A Bound for the Locating Chromatic Numbers of Trees

Let $f$ be a proper $k$-coloring of a connected graph $G$ and $\Pi=(V_1,V_2,\ldots,V_k)$ be an ordered partition of $V(G)$ into the resulting color classes. For a vertex $v$ of $G$, the color code of $v$ with respect to $\Pi$ is defined to be the ordered $k$-tuple $c_{{}_\Pi}(v)=(d(v,V_1),d(v,V_2),\ldots,d(v,V_k)),$ where $d(v,V_i)=\min\{d(v,x): x\in V_i\}, 1\leq i\leq k$. If distinct vertices have distinct color codes, then $f$ is called a locating coloring. The minimum number of colors needed in a locating coloring of $G$ is the locating chromatic number of $G$, denoted by \Cchi_{{}_L}(G). In this paper, we study the locating chromatic numbers of trees. We provide a counter example to a theorem of Gary Chartrand et al. [G. Chartrand, D. Erwin, M.A. Henning, P.J. Slater, P. Zhang, The locating-chromatic number of a graph, Bull. Inst. Combin. Appl. 36 (2002) 89-101] about the locating chromatic numbers of trees. Also, we offer a new bound for the locating chromatic number of trees. Then, by constructing a special family of trees, we show that this bound is best possible.Comment: 9 pages, 3 figure

On the Randic and degree distance indices of the Mycielskian of a graph

In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph into a new graph which is called the Mycielskian of that graph. In this paper we provide some sharp bounds for the Randic index of the Mycielskian graphs. Also, we determine the degree distance index of the Mycielskian of each graph with diameter two.Comment: just thank

Applications of Blockchain Technology beyond Cryptocurrency

Blockchain (BC), the technology behind the Bitcoin crypto-currency system, is considered to be both alluring and critical for ensuring enhanced security and (in some implementations, non-traceable) privacy for diverse applications in many other domains including in the Internet of Things (IoT) eco-system. Intensive research is currently being conducted in both academia and industry applying the Blockchain technology in multifarious applications. Proof-of-Work (PoW), a cryptographic puzzle, plays a vital role in ensuring BC security by maintaining a digital ledger of transactions, which is considered to be incorruptible. Furthermore, BC uses a changeable Public Key (PK) to record the users' identity, which provides an extra layer of privacy. Not only in cryptocurrency has the successful adoption of BC been implemented but also in multifaceted non-monetary systems such as in: distributed storage systems, proof-of-location, healthcare, decentralized voting and so forth. Recent research articles and projects/applications were surveyed to assess the implementation of BC for enhanced security, to identify associated challenges and to propose solutions for BC enabled enhanced security systems

Effective interactions between inclusions in an active bath

We study effective two- and three-body interactions between non-active colloidal inclusions in an active bath of chiral or non-chiral particles, using Brownian Dynamics simulations within a standard, two-dimensional model of disk-shaped inclusions and active particles. In a non-chiral active bath, we first corroborate previous findings on effective two-body repulsion mediated between the inclusions by elucidating the detailed non-monotonic features of the two-body force profiles, including a primary maximum, and a secondary hump at larger separations that was not previously reported. We then show that these features arise directly from the formation, and sequential overlaps, of circular layers (or 'rings') of active particles around the inclusions, as the latter are brought to small surface separations. These rings extend to radial distances of a few active-particle radii from the surface of inclusions, giving the hard-core inclusions relatively thick, soft, repulsive 'shoulders', whose multiple overlaps then enable significant (non-pairwise) three-body forces in both non-chiral and chiral active baths. The resulting three-body forces can even exceed the two-body forces in magnitude and display distinct repulsive and attractive regimes at intermediate to large self-propulsion strengths. In a chiral active bath, we show that, while active particles still tend to accumulate at the immediate vicinity of the inclusions, they exhibit strong depletion from the intervening region between the inclusions, and partial depletion from relatively thick, circular, zones further away from the inclusions. In this case, the effective, predominantly repulsive, interactions between the inclusions turn to active, chirality-induced, depletion-type attractions, acting over an extended range of separations.Comment: 12 pages, 14 figure

Description of clustering of inertial particles in turbulent flows via finite-time Lyapunov exponents

An asymptotic solution is derived for the motion of inertial particles exposed to Stokes drag in an unsteady random flow. This solution provides the finite-time Lyapunov exponents as a function of Stokes number and Lagrangian strain- and rotation-rates autocovariances. The sum of these exponents, which corresponds to a concentration-weighted divergence of particle velocity field, is considered as a measure of clustering. For inertial particles dispersed in an isotropic turbulent flow our analysis predicts maximum clustering at an intermediate Stokes number and minimal clustering at small and large Stokes numbers. Direct numerical simulations are performed for quantitative validation of our analysis, showing a reasonable agreement between the two.Comment: 11 pages and 4 figure

A modal analysis of the behavior of inertial particles in turbulence

The clustering of small heavy inertial particles subjected to Stokes drag in turbulence is known to be minimal at small and large Stokes number and substantial at $\rm St = \mathcal O(1)$. This non-monotonic trend, which has been shown computationally and experimentally, is yet to be explained analytically. In this study, we obtain an analytical expression for the Lyapunov exponents that quantitatively predicts this trend. The sum of the exponents, which is the normalized rate of change of the signed-volume of a small cloud of particles, is correctly predicted to be negative and positive at small and large Stokes numbers, respectively, asymptoting to $\tau Q$ as $\tau \to 0$ and $\tau^{-1/2} |Q|^{1/4}$ as $\tau \to \infty$, where $\tau$ is the particle relaxation time and $Q(\tau)$ is the difference between the norm of the rotation- and strain-rate tensors computed along the particle trajectory. Additionally, the trajectory crossing is predicted only in hyperbolic flows where $Q<0$ for sufficiently inertial particles with a $\tau$ that scales with $|Q|^{-1/2}$. Following the onset of crossovers, a transition from clustering to dispersion is predicted correctly. We show these behaviors are not unique to three-dimensional isotropic turbulence and can be reproduced closely by a one-dimensional mono-harmonic flow, which appears as a fundamental canonical problem in the study of particle clustering. Analysis of this one-dimensional canonical flow shows that the rate of clustering, quantified as the product of the Lyapunov exponent and particle relaxation time, is bounded by $-1/2$, behaving with extreme nonlinearity in the hyperbolic flows and always remaining positive in the elliptic flows. These findings, which are stemmed from our analysis, are corroborated by the direct numerical simulations.Comment: full article, 16 figure

Stable Stair-Climbing of a Quadruped Robot

Synthesizing a stable gait that enables a quadruped robot to climb stairs is the focus of this paper. To this end, first a stable transition from initial to desired configuration is made based on the minimum number of steps and maximum use of the leg workspace to prepare the robot for the movement. Next, swing leg and body trajectories are planned for a successful stair- climbing gait. Afterwards, a stable spinning gait is proposed to change the orientation of the body. We simulate our gait planning algorithms on a model of quadruped robot. The results show that the robot is able to climb up stairs, rotate about its yaw axis, and climb down stairs while its stability is guaranteed.Comment: Proceeding of the 2013 RSI/ISM International Conference on Robotics and Mechatronics, February 13-15, 2013, Tehran, Ira

A Lagrangian Decomposition Algorithm for Robust Green Transportation Location Problem

In this paper, a green transportation location problem is considered with uncertain demand parameter. Increasing robustness influences the number of trucks for sending goods and products, and consequently, makes the air pollution enhance. In this paper, two green approaches are introduced which demand is the main uncertain parameter in both. These approaches are addressed to provide a trade-off between using available trucks and buying new hybrid trucks for evaluating total costs besides air pollution. Due to growing complexity, a Lagrangian decomposition algorithm is applied to find a tight lower bound for each approach. In this propounded algorithm, the main model is decomposed into master and subproblems to speed up convergence with a tight gap. Finally, the suggested algorithm is compared with commercial solver regarding total cost and computational time. Due to computational results for the proposed approach, the Lagrangian decomposition algorithm is provided a close lower bound in less time against commercial solver

On the Design of Universal LDPC Codes

Low-density parity-check (LDPC) coding for a multitude of equal-capacity channels is studied. First, based on numerous observations, a conjecture is stated that when the belief propagation decoder converges on a set of equal-capacity channels, it would also converge on any convex combination of those channels. Then, it is proved that when the stability condition is satisfied for a number of channels, it is also satisfied for any channel in their convex hull. For the purpose of code design, a method is proposed which can decompose every symmetric channel with capacity C into a set of identical-capacity basis channels. We expect codes that work on the basis channels to be suitable for any channel with capacity C. Such codes are found and in comparison with existing LDPC codes that are designed for specific channels, our codes obtain considerable coding gains when used across a multitude of channels.Comment: 5 pages, 2 figures, To appear in Proc. IEEE International Symposium on Information Theory (ISIT 2008), Toronto, Canada, July 200

Novel method for planar microstrip antenna matching impedance

Because all microstrip antennas have to be matched to the standard generator impedance or load, the input impedance matching method for antenna is particularly important. In this paper a new methodology in achieving matching impedance of a planar microstrip antenna for wireless application is described. The method is based on embedding an Interdigital capacitor. The fine results obtained by using a microstrip Interdigital capacitor for matching antenna impedance led to an efficient method to improve array antenna performance. In fact, a substantial saving on the whole surfaces as well as enhancement of the gain, the directivity and the power radiated was achieved.Comment: Submitted to Journal of Telecommunications, see http://sites.google.com/site/journaloftelecommunications/volume-2-issue-2-may-201