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Locally convex (or nondegenerate) curves in the sphere $S^n$ (or the projective space) have been studied for several reasons, including the study of linear ordinary differential equations of order $n+1$. Taking Frenet frames allows us to obtain corresponding curves $\Gamma$ in the group $Spin_{n+1}$; recall that $\Pi: Spin_{n+1} \to Flag_{n+1}$ is the universal cover of the space of flags. Let $L(z_0;z_1)$ be the space of such curves $\Gamma$ with prescribed endpoints $\Gamma(0) = z_0$, $\Gamma(1) = z_1$. The aim of this paper is to determine the homotopy type of the spaces $L_3(z_0;z_1)$ for all $z_0, z_1 \in Spin_4$. Recall that $Spin_4 = S^3 \times S^3 \subset H \times H$, where $H$ is the ring of quaternions. This paper relies heavily on previous publications. The earliest such paper solves the corresponding problem for curves in $S^2$. Another previous result (with B. Shapiro) reduces the problem to $z_0 = 1$ and $z_1 \in Quat_4$ where $Quat_4 \subset Spin_4$ is a finite group of order $16$ with center $Z(Quat_4) = \{(\pm 1,\pm 1)\}$. A more recent paper shows that for $z_1 \in Quat_4 \smallsetminus Z(Quat_4)$ we have a homotopy equivalence $L_3(1;z_1) \approx \Omega Spin_4$. In this paper we compute the homotopy type of $L_3(1;z_1)$ for $z_1 \in Z(Quat_4)$: it is equivalent to the infinite wedge of $\Omega Spin_4$ with an infinite countable family of spheres (as for the case $n = 2$). The structure of the proof is roughly the same as for the case $n = 2$ but some of the steps are much harder. In both cases we construct explicit subsets $Y \subset L_n(z_0;z_1)$ for which the inclusion $Y \subset \Omega Spin_{n+1}(z_0;z_1)$ is a homotopy equivalence. For $n = 3$, this is done by considering the itineraries of such curves. The itinerary of a curve in $L_n(1;z_1)$ is a finite word in the alphabet $S_{n+1} \smallsetminus \{e\}$ of nontrivial permutations., Comment: 34 pages, 4 figures

Locally convex curves in the sphere $S^n$ have been studied for several reasons, including the study of linear ordinary differential equations. Taking Frenet frames obtains corresponding curves $\Gamma$ in the group $Spin_{n+1}$; $\Pi: Spin_{n+1} \to Flag_{n+1}$ is the universal cover of the space of flags. Determining the homotopy type of spaces of such curves $\Gamma$ with prescribed initial and final points appears to be a hard problem. We may focus on $L_n$, the space of locally convex curves $\Gamma: [0,1] \to Spin_{n+1}$ with $\Gamma(0) = 1$, $\Pi(\Gamma(1)) = \Pi(1)$. Convex curves form a contractible connected component of $L_n$; there are $2^{n+1}$ other components, one for each endpoint. The homotopy type of $L_n$ has so far been determined only for $n=2$. This paper is a step towards solving the problem for larger values of $n$. The itinerary of $\Gamma$ belongs to $W_n$, the set of finite words in the alphabet $S_{n+1} \setminus \{e\}$. The itinerary of a curve lists the non open Bruhat cells crossed. Itineraries stratify the space $L_n$. We construct a CW complex $D_n$ which is a kind of dual of $L_n$ under this stratification: the construction is similar to Poincar\'e duality. The CW complex $D_n$ is homotopy equivalent to $L_n$. The cells of $D_n$ are naturally labeled by words in $W_n$; $D_n$ is locally finite. Explicit glueing instructions are described for lower dimensions. We describe an open subset $Y_n \subset L_n$, a union of strata of $L_n$. In each non convex component of $L_n$, the intersection with $Y_n$ is connected and dense. Most connected components of $L_n$ are contained in $Y_n$. For $n > 3$, in the other components the complement of $Y_n$ has codimension at least $2$. The set $Y_n$ is homotopy equivalent to the disjoint union of $2^{n+1}$ copies of $\Omega Spin_{n+1}$. For all $n \ge 2$, all connected components of $L_n$ are simply connected., Comment: 56 pages, 15 figures. Updated references and new results. Minor corrections. Text overlap with arXiv:1810.08632

In this paper, the authors discuss fine-grain power savings obtained through a technique which we call Effective Precision Matching Computation—PreMatch. It works by dynamically selecting arithmetic modules in a system designed with Functional Redundancy. Functional Redundant (FR) systems are those which contain different implementation instances of functional blocks or circuits on chip. The different instances are obtained by using different hardware algorithms and/or precision (bitwidth) of their operands. The selection of modules is completely dynamic and data-driven, i.e., it is done according to the contents of the parameters passed as input to those modules. Basically, according to the values of the operands different instances of adders, multipliers, and so on can be selected to “match” the required precision of the computation. Experiments showed that up to 28.4% energy savings can be achieved with Pre- Match alone, and up 29.3% using both techniques (FR and PreMatch).

There is a high demand for highly skilled professionals that have a broad, systematic knowledge on hardware design, embedded software and also management skills. However, such demands are hard to satisfy in normal university curricula. The QUBE program provides state-of-the-art courses in HW, SW, co-design and MOT (Management of Technology) to fill this curriculum gap and the needs of companies to stay up-to-date on new advanced technologies.