Compact Curves Research Articles

Finite polynomial cohomology with coefficients

We introduce a theory of finite polynomial cohomology with coefficients in this paper. We prove several basic properties and introduce an Abel–Jacobi map with coefficients. As applications, we use this cohomology theory to study the arithmetic of compact Shimura curves over \mathbb{Q} and to simplify the proofs of the works of Darmon–Rotger and Bertolini–Darmon–Prasanna.

Compact Curve Shortening Flow Solutions Out of Non-Compact Curves

Abstract We construct a slingshot, that is a compact, embedded solution to curve shortening flow that comes out of a non compact curve and exists for a finite time.

Lightweight ECC Coprocessor With Resistance Against Power Analysis Attacks Over NIST Prime Fields

This brief proposes a compact and fast elliptic curve cryptography coprocessor using affine coordinates. Although using affine coordinates requires compute-intensive modular inversion (ModInv), the proposed coprocessor can be accelerated using the improved ModInv. In addition, the study proposes a novel point multiplication method that is effective in preventing simple power analysis when affine coordinates are used. Using point randomization, the proposed coprocessor is also resistant to statistical power analysis (PA) such as differential PA (DPA) and correlation PA (CPA), and it requires 60 k gate counts and can perform one elliptic curve point multiplication within 1 ms at the maximum clock frequency.

The Second Variation for Null-Torsion Holomorphic Curves in the 6-Sphere

In the round 6-sphere, null-torsion holomorphic curves are fundamental examples of minimal surfaces. This class of minimal surfaces is quite rich: By a theorem of Bryant, extended by Rowland, every closed Riemann surface may be conformally embedded in the round 6-sphere as a null-torsion holomorphic curve. In this work, we study the second variation of area for compact null-torsion holomorphic curves \(\Sigma \) of genus g and area \(4\pi d\), focusing on the spectrum of the Jacobi operator. We show that if \(g \le 6\), then the multiplicity of the lowest eigenvalue \(\lambda _1 = -2\) is equal to 4d. Moreover, for any genus, we show that the nullity is at least \(2d + 2 - 2g\). These results are likely to have implications for the deformation theory of asymptotically conical associative 3-folds in \({\mathbb {R}}^7\), as studied by Lotay.

Edge States for Generalized Iwatsuka Models: Magnetic Fields Having a Fast Transition Across a Curve

In this paper, we study the localization and propagation properties of the edge states associated with a class of magnetic Laplacians in mathbb {R}^2. We assume that the intensity of the magnetic field has a fast transition along a regular and compact curve Gamma . Our main results extend to a general regular curve the study of the localized eigenfunction obtained when Gamma is a straight line (i.e. Iwatsuka models). Furthermore, we include in our analysis the case of magnetic fields that slowly change along the curve Gamma and we obtain a rigorous and explicit characterization of the asymptotic mass distribution of the edge state along Gamma .

Stein neighbourhoods of bordered complex curves attached to holomorphically convex sets

In this paper we construct open Stein neighbourhoods of compact sets of the form $A\cup K$ in a complex space, where $K$ is a compact holomorphically convex set, $A$ is a compact complex curve with boundary $bA$ of class $\mathscr C^2$ which may intersect $K$, and $A\cap K$ is $\mathscr O(A)$-convex.

Algebraic surfaces determine analyticity of functions

Let f :X rightarrow mathbb {R} be a function defined on a nonsingular real algebraic set X of dimension at least 3. We prove that f is an analytic (resp. a Nash) function whenever the restriction f|_{S} is an analytic (resp. a Nash) function for every nonsingular algebraic surface S subset X whose each connected component is homeomorphic to the unit 2-sphere. Furthermore, the surfaces S can be replaced by compact nonsingular algebraic curves in X, provided that dimX ge 2 and f is of class mathcal {C}^{infty }.

On an inverse curvature flow in two-dimensional space forms

We study the evolution of compact convex curves in two-dimensional space forms. The normal speed is given by the difference of the weighted inverse curvature with the support function, and in the case where the ambient space is the Euclidean plane, is equivalent to the standard inverse curvature flow. We prove that solutions exist for all time and converge exponentially fast in the smooth topology to a standard round geodesic circle. This has a number of consequences: first, to prove the isoperimetric inequality; second, to establish a range of weighted geometric inequalities; and third, to give a counterexample to the $$n=2$$ case of a conjecture of Girão–Pinheiro.

Kuznets curve with parabola-shaped ultra-wideband antenna with defected ground plane for communications

PurposeThis paper aims to state the configuration of the proposed antenna which is competent to many networks such as LTE and X band applications. The experimental study encountered the significance of the proposed antenna.Design/methodology/approachA novel compact Kuznets curve with parabola-shaped quad-band notched antenna is demonstrated in this paper. The presented prototype is ascertained on a composite material composed of woven fiberglass cloth with an epoxy resin binder. The resulting ultra-wideband antenna ranges 3.1–3.54 GHz, 5.17–5.51 GHz, 5.74–6.43 GHz and 6.79–7.60 GHz. To avoid the frequency bands which cause UWB interference,the projected antenna has been incorporated with slotted patch. The proposed antenna design is attained in four steps. The simple circular patch antenna model with defected ground plane is subjected to stepwise progression by including parabola-shaped slot and U shaped slot on the patch to attain four notched bands.FindingsThis projected antenna possesses an optimal bond among simulated and measured outcomes,which is more suitable for the quad notched band applications. Substrate analysis is done by varying substrate material, and notch behavior is presented. The proposed method’s optimum performance in metrics such as return loss, voltage standing wave ratio and radiation pattern varies its frequency range from 2.56 to 7.6 GHz.Originality/valueThe antenna adaptation of the defected ground plane has achieved through the quad notched band with operating frequency ranges 2.56 to 7.6 GHz and with eliminated frequency ranges 3.55–5.16 GHz, 5.52–5.73 GHz, 6.44–6.78 GHz and 7.66–10.6 GHz.

Moduli of wild Higgs bundles on with -actions

AbstractWe study a set $\mathcal{M}_{K,N}$ parameterising filtered SL(K)-Higgs bundles over $\mathbb{C}P^1$ with an irregular singularity at $z = \infty$, such that the eigenvalues of the Higgs field grow like $\vert \lambda \vert \sim \vert z^{N/K} \mathrm{d}z \vert$, where K and N are coprime. $\mathcal{M}_{K,N}$ carries a $\mathbb{C}^\times$-action analogous to the famous $\mathbb{C}^\times$-action introduced by Hitchin on the moduli spaces of Higgs bundles over compact curves. The construction of this $\mathbb{C}^\times$-action on $\mathcal{M}_{K,N}$ involves the rotation automorphism of the base $\mathbb{C}P^1$. We classify the fixed points of this $\mathbb{C}^\times$-action, and exhibit a curious 1-1 correspondence between these fixed points and certain representations of the vertex algebra $\mathcal{W}_K$ ; in particular we have the relation $\mu = {k-1-c_{\mathrm{eff}}}/{12}$ , where $\mu$ is a regulated version of the L2 norm of the Higgs field, and $c_{\mathrm{eff}}$ is the effective Virasoro central charge of the corresponding W-algebra representation. We also discuss a Białynicki–Birula-type decomposition of $\mathcal{M}_{K,N}$ , where the strata are labeled by isomorphism classes of the underlying filtered vector bundles.

The strong Suslin reciprocity law

We prove the strong Suslin reciprocity law conjectured by A. Goncharov. The Suslin reciprocity law is a generalization of the Weil reciprocity law to higher Milnor$K$-theory. The Milnor$K$-groups can be identified with the top cohomology groups of the polylogarithmic motivic complexes; Goncharov's conjecture predicts the existence of a contracting homotopy underlying Suslin reciprocity. The main ingredient of the proof is a homotopy invariance theorem for the cohomology of the polylogarithmic motivic complexes in the ‘next to Milnor’ degree. We apply these results to the theory of scissors congruences of hyperbolic polytopes. For every triple of rational functions on a compact projective curve over$\mathbb {C}$we construct a hyperbolic polytope (defined up to scissors congruence). The hyperbolic volume and the Dehn invariant of this polytope can be computed directly from the triple of rational functions on the curve.

Codimension bounds and rigidity of ancient mean curvature flows by the tangent flow at −∞

Motivated by the limiting behavior of an explicit class of compact ancient curve shortening flows, by adapting the work of Colding–Minicozzi[11], we prove codimension bounds for ancient mean curvature flows by their tangent flow at [Formula: see text]. In the case of the [Formula: see text]-covered circle, we apply this bound to prove a strong rigidity theorem. Furthermore, we extend this paradigm by showing that under the assumption of sufficiently rapid convergence, a compact ancient mean curvature flow is identical to its tangent flow at [Formula: see text].

On inhomogeneous nonholonomic Bilimovich system

We consider a simple motivating example of a non-Hamiltonian dynamical system with time-dependent constraints obtained by imposing rheonomic non-integrable Bilimovich’s constraint on a freely rotating rigid body. Dynamics of this low-dimensional nonlinear nonautonomous dynamic system involves different kinds of stable and unstable attractors, quasi and strange attractors, compact and noncompact invariant attractive curves, etc. To study this cautionary example we apply the Poincaré map method to disambiguate and discover multiscale temporal dynamics, specifically the coarse-grained dynamics resulting from fast-scale nonlinear control via nonholonomic Bilimovich’s constraint.

A Simple and Stable Centeredness Measure for 3D Curve Skeleton Extraction.

Existing methods for extracting 3D curve skeletons mostly suffer from the difficulty of finding the center points of 3D shapes and tedious manual adjustments of the thresholds for pruning spurious branches due to the influence of shape boundary perturbations. In this article, we present a method based on medial surfaces of 3D shapes for the convenient and fast extraction of high-quality curve skeletons. Our main contribution is a simple and stable centeredness measure. It is based on simulating fire propagation via the scheme of inside-out evolution from the interior to the boundary, differentiating it from existing methods that use the scheme of outside-in evolution from the boundary to the interior. Thus, our measure is much more localized, and it can be implemented with a high degree of parallelism. In addition, we propose measures to effectively suppress the influence of details to obtain a stable measurement, and employ minimum set covers to optimize the center points to generate compact skeletons, which enables spurious branches to be effectively excluded without the tedious work of manually adjusting thresholds. Our experiments show the superiority of our method over existing methods, including its convenient generation of clean, compact and centered curve skeletons while running much faster than state-of-the-art methods.

Continuity-preserving tool path generation for minimizing machining errors in five-axis CNC flank milling of ruled surfaces

Five-axis CNC flank machining has been commonly used in the industry for shaping complex geometries. Geometrical errors typically occur in five-axis flank finishing of non-developable surfaces using a cylindrical cutter. Most existing tool path planning methods adjust discrete cutter locations to reduce these errors. An excessive change in the cutter center or axis between consecutive cutter locations may deteriorate the machined surface quality. This study developed a tool path generation method for minimizing geometrical errors on finished surfaces while preserving high-order continuity in the cutter motion. A tool path is described using the moving trajectory of the cutter center and changes in two rotational angles in compact curve representations. An optimization scheme is proposed to search for optimal curve control points and the resulting tool path. A curve subdivision mechanism progressively increases the control points during the search process. Simulation results confirm that the proposed method not only enhances the computational efficiency of tool path generation but also improves the machined surface finish. This study provides a computational approach for precision tool path planning in five-axis CNC flank finishing of ruled surfaces.

Compact Elliptic Curve Scalar Multiplication with a Secure Generality

Compact Elliptic Curve Scalar Multiplication with a Secure Generality

Internet of Things (IoT) based outdoor performance characterisation of solar photovoltaic module

Outdoor performance characterisation of the Solar Photovoltaic (SPV) module is essential to while designing and commissioning a new SPV power plant. The health of SPV plant is monitored using a conventional method which underutilises the workforce and resources used for Operation and Maintenance (O&M) of the SPV power plant. Outdoor performance characterisation of SPV module using reliable, compact, portable, and economical Current-Voltage (I-V) curve tracer having IoT capability and auto sweep capability is presented in this paper. The capacitive load method is used for I-V curve sweep, and the result is compared with the resistive load method. In this paper, the advantages of using a capacitive load method over resistive load method are observed and experimentally validated. The ease of using IoT feature makes this proposed I-V Curve Tracer (IVCT) device more reliable to trace Current-Voltage (I-V) curve and Power-Voltage (P-V) curve for outdoor performance characterisation of SPV module.

On Compact Affine Quaternionic Curves and Surfaces

This paper is devoted to the study of affine quaternionic manifolds and to a possible classification of all compact affine quaternionic curves and surfaces. It is established that on an affine quaternionic manifold there is one and only one affine quaternionic structure. A direct result, based on the celebrated Kodaira Theorem that studies compact complex manifolds in complex dimension 2, states that the only compact affine quaternionic curves are the quaternionic tori and the primary Hopf surface $$S^3\times S^1$$ . As for compact affine quaternionic surfaces, we restrict to the complete ones: the study of their fundamental groups, together with the inspection of all nilpotent hypercomplex simply connected 8-dimensional Lie Groups, identifies a path towards their classification.

Design of Compact Planar B-spline Curves using DP Control Point Selection with Multi-level Error Functions : Towards Usability Improvement in Design of Dynamic Font-based Characters

In this paper, we consider the problem of representing some given planar curves on a two-dimensional plane as so-called compact B-spline curves by using only the dominant control points. In particular, we here develop a method for selecting the dominant control points by introducing the dynamic programming (DP) approach and a new idea of knot points selection using a multi-level error function where the term multi-level means that not only function values of a given curve are considered, but also the derivatives. Moreover, we offer a way to represent compact planar B-spline curves using non-uniform rational B-splines (NURBS). We demonstrate performance with experimental studies using handwriting data. The results of this paper will be useful for improving usability in the design of the so-called dynamic font-based characters.

FPGA Implementation of the ECC Over GF(2 m ) for Small Embedded Applications

In this article, we propose a compact elliptic curve cryptographic core over GF(2 m ). The proposed architecture is based on the Lopez-Dahab projective point arithmetic operations. To achieve efficiency in resources usage, an iterative method that uses a ROM-based state machine is developed for the elliptic curve cryptography (ECC) point doubling and addition operations. The compact ECC core has been implemented using Virtex FPGA devices. The number of the required slices is 2,102 at 321MHz and 6,738 slices at 262MHz for different GF(2 m ). Extensive experiments were conducted to compare our solution to existing methods in the literature. Our compact core consumes less area than all previously proposed methods. It also provides an excellent performance for scalar multiplication. In addition, the ECC core is implemented in ASIC 0.18μm CMOS technology, and the results show excellent performance. Therefore, our proposed ECC core method provides a balance in terms of speed, area, and power consumption. This makes the proposed design the right choice for cryptosystems in limited-resource devices such as cell phones, IP cores of SoCs, and smart cards. Moreover, side-channel attack resistance is implemented to prevent power analysis.