Rational approximation to real points on conics

In this paper we summarize new results on the regularity of optimal controls for dynamic optimization problems with functional inequality constraints, a control constraint expressed in terms of a general closed convex set and a coercive cost function. Recently it has been shown that the linear independence condition on active state constraints, present in the earlier literature, can be replaced by a less restrictive, positive linear independence condition, that requires linear independence merely with respect to non-negative weighting parameters, provided the control constraint set is non-time varying. We show that, if the control constraint set, regarded as a time dependent multifunction, is merely Lipschitz continuous with respect to the time variable, then optimal controls can fail to be Lipschitz continuous. In these circumstance, however, a weaker regularity property (Holder continuity with Holder index 1/2) can be established. On the other hand, Lipschitz continuity of optimal controls is guaranteed for time varying control sets under a positive linear independence hypothesis, when the control constraint sets are described, at each time, by a finite collection of functional inequalities.

For a nonlinear programming problem with a canonical perturbations, we give an elementary proof of the following result: If the Karush---Kuhn---Tucker map is locally single-valued and Lipschitz continuous, then the linear independence condition for the gradients of the active constraints and the strong second-order sufficient optimality condition hold.

Using standard nonlinear programming (NLP) theory, we establish formulas for first and second order directional derivatives of optimal value functions of parametric mathematical programs with complementarity constraints (MPCCs). The main point is that under a linear independence condition on the active constraint gradients, optimal value sensitivity of MPCCs is essentially the same as for nonlinear programs, in spite of the combinatorial nature of the MPCC feasible set. Unlike NLP however, second order directional derivatives of the MPCC optimal value function show combinatorial structure.

Topology identification of biological neural networks has received great attention in recent years. But most existing researches ignored the importance of linear independence condition. Once network occurs inner synchronization, such a condition would inevitably break, which could lead to the failure of topology identification. However, synchronous firing between neurons is unavoidable in biological neural network. This phenomenon makes it difficult to identify the topology of neural networks. In this paper, we proposed a new method to identify Hindmarsh-Rose Neural Networks. By this method, the original neural network can implement inner desynchronization and the topology observers can successfully identify the unknown topology. It is worth mentioning that the our method can ensure that the linear independence condition naturally holds. This means that topology identification will not meet with failure. Finally, the effectiveness of the proposed method is successfully validated by numerical simulations.

Topology identification of complex dynamical networks received extensive attention in the past decade. Most existing studies rely heavily on the linear independence condition (LIC). We find that a critical step in using this condition is not rigorous. Besides, it is difficult to verify this condition. Without regulating the original network, possible identification failure caused by network synchronization cannot be avoided. In this paper, we propose a new method to overcome these shortcomings. We add a regulation mechanism to the original network and construct an auxiliary network consisting of isolated nodes. Along with the outer synchronization between the regulated network and the auxiliary network, we show that the original network can be identified. Our method can avoid identification failure caused by network synchronization. Moreover, we show that there is no need to check the LIC. We finally provide some examples to demonstrate that our method is reliable and has good performances.

The problem of linear independence of the integer translates of μ * B, where μ is a compactly supported distribution and B is an exponential box spline, is considered in this paper. The main result relates the linear independence issue with the distribution of the zeros of the Fourier-Laplace transform, of μ on certain linear manifolds associated with B. The proof of our result makes an essential use of the necessary and sufficient condition derived in [12]. Several applications to specific situations are discussed. Particularly, it is shown that if the support of μ is small enough then linear independence is guaranteed provided that does not vanish at a certain finite set of critical points associated with B. Also, the results here provide a new proof of the linear independence condition for the translates of B itself.

We explicitly compute the exponents of quadratic approximation of two algebraic elements over fields of power series with coefficients in a finite field. We prove that they have well quadratic approximation although their rational approximation is bad.

Let us suppose that ƒ(x, y) is an indefinite binary quadratic form that does not represent zero. If P is the real point (x0, y0) then we may define where the infimum is taken over all integral x, y. The inhomogeneous minimum of the form ƒ is defined where the supremum taken over all real points P, need only extend over some complete set of points, incongruent mod 1.

Many signal processing and machine learning methods share essentially the same linear-in-the-parameter model, with as many parameters as available samples as in kernel-based machines. Sparse approximation is essential in many disciplines, with new challenges emerging in online learning with kernels. To this end, several sparsity measures have been proposed in the literature to quantify sparse dictionaries and constructing relevant ones, the most prolific ones being the distance, the approximation, the coherence and the Babel measures. In this paper, we analyze sparse dictionaries based on these measures. By conducting an eigenvalue analysis, we show that these sparsity measures share many properties, including the linear independence condition and inducing a well-posed optimization problem. Furthermore, we prove that there exists a quasi-isometry between the parameter (i.e., dual) space and the dictionary's induced feature space.

In this paper, a simple design of a linear dispersive space-time-frequency (STF) code is proposed with a design criterion to achieve high rate for frequency-selective channels in terms of multipath when the Partial Interference Cancellation Group (PIC) group decoding is applied at receiver. With an appropriate grouping scheme as well as the PIC group decoding, the proposed STF code is shown to obtain the similar diversity gain as the maximum likelihood (ML) decoding, namely full-dimensional sphere decoding, but have a low decoding complexity. It seems as an intermediate decoding between the ML receiver and zero-forcing (ZF) receiver. The proposed grouping design criterion for the PIC group decoding to achieve full diversity deploying the orthogonal frequency-division multiplexing (OFDM) technique is also an intermediate condition between the loosest ML full rank criterion of codewords and the strongest ZF linear independence condition of the column vectors for the equivalent frequency-selective channel matrix. It can achieves full diversity with the PIC group decoding for any number of sub-carriers and the data rate can be made high. Simulation results show that the proposed STF code can well address the rate-performance-complexity tradeoff of the multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) communication system.

In this paper, we study the relative error in the numerical solution of a linear ordinary differential equation y'(t) = Ay(t), t ≥ 0, where A is a normal matrix. The numerical solution is obtained by using at any step an approximation of the matrix exponential, e.g., a polynomial or a rational approximation. The error of the numerical solution with respect to the exact solution is due to this approximation as well as to a possible perturbation in the initial value. For an unperturbed initial value, we have found: (1) unlike the absolute error, the relative error always grows linearly in time; (2) in the long-time, the contributions to the relative error relevant to non-rightmost eigenvalues of A disappear.

On matrix exponential approximations of ruin probabilities for the classic and Brownian perturbed Cramér–Lundberg processes

Consider the classical problem of rational simultaneous approximation to a point in \({\mathbb {R}}^{n}\). The optimal lower bound on the gap between the induced ordinary and uniform approximation exponents has been established by Marnat and Moshchevitin in 2018. Recently Nguyen, Poels and Roy provided information on the best approximating rational vectors to the points where the gap is close to this minimal value. Combining the latter result with parametric geometry of numbers, we effectively bound the dual linear form exponents in the described situation. As an application, we slightly improve the upper bound for the classical exponent of uniform Diophantine approximation \({\widehat{\lambda }}_{n}(\xi )\), for even \(n\ge 4\). Unfortunately our improvements are small, for \(n=4\) only in the fifth decimal digit. However, the underlying method in principle can be improved with more effort to provide better bounds. We indeed establish reasonably stronger results for numbers which almost satisfy equality in the estimate by Marnat and Moshchevitin. We conclude with consequences on the classical problem of approximation to real numbers by algebraic numbers/integers of uniformly bounded degree.

In 1988 Rieger exhibited a differentiable function having a zero at the golden ratio (−1 + )/2 for which when Newton's method for approximating roots is applied with an initial value x0 = 0, all approximates are so-called “best rational approximates”—in this case, of the form F2n/F2n+1, where Fn denotes the n-th Fibonacci number. Recently this observation was extended by Komatsu to the class of all quadratic irrationals whose continued fraction expansions have period length 2. Here we generalize these observations by producing an analogous result for all quadratic irrationals and thus provide an explanation for these phenomena.

Consider a real matrix $\Theta$ consisting of rows $(\theta_{i,1},\ldots,\theta_{i,n})$, for $1\leq i\leq m$. The problem of making the system linear forms $x_{1}\theta_{i,1}+\cdots+x_{n}\theta_{i,n}-y_{i}$ for integers $x_{j},y_{i}$ small naturally induces an ordinary and a uniform exponent of approximation, denoted by $w(\Theta)$ and $\widehat{w}(\Theta)$ respectively. For $m=1$, a sharp lower bound for the ratio $w(\Theta)/\widehat{w}(\Theta)$ was recently established by Marnat and Moshchevitin. We give a short, new proof of this result upon a hypothesis on the best approximation integer vectors associated to $\Theta$. Our conditional result extends to general $m>1$ (but may not be optimal in this case). Moreover, our hypothesis is always satisfied in particular for $m=1, n=2$ and thereby unconditionally confirms a previous observation of Jarn\'ik. We formulate our results in the more general context of approximation of subspaces of Euclidean spaces by lattices. We further establish criteria upon which a given number $\ell$ of consecutive best approximation vectors are linearly independent. Our method is based on Siegel's Lemma.