A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription. AbstractWe present an internal formalisation of a type heory with dependent types in Type Theory using a special case of higher inductive types from Homotopy Type Theory which we call quotient inductive types (QITs). Our formalisation of type theory avoids referring to preterms or a typability relation but defines directly well typed objects by an inductive definition. We use the elimination principle to define the set-theoretic and logical predicate interpretation. The work has been formalized using the Agda system extended with QITs using postulates.

Quotient inductive-inductive types (QIITs) generalise inductive types in two ways: a QIIT can have more than one sort and the later sorts can be indexed over the previous ones. In addition, equality constructors are also allowed. We work in a setting with uniqueness of identity proofs, hence we use the term QIIT instead of higher inductive-inductive type. An example of a QIIT is the well-typed (intrinsic) syntax of type theory quotiented by conversion. In this paper first we specify finitary QIITs using a domain-specific type theory which we call the theory of signatures. The syntax of the theory of signatures is given by a QIIT as well. Then, using this syntax we show that all specified QIITs exist and they have a dependent elimination principle. We also show that algebras of a signature form a category with families (CwF) and use the internal language of this CwF to show that dependent elimination is equivalent to initiality.

No abstract

No abstract

Flow cytometry enables the sequential determination of calcium levels in millions of stimulated lymphocytes over a short period of time. Current algorithms available are not suitable for the statistical analysis of this large amount of data. The authors aimed to develop a robust algorithm that fits a function to median values of measured data and provides an opportunity for statistical comparison between different calcium-flux measurements. The alteration of calcium signal was monitored in CD41 cells loaded with calcium binding fluorescent dyes and stimulated with phytohemagglutinin; the alteration of calcium signal was monitored for 10 minutes. The authors also reanalyzed published calcium-flux data of CD31 cells and Jurkat cells stimulated with different concentrations of anti-CD3 and thapsigargin. The authors fitted different functions to the medians of data per time unit and identified hormesis function as the best fitting one. On the basis of the optimally fitting function, the authors calculated the most relevant biological descriptors such as starting value, peak, time to reach the maximum, and time to reach 50% of maximum before and after the peak. Statistically significant differences in cell activation kinetics at different stimulatory concentrations were also demonstrated. This approach enables us to characterize the kinetics and distribution of calcium-flux data derived by flow cytometry and may be a reliable tool for the characterization of lymphocyte activation (for details see: http://calciumflux.intralab.eu).'