Prais–Winsten Algorithm for Regression with Second or Higher Order Autoregressive Errors

Dimitrios V. Vougas

doi:10.3390/econometrics9030032

Preserving the Shape of Functions by Applying Multidimensional Schoenberg-Type Operators

Symmetry ◽

10.3390/sym13061016 ◽

2021 ◽

Vol 13 (6) ◽

pp. 1016

Author(s):

Camelia Liliana Moldovan ◽

Radu Păltănea

Keyword(s):

Applied Sciences ◽

Degree Of Approximation ◽

Higher Order ◽

Second Order ◽

Dimensional Case ◽

Moduli Of Continuity ◽

One Dimensional ◽

Multidimensional Generalization ◽

The One

The paper presents a multidimensional generalization of the Schoenberg operators of higher order. The new operators are powerful tools that can be used for approximation processes in many fields of applied sciences. The construction of these operators uses a symmetry regarding the domain of definition. The degree of approximation by sequences of such operators is given in terms of the first and the second order moduli of continuity. Extending certain results obtained by Marsden in the one-dimensional case, the property of preservation of monotonicity and convexity is proved.

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Higher order intrinsic weak differentiability and Sobolev spaces between manifolds

Advances in Calculus of Variations ◽

10.1515/acv-2017-0008 ◽

2019 ◽

Vol 12 (3) ◽

pp. 303-332 ◽

Cited By ~ 1

Author(s):

Alexandra Convent ◽

Jean Van Schaftingen

Keyword(s):

Sobolev Space ◽

Isometric Embedding ◽

Riemannian Metrics ◽

Higher Order ◽

Necessary Condition ◽

Smooth Maps ◽

Homogeneous Sobolev Space ◽

The One ◽

Intrinsic Definition ◽

Sobolev Spaces Between Manifolds

AbstractWe define the notion of higher-order colocally weakly differentiable maps from a manifold M to a manifold N. When M and N are endowed with Riemannian metrics, {p\geq 1} and {k\geq 2}, this allows us to define the intrinsic higher-order homogeneous Sobolev space {\dot{W}^{k,p}(M,N)}. We show that this new intrinsic definition is not equivalent in general with the definition by an isometric embedding of N in a Euclidean space; if the manifolds M and N are compact, the intrinsic space is a larger space than the one obtained by embedding. We show that a necessary condition for the density of smooth maps in the intrinsic space {\dot{W}^{k,p}(M,N)} is that {\pi_{\lfloor kp\rfloor}(N)\simeq\{0\}}. We investigate the chain rule for higher-order differentiability in this setting.

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Exact statistical properties of the Burgers equation

Journal of Fluid Mechanics ◽

10.1017/s0022112000001142 ◽

2000 ◽

Vol 417 ◽

pp. 323-349 ◽

Cited By ~ 52

Author(s):

L. FRACHEBOURG ◽

Ph. A. MARTIN

Keyword(s):

White Noise ◽

Large Distance ◽

Burgers Equation ◽

Higher Order ◽

Statistical Properties ◽

Inviscid Limit ◽

Initial Condition ◽

Spatial Correlations ◽

One Dimensional ◽

The One

The one-dimensional Burgers equation in the inviscid limit with white noise initial condition is revisited. The one- and two-point distributions of the Burgers field as well as the related distributions of shocks are obtained in closed analytical forms. In particular, the large distance behaviour of spatial correlations of the field is determined. Since higher-order distributions factorize in terms of the one- and two- point functions, our analysis provides an explicit and complete statistical description of this problem.

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Effect of ionospheric depth on the ionospheric feedback instability: cutoff and subsequent <I>E</I><sub>||</sub> modes

Annales Geophysicae ◽

10.5194/angeo-28-1777-2010 ◽

2010 ◽

Vol 28 (9) ◽

pp. 1777-1794 ◽

Cited By ~ 2

Author(s):

R. Cosgrove ◽

R. Doe

Keyword(s):

Growth Rate ◽

Electric Fields ◽

Finite Thickness ◽

Higher Order ◽

Plasma Structure ◽

Higher Order Modes ◽

Order Of Magnitude ◽

E Region ◽

The One ◽

Two Fluid

Abstract. The ionospheric feedback instability (IFI), which involves feedback between ionospheric modifications and waves reflected off the magnetosphere, has up to this point been analyzed in terms of field line integrated (FLI) ionospheric quantities, that is, with the assumption that the ionospheric thickness can be ignored. In this work we test this assumption by solving the two-fluid equations for a representative ionospheric slab of finite thickness. We find that the results are for the most part incompatible with a description in terms of FLI quantities, and that their use can easily lead to an order of magnitude overestimation of the growth rate. This occurs because the first eigenmode, which is the one compatible with an FLI description, is cutoff above a certain frequency, leaving only higher order modes with wavelengths along B that are subsumed by the slab. Taking the results at face value, the parallel electric fields associated with the higher order modes are a possible contributor to electron heating and plasma structure in the E-region ionosphere.

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IDENTIFICATION OF COMMUNITY STRUCTURE IN NETWORKS USING HIGHER ORDER NEIGHBORHOOD CONCEPTS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127409024396 ◽

2009 ◽

Vol 19 (08) ◽

pp. 2677-2685 ◽

Cited By ~ 6

Author(s):

ROBERTO F. S. ANDRADE ◽

SUANI T. R. PINHO ◽

THIERRY PETIT LOBÃO

Keyword(s):

Community Structure ◽

Higher Order ◽

Community Structures ◽

Precise Description ◽

Successive Elimination ◽

Modular Networks ◽

Matrix Formula ◽

The One ◽

Individual Nodes

The identification of community structures in networks is investigated within a framework based on the concepts of higher order neighborhoods and neighborhood matrix [Formula: see text]. This procedure is of relevance especially for networks representing evolutionary situations, since several evidences show that they are assembled from pre-existing smaller structures, rather than by the mere adhesion of individual nodes. We proceed within the successive elimination of the links with largest betweenness degree. The effect of erasing a link at step k is quantified by the distance between [Formula: see text] and [Formula: see text], which describe the network neighborhoods prior and after the kth link elimination. For modular networks, this measure is characterized by a very long sequence of sharp peaks, following a much more complete cascade of cluster splitting. The evidences indicate that this method identifies a more precise description of smaller communities splitting than the one based on modularity function.

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The Multi-round Process Matrix

Quantum ◽

10.22331/q-2021-01-20-384 ◽

2021 ◽

Vol 5 ◽

pp. 384

Author(s):

Timothée Hoffreumon ◽

Ognyan Oreshkov

Keyword(s):

Information Exchange ◽

Higher Order ◽

Side Channel ◽

Order Process ◽

Causal Order ◽

Quantum Operations ◽

First Case ◽

The One

We develop an extension of the process matrix (PM) framework for correlations between quantum operations with no causal order that allows multiple rounds of information exchange for each party compatibly with the assumption of well-defined causal order of events locally. We characterise the higher-order process describing such correlations, which we name the multi-round process matrix (MPM), and formulate a notion of causal nonseparability for it that extends the one for standard PMs. We show that in the multi-round case there are novel manifestations of causal nonseparability that are not captured by a naive application of the standard PM formalism: we exhibit an instance of an operator that is both a valid PM and a valid MPM, but is causally separable in the first case and can violate causal inequalities in the second case due to the possibility of using a side channel.

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POLYMORPHISM AND THE OBSTINATE CIRCULARITY OF SECOND ORDER LOGIC: A VICTIMS’ TALE

Bulletin of Symbolic Logic ◽

10.1017/bsl.2017.43 ◽

2018 ◽

Vol 24 (1) ◽

pp. 1-52

Author(s):

PAOLO PISTONE

Keyword(s):

Type Theory ◽

Order Type ◽

Higher Order ◽

Second Order ◽

Order Logic ◽

Higher Order Logic ◽

Related Notion ◽

Foundational Research ◽

The One ◽

Second Order Logic

AbstractThe investigations on higher-order type theories and on the related notion of parametric polymorphism constitute the technical counterpart of the old foundational problem of the circularity (or impredicativity) of second and higher-order logic. However, the epistemological significance of such investigations has not received much attention in the contemporary foundational debate.We discuss Girard’s normalization proof for second order type theory or System F and compare it with two faulty consistency arguments: the one given by Frege for the logical system of the Grundgesetze (shown inconsistent by Russell’s paradox) and the one given by Martin-Löf for the intuitionistic type theory with a type of all types (shown inconsistent by Girard’s paradox).The comparison suggests that the question of the circularity of second order logic cannot be reduced to Russell’s and Poincaré’s 1906 “vicious circle” diagnosis. Rather, it reveals a bunch of mathematical and logical ideas hidden behind the hazardous idea of impredicative quantification, constituting a vast (and largely unexplored) domain for foundational research.

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Extending Integrability of Nonlinear Water Wave Equations: Nonlinear Fourier Analysis of Breather Packets and Rogue Waves at Higher Order

Volume 3: Structures, Safety, and Reliability ◽

10.1115/omae2019-95543 ◽

2019 ◽

Author(s):

Alfred R. Osborne

Keyword(s):

Real Time ◽

Theta Function ◽

Bilinear Form ◽

Wave Equations ◽

Higher Order ◽

Plane Boundary ◽

Integrable Case ◽

Time Analysis ◽

Real Time Analysis ◽

The One

Abstract I suggest a formulation to give approximate spectral solutions of nonintegrable, nonlinear wave equations in 2+1 dimensions. Nonintegrable systems such as the 2+1 NLS, Dysthe and extended Dysthe equations can be approximately integrated by selecting a nearby theta function formulation. I study the subclass of wave equations that are in the form of nonlinear envelope equations for which all members can be reduced to a particular Hirota bilinear form. To find the approximately integrable formulation associated with a nonintegrable equation, I first study the one and two soliton solutions and subsequently extend these to larger numbers of solitons to obtain the Hirota N-soliton solution (for infinite-plane boundary conditions). Subsequently, I address the one and two periodic solutions from the bilinear form, so that I can develop the associated Riemann theta function solution to a nearby integrable case. I discuss how to obtain the higher order breather packets from the point of view of the theta functions. This work is being developed for real time analysis of shipboard radar analysis of ocean waves. Further applications include real time analysis of lidar and synthetic aperture radar (SAR) data taken by airplanes flying over high sea states.

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