scholarly journals Solving an Open Problem About the G-Drazin Partial Order

2020 ◽  
Vol 36 (36) ◽  
pp. 55-66 ◽  
Author(s):  
David Ferreyra ◽  
Marina Lattanzi ◽  
Fabián Levis ◽  
Nestor Thome
Keyword(s):  
Partial Order ◽  
Open Problem ◽  
Additional Condition ◽  
Drazin Inverse ◽  
Necessary Condition ◽  
Converse Implication ◽  
Special Cases

G-Drazin inverses and the G-Drazin partial order for square matrices have been both recently introduced by Wang and Liu. They proved the following implication: if A is below B under the G-Drazin partial order then any G-Drazin inverse of B is also a G-Drazin inverse of A. However, this necessary condition could not be stated as a characterization and the validity (or not) of the converse implication was posed as an open problem. In this paper, we solve completely this problem. We show that the converse, in general, is false and we provide a form to construct counterexamples. We also prove that the converse holds under an additional condition (which is also necessary) as well as for some special cases of matrices.

The theory of the recursively enumerable weak truth-table degrees is undecidable

1992 ◽  
Vol 57 (3) ◽  
pp. 864-874 ◽  
Author(s):  
Klaus Ambos-Spies ◽  
André Nies ◽  
Richard A. Shore
Keyword(s):  
Partial Order ◽  
Open Problem ◽  
Truth Table ◽  
Recursively Enumerable ◽  
Undecidable Theory ◽  
Weak Truth Table Degrees

AbstractWe show that the partial order of -sets under inclusion is elementarily definable with parameters in the semilattice of r.e. wtt-degrees. Using a result of E. Herrmann, we can deduce that this semilattice has an undecidable theory, thereby solving an open problem of P. Odifreddi.

Maximin: a direct approach to sustainability

2006 ◽  
Vol 11 (3) ◽  
pp. 275-300 ◽  
Author(s):  
ROBERT D. CAIRNS ◽  
NGO VAN LONG
Keyword(s):  
Precautionary Principle ◽  
Direct Approach ◽  
Necessary Condition ◽  
Hyperbolic Discount ◽  
Discount Factors ◽  
Discounted Utility ◽  
Special Cases ◽  
Analytic Expression ◽  
Hotelling's Rule ◽  
Competitive Economy

We solve directly a general maximin (sustainment, intergenerational-equity) problem. Because the shadow values of a maximin problem do not correspond to the shadow values from a general discounted-utility solution, they correspond to the prices of only a very special competitive economy. Virtual discount factors for the economy arise. They do not correspond to hyperbolic discount factors. Hartwick's rule is derived and generalized naturally to take into account non-autonomous and non-deterministic features of the economy. Under uncertainty, Hartwick's rule is the analytic expression of a form of precautionary principle. Hotelling's rule is a necessary condition, but may be more complex than has been appreciated in simple models. Some interpretations of strong sustainment are special cases of weak sustainment but, paradoxically, may be more difficult to solve.

Exploring scalar quantum walks on Cayley graphs

2008 ◽  
Vol 8 (1&2) ◽  
pp. 68-81
Author(s):  
O.L. Acevedo ◽  
J. Roland ◽  
N.J. Cerf
Keyword(s):  
Evolution Operator ◽  
Cayley Graphs ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Constructive Method ◽  
Quantum Evolution ◽  
Necessary Condition ◽  
Internal Space ◽  
Special Cases ◽  
Scalar Quantum

A quantum walk, \emph{i.e.}, the quantum evolution of a particle on a graph, is termed \emph{scalar} if the internal space of the moving particle (often called the coin) has dimension one. Here, we study the existence of scalar quantum walks on Cayley graphs, which are built from the generators of a group. After deriving a necessary condition on these generators for the existence of a scalar quantum walk, we present a general method to express the evolution operator of the walk, assuming homogeneity of the evolution. We use this necessary condition and the subsequent constructive method to investigate the existence of scalar quantum walks on Cayley graphs of groups presented with two or three generators. In this restricted framework, we classify all groups -- in terms of relations between their generators -- that admit scalar quantum walks, and we also derive the form of the most general evolution operator. Finally, we point out some interesting special cases, and extend our study to a few examples of Cayley graphs built with more than three generators.

scholarly journals Quantum algorithm for multivariate polynomial interpolation

2018 ◽  
Vol 474 (2209) ◽  
pp. 20170480 ◽  
Author(s):  
Jianxin Chen ◽  
Andrew M. Childs ◽  
Shih-Han Hung
Keyword(s):  
Open Problem ◽  
Polynomial Interpolation ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Query Complexity ◽  
Large Field ◽  
Multivariate Polynomial ◽  
Multivariate Polynomial Interpolation ◽  
Special Cases ◽  
Speed Up

How many quantum queries are required to determine the coefficients of a degree- d polynomial in n variables? We present and analyse quantum algorithms for this multivariate polynomial interpolation problem over the fields F q , R and C . We show that k C and 2 k C queries suffice to achieve probability 1 for C and R , respectively, where k C = ⌈ ( 1 / ( n + 1 ) ) ( n + d d ) ⌉ except for d =2 and four other special cases. For F q , we show that ⌈( d /( n + d ))( n + d d ) ⌉ queries suffice to achieve probability approaching 1 for large field order q . The classical query complexity of this problem is ( n + d d ) , so our result provides a speed-up by a factor of n +1, ( n +1)/2 and ( n + d )/ d for C , R and F q , respectively. Thus, we find a much larger gap between classical and quantum algorithms than the univariate case, where the speedup is by a factor of 2. For the case of F q , we conjecture that 2 k C queries also suffice to achieve probability approaching 1 for large field order q , although we leave this as an open problem.

scholarly journals Implications for Firms with Limited Information to Take Advantage of Reference Price Effect in Competitive Settings

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-16 ◽  
Author(s):  
Junhai Ma ◽  
Zhanbing Guo
Keyword(s):  
Reference Price ◽  
Partial Information ◽  
Necessary Condition ◽  
Adjustment Process ◽  
Flip Bifurcation ◽  
Limited Information ◽  
Dynamic Adjustment ◽  
Internal Reference ◽  
Special Cases ◽  
Price Effects

This paper studies internal reference price effects when competitive firms face reference price effects and make decisions based on partial information, where their decision-making mechanism is modeled by a dynamic adjustment process. It is shown that the evolution of this dynamic adjustment goes to stabilization if both adjustment speeds are small and the complexity of this evolution increases in adjustment speeds. It is proved that the necessary condition for flip bifurcation or Neimark-Sacker bifurcation will occur with the increase of adjustment speed in two special cases. What is more, numerical simulations show that these bifurcations do occur. Then, the impacts of parameters on stability and profits are investigated and some management insights for firms with limited information to take advantage of reference price effects are provided.

scholarly journals Bound Graph Polysemy

10.37236/1521 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Paul J. Tanenbaum
Keyword(s):  
Lower Bound ◽  
Partial Order ◽  
Upper Bound ◽  
Special Cases ◽  
Vertex Set

Bound polysemy is the property of any pair $(G_1, G_2)$ of graphs on a shared vertex set $V$ for which there exists a partial order on $V$ such that any pair of vertices has an upper bound precisely when the pair is an edge in $G_1$ and a lower bound precisely when it is an edge in $G_2$. We examine several special cases and prove a characterization of the bound polysemic pairs that illuminates a connection with the squared graphs.

scholarly journals Additive results for the generalized Drazin inverse

2002 ◽  
Vol 73 (1) ◽  
pp. 115-126 ◽  
Author(s):  
Dragan S. Djordjević ◽  
Yimin Wei
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Drazin Inverse ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Special Cases ◽  
Arbitrary Complex ◽  
Additive Perturbation ◽  
Generalized Drazin Inverse ◽  
Banach Space Operators

AbstractAdditive perturbation results for the generalized Drazin inverse of Banach space operators are presented. Precisely, if Ad denotes the generalized Drazin inverse of a bounded linear operator A on an arbitrary complex Banach space, then in some special cases (A + B)d is computed in terms of Ad and Bd. Thus, recent results of Hartwig, Wang and Wei (Linear Algebra Appl. 322 (2001), 207–217) are extended to infinite dimensional settings with simplified proofs.

scholarly journals On improper integrals of products of logarithmic, power and Bessel functions

1992 ◽  
Vol 45 (3) ◽  
pp. 395-398
Author(s):  
M. Aslam Chaudhry ◽  
M. Ahmad
Keyword(s):  
Integral Representation ◽  
Open Problem ◽  
Bessel Functions ◽  
Euler Constant ◽  
Special Cases ◽  
Improper Integrals

In this paper we have evaluated the integral . A new integral representation of the Euler constant is shown. Some special cases of the result are discussed and an open problem is posed.

scholarly journals A note on computing the generalized inverseA T,S (2)of a matrixA

2002 ◽  
Vol 31 (8) ◽  
pp. 497-507 ◽  
Author(s):  
Xiezhang Li ◽  
Yimin Wei
Keyword(s):  
Iterative Methods ◽  
Half Plane ◽  
Generalized Inverse ◽  
Null Space ◽  
Drazin Inverse ◽  
Numerical Examples ◽  
Special Cases ◽  
Computational Procedures

The generalized inverseA T,S (2)of a matrixAis a{2}-inverse ofAwith the prescribed rangeTand null spaceS. A representation for the generalized inverseA T,S (2)has been recently developed with the conditionσ (GA| T)⊂(0,∞), whereGis a matrix withR(G)=TandN(G)=S. In this note, we remove the above condition. Three types of iterative methods forA T,S (2)are presented ifσ(GA|T)is a subset of the open right half-plane and they are extensions of existing computational procedures ofA T,S (2), including special cases such as the weighted Moore-Penrose inverseA M,N †and the Drazin inverseAD. Numerical examples are given to illustrate our results.

The (b,c)-inverse for products and lower triangular matrices

2017 ◽  
Vol 16 (12) ◽  
pp. 1750222 ◽  
Author(s):  
Yuanyuan Ke ◽  
Zhou Wang ◽  
Jianlong Chen
Keyword(s):  
Sufficient Conditions ◽  
Triangular Matrix ◽  
Drazin Inverse ◽  
Triangular Matrices ◽  
Core Inverse ◽  
Special Cases ◽  
Lower Triangular ◽  
Necessary And Sufficient ◽  
Lower Triangular Matrices ◽  
Dual Core

Let [Formula: see text] be a semigroup and [Formula: see text]. The concept of [Formula: see text]-inverses was introduced by Drazin in 2012. It is well known that the Moore–Penrose inverse, the Drazin inverse, the Bott–Duffin inverse, the inverse along an element, the core inverse and dual core inverse are all special cases of the [Formula: see text]-inverse. In this paper, a new relationship between the [Formula: see text]-inverse and the Bott–Duffin [Formula: see text]-inverse is established. The relations between the [Formula: see text]-inverse of [Formula: see text] and certain classes of generalized inverses of [Formula: see text] and [Formula: see text], and the [Formula: see text]-inverse of [Formula: see text] are characterized for some [Formula: see text], where [Formula: see text]. Necessary and sufficient conditions for the existence of the [Formula: see text]-inverse of a lower triangular matrix over an associative ring [Formula: see text] are also given, and its expression is derived, where [Formula: see text] are regular triangular matrices.

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