An interesting fact regarding the Routh table

2009 ◽  
Vol 223 (5) ◽  
pp. 709-711
Author(s):  
R Landers
Keyword(s):  
Technical Note ◽  
Analysis Tool ◽  
Positive Real ◽  
Interesting Fact ◽  
The Given ◽  
Special Case

The Routh table is a powerful analysis tool used to determine how many roots of a given polynomial have negative, zero, and positive real parts. A special case occurs when a row of zeros is encountered. This indicates an even polynomial may be factored from the given polynomial. It is well known that the even polynomial is in the row above the row of zeros. In this technical note it will be shown that the even polynomial may be factored from all of the polynomials in the rows above the row of zeros.

scholarly journals A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Khalid K. Ali ◽  
Mohamed A. Abd El salam ◽  
Emad M. H. Mohamed
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Order ◽  
Linear Functional ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Fractional Order Differential Equations ◽  
The Given ◽  
Functional Argument ◽  
Special Case

AbstractIn this paper, a numerical technique for a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series is presented. The proposed equation with its linear functional argument represents a general form of delay and advanced nonlinear fractional-order differential equations. The spectral collocation method is extended to study this problem as a discretization scheme, where the fractional derivatives are defined in the Caputo sense. The collocation method transforms the given equation and conditions to algebraic nonlinear systems of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. A general form of the operational matrix to derivatives includes the fractional-order derivatives and the operational matrix of an ordinary derivative as a special case. To the best of our knowledge, there is no other work discussed this point. Numerical examples are given, and the obtained results show that the proposed method is very effective and convenient.

scholarly journals Positivity and Monotonicity Preserving Biquartic Rational Interpolation Spline Surface

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Xinru Liu ◽  
Yuanpeng Zhu ◽  
Shengjun Liu
Keyword(s):  
Sufficient Conditions ◽  
Rational Interpolation ◽  
Rectangular Domain ◽  
Interpolation Spline ◽  
Spline Surface ◽  
Surface Data ◽  
Shape Preserving Interpolation ◽  
The Given ◽  
Surface Construction ◽  
Special Case

A biquartic rational interpolation spline surface over rectangular domain is constructed in this paper, which includes the classical bicubic Coons surface as a special case. Sufficient conditions for generating shape preserving interpolation splines for positive or monotonic surface data are deduced. The given numeric experiments show our method can deal with surface construction from positive or monotonic data effectively.

Holomorphic Functions with Positive Real Part

1982 ◽  
Vol 34 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Eric Sawyer
Keyword(s):  
Unit Ball ◽  
Hardy Spaces ◽  
Convex Cone ◽  
Holomorphic Functions ◽  
Positive Real Part ◽  
Positive Real ◽  
Spaces Of Holomorphic Functions ◽  
Extreme Element ◽  
Image Position ◽  
Special Case

The main purpose of this note is to prove a special case of the following conjecture.Conjecture. If F is holomorphic on the unit ball Bn in Cn and has positive real part, then F is in Hp(Bn) for 0 < p < ½(n + 1).Here Hp(Bn) (0 < p < ∞) denote the usual Hardy spaces of holomorphic functions on Bn. See below for definitions. We remark that the conjecture is known for 0 < p < 1 and that some evidence for it already exists in the literature; for example [1, Theorems 3.11 and 3.15] where it is shown that a particular extreme element of the convex cone of functionsis in Hp(B2) for 0 < p < 3/2.

A Probability Scoring Rule for Simultaneous Events

2019 ◽  
Vol 16 (4) ◽  
pp. 301-313
Author(s):  
Andrew Grant ◽  
David Johnstone ◽  
Oh Kang Kwon
Keyword(s):  
Economic Benefit ◽  
Score Function ◽  
Interesting Property ◽  
Usual Sense ◽  
Power Utility ◽  
Scoring Rule ◽  
The Individual ◽  
Individual Scores ◽  
The Given ◽  
Special Case

We develop a scoring rule tailored to a decision maker who makes simultaneous bets on events that occur at times that require bets to be placed together. The rule proposed captures the economic benefit to a well-defined bettor who acts on one set of probabilities p against a baseline or rival set q. To allow for simultaneous bets, we assume a myopic power utility function with a risk aversion parameter tailored to suit the given user or application. Our score function is “proper” in the usual sense of motivating honesty. Apart from a special case of power utility, namely, log utility, the score is not “local,” which we excuse because a local scoring rule cannot capture the economic result that our score reflects. An interesting property of our rule is that the individual scores from individual events are multiplicative, rather than additive. Probability scores are often added to give a measure of aggregate performance over a set of trials. Our rule is unique in that scores must be multiplied to reach a meaningful aggregate.

Solutions of barotropic trapped waves around seamounts

2010 ◽  
Vol 661 ◽  
pp. 32-44 ◽  
Author(s):  
LUIS ZAVALA SANSÓN
Keyword(s):  
Positive Real Number ◽  
Radial Distance ◽  
Phase Speed ◽  
Trapped Waves ◽  
Positive Real ◽  
Wave Characteristics ◽  
Sharp Cone ◽  
The Waves ◽  
Wide Family ◽  
Special Case

In this paper, solutions of free, barotropic waves around axisymmetric seamounts are derived. Even though this type of oscillation has been studied before, we revisit this problem for two main reasons: (i) the linear, barotropic, shallow-water equations with a rigid lid are now solved with no further approximations, in contrast with previous studies; (ii) the solutions are applied to a wide family of seamounts with profiles proportional to exp(rs), with r being the radial distance from the centre of the mountain and s any positive real number. (Most previous works are restricted to the special case s = 2.) The resulting dispersion relation possesses a remarkable simplicity that reveals a number of wave characteristics, for instance, the discrete wave frequencies and the angular phase speed of the waves around the seamount are easily derived as a function of the seamount shape. By varying the shape parameter one can study trapped waves around flat-topped seamounts or guyots (s > 2) or sharp, cone-shaped topographies (s < 2).

THE BANACH-LIE *-ALGEBRA OF MULTIPLICATION OPERATORS ON A W*-ALGEBRA

2011 ◽  
Vol 04 (02) ◽  
pp. 235-261
Author(s):  
Maysaa Alqurashi ◽  
Najla A. Altwaijry ◽  
C. Martin Edwards ◽  
Christopher S. Hoskin
Keyword(s):  
State Space ◽  
Lie Algebra ◽  
The State ◽  
Multiplication Operators ◽  
Dual Cone ◽  
Positive Real ◽  
Linear Isometry ◽  
Real Linear ◽  
Special Case

The hermitian part [Formula: see text] of the Banach-Lie *-algebra [Formula: see text] of multiplication operators on the W *-algebra A is a unital GM-space, the base of the dual cone in the dual GL-space [Formula: see text] of which is affine isomorphic and weak*-homeomorphic to the state space of [Formula: see text]. It is shown that there exists a Lie *-isomorphism ϕ from the quotient (A ⊕∞ Aop)/K of an enveloping W *-algebra A ⊕∞ Aop of A by a weak*-closed Lie *-ideal K onto [Formula: see text], the restriction to the hermitian part ((A ⊕∞ Aop)/K)h of which is a bi-positive real linear isometry, thereby giving a characterization of the state space of [Formula: see text]. In the special case in which A is a W *-factor this leads to a further identification of the state space of [Formula: see text] in terms of the state space of A. For any W *-algebra A, the Banach-Lie *-algebra [Formula: see text] coincides with the set of generalized derivations of A, and, as an application, a formula is obtained for the norm of an element of [Formula: see text] in terms of a centre-valued 'norm' on A, which is similar to that previously obtained by non-order-theoretic methods.

scholarly journals Oscillation of Two-Dimensional Neutral Delay Dynamic Systems

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.

scholarly journals COMPUTING SIGNED PERMUTATIONS OF POLYGONS

2011 ◽  
Vol 21 (01) ◽  
pp. 87-100
Author(s):  
GREG ALOUPIS ◽  
PROSENJIT BOSE ◽  
ERIK D. DEMAINE ◽  
STEFAN LANGERMAN ◽  
HENK MEIJER ◽  
...  
Keyword(s):  
Mirror Symmetry ◽  
Time Algorithm ◽  
Worst Case ◽  
Signed Permutations ◽  
Np Complete ◽  
Edge List ◽  
The Given ◽  
Special Case ◽  
Planar Polygon

Given a planar polygon (or chain) with a list of edges {e1, e2, e3, …, en-1, en}, we examine the effect of several operations that permute this edge list, resulting in the formation of a new polygon. The main operations that we consider are: reversals which involve inverting the order of a sublist, transpositions which involve interchanging subchains (sublists), and edge-swaps which are a special case and involve interchanging two consecutive edges. When each edge of the given polygon has also been assigned a direction we say that the polygon is signed. In this case any edge involved in a reversal changes direction. We show that a star-shaped polygon can be convexified using O(n2) edge-swaps, while maintaining simplicity, and that this is tight in the worst case. We show that determining whether a signed polygon P can be transformed to one that has rotational or mirror symmetry with P, using transpositions, takes Θ(n log n) time. We prove that the problem of deciding whether transpositions can modify a polygon to fit inside a rectangle is weakly NP-complete. Finally we give an O(n log n) time algorithm to compute the maximum endpoint distance for an oriented chain.

scholarly journals Technical Note—A Special Case of the Transportation Problem

1977 ◽  
Vol 25 (3) ◽  
pp. 525-528 ◽  
Author(s):  
R. Chandrasekaran ◽  
S. Subba Rao
Keyword(s):  
Transportation Problem ◽  
Technical Note ◽  
Special Case

scholarly journals Plains in a Special Case. The Relationship Between the Planes

2021 ◽  
Vol 9 (11) ◽  
pp. 548-550
Author(s):  
Qozaqova Munojat Sharifjanovna
Keyword(s):  
Horizontal Projection ◽  
Straight Lines ◽  
The Given ◽  
The Relationship ◽  
Special Case

Annotation: To develop students' understanding of straight lines and planes and to develop skills and competencies in working on related issues. The listener must complete the given task on A4 paper with the necessary tools. Keywords: straight lines, perpendicular, horizontal projection, frontal projections.

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