Showing papers on "Bounded function published in 2007"
TL;DR: This paper proposes an information theoretic criterion for comparing two partitions, or clusterings, of the same data set, called variation of information (VI), and presents it from an axiomatic point of view, showing that it is the only ''sensible'' criterion for compare partitions that is both aligned to the lattice and convexely additive.
1,527 citations
TL;DR: Several algorithms achieving logarithmic regret are proposed, which besides being more general are also much more efficient to implement, and give rise to an efficient algorithm based on the Newton method for optimization, a new tool in the field.
Abstract: In an online convex optimization problem a decision-maker makes a sequence of decisions, i.e., chooses a sequence of points in Euclidean space, from a fixed feasible set. After each point is chosen, it encounters a sequence of (possibly unrelated) convex cost functions. Zinkevich (ICML 2003) introduced this framework, which models many natural repeated decision-making problems and generalizes many existing problems such as Prediction from Expert Advice and Cover's Universal Portfolios. Zinkevich showed that a simple online gradient descent algorithm achieves additive regret $O(\sqrt{T})$ , for an arbitrary sequence of T convex cost functions (of bounded gradients), with respect to the best single decision in hindsight.
In this paper, we give algorithms that achieve regret O(log?(T)) for an arbitrary sequence of strictly convex functions (with bounded first and second derivatives). This mirrors what has been done for the special cases of prediction from expert advice by Kivinen and Warmuth (EuroCOLT 1999), and Universal Portfolios by Cover (Math. Finance 1:1---19, 1991). We propose several algorithms achieving logarithmic regret, which besides being more general are also much more efficient to implement.
The main new ideas give rise to an efficient algorithm based on the Newton method for optimization, a new tool in the field. Our analysis shows a surprising connection between the natural follow-the-leader approach and the Newton method. We also analyze other algorithms, which tie together several different previous approaches including follow-the-leader, exponential weighting, Cover's algorithm and gradient descent.
1,124 citations
TL;DR: In this paper, the evolution problem is rewritten as a semilinear hyperbolic system in an L∞ space, containing a non-local source term which is discontinuous but has bounded directional variation.
Abstract: This paper is devoted to the continuation of solutions to the Camassa–Holm equation after wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear hyperbolic system in an L∞ space, containing a non-local source term which is discontinuous but has bounded directional variation. For a given initial condition, the Cauchy problem has a unique solution obtained as fixed point of a contractive integral transformation. Returning to the original variables, we obtain a semigroup of global dissipative solutions, defined for every initial data $\bar u\in H^1 ({\mathbb R})$, and continuously depending on the initial data. The new variables resolve all singularities due to possible wave breaking and ensure that energy loss occurs only through wave breaking.
511 citations
TL;DR: In this article, the existence of global bounded classical solutions is proved under the assumption that either the space dimension does not exceed two, or that the logistic damping effect is strong enough.
Abstract: This paper deals with a nonlinear system of two partial differential equations arising in chemotaxis, involving a source term of logistic type The existence of global bounded classical solutions is proved under the assumption that either the space dimension does not exceed two, or that the logistic damping effect is strong enough Also, the existence of global weak solutions is shown under rather mild conditions Secondly, the corresponding stationary problem is studied and some regularity properties are given It is proved that in presence of certain, sufficiently strong logistic damping there is only one nonzero equilibrium, and all solutions of the non-stationary system approach this steady state for large times On the other hand, for small logistic terms some multiplicity and bifurcation results are established
482 citations
TL;DR: It is proved that the metric dimension of G\,\square\,G$ is tied in a strong sense to the minimum order of a so-called doubly resolving set in $G$.
Abstract: A set of vertices $S$ resolves a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. This paper studies the metric dimension of cartesian products $G\,\square\,H$. We prove that the metric dimension of $G\,\square\,G$ is tied in a strong sense to the minimum order of a so-called doubly resolving set in $G$. Using bounds on the order of doubly resolving sets, we establish bounds on $G\,\square\,H$ for many examples of $G$ and $H$. One of our main results is a family of graphs $G$ with bounded metric dimension for which the metric dimension of $G\,\square\,G$ is unbounded.
461 citations
TL;DR: The synthesis of state-feedback controllers is solved in terms of linear programming problem, including the requirement of positiveness of the controller and its extension to uncertain plants.
Abstract: This brief solves some synthesis problems for a class of linear systems for which the state takes nonnegative values whenever the initial conditions are nonnegative. In particular, the synthesis of state-feedback controllers is solved in terms of linear programming problem, including the requirement of positiveness of the controller and its extension to uncertain plants. In addition, the synthesis problem with nonsymmetrical bounds on the stabilizing control is treated
424 citations
01 Dec 2007
TL;DR: This note shows that consensus is reached asymptotically for the first two cases if the undirected interaction graph is connected and for the third case if the directed interaction graph has a directed spanning tree and the gain for velocity matching with the group reference velocity is above a certain bound.
Abstract: This paper extends some existing results in consensus algorithms for double-integrator dynamics. We propose consensus algorithms for double-integrator dynamics in four cases: (i) with a bounded control input, (ii) without relative velocity measurement, (iii) without relative velocity measurement in the presence of a group reference velocity, and (iv) with a bounded control input and with partial access to a group reference state. We show that consensus is reached asymptotically for the first two cases if the undirected interaction graph is connected. We further show that consensus is reached asymptotically for the third case if the directed interaction graph has a directed spanning tree and the gain for velocity matching with the group reference velocity is above a certain bound. We also show that consensus is reached asymptotically for the fourth case if and only if the group reference state flows directly or indirectly to all of the vehicles in the team.
419 citations
TL;DR: It is shown that the problem is equivalent to the finite time stabilization of higher order input-output dynamics with bounded uncertainties ([email protected]?N).
404 citations
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TL;DR: In this paper, a new point of view on weak solutions of the Euler equations is proposed, which describes the motion of an ideal incompressible fluid in the plane with constant velocity and pressure, and gives transparent proofs of several celebrated results of V. Scheffer and A. Shnirelman.
Abstract: In this paper we propose a new point of view on weak solutions of the Euler equations, describing the motion of an ideal incompressible fluid in $\mathbb{R}^n$ with $n\geq 2$. We give a reformulation of the Euler equations as a differential inclusion, and in this way we obtain transparent proofs of several celebrated results of V. Scheffer and A. Shnirelman concerning the non-uniqueness of weak solutions and the existence of energy--decreasing solutions. Our results are stronger because they work in any dimension and yield bounded velocity and pressure.
375 citations
TL;DR: A fixed-point formalization of the well-known analysis of Bianchi is studied, and it is shown how the saturated network analysis can be used to obtain TCP transfer throughputs in some cases.
Abstract: We study a fixed-point formalization of the well-known analysis of Bianchi. We provide a significant simplification and generalization of the analysis. In this more general framework, the fixed-point solution and performance measures resulting from it are studied. Uniqueness of the fixed point is established. Simple and general throughput formulas are provided. It is shown that the throughput of any flow will be bounded by the one with the smallest transmission rate. The aggregate throughput is bounded by the reciprocal of the harmonic mean of the transmission rates. In an asymptotic regime with a large number of nodes, explicit formulas for the collision probability, the aggregate attempt rate, and the aggregate throughput are provided. The results from the analysis are compared with ns2 simulations and also with an exact Markov model of the backoff process. It is shown how the saturated network analysis can be used to obtain TCP transfer throughputs in some cases.
375 citations
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TL;DR: For Riemannian manifolds with a measure (M,g, e^{-f} dvol_g) and a measure σ, σ ≥ 0, the curvature and volume comparison results when the Bakry-Emery Ricci tensor is bounded from below and $f$ is bounded or Ω √ √ r f is bounded was shown in this paper.
Abstract: For Riemannian manifolds with a measure $(M,g, e^{-f} dvol_g)$ we prove mean curvature and volume comparison results when the $\infty$-Bakry-Emery Ricci tensor is bounded from below and $f$ is bounded or $\partial_r f$ is bounded from below, generalizing the classical ones (i.e. when $f$ is constant). This leads to extensions of many theorems for Ricci curvature bounded below to the Bakry-Emery Ricci tensor. In particular, we give extensions of all of the major comparison theorems when $f$ is bounded. Simple examples show the bound on $f$ is necessary for these results.
TL;DR: It is proved that the effect of sampling and quantization nearly vanishes when a low-rank approximation to A + N is computed, which gives high probability bounds on the quality of the approximation both in the Frobenius and the 2-norm.
Abstract: Given a matrix A, it is often desirable to find a good approximation to A that has low rank. We introduce a simple technique for accelerating the computation of such approximations when A has strong spectral features, that is, when the singular values of interest are significantly greater than those of a random matrix with size and entries similar to A. Our technique amounts to independently sampling and/or quantizing the entries of A, thus speeding up computation by reducing the number of nonzero entries and/or the length of their representation. Our analysis is based on observing that the acts of sampling and quantization can be viewed as adding a random matrix N to A, whose entries are independent random variables with zero-mean and bounded variance. Since, with high probability, N has very weak spectral features, we can prove that the effect of sampling and quantization nearly vanishes when a low-rank approximation to A p N is computed. We give high probability bounds on the quality of our approximation both in the Frobenius and the 2-norm.
TL;DR: In this article, the authors consider solutions to the Cauchy problem for the incompressible Euler equations satisfying several additional requirements, like the global and local energy inequalities, and show that, for some bounded compactly supported initial data, none of these admissibility criteria singles out a unique weak solution.
Abstract: We consider solutions to the Cauchy problem for the incompressible Euler equations satisfying several additional requirements, like the global and local energy inequalities. Using some techniques introduced in an earlier paper we show that, for some bounded compactly supported initial data, none of these admissibility criteria singles out a unique weak solution.
As a byproduct we show bounded initial data for which admissible solutions to the p-system of isentropic gas dynamics in Eulerian coordinates are not unique in more than one space dimension.
TL;DR: In this paper, the global C 1, α regularity of the bounded generalized solutions of the variable exponent elliptic equations in divergence form with both Dirichlet and Neumann boundary conditions was studied.
TL;DR: This paper proves that a switched nonlinear system has several useful input-to-state stable (ISS) properties under average dwell-time switching signals if each constituent dynamical system is ISS, and applies this result to stabilization of uncertain nonlinear systems via switching supervisory control.
TL;DR: In this article, the authors studied solutions of the Lane-Emden equation on unbounded domains of RN with N⩾2 and p>1 and proved various classification theorems and Liouville-type results for C2 solutions belonging to one of the following classes: stable solutions, finite Morse index solutions, solutions which are stable outside a compact set, radial solutions and non-negative solutions.
TL;DR: The authors argue that the bounded field-site, rethought as an arbitrary location, becomes an explicitly "partial" and incomplete window onto complexity, and suggest that we could think of boundedness as a productive way of challenging holisms and deferring closure.
Abstract: The article offers a sympathetic critique of the original formulations of multi-local/multi-sited ethnography. The ‘multi-sited imaginary’ values unboundedness and promotes methodological freedom, but it also implies a problematic reconfiguration of holism (on a grander scale). Whereas these formulations were extremely productive in straining against certain methodological rigidities, their very success in breaking down ‘boundaries’ has given rise to new problems in the doing and writing of ethnography. Written from the perspective of a recent Ph.D. graduate and first-time fieldworker, the article suggests we reconsider the value of self-imposed limitations, of boundedness as a methodological tool. What role did the bounded field-site play for its so-called ‘traditional’ practitioners in social/cultural anthropology? What role could it play for anthropologists who have taken on board the precepts of multi-sitedness? Based on a case study from my own fieldwork in Corsica, I argue that we could think of boundedness (paradoxically) as a productive way of challenging holisms and deferring closure. The bounded field-site, rethought as an ‘arbitrary location’, becomes an explicitly ‘partial’ and incomplete window onto complexity.
TL;DR: Criteria for verifying robust stability are formulated as feasibility problems over a set of frequency-dependent linear matrix inequalities and can be equivalently formulated as semi-definite programs (SDP) using Kalman-Yakubovich-Popov lemma.
TL;DR: In this article, the authors analyze the two-dimensional parabolic-elliptic Patlak-Keller-Segel model in the whole Euclidean space R 2 and show local in time existence for any mass of "free-energy solutions", namely weak solutions with some free energy estimates.
Abstract: We analyze the two-dimensional parabolic-elliptic Patlak-Keller-Segel model in the whole Euclidean space R2. Under the hypotheses of integrable initial data with finite second moment and entropy, we first show local in time existence for any mass of "free-energy solutions", namely weak solutions with some free energy estimates. We also prove that the solution exists as long as the entropy is controlled from above. The main result of the paper is to show the global existence of free-energy solutions with initial data as before for the critical mass 8 Π/Χ. Actually, we prove that solutions blow-up as a delta dirac at the center of mass when t→∞ keeping constant their second moment at any time. Furthermore, all moments larger than 2 blow-up as t→∞ if initially bounded.
TL;DR: In this paper, a 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations,..., etc.) is presented in divergence form.
Abstract: We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations,..., etc.) in divergence form. These divergence-free quantities generalize to target manifolds without symmetries the well known conservation laws for weakly harmonic maps into homogeneous spaces. From this form we can recover, without the use of moving frame, all the classical regularity results known for 2-dimensional conformally invariant non-linear elliptic PDE (see [Hel]). It enables us also to establish new results. In particular we solve a conjecture by E. Heinz asserting that the solutions to the prescribed bounded mean curvature equation in arbitrary manifolds are continuous and we solve a conjecture by S. Hildebrandt [Hil1] claiming that critical points of continuously differentiable elliptic conformally invariant Lagrangian in two dimensions are continuous.
TL;DR: In this paper, it was shown that the size of the smallest self-assembly program that builds a shape and the shape's descriptional (Kolmogorov) complexity should be related.
Abstract: The connection between self-assembly and computation suggests that a shape can be considered the output of a self-assembly “program,” a set of tiles that fit together to create a shape. It seems plausible that the size of the smallest self-assembly program that builds a shape and the shape’s descriptional (Kolmogorov) complexity should be related. We show that when using a notion of a shape that is independent of scale, this is indeed so: in the tile assembly model, the minimal number of distinct tile types necessary to self-assemble a shape, at some scale, can be bounded both above and below in terms of the shape’s Kolmogorov complexity. As part of the proof, we develop a universal constructor for this model of self-assembly that can execute an arbitrary Turing machine program specifying how to grow a shape. Our result implies, somewhat counterintuitively, that self-assembly of a scaled-up version of a shape often requires fewer tile types. Furthermore, the independence of scale in self-assembly theory appears to play the same crucial role as the independence of running time in the theory of computability. This leads to an elegant formulation of languages of shapes generated by self-assembly. Considering functions from bit strings to shapes, we show that the running-time complexity, with respect to Turing machines, is polynomially equivalent to the scale complexity of the same function implemented via self-assembly by a finite set of tile types. Our results also hold for shapes defined by Wang tiling—where there is no sense of a self-assembly process—except that here time complexity must be measured with respect to nondeterministic Turing machines.
TL;DR: In this article, the norm of the Hilbert transform as an operator in the weighted space L P R (ω) for 2 < p < oo is bounded by a constant multiple of the first power of the classical Ap characteristic of ω.
Abstract: We show that the norm of the Hilbert transform as an operator in the weighted space L P R (ω) for 2 < p < oo is bounded by a constant multiple of the first power of the classical Ap characteristic of ω. This result is sharp. We also prove a bilinear imbedding theorem with simple conditions.
TL;DR: It is shown that a semidefinite programming (SDP) relaxation for this nonconvex quadratically constrained quadratic program (QP) provides an O(1/\ln(m)$ approximation, which is analogous to a result of Nemirovski e for the real case.
Abstract: We consider the NP-hard problem of finding a minimum norm vector in $n$-dimensional real or complex Euclidean space, subject to $m$ concave homogeneous quadratic constraints. We show that a semidefinite programming (SDP) relaxation for this nonconvex quadratically constrained quadratic program (QP) provides an $O(m^2)$ approximation in the real case and an $O(m)$ approximation in the complex case. Moreover, we show that these bounds are tight up to a constant factor. When the Hessian of each constraint function is of rank $1$ (namely, outer products of some given so-called steering vectors) and the phase spread of the entries of these steering vectors are bounded away from $\pi/2$, we establish a certain “constant factor” approximation (depending on the phase spread but independent of $m$ and $n$) for both the SDP relaxation and a convex QP restriction of the original NP-hard problem. Finally, we consider a related problem of finding a maximum norm vector subject to $m$ convex homogeneous quadratic constraints. We show that an SDP relaxation for this nonconvex QP provides an $O(1/\ln(m))$ approximation, which is analogous to a result of Nemirovski e [Math. Program., 86 (1999), pp. 463-473] for the real case.
09 May 2007
TL;DR: In this article, the authors considered the nonlinear eigenvalue problem of the nonhomogeneous quasilinear problem and proved that 1 0 sufficiently small is an eigen value.
Abstract: We consider the nonlinear eigenvalue problem -div (|∇ u | p(x) - 2 ∇ u ) = λ( u ( q(x) - 2 u in Ω, u = 0 on ∂Ω, where Ω is a bounded open set in R N with smooth boundary and p, q are continuous functions on Ω such that 1 0 sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland's variational principle.
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TL;DR: This article introduces the bounded synthesis approach, which makes it possible to traverse this immense search space in a structured manner and demonstrates that bounded synthesis solves many synthesis problems that were previously considered intractable.
Abstract: The bounded synthesis problem is to construct an implementation that satisfies a given temporal specification and a given bound on the number of states. We present a solution to the bounded synthesis problem for linear-time temporal logic (LTL), based on a novel emptiness-preserving translation from LTL to safety tree automata. For distributed architectures, where standard unbounded synthesis is in general undecidable, we show that bounded synthesis can be reduced to a SAT problem. As a result, we obtain an effective algorithm for the bounded synthesis from LTL specifications in arbitrary architectures. By iteratively increasing the bound, our construction can also be used as a semi-decision procedure for the unbounded synthesis problem.
TL;DR: In this paper, the steady state fractional advection dispersion equation (FADE) on bounded domains in ℝd is discussed and a theoretical framework for the variational solution of FADE is presented.
Abstract: In this article, we discuss the steady state fractional advection dispersion equation (FADE) on bounded domains in ℝd. Fractional differential and integral operators are defined and analyzed. Appropriate fractional derivative spaces are defined and shown to be equivalent to the fractional dimensional Sobolev spaces. A theoretical framework for the variational solution of the steady state FADE is presented. Existence and uniqueness results are proven, and error estimates obtained for the finite element approximation. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 256–281, 2007
TL;DR: In this paper, the authors investigated the boundedness of unimodular Fourier multipliers on modulation spaces and showed that the multipliers with general symbol e i | ξ | α, where α ∈ [ 0, 2 ], are bounded on all modulation spaces, but, in general, fail to be bounded on the usual L p -spaces.
TL;DR: The convergence rate of the deterministic solution algorithm is analysed in terms of the number N of deterministic problems to be solved as both the chaos dimension M and the multiresolution level of the sparse discretization resp.
Abstract: A scalar, elliptic boundary-value problem in divergence form with stochastic diffusion coefficient a (x, co) in a bounded domain D R d is reformulated as a deterministic, infinite-dimensional, parametric problem by separation of deterministic (x ∈ D) and stochastic (ω e Q) variables in a(x,ω) via Karhunen-Loeve or Legendre expansions of the diffusion coefficient. Deterministic, approximate solvers are obtained by projection of this problem into a product probability space of finite dimension M and sparse discretizations of the resulting M-dimensional parametric problem. Both Galerkin and collocation approximations are considered. Under regularity assumptions on the fluctuation of a(x, ω) in the deterministic variable x, the convergence rate of the deterministic solution algorithm is analysed in terms of the number N of deterministic problems to be solved as both the chaos dimension M and the multiresolution level of the sparse discretization resp. the polynomial degree of the chaos expansion increase simultaneously.
TL;DR: This paper develops the idea of min-max robust experiment design for dynamic system identification and proposes a convex optimisation algorithm that can be applied more generally to a discretised approximation to the design problem.
TL;DR: A discretized version of the fractional Laplacian operator -(-delta)(alpha/2) which is well suited to deal with boundary conditions on a finite interval is provided.
Abstract: The fractional Laplacian operator -(-delta)(alpha/2) appears in a wide class of physical systems, including Levy flights and stochastic interfaces. In this paper, we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval. The implementation of boundary conditions is justified by appealing to two physical models, namely, hopping particles and elastic springs. The eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions. Some analytical results concerning the structure of the eigenvalue spectrum are also obtained.