The primary purpose of this study was to evaluate the effect of basic multiplication mastery (0-12) and self-efficacy outcomes for elementary age students attempting to master multiplication facts in a Computer-Assisted Instructional (CAI) environment. Timez Attack (TA), a modern Internet based 3-D multiplication video game, was the computer program used in this study. Four third- and four fourth-grade classes of students at a public charter school received either 12 20-minute Teacher-Led Instructional (TLI), or TA multiplication practice sessions. Pre- and post Math Attitude Survey (MAS), timed multiplication tests, observations, and informal interviews were used to assess and compare TA and TLI's learning environments, performance, and self-efficacy outcomes. Both third- and fourth-grade TA students' level of multiplication mastery improved significantly after intervention. Results from the post-MAS also revealed significantly higher self-efficacy beliefs, and reduced nervousness in learning multiplication facts amongst some TA students. Statistical data analysis revealed no significant performance outcome differences between TLI and TA third-grade classes; however, post-test comparisons between fourth-grade TLI and TA students showed TA students significantly outperforming their comparison group counterparts by answering approximately 50% more problems in a given time and feeling significantly less nervous toward learning new multiplication math facts. The TA program's motivational, self-paced, and self-evaluative features seemed to produce a positive learning environment, which encouraged student learning. Educators should consider using CAI with features similar to TA's to improve students' academic performance and self-efficacy.

In an attempt to examine the effects of journaling and vocabulary strategies on elementary students’ attitudes towards mathematical performance, I embedded reflective journaling and vocabulary strategies into my fourth grade mathematics curriculum. The mathematics content focused on whole number place value, multiplication, and division. My study revealed the positive effects these interventions can have on elementary students’ attitudes towards mathematics.

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In this thesis we develop exact repair schemes capable of repairing or circumventing unavailable servers of a distributed network in the context of distributed storage and secure distributed matrix multiplication. We develop the (Λ, Γ, W, ⊙)-exact repair scheme framework for discussing both of these contexts and develop a multitude of explicit exact repair schemes utilizing decreasing monomial-Cartesian codes (DMC codes). Specifically, we construct novel DMC codes in the form of augmented Cartesian codes and rectangular monomial-Cartesian codes, as well as design exact repair schemes utilizing these constructions inspired by the schemes from Guruswami and Wootters [16] and Chen and Zhang [6]. In the context of distributed storage we demonstrate the existence of both high rate and low bandwidth systems based on these schemes, and we develop two methods to extend them to the l-erasure case. Additionally, we develop a family of hybrid schemes capable of attaining high rates, low bandwidths, and a balance in between which proves to be competitive compared to existing schemes. In the context of secure distributed matrix multiplication we develop similarly impactful schemes which have very competitive communication costs. We also construct an encoding algorithm based on multivariate interpolation and prove it is T-secure. / Doctor of Philosophy / Distributed networks may be thought of as networks of computers and/or servers which are capable of transmitting and receiving data from one another. For many applications it is possible for distributed networks to perform better than the sum of their constituent parts. In this thesis we will focus on the particular applications of distributed storage and secure distributed multiplication. A distributed storage system is a system that is capable of storing a single data file over every server in a distributed network. Distributed storage systems often come with exact repair schemes which are algorithms designed to reconstruct the data from a server in the network given the data from the other servers. In particular, if a server on the network ever fails or is otherwise unavailable an exact repair scheme can be used to repair the lost data from the server and maintain the original file. A distributed matrix multiplication scheme on the other hand is a process by which two matrices stored on a source server can be multiplied using a distributed network of helper servers. Again if a helper server becomes unavailable during this process we may use an exact repair scheme to circumvent this delay. The main goal of this thesis is to develop exact repair schemes for the distributed storage and secure distributed matrix multiplication contexts utilizing a mathematical object known as an evaluation code. We will develop several families of exact repair schemes which may be finely tuned to fit particular situations within these contexts, and we will compare these schemes to the existing schemes in the field.

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Starting with the work of Volker Strassen, algorithms for matrix multiplication have been developed which are time complexity-wise more efficient than the standard algorithm from the definition of multiplication. The general method of the developments has been viewing the bilinear mapping that matrix multiplication is as a three-dimensional tensor, where there is an exact correspondence between time complexity of the multiplication algorithm and tensor rank. The latter can be seen as a generalisation of matrix rank, being the minimum number of terms a tensor can be decomposed as. However, in contrast to matrix rank there is no general method of computing tensor ranks, with many values being unknown for important three-dimensional tensors. To further improve the theoretical bounds of the time complexity of matrix multiplication, support rank of tensors has been introduced, which is the lowest rank of tensors with the same support in some basis. The goal of this master's thesis has been to go through the history of faster matrix multiplication, as well as specifically examining the properties of support rank for general tensors. In regards to the latter, a complete classification of rank structures of support classes is made for the smallest non-degenerate tensor product space in three dimensions. From this, the size of a support can be seen affecting the pool of possible ranks within a support class. At the same time, there is in general no symmetry with regards to support size occurring in the rank structures of the support classes, despite there existing a symmetry and bijection between mirrored supports. Discussions about how to classify support rank structures for larger tensor product spaces are also included. / Från och med forskning gjord av Volker Strassen har flera algoritmer för matrismultiplikation utvecklats som är effektivare visavi tidskomplexitet än standardalgoritmen som utgår från defintionen av multiplikation. Generellt sett har metoden varit att se den bilinjära avbildningen som matrismultiplikation är som en tredimensionell tensor. Där används att det finns en exakt korrespondens mellan multiplikationsalgoritmens tidskomplexitet och tensorrang. Det sistnämnda är ett slags generalisering av matrisrang, och är minsta antalet termer en tensor kan skrivas som. Till skillnad frpn matrisrang finns ingen allmän metod för att beräkna tensorrang, och många värden är okända även för välstuderade och viktiga tensorer. För att hitta fler övre begränsningar på matrismultiplikations tidskomplexitet har stödrang av tensorer införts, som är den lägsta rangen bland tensor med samma stöd i en viss bas. Målet med detta examensarbete har varit att göra en genomgång av historien om snabbare matrismultiplikation, samt att specifikt undersöka egenskaper av stödrang för allmänna tredimensionella tensorer. För det sistnämnda görs en fullständig klassificering av rangstrukturer bland stödklasser för den minsta icke-degenererade tensorprodukten av tre vektorrum. Slutsatser är bl.a. att storleken av ett stöd kan ses påverka antalet möjliga ranger inom en stödklass. Samtidigt finns i allmänhet ingen symmetri med avseende på stödstorlek i stödklassernas rangstrukturer. Detta trots att det finns en symmetri och bijektion mellan speglade stöd. I arbetet ingår även en diskussion om hur stödrangstrukturer skulle kunna klassificeras för större tensorprodukter.

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De plus en plus de constructeurs proposent des accélérateurs de calculs à base de circuits reconfigurables FPGA, cette technologie présentant bien plus de souplesse que le microprocesseur. Valoriser cette flexibilité dans le domaine de l'accélération de calcul flottant en utilisant les langages de description de circuits classiques (VHDL ou Verilog) reste toutefois très difficile, voire impossible parfois. Cette thèse a contribué au développement du logiciel FloPoCo, qui offre aux utilisateurs familiers avec VHDL un cadre C++ de description d'opérateurs arithmétiques génériques adapté au calcul reconfigurable. Ce cadre distingue explicitement la fonctionnalité combinatoire d'un opérateur, et la problématique de son pipeline pour une précision, une fréquence et un FPGA cible donnés. Afin de pouvoir utiliser FloPoCo pour concevoir des opérateurs haute performance en virgule flottante, il a fallu d'abord concevoir des blocs de bases optimisés. Nous avons d'abord développé des additionneurs pipelinés autour des lignes de propagation de retenue rapides, puis, à l'aide de techniques de pavages, nous avons conçu de gros multiplieurs, possiblement tronqués, utilisant des petits multiplieurs. L'évaluation de fonctions élémentaires en flottant implique souvent l'évaluation en virgule fixe d'une fonction. Nous présentons un opérateur générique de FloPoCo qui prend en entrée l'expression de la fonction à évaluer, avec ses précisions d'entrée et de sortie, et construit un évaluateur polynomial optimisé de cette fonction. Ce bloc de base a permis de développer des opérateurs en virgule flottante pour la racine carrée et l'exponentielle qui améliorent considérablement l'état de l'art. Nous avons aussi travaillé sur des techniques de compilation avancée pour adapter l'exécution d'un code C aux pipelines flexibles de nos opérateurs. FloPoCo a pu ainsi être utilisé pour implanter sur FPGA des applications complètes.

We studied the dynamic of nine populations of Alpine ibex ( Capra ibex ibex ) in five protected areas. We showed a strong effect of age on demographic parameters, with a marked decrease of survival after 10-12 years of age. We also found a high variability of old females' reproduction between populations. Contrary to what is expected for a highly dimorphic species like ibex, males survived as well as females except for old individuals. Winter harshness had a negative impact on survival of old individuals but not on reproduction and survival of young. We conclude that ibex have evolved a highly conservative life-history tactic compared to other ungulates studied to now. From a management viewpoint we also showed that under some circumstances ground counts might provide reliable estimates of ibex population trends.