Piecewise Description Research Articles

One-loop power spectrum in ultra slow-roll inflation and implications for primordial black hole dark matter

We apply the in-in formalism to address the question of whether the size of the one-loop spectrum of curvature fluctuations in ultra-slow-roll inflation models designed for producing a large population of primordial black holes implies a breakdown of perturbation theory. We consider a simplified piece-wise description of inflation, in which the ultra-slow-roll phase is preceded and followed by slow-roll phases linked by transitional periods. We work in the δϕ-gauge, including all relevant cubic and quartic interactions and the necessary counterterms to renormalize the ultraviolet divergences, regularized by a cutoff. The ratio of the one-loop to the tree-level contributions to the spectrum of curvature perturbations is controlled by the duration of the ultra-slow-roll phase and of the transitions. Our results indicate that perturbation theory does not necessarily break in well-known models proposed to account for all the dark matter in the form of primordial black holes.

Long-Term Stochastic Modeling of Sheet Pile Corrosion in Coastal Environment from On-Site Measurements

Optimization of maintenance and design of coastal steel infrastructure needs for long-term predictive degradation models. The phenomenon of corrosion in the offshore and coastal environment is very complex due to the stochastic and changing nature of the environment (temperature, water chemical properties) and the multiple involved processes in competition. In the GEROM French project, the objective was to build a database of the residual thickness measured from ultrasonic measurements on sheet piles and piles carried out during 40 years in commercial and military harbors along the French coast. A total amount of 35,460 measurements were gathered. After a detailed analysis of the data and statistical modeling, a probabilistic model of corrosion is proposed for sheet piles. It relies on a piecewise description of the process with the depth depending on the exposed zone and a time-dependent evolution of parameters of the gamma probability density function.

High-Order Layerwise Formulation of Transverse Shear Stress Field for Laminated Composite Beams

This paper presents a high-order layerwise theoretical framework for laminated composite beams based on a piecewise description of the transverse shear-stress field. Linear and quadratic variations in the transverse shear-stress field are assumed in each discrete layer, and compatibility conditions of the shear stress are a priori fulfilled through the introduction of shear-stress variables defined on the layer surfaces. Based on the stress–strain relations, the transverse shear strain is obtained, and the in-plane displacement is determined by integrating the transverse shear strain along the thickness direction. By imposing continuity conditions of the displacements, a displacement field expression that contains only displacement variables is formulated. Moreover, a two-node beam element associated with the present model is developed, where Hermite cubic interpolation functions are used for the transverse displacement variable, and linear interpolation functions are employed for the remaining displacement variables. Comparisons with the exact solution, various displacement-based theories, and high-fidelity finite element models demonstrate the high accuracy of the present model in predicting the stress and displacement distributions. In addition, the introduction of sublayers to improve the modeling accuracy and the influence of the stiffness ratio between layers on the effectiveness of the proposed model are addressed.

A quadratic trigonometric spline for curve modeling.

An imperative curve modeling technique has been established with a view to its applications in various disciplines of science, engineering and design. It is a new spline method using piecewise quadratic trigonometric functions. It possesses error bounds of order 3. The proposed curve model also owns the most favorable geometric properties. The proposed spline method accomplishes C2 smoothness and produces a Quadratic Trigonometric Spline (QTS) with the view to its applications in curve design and control. It produces a C2 quadratic trigonometric alternative to the traditional cubic polynomial spline (CPS) because of having four control points in its piecewise description. The comparison analysis of QTS and CPS verifies the QTS as better alternate to CPS. Also, the time analysis proves QTS computationally efficient than CPS.

A graphical approach to optimal power management for uncertain OFF-Grid PV-FC-electrolyzer-battery hybrid systems

The problem of optimal power management for uncertain off-grid hybrid systems is studied in this paper. A suitable parametrization of the energy sources and sinks is used to obtain a graphical representation of the feasible set along with a direct observation of the solution and its piecewise description. The proposed approach pretends to clarify how the optimal solution migrates from the use of one source to another as the solar radiation changes and how the dynamic response of the sources restrict the feasible set. A power generation system (PGS) is used to study the limitations that the sun power variability and the parametric uncertainty impose on the optimality; being the PGS constituted by photovoltaic (PV) panels, a fuel cell (FC), an electrolyzer (E) and a battery bank. A simple optimal Power Management Strategy (PMS) is proposed as a result of the analysis and its performance is illustrated in a telecom facility for three different locations in Mexico. Results are shown indicating the viability of the strategy for highly variable environments.

A Quadratic Trigonometric Nu Spline with Shape Control

A significant curve modeling technique has been introduced with a view to its applications in geometric modeling, computer graphics and computer-aided design. It is a new spline method using quadratic trigonometric functions with well-controlled shape influences of parameters introduced through geometric continuity of order two. The proposed curve model owns the best possible geometric properties such as convex hull, partition of unity, affine invariance and variation diminishing. A quadratic normally has three control points giving lesser flexibility to one piece of curve. However, a cubic has four control points giving higher flexibility to one piece of curve. In the proposed scheme, we have introduced a quadratic trigonometric with four control points. Thus, the proposed quadratic trigonometric has embedded geometric features of cubic/cubic trigonometric. The proposed spline method, constrained with Nu spline like GC2 smoothness, produces a quadratic trigonometric Nu spline (QTNS) with interesting shape control locally and globally. The method is helpful for a variety of shape effects, like point tension, interval tension or global tension. It also produces a quadratic trigonometric alternative to cubic/cubic trigonometric spline because of having four control points in its piecewise description.

On corner cutting in multi-obstacle avoidance problems

One challenging and not extensively studied issue in obstacle avoidance is the corner cutting problem. Avoidance constraints are usually imposed at the sampling time without regards to the intra-sample behavior of the dynamics. This paper improves upon state of the art by describing a multi-obstacle environment over a hyperplane arrangement scaffolding, provides a piecewise description of the "shadow" regions and represents them into a combined mixed integer and predictive control formulation. Furthermore, over-approximation constraints which reduce to strictly binary formulations are discussed in detail. Illustrative proofs of concept, comparisons with the state of the art and simulation results over a classical multi-obstacle avoidance problem validate the benefits of the proposed approach.

Short-time fractional Fourier methods for signal representation in the time-frequency plane

The analysis of multicomponent chirp signals is of interest in sonar, geophysical, bioacoustic, ultrasonic, and radar applications. This paper examines the interpretation of the fractional Fourier transform (FrFT) of such signals in terms of their representation in the time-frequency plane. The FrFT has been shown to be of value in analyzing and filtering linear chirp signals with various characteristics. The desirability of a short-time variant has been suggested by several authors and this paper introduces new short-time fractional Fourier methods for the piecewise description and parameterization of nonlinear chirp signals. Understanding of a signal’s FrFT as the projection of the joint time-frequency distribution onto a line at angle α to the time axis permits development of these short-time methods. Parameter α provides a means for transform optimization with reference to the entire signal or on a windowed basis. While providing much improved time-frequency resolution over the spectrogram, these techniques do not suffer from cross-term interference seen in Wigner distributions and related methods. Representation of highly nonlinear chirps in the time-frequency plane is demonstrated through analysis of a variety of signals including bat and dolphin chirps, heart sounds, and synthetic signals constructed to highlight particular differences between the techniques.

Quillen stratification for modules

Let G be a finite group and k a fixed algebraically closed field of characteristic p>O. If p is odd, let H, be the subring of H*(G, k) consisting of elements of even degree; following [20-221 we take H, = H*(G, k) if p=2, though one could just as well use the subring of elements of even degree for all p. H, is a finitely generated commutative k-algebra [13], and we let V, denote its associated affine variety Max H,. If M is any finitely generated kG-module, then the cohomology variety V,(M) of M may be defined as the support in V, of the H,-module H*(G, M) if G is a p-group, and in general as the largest support of H*(G, L@ M), where L is any kG-module [4, 91. A module L with each irreducible kc-module as a direct summand will serve. D. Quillen [20-221 proved a number of beautiful results relating k;; to the varieties V, associated with the various elementary abelian p-subgroups E of G, culminating in his stratification theorem [20, 221. This theorem gives a piecewise description of V, almost explicitly in terms of the subgroups E and their normalizers in G. A well-known corollary is that dim V, =max dim V,, where E

A Quillen stratification theorem for modules

Let G be a finite group and k a fixed algebraically closed field of characteristic p > 0. If p is odd, let HG be the subring of //*(G, k) consisting of elements of even degree; take HG = //*(G, k) if p = 2. HG is a finitely generated commutative fc-algebra, and we let VG denote its associated affine variety Max HG. If M is any finitely generated fcG-module, the cohomology variety VG(M) of M may be defined as the support in VG of the HG -module H*(G, M) if G is a pgroup, and in general as the largest support of //*(G, L ® M) where L is any kGmodule. A module L with each irreducible fcG-module as a direct summand will do [3]. D. Quillen [9, 10] proved a number of beautiful results relating VG to the varieties VE associated with the elementary abelian p-subgroups E of Gf culminating in his stratification theorem. This theorem gives a piecewise description of VG in terms of the subgroups E and their normalizers in G. Some of Quillen's results have been extended to the variety VG(M) associated with a fcG-module M [1,4, 5, 6, 7, 8 ] , and the work of Alperin and Evens [2] and Avrunin [3] showed that there was at least a surjection IIE VE(M) —• VG(M). However, the stratification theorem for VG(M) remained elusive, since one still needed to know that a point in VG(M) in the image of a given VE was in fact in the image of VE(M). We announce here a proof of the stratification theorem for VG(M)9 as well as a proof of a conjecture of J. Carlson regarding VE(M) for E an elementary abelian p-subgroup. We are also able to generalize several of Quillen's other results to the module case. For H VG be the transfer map induced by restriction on the cohomology rings. For an elementary abelian p-subgroup E, let V^ = EWF<E tE,F F and let KM = V% n P*0OThen P K.EW = tGEV^ O VG(M). We have the following stratification theorem.