Irregularity and Modular Irregularity Strength of Wheels

It is easily observed that the vertices of a simple graph cannot have pairwise distinct degrees. This means that no simple graph of the order of at least two is, in this way, irregular. However, a multigraph can be irregular. Chartrand et al., in 1988, posed the following problem: in a loopless multigraph, how can one determine the fewest parallel edges required to ensure that all vertices have distinct degrees? This problem is known as the graph labeling problem and, for its solution, Chartrand et al. introduced irregular assignments. The irregularity strength of a graph G is known as the maximal edge label used in an irregular assignment, minimized over all irregular assignments. Thus, the irregularity strength of a simple graph G is equal to the smallest maximum multiplicity of an edge of G in order to create an irregular multigraph from G. In the present paper, we show the existence of a required irregular labeling scheme that proves the exact value of the irregularity strength of wheels. Then, we modify this irregular mapping in six cases and obtain labelings that determine the exact value of the modular irregularity strength of wheels as a natural modification of the irregularity strength.

A Note on Repeated Degrees of Line Graphs

Let $\text{rep}(G)$ be the maximum multiplicity of a vertex degree in graph $G$. It was proven in Caro and West [E-JC, 2009] that if $G$ is an $n$-vertex line graph, then $\text{rep}(G) \geqslant \frac{1}{4} n^{1/3}$. In this note we prove that for infinitely many $n$ there is a $n$-vertex line graph $G$ such that $\text{rep}(G) \leqslant \left(2n\right)^{1/3}$, thus showing that the bound above is asymptotically tight. Previously it was only known that for infinitely many $n$ there is a $n$-vertex line graph $G$ such that $\text{rep}(G) \leqslant \sqrt{4n/3}$ (Caro and West [E-JC, 2009]). Finally we prove that if $G$ is a $n$-vertex line graph, then $\text{rep}(G) \geqslant \left(\left(\frac{1}{2}-o(1)\right)n\right)^{1/3}$.

Existence of Multiple Periodic Solutions for Cubic Nonautonomous Differential Equation

In this article, approaches to estimate the number of periodic solutions of ordinary differential equation are considered. Conditions that allow determination of periodic solutions are discussed. We investigated focal values for first-order differential nonautonomous equation by using the method of bifurcation analysis of periodic solutions from a fine focus Z=0. Keeping in focus the second part of Hilbert’s sixteenth problem particularly, we are interested in detecting the maximum number of periodic solution into which a given solution can bifurcate under perturbation of the coefficients. For some classes like C7,7,C8,5,C8,6,C8,7, eight periodic multiplicities have been observed. The new formulas ξ10 and ϰ10 are constructed. We used our new formulas to find the maximum multiplicity for class C9,2. We have succeeded to determine the maximum multiplicity ten for class C9,2 which is the highest known multiplicity among the available literature to date. Another challenge is to check the applicability of the methods discussed which is achieved by presenting some examples. Overall, the results discussed are new, authentic, and novel in its domain of research.

ASSESSMENT OF THE IMPACT OF THE PAPER ENTERPRISE ON THE ENVIRONMENT

Pulp and paper enterprises are of great importance for sustainable economic and environmental development of certain regions and the economy of Ukraine. Man uses many natural resources during the lifetime, creating a burden on nature. As the world's population increases, this load increases, leading to a shortage of resources and deterioration in the environment. The main activity of Kokhavynska Paper Mill PJSC is the production of sanitary products for the domestic market and export. Rational use of secondary raw materials helps to cut the use of forest resources and reduce the amount of waste paper utilized in landfills. 15291.76 t of pollutants from the sources of the enterprise emissions enter the air each year. A total of 9 standardized and 3 non-standardized substances (greenhouse gases) are released into the atmosphere, namely nitrogen oxides, carbon monoxide, nitrogen (1) oxide (N 2 O) (greenhouse gas), carbon dioxide (greenhouse gas), methane (greenhouse gas), iron and its compounds, manganese and its compounds, hexavalent chromium (expressed in terms of chromium trioxide), gaseous fluorides, hydrogen sulfide, saturated hydrocarbons C12- C19. There is no excess of the established maximum concentration limits at all emission sources at this enterprise. To assess the impact of the enterprise on soils, the content of heavy metals Zn, Cr (VI), Co, Cu, Pb, Mn, and Fe at the border of the sanitary protection zone of the enterprise was determined and the maximum multi plicity of the excess of MPC of heavy metals was calculated. The content of heavy metals in the selected soil samples was determined using a spectrophotometer atomic absorption C-115-M1. Evaluation of soils for heavy metals showed that the soils at the border of the sanitary protection zone of the enterprise are contaminated with heavy metals. The maximum multiplicity of the excess of the MPC of heavy metals in the soil is 1.04 times for lead; 0.43 times for zinc; 0.37 times for chromium; 0.93 times for copper; 1.85 times for manganese; 0.35 times for cobalt. Maximum concentrations of heavy metals exceed their background content: 1.6 times for lead; 1.8 times for zinc; 1.16 times for chromium; 0.77 times for copper; 3.9 times for manganese; 9.7 times for iron; 1.2 times for cobalt.

Rigid Linkages and Partial Zero Forcing

Connections between vital linkages and zero forcing are established. Specifically, the notion of a rigid linkage is introduced as a special kind of unique linkage and it is shown that spanning forcing paths of a zero forcing process form a spanning rigid linkage and thus a vital linkage. A related generalization of zero forcing that produces a rigid linkage via a coloring process is developed. One of the motivations for introducing zero forcing is to provide an upper bound on the maximum multiplicity of an eigenvalue among the real symmetric matrices described by a graph. Rigid linkages and a related notion of rigid shortest linkages are utilized to obtain bounds on the multiplicities of eigenvalues of this family of matrices.

Asymmetry in Identification of Multiplicity Errors in Conceptual Models of Business Processes

ABSTRACT Business rules can be represented by multiplicities in a Unified Modeling Language (UML) class diagram. Diagrams containing erroneous multiplicities may be implemented as an inefficient/ineffective database. System validators must be able to validate such diagrams, including multiplicities, to prevent the implementation of design errors. Prior research reveals conflicting evidence regarding the expected accuracy in validating minimum multiplicities, indicating a need for additional research to further our understanding. Ontology research claims that multiplicities that depict optional participation are ambiguous and lead to poorer understanding and accuracy compared to multiplicities that depict mandatory participation. However, other research has reported better accuracy validating multiplicities that depict optional participation compared to mandatory participation. We conducted an experiment to help resolve this apparent contradiction, and to explore whether any asymmetry exists in accuracy for maximum multiplicity validation. Results indicate an asymmetry for validation of minimum multiplicities such that accuracy is greatest when the underlying semantics represent mandatory participation. Results also indicate an asymmetry for validation of maximum multiplicities such that accuracy is greatest when the underlying semantics represent flexible participation. Given that many business relationships call for optional minimum participation and that many business relationships call for restrictive maximum participation, these error identification asymmetries are cause for concern.