Recent work of C. Fefferman and the first author 8 has demonstrated that the linear system of equations∑j=1MAij(x)Fj(x)=fi(x)(i=1,…,N), has a Cm solution F=(F1,…,FM) if and only if f1,…,fN satisfy a ...certain finite collection of partial differential equations. Here, the Aij are fixed semialgebraic functions.
In this paper, we consider the analogous problem for systems of linear inequalities:∑j=1MAij(x)Fj(x)≤fi(x)(i=1,…,N). Our main result is a negative one, demonstrated by counterexample: the existence of a Cm solution F may not, in general, be determined via an analogous finite set of partial differential inequalities in f1,…,fN.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Simulation from the truncated multivariate normal distribution in high dimensions is a recurrent problem in statistical computing and is typically only feasible by using approximate Markov chain ...Monte Carlo sampling. We propose a minimax tilting method for exact independently and identically distributed data simulation from the truncated multivariate normal distribution. The new methodology provides both a method for simulation and an efficient estimator to hitherto intractable Gaussian integrals. We prove that the estimator has a rare vanishing relative error asymptotic property. Numerical experiments suggest that the scheme proposed is accurate in a wide range of set-ups for which competing estimation schemes fail. We give an application to exact independently and identically distributed data simulation from the Bayesian posterior of the probit regression model.
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BFBNIB, FZAB, GIS, IJS, INZLJ, IZUM, KILJ, NLZOH, NMLJ, NUK, OILJ, PILJ, PNG, SAZU, SBCE, SBMB, UL, UM, UPUK, ZRSKP
Linear inequalities are mathematical expressions that compare two expressions using the inequality symbol, in either be algebraic or numerical or both. However, in solving either of these types some ...student-teachers commit errors that have been backed by associated misconceptions. This research examined these errors and the associated misconceptions thereafter. Guided by two research questions, the researchers adopted the qualitative narrative inquiry design. The purposive sampling was employed to select 15 student-teachers who met the best requirement that fits the purpose, problem, and objective of a qualitative narrative inquiry. The main instruments were interview guides, where the participants and researchers collaborated with each other to ensure that the story was properly told and aligned with linear inequalities through field notes, observations, photos and artefacts. The narrative analysis started with verbatim transcription of the narratives and ended with deductive coding. The results were scanned copies of participants’ sample narratives that were pasted at appropriate places and discussed. Consequently, it was concluded that student-teachers lacked the basic rules, procedural fluency and skills, and formulation of linear inequalities. These errors emanated from misconceived methods and rote memorization. It was therefore recommended that educators imbibe practical and everyday methodologies into the teaching and learning of linear inequalities.
This paper is devoted to the stability analysis of a class of switched nonlinear positive systems with nonlinearities of a sector type. A special construction of common Lyapunov function is proposed ...for the family of subsystems associated with a switched system and conditions of the existence of such a function are derived. The obtained results are used for the absolute stability investigation of switched systems with rank one difference.
•Stability of a class of switched nonlinear positive systems with nonlinearities of a sector type is studied.•Conditions of the existence of a common Lyapunov function of a special form for a family of nonlinear positive systems are derived.•New conditions of absolute stability for nonlinear switched systems with rank one difference are obtained.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
We study randomized variants of two classical algorithms: coordinate descent for systems of linear equations and iterated projections for systems of linear inequalities. Expanding on a recent ...randomized iterated projection algorithm of Strohmer and Vershynin (Strohmer, T., R. Vershynin. 2009. A randomized Kaczmarz algorithm with exponential convergence.
J. Fourier Anal. Appl.
15
262-278) for systems of linear equations, we show that, under appropriate probability distributions, the linear rates of convergence (in expectation) can be bounded in terms of natural linear-algebraic condition numbers for the problems. We relate these condition measures to distances to ill-posedness and discuss generalizations to convex systems under metric regularity assumptions.
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BFBNIB, CEKLJ, INZLJ, IZUM, KILJ, NMLJ, NUK, PILJ, PNG, SAZU, UL, UM, UPUK, ZRSKP
The ellipsoid algorithm is a fundamental algorithm for computing a solution to the system of
m
linear inequalities in
n
variables
(
P
)
:
A
⊤
x
≤
u
when its set of solutions has positive volume. ...However, when
(
P
)
is infeasible, the ellipsoid algorithm has no mechanism for proving that (
P
) is infeasible. This is in contrast to the other two fundamental algorithms for tackling
(
P
)
, namely, the simplex and interior-point methods, each of which can be easily implemented in a way that either produces a solution of
(
P
)
or proves that
(
P
)
is infeasible by producing a solution to the alternative system
(
Alt
)
:
A
λ
=
0
,
u
⊤
λ
<
0
,
λ
≥
0
. This paper develops an oblivious ellipsoid algorithm (OEA) that either produces a solution of
(
P
)
or produces a solution of
(
Alt
)
. Depending on the dimensions and other condition measures, the computational complexity of the basic OEA may be worse than, the same as, or better than that of the standard ellipsoid algorithm. We also present two modified versions of OEA, whose computational complexity is superior to that of OEA when
n
≪
m
. This is achieved in the first modified version by proving infeasibility without producing a solution of
(
Alt
)
, and in the second version by using more memory.
Funding:
J. Lamperski and R. M. Freund were supported by the Air Force Office of Scientific Research Grant FA9550-19-1-0240.
In the smallest cases where there exist nonnegative polynomials that are not sums of squares we present a complete explanation of this distinction. The fundamental reason that the cone of sums of ...squares is strictly contained in the cone of nonnegative polynomials is that polynomials of degree d. For any nonnegative polynomial that is not a sum of squares we can write down a linear inequality coming from a Cayley-Bacharach relation that certifies this fact. We also characterize strictly positive sums of squares that lie on the boundary of the cone of sums of squares and extreme rays of the cone dual to the cone of sums of squares.
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The paper proposes a method for solving systems of linear inequalities. This method determines in a finite number of iterations whether the given system of linear ineqalities has a solution.
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