An algebra of mixed computation

AbstractAlgebraic tools for mixed computation are presented. Some axioms for informational objects, program functions, inputs and outputs are introduced. These axioms are sufficient for the correct mixed computation of some basic program composition forms

Algorithms for computing mixed multiplicities, mixed volumes and sectional Milnor numbers

We give Macaulay2 algorithms for computing mixed multiplicities of ideals in a polynomial ring. This enables us to find mixed volumes of lattice polytopes and sectional Milnor numbers of a hypersurface with an isolated singularity. The algorithms use the defining equations of the multi-Rees algebra of ideals. We achieve this by generalizing a recent result of David A. Cox, Kuei-Nuan Lin, and Gabriel Sosa in. One can also use a Macaulay2 command `reesIdeal' to calculate the defining equations of the Rees algebra. We compare the computation time of our scripts with the scripts already available.Comment: 32 page

Exploring Rates of Growth for Middle School Math Curriculum-Based Measurement

xvi, 136 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number.An examination of evidence-based practices for mathematics reveals that a solid grasp of key algebraic topics is essential for successful transition from concrete to abstract reasoning in mathematics. In addition, experts indicate a need to emphasize formative assessment to allow results to inform instruction. To address the dearth of technically adequate assessments designed to support data based decision making in algebra, this study examined (a) the validity of algebra and mixed computation curriculum-based measurement for predicting mid-year general math and algebra outcomes in 8th grade (b) growth rates for algebra and mixed computation CBM in the fall of 8th grade, (c) whether slope is a significant predictor of general math and algebra outcomes after controlling for initial skill, and (d) whether growth rates differ for pre-algebra and algebra students. Participants were 198 eighth grade pre-algebra ( n = 70) and algebra (n = 128) students from three middle schools in the Pacific Northwest. Results indicate moderate relationships between fall performance on mixed computation and algebra CBM and winter SAT-10 and algebra performance and significant growth across the fall. Growth was not found to predict general math and algebra outcomes after controlling for initial skill. Future studies should examine (a) growth rates over an extended period of time with a larger sample of classrooms, (b) instructional variables that may impact growth across classrooms, and (c) the impact on student performance when data gleaned from the mixed computation and algebra CBM are used to support data based decision making in middle school algebra and pre-algebra classrooms.Committee in charge: Roland Good, Chairperson, Special Education and Clinical Sciences; Elizabeth Ham, Member, Special Education and Clinical Sciences; Leanne Ketterlin Geller, Member, Educational Methodology, Policy, and Leadership; Christopher Phillips, Outside Member, Mathematic

On the geometry of mixed states and the Fisher information tensor

In this paper, we will review the co-adjoint orbit formulation of finite dimensional quantum mechanics, and in this framework, we will interpret the notion of quantum Fisher information index (and metric). Following previous work of part of the authors, who introduced the definition of Fisher information tensor, we will show how its antisymmetric part is the pullback of the natural Kostant-Kirillov-Souriau symplectic form along some natural diffeomorphism. In order to do this, we will need to understand the symmetric logarithmic derivative as a proper 1-form, settling the issues about its very definition and explicit computation. Moreover, the fibration of co-adjoint orbits, seen as spaces of mixed states, is also discussed.Comment: 27 pages; Accepted Manuscrip

The handbook of zonoid calculus

In this work, we present a new method of computation that we call zonoid calculus. It is based on a particular class of convex bodies called zonoids and on a representation of zonoids using random vectors. Concretely, this is a recipe to build multilinear maps on spaces of zonoids from multilinear maps on the underlying vector spaces. We call this recipe the fundamental theorem of zonoid calculus (FTZC). Using this and the wedge product in the exterior algebra we build the zonoid algebra, that is a structure of algebra on the space of convex bodies of the exterior algebra of a vector space. We show how this relates to the notion of mixed volume on one side and to random determinants on the other. This produces new inequalities on expected absolute determinants. We also show how this applies in two detailed examples: fiber bodies and non centered Gaussian determinants. We then use FTZC to produce a new function on zonoids of a complex vector space that we call the mixed J-volume. We uncover a link between the zonoid algebra and the algebra of valuations on convex bodies. We prove that the wedge product of zonoids extends Alesker’s product of smooth valuations. Finally we apply the previous results to integral geometry in two different context. First we show how, in Riemannian homogeneous spaces, the expected volume of random intersections can be computed in the zonoid algebra. We use this to produce a new inequality modelled on the Alexandrov–Fenchel inequality, and to compute formulas for random intersection of real submanifolds in complex projective space. Secondly, we prove how a Kac-Rice type formula can relate to the zonoid algebra and a certain zonoid section. We use this to study the expected volume of random submanifolds given as the zero set of a random function. We again produce an inequality on the densities of expected volume modelled on the Alexandrov–Fenchel inequality, as well as a general Crofton formula in Finsler geometry

Relative cyclic homology of square zero extensions

Let k be a characteristic zero field, C a k-algebra and M a square zero two sided ideal of C. We obtain a new mixed complex, simpler that the canonical one, giving the Hochschild and cyclic homologies of C relative to M. This complex resembles the canonical reduced mixed complex of an augmented algebra. We begin the study of our complex showing that it has a harmonic decomposition like to the one considered by Cuntz and Quillen for the normalized mixed complex of an algebra. We also give new proofs of two theorems of Goodwillie, obtaining an improvement of one of them.Comment: 24 pages. Definitive version, to appear in Crelle Journa