ANCOVA: a heteroscedastic global test when there is curvature and two covariates

For two independent groups, let \(M_j(\mathbf {X})\) be some conditional measure of location for the jth group associated with some random variable Y given \(\mathbf {X}=(X_1, X_2)\). Let \(\Omega =\{\mathbf {X}_1, \ldots , \mathbf {X}_K\}\) be a set of K points to be determined. An extant technique can be used to test \(H_0\): \(M_1(\mathbf {X})=M_2(\mathbf {X})\) for each \(\mathbf {X} \in \Omega \) without making any parametric assumption about \(M_j(\mathbf {X})\). But there are two general reasons to suspect that the method can have relatively low power. The paper reports simulation results on an alternative approach that is designed to test the global hypothesis \(H_0\): \(M_1(\mathbf {X})=M_2(\mathbf {X})\) for all \(\mathbf {X} \in \Omega \). The main result is that the new method offers a distinct power advantage. Using data from the Well Elderly 2 study, it is illustrated that the alternative method can make a practical difference in terms of detecting a difference between two groups.     

Korovkin-Type Approximation Theory in Riesz Spaces

In this paper, we prove the following Riesz spaces’ version of the Korovkin theorem. Let E and F be two Archimedean Riesz spaces with F uniformly complete, let W be a nonempty subset of \(E^{+}\), and let \((T_{n})\) be a given sequence of (r-u)-continuous elements of \(\mathcal {L(}E,F)\), such that \(\left| T_{n}-T_{m}\right| x=\left| (T_{n}-T_{m})x\right| \mathcal {\ }\)for all \(x\in E^{+},\) \(m,n\ge n_{0}\) (for a given \(n_{0}\in \mathbb {N} )\). If the sequence \((T_{n}x)_{n}\) \((r-u)\)-converges for every \(x\in W\), then \((T_{n})\) \((r-u)\)-converges also pointwise on the ideal \(E_{W}\), generated by W, to a linear operator \(S_{0}:E_{W}\rightarrow F\). We also prove a similar Korovkin-type theorem for nets of operators. Some applications for f-algebras and orthomorphisms are presented.     

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 \(C_m\).     

Closed range and nonclosed range adjointable operators on Hilbert $$C^*$$-modules

In this paper, we show that for a positive operator A on a Hilbert \(C^*\)-module \( \mathscr {E} \), the range \( \mathscr {R}(A) \) of A is closed if and only if \( \mathscr {R}(A^\alpha ) \) is closed for all \(\alpha \in (0,1)\cup (1,+\,\infty )\), and this occurs if and only if \( \mathscr {R}(A)=\mathscr {R}(A^\alpha ) \) for all \(\alpha \in (0,1)\cup (1,+\,\infty )\). As an application, we prove that for an adjontable operator A if \(\mathscr {R}(A)\) is nonclosed, then \(\dim \left( \overline{\mathscr {R}(A)}/\mathscr {R}(A)\right) =+\,\infty \). Finally, we show that for an adjointable operator A if \( \overline{\mathscr {R}(A^*) } \) is orthogonally complemented in \( \mathscr {E} \), then under certain coditions there exists an idempotent C and a unique operator X such that \( XAX=X, AXA=CA, AX=C \) and \( XA=P_{A^*} \), where \( P_{A^*} \) is the orthogonal projection of \( \mathscr {E} \) onto \( \overline{\mathscr {R}(A^*)}\).     

Hyperinvariant Subspace Problem for Some Classes of Operators

An n-normal operator may be defined as an \(n \times n\) operator matrix with entries that are mutually commuting normal operators and an operator \(T \in \mathcal {B(H)}\) is quasi-nM-hyponormal (for \(n \in \mathbb {N}\)) if it is unitarily equivalent to an \(n \times n\) upper triangular operator matrix \((T_{ij})\) acting on \(\mathcal {K}^{(n)}\), where \(\mathcal {K}\) is a separable complex Hilbert space and the diagonal entries \(T_{jj}\) \((j = 1,2,\ldots , n)\) are M-hyponormal operators in \(\mathcal {B(K)}\). This is an extended notion of n-normal operators. We prove a necessary and sufficient condition for an \(n \times n\) triangular operator matrix to have Bishop’s property \((\beta )\). This leads us to study the hyperinvariant subspace problem for an \(n \times n\) triangular operator matrix.     

Lengths of words in transformation semigroups generated by digraphs

Given a simple digraph D on n vertices (with \(n\ge 2\)), there is a natural construction of a semigroup of transformations \(\langle D\rangle \). For any edge (a, b) of D, let \(a\rightarrow b\) be the idempotent of rank \(n-1\) mapping a to b and fixing all vertices other than a; then, define \(\langle D\rangle \) to be the semigroup generated by \(a \rightarrow b\) for all \((a,b) \in E(D)\). For \(\alpha \in \langle D\rangle \), let \(\ell (D,\alpha )\) be the minimal length of a word in E(D) expressing \(\alpha \). It is well known that the semigroup \(\mathrm {Sing}_n\) of all transformations of rank at most \(n-1\) is generated by its idempotents of rank \(n-1\). When \(D=K_n\) is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate \(\ell (K_n,\alpha )\), for any \(\alpha \in \langle K_n\rangle = \mathrm {Sing}_n\); however, no analogous non-trivial results are known when \(D \ne K_n\). In this paper, we characterise all simple digraphs D such that either \(\ell (D,\alpha )\) is equal to Howie–Iwahori’s formula for all \(\alpha \in \langle D\rangle \), or \(\ell (D,\alpha ) = n - \mathrm {fix}(\alpha )\) for all \(\alpha \in \langle D\rangle \), or \(\ell (D,\alpha ) = n - \mathrm {rk}(\alpha )\) for all \(\alpha \in \langle D\rangle \). We also obtain bounds for \(\ell (D,\alpha )\) when D is an acyclic digraph or a strong tournament (the latter case corresponds to a smallest generating set of idempotents of rank \(n-1\) of \(\mathrm {Sing}_n\)). We finish the paper with a list of conjectures and open problems.     

Map determined by rank-$$\varvec{}$$ matrice for relatively mall $$\varvec{}$$

Let n and s be integers such that \(1\le s<\frac{n}{2}\), and let \(M_n(\mathbb {K})\) be the ring of all \(n\times n\) matrices over a field \(\mathbb {K}\). Denote by \([\frac{n}{s}]\) the least integer m with \(m\ge \frac{n}{s}\). In this short note, it is proved that if \(g:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\) is a map such that \(g\left( \sum _{i=1}^{[\frac{n}{s}]}A_i\right) =\sum _{i=1}^{[\frac{n}{s}]}g(A_i)\) holds for any \([\frac{n}{s}]\) rank-s matrices \(A_1,\ldots ,A_{[\frac{n}{s}]}\in M_n(\mathbb {K})\), then \(g(x)=f(x)+g(0)\), \(x\in M_n(\mathbb {K})\), for some additive map \(f:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\). Particularly, g is additive if \(char\mathbb {K}\not \mid \left( [\frac{n}{s}]-1\right) \).     

Extremal Problems for Mappings with Generalized Parametric Representation in $${{\mathbb {C}}}^{n}$$

In this paper we are concerned with the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) (\(t\ge 0\)) of normalized biholomorphic mappings on the Euclidean unit ball \(\mathbb {B}^n\) in \({\mathbb {C}}^n\) that can be embedded in normal Loewner chains whose normalizations are given by time-dependent operators \(A\in \widetilde{\mathcal {A}}\), where \(\widetilde{\mathcal {A}}\) is a family of measurable mappings from \([0,\infty )\) into \(L({\mathbb {C}}^n)\) which satisfy certain natural assumptions. In particular, we consider extreme points and support points associated with the compact family \(\widetilde{S}^t_A(\mathbb {B}^n)\), where \(A\in \widetilde{\mathcal {A}}\). We prove that if \(f(z,t)=V(t)^{-1}z+\cdots \) is a normal Loewner chain such that \(V(s)f(\cdot ,s)\in \mathrm{ex}\,\widetilde{S}^s_A(\mathbb {B}^n)\) (resp. \(V(s)f(\cdot ,s)\in \mathrm{supp}\,\widetilde{S}^s_A(\mathbb {B}^n)\)), then \(V(t)f(\cdot ,t)\in \mathrm{ex}\, \widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\) (resp. \(V(t)f(\cdot ,t)\in \mathrm{supp}\,\widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\)), where V(t) is the unique solution on \([0,\infty )\) of the initial value problem: \(\frac{d V}{d t}(t)=-A(t)V(t)\), a.e. \(t\ge 0\), \(V(0)=I_n\). Also, we obtain an example of a bounded support point for the family \(\widetilde{S}_A^t(\mathbb {B}^2)\), where \(A\in \widetilde{\mathcal {A}}\) is a certain time-dependent operator. We also consider the notion of a reachable family with respect to time-dependent linear operators \(A\in \widetilde{\mathcal {A}}\), and obtain characterizations of extreme/support points associated with these families of bounded biholomorphic mappings on \(\mathbb {B}^n\). Useful examples and applications yield that the study of the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) for time-dependent operators \(A\in \widetilde{\mathcal {A}}\) is basically different from that in the case of constant time-dependent linear operators.     

Dimensions of fibers of generic continuous maps

In an earlier paper Buczolich, Elekes, and the author described the Hausdorff dimension of the level sets of a generic real-valued continuous function (in the sense of Baire category) defined on a compact metric space K by introducing the notion of topological Hausdorff dimension. Later on, the author extended the theory for maps from K to \({\mathbb {R}}^n\). The main goal of this paper is to generalize the relevant results for topological and packing dimensions and to obtain new results for sufficiently homogeneous spaces K even in the case case of Hausdorff dimension. Let K be a compact metric space and let us denote by \(C(K,{\mathbb {R}}^n)\) the set of continuous maps from K to \({\mathbb {R}}^n\) endowed with the maximum norm. Let \(\dim _{*}\) be one of the topological dimension \(\dim _T\), the Hausdorff dimension \(\dim _H\), or the packing dimension \(\dim _P\). Define

$$\begin{aligned} d_{*}^n(K)=\inf \left\{ \dim _{*}(K{\setminus } F): F\subset K \text { is } \sigma \text {-compact with } \dim _T F<n\right\} . \end{aligned}$$

We prove that \(d^n_{*}(K)\) is the right notion to describe the dimensions of the fibers of a generic continuous map \(f\in C(K,{\mathbb {R}}^n)\). In particular, we show that \(\sup \{\dim _{*}f^{-1}(y): y\in {\mathbb {R}}^n\} =d^n_{*}(K)\) provided that \(\dim _T K\ge n\), otherwise every fiber is finite. Proving the above theorem for packing dimension requires entirely new ideas. Moreover, we show that the supremum is attained on the left hand side of the above equation. Assume \(\dim _T K\ge n\). If K is sufficiently homogeneous, then we can say much more. For example, we prove that \(\dim _{*}f^{-1}(y)=d^n_{*}(K)\) for a generic \(f\in C(K,{\mathbb {R}}^n)\) for all \(y\in {{\mathrm{int}}}f(K)\) if and only if \(d^n_{*}(U)=d^n_{*}(K)\) or \(\dim _T U<n\) for all open sets \(U\subset K\). This is new even if \(n=1\) and \(\dim _{*}=\dim _H\). It is known that for a generic \(f\in C(K,{\mathbb {R}}^n)\) the interior of f(K) is not empty. We augment the above characterization by showing that \(\dim _T \partial f(K)=\dim _H \partial f(K)=n-1\) for a generic \(f\in C(K,{\mathbb {R}}^n)\). In particular, almost every point of f(K) is an interior point. In order to obtain more precise results, we use the concept of generalized Hausdorff and packing measures, too.

     

Eichler’s commutation relation and some other invariant subspaces of Hecke operators

Let k be an odd positive integer, L a lattice on a regular positive definite k-dimensional quadratic space over \(\mathbb {Q}\), \(N_L\) the level of L, and \(\mathscr {M}(L)\)  be the linear space of \(\theta \)-series attached to the distinct classes in the genus of L. We prove that, for an odd prime \(p|N_L\), if \(L_p=L_{p,1}\,\bot \, L_{p,2}\), where \(L_{p,1}\) is unimodular, \(L_{p,2}\) is (p)-modular, and \(\mathbb {Q}_pL_{p,2}\) is anisotropic, then \(\mathscr {M}(L;p):=\) \(\mathscr {M}(L)\) \(+T_{p^2}.\) \(\mathscr {M}(L)\)  is stable under the Hecke operator \(T_{p^2}\). If \(L_2\) is isometric to \(\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle \) or \(\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle 2\varepsilon \rangle \) or \(\left( \begin{array}{ll}0&{}1\\ 1&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle \) with \(\varepsilon \in \mathbb {Z}_2^{\times }\) and \(\kappa :=\frac{k-1}{2}\), then \(\mathscr {M}(L;2):=T_{2^2}.\mathscr {M}(L)+T_{2^2}^2.\,\mathscr {M}(L)\) is stable under the Hecke operator \(T_{2^2}\). Furthermore, we determine some invariant subspaces of the cusp forms for the Hecke operators.     

A characterization of nonlinear Hölder seminorm preserving bijections

For a compact metric space (X, d) and \(\alpha \in (0,1)\), let \(\mathrm{Lip}^\alpha (X)\) be the linear space of all complex-valued functions f on X satisfying

Open image in new window

and \(\mathrm{lip}^\alpha (X)\) be the subspace of \(\mathrm{Lip}^\alpha (X)\) consisting of functions f with \(\lim \frac{f(x)-f(y)}{d^\alpha (x,y)} =0\) as \(d(x,y) \rightarrow 0\). In this paper, we give a characterization of a bijective map \(T:\mathrm{lip}^\alpha (X)\longrightarrow \mathrm{lip}^\alpha (Y)\), not necessarily linear, which is an isometry with respect to the Hölder seminorm \(L(\cdot )\). It is shown that there exist \(K_0>0\), a surjective map \(\Psi : Y \longrightarrow X\) with \(d^\alpha (y,z)= K_0 \, d^\alpha (\Psi (y),\Psi (z))\) for all \(y,z\in Y\), and a function \(\Lambda : \mathrm{lip}^\alpha (X) \longrightarrow {\mathbb {C}}\) (which is linear or real-linear if T is so) such that either

$$\begin{aligned} Tf(y)= T0(y)+\overline{\tau } K_0\, f(\Psi (y))+\Lambda (f)\quad (f\in \mathrm{lip}^\alpha (X), y\in Y) \end{aligned}$$

or

$$\begin{aligned} Tf(y)= T0(y)+\overline{\tau } K_0 \,\overline{f(\Psi (y))}+ \Lambda (f)\quad (f\in \mathrm{lip}^\alpha (X), y\in Y), \end{aligned}$$

where \(\tau =e^{i\theta }\) for some \(\theta \in [0,\pi )\).

     

Sandwich semigroups in locally small categories II: transformations

Fix sets X and Y, and write \(\mathcal P\mathcal T_{XY}\) for the set of all partial functions \(X\rightarrow Y\). Fix a partial function \({a:Y\rightarrow X}\), and define the operation \(\star _a\) on \(\mathcal P\mathcal T_{XY}\) by \(f\star _ag=fag\) for \(f,g\in \mathcal P\mathcal T_{XY}\). The sandwich semigroup \((\mathcal P\mathcal T_{XY},\star _a)\) is denoted \(\mathcal P\mathcal T_{XY}^a\). We apply general results from Part I to thoroughly describe the structural and combinatorial properties of \(\mathcal P\mathcal T_{XY}^a\), as well as its regular and idempotent-generated subsemigroups, \({\text {Reg}}(\mathcal P\mathcal T_{XY}^a)\) and \(\mathbb E(\mathcal P\mathcal T_{XY}^a)\). After describing regularity, stability and Green’s relations and preorders, we exhibit \({\text {Reg}}(\mathcal P\mathcal T_{XY}^a)\) as a pullback product of certain regular subsemigroups of the (non-sandwich) partial transformation semigroups \(\mathcal P\mathcal T_X\) and \(\mathcal P\mathcal T_Y\), and as a kind of “inflation” of \(\mathcal P\mathcal T_A\), where A is the image of the sandwich element a. We also calculate the rank (minimal size of a generating set) and, where appropriate, the idempotent rank (minimal size of an idempotent generating set) of \(\mathcal P\mathcal T_{XY}^a\), \({\text {Reg}}(\mathcal P\mathcal T_{XY}^a)\) and \(\mathbb E(\mathcal P\mathcal T_{XY}^a)\). The same program is also carried out for sandwich semigroups of totally defined functions and for injective partial functions. Several corollaries are obtained for various (non-sandwich) semigroups of (partial) transformations with restricted image, domain and/or kernel.     

Couplings of Brownian Motions of Deterministic Distance in Model Spaces of Constant Curvature

We consider the model space \(\mathbb {M}^{n}_{K}\) of constant curvature K and dimension \(n\ge 1\) (Euclidean space for \(K=0\), sphere for \(K>0\) and hyperbolic space for \(K<0\)), and we show that given a function \(\rho :[0,\infty )\rightarrow [0, \infty )\) with \(\rho (0)=\mathrm {dist}(x,y)\) there exists a coadapted coupling (X(t), Y(t)) of Brownian motions on \(\mathbb {M}^{n}_{K}\) starting at (x, y) such that \(\rho (t)=\mathrm {dist}(X(t),Y(t))\) for every \(t\ge 0\) if and only if \(\rho \) is continuous and satisfies for almost every \(t\ge 0\) the differential inequality

$$\begin{aligned} -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) \le \rho '(t)\le -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) +\tfrac{2(n-1)\sqrt{K}}{\sin (\sqrt{K}\rho (t))}. \end{aligned}$$

In other words, we characterize all coadapted couplings of Brownian motions on the model space \(\mathbb {M}^{n}_{K}\) for which the distance between the processes is deterministic. In addition, the construction of the coupling is explicit for every choice of \(\rho \) satisfying the above hypotheses.

     

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.     

Constructive Approximation in de Branges–Rovnyak Spaces

In most classical holomorphic function spaces on the unit disk in which the polynomials are dense, a function f can be approximated in norm by its dilates \(f_r(z):=f(rz)~(r<1)\). We show that this is not the case for the de Branges–Rovnyak spaces \(\mathcal{H}(b)\). More precisely, we exhibit a space \(\mathcal{H}(b)\) in which the polynomials are dense and a function \(f\in \mathcal{H}(b)\) such that \(\lim _{r\rightarrow 1^-}\Vert f_r\Vert _{\mathcal{H}(b)}=\infty \). On the positive side, we prove the following approximation theorem for Toeplitz operators on general de Branges–Rovnyak spaces \(\mathcal{H}(b)\). If \((h_n)\) is a sequence in \(H^\infty \) such that \(\Vert h_n\Vert _{H^\infty }\le 1\) and \(h_n(0)\rightarrow 1\), then \(\Vert T_{\overline{h}_n}f-f\Vert _{\mathcal{H}(b)}\rightarrow 0\) for all \(f\in \mathcal{H}(b)\). Using this result, we give the first constructive proof that, if b is a nonextreme point of the unit ball of \(H^\infty \), then the polynomials are dense in \(\mathcal{H}(b)\).     

Random Walks in the Hyperbolic Plane and the Minkowski Question Mark Function

Consider \(G=SL_2(\mathbb {Z})/\{\pm I\}\) acting on the complex upper half plane H by \(h_M(z)=\frac{az\,+\,b}{cz\,+\,d}\) for \(M \in G\). Let \(D=\{z \in H: |z|\ge 1, |\mathfrak {R}(z)|\le 1/2\}\). We consider the set \({\mathcal {E}} \subset G\) with the nine elements M, different from the identity, such that \(\mathrm{tr\,}(MM^T)\le 3\). We equip the tiling of H defined by \(\mathbb {D}=\{h_M(D){:}\, M \in G\}\) with a graph structure where the neighbours are defined by \(h_M(D) \cap h_{M'}(D) \ne \emptyset \), equivalently \(M^{-1}M' \in {\mathcal {E}}\). The present paper studies several Markov chains related to the above structure. We show that the simple random walk on the above graph converges a.s. to a point X of the real line with the same distribution of \(S_2 W^{S_1}\), where \(S_1,S_2,W\) are independent with \(\Pr (S_i=\pm 1)=1/2\) and where W is valued in (0, 1) with distribution \(\Pr (W<w)=\mathbf ? (w)\). Here \(\mathbf ? \) is the Minkowski function. If \(K_1, K_2, \ldots \) are i.i.d with distribution \(\Pr (K_i=n)= 1/2^n\) for \(n=1,2,\ldots \), then \(W= \frac{1}{K_1+\frac{1}{K_2+\ldots }}\): this known result (Isola in Appl Math 5:1067–1090, 2014) is derived again here.     

Linear sets in the projective line over the endomorphism ring of a finite field

Let \({{\mathrm{{PG}}}}(1,E)\) be the projective line over the endomorphism ring \( E={{\mathrm{End}}}_q({\mathbb F}_{q^t})\) of the \({\mathbb F}_q\)-vector space \({\mathbb F}_{q^t}\). As is well known, there is a bijection \(\varPsi :{{\mathrm{{PG}}}}(1,E)\rightarrow {\mathcal G}_{2t,t,q}\) with the Grassmannian of the \((t-1)\)-subspaces in \({{\mathrm{{PG}}}}(2t-1,q)\). In this paper along with any \({\mathbb F}_q\)-linear set L of rank t in \({{\mathrm{{PG}}}}(1,q^t)\), determined by a \((t-1)\)-dimensional subspace \(T^\varPsi \) of \({{\mathrm{{PG}}}}(2t-1,q)\), a subset \(L_T\) of \({{\mathrm{{PG}}}}(1,E)\) is investigated. Some properties of linear sets are expressed in terms of the projective line over the ring E. In particular, the attention is focused on the relationship between \(L_T\) and the set \(L'_T\), corresponding via \(\varPsi \) to a collection of pairwise skew \((t-1)\)-dimensional subspaces, with \(T\in L'_T\), each of which determine L. This leads among other things to a characterization of the linear sets of pseudoregulus type. It is proved that a scattered linear set L related to \(T\in {{\mathrm{{PG}}}}(1,E)\) is of pseudoregulus type if and only if there exists a projectivity \(\varphi \) of \({{\mathrm{{PG}}}}(1,E)\) such that \(L_T^\varphi =L'_T\).     

Lower order eigenvalues of a system of equations of the drifting Laplacian on the metric measure spaces

Let \(\Omega \) be a bounded domain with smooth boundary in an n-dimensional metric measure space \((\mathbb {R}^n, \langle ,\rangle , e^{-\phi }dv)\) and let \(\mathbf {u}=(u^1, \ldots , u^n)\) be a vector-valued function from \(\Omega \) to \(\mathbb {R}^n\). In this paper, we investigate the Dirichlet eigenvalue problem of a system of equations of the drifting Laplacian: \(\mathbb {L}_{\phi } \mathbf {u} + \alpha [ \nabla (\mathrm {div}\mathbf { u}) -\nabla \phi \mathrm {div} \mathbf {u}]= - \widetilde{\sigma } \mathbf {u}\), in \( \Omega \), and \(u|_{\partial \Omega }=0,\) where \(\mathbb {L}_{\phi } = \Delta - \nabla \phi \cdot \nabla \) is the drifting Laplacian and \(\alpha \) is a nonnegative constant. We establish some universal inequalities for lower order eigenvalues of this problem on the metric measure space \((\mathbb {R}^n, \langle ,\rangle , e^{-\phi }dv)\) and the Gaussian shrinking soliton \((\mathbb {R}^n, \langle ,\rangle _{\mathrm {can}}, e^{-\frac{|x|^2}{4}}dv, \frac{1}{2})\). Moreover, we give an estimate for the upper bound of the second eigenvalue of this problem in terms of its first eigenvalue on the gradient product Ricci soliton \((\Sigma \times \mathbb {R}, \langle ,\rangle , e^{-\frac{\kappa t^2}{2}}dv, \kappa )\), where \( \Sigma \) is an Einstein manifold with constant Ricci curvature \(\kappa \).     

Anderson Polymer in a Fractional Brownian Environment: Asymptotic Behavior of the Partition Function

We consider the Anderson polymer partition function

$$\begin{aligned} u(t):=\mathbb {E}^X\left[ e^{\int _0^t \mathrm {d}B^{X(s)}_s}\right] \,, \end{aligned}$$

where \(\{B^{x}_t\,;\, t\ge 0\}_{x\in \mathbb {Z}^d}\) is a family of independent fractional Brownian motions all with Hurst parameter \(H\in (0,1)\), and \(\{X(t)\}_{t\in \mathbb {R}^{\ge 0}}\) is a continuous-time simple symmetric random walk on \(\mathbb {Z}^d\) with jump rate \(\kappa \) and started from the origin. \(\mathbb {E}^X\) is the expectation with respect to this random walk. We prove that when \(H\le 1/2\), the function u(t) almost surely grows asymptotically like \(e^{\lambda t}\), where \(\lambda >0\) is a deterministic number. More precisely, we show that as t approaches \(+\infty \), the expression \(\{\frac{1}{t}\log u(t)\}_{t\in \mathbb {R}^{>0}}\) converges both almost surely and in the \(\hbox {L}^1\) sense to some positive deterministic number \(\lambda \). For \(H>1/2\), we first show that \(\lim _{t\rightarrow \infty } \frac{1}{t}\log u(t)\) exists both almost surely and in the \(\hbox {L}^1\) sense and equals a strictly positive deterministic number (possibly \(+\infty \)); hence, almost surely u(t) grows asymptotically at least like \(e^{\alpha t}\) for some deterministic constant \(\alpha >0\). On the other hand, we also show that almost surely and in the \(\hbox {L}^1\) sense, \(\limsup _{t\rightarrow \infty } \frac{1}{t\sqrt{\log t}}\log u(t)\) is a deterministic finite real number (possibly zero), hence proving that almost surely u(t) grows asymptotically at most like \(e^{\beta t\sqrt{\log t}}\) for some deterministic positive constant \(\beta \). Finally, for \(H>1/2\) when \(\mathbb {Z}^d\) is replaced by a circle endowed with a Hölder continuous covariance function, we show that \(\limsup _{t\rightarrow \infty } \frac{1}{t}\log u(t)\) is a deterministic finite positive real number, hence proving that almost surely u(t) grows asymptotically at most like \(e^{c t}\) for some deterministic positive constant c.

     

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.