Weakly compact weighted composition operators on spaces of Lipschitz functions

We prove that if X is a compact metric space and \({\text {lip}}(X,d)\) has the uniform separation property, then weakly compact weighted composition operators on spaces of Lipschitz functions \({\text {Lip}}(X,d)\) and \({\text {lip}}(X,d)\) are compact.     

Linear extensions of some Baire-one functions

Let X be a Hausdorff topological space, and let \({\mathscr {B}}_1(X)\) denote the space of all real Baire-one functions defined on X. Let A be a nonempty subset of X endowed with the topology induced from X, and let \({\mathscr {F}}(A)\) be the set of functions \(A\rightarrow {\mathbb R}\) with a property \({\mathscr {F}}\) making \({\mathscr {F}}(A)\) a linear subspace of \({\mathscr {B}}_1(A)\). We give a sufficient condition for the existence of a linear extension operator \(T_A:{\mathscr {F}}(A)\rightarrow {\mathscr {F}}(X)\), where \({\mathscr {F}}\) means to be piecewise continuous on a sequence of closed and \(G_\delta \) subsets of X and is denoted by \({\mathscr {P}_0}\). We show that \(T_A\) restricted to bounded elements of \({\mathscr {F}}(A)\) endowed with the supremum norm is an isometry. As a consequence of our main theorem, we formulate the conclusion about existence of a linear extension operator for the classes of Baire-one-star and piecewise continuous functions.     

Regularity of powers of bipartite graphs

Let G be a finite simple graph and I(G) denote the corresponding edge ideal. For all \(s \ge 1\), we obtain upper bounds for \({\text {reg}}(I(G)^s)\) for bipartite graphs. We then compare the properties of G and \(G'\), where \(G'\) is the graph associated with the polarization of the ideal \((I(G)^{s+1} : e_1\cdots e_s)\), where \(e_1,\cdots , e_s\) are edges of G. Using these results, we explicitly compute \({\text {reg}}(I(G)^s)\) for several subclasses of bipartite graphs.     

Bernstein components via the Bernstein center

Let G be a reductive p-adic group. Let \(\Phi \) be an invariant distribution on G lying in the Bernstein center \({\mathcal {Z}}(G)\). We prove that \(\Phi \) is supported on compact elements in G if and only if it defines a constant function on every component of the set \({\text {Irr}}(G)\); in particular, we show that the space of all elements of \({\mathcal {Z}}(G)\) supported on compact elements is a subalgebra of \({\mathcal {Z}}(G)\). Our proof is a slight modification of the argument from Section 2 of Dat (J Reine Angew Math 554:69–103, 2003), where our result is proved in one direction.     

Some results on the annihilators and attached primes of local cohomology modules

Let \((R, \mathfrak {m})\) be a local ring and M a finitely generated R-module. It is shown that if M is relative Cohen–Macaulay with respect to an ideal \(\mathfrak {a}\) of R, then \({\text {Ann}}_R(H_{\mathfrak {a}}^{{\text {cd}}(\mathfrak {a}, M)}(M))={\text {Ann}}_RM/L={\text {Ann}}_RM\) and \({\text {Ass}}_R (R/{\text {Ann}}_RM)\subseteq \{\mathfrak {p}\in {\text {Ass}}_R M|\,\mathrm{cd}(\mathfrak {a}, R/\mathfrak {p})={\text {cd}}(\mathfrak {a}, M)\},\) where L is the largest submodule of M such that \(\mathrm{cd}(\mathfrak {a}, L)< \mathrm{cd}(\mathfrak {a}, M)\). We also show that if \(H^{\dim M}_{\mathfrak {a}}(M)=0\), then \({\text {Att}}_R(H^{\dim M-1}_{\mathfrak {a}}(M))= \{\mathfrak {p}\in {\text {Supp}}(M)|\mathrm{cd}(\mathfrak {a}, R/\mathfrak {p})=\dim M-1\},\) and so the attached primes of \(H^{\dim M-1}_{\mathfrak {a}}(M)\) depend only on \({\text {Supp}}(M)\). Finally, we prove that if M is an arbitrary module (not necessarily finitely generated) over a Noetherian ring R with \(\mathrm{cd}(\mathfrak {a}, M)=\mathrm{cd}(\mathfrak {a}, R/{\text {Ann}}_RM)\), then \({\text {Att}}_R(H^{\mathrm{cd}(\mathfrak {a}, M)}_{\mathfrak {a}}(M))\subseteq \{\mathfrak {p}\in {\text {V}}({\text {Ann}}_RM)|\,\mathrm{cd}(\mathfrak {a}, R/\mathfrak {p})=\mathrm{cd}(\mathfrak {a}, M)\}.\) As a consequence of this, it is shown that if \(\dim M=\dim R\), then \({\text {Att}}_R(H^{\dim M}_{\mathfrak {a}}(M))\subseteq \{\mathfrak {p}\in {\text {Ass}}_R M|\mathrm{cd}(\mathfrak {a}, R/\mathfrak {p})=\dim M\}\).     

Sandwich semigroups in locally small categories I: foundations

Fix (not necessarily distinct) objects i and j of a locally small category S, and write \(S_{ij}\) for the set of all morphisms \(i\rightarrow j\). Fix a morphism \(a\in S_{ji}\), and define an operation \(\star _a\) on \(S_{ij}\) by \(x\star _ay=xay\) for all \(x,y\in S_{ij}\). Then \((S_{ij},\star _a)\) is a semigroup, known as a sandwich semigroup, and denoted by \(S_{ij}^a\). This article develops a general theory of sandwich semigroups in locally small categories. We begin with structural issues such as regularity, Green’s relations and stability, focusing on the relationships between these properties on \(S_{ij}^a\) and the whole category S. We then identify a natural condition on a, called sandwich regularity, under which the set \({\text {Reg}}(S_{ij}^a)\) of all regular elements of \(S_{ij}^a\) is a subsemigroup of \(S_{ij}^a\). Under this condition, we carefully analyse the structure of the semigroup \({\text {Reg}}(S_{ij}^a)\), relating it via pullback products to certain regular subsemigroups of \(S_{ii}\) and \(S_{jj}\), and to a certain regular sandwich monoid defined on a subset of \(S_{ji}\); among other things, this allows us to also describe the idempotent-generated subsemigroup \(\mathbb E(S_{ij}^a)\) of \(S_{ij}^a\). We also study combinatorial invariants such as the rank (minimal size of a generating set) of the semigroups \(S_{ij}^a\), \({\text {Reg}}(S_{ij}^a)\) and \(\mathbb E(S_{ij}^a)\); we give lower bounds for these ranks, and in the case of \({\text {Reg}}(S_{ij}^a)\) and \(\mathbb E(S_{ij}^a)\) show that the bounds are sharp under a certain condition we call MI-domination. Applications to concrete categories of transformations and partial transformations are given in Part II.     

Augmentation quotients for real representation rings of cyclic groups

Denote by \(C_m\) the cyclic group of order m. Let \({\mathcal {R}}(C_m)\) be its real representation ring, and \(\Delta (C_m)\) its augmentation ideal. In this paper, we give an explicit \({\mathbb {Z}}\)-basis for the n-th power \(\Delta ^{n}(C_m)\) and determine the isomorphism class of the n-th augmentation quotient \(\Delta ^n(C_m)/\Delta ^{n+1}(C_m)\) for each positive integer n.     

Improved upper bounds for partial spreads

A partial \((k-1)\)-spread in \({\text {PG}}(n-1,q)\) is a collection of \((k-1)\)-dimensional subspaces with trivial intersection. So far, the maximum size of a partial \((k-1)\)-spread in \({\text {PG}}(n-1,q)\) was known for the cases \(n\equiv 0\pmod k\), \(n\equiv 1\pmod k\), and \(n\equiv 2\pmod k\) with the additional requirements \(q=2\) and \(k=3\). We completely resolve the case \(n\equiv 2\pmod k\) for the binary case \(q=2\).     

Littlewood–Paley Theory for Triangle Buildings

For the natural two-parameter filtration \(\left( {\mathcal {F}_\lambda }: {\lambda \in P}\right) \) on the boundary of a triangle building, we define a maximal function and a square function and show their boundedness on \(L^p(\Omega _0)\) for \(p \in (1, \infty )\). At the end, we consider \(L^p(\Omega _0)\) boundedness of martingale transforms. If the building is of \({\text {GL}}(3, \mathbb {Q}_p)\), then \(\Omega _0\) can be identified with p-adic Heisenberg group.     

A geometric approach to alternating <Emphasis Type="Italic">k</Emphasis>-linear forms

Denote by \({{\mathcal {G}}}_k(V)\) the Grassmannian of the k-subspaces of a vector space V over a field \({\mathbb {K}}\). There is a natural correspondence between hyperplanes H of \({\mathcal {G}}_k(V)\) and alternating k-linear forms on V defined up to a scalar multiple. Given a hyperplane H of \({{\mathcal {G}}_k}(V)\), we define a subspace \(R^{\uparrow }(H)\) of \({{\mathcal {G}}_{k-1}}(V)\) whose elements are the \((k-1)\)-subspaces A such that all k-spaces containing A belong to H. When \(n-k\) is even, \(R^{\uparrow }(H)\) might be empty; when \(n-k\) is odd, each element of \({\mathcal {G}}_{k-2}(V)\) is contained in at least one element of \(R^{\uparrow }(H)\). In the present paper, we investigate several properties of \(R^{\uparrow }(H)\), settle some open problems and propose a conjecture.     

Tempered representations of <Emphasis Type="Italic">p</Emphasis>-adic groups: special idempotents and topology

Let \(G=G(k)\) be a connected reductive group over a p-adic field k. The smooth (and tempered) complex representations of G can be considered as the nondegenerate modules over the Hecke algebra \({\mathcal {H}}={\mathcal {H}}(G)\) and the Schwartz algebra \({\mathcal {S}}={\mathcal {S}}(G)\) forming abelian categories \({\mathcal {M}}(G)\) and \({\mathcal {M}}^t(G)\), respectively. Idempotents \(e\in {\mathcal {H}}\) or \({\mathcal {S}}\) define full subcategories \({\mathcal {M}}_e(G)= \{V : {\mathcal {H}}eV=V\}\) and \({\mathcal {M}}_e^t(G)= \{V : {\mathcal {S}}eV=V\}\). Such an e is said to be special (in \({\mathcal {H}}\) or \({\mathcal {S}}\)) if the corresponding subcategory is abelian. Parallel to Bernstein’s result for \(e\in {\mathcal {H}}\) we will prove that, for special \(e \in {\mathcal {S}}\), \({\mathcal {M}}_e^t(G) = \prod _{\Theta \in \theta _e} {\mathcal {M}}^t(\Theta )\) is a finite direct product of component categories \({\mathcal {M}}^t(\Theta )\), now referring to connected components of the center of \({\mathcal {S}}\). A special \(e\in {\mathcal {H}}\) will be also special in \({\mathcal {S}}\), but idempotents \(e\in {\mathcal {H}}\) not being special can become special in \({\mathcal {S}}\). To obtain conditions we consider the sets \(\mathrm{Irr}^t(G) \subset \mathrm{Irr}(G)\) of (tempered) smooth irreducible representations of G, and we view \(\mathrm{Irr}(G)\) as a topological space for the Jacobson topology defined by the algebra \({\mathcal {H}}\). We use this topology to introduce a preorder on the connected components of \(\mathrm{Irr}^t(G)\). Then we prove that, for an idempotent \(e \in {\mathcal {H}}\) which becomes special in \({\mathcal {S}}\), its support \(\theta _e\) must be saturated with respect to that preorder. We further analyze the above decomposition of \({\mathcal {M}}_e^t(G)\) in the case where G is k-split with connected center and where \(e = e_J \in {\mathcal {H}}\) is the Iwahori idempotent. Here we can use work of Kazhdan and Lusztig to relate our preorder on the support \(\theta _{e_J}\) to the reverse of the natural partial order on the unipotent classes in G. We finish by explicitly computing the case \(G=GL_n\), where \(\theta _{e_J}\) identifies with the set of partitions of n. Surprisingly our preorder (which is a partial order now) is strictly coarser than the reverse of the dominance order on partitions.     

Finite elements in some vector lattices of nonlinear operators

We study the collection of finite elements \(\Phi _{1}\big ({\mathcal {U}}(E,F)\big )\) in the vector lattice \({\mathcal {U}}(E,F)\) of orthogonally additive, order bounded (called abstract Uryson) operators between two vector lattices E and F, where F is Dedekind complete. In particular, for an atomic vector lattice E it is proved that for a finite element in \(\varphi \in {\mathcal {U}}(E,{\mathbb {R}})\) there is only a finite set of mutually disjoint atoms, where \(\varphi \) does not vanish and, for an atomless vector lattice the zero-vector is the only finite element in the band of \(\sigma \)-laterally continuous abstract Uryson functionals. We also describe the ideal \(\Phi _{1}\big ({\mathcal {U}}({\mathbb {R}}^n,{\mathbb {R}}^m)\big )\) for \(n,m\in {\mathbb {N}}\) and consider rank one operators to be finite elements in \({\mathcal {U}}(E,F)\).     

Some results on Artinian cofinite top local cohomology modules

Let \((R,\mathfrak {m})\) be a Noetherian local ring, I be an ideal of R, and M be a finitely generated R-module such that \({\text {H}}_I^t(M)\) is Artinian and I-cofinite, where \(t={\text {cd}}\,(I,M)\). In this paper, we give some equivalent conditions for the property

$$\begin{aligned} {\text {Ann}}\,_R\left( 0:_{{\text {H}}_I^t (M)} \mathfrak {p}\right) =\mathfrak {p}~\text {for all prime ideals }~ \mathfrak {p}\supseteq {\text {Ann}}\,_R{\text {H}}_I^t(M).(*) \end{aligned}$$

Also, we show that if \({\text {H}}_I^t(M)\) satisfies the property \((*)\), then \({\text {H}}_I^t(M)\cong {\text {H}}_{\mathfrak {m}}^t(M/N)\) for some submodule N of M with \({\text {dim}}\,(M/N)=t\).

     

Number fields in fibers: the geometrically abelian case with rational critical values

Let X be an algebraic curve over \({\mathbb {Q}}\) and \({t\in {\mathbb {Q}}(X)}\) a non-constant rational function such that \({{\mathbb {Q}}(X)\ne {\mathbb {Q}}(t)}\). For every \({ n \in {\mathbb {Z}}}\) pick \({P_ n \in X(\bar{{\mathbb {Q}}})}\) such that \({t(P_n)=n}\). We conjecture that, for large N, among the number fields \({\mathbb {Q}}(P_1), \ldots , {\mathbb {Q}}(P_N)\) there are at least cN distinct. We prove this conjecture in the special case when \(\bar{{\mathbb {Q}}}(X)/\bar{{\mathbb {Q}}}(t)\) is an abelian field extension and the critical values of t are all rational. This implies, in particular, that our conjecture follows from a more famous conjecture of Schinzel.     

Covering arrays of strength three from extended permutation vectors

A covering array \(\text{ CA }(N;t,k,v)\) is an \(N\times k\) array such that in every \(N\times t\) subarray each possible t-tuple over a v-set appears as a row of the subarray at least once. The integers t and v are respectively the strength and the order of the covering array. Let v be a prime power and let \({\mathbb {F}}_v\) denote the finite field with v elements. In this work the original concept of permutation vectors generated by a \((t-1)\)-tuple over \({\mathbb {F}}_v\) is extended to vectors generated by a t-tuple over \({\mathbb {F}}_v\). We call these last vectors extended permutation vectors (EPVs). For every prime power v, a covering perfect hash family \(\text{ CPHF }(2;v^2-v+3,v^3,3)\) is constructed from EPVs given by subintervals of a linear feedback shift register sequence over \({\mathbb {F}}_v\). When \(v\in \{7,9,11,13,16,17,19,23,25\}\) the covering array \(\text{ CA }(2v^3-v;3,v^2-v+3,v)\) generated by \(\text{ CPHF }(2;v^2-v+3,v^3,3)\) has less rows than the best-known covering array with strength three, \(v^2-v+3\) columns, and order v. CPHFs formed by EPVs are also constructed using simulated annealing; in this case the results improve the size of eighteen covering arrays of strength three.     

A normal generating set for the Torelli group of a compact non-orientable surface

For a compact surface S, let \({\mathcal {I}}(S)\) denote the Torelli group of S. For a compact orientable surface \(\Sigma \), \({\mathcal {I}}(\Sigma )\) is generated by two types of mapping classes, called bounding simple closed curve maps (BSCC maps) and bounding pair maps (BP maps) (see Powell in Proc Am Math Soc 68:347–350, 1978; Putman in Geom Topol 11:829–865, 2007). For a non-orientable closed surface N, \({\mathcal {I}}(N)\) is generated by BSCC maps and BP maps (see Hirose and Kobayashi in Fund Math 238:29–51, 2017). In this paper, we give an explicit normal generating set for \({\mathcal {I}}(N_g^b)\), where \(N_g^b\) is a genus-g compact non-orientable surface with b boundary components for \(g\ge 4\) and \(b\ge 1\).     

Completely regular clique graphs,II

Let \(\varGamma = (X,R)\) be a connected graph. Then \(\varGamma \) is said to be a completely regular clique graph of parameters (s, c) with \(s\ge 1\) and \(c\ge 1\), if there is a collection \({\mathcal {C}}\) of completely regular cliques of size \(s+1\) such that every edge is contained in exactly c members of \({\mathcal {C}}\). In the previous paper (Suzuki in J Algebr Combin 40:233–244, 2014), we showed, among other things, that a completely regular clique graph is distance-regular if and only if it is a bipartite half of a certain distance-semiregular graph. In this paper, we show that a completely regular clique graph with respect to \({\mathcal {C}}\) is distance-regular if and only if every \({\mathcal {T}}(C)\)-module of endpoint zero is thin for all \(C\in {\mathcal {C}}\). We also discuss the relation between a \({\mathcal {T}}(C)\)-module of endpoint 0 and a \({\mathcal {T}}(x)\)-module of endpoint 1 and study examples of completely regular clique graphs.     

On bigraded regularities of Rees algebra

For any homogeneous ideal I in \(K[x_1,\ldots ,x_n]\) of analytic spread \(\ell \), we show that for the Rees algebra R(I), \({\text {reg}}_{(0,1)}^{\mathrm{syz}}(R(I))={\text {reg}}_{(0,1)}^{\mathrm{T}}(R(I))\). We compute a formula for the (0, 1)-regularity of R(I), which is a bigraded analog of Theorem 1.1 of Aramova and Herzog (Am. J. Math. 122(4) (2000) 689–719) and Theorem 2.2 of Römer (Ill. J. Math. 45(4) (2001) 1361–1376) to R(I). We show that if the defect sequence, \(e_k:= {\text {reg}}(I^k)-k\rho (I)\), is weakly increasing for \(k \ge {\text {reg}}^{\mathrm{syz}}_{(0,1)}(R(I))\), then \({\text {reg}}(I^j)=j\rho (I)+e\) for \(j \ge {\text {reg}}^{\mathrm{syz}}_{(0,1)}(R(I))+\ell \), where \(\ell ={\text {min}}\{\mu (J)~|~ J\subseteq I \text{ a } \text{ graded } \text{ minimal } \text{ reduction } \text{ of } I\}\). This is an improvement of Corollary 5.9(i) of [16].     

On the additivity of block designs

We show that symmetric block designs \({\mathcal {D}}=({\mathcal {P}},{\mathcal {B}})\) can be embedded in a suitable commutative group \({\mathfrak {G}}_{\mathcal {D}}\) in such a way that the sum of the elements in each block is zero, whereas the only Steiner triple systems with this property are the point-line designs of \({\mathrm {PG}}(d,2)\) and \({\mathrm {AG}}(d,3)\). In both cases, the blocks can be characterized as the only k-subsets of \(\mathcal {P}\) whose elements sum to zero. It follows that the group of automorphisms of any such design \(\mathcal {D}\) is the group of automorphisms of \({\mathfrak {G}}_\mathcal {D}\) that leave \(\mathcal {P}\) invariant. In some special cases, the group \({\mathfrak {G}}_\mathcal {D}\) can be determined uniquely by the parameters of \(\mathcal {D}\). For instance, if \(\mathcal {D}\) is a 2-\((v,k,\lambda )\) symmetric design of prime order p not dividing k, then \({\mathfrak {G}}_\mathcal {D}\) is (essentially) isomorphic to \(({\mathbb {Z}}/p{\mathbb {Z}})^{\frac{v-1}{2}}\), and the embedding of the design in the group can be described explicitly. Moreover, in this case, the blocks of \(\mathcal {B}\) can be characterized also as the v intersections of \(\mathcal {P}\) with v suitable hyperplanes of \(({\mathbb {Z}}/p{\mathbb {Z}})^{\frac{v-1}{2}}\).     

Estimates for coefficients of certain <Emphasis Type="Italic">L</Emphasis>-functions

Let \(\pi _{\varphi }\) (or \(\pi _{\psi }\)) be an automorphic cuspidal representation of \(\text {GL}_{2} (\mathbb {A}_{\mathbb {Q}})\) associated to a primitive Maass cusp form \(\varphi \) (or \(\psi \)), and \(\mathrm{sym}^j \pi _{\varphi }\) be the jth symmetric power lift of \(\pi _{\varphi }\). Let \(a_{\mathrm{sym}^j \pi _{\varphi }}(n)\) denote the nth Dirichlet series coefficient of the principal L-function associated to \(\mathrm{sym}^j \pi _{\varphi }\). In this paper, we study first moments of Dirichlet series coefficients of automorphic representations \(\mathrm{sym}^3 \pi _{\varphi }\) of \(\text {GL}_{4}(\mathbb {A}_{\mathbb {Q}})\), and \(\pi _{\psi }\otimes \mathrm{sym}^2 \pi _{\varphi }\) of \(\text {GL}_{6}(\mathbb {A}_{\mathbb {Q}})\). For \(3 \le j \le 8\), estimates for \(|a_{\mathrm{sym}^j \pi _{\varphi }}(n)|\) on average over a short interval have also been established.