Pth Power Research Articles

Estimation of integrated volatility in stochastic volatility models

In the framework of stochastic volatility models we examine estimators for the integrated volatility based on the pth power variation (i.e. the sum of pth absolute powers of the log-returns). We derive consistency and distributional results for the estimators given high-frequency data, especially taking into account what kind of process we may add to our model without affecting the estimate of the integrated volatility. This may on the one hand be interpreted as a possible flexibility in modelling, for example adding jumps or even leaving the framework of semimartingales by adding a fractional Brownian motion, or on the other hand as robustness against model misspecification. We will discuss possible choices of p under different model assumptions and irregularly spaced data. Copyright © 2005 John Wiley & Sons, Ltd.

Some examples on solution structures for weakly nonlinear elliptic equations

We discuss bifurcation problems for weakly nonlinear ellipticelliptic equations involving a pth power term where p is largerthan 1 but close to 1.We show that in a number of cases one canobtain rather complete information about the structure of the setof positive solutions.These results apply to any bounded domain.

A new family of windows――convolution windows and their applications

A new family of windows is constructed by convolutions via a few rectangular windows with same time width and is thus referred to as convolution windows. The expressions of the second-order up to the eighth-order convolution windows in both the time and frequency domains are derived. Their applications in high accuracy harmonic analysis of periodic signals are investigated. Comparisons between the proposed windows and some known windows with the same width shows that, when the synchronous deviation of data sampling is slight, the proposed ones have the least effect of spectral leakage. Therefore, the new windows are well suited for high accuracy harmonic analysis and parameter estimation for periodic signals. The error analysis and computer simulations show that the estimation errors, corresponding to frequency, amplitude and phase of every harmonic component of a signal, are proportional to the pth power of the relative frequency deviation in case of the pth-order convolution window is applied to windowing signal of approximately p cycles. By introducing real time adjustment in sampling interval, the proposed algorithm can adaptively trace signal frequency and lead to less sampling synchronous deviation. The proposed approach has the advantages of easy implementation and high measure precision and can be used in harmonic analysis of quasi-periodic signals whose fundamental frequency drifts slowly with time.

On the Inviscid Limit for Two-Dimensional Incompressible Flow with Navier Friction Condition

In [Nonlinearity, 11 (1998), pp. 1625--1636], Clopeau, Mikelić, and Robert studied the inviscid limit of the two-dimensional incompressible Navier--Stokes equations in a bounded domain subject to Navier friction--type boundary conditions. They proved that the inviscid limit satisfies the incompressible Euler equations, and their result ultimately includes flows generated by bounded initial vorticities. Our purpose in this article is to adapt and, to some extent, simplify their argument in order to include pth power integrable initial vorticities, with p > 2.

Optimal Control of Obstacle for Quasi-linear Elliptic Variational Bilateral Problems

This paper is concerned with an optimal control problem for quasi-linear elliptic variational inequality in which the bilateral obstacles are the control. The cost functional of this optimal control problem is of Lagrange type in which the pth power of Laplacian of the control appears. This feature leads to the fact that it is hard to derive the optimality system for the underlying problem. In this paper, the optimality system is established by utilizing the special structure of the approximate optimality system including the monotonicity of the leading differential operator.

Broad relaxation spectrum and the field theory of glassy dynamics for pinned elastic systems.

We study thermally activated, low-temperature equilibrium dynamics of elastic systems pinned by disorder using one loop functional renormalization group (FRG). Through a series of increasingly complete approximations, we investigate how the field theory reveals the glassy nature of the dynamics, in particular divergent barriers and barrier distributions controling the spectrum of relaxation times. First, we naively assume a single relaxation time tau(k) for each wave vector k, leading to analytical expressions for equilibrium dynamical response and correlations. These exhibit two distinct scaling regimes (scaling variables T k(theta) ln t and t/ tau(k), respectively, with T the temperature, theta the energy fluctuation exponent, and tau(k) approximately e(c k(-theta) /T) ) and are easily extended to quasiequilibrium and aging regimes. A careful study of the dynamical operators encoding for fluctuations of the relaxation times shows that this first approach is unsatisfactory. A second stage of approximation including these fluctuations, based on a truncation of the dynamical effective action to a random friction model, yields a size (L) dependent log-normal distribution of relaxation times (effective barriers centered around Ltheta and of fluctuations approximately L(theta/2) ) and some procedure to estimate dynamical scaling functions. Finally, we study the full structure of the running dynamical effective action within the field theory. We find that relaxation time distributions are nontrivial (broad but not log normal) and encoded in a closed hierarchy of FRG equations divided into levels p=0,1, em leader, corresponding to vertices proportional to the pth power of frequency omega(p). We show how each level p can be solved independently of higher ones, the lowest one (p=0) comprising the statics. A thermal boundary layer ansatz (TBLA) appears as a consistent solution. It extends the one discovered in the statics which was shown to embody droplet thermal fluctuations. Although perturbative control remains a challenge, the structure of the dynamical TBLA which encodes barrier distributions opens the way for deeper understanding of the field theory approach to glasses.

On temporal asymptotics for the pth power viscous reactive gas

In this paper, we investigate the long time behaviour of solutions to the system governing a heat-conductive, viscous reactive pth power gas confined between two parallel plates. For initial-boundary value problems with the end-points held at a prescribed temperature or insulated, we prove the global existence of physically relevant solutions, and establish their rate of convergence to equilibria, for generic initial data. The estimates for different boundary conditions are presented in a unified manner.

Effect of non-bonding molecular orbitals on total π-electron energy

The presence of non-bonding molecular orbitals diminishes the HMO total π-electron energy. A method is elaborated to quantitatively assess this effect. The effect is found to be much smaller than earlier anticipated. Contrary to earlier claims, it depends on the size of the conjugated system and, as a good approximation, decreases as the pth power of the number of vertices of the molecular graph; p≈0.95. The effect is almost insensitive to other structural features present in the conjugated system.

Some properties of the Bézier–Kantorovich type operators

The aim of this paper is to present estimates for the rate of pointwise convergence of the Bézier–Kantorovich modification of the discrete Feller operators in some classes of measurable functions bounded on an interval I, in particular, for functions of bounded pth power variation on I. Our theorems generalize and extend the recent results of Zeng and Piriou (J. Approx. Theory 95(1998) 369; 104(2000) 330) for the kantorovichians of the Bernstein–Bézier operators in the class of functions of bounded variation in the Jordan sense on [0,1].

Choosability of Powers of Circuits

It is proved that ch(G)=χ(G) if G=C n p , the pth power of the circuit graph C n , or if G is a uniform inflation of such a graph. The proof uses the method of Alon and Tarsi. As a corollary, the (a : b)-choosability conjectures hold for all such graphs.

Relations between coal properties and spontaneous combustion parameters : Kaymakci, E. and Didari, V. Turkish Journal of Engineering & Environmental Sciences, 2002, 26, (1), 59–64

Although the least mean pth power (LMP) and normalized LMP (NLMP) algorithms of adaptive Volterra filters outperform the conventional least mean square (LMS) algorithm in the presence of α-stable noise, they still exhibit slow convergence and high steady-state kernel error in nonlinear system identification. To overcome these limitations, an enhanced recursive least mean pth power algorithm with logarithmic transformation (RLogLMP) is proposed in this paper. The proposed algorithm is adjusted to minimize the new cost function with the p-norm logarithmic transformation of the error signal. The logarithmic transformation, which can diminish the significance of outliers under α-stable noise environment, increases the robustness of the proposed algorithm and reduces the steady-state kernel error. Moreover, the proposed method improves the convergence rate by the enhanced recursive scheme. Finally, simulation results demonstrate that the proposed algorithm is superior to the LMP, NLMP, normalized least mean absolute deviation (NLMAD), recursive least squares (RLS) and nonlinear iteratively reweighted least squares (NIRLS) algorithms in terms of convergence rate and steady-state kernel error.

Non-trivial effect of fast vibration on the dynamics of a class of non-linearly damped mechanical systems

The effect of very high-frequency excitation on the slow dynamics of a class of non-linearly damped mechanical oscillators is considered. Two different models of damping namely, piecewise linear and pth power damping are considered. Fast excitation is modelled as triangular, sinusoidal and random base excitation. The effect of fast excitation is theoretically analyzed using the method of direct partition of motion (MDPM) and direct simulation. The method of numerical averaging is also used, where damping characteristics or excitations are not amenable to analytical techniques. Fast excitation has the non-trivial effect of increasing and decreasing the low-velocity damping of hard and soft dampers, respectively. The effect of fast excitation on the transient and steady state slow dynamics of the system is investigated by direct numerical integration of the equation of motion.

On optimal sequential prediction for general processes

This paper considers several aspects of the sequential prediction problem for unbounded, nonstationary processes under pth power loss /spl lscr//sub p/(u,v)=|u-v|/sup p/, 1<p</spl infin/. In the first part, it is shown that Bayes prediction schemes are Cesaro optimal under general conditions, that Cesaro optimal prediction schemes are unique in a natural sense, and that Cesaro optimality is equivalent to a form of weak calibration. Connections between calibration and stronger forms of optimality are considered. Extensions of the existence and uniqueness results to generalized prediction, and prediction from observations with additive noise, are established. For binary processes, it is shown that thresholding an optimal prediction scheme for the squared loss yields an optimal binary prediction scheme for the Hamming loss. In the second part of the paper, it is shown how to construct, from a countable family of prediction schemes, a single composite scheme whose asymptotic performance on any suitable process dominates the performance of each member of the family. The construction is based on aggregating methods for individual binary sequences. Using the construction, some results of Algoet (1994, 1995) on the existence of Cesaro optimal schemes for families of ergodic processes are rederived in a direct way and extended to unbounded processes.

Note on a Fermat-type diophantine equation

Let p>5 be a prime number and ζ a pth root of unity. Let c be an integer divisible only by primes of the form kp−1, (k,p)=1 . Let C p ( i) be the eigenspace of the p-Sylow subgroup of ideal class group C of Q (ζ) corresponding to ω i, ω being the Teichmuller character. In this article we extend the main theorem in Sitaraman (J. Number Theory 80 (2000) 174) and get the following: For any fixed odd positive integer n< p−4, assume: (a) At least one of C p (3), C p (5),…, C p ( n) is non-trivial. (b) C p ( i) =0 for p− n−1⩽ i⩽ p−2. (c) 2 i≢1 mod (p) for 1⩽ i⩽ n+1. Let q be an odd prime such that q≡1 mod (p) , and such that there is a prime ideal Q over q in Q (ζ) whose ideal class is of the form I p J where J is non-trivial, not a pth power and J∈ C p (3)⊕ C p (5)⊕⋯⊕ C p ( n) . For such p and q, if x p + y p = pcz p has a non-trivial solution x,y,z∈ Z , with ( x, y, z)=1, then q∤ pcz p x+y . Let t( n)= n 224 n 4 . If log p>t(n) , then applying a result of Soulé (J. Reine Angew. Math. 517 (1999) 209), we show that the above result holds with only condition (a) because the others are automatically satisfied. We also make a remark about the effect of Soulé's result on the p-divisibility of h p + (the class number of the maximal real subgroup of Q (ζ) ) which is relevant to the existence of integral solutions to x p + y p = pcz p .

Linear complexity of polyphase power residue sequences

The well known family of binary Legendre or quadratic residue sequences can be generalised to the multiple-valued case by employing a polyphase representation. These p-phase sequences, with p prime, also have prime length L, and can be constructed from the index sequence of length L or, equivalently, from the cosets of pth power residues and non-residues modulo-L. The linear complexity of these polyphase sequences is derived and shown to fall into four classes depending on the value assigned to b0, the initial digit of the sequence, and on whether p belongs to the set of pth power residues or not. The characteristic polynomials of the linear feedback shift registers that generate these sequences are also derived.

Convexification, Concavification and Monotonization in Global Optimization

We show in this paper that via certain convexification, concavification and monotonization schemes a nonconvex optimization problem over a simplex can be always converted into an equivalent better-structured nonconvex optimization problem, e.g., a concave optimization problem or a D.C. programming problem, thus facilitating the search of a global optimum by using the existing methods in concave minimization and D.C. programming. We first prove that a monotone optimization problem (with a monotone objective function and monotone constraints) can be transformed into a concave minimization problem over a convex set or a D.C. programming problem via pth power transformation. We then prove that a class of nonconvex minimization problems can be always reduced to a monotone optimization problem, thus a concave minimization problem or a D.C. programming problem.

Root Sets of Polynomials Modulo Prime Powers

A subset R of the integers modulo n is defined to be a root set if it is the set of roots of some polynomial. Using the Chinese Remainder Theorem, the question of finding and counting root sets modn is reduced to finding root sets modulo a prime power. In this paper, we provide a recursive construction for root sets modulo a prime power. We use this recursion to show that the number of root sets modulo pk for fixed k is a polynomial in p, raised to the pth power. Moreover, we show that the leading term of this polynomial is ckp⌊k2/4⌋ where ck=(k2!)−1 if k is even and ck=(k−12!)−1+(k+12!)−1 if k is odd, thus giving an asymptotic estimate on the number of root sets for fixed k. Finally, we generalize these results to arbitrary Dedekind domains.

Estimate for the second-order derivatives of solutions to boundary-value problems of the theory of non-Newtonian liquids

The boundary regularity of solutions to some boundary-value problems describing stationary flow of generalized Newtonian liquids is studied. The dissipative potential is of quadratic growth at infinity. We prove that the second-order derivatives of the solution are pth power summable functions, where p is greater than two. The partial regularity of the strain velocity tensor is established. In the two-dimensional case, the complete regularity of the strain velocity tensor is also proved. Bibliography: 14 titles.

Pth Power Lagrangian Method for Integer Programming

When does there exist an optimal generating Lagrangian multiplier vector (that generates an optimal solution of an integer programming problem in a Lagrangian relaxation formulation), and in cases of nonexistence, can we produce the existence in some other equivalent representation space? Under what conditions does there exist an optimal primal-dual pair in integer programming? This paper considers both questions. A theoretical characterization of the perturbation function in integer programming yields a new insight on the existence of an optimal generating Lagrangian multiplier vector, the existence of an optimal primal-dual pair, and the duality gap. The proposed pth power Lagrangian method convexifies the perturbation function and guarantees the existence of an optimal generating Lagrangian multiplier vector. A condition for the existence of an optimal primal-dual pair is given for the Lagrangian relaxation method to be successful in identifying an optimal solution of the primal problem via the maximization of the Lagrangian dual. The existence of an optimal primal-dual pair is assured for cases with a single Lagrangian constraint, while adopting the pth power Lagrangian method. This paper then shows that an integer programming problem with multiple constraints can be always converted into an equivalent form with a single surrogate constraint. Therefore, success of a dual search is guaranteed for a general class of finite integer programming problems with a prominent feature of a one-dimensional dual search.

Notes on Inequalities with Doubling Weights

Various important weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii, Schur, Remez, etc., inequalities, have been proved recently by Giuseppe Mastroianni and Vilmos Totik under minimal assumptions on the weights. In most of the cases this minimal assumption is the doubling condition. Sometimes, however, as in the weighted Nikolskii inequality, the slightly stronger A ∞condition is used. Throughout their paper the L p norm is studied under the assumption 1⩽ p<∞. In this note we show that their proofs can be modified so that many of their inequalities hold even if 0< p<1. The crucial tool is an estimate for quadrature sums for the pth power (0< p<∞ is arbitrary) of trigonometric polynomials established by Lubinsky, Máté, and Nevai. For technical reasons we discuss only the trigonometric cases.