Planar graphs without intersecting 5-cycles are signed-4-choosable

A signed graph is a pair [Formula: see text], where [Formula: see text] is a graph and [Formula: see text] is a signature of [Formula: see text]. A set [Formula: see text] of integers is symmetric if [Formula: see text] implies that [Formula: see text]. Given a list assignment [Formula: see text] of [Formula: see text], an [Formula: see text]-coloring of a signed graph [Formula: see text] is a coloring [Formula: see text] of [Formula: see text] such that [Formula: see text] for each [Formula: see text] and [Formula: see text] for every edge [Formula: see text]. The signed choice number [Formula: see text] of a graph [Formula: see text] is defined to be the minimum integer [Formula: see text] such that for any [Formula: see text]-list assignment [Formula: see text] of [Formula: see text] and for any signature [Formula: see text] on [Formula: see text], there is a proper [Formula: see text]-coloring of [Formula: see text]. List signed coloring is a generalization of list coloring. However, the difference between signed choice number and choice number can be arbitrarily large. Hu and Wu [Planar graphs without intersecting [Formula: see text]-cycles are [Formula: see text]-choosable, Discrete Math. 340 (2017) 1788–1792] showed that every planar graph without intersecting 5-cycles is 4-choosable. In this paper, we prove that [Formula: see text] if [Formula: see text] is a planar graph without intersecting 5-cycles, which extends the main result of [D. Hu and J. Wu, Planar graphs without intersecting [Formula: see text]-cycles are [Formula: see text]-choosable, Discrete Math. 340 (2017) 1788–1792].

SOPHIA AS A SYMBOLIC FORMULA OF THE EURASIAN COLLEGIALITY IN RUSSIAN RELIGIOUS PHILOSOPHY

The background of the article is connected with the correlation of the image-concept of Sophia as a symbol of the unity of the world in culture with the idea of Eurasianism as collegiality in Russian religious philosophy of the 19th-20th centuries. The purpose of the study is a comparative analysis of the signature formula «Sophia» in the author's receptions of collegiality in Russian religious and philosophical thought through the prism of the idea of Eurasianism as the archetype of the East Slavic Logos. The archetype of Eurasian collegiality as a structural element of world unity is traced from implicit historical origins in the Old Russian Middle Ages to the explication of this concept in the New Time. Research hypothesis - substantiation of the assumption that the Eurasian idea of collegiality as an idea of synthesis of the West and the East goes back in its cultural and historical roots to the archetype of Sophia and undergoes certain transformations from premodern through modern and postmodern to neo-modern, without losing its theoretical relevance and applied significance as a mechanism for integrating traditionalism, universalism and personalism. Research methods: dialectic method, structural and functional analysis, phenomenological description, hermeneutic interpretation, philosophical comparative studies, universal ethics of dialogue and semiotics of culture. The result of the research is a historical tracing of the evolution of sophiology from the times of Ancient Rus through the Age of Enlightenment to late modernity in the context of collegiality of Eurasianism, symbolically embodied in the image-concept of Sophia in Russian religious philosophy with the typology of its main features (messianism, providentialism, mystery, Westernism vs Slavophilism as a dialectical unity). Conclusions: interpretation of the concept of Eurasianism as a manifestation of the Russian religious and philosophical idea of collegiality, and the interpretation of idea of collegiality as a manifestation of the symbolic phenomenon of Sophia, which removes the contradictions of illusory dichotomies (universalism - particularism, modernism - traditionalism, theism - atheism, etc.) in favor of harmonious unity of the world of the West and the East on the civilizational, social, cultural and mental levels.

Slopes of links and signature formulas

We present a new invariant, called slope, of a colored link in an integral homology sphere and use this invariant to complete the signature formula for the splice of two links. We develop a number of ways of computing the slope and a few vanishing results. Besides, we discuss the concordance invariance of the slope and establish its close relation to the Conway polynomials, on the one hand, and to the Kojima–Yamasaki η \eta -function (in the univariate case) and Cochran invariants, on the other hand.

Curvature models of conformally flat Walker (2,2)-manifolds

Four-dimensional conformally flat curvature models of signature [Formula: see text] are considered and complete classification of curvature tensor is obtained. Then, some remarks about Ivanov–Petrova and Walker–Ivanov–Petrova properties are stated.

Unifying structural signature of eukaryotic α-helical host defense peptides

Diversity of α-helical host defense peptides (αHDPs) contributes to immunity against a broad spectrum of pathogens via multiple functions. Thus, resolving common structure–function relationships among αHDPs is inherently difficult, even for artificial-intelligence–based methods that seek multifactorial trends rather than foundational principles. Here, bioinformatic and pattern recognition methods were applied to identify a unifying signature of eukaryotic αHDPs derived from amino acid sequence, biochemical, and three-dimensional properties of known αHDPs. The signature formula contains a helical domain of 12 residues with a mean hydrophobic moment of 0.50 and favoring aliphatic over aromatic hydrophobes in 18-aa windows of peptides or proteins matching its semantic definition. The holistic α-core signature subsumes existing physicochemical properties of αHDPs, and converged strongly with predictions of an independent machine-learning–based classifier recognizing sequences inducing negative Gaussian curvature in target membranes. Queries using the α-core formula identified 93% of all annotated αHDPs in proteomic databases and retrieved all major αHDP families. Synthesis and antimicrobial assays confirmed efficacies of predicted sequences having no previously known antimicrobial activity. The unifying α-core signature establishes a foundational framework for discovering and understanding αHDPs encompassing diverse structural and mechanistic variations, and affords possibilities for deterministic design of antiinfectives.

Free immersions and panelled web 4-manifolds

We show that if a compact, oriented 4-manifold admits a coassociative([Formula: see text])-free immersion into [Formula: see text], then its Euler characteristic [Formula: see text] and signature [Formula: see text] vanish. Moreover, in the spin case, the Gauss map is contractible, so that the immersed manifold is parallelizable. The proof makes use of homotopy theory, in particular, obstruction theory. As a further application, we prove a non-existence result for some infinite families of 4-manifolds that have not been addressed previously. We give concrete examples of parallelizable 4-manifolds with complicated non-simply-connected topology.

The Hecke algebras for the orthogonal group SO(2,3) and the paramodular group of degree 2

In this paper, we consider the integral orthogonal group with respect to the quadratic form of signature [Formula: see text] given by [Formula: see text] for square-free [Formula: see text]. The associated Hecke algebra is commutative and also the tensor product of its primary components, which turn out to be polynomial rings over [Formula: see text] in two algebraically independent elements. The integral orthogonal group is isomorphic to the paramodular group of degree [Formula: see text] and level [Formula: see text], more precisely to its maximal discrete normal extension. The results can be reformulated in the paramodular setting by virtue of an explicit isomorphism. The Hecke algebra of the non-maximal paramodular group inside [Formula: see text] fails to be commutative if [Formula: see text].

Half conformally flat generalized quasi-Einstein manifolds of metric signature (2,2)

We provide classification results for and examples of half conformally flat generalized quasi-Einstein manifolds of signature [Formula: see text]. This analysis leads to a natural equation in affine geometry called the affine quasi-Einstein equation that we explore in further detail.

Switched signed graphs of integer additive set-valued signed graphs

Let [Formula: see text] denote a set of non-negative integers and [Formula: see text] be its power set. An integer additive set-labeling (IASL) of a graph [Formula: see text] is an injective set-valued function [Formula: see text] such that the induced function [Formula: see text] is defined by [Formula: see text], where [Formula: see text] is the sumset of [Formula: see text] and [Formula: see text]. An IASL of a signed graph [Formula: see text] is an IASL of its underlying graph [Formula: see text] together with the signature [Formula: see text] defined by [Formula: see text]. A marking of a signed graph is an injective map [Formula: see text], defined by [Formula: see text] for all [Formula: see text]. Switching of signed graph is the process of changing the sign of the edges in [Formula: see text] whose end vertices have different signs. In this paper, we discuss certain characteristics of the switched signed graphs of certain types of integer additive set-labeled signed graphs.

On κ-reducibility of pseudovarieties of the form V ∗D

This paper deals with the reducibility property of semidirect products of the form [Formula: see text] relatively to graph equation systems, where D denotes the pseudovariety of definite semigroups. We show that if the pseudovariety [Formula: see text] is reducible with respect to the canonical signature [Formula: see text] consisting of the multiplication and the [Formula: see text]-power, then [Formula: see text] is also reducible with respect to [Formula: see text].