Emergence of bi-cluster flocking for the Cucker–Smale model

2016 ◽  
Vol 26 (06) ◽  
pp. 1191-1218 ◽  
Author(s):  
Junghee Cho ◽  
Seung-Yeal Ha ◽  
Feimin Huang ◽  
Chunyin Jin ◽  
Dongnam Ko
Keyword(s):  
Lower Bound ◽  
Asymptotic Convergence ◽  
Far Field ◽  
Differential Inequalities ◽  
Local Velocity ◽  
Velocity Fluctuations ◽  
Spatial Fluctuations ◽  
The Difference ◽  
Field Decay ◽  
Velocity Averages

We present the mathematical analysis of bi-cluster flocking phenomenon for the short-ranged Cucker–Smale model with some well-prepared initial data. For this, we derive a system of differential inequalities for the functionals measuring the local spatial and velocity fluctuations and differences of local velocity averages, and then estimate the upper bound of spatial fluctuations and the lower bound of the difference between local velocity averages. We explicitly present an admissible class of initial configurations leading to the asymptotic emergence of bi-cluster flocking phenomenon. Unlike global flocking (a mono-cluster flocking configuration in velocity), where the convergence rate is always exponential, the asymptotic convergence to bi-cluster flocking configurations is affected by the far-field decay rate of communication weights so that it can be algebraic.

scholarly journals Primitive model electrolytes in the near and far field: Decay lengths from DFT and simulations

2021 ◽  
Vol 154 (12) ◽  
pp. 124504
Author(s):  
P. Cats ◽  
R. Evans ◽  
A. Härtel ◽  
R. van Roij
Keyword(s):  
Far Field ◽  
Primitive Model ◽  
Field Decay

scholarly journals THE PECULIARITIES OF THE SOUND FIELD ENERGY DECAY IN A ROOM WITH THE USE OF DIFFERENT PULSED SOUND SOURCES/GARSO LAUKO ENERGIJOS SLOPIMO PATALPOJE YPATUMAI, NAUDOJANT SKIRTINGUS IMPULSINIUS GARSO ŠALTINIUS

1999 ◽  
Vol 5 (2) ◽  
pp. 135-140
Author(s):  
Vytautas Stauskis
Keyword(s):  
Noise Level ◽  
Sound Field ◽  
Maximum Energy ◽  
Reverberation Time ◽  
Frequency Range ◽  
Sound Sources ◽  
Low Frequencies ◽  
High Frequencies ◽  
The Difference ◽  
Field Decay

The paper deals with the differences between the energy created by four different pulsed sound sources, ie a sound gun, a start gun, a toy gun, and a hunting gun. A knowledge of the differences between the maximum energy and the minimum energy, or the signal-noise ratio, is necessary to correctly calculate the frequency dependence of reverberation time. It has been established by investigations that the maximum energy excited by the sound gun is within the frequency range of 250 to 2000 Hz. It decreases by about 28 dB at the low frequencies. The character of change in the energy created by the hunting gun differs from that of the sound gun. There is no change in the maximum energy within the frequency range of 63–100 Hz, whereas afterwards it increases with the increase in frequency but only to the limit of 2000 Hz. In the frequency range of 63–500 Hz, the energy excited by the hunting gun is lower by 15–30 dB than that of the sound gun. As frequency increases the difference is reduced and amounts to 5–10 dB. The maximum energy of the start gun is lower by 4–5 dB than that of the hunting gun in the frequency range of up to 1000 Hz, while afterwards the difference is insignificant. In the frequency range of 125–250 Hz, the maximum energy generated by the sound gun exceeds that generated by the hunting gun by 20 dB, that by the start gun by 25 dB, and that by the toy gun—by as much as 35 dB. The maximum energy emitted by it occupies a wide frequency range of 250 to 2000 Hz. Thus, the sound gun has an advantage over the other three sound sources from the point of view of maximum energy. Up until 500 Hz the character of change in the direct sound energy is similar for all types of sources. The maximum energy of direct sound is also created by the sound gun and it increases along with frequency, the maximum values being reached at 500 Hz and 1000 Hz. The maximum energy of the hunting gun in the frequency range of 125—500 Hz is lower by about 20 dB than that of the sound gun, while the maximum energy of the toy gun is lower by about 25 dB. The maximum of the direct sound energy generated by the hunting gun, the start gun and the toy gun is found at high frequencies, ie at 1000 Hz and 2000 Hz, while the sound gun generates the maximum energy at 500 Hz and 1000 Hz. Thus, the best results are obtained when the energy is emitted by the sound gun. When the sound field is generated by the sound gun, the difference between the maximum energy and the noise level is about 35 dB at 63 Hz, while the use of the hunting gun reduces the difference to about 20–22 dB. The start gun emits only small quantities of low frequencies and is not suitable for room's acoustical analysis at 63 Hz. At the frequency of 80 Hz, the difference between the maximum energy and the noise level makes up about 50 dB, when the sound field is generated by the sound gun, and about 27 dB, when it is generated by the hunting gun. When the start gun is used, the difference between the maximum signal and the noise level is as small as 20 dB, which is not sufficient to make a reverberation time analysis correctly. At the frequency of 100 Hz, the difference of about 55 dB between the maximum energy and the noise level is only achieved by the sound gun. The hunting gun, the start gun and the toy gun create the decrease of about 25 dB, which is not sufficient for the calculation of the reverberation time. At the frequency of 125 Hz, a sufficiently large difference in the sound field decay amounting to about 40 dB is created by the sound gun, the hunting gun and the start gun, though the character of the sound field curve decay of the latter is different from the former two. At 250 Hz, the sound gun produces a field decay difference of almost 60 dB, the hunting gun almost 50 dB, the start gun almost 40 dB, and the toy gun about 45 dB. At 500 Hz, the sound field decay is sufficient when any of the four sound sources is used. The energy difference created by the sound gun is as large as 70 dB, by the hunting gun 50 dB, by the start gun 52 dB, and by the toy gun 48 dB. Such energy differences are sufficient for the analysis of acoustic indicators. At the high frequencies of 1000 to 4000 Hz, all the four sound sources used, even the toy gun, produce a good difference of the sound field decay and in all cases it is possible to analyse the reverberation process at varied intervals of the sound level decay.

scholarly journals An improved lower bound related to the Furstenberg-Sárközy theorem

10.37236/4656 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Mark Lewko
Keyword(s):  
Lower Bound ◽  
The Other ◽  
The Difference

Let $D(n)$ denote the cardinality of the largest subset of the set $\{1,2,\ldots,n\}$ such that the difference of no pair of distinct elements is a square. A well-known theorem of Furstenberg and Sárközy states that $D(n)=o(n)$. In the other direction, Ruzsa has proven that $D(n) \gtrsim n^{\gamma}$ for $\gamma = \frac{1}{2}\left( 1 + \frac{\log 7}{\log 65} \right) \approx 0.733077$. We improve this to $\gamma = \frac{1}{2}\left( 1 + \frac{\log 12}{\log 205} \right)  \approx 0.733412$.

scholarly journals A lower bound for a remainder term associated with the sum of digits function

1991 ◽  
Vol 34 (1) ◽  
pp. 121-142 ◽  
Author(s):  
D. M. E. Foster
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Upper Bound ◽  
Remainder Term ◽  
Fixed Integer ◽  
Sum Of Digits ◽  
Sum Of Digits Function ◽  
The Difference

For a fixed integer q≧2, every positive integer k = Σr≧0ar(q, k)qr where each ar(q, k)∈{0,1,2,…, q−1}. The sum of digits function α(q, k) Σr≧0ar(q, k) behaves rather erratically but on averaging has a uniform behaviour. In particular if , where n>1, then it is well known that A(q, n)∼½((q − 1)/log q)n logn as n → ∞. For odd values of q, a lower bound is now obtained for the difference 2S(q, n) = A(q, n)−½(q − 1))[log n/log q, where [log n/log q] denotes the greatest integer ≦log n /log q. This complements an upper bound already found.

scholarly journals COMPLEMENTARITY IN GENERIC OPEN QUANTUM SYSTEMS

2010 ◽  
Vol 24 (24) ◽  
pp. 2485-2509 ◽  
Author(s):  
SUBHASHISH BANERJEE ◽  
R. SRIKANTH
Keyword(s):  
Lower Bound ◽  
Uniform Distribution ◽  
Open Quantum Systems ◽  
Quantum Systems ◽  
Atomic Systems ◽  
Information Theoretic ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
The Difference ◽  
Positive Operator Valued Measure

We develop a unified, information theoretic interpretation of the number-phase complementarity that is applicable both to finite-dimensional (atomic) and infinite-dimensional (oscillator) systems, with number treated as a discrete Hermitian observable and phase as a continuous positive operator valued measure (POVM). The relevant uncertainty principle is obtained as a lower bound on entropy excess, X, the difference between the entropy of one variable, typically the number, and the knowledge of its complementary variable, typically the phase, where knowledge of a variable is defined as its relative entropy with respect to the uniform distribution. In the case of finite-dimensional systems, a weighting of phase knowledge by a factor μ (> 1) is necessary in order to make the bound tight, essentially on account of the POVM nature of phase as defined here. Numerical and analytical evidence suggests that μ tends to 1 as the system dimension becomes infinite. We study the effect of non-dissipative and dissipative noise on these complementary variables for an oscillator as well as atomic systems.

The Cramer–Rao lower bound in a non-AWGN model for the affine phase retrieval

2021 ◽  
pp. 2050089
Author(s):  
Zhitao Zhuang ◽  
Kaixin Wang
Keyword(s):  
Lower Bound ◽  
Mean Square Error ◽  
Gaussian Noise ◽  
Phase Retrieval ◽  
Additive White Gaussian Noise ◽  
White Gaussian Noise ◽  
Mean Square ◽  
Cramer Rao Lower Bound ◽  
The Difference

In this paper, we derive the Cramer–Rao lower bound (CRLB) in a non-additive white Gaussian noise (AWGN) model for the affine phase retrieval (APR) and simulate the difference of CRLB and mean square error produced by PhaseLift of phase retrieval and APR in AWGN and non-AWGN cases.

scholarly journals Kesten’s theorem for uniformly recurrent subgroups

2019 ◽  
Vol 40 (10) ◽  
pp. 2778-2787
Author(s):  
MIKOLAJ FRACZYK
Keyword(s):  
Lower Bound ◽  
Cayley Graph ◽  
Spectral Radius ◽  
Short Proof ◽  
Ramanujan Graphs ◽  
Ramanujan Graph ◽  
Random Trajectory ◽  
The Difference

We prove a lower bound on the difference between the spectral radius of the Cayley graph of a group $G$ and the spectral radius of the Schreier graph $H\backslash G$ for any subgroup $H$. As an application, we extend Kesten’s theorem on spectral radii to uniformly recurrent subgroups and give a short proof that the result of Lyons and Peres on cycle density in Ramanujan graphs [Lyons and Peres. Cycle density in infinite Ramanujan graphs. Ann. Probab.43(6) (2015), 3337–3358, Theorem 1.2] holds on average. More precisely, we show that if ${\mathcal{G}}$ is an infinite deterministic Ramanujan graph then the time spent in short cycles by a random trajectory of length $n$ is $o(n)$.

scholarly journals Analysis of an atomistic model for anti-plane fracture

2019 ◽  
Vol 29 (13) ◽  
pp. 2469-2521
Author(s):  
Maciej Buze ◽  
Thomas Hudson ◽  
Christoph Ortner
Keyword(s):  
Pair Potential ◽  
Square Lattice ◽  
Far Field ◽  
Decay Estimates ◽  
Nearest Neighbour ◽  
Stability Of Solutions ◽  
Local Uniqueness ◽  
Field Decay ◽  
Interatomic Pair Potential ◽  
Crack Defect

We develop a model for an anti-plane crack defect posed on a square lattice under an interatomic pair-potential with nearest-neighbour interactions. In particular, we establish existence, local uniqueness and stability of solutions for small loading parameters and further prove qualitatively sharp far-field decay estimates. The latter requires establishing decay estimates for the corresponding lattice Green’s function, which are of independent interest.

Linking invariants of even virtual links

2017 ◽  
Vol 26 (12) ◽  
pp. 1750072 ◽  
Author(s):  
Haruko A. Miyazawa ◽  
Kodai Wada ◽  
Akira Yasuhara
Keyword(s):  
Lower Bound ◽  
The Other ◽  
Virtual Link ◽  
Link Diagram ◽  
Linking Number ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Classical Link ◽  
Virtual Links ◽  
The Difference

A virtual link diagram is even if the virtual crossings divide each component into an even number of arcs. The set of even virtual link diagrams is closed under classical and virtual Reidemeister moves, and it contains the set of classical link diagrams. For an even virtual link diagram, we define a certain linking invariant which is similar to the linking number. In contrast to the usual linking number, our linking invariant is not preserved under the forbidden moves. In particular, for two fused isotopic even virtual link diagrams, the difference between the linking invariants of them gives a lower bound of the minimal number of forbidden moves needed to deform one into the other. Moreover, we give an example which shows that the lower bound is best possible.

NORMAL‐MOVEOUT AND VELOCITY RELATIONS FOR FLAT AND DIPPING BEDS AND FOR LONG OFFSETS

Geophysics ◽  
1969 ◽  
Vol 34 (2) ◽  
pp. 180-195 ◽  
Author(s):  
Robert J. S. Brown
Keyword(s):  
Random Noise ◽  
Seismic Energy ◽  
Long Offset ◽  
Dominant Period ◽  
Exploration Seismology ◽  
Time Average ◽  
The Difference ◽  
Zero Offset ◽  
Velocity Averages ◽  
Velocity Average

Accurate relations between NMO and velocity are needed in modern exploration seismology, especially in long‐offset CDP work, where accurate NMO corrections must be made for stacking, and where several types of velocity averages may be computed with accuracy from NMO data. The velocity average associated with NMO is the time‐rms velocity [Formula: see text]. Even for long offsets the straight‐ray computation using [Formula: see text] is usually adequate, but a closer approximation for horizontal reflectors is obtained by reducing the NMO calculated from [Formula: see text] or reducing the value of [Formula: see text] calculated from NMO by the factor [Formula: see text], where σ is the rms deviation of the velocity from its mean, T is zero‐offset traveltime, and ΔT the NMO. The difference between time‐average and time‐rms velocities is often several percent. For the velocity function [Formula: see text] and for reflectors of arbitrary dip and strike, the NMO is shown to be [Formula: see text] where X is offset, α is emergence angle, and ψ is the angle between the offset direction and the reflector dip direction. The terms that contain angles can be used as a correction ΔΔT to the NMO value computed as if the seismic energy were reflected from a horizontal reflector, even for offset greater than those for which an NMO expression quadratic in offset is accurate. A further approximation gives [Formula: see text], where δT is dip moveout over a spread of length L, and [Formula: see text] is the angle between the receiver line and the dip direction, differing from ψ only if there is substantial perpendicular offset of the source point. An expression for the degradation of the stacked signals in CDP stacks because of NMO errors is given. It is shown that the criterion that the signal‐to‐random‐noise ratio could not be improved by dropping the longest‐offset trace(s) requires that the NMO error be not much larger than one‐quarter of a dominant period.

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