Hall Polynomials and Composition Algebra of Representation Finite Algebras

Let \(\mathcal{A}\) be a representation finite algebra over finite field k such that the indecomposable \(\mathcal{A}\) -modules are determined by their dimension vectors and for each \(M, L \in ind(\mathcal{A})\) and \(N\in mod(\mathcal{A})\) , either \(F^{M}_{N L}=0\) or \(F^{M}_{L N}=0\) . We show that \(\mathcal{A}\) has Hall polynomials and the rational extension of its Ringel–Hall algebra equals the rational extension of its composition algebra. This result extend and unify some known results about Hall polynomials. As a consequence we show that if \(\mathcal{A}\) is a representation finite simply-connected algebra, or finite dimensional k-algebra such that there are no short cycles in \(mod(\mathcal{A})\) , or representation finite cluster tilted algebra, then \(\mathcal{A}\) has Hall polynomials and \(\mathcal{H}(\mathcal{A})\otimes_\mathbb{Z}Q=\mathcal{C}(\mathcal{A})\otimes_\mathbb{Z}Q\) .     

Erdős-Ginzburg-Ziv theorem for finite commutative semigroups

Let \(\mathcal{S}\) be a finite additively written commutative semigroup, and let \(\exp(\mathcal{S})\) be its exponent which is defined as the least common multiple of all periods of the elements in \(\mathcal{S}\) . For every sequence T of elements in \(\mathcal{S}\) (repetition allowed), let \(\sigma(T) \in\mathcal{S}\) denote the sum of all terms of T. Define the Davenport constant \(\mathsf{D}(\mathcal{S})\) of \(\mathcal{S}\) to be the least positive integer d such that every sequence T over \(\mathcal{S}\) of length at least d contains a proper subsequence T′ with σ(T′)=σ(T), and define \(\mathsf{E}(\mathcal{S})\) to be the least positive integer ? such that every sequence T over \(\mathcal{S}\) of length at least ? contains a subsequence T′ with \(|T|-|T'|= \lceil\frac{|\mathcal{S}|}{\exp(\mathcal{S})} \rceil \exp(\mathcal{S})\) and σ(T′)=σ(T). When \(\mathcal{S}\) is a finite abelian group, it is well known that \(\lceil\frac{|\mathcal{S}|}{\exp(\mathcal{S})} \rceil\exp (\mathcal{S})=|\mathcal{S}|\) and \(\mathsf{E}(\mathcal{S})=\mathsf{D}(\mathcal{S})+|\mathcal{S}|-1\) . In this paper we investigate whether \(\mathsf{E}(\mathcal{S})\leq \mathsf{D}(\mathcal{S})+ \lceil\frac{|\mathcal{S}|}{\exp(\mathcal {S})} \rceil \exp(\mathcal{S})-1\) holds true for all finite commutative semigroups \(\mathcal{S}\) . We provide a positive answer to the question above for some classes of finite commutative semigroups, including group-free semigroups, elementary semigroups, and archimedean semigroups with certain constraints.     

From submodule categories to preprojective algebras

Let \(S(n)\) be the category of invariant subspaces of nilpotent operators with nilpotency index at most \(n\) . Such submodule categories have been studied already in 1934 by Birkhoff, they have attracted a lot of attention in recent years, for example in connection with some weighted projective lines (Kussin, Lenzing, Meltzer). On the other hand, we consider the preprojective algebra \(\Pi _n\) of type \(\mathbb {A}_n\) ; the preprojective algebras were introduced by Gelfand and Ponomarev, they are now of great interest, for example they form an important tool to study quantum groups (Lusztig) or cluster algebras (Geiss, Leclerc, Schröer). We are going to discuss the connection between the submodule category \(\mathcal {S}(n)\) and the module category \(\hbox {mod}\;\Pi _{n-1}\) of the preprojective algebra \(\Pi _{n-1}\) . Dense functors \(\mathcal {S}(n) \rightarrow \hbox {mod}\;\Pi _{n-1}\) are known to exist: one has been constructed quite a long time ago by Auslander and Reiten, recently another one by Li and Zhang. We will show that these two functors are full, dense, objective functors with index \(2n\) , thus \(\hbox {mod}\;\Pi _{n-1}\) is obtained from \(\mathcal {S}(n)\) by factoring out an ideal which is generated by \(2n\) indecomposable objects. As a byproduct we also obtain new examples of ideals in triangulated categories, namely ideals \(\mathcal {I}\) in a triangulated category \(\mathcal {T}\) which are generated by an idempotent such that the factor category \(\mathcal {T}/\mathcal {I}\) is an abelian category.     

Non-Lipschitz points and the {\textit{SBV}} regularity of the minimum time function

This paper is devoted to the study of the Hausdorff dimension of the singular set of the minimum time function \(T\) under controllability conditions which do not imply the Lipschitz continuity of \(T\) . We consider first the case of normal linear control systems with constant coefficients in \({\mathbb {R}}^N\) . We characterize points around which \(T\) is not Lipschitz as those which can be reached from the origin by an optimal trajectory (of the reversed dynamics) with vanishing minimized Hamiltonian. Linearity permits an explicit representation of such set, that we call \(\mathcal {S}\) . Furthermore, we show that \(\mathcal {S}\) is countably \(\mathcal {H}^{N-1}\) -rectifiable with positive \(\mathcal {H}^{N-1}\) -measure. Second, we consider a class of control-affine planar nonlinear systems satisfying a second order controllability condition: we characterize the set \(\mathcal {S}\) in a neighborhood of the origin in a similar way and prove the \(\mathcal {H}^1\) -rectifiability of \(\mathcal {S}\) and that \(\mathcal {H}^1(\mathcal {S})>0\) . In both cases, \(T\) is known to have epigraph with positive reach, hence to be a locally \(BV\) function (see Colombo et al.: SIAM J Control Optim 44:2285–2299, 2006; Colombo and Nguyen.: Math Control Relat 3: 51–82, 2013). Since the Cantor part of \(DT\) must be concentrated in \(\mathcal {S}\) , our analysis yields that \(T\) is locally \(SBV\) , i.e., the Cantor part of \(DT\) vanishes. Our results imply also that \(T\) is differentiable outside a \(\mathcal {H}^{N-1}\) -rectifiable set. With small changes, our results are valid also in the case of multiple control input.     

Strongly amenable involutive representations of involutive Banach algebras

Let \(\mathfrak{A }\) be a Banach \(*\) -algebra and let \(\varphi \) be a nonzero self-adjoint character on \(\mathfrak{A }\) . For a   \(*\) -representation \(\pi \) of \(\mathfrak{A }\) on a Hilbert space \(\mathcal{H }\) , we introduce and study strong \(\varphi \) -amenability of \(\pi \) in terms of certain states on the von Neumann algebra of bounded operators on \(\mathcal{H }\) . We then give some characterizations of this notion in terms of certain positive functionals on \(\mathfrak{A }\) . We finally investigate some hereditary properties of strong \(\varphi \) -amenability of Banach algebras.     

Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space

Let \(M\) and \(N\) be two connected smooth manifolds, where \(M\) is compact and oriented and \(N\) is Riemannian. Let \(\mathcal {E}\) be the Fréchet manifold of all embeddings of \(M\) in \(N\) , endowed with the canonical weak Riemannian metric. Let \(\sim \) be the equivalence relation on \(\mathcal {E}\) defined by \(f\sim g\) if and only if \(f=g\circ \phi \) for some orientation preserving diffeomorphism \(\phi \) of \(M\) . The Fréchet manifold \(\mathcal {S}= \mathcal {E}/_{\sim }\) of equivalence classes, which may be thought of as the set of submanifolds of \(N\) diffeomorphic to \(M\) and is called the nonlinear Grassmannian (or Chow manifold) of \(N\) of type \(M\) , inherits from \( \mathcal {E}\) a weak Riemannian structure. We consider the following particular case: \(N\) is a compact irreducible symmetric space and \(M\) is a reflective submanifold of \(N\) (that is, a connected component of the set of fixed points of an involutive isometry of \( N\) ). Let \(\mathcal {C}\) be the set of submanifolds of \(N\) which are congruent to \(M\) . We prove that the natural inclusion of \(\mathcal {C}\) in \(\mathcal {S}\) is totally geodesic.     

The Lattice of Subvarieties of Semilattice Ordered Algebras

This paper is devoted to the semilattice ordered \(\mathcal{V}\) -algebras of the form (A, Ω, +?), where + is a join-semilattice operation and (A, Ω) is an algebra from some given variety \(\mathcal{V}\) . We characterize the free semilattice ordered algebras using the concept of extended power algebras. Next we apply the result to describe the lattice of subvarieties of the variety of semilattice ordered \(\mathcal{V}\) -algebras in relation to the lattice of subvarieties of the variety \(\mathcal{V}\) .     

Neighbour-transitive codes in Johnson graphs

The Johnson graph \(J(v,k)\) has, as vertices, the \(k\) -subsets of a \(v\) -set \(\mathcal {V}\) and as edges the pairs of \(k\) -subsets with intersection of size \(k-1\) . We introduce the notion of a neighbour-transitive code in \(J(v,k)\) . This is a proper vertex subset \(\Gamma \) such that the subgroup \(G\) of graph automorphisms leaving \(\Gamma \) invariant is transitive on both the set \(\Gamma \) of ‘codewords’ and also the set of ‘neighbours’ of \(\Gamma \) , which are the non-codewords joined by an edge to some codeword. We classify all examples where the group \(G\) is a subgroup of the symmetric group \(\mathrm{Sym}\,(\mathcal {V})\) and is intransitive or imprimitive on the underlying \(v\) -set \(\mathcal {V}\) . In the remaining case where \(G\le \mathrm{Sym}\,(\mathcal {V})\) and \(G\) is primitive on \(\mathcal {V}\) , we prove that, provided distinct codewords are at distance at least \(3\) , then \(G\) is \(2\) -transitive on \(\mathcal {V}\) . We examine many of the infinite families of finite \(2\) -transitive permutation groups and construct surprisingly rich families of examples of neighbour-transitive codes. A major unresolved case remains.     

On the group sheaf of \mathcal{A}-symplectomorphisms

This is a part of a further undertaking to affirm that most of classical module theory may be retrieved in the framework of Abstract Differential Geometry (à la Mallios). More precisely, within this article, we study some defining basic concepts of symplectic geometry on free \(\mathcal{A}\) -modules by focussing in particular on the group sheaf of \(\mathcal{A}\) -symplectomorphisms, where \(\mathcal{A}\) is assumed to be a torsion-free PID ?-algebra sheaf. The main result arising hereby is that \(\mathcal{A}\) -symplectomorphisms locally are products of symplectic transvections, which is a particularly well-behaved counterpart of the classical result.     

Characterization for Fock-Type Space via Higher Order Derivatives and its Application

The space of entire functions \(\mathcal {F}_{\alpha }^{\infty }\) is mentioned in the paper of S. Janson, J. Peetre, and R. Rochberg. In this paper, we establish a characterization for the space \(\mathcal {F}_{\alpha }^{\infty }\) by \(n\) -th derivatives of entire functions. As an application of this result, we study the boundedness of Li-Stevi?’s integral operators and estimate essential norms of these operators acting on \(\mathcal {F}_{\alpha }^{\infty }\) . Furthermore we describe complete characterizations for boundedness and compactness of the Volterra-type integral operators on \(\mathcal {F}_{\alpha }^{\infty }\) .     

\mathcal{D}-modules with finite support are semi-simple

Let \((R, \frak{m}, k_{R})\) be a regular local k-algebra satisfying the weak Jacobian criterion, and such that k _R /k is an algebraic field extension. Let \(\mathcal{D}_{R}\) be the ring of k-linear differential operators of R. We give an explicit decomposition of the \(\mathcal{D}_{R}\) -module \(\mathcal{D}_{R}/\mathcal{D}_{R} \frak{m}_{R}^{n+1}\) as a direct sum of simple modules, all isomorphic to \(\mathcal{D}_{R}/\mathcal{D}_{R} \frak{m}\) , where certain “Pochhammer” differential operators are used to describe generators of the simple components.     

Measure of Self-Affine Sets and Associated Densities

Let \(B\) be an \(n\times n\) real expanding matrix and \(\mathcal {D}\) be a finite subset of \(\mathbb {R}^n\) with \(0\in \mathcal {D}\) . The self-affine set \(K=K(B,\mathcal {D})\) is the unique compact set satisfying the set-valued equation \(BK=\bigcup _{d\in \mathcal {D}}(K+d)\) . In the case where \(\#\mathcal D=|\det B|,\) we relate the Lebesgue measure of \(K(B,\mathcal {D})\) to the upper Beurling density of the associated measure \(\mu =\lim _{s\rightarrow \infty }\sum _{\ell _0, \ldots ,\ell _{s-1}\in \mathcal {D}}\delta _{\ell _0+B\ell _1+\cdots +B^{s-1}\ell _{s-1}}.\) If, on the other hand, \(\#\mathcal D<|\det B|\) and \(B\) is a similarity matrix, we relate the Hausdorff measure \(\mathcal {H}^s(K)\) , where \(s\) is the similarity dimension of \(K\) , to a corresponding notion of upper density for the measure \(\mu \) .     

Pro-\mathcal {C} completions of orientable \text{ PD }^3-pairs

Let \({\mathcal {C}}\) be a class of finite groups. We study some sufficient conditions for the pro- \({\mathcal {C}}\) completion of an orientable \(\text{ PD }^3\) -pair over \(\mathbb {Z}\) to be an orientable profinite \(\text{ PD }^3\) -pair over \(\mathbb {F}_p\) . More results are proven for the pro- \(p\) completion of \(\text{ PD }^3\) -pairs.     

Approximate Unitary Equivalence to Skew Symmetric Operators

An operator \(T\) on a complex Hilbert space \(\mathcal {H}\) is called skew symmetric if \(T\) can be represented as a skew symmetric matrix relative to some orthonormal basis for \(\mathcal {H}\) . In this paper, we study the approximation of skew symmetric operators and provide a \(C^*\) -algebra approach to skew symmetric operators. We classify up to approximate unitary equivalence those skew symmetric operators \(T\in \mathcal {B(H)}\) satisfying \(C^*(T)\cap \mathcal {K(H)}=\{0\}\) . This is used to characterize when a unilateral weighted shift with nonzero weights is approximately unitarily equivalent to a skew symmetric operator.     

Triple-consistent social choice and the majority rule

We define generalized (preference) domains \(\mathcal{D}\) as subsets of the hypercube {?1,1} ^D , where each of the D coordinates relates to a yes-no issue. Given a finite set of n individuals, a profile assigns each individual to an element of \(\mathcal{D}\) . We prove that, for any domain \(\mathcal{D}\) , the outcome of issue-wise majority voting φ _m belongs to \(\mathcal{D}\) at any profile where φ _m is well-defined if and only if this is true when φ _m is applied to any profile involving only 3 elements of \(\mathcal{D}\) . We call this property triple-consistency. We characterize the class of anonymous issue-wise voting rules that are triple-consistent, and give several interpretations of the result, each being related to a specific collective choice problem.     

Descent in Locally Presentable Categories

Let \(\mathbb {V}=(VV, \otimes , I)\) be a symmetric monoidal category such that \(\mathcal {V}\) is locally presentable and that all functors \(V\otimes - : \mathcal {V} \rightarrow \mathcal {V}\) for \(V \in \mathcal {V}\) preserve reflexive coequalizers and directed colimits. It is proved that any pure morphism of commutative ??-monoids is an effective descent morphism with respect to the indexed category given by commutative ??-monoids and modules over them. As a by-product, we prove that pure morphisms in a locally presentable category are effective for codescent.     

\mathcal {A}_{p, {\mathbb {E}}} Weights,Maximal Operators,and Hardy Spaces Associated with a Family of General Sets

Suppose that \({\mathbb {E}}:=\{E_r(x)\}_{r\in {\mathcal {I}}, x\in X}\) is a family of open subsets of a topological space \(X\) endowed with a nonnegative Borel measure \(\mu \) satisfying certain basic conditions. We establish an \(\mathcal {A}_{{\mathbb {E}}, p}\) weights theory with respect to \({\mathbb {E}}\) and get the characterization of weighted weak type (1,1) and strong type \((p,p)\) , \(1<p\le \infty \) , for the maximal operator \({\mathcal {M}}_{{\mathbb {E}}}\) associated with \({\mathbb {E}}\) . As applications, we introduce the weighted atomic Hardy space \(H^1_{{\mathbb {E}}, w}\) and its dual \(BMO_{{\mathbb {E}},w}\) , and give a maximal function characterization of \(H^1_{{\mathbb {E}},w}\) . Our results generalize several well-known results.     

Divide and Conquer Roadmap for Algebraic Sets

Let \(\mathrm{R}\) be a real closed field and \(\hbox {D}\subset \mathrm{R}\) an ordered domain. We describe an algorithm that given as input a polynomial \(P \in \hbox {D}[ X_{1} , \ldots ,X_{{ k}} ]\) and a finite set, \(\mathcal {A}= \{ p_{1} , \ldots ,p_{m} \}\) , of points contained in \(V= {\mathrm{{Zer}}} ( P, \mathrm{R}^{{ k}})\) described by real univariate representations, computes a roadmap of \(V\) containing \(\mathcal {A}\) . The complexity of the algorithm, measured by the number of arithmetic operations in \(\hbox {D}\) , is bounded by \(\big ( \sum _{i=1}^{m} D^{O ( \log ^{2} ( k ) )}_{i} +1 \big ) ( k^{\log ( k )} d )^{O ( k\log ^{2} ( k ))}\) , where \(d= \deg ( P )\) and \(D_{i}\) is the degree of the real univariate representation describing the point \(p_{i}\) . The best previous algorithm for this problem had complexity card \(( \mathcal {A} )^{O ( 1 )} d^{O ( k^{3/2} )}\) (Basu et al., ArXiv, 2012), where it is assumed that the degrees of the polynomials appearing in the representations of the points in \(\mathcal {A}\) are bounded by \(d^{O ( k )}\) . As an application of our result we prove that for any real algebraic subset \(V\) of \(\mathbb {R}^{k}\) defined by a polynomial of degree \(d\) , any connected component \(C\) of \(V\) contained in the unit ball, and any two points of \(C\) , there exists a semi-algebraic path connecting them in \(C\) , of length at most \(( k ^{\log (k )} d )^{O ( k\log ( k ) )}\) , consisting of at most \(( k ^{\log (k )} d )^{O ( k\log ( k ) )}\) curve segments of degrees bounded by \(( k ^{\log ( k )} d )^{O ( k \log ( k) )}\) . While it was known previously, by a result of D’Acunto and Kurdyka (Bull Lond Math Soc 38(6):951–965, 2006), that there always exists a path of length \(( O ( d ) )^{k-1}\) connecting two such points, there was no upper bound on the complexity of such a path.     

Approximating minimum-cost edge-covers of crossing biset-families

Part of this paper appeared in the preliminary version [16]. An ordered pair ? = (S, S ₊) of subsets of a groundset V is called a biset if S ? S+; (V S ⁺;V S) is the co-biset of ?. Two bisets \(\hat X,\hat Y\) intersect if ^X X ∩ Y ≠ \(\not 0\) and cross if both X ∩ Y \(\not 0\) and X ⁺ ∪ Y ⁺ ≠= V. The intersection and the union of two bisets \(\hat X,\hat Y\) are defined by \(\hat X \cap \hat Y = (X \cap Y,X^ + \cap Y^ + )\) and \(\hat X \cup \hat Y = (X \cup Y,X^ + \cup Y^ + )\) . A biset-family \(\mathcal{F}\) is crossing (intersecting) if \(\hat X \cap \hat Y,\hat X \cup \hat Y \in \mathcal{F}\) for any \(\hat X,\hat Y \in \mathcal{F}\) that cross (intersect). A directed edge covers a biset ? if it goes from S to V S ⁺. We consider the problem of covering a crossing biset-family \(\mathcal{F}\) by a minimum-cost set of directed edges. While for intersecting \(\mathcal{F}\) , a standard primal-dual algorithm computes an optimal solution, the approximability of the case of crossing \(\mathcal{F}\) is not yet understood, as it includes several NP-hard problems, for which a poly-logarithmic approximation was discovered only recently or is not known. Let us say that a biset-family \(\mathcal{F}\) is k-regular if \(\hat X \cap \hat Y,\hat X \cup \hat Y \in \mathcal{F}\) for any \(\hat X,\hat Y \in \mathcal{F}\) with |V (X∪Y)≥k+1 that intersect. In this paper we obtain an O(log |V|)-approximation algorithm for arbitrary crossing \(\mathcal{F}\) if in addition both \(\mathcal{F}\) and the family of co-bisets of \(\mathcal{F}\) are k-regular, our ratios are: \(O\left( {\log \frac{{|V|}} {{|V| - k}}} \right) \) if |S ⁺ \ S| = k for all \(\hat S \in \mathcal{F}\) , and \(O\left( {\frac{{|V|}} {{|V| - k}}\log \frac{{|V|}} {{|V| - k}}} \right) \) if |S ⁺ \ S| = k for all \(\hat S \in \mathcal{F}\) . Using these generic algorithms, we derive for some network design problems the following approximation ratios: \(O\left( {\log k \cdot \log \tfrac{n} {{n - k}}} \right) \) for k-Connected Subgraph, and O(logk) \(\min \{ \tfrac{n} {{n - k}}\log \tfrac{n} {{n - k}},\log k\} \) for Subset k-Connected Subgraph when all edges with positive cost have their endnodes in the subset.     

Sufficient Conditions for Sampling and Interpolation on the Sphere

We obtain sufficient conditions for arrays of points, \(\mathcal {Z}=\{\mathcal {Z}(L) \}_{L\ge 1}\) , on the unit sphere \(\mathcal {Z}(L)\subset \mathbb {S}^d\) , to be Marcinkiewicz–Zygmund and interpolating arrays for spaces of spherical harmonics. The conditions are in terms of the mesh norm and the separation radius of \(\mathcal {Z}(L)\) .