Explicit Intervals Research Articles

Mutli-Robot Cooperative Object Transportation with Guaranteed Safety and Convergence in Planar Obstacle Cluttered Workspaces via Configuration Space Decomposition

In this work, we consider the autonomous object transportation problem employing a team of mobile manipulators within a compact planar workspace with obstacles. As the object is allowed to translate and rotate and each robot is equipped with a manipulator consisting of one or more moving links, the overall system (object and mobile manipulators) should adapt its shape in a flexible way so that it fulfills the transportation task with safety. To this end, we built a sequence of configuration space cells, each of which defines an allowable set of configurations of the object, as well as explicit intervals for each manipulator’s states. Furthermore, appropriately designed under- and over-approximations of the free configuration space are used in an innovative way to guide the configuration space’s exploration without loss of completeness. In addition, we coupled methodologies based on Reference Governors and Prescribed Performance Control with harmonic maps, in order to design a distributed control law for implementing the transitions specified by the high-level planner, which possesses guaranteed invariance and global convergence properties, thus avoiding the requirement for synchronized motion as inherently dictated by the majority of the related works. Furthermore, the proposed low-level control law does not require continuous information exchange between the robots, which rely only on measurements of the object’s configuration and their own states. Finally, a transportation scenario within a complex warehouse workspace demonstrates the proposed approach and verifies its efficiency.

Critical points of Eisenstein series

For any even integer $k \ge 4$, let $\E_k$ be the normalized Eisenstein series of weight $k$ for $\SL_2(\Z)$. Also let $\D$ be the closure of the standard fundamental domain of the Poincar\'e upper half plane modulo $\SL_2(\Z)$. F.~K.~C.~Rankin and H. P. F. Swinnerton-Dyer showed that all zeros of $\E_k$ in $\D$ are of modulus one. In this article, we study the critical points of $\E_k$, that is to say the zeros of the derivative of $\E_k$. We show that they are simple. We count those belonging to $\D$, prove that they are located on the two vertical edges of $\D$ and produce explicit intervals that separate them. We then count those belonging to $\gamma\D$, for any $\gamma \in \SL_2(\Z)$.

Markov Chain Confidence Intervals and Biases

We derive explicit asymptotic confidence intervals for any Markov chain Monte Carlo (MCMC) algorithm with finite asymptotic variance, started at any initial state, without requiring a Central Limit Theorem nor reversibility nor geometric ergodicity nor any bias bound. We also derive explicit non-asymptotic confidence intervals assuming bounds on the bias or first moment, or alternatively that the chain starts in stationarity. We relate those non-asymptotic bounds to properties of MCMC bias, and show that polynomially ergodicity implies certain bias bounds. We also apply our results to several numerical examples. It is our hope that these results will provide simple and useful tools for estimating errors of MCMC algorithms when CLTs are not available.

Competition of Multiple Species in Advective Environments

We study the effect of changes in flow speed on competition of an arbitrary number of species living in advective environments, such as streams and rivers. We begin with a spatial Lotka-Volterra model which is described by n reaction-diffusion-advection equations with Danckwerts boundary conditions. Using the dominant eigenvalue [Formula: see text] of the diffusion-advection operator subject to boundary conditions, we reduce the model to a system of ordinary differential equations. We impose a "transitive arrangement" of the competitors in terms of their interspecific coefficients and growth rates, which means that in the absence of advection, we have the following situation: for all [Formula: see text], species i out-competes species j, while species j has higher intrinsic growth rate than species i. Changing advection speed in the original spatial model corresponds to changing the value of [Formula: see text] in the spatially implicit model. Considering the cases of the odd and even n separately, we obtain explicit intervals of the values of [Formula: see text] that allow all n species to be present in the habitat (coexistence interval). Stability of this equilibrium is shown for [Formula: see text].

On singularly weighted generalized Laplacian systems and their applications

Abstract We study the homogeneous Dirichlet boundary value problem of generalized Laplacian systems with a singular weight which may not be integrable. Some explicit intervals which correspond to the existence and nonexistence of positive solutions for the system with the finite asymptotic behaviors of the nonlinearities at 0 and ∞ {\infty} are obtained.

Eigenvalues of higher order Sturm-Liouville boundary value problems with derivatives in nonlinear terms

We shall consider the Sturm-Liouville boundary value problem $y^{(m)}(t)+\lambda F (t,y(t),y'(t),\ldots, y^{(q)}(t) )=0$ , $t\in (0,1)$ , $y^{(k)}(0)=0$ , $0\leq k\leq m-3$ , $\zeta y^{(m-2)}(0)-\theta y^{(m-1)}(0)=0$ , $\rho y^{(m-2)}(1)+\delta y^{(m-1)}(1)=0 $ where $m\geq3$ , $1\leq q\leq m-2$ , and $\lambda>0$ . It is noted that the boundary value problem considered has a derivative-dependent nonlinear term, which makes the investigation much more challenging. In this paper we shall develop a new technique to characterize the eigenvalues λ so that the boundary value problem has a positive solution. Explicit eigenvalue intervals are also established. Some examples are included to dwell upon the usefulness of the results obtained.

Existence and multiplicity of positive solutions for nonhomogeneous boundary value problems with fractional q-derivatives

In this paper, we study a class of fractional q-difference equations with nonhomogeneous boundary conditions. By applying the classical tools from functional analysis, sufficient conditions for the existence of single and multiple positive solutions to the boundary value problem are obtained in term of the explicit intervals for the nonhomogeneous term. In addition, some examples to illustrate our results are given.MSC:34A08, 34B18, 39A13.

Eigenvalues of complementary Lidstone boundary value problems

We consider the following complementary Lidstone boundary value problem ( - 1 ) m y ( 2 m + 1 ) ( t ) = λ F ( t , y ( t ) , y ′ ( t ) ) , t ∈ ( 0 , 1 ) y ( 0 ) = 0 , y ( 2 k - 1 ) ( 0 ) = y ( 2 k - 1 ) ( 1 ) = 0 , 1 ≤ k ≤ m where λ > 0. The values of λ are characterized so that the boundary value problem has a positive solution. Moreover, we derive explicit intervals of λ such that for any λ in the interval, the existence of a positive solution of the boundary value problem is guaranteed. Some examples are also included to illustrate the results obtained. Note that the nonlinear term F depends on y' and this derivative dependence is seldom investigated in the literature.AMS Subject Classification: 34B15.

Positive Solutions for Second-Order Nonlinear Ordinary Differential Systems with Two Parameters

By using fixed-point theorem and under suitable conditions, we investigate the existence and multiplicity positive solutions to the following systems: , where are four positive constants and , , and . We derive two explicit intervals of and , such that the existence and multiplicity of positive solutions for the systems is guaranteed.

Confidence Intervals for Difference Between Two Poisson Rates

In this article, we develop four explicit asymptotic two-sided confidence intervals for the difference between two Poisson rates via a hybrid method. The basic idea of the proposed method is to estimate or recover the variances of the two Poisson rate estimates, which are required for constructing the confidence interval for the rate difference, from the confidence limits for the two individual Poisson rates. The basic building blocks of the approach are reliable confidence limits for the two individual Poisson rates. Four confidence interval estimators that have explicit solutions and good coverage levels are employed: the first normal with continuity correction, Rao score, Freeman and Tukey, and Jeffreys confidence intervals. Using simulation studies, we examine the performance of the four hybrid confidence intervals and compare them with three existing confidence intervals: the non-informative prior Bayes confidence interval, the t confidence interval based on Satterthwait's degrees of freedom, and the Bayes confidence interval based on Student's t confidence coefficient. Simulation results show that the proposed hybrid Freeman and Tukey, and the hybrid Jeffreys confidence intervals can be highly recommended because they outperform the others in terms of coverage probabilities and widths. The other methods tend to be too conservative and produce wider confidence intervals. The application of these confidence intervals are illustrated with three real data sets.

Spatio-temporal patterns of Hopf bifurcating periodic oscillations in a pair of identical tri-neuron network loops

Recently, the coupling time delay has been considered as the source of the occurrence of the phase-flip bifurcation in time-delay coupled system. But the analytical results of how the coupling time delay affects this phenomenon is still lacking. In this paper, we consider a pair of identical tri-neuron network coupled with time delay. By using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups, we investigate the spatio-temporal patterns of Hopf bifurcating periodic oscillations induced by the coupling time delay. The explicit intervals of delay and the regions in the plane of the coupling strength and the gain of the inherent response function for the existence of synchronized in-phase or anti-phase oscillation are obtained. Our study show that the coupling time delay does not affect the spatio-temporal patterns of the individual neural loop but it has the significant impact on the spatio-temporal patterns between the two loops. These analytic results are then verified by numerical simulations.

Positive solutions for singular third-order nonhomogeneous boundary value problems

In this paper, we investigate the existence of positive solutions for singular third-order nonhomogeneous boundary value problems. By using a fixed point theorem of cone expansion-compression type due to Krasnosel’skii, we establish various results on the existence or nonexistence of single and multiple positive solutions to the singular boundary problems in the explicit intervals for the nonhomogeneous term. An example is also given to illustrate some of the main results.

Existence of multiple solutions for a second-order difference equation with a parameter

Applying the critical point theorem, we establish existence and multiple solutions for a second-order difference boundary value problem and show the explicit intervals of λ such that the equation has at least 2 N distinct solutions.

Nonnegative doubly periodic solutions for nonlinear telegraph system with twin-parameters

In this paper, by using Krasnosel’skii fixed-point theorem and under suitable conditions, we present the existence and multiplicity of nonnegative doubly periodic solutions for the following system: u tt - u xx + c 1 u t + a 11 ( t , x ) u + a 12 ( t , x ) v = b 1 ( t , x ) f ( t , x , u , v ) , v tt - v xx + c 2 v t + a 21 ( t , x ) u + a 22 ( t , x ) v = b 2 ( t , x ) g ( t , x , u , v ) , where c i > 0 is a constant, a 11 ( t , x ) , a 22 ( t , x ) , b 1 ( t , x ) , b 2 ( t , x ) ∈ C ( R 2 , R + ) , a 12 ( t , x ) , a 21 ( t , x ) ∈ C ( R 2 , R - ) , f ( t , x , u , v ) , g ( t , x , u , v ) ∈ C ( R 2 × R + × R + , R + ) , and a ij , b i , f , g are 2 π -periodic in t and x. We derive two explicit intervals of b 1 ( t , x ) and b 2 ( t , x ) such that for any b 1 ( t , x ) and b 2 ( t , x ) in the two intervals respectively, the existence of at least one solution for the system is guaranteed, and the existence of at least two solutions for b 1 ( t , x ) and b 2 ( t , x ) in appropriate intervals is also discussed.

Eigenvalues of a system of generalized right focal problems with deviating arguments

We consider the following system of third-order three-point generalized right focal boundary value problems u i ‴ ( t ) = λ f i ( t , u 1 ( φ 1 ( t ) ) , u 2 ( φ 2 ( t ) ) , … , u n ( φ n ( t ) ) ) , t ∈ [ a , b ] , u i ( a ) = u i ′ ( z ) = 0 , γ i u i ( b ) + δ i u i ″ ( b ) = 0 , where λ > 0 , i = 1 , 2 , … , n , 1 2 ( a + b ) < z < b , γ i ⩾ 0 , δ i > 0 , and φ i are deviating arguments. Criteria will be developed so that for values of λ that form an interval (bounded or unbounded), the above system has a fixed-sign solution u = ( u 1 , u 2 , … , u n ) , i.e., for each 1 ⩽ i ⩽ n , θ i u i ( t ) ⩾ 0 for t ∈ [ a , b ] , where θ i ∈ { 1 ,- 1 } is fixed. Explicit intervals for such values of λ will also be presented. A numerical example is included to illustrate the results obtained.

Positive solutions for 2p-order and 2q-order nonlinear ordinary differential systems

In this paper, by using Krasnosel'skii fixed point theorem and under suitable conditions, we present the existence of single and multiple positive solutions to the following systems: { ( − 1 ) p u ( 2 p ) = λ a ( t ) f ( u ( t ) , v ( t ) ) , t ∈ [ 0 , 1 ] , ( − 1 ) q v ( 2 q ) = μ b ( t ) g ( u ( t ) , v ( t ) ) , t ∈ [ 0 , 1 ] , u ( 2 i ) ( 0 ) = u ( 2 i ) ( 1 ) = 0 , 0 ⩽ i ⩽ p − 1 , v ( 2 j ) ( 0 ) = v ( 2 j ) ( 1 ) = 0 , 0 ⩽ j ⩽ q − 1 , where λ > 0 , μ > 0 , p , q ∈ N . We derive two explicit intervals of λ and μ such that for any λ and μ in the two intervals respectively, the existence of at least one solution to the systems is guaranteed, and the existence of at least two solutions for λ and μ in appropriate intervals is also discussed.

Two-point right focal eigenvalue problems on time scales

We consider the following right focal boundary value problem: ( - 1 ) n - 1 y Δ n ( t ) = λ ( - 1 ) p + 1 F ( t , y ( σ n - 1 ( t ) ) ) , t ∈ [ a , b ] ∩ T , y Δ i ( a ) = 0 , 0 ⩽ i ⩽ p - 1 y Δ i ( σ ( b ) ) = 0 , p ⩽ i ⩽ n - 1 where λ > 0, n ⩾ 2, 1 ⩽ p ⩽ n − 1 is fixed and T is a time scale. The Green’s function related to the boundary value problem is obtained. The values of λ are characterized so that this problem has a positive solution. In addition, explicit intervals of λ are established. We also include examples to illustrate the usefulness of the results obtained.

On the spectrum of pairwise comparison matrices

This paper deals with the spectrum of symmetrically reciprocal (SR) perturbations of transitive matrices of nonzero complex entries. By introducing the so-called pseudo-circulant matrices, the spectral decomposition of specific SR perturbations of n×n transitive matrices and that of arbitrary 3×3 SR matrices are provided. Closed forms are given for all eigenvalues and their associated eigenvectors of these SR matrices, also known as pairwise comparison matrices (PCMs). To improve the evaluations at a given PCM, as well as the priority ranking of the alternatives, these results are applied to the analytic hierarchy process (AHP) using positive SR matrices. Explicit intervals for the ranges of possible rank reversals between the alternatives are established. The spectrum of certain SR perturbations of transitive matrices with nonzero complex entries is also derived which can be applied in vehicle system dynamics. Three application examples are included as well.

Eigenvalues of a system of Fredholm integral equations

We consider the following system of Fredholm integral equations: u i=λ ∫ 0 1 g i(t,s)P i(s,u 1(s),u 2(s),…,u n(s))ds, tϵ[0,1], 1≤i≤n, where λ > 0. Our aim is to determine those values of λ such that the above system has a constant-sign solution. In addition, explicit intervals for λ will be presented. The generality of the results obtained is illustrated through applications to several well-known boundary value problems. We also extend the above problem to that on the half-line [0, ∞) u i=λ ∫ 0 ∞ g i(t,s)P i(s,u 1(s),u 2(s),…,u n(s))ds, tϵ[0,∞], 1≤i≤n, Finally, both the systems above are extended to the general case when λ is replaced by λ i.

On the spectrum of pairwise comparison matrices

This paper deals with the spectrum of symmetrically reciprocal (SR) perturbations of transitive matrices of nonzero complex entries. By introducing the so-called pseudo-circulant matrices, the spectral decomposition of specific SR perturbations of n× n transitive matrices and that of arbitrary 3×3 SR matrices are provided. Closed forms are given for all eigenvalues and their associated eigenvectors of these SR matrices, also known as pairwise comparison matrices (PCMs). To improve the evaluations at a given PCM, as well as the priority ranking of the alternatives, these results are applied to the analytic hierarchy process (AHP) using positive SR matrices. Explicit intervals for the ranges of possible rank reversals between the alternatives are established. The spectrum of certain SR perturbations of transitive matrices with nonzero complex entries is also derived which can be applied in vehicle system dynamics. Three application examples are included as well.