Rationality of the spectral action for Robertson-Walker metrics and the geometry of the determinant line bundle for the noncommutative two torus

In noncommutative geometry, the geometry of a space is given via a spectral triple $(\mathcal{A,H},D)$. Geometric information, in this approach, is encoded in the spectrum of $D$ and to extract them, one should study spectral functions such as the heat trace \Tr (e^{-tD^2}), the spectral zeta function \Tr(|D|^{-s}) and the spectral action functional, \Tr f(D/\Lambda). The main focus of this thesis is on the methods and tools that can be used to extract the spectral information. Applying the pseudodifferential calculus and the heat trace techniques, in addition to computing the newer terms, we prove the rationality of the spectral action of the Robertson-Walker metrics, which was conjectured by Chamseddine and Connes. In the second part, we define the canonical trace for Connes\u27 pseudodifferential calculus on the noncommutative torus and use it to compute the curvature of the determinant line bundle for the noncommutative torus. In the last chapter, the Euler-Maclaurin summation formula is used to compute the spectral action of a Dirac operator (with torsion) on the Berger spheres $\mathbb{S}^3(T)$

Optimal kinetic parameters and batch modeling for pectin hydrolysis to galacturonic acid with Pectinex Ultra SP-L enzyme

Galacturonic acid is a monosaccharide obtained by pectin hydrolysis and a suitable substrate to produce bioethanol by fermentation. This paper focuses on quantification of citrus pectin hydrolysis to galacturonic acid and provides new, reliable kinetic parameters for the Michaelis-Menten equation when the well-known, commercial Pectinex Ultra SP-L is employed as enzyme. They are: rmax=1.10 g/ (L min), Km=10.42g/L and KIGA=10.05g/L, as obtained with a great accuracy by a non-linear regression method and confirmed by the three classical linearization procedures (Lineweaver-Burk, Langmuir and Eadie-Hofstee). The quantification of product inhibition has been achieved, with its inclusion in the rate equation. A batch reactor model yields a perfect agreement between predictions and experiments, even under conditions different from those on which the parameters had been determined by regressio

A Selberg Trace Formula for GL3(Fp)∖GL3(Fq)/K

Peer reviewed: TrueAcknowledgements: We would like to thank the Fields Institute for organizing the Fields Undergraduate Summer Research Program in 2021. Daksh Aggarwal, Jiyuan Lu, Balazs Németh, and C. Shijia Yu were participants in this program while working on this paper. The project was proposed and supervised by Masoud Khalkhali and Asghar Ghorbanpour.Publication status: PublishedFunder: Fields Institute through the Fields Undergraduate Summer Research Program in 2021Funder: Natural Sciences and Engineering Research Council of CanadaIn this paper, we prove a discrete analog of the Selberg Trace Formula for the group GL3(Fq). By considering a cubic extension of the finite field Fq, we define an analog of the upper half-space and an action of GL3(Fq) on it. To compute the orbital sums, we explicitly identify the double coset spaces and fundamental domains in our upper half space. To understand the spectral side of the trace formula, we decompose the induced representation ρ=IndΓG1 for G=GL3(Fq) and Γ=GL3(Fp).</jats:p

A Selberg Trace Formula for GL₃(<inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">F</mi><mi mathvariant="bold-italic">p</mi></msub></mrow></semantics></math></inline-formula>)∖GL₃(<inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">F</mi><mi mathvariant="bold-italic">q</mi></msub></mrow></semantics></math></inline-formula>)/<i>K</i>

In this paper, we prove a discrete analog of the Selberg Trace Formula for the group GL3(Fq). By considering a cubic extension of the finite field Fq, we define an analog of the upper half-space and an action of GL3(Fq) on it. To compute the orbital sums, we explicitly identify the double coset spaces and fundamental domains in our upper half space. To understand the spectral side of the trace formula, we decompose the induced representation ρ=IndΓG1 for G=GL3(Fq) and Γ=GL3(Fp).</inline-formula