Parameterized complexity of three edge contraction problems with degree constraints

For any graph class \(\mathcal{H}\) , the \(\mathcal{H}\) -Contraction problem takes as input a graph \(G\) and an integer \(k\) , and asks whether there exists a graph \(H\in \mathcal{H}\) such that \(G\) can be modified into \(H\) using at most \(k\) edge contractions. We study the parameterized complexity of \(\mathcal{H}\) -Contraction for three different classes \(\mathcal{H}\) : the class \(\mathcal{H}_{\le d}\) of graphs with maximum degree at most  \(d\) , the class \(\mathcal{H}_{=d}\) of \(d\) -regular graphs, and the class of \(d\) -degenerate graphs. We completely classify the parameterized complexity of all three problems with respect to the parameters \(k\) , \(d\) , and \(d+k\) . Moreover, we show that \(\mathcal{H}\) -Contraction admits an \(O(k)\) vertex kernel on connected graphs when \(\mathcal{H}\in \{\mathcal{H}_{\le 2},\mathcal{H}_{=2}\}\) , while the problem is \(\mathsf{W}[2]\) -hard when \(\mathcal{H}\) is the class of \(2\) -degenerate graphs and hence is expected not to admit a kernel at all. In particular, our results imply that \(\mathcal{H}\) -Contraction admits a linear vertex kernel when \(\mathcal{H}\) is the class of cycles.     

The categories L-\mathbf{Top}_{0} and L-\mathbf{Sob} as epireflective hulls

We show that the category \(L\) - \(\mathbf{Top}_{0}\) of \(T_{0}\) - \(L\) -topological spaces is the epireflective hull of Sierpinski \(L\) -topological space in the category \(L\) - \(\mathbf{Top}\) of \(L\) -topological spaces and the category \(L\) - \(\mathbf{Sob}\) of sober \(L\) -topological spaces is the epireflective hull of Sierpinski \(L\) -topological space in the category \(L\) - \(\mathbf{Top}_{0}\) .     

Universal regular control for generic semilinear systems

We consider discrete-time projective semilinear control systems \(\xi _{t+1} = A(u_t) \cdot \xi _t\) , where the states \(\xi _t\) are in projective space \(\mathbb {R}\hbox {P}^{d-1}\) , inputs \(u_t\) are in a manifold \(\mathcal {U}\) of arbitrary finite dimension, and \(A :\mathcal {U}\rightarrow \hbox {GL}(d,\mathbb {R})\) is a differentiable mapping. An input sequence \((u_0,\ldots ,u_{N-1})\) is called universally regular if for any initial state \(\xi _0 \in \mathbb {R}\hbox {P}^{d-1}\) , the derivative of the time- \(N\) state with respect to the inputs is onto. In this paper, we deal with the universal regularity of constant input sequences \((u_0, \ldots , u_0)\) . Our main result states that generically in the space of such systems, for sufficiently large \(N\) , all constant inputs of length \(N\) are universally regular, with the exception of a discrete set. More precisely, the conclusion holds for a \(C^2\) -open and \(C^\infty \) -dense set of maps \(A\) , and \(N\) only depends on \(d\) and on the dimension of \(\mathcal {U}\) . We also show that the inputs on that discrete set are nearly universally regular; indeed, there is a unique non-regular initial state, and its corank is 1. In order to establish the result, we study the spaces of bilinear control systems. We show that the codimension of the set of systems for which the zero input is not universally regular coincides with the dimension of the control space. The proof is based on careful matrix analysis and some elementary algebraic geometry. Then the main result follows by applying standard transversality theorems.     

Growth rates for persistently excited linear systems

We consider a family of linear control systems \(\dot{x}=Ax+\alpha Bu\) on \(\mathbb {R}^d\) , where \(\alpha \) belongs to a given class of persistently exciting signals. We seek maximal \(\alpha \) -uniform stabilization and destabilization by means of linear feedbacks \(u=Kx\) . We extend previous results obtained for bidimensional single-input linear control systems to the general case as follows: if there exists at least one \(K\) such that the Lie algebra generated by \(A\) and \(BK\) is equal to the set of all \(d\times d\) matrices, then the maximal rate of convergence of \((A,B)\) is equal to the maximal rate of divergence of \((-A,-B)\) . We also provide more precise results in the general single-input case, where the above result is obtained under the simpler assumption of controllability of the pair \((A,B)\) .     

Fault tolerance of hypercubes and folded hypercubes

Let \(G = (V,E)\) be a connected graph. The conditional edge connectivity \(\lambda _\delta ^k(G)\) is the cardinality of the minimum edge cuts, if any, whose deletion disconnects \(G\) and each component of \(G - F\) has \(\delta \ge k\) . We assume that \(F \subseteq E\) is an edge set, \(F\) is called edge extra-cut, if \(G - F\) is not connected and each component of \(G - F\) has more than \(k\) vertices. The edge extraconnectivity \(\lambda _\mathrm{e}^k(G)\) is the cardinality of the minimum edge extra-cuts. In this paper, we study the conditional edge connectivity and edge extraconnectivity of hypercubes and folded hypercubes.     

Approximation of integrable functions based on \phi -transform

Let \(E\) be a bounded subset of real line which contains its infimum and supremum. In this paper, we have defined the \(\phi -\) transform and its inverse, where \(\phi \) is a function from \(E\) into \((0,1]\) . We will have shown that real-valued integrable functions on \([a, b]\) and real-valued continuous functions on \(E\) can be approximated by this transformation with an arbitrary precision.     

Isomorphisms and functors of fuzzy sets and cut systems

Any fuzzy set \(X\) in a classical set \(A\) with values in a complete (residuated) lattice \( Q\) can be identified with a system of \(\alpha \) -cuts \(X_{\alpha }\) , \(\alpha \in Q\) . Analogical results were proved for sets with similarity relations with values in \( Q\) (e.g. \( Q\) -sets), which are objects of two special categories \({\mathbf K}={Set}( Q)\) or \({SetR}( Q)\) of \( Q\) -sets, and for fuzzy sets defined as morphisms from an \( Q\) -set into a special \(Q\) -set \(( Q,\leftrightarrow )\) . These fuzzy sets can be defined equivalently as special cut systems \((C_{\alpha })_{\alpha }\) , called f-cuts. This equivalence then represents a natural isomorphism between covariant functor of fuzzy sets \(\mathcal{F}_{\mathbf K}\) and covariant functor of f-cuts \(\mathcal{C}_{\mathbf K}\) . In this paper, we prove that analogical natural isomorphism exists also between contravariant versions of these functors. We are also interested in relationships between sets of fuzzy sets and sets of f-cuts in an \(Q\) -set \((A,\delta )\) in the corresponding categories \({Set}( Q)\) and \({SetR}( Q)\) , which are endowed with binary operations extended either from binary operations in the lattice \(Q\) , or from binary operations defined in a set \(A\) by the generalized Zadeh’s extension principle. We prove that the resulting binary structures are (under some conditions) isomorphic.     

Greedy routing in small-world networks with power-law degrees

In this paper we study decentralized routing in small-world networks that combine a wide variation in node degrees with a notion of spatial embedding. Specifically, we consider a variant of J. Kleinberg’s grid-based small-world model in which (1) the number of long-range edges of each node is not fixed, but is drawn from a power-law probability distribution with exponent parameter \(\alpha \ge 0\) and constant mean, and (2) the long-range edges are considered to be bidirectional for the purposes of routing. This model is motivated by empirical observations indicating that several real networks have degrees that follow a power-law distribution. The measured power-law exponent \(\alpha \) for these networks is often in the range between 2 and 3. For the small-world model we consider, we show that when \(2 < \alpha < 3\) the standard greedy routing algorithm, in which a node forwards the message to its neighbor that is closest to the target in the grid, finishes in an expected number of \(O(\log ^{\alpha -1} n\cdot \log \log n)\) steps, for any source–target pair. This is asymptotically smaller than the \(O(\log ^2 n)\) steps needed in Kleinberg’s original model with the same average degree, and approaches \(O(\log n)\) as \(\alpha \) approaches 2. Further, we show that when \(0\le \alpha < 2\) or \(\alpha \ge 3\) the expected number of steps is \(O(\log ^2 n)\) , while for \(\alpha = 2\) it is \(O(\log ^{4/3} n)\) . We complement these results with lower bounds that match the upper bounds within at most a \(\log \log n\) factor.     

Aircraft noise scattering prediction using different accelerator architectures

In this work, we present a tool that exploits heterogeneous computing to calculate the noise scattered by an object from the pressure distribution over its surface and its normal derivative. The method mainly deals with a large Matrix–Vector Product where the matrix elements must be calculated on the fly in such a way that the problem fits in main memory. To prove the performance of the heterogeneous implementations, the tool is tested using one NVIDIA K20c GPU, one Intel Xeon Phi 5110P, and two Intel Xeon E5-2650 CPUs. The speedup of the accelerated implementations ranges from \(3\times \) (Xeon Phi) to \(8\times \) (Xeon Phi  \(+\)  K20c) when compared to our parallel CPU code with \(32\) threads. This work, combined with the authors’ previous works for the computation of the acoustic pressure over the obstacle surface, results in a valuable toolset for noise control applications during aircraft design.     

Quantum entropy-typical subspace and universal data compression

The quantum entropy-typical subspace theory is specified. It is shown that any \(\rho ^{\otimes n}\) with von Neumann \(\hbox {entropy}\le h\) can be preserved approximately by the entropy-typical subspace with \(\hbox {entropy}=h\) . This result implies an universal compression scheme for the case that the von Neumann entropy of the source does not exceed \(h\) .     

A computationally efficient importance sampling tracking algorithm

This paper proposes a computationally efficient importance sampling algorithm applicable to computer vision tracking. The algorithm is based on the CONDENSATION algorithm, but it avoids expensive operations that are costly in real-time embedded systems. It also includes a method that reduces the number of particles during execution and a new resampling scheme. Our experiments demonstrate that the proposed algorithm is as accurate as the CONDENSATION algorithm. Depending on the processed sequence, the acceleration with respect to CONDENSATION can reach 7 \(\times \) for 50 particles, 12 \(\times \) for 100 particles and 58 \(\times \) for 200 particles.     

Generation of atomic NOON states via adiabatic passage

We propose a scheme for generating atomic NOON states via adiabatic passage. In the scheme, a double \(\Lambda \) -type three-level atom is trapped in a bimodal cavity, and two sets of \(\Lambda \) -type three-level atoms are translated into and outside of two single-mode cavities, respectively. The three cavities connected by optical fibers are always in vacuum states. After a series of operations and suitable interaction time, we can obtain arbitrary large- \(n\) NOON states of two sets of \(\Lambda \) -type three-level atoms in distant cavities by performing a single projective measurement on the double \(\Lambda \) -type three-level atom. Our scheme is robust against the spontaneous emissions of atoms, the decays of fibers, and photon leakage of cavities, due to the adiabatic elimination of atomic excited states and the application of adiabatic passage.     

Informational correlation between two parties of a quantum system: spin-1/2 chains

We introduce the informational correlation \(E^{AB}\) between two interacting quantum subsystems \(A\) and \(B\) of a quantum system as the number of arbitrary parameters \(\varphi _i\) of a unitary transformation \(U^A\) (locally performed on the subsystem \(A\) ) which may be detected in the subsystem \(B\) by the local measurements. This quantity indicates whether the state of the subsystem \(B\) may be effected by means of the unitary transformation applied to the subsystem \(A\) . Emphasize that \(E^{AB}\ne E^{BA}\) in general. The informational correlations in systems with tensor product initial states are studied in more details. In particular, it is shown that the informational correlation may be changed by the local unitary transformations of the subsystem \(B\) . However, there is some non-reducible part of \(E^{AB}(t)\) which may not be decreased by any unitary transformation of the subsystem \(B\) at a fixed time instant \(t\) . Two examples of the informational correlations between two parties of the four-node spin-1/2 chain with mixed initial states are studied. The long chains with a single initially excited spin (the pure initial state) are considered as well.     

Fault tolerance in distributed systems using fused state machines

Replication is a standard technique for fault tolerance in distributed systems modeled as deterministic finite state machines (DFSMs or machines). To correct \(f\) crash or \(\lfloor f/2 \rfloor \) Byzantine faults among \(n\) different machines, replication requires \(nf\) backup machines. We present a solution called fusion that requires just \(f\) backup machines. First, we build a framework for fault tolerance in DFSMs based on the notion of Hamming distances. We introduce the concept of an ( \(f\) , \(m\) )-fusion, which is a set of \(m\) backup machines that can correct \(f\) crash faults or \(\lfloor f/2 \rfloor \) Byzantine faults among a given set of machines. Second, we present an algorithm to generate an ( \(f\) , \(f\) )-fusion for a given set of machines. We ensure that our backups are efficient in terms of the size of their state and event sets. Third, we use locality sensitive hashing for the detection and correction of faults that incurs almost the same overhead as that for replication. We detect Byzantine faults with time complexity \(O(n f)\) on average while we correct crash and Byzantine faults with time complexity \(O(n \rho f)\) with high probability, where \(\rho \) is the average state reduction achieved by fusion. Finally, our evaluation of fusion on the widely used MCNC’91 benchmarks for DFSMs shows that the average state space savings in fusion (over replication) is 38 % (range 0–99 %). To demonstrate the practical use of fusion, we describe its potential application to two areas: sensor networks and the MapReduce framework. In the case of sensor networks a fusion-based solution can lead to significantly fewer sensor-nodes than a replication-based solution. For the MapReduce framework, fusion can reduce the number of map-tasks compared to replication. Hence, fusion results in considerable savings in state space and other resources such as the power needed to run the backups.     

Some kinds of primitive and non-primitive words

If the length of a primitive word \(p\) is equal to the length of another primitive word \(q\) , then \(p^{n}q^{m}\) is a primitive word for any \(n,m\ge 1\) and \((n,m)\ne (1,1)\) . This was obtained separately by Tetsuo Moriya in 2008 and Shyr and Yu in 1994. In this paper, we prove that if the length of \(p\) is divisible by the length of \(q\) and the length of \(p\) is less than or equal to \(m\) times the length of \(q\) , then \(p^{n}q^{m}\) is a primitive word for any \(n,m\ge 1\) and \((n,m)\ne (1,1)\) . Then we show that if \(uv,u\) are non-primitive words and the length of \(u\) is divisible by the length \(v\) or one of the length of \(u\) and \(uv\) is odd for any two nonempty words \(u\) and \(v\) , then \(u\) is a power of \(v\) .     

Collaborative filtering with information-rich and information-sparse entities

In this paper, we consider a popular model for collaborative filtering in recommender systems. In particular, we consider both the clustering model, where only users (or items) are clustered, and the co-clustering model, where both users and items are clustered, and further, we assume that some users rate many items (information-rich users) and some users rate only a few items (information-sparse users). When users (or items) are clustered, our algorithm can recover the rating matrix with \(\omega (MK \log M)\) noisy entries while \(MK\) entries are necessary, where \(K\) is the number of clusters and \(M\) is the number of items. In the case of co-clustering, we prove that \(K^2\) entries are necessary for recovering the rating matrix, and our algorithm achieves this lower bound within a logarithmic factor when \(K\) is sufficiently large. Extensive simulations on Netflix and MovieLens data show that our algorithm outperforms the alternating minimization and the popularity-among-friends algorithm. The performance difference increases even more when noise is added to the datasets.     

The Local Discontinuous Galerkin Method for the Fourth-Order Euler–Bernoulli Partial Differential Equation in One Space Dimension. Part II: A Posteriori Error Estimation

In this paper new a posteriori error estimates for the local discontinuous Galerkin (LDG) method for one-dimensional fourth-order Euler–Bernoulli partial differential equation are presented and analyzed. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We use the optimal error estimates and the superconvergence results proved in Part I to show that the significant parts of the discretization errors for the LDG solution and its spatial derivatives (up to third order) are proportional to \((k+1)\) -degree Radau polynomials, when polynomials of total degree not exceeding \(k\) are used. These results allow us to prove that the \(k\) -degree LDG solution and its derivatives are \(\mathcal O (h^{k+3/2})\) superconvergent at the roots of \((k+1)\) -degree Radau polynomials. We use these results to construct asymptotically exact a posteriori error estimates. We further apply the results proved in Part I to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivatives at a fixed time \(t\) converge to the true errors at \(\mathcal O (h^{k+5/4})\) rate. We also prove that the global effectivity indices, for the solution and its derivatives up to third order, in the \(L^2\) -norm converge to unity at \(\mathcal O (h^{1/2})\) rate. Our proofs are valid for arbitrary regular meshes and for \(P^k\) polynomials with \(k\ge 1\) , and for periodic and other classical mixed boundary conditions. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates. Finally, we present a local adaptive procedure that makes use of our local a posteriori error estimate.