An official website of the United States government
Abstract Persistent Betti numbers are a major tool in persistent homology, a subfield of topological data analysis. Many tools in persistent homology rely on the properties of persistent Betti numbers considered as a two-dimensional stochastic process$$ (r,s) \mapsto n^{-1/2} (\beta^{r,s}_q ( \mathcal{K}(n^{1/d} \mathcal{X}_n))-\mathbb{E}[\beta^{r,s}_q ( \mathcal{K}( n^{1/d} \mathcal{X}_n))])$$. So far, pointwise limit theorems have been established in various settings. In particular, the pointwise asymptotic normality of (persistent) Betti numbers has been established for stationary Poisson processes and binomial processes with constant intensity function in the so-called critical (or thermodynamic) regime; see Yogeshwaranet al.(Prob. Theory Relat. Fields167, 2017) and Hiraokaet al.(Ann. Appl. Prob.28, 2018). In this contribution, we derive a strong stabilization property (in the spirit of Penrose and Yukich,Ann. Appl. Prob.11, 2001) of persistent Betti numbers, and we generalize the existing results on their asymptotic normality to the multivariate case and to a broader class of underlying Poisson and binomial processes. Most importantly, we show that multivariate asymptotic normality holds for all pairs (r,s),$$0\le r\le s<\infty$$, and that it is not affected by percolation effects in the underlying random geometric graph.
more »
« less
On the Range Assignment in Wireless Sensor Networks for Minimizing the Coverage-Connectivity Cost
This article deals with reliable and unreliable mobile sensors having identical sensing radiusr, communication radiusR, provided thatr≤Rand initially randomly deployedon the planeby dropping them from an aircraft according to general random process. The sensors have to move from their initial random positions to the final destinations to provide greedy pathk1-coverage simultaneously withk2-connectivity. In particular, we are interested in assigning the sensing radiusrand communication radiusRto minimizethe time requiredandthe energy consumptionof transportation cost for sensors to provide the desiredk1-coverage withk2-connectivity. We prove that for both of these optimization problems, the optimal solution is to assign the sensing radius equal tor=k1||E[S]||/2 and the communication radiusR=k2||E[S]||/2, where ||E[S]|| is the characteristic of general random process according to which the sensors are deployed. Whenr<k1||E[S]||/2 orR<k2||E[S]||/ 2, and sensors are reliable, we discover and explainthe sharp increasein the time required and the energy consumption in transportation cost to ensure the desiredk1-coverage withk2-connectivity.
more »
« less
- Award ID(s):
- 2104078
- PAR ID:
- 10634007
- Publisher / Repository:
- ACM
- Date Published:
- Journal Name:
- ACM Transactions on Sensor Networks
- Volume:
- 17
- Issue:
- 4
- ISSN:
- 1550-4859
- Page Range / eLocation ID:
- 1 to 48
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract We conduct 3D magnetohydrodynamic simulations of decaying turbulence in the context of the solar wind. To account for the spherical expansion of the solar wind, we implement the expanding box model. The initial turbulence comprises uncorrelated counterpropagating Alfvén waves and exhibits an isotropic power spectrum. Our findings reveal the consistent generation of negative residual energy whenever nonlinear interactions are present, independent of the normalized cross helicityσcand compressibility. The spherical expansion facilitates this process. The resulting residual energy is primarily distributed in the perpendicular direction, withS2(b) − S2(u) ∝ l⊥or equivalently . HereS2(b) andS2(u) are second-order structure functions of magnetic field and velocity respectively. In most runs,S2(b) develops a scaling relation ( ). In contrast,S2(u) is consistently shallower thanS2(b), which aligns with in situ observations of the solar wind. We observe that the higher-order statistics of the turbulence, which act as a proxy for intermittency, depend on the initialσcand are strongly affected by the expansion effect. Generally, the intermittency is more pronounced when the expansion effect is present. Finally, we find that in our simulations, although the negative residual energy and intermittency grow simultaneously as the turbulence evolves, the causal relation between them seems to be weak, possibly because they are generated on different scales.more » « less
-
Abstract A classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in ann-vertex graph not containingC2k, the cycle of length 2k, isO(n1+1/k). Simonovits established a corresponding supersaturation result forC2k’s, showing that there exist positive constantsC,cdepending only onksuch that everyn-vertex graphGwithe(G)⩾Cn1+1/kcontains at leastc(e(G)/v(G))2kcopies ofC2k, this number of copies tightly achieved by the random graph (up to a multiplicative constant). In this paper we extend Simonovits' result to a supersaturation result ofr-uniform linear cycles of even length inr-uniform linear hypergraphs. Our proof is self-contained and includes ther= 2 case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.more » « less
-
Abstract A new nonheme iron(II) complex, FeII(Me3TACN)((OSiPh2)2O) (1), is reported. Reaction of1with NO(g)gives a stable mononitrosyl complex Fe(NO)(Me3TACN)((OSiPh2)2O) (2), which was characterized by Mössbauer (δ=0.52 mm s−1, |ΔEQ|=0.80 mm s−1), EPR (S=3/2), resonance Raman (RR) and Fe K‐edge X‐ray absorption spectroscopies. The data show that2is an {FeNO}7complex with anS=3/2 spin ground state. The RR spectrum (λexc=458 nm) of2combined with isotopic labeling (15N,18O) reveals ν(N‐O)=1680 cm−1, which is highly activated, and is a nearly identical match to that seen for the reactive mononitrosyl intermediate in the nonheme iron enzyme FDPnor (ν(NO)=1681 cm−1). Complex2reacts rapidly with H2O in THF to produce the N‐N coupled product N2O, providing the first example of a mononuclear nonheme iron complex that is capable of converting NO to N2O in the absence of an exogenous reductant.more » « less
-
Structural characteristics are reported for two thioether–ketones,DtdpeandMtdp[2-({2-[(2-oxo-2-phenylethyl)sulfanyl]ethyl}sulfanyl)-1-phenylethan-1-one, C18H18O2S2, and 2-[(2-oxo-2-phenylethyl)sulfanyl]-1-phenylethan-1-one, C16H14O2S], and for related derivatives, the bis(pyridylhydrazones)DhpkandPrpsb[2-((2E)-2-{(2Z)-2-phenyl-2-[2-(pyridin-2-yl)hydrazin-1-ylidene]ethylidene}hydrazin-1-yl)pyridine, C18H16N6, and 2-[(2Z,12Z)-3,12-diphenyl-14-(pyridin-2-yl)-5,10-dithia-1,2,13,14-tetraazatetradeca-2,12-dien-1-yl]pyridine, C30H32N6S2], as well as for the macrocyclic thiocarbohydrazide derivativeCtrsp[(3E,8Z)-3,9-dimethyl-1,11-dithia-4,5,7,8-tetraazacyclotetradeca-3,8-diene-6-thione, C10H18N4S3]. Three of the five compounds exhibit conformational enantiomerism in the solid state. The occurrence of intra- and intermolecular hydrogen bonding is commented upon through quantum mechanical (DFT) calculations. Weak C—H...S interactions are noted, while stronger N—H...N and N—H...S hydrogen bridges are delineated.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1145/3457408
