ASSOCIATED FORMS OF BINARY QUARTICS AND TERNARY CUBICS

Let \( {\mathcal{Q}}_n^d \) be the vector space of forms of degree d?≥?3 on ? ⁿ , with n?≥?2. The object of our study is the map Φ, introduced in earlier articles by M. Eastwood and the first two authors, that assigns every nondegenerate form in \( {\mathcal{Q}}_n^d \) the so-called associated form, which is an element of \( {{\mathcal{Q}}_n^d}^{\left(d-2\right)*} \). We focus on two cases: those of binary quartics (n?=?2, d?=?4) and ternary cubics (n?=?3, d?=?3). In these situations the map Φ induces a rational equivariant involution on the projective space ?\( \left({\mathcal{Q}}_n^d\right) \), which is in fact the only nontrivial rational equivariant involution on ?\( \left({\mathcal{Q}}_n^d\right) \). In particular, there exists an equivariant involution on the space of elliptic curves with nonvanishing j-invariant. In the present paper, we give a simple interpretation of this involution in terms of projective duality. Furthermore, we express it via classical contravariants.     

On the densities of rational multiples

For two subsets of natural numbers \( A,B \subset \mathbb{N} \), define the set of rational numbers \( \mathcal{M}\left( {A,B} \right) \) with the elements represented by m/n, where m and n are coprime, m is divisible by some a ∈ A, and n is divisible by some b ∈ B. Let I be some interval of positive real numbers and \( \mathcal{F}_x^I \) denote the set of rational numbers m/n ∈ I such that m and n are coprime and n ? x. The analogue to the Erdös–Davenport theorem about multiples is proved: under some constraints on I, the limits \( {{{\sum {\left\{ {\frac{1}{{mn}}:\frac{m}{n} \in \mathcal{F}_x^I \cap \mathcal{M}\left( {A,B} \right)} \right\}} }} \left/ {{\sum {\left\{ {\frac{1}{{mn}}:\frac{m}{n} \in \mathcal{F}_x^I} \right\}} }} \right.} \) exist for all subsets \( A,B \subset \mathbb{N} \) as x → ∞.     

Semigroups of rectangular matrices under a sandwich operation

Let \({\mathcal {M}}_{mn}={\mathcal {M}}_{mn}({\mathbb {F}})\) denote the set of all \(m\times n\) matrices over a field \({\mathbb {F}}\), and fix some \(n\times m\) matrix \(A\in {\mathcal {M}}_{nm}\). An associative operation \(\star \) may be defined on \({\mathcal {M}}_{mn}\) by \(X\star Y=XAY\) for all \(X,Y\in {\mathcal {M}}_{mn}\), and the resulting sandwich semigroup is denoted \({\mathcal {M}}_{mn}^A={\mathcal {M}}_{mn}^A({\mathbb {F}})\). These semigroups are closely related to Munn rings, which are fundamental tools in the representation theory of finite semigroups. We study \({\mathcal {M}}_{mn}^A\) as well as its subsemigroups \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\) and \({\mathcal {E}}_{mn}^A\) (consisting of all regular elements and products of idempotents, respectively), and the ideals of \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\). Among other results, we characterise the regular elements; determine Green’s relations and preorders; calculate the minimal number of matrices (or idempotent matrices, if applicable) required to generate each semigroup we consider; and classify the isomorphisms between finite sandwich semigroups \({\mathcal {M}}_{mn}^A({\mathbb {F}}_1)\) and \({\mathcal {M}}_{kl}^B({\mathbb {F}}_2)\). Along the way, we develop a general theory of sandwich semigroups in a suitably defined class of partial semigroups related to Ehresmann-style “arrows only” categories; we hope this framework will be useful in studies of sandwich semigroups in other categories. We note that all our results have applications to the variants \({\mathcal {M}}_n^A\) of the full linear monoid \({\mathcal {M}}_n\) (in the case \(m=n\)), and to certain semigroups of linear transformations of restricted range or kernel (in the case that \(\hbox {rank}(A)\) is equal to one of m, n).     

Determination of cusp forms by central values of Rankin–Selberg <Emphasis Type="BoldItalic">L</Emphasis>-functions<Superscript>*</Superscript>

Let f be a fixed holomorphic Hecke eigen cusp form of weight k for \( SL\left( {2,{\mathbb Z}} \right) \), and let \( {\mathcal U} = \left\{ {{u_j}:j \geqslant 1} \right\} \) be an orthonormal basis of Hecke–Maass cusp forms for \( SL\left( {2,{\mathbb Z}} \right) \). We prove an asymptotic formula for the twisted first moment of the Rankin–Selberg L-functions \( L\left( {s,f \otimes {u_j}} \right) \) at \( s = \frac{1}{2} \) as u _j runs over \( {\mathcal U} \). It follows that f is uniquely determined by the central values of the family of Rankin–Selberg L-functions \( \left\{ {L\left( {s,f \otimes {u_j}} \right):{u_j} \in {\mathcal U}} \right\} \).     

Semidefinite bounds for nonbinary codes based on quadruples

For nonnegative integers q, n, d, let \(A_q(n,d)\) denote the maximum cardinality of a code of length n over an alphabet [q] with q letters and with minimum distance at least d. We consider the following upper bound on \(A_q(n,d)\). For any k, let \(\mathcal{C}_k\) be the collection of codes of cardinality at most k. Then \(A_q(n,d)\) is at most the maximum value of \(\sum _{v\in [q]^n}x(\{v\})\), where x is a function \(\mathcal{C}_4\rightarrow {\mathbb {R}}_+\) such that \(x(\emptyset )=1\) and \(x(C)=\!0\) if C has minimum distance less than d, and such that the \(\mathcal{C}_2\times \mathcal{C}_2\) matrix \((x(C\cup C'))_{C,C'\in \mathcal{C}_2}\) is positive semidefinite. By the symmetry of the problem, we can apply representation theory to reduce the problem to a semidefinite programming problem with order bounded by a polynomial in n. It yields the new upper bounds \(A_4(6,3)\le 176\), \(A_4(7,3)\le 596\), \(A_4(7,4)\le 155\), \(A_5(7,4)\le 489\), and \(A_5(7,5)\le 87\).     

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.     

The weak asymptotic equivalence and the generalized inverse

In this paper, we discuss the relationship between the weak asymptotic equivalence relation and the generalized inverse in the class \({\mathcal{A}} \) of all nondecreasing and unbounded functions, defined and positive on a half-axis [a,+∞) (a > 0). In the main theorem, we prove a proper characterization of the functional class \({{ORV} \cap \mathcal{A}} \), where ORV is the class of all \({\mathcal{O}} \)-regularly varying functions (in the sense of Karamata).     

Solutions of perturbed Schrödinger equations with critical nonlinearity

We consider the perturbed Schrödinger equation

$\left\{\begin{array}{ll}{- \varepsilon ^2 \Delta u + V(x)u = P(x)|u|^{p - 2} u + k(x)|u|^{2* - 2} u} &; {\text{for}}\, x \in {\mathbb{R}}^N\\ \qquad \qquad \quad {u(x) \rightarrow 0} &; \text{as}\, {|x| \rightarrow \infty} \end{array} \right.$

where \(N\geq 3, \ 2^*=2N/(N-2)\) is the Sobolev critical exponent, \(p\in (2, 2^*)\) , P(x) and K(x) are bounded positive functions. Under proper conditions on V we show that it has at least one positive solution provided that \(\varepsilon\leq{\mathcal{E}}\) ; for any \(m\in{\mathbb{N}}\) , it has m pairs of solutions if \(\varepsilon\leq{\mathcal{E}}_{m}\) ; and suppose there exists an orthogonal involution \(\tau:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}\) such that V(x), P(x) and K(x) are τ -invariant, then it has at least one pair of solutions which change sign exactly once provided that \(\varepsilon\leq{\mathcal{E}}\) , where \({\mathcal{E}}\) and \({\mathcal{E}}_{m}\) are sufficiently small positive numbers. Moreover, these solutions \(u_\varepsilon\to 0\) in \(H^1({\mathbb{R}}^N)\) as \(\varepsilon\to 0\) .

     

A Completion of the Spectrum for the Overlarge Sets of Pure Mendelsohn Triple Systems

A pure Mendelsohn triple system of order v, denoted by PMTS(v), is a pair \((X,\mathcal {B})\) where X is a v-set and \(\mathcal {B}\) is a collection of cyclic triples on X such that every ordered pair of X belongs to exactly one triple of \(\mathcal {B}\) and if \(\langle a,b,c\rangle \in \mathcal {B}\) implies \(\langle c,b,a\rangle \notin \mathcal {B}\). An overlarge set of PMTS(v), denoted by OLPMTS(v), is a collection \(\{(Y{\setminus }\{y_i\},{\mathcal {A}}_i)\}_i\), where Y is a \((v+1)\)-set, \(y_i\in Y\), each \((Y{\setminus }\{y_i\},{\mathcal {A}}_i)\) is a PMTS(v) and these \({\mathcal {A}}_i\)s form a partition of all cyclic triples on Y. It is shown in [3] that there exists an OLPMTS(v) for \(v\equiv 1,3\) (mod 6), \(v>3\), or \(v \equiv 0,4\) (mod 12). In this paper, we shall discuss the existence problem of OLPMTS(v)s for \(v\equiv 6,10\) (mod 12) and get the following conclusion: there exists an OLPMTS(v) if and only if \(v\equiv 0,1\) (mod 3), \(v>3\) and \(v\ne 6\).     

On coherent families of uniformizing elements in some towers of Abelian extensions of local number fields

For a local number field K with the ring of integers \( {\mathcal{O}_K} \), the residue field \( {\mathbb{F}_q} \), and uniformizing π, we consider the Lubin–Tate tower \( {K_\pi } = \bigcap\limits_{n \geqslant 0} {{K_n}} \), where K _n = K(π _n ), f(π0) = 0, and f(π _n +1) = π _n . Here f(X) defines the endomorphism [π] of the Lubin–Tate group. If q ≠ 2, then for any formal power series \( g(X) \in {\mathcal{O}_K}\left[ {\left[ X \right]} \right] \) the following equality holds: \( \sum\limits_{n = 0}^\infty {{\text{SP}}{{{K_n}} \mathord{\left/{\vphantom {{{K_n}} K}} \right.} K}} g\left( {{\pi_n}} \right) = - g(0) \). One has a similar equality in the case q = 2.     

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 \(C_m\).     

EKR Sets for Large <Emphasis Type="Italic">n</Emphasis> and <Emphasis Type="Italic">r</Emphasis>

Let \(\mathcal {A}\subset \left( {\begin{array}{c}[n]\\ r\end{array}}\right) \) be a compressed, intersecting family and let \(X\subset [n]\). Let \(\mathcal {A}(X)=\{A\in \mathcal {A}:A\cap X\ne \emptyset \}\) and \(\mathcal {S}_{n,r}=\left( {\begin{array}{c}[n]\\ r\end{array}}\right) (\{1\})\). Motivated by the Erd?s–Ko–Rado theorem, Borg asked for which \(X\subset [2,n]\) do we have \(|\mathcal {A}(X)|\le |\mathcal {S}_{n,r}(X)|\) for all compressed, intersecting families \(\mathcal {A}\)? We call X that satisfy this property EKR. Borg classified EKR sets X such that \(|X|\ge r\). Barber classified X, with \(|X|\le r\), such that X is EKR for sufficiently large n, and asked how large n must be. We prove n is sufficiently large when n grows quadratically in r. In the case where \(\mathcal {A}\) has a maximal element, we sharpen this bound to \(n>\varphi ^{2}r\) implies \(|\mathcal {A}(X)|\le |\mathcal {S}_{n,r}(X)|\). We conclude by giving a generating function that speeds up computation of \(|\mathcal {A}(X)|\) in comparison with the naïve methods.     

Algebraic properties of Zappa–Szép products of semigroups and monoids

Direct, semidirect and Zappa–Szép products provide tools to decompose algebraic structures, with each being a natural generalisation of its predecessor. In this paper we examine Zappa–Szép products of monoids and semigroups and investigate generalised Greens relations \({\mathcal R}^{*},\, {\mathcal L}^{*},\, \widetilde{\mathcal {R}}_E\) and \(\widetilde{\mathcal {L}}_E\) for these Zappa–Szép products. We consider a left restriction semigroup S with semilattice of projections E and define left and right actions of S on E and E on S, respectively, to form the Zappa–Szép product \(E \bowtie S\). We further investigate properties of \(E \bowtie S\) and show that S is a retract of \(E\bowtie S\). We also find a subset T of \(E \bowtie S\) which is left restriction.     

On the classification of 4-dimensional $$(m,\rho )$$-quasi-Einstein manifolds with harmonic Weyl curvature

In this paper we study four-dimensional \((m,\rho )\)-quasi-Einstein manifolds with harmonic Weyl curvature when \(m\notin \{0,\pm 1,-2,\pm \infty \}\) and \(\rho \notin \{\frac{1}{4},\frac{1}{6}\}\). We prove that a non-trivial \((m,\rho )\)-quasi-Einstein metric g (not necessarily complete) is locally isometric to one of the following: (i) \({\mathcal {B}}^2_\frac{R}{2(m+2)}\times {\mathbb {N}}^2_\frac{R(m+1)}{2(m+2)}\), where \({\mathcal {B}}^2_\frac{R}{2(m+2)}\) is the northern hemisphere in the two-dimensional (2D) sphere \({\mathbb {S}}^2_\frac{R}{2(m+2)}\), \({\mathbb {N}}_\delta \) is a 2D Riemannian manifold with constant curvature \(\delta \), and R is the constant scalar curvature of g. (ii) \({\mathcal {D}}^2_\frac{R}{2(m+2)}\times {\mathbb {N}}^2_\frac{R(m+1)}{2(m+2)}\), where \({\mathcal {D}}^2_\frac{R}{2(m+2)}\) is half (cut by a hyperbolic line) of hyperbolic plane \({\mathbb {H}}^2_\frac{R}{2(m+2)}\). (iii) \({\mathbb {H}}^2_\frac{R}{2(m+2)}\times {\mathbb {N}}^2_\frac{R(m+1)}{2(m+2)}\). (iv) A certain singular metric with \(\rho =0\). (v) A locally conformal flat metric. By applying this local classification, we obtain a classification of the complete \((m,\rho )\)-quasi-Einstein manifolds given the condition of a harmonic Weyl curvature. Our result can be viewed as a local classification of gradient Einstein-type manifolds. A corollary of our result is the classification of \((\lambda ,4+m)\)-Einstein manifolds, which can be viewed as (m, 0)-quasi-Einstein manifolds.     

Diagonalization of the Finite Hilbert Transform on Two Adjacent Intervals

We continue the study of stability of solving the interior problem of tomography. The starting point is the Gelfand–Graev formula, which converts the tomographic data into the finite Hilbert transform (FHT) of an unknown function f along a collection of lines. Pick one such line, call it the x-axis, and assume that the function to be reconstructed depends on a one-dimensional argument by restricting f to the x-axis. Let \(I_1\) be the interval where f is supported, and \(I_2\) be the interval where the Hilbert transform of f can be computed using the Gelfand–Graev formula. The equation to be solved is \(\left. {\mathcal {H}}_1 f=g\right| _{I_2}\), where \({\mathcal {H}}_1\) is the FHT that integrates over \(I_1\) and gives the result on \(I_2\), i.e. \({\mathcal {H}}_1: L^2(I_1)\rightarrow L^2(I_2)\). In the case of complete data, \(I_1\subset I_2\), and the classical FHT inversion formula reconstructs f in a stable fashion. In the case of interior problem (i.e., when the tomographic data are truncated), \(I_1\) is no longer a subset of \(I_2\), and the inversion problems becomes severely unstable. By using a differential operator L that commutes with \({\mathcal {H}}_1\), one can obtain the singular value decomposition of \({\mathcal {H}}_1\). Then the rate of decay of singular values of \({\mathcal {H}}_1\) is the measure of instability of finding f. Depending on the available tomographic data, different relative positions of the intervals \(I_{1,2}\) are possible. The cases when \(I_1\) and \(I_2\) are at a positive distance from each other or when they overlap have been investigated already. It was shown that in both cases the spectrum of the operator \({\mathcal {H}}_1^*{\mathcal {H}}_1\) is discrete, and the asymptotics of its eigenvalues \(\sigma _n\) as \(n\rightarrow \infty \) has been obtained. In this paper we consider the case when the intervals \(I_1=(a_1,0)\) and \(I_2=(0,a_2)\) are adjacent. Here \(a_1 < 0 < a_2\). Using recent developments in the Titchmarsh–Weyl theory, we show that the operator L corresponding to two touching intervals has only continuous spectrum and obtain two isometric transformations \(U_1\), \(U_2\), such that \(U_2{\mathcal {H}}_1 U_1^*\) is the multiplication operator with the function \(\sigma (\lambda )\), \(\lambda \ge (a_1^2+a_2^2)/8\). Here \(\lambda \) is the spectral parameter. Then we show that \(\sigma (\lambda )\rightarrow 0\) as \(\lambda \rightarrow \infty \) exponentially fast. This implies that the problem of finding f is severely ill-posed. We also obtain the leading asymptotic behavior of the kernels involved in the integral operators \(U_1\), \(U_2\) as \(\lambda \rightarrow \infty \). When the intervals are symmetric, i.e. \(-a_1=a_2\), the operators \(U_1\), \(U_2\) are obtained explicitly in terms of hypergeometric functions.     

Uniqueness of self-shrinkers to the degree-one curvature flow with a tangent cone at infinity

Given a smooth, symmetric and homogeneous of degree one function \(f\left( \lambda _{1},\ldots ,\lambda _{n}\right) \) satisfying \(\partial _{i}f>0\quad \forall \,i=1,\ldots , n\), and a properly embedded smooth cone \({\mathcal {C}}\) in \({\mathbb {R}}^{n+1}\), we show that under suitable conditions on f, there is at most one f self-shrinker (i.e. a hypersurface \(\Sigma \) in \({\mathbb {R}}^{n+1}\) satisfying \(f\left( \kappa _{1},\ldots ,\kappa _{n}\right) +\frac{1}{2}X\cdot N=0\), where \(\kappa _{1},\ldots ,\kappa _{n}\) are principal curvatures of \(\Sigma \)) that is asymptotic to the given cone \({\mathcal {C}}\) at infinity.     

Finite elements in some vector lattices of nonlinear operators

We study the collection of finite elements \(\Phi _{1}\big ({\mathcal {U}}(E,F)\big )\) in the vector lattice \({\mathcal {U}}(E,F)\) of orthogonally additive, order bounded (called abstract Uryson) operators between two vector lattices E and F, where F is Dedekind complete. In particular, for an atomic vector lattice E it is proved that for a finite element in \(\varphi \in {\mathcal {U}}(E,{\mathbb {R}})\) there is only a finite set of mutually disjoint atoms, where \(\varphi \) does not vanish and, for an atomless vector lattice the zero-vector is the only finite element in the band of \(\sigma \)-laterally continuous abstract Uryson functionals. We also describe the ideal \(\Phi _{1}\big ({\mathcal {U}}({\mathbb {R}}^n,{\mathbb {R}}^m)\big )\) for \(n,m\in {\mathbb {N}}\) and consider rank one operators to be finite elements in \({\mathcal {U}}(E,F)\).     

Conformally Covariant Differential Operators for the Diagonal Action of $$O(p,\,q)$$ on Real Quadrics

Let \(X=G/P\) be a real projective quadric, where \(G=O(p,\,q)\) and P is a parabolic subgroup of G. Let \((\pi _{\lambda ,\epsilon },\, \mathcal H_{\lambda ,\epsilon })_{ (\lambda ,\epsilon )\in {\mathbb {C}}\times \{\pm \}}\) be the family of (smooth) representations of G induced from the characters of P. For \((\lambda ,\, \epsilon ),\, (\mu ,\, \eta )\in {\mathbb {C}}\times \{\pm \},\) a differential operator \(\mathbf D_{(\mu ,\eta )}^\mathrm{reg}\) on \(X\times X,\) acting G-covariantly from \({\mathcal {H}}_{\lambda ,\epsilon } \otimes {\mathcal {H}}_{\mu , \eta }\) into \({\mathcal {H}}_{\lambda +1,-\epsilon } \otimes {\mathcal {H}}_{\mu +1, -\eta }\) is constructed.     

Infinite dual symmetric inverse monoids

Let \({\mathcal {I}}^*_X\) be the dual symmetric inverse monoid on an infinite set X. We show that \({\mathcal {I}}^*_X\) may be generated by the symmetric group \({\mathcal {S}}_X\) together with two (but no fewer) additional block bijections, and we classify the pairs \(\alpha ,\beta \in {\mathcal {I}}^*_X\) for which \({\mathcal {I}}^*_X\) is generated by \({\mathcal {S}}_X\,\cup \,\{\alpha ,\beta \}\). Among other results, we show that any countable subset of \({\mathcal {I}}^*_X\) is contained in a 4-generated subsemigroup of \({\mathcal {I}}^*_X\), and that the length function on \({\mathcal {I}}^*_X\) (and its finitary power semigroup) is bounded with respect to any generating set.     

Chebyshev polynomials of the second kind via raising operator preserving the orthogonality

It is well known that monic orthogonal polynomial sequences \(\{T_n\}_{n\ge 0}\) and \(\{U_n\}_{n\ge 0}\), the Chebyshev polynomials of the first and second kind, satisfy the relation \(DT_{n+1}=(n+1)U_n\) (\(n\ge 0\)). One can also easily check that the following “inverse” of the mentioned formula holds: \({\mathcal {U}}_{-1}(U_n)=(n+1)T_{n+1}\) (\(n\ge 0\)), where \({\mathcal {U}}_\xi =x(xD+{\mathbb {I}})+\xi D\) with \(\xi \) being an arbitrary nonzero parameter and \({\mathbb {I}}\) representing the identity operator. Note that whereas the first expression involves the operator D which lowers the degree by one, the second one involves \({\mathcal {U}}_\xi \) which raises the degree by one (i.e. it is a “raising operator”). In this paper it is shown that the scaled Chebyshev polynomial sequence \(\{a^{-n}U_n(ax)\}_{n\ge 0}\) where \(a^2=-\xi ^{-1}\), is actually the only monic orthogonal polynomial sequence which is \({\mathcal {U}}_\xi \)-classical (i.e. for which the application of the raising operator \({\mathcal {U}}_\xi \) turns the original sequence into another orthogonal one).