scholarly journals Entropy Inequalities for Lattices

2018 ◽  
Author(s):  
Peter Harremoës
Keyword(s):  
Boolean Lattice ◽  
Functional Dependencies ◽  
Modular Lattices ◽  
Power Set ◽  
Entropy Inequalities ◽  
Lattice Of Subgroups

We study the existence or absence of non-Shannon inequalities for variables that are related by functional dependencies. Although the power-set on four variables is the smallest Boolean lattice with non-Shannon inequalities there exist lattices with many more variables without non-Shannon inequalities. We search for conditions that excludes the existence of non-Shannon inequalities. It is demonstrated that planar modular lattices cannot have non-Shannon inequalities. The existence of non-Shannon inequalities is related to the question of whether a lattice is isomorphic to a lattice of subgroups of a group.

scholarly journals Entropy Inequalities for Lattices

2018 ◽  
Author(s):  
Peter Harremoës
Keyword(s):  
Boolean Lattice ◽  
Functional Dependencies ◽  
Modular Lattices ◽  
Power Set ◽  
Entropy Inequalities ◽  
Lattice Of Subgroups

We study the existence or absence of non-Shannon inequalities for variables that are related by functional dependencies. Although the power-set on four variables is the smallest Boolean lattice with non-Shannon inequalities there exist lattices with many more variables without non-Shannon inequalities. We search for conditions that excludes the existence of non-Shannon inequalities. It is demonstrated that planar modular lattices cannot have non-Shannon inequalities. The existence of non-Shannon inequalities is related to the question of whether a lattice is isomorphic to a lattice of subgroups of a group.

scholarly journals Entropy Inequalities for Lattices

Entropy ◽  
2018 ◽  
Vol 20 (10) ◽  
pp. 784 ◽  
Author(s):  
Peter Harremoës
Keyword(s):  
Group Theory ◽  
Conditional Independence ◽  
Lattice Theory ◽  
Boolean Lattice ◽  
Functional Dependencies ◽  
Modular Lattices ◽  
Special Lattice ◽  
Entropy Inequalities ◽  
Lattice Of Subgroups

We study entropy inequalities for variables that are related by functional dependencies. Although the powerset on four variables is the smallest Boolean lattice with non-Shannon inequalities, there exist lattices with many more variables where the Shannon inequalities are sufficient. We search for conditions that exclude the existence of non-Shannon inequalities. The existence of non-Shannon inequalities is related to the question of whether a lattice is isomorphic to a lattice of subgroups of a group. In order to formulate and prove the results, one has to bridge lattice theory, group theory, the theory of functional dependences and the theory of conditional independence. It is demonstrated that the Shannon inequalities are sufficient for planar modular lattices. The proof applies a gluing technique that uses that if the Shannon inequalities are sufficient for the pieces, then they are also sufficient for the whole lattice. It is conjectured that the Shannon inequalities are sufficient if and only if the lattice does not contain a special lattice as a sub-semilattice.

scholarly journals Decompositions of the Boolean Lattice into Rank-symmetric Chains

10.37236/5328 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
István Tomon
Keyword(s):  
Lower Bound ◽  
Additional Condition ◽  
Boolean Lattice ◽  
Partial Ordering ◽  
Symmetric Chains ◽  
Power Set ◽  
A Chain ◽  
Symmetric Chain

The Boolean lattice $2^{[n]}$ is the power set of $[n]$ ordered by inclusion. A chain $c_{0}\subset\cdots\subset c_{k}$ in $2^{[n]}$ is rank-symmetric, if $|c_{i}|+|c_{k-i}|=n$ for $i=0,\ldots,k$; and it is symmetric, if $|c_{i}|=(n-k)/2+i$. We show that there exist a bijection $$p: [n]^{(\geq n/2)}\rightarrow [n]^{(\leq n/2)}$$ and a partial ordering $<$ on $[n]^{(\geq n/2)}$ satisfying the following properties:$\subset$ is an extension of $<$ on $[n]^{(\geq n/2)}$;if $C\subset [n]^{(\geq n/2)}$ is a chain with respect to $<$, then $p(C)\cup C$ is a rank-symmetric chain in $2^{[n]}$, where $p(C)=\{p(x): x\in C\}$;the poset $([n]^{(\geq n/2)},<)$ has the so called normalized matching property.We show two applications of this result.A conjecture of  Füredi asks if $2^{[n]}$ can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ chains such that the size of any two chains differ by at most 1. We prove an asymptotic version of this conjecture with the additional condition that every chain in the partition is rank-symmetric: $2^{[n]}$ can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ rank-symmetric chains, each of size $\Theta(\sqrt{n})$.Our second application gives a lower bound for the number of symmetric chain partitions of $2^{[n]}$. We show that $2^{[n]}$ has at least $2^{\Omega(2^{n}\log n/\sqrt{n})}$ symmetric chain partitions.

Modified algorithm for fast bandwidth selection for kernel estimates of multidimensional probability densities

2020 ◽  
pp. 9-13
Author(s):  
A. V. Lapko ◽  
V. A. Lapko
Keyword(s):  
Probability Density ◽  
Optimal Parameter ◽  
Random Variables ◽  
Kernel Functions ◽  
Independent Random Variables ◽  
Functional Dependencies ◽  
Approximation Properties ◽  
Probability Density Estimation ◽  
Multidimensional Probability ◽  
Selection Of

An original technique has been justified for the fast bandwidths selection of kernel functions in a nonparametric estimate of the multidimensional probability density of the Rosenblatt–Parzen type. The proposed method makes it possible to significantly increase the computational efficiency of the optimization procedure for kernel probability density estimates in the conditions of large-volume statistical data in comparison with traditional approaches. The basis of the proposed approach is the analysis of the optimal parameter formula for the bandwidths of a multidimensional kernel probability density estimate. Dependencies between the nonlinear functional on the probability density and its derivatives up to the second order inclusive of the antikurtosis coefficients of random variables are found. The bandwidths for each random variable are represented as the product of an undefined parameter and their mean square deviation. The influence of the error in restoring the established functional dependencies on the approximation properties of the kernel probability density estimation is determined. The obtained results are implemented as a method of synthesis and analysis of a fast bandwidths selection of the kernel estimation of the two-dimensional probability density of independent random variables. This method uses data on the quantitative characteristics of a family of lognormal distribution laws.

Migration and family change in Egypt: a comparative approach to social remittances

2013 ◽  
Vol 10 (1) ◽  
pp. 71-80 ◽  
Author(s):  
Lucile Gruntz ◽  
Delphine Pagès-El Karoui
Keyword(s):  
Public Discourse ◽  
Comparative Approach ◽  
Structural Factors ◽  
Host Countries ◽  
Return Policies ◽  
Social Remittances ◽  
Power Set ◽  
Gender Ideals ◽  
Gulf States ◽  
And Gender

Based on two ethnographical studies, our article explores social remittances from France and from the Gulf States, i.e. the way Egyptian migrants and returnees contribute to social change in their homeland with a focus on gender ideals and practices, as well as on the ways families cope with departure, absence and return. Policies in the home and host countries, public discourse, translocal networks, and individual locations within evolving structures of power, set the frame for an analysis of the consequences of migration in Egypt. This combination of structural factors is necessary to grasp the complex negotiations of family and gender norms, as asserted through idealized models, or enacted in daily practices in immigration and back home.

scholarly journals CRISPRi enables isoform-specific loss-of-function screens and identification of gastric cancer-specific isoform dependencies

2021 ◽  
Vol 22 (1) ◽  
Author(s):  
Rebecca Davies ◽  
Ling Liu ◽  
Sheng Taotao ◽  
Natasha Tuano ◽  
Richa Chaturvedi ◽  
...  
Keyword(s):  
Gastric Cancer ◽  
Functional Dependencies ◽  
Loss Of Function ◽  
Genetic Screens ◽  
Transcript Isoforms ◽  
Specific Loss ◽  
Current Loss ◽  
Specific Transcript ◽  
Multiple Promoters ◽  
Poor Patient

Abstract Introduction Genes contain multiple promoters that can drive the expression of various transcript isoforms. Although transcript isoforms from the same gene could have diverse and non-overlapping functions, current loss-of-function methodologies are not able to differentiate between isoform-specific phenotypes. Results Here, we show that CRISPR interference (CRISPRi) can be adopted for targeting specific promoters within a gene, enabling isoform-specific loss-of-function genetic screens. We use this strategy to test functional dependencies of 820 transcript isoforms that are gained in gastric cancer (GC). We identify a subset of GC-gained transcript isoform dependencies, and of these, we validate CIT kinase as a novel GC dependency. We further show that some genes express isoforms with opposite functions. Specifically, we find that the tumour suppressor ZFHX3 expresses an isoform that has a paradoxical oncogenic role that correlates with poor patient outcome. Conclusions Our work finds isoform-specific phenotypes that would not be identified using current loss-of-function approaches that are not designed to target specific transcript isoforms.

Algorithms for the discovery of embedded functional dependencies

2021 ◽  
Author(s):  
Ziheng Wei ◽  
Sven Hartmann ◽  
Sebastian Link
Keyword(s):  
Functional Dependencies

scholarly journals Online Learning Algorithms for the Real-Time Set-Point Tracking Problem

2021 ◽  
Vol 11 (14) ◽  
pp. 6620
Author(s):  
Arman Alahyari ◽  
David Pozo ◽  
Meisam Farrokhifar
Keyword(s):  
Decision Making ◽  
Power Systems ◽  
Real Time ◽  
Demand Response ◽  
Tracking Problem ◽  
Smart Meters ◽  
Set Point ◽  
Point Tracking ◽  
Power Set ◽  
Decision Making Framework

With the recent advent of technology within the smart grid, many conventional concepts of power systems have undergone drastic changes. Owing to technological developments, even small customers can monitor their energy consumption and schedule household applications with the utilization of smart meters and mobile devices. In this paper, we address the power set-point tracking problem for an aggregator that participates in a real-time ancillary program. Fast communication of data and control signal is possible, and the end-user side can exploit the provided signals through demand response programs benefiting both customers and the power grid. However, the existing optimization approaches rely on heavy computation and future parameter predictions, making them ineffective regarding real-time decision-making. As an alternative to the fixed control rules and offline optimization models, we propose the use of an online optimization decision-making framework for the power set-point tracking problem. For the introduced decision-making framework, two types of online algorithms are investigated with and without projections. The former is based on the standard online gradient descent (OGD) algorithm, while the latter is based on the Online Frank–Wolfe (OFW) algorithm. The results demonstrated that both algorithms could achieve sub-linear regret where the OGD approach reached approximately 2.4-times lower average losses. However, the OFW-based demand response algorithm performed up to twenty-nine percent faster when the number of loads increased for each round of optimization.

scholarly journals Simulation of Diffusion Processes in Chemical and Thermal Processing of Machine Parts

Processes ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 698
Author(s):  
Kateryna Kostyk ◽  
Michal Hatala ◽  
Viktoriia Kostyk ◽  
Vitalii Ivanov ◽  
Ivan Pavlenko ◽  
...  
Keyword(s):  
Mathematical Modeling ◽  
Temperature Distribution ◽  
Thermal Treatment ◽  
Diffusion Processes ◽  
Nitrogen Concentration ◽  
The Body ◽  
Functional Dependencies ◽  
Technological Parameters ◽  
Gas Nitriding ◽  
Temperature State

To solve a number of technological issues, it is advisable to use mathematical modeling, which will allow us to obtain the dependences of the influence of the technological parameters of chemical and thermal treatment processes on forming the depth of the diffusion layers of steels and alloys. The paper presents mathematical modeling of diffusion processes based on the existing chemical and thermal treatment of steel parts. Mathematical modeling is considered on the example of 38Cr2MoAl steel after gas nitriding. The gas nitriding technology was carried out at different temperatures for a duration of 20, 50, and 80 h in the SSHAM-12.12/7 electric furnace. When modeling the diffusion processes of surface hardening of parts in general, providing a specifically given distribution of nitrogen concentration over the diffusion layer’s depth from the product’s surface was solved. The model of the diffusion stage is used under the following assumptions: The diffusion coefficient of the saturating element primarily depends on temperature changes; the metal surface is instantly saturated to equilibrium concentrations with the saturating atmosphere; the surface layer and the entire product are heated unevenly, that is, the product temperature is a function of time and coordinates. Having satisfied the limit, initial, and boundary conditions, the temperature distribution equations over the diffusion layer’s depth were obtained. The final determination of the temperature was solved by an iterative method. Mathematical modeling allowed us to get functional dependencies for calculating the temperature distribution over the depth of the layer and studying the influence of various factors on the body’s temperature state of the body.

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