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 \(q=2\).     

Bochner representable operators on Banach function spaces

Let \((E,\Vert \cdot \Vert _E)\) be a Banach function space, \(E'\) the Köthe dual of E and \((X,\Vert \cdot \Vert _X)\) be a Banach space. It is shown that every Bochner representable operator \(T:E\rightarrow X\) maps relatively \(\sigma (E,E')\)-compact sets in E onto relatively norm compact sets in X. If, in particular, the associated norm \(\Vert \cdot \Vert _{E'}\) on \(E'\) is order continuous, then every Bochner representable operator \(T:E\rightarrow X\) is \((\gamma _E,\Vert \cdot \Vert _X)\)-compact, where \(\gamma _E\) stands for the natural mixed topology on E. Applications to Bochner representable operators on Orlicz spaces are given.     

On the existence of closed magnetic geodesics via symplectic reduction

Let (M, g) be a closed Riemannian manifold and \(\sigma \) be a closed 2-form on M representing an integer cohomology class. In this paper, using symplectic reduction, we show how the problem of existence of closed magnetic geodesics for the magnetic flow of the pair \((g,\sigma )\) can be interpreted as a critical point problem for a Rabinowitz-type action functional defined on the cotangent bundle \(T^*E\) of a suitable \(S^1\)-bundle E over M or, equivalently, as a critical point problem for a Lagrangian-type action functional defined on the free loopspace of E. We thenstudy the relation between the stability property of energy hypersurfacesin \((T^*M,dp\wedge dq+\pi ^*\sigma )\) and of the corresponding codimension2 coisotropic submanifolds in \((T^*E,dp\wedge dq)\) arising via symplecticreduction. Finally, we reprove the main result of Asselle and Benedetti (J Topol Anal 8(3):545–570, 2016) in this setting.     

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.     

The crank moments weighted by the parity of cranks

In this note, we introduce the 2kth crank moment \(\mu _{2k}(-1,n)\) weighted by the parity of cranks and show that \((-1)^n \mu _{2k}(-1,n)>0\) for \(n\ge k \ge 0\). When \(k=0\), the inequality \((-1)^n \mu _{2k}(-1,n)>0\) reduces to Andrews and Lewis’s inequality \((-1)^n(M_e(n)-M_o(n))>0\) for \(n\ge 0\), where \(M_e(n)\) (resp. \(M_o(n)\)) denotes the number of partitions of n with even (resp. odd) crank. Several generating functions of \(\mu _{2k}(-1,n)\) are also studied in order to show the positivity of \((-1)^n\mu _{2k}(-1,n)\).     

A Compactness Theorem of the Space of Free Boundary <Emphasis Type="Italic">f</Emphasis>-Minimal Surfaces in Three-Dimensional Smooth Metric Measure Space with Boundary

Let \((M^3,g,e^{-f}d\mu _M)\) be a compact three-dimensional smooth metric measure space with nonempty boundary. Suppose that M has nonnegative Bakry–Émery Ricci curvature and the boundary \(\partial M\) is strictly f-mean convex. We prove that there exists a properly embedded smooth f-minimal surface \(\Sigma \) in M with free boundary \(\partial \Sigma \) on \(\partial M\). If we further assume that the boundary \(\partial M\) is strictly convex, then we prove that \(M^3\) is diffeomorphic to the 3-ball \(B^3\), and a compactness theorem for the space of properly embedded f-minimal surfaces with free boundary in such \((M^3,g,e^{-f}d\mu _M)\), when the topology of these f-minimal surfaces is fixed.     

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.     

The distortion dimension of $$\mathbb $$-rank 1 lattices

Let \(X=G/K\) be a symmetric space of noncompact type and rank \(k\ge 2\). We prove that horospheres in X are Lipschitz \((k-2)\)-connected if their centers are not contained in a proper join factor of the spherical building of X at infinity. As a consequence, the distortion dimension of an irreducible \(\mathbb {Q}\)-rank-1 lattice \(\Gamma \) in a linear, semisimple Lie group G of \(\mathbb R\)-rank k is \(k-1\). That is, given \(m< k-1\), a Lipschitz m-sphere S in (a polyhedral complex quasi-isometric to) \(\Gamma \), and a \((m+1)\)-ball B in X (or G) filling S, there is a \((m+1)\)-ball \(B'\) in \(\Gamma \) filling S such that \({{\mathrm{vol}}}B'\sim {{\mathrm{vol}}}B\). In particular, such arithmetic lattices satisfy Euclidean isoperimetric inequalities up to dimension \(k-1\).     

Heinz mean curvature estimates in warped product spaces $$M\times _{e^{\psi }}N$$

If a graph submanifold (x, f(x)) of a Riemannian warped product space \((M^m\times _{e^{\psi }}N^n,\tilde{g}=g+ e^{2\psi }h)\) is immersed with parallel mean curvature H, then we obtain a Heinz-type estimation of the mean curvature. Namely, on each compact domain D of M, \(m\Vert H\Vert \le \frac{A_{\psi }(\partial D)}{V_{\psi }(D)}\) holds, where \(A_{\psi }(\partial D)\) and \(V_{\psi }(D)\) are the \({\psi }\)-weighted area and volume, respectively. In particular, \(H=0\) if (M, g) has zero-weighted Cheeger constant, a concept recently introduced by Impera et al. (Height estimates for killing graphs. arXiv:1612.01257, 2016). This generalizes the known cases \(n=1\) or \(\psi =0\). We also conclude minimality using a closed calibration, assuming \((M,g_*)\) is complete where \(g_*=g+e^{2\psi }f^*h\), and for some constants \(\alpha \ge \delta \ge 0\), \(C_1>0\) and \(\beta \in [0,1)\), \(\Vert \nabla ^*\psi \Vert ^2_{g_*}\le \delta \), \(\mathrm {Ricci}_{\psi ,g_*}\ge \alpha \), and \({\mathrm{det}}_g(g_*)\le C_1 r^{2\beta }\) holds when \(r\rightarrow +\infty \), where r(x) is the distance function on \((M,g_*)\) from some fixed point. Both results rely on expressing the squared norm of the mean curvature as a weighted divergence of a suitable vector field.     

Families of conditionally approximately convex functions

Let \((G,+)\) be an Abelian topological group, which is also a \(T_{0}\)-space and a Baire space simultaneously, D be an open connected subset of G and \(\alpha : D-D \rightarrow {\mathbb R}\) be a function continuous at zero and such that \(\alpha (0)=0\). We show that if \((f_n)\) is a sequence of continuous functions \(f_n : D \rightarrow {\mathbb R}\) such that \(f_n(z) \le \frac{1}{2} f_n(x)+\frac{1}{2}f(y)+\alpha (x-y)\) for \(n\in {\mathbb N}\) and \(x,y,z\in D\) such that \(2z=x+y\) and if \((f_n)\) is pointwise convergent [bounded] then it is convergent uniformly on compact subsets of D [in the case when G is additionally a separable space, it contains a subsequence which is convergent on compact subsets of D].     

Generating Hyperbolic Singularities in Semitoric Systems Via Hopf Bifurcations

Let \((M,\Omega )\) be a connected symplectic 4-manifold and let \(F=(J,H) :M\rightarrow \mathbb {R}^2\) be a completely integrable system on M with only non-degenerate singularities. Assume that F does not have singularities with hyperbolic blocks and that \(p_1,\ldots ,p_n\) are the focus–focus singularities of F. For each subset \(S=\{i_1,\ldots ,i_j\}\), we will show how to modify F locally around any \(p_i, i \in S\), in order to create a new integrable system \(\widetilde{F}=(J, \widetilde{H}) :M \rightarrow \mathbb {R}^2\) such that its classical spectrum \(\widetilde{F}(M)\) contains j smooth curves of singular values corresponding to non-degenerate transversally hyperbolic singularities of \(\widetilde{F}\). Moreover the focus–focus singularities of \(\widetilde{F}\) are precisely \(p_i\), \(i \in \{1,\ldots ,n\} \setminus S\). The proof is based on Eliasson’s linearization theorem for non-degenerate singularities, and properties of the Hamiltonian Hopf bifurcation.     

Interassociates of the bicyclic semigroup

An interassociate of a semigroup \((S,\cdot )\) is a semigroup \((S, *)\) such that for all \(a, b, c \in S\), \(a\cdot (b*c)=(a\cdot b) *c\) and \(a*(b\cdot c)=(a*b) \cdot c\). We investigate the bicyclic semigroup C and its interassociates. In particular, if p and q are the generators of the bicyclic semigroup and m and n are fixed nonnegative integers, the operation \(a*_{m,n} b= aq^mp^n b\) is known to be an interassociate. We show that for distinct pairs (m, n) and (s, t), the interassociates \((C, *_{m,n})\) and \((C, *_{s,t})\) are not isomorphic. We also generalize a result regarding homomorphisms on C to homomorphisms on its interassociates.     

CLT for Random Walks of Commuting Endomorphisms on Compact Abelian Groups

Let \(\mathcal S\) be an abelian group of automorphisms of a probability space \((X, {\mathcal A}, \mu )\) with a finite system of generators \((A_1, \ldots , A_d).\) Let \(A^{{\underline{\ell }}}\) denote \(A_1^{\ell _1} \ldots A_d^{\ell _d}\), for \({{\underline{\ell }}}= (\ell _1, \ldots , \ell _d).\) If \((Z_k)\) is a random walk on \({\mathbb {Z}}^d\), one can study the asymptotic distribution of the sums \(\sum _{k=0}^{n-1} \, f \circ A^{\,{Z_k(\omega )}}\) and \(\sum _{{\underline{\ell }}\in {\mathbb {Z}}^d} {\mathbb {P}}(Z_n= {\underline{\ell }}) \, A^{\underline{\ell }}f\), for a function f on X. In particular, given a random walk on commuting matrices in \(SL(\rho , {\mathbb {Z}})\) or in \({\mathcal M}^*(\rho , {\mathbb {Z}})\) acting on the torus \({\mathbb {T}}^\rho \), \(\rho \ge 1\), what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on \({\mathbb {T}}^\rho \) after normalization? In this paper, we prove a central limit theorem when X is a compact abelian connected group G endowed with its Haar measure (e.g., a torus or a connected extension of a torus), \(\mathcal S\) a totally ergodic d-dimensional group of commuting algebraic automorphisms of G and f a regular function on G. The proof is based on the cumulant method and on preliminary results on random walks.     

$${\varvec{}}$$-Jorda homomorphisms ad pseudo $${\varvec{}}$$-Jorda homomorphisms o Baach algebras

Let \(n\in \mathbb {N}\), A and B be Banach algebras and let B be a right A-module. We say that a linear mapping \(\varphi :A\longrightarrow B\) is a pseudo n-Jordan homomorphism if there exists an element \(w\in A\) such that \(\varphi (a^nw)=\varphi (a)^n\cdot w\), for every \(a\in A\) and \(n\ge \) 2. In this paper, among other things, we show that under some conditions if a linear mapping \(\varphi \) is a (pseudo) n-Jordan homomorphism, then it is a (pseudo) \((n + 1)\)-Jordan homomorphism. Additionally, we investigate automatic continuity of surjective pseudo n-Jordan homomorphisms under some conditions.     

Complete determination of the zeta function of the Hilbert scheme of <Emphasis Type="Italic">n</Emphasis> points on a two-dimensional torus

We compute the coefficients of the polynomials \(C_n(q)\) defined by the equation

$$\begin{aligned} 1 + \sum _{n\ge 1} \, \frac{C_n(q)}{q^n} \, t^n = \prod _{i\ge 1}\, \frac{(1-t^i)^2}{1-(q+q^{-1})t^i + t^{2i}} \, . \end{aligned}$$

As an application we obtain an explicit formula for the zeta function of the Hilbert scheme of n points on a two-dimensional torus and show that this zeta function satisfies a remarkable functional equation. The polynomials \(C_n(q)\) are divisible by \((q-1)^2\). We also compute the coefficients of the polynomials \(P_n(q) = C_n(q)/(q-1)^2\): each coefficient counts the divisors of n in a certain interval; it is thus a non-negative integer. Finally we give arithmetical interpretations for the values of \(C_n(q)\) and of \(P_n(q)\) at \(q = -1\) and at roots of unity of order 3, 4, 6.

     

Dirichlet and Neumann eigenvalue problems on CR manifolds

We study the properties of Carnot–Carathéodory spaces attached to a strictly pseudoconvex CR manifold M, in a neighborhood of each point \(x \in M\), versus the pseudohermitian geometry of M arising from a fixed positively oriented contact form \(\theta \) on M. The weak Dirichlet problem for the sublaplacian \(\Delta _b\) on \((M, \theta )\) is solved on domains \(\Omega \subset M\) supporting the Poincaré inequality. The solution to Neumann problem for the sublaplacian \(\Delta _b\) on a \(C^{1,1}\) connected \((\epsilon , \delta )\)-domain \(\Omega \subset {{\mathbb {G}}}\) in a Carnot group (due to Danielli et al. in: Memoirs of American Mathematical Society 2006) is revisited for domains in a CR manifold. As an application we prove discreetness of the Dirichlet and Neumann spectra of \(\Delta _b\) on \(\Omega \subset M\) in a Carnot–Carthéodory complete pseudohermitian manifold \((M, \theta )\).     

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 \(L_T^\varphi =L'_T\).     

Continued fractions and irrational rotations

Let \(\alpha \in (0, 1)\) be an irrational number with continued fraction expansion \(\alpha =[0; a_1, a_2, \ldots ]\) and let \(p_n/q_n= [0; a_1, \ldots , a_n]\) be the nth convergent to \(\alpha \). We prove a formula for \(p_nq_k-q_np_k\) \((k<n)\) in terms of a Fibonacci type sequence \(Q_n\) defined in terms of the \(a_n\) and use it to provide an exact formula for \(\{n\alpha \}\) for all n.     

Group divisible designs with large block sizes

For positive integers n, k with \(3\le k\le n\), let \(X=\mathbb {F}_{2^n}\setminus \{0,1\}\), \({\mathcal {G}}=\{\{x,x+1\}:x\in X\}\), and \({\mathcal {B}}_k=\left\{ \{x_1,x_2,\ldots ,x_k\}\!\subset \!X:\sum \limits _{i=1}^kx_i=1,\ \sum \limits _{i\in I}x_i\!\ne \!1\ \mathrm{for\ any}\ \emptyset \!\ne \!I\!\subsetneqq \!\{1,2,\ldots ,k\}\right\} \). Lee et al. used the inclusion–exclusion principle to show that the triple \((X,{\mathcal {G}},{\mathcal {B}}_k)\) is a \((k,\lambda _k)\)-GDD of type \(2^{2^{n-1}-1}\) for \(k\in \{3,4,5,6,7\}\) where \(\lambda _k=\frac{\prod _{i=3}^{k-1}(2^n-2^i)}{(k-2)!}\) (Lee et al. in Des Codes Cryptogr,  https://doi.org/10.1007/s10623-017-0395-8, 2017). They conjectured that \((X,{\mathcal {G}},{\mathcal {B}}_k)\) is also a \((k,\lambda _k)\)-GDD of type \(2^{2^{n-1}-1}\) for any integer \(k\ge 8\). In this paper, we use a similar construction and counting principles to show that there is a \((k,\lambda _k)\)-GDD of type \((q^2-q)^{(q^{n-1}-1)/(q-1)}\) for any prime power q and any integers k, n with \(3\le k\le n\) where \(\lambda _k=\frac{\prod _{i=3}^{k-1}(q^n-q^i)}{(k-2)!}\). Consequently, their conjecture holds. Such a method is also generalized to yield a \((k,\lambda _k)\)-GDD of type \((q^{\ell +1}-q^{\ell })^{(q^{n-\ell }-1)/(q-1)}\) where \(\lambda _k=\frac{\prod _{i=3}^{k-1}(q^n-q^{\ell +i-1})}{(k-2)!}\) and \(k+\ell \le n+1\).     

On the gaps between non-zero Fourier coefficients of cusp forms of higher weight

We show that if a modular cuspidal eigenform f of weight 2k is 2-adically close to an elliptic curve \(E/\mathbb {Q}\), which has a cyclic rational 4-isogeny, then n-th Fourier coefficient of f is non-zero in the short interval \((X, X + cX^{\frac{1}{4}})\) for all \(X \gg 0\) and for some \(c > 0\). We use this fact to produce non-CM cuspidal eigenforms f of level \(N>1\) and weight \(k > 2\) such that \(i_f(n) \ll n^{\frac{1}{4}}\) for all \(n \gg 0\).