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About 793 papers for “Injective function”

THE HARMONIOUS, ODD HARMONIOUS, AND EVEN HARMONIOUS LABELING

Ahmad Lasim, Ikhsanul Halikin, Kristiana Wijaya · BAREKENG JURNAL ILMU MATEMATIKA DAN TERAPAN · 2022

Suppose is a simple and connected graph with edges. A harmonious labeling on a graph is an injective function so that there exists a bijective function where for each An odd harmonious labeling on a graph is an injective function from to non-negative integer set less than so that there is a function where for every An even harmonious labeling on a graph is an injective function so that there is a bijective function where for each . In this paper, we discuss how to build new labeling (harmonious, odd harmonious, even harmonious) based on the existing labeling (harmonious, odd harmonious, even harmonious)

Cited by 3Open access

Weak Integer Additive Set-Indexed Graphs: A Creative Review

Chithra, K P, Germina, K A, Sudev, N K · arXiv (Cornell University) · 2014

For a non-empty ground set $X$, finite or infinite, the {\em set-valuation} or {\em set-labeling} of a given graph $G$ is an injective function $f:V(G) \to \mathcal{P}(X)$, where $\mathcal{P}(X)$ is the power set of the set $X$. A set-indexer of a graph $G$ is an injective set-valued function $f:V(G) \to \mathcal{P}(X)$ such that the function $f^{\ast}:E(G)\to \mathcal{P}(X)-\{\emptyset\}$ defined by $f^{\ast}(uv) = f(u){\ast} f(v)$ for every $uv{\in} E(G)$ is also injective., where $\ast$ is a binary operation on sets. An integer additive set-indexer (IASI) is defined as an injective function $f:V(G)\to \mathcal{P}({\mathbb{N}_0})$ such that the induced function $g_f:E(G) \to \mathcal{P}(\mathbb{N}_0)$ defined by $g_f (uv) = f(u)+ f(v)$ is also injective, where $\mathbb{N}_0$ is the set of all non-negative integers and $\mathcal{P}(\mathbb{N}_0)$ is its power set. A weak IASI is an IASI $f$ such that $|f^+(uv)|= \text{max}(f(u),f(v))$. In this paper, we critically and creatively review the concepts and properties of weak integer additive set-valued graphs.

Cited by 0Open access

Identifying and classifying image transforms

Adnan A. Y. Mustafa · 2003

Image transforms are used extensively in image processing to convert one image form into another form. These transforms are either point-operation transforms or neighborhood-operation transforms. Point-operation transforms can be classified as either injective transforms or non-injective transforms depending on the type of mathematical function applied to the image and the image characteristics. In this paper we present a novel technique that can identify if an image was processed by a point-operation transform and classify the transform as being either injective or non-injective. The mapping function of the transform is also recovered in the process. The technique is applicable to both linear and non-linear transforms. Tests are conducted on real images to show the validity of our technique.

Cited by 2

On Integer Additive SetIndexers of Graphs

N K Sudev, K A Germina · 2016

A set-indexer of a graph G is an injective set-valued function f: V (G) → 2X such that the function f ⊕ : E(G) → 2X − {∅} defined by f⊕(uv) = f(u)⊕f(v) for every uv∈E(G) is also injective, where 2X is the set of all subsets of X and ⊕ is the symmetric difference of sets. An integer additive set-indexer is defined as an injective function f: V (G) → 2N0 such that the induced function gf: E(G) → 2N0 defined by gf (uv) = f(u) + f(v) is also injective. A graph G which admits an IASI is called an IASI graph. An IASI f is said to be a weak IASI if |gf (uv) | = max(|f(u)|, |f(v)|) and an IASI f is said to be a strong IASI if |gf (uv) | = |f(u)||f(v) | for all u, v ∈ V (G). In this paper, we study about certain characteristics of inter additive set-indexers. Key words: Set-indexers, integer additive set-indexers, uniform integer ad-ditive set-indexers, compatible classes, compatible index. AMS Subject Classification: 05C78 1

Cited by 0

Compositional representation of rational functions

Tero Harju, H. C. M. Kleijn, Michel Latteux · RAIRO - Theoretical Informatics and Applications · 1992

The rational functions are shown to coincide with the compositions of endmarkings, morphisms and inverses of injective morphisms. To represent a rational function x we need one ednmarking \x m , two morphisms a u <x 3 and one inverse of an injective morphism <x 2 and then

Cited by 4Open access

(S,d) Magic Labeling of Subdivision of Some Snake Graphs - Paper II

P. SUMATHI -, P. Mala · International Journal For Multidisciplinary Research · 2025

Let G (p,q) be a connected, undirected, simple and non-trivial graph with p vertices and q edges. Let f be an injective function f: V(G) →{ } and g be an injective function g: E(G) →{d,2d,3d....2(q-1)d}.Then the function f is said to be (s, d) magic labeling if f(u)+g(uv)+f(v) is a constant, for all u,v∈V(G) and uv∈E(G). A graph G is called (s,d) magic graph if it admits (s,d) magic labeling.

Cited by 0Open access

A Study on Topological Integer Additive Set-Labeling of Graphs

Sudev Naduvath, K. A. Germina · Electronic Journal of Graph Theory and Applications · 2015

A set-labeling of a graph $G$ is an injective function $f:V(G)\to \mathcal{P}(X)$, where $X$ is a finite set and a set-indexer of $G$ is a set-labeling such that the induced function $f^{\oplus}:E(G)\to \mathcal{P}(X)-\{\emptyset\}$ defined by $f^{\oplus}(uv) = f(u){\oplus}f(v)$ for every $uv{\in} E(G)$ is also injective. Let $G$ be a graph and let $X$ be a non-empty set. A set-indexer $f:V(G)\to \mathcal{P}(X)$ is called a topological set-labeling of $G$ if $f(V(G))$ is a topology of $X$. An integer additive set-labeling is an injective function $f:V(G)\to \mathcal{P}(\mathbb{N}_0)$, whose associated function $f^+:E(G)\to \mathcal{P}(\mathbb{N}_0)$ is defined by $f(uv)=f(u)+f(v), uv\in E(G)$, where $\mathbb{N}_0$ is the set of all non-negative integers and $\mathcal{P}(\mathbb{N}_0)$ is its power set. An integer additive set-indexer is an integer additive set-labeling such that the induced function $f^+:E(G) \to \mathcal{P}(\mathbb{N}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective. In this paper, we extend the concepts of topological set-labeling of graphs to topological integer additive set-labeling of graphs.

Cited by 13Open access

On Integer Additive set-Sequential Graphs

Sudev Naduvath, K. A. Germina · arXiv (Cornell University) · 2014

A set-labeling of a graph $G$ is an injective function $f:V(G)\to \mathcal{P}(X)$, where $X$ is a finite set of non-negative integers and a set-indexer of $G$ is a set-labeling such that the induced function $f^{\oplus}:E(G)\rightarrow \mathcal{P}(X)-\{\emptyset\}$ defined by $f^{\oplus}(uv) = f(u){\oplus}f(v)$ for every $uv{\in} E(G)$ is also injective. A set-indexer $f:V(G)\to \mathcal{P}(X)$ is called a set-sequential labeling of $G$ if $f^{\oplus}(V(G)\cup E(G))=\mathcal{P}(X)-\{\emptyset\}$. A graph $G$ which admits a set-sequential labeling is called a set-sequential graph. An integer additive set-labeling is an injective function $f:V(G)\rightarrow \mathcal{P}(\mathbb{N}_0)$, $\mathbb{N}_0$ is the set of all non-negative integers and an integer additive set-indexer is an integer additive set-labeling such that the induced function $f^+:E(G) \rightarrow \mathcal{P}(\mathbb{N}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective. In this paper, we extend the concepts of set-sequential labeling to integer additive set-labelings of graphs and provide some results on them.

Cited by 7Open access

Globally injective and bijective neural operators

Takashi Furuya, Michael Puthawala, Matti Lassas et al. · arXiv (Cornell University) · 2023

Recently there has been great interest in operator learning, where networks learn operators between function spaces from an essentially infinite-dimensional perspective. In this work we present results for when the operators learned by these networks are injective and surjective. As a warmup, we combine prior work in both the finite-dimensional ReLU and operator learning setting by giving sharp conditions under which ReLU layers with linear neural operators are injective. We then consider the case the case when the activation function is pointwise bijective and obtain sufficient conditions for the layer to be injective. We remark that this question, while trivial in the finite-rank case, is subtler in the infinite-rank case and is proved using tools from Fredholm theory. Next, we prove that our supplied injective neural operators are universal approximators and that their implementation, with finite-rank neural networks, are still injective. This ensures that injectivity is not `lost' in the transcription from analytical operators to their finite-rank implementation with networks. Finally, we conclude with an increase in abstraction and consider general conditions when subnetworks, which may be many layers deep, are injective and surjective and provide an exact inversion from a `linearization.' This section uses general arguments from Fredholm theory and Leray-Schauder degree theory for non-linear integral equations to analyze the mapping properties of neural operators in function spaces. These results apply to subnetworks formed from the layers considered in this work, under natural conditions. We believe that our work has applications in Bayesian UQ where injectivity enables likelihood estimation and in inverse problems where surjectivity and injectivity corresponds to existence and uniqueness, respectively.

Cited by 1Open access

Preclusion of switch behavior in reaction networks with mass-action kinetics

Elisenda Feliu, Carsten Wiuf · arXiv (Cornell University) · 2011

We provide a Jacobian criterion that applies to arbitrary chemical reaction networks taken with mass-action kinetics to preclude the existence of multiple positive steady states within any stoichiometric class for any choice of rate constants. We are concerned with the characterization of injective networks, that is, networks for which the species formation rate function is injective in the interior of the positive orthant within each stoichiometric class. We show that a network is injective if and only if the determinant of the Jacobian of a certain function does not vanish. The function consists of components of the species formation rate function and a maximal set of independent conservation laws. The determinant of the function is a polynomial in the species concentrations and the rate constants (linear in the latter) and its coefficients are fully determined. The criterion also precludes the existence of degenerate steady states. Further, we relate injectivity of a chemical reaction network to that of the chemical reaction network obtained by adding outflow, or degradation, reactions for all species.

Cited by 23Open access

Preclusion of switch behavior in reaction networks with mass-action\n kinetics

Elisenda Feliu, Carsten Wiuf · arXiv (Cornell University) · 2011

We provide a Jacobian criterion that applies to arbitrary chemical reaction\nnetworks taken with mass-action kinetics to preclude the existence of multiple\npositive steady states within any stoichiometric class for any choice of rate\nconstants. We are concerned with the characterization of injective networks,\nthat is, networks for which the species formation rate function is injective in\nthe interior of the positive orthant within each stoichiometric class. We show\nthat a network is injective if and only if the determinant of the Jacobian of a\ncertain function does not vanish. The function consists of components of the\nspecies formation rate function and a maximal set of independent conservation\nlaws. The determinant of the function is a polynomial in the species\nconcentrations and the rate constants (linear in the latter) and its\ncoefficients are fully determined. The criterion also precludes the existence\nof degenerate steady states. Further, we relate injectivity of a chemical\nreaction network to that of the chemical reaction network obtained by adding\noutflow, or degradation, reactions for all species.\n

Cited by 5Open access

(S, D) Magic Labeling of Subdivision of Some Special Trees

P. Sumathi, P. Mala · INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER RESEARCH · 2024

Let G (connected, undirected, simple and non-trivial graph with p vertices and q edges. Let f be an injective function f: V(G) } and g be an injective function g: E(G) .Then the graph G is said to be (s, d) magic labeling if is a constant, for all . A graph G is called (magic graphif it admits magic labeling. In this paper the existence of (s, d) magic labeling of subdivision on some special trees are found.

Cited by 0Open access

A Note On Cohomological Vanishing And The Linear Isoperimetric Inequality

S.M. Gersten · 1996

If G is a finitely presented group, then H 2 (#) (G, A) vanishes for all injective Banach spaces iff the regularized homological area function satisfies the linear isoperimetric inequality. This contrasts with the known result that G is word hyperbolic iff the homological area function satisfies the linear isoperimetric inequality. A closed 3-manifold group G is hyperbolic iff H 2 (#) (G, A) vanishes for all injective Banach spaces A. A vanishing theorem is proved for the fundamental group of a closed Riemannian manifold of negative curvature.

Cited by 0

On the spectrum of randomly aggregate <i>ARMA</i> models

Amina Kadi · Stochastic Analysis and Applications · 2002

The paper is devoted to the zeros of univariate randomly aggregate autoregressive moving-average (ARMA) models. We determine precisely matrix representations for the spectrum numerator coefficients of the randomly aggregate ARMA models. The initial ARMA model poles are simple. We first prove the results when the probability generating function of the random sampling law is injective, then we precise the results when it is not injective.

Cited by 0

Quantifying discontinuity

Henry Adams, Florian Frick, M. Harrison et al. · ArXiv.org · 2025

Given a compact space $X$ that does not admit an embedding (an injective continuous function) into $\mathbb{R}^d$, we study the ''degree'' of discontinuity that any injective function $X \to \mathbb{R}^d$ must have. To this end, we define a scale invariant modulus of discontinuity and obtain general lower bounds, thus obtaining quantified nonembeddability results of Haefliger--Weber type. Moreover, we establish analogous lower bounds for simplicial complexes that do not admit an almost $r$-embedding in $\mathbb{R}^d$, thus obtaining a quantified version of the topological Tverberg theorem.

Cited by 0Open access

Injectivity of non-singular planar maps with disconnecting curves in the eigenvalues space

Marco Sabatini · arXiv (Cornell University) · 2022

Fessler and Gutierrez \cite{Fe,Gu} proved that if a non-singular planar map has Jacobian matrix without eigenvalues in $(0,+\infty)$, then it is injective. We prove that the same holds replacing $(0,+\infty)$ with any unbounded curve disconnecting the upper (lower) complex half-plane. Additionally we prove that a Jacobian map $(P,Q)$ is injective if $P_x + Q_y$ is not a surjective function.

Cited by 0Open access

The Cauchy Problem for Solutions of Elliptic Equations

Николай Тарханов · Medical Entomology and Zoology · 1995

Function spaces pseudodifferential operators in the spaces of distributions on closed sets capacity systems of differential equations with injective (surjective) symbol coarse results on approximation by solutions of a system with surjective symbol approximation in spaces of smooth functions approximation in spaces of Holder functions approximation in spaces of Sobolev functions generalized boundary values of solutions of a system with injective symbol. (Part contents).

Cited by 149