A continuous analogue of Erdős' k-Sperner theorem

2020 ◽  
Vol 484 (2) ◽  
pp. 123754
Author(s):  
Themis Mitsis ◽  
Christos Pelekis ◽  
Václav Vlasák
Keyword(s):  
Continuous Analogue

scholarly journals A continuous analogue and an extension of Radó's formulæ

1996 ◽  
Vol 53 (2) ◽  
pp. 229-233 ◽  
Author(s):  
C.E.M. Pearce ◽  
J. pečarić
Keyword(s):  
Continuous Analogue

A continuous analogue is derived for Radó's comparison formulæ. The analogue is then employed to provide a result which continues Radó's result and interpolates an inequality of Pittenger.

scholarly journals Surface energies emerging in a microscopic, two-dimensional two-well problem

2017 ◽  
Vol 147 (5) ◽  
pp. 1041-1089 ◽  
Author(s):  
Georgy Kitavtsev ◽  
Stephan Luckhaus ◽  
Angkana Rüland
Keyword(s):  
Surface Energy ◽  
Two Dimensional ◽  
Direct Control ◽  
First Order ◽  
Surface Energies ◽  
Energy Scaling ◽  
Continuous Analogue ◽  
Dimensional Shape ◽  
Set Up ◽  
Microscopic Modelling

In this paper we are interested in the microscopic modelling of a two-dimensional two-well problem that arises from the square-to-rectangular transformation in (two-dimensional) shape-memory materials. In this discrete set-up, we focus on the surface energy scaling regime and further analyse the Hamiltonian that was introduced by Kitavtsev et al. in 2015. It turns out that this class of Hamiltonians allows for a direct control of the discrete second-order gradients and for a one-sided comparison with a two-dimensional spin system. Using this and relying on the ideas of Conti and Schweizer, which were developed for a continuous analogue of the model under consideration, we derive a (first-order) continuum limit. This shows the emergence of surface energy in the form of a sharp-interface limiting model as well the explicit structure of the minimizers to the latter.

scholarly journals A Continuous Analogue of Lattice Path Enumeration

10.37236/8788 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Quang-Nhat Le ◽  
Sinai Robins ◽  
Christophe Vignat ◽  
Tanay Wakhare
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lattice Paths ◽  
Lattice Path ◽  
Continuous Version ◽  
Continuous Analog ◽  
Continuous Analogue ◽  
Lattice Path Enumeration ◽  
Multinomial Coefficients ◽  
General Method

Following the work of Cano and Díaz, we consider a continuous analog of lattice path enumeration. This process allows us to define a continuous version of many discrete objects that count certain types of lattice paths. As an example of this process, we define continuous versions of binomial and multinomial coefficients, and describe some identities and partial differential equations that they satisfy. Finally, as an important byproduct of these continuous analogs, we illustrate a general method to recover discrete combinatorial quantities from their continuous analogs, via an application of the Khovanski-Puklikov discretizing Todd operators.  

scholarly journals On the continuous analogue of the Seidel method

2018 ◽  
Vol 20 (4) ◽  
pp. 364-377
Author(s):  
I. V. Boikov ◽  
A. I. Boikova
Keyword(s):  
Differential Equations ◽  
Systems Of Nonlinear Equations ◽  
Algebraic Equations ◽  
Seidel Method ◽  
Linear Algebraic Equations ◽  
Nonlinear Algebraic Equations ◽  
Continuous Analogue ◽  
Solution Methods ◽  
Linear And Nonlinear ◽  
Equations With Delay

Continuous Seidel method for solving systems of linear and nonlinear algebraic equations is constructed in the article, and the convergence of this method is investigated. According to the method discussed, solving a system of algebraic equations is reduced to solving systems of ordinary differential equations with delay. This allows to use rich arsenal of numerical ODE solution methods while solving systems of algebraic equations. The main advantage of the continuous analogue of the Seidel method compared to the classical one is that it does not require all the elements of the diagonal matrix to be non-zero while solving linear algebraic equations’ systems. The continuous analogue has the similar advantage when solving systems of nonlinear equations.

scholarly journals Simultaneous parameter estimation and variable selection via the logit-normal continuous analogue of the spike-and-slab prior

2019 ◽  
Vol 16 (150) ◽  
pp. 20180572 ◽  
Author(s):  
W. Thomson ◽  
S. Jabbari ◽  
A. E. Taylor ◽  
W. Arlt ◽  
D. J. Smith
Keyword(s):  
Parameter Estimation ◽  
Statistical Models ◽  
Prior Distribution ◽  
Biological Data ◽  
Variable Model ◽  
Machine Learning Methods ◽  
Spike And Slab Prior ◽  
Unseen Data ◽  
Continuous Analogue ◽  
Bayesian Prior

We introduce a Bayesian prior distribution, the logit-normal continuous analogue of the spike-and-slab, which enables flexible parameter estimation and variable/model selection in a variety of settings. We demonstrate its use and efficacy in three case studies—a simulation study and two studies on real biological data from the fields of metabolomics and genomics. The prior allows the use of classical statistical models, which are easily interpretable and well known to applied scientists, but performs comparably to common machine learning methods in terms of generalizability to previously unseen data.

Rado's theorem for polymatroids

1975 ◽  
Vol 78 (2) ◽  
pp. 263-281 ◽  
Author(s):  
Colin J. H. McDiarmid
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Rank Function ◽  
Fundamental Importance ◽  
Matroid Theory ◽  
Continuous Analogue ◽  
Finite Set ◽  
Necessary And Sufficient ◽  
Transversal Theory

The theorem of R. Rado (12) to which I refer by the name ‘Rado's theorem for matroids’ gives necessary and sufficient conditions for a family of subsets of a finite set Y to have a transversal independent in a given matroid on Y. This theorem is of fundamental importance in both transversal theory and matroid theory (see, for example, (11)). In (3) J. Edmonds introduced and studied ‘polymatroids’ as a sort of continuous analogue of a matroid. I start this paper with a brief introduction to polymatroids, emphasizing the role of the ‘ground-set rank function’. The main result is an analogue for polymatroids of Rado's theorem for matroids, which I call not unnaturally ‘Rado's theorem for polymatroids’.

Fluxon Centering in Josephson Junctions with Exponentially Varying Width

2012 ◽  
Vol 2 (3) ◽  
pp. 204-213
Author(s):  
E. G. Semerdjieva ◽  
M. D. Todorov
Keyword(s):  
Josephson Junctions ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Static Solution ◽  
Physical Parameters ◽  
Nonlinear Eigenvalue Problems ◽  
Continuous Analogue ◽  
Sturm Liouville ◽  
The Stability ◽  
The Impact

AbstractNonlinear eigenvalue problems for fluxons in long Josephson junctions with exponentially varying width are treated. Appropriate algorithms are created and realized numerically. The results obtained concern the stability of the fluxons, the centering both magnetic field and current for the magnetic flux quanta in the Josephson junction as well as the ascertaining of the impact of the geometric and physical parameters on these quantities. Each static solution of the nonlinear boundary-value problem is identified as stable or unstable in dependence on the eigenvalues of associated Sturm-Liouville problem. The above compound problem is linearized and solved by using of the reliable Continuous analogue of Newton method.

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