The embedding theorems for anisotropic Nikol’skii-Besov spaces with generalized mixed smoothness

The theory of embedding of spaces of differentiable functions studies the important relations of differential (smoothness) properties of functions in various metrics and has a wide application in the theory of boundary value problems of mathematical physics, approximation theory, and other fields of mathematics. In this article, we prove the embedding theorems for anisotropic spaces Nikol’skii-Besov with a generalized mixed smoothness and mixed metric, and anisotropic Lorentz spaces. The proofs of the obtained results are based on the inequality of different metrics for trigonometric polynomials in Lebesgue spaces with mixed metrics and interpolation properties of the corresponding spaces.

A Comparative Study of Three Resolving Parameters of Graphs

Graph theory is one of those subjects that is a vital part of the digital world. It is used to monitor the movement of robots on a network, to debug computer networks, to develop algorithms, and to analyze the structural properties of chemical structures, among other things. It is also useful in airplane scheduling and the study of diffusion mechanisms. The parameters computed in this article are very useful in pattern recognition and image processing. A number d f , w = min d w , t , d w , s is referred as distance between f = t s an edge and w a vertex. d w , f 1 ≠ d w , f 2 implies that two edges f 1 , f 2 ∈ E are resolved by node w ∈ V . A set of nodes A is referred to as an edge metric generator if every two links/edges of Γ are resolved by some nodes of A and least cardinality of such sets is termed as edge metric dimension, e dim Γ for a graph Γ . A set B of some nodes of Γ is a mixed metric generator if any two members of V ∪ E are resolved by some members of B . Such a set B with least cardinality is termed as mixed metric dimension, m dim Γ . In this paper, the metric dimension, edge metric dimension, and mixed metric dimension of dragon graph T n , m , line graph of dragon graph L T n , m , paraline graph of dragon graph L S T n , m , and line graph of line graph of dragon graph L L T n , m have been computed. It is shown that these parameters are constant, and a comparative analysis is also given for the said families of graphs.

The Vertex-Edge Resolvability of Some Wheel-Related Graphs

A vertex w ∈ V H distinguishes (or resolves) two elements (edges or vertices) a , z ∈ V H ∪ E H if d w , a ≠ d w , z . A set W m of vertices in a nontrivial connected graph H is said to be a mixed resolving set for H if every two different elements (edges and vertices) of H are distinguished by at least one vertex of W m . The mixed resolving set with minimum cardinality in H is called the mixed metric dimension (vertex-edge resolvability) of H and denoted by m dim H . The aim of this research is to determine the mixed metric dimension of some wheel graph subdivisions. We specifically analyze and compare the mixed metric, edge metric, and metric dimensions of the graphs obtained after the wheel graphs’ spoke, cycle, and barycentric subdivisions. We also prove that the mixed resolving sets for some of these graphs are independent.

The mixed metric dimension of flower snarks and wheels

New graph invariant, which is called a mixed metric dimension, has been recently introduced. In this paper, exact results of the mixed metric dimension on two special classes of graphs are found: flower snarks J n {J}_{n} and wheels W n {W}_{n} . It is proved that the mixed metric dimension for J 5 {J}_{5} is equal to 5, while for higher dimensions it is constant and equal to 4. For W n {W}_{n} , the mixed metric dimension is not constant, but it is equal to n n when n ≥ 4 n\ge 4 , while it is equal to 4, for n = 3 n=3 .

Interpolation theorem for Nikol’skii-Besov type spaceswith mixed metric

In this paper we study the interpolation properties of Nikol’skii-Besov spaces with a dominant mixed derivative and mixed metric with respect to anisotropic and complex interpolation methods. An interpolation theorem is proved for a weighted discrete space of vector-valued sequences l^α_q(A). It is shown that the Nikol’skii-Besov space under study is a retract of the space l^α_q(Lp). Based on the above results, interpolation theorems were obtained for Nikol’skii-Besov spaces with the dominant mixed derivative and mixed metric.

The Influence of Rate and Accentuation on Subjective Rhythmization

The parsing of undifferentiated tone sequences into groups of qualitatively distinct elements is one of the earliest rhythmic phenomena to have been investigated experimentally (Bolton, 1894). The present study aimed to replicate and extend these findings through online experimentation using a spontaneous grouping paradigm with forced-choice response (from 1 to 12 tones per group). Two types of isochronous sequences were used: equitone sequences, which varied only with respect to signal rate (200, 550, or 950 ms interonset intervals), and accented sequences, in which accents were added every two or three tones to test the effect of induced grouping (duple vs. triple) and accent type (intensity, duration, or pitch). In equitone sequences, participants’ grouping percepts (N = 4,194) were asymmetrical and tempo-dependent, with “no grouping” and groups of four being most frequently reported. In accented sequences, slower rate, induced triple grouping, and intensity accents correlated with increases in group length. Furthermore, the probability of observing a mixed metric type—that is, grouping percepts divisible by both two and three (6 and 12)—was found to be highest in faster sequences with induced triple grouping. These findings suggest that lower-level triple grouping gives rise to binary grouping percepts at higher metrical levels.