Largest initial segments pointwise fixed by automorphisms of models of set theory

Given a model \(\mathcal {M}\) of set theory, and a nontrivial automorphism j of \(\mathcal {M}\), let \(\mathcal {I}_{\mathrm {fix}}(j)\) be the submodel of \(\mathcal {M}\) whose universe consists of elements m of \(\mathcal {M}\) such that \(j(x)=x\) for every x in the transitive closure of m (where the transitive closure of m is computed within \(\mathcal {M}\)). Here we study the class \(\mathcal {C}\) of structures of the form \(\mathcal {I}_{\mathrm {fix}}(j)\), where the ambient model \(\mathcal {M}\) satisfies a frugal yet robust fragment of \(\mathrm {ZFC}\) known as \(\mathrm {MOST}\), and \(j(m)=m\) whenever m is a finite ordinal in the sense of \(\mathcal {M}.\) Our main achievement is the calculation of the theory of \(\mathcal {C}\) as precisely \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\). The following theorems encapsulate our principal results: Theorem A. Every structure in \(\mathcal {C}\) satisfies \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm { Collection}\). Theorem B. Each of the following three conditions is sufficient for a countable structure \(\mathcal {N}\) to be in \(\mathcal {C}\):(a) \(\mathcal {N}\) is a transitive model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).(b) \(\mathcal {N}\) is a recursively saturated model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).(c) \(\mathcal {N}\) is a model of \(\mathrm {ZFC}\). Theorem C. Suppose \(\mathcal {M}\) is a countable recursively saturated model of \(\mathrm {ZFC}\) and I is a proper initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is closed under exponentiation and contains \(\omega ^\mathcal {M}\) . There is a group embedding \(j\longmapsto \check{j}\) from \(\mathrm {Aut}(\mathbb {Q})\) into \(\mathrm {Aut}(\mathcal {M})\) such that I is the longest initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is pointwise fixed by \(\check{j}\) for every nontrivial \(j\in \mathrm {Aut}(\mathbb {Q}).\) In Theorem C, \(\mathrm {Aut}(X)\) is the group of automorphisms of the structure X, and \(\mathbb {Q}\) is the ordered set of rationals.     

On Standard Derived Equivalences of Orbit Categories

Let k be a commutative ring, \(\mathcal {A}\) and \(\mathcal {B}\) – two k-linear categories with an action of a group G. We introduce the notion of a standard G-equivalence from \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}\) to \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\), where \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) is the homotopy category of finitely generated projective \(\mathcal {A}\)-complexes. We construct a map from the set of standard G-equivalences to the set of standard equivalences from \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}\) to \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) and a map from the set of standard G-equivalences from \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}\) to \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) to the set of standard equivalences from \(\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {B}/G)\) to \(\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {A}/G)\), where \(\mathcal {A}/G\) denotes the orbit category. We investigate the properties of these maps and apply our results to the case where \(\mathcal {A}=\mathcal {B}=R\) is a Frobenius k-algebra and G is the cyclic group generated by its Nakayama automorphism ν. We apply this technique to obtain the generating set of the derived Picard group of a Frobenius Nakayama algebra over an algebraically closed field.     

<Emphasis Type="Italic">l</Emphasis>-Groups <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">X</Emphasis>) in continuous logic

In the context of continuous logic, this paper axiomatizes both the class \(\mathcal {C}\) of lattice-ordered groups isomorphic to C(X) for X compact and the subclass \(\mathcal {C}^+\) of structures existentially closed in \(\mathcal {C}\); shows that the theory of \(\mathcal {C}^+\) is \(\aleph _0\)-categorical and admits elimination of quantifiers; establishes a Nullstellensatz for \(\mathcal {C}\) and \(\mathcal {C}^+\); shows that \(C(X)\in \mathcal {C}\) has a prime-model extension in \(\mathcal {C}^+\) just in case X is Boolean; and proves that in a sense relevant to continuous logic, positive formulas admit in \(\mathcal {C}^+\) elimination of quantifiers to positive formulas.     

Isometric multipliers of a vector valued Beurling algebra on a discrete semigroup

Let (S,ω) be a weighted abelian semigroup, let M _ω (S) be the semigroup of ω-bounded multipliers of S, and let \(\mathcal {A}\) be a strictly convex commutative Banach algebra with identity. It is shown that T is an onto isometric multiplier of \(\ell ^{1}(S,\omega , \mathcal {A})\) if and only if there exists an invertible σ ∈ M _ω (S), a unitary point \(a \in \mathcal {A}\), and a k>0 such that \(T(f)= ka{\sum }_{x \in S} f(x)\delta _{\sigma (x)}\) for each \(f={\sum }_{x \in S}f(x)\delta _{x} \in \ell ^{1}(S,\omega ,\mathcal {A})\). It is also shown that an isomorphism from \(\ell ^{1}(S_{1},\omega _{1},\mathcal {A})\) onto \(\ell ^{1}(S_{2},\omega _{2}, \mathcal {B})\) induces an isomorphism from \(M(\ell ^{1}(S_{1},\omega _{1},\mathcal {A}))\), the set of all multipliers of \(\ell ^{1}(S_{1},\omega _{1},\mathcal {A})\), onto \(M(\ell ^{1}(S_{2},\omega _{2},\mathcal {B}))\).     

Geometry of Coadjoint Orbits and Multiplicity-one Branching Laws for Symmetric Pairs

Consider the restriction of an irreducible unitary representation π of a Lie group G to its subgroup H. Kirillov’s revolutionary idea on the orbit method suggests that the multiplicity of an irreducible H-module ν occurring in the restriction π|_H could be read from the coadjoint action of H on \(\mathcal {O}^{G} \cap \text {pr}^{-1}({\mathcal {O}}^{H})\), provided π and ν are ‘geometric quantizations’ of a G-coadjoint orbit \(\mathcal {O}^{G}\) and an H-coadjoint orbit \(\mathcal {O}^{H}\), respectively, where \(\text {pr} \colon \sqrt {-1}\mathfrak {g}^{\ast } \to \sqrt {-1}\mathfrak {h}^{\ast }\) is the projection dual to the inclusion \(\mathfrak {h} \subset \mathfrak {g}\) of Lie algebras. Such results were previously established by Kirillov, Corwin and Greenleaf for nilpotent Lie groups. In this article, we highlight specific elliptic orbits \(\mathcal {O}^{G}\) of a semisimple Lie group G corresponding to highest weight modules of scalar type. We prove that the Corwin–Greenleaf number \(\sharp (\mathcal {O}^{G} \cap \text {pr}^{-1}({\mathcal {O}}^{H}))/H\) is either zero or one for any H-coadjoint orbit \(\mathcal {O}^{H}\), whenever (G,H) is a symmetric pair of holomorphic type. Furthermore, we determine the coadjoint orbits \(\mathcal {O}^{H}\) with nonzero Corwin–Greenleaf number. Our results coincide with the prediction of the orbit philosophy, and can be seen as ‘classical limits’ of the multiplicity-free branching laws of holomorphic discrete series representations (Kobayashi [Progr. Math. 2007]).     

Inflation of Finite Lattices Along All-or-nothing Sets

We introduce a new generalization of Alan Day’s doubling construction. For ordered sets \(\mathcal {L}\) and \(\mathcal {K}\) and a subset \(E \subseteq \ \leq _{\mathcal {L}}\) we define the ordered set \(\mathcal {L} \star _{E} \mathcal {K}\) arising from inflation of \(\mathcal {L}\) along E by \(\mathcal {K}\). Under the restriction that \(\mathcal {L}\) and \(\mathcal {K}\) are finite lattices, we find those subsets \(E \subseteq \ \leq _{\mathcal {L}}\) such that the ordered set \(\mathcal {L} \star _{E} \mathcal {K}\) is a lattice. Finite lattices that can be constructed in this way are classified in terms of their congruence lattices.A finite lattice is binary cut-through codable if and only if there exists a 0?1 spanning chain \(\left \{\theta _{i}\colon 0 \leq i \leq n \right \}\) in \(Con(\mathcal {L})\) such that the cardinality of the largest block of ?? _i /?? _i?1 is 2 for every i with 1≤i≤n. These are exactly the lattices that can be constructed by inflation from the 1-element lattice using only the 2-element lattice. We investigate the structure of binary cut-through codable lattices and describe an infinite class of lattices that generate binary cut-through codable varieties.     

Homomorphisms of infinitely generated analytic sheaves

We prove that every homomorphism \(\mathcal{O}^{E}_{\zeta}\rightarrow\mathcal{O}^{F}_{\zeta}\), with E and F Banach spaces and ζ∈? ^m , is induced by a \(\mathop{\mathrm{Hom}}(E,F)\)-valued holomorphic germ, provided that 1≤m<∞. A similar structure theorem is obtained for the homomorphisms of type \(\mathcal{O}^{E}_{\zeta}\rightarrow\mathcal{S}_{\zeta}\), where \(\mathcal{S}_{\zeta}\) is a stalk of a coherent sheaf of positive depth. We later extend these results to sheaf homomorphisms, obtaining a condition on coherent sheaves which guarantees the sheaf to be equipped with a unique analytic structure in the sense of Lempert–Patyi.     

A spectral method for nonlinear elliptic equations

Let Ω be an open, simply connected, and bounded region in \(\mathbb {R}^{d}\), d ≥ 2, and assume its boundary ?Ω is smooth and homeomorphic to \(\mathbb {S}^{d-1}\). Consider solving an elliptic partial differential equation L u = f(?, u) over Ω with zero Dirichlet boundary value. The function f is a nonlinear function of the solution u. The problem is converted to an equivalent elliptic problem over the open unit ball \(\mathbb {B}^{d}\) in \(\mathbb {R}^{d}\), say \(\widetilde {L}\widetilde {u} =\widetilde {f}(\cdot ,\widetilde {u})\). Then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials \(\widetilde {u} _{n}\) of degree ≤ n that is convergent to \(\widetilde {u}\). The transformation from Ω to \(\mathbb {B}^{d}\) requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For \(u\in C^{\infty } \left (\overline {\Omega }\right ) \) and assuming ?Ω is a C ^∞ boundary, the convergence of \(\left \Vert \widetilde {u} -\widetilde {u}_{n}\right \Vert _{H^{1}}\) to zero is faster than any power of 1/n. The error analysis uses a reformulation of the boundary value problem as an integral equation, and then it uses tools from nonlinear integral equations to analyze the numerical method. Numerical examples illustrate experimentally an exponential rate of convergence. A generalization to ?Δu + γ u = f(u) with a zero Neumann boundary condition is also presented.     

Frame completions with prescribed norms: local minimizers and applications

Let \(\mathcal {F}_{0}=\{f_{i}\}_{i\in \mathbb {I}_{n_{0}}}\) be a finite sequence of vectors in \(\mathbb {C}^{d}\) and let \(\mathbf {a}=(a_{i})_{i\in \mathbb {I}_{k}}\) be a finite sequence of positive numbers, where \(\mathbb {I}_{n}=\{1,\ldots , n\}\) for \(n\in \mathbb {N}\). We consider the completions of \(\mathcal {F}_{0}\) of the form \(\mathcal {F}=(\mathcal {F}_{0},\mathcal {G})\) obtained by appending a sequence \(\mathcal {G}=\{g_{i}\}_{i\in \mathbb {I}_{k}}\) of vectors in \(\mathbb {C}^{d}\) such that ∥g _i ∥² = a _i for \(i\in \mathbb {I}_{k}\), and endow the set of completions with the metric \(d(\mathcal {F},\tilde {\mathcal {F}}) =\max \{ \,\|g_{i}-\tilde {g}_{i}\|: \ i\in \mathbb {I}_{k}\}\) where \(\tilde {\mathcal {F}}=(\mathcal {F}_{0},\,\tilde {\mathcal {G}})\). In this context we show that local minimizers on the set of completions of a convex potential P _φ , induced by a strictly convex function φ, are also global minimizers. In case that φ(x) = x ² then P _φ is the so-called frame potential introduced by Benedetto and Fickus, and our work generalizes several well known results for this potential. We show that there is an intimate connection between frame completion problems with prescribed norms and frame operator distance (FOD) problems. We use this connection and our results to settle in the affirmative a generalized version of Strawn’s conjecture on the FOD.     

Symmetric Pairs and Self-Adjoint Extensions of Operators,with Applications to Energy Networks

We provide a streamlined construction of the Friedrichs extension of a densely-defined self-adjoint and semibounded operator A on a Hilbert space \(\mathcal {H}\), by means of a symmetric pair of operators. A symmetric pair is comprised of densely defined operators \(J: \mathcal {H}_1 \rightarrow \mathcal {H}_2\) and \(K: \mathcal {H}_2 \rightarrow \mathcal {H}_1\) which are compatible in a certain sense. With the appropriate definitions of \(\mathcal {H}_1\) and J in terms of A and \(\mathcal {H}\), we show that \((\textit{JJ}^\star )^{-1}\) is the Friedrichs extension of A. Furthermore, we use related ideas (including the notion of unbounded containment) to construct a generalization of the construction of the Krein extension of A as laid out in a previous paper of the authors. These results are applied to the study of the graph Laplacian on infinite networks, in relation to the Hilbert spaces \(\ell ^2(G)\) and \(\mathcal {H}_{\mathcal {E}}\) (the energy space).     

Equilibrium Diffusion on the Cone of Discrete Radon Measures

Let \({\mathbb {K}(\mathbb {R}^{d})}\) denote the cone of discrete Radon measures on \(\mathbb {R}^{d}\). There is a natural differentiation on \(\mathbb {K}(\mathbb {R}^{d})\): for a differentiable function \(F:\mathbb {K}(\mathbb {R}^{d})\to \mathbb {R}\), one defines its gradient \(\nabla ^{\mathbb {K}}F\) as a vector field which assigns to each \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) an element of a tangent space \(T_{\eta }(\mathbb {K}(\mathbb {R}^{d}))\) to \(\mathbb {K}(\mathbb {R}^{d})\) at point η. Let \(\phi :\mathbb {R}^{d}\times \mathbb {R}^{d}\to \mathbb {R}\) be a potential of pair interaction, and let μ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on \(\mathbb {R}^{d}\). In particular, μ is a probability measure on \(\mathbb {K}(\mathbb {R}^{d})\) such that the set of atoms of a discrete measure \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) is μ-a.s. dense in \(\mathbb {R}^{d}\). We consider the corresponding Dirichlet form

$$\mathcal{E}^{\mathbb{K}}(F,G)={\int}_{\mathbb K(\mathbb{R}^{d})}\langle\nabla^{\mathbb{K}} F(\eta), \nabla^{\mathbb{K}} G(\eta)\rangle_{T_{\eta}(\mathbb{K})}\,d\mu(\eta). $$

Integrating by parts with respect to the measure μ, we explicitly find the generator of this Dirichlet form. By using the theory of Dirichlet forms, we prove the main result of the paper: If d ≥ 2, there exists a conservative diffusion process on \(\mathbb {K}(\mathbb {R}^{d})\) which is properly associated with the Dirichlet form \(\mathcal {E}^{\mathbb {K}}\).

     

Bavrin’s Type Factorization of the Temljakov Operator for Holomorphic Functions in Circular Domains of $$\mathbb {C}^{n}$$

The paper concerns investigations of holomorphic functions of several complex variables with a factorization of their Temljakov transform. Firstly, there were considered some inclusions between the families \(\mathcal {C}_{\mathcal {G}},\mathcal {M}_{\mathcal {G}},\mathcal {N}_{\mathcal {G}},\mathcal {R}_{\mathcal {G}},\mathcal {V}_{\mathcal {G}}\) of such holomorphic functions on complete n-circular domain \(\mathcal {G}\) of \(\mathbb {C}^{n}\) in some papers of Bavrin, Fukui, Higuchi, Michiwaki. A motivation of our investigations is a condensation of the mentioned inclusions by some new families of Bavrin’s type. Hence we consider some families \(\mathcal {K}_{ \mathcal {G}}^{k},k\ge 2,\) of holomorphic functions f :  \(\mathcal {G}\rightarrow \mathbb {C},f(0)=1,\) defined also by a factorization of \( \mathcal {L}f\) onto factors from \(\mathcal {C}_{\mathcal {G}}\) and \(\mathcal {M} _{\mathcal {G}}.\) We present some interesting properties and extremal problems on \(\mathcal {K}_{\mathcal {G}}^{k}\).     

On the horizontal diameter of the unit sphere

For a singular Riemannian foliation \(\mathcal {F}\) on a Riemannian manifold M, a curve is called horizontal if it meets the leaves of \(\mathcal {F}\) perpendicularly. For a singular Riemannian foliation \(\mathcal {F}\) on a unit sphere \(\mathbb {S}^{n}\), we show that if \(\mathcal {F}\) satisfies some properties, then the horizontal diameter of \(\mathbb {S}^{n}\) is \(\pi \), i.e., any two points in \(\mathbb {S}^{n}\) can be connected by a horizontal curve of length \(\le \pi \).     

The Representation Type of Determinantal Varieties

This work is entirely devoted to construct huge families of indecomposable arithmetically Cohen-Macaulay (resp. Ulrich) sheaves \(\mathcal {E}\) of arbitrary high rank on a general standard (resp. linear) determinantal scheme \(X\subset \mathbb {P}^{n}\) of codimension c ≥ 1, n ? c ≥ 1 and defined by the maximal minors of a t × (t + c?1) homogeneous matrix \(\mathcal {A}\). The sheaves \(\mathcal {E}\) are constructed as iterated extensions of sheaves of lower rank. As applications: (1) we prove that any general standard determinantal scheme \(X\subset \mathbb {P}^{n}\) is of wild representation type provided the degrees of the entries of the matrix \(\mathcal {A}\) satisfy some weak numerical assumptions; and (2) we determine values of t, n and n ? c for which a linear standard determinantal scheme \(X\subset \mathbb {P}^{n}\) is of wild representation type with respect to the much more restrictive category of its indecomposable Ulrich sheaves, i.e. X is of Ulrich wild representation type.     

Resilience of ranks of higher inclusion matrices

Let \(n \ge r \ge s \ge 0\) be integers and \(\mathcal {F}\) a family of r-subsets of [n]. Let \(W_{r,s}^{\mathcal {F}}\) be the higher inclusion matrix of the subsets in \({{\mathcal {F}}}\) vs. the s-subsets of [n]. When \(\mathcal {F}\) consists of all r-subsets of [n], we shall simply write \(W_{r,s}\) in place of \(W_{r,s}^{\mathcal {F}}\). In this paper we prove that the rank of the higher inclusion matrix \(W_{r,s}\) over an arbitrary field K is resilient. That is, if the size of \(\mathcal {F}\) is “close” to \({n \atopwithdelims ()r}\) then \({{\mathrm{rank}}}_{K}( W_{r,s}^{\mathcal {F}}) = {{\mathrm{rank}}}_{K}(W_{r,s})\), where K is an arbitrary field. Furthermore, we prove that the rank (over a field K) of the higher inclusion matrix of r-subspaces vs. s-subspaces of an n-dimensional vector space over \({\mathbb {F}}_q\) is also resilient if \(\mathrm{char}(K)\) is coprime to q.     

Reproducing Kernel for the Herglotz Functions in $$\mathbb {R}^n$$ and Solutions of the Helmholtz Equation

The purpose of this article is to extend to \(\mathbb {R}^{n}\) known results in dimension 2 concerning the structure of a Hilbert space with reproducing kernel of the space of Herglotz wave functions. These functions are the solutions of Helmholtz equation in \(\mathbb {R} ^{n}\) that are the Fourier transform of measures supported in the unit sphere with density in \(L^{2}(\mathbb {S}^{n-1})\). As a natural extension of this, we define Banach spaces of solutions of the Helmholtz equation in \(\mathbb {R}^{n}\) belonging to weighted Sobolev type spaces \(\mathcal {H}^{p}\) having in a non local norm that involves radial derivatives and spherical gradients. We calculate the reproducing kernel of the Herglotz wave functions and study in \(\mathcal {H}^{p}\) and in mixed norm spaces, the continuity of the orthogonal projection \(\mathcal {P}\) of \(\mathcal {H}^{2}\) onto the Herglotz wave functions.     

Numerical semigroups whose fractions are of maximal embedding dimension

Each saturated (resp., Arf) numerical semigroup S has the property that each of its fractions \(\frac{S}{k}\) is saturated (resp., Arf), but the property of being of maximal embedding dimension (MED) is not stable under formation of fractions. If S is a numerical semigroup, then S is MED (resp., Arf; resp., saturated) if and only if, for each 2≤k∈?, \(S = \frac{T}{k}\) for infinitely many MED (resp., Arf; resp., saturated) numerical semigroups T. Let \(\mathcal{A}\) (resp., \(\mathcal{F}\)) be the class of Arf numerical semigroups (resp., of numerical semigroups each of whose fractions is of maximal embedding dimension). Then there exists an infinite strictly ascending chain \(\mathcal{A} =\mathcal{C}_{1} \subset\mathcal{C}_{2} \subset\mathcal{C}_{3}\subset \,\cdots\, \subset\mathcal{F}\), where, like \(\mathcal{A}\) and \(\mathcal{F}\), each \(\mathcal{C}_{n}\) is stable under the formation of fractions.     

Recurrent Dirichlet Forms and Markov Property of Associated Gaussian Fields

For the extended Dirichlet space \(\mathcal {F}_{e}\) of a general irreducible recurrent regular Dirichlet form \((\mathcal {E},\mathcal {F})\) on L ²(E;m), we consider the family \(\mathbb {G}(\mathcal {E})=\{X_{u};u\in \mathcal {F}_{e}\}\) of centered Gaussian random variables defined on a probability space \(({\Omega }, \mathcal {B}, \mathbb {P})\) indexed by the elements of \(\mathcal {F}_{e}\) and possessing the Dirichlet form \(\mathcal {E}\) as its covariance. We formulate the Markov property of the Gaussian field \(\mathbb {G}(\mathcal {E})\) by associating with each set A ? E the sub-σ-field σ(A) of \(\mathcal {B}\) generated by X _u for every \(u\in \mathcal {F}_{e}\) whose spectrum s(u) is contained in A. Under a mild absolute continuity condition on the transition function of the Hunt process associated with \((\mathcal {E}, \mathcal {F})\), we prove the equivalence of the Markov property of \(\mathbb {G}(\mathcal {E})\) and the local property of \((\mathcal {E},\mathcal {F})\). One of the key ingredients in the proof is in that we construct potentials of finite signed measures of zero total mass and show that, for any Borel set B with m(B) >?0, any function \(u\in \mathcal {F}_{e}\) with s(u) ? B can be approximated by a sequence of potentials of measures supported by B.     

Efficient construction of 2-chains representing a basis of $H_{2}(\overline {\Omega }, \partial {\Omega }; \mathbb {Z})$

We present an efficient algorithm for the construction of a basis of \(H_{2}(\overline {\Omega },\partial {\Omega };\mathbb {Z})\) via the Poincaré-Lefschetz duality theorem. Denoting by g the first Betti number of \(\overline {\Omega }\) the idea is to find, first g different 1-boundaries of \(\overline {\Omega }\) with supports contained in ?Ω whose homology classes in \(\mathbb {R}^{3} \setminus {\Omega }\) form a basis of \(H_{1}(\mathbb {R}^{3} \setminus {\Omega };\mathbb {Z})\), and then to construct a set of 2-chains in \(\overline {\Omega }\) having these 1-boundaries as their boundaries. The Poincaré-Lefschetz duality theorem ensures that the relative homology classes of these 2-chains in \(\overline {\Omega }\) modulo ?Ω form a basis of \(H_{2}(\overline {\Omega },\partial {\Omega };\mathbb {Z})\). We devise a simple procedure for the construction of the required set of 1-boundaries of \(\overline {\Omega }\) that, combined with a fast algorithm for the construction of 2-chains with prescribed boundary, allows the efficient computation of a basis of \(H_{2}(\overline {\Omega },\partial {\Omega };\mathbb {Z})\) via this very natural approach. Some numerical experiments show the efficiency of the method and its performance comparing with other algorithms.     

Liouville Theorems for a General Class of Nonlocal Operators

In this paper, we study the equation \(\mathcal {L} u=0\) in \(\mathbb {R}^{N}\), where \(\mathcal {L}\) belongs to a general class of nonlocal linear operators which may be anisotropic and nonsymmetric. We classify distributional solutions of this equation, thereby extending and generalizing recent Liouville type theorems in the case where \(\mathcal {L}= (-{\Delta })^{s}\), s ∈ (0, 1) is the classical fractional Laplacian.