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Showing papers in "Mathematische Annalen in 2008"


Journal ArticleDOI
TL;DR: In this paper, a Galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations is presented, which is able to reprove Holder's theorem that the Gamma function satisfies no polynomial differential equation and give general results that imply, for example, that no differential relationship holds among q-hypergeometric equations.
Abstract: We present a Galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations. With this we are able to reprove Holder’s theorem that the Gamma function satisfies no polynomial differential equation and are able to give general results that imply, for example, that no differential relationship holds among solutions of certain classes of q-hypergeometric equations.

152 citations


Journal ArticleDOI
TL;DR: In this paper, an Euler equation on the diffeomorphism group of the circle corresponding to a natural right-invariant Sobolev metric is studied, which describes the evolution of rotators in liquid crystals with external magnetic field and self-interaction.
Abstract: We study an equation lying ‘mid-way’ between the periodic Hunter–Saxton and Camassa–Holm equations, and which describes evolution of rotators in liquid crystals with external magnetic field and self-interaction. We prove that it is an Euler equation on the diffeomorphism group of the circle corresponding to a natural right-invariant Sobolev metric. We show that the equation is bihamiltonian and admits both cusped and smooth traveling-wave solutions which are natural candidates for solitons. We also prove that it is locally well-posed and establish results on the lifespan of its solutions. Throughout the paper we argue that despite similarities to the KdV, CH and HS equations, the new equation manifests several distinctive features that set it apart from the other three.

135 citations


Journal ArticleDOI
TL;DR: In this article, a resonance method was introduced to produce large values of the Riemann zeta function on the critical line, and large and small central values of L-functions.
Abstract: We introduce a resonance method to produce large values of the Riemann zeta-function on the critical line, and large and small central values of L-functions.

127 citations


Journal ArticleDOI
TL;DR: In this paper, the existence of the outer Minkowski content for d-dimensional closed sets with Lipschitz boundary has been shown to be stable under finite unions of sets with positive reach.
Abstract: We find conditions ensuring the existence of the outer Minkowski content for d-dimensional closed sets in \({\mathbb{R}^d}\) , in connection with regularity properties of their boundaries. Moreover, we provide a class of sets (including all sufficiently regular sets) stable under finite unions for which the outer Minkowski content exists. It follows, in particular, that finite unions of sets with Lipschitz boundary and a type of sets with positive reach belong to this class.

119 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that on compact Kahler manifolds solutions to the complex Monge-Ampere equation, with the right-hand side in L p ≥ 1, the problem is Holder continuous.
Abstract: We prove that on compact Kahler manifolds solutions to the complex Monge–Ampere equation, with the right-hand side in L p , p > 1, are Holder continuous.

119 citations


Journal ArticleDOI
TL;DR: In this article, the generalized Korteweg-de Vries equation (gKdV) was considered under general assumptions on f and Q� c��, and it was shown that the family of solitons around Q� c�� is asymptotically stable in some local sense.
Abstract: We consider the generalized Korteweg-de Vries equation (gKdV) $$\partial_t u + \partial_x (\partial_x^2 u + f(u)) = 0, \quad (t, x) \in [0, T) \times {\mathbb{R}},$$ with general C 3 nonlinearity f. Under an explicit condition on f and c > 0, there exists a solution in the energy space H 1 of the type u(t, x) = Q c (x − x 0 − ct), called soliton. In this paper, under general assumptions on f and Q c , we prove that the family of solitons around Q c is asymptotically stable in some local sense in H 1, i.e. if u(t) is close to Q c (for all t ≥ 0), then u(t) locally converges in the energy space to some Q c+ as t → +∞. Note in particular that we do not assume the stability of Q c . This result is based on a rigidity property of the gKdV equation around Q c in the energy space whose proof relies on the introduction of a dual problem. These results extend the main results in Martel (SIAM J. Math. Anal. 38:759–781, 2006); Martel and Merle (J. Math. Pures Appl. 79:339–425, 2000), (Arch. Ration. Mech. Anal. 157:219–254, 2001), (Nonlinearity 1:55–80), devoted to the pure power case.

118 citations


Journal ArticleDOI
TL;DR: In this paper, the authors considered a class of non-local (integro-differential) operators, and established a sharp two-sided heat kernel estimate and derived parabolic Harnack principle for them.
Abstract: It is well-known that there is a deep interplay between analysis and probability theory. For example, for a Markovian infinitesimal generator $${\mathcal{L}}$$ , the transition density function p(t, x, y) of the Markov process associated with $${\mathcal{L}}$$ (if it exists) is the fundamental solution (or heat kernel) of $${\mathcal{L}}$$ . A fundamental problem in analysis and in probability theory is to obtain sharp estimates of p(t, x, y). In this paper, we consider a class of non-local (integro-differential) operators $${\mathcal{L}}$$ on $${\mathbb{R}^d}$$ of the form $$\mathcal{L}u(x) = \lim\limits_{{\varepsilon \downarrow 0}} \int\limits_{\{y\in \mathbb {R}^d: \, |y-x| > \varepsilon\}} (u(y)-u(x)) J(x, y) dy,$$ where $${\displaystyle J(x, y)= \frac{c (x, y)}{|x-y|^{d+\alpha}} {\bf 1}_{\{|x-y| \leq \kappa\}}}$$ for some constant $${\kappa > 0}$$ and a measurable symmetric function c(x, y) that is bounded between two positive constants. Associated with such a non-local operator $${\mathcal{L}}$$ is an $${\mathbb{R}^d}$$ -valued symmetric jump process of finite range with jumping kernel J(x, y). We establish sharp two-sided heat kernel estimate and derive parabolic Harnack principle for them. Along the way, some new heat kernel estimates are obtained for more general finite range jump processes that were studied in (Barlow et al. in Trans Am Math Soc, 2008). One of our key tools is a new form of weighted Poincare inequality of fractional order, which corresponds to the one established by Jerison in (Duke Math J 53(2):503–523, 1986) for differential operators. Using Meyer’s construction of adding new jumps, we also obtain various a priori estimates such as Holder continuity estimates for parabolic functions of jump processes (not necessarily of finite range) where only a very mild integrability condition is assumed for large jumps. To establish these results, we employ methods from both probability theory and analysis extensively.

118 citations


Journal ArticleDOI
TL;DR: In this paper, the authors studied the Riesz transform of resolvent kernels on asymptotically Euclidean conic manifolds and showed that it is polynomial in the number of non-zero eigenvalues of the Laplacian on the boundary.
Abstract: Let $$M^\circ$$ be a complete noncompact manifold and g an asymptotically conic manifold on $$M^\circ$$ , in the sense that $$M^\circ$$ compactifies to a manifold with boundary M in such a way that g becomes a scattering metric on M. A special case that we focus on is that of asymptotically Euclidean manifolds, where the induced metric at infinity is equal to the standard metric on S n−1; such manifolds have an end that can be identified with $${\mathbb{R}}^n \backslash B(R,0)$$ in such a way that the metric is asymptotic in a precise sense to the flat Euclidean metric. We analyze the asymptotics of the resolvent kernel (P + k 2)−1 where $$P = \Delta_g + V$$ is the sum of the positive Laplacian associated to g and a real potential function $$V\in C^{\infty}(M)$$ which vanishes to second order at the boundary (i.e. decays to second order at infinity on $$M^\circ$$ ) and such that $$\Delta_{\partial M}+(n-2)^2/4+V_0 > 0$$ if $$V_0:=(x^{-2}V)|_{\partial M}$$ . Then we show that on a blown up version of $$M^2 \times [0, k_0]$$ the resolvent kernel is conormal to the lifted diagonal and polyhomogeneous at the boundary, and we are able to identify explicitly the leading behaviour of the kernel at each boundary hypersurface. Using this we show that the Riesz transform of P is bounded on $$L^p(M^\circ)$$ for 1 < p < n if $$V_0\equiv 0$$ , and that this range is optimal if $$V ot\equiv 0$$ or if M has more than one end. The result with $$V ot\equiv 0$$ is new even when $$M^\circ = {\mathbb{R}}^n$$ , g is the Euclidean metric and V is compactly supported. When V ≡ 0 with one end, the range of p becomes 1 < p < p max where p max > n depends explicitly on the first non-zero eigenvalue of the Laplacian on the boundary $$\partial M$$ . Our results hold for all dimensions ≥ 3 under the assumption that P has neither zero modes nor a zero-resonance. In the follow-up paper Guillarmou and Hassell (Resolvent at low energy and Riesz transform for Schrodinger operators on asymptotically conic manifolds, preprint) [7] we analyze the same situation in the presence of zero modes and zero-resonances.

117 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that a complete shrinking soliton is compact if and only if it is bounded and moreover, in such a case, it has finite fundamental group.
Abstract: It is shown that a complete shrinking soliton is compact if and only if it is bounded and moreover, in such a case, it has finite fundamental group.

114 citations


Journal ArticleDOI
TL;DR: In this paper, two differential operators on harmonic weak Maass forms of weight 2−k were studied for integers k ≤ 2 and they were shown to have algebraic coefficients for CM forms with vanishing Hecke eigenvalues.
Abstract: For integers k ≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2 − k. The operator ξ2-k (resp. D k-1) defines a map to the space of weight k cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms are expected to have transcendental coefficients, we show that those forms which are “dual” under ξ2-k to newforms with vanishing Hecke eigenvalues (such as CM forms) have algebraic coefficients. Using regularized inner products, we also characterize the image of D k-1.

110 citations


Journal ArticleDOI
Jun Kigami1
TL;DR: In this paper, the standard Dirichlet form and its energy measure,called the Kusuoka measure, on the Sierpinski gasket as aprototype of "measurable Riemannian geometry" were studied.
Abstract: We study the standard Dirichlet form and its energy measure,called the Kusuoka measure, on the Sierpinski gasket as aprototype of “measurable Riemannian geometry”. The shortest pathmetric on the harmonic Sierpinski gasket is shown to be thegeodesic distance associated with the “measurable Riemannianstructure”. The Kusuoka measure is shown to have the volumedoubling property with respect to the Euclidean distance and alsoto the geodesic distance. Li–Yau type Gaussian off-diagonal heatkernel estimate is established for the heat kernel associated withthe Kusuoka measure.

Journal ArticleDOI
TL;DR: A nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of Fomin and Kirillov (Proc Formal Power Series Alg Comb, 1994) to K-theoretic Grassmannian Littlewood-Richardson rule of Buch and Buch (Acta Math 189(1):37-78, 2002) was proposed in this paper.
Abstract: We formulate a nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of Fomin and Kirillov (Proc Formal Power Series Alg Comb, 1994) in the basis of stable Grothendieck polynomials for partitions This gives a common generalization, as well as new proofs of the rule of Fomin and Greene (Discret Math 193:565–596, 1998) for the expansion of the stable Schubert polynomials into Schur polynomials, and the K-theoretic Grassmannian Littlewood–Richardson rule of Buch (Acta Math 189(1):37–78, 2002) The proof is based on a generalization of the Robinson–Schensted and Edelman–Greene insertion algorithms Our results are applied to prove a number of new formulas and properties for K-theoretic quiver polynomials, and the Grothendieck polynomials of Lascoux and Schutzenberger (C R Acad Sci Paris Ser I Math 294(13):447–450, 1982) In particular, we provide the first K-theoretic analogue of the factor sequence formula of Buch and Fulton (Invent Math 135(3):665–687, 1999) for the cohomological quiver polynomials

Journal ArticleDOI
TL;DR: In this paper, it was shown that for a given infinite dimensional Banach space Y the width of Bohr's strip for a Dirichlet series with coefficients an in Y is bounded by 1 − 1/Cot(Y ), where Cot(Y ) denotes the optimal cotype of Y.
Abstract: Bohr showed that the width of the strip (in the complex plane) on which a given Dirichlet seriesan/n s , s ∈ C, converges uniformly but not absolutely, is at most 1/2, and Bohnenblust-Hille that this bound in general is optimal. We prove that for a given infinite dimensional Banach space Y the width of Bohr's strip for a Dirichlet series with coefficients an in Y is bounded by 1 −1/Cot(Y ), where Cot(Y ) denotes the optimal cotype of Y. This estimate even turns out to be optimal, and hence leads to a new characterization of cotype in terms of vector valued Dirichlet series. Mathematics Subject Classification (2000) Primary 32A05; Secondary 46B07 ·

Journal ArticleDOI
TL;DR: In this paper, the authors provide an explicit construction of elements of the middle third Cantor set with any prescribed irrationality exponent, which answers a question posed by Kurt Mahler and provides a solution to the problem.
Abstract: We provide an explicit construction of elements of the middle third Cantor set with any prescribed irrationality exponent. This answers a question posed by Kurt Mahler.

Journal ArticleDOI
TL;DR: In this article, a complete list of normal forms for the two-dimensional metrics that admit a transitive Lie pseudogroup of geodesic-preserving transformations is given, and it is shown that these normal forms are mutually non-isometric.
Abstract: We give a complete list of normal forms for the two-dimensional metrics that admit a transitive Lie pseudogroup of geodesic-preserving transformations and we show that these normal forms are mutually non-isometric. This solves a problem posed by Sophus Lie.

Journal ArticleDOI
TL;DR: In this article, the topological structure of the fixed locus of a non-symplectic automorphism of order 3 is classified and the action on cohomology is determined.
Abstract: In this paper we study K3 surfaces with a non-symplectic automorphism of order 3. In particular, we classify the topological structure of the fixed locus of such automorphisms and we show that it determines the action on cohomology. This allows us to describe the structure of the moduli space and to show that it has three irreducible components.

Journal ArticleDOI
Kunyu Guo1, Kai Wang1
TL;DR: In this article, the essential normality of graded submodules is investigated in the dimension d ǫ = 2, 3 and it is shown that the Arveson conjecture is true.
Abstract: This paper mainly concerns the essential normality of graded submodules. Essentially all of the basic Hilbert modules that have received attention over the years are p-essentially normal—including the d-shift Hilbert module, the Hardy and Bergman modules of the unit ball. Arveson conjectured graded submodules over the unit ball inherit this property and provided motivations to seek an affirmative answer. Some positive results have been obtained by Arveson and Douglas. However, the problem has been resistant. In dimensions d = 2, 3, this paper shows that the Arveson’s conjecture is true. In any dimension, the paper also gives an affirmative answer in the case of the graded principal submodule. Finally, the paper is associated with K-homology invariants arising from graded quotient modules, by which geometry of the quotient modules and geometry of algebraic varieties are connected. In dimensions d = 2, 3, it is shown that K-homology invariants determined by graded quotients are nontrivial. The paper also establishes results on p-smoothness of K-homology elements, and gives an explicit expression for K-homology invariant in dimension d = 2.

Journal ArticleDOI
Mihai Paun1
TL;DR: In this article, it was shown that the Kobayashi conjecture is true for very generic hypersurfaces in the 3D projective space, as long as the degree of the hypersurface is greater than 18.
Abstract: The results we obtain in this article concern the hyperbolicity of very generic hypersurfaces in the 3-dimensional projective space: we show that the Kobayashi conjecture is true in this setting, as long as the degree of the hypersurface is greater than 18.

Journal ArticleDOI
TL;DR: In this article, the weak solution of an elliptic system with discontinuous coefficients in non-smooth domains without using maximal function approach was obtained. But it is assumed that the boundary of a bounded domain is well approximated by hyperplanes at every point and at every scale, and the tensor coefficients belong to BMO space with their BMO semi-norms sufficiently small.
Abstract: We obtain the global W 1,p , 1 < p < ∞, estimate for the weak solution of an elliptic system with discontinuous coefficients in non-smooth domains without using maximal function approach. It is assumed that the boundary of a bounded domain is well approximated by hyperplanes at every point and at every scale, and that the tensor coefficients belong to BMO space with their BMO semi-norms sufficiently small.

Journal ArticleDOI
TL;DR: In this article, the authors consider a class of ultraparabolic differential equations that satisfy the Hormander's hypoellipticity condition and prove that the weak solutions to the equation with measurable coefficients are locally bounded functions.
Abstract: We consider a class of ultraparabolic differential equations that satisfy the Hormander’s hypoellipticity condition and we prove that the weak solutions to the equation with measurable coefficients are locally bounded functions. The method extends the Moser’s iteration procedure and has previously been employed in the case of operators verifying a further homogeneity assumption. Here we remove that assumption by proving some potential estimates and some ad hoc Sobolev type inequalities for solutions.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the intersection of an algebraic curve C lying in a multiplicative torus over Q with the union of all algebraic subgroups of codimension 2 and proved that finiteness holds precisely when C is not contained in a proper subgroup.
Abstract: We study the intersection of an algebraic curve C lying in a multiplicative torus over \({\bar{\mathbb{Q}}}\) with the union of all algebraic subgroups of codimension 2. Finiteness of this set has already been proved by Bombieri, Masser and Zannier under the assumption that C is not contained in a translate of a proper subtorus. Following this result, the question of the minimal hypothesis implying finiteness has been raised by these authors, giving rise to the conjecture~: finiteness holds precisely when C is not contained in a proper subgroup. We prove here this statement which is also a special case of more general conjectures stated independently by Zilber and Pink. Our proof takes its inspiration from an article by Remond and Viada concerning the Zilber-Pink conjecture for curves lying in a power of an elliptic curve. Hence, it relies on a uniform version of the Vojta inequality proven via the generalized Vojta inequality of Remond. The main task is to establish a lower bound for some intersection numbers, here on a whole family of surfaces obtained by blowing up a compactification of C × C.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the elementary obstruction to the existence of 0-cycles of degree 1 on an arbitrary variety X, over an arbitrary field, can be expressed in terms of the Albanese 1-motives associated with dense open subsets of X.
Abstract: We show that the elementary obstruction to the existence of 0-cycles of degree 1 on an arbitrary variety X, over an arbitrary field, can be expressed in terms of the Albanese 1-motives associated with dense open subsets of X. Arithmetic applications are given.

Journal ArticleDOI
TL;DR: In this article, a smooth continuous trace algebra with a Riemannian manifold spectrum X equipped with a smooth action by a discrete group G such that G acts on X properly and isometrically was shown to be KK-theoretically Poincare dual to a twisted group algebras.
Abstract: Let \({\mathcal{A}}\) be a smooth continuous trace algebra, with a Riemannian manifold spectrum X, equipped with a smooth action by a discrete group G such that G acts on X properly and isometrically. Then \({\mathcal{A}}^{-1}\rtimes G \) is KK-theoretically Poincare dual to \(\big(\mathcal A {\hat {\otimes}_{C_0(X)}} C_\tau (X)\big) \rtimes G\) , where \({\mathcal{A}}^{-1}\) is the inverse of \({\mathcal{A}}\) in the Brauer group of Morita equivalence classes of continuous trace algebras equipped with a group action. We deduce this from a strengthening of Kasparov’s duality theorem. As applications we obtain a version of the above Poincare duality with X replaced by a compact G-manifold M and Poincare dualities for twisted group algebras if the group satisfies some additional properties related to the Dirac dual-Dirac method for the Baum- Connes conjecture.

Journal ArticleDOI
TL;DR: In this paper, the authors considered quite general h-pseudodifferential operators on R n with small random perturbations and showed that in the limit h → 0 the eigenvalues are distributed according to a Weyl law with a probabality that tends to 1.
Abstract: We consider quite general h-pseudodifferential operators on R n with small random perturbations and show that in the limit h → 0 the eigenvalues are distributed according to a Weyl law with a probabality that tends to 1. The first author has previously obtained a similar result in dimension 1. Our class of perturbations is different.

Journal ArticleDOI
TL;DR: In this article, a covering lemma of Besicovitch type for metric balls in the setting of Holder quasimetric spaces of homogenous type was established and used to prove a covering theorem for measurable sets.
Abstract: We establish a covering lemma of Besicovitch type for metric balls in the setting of Holder quasimetric spaces of homogenous type and use it to prove a covering theorem for measurable sets. For families of measurable functions, we introduce the notions of power decay, critical density and double ball property and with the aid of the covering theorem we show how these notions are related. Next we present an axiomatic procedure to establish Harnack inequality that permits to handle both divergence and non divergence linear equations.

Journal ArticleDOI
TL;DR: For a commutative noetherian ring A, the authors in this paper compare the support of a complex of A-modules with its cohomology, which leads to a classification of all full subcategories of A -modules which are thick (i.e., closed under taking kernels, cokernels, and extensions) and open under taking arbitrary direct sums.
Abstract: For a commutative noetherian ring A, we compare the support of a complex of A-modules with the support of its cohomology. This leads to a classification of all full subcategories of A-modules which are thick (that is, closed under taking kernels, cokernels, and extensions) and closed under taking arbitrary direct sums. In addition, subcategories of A-modules that are closed under taking submodules, extensions, and direct unions are classified via associated prime ideals.

Journal ArticleDOI
TL;DR: In this article, it was shown that for every separable Banach space X with non-separable dual, the space contains an unconditional family of size Θ(X^{**} ) of size Ϙ(x, n) for trees and finite products of perfect sets of reals.
Abstract: It is shown that for every separable Banach space X with non-separable dual, the space \(X^{**}\) contains an unconditional family of size \(|X^{**}|\) . The proof is based on Ramsey Theory for trees and finite products of perfect sets of reals. Among its consequences, it is proved that every dual Banach space has a separable quotient.

Journal ArticleDOI
TL;DR: The authors characterizes radial Fourier multipliers as acting on radial L p functions, 1 < p < 2d/(d + 1), in terms of Lebesgue space norms for Fourier localized pieces of the convolution kernel.
Abstract: We give characterizations of radial Fourier multipliers as acting on radial L p functions, 1 < p < 2d/(d + 1), in terms of Lebesgue space norms for Fourier localized pieces of the convolution kernel. This is a special case of corresponding results for general Hankel multipliers. Besides L p − L q bounds we also characte- rize weak type inequalities and intermediate inequalities involving Lorentz spaces. Applications include results on interpolation of multiplier spaces.

Journal ArticleDOI
TL;DR: In this article, a non-commutative multivariable analogue of the classical Nevanlinna-Pick interpolation problem for analytic functions with positive real parts on the open unit disc was obtained.
Abstract: In this paper we obtain a noncommutative multivariable analogue of the classical Nevanlinna–Pick interpolation problem for analytic functions with positive real parts on the open unit disc. Given a function \(f : \Lambda \to \mathbb {C}\) , where \(\Lambda\) is an arbitrary subset of the open unit ball \(\mathbb{B}_n:=\{z\in \mathbb {C}^n: \|z\| < 1\}\) , we find necessary and sufficient conditions for the existence of a free holomorphic function g with complex coefficients on the noncommutative open unit ball \([B({\mathcal H})^n]_1\) such that $${\rm Re} \ g \geq 0 \quad {\rm and} \quad g(z)=f(z),\quad z\in \Lambda,$$ where \(B({\mathcal H})\) is the algebra of all bounded linear operators on a Hilbert space \({\mathcal H}\) . The proof employs several results from noncommutative multivariable operator theory and a noncommutative Cayley transform (introduced and studied in the present paper) acting from the set of all free holomorphic functions with positive real parts to the set of all bounded free holomorphic functions. All the results of this paper are obtained in the more general setting of free holomorphic functions with operator-valued coefficients. As consequences, we deduce some results concerning operator-valued analytic interpolation on the unit ball \({\mathbb B}_n\).

Journal ArticleDOI
TL;DR: In this article, the authors consider the problem of Heegaard splitting up to isotopy, i.e. the amalgamation of two Heegaards, and show that if d(S 1, d(Si) ≥ 2(g(M 1 + g(M 2 ) − g(F 2 )) then M has a unique minimal Heegard splitting.
Abstract: Let M be a compact orientable manifold, and F be an essential closed surface which cuts M into two 3-manifolds M1 and M2. Let \({M_{i}=V_{i}\cup_{S_{i}} W_{i}}\) be a Heegaard splitting for i = 1, 2. We denote by d(Si) the distance of \({V_{i}\cup_{S_{i}} W_{i}}\) . If d(S1), d(S2) ≥ 2(g(M1) + g(M2) − g(F)), then M has a unique minimal Heegaard splitting up to isotopy, i.e. the amalgamation of \({V_{1}\cup_{S_{1}} W_{1}}\) and \({V_{2}\cup_{S_{2}} W_{2}}\) .