A Method for Computationally Constructing Eukaryotic Synthetic Signal Peptide Sequences

We developed a computational method for constructing synthetic signal peptides from a base set of signal peptides (SPs) and non-SP sequences. A large number of structured "building blocks", represented as m-step ordered pairs of amino acids, are extracted from the base. Using a straightforward procedure, the building blocks enable the construction of a diverse set of synthetic SPs that could be utilized for industrial and therapeutic purposes. We have validated the proposed methodology using existing sequence prediction platforms such as Signal-BLAST and MULocDeep. In one experiment, 9,555 protein sequences were generated from a large randomly selected set of "building blocks". Signal-BLAST identified 8,444 (88%) of the sequences as signal peptides. In addition, the Signal-BLAST tool predicted that the generated synthetic sequences belonged to 854 distinct eukaryotic organisms. Here, we provide detailed descriptions and results from various experiments illustrating the potential usefulness of the methodology in generating signal peptide protein sequences.

Associations between patterns in comorbid diagnostic trajectories of individuals with schizophrenia and etiological factors

Schizophrenia is a heterogeneous disorder, exhibiting variability in presentation and outcomes that complicate treatment and recovery. To explore this heterogeneity, we leverage the comprehensive Danish health registries to conduct a prospective, longitudinal study from birth of 5432 individuals who would ultimately be diagnosed with schizophrenia, building individual trajectories that represent sequences of comorbid diagnoses, and describing patterns in the individual-level variability. We show that psychiatric comorbidity is prevalent among individuals with schizophrenia (82%) and multi-morbidity occur more frequently in specific, time-ordered pairs. Three latent factors capture 79% of variation in longitudinal comorbidity and broadly relate to the number of co-occurring diagnoses, the presence of child versus adult comorbidities and substance abuse. Clustering of the factor scores revealed five stable clusters of individuals, associated with specific risk factors and outcomes. The presentation and course of schizophrenia may be associated with heterogeneity in etiological factors including family history of mental disorders.

A three solutions theorem for Pucci's extremal operator and its application

In this article we prove a three solution type theorem for the following boundary value problem: \begin{equation*} \label{abs} \begin{cases} -\mathcal{M}_{\lambda,\Lambda}^+(D^2u) =f(u)& \text{in }\Omega,\\ u =0& \text{on }\partial\Omega, \end{cases} \end{equation*} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ and $f\colon [0,\infty]\to[0,\infty]$ is a $C^{\alpha}$ function. This is motivated by the work of Amann \cite{aman} and Shivaji \cite{shivaji1987remark}, where a three solutions theorem has been established for the Laplace operator. Furthermore, using this result we show the existence of three positive solutions to above boundary value by explicitly constructing two ordered pairs of sub and supersolutions when $f$ has a sublinear growth and $f(0)=0.$

The Problem of Quantifying Utility

Purpose of the study. Analysis of the literature shows that the ordinal theory of utility is widespread in the theory of consumer behavior. To analyze consumer preferences, a utility function is used, which characterizes the value of the utility of the consumed goods and services on a scale of order. Moreover, to find the marginal utility of a product, arithmetic operations are used, which are impossible on a scale of order. To allow arithmetic operations, a quantitative analysis of the utility function is required. Consequently, the problem of quantitative measurement of the utility function is relevant.The measurement problem also arises in decision theory. For example, the hierarchy analysis method is a popular method for solving multicriteria problems, but contains an erroneous model of subjective measurement. For this reason, other methods appear in decision-making theory that should replace the method of analyzing hierarchies. The theory of the importance of criteria is being actively developed. However, the theory of the importance of criteria also does not solve the problem of quantitative measurement.For a long time, the problem of measurement has also existed in psychophysics. The existence of two mismatched psychophysical laws contradicts the classical theory of measurements. Recently, a rating solution has been proposed. The equivalence of the basic laws of psychophysics has been proved. In this paper, it is proposed to use the rating method to measure preferences in utility theory and in decision theory.Materials and methods. The domain of the rating is the set of ordered pairs of objects. Moreover, the composition (operation of addition) of objects is defined on the set of ordered pairs. A rating is a number that is assigned during a measurement to an ordered pair of objects.The rating is assumed to preserve the operation of composition of ordered pairs.An arithmetic operation is selected to carry out the measurement. The measurement result must match the result of the arithmetic operation. The result of an arithmetic operation is the difference or ratio of the values of the quantity. The rating values coincide with the result of the arithmetic operation (up to isomorphism).The additivity of the rating is used to check the adequacy of the measurement results. For this, it is assumed that the rating is independent of the measurement method. The theoretical justification for independence is the isomorphism condition. The empirical confirmation of independence is the equivalence of the basic psychophysical laws.Results. The paper presents an axiomatic approach to the measurement problem. Measurement can be carried out in both objective and subjective ways. The axiomatic and classical definition of the rating has been formulated. The axiomatic definition implies the classical definition for a special set of objects. The classic definition is constructive. To check the adequacy of the measurement results, it is enough to compare the ratings obtained by different measurement methods (method of alternatives).Conclusion. The rating method is a quantitative measurement method. The rating method can be used to construct a model of consumer behavior and in decision-making theory.

Mathematical Investigation of Functions

Generally, when the independent variable of a given exponential function is used as an exponent, the function is considered an exponential. Thus, the following can be examples of exponential functions: $f(x) = ab^x + c$, $f(x) = ae^bx + c$, or $f(x) = e^{a^2+bx+c}$. However, deriving functions of these types given the set of ordered pairs is difficult. This study was conducted to derive formulas for the arbitrary constants a ,b, and $c$ of the exponential function $f(x) = ab^x + c$. It applied the inductive method by using definitions of functions to derive the arbitrary constants from the patterns produced. The findings of the study were: a) For linear, given the table of ordered pairs, equal differences in $x$ produce equal first differences in $y$; b) for quadratic, given the table of ordered pairs, equal differences in $x$ produce equal second differences in $y$; and c) for an exponential function, given a table of ordered pairs, equal differences in $x$ produce a common ratio in the first differences in y. The study obtained the following forms: $b=\sqrt[d]{r}$, $a=\frac{q}{b^n {(b^d-1)}}$, $c=p-ab^n$. Since most models developed used the concept of linear and multiple regressions, it is recommended that pattern analysis be used specifically when data are expressed in terms of ordered pairs.

Application of a Diagnostic Methodology of Cardiac Systems Based on the Proportions of Entropy in Normal Patients and with Different Pathologies

<b><i>Introduction:</i></b> Dynamical systems theory, probability, and entropy were the substrate for the development of the diagnostic and predictive methodology of adult heart dynamics. <b><i>Objective:</i></b> To apply a previously developed methodology from dynamical systems, probability, and entropy in both normal and pathological subjects. <b><i>Methods:</i></b> Electrocardiographic records were selected from 30 healthy subjects and 200 with different pathologies, with a length of least 18 h. Numerical attractors from dynamical attractors and the probability of occurrence of ordered pairs of consecutive heart rates were built. A calculation of entropy and its proportions was performed and statistical analysis was conducted. <b><i>Results:</i></b> The normal patients’ heart dynamics were evaluated according to the methodology of entropy proportions, highlighting that there are differences in normal patients with different pathologies. There was maximal level of sensitivity, specificity, and diagnostic agreement. <b><i>Conclusion:</i></b> Proportional entropy constitutes a diagnostic and predictive method of heart systems, and may be useful as a tool to objectively diagnose and perform the follow-up of normal and pathological cases.

Applications of Coupled Fixed Points for Multivalued Maps in the Equilibrium in Duopoly Markets and in Aquatic Ecosystems

We have obtained a new class of ordered pairs of multivalued maps that have pairs of coupled fixed points. We illustrate the main result with two examples that cover a wide range of models. We apply the main result in models in duopoly markets to get a market equilibrium and in aquatic ecosystems, also to get an equilibrium.

Existence of Coupled Best Proximity Points of p-Cyclic Contractions

We generalize the notion of coupled fixed (or best proximity) points for cyclic ordered pairs of maps to p-cyclic ordered pairs of maps. We find sufficient conditions for the existence and uniqueness of the coupled fixed (or best proximity) points. We illustrate the results with an example that covers a wide class of maps.

Four-Fold Formal Concept Analysis Based on Complete Idempotent Semifields

Formal Concept Analysis (FCA) is a well-known supervised boolean data-mining technique rooted in Lattice and Order Theory, that has several extensions to, e.g., fuzzy and idempotent semirings. At the heart of FCA lies a Galois connection between two powersets. In this paper we extend the FCA formalism to include all four Galois connections between four different semivectors spaces over idempotent semifields, at the same time. The result is K¯-four-fold Formal Concept Analysis (K¯-4FCA) where K¯ is the idempotent semifield biasing the analysis. Since complete idempotent semifields come in dually-ordered pairs—e.g., the complete max-plus and min-plus semirings—the basic construction shows dual-order-, row–column- and Galois-connection-induced dualities that appear simultaneously a number of times to provide the full spectrum of variability. Our results lead to a fundamental theorem of K¯-four-fold Formal Concept Analysis that properly defines quadrilattices as 4-tuples of (order-dually) isomorphic lattices of vectors and discuss its relevance vis-à-vis previous formal conceptual analyses and some affordances of their results.