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We generalize valuations on polyhedral cones to valuations on (plane) fans. For fans induced by hyperplane arrangements, we show a correspondence between rotation-invariant valuations and deletion– restriction invariants. In particular, we define a characteristic polynomial for fans in terms of spherical intrinsic volumes and show that it coincides with the usual characteristic polynomial in the case of hyperplane arrangements. This gives a simple deletion–restriction proof of a result of Klivans–Swartz. The metric projection of a cone is a piecewise-linear map, whose underlying fan prompts a generalization of spherical intrinsic volumes to indicator functions. We show that these intrinsic indicators yield valuations that separate polyhedral cones. Applied to hyperplane arrangements, this generalizes a result of Kabluchko on projection volumes. [ABSTRACT FROM AUTHOR]

The mod-2 arithmetic Milnor invariants, introduced by Morishita, provide a decomposition law for primes in canonical Galois extensions of Q with unitriangular Galois groups and contain the Legendre and Rédei symbols as special cases. Morishita further proposed a notion of mod-q arithmetic Milnor invariants, where q is a prime power, for number fields containing the qth roots of unity and satisfying certain class field theory assumptions. We extend this theory from the number field context to general fields, by introducing a notion of a linking invariant for discrete valuations and orderings. We further express it as a Magnus homomorphism coefficient and relate it to Massey product elements in Galois cohomology. [ABSTRACT FROM AUTHOR]

Agroecosystems are socioecosystems where people perceive, value, and benefit from the goods and services provided by nature, while also modifying it according to their preferences and decisions. Due to intensive management, functions, services, and biodiversity have been altered and lost, making the ecological restoration of these landscapes a current priority. To recognize the perceptions and valuations of ecosystem services associated with native forests, and their relationship with the restoration preferences of producers in the Espinal Region, southeast of Córdoba, tools derived from ethnobiological approaches (in-depth interviews with key actors, semi-structured surveys, free listings, Likert scales, and projective tests) were used. Models were created to evaluate the relationship between sociodemographic and productive variables, valuations, and restoration preferences. The regulatory and support services associated with native forests were the most mentioned and valued. Provisioning services were the least valued, particularly by older producers with patches of forest already present on their fields. Agricultural producers preferred to restore areas with low-quality soils and areas with temporary or permanent flooding. Those producers with forest fragments on their fields chose to enrich them, while the rest preferred the implementation of forest buffers. For this purpose, they selected native plant species from the region, which they associated with forest patches and which were also recommended by experts. Including valuations assigned to native forests, age, and the presence of patches owned by the producers being studied, along with economic, contractual, and field characteristics (soil type, presence of flooding), will allow the design of restoration actions adapted to the specific contexts of each producer and, therefore, ensure their success. [ABSTRACT FROM AUTHOR]

For positive integers n ≥ m , let P n , m (x) : = ∑ j = 0 m n j x j = n 0 + n 1 x + ... + n m x m be the truncated binomial expansion of (1 + x) n consisting of all terms of degree ≤ m. It is conjectured that for n > m + 1 , the polynomial P n , m (x) is irreducible. We confirm this conjecture when 2 m ≤ n < (m + 1) 1 0. Also we show for any r ≥ 1 0 and 2 m ≤ n < (m + 1) r + 1 , the polynomial P n , m (x) is irreducible when m ≥ max { 1 0 6 , 2 r 3 }. Under the explicit abc-conjecture, for a fixed m , we give an explicit n 0 , n 1 depending only on m such that ∀ n ≥ n 0 , the polynomial P n , m (x) is irreducible. Further ∀ n ≥ n 1 , the Galois group associated to P n , m (x) is the symmetric group S m. [ABSTRACT FROM AUTHOR]

We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a consequence, we prove that the volume of a valuation is log convex on simplices of quasi-monomial valuations and give a new proof of a theorem of Xu and Zhuang on the uniqueness of normalized volume minimizers. In another direction, we generalize a theorem of Rees on multiplicities of ideals to filtrations and characterize when the Minkowski inequality for filtrations is an equality under mild assumptions. [ABSTRACT FROM AUTHOR]

We conduct a field experiment on conflict in swimming pools. When all lanes are occupied, an actor joins the least crowded lane and asks one of the swimmers to move to another lane. The lane represents a contested scarce resource. We vary the actor's valuation (high and low) for the good through the message they deliver. Also, we take advantage of the natural variation in the number of swimmers to proxy for their valuation. Consistent with theoretical predictions, a swimmer's propensity to engage in conflict increases in scarcity and decreases in the actor's valuation. We complement the results with survey evidence. [ABSTRACT FROM AUTHOR]

General partners (GPs) of private equity (PE) are facing increasing pressure from limited partners (LPs) to make investments that meet environmental, social, and governance (ESG) targets. We develop a tractable model to analyze an LP's valuation of participating in PE investment when the ESG factors are objectives. Integrating the ESG factors makes the LP commit to investing capital in ESG demand spending and choosing sustainable PE, which has an inverted U‐shaped effect on the LP's certainty‐equivalent wealth. At the optimal ESG demand spending, the LP tends to choose more consumption but less public equity investment due to his or her positive expectation about higher certainty‐equivalent wealth from PE investment. However, an endogenous risk‐aversion attitude and exogenous idiosyncratic risk will have an adverse effect and make the LP less eager to integrate the ESG factors for sustainable development. Thus, the optimal ESG demand decreases, and the LP requires a higher break‐even excess return. Our analysis provides new theoretical insights for both PE partners and regulators when integrating ESG factors and sustainable concepts into PE valuation and management. [ABSTRACT FROM AUTHOR]

We extend and prove a conjecture of Benguş–Lasnier on the parametrization of valuations on a polynomial ring by certain spaces of diskoids. [ABSTRACT FROM AUTHOR]

Copyright of Palobra is the property of Universidad de Cartagena and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Given a generically finite local extension of valuation rings V⊂W$V \subset W$, the question of whether W is the localization of a finitely generated V‐algebra is significant for approaches to the problem of local uniformization of valuations using ramification theory. Hagen Knaf proposed a characterization of when W is essentially of finite type over V in terms of classical invariants of the extension of associated valuations. Knaf's conjecture has been verified in important special cases by Cutkosky and Novacoski using local uniformization of Abhyankar valuations and resolution of singularities of excellent surfaces in arbitrary characteristic, and by Cutkosky for valuation rings of function fields of characteristic 0 using embedded resolution of singularities. In this paper, we prove Knaf's conjecture in full generality. [ABSTRACT FROM AUTHOR]

Contemporary reflection on the concept of value oscillates between positions that advocate absolutist realism and positions that propose total relativism. The present paper aims to distance itself from these two ways of reading the topic, rejecting any monolithic conception, and proposing a stratified theory of value. Beginning with an analysis of values that differentiates them from both "goods" and "valuations," the author outlines an understanding of value that must be investigated in its multiple, interconnected layers. The stratification starts from the role of the emotional sphere and comes down to a formal and absolute conception of the concept. The goal of the paper is to outline the different levels of validity of each proposed layer, which must be recognized and differentiated, with the aim of also capturing the difficulties and ambiguities that characterize the transition from one level to another. [ABSTRACT FROM AUTHOR]

Copyright of Revista de Derecho Penal y Criminologia is the property of Editorial UNED and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper characterizes those preferences over bounded infinite utility streams that satisfy the time value of money principle and an additivity property, and the subset of these preferences that, in addition, are either impatient or patient. Based on this characterization, the paper introduces a concept of optimization that is robust to a small imprecision in the specification of the preference, and proves that the set of feasible streams of payoffs of a finite Markov decision process admits such a robust optimization. [ABSTRACT FROM AUTHOR]

The acquisition of the competences of a subject established in the study plan encourages the search for new methods that improve the teaching-learning process, especially in the qualifications that enable the exercise of a profession with legal attributions. Through this text, the results of the application of the peer evaluation system and the co-evaluation of the subject Valuations of Real Estate and Expert reports of the Degree in Building Engineering are explained throughout 5 academic years, allowing to conclude the benefits on the acquisition of skills and improvement of success rates, as well as the implementation of professional powers. [ABSTRACT FROM AUTHOR]

A version of the Hodge–Riemann relations for valuations was recently conjectured and proved in several special cases by [J. Kotrbatý, On Hodge–Riemann relations for translation-invariant valuations, preprint (2020), arXiv:2009.00310]. The Lefschetz operator considered there arises as either the product or the convolution with the mixed volume of several Euclidean balls. Here we prove that in (co-)degree one, the Hodge–Riemann relations persist if the balls are replaced by several different (centrally symmetric) convex bodies with smooth boundary with positive Gauss curvature. While these mixed Hodge–Riemann relations for the convolution directly imply the Aleksandrov–Fenchel inequality, they yield for the dual operation of the product a new inequality. This new inequality strengthens classical consequences of the Aleksandrov–Fenchel inequality for lower-dimensional convex bodies and generalizes some of the geometric inequalities recently discovered by [S. Alesker, Kotrbatý's theorem on valuations and geometric inequalities for convex bodies, preprint (2020), arXiv:2010.01859]. [ABSTRACT FROM AUTHOR]

For integers n , s , b 0 , ... , b n with n ≥ 3 , s ≥ 0 , | b 0 | = | b n | = 1 , let G 1 (x) = G 1 (x , n , s) ≔ n ! ∑ j = 0 n b j (j !) − 1 n + s − j n − j x j . For n ≥ 0 and 0 ≤ s ≤ 92 it is proved in Shorey and Sinha (2022) that, except for finitely many pairs (n , s) , G 1 (x) = G 1 (x , n , s) is either irreducible or linear factor times an irreducible polynomial. If s ≤ 30 , we determine here explicitly the set of pairs (n , s) in the above assertion. This implies a new proof of the result of Nair and Shorey (2015) that G 1 (x) is irreducible for s ≤ 22. [ABSTRACT FROM AUTHOR]

Alesker has proved the existence of a remarkable isomorphism of the space of translation-invariant smooth valuations that has the same functorial properties as the classical Fourier transform. In this paper, we show how to directly describe this isomorphism in terms of the Fourier transform on functions. As a consequence, we obtain simple proofs of the main properties of the Alesker–Fourier transform. One of these properties was previously only conjectured by Alesker and is proved here for the first time. [ABSTRACT FROM AUTHOR]

• Sociomaterialities of extractivism rush and resist the making of resource frontiers. • Communities resist coercive extractivism to defend socionatural multiplicities. • Activism at times disrupts extractivism and at others altogether unmake resources. • Resistance we unearth in Maidan is a process of unmaking gold as a resource. • Maidan's struggle is an institutional experimentation for shaping just socionatures. In October 2013, around 200 protestors from the rural settlement Maidan in Kyrgyzstan clashed violently with the representatives of the exploration company as they brought in the first excavator to construct the mining infrastructure for the 'Shambesai' gold deposit. This paper is an attempt to understand the processes and practices that have led to this escalation and that continue to sustain Maidan's rejection of the gold mine to date. Motivated by state and corporate assertions that attribute such actions primarily to material interests, we engage this resistance to gold extractivism in sociomaterial terms trying to understand more deeply the dynamics of ordinary citizens' activism. Based on multi-stage interdisciplinary research, we trace and reconstruct the socionatural conditions and practices that have culminated in Maidan's decade-long struggle to unmake gold as a resource on their territory. Focusing on resource materialities, their valuations and governance, we present an historico-geographical analysis of making and unmaking of a resource frontier. Against the backdrop of the extractive order that has prevailed in Kyrgyzstan over the last three decades, we understand Maidan's struggle to be a form of situated institutional experimentation for shaping meaningful and just more-than-human socionatures. [ABSTRACT FROM AUTHOR]

This paper assesses the impact of the COVID-19 pandemic on non-financial firms' valuations in the European Union (EU) using a stress testing approach. Notably, the paper investigates the extent to which the COVID-19 may deteriorate non-financial firms' value in the ten EU countries to provide a robust anchor to policymakers in formulating strategic government interventions. We employ a sample of 5342 listed non-financial firms across the selected member states that have consistent analyst coverage from 2010 to 2019. First, we estimate the input sensitivities of free cash flow and residual income models using a random effect panel employed to in-sample data. Second, based on these sensitivities, we compute the model-driven ex-post valuations and compare their robustness with actual price and analyst forecasts for the same period. Finally, we introduce multiple stress scenarios that may emanate from COVID-19, i.e., a decline in expected sales and an increase/decrease in equity cost. Our findings show a significant loss in valuations across all sectors due to a possible reduction in sales and an increase in equity cost. In extreme cases, average firms in some industries may lose up to 60% of their intrinsic value in one year. The results remained consistent regardless of the cash flow or residual income-driven valuation.

In this paper, we carry out in an abstract order context some real subset combinatorial problems. Specifically, let (X , ≤ , c) be a finite poset, where c : X → X is an order-reversing and involutive map such that c (x) ≠ x for each x ∈ X. Let B 2 = { N < P } be the Boolean lattice with two elements and + (X , B 2) the family of all the order-preserving 2-valued maps A : X → B 2 such that A (c (x)) = P if A (x) = N for all x ∈ X. In this paper, we build a family ℬ w + (X) of particular subsets of X , that we call w + -bases on X , and we determine a bijection between the family ℬ w + (X) and the family + (X , B 2). In such a bijection, a w + -basis Ω on X corresponds to a map A ∈ + (X , B 2) whose restriction of A to Ω is the smallest 2-valued partial map on X which has A as its unique extension in + (X , B 2). Next we show how each w + -basis on X becomes, in a particular context, a sub-system of a larger system of linear inequalities, whose compatibility implies the compatibility of the whole system. [ABSTRACT FROM AUTHOR]

This article is concerned with the constants that appear in Harish-Chandra's character formula for stable discrete series of real reductive groups, although it does not require any knowledge about real reductive groups or discrete series. In Harish-Chandra's work the only information we have about these constants is that they are uniquely determined by an inductive property. Later, Goresky-Kottwitz-MacPherson (1997) and Herb (2000) gave different formulas for these constants. In this article, we generalize these formulas to the case of arbitrary finite Coxeter groups (in this setting, discrete series no longer make sense), and give a direct proof that the two formulas agree. We actually prove a slightly more general identity that also implies the combinatorial identity underlying the discrete series character identities of Morel (2011). We deduce this identity from a general abstract theorem giving a way to calculate the alternating sum of the values of a valuation on the chambers of a Coxeter arrangement. We also introduce a ring structure on the set of valuations on polyhedral cones in Euclidean space with values in a fixed ring. This gives a theoretical framework for the valuation appearing in Goresky-Kottwitz-MacPherson's 1997 paper. In an appendix, we extend Herb's notion of 2-structures to pseudo-root systems. [ABSTRACT FROM AUTHOR]

In this article we introduce a family of valuative invariants defined in terms of the p-th moment of the expected vanishing order. These invariants lie between α and δ -invariants. They vary continuously in the big cone and semi-continuously in families. Most importantly, they give sufficient conditions for K-stability of Fano varieties, which generalizes the α and δ -criterions in the literature. They are also related to the d p -geometry of maximal geodesic rays. [ABSTRACT FROM AUTHOR]

Given a minimal model of an elliptic curve, E / K , over a finite extension, K , of Q p for any rational prime, p , and any point P ∈ E (K) of infinite order, we determine precisely min ⁡ (v (ϕ n (P)) , v (ψ n 2 (P))) , where v is a normalised valuation on K and ϕ n (P) and ψ n (P) are polynomials arising from multiplication by n on this model of the curve. [ABSTRACT FROM AUTHOR]

We consider three monads on $\mathsf{Top}$ , the category of topological spaces, which formalize topological aspects of probability and possibility in categorical terms. The first one is the Hoare hyperspace monad H, which assigns to every space its space of closed subsets equipped with the lower Vietoris topology. The second one is the monad V of continuous valuations, also known as the extended probabilistic powerdomain. We construct both monads in a unified way in terms of double dualization. This reveals a close analogy between them and allows us to prove that the operation of taking the support of a continuous valuation is a morphism of monads $V \to H$. In particular, this implies that every H-algebra (topological complete semilattice) is also a V-algebra. We show that V can be restricted to a submonad of $\tau$ -smooth probability measures on $\mathsf{Top}$. By composing these morphisms of monads, we obtain that taking the supports of $\tau$ -smooth probability measures is also a morphism of monads. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to characterize limit key polynomials for a valuation ν on K [ x ]. We consider the set Ψ α of key polynomials for ν of degree α. We set p to be the exponent characteristic of ν. Our first main result (Theorem 1.1) is that if F is a limit key polynomial for Ψ α , then the degree of F is p r α for some r ∈ N. Moreover, in Theorem 1.2, we show that there exist Q ∈ Ψ α and F a limit key polynomial for Ψ α , such that the Q -expansion of F only has terms which are powers of p. [ABSTRACT FROM AUTHOR]

The emotional reactions aroused by the fire that partly destroyed Notre-Dame de Paris in April 2019 can be analyzed as “valuations” in the light of the pragmatic sociology of values, since they provide empirically grounded material allowing for the description and modeling of the actual implementations and effects of valuations. After a quick summary of the recent history of the pragmatic turn in sociology as related to the sociology of valuation, and a short reflection on the relationship between emotions and values, the fire of Notre-Dame de Paris is used as a case study in the light of “axiological sociology”, a model built on value judgments observed in various contexts, including the display of emotions. This article intends to demonstrate both empirically and theoretically how important it is for the social sciences to consider values as an autonomous issue, deserving to be treated as “axiological facts”, as any other kind of social fact.

We prove for a large class of fields F that every proper finite extension of F p y t h , the pythagorean closure of F , is not a pythagorean field. This class of fields contains number fields and fields F that are finitely generated of transcendence degree at least one over some subfield of F. [ABSTRACT FROM AUTHOR]

Copyright of Revista del Museo de Antropología is the property of Museo de Antropologia - IDACOR and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this article, it will be argued that subjective assumptions play a prominent part in the way valuations are conducted and investment decisions are made by financial managers (FMs) from the perspective of agency theory. The problem is that there is a general absence of rules of compliance for financial management. Quantitative calculations are based on theories, models and accepted practices that guide FMs in their decision-making process. The selection criteria that inform these theories, models and practices rely on qualitative assumptions, which are informed by the presuppositions of the FM. The problem is that this understanding framework has traditionally not been assessed because of the assumption that the theories, models and practices are value neutral. Value neutrality generally implies that calculations, processes and projections are based on factual information and rational procedures that are objective. The scrutiny of the assumptions involved in financial management assumes that the individual has philosophical sensitivity. However, this is not traditionally part of the expected competencies of finance professionals. Philosophical competency means that the FM is capable of evaluating decision-making processes in order not to fall into the trap of circular logic or solipsism as highlighted by Gadamer. Maurice Merleau-Ponty identified that the limitations of such an understanding of reality result in a failure of responsibility and accountability. From the philosophy of Merleau-Ponty, there are three principles for philosophical analysis, namely, dialogue, accountability and context. These principles are applied to the purchase of Gourmet Burger Kitchen by Famous Brands in 2016.

In this note, we develop a theory of multiplicities of roots for polynomials over hyperfields and use this to provide a unified and conceptual proof of both Descartes' rule of signs and Newton's "polygon rule". [ABSTRACT FROM AUTHOR]

The emotional reactions aroused by the fire that partly destroyed Notre-Dame de Paris in April 2019 can be analyzed as "valuations" in the light of the pragmatic sociology of values, since they provide empirically grounded material allowing for the description and modeling of the actual implementations and effects of valuations. After a quick summary of the recent history of the pragmatic turn in sociology as related to the sociology of valuation, and a short reflection on the relationship between emotions and values, the fire of Notre-Dame de Paris is used as a case study in the light of "axiological sociology", a model built on value judgments observed in various contexts, including the display of emotions. This article intends to demonstrate both empirically and theoretically how important it is for the social sciences to consider values as an autonomous issue, deserving to be treated as "axiological facts", as any other kind of social fact. [ABSTRACT FROM AUTHOR]

This is an expository paper about the fundamental mathematical notion of starshapedness, emphasizing the geometric, analytical, combinatorial, and topological properties of starshaped sets and their broad applicability in many mathematical fields. The authors decided to approach the topic in a very broad way since they are not aware of any related survey-like publications dealing with this natural notion. The concept of starshapedness is very close to that of convexity, and it is needed in fields like classical convexity, convex analysis, functional analysis, discrete, combinatorial and computational geometry, differential geometry, approximation theory, PDE, and optimization; it is strongly related to notions like radial functions, section functions, visibility, (support) cones, kernels, duality, and many others. We present in a detailed way many definitions of and theorems on the basic properties of starshaped sets, followed by survey-like discussions of related results. At the end of the article, we additionally survey a broad spectrum of applications in some of the above mentioned disciplines. [ABSTRACT FROM AUTHOR]

In this paper we present a matricial result that generalizes Hironaka's game and Perron transforms simultaneously. We also show how one can deduce the various forms in which the algorithm of Perron appears in proofs of local uniformization from our main result. [ABSTRACT FROM AUTHOR]

We extend the model of dynamic leverage and valuation of hedge funds [Lan, Y., N. Wang, and J. Yang. 2013. The Economics of Hedge Funds." Journal of Financial Economics 110: 300–323.] by incorporating model uncertainty. Our theoretical model predicts that concerns about model uncertainty induce risk-neutral managers to behave more endogenously risk-averse and to choose a more conservative leverage strategy. Moreover, it shows that model uncertainty may reduce the valuations of hedge funds, including incentive fees, and total fees for the manager as well as the investors' payoff, while model uncertainty has ambiguous effects on the valuations of managers' management fees. Finally, we find that model uncertainty significantly increases the break-even alpha at the founding of a hedge fund. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to study the structure of the graded algebra associated to a valuation. More specifically, we prove that the associated graded algebra gr v (R) of a subring (R , m) of a valuation ring O v , for which K v : = O v / m v = R / m , is isomorphic to K v t v (R) , where the multiplication is given by a twisting. We show that this twisted multiplication can be chosen to be the usual one in the cases where the value group is free or the residue field is closed by radicals. We also present an example that shows that the isomorphism (with the trivial twisting) does not have to exist. [ABSTRACT FROM AUTHOR]

Copyright of Cultural Sociology is the property of Sage Publications, Ltd. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In science and technology studies today, there is a troubling tendency to portray actors in the biosciences as "cultural dopes" and technology as having monolithic qualities with predetermined outcomes. To remedy this analytical impasse, this article introduces the concept styles of valuation to analyze how actors struggle with valuing technology in practice. Empirically, this article examines how actors in a bioscientific laboratory struggle with valuing the properties and qualities of algorithms in a high-throughput setting and identifies the copresence of several different styles. The question that the actors struggle with is what different configurations of algorithms, devices, and humans are "good bioscience," that is, what do the actors perform as a good distribution of agency between algorithms and humans? A key finding is that algorithms, robots, and humans are valued in multiple ways in the same setting. For the actors, it is not apparent which configuration of agency and devices is more authoritative nor is it obvious which skills and functions should be redistributed to the algorithms. Thus, rather than tying algorithms to one set of values, such as "speed," "precision," or "automation," this article demonstrates the broad utility of attending to the multivalence of algorithms and technology in practice. [ABSTRACT FROM AUTHOR]

Kaplansky's Infinite Abelian Groups, originally published in 1954, has recently been republished. In recognition of its continuing significance after 65 years, the authors illustrate some important ideas and techniques described in the book. In particular, a class of groups is defined for which there are affirmative answers to the author's two famous "Test Problems." [ABSTRACT FROM AUTHOR]

This paper examines the reliability of survey data on business incomes, valuations, and rates of return, which are key inputs for studies of wealth inequality and entrepreneurial choice. We compare survey responses of business owners with available data from administrative tax records, brokered private business sales, and publicly traded company lings and document problems due to nonrepresentative samples and measurement errors across several surveys, subsamples, and years. We find that the discrepancies are economically relevant for the statistics of interest. We investigate reasons for these discrepancies and propose corrections for future survey designs. [ABSTRACT FROM AUTHOR]

Let v be a Krull valuation of a field K with valuation ring R v and K 1 , K 2 be finite separable extensions of K which are linearly disjoint over K. Assume that the integral closure of R v in the composite field K 1 K 2 is a free R v -module. For a given pair of prolongations v 1 , v 2 of v to K 1 , K 2 respectively, it is shown that there exists a unique prolongation w of v to K 1 K 2 which extends both v 1 , v 2. Moreover with S i as the integral closure of R v in K i , if the ring S 1 S 2 is integrally closed and the residue field of v is perfect, then f (w / v) = f (v 1 / v) f (v 2 / v) , where f (v ′ / v) stands for the degree of the residue field of a prolongation v ′ of v over the residue field of v. As an application, it is deduced that if K 1 , K 2 are algebraic number fields which are linearly disjoint over K = K 1 ∩ K 2 , then the number of prime ideals of the ring A K 1 K 2 of algebraic integers of K 1 K 2 lying over a given prime ideal ℘ of A K equals the product of the numbers of prime ideals of A K i lying over ℘ for i = 1 , 2. [ABSTRACT FROM AUTHOR]