Shellability of the higher pinched Veronese posets

The pinched Veronese poset \({\mathcal {V}}^{\bullet }_n\) is the poset with ground set consisting of all nonnegative integer vectors of length \(n\) such that the sum of their coordinates is divisible by \(n\) with exception of the vector \((1,\ldots ,1)\) . For two vectors \(\mathbf {a}\) and \(\mathbf {b}\) in \({\mathcal {V}}^{\bullet }_n\) , we have \(\mathbf {a}\preceq \mathbf {b}\) if and only if \(\mathbf {b}- \mathbf {a}\) belongs to the ground set of \({\mathcal {V}}^{\bullet }_n\) . We show that every interval in \({\mathcal {V}}^{\bullet }_n\) is shellable for \(n \ge 4\) . In order to obtain the result, we develop a new method for showing that a poset is shellable. This method differs from classical lexicographic shellability. Shellability of intervals in \({\mathcal {V}}^{\bullet }_n\) has consequences in commutative algebra. As a corollary, we obtain a combinatorial proof of the fact that the pinched Veronese ring is Koszul for \(n \ge 4\) . (This also follows from a result by Conca, Herzog, Trung, and Valla.)     

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 \(\mathcal {F}_{\alpha }^{\infty }\) .     

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.     

Classical limit of the tensor product of holomorphic discrete series representations

For three coadjoint orbits \(\mathcal {O}_1, \mathcal {O}_2\) and \(\mathcal {O}_3\) in \(\mathfrak {g}^*\) , the Corwin–Greenleaf function \(n(\mathcal {O}_1 \times \mathcal {O}_2, \mathcal {O}_3)\) is given by the number of \(G\) -orbits in \(\{(\lambda , \mu ) \in \mathcal {O}_1 \times \mathcal {O}_2 \, : \, \lambda + \mu \in \mathcal {O}_3 \}\) under the diagonal action. In the case where \(G\) is a simple Lie group of Hermitian type, we give an explicit formula of \(n(\mathcal {O}_1 \times \mathcal {O}_2, \mathcal {O}_3)\) for coadjoint orbits \(\mathcal {O}_1\) and \(\mathcal {O}_2\) that meet \(\left( [\mathfrak {k}, \mathfrak {k}] + \mathfrak {p}\right) ^{\perp }\) , and show that the formula is regarded as the ‘classical limit’ of a special case of Kobayashi’s multiplicity-free theorem (Progr. Math. 2007) in the branching law to symmetric pairs.     

Distributions and the Global Operator of Slice Monogenic Functions

In this paper we consider functions \(f\) defined on an open set \(U\) of the Euclidean space \(\mathbb{R }^{n+1}\) and with values in the Clifford Algebra \(\mathbb{R }_n\) . Slice monogenic functions \(f: U \subseteq \mathbb{R }^{n+1} \rightarrow \mathbb{R }_n\) belong to the kernel of the global differential operator with non constant coefficients given by \( \mathcal{G }=|{\underline{x}}|^2\frac{\partial }{\partial x_0} \ + \ {\underline{x}} \ \sum _{j=1}^n x_j\frac{\partial }{\partial x_j}. \) Since the operator \(\mathcal{G }\) is not elliptic and there is a degeneracy in \( {\underline{x}}=0\) , its kernel contains also less smooth functions that have to be interpreted as distributions. We study the distributional solutions of the differential equation \(\mathcal{G }F(x_0,{\underline{x}})=G(x_0,{\underline{x}})\) and some of its variations. In particular, we focus our attention on the solutions of the differential equation \( ({\underline{x}}\frac{\partial }{\partial x_0} \ - E)F(x_0,{\underline{x}})=G(x_0,{\underline{x}}), \) where \(E= \sum _{j=1}^n x_j\frac{\partial }{\partial x_j}\) is the Euler operator, from which we deduce properties of the solutions of the equation \( \mathcal{G }F(x_0,{\underline{x}})=G(x_0,{\underline{x}})\) .     

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.     

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 \(\mathcal{H}(\mathcal{A})\otimes_\mathbb{Z}Q=\mathcal{C}(\mathcal{A})\otimes_\mathbb{Z}Q\) .     

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.     

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.     

On a subsemigroup of the universal covering of Lie semigroups

Let \(G\) be a connected Lie group and \(S\) a generating Lie semigroup. An important fact is that generating Lie semigroups admit simply connected covering semigroups. Denote by \(\widetilde{S}\) the simply connected universal covering semigroup of \(S\) . In connection with the problem of identifying the semigroup \(\Gamma (S)\) of monotonic homotopy with a certain subsemigroup of the simply connected covering semigroup \(\widetilde{S}\) we consider in this paper the following subsemigroup $$\begin{aligned} \widetilde{S}_{L}=\overline{\left\langle \mathrm {Exp}(\mathbb {L} (S))\right\rangle } \subset \widetilde{S}, \end{aligned}$$ where \(\mathrm {Exp}:\mathbb {L}(S)\rightarrow S\) is the lifting to \( \widetilde{S}\) of the exponential mapping \(\exp :\mathbb {L}(S)\rightarrow S\) . We prove that \(\widetilde{S}_{L}\) is also simply connected under the assumption that the Lie semigroup \(S\) is right reversible. We further comment how this result should be related to the identification problem mentioned above.     

The \bar{\partial }-Equation on an Annulus Between Two Strictly q-Convex Domains with Smooth Boundaries

In this paper, we study the global boundary regularity of the \(\bar{\partial }\) - equation on an annulus domain \(\Omega \) between two strictly \(q\) -convex domains with smooth boundaries in \(\mathbb{C }^n\) for some bidegree. To this finish, we first show that the \(\bar{\partial }\) -operator has closed range on \(L^{2}_{r, s}(\Omega )\) and the \(\bar{\partial }\) -Neumann operator exists and is compact on \(L^{2}_{r,s}(\Omega )\) for all \(r\ge 0\) , \(q\le s\le n-q- 1\) . We also prove that the \(\bar{\partial }\) -Neumann operator and the Bergman projection operator are continuous on the Sobolev space \(W^{k}_{r,s}(\Omega )\) , \(k\ge 0\) , \(r\ge 0\) , and \(q\le s\le n-q-1\) . Consequently, the \(L^{2}\) -existence theorem for the \(\bar{\partial }\) -equation on such domain is established. As an application, we obtain a global solution for the \(\bar{\partial }\) equation with Hölder and \(L^p\) -estimates on strictly \(q\) -concave domain with smooth \(\mathcal C ^2\) boundary in \(\mathbb{C }^n\) , by using the local solutions and applying the pushing out method of Kerzman (Commun Pure Appl Math 24:301–380, 1971).     

Fixed points and homology of superelliptic Jacobians

Let \(\eta : C_{f,N}\rightarrow \mathbb {P}^1\) be a cyclic cover of \(\mathbb {P}^1\) of degree \(N\) which is totally and tamely ramified for all the ramification points. We determine the group of fixed points of the cyclic covering group \({{\mathrm{Aut}}}(\eta )\simeq \mathbb {Z}/ N \mathbb {Z}\) acting on the Jacobian \(J_N:={{\mathrm{Jac}}}(C_{f,N})\) . For each prime \(\ell \) distinct from the characteristic of the base field, the Tate module \(T_\ell J_N\) is shown to be a free module over the ring \(\mathbb {Z}_\ell [T]/(\sum _{i=0}^{N-1}T^i)\) . We also study the subvarieties of \(J_N\) and calculate the degree of the induced polarization on the new part \(J_N^\mathrm {new}\) of the Jacobian.     

Regularity and uniqueness of a class of biharmonic map heat flows

We consider a class of weak solutions of the heat flow of biharmonic maps from \(\Omega \subset \mathbb{R }^n\) to the unit sphere \(\mathbb{S }^L\subset \mathbb{R }^{L+1}\) , that have small renormalized total energies locally at each interior point. For any such a weak solution, we prove the interior smoothness, and the properties of uniqueness, convexity of hessian energy, and unique limit at \(t=\infty \) . We verify that any weak solution \(u\) to the heat flow of biharmonic maps from \(\Omega \) to a compact Riemannian manifold \(N\) without boundary, with \(\nabla ^2 u\in L^q_tL^p_x\) for some \(p>\frac{n}{2}\) and \(q>2\) satisfying (1.12), has small renormalized total energy locally and hence enjoys both the interior smoothness and uniqueness property. Finally, if an initial data \(u_0\in W^{2,r}(\mathbb{R }^n, N)\) for some \(r>\frac{n}{2}\) , then we establish the local existence of heat flow of biharmonic maps \(u\) , with \(\nabla ^2 u\in L^q_tL^p_x\) for some \(p>\frac{n}{2}\) and \(q>2\) satisfying (1.12).     

A construction of nonabelian simple {\acute{\mathrm{e}}}tale fundamental groups

Under the assumption of the generalized Riemann hypothesis (GRH), we show that there is a real quadratic field \(K\) such that the \({\acute{\mathrm{e}}}\) tale fundamental group \(\pi ^{\acute{\mathrm{et}}}_1(\mathrm {Spec}\;\mathcal {O}_K)\) of the spectrum of the ring of integers \(\mathcal {O}_K\) of \(K\) is isomorphic to \(A_5\) . The proof uses standard methods involving Odlyzko bounds, as well as the proof of Serre’s modularity conjecture. To the best of the author’s knowledge, this is the first example of a number field \(K\) for which \(\pi ^{\acute{\mathrm{et}}}_1(\mathrm {Spec}\;\mathcal {O}_K)\) is finite, nonabelian and simple under the assumption of the GRH.     

Classification of finite Morse index solutions for Hénon type elliptic equation -\Delta u = |x|^{\alpha } u_{+}^{p}

In this paper we investigate the non-autonomous elliptic equations \(-\Delta u = |x|^{\alpha } u_{+}^{p}\) in \( \mathbb{R }^{N}\) and in \( \mathbb{R }_+^{N}\) with the Dirichlet boundary condition, with \(N \ge 2\) , \(p>1\) and \(\alpha >-2\) . We consider the weak solutions with finite Morse index and obtain some classification results.     

Limit theorems for von Mises statistics of a measure preserving transformation

For a measure preserving transformation \(T\) of a probability space \((X,\mathcal{F },\mu )\) and some \(d \ge 1\) we investigate almost sure and distributional convergence of random variables of the form $$\begin{aligned} x \rightarrow \frac{1}{C_n} \sum _{0\le i_1,\ldots ,\,i_d where \(C_1, C_2,\ldots \) are normalizing constants and the kernel \(f\) belongs to an appropriate subspace in some \(L_p(X^d\!,\, \mathcal{F }^{\otimes d}\!,\,\mu ^d)\) . We establish a form of the individual ergodic theorem for such sequences. Using a filtration compatible with \(T\) and the martingale approximation, we prove a central limit theorem in the non-degenerate case; for a class of canonical (totally degenerate) kernels and \(d=2\) , we also show that the convergence holds in distribution towards a quadratic form \(\sum _{m=1}^{\infty } \lambda _m\eta ^2_m\) in independent standard Gaussian variables \(\eta _1, \eta _2, \ldots \) .     

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.     

Renewal reward process for T-related fuzzy random variables on (\mathbb {R}^{p}, \mathbb {R}^{q})

In this paper, following our previous studies, we investigate the renewal rewards process with respect to the necessity, credibility, chance measure and the expected value in which the random inter-arrival times and random rewards are characterized as weighted fuzzy numbers under \(t\) -norm-based fuzzy operations on \(\mathbb {R}^{p}\) and \(\mathbb {R}^{q}\,\,p,\,q \ge 1,\) respectively. Many versions of \(T\) -related fuzzy renewal rewards theorems are proved by using the law of large numbers for weighted fuzzy variables on \(\mathbb {R}^{p}\) . An application example is provided to illustrate the utility of the results.     

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 \(\mu \) .     

A canonical structure on the tangent bundle of a pseudo- or para-Kähler manifold

It is a classical fact that the cotangent bundle \(T^* {\mathcal {M}}\) of a differentiable manifold \({\mathcal {M}}\) enjoys a canonical symplectic form \(\Omega ^*\) . If \(({\mathcal {M}},\mathrm{J} ,g,\omega )\) is a pseudo-Kähler or para-Kähler \(2n\) -dimensional manifold, we prove that the tangent bundle \(T{\mathcal {M}}\) also enjoys a natural pseudo-Kähler or para-Kähler structure \(({\tilde{\hbox {J}}},\tilde{g},\Omega )\) , where \(\Omega \) is the pull-back by \(g\) of \(\Omega ^*\) and \(\tilde{g}\) is a pseudo-Riemannian metric with neutral signature \((2n,2n)\) . We investigate the curvature properties of the pair \(({\tilde{\hbox {J}}},\tilde{g})\) and prove that: \(\tilde{g}\) is scalar-flat, is not Einstein unless \(g\) is flat, has nonpositive (resp. nonnegative) Ricci curvature if and only if \(g\) has nonpositive (resp. nonnegative) Ricci curvature as well, and is locally conformally flat if and only if \(n=1\) and \(g\) has constant curvature, or \(n>2\) and \(g\) is flat. We also check that (i) the holomorphic sectional curvature of \(({\tilde{\hbox {J}}},\tilde{g})\) is not constant unless \(g\) is flat, and (ii) in \(n=1\) case, that \(\tilde{g}\) is never anti-self-dual, unless conformally flat.