Computing global offensive alliances in Cartesian product graphs A global offensive alliance in a graph G is a set S of vertices with the property that every vertex not belonging to S has at least one more neighbor in S than it has outside of S. The global offensive alliance number of G, γ o (G), is the minimum cardinality of a global offensive alliance in G. A set S of vertices of a graph G is a dominating set for G if every vertex not belonging to S has at least one neighbor in S. The domination number of G, γ(G), is the minimum cardinality of a dominating set of G. In this work we obtain closed formulas for the global offensive alliance number of several families of Cartesian product graphs, we also prove thatfor any graphs G and H and we show that if G has an efficient dominating set, then γ o (G H) ≥ γ(G)γ o (H). Moreover, we present a Vizing-like conjecture for the global offensive alliance number and we prove it for several families of graphs. Alliances in graphs were described first by Kristiansen et al in [11] , where alliances were classified into defensive, offensive or powerful. After this seminal paper, the issue has been studied intensively. Remarkable examples are the articles [13, 14] , where alliances were generalized to k-alliances, and [4] , where the authors presented the first results on offensive alliances. One of the main motivations of this study is based on the NP-completeness of computing minimum cardinality of (defensive, offensive, powerful) alliances in graphs.. On the other hand, several graphs may be constructible from smaller and simpler components by basic operations like unions, joins, compositions, or multiplications with respect to various products, where properties of the constituents determine the properties of the composite graph. It is therefore desirable to reduce the problem of computing the graph parameters (alliance numbers, for instance) of product graphs, to the problem of computing some parameters of the factor graphs.. Nowadays the study of the behavior of several graph parameters in product graphs has become an interesting topic of research [8, 9] . For instance, we emphasize the Shannon capacity of a graph [15] , which is a certain limiting value involving the vertex independence number of strong product powers of a graph, and Hedetniemi's coloring conjecture for the categorical product [7, 9] , which states that the chromatic number of any categorial product graph is equal to the minimum value between the chromatic numbers of its factors. Also, one of the oldest open problems on domination in graphs is related to Cartesian product graphs. The problem was presented first by Vizing in 1963 [18, 19] . Vizing's conjecture states that the domination number of any Cartesian product graph is greater than or equal to the product of the domination numbers of its factors.. Cartesian product graphs have been much studied in graph theory. Interest in Cartesian product graphs has been increased by the advent of massively parallel computers whose structure is that of the Cartesian product graphs [6, 10] . This not only provides potential applications for the existing theory, but also suggests some new aspects of these graphs that deserve study.. Studies on defensive alliances in graphs product were initiated in [3] , for the case of the torus graph C s C t , and studied further in [2, 16, 20] . Nevertheless the great part of the results in these works are upper bounds on the alliance numbers of Cartesian product graphs. In the present work we obtain closed formulas for the global offensive alliance number of several families of Cartesian product graphs, we obtain new formulas relating the global offensive alliance number of Cartesian product graphs with the domination number and the global offensive alliance number of its factors, we present a Vizing-like conjecture for the global offensive alliance number and we prove it for several families of graphs.. We begin by stating the terminology used. Throughout this article, G = (V, E) denotes a simple graph of order |V | = n. We denote two adjacent vertices u and v by u ∼ v. Given a vertex v ∈ V, the set N(v) = {u ∈ V : u ∼ v} is the open neighborhood of v, and the set. For a nonempty set S ⊆ V , and a vertex v ∈ V , N S (v) denotes the set of neighbors v has in S, i.e., N S (v) = S ∩ N(v). The degree of v in S will be denoted by δ S (v) = |N S (v)|. The complement of a set S in V is denoted by S.. A set S ⊆ V is a dominating set in G if for every vertex v ∈ S, δ S (v) > 0 (every vertex in S is adjacent to at least one vertex in S). The domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set in G [5] . An efficient dominating set is a dominating set. Examples of graphs having an efficient dominating set are the path graphs P n , the cycle graphs C 3k and the cube graph Q 3 .. A nonempty set S ⊆ V is a global offensive alliance in G if. Note that every global offensive alliance is a dominating set. The global offensive alliance number of G, denoted by γ o (G), is defined as the minimum cardinality of a global offensive alliance in G.. A global offensive alliance of cardinality. We recall that given two graphs G and H with set of vertices V 1 = {v 1 , v 2 , ..., v n 1 } and V 2 = {u 1 , u 2 , ..., u n 2 }, respectively, the Cartesian product of G and H is the graph G H = (V, E), where V = V 1 × V 2 and two vertices (v i , u j ) and (v k , u l ) are adjacent in G H if and only if. Given two graphs G = (V 1 , E 1 ), H = (V 2 , E 2 ) and a set X ⊂ V 1 × V 2 of vertices of G H, the projections of X over V 1 and V 2 are denoted by P G (X) and P H (X), respectively. Moreover, given a set C ⊂ V 1 of vertices of G and a vertex Now we establish a Vizing-like conjecture for the global offensive alliance number.. Conjecture 1 (Vizing-like conjecture for the global offensive alliances). For any graphs G and. Below we will prove the conjecture for several families of graphs. Theorem 2. For any graphs G and H,. Moreover, if G has an efficient dominating set, then. Proof. Let V 1 and V 2 be the vertex sets of the graphs G and H, respectively. Let S = {u 1 , ..., u γ(G) } be a dominating set for G.. If W i is not a global offensive alliance in H, then there exists at least a vertex v ∈ W i such that. So, every vertex belonging to A v i has at least one neighbor in. Now, for every i ∈ {1, ..., γ(G)}, let Q i ⊆ W i be the set of vertices of H satisfying the inequality (4). Since W i ∪ Q i is a global offensive alliance in H,. Hence, we have that. and, as a consequence, we have. On the other hand, notice that for each v ∈ V 2 , q v is the number of G(A i , v)-cells for which v satisfies (4), as well as for each i ∈ {1, ..., γ(G)}, |Q i | is the number of vertices of H satisfying inequality (4). Thus,. Now, if q v > |D v |, then we have. which is a contradiction. So, we have q v ≤ |D v | and we obtain. Thus, by (6), (7) and (8) we deduce. Analogously, we obtain that. Therefore, the first result follows. Now, if S = {u 1 , ..., u γ(G) } is an efficient dominating set for G, then for every i ∈ {1, ..., γ(G)}, W i is a global offensive alliance in H. That is, if we suppose that W i is not a global offensive alliance in H, then there exists at least one vertex v ∈ W i which satisfies (4). Thus, every vertex belonging to A v i has at least one neighbor in D v ∩ Π j , for some j = i, which is a contradiction because (u i , v) has no neighbors outside of Π i . As a consequence, |Q i | = 0. So, (3) and (5) . Notice that for the case of star graphs, S 1,n , the central vertex forms an efficient dominating set of minimum cardinality, and it is also a global offensive alliance, then the above theorem leads to the following Vizing-like result for the global offensive alliance number.. Corollary 3. Let S 1,n be a star graph. For any graph H,. As the following remark shows there is no other family of connected graphs containing an efficient dominating set of minimum cardinality which is also a global offensive alliance.

O B I I O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B I I O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O B I I O O O O O O O O O B I I O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O B I I O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion In this paper, we study the time-space fractional order (fractional for simplicity) nonlinear subdiffusion and superdiffusion equations, which can relate the matter flux vector to concentration gradient in the general sense, describing, for example, the phenomena of anomalous diffusion, fractional Brownian motion, and so on. The semi-discrete and fully discrete numerical approximations are both analyzed, where the Galerkin finite element method for the space Riemann-Liouville fractional derivative with order 1 + β ∈ [1, 2] and the finite difference scheme for the time Caputo derivative with order α ∈ (0, 1) (for subdiffusion) and (1, 2) (for superdiffusion) are analyzed, respectively. Results on the existence and uniqueness of the weak solutions, the numerical stability, and the error estimates are presented. Numerical examples are included to confirm the theoretical analysis. During our simulations, an interesting diffusion phenomenon of particles is observed, that is, on average, the diffusion velocity for 0 < α < 1 is slower than that for α = 1, but the diffusion velocity for 1 < α < 2 is faster than that for α = 1. For the spatial diffusion, we have a similar observation. Fractional calculus is considered as the generalization of the classical (or integer order) calculus with a history of at least three hundred years. It can be dated back to Leibniz's letter to L'Hospital, in which the meaning of the one-half order derivative was first discussed [1] . Although fractional calculus has such a long history, its research still stay in the realm of theory, due to the lack of proper mathematical analysis methods and real applications. Until the past several decades, many authors pointed out that fractional derivative and fractional differential equation do have many applications in various fields, such as in viscoelastic mechanics, power-law phenomenon in fluid and complex network, allometric scaling laws in biology and ecology, colored noise, electrode-electrolyte polarization, dielectric polarization, boundary layer effects in ducts, electromagnetic waves, quantitative finance, quantum evolution of complex systems, fractional kinetics, and so on [1] .. At present, there have been studies on approximate methods for fractional differential equations. For example, Dithelm et al., have presented some results on numerical fractional ordinary differential equations [2] [3] [4] [5] . Adomian decomposition method, homotopy perturbation method and differential transform method are other choices when seeking approximate solutions to the fractional ordinary/partial differential equations; for example, see [6] [7] [8] [9] [10] . Recently, a typical fractional difference method and its variants have been utilized to approximate the fractional operator that specially appeared in fractional partial differential equations [11] [12] [13] [14] [15] [16] [17] [18] . On the other hand, the finite element method has also been used to find the variational solution to the space fractional partial differential equations [19] [20] [21] [22] [23] [24] [25] [26] , where the temporal derivative is the classical derivative and the spatial derivative is of fractional order. These are pioneering works on finite element analysis for fractional partial differential equations. However, there are few works to solve the time-space fractional partial differential equation numerically by using the fractional finite-difference-finite-element mixed method, especially for nonlinear problems. The main difficulties are whether the temporal and spatial derivatives are both nonlocal operators and whether or not stable for the iteration with the nonlinear term in the progress of the fractional finite-difference-finiteelement mixed method. In this paper, we use the fractional finite-difference-finite-element mixed method to numerically solve the time-space fractional partial differential equation, where both temporal and spatial derivatives are of fractional orders.. The fractional order diffusion equation as a typical kind of fractional partial differential equation [27] , is a generalization of the classical diffusion equation which can be used to better characterize anomalous diffusion phenomena. Here, we briefly introduce the formulation of the time fractional diffusion equation.. The classical diffusion theory is based on the Fick law J = −κ grad u relating the matter flux vector J to the concentration gradient, where κ is the diffusion conductivity. In combination with the balance equation for mass, the Fick law leads to the classical diffusion equation. where λ is the diffusivity coefficient. Eq. (1) makes an integral part of the classical theory of the diffusive stresses.. For more general cases, the classical Fick law was generalized to characterize the time-nonlocal or/and space-nonlocal dependence between the heat flux vector and the temperature gradient. This generalization resulted in an integrodifferential heat conduction equation [28] , which was then applied to the thermoelasticity theory [29] . Viewing from another angle, let us take a closer look at the fractional Fick law proposed in [30, 31] :. (2). Green and Naghdi [32] used the following simplest form of the kernel function K (t − τ ) = 1, which actually corresponds to the wave equation for the temperature and thermoelasticity without any energy dissipation.. Chandrasekharaiah [33] pointed out that the constitutive equation for the heat flux could also be rewritten in a nonlocal form with the ''short-tail'' exponential time-nonlocal kernel. in which ξ is a non-negative constant, and τ is a variable defined in (0, t).. The time-nonlocal dependence between the flux and the associate gradients with a ''long-tail'' power-law-like kernel can be rewritten in terms of the fractional integral and derivatives, which yields the following time fractional diffusion equation:. The kernel of the constitutive Eq. (2) of (3) is given as. Therefore, one can see that the fractional constitutive relations are read as. and. where I 1−α and I α−1 are fractional integrals [34] defined in the next section.. (3) is usually called ''anomalous diffusion'' or ''fractional diffusion'' [35] . The limiting case α = 0 corresponds to the Helmholtz equation which is associated with the localized diffusion. The subdiffusion regime is characterized by 0 < α < 1.. The ordinary diffusion corresponds to α = 1 and the superdiffusion regime by 1 < α < 2. The limiting case α = 2 corresponds to the wave equation which is called ''ballistic diffusion''.. Here, we mainly focus on constructing and analyzing a kind of efficient numerical scheme for approximately solving time-space fractional nonlinear subdiffusion and superdiffusion equations, respectively. The considered equations read as follows: for 0 < α < 1,. where Ω = (a, b); time T > 0; 0 < β < 1, and 0 < α < 1 is for the subdiffusion case, 1 < α < 2 is for the superdiffusion case. Here the spatial fractional differential operator RL D 1+β is also commonly referred to as anomalous diffusion, where the underlying stochastic process is a Lévy (1 + β)/2-stable flight [36, 37] . And when β = 1, it models a Brownian diffusion process. The nonlinear term B(u)u x can be considered as the general form of uu x . And when B(u) is a linear integral operator of u, it models particle interactions [38] ; when B(u) equals u, it models the Burgers turbulent flow [37, 39] . f is a source term, λ 1 is the diffusivity coefficient, and λ 1 , λ 2 satisfy some suitable conditions for the error estimates.. The rest of this paper is constructed as follows. In Section 2 the preliminary knowledge of fractional derivatives and fractional derivative spaces are introduced. The approximate system of these equations, unconditional stability analysis, existence and uniqueness of the weak solution, and the error estimates of the semi-discrete schemes of these considered diffusion equations are discussed in Section 3. The error estimates of the full discrete schemes for (4) and (5) are studied in Section 4. In Section 5, numerical examples are presented to demonstrate the efficiency of the theoretical results derived in Section 4.

O O O O O O O O B O B O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A Fully Consistent Hidden Semi-Markov Model-Based Speech Recognition System In a hidden Markov model (HMM), state duration probabilities decrease exponentially with time, which fails to adequately represent the temporal structure of speech. One of the solutions to this problem is integrating state duration probability distributions explicitly into the HMM. This form is known as a hidden semi-Markov model (HSMM). However, though a number of attempts to use HSMMs in speech recognition systems have been proposed, they are not consistent because various approximations were used in both training and decoding. By avoiding these approximations using a generalized forward-backward algorithm, a context-dependent duration modeling technique and weighted finite-state transducers (WFSTs), we construct a fully consistent HSMM-based speech recognition system. In a speaker-dependent continuous speech recognition experiment, our system achieved about 9.1% relative error reduction over the corresponding HMM-based system. key words: speech recognition, hidden Markov model, hidden semiMarkov model, weighted finite-state transducer Hidden Markov models (HMMs) ( Fig. 1 (a) ) have formed the basis of many speech recognition systems since the 1970s. The advantages of using HMMs are: i) They can represent speech as probability distributions. ii) They are robust to temporal structure variations. iii) They provide efficient algorithms for estimating their model parameters. However, a number of limitations of HMMs for modeling speech have been reported [1] . One of their major limitations is in duration modeling. In HMMs, state duration probabilities are implicitly modeled by state transition probabilities; state duration probabilities decrease exponentially with time. Geometric distribution calculated by the state transition probabilities of HMMs would be inappropriate state duration probability distribution representation of the temporal structure of speech.. One of the solutions to this problem is to integrate state duration probability distributions explicitly into the HMM. This model is known as a hidden semi-Markov model (HSMM) [2] - [4] , and is illustrated in Fig. 1 (b) . Unlike HMMs, HSMMs have state duration probability distributions. Figure 2 shows the state duration probability distributions of an HMM and an HSMM. Geometric distribution in an HMM would be inappropriate representation of the temporal structure of speech. Although the gamma, Poisson and log Gaussian distributions have been applied to state duration modeling in HMM-based speech recognition, in this paper, we assume that each state duration probability distribution is represented by a Gaussian distribution because Although discrete probability distributions can represent any distributions, the lack of training data could be an issue. In [4] , a smoothing technique is used to prevent over training caused by the lack of training data. On the other hand, the use of continuous probability distributions may remedy such a over training problem. However, it is nessessary to choose an appropriate continuous probability distribution type previously. In this paper, we use continuous probability distributions because it can avoid additional processing such as smoothing.. A variety of attempts to include explicit duration models in speech recognition systems have been reported [6] - [8] . However, they are not fully consistent because various approximations are used in both training and decoding: 1) State duration probability distributions were estimated from statistical variables calculated by the forward-backward algorithm of the HMM, not of the HSMM [9] . 2) State duration probability estimation utilizes a contextindependent model or context-dependent state tying structure of state output probability distributions [7] . 3) State duration models were not applied directly in the decoding process. Instead, the N-best hypotheses generated by the HMMs were rescored [8] .. We propose a fully consistent HSMM-based speech recognition system to overcome the above approximations. For approximation 1), we simultaneously estimate both state output and duration probability distributions based on the HSMM statistics calculated by the generalized forwardbackward algorithm [2] . For approximation 2), state output and duration probability distributions are independently clustered using different phonetic decision trees [9] by a decision-tree-based state clustering technique [10] . For approximation 3), we design an HSMM-native speech decoder based on weighted finite-state transducers (WFSTs) to apply HSMM directly to the input speech. The rest of this paper is organized as follows. Section 2 describes training algorithms for the HSMM, contextclustering for state duration probability distributions, and a speech decoder for the HSMM using WFSTs. Results of our speech recognition experiment are presented in Sect. 3. Finally, concluding remarks and future plans are presented in Sect. 4. In this paper, we constructed a fully consistent HSMMbased speech recognition system and evaluated its performance while avoiding approximations in training, contextclustering, and decoding. The result showed an obvious improvement in phoneme recognition accuracy. Future works include evaluation on speaker independent speech recognition tasks with multi-mixture state output probability distributions and investigation other kind of distributions such as gamma, Poisson and log Gaussian distributions for state duration modeling. To evaluate the performance of the proposed HSMM-based speech recognition system, speaker-dependent continuous phoneme recognition experiments were conducted on the ATR Japanese speech database B set (phonetically-balanced sentences). In all experiments, the speech data was down-sampled from 20 kHz to 16 kHz, windowed at a 5 ms frame rate using a 25 ms Blackman window, and parameterized into 25 mel-cepstral coefficients with a mel-cepstral analysis technique. Static coefficients including the zero-th coefficients and their first and second derivatives were used as feature parameters. 3-state left-to-right structures were used and 118 questions about left and right phonetic contexts were prepared for decision tree construction. Each state output distribution was modeled by a Gaussian distribution with a diagonal covariance matrix. A WFST for decoding was constructed from WFSTs representing chained triphone HMMs and a phoneme network (phoneme-pair grammar) based on the WFST composition and determinization. Maximum and minimum state duration of each HSMM state was limited to ξ i ± σ 2 i × 2.

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Efficient codebooks for fast and accurate low resource ASR systems Nowadays, speech interfaces have become widely employed in mobile devices, thus recognition speed and power consumption are becoming new metrics of Automatic Speech Recognition (ASR) performance.For ASR systems using continuous Hidden Markov Models (HMMs), the computation of the state likelihood is one of the most time consuming parts. Hence, we propose in this paper novel multi-level Gaussian selection techniques to reduce the cost of state likelihood computation. The proposed algorithms are evaluated within the framework of a large vocabulary continuous speech recognition task. The proliferation of mobile devices in daily life has created a great demand for efficient and simple interfaces. In particular, speech recognition being a key element of the conversational interface, there is a significant requirement for low-resource and accurate automatic speech recognition systems. Recent mobile devices (GP S 1 , GSM 2 , P DA 3 , . . .) offer a large set of functionalities but their resources are too limited for accurate continuous speech recognition engines. Indeed, state-of-the-art continuous speech recognition systems use hidden Markov models (HMM) with many tens of thousands of Gaussian distributions to achieve improved recognition. As the performance and the speed of such systems are closely related to the number of HMM Gaussians, reducing the number of Gaussians without decreasing the system performance is of major interest. According to previous studies (Bocchieri (1993) ; Mak et al. (2001) ; ; Gales et al. (1996) ) only a few Gaussians dominate the state likelihood computation. Hence, different techniques were developed to select them. These methods can be divided into two categories : -State based methods : These methods are often applied to systems having acoustic models with a high number of Gaussians per state such as semicontinuous or monophone based systems (Jurgen et al. (1995) ; Woszczyna (1998)).. -Model based methods : Gaussian selection is applied to the distributions belonging to all the states of the acoustic models. This is for example the case of triphone based recognition systems (Bocchieri (1993) ; Mak et al. (2001) ; Gales et al. (1999) ). In this paper, we propose several Gaussian selection techniques that reduce the cost of likelihood computation either in the state or in the model level. The proposed algorithms are evaluated within the framework of a large vocabulary continuous speech recognition task. For all the experiments we use : -parameter vectors with 12 MFCC coefficients, energy, and their first and second derivatives. -40 context independent models trained with 82 hours of the Ester train database ( Galliano et al. (2005) ). -a dictionary containing 118000 words. -a language model formed by 4 millions of bigrams and trigrams.. -an hour of Broadcast News extracted from the Ester test data set.. In order to compare the different systems, several reference systems 32, 64, 80, 128, 180, 256, 220 and 256 Gaussians per state are produced.. Considering the system with 256 Gaussians per state, the clustering algorithm is applied to each Gaussian mixture. Depending on the experiments, either the likelihood loss or the weighted cross entropy based metric is used. The previously tree cutting criteria (fixed, weight based or distance based) are also evaluated.. The Sirocco-Htk large vocabulary speech recognition system was used to compare the performance of the different schemes. This reference system makes use of 40 context independent acoustic models with 3 states each and 512 Gaussians per state. The parameters of these models are estimated on the Ester train database. The same parametrisation and textuel resources (dictionnary and language model) of the previous experiments are employed. Tests are conducted using an hour of Broadcast News extracted from the Ester test data set. The performance is addressed in terms of Word Error Rate (W ER) and percentage of likelihood computation C. The latter is defined as :. For the reference system : W ER = 35.5% and C= 100%. We create a large vocabulary speech recognition system based on Sphinx training/test tools. The acoustic models are cross-word CD with 6108 tied states and 32 Gaussians per state. The parameters of these models are estimated on the Ester train database. Tests are conducted using an hour of Broadcast News extracted from Ester test set. For this reference system all the Gaussian . In all of the experiments, the performance of the proposed methods are compared to those of the initial system which are (W ER, C) = (28.7%, 100%) and to those of an existing sub-vector quantization method (described in Ravishankar et al. (1997) and 3.2). Let us keep in mind that the existing sub-vector quantization (SVQ) method is based on a simultaneous mean and variance vectors clustering to construct the codebook. The K-means algorithm is employed to perform the clustering process by stream.. The parameter vectors are composed of (12mfcc + energy) and their first and second derivatives. Three subdivisions of the parameters vector are tested : -Only one stream of dimension 39 is considered.. -Three streams : (12mfcc+energy) + ∆(12mfcc+energy)+ ∆ ∆(12mfcc+energy).. -Four streams : (energy,∆energy,∆ ∆energy) + (12 mfcc) + (12 ∆(mfcc)) + (12 ∆ ∆ (mfcc)).

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On a Conjecture of Butler and Graham Abstract. Motivated by a hat guessing problem proposed by Iwasawa [6], Butler and Graham [2] made the following conjecture on the existence of certain way of marking the coordinate lines in [k] n : there exists a way to mark one point on each coordinate line in [k] n , so that every point in [k] n is marked exactly a or b times as long as the parameters (a, b, n, k) satisfies that there are non-negative integers s and t such that s + t = k n and as + bt = nk n−1 . In this paper we prove this conjecture for any prime number k. Moreover, we prove the conjecture for the case when a = 0 for general k. In [2] Butler and Graham considered the problem of the existence of certain way of marking coordinate lines in [k] n . A coordinate line in [k] n is the set of k points in which all but one coordinate are fixed and the unfixed coordinate varies over all possibilities. Marking a line 1 means designating a point on that line. They conjectured that Conjecture 1 (Butler, Graham [2] ). There is a marking of the lines in [k] n so that on each line exactly one point is marked and each point is marked either a or b times if and only if there are nonnegative integers s and t satisfying the linear equations s + t = k n and as + bt = nk n−1 .. The "only if" part of the conjecture is straightforward, leaving to be crucial the construction of a marking of lines in [k] n with the desired properties. Buhler, Butler, Graham, and Tressler [1] proved the conjecture for k = 2. Butler and Graham [2] proved the conjecture when n ≤ 5. The main contributions of this paper are i) we prove all the cases when k is an odd prime; ii) we prove the case when a = 0 (without any assumption on k). Theorem 1. For any prime k and 0 ≤ a < b ≤ n, there exists a marking of lines in [k] n so that each point is marked either a times or b times if and only if there are nonnegative integers s and t so that s + t = k n and as + bt = nk n−1 . The motivation of investigating this marking line problem is to reformulate and solve a hat guessing question proposed by Iwasawa [6] . In that game there are several players sitting around a table, each of which is assigned a hat with one of k colors. Each player can see all the colors of others' hat but his/her own. The players try to coordinate a strategy before the game starts, and guess the colors of their own hats simultaneously and independently after the hats are placed on their heads. Their goal is to design a strategy that guarantees exactly either a or b correct guesses. For example, one special case is that either everybody guesses the color correctly or nobody guesses correctly, i.e. a = 0 and b = n.. Several variations of hat guessing game have been considered in the literature. Ebert [4] considered the model that players are allowed to answer "unknown". He showed that in this model there is a perfect strategy for players when n is of the form 2 m − 1. Lenstra and Seroussi [7] studied the case that n is not of such form. Butler, Hajiaghayi, Kleinberg and Leighton [3] considered the worst case that each player can see only part of the others' hats with respect to a sight graph. Feige [5] investigated the average case with a sight graph. Peterson and Stinson [9] investigated the case that each player can see hats in front of him and they guess one by one. Recently Ma, Sun and Yu [8] proposed a variation which allow to answer "unknown" and require at least k correct guesses for winning condition.

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Complex network analysis of teaching practices The application of functional analysis to infer networks in large datasets is potentially helpful to experimenters in various fields. In this paper, we develop a technique to construct networks of statistically significant transitions between variable pairs from a high-dimensional and multiscale dataset of teaching practices observed in Grade 5 and Grade 9 Mathematics classes obtained by the National Institute of Education in Singapore. From the Minimum Spanning Trees (MST) and Planar Maximally Filtered Graphs (PMFG) of the transition networks, we establish that teaching knowledge as truth and teacher-dominated talking serve as hubs for teaching practices in Singapore. These practices reflect a transmissionist model of teaching and learning. We also identify complex teacher-student-teacher-student interaction sequences of teaching practices that are over-represented in the data. In recent years, it has become popular to use networks to visualize and analyze the hierarchical structure within large datasets usually obtained from complex dynamical systems. Networks are especially useful for representing the extent and magnitude of interactions among the many components in complex systems. In such visualizations we show the important pair-wise relationships between variables. These relations may be directed transitions, causation, or activations, or undirected when we examine similarities or cooccurrences. Examples of such usage can be found in finance [ Broadly speaking, there are two approaches to generate networks from data. First, some datasets such as air traffic data [, ] and World Wide Web connectivity data [] contain explicit information about the network structure. In other cases, we often have to infer the interactions among various components or variables by performing functional analysis on the data [] . Examples of this approach include the estimation of an undirected network of stocks based on the Pearson correlations of their daily returns [] , as well as the mapping of a directed functional network of brain regions from fMRI data using Granger causality as a directed measure [] . In both examples, the resulting network provides us deeper insights into the system it represents -clustering of stocks in similar industries in the stock market, and a functional hierarchy in the brain. We believe there is great potential in applying functional analysis to infer networks in large datasets under-utilized by experimenters in various fields.. In , the National Institute of Education (NIE) conducted a large-scale study of teaching practices in Singapore classrooms drawing on a representative national sample of schools. The study had the goal of investigating how teachers teach in Singapore, why they teach the way they do, and the effects of their teaching on student learning. The overall focus of the study was to map and model the logic of pedagogical instruction, the intellectual quality of knowledge work in classrooms, and the impact of instructional practice on student achievement while controlling for student and family characteristics. A total of  English and Mathematics lessons from Grades  and  classes were video-recorded and manually annotated for the presence of some  instructional indicators over the course of each lesson. These indicators focus on a broad range of organisational, instructional and interactional practices in the classroom, the knowledge and cognitive focus of teaching, the types of classroom talk and their knowledge content, how students respond, and the quality of the disciplinary knowledge of the subject-domain (English and Mathematics). The resulting data set is extremely high-dimensional (one dimension for each indicator) with greatly varying activities -some indicators are present up to % of the time while others appear only .% of the time. Using commercial software such as SPSS to analyse the large dataset proved to be computationally intensive, as it explores the combinatorially large space of variables to find statistically significant transitions. In particular, such commercial software was unable to perform complex temporal analysis of classroom pedagogy, which remains a major methodological challenge for educational research [] .. In this paper, we describe a complex-network approach to the functional data collected in the NIE study. Our approach is both empirical and normative in that we focus, in the first instance, on establishing statistically weighted relationships between instructional practice and, in the second, because we assume that not all instructional practices are equal and that from a normative perspective, there is a particular logic to instruction that teachers need to attend to if they are to optimise the intellectual quality of teaching and learning in the classroom. As such, teaching and learning activities do not follow a random sequence but follow a purposeful and deliberate sequence that teachers implement from their lesson plans, and simultaneously adapt in response to their monitoring of students' work. Effective teaching sequences, most often called learning progressions [, ] , should lead to successful learning outcomes while ineffective sequences, perhaps due to structural weaknesses in the transition sequence, may not only have the opposite effect, but may disengage students from active learning in the long term. It follows that the resulting network of activity transitions should contain meaningful structures that correspond to actual teaching practices. The goals of this work are to (i) construct complex networks that typify activity transitions in the classroom, (ii) identify prevalent pairwise transitions and important practices, and (iii) extract sequences of transitions that frequently occur in the classroom. Since the focus of this paper is on the complex network methodology, we illustrate what the method is capable of discovering only for the data of Grade  and Grade  Mathematics. In this work, we presented a technique of constructing transition networks from highdimensional and multiscale data to allow for highly simplified analysis of interdependence and correlation. We applied the technique on teaching practices data obtained by the National Institute of Education in Singapore. We were able to identify and verify the dominant transitions between instructional practices and also pick out motifs of sequences of several practices and activities that are common in many lessons. We established that teaching practices of both Grade  and Grade  Mathematics lessons are organized around Knowledge as Truth and Instructional Activity (IA): Teacher-Dominated Talk hubs which exemplify the transmissionist model of teaching. In addition, a Doing Mathematics Activity hub is present in the Grade  transition network. This suggests that teaching practices at this level have incorporated exploratory elements into the pedagogy, a positive sign towards the goal of st century learning. In contrast, in Grade , disciplinary understanding of mathematics has been replaced by a focus on procedural knowledge, which nevertheless accounts for Singapore's strong performance in international benchmarks. The motifs we extracted from the network highlight cycles of complex teacher-student-teacher-student sequences with great similarity between Grade  and Grade . In future work, this methodology will be employed and modified for more in-depth study of the transitions data. This includes using knowledge of the topology of the transition network to formulate strategies that can direct the flow of teaching activities towards a desired outcome. The transmissionist model of teaching views teaching as a straightforward and unproblematic process of transferring information (knowledge) from the mind of the teacher to the mind of the student [-]. In this model, knowledge is a collection of facts about the world and procedures to solve problems; knowledge is objective, fixed, and external and learning is something akin to accumulating goods -but goods in the form of propositional knowledge, whether factual or conceptual; the goal of schooling is to transmit these facts and procedures from the teacher to the student; teachers know these facts and procedures, and their job is to transmit them to students; learning is an intra-subjective process that goes on inside students' minds as knowledge is transferred or transmitted from the teacher to the student; simpler facts and procedures should be learned first, followed by progressively more complex facts and procedures; the way to determine the success of schooling is to test students to see how many of these facts and procedures they have acquired through well-established assessment procedures. In the transition networks of both Grade  and Grade  Mathematics, Knowledge as Truth and Instructional Activity (IA): Teacher-Dominated Talk are seen as dominant hubs. Knowledge as Truth means that the knowledge that is presented in the lesson is viewed as non-contestable, non-negotiable and objectively valid. As an Instructional Activity (IA), Teacher-Dominated Talk means simply that the teacher does most of the talking in the classroom. These practices operate most effectively in the transmissionist model of teaching, since knowledge is unproblematic and non-contestable, and teachers need only to ensure that they cover the curriculum. Not the least of the opportunity costs for Teacher-Dominated Talk however, is that when students are not given the opportunity to engage in extended and productive discussions during lessons, misconceptions often go undetected, existing conceptual schemas can go unchallenged, students might become disengaged with the learning process, and over time, because they have less agency in the class to speak up, and lose motivation to become good students.. However, as Ball [] points out, mathematics instruction ought to involve a lot more than the transmission of facts and procedures. Above all, mathematics instruction should provide students with rich opportunities to engage in authentic, domain-specific mathematical practices. "Mathematics practices, " she goes on, "focus on the mathematical knowhow, beyond content knowledge, that constitutes expertise in learning and using mathematics [such as] justifying claims, using symbolic notation efficiently, defining terms precisely, and making generalisations are examples of mathematical practices. " Thus, the transmissionist model is insufficient as a teaching and learning model for mathematics. In Grade , the strong presence of Doing Mathematics Activity is an important practice that we hoped to see in mathematics classrooms. Doing Mathematics Activity is inherently disciplinary in nature, and students need to acquire the mathematics-specific disciplinary skills, knowledge and disposition to understand mathematics deeply, debate, and discuss mathematical procedures and concepts, and appreciate the importance and relevance of mathematics. It captures the idea that mathematics learning does not consist purely of learning algorithmic procedures (captured in Procedural Talk and Procedural Knowledge Focus) and content (captured in Factual Talk and Factual Knowledge Focus), but requires students to perform discipline-specific knowledge practices as outlined by Ball. Similarly, Stein and Lane [] characterize 'doing mathematics' as "The use of complex, non-algorithmic thinking to solve a task in which there is not a predictable, well-rehearsed approach or pathway explicitly suggested by the task, task instructions, or a worked out example. 'Doing mathematics' processes are often likened to the processes in which mathematicians engage when solving problems. " These activities are the most demanding of all activities, since they require the students to draw upon a range of mathematical knowledge and procedures/skills to complete some work that has no wellrehearsed way of completing the task. Instead, students are likely to be asked to design, test and justify a new procedure to fit a new kind of problem.. The disappearance of Doing Mathematics Activity as a hub in Grade  suggests that at this level, the focus on what we consider to be a highly-valued disciplinary understand-ing of mathematics has been replaced by a focus on procedural knowledge (Procedural Knowledge Focus) as an effective and efficient means to present such knowledge, such as efficient problem solving methods. Yet, we believe that it is precisely because of this pedagogical approach focused on a highly proceduralised and efficient form of learning abstract mathematics that has made Singapore one of the top performing countries in international assessments that measure students' ability in mathematics. Our observation confirms the PISA  findings on Singapore Mathematics [] which shows that there is a strong relationship between the teaching of Formal Mathematics (as opposed to Applied Mathematics or Word Problems) and student mathematical performance in the PISA tests.. Finally, teaching and learning must be understood as a process. Christie [] talks of the prototypical classroom lesson having a beginning, middle and end pattern. The beginning comprises some form of teacher direction (Curriculum Initiation), followed by the teacher and students' sharing of direction (Curriculum Collaboration/Negotiation) and ending with students' independent activity (Curriculum Closure). In the motifs we identified for Grade  and Grade , the didactic of teacher-student-teacher as well as teacherstudent-teacher-student sequences highlights the complex interaction cycles that occur in mathematics classrooms. Almost all lessons begin, as described above, with a teacherled initiation (Knowledge as Truth, Instructional Activity (IA): Teacher-Dominated Talk, Procedural Talk, Procedural Knowledge Focus). Often, these motifs lead immediately to individual practice work (IA: Individual Seatwork), seemingly replacing the middle phase of Christie's pattern with student's independent activity. Some motifs then terminate with feedback from the teacher in the form of lengthy talk (Teacher Comment/Remark, IA: Teacher-Dominated Talk, Teacher Exposition) following student's practice work. These motifs suggest a teacher-dominated closure in the teaching pattern. However, in other motifs, we discovered a fourth movement which is a return to more practice work (CD: To Practice What Was Learnt) coupled with teacher's monitoring of such practice work (IS:

O O O O B I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


An analytical model to estimate path duration in MANETs Mobile Ad Hoc Networks (MANETs) are strongly impacted by the mobility of the ad hoc nodes. Mobility models help engineers to abstract the changes of position in network users. Different mobility models have already been proposed to characterize the behaviors of mobile devices that are greatly correlated to the conditions of the possible scenarios where networks can be deployed. In the context of mobile ad hoc networks, the Random WayPoint model has received significant attention and it has become one of the most employed mobility patterns. However, a formal description of the time during which a generic N-hop path is valid has not been proposed yet for this model. In this paper, the authors provide an analytical model for link duration as well as an analytical study of path duration in multi-hop wireless networks. The predicted results are compared with measured data and good agreement is reported. Among the multiple scenarios where ubiquitous computing may be present, the Mobile Ad Hoc Networks are receiving significant attention. Mobile Ad hoc Networks (MANET) allow wireless devices to intercommunicate without the use of any pre-existing infrastructure. However, the absence of any centralized entity transfers the routing procedures to the nomadic ad hoc terminals as direct communication is only possible between neighbouring nodes. Thus, distant nodes require the collaboration of intermediate terminals that should forward the source packets to the final destination through multiple links or hops.. The method for selecting the intermediate nodes as well as the response to any link break in the path are the main characteristics that differentiate the existing ad hoc routing protocols. These protocols can be roughly divided into three categories: proactive, reactive and hybrid. In proactive or table driven protocols nodes send messages periodically in order to update the topological information about the network, so that a route is immediately available as soon as a new connection is required. On the contrary, in reactive or on-demand protocols, nodes generate route queries in order to discover a needed route to a particular node. When a response is back-propagated by the network to the origin node, nodes cache the configured routes for a certain period of time. Finally, hybrid protocols combine both strategies as they follow proactive scheme for a restricted closer zone around the sending node while it employs the reactive technique to communicate with nodes out of this defined proactive area.. In general terms, routing protocols requires the configuration of some parameters which greatly influences on the network performance. Usually, the specifications recommend some default values for these parameters that do not follow any experimentally evaluated criterion, so that the values differ in various specifications. Specifically, in the case of reactive protocols, the time that a route is considered to be valid varies depending on the particular algorithm that is being utilized. For instance, in AODV with enabled link layer detection, stored routes become invalid after 10 seconds (the Active Route Timeout) without being utilized [1] . Meanwhile DSR and DYMO contemplate another time constant equal to 3 and 300 seconds (RouteCacheTimeout and RouteTimeout respectively) to suppress paths from the node caches [2] [3] . These time-out periods are applied for all the nodes without taking into consideration the mobility conditions or the route length (estimated by the number of hops). In order to select an appropriate value for this parameter as well as to know its main impacting factors, an analysis of the time that a path is valid should be performed. In this sense, the aim of this work is to present a statistical characterization of the duration of routes in mobile multi-hop wireless networks. This characterization can be of great help for the routing protocols to select stable routes or proper values for the route timeouts. Additionally, the prior assumption of the disruption of a route may be useful to initiate the route discovery procedures earlier, diminishing the delay associated to reactive protocols [4] .. The rest of the paper is organized as follows. Section 2 presents a short survey on related work in the area. Link duration for the Random WayPoint is studied in Section 3. Section 4 explains the analytical model for the path duration. This model is verified by extensive simulations described in section 5. Finally, Section 6 draws the main conclusions of the paper. In this paper authors propose an analytical model for link duration when nodes follow the extendedly used Random WayPoint. This model has been verified by means of simulations across 50 scenarios enclosing different mobility and transmission conditions. From the link model, path duration model is constructed and checked again via simulations. Both ns and Matlab simulations were run. The simulation results are well approximated by the analytical equations. To verify the correctness of the abovementioned analytical results, the software module developed for link duration was extended in order to compute all routes that the nodes could establish based on a connectivity graph. Once the computation of link duration is completed, it is straightforward to extend it in order to calculate the duration of a sequence of links, i.e. a path.. The Figure 2 As shown, simulation data closely follow analytical results. Therefore the analytical model approximates accurately the path duration. As exposed in Figure 2 , path duration decreases when the number of hops that it contains increases. Simulation has been the main method for analyzing the properties of route or path duration in ad hoc networks. One of the first studies concerning the analysis of path duration was due to Bai et al. [5] . Basing on experimental results obtained by simulations, they assume that the lifetime of a path with four or more hops can be approximated by an exponential distribution. However the authors do not consider the fit of any other standard distribution. Moreover, they do not justify the selection of an exponential distribution with any mathematical validation. To cope with this lack, Han et al., basing their work on Palm´s theorem, state that, under some circumstances, the lifetime associated to those paths with a large number of hops converges to an exponential distribution [6] . The previous works present a clear disadvantage as they provide a solution for the analysis of paths which is valid only for routes with a large number of hops. Therefore their study could not be fully applied to usual ad hoc networks and practical MANET applications where the paths only consist of 1 to 4 hops [7] . The importance of short paths is reinforced by the fact that the most relevant protocols utilize the minimum hop count as the metric to select the route in use in order to reduce the effects of the wireless retransmissions on the performance of the network. Anyway, the popularity of the exponential fitting has made it a common approximation in some other works as in [8] [9] .. Most authors have analyzed path duration by means of empirical results. For instance, [10] have shown that the mean residual lifetime of routes depends on the number of hops as well as on the mean Link Duration. On the other hand, [4] analytically proves that the average lifetime of a path decreases with its length.. An analytical study on this issue is due to Tseng et al. They base the analysis of the route lifetime on a spatial discrete model [11] . This study simplifies a MANET into a cellular network composed by hexagonal cells to compute the path availability. In [12] , authors formally describe the distribution function of path duration assuming that nodes move according to a constant velocity model.. In this paper, we develop an analytical framework in order to analyze the factors on the lifetime of a path that is formed by an arbitrary number of hops.

O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Lifetime and coverage guarantees through distributed coordinate-free sensor activation Wireless Sensor Networks are emerging as a key sensing technology, with diverse military and civilian applications. In these networks, a large number of sensors perform distributed sensing of a target field. Each sensor is a small battery-operated device that can sense events of interest in its sensing range and can communicate with neighboring sensors. A sensor cover is a subset of the set of all sensors such that every point in the target field is in the interior of the sensing ranges of at least k different sensors in the subset, where k is a given positive integer. The lifetime of the network is the time from the point the network starts operation until the set of all sensors with non-zero remaining energy does not constitute a sensor cover. An important goal in sensor networks is to design a schedule, that is, a sequence of sensor covers to activate in every time slot, so as to maximize the lifetime of the network. In this paper, we design a polynomial-time, distributed algorithm for maximizing the lifetime of the network and prove that its lifetime is at most a factor O(log n * log nB) lower than the maximum possible lifetime, where n is the number of sensors and B is an upper bound on the initial energy of each sensor. Our algorithm does not require knowledge of the locations of nodes or directional information, which is difficult to obtain in sensor networks. Each sensor only needs to know the distances between adjacent nodes in its transmission range and their sensing radii. In every slot, the algorithm first assigns a weight to each node that is exponential in the fraction of its initial energy that has been used up so far. Then, in a distributed manner, it finds a O(log n) approximate minimum weight sensor cover which it activates in the slot. Our simulations reveal that our algorithm substantially outperforms several existing lifetime maximization algorithms. Recent advances in wireless communications and electronics have enabled the development of low-cost sensor nodes [12] . Each sensor node is capable of sensing specific events in its vicinity and of communicating with adjacent nodes. Thus, for event sensing applications, a large number of sensor nodes are deployed in a distribution area and they collaborate to form an ad-hoc network, referred to as a wireless sensor network (WSN). WSNs have the potential to become the dominant sensing technology in many civilian and military applications, such as intrusion detection, environmental monitoring, object tracking, traffic control, and inventory management. In many of these applications, WSNs need to monitor the target field for detecting events of interest, e.g., entrance of an intruder in an intrusion detection application.. Wide-spread deployment of WSNs in target field monitoring is being deterred by the energy consumed in the monitoring process. The challenge is compounded by the fact that the sensors are battery-powered and owing to size limitations the sensors can only be deployed with low-lifetime batteries, most of which are not rechargeable. Thus, a sensor ceases to function (e.g., monitor) once its battery expires, and oftentimes, sensors whose batteries have expired can not be easily replaced owing to logistics issues such as remoteness or inaccessibility of distribution areas. Thus, the success of the WSN technology is contingent upon developing strategies for intelligently using the available sensors so as to maximize the duration for which the entire target field is monitored by sensors. This duration, referred to as the network lifetime, is an important performance metric for the network as the coverage of the entire target field is essential for reliable detection of events of interest.. Owing to large scale availability of low cost sensors, sensors are often deployed with some redundancy, that is, several locations in the target field can be monitored by multiple sensors. Lifetime of the WSNs can be substantially enhanced by intelligently activating the sensors that monitor the target field at any given time. We seek to maxi-mize the lifetime of sensor networks by designing algorithms that dynamically activate sensors based on their residual energy content. The algorithm we develop is completely distributed, does not need to know the coordinates of any sensor, and provides provable guarantees on the attained lifetimes. We assume that during the system activation every sensor u initially evaluates its distance to each one of its neighbors in N u and it broadcasts these distances d u,v , v ∈ N u , as well as its sensing radius ru to its neighbors. Next, u detects each neighbor v ∈ Nu that intersects with u by using Property 1 in Sub-section 2.4 and their joint intersection points. It also calculates the set Q u of all the intersection points of u's sensing border with the sensing borders of its neighbors. For every intersection point p ∈ Qu, u finds Sp, the set of sensors that cover p. Then, u communicates these sets Sp, p ∈ Q u to its neighbors. This process enables every neighbor v ∈ N u of u to know that a given intersection point p ∈ Q u is included in its sensing range and accordingly to add p to its set Pv of intersection points that it covers (for calculating (i) above). Moreover, the knowledge of each set Sp, p ∈ Pv, allows node v to identify its neighbors w that also cover each point p ∈ P v and update its set T v accordingly (for calculating (ii) above). Node v can also calculate the sets Pv,w = Pv ∩Pw, for each w ∈ Tv (for calculating (iii) above). Thus, to complete our description, we just need to present the process for detecting the set Q u of any given sensor u ∈ S and calculating the set S p for every point p ∈ Q u . A major challenge in the initialization process is determining a unique identification for each intersection point. Since the sensors do not have any location information, the coordinates of the intersection points are unknown and cannot be used as identifiers. To overcome this difficulty, every intersection point of any pair u, v of intersecting sensors is identified by a triplet IP (u, v, i), where u is the sensor with lower id, v is the sensor with higher id, and i = {1, 2} denotes the point index. Since every pair u, v of intersecting sensors have two common intersection points, the node with the lower id, say u, arbitrarily determines the index i of each point. In addition, u also calculates the set Sp i for both points p i = IP (u, v, i), i = {1, 2} and communicates these sets to its neighbors, including node v. This ensures that each calculated set Sp corresponds to a single intersection point that is uniquely defined. We describe the calculation of such sets S p in the next sub-section. Consider a pair u, v of intersecting sensors and let pi = IP (u, v, i), i = {1, 2} denote their intersection points, as depicted in Figure 4 . We now describe a simple method for calculating the covering set Sp i , i = 1, 2. First, note that a simple application of the triangle law on the distance metric and the fact thatR ≥ 2r establishes that S p i ⊆ N u ∩ N v . Now, our calculation proceeds in two steps. In Step 1, we partition Nu ∩ Nv in three sets: nodes that cover (i) none, (ii) only one (iii) both of p1, p2. In Step 2, we identify which nodes among the second set in the above partition cover p 1 (p 2 , respectively). This completes the computation of both S p 1 , S p 2 . In absence of location information, we rely on the Cosine Rule throughout . Let d a,b , da,c and d We designed a distributed, coordinate-free algorithm for attaining high lifetimes in sensor networks, subject to ensuring the k-coverage of the target field during the network lifetime. We proved that the lifetime attained by our algorithm approximates the maximum possible lifetime within a logarithmic approximation factor. Simulation results reveal that our algorithm substantially outperforms other schemes for lifetime maximization.

O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Biomedic Organizations: An intelligent dynamic architecture for KDD Multi-agent system Case-based reasoning Microarray Neural network Case-based planning a b s t r a c tThe application of information technology in the field of biomedicine has become increasingly important over the last several years. This study presents the Intelligent Biomedic Organizations (IBOs) model, an intelligent dynamic architecture for knowledge discovery in biomedical databases. It involves an organizational model specially designed to support medical personnel in their daily tasks and to establish an innovative intelligent system to make classifications and predictions with huge volumes of information. IBO is based on a multi-agent architecture with Web service integration capability. The core of the system is a type of agent that integrates a novel strategy based on a case-based planning mechanism for automatic reorganization. This agent proposes a new reasoning agent model, where the complex processes are modeled as external services. In this sense, the agents act as coordinators of Web services that implement the four stages of the case-based planning cycle. The multi-agent system has been implemented in a real scenario to classify leukemia patients, and the classification strategy includes services such as a novel ESOINN neural network and statistical methods to analyze patient data. The results obtained are presented within this paper and demonstrate the effectiveness of the proposed organizational model. Cancer diagnosis is a field requiring novel automated solutions and tools and the ability to facilitate the early detection, even prediction, of cancerous patterns. The continuous growth of techniques for obtaining cancerous samples, specifically those using microarray technologies, provides a great amount of data. Microarray has become an essential tool in genomic research, making it possible to investigate global genes in all aspects of human disease [24] . Currently, there are several kinds of microarrays such as CGH arrays [28] and expression arrays [2] . Expression arrays contain information about thousands of genes in a patient's samples. The genes are represented by a series of probes that are composed of oligonucleotides. The number of oligonucleotides is up to one million per sample. The large amount of data requiring analysis makes it necessary to use data mining techniques in order to reduce processing time.. There are different approaches for decision support systems, including myGrid [30] , which base their functionality on the creation of Web services that are implemented according to OGSA (Open Grid Services Architecture) [12] . The main disadvantage, however, is that the user must be the responsible for creating the sequence of actions which resolve specific problems. These systems provide methods for solving complex problems in a distributed way through SOA [11] and Grid high-dimensional arrays, it is important to have a good pre-processing technique that can facilitate automatic decision-making about the variables that will be vital for the classification process. In light of these decisions it will be possible to reduce the original dataset.. Case-based reasoning [26] is particularly applicable to this problem domain because it (i) supports a rich and evolvable representation of experiences, problems, solutions and feedback; (ii) provides efficient and flexible ways to retrieve these experiences; and (iii) applies analogical reasoning to solve new problems [18] . CBR systems can be used to propose new solutions or evaluate solutions to avoid potential problems. The research in [1] suggests that analogical reasoning is particularly applicable to the biological domain, in part because biological systems are often homologous (rooted in evolution). In [4] a mixture of experts for case-based reasoning (MOE4CBR) is proposed. It is a method that combines an ensemble of CBR classifiers with spectral clustering and logistic regression, but does not incorporate extraction of knowledge techniques and does not focus on dimensionality reduction. This study has presented the IBO multi-agent architecture and its application to real problems. IBO facilitates task automation by means of intelligent agents capable of autonomously planning the stages of an expression analysis. The characteristics of this novel architecture facilitate an organizational-oriented approach where the dynamics of a real scenario can be captured and modeled into CBP-BDI agents. The agents act as controllers and coordinators of the organization. The complex functionalities of the agents are modeled as distributed services. IBO provides a service oriented approach and facilitates the distribution and management of resources. Moreover, IBO facilitates the distributed execution of complex computational services, reducing the number of crashes in agents. The multi-agent system developed is integrated within Web services, aims to reduce the dimensionality of the original data set, and proposes a novel method of clustering for classifying patients. The multi-agent perspective allows the system to work in a way similar to how human specialists operate in the laboratory, but is able to work with great amounts of data and make decisions automatically, thus reducing significantly both the time required to make a prediction, and the rate of human error arising from confusion. The Bayesian networks make it possible to generate the plans that best adjust to the different case studies, allowing the generation of new plans based on existing information without actually needing an existing memory of plans. If we follow the same procedure as the one established for selecting the parameters, it would be necessary to have extensive memory plans and the definition of a mechanism for carrying out the composition of efficient plans and generating a new and more efficient plan.. The multi agent system simulates the behavior of experts working in a laboratory, making it possible to carry out a data analysis in a distributed manner, as normally done by experts. The system is capable of learning based on previous experiences and of generating new behaviors and, as a result, creates an application that adapts to the information that characterizes the case studies.

O O O O O O O O B I B I B B I B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Assistive mobility devices focusing on Smart Walkers: Classification and review In an aging society it is extremely important to develop devices, which can support and aid the elderly in their daily life. This demands means and tools that extend independent living and promote improved health.Thus, the goal of this article is to review the state of the art in the robotic technology for mobility assistive devices for people with mobility disabilities. The important role that robotics can play in mobility assistive devices is presented, as well as the identification and survey of mobility assistive devices subsystems with a particular focus on the walkers technology. The advances in the walkers' field have been enormous and have shown a great potential on helping people with mobility disabilities. Thus it is presented a review of the available literature of walkers and are discussed major advances that have been made and limitations to be overcome. During the twentieth century the proportion of elderly people had an important rise, and this trend is expected to continue into the twenty-first century. The proportion of population over 60 years old was 8% in 1950, 10% in 2000, and it is estimated to reach 21% in 2050 [1] . Loosing complete or part of mobility, affects not only the ability to walk but also the ability to perform personal tasks, which is a major determinant in life quality and causes dependence of others in daily life.. In an aging society it is extremely important to develop devices, which can support and assist the elderly in their daily life, since their mobility degrades with age. This situation requires a great medical care, incurs large costs and can be fatal in some cases. Elderly tend to have cognitive impairments and experience more serious falls but there is strong evidence that daily exercise may result on fall prevention and postural stability [2] . So, it becomes more and more relevant to find ways and tools to compensate, to improve or to restore and to enhance this mobility.. For several years, researchers have been addressing the needs of persons with mobility disabilities through alternative or augmentative devices [2] . These solutions are selected based on the degree of disability of the user. For the purposes of this review, it will be presented, on the one hand, the alternative devices, which are used in case of total incapacity of mobility. These devices are usually wheelchairs or solutions based on autonomous especial vehicles [3] [4] . On Over the past years, technological advances allowed the incorporation of sensors and actuators in conventional walkers, providing the stability of fourlegged walkers, without affecting the resultant naturalness of the users' gait patterns. Besides, these devices enable to identify the movement intentions of the users and therefore control the mobility assistance accordingly.. We intend to cover all of the major developments in the mobility assistive devices described before, particularly focusing on the walker devices. In this review, it is introduced a brief presentation of the causes and consequences of the gait dysfunctions that lead patients to use mobility assistive devices. Then, it is reviewed the literature regarding the corresponding mobility assistive devices.. Finally, we focus on the literature regarding the walker' devices, presenting a more detailed review about the various existing models to date. It is presented a discussion of this information, summarizing the major accomplishments in the field and identifying the limitations and challenges to be overcome in future researches. On this article, it was made a brief review of the dysfunctions that can affect the mobility, and it was presented a critical review about the assistive mobility devices existing on the literature, giving emphases to the walkers, and more specifically the smart walkers.. The review about the walkers has shown that in the last years their development has witnessed a huge increase due to their enormous potential for rehabilitation and functional compensation. This potential emerges since this device makes use of the residual mobility capacities of users, maintaining and/or ameliorating them.. Thus, it provides the opposite effect of the wheelchairs, which have a disability effect on the mobility of the users.. Evaluating the works that were presented in this article, one can infer that, in general, almost all the smart walkers have an assistive navigation system with sensors that detect obstacles, as well as, a design that has been studied to provide and improve a stable gait. This factor is possible with different designs of the guide platform that can have, for example, a forearm support. Another aspect that increases stability is the fact that electronics and other heavy components are put on the lower base of the walkers, giving a greater balance and stability to the device.. Thus, rollators can be easily augmented with simple and relatively low cost instrumentation technologies to provide a wide range of functionality and gait characteristics as Smart walkers, and avoid to inadequately resort to alternative devices, thus contributing to the maintenance or to the improvement of the physical and cognitive capabilities of the user.

O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O