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Sparse principal component analysis with global support (SPCAgs), is the problem of finding the top-r leading principal components such that all these principal components are linear combinations of a common subset of at most k variables. SPCAgs is a popular dimension reduction tool in statistics that enhances interpretability compared to regular principal component analysis (PCA). Methods for solving SPCAgs in the literature are either greedy heuristics (in the special case of r = 1 ) with guarantees under restrictive statistical models or algorithms with stationary point convergence for some regularized reformulation of SPCAgs. Crucially, none of the existing computational methods can efficiently guarantee the quality of the solutions obtained by comparing them against dual bounds. In this work, we first propose a convex relaxation based on operator norms that provably approximates the feasible region of SPCAgs within a c 1 + c 2 log r = O (log r) factor for some constants c 1 , c 2 . To prove this result, we use a novel random sparsification procedure that uses the Pietsch-Grothendieck factorization theorem and may be of independent interest. We also propose a simpler relaxation that is second-order cone representable and gives a (2 r) -approximation for the feasible region. Using these relaxations, we then propose a convex integer program that provides a dual bound for the optimal value of SPCAgs. Moreover, it also has worst-case guarantees: it is within a multiplicative/additive factor of the original optimal value, and the multiplicative factor is O (log r) or O(r) depending on the relaxation used. Finally, we conduct computational experiments that show that our convex integer program provides, within a reasonable time, good upper bounds that are typically significantly better than the natural baselines. [ABSTRACT FROM AUTHOR]

The article provide language to support discussions around roles of teaching and scholarly work in education. It mentions that all faculty at institutions of higher education should be engaged in some form of scholarly teaching. It also mentions that growth and recognition of scholarship of teaching and learning (SOTL) contributed to the development of discipline-based education research.

While many classes of cutting-planes are at the disposal of integer programming solvers, our scientific understanding is far from complete with regards to cutting-plane selection, i.e., the task of selecting a portfolio of cutting-planes to be added to the LP relaxation at a given node of the branch-and-bound tree. In this paper we review the different classes of cutting-planes available, known theoretical results about their relative strength, important issues pertaining to cut selection, and discuss some possible new directions to be pursued in order to accomplish cutting-plane selection in a more principled manner. Finally, we review some lines of work that we undertook to provide a preliminary theoretical underpinning for some of the issues related to cut selection. [ABSTRACT FROM AUTHOR]

The current best algorithms for the convex body chasing (CBC) problem in online algorithms use the notion of the Steiner point of a convex set. In particular, the algorithm that always moves to the Steiner point of the request set is O(d) competitive for nested CBC, and this is optimal among memoryless algorithms [Bubeck et al.: Chasing nested convex bodies nearly optimally. In: 31st Annual ACM–SIAM Symposium on Discrete Algorithms (Salt Lake City 2020), pp. 1496–1508. SIAM, Philadelphia (2020)]. A memoryless algorithm coincides with the notion of a selector in functional analysis. The Steiner point is noted for being Lipschitz with respect to the Hausdorff metric, and for achieving the minimal Lipschitz constant possible. It is natural to ask whether every selector with this Lipschitz property yields a competitive algorithm for nested CBC. We answer this question in the negative by exhibiting a selector that yields a non-competitive algorithm for nested CBC but is Lipschitz with respect to Hausdorff distance. Furthermore, we show that being Lipschitz with respect to an L p -type analog of the Hausdorff distance is sufficient to guarantee competitiveness if and only if p = 1 . [ABSTRACT FROM AUTHOR]

In this paper, we study how well one can approximate arbitrary polytopes using sparse inequalities. Our motivation comes from the use of sparse cutting-planes in mixed-integer programing (MIP) solvers, since they help in solving the linear programs encountered during branch-&-bound more efficiently. However, how well can we approximate the integer hull by just using sparse cutting-planes? In order to understand this question better, given a polyope $$P$$ (e.g. the integer hull of a MIP), let $$P^k$$ be its best approximation using cuts with at most k non-zero coefficients. We consider $${\text {d}}(P, P^k) = \max _{x \in P^k} \left( \min _{y \in P} \Vert x - y\Vert \right) $$ as a measure of the quality of sparse cuts.In our first result, we present general upper bounds on $${\text {d}}(P, P^k)$$ which depend on the number of vertices in the polytope. Our bounds imply that if $$P$$ has polynomially many vertices, using half sparsity already approximates it very well. Second, we present a lower bound on $${\text {d}}(P, P^k)$$ for random polytopes that show that the upper bounds are quite tight. Third, we show that for a class of hard packing IPs, sparse cutting-planes do not approximate the integer hull well, that is $$d(P, P^k)$$ is large for such instances unless k is very close to n. Finally, we show that using sparse cutting-planes in extended formulations is at least as good as using them in the original polyhedron, and give an example where the former is actually much better. [ABSTRACT FROM AUTHOR]

Branch-and-bound is the workhorse of all state-of-the-art mixed integer linear programming (MILP) solvers. These implementations of branch-and-bound typically use variable branching, that is, the child nodes are obtained via disjunctions of the form x j ≤ ⌊ x ¯ j ⌋ ∨ x j ≥ ⌈ x ¯ j ⌉ , where x ¯ is an optimal solution to the LP corresponding to the parent node. Even though modern MILP solvers are able to solve very large-scale instances efficiently, relatively little attention has been given to understanding why the underlying branch-and-bound algorithm performs so well. In this paper, our goal is to theoretically analyze the performance of the standard variable branching based branch-and-bound algorithm. In order to avoid the exponential worst-case lower bounds, we follow the common idea of considering random instances. More precisely, we consider random integer programs where the entries of the coefficient matrix and the objective function are randomly sampled. Our main result is that with good probability branch-and-bound with variable branching explores only a polynomial number of nodes to solve these instances, for a fixed number of constraints. To the best of our knowledge this is the first known such result for a standard version of branch-and-bound. We believe that this result provides an indication as to why branch-and-bound with variable branching works so well in practice. [ABSTRACT FROM AUTHOR]

A lesson plan is presented for the Science, Biostatistics, and Cancer Education (SBCE) Distributions Module, which is used to develop the statistical literacy of science students and teach them to form evidence-based conclusions based on data.

A general branch-and-bound tree is a branch-and-bound tree which is allowed to use general disjunctions of the form π ⊤ x ≤ π 0 ∨ π ⊤ x ≥ π 0 + 1 , where π is an integer vector and π 0 is an integer scalar, to create child nodes. We construct a packing instance, a set covering instance, and a Traveling Salesman Problem instance, such that any general branch-and-bound tree that solves these instances must be of exponential size. We also verify that an exponential lower bound on the size of general branch-and-bound trees persists even when we add Gaussian noise to the coefficients of the cross-polytope, thus showing that a polynomial-size "smoothed analysis" upper bound is not possible. The results in this paper can be viewed as the branch-and-bound analog of the seminal paper by Chvátal et al. (Linear Algebra Appl 114:455–499, 1989), who proved lower bounds for the Chvátal–Gomory rank. [ABSTRACT FROM AUTHOR]

In this paper we consider the infinite relaxation of the corner polyhedron with 2 rows. For the 1-row case, Gomory and Johnson proved in their seminal paper a sufficient condition for a minimal function to be extreme, the celebrated 2-Slope Theorem. Despite increased interest in understanding the multiple row setting, no generalization of this theorem was known for this case. We present an extension of the 2-Slope Theorem for the case of 2 rows by showing that minimal 3-slope functions satisfying an additional regularity condition are facets (and hence extreme). Moreover, we show that this regularity condition is necessary, unveiling a structure which is only present in the multi-row setting. [ABSTRACT FROM AUTHOR]

We consider the problem of computing efficient strategies for searching in trees. As a generalization of the classical binary search for ordered lists, suppose one wishes to find a (unknown) specific node of a tree by asking queries to its arcs, where each query indicates the endpoint closer to the desired node. Given the likelihood of each node being the one searched, the objective is to compute a search strategy that minimizes the expected number of queries. Practical applications of this problem include file system synchronization and software testing. Here we present a linear time algorithm which is the first constant factor approximation for this problem. This represents a significant improvement over previous O(log n)-approximation. [ABSTRACT FROM AUTHOR]

The article discusses the initiatives of the University of California-Davis to improve the retention of students within specific science, technology, engineering, or mathematics (STEM) fields. Topics mentioned include the establishment of the integrated agriculture, medicine, science, technology, engineering, and mathematics (iAMSTEM) Hub in 2012, analysis on the challenges and opportunities associated with iAMSTEM implementation, and testing of various teaching methods and strategies.

Introduction: Kelvin-Helmholtz (KH) and tearing mode (TM) instabilities are one of the most important mechanisms of solar wind energy, momentum and plasma transport within the magnetosphere. Methods: To investigate the conditions under which KHTM instabilities occur in the Earth environment it is fundamental to combine simultaneous multipoint in situ measurements and MHD simulations. We analyzed data from the THEMIS and Cluster spacecraft considering two Space Weather (SWE) events starting with an M2.0 flare event (hereafter Case-1) that occurred on 21 June 2015 and the most-intensive flare (X9.3) of solar cycle 24 that occurred on 6 September 2017 (hereafter Case-2). Results: Our analysis utilized a 2D MHD model for incompressible and viscous flow. The results from Case-1 indicate the presence of KH and TM instabilities, suggesting existence of observed low-amplitude oscillations at the nose of the magnetopause. However, the MHD simulations for Case-2 did not show any evidence of KH vortices, but did reveal the presence of "magnetic island" structures during a low-shear condition. The reconnection rate derived from the observations is compared with the computed one in the presence of developed instabilities inside the Earth's magnetopause. [ABSTRACT FROM AUTHOR]

While semidefinite programming (SDP) problems are polynomially solvable in theory, it is often difficult to solve large SDP instances in practice. One technique to address this issue is to relax the global positive-semidefiniteness (PSD) constraint and only enforce PSD-ness on smaller k × k principal submatrices—we call this the sparse SDP relaxation. Surprisingly, it has been observed empirically that in some cases this approach appears to produce bounds that are close to the optimal objective function value of the original SDP. In this paper, we formally attempt to compare the strength of the sparse SDP relaxation vis-à-vis the original SDP from a theoretical perspective. In order to simplify the question, we arrive at a data independent version of it, where we compare the sizes of SDP cone and the k -PSD closure, which is the cone of matrices where PSD-ness is enforced on all k × k principal submatrices. In particular, we investigate the question of how far a matrix of unit Frobenius norm in the k -PSD closure can be from the SDP cone. We provide two incomparable upper bounds on this farthest distance as a function of k and n. We also provide matching lower bounds, which show that the upper bounds are tight within a constant in different regimes of k and n. Other than linear algebra techniques, we extensively use probabilistic methods to arrive at these bounds. One of the lower bounds is obtained by observing a connection between matrices in the k -PSD closure and matrices satisfying the restricted isometry property. [ABSTRACT FROM AUTHOR]

We study continuous (strongly) minimal cut generating functions for the model where all variables are integer. We consider both the original Gomory-Johnson setting as well as a recent extension by Yıldız and Cornuéjols (Math Oper Res 41:1381-1403, 2016). We show that for any continuous minimal or strongly minimal cut generating function, there exists an extreme cut generating function that approximates the (strongly) minimal function as closely as desired. In other words, the extreme functions are “dense” in the set of continuous (strongly) minimal functions. [ABSTRACT FROM AUTHOR]

Full strong-branching (henceforth referred to as strong-branching) is a well-known variable selection rule that is known experimentally to produce significantly smaller branch-and-bound trees in comparison to all other known variable selection rules. In this paper, we attempt an analysis of the performance of the strong-branching rule both from a theoretical and a computational perspective. On the positive side for strong-branching, we identify vertex cover as a class of instances where this rule provably works well. In particular, for vertex cover we present: (1) an upper bound on the size of the branch-and-bound tree using strong-branching as a function of the additive integrality gap; (2) show how the Nemhauser-Trotter property of persistency, which can be used as a pre-solve technique for vertex cover, is being recursively and consistently used throughout the strong-branching tree; (3) and finally provide an example of a vertex cover instance where not using strong-branching leads to a tree that has at least exponentially more nodes than the branch-and-bound tree based on strong-branching. On the negative side for strong-branching, we identify another class of instances where the strong-branching tree is exponentially larger than another branch-and-bound tree for solving these instances. On the computational side, we conduct experiments on various types of instances, like the lot-sizing problem and its variants, packing integer programs (IP), covering IPs, chance constrained IPs, vertex cover, etc., to understand how much larger is the size of the strong-branching based branch-and-bound tree in comparison to the optimal branch-and-bound tree. The main take-away from these experiments (on small instances) is that for all these instances the size of the strong-branching tree is within a factor of two of the size of the optimal branch-and-bound tree. [ABSTRACT FROM AUTHOR]

Feasibility pump is a successful primal heuristic for mixed-integer linear programs. The algorithm consists of three main components: rounding fractional solution to a mixed-integer one, projection of infeasible solutions to the linear programming relaxation, and a randomization step used when the algorithm stalls. While many generalizations and improvements to the original Feasibility Pump have been proposed, they mainly focus on the rounding and projection steps. We start a more in-depth study of the randomization step in Feasibility Pump. For that, we propose a new randomization step based on the WalkSAT algorithm for solving instances of the Boolean satisfiability problem. First, we provide theoretical analyses for instances with disjoint equality constraints that show the potential of this randomization step; to the best of our knowledge, this is the first time any theoretical analysis of the running-time of Feasibility Pump or its variants has been conducted, even for a special class of instances. Moreover, we propose a practical version of a new randomization step, and incorporate it into a state-of-the-art Feasibility Pump code. Our experiments suggests that this simple-to-implement modification consistently dominates the standard randomization previously used. [ABSTRACT FROM AUTHOR]

Mixed-integer quadratic programming is the problem of optimizing a quadratic function over points in a polyhedral set where some of the components are restricted to be integral. In this paper, we prove that the decision version of mixed-integer quadratic programming is in NP, thereby showing that it is NP-complete. This is established by showing that if the decision version of mixed-integer quadratic programming is feasible, then there exists a solution of polynomial size. This result generalizes and unifies classical results that quadratic programming is in NP (Vavasis in Inf Process Lett 36(2):73-77 [17]) and integer linear programming is in NP (Borosh and Treybig in Proc Am Math Soc 55:299-304 [1], von zur Gathen and Sieveking in Proc Am Math Soc 72:155-158 [18], Kannan and Monma in Lecture Notes in Economics and Mathematical Systems, vol. 157, pp. 161-172. Springer [9], Papadimitriou in J Assoc Comput Mach 28:765-768 [15]). [ABSTRACT FROM AUTHOR]

A function $$f(x_1, \ldots , x_d)$$ , where each input is an integer from 1 to $$n$$ and output is a real number, is Lipschitz if changing one of the inputs by 1 changes the output by at most 1. In other words, Lipschitz functions are not very sensitive to small changes in the input. Our main result is an efficient tester for the Lipschitz property of functions $$f: [n]^d \rightarrow \delta {\mathbb {Z}}$$ , where $$\delta \in (0,1]$$ and $$\delta {\mathbb {Z}}$$ is the set of integer multiples of $$\delta $$ . A property tester is given an oracle access to a function $$f$$ and a proximity parameter $$\epsilon $$ , and it has to distinguish, with high probability, functions that have the property from functions that differ on at least an $$\epsilon $$ fraction of values from every function with the property. The Lipschitz property was first studied by Jha and Raskhodnikova (FOCS'11) who motivated it by applications to data privacy and program verification. They presented efficient testers for the Lipschitz property of functions on the domains $$\{0,1\}^d$$ and $$[n]$$ . Our tester for functions on the more general domain $$[n]^d$$ runs in time $$O(d^{1.5} n\log n)$$ for constant $$\epsilon $$ and $$\delta $$ . The main tool in the analysis of our tester is a smoothing procedure that makes a function Lipschitz by modifying it at a few points. Its analysis is already nontrivial for the 1-dimensional version, which we call Bubble Smooth, in analogy to Bubble Sort. In one step, Bubble Smooth modifies two values that violate the Lipschitz property, namely, differ by more than 1, by transferring $$\delta $$ units from the larger to the smaller. We define a transfer graph to keep track of the transfers, and use it to show that the $$\ell _1$$ distance between $$f$$ and BubbleSmooth $$(f)$$ is at most twice the $$\ell _1$$ distance from $$f$$ to the nearest Lipschitz function. Bubble Smooth has several other important properties that allow us to obtain a dimension reduction, i.e., a reduction from testing functions on multidimensional domains to testing functions on one-dimensional domains. Our dimension reduction incurs only a small multiplicative overhead in the running time and thus avoids the exponential dependence on the dimension. [ABSTRACT FROM AUTHOR]

We investigate the information complexity of mixed-integer convex optimization under different types of oracles. We establish new lower bounds for the standard first-order oracle, improving upon the previous best known lower bound. This leaves only a lower order linear term (in the dimension) as the gap between the lower and upper bounds. This is derived as a corollary of a more fundamental “transfer” result that shows how lower bounds on information complexity of continuous convex optimization under different oracles can be transferred to the mixed-integer setting in a black-box manner. Further, we (to the best of our knowledge) initiate the study of, and obtain the first set of results on, information complexity under oracles that only reveal partial first-order information, e.g., where one can only make a binary query over the function value or subgradient at a given point. We give algorithms for (mixed-integer) convex optimization that work under these less informative oracles. We also give lower bounds showing that, for some of these oracles, every algorithm requires more iterations to achieve a target error compared to when complete first-order information is available. That is, these oracles are provably less informative than full first-order oracles for the purpose of optimization. [ABSTRACT FROM AUTHOR]

• Definition of the Time Constrained Learning task. • Design of TCT a practical algorithm for the Time Constrained Learning task. • Experimental evidence performs better than some competitors for the task. • Provable guarantees that our algorithm is never much worse than random sampling. • Provable guarantees that TCT is much better than random sampling for some settings. Consider a scenario in which we have a huge labeled dataset D and a limited time to train a given learner using D. Since we may not be able to use the whole dataset, how should we proceed? We propose TCT, an algorithm for this task, whose design relies on principles from Machine Teaching. We present an experimental study involving 5 different learners and 20 datasets where we show that TCT consistently outperforms alternative teaching/training methods, namely: (1) Training over batches of random samples, until the time limit is reached; (2) The state-of-the-art Machine Teaching algorithm for black-box learners proposed in [Dasgupta et al., ICML 19], and (3) Stochastic Gradient Descent (when applicable). While our work is primarily practical, we also show that a stripped-down version of TCT has provable guarantees. [ABSTRACT FROM AUTHOR]

In this paper, we present lower bounds on the rank of the split closure, the multibranch closure, and the lattice-free closure for packing sets as a function of the integrality gap. We also provide a similar lower bound on the split rank of covering polyhedra. These results indicate that whenever the integrality gap is high, these classes of cutting planes must necessarily be applied for many rounds in order to obtain the integer hull. [ABSTRACT FROM AUTHOR]

In this paper, we present lower bounds on the rank of the split closure, the multibranch closure, and the lattice-free closure for packing sets as a function of the integrality gap. We also provide a similar lower bound on the split rank of covering polyhedra. These results indicate that whenever the integrality gap is high, these classes of cutting planes must necessarily be applied for many rounds in order to obtain the integer hull. [ABSTRACT FROM AUTHOR]

In this paper, we study the strength of Chvátal-Gomory (CG) cuts and more generally aggregation cuts for packing and covering integer programs (IPs). Aggregation cuts are obtained as follows: given an IP formulation, we first generate a single implied inequality using aggregation of the original constraints, then obtain the integer hull of the set defined by this single inequality with variable bounds, and finally use the inequalities describing the integer hull as cutting-planes. Our first main result is to show that for packing and covering IPs, the CG and aggregation closures can be 2-approximated by simply generating the respective closures for each of the original formulation constraints, without using any aggregations. On the other hand, we use computational experiments to show that aggregation cuts can be arbitrarily stronger than cuts from individual constraints for general IPs. The proof of the above stated results for the case of covering IPs with bounds require the development of some new structural results, which may be of independent interest. Finally, we examine the strength of cuts based on k different aggregation inequalities simultaneously, the so-called multi-row cuts, and show that every packing or covering IP with a large integrality gap also has a large k-aggregation closure rank. In particular, this rank is always at least of the order of the logarithm of the integrality gap. [ABSTRACT FROM AUTHOR]

We investigate how well the graph of a bilinear function $$b{:}\;[0,1]^n\rightarrow \mathbb {R}$$ can be approximated by its McCormick relaxation. In particular, we are interested in the smallest number c such that the difference between the concave upper bounding and convex lower bounding functions obtained from the McCormick relaxation approach is at most c times the difference between the concave and convex envelopes. Answering a question of Luedtke, Namazifar and Linderoth, we show that this factor c cannot be bounded by a constant independent of n. More precisely, we show that for a random bilinear function b we have asymptotically almost surely $$c\geqslant \sqrt{n}/4$$ . On the other hand, we prove that $$c\leqslant 600\sqrt{n}$$ , which improves the linear upper bound proved by Luedtke, Namazifar and Linderoth. In addition, we present an alternative proof for a result of Misener, Smadbeck and Floudas characterizing functions b for which the McCormick relaxation is equal to the convex hull. [ABSTRACT FROM AUTHOR]

We study the following tree search problem: in a given tree T=( V, E) a vertex has been marked and we want to identify it. In order to locate the marked vertex, we can use edge queries. An edge query e asks in which of the two connected components of T∖ e the marked vertex lies. The worst-case scenario where one is interested in minimizing the maximum number of queries is well understood, and linear time algorithms are known for finding an optimal search strategy. Here we study the more involved average-case analysis: A function w: V→ℝ is given which measures the likelihood for a vertex to be the one marked, and we seek to determine the strategy (decision tree) that minimizes the weighted average number of queries. In a companion paper we prove that the above tree search problem is $\mathcal {NP}$-complete even for the class of trees of bounded diameter or bounded degree. Here, we match this complexity result with a tight algorithmic analysis of the bounded degree instances. We show that any optimal strategy (i.e., one that minimizes the weighted average number of queries) performs at most O(Δ( T)(log| V|+log( w( T)/ w))) queries in the worst case, where w( T) is the sum of the likelihoods of the vertices of T, w is the minimum positive likelihood over the vertices of T and Δ( T) is the maximum degree of T. We combine this result with a non-trivial exponential time algorithm to provide an FPTAS for trees with bounded degree. We also show that for unbounded instances a natural greedy strategy attains a 1.62-approximation, improving upon the best known 14-approximation guarantee, previously provided by two of the authors. [ABSTRACT FROM AUTHOR]

We prove that any minimal valid function for the k-dimensional infinite group relaxation that is continuous piecewise linear with at most k + 1 slopes and does not factor through a linear map with nontrivial kernel is extreme. This generalizes a theorem of Gomory and Johnson for k = 1 and of Cornuéjols and Molinaro for k = 2. [ABSTRACT FROM AUTHOR]

Abstract: We study the problem of minimizing the weighted average number of queries to identify an initially unknown object in a poset. We show that for general posets, there cannot be any -approximation algorithm unless . When the Hasse diagram of the partially ordered set has the structure of a tree, the problem is equivalent to the following tree search problem: in a given rooted tree a node has been marked and we want to identify it. In order to locate the marked node, we can perform node queries. A node query asks whether the marked node lies in the subtree rooted at . A function is given which defines the likelihood for a node to be the one marked, and we want the strategy that minimizes the expected number of queries. Prior to this paper the complexity of this problem had remained open. We prove that the above tree search problem is -complete even for the class of trees with diameter at most 4. This results in a complete characterization of the complexity of the problem with respect to the diameter size. In fact, for diameter not larger than 3 we show that the problem is polynomially solvable using a dynamic programming approach. In addition we prove that the problem is -complete even for the class of trees of maximum degree at most 16. To the best of our knowledge, the only known result in this direction is that the tree search problem is solvable in time for trees with degree at most 2 (paths). Our results sharply contrast with those for the variant of the problem where one is interested in minimizing the maximum number of queries. In fact, for the worst case scenario, linear time algorithms are known for finding an optimal search strategy [K. Onak, P. Parys, Generalization of binary search: searching in trees and forest-like partial orders, in: FOCS’06: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Washington, DC, USA, 2006, pp. 379–388; S. Mozes, K. Onak, O. Weimann, Finding an optimal tree searching strategy in linear time, in: SODA’08: Proceedings of the Nineteenth Annual ACM–SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2008, pp. 1096–1105]. [Copyright &y& Elsevier]

In this paper we report the development of two attachments to a commercial cell phone that transform the phone's integrated lens and image sensor into a 350× microscope and visible-light spectrometer. The microscope is capable of transmission and polarized microscopy modes and is shown to have 1.5 micron resolution and a usable field-of-view of ∼150×150 ηm with no image processing, and approximately 350×350 ηm when post-processing is applied. The spectrometer has a 300 nm bandwidth with a limiting spectral resolution of close to 5 nm. We show applications of the devices to medically relevant problems. In the case of the microscope, we image both stained and unstained blood-smears showing the ability to acquire images of similar quality to commercial microscope platforms, thus allowing diagnosis of clinical pathologies. With the spectrometer we demonstrate acquisition of a white-light transmission spectrum through diffuse tissue as well as the acquisition of a fluorescence spectrum. We also envision the devices to have immediate relevance in the educational field. [ABSTRACT FROM AUTHOR]

Presents information on the development and testing of topical modules for the first two years of the college chemistry curriculum by the ChemLinks Coalition and Modular Chemistry Consortium, which is based at Beloit College and the University of California-Berkeley. Evaluation of the modular approach; Insight on the use of multimedia in the modular approach.