scholarly journals On the Volume of Sections of the Cube

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Grigory Ivanov ◽  
Igor Tsiutsiurupa
Keyword(s):  
Upper Bound ◽  
Dimensional Subspace ◽  
Standard Basis ◽  
Necessary Condition ◽  
Maximal Volume ◽  
First Order ◽  
Planar Section ◽  
Dimensional Cube

Abstract We study the properties of the maximal volume k-dimensional sections of the n-dimensional cube [−1, 1] n . We obtain a first order necessary condition for a k-dimensional subspace to be a local maximizer of the volume of such sections, which we formulate in a geometric way. We estimate the length of the projection of a vector of the standard basis of ℝ n onto a k-dimensional subspace that maximizes the volume of the intersection. We find the optimal upper bound on the volume of a planar section of the cube [−1, 1] n , n ≥ 2.

Coercion and the Nature of Law

2020 ◽  
Author(s):  
Kenneth Einar Himma
Keyword(s):  
Conceptual Analysis ◽  
Existence Condition ◽  
Necessary Condition ◽  
First Order ◽  
Normative System ◽  
Motivating Reasons

COERCION AND THE NATURE OF LAW argues that it is a conceptually necessary condition for something to count as a system of law according to our conceptual practices that it authorizes the imposition of coercive sanctions for violations of some mandatory norms governing non-official behavior (the Coercion Thesis). The book begins with an explication of the modest approach to conceptual analysis that is deployed throughout. The remainder of the book is concerned to show that an institutional normative system is not reasonably contrived to do anything that law must be able to do for us to make sense of why we adopt systems of law to regulate non-official behavior unless we assume that mandatory norms governing that behavior are backed by the threat of a sovereign; an institutional normative system that satisfies every other plausible existence condition for law is not reasonably contrived to give rise to either objective or subjective first-order motivating reasons to comply with mandatory norms governing non-official behavior unless they are backed by the threat of a coercive sanction. Law’s presumed conceptual normativity can be explained only by the Coercion Thesis.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

A Difference Method for Plane Problems in Magnetoelastodynamics

1972 ◽  
Vol 39 (3) ◽  
pp. 689-695 ◽  
Author(s):  
W. W. Recker
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Method ◽  
Difference Method ◽  
Necessary Condition ◽  
Two Dimensional ◽  
Symmetric Hyperbolic System ◽  
Von Neumann ◽  
First Order ◽  
Independent Variables

The two-dimensional equations of magnetoelastodynamics are considered as a symmetric hyperbolic system of linear first-order partial-differential equations in three independent variables. The characteristic properties of the system are determined and a numerical method for obtaining the solution to mixed initial and boundary-value problems in plane magnetoelastodynamics is presented. Results on the von Neumann necessary condition are presented. Application of the method to a problem which has a known solution provides further numerical evidence of the convergence and stability of the method.

scholarly journals A gradient technique for an optimal control problem governed by a system of nonlinear first order partial differential equations

1995 ◽  
Vol 36 (3) ◽  
pp. 261-273 ◽  
Author(s):  
Mohammad A. Kazemi
Keyword(s):  
Optimal Control ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Optimal Control Problems ◽  
Fréchet Derivative ◽  
Necessary Condition ◽  
Control Problems ◽  
Partial Differential ◽  
Frechet Derivative ◽  
First Order

AbstractIn this paper a class of optimal control problems with distributed parameters is considered. The governing equations are nonlinear first order partial differential equations that arise in the study of heterogeneous reactors and control of chemical processes. The main focus of the present paper is the mathematical theory underlying the algorithm. A conditional gradient method is used to devise an algorithm for solving such optimal control problems. A formula for the Fréchet derivative of the objective function is obtained, and its properties are studied. A necessary condition for optimality in terms of the Fréchet derivative is presented, and then it is shown that any accumulation point of the sequence of admissible controls generated by the algorithm satisfies this necessary condition for optimality.

scholarly journals BILLIARD KNOTS IN A CYLINDER

1999 ◽  
Vol 08 (03) ◽  
pp. 353-366 ◽  
Author(s):  
C. LAMM ◽  
D. OBERMEYER
Keyword(s):  
Upper Bound ◽  
Necessary Condition

We define cylinder knots as billiard knots in a cylinder. We present a necessary condition for cylinder knots: after dividing cylinder knots by possible rotational symmetries we obtain ribbon knots. We obtain an upper bound for the number of cylinder knots with two fixed parameters (out of three). In addition we prove that rosette knots are cylinder knots.

EMBEDDINGS OF FREE TOPOLOGICAL VECTOR SPACES

2019 ◽  
Vol 101 (2) ◽  
pp. 311-324
Author(s):  
ARKADY LEIDERMAN ◽  
SIDNEY A. MORRIS
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Dimensional Subspace ◽  
Vector Subspace ◽  
Compact Metric Space ◽  
Topological Vector Spaces ◽  
Finite Dimensional ◽  
Bernstein Set ◽  
Dimensional Cube ◽  
Open Question

It is proved that the free topological vector space $\mathbb{V}([0,1])$ contains an isomorphic copy of the free topological vector space $\mathbb{V}([0,1]^{n})$ for every finite-dimensional cube $[0,1]^{n}$, thereby answering an open question in the literature. We show that this result cannot be extended from the closed unit interval $[0,1]$ to general metrisable spaces. Indeed, we prove that the free topological vector space $\mathbb{V}(X)$ does not even have a vector subspace isomorphic as a topological vector space to $\mathbb{V}(X\oplus X)$, where $X$ is a Cook continuum, which is a one-dimensional compact metric space. This is also shown to be the case for a rigid Bernstein set, which is a zero-dimensional subspace of the real line.

Bifurcation of Limit Cycles in a Near-Hamiltonian System with a Cusp of Order Two and a Saddle

2016 ◽  
Vol 26 (11) ◽  
pp. 1650180 ◽  
Author(s):  
Ali Bakhshalizadeh ◽  
Hamid R. Z. Zangeneh ◽  
Rasool Kazemi
Keyword(s):  
Asymptotic Expansion ◽  
Hamiltonian System ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Bifurcation Of Limit Cycles ◽  
Heteroclinic Loop ◽  
First Order ◽  
Period Annulus

In this paper, the asymptotic expansion of first-order Melnikov function of a heteroclinic loop connecting a cusp of order two and a hyperbolic saddle for a planar near-Hamiltonian system is given. Next, we consider the limit cycle bifurcations of a hyper-elliptic Liénard system with this kind of heteroclinic loop and study the least upper bound of limit cycles bifurcated from the period annulus inside the heteroclinic loop, from the heteroclinic loop itself and the center. We find that at most three limit cycles can be bifurcated from the period annulus, also we present different distributions of bifurcated limit cycles.

FEEDBACK LOOPS, STABILITY AND MULTISTATIONARITY IN DYNAMICAL SYSTEMS

1995 ◽  
Vol 03 (02) ◽  
pp. 409-413 ◽  
Author(s):  
ERIK PLAHTE ◽  
THOMAS MESTL ◽  
STIG W. OMHOLT
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Negative Feedback ◽  
Feedback Loop ◽  
Feedback Loops ◽  
Necessary Condition ◽  
Negative Feedback Loop ◽  
Positive Feedback Loop ◽  
First Order ◽  
First Order Differential Equations

By fairly simple considerations of stability and multistationarity in nonlinear systems of first order differential equations it is shown that under quite mild restrictions a negative feedback loop is a necessary condition for stability, and that a positive feedback loop is a necessary condition for multistationarity.

A comparative study of two-phase flow models relevant to bubble column dynamics

1999 ◽  
Vol 394 ◽  
pp. 73-96 ◽  
Author(s):  
P. D. MINEV ◽  
U. LANGE ◽  
K. NANDAKUMAR
Keyword(s):  
Bubble Column ◽  
Bubbly Flows ◽  
Necessary Condition ◽  
Qualitative Behaviour ◽  
Two Phase ◽  
Weakly Nonlinear ◽  
Flow Models ◽  
First Order ◽  
Column Dynamics ◽  
Permanent Wave

Multiphase flow modelling is still a major challenge in fluid dynamics and, although many different models have been derived, there is no clear evidence of their relevance to certain flow situations. That is particularly valid for bubbly flows, because most of the studies have considered the case of fluidized beds. In the present study we give a general formulation to five existing models and study their relevance to bubbly flows. The results of the linear analysis of those models clearly show that only two of them are applicable to that case. They both show a very similar qualitative linear stability behaviour. In the subsequent asymptotic analysis we derive an equation hierarchy which describes the weakly nonlinear stability of the models. Their qualitative behaviour up to first order with respect to the small parameter is again identical. A permanent-wave solution of the first two equations of the hierarchy is found. It is shown, however, that the permanent-wave (soliton) solution is very unlikely to occur for the most common case of gas bubbles in water. The reason is that the weakly nonlinear equations are unstable due to the low magnitude of the bulk modulus of elasticity. Physically relevant stabilization can eventually be achieved using some available experimental data. Finally, a necessary condition for existence of a fully nonlinear soliton is derived.

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