scholarly journals A Structure Theorem for Level Sets of Multiplicative Functions and Applications

2018 ◽  
Vol 2020 (5) ◽  
pp. 1300-1345 ◽  
Author(s):  
Vitaly Bergelson ◽  
Joanna Kułaga-Przymus ◽  
Mariusz Lemańczyk ◽  
Florian K Richter
Keyword(s):  
Level Set ◽  
Multiplicative Function ◽  
Level Sets ◽  
Structure Theorem ◽  
Almost Periodic ◽  
Multiplicative Functions ◽  
Multiple Recurrence ◽  
Random Part

Abstract Given a level set E of an arbitrary multiplicative function f, we establish, by building on the fundamental work of Frantzikinakis and Host [14, 15], a structure theorem that gives a decomposition of $1_{E}$ into an almost periodic and a pseudo-random part. Using this structure theorem together with the technique developed by the authors in [3], we obtain the following result pertaining to polynomial multiple recurrence.

scholarly journals Bipolar Fuzzy Relations

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1044 ◽  
Author(s):  
Jeong-Gon Lee ◽  
Kul Hur
Keyword(s):  
Equivalence Class ◽  
Level Set ◽  
Equivalence Relation ◽  
Level Sets ◽  
Equivalence Classes ◽  
Fuzzy Relation ◽  
Fuzzy Relations ◽  
Transitive Relation ◽  
Fuzzy Partition

We introduce the concepts of a bipolar fuzzy reflexive, symmetric, and transitive relation. We study bipolar fuzzy analogues of many results concerning relationships between ordinary reflexive, symmetric, and transitive relations. Next, we define the concepts of a bipolar fuzzy equivalence class and a bipolar fuzzy partition, and we prove that the set of all bipolar fuzzy equivalence classes is a bipolar fuzzy partition and that the bipolar fuzzy equivalence relation is induced by a bipolar fuzzy partition. Finally, we define an ( a , b ) -level set of a bipolar fuzzy relation and investigate some relationships between bipolar fuzzy relations and their ( a , b ) -level sets.

scholarly journals On the Correlation Between Nodal and Nonzero Level Sets for Random Spherical Harmonics

2020 ◽  
Author(s):  
Domenico Marinucci ◽  
Maurizia Rossi
Keyword(s):  
Level Set ◽  
Spherical Harmonics ◽  
Level Sets ◽  
Partial Correlation

AbstractWe study the correlation between the nodal length of random spherical harmonics and the length of a nonzero level set. We show that the correlation is asymptotically zero, while the partial correlation after removing the effect of the random $$L^2$$ L 2 -norm of the eigenfunctions is asymptotically one.

scholarly journals The Lyapunov spectrum of some parabolic systems

2009 ◽  
Vol 29 (3) ◽  
pp. 919-940 ◽  
Author(s):  
KATRIN GELFERT ◽  
MICHAŁ RAMS
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Level Set ◽  
Level Sets ◽  
Parabolic Systems ◽  
Lyapunov Spectrum ◽  
Interval Maps ◽  
Set Of Points

AbstractWe study the Hausdorff dimension for Lyapunov exponents for a class of interval maps which includes several non-hyperbolic situations. We also analyze the level sets of points with given lower and upper Lyapunov exponents and, in particular, with zero lower Lyapunov exponent. We prove that the level set of points with zero exponent has full Hausdorff dimension, but carries no topological entropy.

scholarly journals The Geometry of the Weak Lefschetz Property and Level Sets of Points

2008 ◽  
Vol 60 (2) ◽  
pp. 391-411 ◽  
Author(s):  
Juan C. Migliore
Keyword(s):  
Level Set ◽  
Level Sets ◽  
Hilbert Function ◽  
Worst Case ◽  
Weak Lefschetz Property ◽  
Lefschetz Property ◽  
Strong Lefschetz Property ◽  
Set Of Points

AbstractIn a recent paper, F. Zanello showed that level Artinian algebras in 3 variables can fail to have the Weak Lefschetz Property (WLP), and can even fail to have unimodal Hilbert function. We show that the same is true for the Artinian reduction of reduced, level sets of points in projective 3-space. Our main goal is to begin an understanding of how the geometry of a set of points can prevent its Artinian reduction from having WLP, which in itself is a very algebraic notion. More precisely, we produce level sets of points whose Artinian reductions have socle types 3 and 4 and arbitrary socle degree ≥ 12 (in the worst case), but fail to have WLP. We also produce a level set of points whose Artinian reduction fails to have unimodal Hilbert function; our example is based on Zanello's example. Finally, we show that a level set of points can have Artinian reduction that has WLP but fails to have the Strong Lefschetz Property. While our constructions are all based on basic double G-linkage, the implementations use very different methods.

scholarly journals Some characterizations of totients

1996 ◽  
Vol 19 (2) ◽  
pp. 209-217 ◽  
Author(s):  
Pentti Haukkanen
Keyword(s):  
Multiplicative Function ◽  
Arithmetical Function ◽  
Multiplicative Functions ◽  
Dirichlet Convolution

An arithmetical function is said to be a totient if it is the Dirichlet convolution between a completely multiplicative function and the inverse of a completely multiplicative function. Euler's phi-function is a famous example of a totient. All completely multiplicative functions are also totients. There is a large number of characterizations of completely multiplicative functions in the literature, while characterizations of totients have not been widely studied in the literature. In this paper we present several arithmetical identities serving as characterizations of totients. We also introduce a new concrete example of a totient.

scholarly journals Correlations of multiplicative functions and applications

2017 ◽  
Vol 153 (8) ◽  
pp. 1622-1657 ◽  
Author(s):  
Oleksiy Klurman
Keyword(s):  
Asymptotic Formula ◽  
Multiplicative Function ◽  
Divided Difference ◽  
Circle Method ◽  
Partial Sums ◽  
Multiplicative Functions ◽  
Image Position ◽  
Bounded Partial Sums

We give an asymptotic formula for correlations $$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}f_{1}(P_{1}(n))f_{2}(P_{2}(n))\cdots f_{m}(P_{m}(n)),\end{eqnarray}$$ where $f,\ldots ,f_{m}$ are bounded ‘pretentious’ multiplicative functions, under certain natural hypotheses. We then deduce several desirable consequences. First, we characterize all multiplicative functions $f:\mathbb{N}\rightarrow \{-1,+1\}$ with bounded partial sums. This answers a question of Erdős from $1957$ in the form conjectured by Tao. Second, we show that if the average of the first divided difference of the multiplicative function is zero, then either $f(n)=n^{s}$ for $\operatorname{Re}(s)<1$ or $|f(n)|$ is small on average. This settles an old conjecture of Kátai. Third, we apply our theorem to count the number of representations of $n=a+b$, where $a,b$ belong to some multiplicative subsets of $\mathbb{N}$. This gives a new ‘circle method-free’ proof of a result of Brüdern.

scholarly journals Development of a Computer Simulation Model for Blowing Glass Containers

2006 ◽  
Author(s):  
C. G. Giannopapa
Keyword(s):  
Level Set ◽  
Level Sets ◽  
Energy Exchange ◽  
Mesh Size ◽  
Computer Simulation Model ◽  
Time Step ◽  
Process Step ◽  
Material Parameters ◽  
Glass Containers ◽  
Container Manufacturing

In glass container manufacturing (e.g. production of glass bottles and jars) an important process step is the blowing of the final product. This process is fast and is characterized by large deformations and the interaction of a hot glass fluid that gets into contact with a colder metal, the mould. The objective of this paper is to create a robust finite element model to be used for industrial purposes that accurately captures the blowing step of glass containers. The model should be able to correctly represent the flow of glass and the energy exchange during the process. For tracking the geometry of the deforming inner and outer interface of glass, level set technique is applied on structured and unstructured fixed mesh. At each time step the coupled problem of flow and energy exchange is solved by the model. Here the flow problem is only solved for the domain enclosed by the mould, whereas in the energy calculations, the mould domain is also taken into account in the computations. For all the calculations the material parameters (like viscosity) are based on the glass position, i.e. the position of the level sets. The velocity distribution, as found from this solution procedure, is then used to update the two level sets by means of solving a convection equation. A re-initialization algorithm is applied after each time step in order to let the level sets re-attain the property of being a signed distance function. The model is validated by several examples focusing on both the overall behavior (such as conservation of mass and energy) and the local behavior of the flow (such as glass-mould contact) and temperature distributions for different mesh size, time step, level set settings and material parameters.

scholarly journals Level Set Segmentation: Active Contours without edge

2009 ◽  
Author(s):  
Kishore Mosaliganti ◽  
Benjamin Smith ◽  
Arnaud Gelas ◽  
Alexandre Gouaillard ◽  
sean megason
Keyword(s):  
Level Set ◽  
Level Sets ◽  
Active Contours ◽  
Level Set Methods ◽  
Constant Intensity ◽  
Piecewise Constant ◽  
Gradient Information ◽  
Cardiac Image ◽  
Contour Evolution ◽  
Processing Framework

An Insight Toolkit (ITK) processing framework for segmentation using active contours without edges is presented in this paper. Our algorithm is based on the work of Chan and Vese [1] that uses level- sets to accomplish region segmentation in images with poor or no gradient information. The basic idea is to partion the image into two piecewise constant intensity regions. This work is in contrast to the level-set methods currently available in ITK which necessarily require gradient information. Similar to those methods, the methods presented in this paper are also made efficient using a sparse implementation strategy that solves the contour evolution PDE at the level-set boundary. The framework consists of 6 new ITK filters that inherit in succession from itk::SegmentationFilter. We include 2D/3D example code, parameter settings and show the results generated on a 2D cardiac image.

scholarly journals level set segmentation using coupled active surfaces

2009 ◽  
Author(s):  
Kishore Mosaliganti ◽  
Benjamin Smith ◽  
Arnaud Gelas ◽  
Alexandre Gouaillard ◽  
sean megason
Keyword(s):  
Level Set ◽  
Level Sets ◽  
Active Contours ◽  
Level Set Methods ◽  
Multiple Objects ◽  
Constant Intensity ◽  
Piecewise Constant ◽  
Gradient Information ◽  
Cardiac Image ◽  
Contour Evolution

An Insight Toolkit (ITK) processing framework for simultaneous segmentation of multiple objects using active contours without edges is presented in this paper. These techniques are also popularly referred to as multiphase methods. Earlier, we had an implemented the Chan and Vese [1] algorithm that uses level- sets to accomplish region segmentation in images with poor or no gradient information. The current work extends that submission to use multiple level sets that evolve concurrently. The basic idea is to partion the image into several sets of piecewise constant intensity regions. This work is in contrast to the level-set methods currently available in ITK which necessarily require gradient information and also necessarily segment a single object-of-interest. Similar to those methods, the methods presented in this paper are also made efficient using a sparse implementation strategy that solves the contour evolution PDE at the level-set boundary. This work does not introduce any new filter but extends the earlier submitted to filters to process multiple objects. We include 2D/3D example code, parameter settings and show the results generated on a 2D cardiac image.

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