Optimal Lower Bound Research Articles

An isoperimetric result for the fundamental frequency via domain derivative

The Faber–Krahn deficit \(\delta \lambda \) of an open bounded set \(\Omega \) is the normalized gap between the values that the first Dirichlet Laplacian eigenvalue achieves on \(\Omega \) and on the ball having same measure as \(\Omega \). For any given family of open bounded sets of \(\mathbb R ^N\) (\(N\ge 2\)) smoothly converging to a ball, it is well known that both \(\delta \lambda \) and the isoperimetric deficit \(\delta P\) are vanishing quantities. It is known as well that, at least for convex sets, the ratio \(\frac{\delta P}{\delta \lambda }\) is bounded by below by some positive constant (Brandolini et al., Arch Math (Basel) 94(4): 391–400, 2010; Payne and Weinberger, J Math Anal Appl 2:210–216, 1961), and in this note, using the technique of the shape derivative, we provide the explicit optimal lower bound of such a ratio as \(\delta P\) goes to zero.

An Algorithm for Finding Optimal Lower Bounds on Ramsey Numbers Based on Cyclic Graphs

An Algorithm for Finding Optimal Lower Bounds on Ramsey Numbers Based on Cyclic Graphs

Open shop scheduling problem to minimize makespan with release dates

The scheduling problem of open shop to minimize makespan with release dates is investigated in this paper. Unlike the usual researches to confirm the conjecture that the tight worst-case performance ratio of the Dense Schedule (DS) is 2−1/m, where m is the number of machines, the asymptotic optimality of the DS is proven when the problem scale tends to infinity. Furthermore, an on-line heuristic based on DS, Dynamic Shortest Processing Time-Dense Schedule, is presented to deal with the off-line and on-line versions of this problem. At the end of the paper, an asymptotically optimal lower bound is provided and the results of numerical experiments show the effectiveness of the heuristic.

Optimal Lower Bounds for Anonymous Scheduling Mechanisms

We consider the problem of designing truthful mechanisms on m unrelated machines, to minimize some optimization goal. Nisan and Ronen [Nisan, N., A. Ronen. 2001. Algorithmic mechanism design. Games Econom. Behav. 35 166–196] consider the specific goal of makespan minimization, and show a lower bound of 2, and an upper bound of m. This large gap inspired many attempts that yielded positive results for several special cases, but very partial success for the general case: the lower bound was slightly increased to 2.61 by Christodoulou et al. [Christodoulou, G., E. Koutsoupias, A. Kovács. 2010. Mechanism design for fractional scheduling on unrelated machines. ACM Trans. Algorithms (TALG) 6(2) 1–18] and Koutsoupias and Vidali [Koutsoupias, E., A. Vidali. 2007. A lower bound of 1+phi for truthful scheduling mechanisms. Proc. 32nd Internat. Sympos. Math. Foundations Comput. Sci. (MFCS)], while the best upper bound remains unchanged. In this paper we show the optimal lower bound on truthful anonymous mechanisms: no such mechanism can guarantee an approximation ratio better than m. Moreover, our proof yields similar optimal bounds for two other optimization goals: the sum of completion times and the lp norm of the schedule.

Optimal lower bound of the resonance widths for a Helmholtz tube-shaped resonator

The study of the resonances of theHelmholtz resonator has been broadly described in previous works (see [11] and references therein). Here, for a simple 2-dimensional resonator in the shape of a tube, we analyze the transition zone where oscillations start to appear. Following a careful analysis, we obtain an optimal lower bound of the width of the resonances.

Homogeneous polynomials and extensions of Hardy‐Hilbert's inequality

AbstractIf L is a continuous symmetric n‐linear form on a real or complex Hilbert space and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\widehat{L}$\end{document} is the associated continuous n‐homogeneous polynomial, then \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Vert L\Vert =\big \Vert \widehat{L}\big \Vert$\end{document}. We give a simple proof of this well‐known result, which works for both real and complex Hilbert spaces, by using a classical inequality due to S. Bernstein for trigonometric polynomials. As an application, an open problem for the optimal lower bound of the norm of a homogeneous polynomial, which is a product of linear forms, is related to the so‐called permanent function of an n × n positive definite Hermitian matrix. We have also derived generalizations of Hardy‐Hilbert's inequality.

On-line approximate string matching with bounded errors

We introduce a new dimension to the widely studied on-line approximate string matching problem, by introducing an error threshold parameter ϵ so that the algorithm is allowed to miss occurrences with probability ϵ. This is particularly appropriate for this problem, as approximate searching is used to model many cases where exact answers are not mandatory. We show that the relaxed version of the problem allows us breaking the average-case optimal lower bound of the classical problem, achieving average case O(nlogσm/m) time with any ϵ=poly(k/m), where n is the text size, m the pattern length, k the number of differences for edit distance, and σ the alphabet size. Our experimental results show the practicality of this novel and promising research direction. Finally, we extend the proposed approach to the multiple approximate string matching setting, where the approximate occurrence of r patterns are simultaneously sought. Again, we can break the average-case optimal lower bound of the classical problem, achieving average case O(nlogσ(rm)/m) time with any ϵ=poly(k/m).

Two-sided bounds of the discretization error for finite elements

We derive an optimal lower bound of the interpolation error for linear finite elements on a bounded two-dimensional domain. Using the supercloseness between the linear interpolant of the true solution of an elliptic problem and its finite element solution on uniform partitions, we further obtain two-sided ap rioribounds of the discretization error by means of the interpolation error. Two-sided bounds for bilinear finite elements are given as well. Numerical tests illustrate our theoretical analysis.

A note on global existence of weak solutions to the compressible magnetohydrodynamic equations with Coulomb force

In this note, for the case of (9+33)/12<γ⩽4/3, we prove the existence of global-in-time finite energy weak solution of the equations of a two-dimensional magnetohydrodynamics with Coulomb force, where γ denotes the adiabatic exponent. The value (9+33)/12 is the optimal lower bound of γ to establish global-in-time finite energy weak solution under current frame.

On the Seshadri constants of adjoint line bundles

In the present paper we study the possible values of Seshadri constants. While in general every positive rational number appears as the local Seshadri constant of some ample line bundle, we point out that for adjoint line bundles there are explicit lower bounds depending only on the dimension of the underlying variety. In the surface case, where the optimal lower bound is 1/2, we characterize all possible values in the range between 1/2 and 1—there are surprisingly few. As expected, one obtains even more restrictive results for the Seshadri constants of adjoints of very ample line bundles. Finally, we study Seshadri constants of adjoint line bundles in the multi-point setting.

Resonances for Manifolds Hyperbolic Near Infinity: Optimal Lower Bounds on Order of Growth

Resonances for Manifolds Hyperbolic Near Infinity: Optimal Lower Bounds on Order of Growth

Lower Bounds to the Performance of Bit Synchronization for Bandwidth Efficient Pulse-Shaping

Bandwidth efficient signaling requires using pulse-shaping that inherently has intersymbol interference (ISI) except when the timing is perfect. Several lower bounds to the mean-square-error (MSE) of non-data-aided bit-synchronizers are derived for this case. The optimal lower bound is derived using the maximum likelihood (ML) criterion for a sequence of binary pulse amplitude modulated pulses in the presence of ISI and Gaussian noise. This lower bound is used as a benchmark to evaluate the performance of other synchronizers in a practical scenario. It is shown that a previous lower bound based on the ISI-free ML synchronizer cannot be used to lower bound the MSE of bit-synchronizers. A detection theory bound (DTB) (also called Ziv-Zakai bound) is applied to the symbol timing recovery problem in the presence of ISI and it is shown that this bound is a tight lower bound on the MSE of the ML synchronizer. A simple lower bound on this DTB is derived and it is shown that the simple bound is almost as tight as the well known modified Cramer-Rao bound (MCRB) at moderate values of SNR, while it does not suffer from the shortcomings of the MCRB at small values of SNR.

Optimal Lower Bounds on Local Stress inside Random Media

A methodology is presented for bounding the higher $L^p$ norms, $2\leq p\leq\infty$, of the local stress inside random media. We present optimal lower bounds that are given in terms of the applied loading and volume fractions for random two phase composites. These bounds provide a means to measure load transfer across length scales relating the excursions of the local fields to applied loads. These results deliver tight upper bounds on the macroscopic strength domains for statistically defined heterogeneous media.

Optimal lower bound of the grow-up rate for a supercritical parabolic equation

We study the behavior of solutions of the Cauchy problem for a supercritical semilinear parabolic equation which approach a singular steady state from below as t → ∞ . It is known that the grow-up rate of such solutions depends on the spatial decay rate of initial data. We give an optimal lower bound on the grow-up rate by using a comparison technique based on a formal asymptotic analysis.

Asymptotically Optimal Lower Bounds for the Condition Number of a Real Vandermonde Matrix

Lower bounds on the condition number $\min\kappa_p(V)$ of a real Vandermonde matrix V are established in terms of the dimension n or n and the largest absolute value among all nodes that define the Vandermonde matrix. All bounds here are asymptotically sharp, similar to those in Beckermann (Numer. Math., 85 (2000), pp. 553–577), but bounds here are sharper and cover more cases. Also, qualitative behaviors of $\min\kappa_p(V)$, as well as nearly optimally conditioned real Vandermonde matrices, as functions of the largest absolute value among all nodes are obtained.

Communication complexity towards lower bounds on circuit depth

Karchmer, Raz, and Wigderson (1995) discuss the circuit depth complexity of n-bit Boolean functions constructed by composing up to d = log n/log log n levels of k = log n-bit Boolean functions. Any such function is in AC1 . They conjecture that circuit depth is additive under composition, which would imply that any (bounded fan-in) circuit for this problem requires $ dk \in \Omega({\rm log}^2 n/{\rm log\,log}\,n) $ depth. This would separate AC1 from NC1. They recommend using the communication game characterization of circuit depth. In order to develop techniques for using communication complexity to prove circuit depth lower bounds, they suggest an intermediate communication complexity problem which they call the Universal Composition Relation. We give an almost optimal lower bound of dk—O(d 2(k log k)1/2) for this problem. In addition, we present a proof, directly in terms of communication complexity, that there is a function on k bits requiring $ \Omega(k) $ circuit depth. Although this fact can be easily established using a counting argument, we hope that the ideas in our proof will be incorporated more easily into subsequent arguments which use communication complexity to prove circuit depth bounds.

On Ricci curvature of certain complex bundles

We study a class of compact complex manifolds, with positive first Chern class C 1 , fibered over products of projective spaces. When the spaces of the basis don't have the same dimension, we know that these bundles cannot carry Einstein–Kähler metrics. In this article, we construct a metric belonging to C 1 , with an optimal lower bound of the Ricci curvature.

Bounds for Dispersers, Extractors, and Depth-Two Superconcentrators

We show that the size of the smallest depth-two $N$-superconcentrator is $$ \Theta(N\log^2 N/\log\log N). $$ Before this work, optimal bounds were known for all depths except two. For the upper bound, we build superconcentrators by putting together a small number of disperser graphs; these disperser graphs are obtained using a probabilistic argument. For obtaining lower bounds, we present two different methods. First, we show that superconcentrators contain several disjoint disperser graphs. When combined with the lower bound for disperser graphs of Kovari, Sos, and Turan, this gives an almost optimal lower bound of $\Omega( N (\log N/\log \log N)^2)$ on the size of $N$-superconcentrators. The second method, based on the work of Hansel, gives the optimal lower bound. The method of Kovari, Sos, and Turan can be extended to give tight lower bounds for extractors, in terms of both the number of truly random bits needed to extract one additional bit and the unavoidable entropy loss in the system. If the input is an $n$-bit source with min-entropy $k$ and the output is required to be within a distance of $\epsilon$ from uniform distribution, then to extract even one additional bit, one must invest at least $\log(n-k) + 2\log(1/\epsilon) - O(1)$ truly random bits; to obtain $m$ output bits one must invest at least $m-k+2\log(1/\epsilon)-O(1)$. Thus, there is a loss of $2\log(1/\epsilon)$ bits during the extraction. Interestingly, in the case of dispersers this loss in entropy is only about $\log\log (1/\epsilon)$.

On the optimal bounds for the effective conductivity of isotropic quasi-symmetric multiphase media

The bounds on the effective conductivity of isotropic multiphase media with certain symmetry among the phases' geometry are examined. The bounds of Phan-Thien and Milton and those of Pham partly coincide, partly differ. Wherever they differ, the bounds of Phan-Thien and Milton are more restrictive. Specifically, the bounds for the subclass of spherical cell materials are identical, while the bounds of Phan-Thien and Milton for the subclass of platelet cell materials, which coincide with those of Pham in the case of two-component materials, are tighter in the general multicomponent case. Phan-Thien-Milton upper bound on the conductivity of multicomponent platelet cell materials is attained by a geometric model of Pham, therefore is verified to be the optimal one. Subsequently, for the whole class of isotropic quasi-symmetric media, the respective bound is optimal over a range of parameters. An optimal lower bound is conjectured. The differential scheme is used to construct certain spherical and platelet cell models.

Approximating Threshold Circuits by Rational Functions

Motivated by the problem of understanding the limitations of threshold networks for representing boolean functions, we consider size-depth trade-offs for threshold circuits that compute the parity function. Using a fundamental result in the theory of rational approximation, we show how to approximate small threshold circuits by rational functions of low degree. We apply this result to establish an almost optimal lower bound of Ω(n2/ln2n) on the number of edges of any depth-2 threshold circuit with polynomially bounded weights that computes the parity function. We also prove that any depth-3 threshold circuit with polynomially bounded weights requires Ω(n1.2/ln5/3n) edges to compute parity. On the other hand, we give a construction of a depth d threshold circuit that computes parity with n1+1/Θ(φd) edges where φ = (1 + √5)/2 is the golden ratio. We conjecture that there are no linear size bounded depth threshold circuits for computing parity.