scholarly journals MRA-Wavelet subspace architecture for logic, probability, and symbolic sequence processing

2014 ◽  
Author(s):  
Daniel J. Greenhoe
Keyword(s):  
Boolean Lattice ◽  
Symbolic Sequence ◽  
Sequence Processing ◽  
Linear Subspaces ◽  
Genomic Signal Processing ◽  
Set Inclusion ◽  
Symbolic Sequences ◽  
Special Lattice ◽  
Boolean Lattices ◽  
Selection Of

The linear subspaces of a multiresolution analysis (MRA) and the linear subspaces of the wavelet analysis induced by the MRA, together with the set inclusion relation, form a very special lattice of subspaces which herein is called a "primorial lattice". This paper introduces an operator R that extracts a set of 2^{N-1} element Boolean lattices from a 2^N element Boolean lattice. Used recursively, a sequence of Boolean lattices with decreasing order is generated---a structure that is similar to an MRA. A second operator, which is a special case of a "difference operator", is introduced that operates on consecutive Boolean lattices L_2^n and L_2^{n-1} to produce a sequence of orthocomplemented lattices. These two sequences, together with the subset ordering relation, form a primorial lattice P. A logic or probability constructed on a Boolean lattice L_2^N likewise induces a primorial lattice P. Such a logic or probability can then be rendered at N different "resolutions" by selecting any one of the N Boolean lattices in P and at N different "frequencies" by selecting any of the N different orthocomplemented lattices in P. Furthermore, P can be used for symbolic sequence analysis by projecting sequences of symbols onto the sublattices in P using one of three lattice projectors introduced. P can be used for symbolic sequence processing by judicious rejection and selection of projected sequences. Examples of symbolic sequences include sequences of logic values, sequences of probabilistic events, and genomic sequences (as used in "genomic signal processing").

scholarly journals MRA-Wavelet subspace architecture for logic, probability, and symbolic sequence processing

2014 ◽  
Author(s):  
Daniel J. Greenhoe
Keyword(s):  
Boolean Lattice ◽  
Symbolic Sequence ◽  
Sequence Processing ◽  
Linear Subspaces ◽  
Genomic Signal Processing ◽  
Set Inclusion ◽  
Symbolic Sequences ◽  
Special Lattice ◽  
Boolean Lattices ◽  
Selection Of

The linear subspaces of a multiresolution analysis (MRA) and the linear subspaces of the wavelet analysis induced by the MRA, together with the set inclusion relation, form a very special lattice of subspaces which herein is called a "primorial lattice". This paper introduces an operator R that extracts a set of 2^{N-1} element Boolean lattices from a 2^N element Boolean lattice. Used recursively, a sequence of Boolean lattices with decreasing order is generated---a structure that is similar to an MRA. A second operator, which is a special case of a "difference operator", is introduced that operates on consecutive Boolean lattices L_2^n and L_2^{n-1} to produce a sequence of orthocomplemented lattices. These two sequences, together with the subset ordering relation, form a primorial lattice P. A logic or probability constructed on a Boolean lattice L_2^N likewise induces a primorial lattice P. Such a logic or probability can then be rendered at N different "resolutions" by selecting any one of the N Boolean lattices in P and at N different "frequencies" by selecting any of the N different orthocomplemented lattices in P. Furthermore, P can be used for symbolic sequence analysis by projecting sequences of symbols onto the sublattices in P using one of three lattice projectors introduced. P can be used for symbolic sequence processing by judicious rejection and selection of projected sequences. Examples of symbolic sequences include sequences of logic values, sequences of probabilistic events, and genomic sequences (as used in "genomic signal processing").

scholarly journals Chains in generalized Boolean lattices

1976 ◽  
Vol 21 (2) ◽  
pp. 234-240
Author(s):  
Richard D. Byrd ◽  
Roberto A. Mena
Keyword(s):  
Distributive Lattice ◽  
Boolean Lattice ◽  
Boolean Lattices ◽  
A Chain

A chain C in a distributive lattice L is called strongly maximal in L if and only if for any homomorphism φ of L onto a distributive lattice K, the chain (Cφ)0 is maximal in K, where (Cφ)0 = Cφ if 0 ∉ K, and (Cφ)0 = Cφ ∪ {0}, otherwise. Gratzer (1971, Theorem 28) states that if B is a generalized Boolean lattice R-generated by L and C is a chain in L, then C R-generates B if and only if C is strongly maximal in L. In this note (Theorem 4.6), we prove the following assertion, which is not far removed from Gratzer's statement: let B be a generalized Boolean lattice R-generated by L and C be a chain in L. If 0 ∈ L, then C generates B if and only if C is strongly maximal in L. If 0 ∉ L, then C generates B if and only if C is strongly maximal in L and [C)L = L. In Section 5 (Example 5.1) a counterexample to Gratzer's statement is provided.

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 Chunking improves symbolic sequence processing and relies on working memory gating mechanisms

2016 ◽  
Vol 23 (3) ◽  
pp. 108-112 ◽  
Author(s):  
Oleg Solopchuk ◽  
Andrea Alamia ◽  
Etienne Olivier ◽  
Alexandre Zénon
Keyword(s):  
Working Memory ◽  
Symbolic Sequence ◽  
Sequence Processing ◽  
Gating Mechanisms

scholarly journals Efficacy and safety of triazoles versus echinocandins in the treatment of invasive aspergillosis: A meta-analysis

2020 ◽  
Vol 77 (9) ◽  
pp. 974-985
Author(s):  
Sanja Uzelac ◽  
Radica Zivkovic-Zaric ◽  
Milan Radovanovic ◽  
Goran Rankovic ◽  
Slobodan Jankovic
Keyword(s):  
Clinical Trials ◽  
Invasive Aspergillosis ◽  
Therapeutic Option ◽  
Meta Analysis ◽  
Treatment Success ◽  
Poor Quality ◽  
Efficacy And Safety ◽  
Set Inclusion ◽  
Selection Of

Backgroun/Aim. Although majority of guidelines recommend triazoles (voriconazole, posaconazole, itraconazole and isavuconazole) as first-line therapeutic option for treatment of invasive aspergillosis, echinocandins (caspofungin, micafungin and anidulafungin) are also used for this purpose. However, head-to-head comparison of triazoles and echinocandins for invasive aspergillosis was rarely target of clinical trials. The aim of this meta-analysis was to compare efficacy and safety of triazoles and echinocandins when used for treatment of patients with invasive aspergillosis. Methods. This meta-analysis was based on systematic search of literature and selection of high-quality evidence according to pre-set inclusion and exclusion criteria. The literature search was made for comparison of treatment with any of triazoles (isavuconazole, itraconazole, posaconazole or voriconazole) versus any of echinocandins (caspofungin, anidulafungin or micafungin). The effects of triazoles (itraconazole, posaconazole or voriconazole) and echinocandins (caspofungin, anidulafungin or micafungin) were summarized using RevMan 5.3.5 software, and heterogeneity assessed by the Cochrane Q test and I? values. Several types of bias were assessed, and publication bias was shown by the funnel plot and Egger?s regression. Results. Two clinical trials and three cohort studies were included in this meta-analysis. Mortality in patients with invasive aspergillosis who were treated with triazoles was significantly lower than in patients treated with echinocandins [odds ratio 0.29 (0.13, 0.67)], and rate of favorable response (overall treatment success) 12 weeks after the therapy onset was higher in patients treated with triazoles [3.05 (1.52, 6.13)]. On the other hand, incidence of adverse events was higher with triazoles than with echinocandins in patients treated for invasive aspergillosis [3.75 (0.89, 15.76)], although this difference was not statistically significant. Conclusion.Triazoles (voriconazole in the first place) could be considered as more effective and somewhat less safe therapeutic option than echinocandins for invasive aspergillosis: However, due to poor quality of studies included in this meta-analysis, definite conclusion should await results of additional, well designed clinical trials.

scholarly journals About Similarity Measures of Components Arrangement of Naturally Ordered Data Arrays

2019 ◽  
Vol 18 (2) ◽  
pp. 471-503
Author(s):  
Alexander Gumenyuk ◽  
Artemiy Skiba ◽  
Nikolay Pozdnichenko ◽  
Stanislav Shpynov
Keyword(s):  
Probabilistic Models ◽  
Geometric Mean ◽  
A Priori ◽  
Similarity Measures ◽  
Mathematical Linguistics ◽  
Distance Measures ◽  
Numerical Sequence ◽  
Symbolic Sequences ◽  
Ordered Data ◽  
Selection Of

At present, adequate mathematical tools are not used to analyze the arrangement of components in arrays of naturally ordered data of a different nature, including words or letters in texts, notes in musical compositions, symbols in sign sequences, monitoring data, numbers representing ordered measurement results, components in genetic texts. Therefore, it is difficult or impossible to measure and compare the order of messages allocated in long information chains. The main approaches for comparing symbol sequences are using probabilistic models and statistical tools, pairwise and multiple alignment, which makes it possible to determine the degree of similarity of sequences using edit distance measures. The application of pseudospectral and fractal representation of symbolic sequences is somewhat exotic. "The curse of a priori unconscious knowledge" of the obvious orderliness of the sequence should be especially noticed, as it is widespread in mathematical linguistics, bioinformatics (mathematical biology), and other similar fields of science. The noted approaches almost do not pay attention to the study and detection of the patterns of the specific arrangement of all symbols, words, and components of data sets that constitute a separate sequence. The object of study in our works is a specifically organized numerical tuple – the arrangement of components (order) in symbolic or numerical sequence. The intervals between the closest identical components of the order are used as the basis for the quantitative representation of the chain arrangement. Multiplying all the intervals or summing their logarithms allows one to get numbers that uniquely reflect the arrangement of components in a particular sequence. These numbers, allow us to obtain a whole set of normalized characteristics of the order, among which the geometric mean interval and its logarithm. Such characteristics surprisingly accurately reflect the arrangement of the components in the symbolic sequences. In this paper, we present an approach for quantitative comparing the arrangement of arrays of naturally ordered data (information chains) of an arbitrary nature. The measures of similarity/distinction and procedure of comparison of the chain order, based on the selection of a list of equal and similar by the order characteristics of the subsequences (components), are proposed. Rank distributions are used for faster selection of a list of matching components. The paper presents a toolkit for comparing the order of information chains and demonstrates some of its applications for studying the structure of nucleotide sequences.

scholarly journals The Boolean Rainbow Ramsey Number of Antichains, Boolean Posets and Chains

10.37236/9034 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Hong-Bin Chen ◽  
Yen-Jen Cheng ◽  
Wei-Tian Li ◽  
Chia-An Liu
Keyword(s):  
Lower Bounds ◽  
Boolean Lattice ◽  
Ramsey Number ◽  
Upper And Lower Bounds ◽  
Ramsey Type ◽  
Boolean Lattices

Motivated by the paper, Boolean lattices: Ramsey properties and embeddings Order, 34 (2) (2017), of Axenovich and Walzer, we study the Ramsey-type problems on the Boolean lattices. Given posets $P$ and $Q$, we look for the smallest Boolean lattice $\mathcal{B}_N$ such that any coloring of elements of $\mathcal{B}_N$ must contain a monochromatic $P$ or a rainbow $Q$ as an induced subposet. This number $N$ is called the Boolean rainbow Ramsey number of $P$ and $Q$ in the paper. Particularly, we determine the exact values of the Boolean rainbow Ramsey number for $P$ and $Q$ being the antichains, the Boolean posets, or the chains. From these results, we also derive some general upper and lower bounds of the Boolean rainbow Ramsey number for general $P$ and $Q$ in terms of the poset parameters.

scholarly journals Representation of Boolean hypolattices

1981 ◽  
Vol 24 (3) ◽  
pp. 389-404 ◽  
Author(s):  
John Boris Miller
Keyword(s):  
Representation Theorem ◽  
Boolean Lattice ◽  
Principal Result ◽  
Direct Sum ◽  
Boolean Lattices ◽  
Stone Representation ◽  
Relative Convexity ◽  
Small Product

The principal result is a representation theorem for relatively-distributive, relatively complemented hypolattices with zero, generalizing the Stone representation theorem for a Boolean lattice. It uses the small product of a family of Boolean lattices which are maximal sublattices of the hypolattice. The paper also characterizes the maximal sublattices when the hypolattice is coherent; and it gives several examples of hypolattices, including hypolattices of subgroups and of ideals by direct sum, and examples from relative convexity, relative closure, and cofinality.

scholarly journals The Complete cd-Index of Boolean Lattices

10.37236/5100 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Neil J.Y. Fan ◽  
Liao He
Keyword(s):  
Coxeter Group ◽  
Nonnegative Integer ◽  
Boolean Lattice ◽  
Combinatorial Interpretation ◽  
Euler Numbers ◽  
Combinatorial Interpretations ◽  
Bruhat Graph ◽  
Boolean Lattices ◽  
One To One ◽  
Entringer Numbers

Let $[u,v]$ be a Bruhat interval of a Coxeter group such that the Bruhat graph $BG(u,v)$ of $[u,v]$ is isomorphic to a Boolean lattice. In this paper, we provide a combinatorial explanation for the coefficients of the complete cd-index of $[u,v]$. Since in this case the complete cd-index and the cd-index of $[u,v]$ coincide, we also obtain a new combinatorial interpretation for the coefficients of the cd-index of Boolean lattices. To this end, we label an edge in $BG(u,v)$ by a pair of nonnegative integers and show that there is a one-to-one correspondence between such sequences of nonnegative integer pairs and Bruhat paths in $BG(u,v)$. Based on this labeling, we construct a flip $\mathcal{F}$ on the set of Bruhat paths in $BG(u,v)$, which is an involution that changes the ascent-descent sequence of a path. Then we show that the flip $\mathcal{F}$ is compatible with any given reflection order and also satisfies the flip condition for any cd-monomial $M$. Thus by results of Karu, the coefficient of $M$ enumerates certain Bruhat paths in $BG(u,v)$, and so can be interpreted as the number of certain sequences of nonnegative integer pairs. Moreover, we give two applications of the flip $\mathcal{F}$. We enumerate the number of cd-monomials in the complete cd-index of $[u,v]$ in terms of Entringer numbers, which are refined enumerations of Euler numbers. We also give a refined enumeration of the coefficient of d${}^n$ in terms of Poupard numbers, and so obtain new combinatorial interpretations for Poupard numbers and reduced tangent numbers.

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