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Efficient state preparation is essential for implementing efficient quantum algorithms. Whilst several techniques for low-cost state preparation exist, this work facilitates further classes of states, whose amplitudes are well approximated by piecewise polynomials. We show how such states can be efficiently prepared using a piecewise Quantum Singular Value Transformation along with a new piecewise linear diagonal block encoding. We illustrate this with the explicit examples of $\sqrt{x}|x\rangle$ and $\log x|x\rangle$. Further, our technique reduces the cost of window boosted Quantum Phase Estimation by efficiently preparing the B-spline window state. We demonstrate this window state requires 100 times fewer T-gates to prepare than the state-of-the-art Kaiser window state, and we show that the B-spline window replicates the Kaiser window's exponential reduction in tail probability for QPE., Comment: 9 pages, comments welcome

We explore the important task of applying a phase $exp(i f(x))$ to a computational basis state $\left| x \right>$. The closely related task of rotating a target qubit by an angle depending on $f(x)$ is also studied. Such operations are key in many quantum subroutines, and often the function $f$ can be well-approximated by a piecewise linear composition. Examples range from the application of diagonal Hamiltonian terms (such as the Coulomb interaction) in grid-based many-body simulation, to derivative pricing algorithms. Here we exploit a parallelisation of the piecewise approach so that all constituent elementary rotations are performed simultaneously, that is, we achieve a total rotation depth of one. Moreover, we explore the use of recursive catalyst 'towers' to implement these elementary rotations efficiently. Depending on the choice of implementation strategy, we find a depth as low as $O(log n + log S)$ for a register of $n$ qubits and a piecewise approximation of $S$ sections. In the limit of multiple repetitions of the oracle, each instance has an $O(S \cdot n)$ T-count.

A class of quasi-distribution evaluation criteria based on piecewise Bezier curves is proposed to address the issue of the inability to objectively evaluate finite element models. During the optimization design of mechanical parts, finite element modeling is performed on their stress deformation, and the mesh node shape variable values are converted into distribution histogram data for piecewise Bezier curve fitting. Being dealt with area normalization method, the fitting curve could be regarded as a kind of probability density function (PDF), and its variance could be used to evaluate the finite element modeling results. The situation with the minimum variance is the optimal choice for overall deformation. Numerical experiments have indicated that the new method demonstrated the intrinsic characteristics of the finite element models of difference mechanical parts. As an objective appraisal method for evaluating finite element models, it is both effective and feasible.

We prove piecewise Sobolev regularity of vector fields that have piecewise regular curl and divergence, but may fail to be globally continuous. The main idea behind our approach is to employ recently developed parametrices for the curl-operator and the regularity theory of Poisson transmission problems. We conclude our work by applying our findings to the heterogeneous time-harmonic Maxwell equations with either a) impedance, b) natural or c) essential boundary conditions and providing wavenumber-explicit piecewise regularity estimates for these equations.

Identifying breakpoints in piecewise regression is critical in enhancing the reliability and interpretability of data fitting. In this paper, we propose novel algorithms based on the greedy algorithm to accurately and efficiently identify breakpoints in piecewise polynomial regression. The algorithm updates the breakpoints to minimize the error by exploring the neighborhood of each breakpoint. It has a fast convergence rate and stability to find optimal breakpoints. Moreover, it can determine the optimal number of breakpoints. The computational results for real and synthetic data show that its accuracy is better than any existing methods. The real-world datasets demonstrate that breakpoints through the proposed algorithm provide valuable data information., Comment: 13 pages, 6 figures

In the domain of combat simulations, the training and deployment of deep reinforcement learning (RL) agents still face substantial challenges due to the dynamic and intricate nature of such environments. Unfortunately, as the complexity of the scenarios and available information increases, the training time required to achieve a certain threshold of performance does not just increase, but often does so exponentially. This relationship underscores the profound impact of complexity in training RL agents. This paper introduces a novel approach that addresses this limitation in training artificial intelligence (AI) agents using RL. Traditional RL methods have been shown to struggle in these high-dimensional, dynamic environments due to real-world computational constraints and the known sample inefficiency challenges of RL. To overcome these limitations, we propose a method of localized observation abstraction using piecewise linear spatial decay. This technique simplifies the state space, reducing computational demands while still preserving essential information, thereby enhancing AI training efficiency in dynamic environments where spatial relationships are often critical. Our analysis reveals that this localized observation approach consistently outperforms the more traditional global observation approach across increasing scenario complexity levels. This paper advances the research on observation abstractions for RL, illustrating how localized observation with piecewise linear spatial decay can provide an effective solution to large state representation challenges in dynamic environments.

For love dynamical models, a new idea combining piecewise concept for integer-order, stochastic, and fractional derivatives is presented in order to capture the chaos and several crossover emotional scenerios. Under the assumptions of linear growth and Lipschitz condition, the fixed-point theorem explain the uniqueness and existence to the models under the investigation. The piecewise derivatives were approximated utilising the Lagrange interpolation method, and the computer results were demonstrated numerically for several values of order $\alpha$. It was observed that the recently presented new idea in love dynamical models can represent disordered emotional patterns in passionate loving partnerships.

This study uses the Lotka Volterra Predator-Prey model to offer a notion of piecewise patterns for the various piecewise derivatives. Using the piecewise derivatives, we produced numerical solutions that are referred to as the Adams-Bashforth method. The computer results show piecewise patterns in the Lotka Volterra Predator-Prey model's real-world behaviours. The Lotka-Volterra model looks into the relationships between competition and abundance between two competing species. Changes in the abundance of one species are modelled as a function of the abundance of its competitors, but the competitive mechanism is given and evaluated. This notion led some scholars to label certain mathematical expressions as "phenomenological" and to propose a different theoretical framework that gives resources special consideration. The Lotka-Volterra model, often called the predator-prey model or the Lotka-Volterra model, is a nonlinear mathematical expression that is frequently used to analyse the dynamical behaviours of biological systems in which two species interact, one as a predator and the other as prey. The variation in the populations over time is illustrated by the mathematical statement., Comment: 20

This study proposes a simple and novel class of stretch-limiting constitutive relations for perfectly flexible elastic strings drawing from modern advances in constitutive theory for elastic bodies. We investigate strings governed by constitutive relations where stretch is a bounded, piecewise linear function of tension, extending beyond the traditional Cauchy elasticity framework. Our analysis includes explicit solutions for both catenaries and longitudinal, piecewise constant stretched motions., Comment: 13 pages, 2 figures

In applied time-to-event analysis, a flexible parametric approach is to model the hazard rate as a piecewise constant function of time. However, the change points and values of the piecewise constant hazard are usually unknown and need to be estimated. In this paper, we develop a fully data-driven procedure for piecewise constant hazard estimation. We work in a general counting process framework which nests a wide range of popular models in time-to-event analysis including Cox's proportional hazards model with potentially high-dimensional covariates, competing risks models as well as more general multi-state models. To construct our estimator, we set up a regression model for the increments of the Breslow estimator and then use fused lasso techniques to approximate the piecewise constant signal in this regression model. In the theoretical part of the paper, we derive the convergence rate of our estimator as well as some results on how well the change points of the piecewise constant hazard are approximated by our method. We complement the theory by both simulations and a real data example, illustrating that our results apply in rather general event histories such as multi-state models.

Piecewise linearization is a key technique for solving nonlinear problems in transportation network design and other optimization fields, in which generating breakpoints is a fundamental task. This paper proposes an optimal breakpoint selection method, rotational adjusting method (RAM), to minimize the approximation error between the original function and the piecewise linear function with limited number of pieces, applicable to both convex or concave function. RAM rotationally adjusts the location of breakpoints based on its adjacent breakpoints, and the optimal positions would be reached after several iterations. The optimality of the method is proved. Numerical experiments are conducted on the logarithmic function., Comment: 10 pages

In this paper, we are concerned with divergence form, higher-order parabolic systems in a cylindrical domain with a finite number of subdomains. We establish $L_\infty$ and Schauder estimates of solutions when the leading coefficients and the non-homogeneous term exhibit piecewise Dini mean oscillation and piecewise H\"{o}lder continuity, respectively. To the best of our knowledge, our results are new for higher-order elliptic and parabolic systems., Comment: 31 pages

The aim of this paper is to explore the relationship between invariant cones and nonlinear normal modes in piecewise linear mechanical systems. As a key result, we extend the invariant cone concept, originally established for homogeneous piecewise linear systems, to a class of inhomogeneous continuous piecewise linear systems. The inhomogeneous terms can be constant and/or time-dependent, modeling nonsmooth mechanical systems with a clearance gap and external harmonic forcing, respectively. Using an augmented state vector, a modified invariant cone problem is formulated and solved to compute the nonlinear normal modes, understood as periodic solutions of the underlying conservative dynamics. An important contribution is that invariant cones of the underlying homogeneous system can be regarded as a singularity in the theory of nonlinear normal modes of continuous piecewise linear systems. In addition, we use a similar methodology to take external harmonic forcing into account. We illustrate our approach using numerical examples of mechanical oscillators with a unilateral elastic contact. The resulting backbone curves and frequency response diagrams are compared to the results obtained using the shooting method and brute force time integration., Comment: submitted to International Journal of Non-Linear Mechanics

In this paper, we study the number of limit cycles of a piecewise smooth differential system separated by one or two parallel straight lines or rays formed by a nilpotent center or degenerate center and linear saddle. Piecewise linear differential systems separated by one or two parallel straight lines with one of the subsystems of type nilpotent center and other subsystems of type linear saddle can have at most two limit cycles and there are systems in these classes having one limit cycle. The limit cycle in particular consists of saddle separatrices of the subsystem., Comment: 19 pages, 4 figures

Piecewise regression is a versatile approach used in various disciplines to approximate complex functions from limited, potentially noisy data points. In control, piecewise regression is, e.g., used to approximate the optimal control law of model predictive control (MPC), the optimal value function, or unknown system dynamics. Neural networks are a common choice to solve the piecewise regression problem. However, due to their nonlinear structure, training is often based on gradient-based methods, which may fail to find a global optimum or even a solution that leads to a small approximation error. To overcome this problem and to find a global optimal solution, methods based on mixed-integer programming (MIP) can be used. However, the known MIP-based methods are either limited to a special class of functions, e.g., convex piecewise affine functions, or they lead to complex approximations in terms of the number of regions of the piecewise defined function. Both complicate a usage in the framework of control. We propose a new MIP-based method that is not restricted to a particular class of piecewise defined functions and leads to functions that are fast to evaluate and can be used within an optimization problem, making them well suited for use in control., Comment: 12 pages, 3 figures, 1 table, published in the proceedings of Machine Learning Research vol 242

Inspired by piecewise polynomiality results of double Hurwitz numbers, Ardila and Brugall\'e introduced an enumerative problem which they call double Gromov--Witten invariants of Hirzebruch surfaces. These invariants serve as a two-dimensional analogue and satisfy a similar piecewise polynomial structure. More precisely, they introduced the enumeration of curves in Hirzebruch surfaces satisfying point conditions and tangency conditions on the two parallel toric boundaries. These conditions are stored in four partitions and the resulting invariants are piecewise polynomial in their entries. Moreover, they found that these expressions also behave polynomially with respect to the parameter determining the underlying Hirzebruch surfaces. Based on work of Ardila and Block, they proposed that such a polynomiality could also hold while changing between more general toric surfaces corresponding to $h$-transverse polygons. In this work, we answer this question affirmatively. Moreover, we express the resulting invariants for $h$-transverse polygons as matrix elements in the two-dimensional bosonic Fock space., Comment: 24 pages, 8 figures, 3 tables. arXiv admin note: text overlap with arXiv:1412.4563 by other authors