8-ranks of class groups of quadratic number fields and their densities

For \(F = \mathbb{Q}\left( {\sqrt {\varepsilon pq} } \right)\), ? ∈ {±1, ±2}, primes ?p ≡ q ≡ 1 mod 4, we give the necessary and sufficient conditions for 8-ranks of narrow class groups of F equal to 1 or 2 such that we can calculate their densities. All results are stated in terms of congruence relations of p, q modulo 2 ⁿ , the quartic residue symbol \(\left( {\frac{p}{q}} \right)_4\) and binary quadratic forms such as q ^h(?2p)/4 = x ² + 2py ², where h(?2p) is the class number of \(\mathbb{Q}\left( {\sqrt { - 2p} } \right)\). The results are very useful for numerical computations.     

Canonical Representations and Overgroups for Hyperboloids

For the hyperboloid \(X = G/H\), where G = SO₀(p, q) and H = SO₀(p, q ? 1), we define canonical representations R_λ,ν λ ∈ ?, ν = 0, 1, as the restrictions to G of representations \(\tilde R\lambda ,\nu\), associated with a cone, of the group \(\tilde G = \operatorname{SO} _0 (p + 1,q)\). They act on functions on the direct product Ω of two spheres of dimensions p ? 1 and q ? 1. The manifold Ω contains two copies of \(X\) as open G-orbits. We explicitly describe the interaction of the Lie operators of the group \({\tilde G}\) in \(\tilde R\lambda ,\nu\) with the Poisson and Fourier transforms associated with the canonical representations. These transforms are operators intertwining the representations R_λ,ν with representations of G associated with a cone.     

Quantum MDS codes with relatively large minimum distance from Hermitian self-orthogonal codes

It has become common knowledge that constructing q-ary quantum MDS codes with minimum distance bigger than \(q/2+1\) is significantly more difficult than constructing those with minimum distance less than or equal to \(q/2+1\). Despite of various constructions of q-ary quantum MDS codes, all known q-ary quantum MDS codes have minimum distance bounded by \(q/2+1\) except for some lengths. The purpose of the current paper is to provide some new q-ary quantum MDS codes with minimum distance bigger than \(q/2+1\). In this paper, we provide several classes of quantum MDS codes with minimum distance bigger than \(q/2+1\). For instance, some examples in these classes include q-ary \([n,n-2k, k+1]\)-quantum MDS codes for cases: (i) \(q\equiv -1\bmod {5}, n=(q^2+4)/5\) and \(1\le k\le (3q-2)/5\); (ii) \(q\equiv -1\bmod {7}, n=(q^2+6)/7\) and \(1\le k\le (4q-3)/7\); (iii) \(2|q, q\equiv -1\bmod {3}, n=2(q^2-1)/3\) and \(1\le k\le (2q-1)/3\); and (iv) \(2|q, q\equiv -1\bmod {5}, n=2(q^2-1)/5\) and \(1\le k\le (3q-2)/5\).     

The non-abelian tensor square of residually finite groups

Let m, n be positive integers and p a prime. We denote by \(\nu (G)\) an extension of the non-abelian tensor square \(G \otimes G\) by \(G \times G\). We prove that if G is a residually finite group satisfying some non-trivial identity \(f \equiv ~1\) and for every \(x,y \in G\) there exists a p-power \(q=q(x,y)\) such that \([x,y^{\varphi }]^q = 1\), then the derived subgroup \(\nu (G)'\) is locally finite (Theorem A). Moreover, we show that if G is a residually finite group in which for every \(x,y \in G\) there exists a p-power \(q=q(x,y)\) dividing \(p^m\) such that \([x,y^{\varphi }]^q\) is left n-Engel, then the non-abelian tensor square \(G \otimes G\) is locally virtually nilpotent (Theorem B).     

Estimates of the Root Functions of a One-Dimensional Schrödinger Operator with a Strong Boundary Singularity

For any operator defined by the differential operation Lu = ?u″ + q(x)u on the interval G = (0, 1) with complex-valued potential q(x) locally integrable on G and satisfying the inequalities \(\int_{{x_1}}^{{x_2}} {\zeta |(q(\zeta ))|d\zeta \leqslant ln({x_1}/{x_2})} \) and \(\int_{{x_1}}^{{x_2}} {\zeta |(q(1 - \zeta ))|d\zeta \leqslant \gamma ln({x_1}/{x_2})} \) with some constant γ for all sufficiently small 0 < x₁ < x₂, we estimate the norms of root functions in the Lebesgue spaces L _p (G), 1 ≤ p < ∞. We show that for sufficiently small γ these norms satisfy the same estimates asymptotic in the spectral parameter as in the unperturbed case.     

New Rates of Convergence for the Iterates of Some Positive Linear Operators

In this paper, we give quantitative results for the convergence of the iterates of some positive linear operators which preserve the functions \(e_0\) and \(e_1\), respectively, \(e_0\) and \(e_2\). We obtain estimates in terms of moduli of smoothness and we improve some previous results. We give examples for the q-Bernstein, the q-Bernstein-Durrmeyer, the q-Stancu, the q-Meyer-Konig–Zeller, the King type of the q-Bernstein and the King type of the q-genuine-Bernstein-Durrmeyer operators. We observe that some q-operators (\(0<q<1\)) provide better convergence for the iterates than the corresponding classical operators (\(q=1\)).     

A Higgs bundle on a Hermitian symmetric space

Let \(M := \Gamma\backslash G/K\) be the quotient of an irreducible Hermitian symmetric space G/K by a torsionfree cocompact lattice \(\Gamma\subset G\) . There is a natural flat principal G-bundle over the compact Kähler manifold M which is constructed from the principal Γ-bundle over M defined by the quotient map \(G/K\longrightarrow M\) . We construct the principal G-Higgs bundle over M corresponding to this flat G-bundle. This principal G-Higgs bundle is rigid if \({\rm dim}_\mathbb{C} M\,\geq\,2\) .     

Complexities of normal bases constructed from Gauss periods

Let q be a power of a prime p, and let \(r=nk+1\) be a prime such that \(r\not \mid q\), where n and k are positive integers. Under a simple condition on q, r and k, a Gauss period of type (n, k) is a normal element of \({\mathbb {F}}_{q}^{n}\) over \({\mathbb {F}}_q\); the complexity of the resulting normal basis of \({\mathbb {F}}_{q}^{n}\) over \({\mathbb {F}}_q\) is denoted by C(n, k; p). Recent works determined C(n, k; p) for \(k\le 7\) and all qualified n and q. In this paper, we show that for any given \(k>0\), C(n, k; p) is given by an explicit formula except for finitely many primes \(r=nk+1\) and the exceptional primes are easily determined. Moreover, we describe an algorithm that allows one to compute C(n, k; p) for the exceptional primes \(r=nk+1\). Our numerical results cover C(n, k; p) for \(k\le 20\) and all qualified n and q.     

ON THE COADJOINT ORBITS OF MAXIMAL UNIPOTENT SUBGROUPS OF REDUCTIVE GROUPS

Let G be a simple algebraic group defined over an algebraically closed field of characteristic 0 or a good prime for G. Let U be a maximal unipotent subgroup of G and \( \mathfrak{u} \) its Lie algebra. We prove the separability of orbit maps and the connectedness of centralizers for the coadjoint action of U on (certain quotients of) the dual \( \mathfrak{u} \)* of \( \mathfrak{u} \). This leads to a method to give a parametrization of the coadjoint orbits in terms of so-called minimal representatives which form a disjoint union of quasi-affine varieties. Moreover, we obtain an algorithm to explicitly calculate this parametrization which has been used for G of rank at most 8, except E₈.When G is defined and split over the field of q elements, for q the power of a good prime for G, this algorithmic parametrization is used to calculate the number k(U(q); \( \mathfrak{u} \)*(q)) of coadjoint orbits of U(q) on \( \mathfrak{u} \)*(q). Since k(U(q), \( \mathfrak{u} \)*(q)) coincides with the number k(U(q)) of conjugacy classes in U(q), these calculations can be viewed as an extension of the results obtained in [11]. In each case considered here there is a polynomial h(t) with integer coefficients such that for every such q we have k(U(q)) = h(q). We also explain implications of our results for a parametrization of the irreducible complex characters of U(q).     

Characterization of finite $$\varvec{}$$-grous by their Schur multilier

Let G be a finite p-group of order \(p^n\) and M(G) be its Schur multiplier. It is a well known result by Green that \(|M(G)|= p^{\frac{1}{2}n(n-1)-t(G)}\) for some \(t(G) \ge 0\). In this article, we classify non-abelian p-groups G of order \(p^n\) for \(t(G)=\log _p(|G|)+1\).     

Around <Emphasis Type="Italic">q</Emphasis>-Appell polynomial sequences

First we show that the quadratic decomposition of the Appell polynomials with respect to the q-divided difference operator is supplied by two other Appell sequences with respect to a new operator \(\mathcal{M}_{q;q^{-\varepsilon}}\), where ε represents a complex parameter different from any negative even integer number. While seeking all the orthogonal polynomial sequences invariant under the action of \(\mathcal{M}_{\sqrt{q};q^{-\varepsilon/2}}\) (the \(\mathcal{M}_{\sqrt{q};q^{-\varepsilon/2}}\)-Appell), only the Wall q-polynomials with parameter q ^ε/2+1 are achieved, up to a linear transformation. This brings a new characterization of these polynomial sequences.     

Nodal sets of smooth functions with finite vanishing order and <Emphasis Type="Italic">p</Emphasis>-sweepouts

We show that on a compact Riemannian manifold (M, g), nodal sets of linear combinations of any \(p+1\) smooth functions form an admissible p-sweepout provided these linear combinations have uniformly bounded vanishing order. This applies in particular to finite linear combinations of Laplace eigenfunctions. As a result, we obtain a new proof of the Gromov, Guth, Marques–Neves upper bounds on the min–max p-widths of M. We also prove that close to a point at which a smooth function on \(\mathbb {R}^{n+1}\) vanishes to order k, its nodal set is contained in the union of \(k\,W^{1,p}\) graphs for some \(p > 1\). This implies that the nodal set is locally countably n-rectifiable and has locally finite \(\mathcal {H}^n\) measure, facts which also follow from a previous result of Bär. Finally, we prove the continuity of the Hausdorff measure of nodal sets under heat flow.     

Cohomogeneity one Kähler and Kähler–Einstein manifolds with one singular orbit I

Let M be a cohomogeneity one manifold of a compact semisimple Lie group G with one singular orbit \(S_0 = G/H\). Then M is G-diffeomorphic to the total space \(G \times _H V\) of the homogeneous vector bundle over \(S_0\) defined by a sphere transitive representation of G in a vector space V. We describe all such manifolds M which admit an invariant Kähler structure of standard type. This means that the restriction \(\mu : S = Gx = G/L \rightarrow F = G/K \) of the moment map of M to a regular orbit \(S=G/L\) is a holomorphic map of S with the induced CR structure onto a flag manifold \(F = G/K\), where \(K = N_G(L)\), endowed with an invariant complex structure \(J^F\). We describe all such standard Kähler cohomogeneity one manifolds in terms of the painted Dynkin diagram associated with \((F = G/K,J^F)\) and a parameterized interval in some T-Weyl chamber. We determine which of these manifolds admit invariant Kähler–Einstein metrics.     

On the periodicity of continued fractions in elliptic fields

Article [1] raised the question of the finiteness of the number of square-free polynomials f ∈ ?[h] of fixed degree for which \(\sqrt f \) has periodic continued fraction expansion in the field ?((h)) and the fields ?(h)(\(\sqrt f \)) are not isomorphic to one another and to fields of the form ?(h)\(\left( {\sqrt {c{h^n} + 1} } \right)\), where c ∈ ?* and n ∈ ?. In this paper, we give a positive answer to this question for an elliptic field ?(h)(\(\sqrt f \)) in the case deg f = 3.     

Reconstruction of the potential of the Sturm–Liouville operator from a finite set of eigenvalues and normalizing constants

It is well known that the potential q of the Sturm–Liouville operator L_y = ?y? + q(x)y on the finite interval [0, π] can be uniquely reconstructed from the spectrum \(\left\{ {{\lambda _k}} \right\}_1^\infty \) and the normalizing numbers \(\left\{ {{\alpha _k}} \right\}_1^\infty \) of the operator L_D with the Dirichlet conditions. For an arbitrary real-valued potential q lying in the Sobolev space \(W_2^\theta \left[ {0,\pi } \right],\theta > - 1\), we construct a function q_N providing a 2N-approximation to the potential on the basis of the finite spectral data set \(\left\{ {{\lambda _k}} \right\}_1^N \cup \left\{ {{\alpha _k}} \right\}_1^N\). The main result is that, for arbitrary τ in the interval ?1 ≤ τ < θ, the estimate \({\left\| {q - \left. {{q_N}} \right\|} \right._\tau } \leqslant C{N^{\tau - \theta }}\) is true, where \({\left\| {\left. \cdot \right\|} \right._\tau }\) is the norm on the Sobolev space \(W_2^\tau \). The constant C depends solely on \({\left\| {\left. q \right\|} \right._\theta }\).     

Exponential growth rates of free and amalgamated products

We prove that there is a gap between \(\sqrt 2 and\left( {1 + \sqrt 5 } \right)/2\) for the exponential growth rate of nontrivial free products. For amalgamated products G = A_*CB with ([A: C] ? 1)([B: C] ? 1) ≥ 2, we show that an exponential growth rate lower than \(\sqrt 2 \) can be achieved. Indeed, there are infinitely many amalgamated products for which the exponential growth rate is equal to ψ ≈ 1.325, where ψ is the unique positive root of the polynomial z³?z?1. One of these groups is \(PGL\left( {2,\mathbb{Z}} \right) \cong \left( {{C_2} \times {C_2}} \right){*_{{C_2}}}{D_6}\). However, under some natural conditions the lower bound can be put up to \(\sqrt 2 \). This answers two questions by Avinoam Mann [The growth of free products, Journal of Algebra 326, no. 1 (2011), 208–217]. We also prove that ψ is a lower bound for the minimal growth rates of a large class of Coxeter groups, including cofinite non-cocompact planar hyperbolic groups, which strengthens a result obtained earlier by William Floyd, who considered only standard Coxeter generators.     

The primitive idempotents and weight distributions of irreducible constacyclic codes

Let \({\mathbb {F}}_q\) be a finite field with q elements such that \(l^v||(q^t-1)\) and \(\gcd (l,q(q-1))=1\), where l, t are primes and v is a positive integer. In this paper, we give all primitive idempotents in a ring \(\mathbb F_q[x]/\langle x^{l^m}-a\rangle \) for \(a\in {\mathbb {F}}_q^*\). Specially for \(t=2\), we give the weight distributions of all irreducible constacyclic codes and their dual codes of length \(l^m\) over \({\mathbb {F}}_q\).     

Euclidean and Hermitian LCD MDS codes

Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual is trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. On the other hand, being optimal codes, maximum distance separable codes (abbreviated MDS) are of much interest from many viewpoints due to their theoretical and practical properties. However, little work has been done on LCD MDS codes. In particular, determining the existence of q-ary [n, k] LCD MDS codes for various lengths n and dimensions k is a basic and interesting problem. In this paper, we firstly study the problem of the existence of q-ary [n, k] LCD MDS codes and solve it for the Euclidean case. More specifically, we show that for \(q>3\) there exists a q-ary [n, k] Euclidean LCD MDS code, where \(0\le k \le n\le q+1\), or, \(q=2^{m}\), \(n=q+2\) and \(k= 3 \text { or } q-1\). Secondly, we investigate several constructions of new Euclidean and Hermitian LCD MDS codes. Our main techniques in constructing Euclidean and Hermitian LCD MDS codes use some linear codes with small dimension or codimension, self-orthogonal codes and generalized Reed-Solomon codes.     

On finite groups with automorphisms whose fixed points are Engel

The main result of the paper is the following theorem. Let q be a prime, n a positive integer, and A an elementary abelian group of order q². Suppose that A acts coprimely on a finite group G and assume that for each \({a \in A^{\#}}\) every element of C_G(a) is n-Engel in G. Then the group G is k-Engel for some \({\{n,q\}}\)-bounded number k.     

A class of constacyclic codes over a finite field-II

Let \(\mathbb{F}_q\) be a finite field with q = p ^m elements, where p is any prime and m ≥ 1. In this paper, we explicitly determine all the μ-constacyclic codes of length ? ⁿ over \(\mathbb{F}_q\), where ? is an odd prime coprime to p and the order of μ is a power of ?. All the repeated-root λ- constacyclic codes of length ? ⁿ p ^s over \(\mathbb{F}_q\) are also determined for any nonzero λ in \(\mathbb{F}_q\). As examples all the λ-constacyclic codes of length 3 ⁿ p ^s over \(\mathbb{F}_q\) for p = 5, 7, 11, 19 for n ≥ 1, s ≥ 1 are derived. We also obtain all the self-orthogonal negacyclic codes of length ? ⁿ over \(\mathbb{F}_q\) when q is odd prime power and give some illustrative examples.