Bauer–Furuta invariants under $${\mathbb{Z}_2}$$ -actions

S. Bauer and M. Furuta defined a stable cohomotopy refinement of the Seiberg–Witten invariants. In this paper, we prove a vanishing theorem of Bauer–Furuta invariants for 4-manifolds with smooth -actions. As an application, we give a constraint on smooth -actions on homotopy K3#K3, and construct a nonsmoothable locally linear -action on K3#K3. We also construct a nonsmoothable locally linear -action on K3.      

Epsilon–delta surgery over $${mathbb{Z}}$$

The purpose of this paper is to prove a controlled surgery exact sequence, including a stability theorem, as used in the construction of exotic homology manifolds. The approach is to show that this result is a formal consequence of the Chapman–Ferry Alpha-approximation theorem.     

Lattices from codes over $$\mathbb {Z}_q$$: generalization of constructions D, $$ D'$$ D′ and $$\overline{ D}$$ D¯

In this paper, we extend the lattice Constructions D, \(D'\) and \(\overline{D}\) (this latter is also known as Forney’s code formula) from codes over \(\mathbb {F}_p\) to linear codes over \(\mathbb {Z}_q\), where \(q \in \mathbb {N}\). We define an operation in \(\mathbb {Z}_q^n\) called zero-one addition, which coincides with the Schur product when restricted to \(\mathbb {Z}_2^n\) and show that the extended Construction \(\overline{D}\) produces a lattice if and only if the nested codes are closed under this addition. A generalization to the real case of the recently developed Construction \(A'\) is also derived and we show that this construction produces a lattice if and only if the corresponding code over \(\mathbb {Z}_q[X]/X^a\) is closed under a shifted zero-one addition. One of the motivations for this work is the recent use of q-ary lattices in cryptography.     

Weighted Reed–Muller codes revisited

We consider weighted Reed–Muller codes over point ensemble S ₁ × · · · × S _m where S _i needs not be of the same size as S _j . For m = 2 we determine optimal weights and analyze in detail what is the impact of the ratio |S ₁|/|S ₂| on the minimum distance. In conclusion the weighted Reed–Muller code construction is much better than its reputation. For a class of affine variety codes that contains the weighted Reed–Muller codes we then present two list decoding algorithms. With a small modification one of these algorithms is able to correct up to 31 errors of the [49,11,28] Joyner code.     

Periodic maximal surfaces in the Lorentz–Minkowski space $${\mathbb{L}^3}$$

A maximal surface with isolated singularities in a complete flat Lorentzian 3-manifold

{\mathbb{Z}_2}-bordism and the Borsuk–Ulam Theorem

The purpose of this work is to classify, for given integers \({m,\, n\geq 1}\), the bordism class of a closed smooth \({m}\)-manifold \({X^m}\) with a free smooth involution \({\tau}\) with respect to the validity of the Borsuk–Ulam property that for every continuous map \({\phi : X^m \to \mathbb{R}^n}\) there exists a point \({x\in X^m}\) such that \({\phi (x)=\phi (\tau (x))}\). We will classify a given free \({\mathbb{Z}_2}\)-bordism class \({\alpha}\) according to the three possible cases that (a) all representatives \({(X^m, \tau)}\) of \({\alpha}\) satisfy the Borsuk–Ulam property; (b) there are representatives \({({X_{1}^{m}}, \tau_1)}\) and \({({X_{2}^{m}}, \tau_2)}\) of \({\alpha}\) such that \({({X_{1}^{m}}, \tau_1)}\) satisfies the Borsuk–Ulam property but \({({X_{2}^{m}}, \tau_2)}\) does not; (c) no representative \({(X^m, \tau)}\) of \({\alpha}\) satisfies the Borsuk–Ulam property.     

${\mathbb{Z}}/3{\mathbb{Z}}$-量子群的Gr\"{o}bner-Shirshov 基

通过计算合成, 我们证明了Yamane 给出的关系是 ${\mathbb{Z}}/3{\mathbb{Z}}$-量子群的一个Gr\"{o}bner-Shirshov 基.     

Minimal codewords in Reed–Muller codes

Minimal codewords were introduced by Massey (Proceedings of the 6th Joint Swedish-Russian International Workshop on Information Theory, pp 276–279, 1993) for cryptographical purposes. They are used in particular secret sharing schemes, to model the access structures. We study minimal codewords of weight smaller than 3 · 2 ^m−r in binary Reed–Muller codes RM(r, m) and translate our problem into a geometrical one, using a classification result of Kasami and Tokura (IEEE Trans Inf Theory 16:752–759, 1970) and Kasami et al. (Inf Control 30(4):380–395, 1976) on Boolean functions. In this geometrical setting, we calculate numbers of non-minimal codewords. So we obtain the number of minimal codewords in the cases where we have information about the weight distribution of the code RM(r, m). The presented results improve previous results obtained theoretically by Borissov et al. (Discrete Appl Math 128(1), 65–74, 2003), and computer aided results of Borissov and Manev (Serdica Math J 30(2-3), 303–324, 2004). This paper is in fact an extended abstract. Full proofs can be found on the arXiv.     

ON HEYDE’S THEOREM ON THE GROUP $$\mathbb{R}$$ × $$\mathbb{T}$$

Doklady Mathematics - According to the well-knows Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of...     

Explicit Inversion Formulas for the X–Ray Transform in {\mathbb{Z}^n}

In the present paper, we state and prove explicit inversion formulas for the X–Ray transform in the lattice ${\mathbb{Z}^n}$ by using certain arithmetical geometrical techniques.     

Z∗$$ {\mathcal{Z}}^{\ast } $$-Semilocal Modules and the Proper Class ℛS$$ \mathrm{\mathcal{R}}\mathcal{S} $$

Ukrainian Mathematical Journal - Over an arbitrary ring, a module M is said to be $$ {\mathcal{Z}}^{\ast } $$-semilocal if every submodule U of M has a $$ {\mathcal{Z}}^{\ast } $$ -supplement V in...     

Poletsky–Stessin Hardy Spaces on Complex Ellipsoids in $$\mathbb {C}^{n}$$

We study Poletsky–Stessin Hardy spaces on complex ellipsoids in \(\mathbb {C}^{n}\). Different from one variable case, classical Hardy spaces are strictly contained in Poletsky–Stessin Hardy spaces on complex ellipsoids so boundary values are not automatically obtained in this case. We have showed that functions belonging to Poletsky–Stessin Hardy spaces have boundary values and they can be approached through admissible approach regions in the complex ellipsoid case. Moreover, we have obtained that polynomials are dense in these spaces. We also considered the composition operators acting on Poletsky–Stessin Hardy spaces on complex ellipsoids and gave conditions for their boundedness and compactness.     

The Intrinsic π-Operator on Domain Manifolds in $${\mathbb{C}^{n+1}}$$

The main aim of this article is to study the hypercomplex π-operator over \mathbbCⁿ⁺¹{\mathbb{C}^{n+1}} via real, compact, n + 1-dimensional manifolds called domain manifolds. We introduce an intrinsic Dirac operator for such types of domain manifolds and define an intrinsic π-operator, study its mapping properties and introduce a Clifford–Beltrami equation in this context.     

Schatten class Hankel and $$\overline{\partial }$$ ∂ ¯ -Neumann operators on pseudoconvex domains in $$\mathbb {C}^{n}$$ C n

Monatshefte für Mathematik - Let $$\Omega $$ be a $$C^2$$ -smooth bounded pseudoconvex domain in $$\mathbb {C}^n$$ for $$n\ge 2$$ and let $$\varphi $$ be a holomorphic function on $$\Omega $$...     

The Iwasawa invariant $$\mu$$ μ vanishes for $$\mathbb{Z}_{2}$$ Z 2 -extensions of certain real biquadratic fields

Acta Mathematica Hungarica - For a real biquadratic field, we denote by $$\lambda$$ , $$\mu$$ and $$\nu$$ the Iwasawa invariants of cyclotomic $$\mathbb{Z}_{2}$$ -extension of $$k$$ . We give...     

Quasi–Monte Carlo rules for numerical integration over the unit sphere {\mathbb{S}^2}

We study numerical integration on the unit sphere ${\mathbb{S}^2 \subseteq\mathbb{R}^3}$ using equal weight quadrature rules, where the weights are such that constant functions are integrated exactly. The quadrature points are constructed by lifting a (0, m, 2)-net given in the unit square [0, 1]² to the sphere ${\mathbb{S}^2}$ by means of an area preserving map. A similar approach has previously been suggested by Cui and Freeden [SIAM J Sci Comput 18(2):595–609, 1997]. We prove three results. The first one is that the construction is (almost) optimal with respect to discrepancies based on spherical rectangles. Further we prove that the point set is asymptotically uniformly distributed on ${\mathbb{S}^2}$ . And finally, we prove an upper bound on the spherical cap L ₂-discrepancy of order N ^?1/2(log N)^1/2 (where N denotes the number of points). This improves upon the bound on the spherical cap L ₂-discrepancy of the construction by Lubotzky, Phillips and Sarnak [Commun Pure Appl Math 39(S, suppl):S149–S186, 1986] by a factor of ${\sqrt{\log N}}$ . Numerical results suggest that the (0, m, 2)-nets lifted to the sphere ${\mathbb{S}^2}$ have spherical cap L ₂-discrepancy converging with the optimal order of N ^?3/4.     

On the average number of subgroups of the group $${\mathbb {Z}}_m \times {\mathbb {Z}}_n$$ Z m × Z n with k-th power sequence

The Ramanujan Journal - Let $$\mathbb {Z}_{n}$$ be the additive group of residue classes modulo n. Let s(m, n) denote the total number of subgroups of the group $$\mathbb {Z}_{m} \times...     

An asymptotic version of Dumnicki’s algorithm for linear systems in $${\mathbb{CP}^2}$$

Using Dumnicki’s approach to showing non-specialty of linear systems consisting of plane curves with prescribed multiplicities in sufficiently general points on we develop an asymptotic method to determine lower bounds for Seshadri constants of general points on . With this method we prove the lower bound for 10 general points on .