Necessary and sufficient conditions for a graph to be three-terminal series-parallel-cascade

A “three-terminal series-parallel-cascade graph” is defined as a three-terminal graph which is constructed by means of cascade connections in addition to series and parallel connections which were used in constructing a three-terminal series-parallel graph in our previous paper. Some properties of the graph are presented, and a theorem of the Kuratowski type is given stating that a three-terminal nonseparable graph is three-terminal series-parallel-cascade if and only if none of certain three graphs can be obtained from it by opening or shorting some of the edges. This theorem characterizes a three-terminal series-parallel-cascade graph completely, and clarifies its structual limitation.     

Necessary conditions and a sufficient condition for the Fischer-Hadamard inequalities

It is well known that a matrix, all of whose principal minors are positive, satisfies the Fischer-Hadamard inequalities if and only if it is weakly sign symmetric. In this paper we consider the general case of matrices whose principal minors may be nonpositive. Necessary conditions and a sufficient condition for the Fischer-Hadamard inequalities to hold are given in the general case.     

Necessary and sufficient conditions for the Robbins-Monro method

This paper examines the relation between convergence of the Robbins-Monro iterates $\text{X}{}_{\mathrm{n+1}}\text{= X}{}_{\mathrm{n}}\text{?a}{}_{\mathrm{n}}\text{?(X}{}_{\mathrm{n}}\text{)+a}{}_{\mathrm{n}}\text{ξ}{}_{\mathrm{n}}\text{, ?(θ)=0}$, and the laws of large numbers S_n=a_nΣ^n?1_j=0ξ_j→0 as n→+∞. If a_n is decreasing at least as rapidly as c/n, then X_n→θw.p. 1 (resp. in L_p, p?1) implies S_n→0 w.p. 1 (resp. in L_p, p?1) as n→+∞. If a_n is decreasing at least as slowly as c?n and lim_n→+∞a_n=0, then S_n→0 w.p. 1 (resp. in L_p, p?2) implies X_n→θw.p. 1 (resp. in L_p, p?2) as n →+∞. Thus, there is equivalence in the frequently examined case $\text{a}{}_{\mathrm{n}}\text{?c?n}$. Counter examples show that the LLN must have the form of S_n, that the rate of decrease conditions are sharp, that the weak LLN is neither necessary nor sufficient for the convergence in probability of X_n to θ when $\text{a}{}_{\mathrm{n}}\text{?c?n}$.     

Necessary and sufficient conditions for hyperplane transversals

We prove that a finite family ={B ₁,B ₂, ...,B _n } of connected compact sets in ^d has a hyperplane transversal if and only if for somek there exists a set of pointsP={p ₁,p ₂, ...,p _n } (i.e., ak-dimensional labeling of the family) which spans ^k and everyk+2 sets of are met by ak-flat consistent with the order type ofP. This is a common generalization of theorems of Hadwiger, Katchalski, Goodman-Pollack and Wenger.Supported in part by NSF grant DMS-8501947 and CCR-8901484, NSA grant MDA904-89-H-2030, and the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), a National Science Foundation Science and Technology Center, under NSF grant STC88-09648.Supported by the National Science and Engineering Research Council of Canada and DIMACS.     

Necessary and sufficient conditions for bang-bang control

Necessary and sufficient conditions for the optimal control to be bang-bang are presented for a nonlinear system. The payoff, which is not necessarily quadratic, is assumed to be described by a Hilbert-space norm and to be differentiable and convex. The results are extensions of Ref. 1 to the case of nonlinear systems.     

Necessary and sufficient conditions for multivariate majorization

Let X and Y be m × n matrices whose elements are in K, a real or complex field. We obtain necessary and sufficient conditions for the existence of a matrix A belonging to the convex hull of a certain subgroup of the general linear group GL_n(K) such that X = YA, which unite and generalize several known results concerning majorization.     

Necessary and sufficient conditions for quadratic minimality

We provide necessary and sufficient conditions for a (non-convex) quadratic function to take a local minimum over a convex set. Various limiting examples are given.     

Necessary and sufficient conditions for interval-valued differentiability

This paper presents necessary and sufficient conditions for generalized Hukuhara differentiability of interval-valued functions and counterexamples of some equivalences previously presented in the literature, for which important results are based on. Moreover, applications of interval generalized Hukuhara differentiability are presented.     

Necessary and sufficient conditions for a minimum of mixed order

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 3, pp. 415–419, March, 1989.     

Necessary and sufficient conditions for finiteness of lifetime

We give a geometric characterization for the finiteness of conditioned Brownian motion for a general class of simply connected domains, extending previous results and exhibit some new examples of domains with infinite area and finite lifetime.I would like to thank Professor Rodrigo Bañuelos, my academic advisor, for his help and guidance on this paper which is part of my Ph.D. thesis.     

Necessary and sufficient conditions for stability of effectivity functions

An effectivity functionE assigns to every coalitionS of players a familyE (S) of subsetsB of an outcome setA such thatS can force the outcome to belong to any of the setsB inE (S). The effectivity functionE is stable if for every preference profile there is an outcomex with the property that there is no coalitionS and subsetB ofA such thatB εE (S) and each player inS prefers everyy εB tox. The paper gives a necessary and sufficient condition for an effectivity function to be stable.     

Necessary and sufficient conditions for expansions of Wilson type

We consider expansions of the type arising from Wilson bases. We characterize such expansions for L^2（R）. As an application, we see that such an expansion must be orthonormal, in contrast to the case of wavelet expansions generated by translations and dilation.     

Necessary and sufficient conditions for convergence of attainment times

Translated fromProblemy Ustoichivosti Stokhasticheskikh Modelei. Trudy Seminara, 1988, pp. 129–137.     

Necessary and sufficient conditions for conservativeness of dynamical semigroups

Dynamical semigroups constitute a quantum-mechanical generalization of Markov semigroups, a concept familiar from the theory of stochastic processes. Let be a Hilbert space andA a von Neumann algebra. A dynamical semigroup P_t is a -weakly continuous one-parameter semigroup of completely positive maps ofA into itself. A semigroup P_t possessing the property of preserving the identityIA is said to be conservative and its infinitesimal operator L[·] is said to be regular. The present paper studies necessary and sufficient conditions for strongly continuous dynamical semigroups to be conservative. It is shown that under certain additional assumptions one can formulate necessary and sufficient conditions which are analogous to Feller's condition for regularity of a diffusion process: the equation P=L[P] has no solutions inA ₊. Using a Jensen-type inequality for completely positive maps, constructive sufficient conditions are obtained for conservativeness, in the form of inequalities for commutators. The restriction of a dynamical subgroup to an Abelian subalgebra of (R ⁿ ) yields a series of new regularity conditions for both diffusion and jump processes.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 36, pp. 149–184, 1990.     

Necessary and sufficient consistency conditions for a recursive kernel regression estimate

A recursive kernel estimate ∑_{i = 1}ⁿ Y_iK⧸(x − X_i)h_i)⧸∑_{j = 1}ⁿ K((x − X_j)⧸h_j) of a regression m(x) = E{Y|X = x} calculated from independent observations (X₁, Y₁),…, (X_n, Y_n) of a pair (X, Y) of random variables is examined. ForE|Y|^{1 + δ} < ∞, δ > 0, the estimate is weakly pointwise consistent for almost all (μ) x ∈ R^d, μ is the probability measure of X, if and only if∑_i−1ⁿ h_i^d I_{hi > ɛ } ⧸ ∑_{j = 1}ⁿ h_j^d → 0 as n → ∞, all ɛ > 0, and∑_{i = 1}^∞ h_i^d = ∞, d is the dimension of X. For E|Y|^{1 + δ} < ∞, δ > 0, the estimate is strongly pointwise consistent for almost all (μ) x ∈ R^d, if and only if the same conditions hold. ForE|Y|^{1 + δ} < ∞, δ > 0, weak and strong consistency are equivalent. Similar results are given for complete convergence.     

Necessary and sufficient conditions for expansions of Gabor type

In this paper, we give necessary and sufficient conditions for two families of Gabor functions of a certain type to yield a reproducing identity on L^2（R^n）. As applications, we characterize when such families yield orthonormal or bi-orthogonal expansions. We also obtain a generalization of the Balian-Low theorem for general reprodueing identities （not necessary coming from a frame）.