Systems of polynomials with at least one positive real zero

In this paper, we prove several theorems on systems of polynomials with at least one positive real zero based on the theory of conceive polynomials. These theorems provide sufficient conditions for systems of multivariate polynomials admitting at least one positive real zero in terms of their Newton polytopes and combinatorial structure. Moreover, a class of polynomials attaining their global minimums in the first quadrant are given, which is useful in polynomial optimization.

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A Formal Account of Structuring Motor Actions With Sensory Prediction for a Naive Agent

Frontiers in Robotics and AI ◽

10.3389/frobt.2020.561660 ◽

2020 ◽

Vol 7 ◽

Author(s):

Jean-Merwan Godon ◽

Sylvain Argentieri ◽

Bruno Gas

Keyword(s):

Sufficient Conditions ◽

Combinatorial Structure ◽

Prediction Function ◽

Formal Framework ◽

Sensory Predictions ◽

Spatial Displacements ◽

Motor Actions ◽

Sensory Prediction ◽

Sensorimotor Contingency

For naive robots to become truly autonomous, they need a means of developing their perceptive capabilities instead of relying on hand crafted models. The sensorimotor contingency theory asserts that such a way resides in learning invariants of the sensorimotor flow. We propose a formal framework inspired by this theory for the description of sensorimotor experiences of a naive agent, extending previous related works. We then use said formalism to conduct a theoretical study where we isolate sufficient conditions for the determination of a sensory prediction function. Furthermore, we also show that algebraic structure found in this prediction can be taken as a proxy for structure on the motor displacements, allowing for the discovery of the combinatorial structure of said displacements. Both these claims are further illustrated in simulations where a toy naive agent determines the sensory predictions of its spatial displacements from its uninterpreted sensory flow, which it then uses to infer the combinatorics of said displacements.

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Positive real lemma: necessary and sufficient conditions for the existence of solutions under virtually no assumptions

IEEE Transactions on Automatic Control ◽

10.1109/tac.2005.847036 ◽

2005 ◽

Vol 50 (5) ◽

pp. 720-724 ◽

Cited By ~ 7

Author(s):

A. Ferrante

Keyword(s):

Existence Of Solutions ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Positive Real ◽

Necessary And Sufficient

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On a partial theta function and its spectrum

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210515000591 ◽

2016 ◽

Vol 146 (3) ◽

pp. 609-623 ◽

Cited By ~ 6

Author(s):

Vladimir Petrov Kostov

Keyword(s):

Entire Function ◽

Theta Function ◽

Local Minimum ◽

Real Zero ◽

Positive Real ◽

The Real ◽

Real Zeros ◽

Partial Theta Function

The bivariate series defines a partial theta function. For fixed q (∣q∣ < 1), θ(q, ·) is an entire function. For q ∈ (–1, 0) the function θ(q, ·) has infinitely many negative and infinitely many positive real zeros. There exists a sequence of values of q tending to –1+ such that has a double real zero (the rest of its real zeros being simple). For k odd (respectively, k even) has a local minimum (respectively, maximum) at , and is the rightmost of the real negative zeros of (respectively, for k sufficiently large is the second from the left of the real negative zeros of ). For k sufficiently large one has . One has and .

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On the rate of convergence of probabilities of moderate deviations

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700006029 ◽

1970 ◽

Vol 11 (1) ◽

pp. 91-94 ◽

Cited By ~ 2

Author(s):

V. K. Rohatgi

Keyword(s):

Sufficient Conditions ◽

Moderate Deviations ◽

Independent Random Variables ◽

Standard Normal Distribution ◽

Positive Real ◽

Moment Conditions ◽

Standard Normal ◽

Image Position ◽

Necessary And Sufficient ◽

Asymptotic Forms

Let {Xn: n ≧ 1} be a sequence of independent random variables and write Letand let . Suppose that converges in law to the standard normal distribution (see [5, 280] for necessary and sufficient conditions). Let {xn} be a monotonic sequence of positive real numbers such that xn → ∞ as n → ∞. Then as n → ∞ for all ε > 0. [6] Rubin and Sethuraman call probabilities of the form probabilities of moderate deviations and obtain asymptotic forms for such probabilities under appropriate moment conditions.

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Oscillation of Two-Dimensional Neutral Delay Dynamic Systems

Advances in Mathematical Physics ◽

10.1155/2013/871961 ◽

2013 ◽

Vol 2013 ◽

pp. 1-7

Author(s):

Xinli Zhang ◽

Shanliang Zhu

Keyword(s):

Time Scales ◽

Dynamic Systems ◽

Neutral Type ◽

Sufficient Conditions ◽

Real Sequence ◽

Two Dimensional ◽

Positive Real ◽

Neutral Delay ◽

All Solutions ◽

Special Case

We consider a class of nonlinear two-dimensional dynamic systems of the neutral type(x(t)-a(t)x(τ1(t)))Δ=p(t)f1(y(t)),yΔ(t)=-q(t)f2(x(τ2(t))).We obtain sufficient conditions for all solutions of the system to be oscillatory. Our oscillation results whena(t)=0improve the oscillation results for dynamic systems on time scales that have been established by Fu and Lin (2010), since our results do not restrict to the case wheref(u)=u. Also, as a special case when𝕋=ℝ, our results do not requireanto be a positive real sequence. Some examples are given to illustrate the main results.

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