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.     

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.     

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).     

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.     

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.     

On the Largest Graph-Lagrangian of 3-Graphs with Fixed Number of Edges

The Graph-Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. In most applications, we need an upper bound for the Graph-Lagrangian of a hypergraph. Frankl and Füredi conjectured that the \({r}\) -graph with \(m\) edges formed by taking the first \(\textit{m}\) sets in the colex ordering of the collection of all subsets of \({\mathbb N}\) of size \({r}\) has the largest Graph-Lagrangian of all \(r\) -graphs with \(m\) edges. In this paper, we show that the largest Graph-Lagrangian of a class of left-compressed \(3\) -graphs with \(m\) edges is at most the Graph-Lagrangian of the \(\mathrm 3 \) -graph with \(m\) edges formed by taking the first \(m\) sets in the colex ordering of the collection of all subsets of \({\mathbb N}\) of size \({3}\) .     

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 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.     

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.     

On the number of distinct prime factors of nj+a^hk

Let \(\omega (n)\) denote the number of distinct prime factors of \(n\) . Then for any given \(K\ge 2\) , small \(\epsilon >0\) and sufficiently large (only depending on \(K\) and \(\epsilon \) ) \(x\) , there exist at least \(x^{1-\epsilon }\) integers \(n\in [x,(1+K^{-1})x]\) such that \(\omega (nj\pm a^hk)\ge (\log \log \log x)^{\frac{1}{3}-\epsilon }\) for all \(2\le a\le K\) , \(1\le j,k\le K\) and \(0\le h\le K\log x\) .     

Large collections of curves pairwise intersecting exactly once

Let \(\Omega =(\omega _{j})_{j\in I}\) be a maximum size collection of pairwise non-isotopic simple closed curves on the closed, orientable, genus \(g\) surface \(S_{g}\) , such that \(\omega _{i}\) and \(\omega _{j}\) intersect exactly once for \(i\ne j\) . We show that for \(g\ge 3\) , there exists atleast two such collections up to the action of the mapping class group, answering a question posed by Malestein, Rivin and Theran. As a consequence, we show that the automorphism group of the systole graph for \(S_{g}, g\ge 3\) (whose vertices are isotopy classes of simple closed curves, and whose edges correspond to pairs of curve intersecting once) does not act transitively on maximal complete subgraphs.     

A study of saturated tensor cone for symmetrizable Kac–Moody algebras

Let \(\mathfrak {g}\) be a symmetrizable Kac-Moody Lie algebra with the standard Cartan subalgebra \(\mathfrak {h}\) and the Weyl group \(W\) . Let \(P_+\) be the set of dominant integral weights. For \(\lambda \in P_+\) , let \(L(\lambda )\) be the integrable, highest weight (irreducible) representation of \(\mathfrak {g}\) with highest weight \(\lambda \) . For a positive integer \(s\) , define the saturated tensor semigroup as $$\begin{aligned} \Gamma _s:= \{(\lambda _1, \dots , \lambda _s,\mu )\in P_+^{s+1}: \exists \, N\ge 1 \,\text {with}\,L(N\mu )\subset L(N\lambda _1)\otimes \dots \otimes L(N\lambda _s)\}. \end{aligned}$$ The aim of this paper is to begin a systematic study of \(\Gamma _s\) in the infinite dimensional symmetrizable Kac-Moody case. In this paper, we produce a set of necessary inequalities satisfied by \(\Gamma _s\) . These inequalities are indexed by products in \(H^*(G^{\mathrm{min }}/B; \mathbb {Z})\) for \(B\) the standard Borel subgroup, where \(G^{\mathrm{min }}\) is the ‘minimal’ Kac-Moody group with Lie algebra \(\mathfrak {g}\) . The proof relies on the Kac-Moody analogue of the Borel-Weil theorem and Geometric Invariant Theory (specifically the Hilbert-Mumford index). In the case that \(\mathfrak {g}\) is affine of rank 2, we show that these inequalities are necessary and sufficient. We further prove that any integer \(d>0\) is a saturation factor for \(A^{(1)}_1\) and 4 is a saturation factor for \(A^{(2)}_2\) .     

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 \(\mu \) .     

Structure of \mathrm {Gal}(\mathbb {k}_2^{(2)}/\mathbb {k}) for some fields \mathbb {k}=\mathbb {Q}(\sqrt{2p_1p_2},i) with \mathbf {C}l_2(\mathbb {k})\simeq (2, 2, 2)

Let \(p_1 \equiv p_2 \equiv 5\pmod 8\) be different primes. Put \(i=\sqrt{-1}\) and \(d=2p_1p_2\) , then the bicyclic biquadratic field \(\mathbb {k}=\mathbb {Q}(\sqrt{d},i)\) has an elementary abelian 2-class group of rank \(3\) . In this paper we determine the nilpotency class, the coclass, the generators and the structure of the non-abelian Galois group \(\mathrm {Gal}(\mathbb {k}_2^{(2)}/\mathbb {k})\) of the second Hilbert 2-class field \(\mathbb {k}_2^{(2)}\) of \(\mathbb {k}\) . We study the capitulation problem of the 2-classes of \(\mathbb {k}\) in its seven unramified quadratic extensions \(\mathbb {K}_i\) and in its seven unramified bicyclic biquadratic extensions \(\mathbb {L}_i\) .     

On the rank of incidence matrices in projective Hjelmslev spaces

Let \(R\) be a finite chain ring with \(|R|=q^m\) , \(R/{{\mathrm{Rad}}}R\cong \mathbb {F}_q\) , and let \(\Omega ={{\mathrm{PHG}}}({}_RR^n)\) . Let \(\tau =(\tau _1,\ldots ,\tau _n)\) be an integer sequence satisfying \(m=\tau _1\ge \tau _2\ge \cdots \ge \tau _n\ge 0\) . We consider the incidence matrix of all shape \(\varvec{m}^s=(\underbrace{m,\ldots ,m}_s)\) versus all shape \(\tau \) subspaces of \(\Omega \) with \(\varvec{m}^s\preceq \tau \preceq \varvec{m}^{n-s}\) . We prove that the rank of \(M_{\varvec{m}^s,\tau }(\Omega )\) over \(\mathbb {Q}\) is equal to the number of shape \(\varvec{m}^s\) subspaces. This is a partial analog of Kantor’s result about the rank of the incidence matrix of all \(s\) dimensional versus all \(t\) dimensional subspaces of \({{\mathrm{PG}}}(n,q)\) . We construct an example for shapes \(\sigma \) and \(\tau \) for which the rank of \(M_{\sigma ,\tau }(\Omega )\) is not maximal.     

The Fourier transform of multiradial functions

We obtain an exact formula for the Fourier transform of multiradial functions, i.e., functions of the form \(\varPhi (x)=\phi (|x_1|, \dots , |x_m|), x_i\in \mathbf R^{n_i}\) , in terms of the Fourier transform of the function \(\phi \) on \(\mathbf R^{r_1}\times \cdots \times \mathbf R^{r_m}\) , where \(r_i\) is either \(1\) or \(2\) .     

Histograms for stationary linear random fields

Denote the integer lattice points in the \(N\) -dimensional Euclidean space by \(\mathbb {Z}^N\) and assume that \(X_\mathbf{n}\) , \(\mathbf{n} \in \mathbb {Z}^N\) is a linear random field. Sharp rates of convergence of histogram estimates of the marginal density of \(X_\mathbf{n}\) are obtained. Histograms can achieve optimal rates of convergence \(({\hat{\mathbf{n}}}^{-1} \log {\hat{\mathbf{n}}})^{1/3}\) where \({\hat{\mathbf{n}}}=n_1 \times \cdots \times n_N\) . The assumptions involved can easily be checked. Histograms appear to be very simple and good estimators from the point of view of uniform convergence.