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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]

For subsets of we consider the local right upper porosity and the local right lower porosity as elements of a cluster set of all porosity numbers. The use of a scaling function provides an extension of the concept of porosity numbers on subsets of . The main results describe interconnections between porosity numbers of a set, features of the scaling functions, and the geometry of so-called pretangent spaces to this set. [ABSTRACT FROM AUTHOR]

Let Ω be a planar Jordan domain. We consider double-dome-like surfaces Σ defined by graphs of functions of dist(., ∂Ω) over Ω. The goal is tofind the right conditions on the geometry of the base Ω and the growth of the height so that Σ is a quasisphere or is quasisymmetric to S2. An internal uniform chord-arc condition on the constant distance sets to ∂Ω, coupled with a mild growth condition on the height, gives a close-to-sharp answer. Our method also produces new examples of quasispheres in Rn, for any n ≥ 3. [ABSTRACT FROM AUTHOR]

Let Ω be a planar Jordan domain. We consider double-dome-like surfaces Σ defined by graphs of functions of dist(., ∂Ω) over Ω. The goal is tofind the right conditions on the geometry of the base Ω and the growth of the height so that Σ is a quasisphere or is quasisymmetric to S2. An internal uniform chord-arc condition on the constant distance sets to ∂Ω, coupled with a mild growth condition on the height, gives a close-to-sharp answer. Our method also produces new examples of quasispheres in Rn, for any n ≥ 3. [ABSTRACT FROM AUTHOR]

We introduce a quasisymmetric refinement of Stanley's chromatic symmetric function. We derive refinements of both Gasharov's Schur-basis expansion of the chromatic symmetric function and Chow's expansion in Gessel's basis of fundamental quasisymmetric functions. We present a conjectural refinement of Stanley's power sum basis expansion, which we prove in special cases. We describe connections between the chromatic quasisymmetric function and both the q -Eulerian polynomials introduced in our earlier work and, conjecturally, representations of symmetric groups on cohomology of regular semisimple Hessenberg varieties, which have been studied by Tymoczko and others. We discuss an approach, using the results and conjectures herein, to the e -positivity conjecture of Stanley and Stembridge for incomparability graphs of ( 3 + 1 ) -free posets. [ABSTRACT FROM AUTHOR]

We introduce a class of metrics on Rn generalizing the classical Grushin plane. These are length metrics defined by the line element ds = dE(., Y )-βdsE for a closed nonempty subset Y ⊂ Rn and β ϵ [0, 1). We prove, assuming a Hölder condition on the metric, that these spaces are quasisymmetrically equivalent to Rn and can be embedded in some larger Euclidean space under a bi-Lipschitz map. Our main tool is an embedding characterization due to Seo, which we strengthen by removing the hypothesis of uniform perfectness. In the two-dimensional case, we give another proof of bi-Lipschitz embeddability based on growth bounds on sectional curvature. [ABSTRACT FROM AUTHOR]

The representation theory of 0-Hecke–Clifford algebras as a degenerate case is not semisimple and also with rich combinatorial meaning. Bergeron, Hivert and Thibon have proved that the Grothendieck ring of the category of finitely generated supermodules of 0-Hecke–Clifford algebras is isomorphic to the algebra of peak quasisymmetric functions defined by Stembridge. In this paper we further study the category of finitely generated projective supermodules and clarify the correspondence between it and the peak algebra of symmetric groups. In particular, two kinds of restriction rules for induced projective supermodules are obtained. After that, we consider the corresponding Heisenberg double and its Fock representation to prove that the ring of peak quasisymmetric functions is free over its subring of symmetric functions spanned by Schur's Q-functions. [ABSTRACT FROM AUTHOR]

In this paper, we try to describe the relationship between the differentiability of a quasisymmetric homeomorphism and the local Hausdorff dimension of the quasiline at a point. [ABSTRACT FROM AUTHOR]

We show that if $${f\colon X\to Y}$$ is a quasisymmetric mapping between Ahlfors regular spaces, then $${dim_H f(E)\leq dim_H E}$$ for 'almost every' bounded Ahlfors regular set $${E\subseteq X}$$ . If additionally, $${X}$$ and $${Y}$$ are Loewner spaces then $${dim_H f(E)=dim_H E}$$ for 'almost every" Ahlfors regular set $${E\subset X}$$ . The precise statements of these results are given in terms of Fuglede's modulus of measures. As a corollary of these general theorems we show that if $${f}$$ is a quasiconformal map of $${\mathbb{R}^N}$$ , $${N\geq 2}$$ , then for Lebesgue a.e. $${y\in\mathbb{R}^N}$$ we have $${dim_H f(y+E) = dim_H E}$$ . A similar result holds for Carnot groups as well. For planar quasiconformal maps, our general estimates imply that if $${E \subset {\mathbb{R}}}$$ is Ahlfors $${d}$$ -regular, $${d < 1}$$ , then some component of $${f(E \times {\mathbb{R}})}$$ has dimension at most $${2/(d+1)}$$ , and we construct examples to show this bound is sharp. In addition, we show there is a $${1}$$ -dimensional set $${S\subseteq \mathbb R}$$ and planar quasiconformal map $${f}$$ such that $${f({\mathbb{R}} \times S)}$$ contains no rectifiable sub-arcs. These results generalize work of Balogh et al. (J Math Pures Appl (2)99:125-149, 2013) and answer questions posed in Balogh et al. (J Math Pures Appl (2)99:125-149, 2013) and Capogna et al. (Mapping theory in metric spaces. , 2016). [ABSTRACT FROM AUTHOR]

In this paper, we mainly consider the relationship between the coarsely quasihyperbolic mappings and the quasisymmetric mappings, and show that a homeomorphism is a coarsely quasihyperbolic mapping which implies that it is quasisymmetric in the quasihyperbolic metric between two suitable metric spaces. [ABSTRACT FROM AUTHOR]

This paper studies the equivalence of the quasisymmetric mappings on non-connected sets. We introduce a generalized form of weak quasisymmetry and prove that, on a uniformly perfect metric space, a generalized weakly quasisymmetric mapping is quasisymmetric. We further improve this result on Cantor-like sets satisfying the small gap condition. [ABSTRACT FROM AUTHOR]

Given a quasisymmetric homeomorphism, we introduce the concept of quasisymmetric exponent and explore its relations to other conformal invariants. As a consequence, we establish a necessary and sufficient condition on the equivalence of the dilatation and the maximal dilatation of a quasisymmetric homeomorphism by using the quasisymmetric exponent. A classification on the elements of the universal Teichmiiller space is obtained by using this necessary and sufficient condition. [ABSTRACT FROM AUTHOR]

It is proved that sense preserving continuous mappings f : D → Rn of a domain D in Rn, n = 2, satisfying some general inequalities for p-modulus of families of curves are open and discrete. [ABSTRACT FROM AUTHOR]

In this paper, we consider Väisälä's problem and obtain that a homeomorphism which is both semi-local M-QH and semi-local η-QS between two suitable metric spaces is an M1-QH map. [ABSTRACT FROM AUTHOR]

Suppose that f : D → D ′ is a quasiconformal mapping, where D and D ′ are domains in R n , and that D is a broad domain. We show that for every arcwise connected subset A in D , the weak quasisymmetry of the restriction f | A : A → f ( A ) implies its quasisymmetry. As a consequence, we see that the answer to one of the open problems raised by Heinonen from 1989 is affirmative, under the additional condition that A is arcwise connected. [ABSTRACT FROM AUTHOR]

The classical Littlewood–Richardson rule is a rule for computing coefficients in many areas, and comes in many guises. In this paper we prove two Littlewood–Richardson rules for symmetric skew quasisymmetric Schur functions that are analogous to the famed version of the classical Littlewood–Richardson rule involving Yamanouchi words. Furthermore, both our rules contain this classical Littlewood–Richardson rule as a special case. We then apply our rules to combinatorially classify symmetric skew quasisymmetric Schur functions. This answers affirmatively a conjecture of Bessenrodt, Luoto and van Willigenburg. [ABSTRACT FROM AUTHOR]

In the 1960s Bers showed how to uniformize simultaneously two Riemann surfaces of the same finite analytic type by using a single quasi-Fuchsian group of the first kind. In this paper, we show how to uniformize simultaneously two uniformly quasisymmetric circle endomorphisms of the same degree by a unique normalized branched covering of the Riemann sphere of the same degree such that this branched covering has a unique normalized quasicircle as an invariant limit set. We use this simultaneous uniformization to define a transformation between their spaces of probability invariant measures and formulate several equivalent conjectures to the uniqueness conjecture for symmetric invariant probability measures. In a subsequent paper, we study these conjectures. [ABSTRACT FROM PUBLISHER]

We show that the construction of quasi-symmetric designs with parameters 2- $$(q^3, q^{2}(q-1)/2, q(q^3 -q^2 -2)/4)$$ and block intersection numbers $$q^{2}(q-2)/4$$ and $$q^{2}(q-1)/4$$ (where $$q \ge 4$$ is a power of 2) given by Blokhuis and Haemers (J Stat Plan Inference 95:117-119, ) leads to exponential numbers of such designs. For $$q=4$$ , there are already at least 28,844 isomorphism classes. [ABSTRACT FROM AUTHOR]

If X is a set, the fix-set quasiorder on a group of permutations of X is the quasiorder induced by containment of the fix-sets of elements of S. Axioms for such quasiorders on groups have previously been given. We generalise these to allow non-faithful group actions, the resulting abstract quasiorders being called fix-orders. We characterise the possible fix-orders on a given group G in terms of certain families of subgroups of G. The special case in which the members of the defining family of subgroups are all normal is considered. Software is used to construct and analyse the lattices of fix-orders of many small finite groups. [ABSTRACT FROM AUTHOR]

Following the lead of Stanley and Gessel, we consider a linear map which associates to an acyclic directed graph (or a poset) a quasi-symmetric function. The latter is naturally defined as a multivariate generating series of nondecreasing functions on the graph. We describe the kernel of this linear map by using a simple combinatorial operation that we call cyclic inclusion-exclusion. Our result also holds for the natural noncommutative analogue and for the commutative and noncommutative restrictions to bipartite graphs. An application to the theory of Kerov character polynomials is given. [ABSTRACT FROM AUTHOR]

This paper studies the quasisymmetric mappings on Moran sets. We introduce a generalized form of weak quasisymmetry and prove that, on Moran set satisfying the small gap condition, a generalized weakly quasisymmetric mapping is quasisymmetric. We further give a criterion for the quasisymmetry of mappings between Moran sets with some regular structure. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the quasihyperbolic metrics are quasiinvariant under a quasisymmetric mapping between two suitable metric spaces. Meanwhile, we also show that quasi-invariance of the quasihyperbolic metrics implies that the corresponding map is quasiconformal. At the end of this paper, as an application of these theorems, we prove that the composition of two quasisymmetric mappings in metric spaces is a quasiconformal mapping. [ABSTRACT FROM AUTHOR]

We study the singularity of solutions of the Schrödinger equation with a finite potential at the point k = 0. In the case of delta-type potentials, we show that the nature of this singularity is automodel in a certain sense. We discuss using the obtained results to construct an approximate solution of the inverse scattering problem on the whole axis. For this, we introduce the concept of a quasisymmetric polynomial associated with a given curve. [ABSTRACT FROM AUTHOR]

We classify the pairwise transitive 2-designs, that is, 2-designs such that a group of automorphisms is transitive on the following five sets of ordered pairs: point-pairs, incident point-block pairs, non-incident point-block pairs, intersecting block-pairs and non-intersecting block-pairs. These 2-designs fall into two classes: the symmetric ones and the quasisymmetric ones. The symmetric examples include the symmetric designs from projective geometry, the 11-point biplane, the Higman–Sims design, and designs of points and quadratic forms on symplectic spaces. The quasisymmetric examples arise from affine geometry and the point-line geometry of projective spaces, as well as several sporadic examples. [ABSTRACT FROM AUTHOR]

Let Γ be a Fuchsian group. Any [μ] in the Teichmüller space T(Γ)determines a quasi-circle fμ(∂D). In this paper, we prove that, for any Fuchsian group Γ of the second kind, the Hausdorff dimension ... is not a real analytic function in T(Γ). [ABSTRACT FROM AUTHOR]

We construct indecomposable modules for the 0-Hecke algebra whose characteristics are the dual immaculate basis of the quasi-symmetric functions. [ABSTRACT FROM AUTHOR]

We demonstrate that the complex plane and a class of generalized Grushin planes Gr, where r is a function satisfying specific requirements, are quasisymmetrically equivalent. Then using conjugation we are able to develop an analytic definition of quasisymmetry for homeomorphisms on Gr spaces. In the last section we show our analytic definition of quasisymmetry is consistent with earlier notions of conformal mappings on the Grushin plane. This leads to several characterizations of conformal mappings on the generalized Grushin planes. [ABSTRACT FROM AUTHOR]

We investigate the uniform piecewise linearizing question for a family of Lorenz maps. Let f be a piecewise linear Lorenz map with different slopes and positive topological entropy, we show that f is conjugate to a linear mod one transformation and the conjugacy admits a dichotomy: it is either bi-Lipschitz or singular depending on whether f is renormalizable or not. f is renormalizable if and only if its rotation interval degenerates to be a rational point. Furthermore, if the endpoints are periodic points with the same rotation number, then the conjugacy is quasisymmetric. [ABSTRACT FROM AUTHOR]

We investigate a quasisymmetrically invariant counterpart of the topological Hausdorff dimension of a metric space. This invariant, called the topological conformal dimension, gives a lower bound on the topological Hausdorff dimension of quasisymmetric images of the space. We obtain results concerning the behavior of this quantity under products and unions, and compute it for some classical fractals. The range of possible values of the topological conformal dimension is also considered, and we show that this quantity can be fractional. [ABSTRACT FROM AUTHOR]

The boundary of the Siegel disk of a quadratic polynomial with an irrationally indifferent fixed point and the rotation number whose continued fraction expansion is preperiodic has been observed to be self-similar with a certain scaling ratio. The restriction of the dynamics of the quadratic polynomial to the boundary of the Siegel disk is known to be quasisymmetrically conjugate to the rigid rotation with the same rotation number. The geometry of this self-similarity is universal for a large class of holomorphic maps. A renormalization explanation of this universality has been proposed in the literature. In this paper we provide an estimate on the quasisymmetric constant of the conjugacy, and use it to prove bounds on the scaling ratio $${\lambda}$$ of the form where s is the period of the continued fraction, and $${\alpha \in (0, 1)}$$ depends on the rotation number in an explicit way, while C > 1, $${\delta \in (0, 1)}$$ and $${\gamma \in (0, 1)}$$ depend only on the maximum of the integers in the continued fraction expansion of the rotation number. [ABSTRACT FROM AUTHOR]

We present a one-parameter family of models for square contingency tables that interpolates between the classical quasisymmetry model and its Pearsonian analogue. Algebraically, this corresponds to deformations of toric ideals associated with graphs. Our discussion of the statistical issues centers around maximum likelihood estimation. [ABSTRACT FROM AUTHOR]

We define the notion of generalised Cayley-Abels graph for compactly generated locally compact groups in terms of quasi-actions. This extends the notion of Cayley-Abels graph of a compactly generated totally disconnected locally compact group, studied in particular by Krön and Möller under the name of rough Cayley graph (and relative Cayley graph). We construct a generalised Cayley-Abels graph for any compactly generated locally compact group using quasi-lattices and show uniqueness up to quasi-isometry. A class of examples is given by the Cayley graphs of cocompact lattices in compactly generated groups. As an application, we show that a compactly generated group has polynomial growth if and only if its generalised Cayley-Abels graph has polynomial growth (same for intermediate and exponential growth). Moreover, a unimodular compactly generated group is amenable if and only if its generalised Cayley-Abels graph is amenable as a metric space. [ABSTRACT FROM AUTHOR]

We introduce Schottky maps--conformal maps between relative Schottky sets--and study their local rigidity properties. This continues the investigations of relative Schottky sets initiated in the author's earlier work entitled Planar relative Schottky sets and quasisymmetric maps, Proc. Lond. Math. Soc. (3) 104 (2012), no. 3, 455-485. Besides being of independent interest, the latter and current works provide key ingredients in the forthcoming proof of quasisymmetric rigidity of Sierpiński carpet Julia sets of rational functions. [ABSTRACT FROM AUTHOR]

We prove that uniform Cantor sets of Hausdorff dimension 1 are all quasisymmetrically minimal for Hausdorff dimension. An analog of this result for packing dimension is also obtained. From the proof a general sufficient condition for minimal Cantor sets can be formulated. [ABSTRACT FROM AUTHOR]

The admissible parameters of symmetric $$(v,k,\lambda )$$ designs satisfying $$v=4(k-\lambda )+2$$ are shown to correspond with the solutions of a certain Pell equation. We then determine the feasible parameters of such designs that could have a quasi-symmetric residual design with respect to a block, and classify them into two possible families. Finally, we consider the feasible parameters of symmetric designs with inner balance as defined by Nilson and Heidtmann (Des. Codes Cryptogr. doi:, ()), and show that (with one exception) they must all belong to one of these families. [ABSTRACT FROM AUTHOR]

Suppose that E and E′ denote real Banach spaces with dimension at least 2 and that D ⊆ E and D′ gl E′ are uniform domains with homogeneously dense boundaries. We consider the class of all φ-FQC (freely (φ-quasiconformal) maps of D onto D′ with bilipschitz boundary values. We show that the maps of this class are n-quasisymmetric. As an application, we show that if D is bounded, then maps of this class satisfy a two sided Hölder condition. Moreover, replacing the class φ-FQC by the smaller class of M-QH maps, we show that M -QH maps with bilipschitz boundary values are bilipschitz. Finally, we show that if / is a φ-FQC map which maps D onto itself with identity boundary values, then there is a constant C, depending only on the function φ, such that for all x e D, the quasihyperbolic distance satisfies kD{x, f(x)) ≤ C. [ABSTRACT FROM AUTHOR]

We define (αn)-regular sets in uniformly perfect metric spaces. This definition is quasisymmetrically invariant and the construction resembles generalized dyadic cubes in metric spaces. For these sets we then determine the necessary and sufficient conditions to be fat (or thin). In addition we discuss restrictions of doubling measures to these sets, and, in particular, give a sufficient condition to retain at least some of the restricted measures doubling on the set. Our main result generalizes and extends analogous results that were previously known to hold on the real line. [ABSTRACT FROM AUTHOR]

We prove a general criterion for a metric space to have conformal dimension one. The conditions are stated in terms of the existence of enough local cut points in the space. We then apply this criterion to the boundaries of hyperbolic groups and show an interesting relationship between conformal dimension and some canonical splittings of the group. [ABSTRACT FROM AUTHOR]

Abstract: Let be an algebraically closed field which is complete with respect to a non-trivial non-Archimedean absolute value . We study metric properties of the limit set Λ of a semigroup G generated by a finite set of contractive analytic functions on . We prove that the limit set Λ of G is uniformly perfect if the derivative of each generating function of G does not vanish on . Furthermore, we show that if each coefficient of the generating functions is in the field of p-adic numbers, or the limit set Λ satisfies the strong open set condition, then Λ has the doubling property. This yields that the limit set Λ is quasisymmetrically equivalent to the space of 2-adic integers. We also give a counterexample to show that not all limit sets have the doubling property. The Berkovich space is introduced to study the limit set Λ, and we prove that the limit set Λ has a positive capacity in the Berkovich space which yields that there exists an equilibrium measure μ whose support is contained in the limit set Λ. We also show that if the semigroup is generated by a countable set of contractive analytic functions, then its limit set Λ can be non-compact. However, if coefficients of the generating functions lie in , then the limit set Λ is compact. [Copyright &y& Elsevier]

We obtain Dini conditions that guarantee that an asymptotically conformal quasisphere is rectifiable. In particular, we show that for any ε > 0 integrability of (ess sup1-t<|x|<1+t Kf (x)-1)2-εdt/t implies that the image of the unit sphere under a global quasiconformal homeomorphism f is rectifiable. We also establish estimates for the weak quasisymmetry constant of a global K-quasiconformal map in neighborhoods with maximal dilatation close to 1. [ABSTRACT FROM AUTHOR]

Haglund, Luoto, Mason, and van Willigenburg introduced a basis for quasisymmetric functions, called the quasisymmetric Schur function basis, generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through reverse column-strict tableaux. We introduce a new basis for quasisymmetric functions, called the row-strict quasisymmetric Schur function basis, generated combinatorially through fillings of composition diagrams in much the same way as quasisymmetic Schur functions are generated through fillings of composition diagrams. We describe the relationship between this new basis and other known bases for quasisymmetric functions, as well as its relationship to Schur polynomials. We obtain a refinement of the omega transform operator as a result of these relationships. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove that if and for any , or if , the Moran set on the line with packing dimension 1 are quasisymmetrically packing-minimal. [Copyright &y& Elsevier]

Rotation is favorable for confinement, but a stellarator can rotate at high speeds if and only if it is sufficiently close to quasisymmetry. This article investigates how close it needs to be. For a magnetic field B = B0 + αB1, where B0 is quasisymmetric, αB1 is a deviation from quasisymmetry, and α ≪ 1, the stellarator can rotate at high velocities if α < ϵ1/2, with ϵ the ion Larmor radius over the characteristic variation length of B0. The cases in which this result may break down are discussed. If the stellarator is sufficiently quasisymmetric in the above sense, the rotation profile, and equivalently, the long-wavelength radial electric field, are not set neoclassically; instead, they can be affected by turbulent transport. Their computation requires the O(ϵ2) pieces of both the turbulent and the long-wavelength components of the distribution function. This article contains the first step towards a formulation to calculate the rotation profile by providing the equations determining the long-wavelength components of the O(ϵ2) pieces. [ABSTRACT FROM AUTHOR]

Abstract: We introduce explicit combinatorial interpretations for the coefficients in some of the transition matrices relating to skew Hall–Littlewood polynomials and Hivertʼs quasisymmetric Hall–Littlewood polynomials . More specifically, we provide: [1.] the G-expansions of the Hall–Littlewood polynomials , the monomial quasisymmetric polynomials , the quasisymmetric Schur polynomials , and the peak quasisymmetric functions ; [2.] an expansion of in terms of the ʼs. The F-expansion of is facilitated by introducing starred tableaux. [Copyright &y& Elsevier]

Our goal is to deal with a family of quasi-symmetric semiclassical orthogonal polynomial sequences of class two through the study of the differential functional equation fulfilled by its corresponding regular form. Up to a linear transformation, we determine all polynomial sequences of this family. The recurrence coefficients and integral representations are established. [ABSTRACT FROM PUBLISHER]

In neoclassical calculations for stellarators, it is customary to retain poloidal E × B precession in the kinetic equation, but to neglect other electric field terms of the same formal magnitude, causing non-conservation of total energy (kinetic plus potential) and magnetic moment. Here we consider the effects of retaining these terms in several types of optimized stellarators. It is challenging to maintain the full conservation laws in analytical calculations for a finite temperature gradient, but it is possible for low collisionality in the case of perfectly quasisymmetric plasmas, or more generally, for omnigenous plasmas. For the omnigenous calculation, we develop a new ordering that allows an expansion about the quasisymmetric solution. Even though canonical angular momentum is not conserved in an omnigenous nonsymmetric plasma, it is still possible to define an effective canonical angular momentum for a corresponding quasisymmetric system, and this quantity is useful as a radial variable in the analysis of the omnigenous plasma. Applying these observations and techniques to the drift-kinetic equation, electric-field-driven modifications to the omnigenous distribution function are derived, including a new term which has no analogue in previous neoclassical calculations for omnigenous or quasisymmetric stellarators. [ABSTRACT FROM AUTHOR]

Malvenuto and Reutenauer (C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra177 (1995), 967–982) showed how the total symmetric group ring ⊕nZΣn could be made into a Hopf algebra with a very nice structure which admitted the Solomon descent algebra as a sub-Hopf algebra. To do this they replaced the group multiplication by a convolution product, thus distancing their structure from the group structure of Σn. In this paper we examine what is possible if we keep to the group multiplication, and we also consider the question for more general families of groups. We show that a Hopf algebra structure is not possible, but cocommutative and non-cocommutative counital bialgebras can be obtained, arising from certain diagrams of group homomorphisms. In the case of the symmetric groups we note that all such structures are weak in the sense that the dual algebras have many zero-divisors, but structures which respect descent sums can be found. [ABSTRACT FROM PUBLISHER]