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.      

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.     

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...     

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.     

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     

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.     

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.     

On certain degenerate Whittaker Models for cuspidal representations of $${\mathrm{GL}_{k \cdot n}(\mathbb {F}_q)}$$ GL k · n ( F q )

Mathematische Zeitschrift - Let $$\pi $$ be an irreducible cuspidal representation of $$\mathrm {GL}_{kn}(\mathbb {F}_q)$$ . Assume that $$\pi = \pi _{\theta }$$ , corresponds to a regular...     

Blowup relations on $$mathbb{C}^2/mathbb{Z}_2$$ from Nakajima–Yoshioka blowup relations

Theoretical and Mathematical Physics - Using an elementary algebraic approach, from the Nakajima–Yoshioka blowup relations, we obtain bilinear relations for the Nekrasov partition functions...     

$\mathbb{Z}_+^k$-作用的方向原像熵

In this paper,a new type of entropy,directional preimage entropy including topological and measure theoretic versions for■-actions,is introduced.Some of their properties including relationships and the invariance are obtained.Moreover,several systems including■-actions generated by the expanding maps,■-actions defined on finite graphs and some infinite graphs with zero directional preimage branch entropy are studied.     

Orbit inequivalent actions for groups containing a copy of $\mathbb{F}_{2}$

We prove that if a countable group Γ contains a copy of \mathbbF₂\mathbb{F}_{2}, then it admits uncountably many non orbit equivalent actions.     

Schiffer’s Problem and an Isoperimetric Inequality for the First Buckling Eigenvalue of Domains on $$\mathbb{S}^{2}$$

We consider the overdetermined eigenvalue problem on a sufficiently regular connected open domain Ω on the 2-sphere :

Equidistribution of saddle periodic points for Hénon-type automorphisms of $$\mathbb {C}^k$$

In this paper, we prove the equidistribution of saddle periodic points for Hénon-type automorphisms of \(\mathbb {C}^k\) with respect to its equilibrium measure. A general strategy to obtain equidistribution properties in any dimension is presented. It is based on our recent theory of densities for positive closed currents. Several fine properties of dynamical currents are also proved.     

Bavrin’s Type Factorization of the Temljakov Operator for Holomorphic Functions in Circular Domains of $$\mathbb {C}^{n}$$

The paper concerns investigations of holomorphic functions of several complex variables with a factorization of their Temljakov transform. Firstly, there were considered some inclusions between the families \(\mathcal {C}_{\mathcal {G}},\mathcal {M}_{\mathcal {G}},\mathcal {N}_{\mathcal {G}},\mathcal {R}_{\mathcal {G}},\mathcal {V}_{\mathcal {G}}\) of such holomorphic functions on complete n-circular domain \(\mathcal {G}\) of \(\mathbb {C}^{n}\) in some papers of Bavrin, Fukui, Higuchi, Michiwaki. A motivation of our investigations is a condensation of the mentioned inclusions by some new families of Bavrin’s type. Hence we consider some families \(\mathcal {K}_{ \mathcal {G}}^{k},k\ge 2,\) of holomorphic functions f :  \(\mathcal {G}\rightarrow \mathbb {C},f(0)=1,\) defined also by a factorization of \( \mathcal {L}f\) onto factors from \(\mathcal {C}_{\mathcal {G}}\) and \(\mathcal {M} _{\mathcal {G}}.\) We present some interesting properties and extremal problems on \(\mathcal {K}_{\mathcal {G}}^{k}\).     

Stable spike clusters for the precursor Gierer–Meinhardt system in $$\mathbb {R}^2$$

We consider the Gierer–Meinhardt system with small inhibitor diffusivity, very small activator diffusivity and a precursor inhomogeneity. For any given positive integer k we construct a spike cluster consisting of k spikes which all approach the same nondegenerate local minimum point of the precursor inhomogeneity. We show that this spike cluster can be linearly stable. In particular, we show the existence of spike clusters for spikes located at the vertices of a polygon with or without centre. Further, the cluster without centre is stable for up to three spikes, whereas the cluster with centre is stable for up to six spikes. The main idea underpinning these stable spike clusters is the following: due to the small inhibitor diffusivity the interaction between spikes is repulsive, and the spikes are attracted towards the local minimum point of the precursor inhomogeneity. Combining these two effects can lead to an equilibrium of spike positions within the cluster such that the cluster is linearly stable.     

A short proof of Cartan’s Nullstellensatz for entire functions in $${mathbb{C}^n}$$

Using the fact that the maximal ideals in the polydisk algebra are given by the kernels of point evaluations, we derive a simple formula that gives a solution to the Bézout equation in the space of all entire functions of several complex variables. Thus a short and easy analytic proof of Cartan’s Nullstellensatz is obtained.     

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 Teichmüller distance between finite index subgroups of $${PSL_2(\mathbb{Z})}$$

For a given , we show that there exist two finite index subgroups of which are -quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any there are two finite regular covers of the Modular once punctured torus T ₀ (or just the Modular torus) and a -quasiconformal map between them that is not homotopic to a conformal map. As an application of the above results, we show that the orbit of the basepoint in the Teichmüller space T(S ^p ) of the punctured solenoid S ^p under the action of the corresponding Modular group (which is the mapping class group of S ^p [6], [7]) has the closure in T(S ^p ) strictly larger than the orbit and that the closure is necessarily uncountable.