Quasi Order Research Articles

Higman’s Lemma is Stronger for Better Quasi Orders

AbstractWe prove that Higman’s lemma is strictly stronger for better quasi orders than for well quasi orders, within the framework of reverse mathematics. In fact, we show a stronger result: the infinite Ramsey theorem (for tuples of all lengths) is equivalent to the statement that any array $$[\mathbb N]^{n+1}\rightarrow \mathbb N^n\times X$$ [ N ] n + 1 → N n × X for a well order X and $$n\in \mathbb N$$ n ∈ N is good, over the base theory $$\mathsf {RCA_0}$$ RCA 0 .

On the measurement of electoral volatility

Electoral volatility measures the degree of vote switching between political parties in two consecutive elections. Political scientists use this as an indicator of party system (in)stability. Pedersen (1979) states that volatility should increase when the number of parties changes and/or relevant parties experience vote transfer between elections. However, his proposed functional form of measuring volatility does not always respond to these changes. To address these limitations, we introduce a class of additively separable electoral volatility measures which are responsive to changes in both the number of parties and their vote shares. We present a set of axioms that are both necessary and sufficient to characterize the proposed class of indices, making the structure of the indices more transparent. The paper also introduces two quasi orders which can rank party systems in terms of all electoral volatility indices satisfying certain intuitively reasonable axioms. Finally, applications of the proposed class of indices and the quasi orders are provided using data from Indian state elections.

On the logical strength of the better quasi order with three elements

The notion of better quasi order ( B Q O \mathsf {BQO} ), due to Nash-Williams, is very fruitful mathematically and intriguing from the standpoint of logic, due to several long-standing open problems. In the present paper, we make a significant step towards one of these: Let 3 \mathbf 3 be the antichain with three elements. We show that arithmetical recursion along the natural numbers ( A C A 0 + \mathsf {ACA}_0^+ ) follows from 3 \mathbf 3 being B Q O \mathsf {BQO} , over the base theory R C A 0 \mathsf {RCA_0} from reverse mathematics. Also over the latter, we deduce arithmetical transfinite recursion ( A T R 0 \mathsf {ATR}_0 ) from the assumption that 3 \mathbf 3 is Δ 2 0 - B Q O \Delta ^0_2\text {-}\mathsf {BQO} , which plays a role in work of Montalbán.

Labelled well-quasi-order for permutation classes

While the theory of labelled well-quasi-order has received significant attention in the graph setting, it has not yet been considered in the context of permutation patterns. We initiate this study here, and show how labelled well quasi order provides a lens through which to view and extend previous well-quasi-order results in the permutation patterns literature. Connections to the graph setting are emphasised throughout. In particular, we establish that a permutation class is labelled well-quasi-ordered if and only if its corresponding graph class is also labelled well-quasi-ordered.Mathematics Subject Classifications: 05A05, 06A07Keywords: Labelled well-quasi-order, permutation patterns, well-quasi-order

DeepNEC: a novel alignment-free tool for the identification and classification of nitrogen biochemical network-related enzymes using deep learning.

Nitrogen is essential for life and its transformations are an important part of the global biogeochemical cycle. Being an essential nutrient, nitrogen exists in a range of oxidation states from +5 (nitrate) to -3 (ammonium and amino-nitrogen), and its oxidation and reduction reactions catalyzed by microbial enzymes determine its environmental fate. The functional annotation of the genes encoding the core nitrogen network enzymes has a broad range of applications in metagenomics, agriculture, wastewater treatment and industrial biotechnology. This study developed an alignment-free computational approach to determine the predicted nitrogen biochemical network-related enzymes from the sequence itself. We propose deepNEC, a novel end-to-end feature selection and classification model training approach for nitrogen biochemical network-related enzyme prediction. The algorithm was developed using Deep Learning, a class of machine learning algorithms that uses multiple layers to extract higher-level features from the raw input data. The derived protein sequence is used as an input, extracting sequential and convolutional features from raw encoded protein sequences based on classification rather than traditional alignment-based methods for enzyme prediction. Two large datasets of protein sequences, enzymes and non-enzymes were used to train the models with protein sequence features like amino acid composition, dipeptide composition (DPC), conformation transition and distribution, normalized Moreau-Broto (NMBroto), conjoint and quasi order, etc. The k-fold cross-validation and independent testing were performed to validate our model training. deepNEC uses a four-tier approach for prediction; in the first phase, it will predict a query sequence as enzyme or non-enzyme; in the second phase, it will further predict and classify enzymes into nitrogen biochemical network-related enzymes or non-nitrogen metabolism enzymes; in the third phase, it classifies predicted enzymes into nine nitrogen metabolism classes; and in the fourth phase, it predicts the enzyme commission number out of 20 classes for nitrogen metabolism. Among all, the DPC + NMBroto hybrid feature gave the best prediction performance (accuracy of 96.15% in k-fold training and 93.43% in independent testing) with an Matthews correlation coefficient (0.92 training and 0.87 independent testing) in phase I; phase II (accuracy of 99.71% in k-fold training and 98.30% in independent testing); phase III (overall accuracy of 99.03% in k-fold training and 98.98% in independent testing); phase IV (overall accuracy of 99.05% in k-fold training and 98.18% in independent testing), the DPC feature gave the best prediction performance. We have also implemented a homology-based method to remove false negatives. All the models have been implemented on a web server (prediction tool), which is freely available at http://bioinfo.usu.edu/deepNEC/.

Controlling a random population

Bertrand et al. introduced a model of parameterised systems, where each agent is represented by a finite state system, and studied the following control problem: for any number of agents, does there exist a controller able to bring all agents to a target state? They showed that the problem is decidable and EXPTIME-complete in the adversarial setting, and posed as an open problem the stochastic setting, where the agent is represented by a Markov decision process. In this paper, we show that the stochastic control problem is decidable. Our solution makes significant uses of well quasi orders, of the max-flow min-cut theorem, and of the theory of regular cost functions. We introduce an intermediate problem of independence interest called the sequential flow problem and study its complexity.

Study of the Synthesis and Characterization of Poly(acrylic acid-co-acrylamide)-kaolinite Composite Reference to Adsorption of Cd and Pb from Aqueous Solutions

A novel poly(acrylic acid-co-acrylamide)/ kaolinite composite was produced by copolymerization of partially neutralized acrylic acid and acrylamide with sawdust for crosslinking applying Bis-acrylamide having linear formulae (C₇H₁₀N₂O₂.), (H₂C=CHCONH) 2CH₂, potassium persulfate, K2S2O8 as initiator and sodium sulphite used as accelerator to reduce the polymerization time. Physiognomy of the composite was performed by techniques viz., Fourier Transform Infrared Spectroscopy, X-Radiation Diffractometry (Crystallography), Structural-Equation-Modeling, Energy-Dispersal X-Radiation-Analysis and Brunauer-Emmett-Teller methods. Divalent cadmium and lead electrolytes (adsorbate) adsorption on the adsorbent (composite) in aqueous solution was studied in batch experiments optimizing reaction parameters like “IA” of equilibrium amount, “CT” (wet time), neutrality (alkalinity and acidity) and calefaction (thermal reading). The quasi 2nd order kinetic archetypal ensued by AK. Adsorption isotherrms were applied to the empirical statistics which fitted well with a) Irving-Langmuir-isotherm and b) Freundlich adsorption isotherm. The supreme adsorption dimensions, qm (from Irving-Langmuir isotherm) at 25oC, were 28.57 mg g-1 and 32.25 mg g-1 for divalent cadmium and lead electrolytes, respectively. Data of activation energy (12.148 kJ mol-1 and 7.165 kJ mol-1 for divalent cadmium and lead electrolytes respectively) indicated physio-absorption. Results of thermodynamic parameters like ∆Go, ∆Ho and ∆So for both divalent cadmium and lead electrolytes suggested the phenomenon of the endothermic and spontaneity nature of the process. The newly synthesized composite attributed to its cost effective and environment friendly nature of the synthesis which will be deemed as a capable substance for the exclusion of adsorbent divalent cadmium and lead electrolytes from aqueous suspensions.

Quantization in *-Algebras II: Archimedeanization

Given a matrix quasi ordered *-vector space $${\mathcal {X}}$$ , we describe the Archimedeanization process of $${\mathcal {X}}$$ and prove some isomorphism theorems. Then, we apply these results to construct a new proof for the algebraic analog of Arveson’s extension theorem. Also we obtain an extension of Kadison’s theorem which realizes a quasi order gauged vector space as a subspace of a commutative $$C^*$$ -algebra.

A pseudo restricted MLE under multivariate order restrictions and its algorithm

It is shown in this paper that a quasi order for the vectors in Rp is a cone induced if and only if the order is preservable under limits and under linear combinations with non-negative coefficients. For the mean vectors in MANOVA subject to the restriction of simple ordering, a pseudo restricted MLE is proposed. This estimator is a matrix projection onto a closed convex set inside the restricted domain. An algorithm for the pseudo restricted MLE is developed, that computes the matrix projections using only vector projections.

Well quasi orders in a categorical setting

This article describes well quasi orders as a category, focusing on limits and colimits. In particular, while quasi orders with monotone maps form a category which is finitely complete, finitely cocomplete, and with exponentiation, the full subcategory of well quasi orders is finitely complete and cocomplete, but with no exponentiation. It is interesting to notice how finite antichains and finite proper descending chains interact to induce this structure in the category: in fact, the full subcategory of quasi orders with finite antichains has finite colimits but no products, while the full subcategory of well founded quasi orders has finite limits but no coequalisers. Moreover, the article characterises when exponential objects exist in the category of well quasi orders and well founded quasi orders. This completes the systematic description of the fundamental constructions in the categories of quasi orders, well founded quasi orders, quasi orders with finite antichains, and well quasi orders.

Approximate solutions and scalarization in set-valued optimization

Certain notions of approximate weak efficient solutions are considered for a set-valued optimization problem based on vector and set criteria approaches. For approximate solutions based on the vector approach, a characterization is provided in terms of an extended Gerstewitz’s function. For the set approach case, two notions of approximate weak efficient solutions are introduced using a lower and an upper quasi order relations for sets and further compactness and stability aspects are discussed for these approximate solutions. Existence and scalarization using a generalized Gerstewitz’s function are also established for approximate solutions, based on the lower set order relation.

On Martin's Axiom and Forms of Choice

Martin's Axiom is the statement that for every well‐ordered cardinal , the statement holds, where is “if is a c.c.c. quasi order and is a family of dense sets in P, then there is a ‐generic filter of P”. In , the fragment is provable, but not in general in . In this paper, we investigate the interrelation between and various choice principles.In the choiceless context, it makes sense to drop the requirement that the cardinal κ be well‐ordered, and we can define for any (not necessarily well‐ordered) cardinal the statement to be “if is a c.c.c. quasi order with , and is a family of dense sets in P, then there is a ‐generic filter of P”. We then define to be the statement that for every (not necessarily well‐ordered) cardinal , we have that holds. We then investigate the set‐theoretic strength of the principle .

$EL^2$–hyperstructures Derived from (Partially) Quasi Ordered Hyperstructures

In this paper, we introduce a new class of (semi)hypergroup from a given (partially) quasi-ordered (semi)hypergroup as a generaliza- tion of El{hyperstructures. Then, we study some basic properties and important elements belong to this class.

On the Completion of Rough Sets System Determined by Arbitrary Binary Relations

In this paper, a solution is given to the problem proposed by Jarvinen in [8]. A smallest completion of the rough sets system determined by an arbitrary binary relation is given. This completion, in the case of a quasi order, coincides with the rough sets system which is a Nelson algebra. Further, the algebraic properties of this completion has been studied.

Scalarization in set optimization with solid and nonsolid ordering cones

This paper focuses on characterizations via scalarization of several kinds of minimal solutions of set-valued optimization problems, where the objective values are compared through relations between sets (set optimization). For this aim we follow an axiomatic approach based on general order representation and order preservation properties, which works in any abstract set ordered by a quasi order (i.e., reflexive and transitive) relation. Then, following this approach, we study a recent Gerstewitz scalarization mapping for set-valued optimization problems with $$K$$ K -proper sets and a solid ordering cone $$K$$ K . In particular we show a dual minimax reformulation of this scalarization. Moreover, in the setting of normed spaces ordered by non necessarily solid ordering cones, we introduce a new scalarization functional based on the so-called oriented distance. Using these scalarization mappings, we obtain necessary and sufficient optimality conditions in set optimization. Finally, whenever the ordering cone is solid, by considering suitable generalized Chebyshev norms with appropriate parameters, we show that the three scalarizations studied in the present work are coincident.

LOCAL MAGNETIC BEHAVIOR OF 54Fe in EuFe2As2 AND Eu0.5K0.5Fe2As2: MICROSCOPIC STUDY USING TIME DIFFERENTIAL PERTURBED ANGULAR DISTRIBUTION (TDPAD) SPECTROSCOPY

In this paper, we report the time differential perturbed angular distribution measurements of 54 Fe on a polycrystalline EuFe 2 As 2 and Eu 0.5 K 0.5 Fe 2 As 2. The hyperfine field and nuclear spin-relaxation rate are strongly temperature dependent in the paramagnetic state suggesting strong spin fluctuation in the parent compound. The local susceptibility show Curie–Weiss-like temperature dependence and Korringa-like relaxation in the tetragonal phase indicating the presence of local moment. In the orthorhombic phase, the hyperfine field behavior suggesting quasi two-dimensional magnetic ordering. The experimental results are in a good agreement with first-principle calculations based on density functional theory.

Algebra of Rough Sets based on Quasi Order

In this paper, we give a characterization theorem for rough sets based on quasi order. We obtain an algebra on the rough sets system determined by a quasi order which is the generalization of the algebra of rough sets system determined by an equivalence relation given in [1]. The properties of this algebra are abstracted at various levels and define new class of algebras. Further we give a representation theorem for the new class of algebras.

On Abelian and Related Fuzzy Subsets of Groupoids

We introduce the notion of abelian fuzzy subsets on a groupoid, and we observe a variety of consequences which follow. New notions include, among others, diagonal symmetric relations, several types of quasi orders, convex sets, and fuzzy centers, some of whose properties are also investigated.

A Method of Representing Rough Sets System Determined by Quasi Orders

In this paper, we study about the ordered structure of rough sets determined by a quasi order. A characterization theorem for rough sets of an approximation space (U, R) based on a quasi order R is given in Nagarajan and Umadevi (2010). Then using the characterization of rough sets determined by a quasi order, its rough sets system is represented by a new construction. This construction is generalized and abstracted into a new method of constructing Kleene based algebraic structures from dually isomorphic distributive lattices. Then by using different varieties of distributive lattices, we obtain various Kleene based algebraic structures. By this construction, we give various algebraic structures to the rough sets system determined by a quasi order R.

Guest Editorial: Special Issue on Parameterized and Exact Computation, Part I

The papers in this issue form the first part of the two parts special issue in the area of parameterized and exact exponential algorithms and complexity. While some of these are selected and extended papers from International Symposium on Parameterized and Exact Computation (IPEC) 2010, the rest were selected from the submissions received for our call for the special issue in the area. The two parts reflect the volume of high quality submissions we received, which in turn reflects the high growth of the area in recent times. The papers in this issue represent the subareas of recent growth in the area including variations of RobertsonSeymour and Courcelle’s theorems, kernelization and structural parameterization. In parameterized (or multivariate) complexity, the aim is to design and analyse algorithms focussing on various parameters in the input—besides the standard input size. In particular, for hard problems, the aim is to confine the combinatorial explosion of the running time of the algorithms to small parameters in the input. Besides giving rise to new and interesting algorithm design techniques and complexity paradigms, the area has been highly successful in domains (like computational biology and artificial intelligence) where small parameters govern practical situations. While most of the initial work in the area considered the size of the solution as the parameter, there has been a lot of work in recent times to consider an ecology of parameters including various ‘structural’ parameters. The size of the vertex cover (which measures how far the graph is from an edgeless graph) and the feedback vertex set (which measures how far the graph is from a forest) are such parameters especially on graph problems. The paper ‘Well Quasi orders in subclasses of bounded treewidth