Academic literature on the topic 'Path homomorphism'
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Journal articles on the topic "Path homomorphism"
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Nešetřil, Jaroslav, and Xuding Zhu. "Path homomorphisms." Mathematical Proceedings of the Cambridge Philosophical Society 120, no. 2 (1996): 207–20. http://dx.doi.org/10.1017/s0305004100074806.
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AbstractWe investigate homomorphisms between finite oriented paths. We demonstrate the surprising richness of this perhaps simplest case of homomorphism between graphs by proving the density theorem for oriented paths. As a consequence every two dimensional countable poset is represented finite paths and their homomorphisms, and every finite dimensional poset is represented finite oriented trees and their homomorphisms. We then consider related problems of universal representability and extendability and on-line representability.
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Bagchi, Susmit. "Interactions between Homotopy and Topological Groups in Covering (C, R) Space Embeddings." Symmetry 13, no. 8 (2021): 1421. http://dx.doi.org/10.3390/sym13081421.
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The interactions between topological covering spaces, homotopy and group structures in a fibered space exhibit an array of interesting properties. This paper proposes the formulation of finite covering space components of compact Lindelof variety in topological (C, R) spaces. The covering spaces form a Noetherian structure under topological injective embeddings. The locally path-connected components of covering spaces establish a set of finite topological groups, maintaining group homomorphism. The homeomorphic topological embedding of covering spaces and base space into a fibered non-compact topological (C, R) space generates two classes of fibers based on the location of identity elements of homomorphic groups. A compact general fiber gives rise to the discrete variety of fundamental groups in the embedded covering subspace. The path-homotopy equivalence is admitted by multiple identity fibers if, and only if, the group homomorphism is preserved in homeomorphic topological embeddings. A single identity fiber maintains the path-homotopy equivalence in the discrete fundamental group. If the fiber is an identity-rigid variety, then the fiber-restricted finite and symmetric translations within the embedded covering space successfully admits path-homotopy equivalence involving kernel. The topological projections on a component and formation of 2-simplex in fibered compact covering space embeddings generate a prime order cyclic group. Interestingly, the finite translations of the 2-simplexes in a dense covering subspace assist in determining the simple connectedness of the covering space components, and preserves cyclic group structure.
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Hazrat, Roozbeh, and Lia Vaš. "K-theory classification of graded ultramatricial algebras with involution." Forum Mathematicum 31, no. 2 (2019): 419–63. http://dx.doi.org/10.1515/forum-2017-0268.
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AbstractWe consider a generalization {K_{0}^{\operatorname{gr}}(R)} of the standard Grothendieck group {K_{0}(R)} of a graded ring R with involution. If Γ is an abelian group, we show that {K_{0}^{\operatorname{gr}}} completely classifies graded ultramatricial {*}-algebras over a Γ-graded {*}-field A such that (1) each nontrivial graded component of A has a unitary element in which case we say that A has enough unitaries, and (2) the zero-component {A_{0}} is 2-proper ({aa^{*}+bb^{*}=0} implies {a=b=0} for any {a,b\in A_{0}}) and {*}-pythagorean (for any {a,b\in A_{0}} one has {aa^{*}+bb^{*}=cc^{*}} for some {c\in A_{0}}). If the involutive structure is not considered, our result implies that {K_{0}^{\operatorname{gr}}} completely classifies graded ultramatricial algebras over any graded field A. If the grading is trivial and the involutive structure is not considered, we obtain some well-known results as corollaries. If R and S are graded matricial {*}-algebras over a Γ-graded {*}-field A with enough unitaries and {f:K_{0}^{\operatorname{gr}}(R)\to K_{0}^{\operatorname{gr}}(S)} is a contractive {\mathbb{Z}[\Gamma]}-module homomorphism, we present a specific formula for a graded {*}-homomorphism {\phi:R\to S} with {K_{0}^{\operatorname{gr}}(\phi)=f}. If the grading is trivial and the involutive structure is not considered, our constructive proof implies the known results with existential proofs. If {A_{0}} is 2-proper and {*}-pythagorean, we also show that two graded {*}-homomorphisms {\phi,\psi:R\to S} are such that {K_{0}^{\operatorname{gr}}(\phi)=K_{0}^{\operatorname{gr}}(\psi)} if and only if there is a unitary element u of degree zero in S such that {\phi(r)=u\psi(r)u^{*}} for any {r\in R}. As an application of our results, we show that the graded version of the Isomorphism Conjecture holds for a class of Leavitt path algebras: if E and F are countable, row-finite, no-exit graphs in which every infinite path ends in a sink or a cycle and K is a 2-proper and {*}-pythagorean field, then the Leavitt path algebras {L_{K}(E)} and {L_{K}(F)} are isomorphic as graded rings if any only if they are isomorphic as graded {*}-algebras. We also present examples which illustrate that {K_{0}^{\operatorname{gr}}} produces a finer invariant than {K_{0}}.
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Pakdaman, Ali, Hamid Torabi, and Behrooz Mashayekhy. "On locally 1-connectedness of quotient spaces and its applications to fundamental groups." Filomat 28, no. 1 (2014): 27–35. http://dx.doi.org/10.2298/fil1401027p.
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Let X be a locally 1-connected metric space and A1,A2,...,An be connected, locally path connected and compact pairwise disjoint subspaces of X. In this paper, we show that the quotient space X/(A1,A2,..., An) obtained from X by collapsing each of the sets Ai?s to a point, is also locally 1-connected. Moreover, we prove that the induced continuous homomorphism of quasitopological fundamental groups is surjective. Finally, we give some applications to find out some properties of the fundamental group of the quotient space X/(A1,A2,...,An).
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Bridges, Douglas, and Matthew Hendtlass. "Continuous isomorphisms from R onto a complete abelian group." Journal of Symbolic Logic 75, no. 3 (2010): 930–44. http://dx.doi.org/10.2178/jsl/1278682208.
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AbstractThis paper provides a Bishop-style constructive analysis of the contrapositive of the statement that a continuous homomorphism of R onto a compact abelian group is periodic. It is shown that, subject to a weak locatedness hypothesis, if G is a complete (metric) abelian group that is the range of a continuous isomorphism from R, then G is noncompact. A special case occurs when G satisfies a certain local path-connectedness condition at 0. A number of results about one-one and injective mappings are proved en route to the main theorem. A Brouwerian example shows that some of our results are the best possible in a constructive framework.
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Phillips, John. "Self-Adjoint Fredholm Operators And Spectral Flow." Canadian Mathematical Bulletin 39, no. 4 (1996): 460–67. http://dx.doi.org/10.4153/cmb-1996-054-4.
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AbstractWe study the topology of the nontrivial component, , of self-adjoint Fredholm operators on a separable Hilbert space. In particular, if {Bt} is a path of such operators, we can associate to {Bt} an integer, sf({Bt}), called the spectral flow of the path. This notion, due to M. Atiyah and G. Lusztig, assigns to the path {Bt} the net number of eigenvalues (counted with multiplicities) which pass through 0 in the positive direction. There are difficulties in making this precise — the usual argument involves looking at the graph of the spectrum of the family (after a suitable perturbation) and then counting intersection numbers with y = 0.We present a completely different approach using the functional calculus to obtain continuous paths of eigenprojections (at least locally) of the form . The spectral flow is then defined as the dimension of the nonnegative eigenspace at the end of this path minus the dimension of the nonnegative eigenspace at the beginning. This leads to an easy proof that spectral flow is a well-defined homomorphism from the homotopy groupoid of onto Z. For the sake of completeness we also outline the seldom-mentioned proof that the restriction of spectral flow to is an isomorphism onto Z.
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EVANS, DAVID E., and JEREMY D. GOULD. "DIMENSION GROUPS AND EMBEDDINGS OF GRAPH ALGEBRAS." International Journal of Mathematics 05, no. 03 (1994): 291–327. http://dx.doi.org/10.1142/s0129167x94000188.
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If Γ is a graph, with distinguished vertex *, let A(Γ) denote the non-commutative path algebra on the space [Formula: see text] of semi-infinite paths in Γ beginning at *. We discuss embeddings A(Γ1) → A(Γ2) of AF algebras associated with graphs Γ1 and Γ2 from a dimension group point of view. For certain infinite T-shaped graphs, we have K0(A(Γ)) ≅ ℤ [t], with positive cone identified with {0}∪ {P ∈ ℤ [t]: P (λ) > 0, λ ∈ (0, γ]}, where γ = γ (Γ) =||Γ||−2 < 1/4. Hence for certain graphs there exists a unital homomorphism A(Γ1) → A(Γ2) if ||Γ1|| ≤ ||Γ2||. For certain finite T-shaped graphs K0 (A(Γ)) ≅ ℤ [t]/<Q> where <Q> denotes the ideal generated by a polynomial Q=Q(Γ) which is essentially the characteristic polynomial of the graph Γ and positive cone identified with {0}∪ {f + <Q>: f(γ) > 0} where γ = γ(Γ) = ||Γ||-2. Hence there exists a unital homomorphism A(Γ1) → A(Γ2) if ||Γ1|| = ||Γ2||, and Q(Γ2) divides Q(Γ1). The structure of K0(A(Γ)) as an ordered ring is related to the fusion rules of rational conformal field theory.
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EVANS, DAVID E., and JEREMY D. GOULD. "DIMENSION GROUPS, EMBEDDINGS AND PRESENTATIONS OF AF ALGEBRAS ASSOCIATED TO SOLVABLE LATTICE MODELS." Modern Physics Letters A 04, no. 20 (1989): 1883–90. http://dx.doi.org/10.1142/s0217732389002136.
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If Γ is a graph, with distinguished vertex *, let A(Γ) denote the non-commutative path algebra on the space [Formula: see text] of semi-infinite paths in Γ beginning at *. Embeddings A(Γ1)→A(Γ2) of non-commutative AF algebras associated with graphs Γ1 and Γ2 are discussed from a dimension group point of view. For certain infinite T-shaped graphs, we have K0(A(Γ))≃ ℤ[t], with positive cone identified with {0}∪{P∈ℤ(t): P(λ)>0, λ∈(0,γ]}, where γ=γ(Γ)= ||Γ||−2<1/4. Hence for certain graphs there exists a unital homomorphism A(Γ1)→A(Γ2) if ||Γ1||=||Γ2||. For certain finite T-shaped graphs K0(A(Γ))≃ℤ[t]/<Q> where <Q> denotes the ideal generated by a polynomial Q=Q(Γ) which is essentially the characteristic polynomial of the graph Γ, and positive cone identified with {0}∪{f+<Q>: f(γ)>0} where γ=γ(Γ)=||Γ||−2. Hence there exists a unital homomorphism A(Γ1)→A(Γ2) if ||Γ1||=||Γ2||, and Q(Γ1) divides Q(Γ2). The structure of K0(A(Γ)) as an ordered ring is related to the fusion rules of rational conformal field theory. Moreover, for these T-shaped graphs there is an algebraic presentation which further illuminates the above embeddings. This presentation involves a new projection and a new relation in addition to those of Temperley-Lieb, and gives a rigidity above index four.
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Csikvári, Péter, and Zhicong Lin. "Homomorphisms of Trees into a Path." SIAM Journal on Discrete Mathematics 29, no. 3 (2015): 1406–22. http://dx.doi.org/10.1137/140993995.
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Kun, Gábor, and Claude Tardif. "Homomorphisms of random paths." European Journal of Combinatorics 31, no. 3 (2010): 688–93. http://dx.doi.org/10.1016/j.ejc.2009.09.003.
Full textDissertations / Theses on the topic "Path homomorphism"
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Ferrão, Inês Margarida Neto. "Homomorfismos de grafos por caminhos e o problema da imersão de redes." Master's thesis, 2019. http://hdl.handle.net/10316/87951.
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Dissertação de Mestrado em Matemática apresentada à Faculdade de Ciências e Tecnologia<br>In this work we begin by adressing the classic problem of graph homomorphism and then, a variation thereof, the path homomorphism problem. We give a formulation to this last problem as an integer linear program and present some numerical tests that we performed to test its behavior and the performance of IBM ILOG CPLEX Optimization Studio for solving it. We then approach the network embedding problem as a specialization of the path homomorphism problem and present further numerical tests. Finally, we apply the developed model to the problem of locating processors on the periphery of a mobile network, taking into account users' mobility patterns. We performed new numerical tests and end by showing an application of the model to a realistic case. .<br>Neste trabalho começamos por abordar o problema clássico do homomorfismo de grafos e em seguida,uma sua variação, o problema do homomorfismo por caminhos. Formulamos este último problemacomo programa linear inteiro e apresentamos alguns testes numéricos que realizamos para estudar oseu comportamento e performance do IBM ILOG CPLEX Otimization Studio na sua resolução.Seguidamente abordamos o problema da imersão de redes como sendo uma especialização doproblema do homomorfismo por caminhos e apresentamos mais alguns testes numéricos.Por fim, aplicamos o modelo desenvolvido ao problema da localização de processadores naperiferia de uma rede móvel, tendo em conta os padrões de mobilidade dos utilizadores. Realizamosnovos testes numéricos e ainda mostramos uma aplicação do modelo desenvolvido a um caso realista. .
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Mailhot, Pierre-Alexandre. "Extension de l'homomorphisme de Calabi aux cobordismes lagrangiens." Thèse, 2019. http://hdl.handle.net/1866/23795.
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Ce mémoire traite de la construction d’un nouvel invariant des cobordismes lagrangiens. Cette construction est inspirée des travaux récents de Solomon dans lesquels une extension de l’homomorphisme de Calabi aux chemins lagrangiens exacts est donnée. Cette extension fut entre autres motivée par le fait que le graphe d’une isotopie hamiltonienne est un chemin lagrangien exact. Nous utilisons la suspension lagrangienne, qui associe à chaque chemin lagrangien exact un cobordisme lagrangien, pour étendre la construction de Solomon aux cobordismes lagrangiens. Au premier chapitre nous donnons une brève exposition des propriétés élémentaires des variétés symplectiques et des sous-variétés lagrangiennes. Le second chapitre traite du groupe des difféomorphismes hamiltoniens et des propriétés fondamentales de l’homomorphisme de Calabi. Le chapitre 3 est dédié aux chemins lagrangiens, l’invariant de Solomon et ses points critiques. Au dernier chapitre nous introduisons la notion de cobordisme lagrangien et construisons le nouvel invariant pour finalement analyser ses points critiques et l’évaluer sur la trace de la chirurgie de deux courbes sur le tore. Dans le cadre de ce calcul, nous serons en mesure de borner la valeur du nouvel invariant en fonction de l’ombre du cobordisme, une notion récemment introduite par Cornea et Shelukhin.<br>In this master's thesis, we construct a new invariant of Lagrangian cobordisms. This construction is inspired by the recent works of Solomon in which an extension of the Calabi homomorphism to exact Lagrangian paths is given. Solomon's extension was motivated by the fact that the graph of any Hamiltonian isotopy is an exact Lagrangian path. We use the Lagrangian suspension construction, which associates to every exact Lagrangian path a Lagrangian cobordism, to extend Solomon's invariant to Lagrangian cobordisms. In the first chapter, we give a brief introduction to the elementary properties of symplectic manifolds and their Lagrangian submanifolds. In the second chapter, we present an introduction to the group of Hamiltonian diffeomorphisms and discuss the fundamental properties of the Calabi homomorphism. Chapter 3 is dedicated to Lagrangian paths, Solomon's invariant and its critical points. In the last chapter, we introduce the notion of Lagrangian cobordism and we construct the new invariant. We analyze its critical points and evaluate it on the trace of the Lagrangian surgery of two curves on the torus. In this setting we further bound the new invariant in terms of the shadow of the cobordism, a notion recently introduced by Cornea and Shelukhin.
Books on the topic "Path homomorphism"
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McDuff, Dusa, and Dietmar Salamon. The group of symplectomorphisms. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198794899.003.0011.
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This chapter discusses the basic properties of the group of symplectomorphisms of a compact connected symplectic manifold and its subgroup of Hamiltonian symplectomorphisms. It begins by showing that the group of symplectomorphisms is locally path-connected and then moves on to the flux homomorphism. The main result here is a theorem of Banyaga that characterizes the Hamiltonian symplectomorphisms in terms of the flux homomorphism. In the noncompact case there is another interesting homomorphism, called the Calabi homomorphism, that takes values in the reals and may be defined on the universal cover of the group of Hamiltonian symplectomorphisms. The chapter ends with a brief comparison of the topological properties of the group of symplectomorphisms with those of the group of diffeomorphisms.
Book chapters on the topic "Path homomorphism"
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Beaudou, Laurent, Florent Foucaud, Florent Madelaine, Lhouari Nourine, and Gaétan Richard. "Complexity of Conjunctive Regular Path Query Homomorphisms." In Computing with Foresight and Industry. Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-22996-2_10.
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Hubička, J., J. Nešetřil, P. Oviedo, and O. Serra. "On the Homomorphism Order of Oriented Paths and Trees." In Trends in Mathematics. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-83823-2_118.
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Kaosar, Golam, and Xun Yi. "Secure Two-Party Association Rule Mining Based on One-Pass FP-Tree." In Privacy Solutions and Security Frameworks in Information Protection. IGI Global, 2013. http://dx.doi.org/10.4018/978-1-4666-2050-6.ch006.
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Frequent Path tree (FP-tree) is a popular method to compute association rules and is faster than Apriori-based solutions in some cases. Association rule mining using FP-tree method cannot ensure entire privacy since frequency of the itemsets are required to share among participants at the first stage. Moreover, FP-tree method requires two scans of database transactions which may not be the best solution if the database is very large or the database server does not allow multiple scans. In addition, one-pass FP-tree can accommodate continuous or periodically changing databases without restarting the process as opposed to a regular FP-tree based solution. In this paper, the authors propose a one-pass FP-tree method to perform association rule mining without compromising any data privacy among two parties. A fully homomorphic encryption system over integer numbers is applied to ensure secure computation among two data sites without disclosing any number belongs to themselves.