scholarly journals Linear lambda terms as invariants of rooted trivalent maps

2016 ◽  
Vol 26 ◽  
Author(s):  
NOAM ZEILBERGER
Keyword(s):  
Lambda Calculus ◽  
Equivalence Classes ◽  
The Other ◽  
Missing Information ◽  
Isomorphism Classes ◽  
Diagrammatic Representation ◽  
Four Color Theorem ◽  
Easy Direction ◽  
General Correspondence ◽  
Free Edges

AbstractThe main aim of the paper is to give a simple and conceptual account for the correspondence (originally described by Bodini, Gardy, and Jacquot) between α-equivalence classes of closed linear lambda terms and isomorphism classes of rooted trivalent maps on compact-oriented surfaces without boundary, as an instance of a more general correspondence between linear lambda terms with a context of free variables and rooted trivalent maps with a boundary of free edges. We begin by recalling a familiar diagrammatic representation for linear lambda terms, while at the same time explaining how such diagrams may be read formally as a notation for endomorphisms of a reflexive object in a symmetric monoidal closed (bi)category. From there, the “easy” direction of the correspondence is a simple forgetful operation which erases annotations on the diagram of a linear lambda term to produce a rooted trivalent map. The other direction views linear lambda terms as complete invariants of their underlying rooted trivalent maps, reconstructing the missing information through a Tutte-style topological recurrence on maps with free edges. As an application in combinatorics, we use this analysis to enumerate bridgeless rooted trivalent maps as linear lambda terms containing no closed proper subterms, and conclude by giving a natural reformulation of the Four Color Theorem as a statement about typing in lambda calculus.

scholarly journals Invariant connections and invariant holomorphic bundles on homogeneous manifolds

2014 ◽  
Vol 12 (1) ◽  
pp. 1-13
Author(s):  
Indranil Biswas ◽  
Andrei Teleman
Keyword(s):  
Lie Group ◽  
Complex Structure ◽  
Differentiable Manifold ◽  
Equivalence Classes ◽  
Transitive Action ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Group A ◽  
Holomorphic Bundles ◽  
Invariant Gauge

AbstractLet X be a differentiable manifold endowed with a transitive action α: A×X→X of a Lie group A. Let K be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms of explicit finite dimensional quotients, of three classes of objects:equivalence classes of α-invariant K-connections on X α-invariant gauge classes of K-connections on X, andα-invariant isomorphism classes of pairs (Q,P) consisting of a holomorphic Kℂ-bundle Q → X and a K-reduction P of Q (when X has an α-invariant complex structure).

On the Ciliary Mechanisms and Interrelationships of Lamellibranches

1943 ◽  
Vol s2-84 (334) ◽  
pp. 187-256
Author(s):  
DAPHNE ATKINS
Keyword(s):  
The Other ◽  
Muscle Fibres ◽  
Frontal Surface ◽  
Posterior Portion ◽  
Food Groove ◽  
Longitudinal Muscles ◽  
Free Edges

In the gill axes of the Microciliobranchia the most important muscles are longitudinal and transverse. The longitudinal muscles are: (a) those extending from one extremity of the gill axis to the other, inserted on the shell anteriorly, and (b) those in the free posterior portion of the axis, inserted on the shell where the axis becomes attached. Together these muscles act as branchial retractors. Withdrawal of the gills prevents (a) their being caught and crushed by the edges of the shell when the valves are suddenly closed, and (b) excessive fouling with sudden intake of muddy or noxious water. The transverse muscles below the chitinous structure arching the axial food groove serve to draw the demibranchs of a gill together, while those above the arch serve to separate them. Such swaying movements of the demibranchs serve to rid them of unwanted material. In the demibranchs are:--(1) muscles of the free edges. These include (a) muscles responsible for movements of the walls of the food grooves, and (b) longitudinal muscles, which effect antero-posterior contraction and assist the longitudinal muscles of the axis in retraction of the gills; (2) vertical muscles of the demibranchs, found chiefly in the Pteriacea, and responsible for dorso-ventral contraction of the demibranchs; (3) muscles of the interlamellar junctions serving to draw the two lamellae of a demibranch together, expelling the contained water; (4) horizontal muscles of the lamellae, present in forms with plicate and heterorhabdic gills and effecting by their action changes in shape of the frontal surface of the principal filaments and movements of the plicae important in connexion with the ciliary sorting mechanism; their contraction increases the folding of the lamellae and decreases the length of the gill: and (5) fine muscle-fibres forming the intrafilamentar ‘septum’.

If vision is “veridical hallucination,” what keeps it veridical?

2003 ◽  
Vol 26 (4) ◽  
pp. 426-427
Author(s):  
Peter Ulric Tse
Keyword(s):  
Visual System ◽  
Reference Frame ◽  
Physical Stability ◽  
The Other ◽  
Data Loss ◽  
Missing Information

If perception is constructed, what keeps perception from becoming mere hallucination unlinked to world events? The visual system has evolved two strategies to anchor itself and correct its errors. One involves completing missing information on the basis of knowledge about what most likely exists in the scene. For example, the visual system fills in information only in cases where it might be responsible for the data loss. The other strategy involves exploiting the physical stability of the environment as a reference frame with respect to which the eyes and body can move.

Regent Results in Comma-Free Codes

1963 ◽  
Vol 15 ◽  
pp. 178-187 ◽  
Author(s):  
B. H. Jiggs
Keyword(s):  
Equivalence Class ◽  
Cyclic Permutation ◽  
Equivalence Classes ◽  
The Other ◽  
Complete Class

A set D of k-letter words is called a comma-free dictionary (2), if whenever (a1a2 . . . ak) and (b1b2 . . . bk) are in D, the "overlaps" (a2a3 . . . akb1), (a3a4 . . . akb1b2), . . . , (akb1 . . . bk-1) are not in D. We say that two k-letter words are in the same equivalence class if one is a cyclic permutation of the other. An equivalence class is called complete if it contains k distinct members. Comma-freedom is violated if we choose words from incomplete equivalence classes, or if more than one word is chosen from the same complete class.

scholarly journals The computational content of atomic polymorphism

2018 ◽  
Vol 27 (5) ◽  
pp. 625-638
Author(s):  
Gilda Ferreira ◽  
Vasco T Vasconcelos
Keyword(s):  
Lambda Calculus ◽  
The Other ◽  
Typed Lambda Calculus ◽  
Polymorphic System

AbstractWe show that the number-theoretic functions definable in the atomic polymorphic system (${\mathbf{F}}_{\mathbf{at}}$) are exactly the extended polynomials. Two proofs of the above result are presented: one, reducing the functions’ definability problem in ${\mathbf{F}}_{\mathbf{at}}$ to definability in the simply typed lambda calculus ($\lambda ^{\rightarrow }$) and the other, directly adapting Helmut Schwichtenberg’s strategy for definability in $\lambda ^{\rightarrow }$ to the atomic polymorphic setting. The uniformity granted in the polymorphic system, when compared with the simply typed lambda calculus, is emphasized.

scholarly journals Isomorphism classes for higher order tangent bundles

2017 ◽  
Vol 17 (2) ◽  
pp. 175-189 ◽  
Author(s):  
Ali Suri
Keyword(s):  
Vector Bundle ◽  
Linear Connection ◽  
Equivalence Classes ◽  
Banach Manifold ◽  
Invariant Vector ◽  
Isomorphism Classes ◽  
Connection Map ◽  
Tangent Bundles ◽  
Order Tangent ◽  
Bundle Structure

AbstractThe tangent bundle TkM of order k of a smooth Banach manifold M consists of all equivalence classes of curves that agree up to their accelerations of order k. In previous work the author proved that TkM, 1 ≤ k ≤∞, admits a vector bundle structure on M if and only if M is endowed with a linear connection, or equivalently if a connection map on TkM is defined. This bundle structure depends heavily on the choice of the connection. In this paper we ask about the extent to which this vector bundle structure remains isomorphic. To this end we define the k-th order differential Tkg : TkM ⟶ TkN for a given differentiable map g between manifolds M and N. As we shall see, Tkg becomes a vector bundle morphism if the base manifolds are endowed with g-related connections. In particular, replacing a connection with a g-related one, where g : M ⟶ M is a diffeomorphism, one obtains invariant vector bundle structures. Finally, using immersions on Hilbert manifolds, convex combinations of connection maps and manifolds of Cr maps we offer three examples for our theory, showing its interaction with known problems such as the Sasaki lift of metrics.

scholarly journals Analisis Satuan Acara Perkuliahan Mata Kuliah Korespondensi Bisnis Jepang di 7 Perguruan Tinggi Se-Jabodetabek

Humaniora ◽  
2010 ◽  
Vol 1 (2) ◽  
pp. 469
Author(s):  
Elisa Carolina Marion
Keyword(s):  
Research Method ◽  
The Other ◽  
Japanese Language ◽  
Language Study ◽  
The Subject ◽  
General Correspondence ◽  
Business Correspondence

Japanese Business Correspondence is a course that is given by 7 Japanese language study institutions all around Jabodetabek (Jakarta, Bogor, Depok, Tangerang, and Bekasi). Each institution has their own uniqueness which can be seen in course unit composition. The research method used in this research is survey. The result of this research is that most institutions teach the general correspondence rather than business correspondence. It is also found that there are 2 institutions having uniqueness compared with the other five institutions. The two institutions are STBA LIA and BINUS University. The speciality of STBA LIA is the subject related to verbal correspondence. Meanwhile, BINUS University is more specialized in email-format correspondence. 

scholarly journals Connected Chord Diagrams and Bridgeless Maps

10.37236/7400 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Julien Courtiel ◽  
Karen Yeats ◽  
Noam Zeilberger
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Lambda Calculus ◽  
The Other ◽  
Combinatorial Proof ◽  
Quantum Field ◽  
Technical Parameter ◽  
Simple Application ◽  
Chord Diagrams ◽  
Combinatorial Maps

We present a surprisingly new connection between two well-studied combinatorial classes: rooted connected chord diagrams on one hand, and rooted bridgeless combinatorial maps on the other hand. We describe a bijection between these two classes, which naturally extends to indecomposable diagrams and general rooted maps. As an application, this bijection provides a simplifying framework for some technical quantum field theory work realized by some of the authors. Most notably, an important but technical parameter naturally translates to vertices at the level of maps. We also give a combinatorial proof to a formula which previously resulted from a technical recurrence, and with similar ideas we prove a conjecture of Hihn. Independently, we revisit an equation due to Arquès and Béraud for the generating function counting rooted maps with respect to edges and vertices, giving a new bijective interpretation of this equation directly on indecomposable chord diagrams, which moreover can be specialized to connected diagrams and refined to incorporate the number of crossings. Finally, we explain how these results have a simple application to the combinatorics of lambda calculus, verifying the conjecture that a certain natural family of lambda terms is equinumerous with bridgeless maps.

scholarly journals Nano topology induced by Lattices

2019 ◽  
Vol 38 (5) ◽  
pp. 197-204
Author(s):  
M. Lellis Thivagar ◽  
V. Sutha Devi
Keyword(s):  
Lower Bound ◽  
19Th Century ◽  
Upper Bound ◽  
Lattice Theory ◽  
Ordered Set ◽  
Equivalence Classes ◽  
The Other ◽  
Five Elements ◽  
Lower And Upper Approximations ◽  
The 19Th Century

Lattice is a partially ordered set in which all finite subsets have a least upper bound and greatest lower bound. Dedekind worked on lattice theory in the 19th century. Nano topology explored by Lellis Thivagar et.al. can be described as a collection of nano approximations, a non-empty finite universe and empty set for which equivalence classes are buliding blocks. This is named as Nano topology, because of its size and what ever may be the size of universe it has atmost five elements in it. The elements of Nano topology are called the Nano open sets. This paper is to study the nano topology within the context of lattices. In lattice, there is a special class of joincongruence relation which is defined with respect to an ideal. We have defined the nano approximations of a set with respect to an ideal of a lattice. Also some properties of the approximations of a set in a lattice with respect to ideals are studied. On the other hand, the lower and upper approximations have also been studied within the context various algebraic structures.

Parameterizing higher-order processes on names and processes

2019 ◽  
Vol 53 (3-4) ◽  
pp. 153-206
Author(s):  
Xian Xu
Keyword(s):  
Lambda Calculus ◽  
Higher Order ◽  
The Other ◽  
First Order ◽  
Full Abstraction

Parameterization extends higher-order processes with the capability of abstraction and application (like those in lambda-calculus). As is well-known, this extension is strict, meaning that higher-order processes equipped with parameterization are strictly more expressive than those without parameterization. This paper studies strictly higher-order processes (i.e., no name-passing) with two kinds of parameterization: one on names and the other on processes themselves. We present two main results. One is that in presence of parameterization, higher-order processes can interpret first-order (name-passing) processes in a quite elegant fashion, in contrast to the fact that higher-order processes without parameterization cannot encode first-order processes at all. We present two such encodings and analyze their properties in depth, particularly full abstraction. In the other result, we provide a simpler characterization of the standard context bisimilarity for higher-order processes with parameterization, in terms of the normal bisimilarity that stems from the well-known normal characterization for higher-order calculus. As a spinoff, we show that the bisimulation up-to context technique is sound in the higher-order setting with parameterization.

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