scholarly journals Counting the dimension of splines of mixed smoothness

2021 ◽  
Vol 47 (1) ◽  
Author(s):  
Deepesh Toshniwal ◽  
Michael DiPasquale
Keyword(s):  
Space Dimension ◽  
Homological Algebra ◽  
Sufficient Conditions ◽  
Spline Space ◽  
Polynomial Splines ◽  
Bivariate Polynomial ◽  
Polygonal Meshes ◽  
Specific Setting ◽  
Mixed Smoothness

AbstractIn this paper, we study the dimension of bivariate polynomial splines of mixed smoothness on polygonal meshes. Here, “mixed smoothness” refers to the choice of different orders of smoothness across different edges of the mesh. To study the dimension of spaces of such splines, we use tools from homological algebra. These tools were first applied to the study of splines by Billera (Trans. Am. Math. Soc. 310(1), 325–340, 1988). Using them, estimation of the spline space dimension amounts to the study of the Billera-Schenck-Stillman complex for the spline space. In particular, when the homology in positions 1 and 0 of this complex is trivial, the dimension of the spline space can be computed combinatorially. We call such spline spaces “lower-acyclic.” In this paper, starting from a spline space which is lower-acyclic, we present sufficient conditions that ensure that the same will be true for the spline space obtained after relaxing the smoothness requirements across a subset of the mesh edges. This general recipe is applied in a specific setting: meshes of arbitrary topologies. We show how our results can be used to compute the dimensions of spline spaces on triangulations, polygonal meshes, and T-meshes with holes.

scholarly journals Polynomial spline spaces of non-uniform bi-degree on T-meshes: combinatorial bounds on the dimension

2021 ◽  
Vol 47 (1) ◽  
Author(s):  
Deepesh Toshniwal ◽  
Bernard Mourrain ◽  
Thomas J. R. Hughes
Keyword(s):  
Space Dimension ◽  
Sufficient Conditions ◽  
General Setting ◽  
Spline Space ◽  
Mesh Adaptivity ◽  
Polynomial Splines ◽  
Theoretical Understanding ◽  
Polynomial Degree ◽  
Local Polynomial ◽  
Degree Adaptivity

AbstractPolynomial splines are ubiquitous in the fields of computer-aided geometric design and computational analysis. Splines on T-meshes, especially, have the potential to be incredibly versatile since local mesh adaptivity enables efficient modeling and approximation of local features. Meaningful use of such splines for modeling and approximation requires the construction of a suitable spanning set of linearly independent splines, and a theoretical understanding of the spline space dimension can be a useful tool when assessing possible approaches for building such splines. Here, we provide such a tool. Focusing on T-meshes, we study the dimension of the space of bivariate polynomial splines, and we discuss the general setting where local mesh adaptivity is combined with local polynomial degree adaptivity. The latter allows for the flexibility of choosing non-uniform bi-degrees for the splines, i.e., different bi-degrees on different faces of the T-mesh. In particular, approaching the problem using tools from homological algebra, we generalize the framework and the discourse presented by Mourrain (Math. Comput. 83(286):847–871, 2014) for uniform bi-degree splines. We derive combinatorial lower and upper bounds on the spline space dimension and subsequently outline sufficient conditions for the bounds to coincide.

scholarly journals On hyperbolic systems with entropy velocity covariant under the action of a group

2015 ◽  
Vol 12 (04) ◽  
pp. 763-785
Author(s):  
François Dubois
Keyword(s):  
Conservation Laws ◽  
Space Dimension ◽  
Hyperbolic Systems ◽  
Sufficient Conditions ◽  
Space Time ◽  
Systems Of Conservation Laws ◽  
Time Transformations ◽  
One Space Dimension

For hyperbolic systems of conservation laws in one space dimension endowed with a mathematical entropy, we define the notion of entropy velocity and we give sufficient conditions for such a system to be covariant under the action of a group of space-time transformations. These conditions naturally introduce a representation of the group in the space of states. We construct such hyperbolic systems from the knowledge of data on the manifold of null velocity. We apply these ideas for Galileo, Lorentz, and circular groups and, in particular, focus on nontrivial examples of [Formula: see text] systems of conservation laws.

scholarly journals Necessary and Sufficient Conditions of Optimality for a Damped Hyperbolic Equation in One-Space Dimension

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Ismail Kucuk ◽  
Kenan Yildirim
Keyword(s):  
Maximum Principle ◽  
Space Dimension ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Distributed Parameter ◽  
Hyperbolic Partial Differential Equation ◽  
The Maximum Principle ◽  
Convexity Assumptions ◽  
Necessary And Sufficient ◽  
One Space Dimension

The present paper deals with the necessary optimality condition for a class of distributed parameter systems in which the system is modeled in one-space dimension by a hyperbolic partial differential equation subject to the damping and mixed constraints on state and controls. Pontryagin maximum principle is derived to be a necessary condition for the controls of such systems to be optimal. With the aid of some convexity assumptions on the constraint functions, it is obtained that the maximum principle is also a sufficient condition for the optimality.

Error bounds for minimal energy bivariate polynomial splines

2002 ◽  
Vol 93 (2) ◽  
pp. 315-331 ◽  
Author(s):  
Manfred von Golitschek ◽  
Ming-Jun Lai ◽  
Larry L. Schumaker
Keyword(s):  
Error Bounds ◽  
Minimal Energy ◽  
Polynomial Splines ◽  
Bivariate Polynomial

scholarly journals Dimension of polynomial splines of mixed smoothness on T-meshes

2020 ◽  
Vol 80 ◽  
pp. 101880
Author(s):  
Deepesh Toshniwal ◽  
Nelly Villamizar
Keyword(s):  
Polynomial Splines ◽  
Mixed Smoothness

MONOTONE APPROXIMATION OF AGGREGATION OPERATORS USING LEAST SQUARES SPLINES

2002 ◽  
Vol 10 (06) ◽  
pp. 659-676 ◽  
Author(s):  
G. BELIAKOV
Keyword(s):  
Least Squares ◽  
Fuzzy Systems ◽  
Sufficient Conditions ◽  
Aggregation Operators ◽  
Linear Functions ◽  
Membership Functions ◽  
Polynomial Splines ◽  
Monotone Approximation ◽  
Necessary And Sufficient ◽  
Linear Restrictions

The need for monotone approximation of scattered data often arises in many problems of regression, when the monotonicity is semantically important. One such domain is fuzzy set theory, where membership functions and aggregation operators are order preserving. Least squares polynomial splines provide great flexbility when modeling non-linear functions, but may fail to be monotone. Linear restrictions on spline coefficients provide necessary and sufficient conditions for spline monotonicity. The basis for splines is selected in such a way that these restrictions take an especially simple form. The resulting non-negative least squares problem can be solved by a variety of standard proven techniques. Additional interpolation requirements can also be imposed in the same framework. The method is applied to fuzzy systems, where membership functions and aggregation operators are constructed from empirical data.

scholarly journals Sample paths of white noise in spaces with dominating mixed smoothness

2021 ◽  
Vol 15 (3) ◽  
Author(s):  
Felix Hummel
Keyword(s):  
Heat Equation ◽  
White Noise ◽  
Space Dimension ◽  
Regularity Of Solutions ◽  
Time Direction ◽  
Sample Paths ◽  
Regularity Results ◽  
Boundary Noise ◽  
Dominating Mixed Smoothness ◽  
Mixed Smoothness

AbstractThe sample paths of white noise are proved to be elements of certain Besov spaces with dominating mixed smoothness. Unlike in isotropic spaces, here the regularity does not get worse with increasing space dimension. Consequently, white noise is actually much smoother than the known sharp regularity results in isotropic spaces suggest. An application of our techniques yields new results for the regularity of solutions of Poisson and heat equation on the half space with boundary noise. The main novelty is the flexible treatment of the interplay between the singularity at the boundary and the smoothness in tangential, normal and time direction.

Some Contributions to the Theory of Near-Critical Bisexual Branching Processes

2007 ◽  
Vol 44 (02) ◽  
pp. 492-505
Author(s):  
M. Molina ◽  
M. Mota ◽  
A. Ramos
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Positive Probability ◽  
Limit Laws ◽  
Bisexual Branching Processes ◽  
Probabilistic Evolution

We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to other classes of near-critical bisexual branching processes.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Normal approximations to discrete unimodal distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray
Keyword(s):  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Hypergeometric Distribution ◽  
Death Process ◽  
Normal Approximations ◽  
Discrete Random Variables ◽  
Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

Sign in / Sign up

Export Citation Format

Share Document