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### Approximate Representations and Approximate Homomorphisms [article]

Cristopher Moore, Alexander Russell
2010 arXiv   pre-print
As an application, we bound the extent to which a function f : G -> H can be an approximate homomorphism where H is another finite group.  ...  We show that if H's representations are significantly smaller than G's, no such f can be much more homomorphic than a random function.  ...  Approximate homomorphisms In this section we prove Theorems 2 and 3, bounding the extent to which a function from one finite group to another can act like a homomorphism. Proof of Theorem 2.  ...

### On Approximate Cubic Homomorphisms

M. Eshaghi Gordji, M. Bavand Savadkouhi
On the other hand, we are looking for situations when the homomorphisms are stable, that is, if a mapping is an approximate homomorphism, then there exists an exact homomorphism near it.  ...  The topic of approximate homomorphisms was studied by a number of mathematicians, see 15-22 and references therein. Jun and Kim 1 introduced the following functional equation: for all a, b ∈ R.  ...  Advances in Difference Equations Furthermore, 2.47 Hence T is cubic homomorphism.  ...

### Approximate homomorphisms on lattices [article]

Roman Badora, Tomasz Kochanek, Barbara Przebieracz
2017 arXiv   pre-print
One of them involves estimates by quite general error functions; the other deals with approximate (join) homomorphisms in terms of certain systems of lattice neighborhoods.  ...  As a corollary, we obtain a stability result for approximately monotone functions.  ...  We will see that this theorem can be easily derived from our results on approximate lattice homomorphisms.  ...

### On approximate cubic homomorphisms [article]

M. Eshaghi Gordji, M. Bavand Savadkouhi
2009 arXiv   pre-print
In the other hand we are looking for situations when the homomorphisms are stable, i.e., if a mapping is an approximate homomorphism, then there exists an exact homomorphism near it.  ...  The topic of approximate homomorphisms was studied by a number of mathematicians, see [2, 3, 10, 14, 15, 19] and references therein.  ...

### On approximate ring homomorphisms

2002 Journal of Mathematical Analysis and Applications
We prove the stability of a generalization of the Bourgin's result on approximate ring homomorphisms.  ...  For ring homomorphisms Bourgin (see  ) proved the following. Theorem B. Let ε and δ be nonnegative real numbers.  ...  To prove the uniqueness property of h, assume that h * is another ring homomorphism with f (x)−h * (x) ε, x ∈ R.  ...

### Homomorphisms of Approximation Spaces

Zhaohao Wang, Lan Shu, Xiuyong Ding
2012 Journal of Applied Mathematics
The notion of the homomorphism of approximation spaces is introduced. Some properties of homomorphism are investigated, and some characterizations of homomorphism are given.  ...  Furthermore, the notion of approximation subspace of approximation spaces is introduced. The relations between approximation subspaces and homomorphisms are studied.  ...  Conclusions In this paper, we present the notion of the homomorphism of approximation spaces. A homomorphism may be viewed as a special mapping between two approximation spaces.  ...

### Approximate C^*-Ternary Ring Homomorphisms [article]

2006 arXiv   pre-print
In this paper, we establish the generalized Hyers--Ulam--Rassias stability of C^*-ternary ring homomorphisms associated to the Trif functional equation d · C_d-2^l-2 f(x_1+...  ...  By an approximate C * -ternary ring homomorphism associated to the Trif equation we mean a mapping f : A → B for which there exists a certain control function ϕ : A d+3 → [0, ∞) such that if D µ f (x 1  ...  Therefore q n T ([xyz]) = T ([xy(q n z)]) = [T (x)T (y)f (q n z)], and so T [xyz]) = lim n→∞ 1 q n [T (x)T (y)f (q n z)] = T (x)T (Suppose that f : A → B is an approximate C * -ternary ring homomorphism  ...

### Removal lemmas and approximate homomorphisms [article]

Jacob Fox, Yufei Zhao
2022 arXiv   pre-print
most M vertices (here an ϵ-approximate homomorphism is a map V(G) → V(F) where all but at most ϵ |V(G)|^2 edges of G are mapped to edges of F).  ...  One such variant, which we call the triangle-free lemma, states that for each ϵ>0 there exists M such that every triangle-free graph G has an ϵ-approximate homomorphism to a triangle-free graph F on at  ...  Suppose φ : V (G ′ ) → V (F ) is an ǫ-approximate homomorphism and F is H-homomorphism-free.  ...

### Approximation of Minimum Cost Homomorphisms [chapter]

Pavol Hell, Monaldo Mastrolilli, Mayssam Mohammadi Nevisi, Arash Rafiey
2012 Lecture Notes in Computer Science
algorithm, and otherwise the minimum cost homomorphism problem to H is not approximable.  ...  We prove that if H is a co-circular arc bigraph then the minimum cost homomorphism problem to H admits a polynomial time constant ratio approximation algorithm; otherwise the minimum cost homomorphism  ...  Introduction We study the approximability of the minimum cost homomorphism problem, introduced below. A c-approximation algorithm produces a solution of cost at most c times the minimum cost.  ...

### Approximate homomorphisms on JB∗-triples

Chun-Gil Park
2005 Journal of Mathematical Analysis and Applications
It is shown that for an approximate homomorphism f : J → B of a JB * -triple J into a JB * -triple B, there exists a unique * -homomorphism H : J → B near the approximate homomorphism.  2004 Elsevier  ...  In this paper, we prove that for an approximate homomorphism f : J → B of a JB *triple J into a JB * -triple B, there exists a unique * -homomorphism H : J → B near the approximate homomorphism.  ...  We are going to show the Cauchy-Rassias stability of * -homomorphisms on JB * -triples.  ...

### Bootstrapping for Approximate Homomorphic Encryption [chapter]

Jung Hee Cheon, Kyoohyung Han, Andrey Kim, Miran Kim, Yongsoo Song
2018 Lecture Notes in Computer Science
This paper extends the leveled homomorphic encryption scheme for an approximate arithmetic of Cheon et al.  ...  We exploit a scaled sine function as an approximation of the modular reduction operation and present an efficient evaluation strategy.  ...  In the EvalExp step, we take two ciphertexts to be homomorphically evaluated by the approximate polynomial P r (·) of the complex exponential polynomial.  ...

### Approximate Homomorphisms of Ternary Semigroups

M. Amyari, M. S. Moslehian
2006 Letters in Mathematical Physics
A mapping f:(G_1,[ ]_1)→ (G_2,[ ]_2) between ternary semigroups will be called a ternary homomorphism if f([xyz]_1)=[f(x)f(y)f(z)]_2.  ...  In addition, we establish the superstability of ternary homomorphisms into Banach algebras endowed with multiplicative norms.  ...  For an extensive account on approximately homomorphisms in the binary mode we refer the reader to  .  ...

### Accelerating Homomorphic Encryption using Approximate Computing Techniques

Shabnam Khanna, Ciara Rafferty
2020 Proceedings of the 17th International Joint Conference on e-Business and Telecommunications
All of the approximate functions, as given in Equations (1-8) , (11), (13) and (14) , are homomorphically evaluated.  ...  Here SEAL is used to measure the effect the proposed approximations have on homomorphic evaluations. Further generic software optimisations could improve performance.  ...

### Approximate -Lie Homomorphisms and Jordan -Lie Homomorphisms on -Lie Algebras

M. Eshaghi Gordji, G. H. Kim
2012 Abstract and Applied Analysis
Using fixed point methods, we establish the stability of -Lie homomorphisms and Jordan -Lie homomorphisms on -Lie algebras associated to the following generalized Jensen functional equation .  ...  Under what condition does there exist a homomorphism near an approximate homomorphism?  ...  A generalized version of the theorem of Hyers for approximately linear mappings was presented by Rassias 12 in 1978 by considering the case when inequality 1.4 is unbounded.  ...

### Beurling moving averages and approximate homomorphisms

N.H. Bingham, A.J. Ostaszewski
2016 Indagationes mathematicae
Extensive use of group structures leads to an algebraicization not previously encountered here, and to the approximate homomorphisms of the title.  ...  The theory of regular variation, in its Karamata and Bojani\'c-Karamata/de Haan forms, is long established and makes essential use of homomorphisms.  ...  1 a F (x)= (x) (a 2 A); (ker) hence we refer to K F here and above as the regular kernel of F -the homomorphism approximating F of our title.  ...
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