292 research outputs found
ΠΠ‘ΠΠΠΠ’ΠΠ’ΠΠ§ΠΠ‘ΠΠΠ ΠΠΠΠΠΠΠΠΠ Π ΠΠΠΠ‘Π’ΠΠ ΠΠ«Π₯ Π ΠΠ‘Π‘Π’ΠΠ―ΠΠΠ Π ΠΠ ΠΠ€ΠΠ₯ ΠΠΠΠ
In the present paper, we prove asymptotically exact bounds for resistance distances in families of Cayley graphs that either have a girth of more than 4 or are free of subgraphs K2,t, assuming that the growth function is at least subexponential, and either the diameter or the inverse value of the spectral gap are polynomial with respect to degrees of a graph.Π Π½Π°ΡΡΠΎΡΡΠ΅ΠΉ ΡΠ°Π±ΠΎΡΠ΅ Π΄ΠΎΠΊΠ°Π·Π°Π½Ρ Π°ΡΠΈΠΌΠΏΡΠΎΡΠΈΡΠ΅ΡΠΊΠΈ ΡΠΎΡΠ½ΡΠ΅ ΠΎΡΠ΅Π½ΠΊΠΈ Π΄Π»Ρ ΡΠ΅Π·ΠΈΡΡΠΎΡΠ½ΡΡ
ΡΠ°ΡΡΡΠΎΡΠ½ΠΈΠΉ Π² Π½Π΅ΠΊΠΎΡΠΎΡΡΡ
ΡΠ΅ΠΌΠ΅ΠΉΡΡΠ²Π°Ρ
Π³ΡΠ°ΡΠΎΠ² ΠΡΠ»ΠΈ ΠΏΡΠΈ ΡΡΠ»ΠΎΠ²ΠΈΠΈ, ΡΡΠΎ ΡΡΠ½ΠΊΡΠΈΡ ΡΠΎΡΡΠ° ΡΠ²Π»ΡΠ΅ΡΡΡ ΠΊΠ°ΠΊ ΠΌΠΈΠ½ΠΈΠΌΡΠΌ ΡΡΠ±ΡΠΊΡΠΏΠΎΠ½Π΅Π½ΡΠΈΠ°Π»ΡΠ½ΠΎΠΉ, Π° Π΄ΠΈΠ°ΠΌΠ΅ΡΡ Π»ΠΈΠ±ΠΎ ΠΎΠ±ΡΠ°ΡΠ½Π°Ρ Π²Π΅Π»ΠΈΡΠΈΠ½Π° ΠΊ ΡΠΏΠ΅ΠΊΡΡΠ°Π»ΡΠ½ΠΎΠΌΡ ΠΏΡΠΎΠ±Π΅Π»Ρ ΠΏΠΎΠ»ΠΈΠ½ΠΎΠΌΠΈΠ°Π»ΡΠ½Ρ ΠΏΠΎ ΡΡΠ΅ΠΏΠ΅Π½ΠΈ Π³ΡΠ°ΡΠ°.Β
Some applications of noncommutative groups and semigroups to information security
We present evidence why the Burnside groups of exponent 3 could be a good candidate for a platform group for the HKKS semidirect product key exchange protocol. We also explore hashing with matrices over SL2(Fp), and compute bounds on the girth of the Cayley graph of the subgroup of SL2(Fp) for specific generators A, B. We demonstrate that even without optimization, these hashes have comparable performance to hashes in the SHA family
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
ΠΠ½Π°Π»ΠΎΠ³ ΡΠ΅ΠΎΡΠ΅ΠΌΡ ΠΠ»ΡΠ΄ΡΡΠ° ΠΎ Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ ΠΏΠ΅ΡΠ΅ΠΌΠ΅ΡΠΈΠ²Π°Π½ΠΈΡ Π΄Π»Ρ Π³ΡΡΠΏΠΏ ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠ½ΡΡ ΠΎΡΡΠ°ΠΆΠ΅Π½ΠΈΠΉ
The subject of this paper is the mixing time of random walks on minimal Cayley graphs of complex reflection groups G(m,1,n). The key role in estimating it is played by the coupling of distributions, which has been used before for the same task on symmetric groups. The difficulty with its adaptation for the current case is that there are now two components in a walk, which are to be coupled, and they influence each otherβs behaviour. To solve this problem, random walks are split into several blocks for each of which the time needed for their states to match is estimated separately. The result is upper and lower bounds on mixing times of random walks on complex reflection groups, analogous to those obtained by Aldous for a symmetric group.ΠΡΡΠ»Π΅Π΄ΡΠ΅ΡΡΡ Π²ΡΠ΅ΠΌΡ ΠΏΠ΅ΡΠ΅ΠΌΠ΅ΡΠΈΠ²Π°Π½ΠΈΡ ΡΠ»ΡΡΠ°ΠΉΠ½ΡΡ
Π±Π»ΡΠΆΠ΄Π°Π½ΠΈΠΉ Π½Π° ΠΌΠΈΠ½ΠΈΠΌΠ°Π»ΡΠ½ΡΡ
Π³ΡΠ°ΡΠ°Ρ
ΠΡΠ»ΠΈ Π³ΡΡΠΏΠΏ ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠ½ΡΡ
ΠΎΡΡΠ°ΠΆΠ΅Π½ΠΈΠΉ G(m,1,n). ΠΠ»ΡΡΠ΅Π²ΡΡ ΡΠΎΠ»Ρ ΠΏΡΠΈ ΡΡΠΎΠΌ ΠΈΠ³ΡΠ°Π΅Ρ Π°Π΄Π°ΠΏΡΠ°ΡΠΈΡ ΠΌΠ΅ΡΠΎΠ΄Π° ΡΠΊΠ»Π΅ΠΈΠ²Π°Π½ΠΈΡ ΡΠ°ΡΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΠΉ, ΠΏΡΠΈΠΌΠ΅Π½ΡΠ²ΡΠ΅Π³ΠΎΡΡ ΡΠ°Π½Π΅Π΅ Π΄Π»Ρ ΡΠΈΠΌΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΎΠΉ Π³ΡΡΠΏΠΏΡ. Π‘Π»ΠΎΠΆΠ½ΠΎΡΡΡ Π°Π΄Π°ΠΏΡΠ°ΡΠΈΠΈ Π·Π°ΠΊΠ»ΡΡΠ°Π΅ΡΡΡ Π² ΡΠΎΠΌ, ΡΡΠΎ Ρ ΠΎΠ±ΠΎΠ±ΡΠ΅Π½ΠΈΠ΅ΠΌ Π² ΡΠ»ΡΡΠ°ΠΉΠ½ΠΎΠΌ Π±Π»ΡΠΆΠ΄Π°Π½ΠΈΠΈ ΠΏΠΎΡΠ²Π»ΡΡΡΡΡ Π΄Π²Π΅ ΠΊΠΎΠΌΠΏΠΎΠ½Π΅Π½ΡΡ, ΠΊ ΠΊΠΎΡΠΎΡΡΠΌ Π½ΡΠΆΠ½ΠΎ ΠΏΡΠΈΠΌΠ΅Π½ΡΡΡ ΡΠΊΠ»Π΅ΠΈΠ²Π°Π½ΠΈΠ΅, ΠΈ ΡΡΠΈ ΠΊΠΎΠΌΠΏΠΎΠ½Π΅Π½ΡΡ Π²Π»ΠΈΡΡΡ Π½Π° ΠΎΠ±ΠΎΡΠ΄Π½ΠΎΠ΅ ΠΏΠΎΠ²Π΅Π΄Π΅Π½ΠΈΠ΅. ΠΠ»Ρ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΡΡΠΎΠΉ ΠΏΡΠΎΠ±Π»Π΅ΠΌΡ ΡΠ»ΡΡΠ°ΠΉΠ½ΡΠ΅ Π±Π»ΡΠΆΠ΄Π°Π½ΠΈΡ ΡΠ°Π·Π±ΠΈΠ²Π°ΡΡΡΡ Π½Π° Π½Π΅ΡΠΊΠΎΠ»ΡΠΊΠΎ Π±Π»ΠΎ- ΠΊΠΎΠ², Π΄Π»Ρ ΠΊΠ°ΠΆΠ΄ΠΎΠ³ΠΎ ΠΈΠ· ΠΊΠΎΡΠΎΡΡΡ
Π΄Π°ΡΡΡΡ ΠΎΡΠ΄Π΅Π»ΡΠ½ΡΠ΅ ΠΎΡΠ΅Π½ΠΊΠΈ Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ, Π½Π΅ΠΎΠ±Ρ
ΠΎΠ΄ΠΈΠΌΠΎΠ³ΠΎ Π΄Π»Ρ ΡΠΎΠ²ΠΏΠ°Π΄Π΅Π½ΠΈΡ ΡΠΎΡΡΠΎΡΠ½ΠΈΠΉ. ΠΠΎΠΊΠ°Π·Π°Π½Ρ ΠΎΡΠ΅Π½ΠΊΠΈ ΡΠ²Π΅ΡΡ
Ρ ΠΈ ΡΠ½ΠΈΠ·Ρ Π½Π° Π²ΡΠ΅ΠΌΡ ΠΏΠ΅ΡΠ΅ΠΌΠ΅ΡΠΈΠ²Π°Π½ΠΈΡ ΡΠ»ΡΡΠ°ΠΉΠ½ΡΡ
Π±Π»ΡΠΆΠ΄Π°Π½ΠΈΠΉ Π½Π° Π³ΡΡΠΏΠΏΠ°Ρ
ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠ½ΡΡ
ΠΎΡΡΠ°ΠΆΠ΅Π½ΠΈΠΉ, Π°Π½Π°Π»ΠΎΠ³ΠΈΡΠ½ΡΠ΅ ΠΎΡΠ΅Π½ΠΊΠ°ΠΌ ΠΠ»ΡΠ΄ΡΡΠ° Π΄Π»Ρ ΡΠΈΠΌΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΎΠΉ Π³ΡΡΠΏΠΏΡ
Localization and Optimization Problems for Camera Networks
In the framework of networked control systems, we focus on networks of autonomous
PTZ cameras. A large set of cameras communicating each other through a network
is a widely used architecture in application areas like video surveillance, tracking and motion.
First, we consider relative localization in sensor networks, and we tackle the issue of
investigating the error propagation, in terms of the mean error on each component of the
optimal estimator of the position vector. The relative error is computed as a function of the
eigenvalues of the network: using this formula and focusing on an exemplary class of networks
(the Abelian Cayley networks), we study the role of the network topology and the dimension
of the networks in the error characterization. Second, in a network of cameras one of the
most crucial problems is calibration. For each camera this consists in understanding what is
its position and orientation with respect to a global common reference frame. Well-known
methods in computer vision permit to obtain relative positions and orientations of pairs
of cameras whose sensing regions overlap. The aim is to propose an algorithm that, from
these noisy input data makes the cameras complete the calibration task autonomously, in a
distributed fashion. We focus on the planar case, formulating an optimization problem over
the manifold SO(2). We propose synchronous deterministic and distributed algorithms that
calibrate planar networks exploiting the cycle structure of the underlying communication
graph. Performance analysis and numerical experiments are shown. Third, we propose a
gossip-like randomized calibration algorithm, whose probabilistic convergence and numerical
studies are provided. Forth and finally, we design surveillance trajectories for a network of
calibrated autonomous cameras to detect intruders in an environment, through a continuous
graph partitioning problem
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