Extension of Monotonic Functions and Representation of Preferences

2021 ◽  
Author(s):  
Özgür Evren ◽  
Farhad Hüsseinov
Keyword(s):  
Dominance Relation ◽  
Translation Invariance ◽  
Compact Set ◽  
Continuous Real Function ◽  
Translation Invariant ◽  
Extension Theorems ◽  
Topological Vector Spaces ◽  
Dominance Relations ◽  
Universal Space ◽  
Vector Structure

Consider a dominance relation (a preorder) ≿ on a topological space X, such as the greater than or equal to relation on a function space or a stochastic dominance relation on a space of probability measures. Given a compact set K ⊆ X, we study when a continuous real function on K that is strictly monotonic with respect to ≿ can be extended to X without violating the continuity and monotonicity conditions. We show that such extensions exist for translation invariant dominance relations on a large class of topological vector spaces. Translation invariance or a vector structure are no longer needed when X is locally compact and second countable. In decision theoretic exercises, our extension theorems help construct monotonic utility functions on the universal space X starting from compact subsets. To illustrate, we prove several representation theorems for revealed or exogenously given preferences that are monotonic with respect to a dominance relation.

Incremental Approach for Updating Approximations of Variable Precision Rough Set Based on Dominance Relations

2014 ◽  
Vol 631-632 ◽  
pp. 49-52
Author(s):  
Yan Li ◽  
Jia Jia Hou ◽  
Xiao Qing Liu
Keyword(s):  
Information System ◽  
Rough Set ◽  
Dominance Relation ◽  
Incremental Updating ◽  
Dominance Relations ◽  
Incremental Approach ◽  
Variable Precision ◽  
Variable Precision Rough Set ◽  
Over Time

Variable precision rough set (VPRS) based on dominance relation is an extension of traditional rough set by which can handle preference-ordered information flexibly. This paper focuses on the maintenance of approximations in dominance based VPRS when the objects in an information system vary over time. The incremental updating principles are given as inserting or deleting an object, and some experimental evaluations validates the effectiveness of the proposed method.

scholarly journals An infinite particle system with additive interactions

1979 ◽  
Vol 11 (02) ◽  
pp. 355-383 ◽  
Author(s):  
Richard Durrett
Keyword(s):  
Necessary Conditions ◽  
Particle System ◽  
Particle Systems ◽  
Compact Set ◽  
Translation Invariant ◽  
Limiting Behavior ◽  
Branching Random Walks ◽  
Sufficient And Necessary Conditions ◽  
Infinite Particle ◽  
Number Of Particles

The models under consideration are a class of infinite particle systems which can be written as a superposition of branching random walks. This paper gives some results about the limiting behavior of the number of particles in a compact set ast→ ∞ and also gives both sufficient and necessary conditions for the existence of a non-trivial translation-invariant stationary distribution.

scholarly journals Kinetics of Recovery of the Dark-adapted Salamander Rod Photoresponse

1998 ◽  
Vol 111 (1) ◽  
pp. 7-37 ◽  
Author(s):  
S. Nikonov ◽  
N. Engheta ◽  
E.N. Pugh
Keyword(s):  
Theoretical Analysis ◽  
Time Constant ◽  
Translation Invariance ◽  
Flash Intensity ◽  
Translation Invariant ◽  
Calcium Dependent ◽  
The Difference ◽  
Analytical Expressions ◽  
Kinetics Of ◽  
Flash Response

The kinetics of the dark-adapted salamander rod photocurrent response to flashes producing from 10 to 105 photoisomerizations (Φ) were investigated in normal Ringer's solution, and in a choline solution that clamps calcium near its resting level. For saturating intensities ranging from ∼102 to 104 Φ, the recovery phases of the responses in choline were nearly invariant in form. Responses in Ringer's were similarly invariant for saturating intensities from ∼103 to 104 Φ. In both solutions, recoveries to flashes in these intensity ranges translated on the time axis a constant amount (τc) per e-fold increment in flash intensity, and exhibited exponentially decaying “tail phases” with time constant τc. The difference in recovery half-times for responses in choline and Ringer's to the same saturating flash was 5–7 s. Above ∼104 Φ, recoveries in both solutions were systematically slower, and translation invariance broke down. Theoretical analysis of the translation-invariant responses established that τc must represent the time constant of inactivation of the disc-associated cascade intermediate (R*, G*, or PDE*) having the longest lifetime, and that the cGMP hydrolysis and cGMP-channel activation reactions are such as to conserve this time constant. Theoretical analysis also demonstrated that the 5–7-s shift in recovery half-times between responses in Ringer's and in choline is largely (4–6 s) accounted for by the calcium-dependent activation of guanylyl cyclase, with the residual (1–2 s) likely caused by an effect of calcium on an intermediate with a nondominant time constant. Analytical expressions for the dim-flash response in calcium clamp and Ringer's are derived, and it is shown that the difference in the responses under the two conditions can be accounted for quantitatively by cyclase activation. Application of these expressions yields an estimate of the calcium buffering capacity of the rod at rest of ∼20, much lower than previous estimates.

Linear Programming Performance Bounds for Markov Chains With Polyhedrally Translation Invariant Probabilities and Applications to Unreliable Manufacturing Systems and Enhanced Wafer Fab Models

2002 ◽  
Author(s):  
James R. Morrison ◽  
P. R. Kumar
Keyword(s):  
Markov Chains ◽  
Queueing Networks ◽  
Manufacturing Systems ◽  
Transition Probabilities ◽  
Invariance Property ◽  
Translation Invariance ◽  
Performance Bounds ◽  
Translation Invariant ◽  
Wafer Fab ◽  
Index Policies

Our focus is on a class of Markov chains which have a polyhedral translation invariance property for the transition probabilities. This class can be used to model several applications of interest which feature complexities not found in usual models of queueing networks, for example failure prone manufacturing systems which are operating under hedging point policies, or enhanced wafer fab models featuring batch tools and setups or affine index policies. We present a new family of performance bounds which is more powerful both in expressive capability as well as the quality of the bounds than some earlier approaches.

Construction of a translation-invariant U(N) damped noncommutative gauge model

2020 ◽  
Vol 35 (22) ◽  
pp. 2050118
Author(s):  
Ouahiba Toumi ◽  
Smain Kouadik
Keyword(s):  
Covariant Derivative ◽  
Torsion Tensor ◽  
Tree Level ◽  
Dirac Field ◽  
Translation Invariance ◽  
Group Model ◽  
Translation Invariant ◽  
Yang Mills ◽  
Interaction Term ◽  
Popov Ghost

We have built a noncommutative unitary gauge group model preserving translation invariance. It describes the interaction of the Dirac field with the gauge field. The interaction term is expanded as a power series resulting from the introduction of the inverse covariant derivative. The consistency of the model is sustained by the fact that the Ward identity holds at tree level. The pure Yang–Mills action, including the fixing term and the Faddeev–Popov ghost term were constructed. It is striking that the commutator of our covariant derivative contained the torsion tensor, in addition to the field strength from which the Yang–Mills action was built.

PERFECT-TRANSLATION-INVARIANT CUSTOMIZABLE COMPLEX DISCRETE WAVELET TRANSFORM

2013 ◽  
Vol 11 (04) ◽  
pp. 1360003 ◽  
Author(s):  
HIROSHI TODA ◽  
ZHONG ZHANG ◽  
TAKASHI IMAMURA
Keyword(s):  
Wavelet Transform ◽  
Discrete Wavelet Transform ◽  
Frequency Domain ◽  
Wavelet Transforms ◽  
Wavelet Packet ◽  
Variable Density ◽  
Discrete Wavelet ◽  
Translation Invariance ◽  
Translation Invariant ◽  
High Degree

The theorems, giving the condition of perfect translation invariance for discrete wavelet transforms, have already been proven. Based on these theorems, the dual-tree complex discrete wavelet transform, the 2-dimensional discrete wavelet transform, the complex wavelet packet transform, the variable-density complex discrete wavelet transform and the real-valued discrete wavelet transform, having perfect translation invariance, were proposed. However, their customizability of wavelets in the frequency domain is limited. In this paper, also based on these theorems, a new type of complex discrete wavelet transform is proposed, which achieves perfect translation invariance with high degree of customizability of wavelets in the frequency domain.

ON DOMINANCE RELATIONS IN DISJUNCTIVE SET-VALUED ORDERED INFORMATION SYSTEMS

2010 ◽  
Vol 09 (01) ◽  
pp. 9-33 ◽  
Author(s):  
Y. H. QIAN ◽  
J. Y. LIANG ◽  
P. SONG ◽  
C. Y. DANG
Keyword(s):  
Decision Making ◽  
Information System ◽  
Information Systems ◽  
Rough Sets ◽  
Semantic Interpretation ◽  
Dominance Relation ◽  
New Approach ◽  
Dominance Relations ◽  
Intelligent Decision Making ◽  
Intelligent Decision

Set-valued information systems are generalized models of single-valued information systems. Its semantic interpretation can be classified into two categories: disjunctive and conjunctive. We focus on the former in this paper. By introducing four types of dominance relations to the disjunctive set-valued information systems, we establish a dominance-based rough sets approach, which is mainly based on the substitution of the indiscernibility relation by the dominance relations. Furthermore, we develop a new approach to sorting for objects in disjunctive set-valued ordered information systems, which is based on the dominance class of an object induced by a dominance relation. Finally, we propose criterion reductions of disjunctive set-valued ordered information systems that eliminate only those information that are not essential from the ordering of objects. The approaches show how to simplify a disjunctive set-valued ordered information system. Throughout this paper, we establish in detail the interrelationships among the four types of dominance relations, which include corresponding dominance classes, rough sets approaches, sorting for objects and criterion reductions. These results give a kind of feasible approaches to intelligent decision making in disjunctive set-valued ordered information systems.

scholarly journals Extensions of $C^*$-algebras and translation invariant asymptotic homomorphisms

2007 ◽  
Vol 100 (1) ◽  
pp. 131
Author(s):  
V. Manuilov ◽  
K. Thomsen
Keyword(s):  
The Other ◽  
Translation Invariance ◽  
Homotopy Classes ◽  
Translation Invariant ◽  
C Algebras

Let $A$, $B$ be $C^*$-algebras; $A$ separable, $B$ $\sigma$-unital and stable. We introduce a notion of translation invariance for asymptotic homomorphisms from $SA=C_0(\mathsf{R})\otimes A$ to $B$ and show that the Connes-Higson construction applied to any extension of $A$ by $B$ is homotopic to a translation invariant asymptotic homomorphism. In the other direction we give a construction which produces extensions of $A$ by $B$ out of such a translation invariant asymptotic homomorphism. This leads to our main result; that the homotopy classes of extensions coincide with the homotopy classes of translation invariant asymptotic homomorphisms.

scholarly journals Borsuk’s antipodal fixed points theorems for compact or condensing set-valued maps

2018 ◽  
Vol 7 (3) ◽  
pp. 307-311 ◽  
Author(s):  
Najla Altwaijry ◽  
Souhail Chebbi ◽  
Hakim Hammami ◽  
Pascal Gourdel
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Points ◽  
Large Class ◽  
Point Theorem ◽  
Vector Spaces ◽  
Compact Set ◽  
Topological Vector Spaces ◽  
Locally Convex

AbstractWe give a generalized version of the well-known Borsuk’s antipodal fixed point theorem for a large class of antipodally approachable condensing or compact set-valued maps defined on closed subsets of locally convex topological vector spaces. These results contain corresponding results obtained in the literature for compact set-valued maps with convex values.

scholarly journals Translation invariance and finite additivity in a probability measure on the natural numbers

2002 ◽  
Vol 29 (10) ◽  
pp. 585-589 ◽  
Author(s):  
Robert Gardner ◽  
Robert Price
Keyword(s):  
Probability Measure ◽  
Translation Invariance ◽  
Finite Additivity ◽  
Natural Numbers ◽  
Translation Invariant ◽  
Finitely Additive Probability ◽  
Power Set ◽  
Additive Probability ◽  
Two Envelopes

Inspired by the “two envelopes exchange paradox,” a finitely additive probability measuremon the natural numbers is introduced. The measure is uniform in the sense thatm({i})=m({j})for alli,j∈ℕ. The measure is shown to be translation invariant and has such desirable properties asm({i∈ℕ|i≡0(mod2)})=1/2. For anyr∈[0,1], a setAis constructed such thatm(A)=r; however,mis not defined on the power set ofℕ. Finally, a resolution to the two envelopes exchange paradox is presented in terms ofm.

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