Let G be a graph with vertex set V(G), and let d(x,y) denote the length of a shortest path between nodes x and y in G. For a positive integer k and for distinct x,y∈V(G), let dk(x,y)=min{d(x,y),k+1} ...and Rk{x,y}={z∈V(G):dk(x,z)≠dk(y,z)}. A subset S⊆V(G) is a k-truncated resolving set of G if |S∩Rk{x,y}|≥1 for any pair of distinct x,y∈V(G). The k-truncated metric dimension, dimk(G), of G is the minimum cardinality over all k-truncated resolving sets of G, and the usual metric dimension is recovered when k+1 is at least the diameter of G. We obtain some general bounds for k-truncated metric dimension. For all k≥1, we characterize connected graphs G of order n with dimk(G)=n−2 and dimk(G)=n−1. For all j,k≥1, we find the maximum possible order, degree, clique number, and chromatic number of any graph G with dimk(G)=j. We determine dimk(G) when G is a cycle or a path. We also examine the effect of vertex or edge deletion on the truncated metric dimension of graphs, and study various problems related to the truncated metric dimension of trees.
The graph-based random walk model of fractal diffusion is limited in its use to the connected sets and does not allow for direct fractal dimension estimation based on observed data. We discuss a task ...of directly obtaining accurate fractal dimension estimates and propose butterfly diffusion as an alternative approach using an explicit relation between walk and fractal dimensions. The validity of the presented approach is evaluated and statistical properties of the dimension estimates are presented. Through experiments on self-similar sets, we demonstrate the effectiveness of this approach in producing unbiased dimension estimates, offering a promising tool for fractal analysis and Monte Carlo simulations. The estimate of fractal dimension can be also created from spectral dimension, but this approach is less general and less accurate.
•Butterfly diffusion as a resampled random walk.•Introduced simple relation between walk and fractal dimensions.•Novel data-driven dimension estimation procedure.•Statistical verification using Monte Carlo simulations on self-similar structures.
Mixed metric dimension of graphs Kelenc, Aleksander; Kuziak, Dorota; Taranenko, Andrej ...
Applied mathematics and computation,
12/2017, Volume:
314
Journal Article
Peer reviewed
Open access
Let G=(V,E) be a connected graph. A vertex w ∈ V distinguishes two elements (vertices or edges) x, y ∈ E ∪ V if dG(w, x) ≠ dG(w, y). A set S of vertices in a connected graph G is a mixed metric ...generator for G if every two distinct elements (vertices or edges) of G are distinguished by some vertex of S. The smallest cardinality of a mixed metric generator for G is called the mixed metric dimension and is denoted by dimm(G). In this paper we consider the structure of mixed metric generators and characterize graphs for which the mixed metric dimension equals the trivial lower and upper bounds. We also give results about the mixed metric dimension of some families of graphs and present an upper bound with respect to the girth of a graph. Finally, we prove that the problem of determining the mixed metric dimension of a graph is NP-hard in the general case.
This study investigates into teachers’ perceptions in order to find out whether they are thinking beyond the concept of method. Specifically speaking, the underlying goal is to diagnose whether ...teachers are context sensitive and are aware of the uniqueness of each specific teaching-learning context, that is, whether they are able to differentiate between the different requirements of one teaching context from another. Instead of exploring teachers’ views on the term’s method, postmethod, which is a dominant pattern in the literature of postmethod studies, teachers’ perceptions of postmethod teaching strategies (Stern, 1992) were explored across two proficiency levels of elementary and intermediate as a kind of contextual variable. To this end, a questionnaire tapped into three teachers’ perceived effectiveness of each of the two distinctive strategies of each dimension across two levels of intermediate and elementary. The statistical analysis didn’t reveal any significant difference in teachers’ perceptions across the two proficiency levels. This indicated that teachers didn’t consider the requirements of context but rather they adhered to just one method at both levels. Therefore, these teachers were not thinking beyond the framework of the method concept and were not oriented toward postmethod pedagogy in their thoughts.
•Combining HFD and NCDF with round function, dispersion HFD (DHFD) was proposed, and the revised metric solved the problem of traditional HFD being unable to handle signal outliers.•ODHFD and ORCMHFD ...were proposed by using intelligent optimization algorithms and refine composite multi-scale processing to solve the problem of DHFD parameter selection and inability to represent signal information at multiple scales.•Experimental results have demonstrated the validity of this metrics, and the results show that ODHFD and ORCMDHFD have the best signal stability and optimal signal separation.
Higuchi fractal dimension (HFD), as a classic nonlinear dynamic metric, which is commonly used to detect signal dynamic changes. However, it is difficult for HFD to process signal outliers. To address this issue, dispersion HFD (DHFD) is proposed, which improves the signal complexity representation ability of HFD by introducing the normal cumulative distribution function and round function in dispersion entropy. Nevertheless, the parameter selection of DHFD can affect the complexity value. Therefore, an optimized dispersion HFD (ODHFD) is proposed, which solves the threshold selection problem of DHFD and can more effectively reflect the complexity of the signal. In addition, an optimized refined composite DHFD (ORCMDHFD) has been proposed, which can more comprehensively reflect the complexity information for the signal at multiple scales. The simulation experiment results show that DHFD has a smaller standard deviation than HFD when calculating white noise signal complexity, and DHFD have the least dependence on signal length compared to other metrics, as well as RCMDHFD has the best separability for simulated noise signals. Actual experiments have shown that ODHFD and ORCMDHFD is superior to other entropy and fractal dimension metrics in distinguishing ship radiated noise and mechanical fault signals, and has broad application prospects in the field of signal analysis.
We introduce a new dimension spectrum motivated by the Assouad dimension; a familiar notion of dimension which, for a given metric space, returns the minimal exponent α⩾0 such that for any pair of ...scales 0<r<R, any ball of radius R may be covered by a constant times (R/r)α balls of radius r. To each θ∈(0,1), we associate the appropriate analogue of the Assouad dimension with the restriction that the two scales r and R used in the definition satisfy logR/logr=θ. The resulting ‘dimension spectrum’ (as a function of θ) thus gives finer geometric information regarding the scaling structure of the space and, in some precise sense, interpolates between the upper box dimension and the Assouad dimension. This latter point is particularly useful because the spectrum is generally better behaved than the Assouad dimension. We also consider the corresponding ‘lower spectrum’, motivated by the lower dimension, which acts as a dual to the Assouad spectrum.
We conduct a detailed study of these dimension spectra; including analytic, geometric, and measureability properties. We also compute the spectra explicitly for some common examples of fractals including decreasing sequences with decreasing gaps and spirals with sub-exponential and monotonic winding. We also give several applications of our results, including: dimension distortion estimates under bi-Hölder maps for Assouad dimension and the provision of new bi-Lipschitz invariants.
Gorenstein weak global dimension is symmetric Christensen, Lars Winther; Estrada, Sergio; Thompson, Peder
Mathematische Nachrichten,
November 2021, Volume:
294, Issue:
11
Journal Article
Peer reviewed
Open access
We study the Gorenstein weak global dimension of associative rings and its relation to the Gorenstein global dimension. In particular, we prove the conjecture that the Gorenstein weak global ...dimension is a left‐right symmetric invariant – just like the (absolute) weak global dimension.
Let G be a graph with vertex set V(G) and edge set E(G), and let d(u,w) denote the length of a u−w geodesic in G. For any vertex v∈V(G) and any edge e=xy∈E(G), let d(e,v)=min{d(x,v),d(y,v)}. For any ...distinct edges e1,e2∈E(G), let R{e1,e2}={z∈V(G):d(z,e1)≠d(z,e2)}. Kelenc, Tratnik and Yero Discrete Appl. Math. 251 (2018) 204-220 introduced the notion of an edge resolving set and the edge dimension of a graph: A vertex subset S⊆V(G) is an edge resolving set of G if |S∩R{e1,e2}|≥1 for any distinct edges e1,e2∈E(G), and the edge dimension, edim(G), of G is the minimum cardinality among all edge resolving sets of G.
For a function g defined on V(G) and for U⊆V(G), let g(U)=∑s∈Ug(s). A real-valued function g:V(G)→0,1 is an edge resolving function of G if g(R{e1,e2})≥1 for any distinct edges e1,e2∈E(G). The fractional edge dimension, edimf(G), of G is min{g(V(G)):g is an edge resolving function of G}. Note that edimf(G) reduces to edim(G) if the codomain of edge resolving functions is restricted to {0,1}.
In this paper, we introduce and study the fractional edge dimension of graphs, and we obtain some general results on the edge dimension of graphs. We show that there exist two non-isomorphic graphs on the same vertex set with the same edge metric coordinates. We construct two graphs G and H such that H⊂G and both edim(H)−edim(G) and edimf(H)−edimf(G) can be arbitrarily large. We show that a graph G with edim(G)=2 cannot have K5 or K3,3 as a subgraph, and we construct a non-planar graph H satisfying edim(H)=2. It is easy to see that, for any connected graph G of order at least three, 1≤edimf(G)≤|V(G)|2; we characterize graphs G satisfying edimf(G)=1 and examine some graph classes satisfying edimf(G)=|V(G)|2. We also determine the fractional edge dimension for some classes of graphs.
Space-time adaptive processing (STAP) for airborne radar with co-prime arrays is shown to have excellent superiority compared to traditional STAP with uniform linear array radar. However, high ...arithmetic computational complexity and large amount of training data are required in this approach. This motivates the authors to present a new approach which is relatively low computational load and fast convergence with satisfactory performance. Specifically, a reduced-dimension transformation in Doppler domain is incorporated into the existing approach. The reduced-dimension interference covariance matrix and target steering vector are then achieved by performing the reduced-dimension process, and hence two reduced-dimension STAP filters are designed using the derived interference covariance matrix and target-steering vector. Numerical simulations are carried out to reveal the superiority of the proposed approach.
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