scholarly journals Expected Logarithm and Negative Integer Moments of a Noncentral χ2-Distributed Random Variable

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 1048
Author(s):  
Stefan Moser
Keyword(s):  
Closed Form ◽  
Lower Bounds ◽  
Degrees Of Freedom ◽  
Random Variable ◽  
Negative Integer ◽  
Upper And Lower Bounds ◽  
Basic Properties

Closed-form expressions for the expected logarithm and for arbitrary negative integer moments of a noncentral χ2-distributed random variable are presented in the cases of both even and odd degrees of freedom. Moreover, some basic properties of these expectations are derived and tight upper and lower bounds on them are proposed.

scholarly journals On the mean and dispersion of the Moore-Penrose generalized inverse of a Wishart matrix

2020 ◽  
Vol 36 (36) ◽  
pp. 124-133
Author(s):  
Shinpei Imori ◽  
Dietrich Von Rosen
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Degrees Of Freedom ◽  
Generalized Inverse ◽  
Upper And Lower Bounds ◽  
Exact Results ◽  
Identity Matrix ◽  
Wishart Matrix ◽  
The Mean ◽  
Scale Matrix

The Moore-Penrose inverse of a singular Wishart matrix is studied. When the scale matrix equals the identity matrix the mean and dispersion matrices of the Moore-Penrose inverse are known. When the scale matrix has an arbitrary structure no exact results are available. The article complements the existing literature by deriving upper and lower bounds for the expectation and an upper bound for the dispersion of the Moore-Penrose inverse. The results show that the bounds become large when the number of rows (columns) of the Wishart matrix are close to the degrees of freedom of the distribution.

scholarly journals Minimum survivable graphs with bounded distance increase

2003 ◽  
Vol Vol. 6 no. 1 ◽  
Author(s):  
Selma Djelloul ◽  
Mekkia Kouider
Keyword(s):  
Fault Tolerance ◽  
Lower Bounds ◽  
Interconnection Networks ◽  
Negative Integer ◽  
Upper And Lower Bounds ◽  
Minimum Size ◽  
Np Hard ◽  
Bounded Distance ◽  
Minimum Number ◽  
International Audience

International audience We study in graphs properties related to fault-tolerance in case a node fails. A graph G is k-self-repairing, where k is a non-negative integer, if after the removal of any vertex no distance in the surviving graph increases by more than k. In the design of interconnection networks such graphs guarantee good fault-tolerance properties. We give upper and lower bounds on the minimum number of edges of a k-self-repairing graph for prescribed k and n, where n is the order of the graph. We prove that the problem of finding, in a k-self-repairing graph, a spanning k-self-repairing subgraph of minimum size is NP-Hard.

scholarly journals Bounds for the variance of the busy period of the M/G/∞ queue

1984 ◽  
Vol 16 (4) ◽  
pp. 929-932 ◽  
Author(s):  
M. F. Ramalhoto
Keyword(s):  
Distribution Function ◽  
Lower Bounds ◽  
Service Time ◽  
Time Distribution ◽  
Busy Period ◽  
Random Variable ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Service Time Distribution ◽  
Lower And Upper Bounds

Some bounds for the variance of the busy period of an M/G/∞ queue are calculated as functions of parameters of the service-time distribution function. For any type of service-time distribution function, upper and lower bounds are evaluated in terms of the intensity of traffic and the coefficient of variation of the service time. Other lower and upper bounds are derived when the service time is a NBUE, DFR or IMRL random variable. The variance of the busy period is also related to the variance of the number of busy periods that are initiated in (0, t] by renewal arguments.

scholarly journals Bounds for the variance of the busy period of the M/G/∞ queue

1984 ◽  
Vol 16 (04) ◽  
pp. 929-932
Author(s):  
M. F. Ramalhoto
Keyword(s):  
Distribution Function ◽  
Lower Bounds ◽  
Service Time ◽  
Time Distribution ◽  
Busy Period ◽  
Random Variable ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Service Time Distribution ◽  
Lower And Upper Bounds

Some bounds for the variance of the busy period of an M/G/∞ queue are calculated as functions of parameters of the service-time distribution function. For any type of service-time distribution function, upper and lower bounds are evaluated in terms of the intensity of traffic and the coefficient of variation of the service time. Other lower and upper bounds are derived when the service time is a NBUE, DFR or IMRL random variable. The variance of the busy period is also related to the variance of the number of busy periods that are initiated in (0, t] by renewal arguments.

scholarly journals Optimal L(h,k)-Labeling of Regular Grids

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Tiziana Calamoneri
Keyword(s):  
Lower Bounds ◽  
Negative Integer ◽  
Upper And Lower Bounds ◽  
Vertex Coloring ◽  
Small Interval ◽  
Regular Grids ◽  
Special Cases ◽  
Hexagonal Grids ◽  
International Audience ◽  
The Difference

International audience The L(h, k)-labeling is an assignment of non negative integer labels to the nodes of a graph such that 'close' nodes have labels which differ by at least k, and 'very close' nodes have labels which differ by at least h. The span of an L(h,k)-labeling is the difference between the largest and the smallest assigned label. We study L(h, k)-labelings of cellular, squared and hexagonal grids, seeking those with minimum span for each value of k and h ≥ k. The L(h,k)-labeling problem has been intensively studied in some special cases, i.e. when k=0 (vertex coloring), h=k (vertex coloring the square of the graph) and h=2k (radio- or λ -coloring) but no results are known in the general case for regular grids. In this paper, we completely solve the L(h,k)-labeling problem on regular grids, finding exact values of the span for each value of h and k; only in a small interval we provide different upper and lower bounds.

scholarly journals On closed-form tight bounds and approximations for the median of a gamma distribution

PLoS ONE ◽  
2021 ◽  
Vol 16 (5) ◽  
pp. e0251626
Author(s):  
Richard F. Lyon
Keyword(s):  
Lower Bound ◽  
Closed Form ◽  
Lower Bounds ◽  
Gamma Distribution ◽  
Upper Bound ◽  
Entire Range ◽  
Shape Parameter ◽  
Upper And Lower Bounds ◽  
Elementary Functions ◽  
Asymptotic Analyses

The median of a gamma distribution, as a function of its shape parameter k, has no known representation in terms of elementary functions. In this work we use numerical simulations and asymptotic analyses to bound the median, finding bounds of the form 2−1/k(A + Bk), including an upper bound that is tight for low k and a lower bound that is tight for high k. These bounds have closed-form expressions for the constant parameters A and B, and are valid over the entire range of k > 0, staying between 48 and 55 percentile. Furthermore, an interpolation between these bounds yields closed-form expressions that more tightly bound the median, with absolute and relative margins to both upper and lower bounds approaching zero at both low and high values of k. These bound results are not supported with analytical proofs, and hence should be regarded as conjectures. Simple approximation expressions between the bounds are also found, including one in closed form that is exact at k = 1 and stays between 49.97 and 50.03 percentile.

Some inequalities on variances

1974 ◽  
Vol 11 (2) ◽  
pp. 409-412 ◽  
Author(s):  
Matthew Goldstein
Keyword(s):  
Lower Bounds ◽  
Random Variable ◽  
Upper And Lower Bounds ◽  
Negative Function

Sharp upper and lower bounds for the variance of a non-negative function of a non-negative random variable are obtained under rather weak hypotheses regarding the function. Comparisons between bounds are made and some specific examples are considered.

Torsion of Cylindrical and Prismatic Bars in the Presence of Steady Creep

1958 ◽  
Vol 25 (2) ◽  
pp. 214-218
Author(s):  
S. A. Patel ◽  
B. Venkatraman ◽  
P. G. Hodge
Keyword(s):  
Closed Form ◽  
Lower Bounds ◽  
Cross Sections ◽  
Creep Behavior ◽  
Upper And Lower Bounds ◽  
Elastic Analysis ◽  
Steady Creep ◽  
Pure Torsion ◽  
Closed Form Solutions ◽  
Creep Problem

Abstract This paper is concerned with the steady creep behavior of cylindrical and prismatic bars in which the deformations are caused by pure torsion. The creep problem is first reduced to one in nonlinear elasticity by means of the elastic analog. The elastic analysis is then carried out by means of the principles of minimum energies. These principles yield upper and lower bounds on the angle of twist. Closed-form solutions also are presented for some cross sections.

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