Sudev Naduvath · arXiv (Cornell University) · 2017
Let $X$ be a non-empty ground set and $\mathscr{P}(X)$ be its power set. A set-labeling (or a set-valuation) of a graph $G$ is an injective set-valued function $f:V(G)\to \mathscr{P}(X)$ such that the induced function $f^*:E(G) \to \mathscr{P}(X)$ is defined by $f^*(uv)=f(u)\ast f(v)$, where $f(u)\ast f(v)$ is a binary operation of the sets $f(u)$ and $f(v)$. A graph which admits a set-labeling is known to be a set-labeled graph. A set-labeling $f$ of a graph $G$ is said to be a set-indexer of $G$ if the associated function $f^*$ is also injective. In this paper, we introduce a new notion namely product set-labeling of graphs as an injective set-valued function $f:V(G)\to \mathscr{P}(\mathbb{N})$ such that the induced edge-function $f^*:V(G)\to \mathscr{P}(\mathbb{N})$ is defined as $^*f(uv)=f(u)\ast f(v) \forall\ uv\in E(G)$, where $f(u)\ast f(v)$ is the product set of the set-labels $f(u)$ and $f(v)$, where $\mathbb{N}$ is the set of all positive integers and discuss certain properties of the graphs which admit this type of set-labeling.
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Raymond Mortini · arXiv (Cornell University) · 2014
We present necessary and sufficient conditions on planar compacta $K$ and continuous functions $f$ on $K$ in order that $f$ generates the algebras $P(K), R(K), A(K)$ or $C(K)$. We also unveil quite surprisingly simple examples of non-polynomial convex compacta $K\subseteq\mathbb C$ and $f\in P(K)$ with the property that $f \in P(K)$ is a homeomorphism, but for which $f^{-1}\notin P(f(K))$. As a consequence, such functions do not admit injective holomorphic extensions to the interior of the polynomial convex hull $\widehat K$. On the other hand, it will be shown that the restriction $f^*|_G$ of the Gelfand-transform $f^*$ of an injective function $f\in P(K)$ is injective on every regular, bounded complementary component $G$ of $K$. A necessary and sufficient condition in terms of the behaviour of $f$ on the outer boundary of $K$ is given in order $f$ admits a holomorphic injective extension to $\widehat K$. We also include some results on the existence of continuous logarithms on punctured compacta containing the origin in their boundary.
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Peter Bradshaw, Alexander Clow, Jingwei Xu · European Journal of Combinatorics · 2025
Given a graph G , an injective edge-coloring of G is a function ψ : E ( G ) → N such that if ψ ( e ) = ψ ( e ′ ) , then no third edge joins an endpoint of e and an endpoint of e ′ . The injective chromatic index of a graph G , written χ inj ′ ( G ) , is the minimum number of colors needed for an injective edge coloring of G . In this paper, we investigate the injective chromatic index of certain classes of degenerate graphs. First, we show that if G is a d -degenerate graph of maximum degree Δ , then χ inj ′ ( G ) = O ( d 3 log Δ ) . Next, we show that if G is a graph of Euler genus g , then χ inj ′ ( G ) ≤ ( 3 + o ( 1 ) ) g , which is tight when G is a clique. Finally, we show that the oriented chromatic number of a graph is at most exponential in its injective chromatic index. Using this fact, we prove that the oriented chromatic number of a graph embedded on a surface of Euler genus g has oriented chromatic number at most O ( g 6400 ) , improving the previously known upper bound of 2 O ( g 1 2 + ɛ ) and resolving a conjecture of Aravind and Subramanian.
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P. Sumathi, P. Mala · Springer Link (Chiba Institute of Technology) · 2023
\nLet G(p, q) be a connected, undirected, simple and non-trivial graph with q nodes and q lines. Let f be an injective function f: V(G) →{ s, s + d, s + 2d,.....s + (q +1)d } and g be an injective function g: E(G) → {d,2d,3d,… 2(q-1)d}.Then the graph G is said to be (s, d) magic labeling if f(u) + g(uv) + f(v) is a constant, for all u, v; ∈ V(G). A graph G is called (s, d) magic graph if it admits (s, d) magic labeling. In this paper the existence of (s, d) magic labeling in some ladder graphs are found.\n
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P. Sumathi, P. Mala · E3S Web of Conferences · 2023
Let G(p, q) be a connected, undirected, simple and non-trivial graph with q nodes and q lines. Let f be an injective function f : V(G) →{ s , s + d , s + 2 d ,..... s + ( q +1) d } and g be an injective function g : E(G) → {d,2d,3d,… 2(q-1)d}.Then the graph G is said to be ( s, d ) magic labeling if f ( u ) + g ( uv ) + f ( v ) is a constant, for all u, v; ∈ V ( G ). A graph G is called ( s, d ) magic graph if it admits ( s, d ) magic labeling. In this paper the existence of ( s, d ) magic labeling in some ladder graphs are found.
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N. T. Nemesh · Moscow University Mathematics Bulletin · 2016
The paper presents a complete description of topologically injective, topologically surjective, isometric and coisometric multiplication operators by a function acting between L p -spaces of σ-finite measure spaces. It is proved that all such operators are invertible from the right and from the left. As a corollary it is proved that in the category consisting of L p -spaces for all p ∈ [1,+∞] considered as left Banach modules over the algebra of bounded measurable functions, all objects are metrically and topologically projective, injective, and flat modules.
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Francine Blanchet-Sadri, Bob Chen, Sinziana Munteanu · Springer Link (Chiba Institute of Technology) · 2013
\n Most of the constructions of infinite words having polynomial subword complexity are quite complicated, e.g., sequences of Toeplitz, sequences defined by billiards in the cube, etc. In this paper, we describe a simple method for constructing infinite words w over a binary alphabet { a,b } with polynomial subword complexity pw. Assuming w contains an infinite number of a’s, our method is based on the gap function which gives the distances between consecutive b’s. It is known that if the gap function is injective, we can obtain at most quadratic subword complexity, and if the gap function is blockwise injective, we can obtain at most cubic subword complexity. Here, we construct infinite binary words w such that pw(n) = Θ(nβ) for any real number β > 1.\n
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P. Jeyanthi, R. Gomathi, Gee-Choon Lau et al. · Proyecciones (Antofagasta) · 2023
A graph G is analytic odd mean if there exist an injective function f : V → {0, 1, 3, . . . , 2q − 1} with an induced edge labeling f∗ : E → Z such that for each edge uv with f(u) < f(v), is injective. Clearly the values of f∗ are odd. We say that f is an analytic odd mean labeling of G. In this paper, we show that the union and identification of some graphs admit analytic odd mean labeling by using the operation of joining of two graphs by an edge.
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Dariusz Partyka, Ken-ichi Sakan · Computational Methods and Function Theory · 2014
In 1958, E. Heinz obtained a lower bound for $$|\partial _x F|^2+|\partial _y F|^2$$ , where $$F$$ is a one-to-one harmonic mapping of the unit disk onto itself keeping the origin fixed. We show various variants of Heinz’s inequality in the case where $$F$$ is the Poisson integral of a function of bounded variation in the unit circle. In particular, we obtain such inequalities for $$F$$ when it is a locally injective quasiregular mapping or an injective mapping of the unit disk onto a bounded convex domain in the complex plane.
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Siegfried Meister · Zenodo (CERN European Organization for Nuclear Research) · 2025
This paper delivers a mathematical takedown of Advaita Vedānta's core explanatory structure (adhyāsa, two truths, sākṣin). Formal Result: Superimposition(S,ℳ) ≡ ∃f:ℳ↪S (injective) ⊢ S≠ℳ ∧ ∃L:IdentityLock(S) Advaita Collapse: Brahman nirguṇa ∧ Superimposition(Brahman,Māyā) ⊢ False Charta Research Sequence (Panel IX):Buddhism (groundlessness) → Advaita (performative contradiction) → Ur-Matrix (coherent resolution) Key Contributions: - First metalogical formalization of adhyāsa as injective function- Mathematical proof of **Identity Lock dependency**- Systematic critique linking Nāgārjuna → Advaita → Ur-Matrix- Visual geometry of metaphysical collapse Part of Charta Research Program – DOI-links to related works in paper. Metalogical Collapse Verified.
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Siegfried Meister · Zenodo (CERN European Organization for Nuclear Research) · 2025
This paper delivers a mathematical takedown of Advaita Vedānta's core explanatory structure (adhyāsa, two truths, sākṣin). Formal Result: Superimposition(S,ℳ) ≡ ∃f:ℳ↪S (injective) ⊢ S≠ℳ ∧ ∃L:IdentityLock(S) Advaita Collapse: Brahman nirguṇa ∧ Superimposition(Brahman,Māyā) ⊢ False Charta Research Sequence (Panel IX):Buddhism (groundlessness) → Advaita (performative contradiction) → Ur-Matrix (coherent resolution) Key Contributions: - First metalogical formalization of adhyāsa as injective function- Mathematical proof of **Identity Lock dependency**- Systematic critique linking Nāgārjuna → Advaita → Ur-Matrix- Visual geometry of metaphysical collapse Part of Charta Research Program – DOI-links to related works in paper. Metalogical Collapse Verified.
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Sudev, N K, Germina, K A · arXiv (Cornell University) · 2014
An integer additive set-indexer is defined as an injective function $f:V(G)\rightarrow 2^{\mathbb{N}_0}$ such that the induced function $g_f:E(G) \rightarrow 2^{\mathbb{N}_0}$ defined by $g_f (uv) = f(u)+ f(v)$ is also injective, where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. If $g_f(uv)=k~\forall~uv\in E(G)$, then $f$ is said to be a $k$-uniform integer additive set-indexer. An integer additive set-indexer $f$ is said to be a strong integer additive set-indexer if $|g_f(uv)|=|f(u)|.|f(v)|~\forall ~ uv\in E(G)$. In this paper, we study the characteristics of certain graph classes and graph powers that admit strong integer additive set-indexers.
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DEY, S, GOHM, R, HARIA, KJ · DSpace (IIT Bombay) · 2018
We prove that the symbol of the characteristic function of a minimal contractive lifting is an injective map and that the converse also holds, using explicit computation and functional models. We discuss when the characteristic function of a lifting is a polynomial and present a series representation for the characteristic functions of liftings.
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Marcelo Montenegro, Adilson E. Presoto · Arkiv för matematik · 2017
Let F :R N R N be a locally Lipschitz continuous function. We prove that F is a global homeomorphism or only injective, under suitable assumptions on the subdifferential F (x). We use variational methods, nonsmooth inverse function theorem and extensions of the Hadamard-Levy Theorem. We also address questions on the Markus-Yamabe conjecture.
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Andrew Zammit‐Mangion, Tin Lok James Ng, Quan Vu et al. · Journal of the American Statistical Association · 2021
Spatial processes with nonstationary and anisotropic covariance structure are often used when modeling, analyzing, and predicting complex environmental phenomena. Such processes may often be expressed as ones that have stationary and isotropic covariance structure on a warped spatial domain. However, the warping function is generally difficult to fit and not constrained to be injective, often resulting in “space-folding.” Here, we propose modeling an injective warping function through a composition of multiple elemental injective functions in a deep-learning framework. We consider two cases; first, when these functions are known up to some weights that need to be estimated, and, second, when the weights in each layer are random. Inspired by recent methodological and technological advances in deep learning and deep Gaussian processes, we employ approximate Bayesian methods to make inference with these models using graphics processing units. Through simulation studies in one and two dimensions we show that the deep compositional spatial models are quick to fit, and are able to provide better predictions and uncertainty quantification than other deep stochastic models of similar complexity. We also show their remarkable capacity to model nonstationary, anisotropic spatial data using radiances from the MODIS instrument aboard the Aqua satellite.
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Rare & Special e-Zone (The Hong Kong University of Science and Technology) · 2002
Injective functions and measure of images by Chiang Lo Fung thesis 2002 vii, 22 leaves : ill. ; 30 cm Function is a popular and important part of Mathematics. This paper will mainly focus on the continuous functions and will…Read more ›
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Pere Ara · Pacific Journal of Mathematics · 1987
In this paper we obtain a characterization of those * -regular rings whose matrix rings are *-regular satisfying LP ~ RP. This result allows us to obtain a structure theorem for the *-regular self-injective rings of type I which satisfy LP ~ RP matricially. Also, we are concerned with pseudo-rank functions and their corresponding metric completions. We show, amongst other things, that the LP ~ RP axiom extends from a unit-regular * -regular ring to its completion with respect to a pseudo-rank function. Finally, we show that the property LP ~ RP holds for some large classes of *-regular self-injective rings of type II.
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Matthew Gwynne · 2009
Given two finite sets, questions about the existence of different types of functions between these two sets are easy to solve, as there are finitely many such functions and so one many simply enumerate them. Further more, if one has two injective functions then it is intuitively obvious that both such functions are bijections (although not necessarily the inverses of each other). However, when one extends this to the infinite, the question becomes much more problematic. For instance, consider the function f: N → N, where f(x) = 2 ∗ x. This function is injective, from N to N and also trivially in the reverse direction, but f itself is not a bijection. This is a shame, as such injective functions are often much easier to find than a bijection between two sets. However, it turns out that the existence of such functions is sufficient to prove the existence of a bijection: Theorem. (Cantor-Schröder-Bernstein Theorem) Given two arbitrary sets A and B, if two injections f: A → B and g: B → A exist, then there exists a bijection h: A → B. This was originally taken as a corollary by Cantor of the well-ordering theorem,
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Sudev Naduvath, K. A. Germina · arXiv (Cornell University) · 2015
Let $\N$ denote the set of all non-negative integers and $\cP(\N)$ be its power set. An integer additive set-labeling (IASL) of a graph $G$ is an injective set-valued function $f:V(G)\to \cP(\N)-\{\emptyset\}$ such that the induced function $f^+:E(G) \to \cP(\N)-\{\emptyset\}$ is defined by $f^+ (uv) = f(u)+ f(v)$, where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. A graph which admits an IASL is usually called an IASL-graph. An IASL $f$ of a graph $G$ is said to be an integer additive set-indexer (IASI) of $G$ if the associated function $f^+$ is also injective. In this paper, we define the notion of integer additive set-labeling of signed graphs and discuss certain properties of signed graphs which admits certain types of integer additive set-labelings.
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Ade Ima Afifa Himayati · Jurnal Karya Pendidikan Matematika · 2020
Injective partial linear transformations in the vector space V and the function composition operations form semigroups called semigroups of injective partial linear transformations denoted by I (V). Suppose Z is a subspace in a vector space V, then the set of all members of I (V) whose function results are contained in Z to the function composition operation is also a semigroup denoted by I (V, Z) which is called a semigroup of injectable partial linear transformations with range restrictions. . In this article, we will examine how the characteristics of regular elements and the characteristics of two set members in I (V, Z) are related to Green using the literature study method. The results of the research in this article are the necessary and sufficient conditions needed so that any element is called a regular element and the characteristics of the two members of the set I (V, Z) are related.
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