scholarly journals A bin-packing system for objects with sizes from a finite set: Necessary and sufficient conditions for stability and some applications

1986 ◽  
Author(s):  
C. A. Courcoubetis ◽  
R. R. Weber
Keyword(s):  
Bin Packing ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finite Set ◽  
Necessary And Sufficient ◽  
Packing System

Rado's theorem for polymatroids

1975 ◽  
Vol 78 (2) ◽  
pp. 263-281 ◽  
Author(s):  
Colin J. H. McDiarmid
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Rank Function ◽  
Fundamental Importance ◽  
Matroid Theory ◽  
Continuous Analogue ◽  
Finite Set ◽  
Necessary And Sufficient ◽  
Transversal Theory

The theorem of R. Rado (12) to which I refer by the name ‘Rado's theorem for matroids’ gives necessary and sufficient conditions for a family of subsets of a finite set Y to have a transversal independent in a given matroid on Y. This theorem is of fundamental importance in both transversal theory and matroid theory (see, for example, (11)). In (3) J. Edmonds introduced and studied ‘polymatroids’ as a sort of continuous analogue of a matroid. I start this paper with a brief introduction to polymatroids, emphasizing the role of the ‘ground-set rank function’. The main result is an analogue for polymatroids of Rado's theorem for matroids, which I call not unnaturally ‘Rado's theorem for polymatroids’.

scholarly journals Square roots in finite full transformation semigroups

1982 ◽  
Vol 23 (2) ◽  
pp. 137-149 ◽  
Author(s):  
Mary Snowden ◽  
J. M. Howie
Keyword(s):  
Transformation Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary Condition ◽  
Square Root ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Square Roots ◽  
Finite Set ◽  
Necessary And Sufficient

Let X be a finite set and let (X) be the full transformation semigroup on X, i.e. the set of all mappings from X into X, the semigroup operation being composition of mappings. This paper aims to characterize those elements of (X) which have square roots. An easily verifiable necessary condition, that of being quasi-square, is found in Theorem 2, and in Theorems 4 and 5 we find necessary and sufficient conditions for certain special elements of (X). The property of being compatibly amenable is shown in Theorem 7 to be equivalent for all elements of (X) to the possession of a square root.

On conditions of efficiency of a solution of a multicriteria discrete problem

2002 ◽  
Vol 12 (2) ◽  
Author(s):  
V.A. Emelichev ◽  
A.V. Pashkevich
Keyword(s):  
Wide Class ◽  
Pareto Optimality ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Discrete Problem ◽  
Monotone Functions ◽  
Additive Method ◽  
Finite Set ◽  
Necessary And Sufficient

AbstractFor a wide class of multicriteria (vector) optimisation problems with a finite set of vector constraints, basing on the additive method of aggregating special monotone functions of partial criteria, we obtain necessary and sufficient conditions of efficiency (Pareto-optimality) of a solution.

Necessary and sufficient conditions for stability of a bin-packing system

1986 ◽  
Vol 23 (4) ◽  
pp. 989-999 ◽  
Author(s):  
C. Courcoubetis ◽  
R. R. Weber
Keyword(s):  
Bin Packing ◽  
Sufficient Conditions ◽  
Empty Space ◽  
Renewal Processes ◽  
Service Facility ◽  
Packing Algorithm ◽  
Arrival Processes ◽  
Necessary And Sufficient ◽  
Convex Polyhedral Cone ◽  
Packing System

Objects of various integer sizes, o1, · ··, on, are to be packed together into bins of size N as they arrive at a service facility. The number of objects of size oi which arrive by time t is , where the components of are independent renewal processes, with At/t → λ as t → ∞. The empty space in those bins which are neither empty nor full at time t is called the wasted space and the system is declared stabilizable if for some finite B there exists a bin-packing algorithm whose use guarantees the expected wasted space is less than B for all t. We show that the system is stabilizable if the arrival processes are Poisson and λ lies in the interior of a certain convex polyhedral cone Λ. In this case there exists a bin-packing algorithm which stabilizes the system without needing to know λ. However, if λ lies on the boundary of Λ the wasted space grows as and if λ is exterior to Λ it grows as O(t); these conclusions hold even if objects may be repacked as often as desired.

scholarly journals Weak convergence of first-rare-event times for semi-Markov processes

2020 ◽  
Author(s):  
AISDL
Keyword(s):  
Weak Convergence ◽  
Markov Processes ◽  
Sufficient Conditions ◽  
Rare Event ◽  
Necessary And Sufficient Conditions ◽  
Finite Set ◽  
In Series ◽  
Event Times ◽  
Semi Markov Processes ◽  
Necessary And Sufficient

Necessary and sufficient conditions for weak convergence of first-rareevent times for semi-Markov processes with finite set of states in series of schemes are obtained.

Necessary and sufficient conditions for stability of a bin-packing system

1986 ◽  
Vol 23 (04) ◽  
pp. 989-999
Author(s):  
C. Courcoubetis ◽  
R. R. Weber
Keyword(s):  
Bin Packing ◽  
Sufficient Conditions ◽  
Empty Space ◽  
Renewal Processes ◽  
Service Facility ◽  
Packing Algorithm ◽  
Arrival Processes ◽  
Necessary And Sufficient ◽  
Convex Polyhedral Cone ◽  
Packing System

Objects of various integer sizes, o 1, · ··, on, are to be packed together into bins of size N as they arrive at a service facility. The number of objects of size oi which arrive by time t is , where the components of are independent renewal processes, with At /t → λ as t → ∞. The empty space in those bins which are neither empty nor full at time t is called the wasted space and the system is declared stabilizable if for some finite B there exists a bin-packing algorithm whose use guarantees the expected wasted space is less than B for all t. We show that the system is stabilizable if the arrival processes are Poisson and λ lies in the interior of a certain convex polyhedral cone Λ. In this case there exists a bin-packing algorithm which stabilizes the system without needing to know λ. However, if λ lies on the boundary of Λ the wasted space grows as and if λ is exterior to Λ it grows as O(t); these conclusions hold even if objects may be repacked as often as desired.

scholarly journals On the Modulus of Continuity of Mappings Between Euclidean Spaces

2013 ◽  
Vol 112 (1) ◽  
pp. 147
Author(s):  
Dieudonné Agbor ◽  
Jan Boman
Keyword(s):  
Modulus Of Continuity ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Euclidean Spaces ◽  
The Real ◽  
Lipschitz Continuous ◽  
Finite Set ◽  
Necessary And Sufficient

Let $f$ be a function from $\mathbf{R}^p$ to $\mathbf{R}^q$ and let $\Lambda$ be a finite set of pairs $(\theta, \eta) \in \mathbf{R}^p \times \mathbf{R}^q$. Assume that the real-valued function $\langle\eta, f(x)\rangle$ is Lipschitz continuous in the direction $\theta$ for every $(\theta, \eta) \in \Lambda$. Necessary and sufficient conditions on $\Lambda$ are given for this assumption to imply each of the following: (1) that $f$ is Lipschitz continuous, and (2) that $f$ is continuous with modulus of continuity $\le C\epsilon |{\log \epsilon}|$.

scholarly journals On involutive division on monoids

2021 ◽  
Vol 29 (4) ◽  
pp. 387-398
Author(s):  
Oleg K. Kroytor ◽  
Mikhail D. Malykh
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Completely Continuous ◽  
Completely Continuous Operators ◽  
Finite Set ◽  
Definition Of ◽  
Original Domain ◽  
Necessary And Sufficient ◽  
Continuous Operators ◽  
Involutive Division

We consider an arbitrary monoid MM, on which an involutive division is introduced, and the set of all its finite subsets SetMM. Division is considered as a mapping d:SetMM{d:SetM \times M}, whose image d(U,m){d(U,m)} is the set of divisors of mm in UU. The properties of division and involutive division are defined axiomatically. Involutive division was introduced in accordance with the definition of involutive monomial division, introduced by V.P. Gerdt and Yu.A. Blinkov. New notation is proposed that provides brief but explicit allowance for the dependence of division on the SetMM element. The theory of involutive completion (closures) of sets is presented for arbitrary monoids, necessary and sufficient conditions for completeness (closedness) - for monoids generated by a finite set XX. The analogy between this theory and the theory of completely continuous operators is emphasized. In the last section, we discuss the possibility of solving the problem of replenishing a given set by successively expanding the original domain and its connection with the axioms used in the definition of division. All results are illustrated with examples of Thomas monomial division.

FURTHER RESULTS ON THE CONTINUOUS REPRESENTABILITY OF SEMIORDERS

2013 ◽  
Vol 21 (05) ◽  
pp. 675-694 ◽  
Author(s):  
A. ESTEVAN ◽  
J. GUTIÉRREZ GARCÍA ◽  
E. INDURÁIN
Keyword(s):  
Topological Space ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Positive Direction ◽  
Positive Threshold ◽  
Finite Set ◽  
Total Preorders ◽  
Necessary And Sufficient

We study necessary and sufficient conditions for the continuous Scott-Suppes representability of a semiorder through a continuous real-valued map and a strictly positive threshold. In the general case of a semiorder defined on topological space, we find several necessary conditions for the continuous representability. These necessary conditions are not sufficient, in general. As a matter of fact, the analogous of the classical Debreu's lemma for the continuous representability of total preorders is no longer valid for semiorders. However, and in a positive direction, we show that if the set is finite those conditions are indeed sufficient. In particular, we characterize the continuous Scott-Suppes representability of semiorders defined on a finite set endowed with a topology.

Necessary and sufficient conditions for stability of a bin-packing system

1986 ◽  
Vol 23 (04) ◽  
pp. 989-999 ◽  
Author(s):  
C. Courcoubetis ◽  
R. R. Weber
Keyword(s):  
Bin Packing ◽  
Sufficient Conditions ◽  
Empty Space ◽  
Renewal Processes ◽  
Service Facility ◽  
Packing Algorithm ◽  
Arrival Processes ◽  
Necessary And Sufficient ◽  
Convex Polyhedral Cone ◽  
Packing System

Objects of various integer sizes,o1, · ··,on,are to be packed together into bins of sizeNas they arrive at a service facility. The number of objects of sizeoiwhich arrive by timetis, where the components ofare independent renewal processes, withAt/t → λast → ∞. The empty space in those bins which are neither empty nor full at timetis called thewasted spaceand the system is declaredstabilizableif for some finiteBthere exists a bin-packing algorithm whose use guarantees the expected wasted space is less thanBfor allt.We show that the system is stabilizable if the arrival processes are Poisson and λ lies in the interior of a certain convex polyhedral cone Λ. In this case there exists a bin-packing algorithm which stabilizes the system without needing to know λ. However, if λ lies on the boundary of Λ the wasted space grows asand if λ is exterior to Λ it grows asO(t); these conclusions hold even if objects may be repacked as often as desired.

Sign in / Sign up

Export Citation Format

Share Document