Multivaluedness in Networks: Theory

Common Input ◽

Generalized Systems ◽

Fundamental Phenomenon ◽

<div><div>This brief note reports the fundamental phenomenon of implicit multivaluedness exhibited from one output to the other of two node-systems with a common input—referred to as counter-cascaded1 systems—under the appropriate conditions. The novel concepts of immanence and transcendence are introduced upon which the formulation and prove of a necessary and sufﬁcient condition for multivaluedness are based; this is the main result of this note. Next, subsequent consequences of this result are presented. Among these is the fact that this result also holds for cascaded generalized systems. </div><div><br></div><div>The novel application of structural complexity reduction in directed networks presented next, demonstrates the utility of multivaluedness and is itself a contribution to the theory of signals and systems.</div><div><br></div><div>The signiﬁcance of the work presented here is that it contributes toward the theory of systems and networks as well as toward the arsenal of tools for studying networks.</div></div>

Multivaluedness in Networks: Theory

10.36227/techrxiv.11921094 ◽

2020 ◽

Author(s):

Anton van Wyk

Keyword(s):

Structural Complexity ◽

Complexity Reduction ◽

The Other ◽

Sufficient Condition ◽

The Novel ◽

Common Input ◽

Generalized Systems ◽

Fundamental Phenomenon ◽

<div><div>This brief note reports the fundamental phenomenon of implicit multivaluedness exhibited from one output to the other of two node-systems with a common input—referred to as counter-cascaded1 systems—under the appropriate conditions. The novel concepts of immanence and transcendence are introduced upon which the formulation and prove of a necessary and sufﬁcient condition for multivaluedness are based; this is the main result of this note. Next, subsequent consequences of this result are presented. Among these is the fact that this result also holds for cascaded generalized systems. </div><div><br></div><div>The novel application of structural complexity reduction in directed networks presented next, demonstrates the utility of multivaluedness and is itself a contribution to the theory of signals and systems.</div><div><br></div><div>The signiﬁcance of the work presented here is that it contributes toward the theory of systems and networks as well as toward the arsenal of tools for studying networks.</div></div>

On Approximation of the Perturbed Inclusion

Georgian Mathematical Journal ◽

10.1515/gmj.2007.253 ◽

2007 ◽

Vol 14 (2) ◽

pp. 253-267

Author(s):

Alexander I. Bulgakov ◽

Anna A. Grigorenko ◽

Anatoliy I. Korobko

Keyword(s):

Stable Process ◽

Approximate Solutions ◽

Continuous Functions ◽

The Other ◽

Sufficient Condition ◽

Space Of Continuous Functions ◽

Linear Integral ◽

The Right ◽

Abstract The paper is concerned with the so-called perturbed inclusion in the space of continuous functions. The right-hand side of the inclusion is represented by an algebraic sum of the values of two multi-valued maps, one of which consists of compacts and the other is not necessarily closed-valued and is a composition of a linear integral operator and multimap convex-valued with respect to switching. For such an inclusion it is proved that approximation in the space of summable functions of the values of a multimap convex-valued with respect to switching is not always a stable process. The necessary and sufficient condition for the closure of the set of approximate solutions to converge to the closure of the set of solutions for perturbed inclusion is derived.

Characterization of ℓ1 minimizer in one-bit compressed sensing

Analysis and Applications ◽

10.1142/s0219530519500131 ◽

2019 ◽

Vol 17 (06) ◽

pp. 1005-1021 ◽

Cited By ~ 1

Author(s):

Ning Bi ◽

Jun Tan

Keyword(s):

Compressive Sensing ◽

Information Science ◽

Convex Sets ◽

Sufficient Condition ◽

Relaxation Model ◽

The One ◽

Necessary And Sufficient ◽

Theoretical Results

The conception of one-bit compressive sensing (one-bit CS) was first introduced in 2008 by Boufounos and Baraniuk [1-Bit compressive sensing, in Proc. Conf. on Information Science and Systems (CISS, Princeton, NJ, 2008)]. Since then, many efficient algorithms have been developed for dealing with the one-bit CS problem. However, few theoretical results are available on one-bit CS. In this paper, we focus on one-bit CS theory with its relaxation model [Formula: see text] and present a necessary and sufficient condition such that the signal [Formula: see text] is the unique [Formula: see text] minimizer in noiseless one-bit CS (Theorem 3). Moreover, by using the improved separation theorem of convex sets (Theorem 6), we completely characterize the [Formula: see text] minimizer in one-bit CS (Theorem 2). Finally, as an application of Theorem 2, the [Formula: see text] minimizer for the considered model can, in general, be a non-sparse vector.

On representation of Poisson mixtures as Poisson sums and a characterization of the gamma distribution

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100053937 ◽

1977 ◽

Vol 82 (2) ◽

pp. 297-300 ◽

Cited By ~ 1

Author(s):

A. V. Godambe

Keyword(s):

Gamma Distribution ◽

Exponential Type ◽

Similar Condition ◽

Sufficient Condition ◽

Poisson Mixture ◽

Mixing Distribution ◽

Poisson Mixtures ◽

AbstractA necessary and sufficient condition for a Poisson mixture with an exponential type mixing distribution to be equivalently represented as a Poisson sum is obtained. The problem of deriving a similar condition under any mixing distribution on (0, ∞) is discussed. Finally, a characterization of the gamma distribution is obtained.

A characterization of the Poisson process

Journal of Applied Probability ◽

10.2307/3212584 ◽

1974 ◽

Vol 11 (1) ◽

pp. 72-85 ◽

Cited By ~ 13

Author(s):

S. M. Samuels

Keyword(s):

Poisson Process ◽

Renewal Process ◽

Poisson Processes ◽

Renewal Processes ◽

Sufficient Condition ◽

Complete Proof ◽

Theorem: A necessary and sufficient condition for the superposition of two ordinary renewal processes to again be a renewal process is that they be Poisson processes.A complete proof of this theorem is given; also it is shown how the theorem follows from the corresponding one for the superposition of two stationary renewal processes.

Equipartitioning and balancing points of polygons

Pythagoras ◽

10.4102/pythagoras.v0i71.2 ◽

2010 ◽

Vol 0 (71) ◽

Author(s):

Shunmugam Pillay ◽

Poobhalan Pillay

Keyword(s):

General Theorem ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

The Other ◽

Sufficient Condition ◽

Centre Of Mass ◽

The centre of mass G of a triangle has the property that the rays to the vertices from G sweep out triangles having equal areas. We show that such points, termed equipartitioning points in this paper, need not exist in other polygons. A necessary and sufficient condition for a quadrilateral to have an equipartitioning point is that one of its diagonals bisects the other. The general theorem, namely, necessary and sufficient conditions for equipartitioning points for arbitrary polygons to exist, is also stated and proved. When this happens, they are in general, distinct from the centre of mass. In parallelograms, and only in them, do the two points coincide.

Orthogonal Fractional Factorial Plans

Calcutta Statistical Association Bulletin ◽

10.1177/0008068319800303 ◽

1980 ◽

Vol 29 (3-4) ◽

pp. 143-160 ◽

Cited By ~ 6

Author(s):

Rahul Mukerjee

Keyword(s):

Broad Class ◽

Existence Results ◽

Fractional Factorial ◽

The Other ◽

Sufficient Condition ◽

Equal Frequency ◽

Other Hand ◽

Necessary And Sufficient ◽

Generally True

This paper shows that the criterion of proportional frequency for (unblocked) orthogonal fractional factorial plans, as suggested by some previous authors, is not generally true. On the other hand, the criterion of equal frequency has been established as a necessary and sufficient condition in the general case. Some other properties of orthogonal fractional factorial plans have been investigated. A necessary and sufficient condition for designs involving two or more blocks has also been presented. A broad class of non-existence results follow.

Uncertainty Regarding Interpretation of the “Negligence Rule” and Its Implications for the Efficiency of Outcomes

Asian Journal of Law and Economics ◽

10.1515/ajle-2015-0031 ◽

2016 ◽

Vol 7 (2) ◽

Author(s):

Satish K. Jain

Keyword(s):

Subjective Probability ◽

The Other ◽

Sufficient Condition ◽

Subjective Probabilities ◽

Liability Rule ◽

Important Conclusion ◽

Standard Version ◽

Necessary And Sufficient ◽

Negligence Rule

AbstractThere are two ways that the negligence rule is interpreted. Under one interpretation a negligent injurer is liable for the entire harm to the victim; and under the other interpretation a negligent injurer is liable only for that part of the harm which can be ascribed to his negligence. Both these versions are efficient. However, if there is uncertainty regarding whether the court will be employing the full liability version or the incremental liability version for determining the liability of a negligent injurer, notwithstanding the fact that both the versions are efficient, inefficiency is possible. It is shown in the paper that a necessary and sufficient condition for efficiency in all cases is that the subjective probability with which the injurer expects the standard version to be employed must be greater than or equal to the subjective probability with which the victim expects the standard version to be employed. For the subset of applications without complementarities in the cares of the two parties and which are such that the total social costs are minimized at a unique care-configuration, it is shown that efficiency obtains regardless of the subjective probabilities with which the parties expect the two versions. One very important conclusion that emerges from the analysis of this paper is that when courts employ more than one liability rule, even if all the employed rules are efficient, the efficiency of all outcomes cannot be taken for granted merely on the ground of the efficiency of the employed rules.

On weak convergence within the hnbue family of life distributions

Journal of Applied Probability ◽

10.1017/s0021900200028850 ◽

1984 ◽

Vol 21 (03) ◽

pp. 654-660 ◽

Cited By ~ 4

Author(s):

Sujit K. Basu ◽

Manish C. Bhattacharjee

Keyword(s):

Weak Convergence ◽

Exponential Distribution ◽

Limiting Distribution ◽

Sufficient Condition ◽

Moment Sequence ◽

Positive Order ◽

Life Distributions ◽

We show that the HNBUE family of life distributions is closed under weak convergence and that weak convergence within this family is equivalent to convergence of each moment sequence of positive order to the corresponding moment of the limiting distribution. A necessary and sufficient condition for weak convergence to the exponential distribution is given, based on a new characterization of exponentials within the HNBUE family of life distributions.