We consider a class of non-conjugate priors as a mixing family of distributions for a parameter (e.g., Poisson or gamma rate, inverse scale or precision of an inverse-gamma, inverse variance of a normal distribution) of an exponential subclass of discrete and continuous data distributions. The prior class is proper, nonzero at the origin (unlike the gamma and inverted beta priors with shape parameter less than one and Jeffreys prior for a Poisson rate), and is easy to generate random numbers from. The prior class also provides flexibility in capturing a wide array of prior beliefs (right-skewed and left-skewed) as modulated by a bounded parameter $\alpha \in (0, 1).$ The resulting posterior family in the single-parameter case can be expressed in closed-form and is proper, making calibration unnecessary. The mixing induced by the inverse stable family results to a marginal prior distribution in the form of a generalized Mittag-Leffler function, which covers a broad array of distributional shapes. We derive closed-form expressions of some properties like the moment generating function and moments. We propose algorithms to generate samples from the posterior distribution and calculate the Bayes estimators for real data analysis. We formulate the predictive prior and posterior distributions. We test the proposed Bayes estimators using Monte Carlo simulations. The extension to hierarchical modeling and inverse variance components models is straightforward. We can find $\alpha$ (which acts like a smoothing parameter) values for which the inverse stable can provide better shrinkage than the inverted beta prior in many cases. We illustrate the methodology using a real data set, introduce a hyperprior density for the hyperparameters, and extend the model to a heavy-tailed distribution.

Data from domains such as social networks, healthcare, finance, and cybersecurity can be represented as graph-structured information. Given the sensitive nature of this data and their frequent distribution among collaborators, ensuring secure and attributable sharing is essential. Graph watermarking enables attribution by embedding user-specific signatures into graph-structured data. While prior work has addressed random perturbation attacks, the threat posed by adversaries leveraging structural properties through community detection remains unexplored. In this work, we introduce a cluster-aware threat model in which adversaries apply community-guided modifications to evade detection. We propose two novel attack strategies and evaluate them on real-world social network graphs. Our results show that cluster-aware attacks can reduce attribution accuracy by up to 80% more than random baselines under equivalent perturbation budgets on sparse graphs. To mitigate this threat, we propose a lightweight embedding enhancement that distributes watermark nodes across graph communities. This approach improves attribution accuracy by up to 60% under attack on dense graphs, without increasing runtime or structural distortion. Our findings underscore the importance of cluster-topological awareness in both watermarking design and adversarial modeling.

Information can be expressed in multiple formats including natural language, images, and motions. Human intelligence usually faces little difficulty to convert from one format to another format, which often shows a true understanding of encoded information. Moreover, such conversions have broad application in many real-world applications. In this paper, we present a text visualization system that can visualize free text with animations. Our system is illustrated by visualizing example sentences of elementary Physics laws.

When sharing sensitive relational databases with other parties, a database owner aims to (i) have privacy guarantees for the database entries, (ii) have liability guarantees (via fingerprinting) in case of unauthorized sharing of its database by the recipients, and (iii) provide a high quality (utility) database to the recipients. We observe that sharing a relational database with privacy and liability guarantees are orthogonal objectives. The former can be achieved by injecting noise into the database to prevent inference of the original data values, whereas, the latter can be achieved by hiding unique marks inside the database to trace malicious parties (data recipients) who redistribute the data without the authorization. We achieve these two objectives simultaneously by proposing a novel entry-level differentially-private fingerprinting mechanism for relational databases. At a high level, the proposed mechanism fulfills the privacy and liability requirements by leveraging the randomization nature that is intrinsic to fingerprinting and achieves desired entry-level privacy guarantees. To be more specific, we devise a bit-level random response scheme to achieve differential privacy guarantee for arbitrary data entries when sharing the entire database, and then, based on this, we develop an $\epsilon$-entry-level differentially-private fingerprinting mechanism. Next, we theoretically analyze the relationships between privacy guarantee, fingerprint robustness, and database utility by deriving closed form expressions. The outcome of this analysis allows us to bound the privacy leakage caused by attribute inference attack and characterize the privacy-utility coupling and privacy-fingerprint robustness coupling. Furthermore, we also propose a SVT-based solution to control the cumulative privacy loss when fingerprinted copies of a database are shared with multiple recipients.

Ambient air pollution poses significant health and environmental challenges. Exposure to high concentrations of PM$_{2.5}$ have been linked to increased respiratory and cardiovascular hospital admissions, more emergency department visits and deaths. Traditional air quality monitoring systems such as EPA-certified stations provide limited spatial and temporal data. The advent of low-cost sensors has dramatically improved the granularity of air quality data, enabling real-time, high-resolution monitoring. This study exploits the extensive data from PurpleAir sensors to assess and compare the effectiveness of various statistical and machine learning models in producing accurate hourly PM$_{2.5}$ maps across California. We evaluate traditional geostatistical methods, including kriging and land use regression, against advanced machine learning approaches such as neural networks, random forests, and support vector machines, as well as ensemble model. Our findings enhanced the predictive accuracy of PM2.5 concentration by correcting the bias in PurpleAir data with an ensemble model, which incorporating both spatiotemporal dependencies and machine learning models.

Objective: The aim of this study was to build an effective co-reference resolution system tailored for the biomedical domain. Materials and Methods: Experiment materials used in this study is provided by the 2011 i2b2 Natural Language Processing Challenge. The 2011 i2b2 challenge involves coreference resolution in medical documents. Concept mentions have been annotated in clinical texts, and the mentions that co-refer in each document are to be linked by coreference chains. Normally, there are two ways of constructing a system to automatically discover co-referent links. One is to manually build rules for co-reference resolution, and the other category of approaches is to use machine learning systems to learn automatically from training datasets and then perform the resolution task on testing datasets. Results: Experiments show the existing co-reference resolution systems are able to find some of the co-referent links, and our rule based system performs well finding the majority of the co-referent links. Our system achieved 89.6% overall performance on multiple medical datasets. Conclusion: The experiment results show that manually crafted rules based on observation of training data is a valid way to accomplish high performance in this coreference resolution task for the critical biomedical domain.

We define a certain merging operation that given two $d$-polytopes $P$ and $Q$ such that $P$ has a simplex facet $F$ and $Q$ has a simple vertex $v$ produces a new $d$-polytope $P\hspace{0.1em}\triangleright Q$ with $f_0(P)+f_0(Q)-(d+1)$ vertices. We show that if for some $1\leq i\leq d-1$, $P$ and $Q$ are $(d-i)$-simplicial $i$-simple $d$-polytopes, then so is $P\hspace{0.1em}\triangleright Q$. We then use this operation to construct new families of $(d-i)$-simplicial $i$-simple $d$-polytopes. Specifically, we prove that for all $2\leq i \leq d-2\leq 6$ with the exception of $(i,d)=(3,8)$ and $(5,8)$, there is an infinite family of $(d-i)$-simplicial $i$-simple $d$-polytopes; furthermore, for all $2\leq i\leq 4$, there is an infinite family of self-dual $i$-simplicial $i$-simple $2i$-polytopes. Finally, we show that for any $d\geq 4$, there are $2^{\Omega(N)}$ combinatorial types of $(d-2)$-simplicial $2$-simple $d$-polytopes with at most $N$ vertices.

We describe a new method for finding analytic solutions to some initial-boundary problems for partial differential equations with constant coefficients. The method is based on expanding the denominator of the Laplace transformed Green's function of the problem into a convergent geometric series. If the denominator is a linear combination of exponents with real powers one obtains a closed form solution as a sum with finite but time dependent number of terms. We call it a d'Alembert sum. This representation is computationally most effective for small evolution times, but it remains valid even when the system of eigenmodes is incomplete and the eigenmode expansion is unavailable. Moreover, it simplifies in such cases. In vibratory problems d'Alembert sums represent superpositions of original and partially reflected traveling waves. They generalize the d'Alembert type formulas for the wave equation, and reduce to them when original waves can undergo only finitely many reflections in the entire course of evolution. The method is applied to vibrations of a bar with dampers at each end and at some internal point. The results are illustrated by computer simulations and comparisons to modal and FEM solutions.

In this paper, we study (zero) forcing sets which induce connected subgraphs of a graph. The minimum cardinality of such a set is called the connected forcing number of the graph. We provide sharp upper and lower bounds on the connected forcing number in terms of the minimum degree, maximum degree, girth, and order of the graph.

Word sense disambiguation (WSD) is one of the main challenges in Computational Linguistics. TreeMatch is a WSD system originally developed using data from SemEval 2007 Task 7 (Coarse-grained English All-words Task) that has been adapted for use in SemEval 2010 Task 17 (All-words Word Sense Disambiguation on a Specific Domain). The system is based on a fully unsupervised method using dependency knowledge drawn from a domain specific knowledge base that was built for this task. When evaluated on the task, the system precision performs above the Most Frequent Selection baseline.

We define a certain merging operation that given two $d$-polytopes $P$ and $Q$ such that $P$ has a simplex facet $F$ and $Q$ has a simple vertex $v$ produces a new $d$-polytope $P\hspace{0.1em}\triangleright Q$ with $f_0(P)+f_0(Q)-(d+1)$ vertices. We show that if for some $1\leq i\leq d-1$, $P$ and $Q$ are $(d-i)$-simplicial $i$-simple $d$-polytopes, then so is $P\hspace{0.1em}\triangleright Q$. We then use this operation to construct new families of $(d-i)$-simplicial $i$-simple $d$-polytopes. Specifically, we prove that for all $2\leq i \leq d-2\leq 6$ with the exception of $(i,d)=(3,8)$ and $(5,8)$, there is an infinite family of $(d-i)$-simplicial $i$-simple $d$-polytopes; furthermore, for all $2\leq i\leq 4$, there is an infinite family of self-dual $i$-simplicial $i$-simple $2i$-polytopes. Finally, we show that for any $d\geq 4$, there are $2^{\Omega(N)}$ combinatorial types of $(d-2)$-simplicial $2$-simple $d$-polytopes with at most $N$ vertices.

We argue that traditional formulations of the reduction thesis that tie it to privileged relational operations do not suffice for Peirce's justification of the categories, and invite the charge of gerrymandering to make it come out as true. We then develop a more robust invariant formulation of the thesis by explicating the use of triads in any relational operations, which is immune to that charge. The explication also allows us to track how Thirdness enters the structure of higher order relations, and even propose a numerical measure of it. Our analysis reveals new conceptual phenomena when negation or disjunction are used to compound relations.

We introduce discrete analogues of the exponential, sine, and cosine functions. Then using a discrete trigonometric version of a non-polynomial divided difference, we define discrete analogues of the trigonometric B-splines. We derive a two-term recurrence relation, a two-term formula for the discrete derivative, and two variants of the Marsden identity for these discrete trigonometric B-splines. Since the classical exponential, sine, and cosine functions are limiting cases of their discrete analogues, we conclude that many of the standard results for classical polynomial B-splines extend naturally both to trigonometric B-splines and to discrete trigonometric B-splines.

A binocular system developed by the author in terms of projective Fourier transform (PFT) of the conformal camera, which numerically integrates the head, eyes, and visual cortex, is used to process visual information during saccadic eye movements. Although we make three saccades per second at the eyeball's maximum speed of 700 deg/sec, our visual system accounts for these incisive eye movements to produce a stable percept of the world. This visual constancy is maintained by neuronal receptive field shifts in various retinotopically organized cortical areas prior to saccade onset, giving the brain access to visual information from the saccade's target before the eyes' arrival. It integrates visual information acquisition across saccades. Our modeling utilizes basic properties of PFT. First, PFT is computable by FFT in complex logarithmic coordinates that approximate the retinotopy. Second, a translation in retinotopic (logarithmic) coordinates, modeled by the shift property of the Fourier transform, remaps the presaccadic scene into a postsaccadic reference frame. It also accounts for the perisaccadic mislocalization observed by human subjects in laboratory experiments. Because our modeling involves cross-disciplinary areas of conformal geometry, abstract and computational harmonic analysis, computational vision, and visual neuroscience, we include the corresponding background material and elucidate how these different areas interwove in our modeling of primate perception. In particular, we present the physiological and behavioral facts underlying the neural processes related to our modeling. We also emphasize the conformal camera's geometry and discuss how it is uniquely useful in the intermediate-level vision computational aspects of natural scene understanding.

Wittgenstein's paradoxical theses that unproved propositions are meaningless, proofs form new concepts and rules, and contradictions are of limited concern, led to a variety of interpretations, most of them centered on the rule-following skepticism. We argue that his intuitions rather reflect resistance to treating meaning as fixed content, and are better understood in the light of C.S. Peirce's distinction between corollarial and theorematic proofs. We show how Peirce's insight that "all necessary reasoning is diagrammatic", vindicated in modern epistemic logic and semantic information theory, helps explain the paradoxical ability of deduction to generate new knowledge and meaning.

Ambient air pollution measurements from regulatory monitoring networks are routinely used to support epidemiologic studies and environmental policy decision making. However, regulatory monitors are spatially sparse and preferentially located in areas with large populations. Numerical air pollution model output can be leveraged into the inference and prediction of air pollution data combining with measurements from monitors. Nonstationary covariance functions allow the model to adapt to spatial surfaces whose variability changes with location like air pollution data. In the paper, we employ localized covariance parameters learned from the numerical output model to knit together into a global nonstationary covariance, to incorporate in a fully Bayesian model. We model the nonstationary structure in a computationally efficient way to make the Bayesian model scalable.

Database fingerprinting have been widely adopted to prevent unauthorized sharing of data and identify the source of data leakages. Although existing schemes are robust against common attacks, like random bit flipping and subset attack, their robustness degrades significantly if attackers utilize the inherent correlations among database entries. In this paper, we first demonstrate the vulnerability of existing database fingerprinting schemes by identifying different correlation attacks: column-wise correlation attack, row-wise correlation attack, and the integration of them. To provide robust fingerprinting against the identified correlation attacks, we then develop mitigation techniques, which can work as post-processing steps for any off-the-shelf database fingerprinting schemes. The proposed mitigation techniques also preserve the utility of the fingerprinted database considering different utility metrics. We empirically investigate the impact of the identified correlation attacks and the performance of mitigation techniques using real-world relational databases. Our results show (i) high success rates of the identified correlation attacks against existing fingerprinting schemes (e.g., the integrated correlation attack can distort 64.8\% fingerprint bits by just modifying 14.2\% entries in a fingerprinted database), and (ii) high robustness of the proposed mitigation techniques (e.g., with the mitigation techniques, the integrated correlation attack can only distort $3\%$ fingerprint bits).

We show that a number of models in virus dynamics, epidemiology and plant biology can be presented as ``damped" versions of the Lotka-Volterra predator-prey model, by analogy to the damped harmonic oscillator. The analogy deepens with the use of Lyapunov functions, which allow us to characterize their dynamics and even make some estimates.

We investigate longitudinal vibrations of a bar subjected to viscous boundary conditions at each end, and an internal damper at an arbitrary point along the bar's length. The system is described by four independent parameters and exhibits a variety of behaviors including rigid motion, super stability/instability and zero damping. The solution is obtained by applying the Laplace transform to the equation of motion and computing the Green's function of the transformed problem. This leads to an unconventional eigenvalue-like problem with the spectral variable in the boundary conditions. The eigenmodes of the problem are necessarily complex-valued and are not orthogonal in the usual inner product. Nonetheless, in generic cases we obtain an explicit eigenmode expansion for the response of the bar to initial conditions and external force. For some special values of parameters the system of eigenmodes may become incomplete, or no non-trivial eigenmodes may exist at all. We thoroughly analyze physical and mathematical reasons for this behavior and explicitly identify the corresponding parameter values. In particular, when no eigenmodes exist, we obtain closed form solutions. Theoretical analysis is complemented by numerical simulations, and analytic solutions are compared to computations using finite elements.