The Vectorial λ\lambda-Calculus

We describe a type system for the linear-algebraic $\lambda$-calculus. The type system accounts for the linear-algebraic aspects of this extension of $\lambda$-calculus: it is able to statically describe the linear combinations of terms that will be obtained when reducing the programs. This gives rise to an original type theory where types, in the same way as terms, can be superposed into linear combinations. We prove that the resulting typed $\lambda$-calculus is strongly normalising and features weak subject reduction. Finally, we show how to naturally encode matrices and vectors in this typed calculus.Comment: Long and corrected version of arXiv:1012.4032 (EPTCS 88:1-15), to appear in Information and Computatio

Asymptotic properties of the solutions of a differential equation appearing in QCD

We establish the asymptotic behaviour of the ratio $h^\prime(0)/h(0)$ for $\lambda\rightarrow\infty$, where $h(r)$ is a solution, vanishing at infinity, of the differential equation $h^{\prime\prime}(r) = i\lambda \omega (r) h(r)$ on the domain $0 \leq r <\infty$ and $\omega (r) = (1-\sqrt{r} K_1(\sqrt{r}))/r$. Some results are valid for more general $\omega$'s.Comment: 6 pages, late

Concrete resource analysis of the quantum linear system algorithm used to compute the electromagnetic scattering cross section of a 2D target

We provide a detailed estimate for the logical resource requirements of the quantum linear system algorithm (QLSA) [Phys. Rev. Lett. 103, 150502 (2009)] including the recently described elaborations [Phys. Rev. Lett. 110, 250504 (2013)]. Our resource estimates are based on the standard quantum-circuit model of quantum computation; they comprise circuit width, circuit depth, the number of qubits and ancilla qubits employed, and the overall number of elementary quantum gate operations as well as more specific gate counts for each elementary fault-tolerant gate from the standard set {X, Y, Z, H, S, T, CNOT}. To perform these estimates, we used an approach that combines manual analysis with automated estimates generated via the Quipper quantum programming language and compiler. Our estimates pertain to the example problem size N=332,020,680 beyond which, according to a crude big-O complexity comparison, QLSA is expected to run faster than the best known classical linear-system solving algorithm. For this problem size, a desired calculation accuracy 0.01 requires an approximate circuit width 340 and circuit depth of order $10^{25}$ if oracle costs are excluded, and a circuit width and depth of order $10^8$ and $10^{29}$, respectively, if oracle costs are included, indicating that the commonly ignored oracle resources are considerable. In addition to providing detailed logical resource estimates, it is also the purpose of this paper to demonstrate explicitly how these impressively large numbers arise with an actual circuit implementation of a quantum algorithm. While our estimates may prove to be conservative as more efficient advanced quantum-computation techniques are developed, they nevertheless provide a valid baseline for research targeting a reduction of the resource requirements, implying that a reduction by many orders of magnitude is necessary for the algorithm to become practical.Comment: 37 pages, 40 figure

: Tubulin dimer binding proteins

International audienceMicrotubules play an essential role in eukaryotic cells, where they perform a wide variety of functions. In this paper, we describe the characterization of proteins associated to tubulin dimer in its native form, using affinity chromatography and mass spectrometry. We used an immunoaffinity column with coupled-monoclonal antibody directed against the alpha-tubulin C-terminus. Tubulin was first loaded onto the column, then interphase and mitotic cell lysates were chromatographed. Tubulin-binding proteins were eluted using a peptide mimicking the alpha-tubulin C-terminus. Elution fractions were analyzed by SDS-PAGE, and a total of 14 proteins were identified with high confidence by mass spectrometry. These proteins could be grouped in four classes: known tubulin-binding proteins, one microtubule-associated protein, heat shock proteins, and proteins that were not shown previously to bind tubulin dimer or microtubules

The Absence of Positive Energy Bound States for a Class of Nonlocal Potentials

We generalize in this paper a theorem of Titchmarsh for the positivity of Fourier sine integrals. We apply then the theorem to derive simple conditions for the absence of positive energy bound states (bound states embedded in the continuum) for the radial Schr\"odinger equation with nonlocal potentials which are superposition of a local potential and separable potentials.Comment: 23 page

Completeness of algebraic CPS simulations

The algebraic lambda calculus and the linear algebraic lambda calculus are two extensions of the classical lambda calculus with linear combinations of terms. They arise independently in distinct contexts: the former is a fragment of the differential lambda calculus, the latter is a candidate lambda calculus for quantum computation. They differ in the handling of application arguments and algebraic rules. The two languages can simulate each other using an algebraic extension of the well-known call-by-value and call-by-name CPS translations. These simulations are sound, in that they preserve reductions. In this paper, we prove that the simulations are actually complete, strengthening the connection between the two languages.Comment: In Proceedings DCM 2011, arXiv:1207.682

Semantics of a Typed Algebraic Lambda-Calculus

Algebraic lambda-calculi have been studied in various ways, but their semantics remain mostly untouched. In this paper we propose a semantic analysis of a general simply-typed lambda-calculus endowed with a structure of vector space. We sketch the relation with two established vectorial lambda-calculi. Then we study the problems arising from the addition of a fixed point combinator and how to modify the equational theory to solve them. We sketch an algebraic vectorial PCF and its possible denotational interpretations