Cross-connections of the singular transformation semigroup

Cross-connection is a construction of regular semigroups using certain categories called normal categories which are abstractions of the partially ordered sets of principal left (right) ideals of a semigroup. We describe the cross-connections in the semigroup $Sing(X)$ of all non-invertible transformations on a set $X$. The categories involved are characterized as the powerset category $\mathscr{P}(X)$ and the category of partitions $\Pi(X)$. We describe these categories and show how a permutation on $X$ gives rise to a cross-connection. Further we prove that every cross-connection between them is induced by a permutation and construct the regular semigroups that arise from the cross-connections. We show that each of the cross-connection semigroups arising this way is isomorphic to $Sing(X)$. We also describe the right reductive subsemigroups of $Sing(X)$ with the category of principal left ideals isomorphic to $\mathscr{P}(X)$. This study sheds light into the more general theory of cross-connections and also provides an alternate way of studying the structure of $Sing(X)$

The Brown dwarf Atmosphere Monitoring (BAM) Project I: The largest near-IR monitoring survey of L- & T-dwarfs

Using SofI on the 3.5m New Technology Telescope, we have conducted an extensive near-infrared monitoring survey of an unbiased sample of 69 brown dwarfs spanning the L0 and T8 spectral range, with at least one example of each spectral type. Each target was observed for a 2-4 hour period in the Js-band, and the median photometric precision of the data is ~0.7%. A total of 14 brown dwarfs were identified as variables with min-to-max amplitudes ranging from 1.7% to 10.8% over the observed duration. All variables satisfy a statistical significance threshold with a p-value <5% based on comparison with the median reference star light curve. Approximately half of the variables show sinusoidal amplitude variations similar to 2M2139, and the remainder shows short timescale evolving light curves similar to SIMP0136. The L/T transition has been suggested to be a region of a higher degree of variability if patchy clouds are present and this survey was designed to test the patchy cloud model with photometric monitoring of both the L/T transition and non-transition brown dwarfs. Considering the targets identified as variable with the same statistical threshold, the measured variability frequency of 13^{+10}_{-4}% for the L7 -- T4 transition region is indistinguishable from that of the earlier spectral types (32^{+11}_{-8}%), the later spectral types (13^{+10}_{-4}%), or the combination of all non-transition region brown dwarfs (21^{+7}_{-5}%). The variables are not concentrated at the transition, in a specific colour, or with binaries. We note that of the systems previously monitored for variability only ~60% maintained the state of variability (variable or constant), with the remaining switching states. The 14 variables include nine newly identified variables that will provide important systems for follow-up multi-wavelength monitoring to further investigate brown dwarf atmosphere physics.Comment: Accepted for publication in the Astronomy & Astrophysics. 15 pages, 13 figure

Formal semantics for LIPS (Language for Implementing Parallel/distributed Systems)

This thesis presents operational semantics and an abstract machine for a point-to-point asynchronous message passing language called LIPS (Language for Implementing Parallel/ distributed Systems). One of the distinctive features of LIPS is its capability to handle computation and communication independently. Taking advantage of this capability, a two steps strategy has been adopted to define the operational semantics. The two steps are as follows: • A big-step semantics with single-step re-writes is used to relate the expressions and their evaluated results (computational part of LIPS). • The developed big-step semantics has been extended with Structural Operational Semantics (SOS) to describe the asynchronous message passing of LIPS (communication part of LIPS). The communication in LIPS has been implemented using Asynchronous Message Passing System (AMPS). It makes use of very simple data structures and avoids the use of buffers. While operational semantics is used to specify the meaning of programs, abstract machines are used to provide intermediate representation of the language's implementation. LIPS Abstract Machine (LAM) is defined to execute LIPS programs. The correctness of the execution of the LIPS program/expression written using the operational semantics is verified by comparing it with its equivalent code generated using the abstract machine. Specification of Asynchronous Communicating Systems (SACS) is a process algebra developed to specify the communication in LIPS programs. It is an asynchronous variant of Synchronous Calculus of Communicating Systems (SCCS). This research presents the SOS for SACS and looks at the bisimulation equivalence properties for SACS which can be used to verify the behaviour of a specified process. An implementation is said to be complete when it is equivalent to its specifications. SACS has been used for the high level specification of the communication part of LIPS programs and is implemented using AMPS. This research proves that SACS and AMPS are equivalent by defining a weak bisimulation equivalence relation between the SOS of both SACS and AMPS

Rationing in IPOs

We provide a model of bookbuilding in IPOs, in which the issuer can choose to ration shares. We consider two allocation rules. Under share dispersion, before informed investors submit their bids, they know that, in the aggregate, winning bidders will receive only a fraction of their demand. We demonstrate that this mitigates the winner’s curse, that is, the incentive of bidders to shade their bids. It leads to more aggressive bidding, to the extent that rationing can be revenue-enhancing. In a parametric example, we characterize bid and revenue functions, and the optimal degree of rationing. We show that, when investors’ information is diffuse, maximal rationing is optimal. Conversely, when their information is concentrated, the seller should not ration shares. We determine the optimal degree of rationing in a class of credible mechanisms. Our model reconciles the documented anomaly that higher bidders in IPOs do not necessarily receive higher allocations.IPOs