Kai F. Wehmeier · Linguistics and Philosophy · 2024
Abstract Given any set E of expressions freely generated from a set of atoms by syntactic operations, there exist trivially compositional functions on E (to wit, the injective and the constant functions), but also plenty of non-trivially compositional functions. Here we show that within the space of non-injective functions (and so a fortiori within the space of non-injective and non-constant functions), compositional functions are not sufficiently abundant in order to generate the consequence relation of every propositional logic. Logical consequence relations thus impose substantive constraints on the existence of compositional functions when coupled with the condition of non-injectivity (though not without it). We ask how the apriori exclusion of injective functions from the search space might be justified, and we discuss the prospects of claims to the effect that any function can be “encoded” in a compositional one.
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Paweł Rzążewski · arXiv (Cornell University) · 2013
For graphs $G$ and $H$, a homomorphism from $G$ to $H$ is a function $φ\colon V(G) \to V(H)$, which maps vertices adjacent in $G$ to adjacent vertices of $H$. A homomorphism is locally injective if no two vertices with a common neighbor are mapped to a single vertex in $H$. Many cases of graph homomorphism and locally injective graph homomorphism are NP-complete, so there is little hope to design polynomial-time algorithms for them. In this paper we present an algorithm for graph homomorphism and locally injective homomorphism working in time $\mathcal{O}^*((b + 2)^{|V(G)|})$, where $b$ is the bandwidth of the complement of $H$.
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Jukka A. Koskinen · Fundamenta Informaticae · 1999
This paper is a cryptographically motivated study of the knapsack, its subset sum function and its inverse relation, the decipherment function. The novelty is that the subset sum function is not assumed to be injective. Instead, various forms of “jec
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Adam Przestacki · Bulletin of the Belgian Mathematical Society - Simon Stevin · 2018
In 2011 Kenessey and Wengenroth gave a full description of closed range composition operators $C_\psi:C^{\infty}(\mathbb{R}^d)\toC^{\infty}(\mathbb{R})$, $F\mapsto F\circ\psi$, corresponding to smooth injective symbols $\psi:\mathbb{R}\to\mathbb{R}^d$. In 2012 Przestacki gave a sufficient condition for the range of $C_\psi$ to be closed in case if $\psi:\mathbb{R}\to\mathbb{R}$ is a smooth not necessarily injective symbol. Using their ideas we give a sufficient condition ensuring that the range of $C_\psi$ is closed when $\psi:\mathbb{R}\to\mathbb{R}^d$ is a smooth not necessarily injective function.
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Rahul Sarkar · arXiv (Cornell University) · 2022
The unique prime factorization theorem is used to show the existence of a function on a countable set $\mathcal{X}$ so that the sum aggregator function is injective on all multisets of $\mathcal{X}$ of finite size.
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Sudev Naduvath, K. A. Germina · Journal of Advanced Research in Pure Mathematics · 2015
An integer additive set-indexer is an injective function $f:V(G)\to 2^{\mathbb{N}_0}$ such that the induced function $g_f:E(G) \to 2^{\mathbb{N}_0}$ defined by $g_f (uv) = f(u)+ f(v)$ is also injective. A graph $G$ which admits an IASI is called an IASI graph. An arithmetic integer additive set-indexer is an integer additive set-indexer $f$, under which the set-labels of all elements of a given graph $G$ are arithmetic progressions. In this paper, we discuss about admissibility of arithmetic integer additive set-indexers by certain graph operations and certain products of graphs.
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Marco Sabatini · Topological Methods in Nonlinear Analysis · 2023
Fessler and Gutierrez \cite{Fe}, \cite{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 $\partial P/\partial x + \partial Q/\partial y$ is not a surjective function.
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P.Sugirtha, R.Vasuki, J.Venkateswari · Zenodo (CERN European Organization for Nuclear Research) · 2014
A graph G with p vertices and q edges is said to have a skolem difference odd mean labeling if there exists an injective function f.
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Paweł Rzążewski · arXiv (Cornell University) · 2013
For graphs $G$ and $H$, a homomorphism from $G$ to $H$ is a function $\\varphi\n\\colon V(G) \\to V(H)$, which maps vertices adjacent in $G$ to adjacent vertices\nof $H$. A homomorphism is locally injective if no two vertices with a common\nneighbor are mapped to a single vertex in $H$. Many cases of graph homomorphism\nand locally injective graph homomorphism are NP-complete, so there is little\nhope to design polynomial-time algorithms for them. In this paper we present an\nalgorithm for graph homomorphism and locally injective homomorphism working in\ntime $\\mathcal{O}^*((b + 2)^{|V(G)|})$, where $b$ is the bandwidth of the\ncomplement of $H$.\n
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A. Lourdusamy, M. Seenivasan · Proyecciones (Antofagasta) · 2017
A vertex-mean labeling of a (p, q) graph G = (V, E) is defined as an injective function f : E → {0, 1, 2, ..., q*} , q* = max (p,q) such that the function fV : V → N defined by the rule
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Sebastian M. Cioabă, M. Ram Murty · Hindustan Book Agency · 2009
Combinatorics is the study of finite sets. To define finite sets, we need the notion of bijective function. Given two sets X and Y, a function f : X → Y is injective or one-to-one if f(a) ≠ f(b) for any a, b ∈ X with a ≠ b. A function f : X → Y is surjective or onto if for any y ∈ Y, there exist x ∈ X such that f(x) = y. A function is bijective if it is injective and surjective. A function f : X → Y is invertible if there exists a function g : Y → X such that f(x) = y if and only if g(y) = x. If g exists, it is called the inverse of f and it is usually denoted by f−1. We leave as an exercise the fact that a function is bijective if and only if it is invertible.
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Jarnicki, Witold · Jagiellonian University Repository (Jagiellonian University) · 2001
Abstract. We give a characterization of all cartesian products D1×D2 ⊂ C2 for which the Lempert function and the injective Lempert function coincide. In particular, we show that there exist domains in C2 for which they are different. 1. Introduction. The
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Amedeo Altavilla · Advances in Geometry · 2018
Abstract In this paper we show that the real differential of any injective slice regular function is everywhere invertible. The result is a generalization of a theorem proved by G. Gentili, S. Salamon and C. Stoppato and it is obtained thanks, in particular, to some new information regarding the first coefficients of a certain polynomial expansion for slice regular functions (called spherical expansion ), and to a new general result which says that the slice derivative of any injective slice regular function is different from zero. A useful tool proven in this paper is a new formula that relates slice and spherical derivatives of a slice regular function. Given a slice regular function, part of its singular set is described as the union of surfaces on which it results to be constant.
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M. Subbulakshmi, S B Chandrakala, G. Siva Prijith · International Journal of Research and Innovation in Applied Science · 2025
Bitopological harmonious labeling for a graph G=(V(G),E(G)) with n vertices, is an injective function f:V(G)→2^X, where X is any non – empty set such that |X|=m,m< n and {f(V(G))} forms a topology on X, that induces an injective function f^*: E(G) → 2^(X^* ), defined by f^* (uv) = f(u)∩f(v) for every uv∈E(G) such that {f^* (E(G))} forms a topology on X^* where X^*=X\{1,2,….,m}. A graph that admits bitopological harmonious labeling is called a bitopological harmonious graph. In this paper, we discuss bitopological harmonious labeling of some star related graphs.
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Dolors Herbera, Michal Hrbek, Giovanna Le Gros · arXiv (Cornell University) · 2024
For a commutative noetherian ring $R$, we classify all the hereditary cotorsion pairs cogenerated by pure-injective modules of finite injective dimension. The classification is done in terms of integer-valued functions on the spectrum of the ring. Each such function gives rise to a system of local depth conditions which describes the left-hand class in the corresponding cotorsion pair. Furthermore, we show that these cotorsion pairs correspond by explicit duality to hereditary Tor-pairs generated by modules of finite flat dimension.
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Sudev Naduvath, K. A. Germina · International Journal of Mathematics and Soft Computing · 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 $f^+:E(G) \rightarrow 2^{\mathbb{N}_0}$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective. A graph $G$ which admits an IASI is called an IASI graph. An arithmetic integer additive set-indexer is an integer additive set-indexer $f$, under which the set-labels of all elements of a given graph $G$ are arithmetic progressions. In this paper, we discuss about admissibility of arithmetic integer additive set-indexers by certain associated graphs of the given graph $G$, like line graph, total graph, etc.
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Bart Bogaerts, Balder ten Cate, Brett McLean et al. · arXiv (Cornell University) · 2023
We investigate a number of semantically defined fragments of Tarski's algebra of binary relations, including the function-preserving fragment. We address the question whether they are generated by a finite set of operations. We obtain several positive and negative results along these lines. Specifically, the homomorphism-safe fragment is finitely generated (both over finite and over arbitrary structures). The function-preserving fragment is not finitely generated (and, in fact, not expressible by any finite set of guarded second-order definable function-preserving operations). Similarly, the total-function-preserving fragment is not finitely generated (and, in fact, not expressible by any finite set of guarded second-order definable total-function-preserving operations). In contrast, the forward-looking function-preserving fragment is finitely generated by composition, intersection, antidomain, and preferential union. Similarly, the forward-and-backward-looking injective-function-preserving fragment is finitely generated by composition, intersection, antidomain, inverse, and an `injective union' operation.
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Bart Bogaerts, Balder ten Cate, Brett McLean et al. · Logical Methods in Computer Science · 2024
We investigate a number of semantically defined fragments of Tarski's algebra of binary relations, including the function-preserving fragment. We address the question whether they are generated by a finite set of operations. We obtain several positive and negative results along these lines. Specifically, the homomorphism-safe fragment is finitely generated (both over finite and over arbitrary structures). The function-preserving fragment is not finitely generated (and, in fact, not expressible by any finite set of guarded second-order definable function-preserving operations). Similarly, the total-function-preserving fragment is not finitely generated (and, in fact, not expressible by any finite set of guarded second-order definable total-function-preserving operations). In contrast, the forward-looking function-preserving fragment is finitely generated by composition, intersection, antidomain, and preferential union. Similarly, the forward-and-backward-looking injective-function-preserving fragment is finitely generated by composition, intersection, antidomain, inverse, and an `injective union' operation.
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Benjamin Antieau · 2016
While the rational numbers Q are dense in the real numbers R, it seems like there are many, many more real numbers than rational. For instance, Q feels discrete, while R feels continuous. More concretely, every decimal expansion of a rational number is eventually repeating and conversely. There is no such restriction for irrational numbers, and certainly “most ” decimal expansions are not repeating. The goal of these lectures is to make this idea precise and to prove in a rigorous way that there are more elements of R than there are of Q. References for this material can be found in [1, Chapter 1] and [3, Chapter 2]. Recall that if A and B are sets, a function f: A → B assigns to every element x of A a unique element f (x) of B. A function f is • injective if f (x) = f (y) implies x = y; • surjective if for every z ∈ B there exists x ∈ A such that f (x) = z; • bijective if it is injective and surjective. Note that an injective function is also called one-to-one, while a surjective function is also called onto. A set with only finitely many elements is called finite. A non-finite set is called infinite. Another
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Andreas Defant, Carsten Michels · ArXiv.org · 1999
We prove a complex interpolation formula for the injective tensor product of vector-valued Banach function spaces satisfying certain geometric assumptions. This result unifies results of Kouba, and moreover, our approach offers an alternate proof of Kouba's interpolation formula for scalar-valued Banach function spaces.
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