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Unfortunately, it still seems that it’s not commonly understood that . But these are the foundation for being able to lean into informal leadership in software engineering— something that will be vital for progressing in your career. soft skills in software engineering are critical But what is informal leadership? Before we get there, let’s start with leadership itself: Our ability to influence and guide others. Let’s dive into what this leadership business is all about. Formal Leadership in Software Engineering In software engineering, among other professions, many often consider those in higher-up roles to be the “leaders”. The CEO, the VPs, the executives… Those are the leaders within the company. The perception of those in leadership extends to those in management roles as well. These are the individuals who are leading specific teams, after all, so it makes sense that they’re also leaders. What I’d like to draw your attention to is that the roles these individuals have by definition include some type of leadership expectation. But leadership itself is a trait that one can possess, and on its own is not a title. So for these individuals, we say that they are in “formal leadership” positions because as non-individual contributors (non-ICs), we expect that their position requires a demonstration of leadership to be effective. Informal Leadership in Software Engineering This brings us to informal leadership then, which is nearly identical but the expectation of this characteristic is not intrinsically built into the role being performed as an IC. Informal leadership in software engineering still looks like influencing and guiding. It means being able to help offer clarity and reduce chaos for your teammates and colleagues. Leveling up the more junior software engineers by helping them areas. Helping organize and drive progress on project deliverables… understand more complex The list goes on. If you’re a more junior software engineer, one of the best ways that I can draw attention to good informal leadership (generally speaking, not as a rule), is for you to think about the more senior software engineers that you love to work with. What kinds of characteristics do they have? What makes you enjoy working with them? How do they approach problem-solving with you? How do they make you feel when it comes to accomplishing your work? Odds are these more senior software engineers are making you feel safe, ensuring that you can progress in your work, both challenging and encouraging you, and inspiring you to do your best work. Why Informal Leadership Matters As we grow in our software engineering careers, a few things are almost certain: You’ll work on higher-impact areas The scope of the areas you work in will be wider The surface area of individuals and teams you work with will increase Your ability to deliver on these three facets will require you to be proficient at informal leadership. To work on the higher-impact areas, you’ll need to be working with people, guiding them to progress on these deliverables. And because the scope of impact is broader, you’ll need more people to be guiding to facilitate this! Both of these tie into the final point which is that your surface area of influence has increased now so effectively leading is critical. You rarely get promoted and advance in your software engineering career and THEN get told to increase your informal leadership. It’s often the reverse order. If you’d like to see growth in your role and your responsibilities, you’ll likely need to demonstrate these informal leadership traits first. What to Focus On For Informal Leadership Remember what I was saying earlier about emulating more senior software engineers that you look up to? Odds are, they’re doing things that encompass good informal leadership because you can feel it in your work with them. More specifically though, here are some things to focus on: Help more junior software engineers problem solve. Don’t give them answers or do work for them, but truly enable them to solve problems more effectively by helping them get unblocked. Drive clarity in the projects you work on with others. Who is working on which part? Which parts are blocked? Is everyone aware of the next priority? What about the risks? Create a . Software engineering involves some amount of risk, especially when we’re pushing boundaries. Make sure that you’re helping create an environment that’s not only safe to push technical boundaries but safe to ask questions without fearing you look stupid. safe place for other software engineers Strive for collaboration. Remove silos between teammates and between other teams. Actively listen and make sure others have a chance to share their perspective. Don’t just wait for a chance to interject. On the other hand, if you want to be great at informal leadership in software engineering: this video walks through a handful of things you should avoid https://www.youtube.com/watch?v=lB4qjyRYOzk&embedable=true Summarizing Informal Leadership in Software Engineering Informal leadership in software engineering is not only something that levels up an organization, it’s your key to leveling up your career. In this article, I explained that “higher up” positions from C-Level, through VPs, and down to front-line of them for formal leadership. However, we can have (and should have) informal leadership at the individual contributor (IC) level as well! managers have expectations Informal leadership in software engineering allows you to have a broader and greater impact in the areas that you’re involved in — including technical deliverables as well as a circle of influence. It’s likely not going to show up for you AFTER you become senior, but it’s much more likely that you’ll need to be skilling this up before you get promoted to senior. If you’re interested in more learning opportunities, and ! subscribe to my free weekly newsletter check out my YouTube channel Also published here.

Why Informal Leadership is Critical in Software Engineering

By , Educating leaders in health informatics since 1997. on . University of Texas School of Biomedical Informatics Originally published Quora With the growth in technology and increased patient data, medical informatics has become a vast field of study. As medical informatics progressed, researchers have identified core areas that will improve and have an impact on the current state of healthcare delivery. A few of these core research areas are listed below: Natural Language Processing Natural Language Processing (NLP) technology is designed to derive meaningful and actionable data from freely written text. NLP technology continues to be refined and provides countless benefits. Through effective NLP, health care professionals are doing more than saving time. They are communicating more effectively, identifying crucial data, and pushing the boundaries of health care forward. Precession Medicine The healthcare IT field, and world in general, is full of hot trends and buzzwords. One important and popular trend that is continuously gaining more traction is the practice of “precision medicine.” Advances in health care research and the industry as a whole have led to more providers and hospitals utilizing precision medicine treatments. Wearable technologies In 2016, we were no longer limited to a few options in terms of smart watches. Most first generation designs have since been replaced by second and third generation versions with many improvements. As the market for wearable devices grows, we will more than likely find more wearable smart devices clinging to our bodies. Augmented Reality Virtual and augmented reality (VR and AR, respectively) has been an area of intrigue for some time now. Initially it was little more than a novelty but, the technology is improving by leaps and bounds. Naturally, its practical applications have also including everything from improved physician training, to pain management. As this technology is further refined and more widely adopted, there is massive potential for truly revolutionized care. Apart from these areas, research work is being carried out in health mobile applications, artificial intelligence, health care big data analytics and large EHR technologies. By , Educating leaders in health informatics since 1997. on . University of Texas School of Biomedical Informatics Originally published Quora For more trending tech answers from , visit . Quora HackerNoon.com/Quora

The 4 Core Research Areas of Medical Informatics

How Mamba and Hyena Are Changing the Way AI Learns and Remembers

What Drives Inflation?

Author:
(1) David Staines. Table of Links Abstract 1 Introduction 2 Mathematical Arguments 3 Outline and Preview 4 Calvo Framework and 4.1 Household’s Problem 4.2 Preferences 4.3 Household Equilibrium Conditions 4.4 Price-Setting Problem 4.5 Nominal Equilibrium Conditions 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.8 Recursive Equilibrium 5 Existing Solutions 5.1 Singular Phillips Curve 5.2 Persistence and Policy Puzzles 5.3 Two Comparison Models 5.4 Lucas Critique 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6.2 Equilibrium Construction 6.3 Literature Comparison 6.4 Equilibrium Analysis 7 General Linearized Phillips Curve 7.1 Slope Coefficients 7.2 Error Coefficients 8 Existence Results and 8.1 Main Results 8.2 Key Proofs 8.3 Discussion 9 Bifurcation Analysis 9.1 Analytic Aspects 9.2 Algebraic Aspects (I) Singularities and Covers 9.3 Algebraic Aspects (II) Homology 9.4 Algebraic Aspects (III) Schemes 9.5 Wider Economic Interpretations 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10.2 Econometric Duality 10.3 Coefficient Properties 10.4 Microeconomic Interpretation 11 Policy Rule 12 Conclusions and References Appendices A Proof of Theorem 2 and A.1 Proof of Part (i) A.2 Behaviour of ∆ A.3 Proof Part (iii) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.3 Price Dispersion (4.5) B.4 Cost Minimization (4.6) and (10.4) B.5 Consolidation (4.8) C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C.2 Extending No Persistence D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D.2 Profits and Long-Run Growth E Slopes and Eigenvalues and E.1 Slope Coefficients E.2 Linearized DSGE Solution E.3 Eigenvalue Conditions E.4 Rouche’s Theorem Conditions F Abstract Algebra and F.1 Homology Groups F.2 Basic Categories F.3 De Rham Cohomology F.4 Marginal Costs and Inflation G Further Keynesian Models and G.1 Taylor Pricing G.2 Calvo Wage Phillips Curve G.3 Unconventional Policy Settings H Empirical Robustness and H.1 Parameter Selection H.2 Phillips Curve I Additional Evidence and I.1 Other Structural Parameters I.2 Lucas Critique I.3 Trend Inflation Volatility F.3 De Rham Cohomology This last mathematical stanza seeks to explain the power of the linear approximation in the most general setting, with a view to justifying major results in the text. This includes the linearization step from (3)-(5) and Decomposition 1, specific to Calvo, and the general bifurcation result Theorem 7. The first subsubsection sets out preliminaries of differential topology and cohomology. The second states the main result. The best textbook here is probably Tu [2011]; some readers may find the informal presentation in Stone and Goldbart [2009] helpful. F.3.1 Preliminaries denote the intersection of domains by and its map by the two chart maps to the two images It follows that the transition map between the two charts is the map between the two images of this intersection under the two chart maps. A Differential Form is a coordinate-independent approach to calculus. It has its roots in physics (see Stone and Goldbart [2009]). It differs from the approach to integration in the rest of the paper because it is orientation-dependent, so integrals can be zero or negative for non-zero functions. In general, for a differential form ω integrated over a manifold M and the same manifold but with opposite orientation M′ then A k form is an object that can be integrated over a k dimensional oriented manifold and is homogeneous of degree k in the coordinate differentials. A one form can be integrated over an oriented interval [a, b] in the domain of f: Similarly, the expression f(x, y, z) dx ∧ dy + g(x, y, z) dz ∧ dx + h(x, y, z) dy ∧ dz is a 2-form, that has the following integral over an oriented surface S likewise, the 3-form f(x, y, z) dx∧dy∧dz is called a volume element, integrated over an oriented region of three dimensional space. ∧ is called the wedge or Exterior Product. It is an alternating product, in the sense that The exterior derivative operator d sends k forms to k+1 forms, so perimeters go to areas, surface areas to enclosed volumes etc. [120] The generalized Stokes theorem states that the integral of a differential form ω over the boundary of some orientable manifold Ω equals the integral of its exterior derivative. This means that the volume of a sphere is equal to the surface area of the family of spheres covering the circle, as their diameters become arbitrarily small. In general, we can approximate the integral over k + 1- form arbitrarily well, by summing over the boundary of its k form. This constitutes a deep extension of the fundamental theorem of calculus. Lastly, returning to algebraic topology, it is necessary to introduce Cohomology in general, specifically the Singular Cohomology. Cohomology is the dual construction to homology. It arises from Cochains, which are functions defined on the group of chains in homology theory. Cohomology develops from the geometry of functions and their pullbacks. Taking two spaces X and Y and a function F on Y , any mapping f : X → Y will lead to another F ◦f called the pullback of F on X under f. Typically, there is interest in the relation between continuity and differentiability properties of the functions. Leading cohomology theories have a product called the Cup Product, which gives them a ring structure. For this reason cohomology can provide richer algebraic invariants. The singular cohomology is a powerful topological invariant, associating a graded-commutative ring to any topological space.121 Decomposition 1 exemplifies this structure. Every continuous map f : X → Y creates a homomorphism from the cohomology ring of Y to that of X; this puts strong restrictions on the behavior of maps between the two spaces. Many properties of cohomology are only minor variants of homology: • Two homotopic maps from X to Y induce the same homomorphism on cohomology (as they do on homology.) • If a space X is the union of open subsets U and V , then there is a long exact sequence Informally, these properties allow cohomology calculations to be broken down over suitable partitions. On the other hand, cohomology has a special structure absent in homology. For any topological space X and commutative ring R, there is a bilinear123 map called the Cup Product. F.3.2 Formalization and Application The De Rham Cohomology is a technique common to both algebraic and differential topology. It is a powerful tool capable of expressing basic topological information about smooth manifolds, making it easy to compute concrete representations of cohomology classes. It is a cohomology theory based on the behavior of differential forms. The key distinction is between exact and closed forms. Here, an Exact Form is is a differential form α, that is the exterior derivative of another differential form β, whilst a closed form is a differential form α whose exterior derivative is zero (dα = 0). Thus an exact form is an image of d and a closed form is a kernel of d. Every exact form is closed but a closed form need not be exact. [125] Poincaré’s lemma ensures it is true for contractible spaces. However, there is a critical connection between the existence of exactness and the existence of holes. From this standpoint, the De Rham cohomology measures how the Fundamental Theorem of Calculus [126] fails on general manifolds in higher dimensions. The De Rham complex is the cochain complex of differential forms on a smooth manifold M with the exterior as its differential: The exterior product endows the direct sum of groups with a ring structure. A further consequence of the theorem is that two cohomology rings are isomorphic, when viewed as graded rings, where the analagous product for the singular cohomology is the cup product. At last, we come to the application, the cohomology computation for our main model. Proposition 32. The Calvo Phillips curve model has the following De Rham cohomology representations In conjunction with Theorem 9, this formally justifies the use of the linear approximations (3)-(5) to represent the non-binding constraint on the representative firm or social planner. This paper is available on arxiv under CC 4.0 license. Author: (1) David Staines. Author: Author: (1) David Staines. Table of Links Abstract Abstract 1 Introduction 1 Introduction 2 Mathematical Arguments 2 Mathematical Arguments 3 Outline and Preview 3 Outline and Preview 4 Calvo Framework and 4.1 Household’s Problem 4 Calvo Framework and 4.1 Household’s Problem 4.2 Preferences 4.2 Preferences 4.3 Household Equilibrium Conditions 4.3 Household Equilibrium Conditions 4.4 Price-Setting Problem 4.4 Price-Setting Problem 4.5 Nominal Equilibrium Conditions 4.5 Nominal Equilibrium Conditions 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.8 Recursive Equilibrium 4.8 Recursive Equilibrium 5 Existing Solutions 5 Existing Solutions 5.1 Singular Phillips Curve 5.1 Singular Phillips Curve 5.2 Persistence and Policy Puzzles 5.2 Persistence and Policy Puzzles 5.3 Two Comparison Models 5.3 Two Comparison Models 5.4 Lucas Critique 5.4 Lucas Critique 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6.2 Equilibrium Construction 6.2 Equilibrium Construction 6.3 Literature Comparison 6.3 Literature Comparison 6.4 Equilibrium Analysis 6.4 Equilibrium Analysis 7 General Linearized Phillips Curve 7 General Linearized Phillips Curve 7.1 Slope Coefficients 7.1 Slope Coefficients 7.2 Error Coefficients 7.2 Error Coefficients 8 Existence Results and 8.1 Main Results 8 Existence Results and 8.1 Main Results 8.2 Key Proofs 8.2 Key Proofs 8.3 Discussion 8.3 Discussion 9 Bifurcation Analysis 9 Bifurcation Analysis 9.1 Analytic Aspects 9.1 Analytic Aspects 9.2 Algebraic Aspects (I) Singularities and Covers 9.2 Algebraic Aspects (I) Singularities and Covers 9.3 Algebraic Aspects (II) Homology 9.3 Algebraic Aspects (II) Homology 9.4 Algebraic Aspects (III) Schemes 9.4 Algebraic Aspects (III) Schemes 9.5 Wider Economic Interpretations 9.5 Wider Economic Interpretations 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10.2 Econometric Duality 10.2 Econometric Duality 10.3 Coefficient Properties 10.3 Coefficient Properties 10.4 Microeconomic Interpretation 10.4 Microeconomic Interpretation 11 Policy Rule 11 Policy Rule 12 Conclusions and References 12 Conclusions and References Appendices Appendices A Proof of Theorem 2 and A.1 Proof of Part (i) A Proof of Theorem 2 and A.1 Proof of Part (i) A.2 Behaviour of ∆ A.2 Behaviour of ∆ A.3 Proof Part (iii) A.3 Proof Part (iii) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.3 Price Dispersion (4.5) B.3 Price Dispersion (4.5) B.4 Cost Minimization (4.6) and (10.4) B.4 Cost Minimization (4.6) and (10.4) B.5 Consolidation (4.8) B.5 Consolidation (4.8) C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C.2 Extending No Persistence C.2 Extending No Persistence D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D.2 Profits and Long-Run Growth D.2 Profits and Long-Run Growth E Slopes and Eigenvalues and E.1 Slope Coefficients E Slopes and Eigenvalues and E.1 Slope Coefficients E.2 Linearized DSGE Solution E.2 Linearized DSGE Solution E.3 Eigenvalue Conditions E.3 Eigenvalue Conditions E.4 Rouche’s Theorem Conditions E.4 Rouche’s Theorem Conditions F Abstract Algebra and F.1 Homology Groups F Abstract Algebra and F.1 Homology Groups F.2 Basic Categories F.2 Basic Categories F.3 De Rham Cohomology F.3 De Rham Cohomology F.4 Marginal Costs and Inflation F.4 Marginal Costs and Inflation G Further Keynesian Models and G.1 Taylor Pricing G Further Keynesian Models and G.1 Taylor Pricing G.2 Calvo Wage Phillips Curve G.2 Calvo Wage Phillips Curve G.3 Unconventional Policy Settings G.3 Unconventional Policy Settings H Empirical Robustness and H.1 Parameter Selection H Empirical Robustness and H.1 Parameter Selection H.2 Phillips Curve H.2 Phillips Curve I Additional Evidence and I.1 Other Structural Parameters I Additional Evidence and I.1 Other Structural Parameters I.2 Lucas Critique I.2 Lucas Critique I.3 Trend Inflation Volatility I.3 Trend Inflation Volatility F.3 De Rham Cohomology This last mathematical stanza seeks to explain the power of the linear approximation in the most general setting, with a view to justifying major results in the text. This includes the linearization step from (3)-(5) and Decomposition 1, specific to Calvo, and the general bifurcation result Theorem 7. The first subsubsection sets out preliminaries of differential topology and cohomology. The second states the main result. The best textbook here is probably Tu [2011]; some readers may find the informal presentation in Stone and Goldbart [2009] helpful. F.3.1 Preliminaries F.3.1 Preliminaries denote the intersection of domains by and its map by the two chart maps to the two images It follows that the transition map between the two charts is the map between the two images of this intersection under the two chart maps. A Differential Form is a coordinate-independent approach to calculus. It has its roots in physics (see Stone and Goldbart [2009]). It differs from the approach to integration in the rest of the paper because it is orientation-dependent, so integrals can be zero or negative for non-zero functions. In general, for a differential form ω integrated over a manifold M and the same manifold but with opposite orientation M′ then Differential Form A k form is an object that can be integrated over a k dimensional oriented manifold and is homogeneous of degree k in the coordinate differentials. A one form can be integrated over an oriented interval [a, b] in the domain of f: Similarly, the expression f(x, y, z) dx ∧ dy + g(x, y, z) dz ∧ dx + h(x, y, z) dy ∧ dz is a 2-form, that has the following integral over an oriented surface S likewise, the 3-form f(x, y, z) dx∧dy∧dz is called a volume element, integrated over an oriented region of three dimensional space. ∧ is called the wedge or Exterior Product . It is an alternating product, in the sense that Exterior Product The exterior derivative operator d sends k forms to k+1 forms, so perimeters go to areas, surface areas to enclosed volumes etc. [120] The generalized Stokes theorem states that the integral of a differential form ω over the boundary of some orientable manifold Ω equals the integral of its exterior derivative. This means that the volume of a sphere is equal to the surface area of the family of spheres covering the circle, as their diameters become arbitrarily small. In general, we can approximate the integral over k + 1- form arbitrarily well, by summing over the boundary of its k form. This constitutes a deep extension of the fundamental theorem of calculus. Lastly, returning to algebraic topology, it is necessary to introduce Cohomology in general, specifically the Singular Cohomology . Cohomology is the dual construction to homology. It arises from Cochains , which are functions defined on the group of chains in homology theory. Cohomology Singular Cohomology Cochains Cohomology develops from the geometry of functions and their pullbacks. Taking two spaces X and Y and a function F on Y , any mapping f : X → Y will lead to another F ◦f called the pullback of F on X under f. Typically, there is interest in the relation between continuity and differentiability properties of the functions. Leading cohomology theories have a product called the Cup Product, which gives them a ring structure. For this reason cohomology can provide richer algebraic invariants. The singular cohomology is a powerful topological invariant, associating a graded-commutative ring to any topological space.121 Decomposition 1 exemplifies this structure. Every continuous map f : X → Y creates a homomorphism from the cohomology ring of Y to that of X; this puts strong restrictions on the behavior of maps between the two spaces. Many properties of cohomology are only minor variants of homology: • Two homotopic maps from X to Y induce the same homomorphism on cohomology (as they do on homology.) • If a space X is the union of open subsets U and V , then there is a long exact sequence Informally, these properties allow cohomology calculations to be broken down over suitable partitions. On the other hand, cohomology has a special structure absent in homology. For any topological space X and commutative ring R, there is a bilinear123 map called the Cup Product. F.3.2 Formalization and Application F.3.2 Formalization and Application The De Rham Cohomology is a technique common to both algebraic and differential topology. It is a powerful tool capable of expressing basic topological information about smooth manifolds, making it easy to compute concrete representations of cohomology classes. It is a cohomology theory based on the behavior of differential forms. The key distinction is between exact and closed forms. Here, an Exact Form is is a differential form α, that is the exterior derivative of another differential form β, whilst a closed form is a differential form α whose exterior derivative is zero (dα = 0). Thus an exact form is an image of d and a closed form is a kernel of d. Every exact form is closed but a closed form need not be exact. [125] Poincaré’s lemma ensures it is true for contractible spaces. However, there is a critical connection between the existence of exactness and the existence of holes. From this standpoint, the De Rham cohomology measures how the Fundamental Theorem of Calculus [126] fails on general manifolds in higher dimensions. The De Rham complex is the cochain complex of differential forms on a smooth manifold M with the exterior as its differential: The exterior product endows the direct sum of groups with a ring structure. A further consequence of the theorem is that two cohomology rings are isomorphic, when viewed as graded rings, where the analagous product for the singular cohomology is the cup product. At last, we come to the application, the cohomology computation for our main model. Proposition 32. The Calvo Phillips curve model has the following De Rham cohomology representations In conjunction with Theorem 9, this formally justifies the use of the linear approximations (3)-(5) to represent the non-binding constraint on the representative firm or social planner. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

A Simple Guide to De Rham Cohomology and Its Mathematical Applications

What is Category Theory?

Author:
(1) David Staines. Table of Links Abstract 1 Introduction 2 Mathematical Arguments 3 Outline and Preview 4 Calvo Framework and 4.1 Household’s Problem 4.2 Preferences 4.3 Household Equilibrium Conditions 4.4 Price-Setting Problem 4.5 Nominal Equilibrium Conditions 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.8 Recursive Equilibrium 5 Existing Solutions 5.1 Singular Phillips Curve 5.2 Persistence and Policy Puzzles 5.3 Two Comparison Models 5.4 Lucas Critique 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6.2 Equilibrium Construction 6.3 Literature Comparison 6.4 Equilibrium Analysis 7 General Linearized Phillips Curve 7.1 Slope Coefficients 7.2 Error Coefficients 8 Existence Results and 8.1 Main Results 8.2 Key Proofs 8.3 Discussion 9 Bifurcation Analysis 9.1 Analytic Aspects 9.2 Algebraic Aspects (I) Singularities and Covers 9.3 Algebraic Aspects (II) Homology 9.4 Algebraic Aspects (III) Schemes 9.5 Wider Economic Interpretations 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10.2 Econometric Duality 10.3 Coefficient Properties 10.4 Microeconomic Interpretation 11 Policy Rule 12 Conclusions and References Appendices A Proof of Theorem 2 and A.1 Proof of Part (i) A.2 Behaviour of ∆ A.3 Proof Part (iii) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.3 Price Dispersion (4.5) B.4 Cost Minimization (4.6) and (10.4) B.5 Consolidation (4.8) C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C.2 Extending No Persistence D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D.2 Profits and Long-Run Growth E Slopes and Eigenvalues and E.1 Slope Coefficients E.2 Linearized DSGE Solution E.3 Eigenvalue Conditions E.4 Rouche’s Theorem Conditions F Abstract Algebra and F.1 Homology Groups F.2 Basic Categories F.3 De Rham Cohomology F.4 Marginal Costs and Inflation G Further Keynesian Models and G.1 Taylor Pricing G.2 Calvo Wage Phillips Curve G.3 Unconventional Policy Settings H Empirical Robustness and H.1 Parameter Selection H.2 Phillips Curve I Additional Evidence and I.1 Other Structural Parameters I.2 Lucas Critique I.3 Trend Inflation Volatility F Abstract Algebra This section supplies an introductory primer on homological algebra, De Rham cohomology and category theory, complimentary to the informal discussion in the text. Each occupies a subsection. At the end, a short part is devoted to additional details of a calculation from Proposition 20, which applies the methods here to the ultimate goal of the paper. F.1 Homology Groups This part develops the theory of singular homology to support the claims and analysis in the main text. Other popular homology constructions are discussed in Jacobson [1980]. Informally, the singular homology is built by a succession of mappings between n dimensional blocks and the paths around their boundaries. F.1.1 Singular Simplex its vertices are the k standard unit vectors and the origin. [112] Familiar examples include: F.1.2 Singular Chain Complex F.1.3 Homotopy Invariance for all n ≥ 0. This means that homology groups are topologically invariants. In particular, if X is a connected contractible space, then all its homology groups are 0 except I finish with a sketch of the proof of homotopy invariance of singular homology. A continuous map f : X → Y induces a homomorphism It is easy to see that where f♯ is a chain map which descends to homomorphisms on homology It is left to show that, if f and g are homotopically equivalent, then f∗ = g∗. This implies that if f is a homotopy equivalence then f∗ is an isomorphism. Let F : X × [0, 1] → Y be a homotopy, that takes f to g on the level of chains. It defines a homomorphism Therefore, they are homologous, which settles the claim. This paper is available on arxiv under CC 4.0 license. Author: (1) David Staines. Author: Author: (1) David Staines. Table of Links Abstract Abstract 1 Introduction 1 Introduction 2 Mathematical Arguments 2 Mathematical Arguments 3 Outline and Preview 3 Outline and Preview 4 Calvo Framework and 4.1 Household’s Problem 4 Calvo Framework and 4.1 Household’s Problem 4.2 Preferences 4.2 Preferences 4.3 Household Equilibrium Conditions 4.3 Household Equilibrium Conditions 4.4 Price-Setting Problem 4.4 Price-Setting Problem 4.5 Nominal Equilibrium Conditions 4.5 Nominal Equilibrium Conditions 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.8 Recursive Equilibrium 4.8 Recursive Equilibrium 5 Existing Solutions 5 Existing Solutions 5.1 Singular Phillips Curve 5.1 Singular Phillips Curve 5.2 Persistence and Policy Puzzles 5.2 Persistence and Policy Puzzles 5.3 Two Comparison Models 5.3 Two Comparison Models 5.4 Lucas Critique 5.4 Lucas Critique 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6.2 Equilibrium Construction 6.2 Equilibrium Construction 6.3 Literature Comparison 6.3 Literature Comparison 6.4 Equilibrium Analysis 6.4 Equilibrium Analysis 7 General Linearized Phillips Curve 7 General Linearized Phillips Curve 7.1 Slope Coefficients 7.1 Slope Coefficients 7.2 Error Coefficients 7.2 Error Coefficients 8 Existence Results and 8.1 Main Results 8 Existence Results and 8.1 Main Results 8.2 Key Proofs 8.2 Key Proofs 8.3 Discussion 8.3 Discussion 9 Bifurcation Analysis 9 Bifurcation Analysis 9.1 Analytic Aspects 9.1 Analytic Aspects 9.2 Algebraic Aspects (I) Singularities and Covers 9.2 Algebraic Aspects (I) Singularities and Covers 9.3 Algebraic Aspects (II) Homology 9.3 Algebraic Aspects (II) Homology 9.4 Algebraic Aspects (III) Schemes 9.4 Algebraic Aspects (III) Schemes 9.5 Wider Economic Interpretations 9.5 Wider Economic Interpretations 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10.2 Econometric Duality 10.2 Econometric Duality 10.3 Coefficient Properties 10.3 Coefficient Properties 10.4 Microeconomic Interpretation 10.4 Microeconomic Interpretation 11 Policy Rule 11 Policy Rule 12 Conclusions and References 12 Conclusions and References Appendices Appendices A Proof of Theorem 2 and A.1 Proof of Part (i) A Proof of Theorem 2 and A.1 Proof of Part (i) A.2 Behaviour of ∆ A.2 Behaviour of ∆ A.3 Proof Part (iii) A.3 Proof Part (iii) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.3 Price Dispersion (4.5) B.3 Price Dispersion (4.5) B.4 Cost Minimization (4.6) and (10.4) B.4 Cost Minimization (4.6) and (10.4) B.5 Consolidation (4.8) B.5 Consolidation (4.8) C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C.2 Extending No Persistence C.2 Extending No Persistence D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D.2 Profits and Long-Run Growth D.2 Profits and Long-Run Growth E Slopes and Eigenvalues and E.1 Slope Coefficients E Slopes and Eigenvalues and E.1 Slope Coefficients E.2 Linearized DSGE Solution E.2 Linearized DSGE Solution E.3 Eigenvalue Conditions E.3 Eigenvalue Conditions E.4 Rouche’s Theorem Conditions E.4 Rouche’s Theorem Conditions F Abstract Algebra and F.1 Homology Groups F Abstract Algebra and F.1 Homology Groups F.2 Basic Categories F.2 Basic Categories F.3 De Rham Cohomology F.3 De Rham Cohomology F.4 Marginal Costs and Inflation F.4 Marginal Costs and Inflation G Further Keynesian Models and G.1 Taylor Pricing G Further Keynesian Models and G.1 Taylor Pricing G.2 Calvo Wage Phillips Curve G.2 Calvo Wage Phillips Curve G.3 Unconventional Policy Settings G.3 Unconventional Policy Settings H Empirical Robustness and H.1 Parameter Selection H Empirical Robustness and H.1 Parameter Selection H.2 Phillips Curve H.2 Phillips Curve I Additional Evidence and I.1 Other Structural Parameters I Additional Evidence and I.1 Other Structural Parameters I.2 Lucas Critique I.2 Lucas Critique I.3 Trend Inflation Volatility I.3 Trend Inflation Volatility F Abstract Algebra This section supplies an introductory primer on homological algebra, De Rham cohomology and category theory, complimentary to the informal discussion in the text. Each occupies a subsection. At the end, a short part is devoted to additional details of a calculation from Proposition 20, which applies the methods here to the ultimate goal of the paper. F.1 Homology Groups This part develops the theory of singular homology to support the claims and analysis in the main text. Other popular homology constructions are discussed in Jacobson [1980]. Informally, the singular homology is built by a succession of mappings between n dimensional blocks and the paths around their boundaries. F.1.1 Singular Simplex F.1.1 Singular Simplex its vertices are the k standard unit vectors and the origin. [112] Familiar examples include: F.1.2 Singular Chain Complex F.1.2 Singular Chain Complex F.1.3 Homotopy Invariance F.1.3 Homotopy Invariance for all n ≥ 0. This means that homology groups are topologically invariants. In particular, if X is a connected contractible space, then all its homology groups are 0 except I finish with a sketch of the proof of homotopy invariance of singular homology. A continuous map f : X → Y induces a homomorphism It is easy to see that where f♯ is a chain map which descends to homomorphisms on homology It is left to show that, if f and g are homotopically equivalent, then f∗ = g∗. This implies that if f is a homotopy equivalence then f∗ is an isomorphism. Let F : X × [0, 1] → Y be a homotopy, that takes f to g on the level of chains. It defines a homomorphism Therefore, they are homologous, which settles the claim. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

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