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Quasi-symmetric designs are the special type of 2 -designs with two block intersection numbers x and y. In this paper, we study the automorphism groups of quasi-symmetric designs with block intersection numbers 0 and 2. We prove that for a quasi-symmetric design with x = 0 and y = 2 satisfying that λ does not divide r , if G ≤ A u t () is flag-transitive, then G must be point-primitive. We further obtain that G is either of affine type or almost simple type by the O'Nan Scott Theorem. Moreover, when the socle of G is sporadic, is a unique 2 - (2 2 , 6 , 5) design and G = M 2 2 or M 2 2 : 2. [ABSTRACT FROM AUTHOR]

The class of quasisymmetric mappings on the real axis was first introduced by Beurling and Ahlfors in 1956. In 1980 Tukia and Väisälä considered these mappings between general metric spaces. In our paper we generalize the concept of a quasisymmetric mapping to the case of general semimetric spaces and study some properties of these mappings. In particular, conditions under which quasisymmetric mappings preserve triangle functions, Ptolemy's inequality and the relation "to lie between" are found. Considering quasisymmetric mappings between semimetric spaces with different triangle functions we give a new estimate for the ratio of diameters of two subsets, which are images of two bounded subsets. This result generalizes the well-known Tukia-Väisälä inequality. Moreover, we study connections between quasisymmetric mappings and weak similarities which form a special class of mappings between semimetric spaces. [ABSTRACT FROM AUTHOR]

The existence of symmetries in complex networks has a significant effect on network dynamic behaviour. Nevertheless, beyond topological symmetry, one should consider the fact that real-world networks are exposed to fluctuations or errors, as well as mistaken insertions or removals. Therefore, the resulting approximate symmetries remain hidden to standard symmetry analysis—fully accomplished by discrete algebra software. There have been a number of attempts to deal with approximate symmetries. In the present work we provide an alternative notion of these weaker symmetries, which we call 'quasi-symmetries'. Differently from other definitions, quasi-symmetries remain free to impose any invariance of a particular network property and they are obtained from the phase differences at the steady-state configuration of an oscillatory dynamical model: the Kuramoto–Sakaguchi model. The analysis of quasi-symmetries unveils otherwise hidden real-world networks attributes. On the one hand, we provide a benchmark to determine whether a network has a more complex pattern than that of a random network with regard to quasi-symmetries, namely, if it is structured into separate quasi-symmetric groups of nodes. On the other hand, we define the 'dual-network', a weighted network (and its corresponding binnarized counterpart) that effectively encodes all the information of quasi-symmetries in the original network. The latter is a powerful instrument for obtaining worthwhile insights about node centrality (obtaining the nodes that are unique from that act as imitators with respect to the others) and community detection (quasi-symmetric groups of nodes). [ABSTRACT FROM AUTHOR]

Quasisymmetric stellarators are appealing intellectually and as fusion reactor candidates since the guiding-centre particle trajectories and neoclassical transport are isomorphic to those in a tokamak, implying good confinement. Previously, quasisymmetric magnetic fields have been identified by applying black-box optimization algorithms to minimize symmetry-breaking Fourier modes of the field strength $B$. Here, instead, we directly construct magnetic fields in cylindrical coordinates that are quasisymmetric to leading order in the distance from the magnetic axis, without using optimization. The method involves solution of a one-dimensional nonlinear ordinary differential equation, originally derived by Garren & Boozer (Phys. Fluids B, vol. 3, 1991, p. 2805). We demonstrate the usefulness and accuracy of this optimization-free approach by providing the results of this construction as input to the codes VMEC and BOOZ_XFORM, confirming the purity and scaling of the magnetic spectrum. The space of magnetic fields that are quasisymmetric to this order is parameterized by the magnetic axis shape along with three other real numbers, one of which reflects the on-axis toroidal current density, and another one of which is zero for stellarator symmetry. The method here could be used to generate good initial conditions for conventional optimization, and its speed enables exhaustive searches of parameter space. [ABSTRACT FROM AUTHOR]

Abstract Given a subgroup G of the symmetric group S n , the cycle index polynomial cyc G is the average of the power-sum symmetric polynomials indexed by the cycle types of permutations in G. By Pólya's Theorem, the monomial expansion of cyc G is the generating function for weighted colorings of n objects, where we identify colorings related by one of the symmetries in G. This paper develops combinatorial formulas for the fundamental quasisymmetric expansions and Schur expansions of certain cycle index polynomials. We give explicit bijective proofs based on standardization algorithms applied to equivalence classes of colorings. Subgroups studied here include Young subgroups of S n , the alternating groups A n , direct products, conjugate subgroups, and certain cyclic subgroups of S n generated by (1 , 2 , ... , k). The analysis of these cyclic subgroups when k is prime reveals an unexpected connection to perfect matchings on a hypercube with certain vertices identified. [ABSTRACT FROM AUTHOR]

We introduce a complex Banach manifold structure on the space of normalized symmetric homeomorphisms on the real line. [ABSTRACT FROM AUTHOR]

Let S p be the standard Sierpiński carpet and QS (S p) the group of quasisymmetric maps of S p onto itself, where p ≥ 3 is odd. Mario Bonk and Sergei Merenkov proved that QS (S p) is finite dihedral and that QS (S 3) is the isometry group of S 3 . They conjectured that QS (S p) is the isometry group of S p for any odd p ≥ 3. In this paper, we shall verify this for p = 5. [ABSTRACT FROM AUTHOR]

We show that quasi-isometries between uniformly discrete bounded geometry spaces that satisfy linear isoperimetric inequalities are within bounded distance to bilipschitz equivalences. We apply this result to regularly branching trees and hyperbolic fillings of compact, Ahlfors regular metric spaces. [ABSTRACT FROM AUTHOR]

In this note we derive skew Pieri rules in the spirit of Assaf–McNamara for skew quasisymmetric Schur functions using the Hopf algebraic techniques of Lam–Lauve–Sottile, and recover the original rules of Assaf–McNamara as a special case. We then apply these techniques a second time to obtain skew Pieri rules for skew noncommutative Schur functions. [ABSTRACT FROM AUTHOR]

Richard P. Stanley defined the chromatic symmetric function of a simple graph and has conjectured that every tree is determined by its chromatic symmetric function. Recently, Takahiro Hasebe and the author proved that the order quasisymmetric functions, which are analogs of the chromatic symmetric functions, distinguish rooted trees. In this paper, using a similar method, we prove that the chromatic symmetric functions distinguish trivially perfect graphs. Moreover, we also prove that claw-free cographs, that is, (K1,3,P4)-free graphs belong to a known class of e-positive graphs. [ABSTRACT FROM AUTHOR]

In this paper, we construct the weak version of peak quasisymmetric functions inside the Hopf algebra of weak composition quasisymmetric functions (WCQSym) defined by Guo, Thibon and Yu. Weak peak quasisymmetric functions (WPQSym) are studied in several aspects. First we find a natural basis of WPQSym lifting peak functions introduced by Stembridge. Then we confirm that WPQSym is a Hopf subalgebra of WCQSym by giving explicit multiplication, comultiplication and antipode formulas. By extending Stembridge's descent-to-peak maps, we also show that WPQSym is a Hopf quotient of WCQSym. On the other hand, we prove that WPQSym embeds as a Rota–Baxter subalgebra of WCQSym, thus of the free commutative Rota–Baxter algebra of weight 1 on one generator. Moreover, WPQSym can also be a Rota–Baxter quotient of WCQSym. [ABSTRACT FROM AUTHOR]

Let f, g : S1 →S1 be two C3 critical homeomorphisms of the circle with the same irrational rotation number and the same (finite) number of critical points, all of which are assumed to be non-flat, of power-law type. In this paper we prove that if h : S1→S1 is a topological conjugacy between f and g and h maps the critical points of f to the critical points of g, then h is quasisymmetric. When the power-law exponents at all critical points are integers, this result is a special case of a general theorem recently proved by T. Clark and S. van Strien preprint, 2014. However, unlike their proof, which relies on heavy complex-analytic machinery, our proof uses purely realvariable methods and is valid for non-integer critical exponents as well. We do not require h to preserve the power-law exponents at corresponding critical points. [ABSTRACT FROM AUTHOR]

Abstract: Given a compact set E ⊂ R d − 1, d ⩾ 1, write K E : = [ 0 , 1 ] × E ⊂ R d. A theorem of Bishop and Tyson states that any set of the form K E is minimal for conformal dimension: If ( X , d ) is a metric space and f : K E → ( X , d ) is a quasisymmetric homeomorphism, then dim H f ( K E ) ⩾ dim H K E .We prove that the measure‐theoretic analogue of the result is not true. For any d ⩾ 2 and 0 ⩽ s < d − 1, there exist compact sets E ⊂ R d − 1 with 0 < H s ( E ) < ∞ such that the conformal dimension of ν, the restriction of the ( 1 + s )‐dimensional Hausdorff measure on K E is zero. More precisely, for any ε > 0, there exists a quasisymmetric embedding F : K E → R d such that dim H F ♯ ν < ε. [ABSTRACT FROM AUTHOR]

We introduce Schur multiple zeta functions which interpolate both the multiple zeta and multiple zeta-star functions of the Euler–Zagier type combinatorially. We first study their basic properties including a region of absolute convergence and the case where all variables are the same. Then, under an assumption on variables, some determinant formulas coming from theory of Schur functions such as the Jacobi–Trudi, Giambelli and dual Cauchy formula are established with the help of Macdonald's ninth variation of Schur functions. Moreover, we investigate the quasi-symmetric functions corresponding to the Schur multiple zeta functions. We obtain the similar results as above for them and, furthermore, describe the images of them by the antipode of the Hopf algebra of quasi-symmetric functions explicitly. Finally, we establish iterated integral representations of the Schur multiple zeta values of ribbon type, which yield a duality for them in some cases. [ABSTRACT FROM AUTHOR]

Since the early work of Richard Stanley, it has been observed that several permutation statistics have a remarkable property with respect to shuffles of permutations. We formalize this notion of a shuffle-compatible permutation statistic and introduce the shuffle algebra of a shuffle-compatible permutation statistic, which encodes the distribution of the statistic over shuffles of permutations. This paper develops a theory of shuffle-compatibility for descent statistics—statistics that depend only on the descent set and length—which has close connections to the theory of P -partitions, quasisymmetric functions, and noncommutative symmetric functions. We use our framework to prove that many descent statistics are shuffle-compatible and to give explicit descriptions of their shuffle algebras, thus unifying past results of Stanley, Gessel, Stembridge, Aguiar–Bergeron–Nyman, and Petersen. [ABSTRACT FROM AUTHOR]

Applying the Bonk-Kleiner characterization of Ahlfors 2-regular quasispheres, we show that a metric two-sphere X is a quasisphere if and only if X is linearly locally connected and carries a weak metric doubling measure, i.e., a measure that deforms the metric on X without much shrinking. [ABSTRACT FROM AUTHOR]

We describe a combinatorial formula for the coefficients when the dual immaculate quasisymmetric functions are decomposed into Young quasisymmetric Schur functions. We prove this using an analogue of Schensted insertion. Using this result, we give necessary and sufficient conditions for a dual immaculate quasisymmetric function to be symmetric. Moreover, we show that the product of a Schur function and a dual immaculate quasisymmetric function expands positively in the Young quasisymmetric Schur basis. We also discuss the decomposition of the Young noncommutative Schur functions into the immaculate functions. Finally, we provide a Remmel–Whitney-style rule to generate the coefficients of the decomposition of the dual immaculates into the Young quasisymmetric Schurs algorithmically and an analogous rule for the decomposition of the dual bases. [ABSTRACT FROM AUTHOR]

We prove that any weakly acausal curve Γ in the boundary of anti-de Sitter (2+1)-space is the asymptotic boundary of two spacelike K-surfaces, one of which is past-convex and the other future-convex, for every K ∊ (-∞,-1). The curve Γ is the graph of a quasisymmetric homeomorphism of the circle if and only if the K-surfaces have bounded principal curvatures. Moreover in this case a uniqueness result holds. The proofs rely on a well-known correspondence between spacelike surfaces in anti-de Sitter space and area-preserving diffeomorphisms of the hyperbolic plane. In fact, an important ingredient is a representation formula, which reconstructs a spacelike surface from the associated area-preserving diffeomorphism. Using this correspondence we then deduce that, for any fixed θ ∊ (0, π), every quasisymmetric homeomorphism of the circle admits a unique extension which is a θ-landslide of the hyperbolic plane. These extensions are quasiconformal. [ABSTRACT FROM AUTHOR]

We give a description of the group of all quasisymmetric self-maps of the Julia set of f(z) = z2−1 that have orientation preserving homeomorphic extensions to the whole plane. More precisely, we prove that this group is the uniform closure of the group generated by the Thompson group of the unit circle and an inversion. Moreover, this result is quantitative in the sense that distortions of the approximating maps are uniformly controlled by the distortion of the given map. [ABSTRACT FROM AUTHOR]

We give a geometric characterization of the sets E ⊂ R for which every quasisymmetric embedding f : E → Rn extends to a quasisymmetric embedding f : R → RN for some N ≥ n. [ABSTRACT FROM AUTHOR]

We introduce the quasi-key basis of the polynomial ring which contains the quasi-Schur polynomials of Haglund, Luoto, Mason and van Willigenburg. We prove that stable limits of quasi-key polynomials are quasi-Schur functions, thus lifting the quasi-Schur basis of quasisymmetric polynomials to the full polynomial ring. The new tool we introduce for this purpose is the combinatorial model of Kohnert tableaux. We use this model to prove that key polynomials expand positively in quasi-key polynomials which in turn expand positively in fundamental slide polynomials introduced earlier by the authors. We give simple combinatorial formulas for these expansions in terms of Kohnert tableaux, lifting the parallel expansions of a Schur function into quasi-Schur functions into fundamental quasisymmetric functions. [ABSTRACT FROM AUTHOR]

We introduce a quasisymmetric generalization of Berele and Regev's hook Schur functions and prove that these new quasisymmetric hook Schur functions decompose the hook Schur functions in a natural way. We examine the combinatorics of the quasisymmetric hook Schur functions, providing a relationship to Gessel's fundamental quasisymmetric functions and an analogue of the Robinson-Schensted-Knuth algorithm. We also prove that the multiplication of quasisymmetric hook Schur functions with hook Schur functions behaves the same as the multiplication of quasisymmetric Schur functions with Schur functions. [ABSTRACT FROM AUTHOR]

In this paper, we prove that a large class of homogeneous perfect sets of Hausdorff dimension 1 is quasisymmetrically Hausdorff minimal. We also obtain the similar result for quasisymmetrically packing minimality. [ABSTRACT FROM AUTHOR]

We consider a class of iterated function systems (IFSs) of contracting similarities of Rn, introduced by Hutchinson, for which the invariant set possesses a natural Hölder continuous parameterization by the unit interval. When such an invariant set is homeomorphic to an interval, we give necessary conditions in terms of the similarities alone for it to possess a quasisymmetric (and as a corollary, bi-Hölder) parameterization. We also give a related necessary condition for the invariant set of such an IFS to be homeomorphic to an interval. [ABSTRACT FROM AUTHOR]

Bonk and Kleiner showed that any metric sphere which is Ahlfors 2-regular and linearly locally contractible is quasisymmetrically equivalent to the standard sphere in a quantitative way. We extend this result to arbitrary metric compact orientable surfaces. [ABSTRACT FROM AUTHOR]

Let δ be a class of finite groups, p a prime and π a set of some primes. A finite group G is called a p-quasi-δ-group (respectively, by π-quasi-F-group) provided that for every δ-eccentric G-chief factor H/K of order divisible by p (respectively, by at least one prime in π), the automorphisms of H/K induced by all elements of G are inner. In this paper, we obtain the characterizations of p-quasi-δ-groups and π-quasi-δ-groups, which give a positive answer to an open problem in the book [3]. [ABSTRACT FROM AUTHOR]

This work aims to discuss the static analysis and reanalysis of near-regular (quasi-symmetric) structures and to develop further the analysis of these structures based on the principle of general connect (coupling) and the group representation theory with symmetry groups C 3v and C 1v . During the analysis, only one-dimensional irreducible representations of different symmetry groups are taken into account, and we investigate the effect of these to the static analysis and to local modification method, as well. For these structures any general load can be decomposed into cyclic and asymmetric load, furthermore at most four symmetry components SSS , AAA (C 3v ) and S , A (C 1v ) ( S – symmetric component, A – antimetric component) are used in this paper. The mentioned methods were applied for static analysis with two perpendicular symmetry planes; this paper examines the effect of three symmetry planes and considers the applicability in case of more-than-three symmetry planes for analysis. [ABSTRACT FROM AUTHOR]

In an important paper [24] , Reimann showed that the flow mappings of a continuous vector field of Zygmund class Λ ⁎ are quasisymmetric homeomorphisms. In this paper, we will discuss the flow mappings when the vector field belongs to the smooth Zygmund class λ ⁎ or the Sobolev class H 3 2 . [ABSTRACT FROM AUTHOR]

We analyze a costly voting model with multiple candidates under plurality rule. In equilibrium, the set of candidates is partitioned into a set of “relevant candidates” (which contains at least two candidates) and the remaining candidates. All relevant candidates receive votes and have an equal chance of winning, independent of their popular support levels. The remaining candidates do not receive any votes. Furthermore, all voters who cast votes do so for their most preferred candidate, i.e., there is no “strategic voting.” [ABSTRACT FROM AUTHOR]

This paper completes the enumeration of quasi-symmetric 2- ( 64 , 24 , 46 ) designs supported by the dual code C ⊥ of the binary linear code C spanned by the lines of A G ( 3 , 4 ) , initiated in Rodrigues and Tonchev (2015). It is shown that C ⊥ supports exactly 30,264 nonisomorphic quasi-symmetric 2- ( 64 , 24 , 46 ) designs. The automorphism groups of the related strongly regular graphs are computed. [ABSTRACT FROM AUTHOR]

The ring of quasisymmetric functions is free over the ring of symmetric functions. This result was previously proved by M. Hazewinkel combinatorially through constructing a polynomial basis for quasisymmetric functions. The recent work by A. Savage and O. Yacobi on representation theory provides a new proof to this result. In this paper, we proved that under certain conditions, the positive part of a Heisenberg double is free over the positive part of the corresponding projective Heisenberg double. Examples satisfying the above conditions are discussed. [ABSTRACT FROM AUTHOR]

For a fixed integer, letbe anm-connected region in the Riemann spherewhose complementis a union ofmdisjoint closed disksand letbe quasisymmetric mappings defined onfor. We construct discrete conformal welding forbased on the circle packing approach. We show that the discrete conformal welding mappings induced by circle packings converge uniformly on compact subsets to their continuous counterparts and that the corresponding discrete conformal welding curves converge uniformly to quasicircles determined by. This gives a constructive proof of the existence and uniqueness theorem for conformal welding of finitely connected regions. [ABSTRACT FROM AUTHOR]

I show that Lp-Lq estimates for the Kakeya maximal function yield lower bounds for the conformal dimension of Kakeya sets, and upper bounds for how much quasisymmetries can increase the Hausdorff dimension of line segments inside Kakeya sets. Combining the known Lp-Lq estimates of Wolff and Katz-Tao with the main result of the paper, the conformal dimension of Kakeya sets in Rn is at least max{(n+2)/2, (4n+3)/7}. Moreover, if f is a quasisymmetry from a Kakeya set K ⊂ Rn onto any at most n-dimensional metric space, the f -image of a.e. line segment inside K has dimension at most min{2n/(n + 2), 7n/(4n + 3)}. The Kakeya maximal function conjecture implies that the bounds can be improved to n and 1, respectively. [ABSTRACT FROM AUTHOR]

The main result in this paper is that there is an example of a conjugacy between two expanding Blaschke products on the circle which preserve the Lebesgue measure such that this conjugacy is symmetric at one point but not symmetric on the whole unit circle. Since the proof uses a symmetric rigidity result in a work by Y. Jiang, we use martingale sequences for uniformly quasisymmetric circle endomorphisms developed in an earlier work of the author to give a simple proof. Furthermore, we give a detailed proof of the result in that prior work of the author that the limiting martingale is invariant under symmetric conjugacy. [ABSTRACT FROM AUTHOR]

It is known that a sense preserving homeomorphism h of the unit circle induces a BMO map Ph by pull-back if and only if it is strongly quasisymmetric. In this note, we will discuss the compactness of the projection operator Ph- sending a BMOA function Φ to the anti-holomorphic part of PhΦ. [ABSTRACT FROM AUTHOR]

If H is a connected, graded Hopf algebra, then Takeuchi's formula can be used to compute its antipode. However, there is usually massive cancellation in the result. We show how sign-reversing involutions can sometimes be used to obtain cancellation-free formulas. We apply this idea to nine different examples. We rederive known formulas for the antipodes in the Hopf algebra of polynomials, the shuffle Hopf algebra, the Hopf algebra of quasisymmetric functions in both the monomial and fundamental bases, the Hopf algebra of multi-quasisymmetric functions in the fundamental basis, and the incidence Hopf algebra of graphs. We also find cancellation-free expressions for particular values of the antipode in the immaculate basis for the noncommutative symmetric functions as well as the Malvenuto–Reutenauer and Poirier–Reutenauer Hopf algebras, some of which are the first of their kind. We include various conjectures and suggestions for future research. [ABSTRACT FROM AUTHOR]

We introduce the new class of Jordan arcs (curves) of bounded rotation which includes all arcs (curves) of bounded turning. We prove that if the boundary of a Jordan domain has bounded rotation everywhere but possibly one singular point then every quasimöbius embedding of this domain extends to a quasiconformal automorphism of the entire plane. [ABSTRACT FROM AUTHOR]

Using uniformization, Cantor type sets can be regarded as boundaries of rooted trees. In this setting, we show that the trace of a first-order Sobolev space on the boundary of a regular rooted tree is exactly a Besov space with an explicit smoothness exponent. Further, we study quasisymmetries between the boundaries of two trees, and show that they have rough quasiisometric extensions to the trees. Conversely, we show that every rough quasiisometry between two trees extends as a quasisymmetry between their boundaries. In both directions we give sharp estimates for the involved constants. We use this to obtain quasisymmetric invariance of certain Besov spaces of functions on Cantor type sets. [ABSTRACT FROM AUTHOR]

For every sense-preserving self-homeomorphism of the real axis, Hubbard constructed an extension that is a self-homeomorphism of the upper half-plane by triangulation. It is natural to ask if such extensions of quasisymmetric homeomorphisms of the real axis are all quasiconformal. Furthermore, for what sense-preserving selfhomeomorphisms are such extensions David mappings? In this article, a sufficient and necessary condition for such extensions to be quasiconformal and a sufficient condition for such extensions to be David mappings are given. [ABSTRACT FROM AUTHOR]

This paper describes an initial description of the resilient divertor properties of quasi-symmetric (QS) stellarators using the HSX (Helically Symmetric eXperiment) configuration as a test-case. Divertors in high-performance QS stellarators will need to be resilient to changes in plasma configuration that arise due to evolution of plasma pressure profiles and bootstrap currents for divertor design. Resiliency is tested by examining the changes in strike point patterns from the field line following, which arise due to configurational changes. A low strike point variation with high configuration changes corresponds to high resiliency. The HSX edge displays resilient properties with configuration changes arising from the (1) wall position, (2) plasma current, and (3) external coils. The resilient behavior is lost if large edge islands intersect the wall structure. The resilient edge properties are corroborated by heat flux calculations from the fully 3-D plasma simulations using EMC3-EIRENE. Additionally, the strike point patterns are found to correspond to high curvature regions of magnetic flux surfaces. [ABSTRACT FROM AUTHOR]

We give a new characterization of the peak subalgebra of the algebra of quasisymmetric functions and use this to construct a new basis for this subalgebra. As an application of these results we obtain a combinatorial formula for the Kazhdan–Lusztig polynomials which holds in complete generality and is simpler and more explicit than any existing one. We point out that, in a certain sense, this formula cannot be simplified. [ABSTRACT FROM AUTHOR]

Let G be a connected non-regular non-bipartite graph whose adjacency ma- trix has spectrum ρ, μ(k); λ(l), where k, l ∊ IN and ρ > μ > λ. We show that if μ is non-main then δ(G) ≥1 + μ - λ λ, with equality if and only if G is of one of three types, derived from a strongly regular graph, a symmetric design or a quasi-symmetric design (with appropriate parameters in each case). [ABSTRACT FROM AUTHOR]

A 0-Hecke algebra is a deformation of the group algebra of a Coxeter group. Based on work of Norton and Krob-Thibon, we introduce a tableau approach to the representation theory of 0-Hecke algebras of type A, which resembles the classic approach to the representation theory of symmetric groups by Young tableaux and tabloids. We extend this approach to types B and D, and obtain a correspondence between the representation theory of 0-Hecke algebras of types B and D and quasisymmetric functions and noncommutative symmetric functions of types B and D. Other applications are also provided. [ABSTRACT FROM AUTHOR]

Due to solid and broadband nature, pentamode acoustic cloak is more promising for engineering applications. A simple algorithm based on an elasticity equation is proposed to obtain quasi-symmetric mapping gradient and in turn the characteristic stress for arbitrary shape cloaks. A high degree of symmetry of the obtained mapping gradient and nearly perfect cloaking effect of the designed pentamode cloaks are confirmed by numerical examples. The proposed method paves the way to design more complicated transformation devices with pentamode materials. [ABSTRACT FROM AUTHOR]

We prove that if ξ is a quasisymmetric homeomorphism between Sierpiński carpets that are Julia sets of postcritically-finite rational maps, then ξ is the restriction of a Möbius transformation. This implies that the group of quasisymmetric homeomorphisms of a Sierpiński carpet Julia set of a postcritically-finite rational map is finite. [ABSTRACT FROM AUTHOR]

Douady–Earle extension provides a quasiconformal extension of a quasisymmetric homeomorphism to the unit disk and induces a continuous self-map σ of Beltrami coefficients. In this note, we show that σ is also continuous (under a stronger topology) on those Beltrami coefficients which induce vanishing Carleson measures and thus project to the VMO-Teichmüller space. An immediate consequence is that the VMO-Teichmüller space is contractible. [ABSTRACT FROM AUTHOR]

We show that fiber-preserving quasisymmetric maps are biLipschitz. As an application, we show that quasisymmetric maps on Carnot groups with reducible first stratum are biLipschitz. [ABSTRACT FROM AUTHOR]

We show that an ε-power-quasisymmetric map f: A → Rn can be extended to a Cε-power-quasisymmetric map F: Rn → Rn if A ⊂ Rn satisfies a geometric thickness condition and ε is small enough. The constant C depends on c and n only. [ABSTRACT FROM AUTHOR]