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...     

Modified Alternating $$\vec{k}$$–generators

This paper deals with a class of pseudorandom bit generators – modified alternating –generators. This class is constructed similarly to the class of alternating step generators. Three subclasses of are distinguished, namely linear, mixed and nonlinear generators. The main attention is devoted to the subclass of linear and mixed generators generating periodic sequences with maximal period lengths. A necessary and sufficient condition for all sequences generated by the linear generators of to be with maximal period lengths is formulated. Such sequences have good statistical properties, such as distribution of zeroes and ones, and large linear complexity. Two methods of cryptanalysis of the proposed generators are given. Finally, three new classes of modified alternating –generators, designed especially to be more secure, are presented.     

Certain Somos’s $$\varvec{}$$–$$\varvec{Q}$$Q type Dedekind $$\varvec{\eta }$$η-function identities

In this paper, we provide a new proof for the Dedekind \(\eta \)-function identities discovered by Somos. During this process, we found two new Dedekind \(\eta \)-function identities. Furthermore, we extract interesting partition identities from some of the \(\eta \)-function identities.     

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 $$\Omega $$...     

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.     

Projective Compactification of $$\mathbb {R}^{1,1}$$ and Its Möbius Geometry

We examine the semi-Riemannian manifold \(\mathbb {R}^{1,1}\), which is realized as the split complex plane, and its conformal compactification as an analogue of the complex plane and the Riemann sphere. We also consider conformal maps on the compactification and study some of their basic properties.     

ANNIHILATORS OF HIGHEST WEIGHT $$ \mathfrak{s}\mathfrak{l} $$(∞)-MODULES

We give a criterion for the annihilator in U(\( \mathfrak{s}\mathfrak{l} \)(∞)) of a simple highest weight \( \mathfrak{s}\mathfrak{l} \)(∞)-module to be nonzero. As a consequence we show that, in contrast with the case of \( \mathfrak{s}\mathfrak{l} \)(n), the annihilator in U(\( \mathfrak{s}\mathfrak{l} \)(∞)) of any simple highest weight \( \mathfrak{s}\mathfrak{l} \)(∞)-module is integrable, i.e., coincides with the annihilator of an integrable \( \mathfrak{s}\mathfrak{l} \)(∞)-module. Furthermore, we define the class of ideal Borel subalgebras of \( \mathfrak{s}\mathfrak{l} \)(∞), and prove that any prime integrable ideal in U(\( \mathfrak{s}\mathfrak{l} \)(∞)) is the annihilator of a simple \( \mathfrak{b} \) ⁰-highest weight module, where \( \mathfrak{b} \) ⁰ is any fixed ideal Borel subalgebra of \( \mathfrak{s}\mathfrak{l} \)(∞). This latter result is an analogue of the celebrated Duoflo Theorem for primitive ideals.     

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...     

Surfaces with $${K^2=2\mathcal{X}-2}$$ and pg ≥ 5

This note describes minimal surfaces S of general type satisfying p _g  ≥ 5 and K ² = 2p _g . For p _g  ≥ 8 the canonical map of such surfaces is generically finite of degree 2 and the bulk of the paper is a complete characterization of such surfaces with non birational canonical map. It turns out that if p _g  ≥ 13, S has always an (unique) genus 2 fibration, whose non 2-connected fibres can be characterized, whilst for p _g  ≤ 12 there are two other classes of such surfaces with non birational canonical map.     

A primal-dual homotopy algorithm for $$\ell _{1}$$-minimization with $$\ell _{\infty }$$ℓ∞-constraints

In this paper we propose a primal-dual homotopy method for \(\ell _1\)-minimization problems with infinity norm constraints in the context of sparse reconstruction. The natural homotopy parameter is the value of the bound for the constraints and we show that there exists a piecewise linear solution path with finitely many break points for the primal problem and a respective piecewise constant path for the dual problem. We show that by solving a small linear program, one can jump to the next primal break point and then, solving another small linear program, a new optimal dual solution is calculated which enables the next such jump in the subsequent iteration. Using a theorem of the alternative, we show that the method never gets stuck and indeed calculates the whole path in a finite number of steps. Numerical experiments demonstrate the effectiveness of our algorithm. In many cases, our method significantly outperforms commercial LP solvers; this is possible since our approach employs a sequence of considerably simpler auxiliary linear programs that can be solved efficiently with specialized active-set strategies.     

Optimal software-implemented Itoh–Tsujii inversion for $$\mathbb {F}_{2^{m}}$$

Field inversion in \(\mathbb {F}_{2^{m}}\) dominates the cost of modern software implementations of certain elliptic curve cryptographic operations, such as point encoding/hashing into elliptic curves (Brown et al. in: Submission to NIST, 2008; Brown in: IACR Cryptology ePrint Archive 2008:12, 2008; Aranha et al. in: Cryptology ePrint Archive, Report 2014/486, 2014) Itoh–Tsujii inversion using a polynomial basis and precomputed table-based multi-squaring has been demonstrated to be highly effective for software implementations (Taverne et al. in: CHES 2011, 2011; Oliveira et al. in: J Cryptogr Eng 4(1):3–17, 2014; Aranha et al. in: Cryptology ePrint Archive, Report 2014/486, 2014), but the performance and memory use depend critically on the choice of addition chain and multi-squaring tables, which in prior work have been determined only by suboptimal ad-hoc methods and manual selection. We thoroughly investigated the performance/memory tradeoff for table-based linear transforms used for efficient multi-squaring. Based upon the results of that investigation, we devised a comprehensive cost model for Itoh–Tsujii inversion and a corresponding optimization procedure that is empirically fast and provably finds globally-optimal solutions. We tested this method on eight binary fields commonly used for elliptic curve cryptography; our method found lower-cost solutions than the ad-hoc methods used previously, and for the first time enables a principled exploration of the time/memory tradeoff of inversion implementations.     

A proof of the Anderson–Badawi $$ \mathbf{rad}\varvec{(I)}^{\varvec{n}} \varvec{\subseteq } {\varvec{I}}$$ formula for $$\varvec{n}$$n-absorbing ideals

In [1], Anderson and Badawi conjectured that \(\mathrm{rad}(I)^n \subseteq I\) for every n-absorbing ideal I of a commutative ring. In this article, we prove their conjecture. We also prove related conjectures for radical ideals.     

System Representations for the Paley–Wiener Space $$\mathcal {PW}_{\pi }^2$$

In this paper we study the approximation of stable linear time-invariant systems for the Paley–Wiener space \(\mathcal {PW}_{\pi }^2\), i.e., the set of bandlimited functions with finite \(L^2\)-norm, by convolution sums. It is possible to use either, the convolution sum where the time variable is in the argument of the bandlimited impulse response, or the convolution sum where the time variable is in the argument of the function, as an approximation process. In addition to the pointwise and uniform convergence behavior, the convergence behavior in the norm of the considered function space, i.e. the \(L^2\)-norm in our case, is important. While it is well-known that both convolution sums converge uniformly on the whole real axis, the \(L^2\)-norm of the second convolution sum can be divergent for certain functions and systems. We show that the there exist an infinite dimensional closed subspace of functions and an infinite dimensional closed subspace of systems, such that for any pair of function and system from these two sets, we have norm divergence.     

The Aronszajn–Donoghue Theory for Rank One Perturbations of the $$\mathcal{H}_{-2} {\text{-Class}}$$

A singular rank one perturbation of a self-adjoint operator A in a Hilbert space is considered, where and but with the usual A–scale of Hilbert spaces. A modified version of the Aronszajn-Krein formula is given. It has the form where F denotes the regularized Borel transform of the scalar spectral measure of A associated with . Using this formula we develop a variant of the well known Aronszajn–Donoghue spectral theory for a general rank one perturbation of the class.Submitted: March 14, 2002 Revised: December 15, 2002     

Discreteness criteria for Möbius groups acting on $$\mathcal{O}\mathcal{L}_p $$

In this paper, three new discreteness criteria for Möbius groups acting on\(\bar R^{n * } \) are obtained; they are generalizations of known results using the information of two-generator subgroups.     

On the exponential diophantine equation $$\left( am^{2}+1\right) ^{x}+\left( bm^{2}-1\right) ^{y}=(cm)^{z}$$ with $$ c\mid m $$c∣m

Let \(a,\ b,\ c,\ m\) be positive integers such that \(a+b=c^2, 2\mid a, 2\not \mid c\) and \(m>1\). In this paper we prove that if \(c\mid m \) and \(m>36c^3 \log c\), then the equation \((am^2+1)^x+(bm^2-1)^y=(cm)^z\) has only the positive integer solution \((x,\ y,\ z)\)=\((1,\ 1,\ 2)\).     

The Hardy–Littlewood–Pólya inequality of majorization in the context of $$\mathbf {\omega }$$ – $$\textbf{m}$$ –star-convex functions

Aequationes mathematicae - The Hardy–Littlewood–Pólya inequality of majorization is extended for $$\mathbf {\omega }$$ – $$\textbf{m}$$ –star-convex functions to the...     

On $$ \mathcal{C} $$*-sets and decompositions of continuous and ηζ-continuous functions

The main purpose of this paper is to introduce the concepts of *-sets, *-continuous functions and to obtain new decompositions of continuous and ηζ-continuous functions. Moreover, properties of *-sets and some properties of -sets are discussed.