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### Non-additive Bounded Sets of Uniqueness in ℤ n [chapter]

Sara Brunetti, Paolo Dulio, Carla Peri
2014 Lecture Notes in Computer Science
In this paper we compute the proportion of non-additive sets of uniqueness with respect to additive sets in a given grid A ⊂ Z n , in the important case when d coordinate directions are employed.  ...  It turns out that d + 1 represents the minimal number of directions one needs in Z n (n ≥ d ≥ 3), under the requirement that such directions span a d-dimensional subspace of Z n .  ...  Non-additive Bounded Set of Uniqueness In this section we study non-additive sets of uniqueness contained in a given grid A, in the important case when S contains the coordinate directions.  ...

### On the Non-additive Sets of Uniqueness in a Finite Grid [chapter]

Sara Brunetti, Paolo Dulio, Carla Peri
2013 Lecture Notes in Computer Science
Between these two opposite situations there are also the non-additive sets of uniqueness, which deserve interest in Discrete Tomography, since their unique reconstruction cannot be derived via the additivity  ...  Discrete lattice sets which are additive with respect to a given set S of lattice directions are uniquely determined by X-rays in the direction of S.  ...  Non-additive Sets of Uniqueness By means of Theorem 1 we can check if a set of four directions S uniquely determines bounded sets in a grid A = {(i, j) ∈ Z 2 : 0 ≤ i < m, 0 ≤ j < n}.  ...

### Representing Yosida-Hewitt decompositions for classical and non-commutative vector measures

J.K. Brooks, J.D. Maitland Wright
2001 Expositiones mathematicae
Let m be a bounded, real valued measure on a field of sets. Then, by the Yosida-Hewitt theorem, m has a unique decomposition into the sum of a countably additive and a singular measure.  ...  From this we obtain a simple formula for the countably additive part of a (strongly bounded) vector measure.  ...  Let N be the set of natural numbers. Let Q(N) be the collection of all non-empty finite subsets of N, and partially order Q(N) by set inclusion.  ...

### Entropy Amplification Property and the Loss for Writing on Dirty Paper

Aaron S. Cohen, Ram Zamir
2008 IEEE Transactions on Information Theory
The objective of this work is to show that even in the basic input-constrained additive state setting, Y = X + S + Z, the encoder is sometimes unable to use the SI efficiently if the receiver does not  ...  Thus almost 100 % of the available capacity is lost in WDP in the presence of difference-set noise.  ...  Acknowledgement The authors are indebted to Dave Forney for referring them to difference set theory. They also thank Yuval Kochman for simplifying the proof of Theorem 1.  ...

### On approximately additive functions

Janusz Brzdęk
2011 Journal of Mathematical Analysis and Applications
Note that F = k∈N kV 1 m > m 1 1 Remark 4 . 4 In view of the example given in Remark 1, the boundedness of C in Theorem 10 is necessary for the uniqueness and continuity of A.  ...  The example given in Remark 1 shows that the boundedness of C is necessary for the uniqueness of A in Proposition 8.  ...

### On the local k-elasticities of Puiseux monoids [article]

Marly Gotti
2018 arXiv   pre-print
If M is an atomic monoid and x is a nonzero non-unit element of M, then the set of lengths L(x) of x is the set of all possible lengths of factorizations of x, where the length of a factorization is the  ...  In a recent paper, F. Gotti and C. O'Neil studied the sets of elasticities R(P) := {L(x)/L(x) : x ∈ P} of Puiseux monoids P.  ...  Because many of the factorization-related questions on integral domains are independent of the ring additive structure, in the last few decades the study of the phenomenon of non-unique factorization has  ...

### Some Fubini Theorems on product σ-algebras for non-additive measures

Alain Chateauneuf, Jean-Philippe Lefort
2008 International Journal of Approximate Reasoning
Since the seminal paper of Ghirardato, it is known that Fubini Theorem for non-additive measures can be available only for functions defined as "slice-comonotonic".  ...  We give different assumptions that provide such Fubini Theorems in the framework of product r-algebras.  ...  Some Fubini Theorems on iterated integrals for non-additive measures on product r-algebras R N f dv ¼ P A;A finite mðAÞ Á min f ðAÞ. So Z Z f dv 1 dv 2 ¼ Z N Z N f ð:; x 2 Þdv 2 !  ...

### Sharing a River with Downstream Externalities

Sarina Steinmann, Ralph Winkler
2019 Games
In addition, we show that this result holds true for numerous extensions of our model.  ...  We consider the problem of efficient emission abatement in a multi polluter setting, where agents are located along a river in which net emissions accumulate and induce negative externalities to downstream  ...  For the grand coalition N = S z , the non-cooperative core upper bounds are satisfied.  ...

### Page 234 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 63, Issue 4 [page]

1957 American Mathematical Society. Bulletin of the American Mathematical Society
Each non-negative Baire measure ¢ on @® has a unique decomposition ¢=¢c+¢p into a non-negative Baire measure ¢c¢ identically zero on Baire sets of the first category, plus a Baire measure ¢p minimal on  ...  Using the complex methods introduced in previous papers the following typical result is obtained: The two conditions (i) f(z)=e*) >. _, saz*/n!=o(1) as z@, |z| —Rez=O(1), (ii) S07_, svse=0(N?)  ...

### Robust Control of Linear Systems With Disturbances Bounded in a State Dependent Set

Reza Ghaemi, Ilya V. Kolmanovsky, Jing Sun
2011 IEEE Transactions on Automatic Control
This technical note examines attractiveness and minimality of invariant sets for linear systems subject to additive disturbances confined in a state-dependent set.  ...  In many practical applications, the disturbance may evolve in a compact set while being generated by a dynamic process with a given model and bounded input.  ...  Rakovic for his help, constructive comments, and suggestion of several steps in this development.  ...

### On Statistical Methods in the Theory of Almost-Periodic Functions

E. K. Haviland
1933 Proceedings of the National Academy of Sciences of the United States of America
Let pi, (P2, ... be a sequence of monotone absolutely additive set functions of uniformly bounded variation in J: (-M < x _ Ml; -M < y < M) converging to the monotone absolutely additive set function so  ...  From a continuum of monotone absolutely additive set functions (T(E) of uniformly bounded variation it is possible to select a sequence Ipo,(E) I such that as n becomes infinite (p"(R) approaches the limit  ...

### Non-commutative automorphisms of bounded non-commutative domains

John E. McCarthy, Richard M. Timoney
2016 Proceedings of the Royal Society of Edinburgh. Section A Mathematics
We establish rigidity (or uniqueness) theorems for non-commutative (NC) automorphisms that are natural extensions of classical results of H. Cartan and are improvements of recent results.  ...  We apply our results to NC domains consisting of unit balls of rectangular matrices.  ...  Unless an additional hypothesis of continuity (or boundedness) is added, nc-functions can behave badly. Example 3.  ...

### Some Aspects in n-dimensional almost Periodic functions iii

Vernor Arguedas Troyo, Edwin Castro
2011 Revista de Matemática: Teoría y Aplicaciones
For this functions we find some uniqueness sets in R and R n . The paper finishes analizing the relation of this functions and the function sinc.  ...  The properties of almost periodical functions and some new results have been published in [CA1], [CA2] and [CA3] In this paper we show some new definitions in order to analize some singularities.  ...  The above statements can be formulated in R N , for example: {sin(n), nZ N } is dense in [−1, 1]. Definition 1 Let G be any discreet non trivial additive group of R N .  ...

### An Addition Theorem and Its Arithmetical Application

Gregory Freiman, Alfred Geroldinger
2000 Journal of Number Theory
Almost arithmetical multiprogressions appear as sets of lengths in rings of algebraic integers, and the addition theorem will be applied to an arithmetical situation again. Academic Press  ...  We prove an addition theorem stating that arbitrary sumsets of such sets are of the same type again.  ...  The best tool investigated so far for describing the non-uniqueness is the system of sets of lengths. We briefly recall some terminology.  ...

### On stochastic set functions. I

A. Prékopa
1956 Acta Mathematica Hungarica
Boundedness of sets of quantiles Let Z be an arbitrary set and ξ z (zZ) a family of random variables. Denote by Q(λ, z) a λ-quantile of the variable ξ z .  ...  If the quantiles Q(λ, z) can be chosen in such a way that the set {Q(λ, z), zZ} is bounded from below (from above), then we have If the quantiles Q(λ, z) can be chosen in such a way that the set {Q(  ...  choice of the quantiles Q(λ, z) does not matter as in this case the boundedness (unboundedness) of the sets {Q(λ, z), zZ} with a special choice of the quantiles Q(λ, z) implies the boundedness of the  ...
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