On Star–Wheel Ramsey Numbers

For two given graphs \(G_1\) and \(G_2\), the Ramsey number \(R(G_1,G_2)\) is the least integer r such that for every graph G on r vertices, either G contains a \(G_1\) or \(\overline{G}\) contains a \(G_2\). In this note, we determined the Ramsey number \(R(K_{1,n},W_m)\) for even m with \(n+2\le m\le 2n-2\), where \(W_m\) is the wheel on \(m+1\) vertices, i.e., the graph obtained from a cycle \(C_m\) by adding a vertex v adjacent to all vertices of the \(C_m\).     

On left-invariant Hörmander operators in $ {\mathbb{R}^N} $. applications to Kolmogorov–Fokker–Planck equations

If \( \mathcal{L} = \sum\limits_{j = 1}^m {X_j^2} + {X_0} \) is a Hörmander partial differential operator in \( {\mathbb{R}^N} \), we give sufficient conditions on the \( {X_{{j^{\text{S}}}}} \) for the existence of a Lie group structure \( \mathbb{G} = \left( {{\mathbb{R}^N},*} \right) \), not necessarily nilpotent, such that \( \mathcal{L} \) is left invariant on \( \mathbb{G} \). We also investigate the existence of a global fundamental solution Γ for \( \mathcal{L} \), providing results that ensure a suitable left-invariance property of Γ. Examples are given for operators \( \mathcal{L} \) to which our results apply: some are new; some have appeared in recent literature, usually quoted as Kolmogorov–Fokker–Planck-type operators. Nontrivial examples of homogeneous groups are also given.     

A method to find generators of a semi-simple Lie group via the topology of its flag manifolds

In this paper we continue to develop the topological method to get semigroup generators of semi-simple Lie groups. Consider a subset \(\Gamma \subset G\) that contains a semi-simple subgroup \(G_{1}\) of G. If one can show that \( \Gamma \) does not leave invariant a contractible subset on any flag manifold of G, then \(\Gamma \) generates G if \(\mathrm {Ad}\left( \Gamma \right) \) generates a Zariski dense subgroup of the algebraic group \(\mathrm {Ad}\left( G\right) \). The proof is reduced to check that some specific closed orbits of \(G_{1}\) in the flag manifolds of G are not trivial in the sense of algebraic topology. Here, we consider three different cases of semi-simple Lie groups G and subgroups \(G_{1}\subset G\).     

Infra-nilmanifolds modeled on the group of uni-triangular matrices

Let \({\mathcal {N}}_m\) be the group of \(m\times m\) upper triangular real matrices with all the diagonal entries 1. Then it is an \((m-1)\)-step nilpotent Lie group, diffeomorphic to \({\mathbb {R}}^{\frac{1}{2} m(m-1)}\). It contains all the integer matrices as a lattice \(\Gamma _m\). The automorphism group of \({\mathcal {N}}_m \ (m\ge 4)\) turns out to be extremely small. In fact, \(\mathrm {Aut}({\mathcal {N}})=\mathcal {I} \rtimes \mathrm {Out}({\mathcal {N}})\), where \(\mathcal {I}\) is a connected, simply connected nilpotent Lie group, and \(\mathrm {Out}({\mathcal {N}})={{\tilde{K}}}={(\mathbb {R}^*)^{m-1}\rtimes \mathbb {Z}_2}\). With a nice left-invariant Riemannian metric on \({\mathcal {N}}\), the isometry group is \(\mathrm {Isom}({\mathcal {N}})= {\mathcal {N}} \rtimes K\), where \(K={(\mathbb {Z}_2)^{m-1}\rtimes \mathbb {Z}_2}\subset {{\tilde{K}}}\) is a maximal compact subgroup of \(\mathrm {Aut}({\mathcal {N}})\). We prove that, for odd \(m\ge 4\), there is no infra-nilmanifold which is essentially covered by the nilmanifold \(\Gamma _m\backslash {\mathcal {N}}_m\). For \(m=2n\ge 4\) (even), there is a unique infra-nilmanifold which is essentially (and doubly) covered by the nilmanifold \(\Gamma _m\backslash {\mathcal {N}}_m\).     

Families of optimal packings in real and complex Grassmannian spaces

A construction based on a \(4l \times 4l\) Hadamard matrix leads to a new family of optimal orthoplex packings in Grassmannian spaces \(G_{\mathbb {R}}(8l, 4l)\) and \(G_{\mathbb {C}}(4l, 2l)\). A related construction gives an optimal simplex packings in \(G_{\mathbb {R}}(8 l-1, 4 l - 1)\) and \(G_{\mathbb {R}}(8l-1, 4l)\) with the additional assumption of an \(8l \times 8l\) skew Hadamard matrix and a related 1-factorization of a complete graph. A construction of a maximal optimal simplex packings in \(G_{\mathbb {C}}(2l-1, l- 1)\) and \(G_{\mathbb {C}}(2l-1,l)\) is given.     

Complete classification of pseudo <Emphasis Type="Italic">H</Emphasis>-type Lie algebras: I

Let \({\mathscr {N}}\) be a 2-step nilpotent Lie algebra endowed with a non-degenerate scalar product \(\langle .\,,.\rangle \), and let \({\mathscr {N}}=V\oplus _{\perp }Z\), where Z is the centre of the Lie algebra and V its orthogonal complement. We study classification of the Lie algebras for which the space V arises as a representation space of the Clifford algebra \({{\mathrm{{\mathrm{Cl}}}}}({\mathbb {R}}^{r,s})\), and the representation map \(J:{{\mathrm{{\mathrm{Cl}}}}}({\mathbb {R}}^{r,s})\rightarrow {{\mathrm{End}}}(V)\) is related to the Lie algebra structure by \(\langle J_zv,w\rangle =\langle z,[v,w]\rangle \) for all \(z\in {\mathbb {R}}^{r,s}\) and \(v,w\in V\). The classification depends on parameters r and s and is completed for the Clifford modules V having minimal possible dimension, that are not necessary irreducible. We find necessary conditions for the existence of a Lie algebra isomorphism according to the range of the integer parameters \(0\le r,s<\infty \). We present a constructive proof for the isomorphism maps for isomorphic Lie algebras and determine the class of non-isomorphic Lie algebras.     

On the Analytic Structure of Commutative Nilmanifolds

In the classification theorems of Vinberg and Yakimova for commutative nilmanifolds, the relevant nilpotent groups have a very surprising analytic property. The manifolds are of the form \(G/K = N\rtimes K/K\) where, in all but three cases, the nilpotent group \(N\) has irreducible unitary representations whose coefficients are square integrable modulo the center \(Z\) of \(N\). Here we show that, in those three “exceptional” cases, the group \(N\) is a semidirect product \(N_{1}\rtimes \mathbb {R}\) or \(N_{1}\rtimes \mathbb {C}\) where the normal subgroup \(N_{1}\) contains the center \(Z\) of \(N\) and has irreducible unitary representations whose coefficients are square integrable modulo \(Z\). This leads directly to explicit harmonic analysis and Fourier inversion formulae for commutative nilmanifolds.     

On the partition dimension of two-component graphs

In this paper, we continue investigating the partition dimension for disconnected graphs. We determine the partition dimension for some classes of disconnected graphs G consisting of two components. If \(G=G_1 \cup G_2\), then we give the bounds of the partition dimension of G for \(G_1 = P_n\) or \(G_1=C_n\) and also for \(pd(G_1)=pd(G_2)\).     

Spectral synthesis for coadjoint orbits of nilpotent Lie groups

We determine the space of primary ideals in the group algebra \(L^{1}(G) \) of a connected nilpotent Lie group by identifying for every \(\pi \in \widehat{G} \) the family \(\mathcal I^\pi \) of primary ideals with hull \(\{\pi \} \) with the family of invariant subspaces of a certain finite dimensional sub-space \(\mathcal P_Q^\pi \) of the space of polynomials \(\mathcal P(G) \) on G.     

Reflecting Diffusion Processes on Manifolds Carrying Geometric Flow

Let \(L_t:=\Delta _t+Z_t\) for a \(C^{\infty }\)-vector field Z on a differentiable manifold M with boundary \(\partial M\), where \(\Delta _t\) is the Laplacian operator, induced by a time dependent metric \(g_t\) differentiable in \(t\in [0,T_\mathrm {c})\). We first establish the derivative formula for the associated reflecting diffusion semigroup generated by \(L_t\). Then, by using parallel displacement and reflection, we construct the couplings for the reflecting \(L_t\)-diffusion processes, which are applied to gradient estimates and Harnack inequalities of the associated heat semigroup. Finally, as applications of the derivative formula, we present a number of equivalent inequalities for a new curvature lower bound and the convexity of the boundary. These inequalities include the gradient estimates, Harnack inequalities, transportation-cost inequalities and other functional inequalities for diffusion semigroups.     

Left annihilator of generalized derivations on Lie ideals in prime rings

Let \(R\) be a prime ring, \(L\) a noncentral Lie ideal of \(R\), \(F\) a generalized derivation with associated nonzero derivation \(d\) of \(R\). If \(a\in R\) such that \(a(d(u)^{l_1} F(u)^{l_2} d(u)^{l_3} F(u)^{l_4} \ldots F(u)^{l_k})^{n}=0\) for all \(u\in L\), where \(l_1,l_2,\ldots ,l_k\) are fixed non negative integers not all are zero and \(n\) is a fixed integer, then either \(a=0\) or \(R\) satisfies \(s_4\), the standard identity in four variables.     

Toric Aspects of the First Eigenvalue

In this paper we study the smallest non-zero eigenvalue \(\lambda _1\) of the Laplacian on toric Kähler manifolds. We find an explicit upper bound for \(\lambda _1\) in terms of moment polytope data. We show that this bound can only be attained for \(\mathbb C\mathbb P^n\) endowed with the Fubini–Study metric and therefore \(\mathbb C\mathbb P^n\) endowed with the Fubini–Study metric is spectrally determined among all toric Kähler metrics. We also study the equivariant counterpart of \(\lambda _1\) which we denote by \(\lambda _1^T\). It is the smallest non-zero eigenvalue of the Laplacian restricted to torus-invariant functions. We prove that \(\lambda _1^T\) is not bounded among toric Kähler metrics thus generalizing a result of Abreu–Freitas on \(S^2\). In particular, \(\lambda _1^T\) and \(\lambda _1\) do not coincide in general.     

First eigenvalues of geometric operators under the Yamabe flow

Suppose \((M,g_0)\) is a compact Riemannian manifold without boundary of dimension \(n\ge 3\). Using the Yamabe flow, we obtain estimate for the first nonzero eigenvalue of the Laplacian of \(g_0\) with negative scalar curvature in terms of the Yamabe metric in its conformal class. On the other hand, we prove that the first eigenvalue of some geometric operators on a compact Riemannian manifold is nondecreasing along the unnormalized Yamabe flow under suitable curvature assumption. Similar results are obtained for manifolds with boundary and for CR manifold.     

Negative Ricci curvature on some non-solvable Lie groups

We show that for any non-trivial representation \((V, \pi )\) of \(\mathfrak {u}(2)\) with the center acting as multiples of the identity, the semidirect product \(\mathfrak {u}(2) \ltimes _\pi V\) admits a metric with negative Ricci curvature that can be explicitly obtained. It is proved that \(\mathfrak {u}(2) \ltimes _\pi V\) degenerates to a solvable Lie algebra that admits a metric with negative Ricci curvature. An n-dimensional Lie group with compact Levi factor \(\mathrm {SU}(2)\) admitting a left invariant metric with negative Ricci is therefore obtained for any \(n \ge 7\).     

Bumpy metrics on spheres and minimal index growth

The existence of two geometrically distinct closed geodesics on an n-dimensional sphere \(S^n\) with a non-reversible and bumpy Finsler metric was shown independently by Duan and Long [7] and the author [25]. We simplify the proof of this statement by the following observation: If for some \(N \in \mathbb {N}\) all closed geodesics of index \(\le \)N of a non-reversible and bumpy Finsler metric on \(S^n\) are geometrically equivalent to the closed geodesic c, then there is a covering \(c^r\) of minimal index growth, i.e.,

$$\begin{aligned} \mathrm{ind}(c^\mathrm{rm})=m \,\mathrm{ind}(c^r)-(m-1)(n-1), \end{aligned}$$

for all \(m \ge 1\) with \(\mathrm{ind}\left( c^\mathrm{rm}\right) \le N.\) But this leads to a contradiction for \(N =\infty \) as pointed out by Goresky and Hingston [13]. We also discuss perturbations of Katok metrics on spheres of even dimension carrying only finitely many closed geodesics. For arbitrarily large \(L>0\), we obtain on \(S^2\) a metric of positive flag curvature carrying only two closed geodesics of length \(<L\) which do not intersect.

     

Universal inequalities on complete noncompact smooth metric measure spaces

In this paper, we obtain universal inequalities for the eigenvalues of the Dirichlet problem and clamped plate problem of drifting Laplacian on (\(n+1\))-dimensional (\(n\ge 4\)) complete noncompact simply connected smooth metric measure spaces which meet some conditions of the sectional curvature and radial weighted Ricci curvature.     

Equitorsion holomorphically projective mappings of generalized Kählerian space of the first kind

In this paper we define generalized Kählerian spaces of the first kind (\(G_1^K N\)) given by (2.1)–(2.3). For them we consider hollomorphically projective mappings with invariant complex structure. Also, we consider equitorsion geodesic mapping between these two spaces (\(G_1^K N\) and \(G_1^{\bar K} N\)) and for them we find invariant geometric objects.     

A Positivstellensatz for forms on the positive orthant

We shall extend the research on power structure of finite p-groups in Mann (J Algebra 42:121–135, 1976) to locally nilpotentp-groups. Firstly, we obtain that a locally nilpotent \(P_i\)-group G with \(|G:\mho _1(G)|< \infty \) is an extension of a divisible abelian group by a finite p-group. Next we get the structure of infinite locally nilpotent p-groups which are not \(P_i\)-groups, but all of whose proper infinite subgroups are \(P_i\)-groups. Finally, we show that locally nilpotent \(P_i\)-groups with all subgroups subnormal are nilpotent.     

Gravitational Field Equations on Fefferman Space–Times

The total space \({\mathfrak M} \approx {\mathbb H}_1 \times S^1\) of the canonical circle bundle over the 3-dimensional Heisenberg group \({\mathbb H}_1\) is a space–time with the Lorentzian metric \(F_{\theta _0}\) (Fefferman’s metric) associated to the canonical Tanaka–Webster flat contact form \(\theta _0\) on \({\mathbb H}_1\). The matter and energy content of \(\mathfrak M\) is described by the energy-momentum tensor \({T}_{\mu \nu }\) (the trace-less Ricci tensor of \(F_{\theta _0}\)) as an effect of the non flat nature of Feferman’s metric \(F_{\theta _0}\). We study the gravitational field equations \(R_{\mu \nu } - (1/2) \, R \, g_{\mu \nu } = {T}_{\mu \nu }\) on \({\mathfrak M}\). We consider the first order perturbation \(g = F_{\theta _0} + \epsilon \, h\), \(\epsilon<< 1\), and linearize the field equations about \(F_{\theta _0}\). We determine a Lorentzian metric g on \({\mathfrak M}\) which solves the linearized field equations corresponding to a diagonal perturbation h.     

Representations and Corresponding Operators Induced by Hecke Algebras

In this paper, by establishing free-probabilistic models on the Hecke algebras \(\mathcal {H}(G_{p})\), we construct canonical free probability spaces \((\mathcal {H}(G_{p}), \psi _{p})\), where \(G_{p} = GL_{2}(\mathbb {Q} _{p})\), for primes \(p\). Dependent upon such free-probabilistic structures, we study corresponding representations of \(\mathcal {H}(G_{p})\), and consider spectral properties of operators realized under representations.