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This study investigates the reconstruction of hyperspectral signatures from RGB data to enhance surgical imaging, utilizing the publicly available HeiPorSPECTRAL dataset from porcine surgery and an in-house neurosurgery dataset. Various architectures based on convolutional neural networks (CNNs) and transformer models are evaluated using comprehensive metrics. Transformer models exhibit superior performance in terms of RMSE, SAM, PSNR and SSIM by effectively integrating spatial information to predict accurate spectral profiles, encompassing both visible and extended spectral ranges. Qualitative assessments demonstrate the capability to predict spectral profiles critical for informed surgical decision-making during procedures. Challenges associated with capturing both the visible and extended hyperspectral ranges are highlighted using the MAE, emphasizing the complexities involved. The findings open up the new research direction of hyperspectral reconstruction for surgical applications and clinical use cases in real-time surgical environments., Comment: 10 pages, 4 figures, 3 tables

This paper introduces a novel, non-deterministic method for embedding data in low-dimensional Euclidean space based on computing realizations of a Gaussian process depending on the geometry of the data. This type of embedding first appeared in (Adler et al, 2018) as a theoretical model for a generic manifold in high dimensions. In particular, we take the covariance function of the Gaussian process to be the heat kernel, and computing the embedding amounts to sketching a matrix representing the heat kernel. The Karhunen-Lo\`eve expansion reveals that the straight-line distances in the embedding approximate the diffusion distance in a probabilistic sense, avoiding the need for sharp cutoffs and maintaining some of the smaller-scale structure. Our method demonstrates further advantage in its robustness to outliers. We justify the approach with both theory and experiments., Comment: 28 pages

The purpose of this paper is to address a manifold-based version of Whitney's extension problem: Given a compact set $E\subset\mathbb{R}^n$, how can we tell if there exists a $d$-dimensional, $C^m$-smooth manifold $\mathcal{M}\supset E$? We provide an answer for compact manifolds with boundary in terms of a Glaeser refinement much like that used in the solution of the classical Whitney extension problem and a topological condition. This condition is the existence of a continuous selection for Grassmannian-valued functions, meant to reflect the collection of possible tangent spaces. We demonstrate the necessity of this condition in general and its non-redundancy in an example, while also showing it need not be checked when $d=1$., Comment: 37 pages, updated with a new example and some minor edits

The success of algorithms in the analysis of high-dimensional data is often attributed to the manifold hypothesis, which supposes that this data lie on or near a manifold of much lower dimension. It is often useful to determine or estimate the dimension of this manifold before performing dimension reduction, for instance. Existing methods for dimension estimation are calibrated using a flat unit ball. In this paper, we develop CA-PCA, a version of local PCA based instead on a calibration of a quadratic embedding, acknowledging the curvature of the underlying manifold. Numerous careful experiments show that this adaptation improves the estimator in a wide range of settings., Comment: Updates to calculations and text after small error found in computation of constant used in main algorithm. Experiments rerun, conclusions remain the same. Some new experiments also added

Recent work of C. Fefferman and the first author has demonstrated that the linear system of equations \begin{equation*} \sum_{j=1}^M A_{ij}(x)F_j(x)=f_i(x)\hspace{.2in} (i=1,...,N), \end{equation*} has a $C^m$ solution $F=(F_1,...,F_M)$ if and only if $f_1,...,f_N$ satisfy a certain finite collection of partial differential equations. Here, the $A_{ij}$ are fixed semialgebraic functions. In this paper, we consider the analogous problem for systems of linear inequalities: \begin{equation*} \sum_{j=1}^M A_{ij}(x)F_j(x)\le f_i(x)\hspace{.2in} (i=1,...,N). \end{equation*} Our main result is a negative one, demonstrated by counterexample: the existence of a $C^m$ solution $F$ may not, in general, be determined via an analogous finite set of partial differential inequalities in $f_1,...,f_N$., Comment: 14 pages, updated version fixes a few typos

Whitney's extension problem asks the following: Given a compact set $E\subset\mathbb{R}^n$ and a function $f:E\to \mathbb{R}$, how can we tell whether there exists $F\in C^m(\mathbb{R}^n)$ such that $F=f$ on $E$? A 2006 theorem of Charles Fefferman \cite{F06} answers this question in its full generality. In this paper, we establish a version of this theorem adapted for variants of the Whitney extension problem, including nonnegative extensions and the smooth selection problems. Among other things, we generalize the Finiteness Principle for smooth selection by Fefferman-Israel-Luli \cite{FIL16} to the setting of infinite sets. Our main result is stated in terms of the iterated Glaeser refinement of a bundle formed by taking potential Taylor polynomials at each point of $E$. In particular, we show that such bundles (and any bundles with closed, convex fibers) stabilize after a bounded number of Glaeser refinements, thus strengthening the previous results of Glaeser, Bierstone-Milman-Paw{\l}ucki, and Fefferman which only hold for bundles with affine fibers., Comment: 59 pages. The second version corrected a minor mistake in the previous version and some typos

In this paper, we improve the finiteness constant for the finiteness principles for $C^m(\mathbb{R}^n,\mathbb{R}^d)$ and $C^{m-1,1}(\mathbb{R}^n,\mathbb{R}^D)$ selection proven by Fefferman, Israel, and the second author and extend the more general shape fields finiteness principle to the vector-valued case., Comment: 17 pages

Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this paper, we use the expansion method to prove a quantitative version of this characterization., Comment: 22 pages

We consider an oscillatory integral operator with Loomis-Whitney multilinear form. The phase is real analytic in a neighborhood of the origin in $\mathbb{R}^d$ and satisfies a nondegeneracy condition related to its Newton polyhedron. Maximal decay is obtained for this operator in certain cases, depending on the Newton polyhedron of the phase and the given Lebesgue exponents. Our estimates imply volumes of sublevel sets of such real analytic functions are small relative to the product of areas of projections onto coordinate hyperplanes.

In this paper, we prove $L^p$ decay estimates for multilinear oscillatory integrals in $\mathbb{R}^2$, establishing sharpness through a scaling argument. The result in this paper is a generalization of the previous work by Gressman and Xiao (2016)., Comment: 7 pages

In this paper, we prove an $L^2-L^2-L^2$ decay estimate for a trilinear oscillatory integral of convolution type in $\mathbb{R}^d,$ which recovers the earlier result of Li (2013) when $d=1.$ We discuss the sharpness of our result in the $d=2$ case. Our main hypothesis has close connections to the property of simple nondegeneracy studied by Christ, Li, Tao and Thiele (2005)., Comment: Application of Theorem 2.1 in the proof of Theorem 1.2 assumed uniformity of the constant C which does not hold in general; in reality, there may be dependence on the phase

One may define a trilinear convolution form on the sphere involving two functions on the sphere and a monotonic function on the interval $[-1,1]$. A symmetrization inequality of Baernstein and Taylor states that this form is maximized when the two functions on the sphere are replaced with their nondecreasing symmetric rearrangements. In the case of indicator functions, we show that under natural hypotheses, the symmetric rearrangements are the only maximizers up to symmetry by establishing a sharpened inequality., Comment: Error found in reduction to small perturbations, the proof that Prop 2.1 implies Theorem 1.2

Consider the trilinear form for twisted convolution on $\mathbb{R}^{2d}$: \begin{equation*} \mathcal{T}_t(\mathbf{f}):=\iint f_1(x)f_2(y)f_3(x+y)e^{it\sigma(x,y)}dxdy,\end{equation*} where $\sigma$ is a symplectic form and $t$ is a real-valued parameter. It is known that in the case $t\neq0$ the optimal constant for twisted convolution is the same as that for convolution, though no extremizers exist. Expanding about the manifold of triples of maximizers and $t=0$ we prove a sharpened inequality for twisted convolution with an arbitrary antisymmetric form in place of $\sigma$., Comment: 13 pages

A symmetrization inequality of Rogers and of Brascamp-Lieb-Luttinger states that for a certain class of multilinear integral expressions, among tuples of sets of prescribed Lebesgue measures, tuples of balls centered at the origin are among the maximizers. Under natural hypotheses, we characterize all maximizing tuples for these inequalities for dimensions strictly greater than 1. We establish a sharpened form of the inequality.

The H\"older-Brascamp-Lieb inequalities are a collection of multilinear inequalities generalizing a convolution inequality of Young and the Loomis-Whitney inequalities. The full range of exponents was classified in Bennett et al. (2008). In a setting similar to that of Ivanisvili and Volberg (2015), we introduce a notion of size for these inequalities which generalizes $L^p$ norms. Under this new setup, we then determine necessary and sufficient conditions for a generalized H\"older-Brascamp-Lieb type inequality to hold and establish sufficient conditions for extremizers to exist when the underlying linear maps match those of the convolution inequality of Young., Comment: November update to clarify an argument in Section 3

A circular arc graph is the intersection graph of a collection of connected arcs on the circle. We solve a Tur'an-type problem for circular arc graphs: for n arcs, if m and M are the minimum and maximum number of arcs that contain a common point, what is the maximum number of edges the circular arc graph can contain? We establish a sharp bound and produce a maximal construction. For a fixed m, this can be used to show that if the circular arc graph has enough edges, there must be a point that is covered by at least M arcs. In the case m=0, we recover results for interval graphs established by Abbott and Katchalski (1979). We suggest applications to voting situations with interval or circular political spectra., Comment: 18 pages, 8 figures, related papers at http://www.math.hmc.edu/~su/papers.html

In 1999 the Texas Legislature provided for the creation of the Master Reading Teacher (MRT) Certificate to address several issues. MRTs primary duties are to: serve as reading teachers; consult with and mentor other teachers; and provide research-based information and training. In September 2001 the Texas Education Agency asked Dr. Jan Hasbrouck to conduct an evaluation of MRT training that occurred during the 2000-2001 academic year. There were five evaluation questions, but this paper discusses only two questions: To what extent have changes in service delivery due to MRT training had a positive effect on student standardized reading test scores (both for students directly taught by MRT program graduates and for students indirectly influenced by MRT program graduates); and From perceptions of both administrators and MRTs, what are the MRT program's greatest strengths and weaknesses? The paper states that the evaluation consisted of four activities: (1) statewide mail survey of all MRTs currently receiving the state stipend for MRTs serving on "high-need" campuses; (2) mail survey of the administrators of MRTs drawing a state stipend; (3) telephone follow-up survey of selected MRTs and administrators; and (4) Texas Assessment of Academic Skills (TAAS, the criterion referenced tests required of Texas public school students from the third-tenth grade) data collection by selected administrators. According to the paper, evaluators gained the impression of a moderately strong beginning of the MRT program, but a beginning that fell short of its potential. With better networking and improved guidelines on program implementation, the prediction is for stronger programs. (NKA)

Low-temperature three-terminal transport measurements through a thiol end-capped Pi -conjugated molecule have been carried out. Electronic excitations, including zero and finite-bias Kondo-effects have been observed and studied as a function of magnetic field. Using a simplified two-orbital model we have accounted for the spin and the electronic configuration of the first four charge states of the molecule. The charge-dependent couplings to gate, source and drain electrodes suggest a scenario in which charges and spins are localized at the ends of the molecule, close to the electrodes., Comment: Supporting information avalaible at http://pubs.acs.org

We provide a framework for reasoning about information-hiding requirements in multiagent systems and for reasoning about anonymity in particular. Our framework employs the modal logic of knowledge within the context of the runs and systems framework, much in the spirit of our earlier work on secrecy [Halpern and O'Neill 2002]. We give several definitions of anonymity with respect to agents, actions, and observers in multiagent systems, and we relate our definitions of anonymity to other definitions of information hiding, such as secrecy. We also give probabilistic definitions of anonymity that are able to quantify an observer s uncertainty about the state of the system. Finally, we relate our definitions of anonymity to other formalizations of anonymity and information hiding, including definitions of anonymity in the process algebra CSP and definitions of information hiding using function views., Comment: Replacement. 36 pages. Full version of CSFW '03 paper, submitted to JCS. Made substantial changes to Section 6; added references throughout

We introduce a general framework for reasoning about secrecy and privacy requirements in multiagent systems. Our definitions extend earlier definitions of secrecy and nondeducibility given by Shannon and Sutherland. Roughly speaking, one agent maintains secrecy with respect to another if the second agent cannot rule out any possibilities for the behavior or state of the first agent. We show that the framework can handle probability and nondeterminism in a clean way, is useful for reasoning about asynchronous systems as well as synchronous systems, and suggests generalizations of secrecy that may be useful for dealing with issues such as resource-bounded reasoning. We also show that a number of well-known attempts to characterize the absence of information flow are special cases of our definitions of secrecy., Comment: This is an extended version of a paper that appears in the Proceedings of the 15th IEEE Computer Security Foundations Workshop}, 2002, pp. 32-46