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In this paper, by means of the Faà di Bruno formula, with the help of explicit formulas for partial Bell polynomials at specific arguments of two specific sequences generated by derivatives at the origin of the inverse sine and inverse cosine functions, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes power series expansions for real powers of the inverse cosine (sine) functions and the inverse hyperbolic cosine (sine) functions. By comparing different series expansions for the square of the inverse cosine function and for the positive integer power of the inverse sine function, the author not only finds infinite series representations of the circular constant π and its real powers, but also derives several combinatorial identities involving central binomial coefficients and the Stirling numbers of the first kind.

A version of a formula deduced by Bretschneider in the mid nineteenth century not only allows the computation of areas of regular and irregular planar quadrilaterals but also provides a more thorough understanding of the geometry of such figures.

GeoGebra is an extremely powerful tool for mathematics teaching and learning. In this article, we share how to create a GeoGebra worksheet that can be used to display dynamically changing quantities. This worksheet can support students as they make meaning of the inverse cosine function.

In this article, we propose another form of ten similarity measures by considering the function of membership degree, non-membership degree, and indeterminacy membership degree between the q-ROFSs on the basis of the traditional cosine similarity measures and cotangent similarity measures. Then, we utilize our presented ten similarity measures and ten weighted similarity measures between q-ROFSs to deal with multiple attribute decision-making (MADM) problems including pattern recognition and scheme selection. Finally, two numerical examples are provided to illustrate the scientific and effective of the similarity measures for pattern recognition and scheme selection.

The main purpose of this paper is to investigate degenerate C-(ultra)distribution cosine functions in the setting of barreled sequentially complete locally convex spaces. In our approach, the infinitesimal generator of a degenerate C-(ultra)distribution cosine function is a multivalued linear operator and the regularizing operator C is not necessarily injective. We provide a few important theoretical novelties, considering also exponential subclasses of degenerate C-(ultra)distribution cosine functions. nema

In this article, we propose a new explanation for why certain cultural products outperform their peers to achieve widespread success. We argue that products' position in feature space significantly predicts their popular success. Using tools from computer science, we construct a novel dataset allowing us to examine whether the musical features of nearly 27,000 songs from Billboard's Hot 100 charts predict their levels of success in this cultural market. We find that, in addition to artist familiarity, genre affiliation, and institutional support, a song's perceived proximity to its peers influences its position on the charts. Contrary to the claim that all popular music sounds the same, we find that songs sounding too much like previous and contemporaneous productions—those that are highly typical—are less likely to succeed. Songs exhibiting some degree of optimal differentiation are more likely to rise to the top of the charts. These findings offer a new perspective on success in cultural markets by specifying how content organizes product competition and audience consumption behavior.

Modern coexistence theory and contemporary niche theory represent parallel frameworks for understanding the niche's role in species coexistence. Despite increasing prominence and shared goals, their compatibility and complementarity have received little attention. This paucity of overlap not only presents an obstacle to newcomers to the field, but it also precludes further conceptual advances at their interface. Here, we present a synthetic treatment of the two frameworks. We review their main concepts and explore their theoretical and empirical relationship, focusing on how the resource supply ratio, impact niche, and requirement niche of contemporary niche theory translate into the stabilizing and equalizing processes of modern coexistence theory. We show, for a general consumer-resource model, that varying resource supply ratios reflects an equalizing process; varying impact niche overlap reflects a stabilizing process; and varying requirement niche overlap may be both stabilizing and equalizing, but has no qualitative effect on coexistence. These generalizations provide mechanistic insight into modern coexistence theory, while also clarifying the role of contemporary niche theory's impacts and requirements in mediating coexistence. From an empirical perspective, we recommend a hierarchical approach, in which quantification of the strength of stabilizing mechanisms is used to guide more focused investigation into the underlying niche factors determining species coexistence. Future research that considers alternative assumptions, including different forms of species interaction, spatiotemporal heterogeneity, and priority effects, would facilitate a more complete synthesis of the two frameworks.

Letαbe a nonnegative number, andC:X→Xa bounded linear operator on a Banach spaceX. In this paper, we shall deduce some basic properties of a nondegenerate localα-times integratedC-cosine function onXand some generation theorems of localα-times integratedC-cosine functions onXwith or without the nondegeneracy, which can be applied to obtain some equivalence relations between the generation of a nondegenerate localα-times integratedC-cosine function onXwith generatorAand the unique existence of solutions of the abstract Cauchy problem: ACP ( A , f , x , y ) { u ″ ( t ) = A u ( t ) + f ( t ) for t ∈ ( 0 , T 0 ) , u ( 0 ) = x , u ′ ( 0 ) = y , just as the case ofα-times integratedC-cosine function whenC:X→Xis injective andA: D(A) ⊂X→Xa closed linear operator inXsuch thatCA⊂AC. Here 0

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