There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

■ Abstract Comparative modeling predicts the three-dimensional structure of a given protein sequence (target) based primarily on its alignment to one or more pro- teins of known structure (templates). The prediction process consists of fold assign- ment, target-template alignment, model building, and model evaluation. The number of protein sequences that can be modeled and the accuracy of the predictions are in- creasing steadily because of the growth in the number of known protein structures and because of the improvements in the modeling software. Further advances are nec- essary in recognizing weak sequence-structure similarities, aligning sequences with structures, modeling of rigid body shifts, distortions, loops and side chains, as well as detecting errors in a model. Despite these problems, it is currently possible to model with useful accuracy significant parts of approximately one third of all known protein sequences. The use of individual comparative models in biology is already rewarding and increasingly widespread. A major new challenge for comparative modeling is the integration of it with the torrents of data from genome sequencing projects as well as from functional and structural genomics. In particular, there is a need to develop an automated, rapid, robust, sensitive, and accurate comparative modeling pipeline applicable to whole genomes. Such large-scale modeling is likely to encourage new kinds of applications for the many resulting models, based on their large number and completeness at the level of the family, organism, or functional network.

Comparative protein structure modeling of genes and genomes

The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose +complex conjugate) is replaced by the physically transparent condition of space?time reflection ( ) symmetry. If H has an unbroken symmetry, then the spectrum is real. Examples of -symmetric non-Hermitian quantum-mechanical Hamiltonians are and . Amazingly, the energy levels of these Hamiltonians are all real and positive!Does a -symmetric Hamiltonian H specify a physical quantum theory in which the norms of states are positive and time evolution is unitary? The answer is that if H has an unbroken symmetry, then it has another symmetry represented by a linear operator . In terms of , one can construct a time-independent inner product with a positive-definite norm. Thus, -symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution.The Lee model provides an excellent example of a -symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm 'ghost' state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. The correct interpretation of the ghost is simply that the non-Hermitian Lee-model Hamiltonian is -symmetric. The operator for the Lee model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.

https://iopscience.iop.org/article/10.1088/0034-4885/70/6/R03/pdf

Making sense of non-Hermitian Hamiltonians

Evolutionary games on graphs

Dealloying is a common corrosion process during which an alloy is 'parted' by the selective dissolution of the most electrochemically active of its elements. This process results in the formation of a nanoporous sponge composed almost entirely of the more noble alloy constituents. Although considerable attention has been devoted to the morphological aspects of the dealloying process, its underlying physical mechanism has remained unclear. Here we propose a continuum model that is fully consistent with experiments and theoretical simulations of alloy dissolution, and demonstrate that nanoporosity in metals is due to an intrinsic dynamical pattern formation process. That is, pores form because the more noble atoms are chemically driven to aggregate into two-dimensional clusters by a phase separation process (spinodal decomposition) at the solid-electrolyte interface, and the surface area continuously increases owing to etching. Together, these processes evolve porosity with a characteristic length scale predicted by our continuum model. We expect that chemically tailored nanoporous gold made by dealloying Ag-Au should be suitable for sensor applications, particularly in a biomaterials context.

/pdf/evolution-of-nanoporosity-in-dealloying-1vysl3gdw0.pdf

Evolution of nanoporosity in dealloying

The traditional description of high-energy small-angle scattering in QCD has two components — a soft Pomeron Regge pole for the tensor glueball, and a hard BFKL Pomeron in leading order at weak coupling. On the basis of gauge/string duality, we present a coherent treatment of the Pomeron. In large-N QCD-like theories, we use curved-space string-theory to describe simultaneously both the BFKL regime and the classic Regge regime. The problem reduces to finding the spectrum of a single j-plane Schrodinger operator. For ultraviolet-conformal theories, the spectrum exhibits a set of Regge trajectories at positive t, and a leading j-plane cut for negative t, the cross-over point being model-dependent. For theories with logarithmically-running couplings, one instead finds a discrete spectrum of poles at all t, where the Regge trajectories at positive t continuously become a set of slowly-varying and closely-spaced poles at negative t. Our results agree with expectations for the BFKL Pomeron at negative t, and with the expected glueball spectrum at positive t, but provide a framework in which they are unified. Effects beyond the single Pomeron exchange are briefly discussed.

https://conservancy.umn.edu/bitstream/handle/11299/130083/strassler.pdf;sequence=1

The Pomeron and gauge/string duality

Solving Lattice QCD systems of equations using mixed precision solvers on GPUs

Glueball spectrum for QCD from AdS supergravity duality

We introduce a class of models for the motion of a boundary between time-dependent phase domains in which the interface itself satisfies an equation of motion. The intended application is to systems for which competing stabilizing and destabilizing forces act on the phase boundary to produce irregular or patterned structures, such as those which occur in solidification. We discuss the kinematics of moving interfaces in two or more dimensions in terms of their intrinsic geometric properties. We formulate local equations of motion as tractable simplifications of the complex nonlocal dynamics that govern moving-interface problems. Special solutions for dendritic crystal growth and their stability are analyzed in some detail.

Geometrical models of interface evolution

QCD is constructed as a lattice gauge theory in which the elements of the link matrices are represented by non-commuting operators acting in a Hilbert space The resulting quantum link model for QCD is formulated with a fifth Euclidean dimension, whose extent resembles the inverse gauge coupling of the resulting four-dimensional theory after dimensional reduction The inclusion of quarks is natural in Shamir{close_quote}s variant of Kaplan{close_quote}s fermion method, which does not require fine-tuning to approach the chiral limit A rishon representation in terms of fermionic constituents of the gluons is derived and the quantum link Hamiltonian for QCD with a U(N) gauge symmetry is expressed in terms of glueball, meson and constituent quark operators The new formulation of QCD is promising both from an analytic and from a computational point of view {copyright} {ital 1999} {ital The American Physical Society}

Richard C. Brower

Papers

The Pomeron and gauge/string duality

Solving Lattice QCD systems of equations using mixed precision solvers on GPUs

Glueball spectrum for QCD from AdS supergravity duality

Geometrical models of interface evolution

QCD as a quantum link model