A discrete Fourier transform associated with the affine Hecke algebra

We introduce an explicit representation of the double affine Hecke algebra (of type _A1$A_1$) at q=1$q=1$ that gives rise to a periodic counterpart of a well-known Fourier transform associated with the affine Hecke algebra.     

Operator norm attainment and inner product spaces

In this paper we prove that a finite dimensional real normed linear space X$\mathbb{X}$ is an inner product space iff for any linear operator T   on X$\mathbb{X}$, T   attains its norm at _e1,_e2∈_SX$e_1,e_2\in S_{\mathbb{X}}$ implies T   attains its norm at span{_e1,_e2}∩_SX$\mathit{span}\{e_1,e_2\}\cap S_{\mathbb{X}}$.     

Dualizability of automatic algebras

We make a start on one of George McNulty's Dozen Easy Problems  : “Which finite automatic algebras are dualizable?” We give some necessary and some sufficient conditions for dualizability. For example, we prove that a finite automatic algebra is dualizable if its letters act as an abelian group of permutations on its states. To illustrate the potential difficulty of the general problem, we exhibit an infinite ascending chain _A1?_A2?_A3??${\mathbf{A}}_1?{\mathbf{A}}_2?{\mathbf{A}}_3??$ of finite automatic algebras that are alternately dualizable and non-dualizable.     

Spokes,mirrors and alternating projections

We give an elementary iterative construction of an example, similar to that of Hundal, of a cone and a half-space in the Hilbert space _?2${?}_2$, which intersect at the origin, but such that the corresponding sequence of alternating nearest point mappings (although weakly convergent) does not converge in norm.     

On some geometric properties of generalized Orlicz–Lorentz function spaces

In this article we continue investigations concerning generalized Orlicz–Lorentz function spaces _Λφ${\Lambda }_{\phi }$ initiated in the papers (Foralewski, 2011)  and  (cf. also (Foralewski, 2008)  and ). First, it is shown that modular _?φ${?}_{\phi }$ is orthogonally subadditive. Next, monotonicity properties are considered. In order to get sufficient conditions for uniform monotonicity of the space _Λφ${\Lambda }_{\phi }$ a strong condition of _Δ2${\Delta }_2$ type and the notion of regularity of the generated Musielak–Orlicz function φ$\phi$ are introduced. Finally, criteria for non-squareness of _Λφ${\Lambda }_{\phi }$ are presented.     

Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians

We classify the real hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians _SU2,m/S(_U2⋅_Um)${\mathit{SU}}_{2,m}/S(U_2\cdot U_m)$, m?2$m?2$. Each can be described as a tube over a totally geodesic _SU2,m−1/S(_U2⋅_Um−1)${\mathit{SU}}_{2,m-1}/S(U_2\cdot U_{m-1})$ in _SU2,m/S(_U2⋅_Um)${\mathit{SU}}_{2,m}/S(U_2\cdot U_m)$ or a horosphere whose center at infinity is singular.     

On the eigenfunction expansions associated with semilinear Sturm–Liouville-type problems

We consider semilinear second-order ordinary differential equations, mainly autonomous, in the form −u^″=f(u)+λu$-u^{″}=f\left(u\right)+\lambda u$, supplied with different sets of standard boundary conditions. Here λ$\lambda$ is a real constant or it plays the role of a spectral parameter. Mainly, we study problems in the interval (0,1)$(0,1)$. It is shown that in this case each problem that we deal with has an infinite sequence of solutions or eigenfunctions. Our aim in the present article is to review recent results on basis properties of sequences of these solutions or eigenfunctions. In a number of cases, it is proved that such a system is a basis in _L2$L_2$ (in addition, a Riesz or Bari basis). In addition, we briefly consider a problem for the half-line (0,∞)$(0,\infty )$. In this case, the spectrum of the problem fills a half-line and an analog of the expansions into the Fourier integral is obtained. The proofs are mainly based on the Bari theorem and, in addition, on our general result on sufficient conditions for a sequence of functions to be a Riesz basis in _L2$L_2$.     

On an open problem concerning total domination critical graphs

A graph G   with no isolated vertex is total domination vertex critical if for any vertex v$v$ of G   that is not adjacent to a vertex of degree one, the total domination number of G-v$G-v$ is less than the total domination number of G  . We call these graphs _γt${\gamma }_t$-critical. If such a graph G has total domination number k, we call it k  -_γt${\gamma }_t$-critical. We verify an open problem of k  -_γt${\gamma }_t$-critical graphs and obtain some results on the characterization of total domination critical graphs of order n=Δ(G)(_γt(G)-1)+1$n=\Delta (G)({\gamma }_t(G)-1)+1$.