Academic literature on the topic 'Fixed points of morphisms'

We study a fundamental combinatorial problem on morphisms in free semigroups: With regard to any string α over some alphabet we ask for the existence of a morphism σ such that σ(α) is unambiguous, i.e. there is no morphism τ with τ(i) ≠ σ(i) for some symbol i in α and, nevertheless, τ(α) = σ(α). As a consequence of its elementary nature, this question shows a variety of connections to those topics in discrete mathematics which are based on finite strings and morphisms such as pattern languages, equality sets and, thus, the Post Correspondence Problem. Our studies demonstrate that the existence of unambiguous morphic images essentially depends on the structure of α: We introduce a partition of the set of all finite strings into those that are decomposable (referred to as prolix) in a particular manner and those that are indecomposable (called succinct). This partition, that is also known to be of major importance for the research on pattern languages and on finite fixed points of morphisms, allows to formulate our main result according to which a string α can be mapped by an injective morphism onto an unambiguous image if and only if α is succinct.

A morphism $\sigma$ is \emph{(strongly) unambiguous} with respect to a word $\alpha$ if there is no other morphism $\tau$ that maps $\alpha$ to the same image as $\sigma$. Moreover, $\sigma$ is said to be \emph{weakly unambiguous} with respect to a word $\alpha$ if $\sigma$ is the only \emph{nonerasing} morphism that can map $\alpha$ to $\sigma(\alpha)$, i.\,e., there does not exist any other nonerasing morphism $\tau$ satisfying $\tau(\alpha) = \sigma(\alpha)$. In the first main part of the present thesis, we wish to characterise those words with respect to which there exists a weakly unambiguous \emph{length-increasing} morphism that maps a word to an image that is strictly longer than the word. Our main result is a compact characterisation that holds for all morphisms with ternary or larger target alphabets. We also comprehensively describe those words that have a weakly unambiguous length-increasing morphism with a unary target alphabet, but we have to leave the problem open for binary alphabets, where we can merely give some non-characteristic conditions. \par The second main part of the present thesis studies the question of whether, for any given word, there exists a strongly unambiguous \emph{1-uniform} morphism, i.\,e., a morphism that maps every letter in the word to an image of length $1$. This problem shows some connections to previous research on \emph{fixed points} of nontrivial morphisms, i.\,e., those words $\alpha$ for which there is a morphism $\phi$ satisfying $\phi(\alpha) = \alpha$ and, for a symbol $x$ in $\alpha$, $\phi(x) \neq x$. Therefore, we can expand our examination of the existence of unambiguous morphisms to a discussion of the question of whether we can reduce the number of different symbols in a word that is not a fixed point such that the resulting word is again not a fixed point. This problem is quite similar to the setting of Billaud's Conjecture, the correctness of which we prove for a special case.

Constructive mathematics is mathematics with intuitionistic logic (together with some appropriate, predicative, foundation)-it is often crudely characterised as mathematics without the law of excluded middle. The intuitionistic interpretation of the connectives and quantifiers ensure that constructive proofs contain an inherent algorithm which realises the computational content of the result it proves, and, in contrast to results from computable mathematics, these inherent algorithms come with fixed rates of convergence. The value of a constructive proof lies in the vast array of models for constructive mathematics. Realizability models and the interpretation of constructive ZF set theory into Martin Löf type theory allows one to view constructive mathematics as a high level programing language, and programs have been extracted and implemented from constructive proofs. Other models, including topological forcing models, of constructive set theory can be used to prove metamathematical results, for example, guaranteeing the (local) continuity of functions or algorithms. In this thesis we have highlighted any use of choice principles, and those results which do not require any choice, in particular, are valid in any topos. This thesis looks at what can and cannot be done in the study of the fundamental fixed point theorems from analysis, and gives some applications to mathematical economics where value is given to computability.

In this thesis, we first obtain coincidence and common fixed point theorems for a pair of generalized non-expansive type mappings in a normed space. Then we discuss two types of convergence theorems, namely, the convergence of Mann iteration procedures and the convergence and stability of fixed points. In addition, we discuss the viscosity approximations generated by (ψ ,ϕ)-weakly contractive mappings and a sequence of non-expansive mappings and then establish Browder and Halpern type convergence theorems on Banach spaces. With regard to iteration procedures, we obtain a result on the convergence of Mann iteration for generalized non-expansive type mappings in a Banach space which satisfies Opial's condition. And, in the case of stability of fixed points, we obtain a number of stability results for the sequence of (ψ,ϕ)- weakly contractive mappings and the sequence of their corresponding fixed points in metric and 2-metric spaces. We also present a generalization of Fraser and Nadler type stability theorems in 2-metric spaces involving a sequence of metrics.

The focus of this thesis is a study of the extension properties of local analytic conjugacies between simple parabolic fixed points. Any given conjugacy itself will generally not have an extension to the immediate basin. However, we show that if both maps belong to a suitable class (which includes polynomial-like maps and rational maps with a simply connected parabolic basin) then for all n large enough g on o X does have an analytic extension to the immediate parabolic basin. We begin by studying qualitative models for the dynamics near a parabolic fixed point, leading us to the Parabolic Flower Theorem. We then construct Fatou coordinates, which conjugate f to the unit translation, and study extension and properties of these maps. By restricting ourselves to the case when the restriction of f to its parabolic basin is a proper map with finitely many critical points we are able to study covering properties of these extended Fatou coordinates. We also introduce the horn map and lifted horn maps and show that the former is a complete invariant of the local analytic conjugacy class. Working from the covering properties of the horn map, we develop an intuition for how critical orbits of two maps f and g with locally conjugate simple parabolic fixed points should be related. In our main theorem, Theorem 3.1.10, we show that if both maps have a proper parabolic basin and is a local analytic conjugacy from (f; z0) to (g; w0) then for all n large enough, the map g on o X has an analytic extension along any curve starting in a region near z0 contained in the basin of z0. Under the additional assumption that the immediate basin is simply connected we can then conclude that the map Xn := g on o X has an analytic extension to a semi-conjugacy between the immediate basins whenever n is large enough.

Renormalization was popularised in the 1940s following the appearance of non- sensical infinities in the calculation of the self-energy of the electron. Notably this led to Quantum Electrodynamics becoming a fully renormalizable quantum field theory. One useful tool that emerges from the technical aspects of renormal- ization is the Renormalization Group. In particular, the β-function defines the variation of the coupling constants with energy. The vanishing of the β-function at a particular value of the coupling is known as a fixed point, the location of which can be found using perturbation theory. Properties of quantum field the- ories such as ultraviolet behaviour can be studied using these fixed points. The calculation of two different types of fixed points forms the spine of this thesis. In Part I the d-dimensional Wilson-Fisher fixed point is used to connect scalar quantum field theories in different space-time dimensions. Specifically we look at dimensions greater than four and explore the property of universality through the Vasil'ev large N expansion. Different universality classes are examined, the first contains φ4 theory with O(N) symmetry while another incorporates O(N)×O(m) Landau-Ginzburg-Wilson theory. In the latter we perform a full fixed point sta- bility analysis and conformal window search which determines where conformal symmetry is present. Part I develops techniques that may later be applicable to calculations involving beyond the Standard Model physics including asymptotic safety, quantum gravity and emergent symmetries. Part II focuses on the non-trivial Banks-Zaks fixed point of four dimensional Quantum Chromodynamics. Using a variety of colour groups and representations we calculate the location of the fixed point and corresponding critical exponents to pinpoint exactly where the true value of the conformal window lies. Additionally a number of different renormalization schemes are used, including the momentum subtraction (MOM) and interpolating momentum subtraction (iMOM) schemes. This allows us to study where in the conformal window scheme dependence is most apparent. Both the Landau gauge and maximal abelian gauge are utilized to extend the analysis. Throughout this thesis we compare and contrast perturbative results with non-perturbative calculations such as those performed in lattice.

Cette thèse traite de la theorie de la preuve pour les logiques a points fixes, telles que le μ-calcul, lalogique lineaire a points fixes, etc. ces logiques sont souvent munies de systèmes de preuves finitairesavec des règles d’induction à la Park. Il existe néanmoins d’autres sytèmes de preuves pour leslogiques à points fixes, qui reposent sur la notion de preuve infinitaire, mais qui sont beaucoupmoins developpés dans la litterature. L’objectif de cette thèse est de pallier à cette lacune dansl’état de l’art, en developpant la théorie de la preuve infnitaire pour les logiques a points fixes,avec deux domaines d’application en vue: les langages de programmation avec types de données(co)inductifs et la vérification des systèmes réactifs.Cette thèse contient trois partie. Dans la première, on rappelle les deux principales approchespour obtenir des systèmes de preuves pour les logiques à points fixes: les systèmes finitaires avecrègle explicite d’induction et les systèmes finitaires, puis on montre comment les deux approchesse relient. Dans la deuxième partie, on argumente que les preuves infinitaires ont effectivement unréel statut preuve-theorique, en montrant que la logique lineaire additive multiplicative avec pointsfixes admet les propriétés d’élimination des coupures et de focalisation. Dans la troisième partie,on utilise nos developpements sur les preuves infinitaires pour monter de manière constructive lacomplétude du μ-calcul lineaire relativement à l’axiomatisation de Kozen
The subject of this thesis is the proof theory of logics with fixed points, such as the μ-calculus,linear-logic with fixed points, etc. These logics are usually equipped with finitary deductive systemsthat rely on Park’s rules for induction. other proof systems for these logics exist, which relyon infinitary proofs, but they are much less developped. This thesis contributes to reduce thisdeficiency by developing the infinitary proof-theory of logics with fixed points, with two domainsof application in mind: programming languages with (co)inductive data types and verification ofreactive systems.This thesis contains three parts. In the first part, we recall the two main approaches to theproof theory for logics with fixed points: the finitary and the infinitary one, then we show theirrelationships. In the second part, we argue that infinitary proofs have a true proof-theoreticalstatus by showing that the multiplicative additive linear-logic with fixed points admits focalizationand cut-elimination. In the third part, we apply our proof-theoretical investigations to obtain aconstructive proof of completeness for the linear-time μ-calculus w.r.t. Kozen’s axiomatization

In 2014, Colombia implemented a policy that added flexibilization to labor contracts for part-time workers that reduced the quasi-fixed costs of employing formal workers. We find that the reform increased the probability of entering the formal sector within the targeted population: low-wage, part-time workers. We use administrative employer-employee matched data and leverage variation across cities and industries in demand for part-time work before the reform. We find that, after the tax reform, the change in the total number of formal workers is 6 percentage points higher at firms that use the new contracts relative to their counterparts that choose not to hire low-wage, formal, part-time workers under the new tax form. Mean daily wages temporarily declined after the reform.

The first-mile, last-mile problem is a significant deterrent for potential transit riders, especially in suburban neighborhoods with low density. Transit agencies have typically sought to solve this problem by adding parking spaces near transit stations and adding stops to connect riders to fixed-route transit. However, these measures are often only short-term solutions. In the last few years, transit agencies have tested whether new mobility services, such as ridehailing, ridesharing, and microtransit, can offer fast, reliable connections to and from transit stations. However, there is limited research that evaluates the potential impacts of these projects. Concurrently, there is growing interest in the future of automated vehicles (AVs) and the potential of AVs to solve this first-mile problem by reducing the cost of providing these new mobility services to promote access to transit. This paper expands upon existing research to model the simulate the travel and revenue impacts of a fleet of automated vehicles that provide transit access services in the San Francisco Bay Area offered over a range of fares. The model simulates a fleet of AVs for first-mile transit access at different price points for three different service models (door-to-door ridehailing and ridesharing and meeting point ridesharing services). These service models include home-based drop-off and pick-up for single passenger service (e.g., Uber and Lyft), home-based drop-off and pick-up for multi-passenger service (e.g., microtransit), and meeting point multi-passenger service (e.g., Via).

Macroeconomic summary Several factors contributed to an increase in projected inflation on the forecast horizon, keeping it above the target rate. These included inflation in December that surpassed expectations (5.62%), indexation to higher inflation rates for various baskets in the consumer price index (CPI), a significant real increase in the legal minimum wage, persistent external and domestic inflationary supply shocks, and heightened exchange rate pressures. The CPI for foods was affected by the persistence of external and domestic supply shocks and was the most significant contributor to unexpectedly high inflation in the fourth quarter. Price adjustments for fuels and certain utilities can explain the acceleration in inflation for regulated items, which was more significant than anticipated. Prices in the CPI for goods excluding food and regulated items also rose more than expected. This was partly due to a smaller effect on prices from the national government’s VAT-free day than anticipated by the technical staff and more persistent external pressures, including via peso depreciation. By contrast, the CPI for services excluding food and regulated items accelerated less than expected, partly reflecting strong competition in the communications sector. This was the only major CPI basket for which prices increased below the target inflation rate. The technical staff revised its inflation forecast upward in response to certain external shocks (prices, costs, and depreciation) and domestic shocks (e.g., on meat products) that were stronger and more persistent than anticipated in the previous report. Observed inflation and a real increase in the legal minimum wage also exceeded expectations, which would boost inflation by affecting price indexation, labor costs, and inflation expectations. The technical staff now expects year-end headline inflation of 4.3% in 2022 and 3.4% in 2023; core inflation is projected to be 4.5% and 3.6%, respectively. These forecasts consider the lapse of certain price relief measures associated with the COVID-19 health emergency, which would contribute to temporarily keeping inflation above the target on the forecast horizon. It is important to note that these estimates continue to contain a significant degree of uncertainty, mainly related to the development of external and domestic supply shocks and their ultimate effects on prices. Other contributing factors include high price volatility and measurement uncertainty related to the extension of Colombia’s health emergency and tax relief measures (such as the VAT-free days) associated with the Social Investment Law (Ley de Inversión Social). The as-yet uncertain magnitude of the effects of a recent real increase in the legal minimum wage (that was high by historical standards) and high observed and expected inflation, are additional factors weighing on the overall uncertainty of the estimates in this report. The size of excess productive capacity remaining in the economy and the degree to which it is closing are also uncertain, as the evolution of the pandemic continues to represent a significant forecast risk. margin, could be less dynamic than expected. And the normalization of monetary policy in the United States could come more quickly than projected in this report, which could negatively affect international financing costs. Finally, there remains a significant degree of uncertainty related to the duration of supply chocks and the degree to which macroeconomic and political conditions could negatively affect the recovery in investment. The technical staff revised its GDP growth projection for 2022 from 4.7% to 4.3% (Graph 1.3). This revision accounts for the likelihood that a larger portion of the recent positive dynamic in private consumption would be transitory than previously expected. This estimate also contemplates less dynamic investment behavior than forecast in the previous report amid less favorable financial conditions and a highly uncertain investment environment. Third-quarter GDP growth (12.9%), which was similar to projections from the October report, and the fourth-quarter growth forecast (8.7%) reflect a positive consumption trend, which has been revised upward. This dynamic has been driven by both public and private spending. Investment growth, meanwhile, has been weaker than forecast. Available fourth-quarter data suggest that consumption spending for the period would have exceeded estimates from October, thanks to three consecutive months that included VAT-free days, a relatively low COVID-19 caseload, and mobility indicators similar to their pre-pandemic levels. By contrast, the most recently available figures on new housing developments and machinery and equipment imports suggest that investment, while continuing to rise, is growing at a slower rate than anticipated in the previous report. The trade deficit is expected to have widened, as imports would have grown at a high level and outpaced exports. Given the above, the technical staff now expects fourth-quarter economic growth of 8.7%, with overall growth for 2021 of 9.9%. Several factors should continue to contribute to output recovery in 2022, though some of these may be less significant than previously forecast. International financial conditions are expected to be less favorable, though external demand should continue to recover and terms of trade continue to increase amid higher projected oil prices. Lower unemployment rates and subsequent positive effects on household income, despite increased inflation, would also boost output recovery, as would progress in the national vaccination campaign. The technical staff expects that the conditions that have favored recent high levels of consumption would be, in large part, transitory. Consumption spending is expected to grow at a slower rate in 2022. Gross fixed capital formation (GFCF) would continue to recover, approaching its pre-pandemic level, though at a slower rate than anticipated in the previous report. This would be due to lower observed GFCF levels and the potential impact of political and fiscal uncertainty. Meanwhile, the policy interest rate would be less expansionary as the process of monetary policy normalization continues. Given the above, growth in 2022 is forecast to decelerate to 4.3% (previously 4.7%). In 2023, that figure (3.1%) is projected to converge to levels closer to the potential growth rate. In this case, excess productive capacity would be expected to tighten at a similar rate as projected in the previous report. The trade deficit would tighten more than previously projected on the forecast horizon, due to expectations of an improved export dynamic and moderation in imports. The growth forecast for 2022 considers a low basis of comparison from the first half of 2021. However, there remain significant downside risks to this forecast. The current projection does not, for example, account for any additional effects on economic activity resulting from further waves of COVID-19. High private consumption levels, which have already surpassed pre-pandemic levels by a large margin, could be less dynamic than expected. And the normalization of monetary policy in the United States could come more quickly than projected in this report, which could negatively affect international financing costs. Finally, there remains a significant degree of uncertainty related to the duration of supply chocks and the degree to which macroeconomic and political conditions could negatively affect the recovery in investment. External demand for Colombian goods and services should continue to recover amid significant global inflation pressures, high oil prices, and less favorable international financial conditions than those estimated in October. Economic activity among Colombia’s major trade partners recovered in 2021 amid countries reopening and ample international liquidity. However, that growth has been somewhat restricted by global supply chain disruptions and new outbreaks of COVID-19. The technical staff has revised its growth forecast for Colombia’s main trade partners from 6.3% to 6.9% for 2021, and from 3.4% to 3.3% for 2022; trade partner economies are expected to grow 2.6% in 2023. Colombia’s annual terms of trade increased in 2021, largely on higher oil, coffee, and coal prices. This improvement came despite increased prices for goods and services imports. The expected oil price trajectory has been revised upward, partly to supply restrictions and lagging investment in the sector that would offset reduced growth forecasts in some major economies. Elevated freight and raw materials costs and supply chain disruptions continue to affect global goods production, and have led to increases in global prices. Coupled with the recovery in global demand, this has put upward pressure on external inflation. Several emerging market economies have continued to normalize monetary policy in this context. Meanwhile, in the United States, the Federal Reserve has anticipated an end to its asset buying program. U.S. inflation in December (7.0%) was again surprisingly high and market average inflation forecasts for 2022 have increased. The Fed is expected to increase its policy rate during the first quarter of 2022, with quarterly increases anticipated over the rest of the year. For its part, Colombia’s sovereign risk premium has increased and is forecast to remain on a higher path, to levels above the 15-year-average, on the forecast horizon. This would be partly due to the effects of a less expansionary monetary policy in the United States and the accumulation of macroeconomic imbalances in Colombia. Given the above, international financial conditions are projected to be less favorable than anticipated in the October report. The increase in Colombia’s external financing costs could be more significant if upward pressures on inflation in the United States persist and monetary policy is normalized more quickly than contemplated in this report. As detailed in Section 2.3, uncertainty surrounding international financial conditions continues to be unusually high. Along with other considerations, recent concerns over the potential effects of new COVID-19 variants, the persistence of global supply chain disruptions, energy crises in certain countries, growing geopolitical tensions, and a more significant deceleration in China are all factors underlying this uncertainty. The changing macroeconomic environment toward greater inflation and unanchoring risks on inflation expectations imply a reduction in the space available for monetary policy stimulus. Recovery in domestic demand and a reduction in excess productive capacity have come in line with the technical staff’s expectations from the October report. Some upside risks to inflation have materialized, while medium-term inflation expectations have increased and are above the 3% target. Monetary policy remains expansionary. Significant global inflationary pressures and the unexpected increase in the CPI in December point to more persistent effects from recent supply shocks. Core inflation is trending upward, but remains below the 3% target. Headline and core inflation projections have increased on the forecast horizon and are above the target rate through the end of 2023. Meanwhile, the expected dynamism of domestic demand would be in line with low levels of excess productive capacity. An accumulation of macroeconomic imbalances in Colombia and the increased likelihood of a faster normalization of monetary policy in the United States would put upward pressure on sovereign risk perceptions in a more persistent manner, with implications for the exchange rate and the natural rate of interest. Persistent disruptions to international supply chains, a high real increase in the legal minimum wage, and the indexation of various baskets in the CPI to higher inflation rates could affect price expectations and push inflation above the target more persistently. These factors suggest that the space to maintain monetary stimulus has continued to diminish, though monetary policy remains expansionary. 1.2 Monetary policy decision Banco de la República’s board of directors (BDBR) in its meetings in December 2021 and January 2022 voted to continue normalizing monetary policy. The BDBR voted by a majority in these two meetings to increase the benchmark interest rate by 50 and 100 basis points, respectively, bringing the policy rate to 4.0%.