistic approaches. The concept of connectome used in recent years to describe neuronal connections from which brain functions originate exemplifies this minimalistic approach, because it only describes structural and functional connections but does not look at brain functions in a holistic view. For this reason, we propose the concept of collectome — to replace the concept of connectome — that describes a homeomorphic and homotopic neuronal framework that has a bicontinuous style of work from micro to macroscale which is based on fractal rules. [ABSTRACT FROM AUTHOR]

Homotopic functional connectivity reflects the degree of synchrony in spontaneous activity between homologous voxels in the two hemispheres. Previous studies have associated increased brain homotopy and decreased white matter integrity with performance decrements on different cognitive tasks across the life-span. Here, we correlated functional homotopy, both at the whole-brain level and specifically in fronto-parietal network nodes, with task-switching performance in young adults. Cue-to-target intervals (CTI: 300 vs. 1200 ms) were manipulated on a trial-by-trial basis to modulate cognitive demands and strategic control. We found that mixing costs, a measure of task-set maintenance and monitoring, were significantly correlated to homotopy in different nodes of the fronto-parietal network depending on CTI. In particular, mixing costs for short CTI trials were smaller with lower homotopy in the superior frontal gyrus, whereas mixing costs for long CTI trials were smaller with lower homotopy in the supramarginal gyrus. These results were specific to the fronto-parietal network, as similar voxel-wise analyses within a control language network did not yield significant correlations with behavior. These findings extend previous literature on the relationship between homotopy and cognitive performance to task-switching, and show a dissociable role of homotopy in different fronto-parietal nodes depending on task demands. [ABSTRACT FROM AUTHOR]

Travelling waves (TWs) instabilities to a degenerate diffusion problem with heterogeneous Fisher-KPP problem have not been previously analysed. The intention along this paper is to study existence, uniqueness and TW instability for a high order diffusion heterogeneous reaction Fisher-KPP problem with advection. The TW profiles are obtained analytically in the proximity of the stationary points, making use of the geometric perturbation theory. In addition, we examine a characterization of a local in time positive inner region where the TW behaves monotonically in contrast with an outer region of instabilities. Furthermore, a numerical exercise determines an accurate estimation of a local time to ensure the existence of the positive inner region, given a certain TW propagation speed. [ABSTRACT FROM AUTHOR]

Algebraic topology has grown over the last decades. This paper presents the essentials of topological spaces, continuous functions and homotopy. The human heart is assumed to be topologically equivalent to the one-sphere. When the human heart undergoes a stimulus, the time in the beat cycle is mapped to the time it recovers from the stimulus. This work concluded with the computation of the stereographic projection and its applications. [ABSTRACT FROM AUTHOR]

A model category has two weak factorizations, a pair of cofibrations and trivial fibrations and a pair of trivial cofibrations and fibrations. Then the class of weak equivalences is the set W consisting of the morphisms that can be decomposed into trivial cofibrations followed by trivial fibrations. One can build a model category out of such two weak factorizations by defining the class of weak equivalences by W as long as it satisfies the two out of three property. In this note we show that given a category with two weak factorizations, if every object is fibrant and cofibrant, W satisfies the two out of three property if and only if W is closed under the homotopies. [ABSTRACT FROM AUTHOR]

This paper is a study of Bousfield-Segal spaces, a notion introduced by Julie Bergner drawing on ideas about Eilenberg- Mac Lane objects due to Bousfield. In analogy to Rezk's Segal spaces, they are defined in such a way that Bousfield-Segal spaces naturally come equipped with a homotopy-coherent fraction operation in place of a composition. In this paper we show that Bergner's model structure for Bousfield-Segal spaces in fact can be obtained from the model structure for Segal spaces both as a localization and a colocalization. We thereby prove that Bousfield-Segal spaces really are Segal spaces, and that they characterize exactly those with invertible arrows. We note that the complete Bousfield-Segal spaces are precisely the homotopically constant Segal spaces, and deduce that the associated model structure yields a model for both ∞-groupoids and Homotopy Type Theory. [ABSTRACT FROM AUTHOR]