Athanassios S. Fokas

Medical image reconstruction began with an exact mathematical result. In 1917, Radon showed how to recover a function from its line integrals, providing the mathematical basis for computed tomography, positron emission tomography, and single-photon emission computed tomography. This review follows the development of reconstruction from analytical inversion to modern deep learning. We first present the Radon transform, its attenuated and three-dimensional generalizations, and the corresponding inversion formulae, including filtered backprojection. We then review iterative and optimization-based methods, including maximum-likelihood expectation-maximization, variational regularization, and compressed sensing. The discussion proceeds to deep learning, covering convolutional and transformer architectures, algorithm unrolling, physics-informed networks, neural operators, and generative diffusion priors. Our central argument is that analytical, iterative, and learned reconstruction are different formulations of the same inverse problem rather than competing paradigms. Many learned components derive from classical formulations, and many classical operators admit learnable counterparts. We conclude by discussing open challenges related to theoretical guarantees, trustworthiness, data efficiency, federated learning, and regulatory validation. The most reliable learned systems are likely to retain the mathematical and physical structure of established reconstruction methods.

A simple formula is derived for the difference of the functions defining the error terms in Atkinson’s formula and in the formula for the Dirichlet divisor problem. Namely, this difference equals π/2 plus a function that is simply related to the square of the Riemann zeta function. Specifically, the large-t leading-order asymptotics of this function is expressed in terms of a double sum that differs from the large-t asymptotics of (ζ(1/2+it))2 only in the factor of 1/ln⁡(2πmn/t). Interestingly, to the leading order for large t, the derivative of the above function coincides with the square of the absolute value of the Riemann zeta function. This remarkable fact implies a very simple formula for the derivative of the function defining the error term in the Dirichlet divisor problem.

In this paper, we solve explicitly and analyze rigorously inhomogeneous initial-boundary-value problems (IBVP) for several fourth-order variations of the traditional diffusion equation and the associated linearized Cahn–Hilliard (C-H) model (also Kuramoto–Sivashinsky equation), formulated in the spatiotemporal quarter-plane. Such models are of relevance to heat-mass transfer phenomena, solid-fluid dynamics and the applied sciences. In particular, we derive formally effective solution representations, justifying a posteriori their validity. This includes the reconstruction of the prescribed initial and boundary data, which requires careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula is utilized to rigorously deduce the solution’s regularity and asymptotic properties near the boundaries of the domain, including uniform convergence, eventual (long-time) periodicity under (eventually) periodic boundary conditions, and null noncontrollability. Importantly, this analysis is indispensable for exploring the (non)uniqueness of the problem’s solution and a new counter-example is constructed. Our work is based on the synergy between: (i) the well-known Fokas unified transform method and (ii) a new approach recently introduced for the rigorous analysis of the Fokas method and for investigating qualitative properties of linear evolution partial differential equations (PDE) on semi-infinite strips. Since only up to third-order evolution PDE have been investigated within this novel framework to date, we present our analysis and results in an illustrative manner and in order of progressively greater complexity, for the convenience of readers. The solution formulae established herein are expected to find utility in well-posedness and asymptotics studies for nonlinear counterparts too.

The central and autonomic communications affecting cognition, emotions, and visceral functions, are determined by the interaction of unconscious and conscious processes. In this regard, we discuss two basic hypotheses. First, unconscious and conscious processes form a dynamic organizational continuum. Second, emotions, which are unconscious forms of feeling, characterize the response of an organism to any internal change disturbing homeostasis or to any change in the exterior of the organism detected via specialized sensory probes. The former hypothesis is illustrated by discussing aspects of visual perception. The validity of the second hypothesis is supported by discussing interactions between unconscious and conscious process necessary for maintaining energy and water balance.

The process of ‘comprehending how we comprehend’, requires elucidating fundamental cognitive mechanisms used by the brain. In these regards, this article discusses two basic hypotheses. First, unconscious and conscious processes form a continuum. Second, humans have a predisposition to construct real versions of mental images and of unconscious structures, as well as to assign to them specific symbols. In addition, it is claimed that our process of comprehending is aided by invoking certain concepts which are important because, apparently, they reflect fundamental cognitive mechanisms. Among them are the notions of continuity, associations, abstraction, reduction, unification, and generalization, as well as the dialectic pairs of local versus global processes and simplicity versus complexity.

The transformative achievements of deep learning have led several scholars to raise the question of whether artificial intelligence (AI) can reach and then surpass the level of human thought. Here, after addressing methodological problems regarding the possible answer to this question, it is argued that the definition of intelligence proposed by proponents of the AI as “the ability to accomplish complex goals,” is appropriate for machines but does not capture the essence of human thought. After discussing the differences regarding understanding between machines and the brain, as well as the importance of subjective experiences, it is emphasized that most proponents of the eventual superiority of AI ignore the importance of the body proper on the brain, the laterization of the brain, and the vital role of the glia cells. By appealing to the incompleteness theorem of Gödel’s and to the analogous result of Turing regarding computations, it is noted that consciousness is much richer than both mathematics and computations. Finally, and perhaps most importantly, it is stressed that artificial algorithms attempt to mimic only the conscious function of parts of the cerebral cortex, ignoring the fact that, not only every conscious experience is preceded by an unconscious process but also that the passage from the unconscious to consciousness is accompanied by loss of information.