Tight Sufficient Condition Research Articles

Fractional matching, factors and spectral radius in graphs involving minimum degree

A fractional matching of a graph G is a function f:E(G)→[0,1] such that for any v∈V(G), ∑e∈EG(v)f(e)≤1, where EG(v)={e∈E(G):eis incident withvinG}. The fractional matching number of G is μf(G)=max{∑e∈E(G)f(e):f is a fractional matching of G}. Let k∈(0,n) is an integer. In this paper, we prove a tight lower bound of the spectral radius to guarantee μf(G)>n−k2 in a graph with minimum degree δ, which implies the result on the fractional perfect matching due to Fan et al. (2022) [6].For a set {A,B,C,…} of graphs, an {A,B,C,…}-factor of a graph G is defined to be a spanning subgraph of G each component of which is isomorphic to one of {A,B,C,…}. We present a tight sufficient condition in terms of the spectral radius for the existence of a {K2,{Ck}}-factor in a graph with minimum degree δ, where k≥3 is an integer. Moreover, we also provide a tight spectral radius condition for the existence of a {K1,1,K1,2,…,K1,k}-factor with k≥2 in a graph with minimum degree δ, which generalizes the result of Miao et al. (2023) [10].

The Q-index and Connectivity of Graphs

A connected graph G is said to be k-connected if it has more than k vertices and remains connected whenever fewer than k vertices are deleted. In this paper, for a connected graph G with sufficiently large order, we present a tight sufficient condition for G with fixed minimum degree to be k-connected based on the Q-index. Our result can be viewed as a spectral counterpart of the corresponding Dirac-type condition.

An asymptotic decoupling approach for adaptive control with unmeasurable coupled dynamics

SummaryWhile adaptive control methods have the capability to suppress the effect of system uncertainties without excessive reliance on dynamical system models, their stability can be adversely affected in the presence of coupled dynamics. Motivated by this standpoint, the contribution of this article is a decoupling approach for model reference adaptive control algorithms. The key feature of the proposed framework is that it guarantees asymptotic convergence between the trajectories of an uncertain dynamical system and a given reference model without relying on any measurements from the coupled dynamics under a tight sufficient stability condition. We also provide a generalization to address the uncertainty in the control effectiveness matrix, where the resulting sufficient stability condition in this case relies on linear matrix inequalities. Finally, numerical examples are provided to illustrate the efficacy of the presented theoretical results.

Identifiability of Multichannel Blind Deconvolution and Nonconvex Regularization Algorithm

In this paper, we consider the multichannel blind deconvolution problem, where we observe the output of channels $\mathbf {h}_{i}\in \mathbb {R}^n$ ( $i=1,...,N$ ) that all convolve with the same unknown input signal $\mathbf {x}\in \mathbb {R}^n$ . We wish to estimate the input signal and blur kernels simultaneously. Existing theoretical results showed that the original inputs are identifiable under subspace assumptions. However, the subspaces discussed before were randomly or generically chosen. Here, we propose deterministic subspace assumption, which is widely used in practice, and give some theoretical results. First of all, we derive tight sufficient condition for identifiability of signal and convolution kernels, which is only violated on a set of Lebesgue measure zero. Then, we present a nonconvex regularization algorithm by a lifting method and approximate the rank-one constraint via the difference of nuclear norm and Frobenius norm. The global minimizer of the proposed nonconvex algorithm is rank-one matrix under mild conditions on parameters and noise level. The stability result is also shown under the assumption that the inputs lie in a compact set. Besides, the computation of our regularization model is carried out and any limit point of iterations converges to a stationary point of our model. Finally, we provide numerical experiments to show that our nonconvex regularization model outperforms convex relaxation models, such as nuclear norm minimization and some nonconvex methods, such as alternating minimization method and spectral method.

Enhancing the sufficient condition of sparsity pattern recovery

The task of recovering a sparse signal from its noisy observations can be divided into two stages; recovering the positions of the nonzero entries or support and estimating their values. In this paper, fundamental limits of sparsity pattern recovery is investigated. Although this problem has been addressed in the recent literature, there is still a gap between those results and the exact limits. In this work, first a tight upper bound is derived on the error probability of the exact sparsity pattern recovery when the Maximum Likelihood (ML) decoder is exploited. Next, a sharp sufficient condition for perfect recovery of the support from noisy projections is derived. In their analyses, the authors treat random sampling matrix with standard normal entries and Gaussian noise. It is shown that the sufficient number of observations for perfect support recovery depends on the noise variance, the minimum nonzero entry of the sparse vector, and the sparsity parameter (k) which is in agreement with previous findings. The proposed upper bound is highly close to the exact probability of error, leading to tight sufficient condition which also matches the growth rate of the necessary conditions of the previous established works. More precisely, the proposed sufficient condition and the previously-established necessary condition on the number of samples are both O(k) (linear in k). The results are proved and supported both theoretically and by simulations.

Signless Laplacian spectral radius and Hamiltonicity of graphs with large minimum degree

ABSTRACTIn this paper, we establish a tight sufficient condition for the Hamiltonicity of graphs with large minimum degree in terms of the signless Laplacian spectral radius and characterize all extremal graphs. Moreover, we prove a similar result for balanced bipartite graphs. Additionally, we construct infinitely many graphs to show that results proved in this paper give new strength for one to determine the Hamiltonicity of graphs.

Spectral radius and k-connectedness of a graph

A connected graph G is said to be k-connected if it has more than k vertices and remains connected whenever fewer than k vertices are deleted. In this paper, we present a tight sufficient condition for a connected graph with fixed minimum degree to be k-connected based on its spectral radius, for sufficiently large order.

A Tight Sufficient Condition for Recursive Formulation of Dynamic Implementation Problems

This note explores conditions that admit recursive representation for a class of dynamic mechanism design problems. We derive a tight sufficient condition, called the common state property (CSP), which ensures that temporary incentive constraints guarantee implementability, and so allows to characterize the principal's problem recursively. The condition imposes no restrictions on agent's preferences and only concerns the properties of the evolution of private information.

Pay-as-Bid: Selling Divisible Goods to Informed Bidders

Pay-as-bid is the most popular auction format for selling treasury securities. We prove the uniqueness of pure-strategy Bayesian-Nash equilibria in pay-as-bid auctions where symmetrically-informed bidders face uncertain supply, and we establish a tight sufficient condition for the existence of this equilibrium. Equilibrium bids have a convenient separable representation: the bid for any unit is a weighted average of marginal values for larger quantities. With optimal supply and reserve price, the pay-as-bid auction is revenue-equivalent to the uniform-price auction.

Regular type distributions in mechanism design and $$\rho $$ -concavity

Some of the best-known results in mechanism design depend critically on Myerson’s (Math Oper Res 6:58–73, 1981) regularity condition. For example, the second-price auction with reserve price is revenue maximizing only if the type distribution is regular. This paper offers two main findings. First, a new interpretation of regularity is developed—similar to that of a monotone hazard rate—in terms of being the next to fail. Second, using expanded concepts of concavity, a tight sufficient condition is obtained for a density to define a regular distribution. New examples of regular distributions are identified. Applications are discussed.

Optimal Design and p-Concavity

Some of the most beautiful results in mechanism design depend crucially on Myerson?s (1981) regularity condition. E.g., the second-price auction with reserve price is revenue maximizing only if the type distribution is regular. This paper offers two main results. First, an interpretation of regularity is developed in terms of being the next to fail. Second, using expanded concepts of concavity, a tight sufficient condition on the density function is formulated. New examples of parameterized distributions are shown to be regular. Applications include standard design problems, optimal reserve prices, the analysis of bidding data, and multidimensional types.

Long-range out-of-sample properties of autoregressive neural networks

We consider already-trained discrete autoregressive neural networks in their most general representations, with the exclusion of time-varying input though, and we provide tight sufficient condition...

A tight sufficient condition for Radner–Stiglitz nonconcavity in the value of information

This paper deals with the existence of a nonconcavity in the value of information, as was first explained by Radner and Stiglitz [A nonconcavity in the value of information, in: M. Boyer, R.E. Kihlstrom (Eds.), Bayesian Models in Economic Theory, Elsevier Science Publishers, Amsterdam, 1984, pp. 33–52 (Chapter 3)]. After defining infinitesimal information distance variation IIDV , we find that IIDV = 0 is sufficient for a zero marginal value of information at the null. This is a condition only on the information structure and in particular is independent of the decision maker's preferences. This condition is tight: when IIDV > 0 , there exists a payoff function for which the marginal value of information at the null is positive under general assumptions.

Learning buyers' valuation distribution in posted-price selling

A dynamic pricing model is studied where a seller of an asset faces a sequence of potential buyers whose valuation distribution is unknown to the seller. The seller learns more about the distribution in the selling process and becomes less optimistic as the object remains unsold. We characterize the optimal posted prices which incorporate updated beliefs every period, and derive a rather tight sufficient condition under which these prices decline over time. An example is provided where the optimal prices can actually increase over time if the condition is violated.

The inverse M-matrix problem

One aspect of the inverse M-matrix problem can be posed as follows. Given a positive n × n matrix A=( a i j ) which has been scaled to have unit diagonal elements and off-diagonal elements which satisfy 0 < y ⩽ a i j ⩽ x < 1, what additional element conditions will guarantee that the inverse of A exists and is an M-matrix? That is, if A −1= B=( b i j ), then b ii > 0 and b i j ⩽ 0 for i≠ j. If n=2 or x= y no further conditions are needed, but if n ⩾ 3 and y < x, then the following is a tight sufficient condition. Define an interpolation parameter s via x 2= sy+(1− s) y 2; then B is an M-matrix if s −1 ⩾ n−2. Moreover, if all off-diagonal elements of A have the value y except for a i j = a j j = x when i= n−1, n and 1 ⩽ j ⩽ n−2, then the condition on both necessary and sufficient for B to be an M-matrix.