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.     

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.     

Examples of Ricci-Mean Curvature Flows

Let \(\pi :{\mathbb {P}}({\mathcal {O}}(0)\oplus {\mathcal {O}}(k))\rightarrow {\mathbb {P}}^{n-1}\) be a projective bundle over \({\mathbb {P}}^{n-1}\) with \(1\le k \le n-1\). We denote \({\mathbb {P}}({\mathcal {O}}(0)\oplus {\mathcal {O}}(k))\) by \(N_{k}^{n}\) and endow it with the U(n)-invariant gradient shrinking Kähler Ricci soliton structure constructed by Cao (Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), A K Peters, Wellesley, 1996) and Koiso (Recent topics in differential and analytic geometry. Advanced studies in pure mathematics, Boston, 1990). In this paper, we show that lens space \(L(k\, ;1)(r)\) with radius r embedded in \(N_{k}^{n}\) is a self-similar solution. We also prove that there exists a pair of critical radii \(r_{1}<r_{2}\), which satisfies the following. The lens space \(L(k\, ;1)(r)\) is a self-shrinker if \(r<r_{2}\) and self-expander if \(r_{2}<r\), and the Ricci-mean curvature flow emanating from \(L(k\, ;1)(r)\) collapses to the 0-section of \(\pi \) if \(r<r_{1}\) and to the \(\infty \)-section of \(\pi \) if \(r_{1}<r\). This paper gives explicit examples of Ricci-mean curvature flows.     

A generalization of the normal rational curve in $$\mathop {\mathrm{PG}}(d,q^n)$$ and its associated non-linear MRD codes

Let A and B be two points of \(\mathop {\mathrm{PG}}(d,q^n)\) and let \(\Phi \) be a collineation between the stars of lines with vertices A and B, that does not map the line AB into itself. In this paper we prove that if \(d=2\) or \(d\ge 3\) and the lines \(\Phi ^{-1}(AB), AB, \Phi (AB) \) are not in a common plane, then the set \(\mathcal{C}\) of points of intersection of corresponding lines under \(\Phi \) is the union of \(q-1\) scattered \({\mathbb {F}}_{q}\)-linear sets of rank n together with \(\{A,B\}\). As an application we will construct, starting from the set \(\mathcal{C}\), infinite families of non-linear \((d+1, n, q;d-1)\)-MRD codes, \(d\le n-1\), generalizing those recently constructed in Cossidente et al. (Des Codes Cryptogr 79:597–609, 2016) and Durante and Siciliano (Electron J Comb, 2017).     

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.     

An Invariant Subspace Theorem and Invariant Subspaces of Analytic Reproducing Kernel Hilbert Spaces - II

This paper is a follow-up contribution to our work (Sarkar in J Oper Theory, 73:433–441, 2015) where we discussed some invariant subspace results for contractions on Hilbert spaces. Here we extend the results of (Sarkar in J Oper Theory, 73:433–441, 2015) to the context of n-tuples of bounded linear operators on Hilbert spaces. Let \(T = (T_1, \ldots , T_n)\) be a pure commuting co-spherically contractive n-tuple of operators on a Hilbert space \({\mathcal {H}}\) and \({\mathcal {S}}\) be a non-trivial closed subspace of \({\mathcal {H}}\). One of our main results states that: \({\mathcal {S}}\) is a joint T-invariant subspace if and only if there exists a partially isometric operator \(\Pi \in {\mathcal {B}}(H^2_n({\mathcal {E}}), {\mathcal {H}})\) such that \({\mathcal {S}}= \Pi H^2_n({\mathcal {E}})\), where \(H^2_n\) is the Drury–Arveson space and \({\mathcal {E}}\) is a coefficient Hilbert space and \(T_i \Pi = \Pi M_{z_i}\), \(i = 1, \ldots , n\). In particular, it follows that a shift invariant subspace of a “nice” reproducing kernel Hilbert space over the unit ball in \({{\mathbb {C}}}^n\) is the range of a “multiplier” with closed range. Our work addresses the case of joint shift invariant subspaces of the Hardy space and the weighted Bergman spaces over the unit ball in \({{\mathbb {C}}}^n\).     

Walk-Powers and Homomorphism Bounds of Planar Signed Graphs

As an extension of the Four-Color Theorem it is conjectured by the first author that every planar graph of odd-girth at least \(2k+1\) admits a homomorphism to the projective cube of dimension 2k, i.e., the Cayley graph \({\mathcal {PC}}(2k)=({\mathbb {Z}}_2^{2k}, \{e_1, e_2,\) \(\ldots ,e_{2k}, J\})\) where the \(e_i\)’s are the standard basis vectors of \({\mathbb {Z}}_2^d\) and J is the all 1 vector. Noting that \({\mathcal {PC}}(2k)\) itself is of odd-girth \(2k+1\), in this work we show that if the conjecture is true, then \({\mathcal {PC}}(2k)\) is an optimal such graph both with respect to the number of vertices and the number of edges. The result is obtained using the notion of walk-power of graphs and their clique numbers. An analogous result is proved for signed bipartite planar graphs of unbalanced-girth 2k. The work is presented in the uniform framework of planar consistent signed graphs.     

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 \({\mathcal {U}}(E,F)\).     

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.     

Structure theorem of the generator of a norm continuous completely positive semigroup: an alternative proof using Bures distance

Here we present an alternative proof using Bures distance that the generator L of a norm continuous completely positive semigroup acting on a \(C^*\)-algebra \({\mathcal {B}}\subset \mathcal B(H)\) has the form \( L(b) = \Psi (b) + k^*b+bk\), \(b\in {\mathcal {B}}\) for some completely positive map \(\Psi :{\mathcal {B}}\rightarrow {\mathcal {B}}(H)\) and \(k\in {\mathcal {B}}(H)\).     

Semigroups of rectangular matrices under a sandwich operation

Let \({\mathcal {M}}_{mn}={\mathcal {M}}_{mn}({\mathbb {F}})\) denote the set of all \(m\times n\) matrices over a field \({\mathbb {F}}\), and fix some \(n\times m\) matrix \(A\in {\mathcal {M}}_{nm}\). An associative operation \(\star \) may be defined on \({\mathcal {M}}_{mn}\) by \(X\star Y=XAY\) for all \(X,Y\in {\mathcal {M}}_{mn}\), and the resulting sandwich semigroup is denoted \({\mathcal {M}}_{mn}^A={\mathcal {M}}_{mn}^A({\mathbb {F}})\). These semigroups are closely related to Munn rings, which are fundamental tools in the representation theory of finite semigroups. We study \({\mathcal {M}}_{mn}^A\) as well as its subsemigroups \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\) and \({\mathcal {E}}_{mn}^A\) (consisting of all regular elements and products of idempotents, respectively), and the ideals of \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\). Among other results, we characterise the regular elements; determine Green’s relations and preorders; calculate the minimal number of matrices (or idempotent matrices, if applicable) required to generate each semigroup we consider; and classify the isomorphisms between finite sandwich semigroups \({\mathcal {M}}_{mn}^A({\mathbb {F}}_1)\) and \({\mathcal {M}}_{kl}^B({\mathbb {F}}_2)\). Along the way, we develop a general theory of sandwich semigroups in a suitably defined class of partial semigroups related to Ehresmann-style “arrows only” categories; we hope this framework will be useful in studies of sandwich semigroups in other categories. We note that all our results have applications to the variants \({\mathcal {M}}_n^A\) of the full linear monoid \({\mathcal {M}}_n\) (in the case \(m=n\)), and to certain semigroups of linear transformations of restricted range or kernel (in the case that \(\hbox {rank}(A)\) is equal to one of m, n).     

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.     

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.     

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 \(q=2\).     

Extension sets,affine designs,and Hamada’s conjecture

We introduce the notion of an extension set for an affine plane of order q to study affine designs \({\mathcal {D}}'\) with the same parameters as, but not isomorphic to, the classical affine design \({\mathcal {D}} = \mathrm {AG}_2(3,q)\) formed by the points and planes of the affine space \(\mathrm {AG}(3,q)\) which are very close to this geometric example in the following sense: there are blocks \(B'\) and B of \({\mathcal {D}'}\) and \({\mathcal {D}}\), respectively, such that the residual structures \({\mathcal {D}}'_{B'}\) and \({\mathcal {D}}_B\) induced on the points not in \(B'\) and B, respectively, agree. Moreover, the structure \({\mathcal {D}}'(B')\) induced on \(B'\) is the q-fold multiple of an affine plane \({\mathcal {A}}'\) which is determined by an extension set for the affine plane \(B \cong AG(2,q)\). In particular, this new approach will result in a purely theoretical construction of the two known counterexamples to Hamada’s conjecture for the case \(\mathrm {AG}_2(3,4)\), which were discovered by Harada et al. [7] as the result of a computer search; a recent alternative construction, again via a computer search, is in [23]. On the other hand, we also prove that extension sets cannot possibly give any further counterexamples to Hamada’s conjecture for the case of affine designs with the parameters of some \(\mathrm {AG}_2(3,q)\); thus the two counterexamples for \(q=4\) might be truly sporadic. This seems to be the first result which establishes the validity of Hamada’s conjecture for some infinite class of affine designs of a special type. Nevertheless, affine designs which are that close to the classical geometric examples are of interest in themselves, and we provide both theoretical and computational results for some particular types of extension sets. Specifically, we obtain a theoretical construction for one of the two affine designs with the parameters of \(\mathrm {AG}_2(3,3)\) and 3-rank 11 and for an affine design with the parameters of \(\mathrm {AG}_2(3,4)\) and 2-rank 17 (in both cases, just one more than the rank of the classical example).     

Relative Cartier divisors and Laurent polynomial extensions

If \(i:A\subset B\) is a commutative ring extension, we show that the group \({\mathcal I}(A,B)\) of invertible A-submodules of B is contracted in the sense of Bass, with \(L{\mathcal I}(A,B)=H^0_{\mathrm {et}}(A,i_*{\mathbb Z}/{\mathbb Z})\). This gives a canonical decomposition for \({\mathcal I}(A[t,\frac{1}{t}],B[t,\frac{1}{t}])\).     

A Tight Upper Bound on Acquaintance Time of Graphs

In this note we confirm a conjecture raised by Benjamini et al. (SIAM J Discrete Math 28(2):767–785, 2014) on the acquaintance time of graphs, proving that for all graphs G with n vertices it holds that \(\mathcal {AC}(G) = O(n^{3/2})\). This is done by proving that for all graphs G with n vertices and maximum degree \(\varDelta \) it holds that \(\mathcal {AC}(G) \le 20 \varDelta n\). Combining this with the bound \(\mathcal {AC}(G) \le O(n^2/\varDelta )\) from Benjamini et al. (SIAM J Discrete Math 28(2):767–785, 2014) gives the uniform upper bound of \(O(n^{3/2})\) for all n-vertex graphs. This bound is tight up to a multiplicative constant. We also prove that for the n-vertex path \(P_n\) it holds that \(\mathcal {AC}(P_n)=n-2\). In addition we show that the barbell graph \(B_n\) consisting of two cliques of sizes \({\lceil n/2\rceil }\) and \({\lfloor n/2\rfloor }\) connected by a single edge also has \(\mathcal {AC}(B_n) = n-2\). This shows that it is possible to add \(\varOmega (n^2\)) edges a graph without changing its \(\mathcal {AC}\) value.     

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].     

Algebras of Convolution-Type Operators with Piecewise Slowly Oscillating Data on Weighted Lebesgue Spaces

Let \({\mathcal B}_{p,w}\) be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space \(L^p(\mathbb {R},w)\), where \(p\in (1,\infty )\) and w is a Muckenhoupt weight. We study the Banach subalgebra \(\mathfrak {A}_{p,w}\) of \({\mathcal B}_{p,w}\) generated by all multiplication operators aI (\(a\in \mathrm{PSO}^\diamond \)) and all convolution operators \(W^0(b)\) (\(b\in \mathrm{PSO}_{p,w}^\diamond \)), where \(\mathrm{PSO}^\diamond \subset L^\infty (\mathbb {R})\) and \(\mathrm{PSO}_{p,w}^\diamond \subset M_{p,w}\) are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of \(\mathbb {R}\cup \{\infty \}\), and \(M_{p,w}\) is the Banach algebra of Fourier multipliers on \(L^p(\mathbb {R},w)\). For any Muckenhoupt weight w, we study the Fredholmness in the Banach algebra \({\mathcal Z}_{p,w}\subset \mathfrak {A}_{p,w}\) generated by the operators \(aW^0(b)\) with slowly oscillating data \(a\in \mathrm{SO}^\diamond \) and \(b\in \mathrm{SO}^\diamond _{p,w}\). Then, under some condition on the weight w, we complete constructing a Fredholm symbol calculus for the Banach algebra \(\mathfrak {A}_{p,w}\) in comparison with Karlovich and Loreto Hernández (Integr. Equations Oper. Theory 74:377–415, 2012) and Karlovich and Loreto Hernández (Integr. Equations Oper. Theory 75:49–86, 2013) and establish a Fredholm criterion for the operators \(A\in \mathfrak {A}_{p,w}\) in terms of their symbols. A new approach to determine local spectra is found.