On homogeneous hypersurfaces in $${\mathbb {C}}^3$$

We consider a family \(M_t^n\), with \(n\geqslant 2\), \(t>1\), of real hypersurfaces in a complex affine n-dimensional quadric arising in connection with the classification of homogeneous compact simply connected real-analytic hypersurfaces in  \({\mathbb {C}}^n\) due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the embeddability of \(M_t^n\) in  \({\mathbb {C}}^n\) for \(n=3,7\). In our earlier article we showed that \(M_t^7\) is not embeddable in  \({\mathbb {C}}^7\) for every t and that \(M_t^3\) is embeddable in  \({\mathbb {C}}^3\) for all \(1<t<1+10^{-6}\). In the present paper, we improve on the latter result by showing that the embeddability of \(M_t^3\) in fact takes place for \(1<t<\sqrt{(2+\sqrt{2})/3}\). This is achieved by analyzing the explicit totally real embedding of the sphere \(S^3\) in \({\mathbb {C}}^3\) constructed by Ahern and Rudin. For \(t\geqslant {\sqrt{(2+\sqrt{2})/3}}\), the problem of the embeddability of \(M_t^3\) remains open.     

Magnetic Trajectories in an Almost Contact Metric Manifold $${\mathbb{R}^{2N+1}}$$

In this paper we classify magnetic trajectories γ in \({{\mathbb{R}}^{2N+1}}\) endowed with a canonical quasi-Sasakian structure, corresponding to a magnetic field proportional to the fundamental 2-form. We prove that they are helices of order 5 and we show that there exists a totally geodesic \({{\mathbb{R}}^5}\) in \({\mathbb{R}^{2N+1}}\) such that γ lies in \({{\mathbb{R}}^5}\). Moreover, the quasi-Sasakian structure of \({{\mathbb{R}}^5}\) is that induced from the ambient manifold.     

Rotational Surfaces in $$S^{3}$$ with Constant Mean Curvature

There is a two-parametric family of rotational symmetric CMC surfaces; more precisely, for every real number H and every \(C\ge 2(H+\sqrt{1+H^2})\) there is a rotational symmetry surface \(\Sigma _{H,C}\) with mean curvature H. Perdomo (Asian J Math 14:73–108, 2010) showed that for every H between \(\cot \left( \frac{\pi }{m}\right) \) and \(\frac{m^2-2}{2\sqrt{m^2-1}}\) there exists an embedded rotational symmetric example with non-constant principal curvatures that is invariant under the cyclic group \(Z_m\). Recently Andrews and Li (J Differ Geom 99:169–189, 2015) showed that these embedded CMC tori are the only embedded genus 1 surfaces with CMC on the sphere. In this paper we complete the study of this family of CMC surfaces and we show that for every integer \(m>2\), there is a properly immersed example in this family that contains a great circle and is invariant under the cyclic group \(Z_m\). We will say that these examples contain the axis of symmetry. We also show that every non-isoparametric surface \(\Sigma _{H,C}\) is either properly immersed and invariant under the cyclic group \(Z_m\) for some integer \(m>1\) or it is dense in the region bounded by two isoparametric tori if the surface \(\Sigma _{H,C}\) does not contain the axis of symmetry or it is dense in the region bounded by a totally umbilical surface if the surface \(\Sigma _{H,C}\) contains the axis of symmetry.     

Globally optimized packings of non-uniform size spheres in $$\mathbb {R}^{d}$$: a computational study

In this work we discuss the following general packing problem: given a finite collection of d-dimensional spheres with (in principle) arbitrarily chosen radii, find the smallest sphere in \(\mathbb {R}^{d}\) that contains the given d-spheres in a non-overlapping arrangement. Analytical (closed-form) solutions cannot be expected for this very general problem-type: therefore we propose a suitable combination of constrained nonlinear optimization methodology with specifically designed heuristic search strategies, in order to find high-quality numerical solutions in an efficient manner. We present optimized sphere configurations with up to \(n = 50\) spheres in dimensions \(d = 2, 3, 4, 5\). Our numerical results are on average within 1% of the entire set of best known results for a well-studied model-instance in \(\mathbb {R}^{2}\), with new (conjectured) packings for previously unexplored generalizations of the same model-class in \(\mathbb {R}^{d}\) with \(d= 3, 4, 5.\) Our results also enable the estimation of the optimized container sphere radii and of the packing fraction as functions of the model instance parameters n and 1 / n, respectively. These findings provide a general framework to define challenging packing problem-classes with conjectured numerical solution estimates.     

The log-behavior of $$\root n \of {p(n)}$$ and $$\root n \of {p(n)/n}$$p(n)/nn

Let p(n) denote the partition function and let \(\Delta \) be the difference operator with respect to n. In this paper, we obtain a lower bound for \(\Delta ^2\log \root n-1 \of {p(n-1)/(n-1)}\), leading to a proof of a conjecture of Sun on the log-convexity of \(\{\root n \of {p(n)/n}\}_{n\ge 60}\). Using the same argument, it can be shown that for any real number \(\alpha \), there exists an integer \(n(\alpha )\) such that the sequence \(\{\root n \of {p(n)/n^{\alpha }}\}_{n\ge n(\alpha )}\) is log-convex. Moreover, we show that \(\lim \limits _{n \rightarrow +\infty }n^{\frac{5}{2}}\Delta ^2\log \root n \of {p(n)}=3\pi /\sqrt{24}\). Finally, by finding an upper bound for \(\Delta ^2 \log \root n-1 \of {p(n-1)}\), we establish an inequality on the ratio \(\frac{\root n-1 \of {p(n-1)}}{\root n \of {p(n)}}\).     

On regular Stein neighborhoods of a union of two maximally totally real subspaces in $$\mathbb {C}^n$$

We construct regular Stein neighborhoods of a union of two maximally totally real subspaces \(M=(A+iI)\mathbb {R}^n\) and \(N=\mathbb {R}^n\) in \(\mathbb {C}^n\), provided that the eigenvalues of the real \(n \times n\) matrix A are sufficiently small. This result is applied to provide regular Stein neighborhoods of an immersed totally real n-manifold in a complex n-manifold, with only finitely many double points, and such that the union of the tangent spaces at each double point in some local coordinates coincides with \(M\cup N\), described above.     

Proper Holomorphic Legendrian Curves in $$SL_2(\mathbb {C})$$

In this paper, we prove that every open Riemann surface properly embeds in the special linear group \(SL_2(\mathbb {C})\) as a holomorphic Legendrian curve, where \(SL_2(\mathbb {C})\) is endowed with its standard contact structure. As a consequence, we derive the existence of proper, weakly complete, flat fronts with arbitrary complex structure in the real hyperbolic space \(\mathbb {H}^3\).     

Improving algebraic attacks on stream ciphers based on linear feedback shift register over $$\mathbb {F}_{2^k}$$

In this paper we investigate univariate algebraic attacks on filter generators over extension fields \(\mathbb {F}_q=\mathbb {F}_{2^n}\) with focus on the Welch–Gong (WG) family of stream ciphers. Our main contribution is to reduce the general algebraic attack complexity on such cipher by proving new and lower bounds for the spectral immunity of such ciphers. The spectral immunity is the univariate analog of algebraic immunity and instead of measuring degree of multiples of a multivariate polynomial, it measures the minimum number of nonzero coefficients of a multiple of a univariate polynomial. In particular, there is an algebraic degeneracy in these constructions, which, when combined with attacks based on low-weight multiples over \(\mathbb {F}_q\), provides much more efficient attacks than over \(\mathbb {F}_2\). With negligible computational complexity, our best attack breaks the primitive WG-5 if given access to 4 kilobytes of keystream, break WG-7 if given access to 16 kilobytes of keystream and break WG-8 if given access to half a megabyte of keystream. Our best attack on WG-16 targeted at 4G-LTE is less practical, and requires \(2^{103}\) computational complexity and \(2^{61}\) bits of keystream. In all instances, we significantly lower both keystream and computational complexity in comparison to previous estimates. On a side note, we resolve an open problem regarding the rank of a type of equation systems used in algebraic attacks.     

The finite Littlewood problem in $${{\mathbb {F}}}_p$$

Let p be a large prime number and f(x) be an integer-valued function defined in \({\mathbb F}_p\). The Littlewood problem in \({{\mathbb {F}}}_p\) is to establish non-trivial lower bounds for the \(\ell _1\) norm of exponential sums involving f(x). In the present paper, we establish new lower bounds for exponential sums including polynomials, powers of any primitive root and subgroups of \(\mathbb {F}_p^*.\)     

Homotopy type of neighborhood complexes of Kneser graphs, $$\varvec{KG_{2,k}}$$

Schrijver (Nieuw Archief voor Wiskunde, 26(3) (1978) 454–461) identified a family of vertex critical subgraphs of the Kneser graphs called the stable Kneser graphs \(SG_{n,k}\). Björner and de Longueville (Combinatorica 23(1) (2003) 23–34) proved that the neighborhood complex of the stable Kneser graph \(SG_{n,k}\) is homotopy equivalent to a k-sphere. In this article, we prove that the homotopy type of the neighborhood complex of the Kneser graph \(KG_{2,k}\) is a wedge of \((k+4)(k+1)+1\) spheres of dimension k. We construct a maximal subgraph \(S_{2,k}\) of \(KG_{2,k}\), whose neighborhood complex is homotopy equivalent to the neighborhood complex of \(SG_{2,k}\). Further, we prove that the neighborhood complex of \(S_{2,k}\) deformation retracts onto the neighborhood complex of \(SG_{2,k}\).     

On Willmore Legendrian surfaces in $$\mathbb {S}^5$$ and the contact stationary Legendrian Willmore surfaces

In this paper we study Willmore Legendrian surfaces (that is Legendrian surfaces which are critical points of the Willmore functional). We use an equality proved in Luo (arXiv:1211.4227v6) to get a relation between Willmore Legendrian surfaces and contact stationary Legendrian surfaces in \(\mathbb {S}^5\), and then we use this relation to prove a classification result for Willmore Legendrian spheres in \(\mathbb {S}^5\). We also get an integral inequality for Willmore Legendrian surfaces and in particular we prove that if the square length of the second fundamental form of a Willmore Legendrian surface in \(\mathbb {S}^5\) belongs to [0, 2], then it must be 0 and L is totally geodesic or 2 and L is a flat minimal Legendrian tori, which generalizes the result of Yamaguchi et al. (Proc Am Math Soc 54:276–280, 1976). We also study variation of the Willmore functional among Legendrian surfaces in 5-dimensional Sasakian manifolds. Let \(\Sigma \) be a closed surface and \((M,\alpha ,g_\alpha ,J)\) a 5-dimensional Sasakian manifold with a contact form \(\alpha \), an associated metric \(g_\alpha \) and an almost complex structure J. Assume that \(f:\Sigma \mapsto M\) is a Legendrian immersion. Then f is called a contact stationary Legendrian Willmore surface (in short, a csL Willmore surface) if it is a critical point of the Willmore functional under contact deformations. To investigate the existence of csL Willmore surfaces we introduce a higher order flow which preserves the Legendre condition and decreases the Willmore energy. As a first step we prove that this flow is well posed if \((M,\alpha ,g_\alpha ,J)\) is a Sasakian Einstein manifold, in particular \(\mathbb {S}^5\).     

Cartoon Approximation with $$\alpha $$-Curvelets

It is well-known that curvelets provide optimal approximations for so-called cartoon images which are defined as piecewise \(C^2\)-functions, separated by a \(C^2\) singularity curve. In this paper, we consider the more general case of piecewise \(C^\beta \)-functions, separated by a \(C^\beta \) singularity curve for \(\beta \in (1,2]\). We first prove a benchmark result for the possibly achievable best N-term approximation rate for this more general signal model. Then we introduce what we call \(\alpha \)-curvelets, which are systems that interpolate between wavelet systems on the one hand (\(\alpha = 1\)) and curvelet systems on the other hand (\(\alpha = \frac{1}{2}\)). Our main result states that those frames achieve this optimal rate for \(\alpha = \frac{1}{\beta }\), up to \(\log \)-factors.     

A transmission problem on $${\mathbb{R}^{2}}$$ with critical exponential growth

In this work we prove the existence of a nontrivial solution for a transmission problem on \({\mathbb{R}^{2}}\) with critical exponential growth, that is, the nonlinearity behaves like exp(α₀ s ²) as |s| → ∞, for some α₀ > 0.     

On MDS convolutional codes over $${\mathbb {Z}}_{p^{r}}$$

Maximum distance separable (MDS) convolutional codes are characterized through the property that the free distance meets the generalized Singleton bound. The existence of free MDS convolutional codes over \({\mathbb {Z}}_{p^{r}}\) was recently discovered in Oued and Sole (IEEE Trans Inf Theory 59(11):7305–7313, 2013) via the Hensel lift of a cyclic code. In this paper we further investigate this important class of convolutional codes over \({\mathbb {Z}}_{p^{r}}\) from a new perspective. We introduce the notions of p-standard form and r-optimal parameters to derive a novel upper bound of Singleton type on the free distance. Moreover, we present a constructive method for building general (non necessarily free) MDS convolutional codes over \({\mathbb {Z}}_{p^{r}}\) for any given set of parameters.     

An arc-search $${\mathcal {O}}(nL)$$ infeasible-interior-point algorithm for linear programming

Arc-search interior-point methods have been proposed to capture the curvature of the central path using an approximation based on ellipse. Yang et al. (J Appl Math Comput 51(1–2):209–225, 2016) proved that an arc-search algorithm has the computational order of \({\mathcal {O}}(n^{5/4}L)\). In this paper, we propose an arc-search infeasible-interior-point algorithms and discuss its convergence analysis. We improve the polynomial bound from \({\mathcal {O}}(n^{5/4}L)\) to \({\mathcal {O}}(nL)\), which is at least as good as the best existing bound for infeasible-interior-point algorithms for linear programming. Numerical results indicate that the proposed method solved LP instances faster than the existing \({\mathcal {O}}(n^{5/4}L)\) method.     

On codes over $$\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}$$

In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring \(R=\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}\), where \(v^{3}=v\), for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over \(\mathbb {F}_q\) and extend these to codes over R.     

Set-theoretic complete intersection monomial curves in $${\mathbb{P}^n}$$

In this paper, we give a sufficient numerical criterion for a monomial curve in a projective space to be a set-theoretic complete intersection. Our main result generalizes a similar statement proven by Keum for monomial curves in three-dimensional projective space. We also prove that there are infinitely many set-theoretic complete intersection monomial curves in the projective n?space for any suitably chosen n ? 1 integers. In particular, for any positive integers p, q, where gcd(p, q) = 1, the monomial curve defined by p, q, r is a set-theoretic complete intersection for every \({r \geq pq( q - 1)}\).     

Aspects of the Segre variety $${\mathcal{S}_{1,1,1}(2)}$$

We consider various aspects of the Segre variety \({\mathcal{S}:=\mathcal{S} _{1,1,1}(2)}\) in PG(7, 2), whose stabilizer group \({\mathcal{G}_{\mathcal{S}}<{\rm GL}(8,2)}\) has the structure \({\mathcal{N}\rtimes{\rm Sym}(3),}\) where \({\mathcal{N} :={\rm GL}(2,2)\times{\rm GL}(2,2)\times{\rm GL} (2,2).}\) In particular we prove that \({\mathcal{S}}\) determines a distinguished Z ₃-subgroup \({\mathcal{Z}<{\rm GL}(8,2)}\) such that \({A\mathcal{Z}A^{-1}=\mathcal{Z},}\) for all \({A\in\mathcal{G}_{\mathcal{S}},}\) and in consequence \({\mathcal{S}}\) determines a \({\mathcal{G}_{\mathcal{S}}}\)-invariant spread of 85 lines in PG(7, 2). Furthermore we see that Segre varieties \({\mathcal{S}_{1,1,1}(2)}\) in PG(7, 2) come along in triplets \({\{\mathcal{S},\mathcal{S}^{\prime},\mathcal{S}^{\prime\prime}\}}\) which share the same distinguished Z ₃-subgroup \({\mathcal{Z}<{\rm GL}(8,2).}\) We conclude by determining all fifteen \({\mathcal{G}_{\mathcal{S}}}\)-invariant polynomial functions on PG(7, 2) which have degree < 8, and their relation to the five \({\mathcal{G}_{\mathcal{S}}}\)-orbits of points in PG(7, 2).     

On the reflexivity of $$\mathcal {P}_{w}(^{n}E;F)$$

In this paper we prove that if E and F are reflexive Banach spaces and G is a closed linear subspace of the space \(\mathcal {L}_{K}(E;F)\) of all compact linear operators from E into F, then G is either reflexive or non-isomorphic to a dual space. This result generalizes (Israel J Math 21:38-49, 1975, Theorem 2) and gives the solution to a problem posed by Feder (Ill J Math 24:196-205, 1980, Problem 1). We also prove that if E and F are reflexive Banach spaces, then the space \(\mathcal {P}_{w}(^{n}E;F)\) of all n-homogeneous polynomials from E into F which are weakly continuous on bounded sets is either reflexive or non-isomorphic to a dual space.     

A remark on the ring of algebraic integers in $$\mathbb{Q}(\sqrt { - d} )$$

It is well-known that the rings O_d of algebraic integers in \(\mathbb{Q}(\sqrt { - d} )\) for d = 19, 43, 67, and 163 are principal ideal domains but not Euclidean. In this article we shall provide a method, based on a result of P. M. Cohn, to construct explicitly pairs (b, a) of integers in O_d for d = 19, 43, 67, and 163 such that, in O_d, there exists no terminating division chain of finite length starting from the pairs (b, a). That is, a greatest common divisor of the pairs (b, a) exists in O_d but it can not be obtained by applying a terminating division chain of finite length starting from (b, a). Furthermore, for squarefree positive integer d ? {1, 2, 3, 7, 11, 19, 43, 67, 163}, we shall also construct pairs (b, a) of integers in O_d which generate O_d but have no terminating division chain of finite length. It is of interest to note that our construction provides a short alternative proof of a theorem of Cohn which is related to the concept of GE₂-rings.