The patch topology in univalent foundations

Recent innovation in the design of type theories—foundational systems of mathematics with a focus on constructivity—has produced the concept of interval variable, which can be used to capture relations between objects that carry computational content. We examine two such relationships in type theory: equality, in particularquotients, and arbitrary relations as applied in parametricity interpretations. Cubical type theory, a system using an interval-based formulation of equality, enables a permissive kind of content-carrying equality that includes in particularisomorphism. Cubical type theory provides a constructive interpretation of homotopy type theory and the Univalent Foundations, formalisms that introduced the idea of isomorphism as equality but which lack intrinsic computational meaning. The cubical approach to equality also rectifies long-standing deficiencies in the behavior ofquotients in type theory. We realize a system of generalized quotients for cubical type theory originally conceived in homotopy type theory, called higher inductive types, that merges the concepts of inductive type and quotient. Such a mutual generalization is particularly essential in the contentful equality setting, but also has significant applications to ordinary mathematics. Parametricity is, among other things, a proof technique for deriving propertiesof constructions performed in type theories, based on the idea that constructions preserve all relations between objects. Traditionally a meta-theoretical tool, recentwork has shown that parametricity properties can be integrated into a type theory itself using an interval-based system. We develop internal parametricity on top ofcubical type theory, examining the similarities and distinctions between the two applications of intervals, finding that a background of cubical equality improves thebehavior of internal parametricity, and applying internal parametricity as a tool to solve difficult problems in cubical type theory. We introduce a system of cohesionmodalities to express the interaction between parametric and non-parametric type theory, enabling the use of parametricity results in a non-parametric setting.

Recent discoveries have been made connecting abstract homotopy theory and the field of type theory from logic and theoretical computer science. This has given rise to a new field, which has been christened homotopy type theory . In this direction, Vladimir Voevodsky observed that it is possible to model type theory using simplicial sets and that this model satisfies an additional property, called the Univalence Axiom , which has a number of striking consequences. He has subsequently advocated a program, which he calls univalent foundations , of developing mathematics in the setting of type theory with the Univalence Axiom and possibly other additional axioms motivated by the simplicial set model. Because type theory possesses good computational properties, this program can be carried out in a computer proof assistant. In this paper we give an introduction to homotopy type theory in Voevodsky’s setting, paying attention to both theoretical and practical issues. In particular, the paper serves as an introduction to both the general ideas of homotopy type theory as well as to some of the concrete details of Voevodsky’s work using the well-known proof assistant Coq. The paper is written for a general audience of mathematicians with basic knowledge of algebraic topology; the paper does not assume any preliminary knowledge of type theory, logic, or computer science. Because a defining characteristic of Voevodsky’s program is that the Coq code has fundamental mathematical content, and many of the mathematical concepts which are efficiently captured in the code cannot be explained in standard mathematical English without a lengthy detour through type theory, the later sections of this paper (beginning with Section \ref{sec2}) make use of code; however, all notions are introduced from the beginning and in a self-contained fashion.

: Two experiments were performed to evaluate multitracking (branching) in a linear program. In experiment one, the multitracking consisted of providing additional cues at each frame for use by those students who felt unsure of their response. Results indicated no significant difference in efficiency between the regular linear program and the multitrack program. In experiment two, the multitracking consisted of large frames followed by more detailed frames whenever the student made an error. Large frames were developed by combining an average of three small frames. Again, the results indicated no difference in instructional efficiency between the regular linear program and the multitrack program. Although more errors were made on the largestep branching program, performance on criterion tests was as good as for the regular small-step linear program. Although branching seems a reasonable way to accommodate individual differences, the two methods attempted in this research did not show an advantage. More promising methods of branching might be (a) less frequent branches, at critical points in the program, and (b) large-step frames followed by special remedial frames, rather than by mere repetition of parts of the original large frame. (Author)

This is a short overview of an experimental library of Mathematics formalized in the Coq proof assistant using the univalent interpretation of the underlying type theory of Coq. I started to work on this library in February 2010 in order to gain experience with formalization of Mathematics in a constructive type theory based on the intuition gained from the univalent models (see Kapulkinet al.2012).

Experiments were conducted to evaluate the effects of dietary energy density or genetic background on protein and fat gain of growing cattle. In Exp. 1, 24 Limousin steers were used in a growing-finishing trial. A 2 X 2 factorial arrangement was used with steers randomly allotted to four treatment combinations and fed the following diets: 80% concentrate, high moisture corn-corn silage diet (HI) or a corn silage diet (LO) during both the growing (GRO) and(or) the finishing (FIN) phases. Body composition for both experiments was determined by a deuterium oxide dilution technique. Empty body weight gains were greater (P less than .05) for HI during GRO, FIN and the total trial. Daily protein gains (DPG) were greater (P less than .05) for HI during GRO and FIN, while cattle receiving HI during at least FIN had the greatest (P less than .05) overall DPG. Daily fat gains (DFG) followed the pattern of DPG, being more rapid (P less than .05) for cattle fed HI during either GRO or FIN. Cattle fed the HI diet also tended to be more energetically efficient. In Exp. 2, large frame (LG) and small frame (SM) cattle were used for the evaluation of frame size effects on protein and fat deposition. Steers were individually fed an 80% concentrate, corn-based diet during the entire trial. Average daily gains and daily dry matter intake (P less than .05, P less than .01) were greater for LG, while feed efficiency was similar for both cattle types. Large cattle had greater (P less than .05) DPG than SM cattle, however, DFG were not different. Small frame steers were energetically more efficient (P less than .05), apparently due to composition of gain difference.

Reviewed by: Elegy for Theory by D. N. Rodowick Heather Ashley Hayes D. N. Rodowick, Elegy for Theory Cambridge, MA: Harvard University Press, 2014, xv + 269 pp. What is “theory”? Academics structured much of the theory wars of the 1970s and 1980s to address this question, and more urgently, to settle the dispute about whether forays into literary theory were academically innovative and sound as opposed to faddish or obscure. By the 2000s, controversy over theory’s presence in studying literature, film, and more seemed to have diffused. D. N. Rodowick, author of Elegy for Theory, stood at the center of these theory wars, founding the Yale Film and Media Studies Program. His first book, The Crisis of Political Modernism, offered a critical analysis of the development of film theory. Considering the legacy of French feminism à la Irigaray and Kristeva in relation to emerging cultural studies approaches to film and media, Rodowick stakes his claim to theory with keen historical approach. Elegy for Theory remains committed to the intellectual history of film studies, placing that history in a larger frame through which theory’s evolving applications to film are demonstrative of a more general trend across all arts and humanities, as opposed to theory’s general location in the history of philosophy: “it is not possible, in either the history of philosophy or in the philosophy of science, to attribute a stable sense to theory that leads inexorably toward a perfectible concept around which a final consensus can be achieved” (xiii). Rodowick lays out the impetus to do his work clearly: “Every violent conceptual struggle is followed by efforts to provide a venerable ancestry for newly won definitions of the concept. New concepts of theory are thus often guided by present interest through forces of retrojection that make the unfamiliar familiar again” (xiv). He closes his introduction to the book with a far-reaching call for the work to come: “What I offer here is not a history of contemporary theory, then, but rather something like the elements of a historiography of concepts, enunciative modalities, and discursive formations in which developments in the academic study of film might stand, pars pro toto, for the vicissitudes of theory in the humanities more generally” (xv). It is notable here that Rodowick also reveals in his introduction that Elegy for Theory was originally meant as a single project but now will serve as only the first volume to a two volume series, the subsequent volume of which is to be called Philosophy’s Artful Conversation. The association and disassociation of theory from ethics and philosophy is an argument one would expect in Elegy for Theory, yet Rodowick directs the attention for that debate to the upcoming, unpublished, companion volume to Elegy. Rather, this work’s first half begins by offering a genealogy of theory (by way of Foucault and Nietzsche) that not only traces its origin points but also “return[s] to a historical sense of [theory’s] discontinuities as a concept and as an activity—not retracing a line, completing a circle, or constructing a frame, but rather following [End Page 384] theory’s complex web of derivations” (6). Drawing heavily from Kant, Hegel, and discussions of aesthetics, Rodowick points to the ways in which modernity’s sense of the aesthetic can be understood as what John Ruskin, in 1846, termed the theoretic. He cites Hegel’s “Introduction” to his Lectures on Aesthetics (1823–1829) as the space in which one finds “the various and contested discursive paths and thickets through which the history and theory of art would gradually emerge by the end of the nineteenth century” (38–39). His close reading of Hegel through the first third of Elegy is compelling, leading Rodowick to a place where he argues art first opens itself to theoretical consideration. Elegy is most potent when Rodowick is working in his domain of film theory and criticism. Extending his own argument from The Virtual Life of Film, he reasserts here that a critical consequence of the rapid and ever changing development of electronic and digital media forms begs the very question: What is cinema? Noting that “our contemporary picture of film theory is ineluctably tied...

A new approach to fuzzy topological spaces and fuzzy perfect mappings

Effects of USDA feeder cattle frame and muscle grades on stocker and feeder cattle performance and profitability

The presence of an additional external upright frame was studied in three experiments to separate the role of visuovestibular, global, and local mechanisms in the rod-and-frame illusion (RFI). In the first experiment, carried out in a dark room, the external frame surrounded a large tilted frame. Rod-setting errors to the vertical were abolished with the additional-frame condition (at 22 degrees inner-frame tilt) confirming earlier findings. However, small, residual direct (at 11 degrees inner-frame tilt) and indirect effects (at 33 degrees inner-frame tilt) were still present, indicating the persistence of global visual processing. In the second experiment, the RFI in the dark was compared with the RFI with the lights on. Turning the light on abolished the effect at 22 degrees and 33 degrees frame tilt; however, a small direct effect was maintained at 11 degrees frame tilt. These two studies indicate that the addition of veridical vertical information abolishes the effect owing to visuovestibular mechanisms. In the third experiment, a small rod and frame was used with the lights on (a condition abolishing visual-vestibular interaction). In the case of a small gap between the rod and the inducing frame (a condition which maximises local processing), the effect of the outer upright frame was negligible; this indicates that the additional frame had no effect on local processing. In the case of a large gap (a condition which minimises local processing), the external square reduced the illusion, indicating its modulating effect on visual global processing. Overall, an upright external frame exerts a differential influence depending on which mechanisms contribute most to the RFI in a given experimental condition.

Dual-band tunable electromagnetically induced transparency in vanadium dioxide-based miniaturized terahertz metasurfaces

Muscle Protein Turnover in Cattle of Differing Genetic Backgrounds as Measured by Urinary Nr-Methylhistidine Excretion

The principal purpose of this article is to provide a conceptual framework and heuristic model for considering the existence, magnitude, and consequences of context effects. This purpose is addressed through an extension of some concepts in generalizability theory. In particular, distinctions are drawn between different types of facets and different types of universes. For example, sets of items are distinguished from conditions of measurement typically associated with context effects (e.g., item sequence). In addition, a validity-defining universe of generalization is distinguished from a universe of allowable observations associated with a standardized measurement proce- dure and certain fixed conditions of measurement. Other fixed conditions of measurement are also considered in order to examine context effects involved in applications such as the use of so-called "scrambled" test forms and item or section preequating. It is concluded that context effects are often misunder- stood or masked, that current measurement models have rather serious limitations for examining context effects, and that the importance and magnitude of context effects need to be evaluated in context.

Constructs that involve taking a quotient are commonplace in mathematics. Here I will consider how they are treated within dependent type theory. The notion of quotient type has its origins in the Nuprl theorem-proving system [R. L. Constable et al., 1986] for extensional type theory. Later Hofmann formulated a version for intensional type theory in his thesis [M. Hofmann, 1995]. This depends on having a pre-existing notion of intensional identity type. Hofmann used Martin-Löf’s notion, the indexed family inductively generated from proofs of reflexivity [B. Nordström et al., 1990]. The recent homotopical view of identity in terms of path types [The Univalent Foundations Program, 2013] gives a more liberal perspective and has brought with it the notion of higher inductive type (HIT) [P. L. Lumsdaine and M. Shulman, 2019], subsuming both inductive and quotient types. Inductively defined indexed families of types (in all their various forms) are perhaps the most useful concept that dependent type theory has contributed to the practice of computer assistance for formalizing mathematical proofs. However, it is often the case that a particular application of such types needs not only to inductively generate a collection of objects, but also to make identifications between the objects. In classical mathematics one can first generate and then identify, using the Axiom of Choice to lift infinitary constructions to the quotient. HITs can allow one to avoid such non-constructive uses of choice by inter-twining generation and identification. Perhaps more important than the constructive/non-constructive issue is that simultaneously declaring how to generate and how to identify can be a very natural way of defining some construct from the user’s point of view. This is why HITs promise to be so useful, once we have robust and convenient mechanisms in theorem-proving systems for defining HITs and defining functions out of HITs. Although some HITs have been axiomatized in various systems, at the moment the only system I know of with built in support for defining quite general forms of HIT and using them is the implementation of cubical type theory [C. Cohen et al., 2018] within recent versions of the Agda system [A. Vezzosi et al., 2019]. The higher dimensional aspect of identity in cubical type theory is fascinating; nevertheless, the simpler one-dimensional version of identity, in which one has uniqueness of identity proofs (UIP), is adequate for many applications. Although some regard UIP as a bug of early versions of Agda with it’s original form of dependent pattern matching [T. Coquand, 1992], it is by choice a feature of the Lean prover [J. Avigad et al., 2020]. Altenkirch and Kaposi [T. Altenkirch and A. Kaposi, 2016] have termed the one-dimensional version of HITs quotient inductive types (QITs) and they promise to be very useful even in the setting of type theory with UIP. In this talk I survey some of these developments, including a recent reduction of QITs to quotients [M. P. Fiore et al., 2020], and the prospects for better support in theorem-provers for quotient constructions.

Elbow breadth as a measure of frame size for US males and females

In March of 1921 — I had just completed my first semester at the university, studying physics - H. Hahn joined the mathematics faculty. His first release was the announcement of a seminar on the curve concept (Neueres uber den Kurvenbegriff), one two-hour meeting weekly during the spring semester. I hesitated, but finally mustered up my courage to audit the first session. There, without introduction, Hahn formulated the problem of making precise the idea of curve that everyone has but that no-one had been able to articulate. Various attempts had been made. G. Cantor thought of defining curves as one-to-one images of a segment; C. Jordan then defined them as its images by continuous mappings. But Cantor himself proved the first, and Peano showed the second, of these definitions to be unacceptable since both include squares and even cubes, which no one calls curves. Thereafter arcs were studied — topological images of a segment; that is, obtainable from it by mappings that are both one-to-one and continuous. Among the arcs one does not find squares or cubes, but neither does one find ellipses or lemniscates. Hence more complicated objects, called graphs, were built up by joining arcs in certain ways; but even this concept was too narrow. Other attempts were unsatisfactory for other reasons. So, Hahn said, the seminar would culminate with the realization that the problem was still unsolved and that a satisfactory definition of the curve concept still was unknown.1 In conclusion of the first session, Hahn presented Cantor’s most elementary concepts of point set theory.