The cardinality of the augmentation category of a Legendrian link

We introduce a notion of cardinality for the augmentation category associated to a Legendrian knot or link in standard contact R^3. This `homotopy cardinality' is an invariant of the category and allows for a weighted count of augmentations, which we prove to be determined by the ruling polynomial of the link. We present an application to the augmentation category of doubly Lagrangian slice knots.Comment: 15 page

Augmentations are Sheaves

We show that the set of augmentations of the Chekanov-Eliashberg algebra of a Legendrian link underlies the structure of a unital A-infinity category. This differs from the non-unital category constructed in [BC], but is related to it in the same way that cohomology is related to compactly supported cohomology. The existence of such a category was predicted by [STZ], who moreover conjectured its equivalence to a category of sheaves on the front plane with singular support meeting infinity in the knot. After showing that the augmentation category forms a sheaf over the x-line, we are able to prove this conjecture by calculating both categories on thin slices of the front plane. In particular, we conclude that every augmentation comes from geometry.Comment: 109 pages; v2: added Legendrian mirror example in section 4.4.4, corrected typos and other minor changes; v3: accepted versio

A COEVOLUTIONARY APPROACH TO UNDERSTANDING THE PARADOX OF SOCIAL PRESSURES VERSUS ECONOMIC EFFICIENCY ACROSS THE WORLD'S FOOD CHAINS

Institutional and Behavioral Economics,

Point-free Construction of Real Exponentiation

We define a point-free construction of real exponentiation and logarithms, i.e. we construct the maps $\exp\colon (0, \infty)\times \mathbb{R} \rightarrow \!(0,\infty),\, (x, \zeta) \mapsto x^\zeta$ and $\log\colon (1,\infty)\times (0, \infty) \rightarrow\mathbb{R},\, (b, y) \mapsto \log_b(y)$, and we develop familiar algebraic rules for them. The point-free approach is constructive, and defines the points of a space as models of a geometric theory, rather than as elements of a set -- in particular, this allows geometric constructions to be applied to points living in toposes other than Set. Our geometric development includes new lifting and gluing techniques in point-free topology, which highlight how properties of $\mathbb{Q}$ determine properties of real exponentiation. This work is motivated by our broader research programme of developing a version of adelic geometry via topos theory. In particular, we wish to construct the classifying topos of places of $\mathbb{Q}$, which will provide a geometric perspective into the subtle relationship between $\mathbb{R}$ and $\mathbb{Q}_p$, a question of longstanding number-theoretic interest.Comment: Editorial and expository changes from previous version. Accepted, Logical Methods in Computer Scienc

A Point-Free Look at Ostrowski's Theorem and Absolute Values

This paper investigates the absolute values on $\mathbb{Z}$ valued in the upper reals (i.e. reals for which only a right Dedekind section is given). These necessarily include multiplicative seminorms corresponding to the finite prime fields $\mathbb{F}_p$. As an Ostrowski-type Theorem, the space of such absolute values is homeomorphic to a space of prime ideals (with co-Zariski topology) suitably paired with upper reals in the range $[-\infty, 1]$, and from this is recovered the standard Ostrowski's Theorem for absolute values on $\mathbb{Q}$. Our approach is fully constructive, using, in the topos-theoretic sense, geometric reasoning with point-free spaces, and that calls for a careful distinction between Dedekinds vs. upper reals. This forces attention on topological subtleties that are obscured in the classical treatment. In particular, the admission of multiplicative seminorms points to connections with Berkovich and adic spectra. The results are also intended to contribute to characterising a (point-free) space of places of $\mathbb{Q}$

Dekomposisi dan Rekombinasi Pengacakan Citra Digital dengan Logistic Mapping

Beberapa citra digital membutuhkan privasi dan kerahasiaan, seperti citra medis, citra diagnosa medis jarak jauh, citra rahasia melalui komunikasi internet, atau citra rahasia kemiliteran. Salah satu cara untuk mengamankan informasi di dalam citra digital adalah dengan melakukan pengacakan (scrambling). Penelitian ini mengacak nilai piksel citra digital dengan mengubah nilai piksel dari sistem bilangan desimal menjadi bilangan basis empat (kuartener), kemudian mengurai (dekomposisi) keempat bit kuartener dan melakukan pengacakan terhadap keempat posisi bit berdasarkan pada bilangan acak yang dihasilkan oleh algoritma logistic mapping, kemudian bit hasil pengacakan digabungkan kembali (rekombinasi) untuk menghasilkan nilai piksel baru. Logistic mapping merupakan penghasil bilangan acak yang mampu menghasilkan deretan bilangan yang acak berdasarkan nilai kunci µ (3.569945 < µ < 4) dan nilai awal x0 (0 < x0 < 1). Hasil penelitian ini dapat melakukan pengacakan terhadap citra digital dengan dekomposisi dan rekombinasi nilai piksel berdasarkan pada nilai acak yang dihasilkan oleh algoritma logistic mapping. Hasil pengujian menunjukkan bahwa pasangan kunci-1 (µ1, x1) memiliki sensitivitas paling tinggi dalam mengacak citra, kemudian diikuti oleh pasangan kunci-2 (µ2, x2), pasangan kunci-3 (µ3, x3) dan pasangan kunci-4 (µ4, x4)

Thriving in a colder and more challenging climate

Hawkridge, D., Ng, K., & Verjans, S. (Eds.) (2011). Thriving in a colder and more challenging climate. The 18th annual conference of the Association for Learning Technology (ALT-C 2011). September, 6-8, 2011, University of Leeds, England, UK. URI:http://repository.alt.ac.uk/2159Here are the proceedings of the 2011 ALT Conference ‘‘Thriving in a colder and more challenging climate’’. Proceedings papers report on a piece of research, possibly in its early stages, or they can be ‘‘thoughtpieces’’ which state a point of view or summarise an area of work, perhaps giving new insights. The conference has six themes: . Research and rigour: creating, marshalling and making effective use of evidence . Making things happen: systematic design, planning and implementation . Broad tents and strange bedfellows: collaborating, scavenging and sharing to increase value . At the sharp end: enabling organisations and their managers to solve business, pedagogic and technical challenges . Teachers of the future: understanding and influencing the future role and practices of teachers . Preparing for a thaw: looking ahead to a time beyond the disruptive discontinuities of the next few years. Interestingly, there were very few proposals for the conference as a whole against the sixth theme: and no proceedings papers. Perhaps the thaw is still perceived as being too far away to warrant any preparation yet!Association for learning technolog