scholarly journals Pattanaik’s axioms and existence of preferred with probability at least half winners

2021 ◽  
Vol 31 (2) ◽  
Author(s):  
Somdeb Lahiri
Keyword(s):  
Probability Distribution ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Positive Probability ◽  
Linear Orders ◽  
State Dependent ◽  
Fixed Set ◽  
States Of Nature

In this paper we show that three conditions due to Pattanaik, when satisfied by a given profile of state-dependent preferences (linear orders) on a given and fixed set of alternatives and a probability distribution with which the various states of nature occur, are individually sufficient, for the non-emptiness of the set of alternative(s) which are individually preferred to all alternatives other than itself with probability at least half. Prior to this, we show that since each axiom individually implies Sen-Coherence, then, as a consequence of a result obtained earlier, each axiom along with asymmetry of the ‘preferred with at probability at least half” relation implies the transitivity of the relation. All the sufficient conditions discussed here are required to be applicable at least to all those otherwise relevant events that have positive probability. This observation also applies to a sufficient condition for the non-emptiness of the set of alternative(s) which are individually preferred to all alternatives other than itself with probability at least half, called Generalised Sen Coherence introduced and discussed in earlier research.

scholarly journals A Note on ‘Improved Fréchet Bounds and Model-Free Pricing of Multi-Asset Options’ by Tankov (2011)

2012 ◽  
Vol 49 (3) ◽  
pp. 866-875 ◽  
Author(s):  
Carole Bernard ◽  
Xiao Jiang ◽  
Steven Vanduffel
Keyword(s):  
Lower Bound ◽  
Portfolio Selection ◽  
Sufficient Conditions ◽  
Optimal Investment ◽  
Sufficient Condition ◽  
Investment Strategies ◽  
Model Free ◽  
State Dependent ◽  
Bivariate Copula ◽  
Fréchet Bounds

Tankov (2011) improved the Fréchet bounds for a bivariate copula when its values on a compact subset of [0, 1]2 are given. He showed that the best possible bounds are quasi-copulas and gave a sufficient condition for these bounds to be copulas. In this note we give weaker sufficient conditions to ensure that the bounds are copulas. We also show how this can be useful in portfolio selection. It turns out that finding a copula as a lower bound plays a key role in determining optimal investment strategies explicitly for investors with some type of state-dependent constraints.

scholarly journals A Note on ‘Improved Fréchet Bounds and Model-Free Pricing of Multi-Asset Options’ by Tankov (2011)

2012 ◽  
Vol 49 (03) ◽  
pp. 866-875 ◽  
Author(s):  
Carole Bernard ◽  
Xiao Jiang ◽  
Steven Vanduffel
Keyword(s):  
Lower Bound ◽  
Portfolio Selection ◽  
Sufficient Conditions ◽  
Optimal Investment ◽  
Sufficient Condition ◽  
Investment Strategies ◽  
Model Free ◽  
State Dependent ◽  
Bivariate Copula ◽  
Fréchet Bounds

Tankov (2011) improved the Fréchet bounds for a bivariate copula when its values on a compact subset of [0, 1]2 are given. He showed that the best possible bounds are quasi-copulas and gave a sufficient condition for these bounds to be copulas. In this note we give weaker sufficient conditions to ensure that the bounds are copulas. We also show how this can be useful in portfolio selection. It turns out that finding a copula as a lower bound plays a key role in determining optimal investment strategies explicitly for investors with some type of state-dependent constraints.

Some Contributions to the Theory of Near-Critical Bisexual Branching Processes

2007 ◽  
Vol 44 (02) ◽  
pp. 492-505
Author(s):  
M. Molina ◽  
M. Mota ◽  
A. Ramos
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Positive Probability ◽  
Limit Laws ◽  
Bisexual Branching Processes ◽  
Probabilistic Evolution

We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to other classes of near-critical bisexual branching processes.

scholarly journals Noetherian properties in composite generalized power series rings

2020 ◽  
Vol 18 (1) ◽  
pp. 1540-1551
Author(s):  
Jung Wook Lim ◽  
Dong Yeol Oh
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generalized Power Series ◽  
Ordered Monoid ◽  
Power Series Rings ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

Abstract Let ({\mathrm{\Gamma}},\le ) be a strictly ordered monoid, and let {{\mathrm{\Gamma}}}^{\ast }\left={\mathrm{\Gamma}}\backslash \{0\} . Let D\subseteq E be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D. Set \begin{array}{l}D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {E}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(0)\in D\right\}\hspace{.5em}\text{and}\\ \hspace{0.2em}D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {D}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(\alpha )\in I,\hspace{.5em}\text{for}\hspace{.25em}\text{all}\hspace{.5em}\alpha \in {{\mathrm{\Gamma}}}^{\ast }\right\}.\end{array} In this paper, we give necessary conditions for the rings D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively ordered, and sufficient conditions for the rings D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively totally ordered. Moreover, we give a necessary and sufficient condition for the ring D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively totally ordered. As corollaries, we give equivalent conditions for the rings D+({X}_{1},\ldots ,{X}_{n})E{[}{X}_{1},\ldots ,{X}_{n}] and D+({X}_{1},\ldots ,{X}_{n})I{[}{X}_{1},\ldots ,{X}_{n}] to be Noetherian.

Embedded renewal processes in the GI/G/s queue

1972 ◽  
Vol 9 (3) ◽  
pp. 650-658 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Limit Theorems ◽  
Sufficient Conditions ◽  
Interarrival Time ◽  
Renewal Processes ◽  
Positive Probability ◽  
Functional Limit ◽  
Light Traffic ◽  
Functional Limit Theorems ◽  
Cumulative Processes ◽  
Rule Out

The stable GI/G/s queue (ρ < 1) is sometimes studied using the “fact” that epochs just prior to an arrival when all servers are idle constitute an embedded persistent renewal process. This is true for the GI/G/1 queue, but a simple GI/G/2 example is given here with all interarrival time and service time moments finite and ρ < 1 in which, not only does the system fail to be empty ever with some positive probability, but it is never empty. Sufficient conditions are then given to rule out such examples. Implications of embedded persistent renewal processes in the GI/G/1 and GI/G/s queues are discussed. For example, functional limit theorems for time-average or cumulative processes associated with a large class of GI/G/s queues in light traffic are implied.

SOME SUFFICIENT CONDITIONS OF LEARNABILITY IN THE LIMIT FROM POSITIVE DATA

2000 ◽  
Vol 11 (03) ◽  
pp. 515-524
Author(s):  
TAKESI OKADOME
Keyword(s):  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Conditions For Learning ◽  
Positive Data ◽  
Necessary And Sufficient ◽  
Learning In The Limit

The paper deals with learning in the limit from positive data. After an introduction and overview of earlier results, we strengthen a result of Sato and Umayahara (1991) by establishing a necessary and sufficient condition for the satisfaction of Angluin's (1980) finite tell-tale condition. Our other two results show that two notions introduced here, the finite net property and the weak finite net property, lead to sufficient conditions for learning in the limit from positive data. Examples not solvable by earlier methods are also given.

Separability criteria based on the Bloch representation of density matrices

2007 ◽  
Vol 7 (7) ◽  
pp. 624-638
Author(s):  
J. de Vicente
Keyword(s):  
Sufficient Conditions ◽  
Quantum Systems ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Pure Product ◽  
Separability Problem ◽  
Arbitrary Dimensions ◽  
Necessary And Sufficient ◽  
Bloch Representation

We study the separability of bipartite quantum systems in arbitrary dimensions using the Bloch representation of their density matrix. This approach enables us to find an alternative characterization of the separability problem, from which we derive a necessary condition and sufficient conditions for separability. For a certain class of states the necessary condition and a sufficient condition turn out to be equivalent, therefore yielding a necessary and sufficient condition. The proofs of the sufficient conditions are constructive, thus providing decompositions in pure product states for the states that satisfy them. We provide examples that show the ability of these conditions to detect entanglement. In particular, the necessary condition is proved to be strong enough to detect bound entangled states.

The Maxslope Taxometric Procedure: Mathematical Derivation, Parameter Estimation, Consistency Tests

2004 ◽  
Vol 95 (2) ◽  
pp. 517-550 ◽  
Author(s):  
William M. Grove
Keyword(s):  
Parameter Estimation ◽  
Nonlinear Regression ◽  
Classification Accuracy ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Indicator Variable ◽  
Regression Slope ◽  
The Relationship ◽  
Mathematical Derivation ◽  
Made In

This article first explains concepts in taxometrics, including the meaning of “taxon” in relation to taxometric procedures. It then mathematically develops the MAXSLOPE procedure of Grove and Meehl which relies on nonlinear regression of one taxometric indicator variable on another. Sufficient conditions for MAXSLOPE's validity are set forth. The relationship between the point of maximum regression slope (MAXSLOPE point) and the HITMAX cut, i.e., the point on a variable which, if used as a diagnostic cut-off score, yields maximum classification accuracy, is analyzed. A sufficient condition is given for the MAXSLOPE point to equal the HITMAX cut; however, most distributions have different MAXSLOPE and HITMAX points. Equations and an algorithm are spelled out for making a graphical test for the existence of a taxon, estimating taxometric parameters, and conducting consistency tests; the latter serve as stringent checks on the validity of a taxonic conjecture. The plausibility of assumptions made, in deriving MAXSLOPE equations, is discussed, and the qualitative effects of violations of these assumptions are explained.

scholarly journals Equipartitioning and balancing points of polygons

Pythagoras ◽  
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 ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

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.

Asymptotic stability of heteroclinic cycles in systems with symmetry. II

2004 ◽  
Vol 134 (6) ◽  
pp. 1177-1197 ◽  
Author(s):  
Martin Krupa ◽  
Ian Melbourne
Keyword(s):  
Asymptotic Stability ◽  
Transition Matrix ◽  
Sufficient Conditions ◽  
Systematic Investigation ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Heteroclinic Cycles ◽  
Higher Dimensional ◽  
Necessary And Sufficient

Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R3.Field and Swift, and Hofbauer, considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique.In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.

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