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The problem of counting polymer coverings on the rectangular lattices is investigated. In this model, a linear rigid polymer covers $k$ adjacent lattice sites such that no two polymers occupy a common site. Those unoccupied lattice sites are considered as monomers. We prove that for a given number of polymers ($k$-mers), the number of arrangements for the polymers on two-dimensional rectangular lattices satisfies simple recurrence relations. These recurrence relations are quite general and apply for arbitrary polymer length ($k$) and the width of the lattices ($n$). The well-studied monomer-dimer problem is a special case of the monomer-polymer model when $k=2$. It is known the enumeration of monomer-dimer configurations in planar lattices is #P-complete. The recurrence relations shown here have the potential for hints for the solution of long-standing problems in this class of computational complexity.

The pattern $(k_1, k_2, \dots, k_\ell)$ is defined to have at least $k_1$ consecutive $1$'s followed by at least $k_2$ consecutive $2$'s, $\dots$, followed by at least $k_\ell$ consecutive $\ell$'s. By iteratively applying the method that was developed previously to decouple the combinatorial complexity involved in studying complicated patterns in random sequences, the distribution of pattern $(k_1, k_2, \dots, k_\ell)$ is derived for arbitrary $\ell$. Numerical examples are provided to illustrate the results.

In the "classical" adventitious angle problem, for a given set of three angles $a$, $b$, and $c$ measured in integral degrees in an isosceles triangle, a fourth angle $\theta$ (the derived angle), also measured in integral degrees, is sought. We generalize the problem to find $\theta$ in fractional degrees. We show that the triplet $(a, b, c) = (45^\circ, 45^\circ, 15^\circ)$ is the only combination that leads to $\theta = 7\frac{1}{2}^\circ$ as the fractional derived angle.

The distributions of the $m$-th longest runs of multivariate random sequences are considered. For random sequences made up of $k$ kinds of letters, the lengths of the runs are sorted in two ways to give two definitions of run length ordering. In one definition, the lengths of the runs are sorted separately for each letter type. In the second definition, the lengths of all the runs are sorted together. Exact formulas are developed for the distributions of the m-th longest runs for both definitions. The derivations are based on a two-step method that is applicable to various other runs-related distributions, such as joint distributions of several letter types and multiple run lengths of a single letter type.

By using the matrix formulation of the two-step approach to distributions of patterns in random sequences, recurrence and explicit formulas for the generating functions of successions in random permutations of arbitrary multisets are derived. Explicit formulas for the mean and variance are also obtained.

By using the matrix formulation of the two-step approach to the distributions of runs, a recursive relation and an explicit expression are derived for the generating function of the joint distribution of rises and falls for multivariate random sequences in terms of generating functions of individual letters, from which the generating functions of the joint distribution of rises, falls, and number of runs are obtained. An explicit formula for the joint distribution of rises and falls with arbitrary specification is also obtained.

We study the dissipative bi-stable Duffing oscillator with equal energy wells and observe fractal patterns in the parameter space of driving frequency, forcing amplitude, and damping ratio. Our numerical investigation reveals the Hausdorff fractal dimension of the boundaries that separate the oscillator's intra-well and inter-well behaviors. Furthermore, we categorize the inter-well behaviors as three steady-state types: switching, reverting, and vacillating. While fractal patterns in the phase space are well-known and heavily studied, our results point to a new research direction about fractal patterns in the parameter space. Another implication of this study is that the vibration of a continuous bi-stable system modeled using a single-mode approximation also manifests fractal patterns in the parameter space. In addition, our findings can guide the design of next-generation bi-stable and multi-stable mechanical metamaterials.

Radionuclide Identification Devices (RIDs) or Backpack Radiation Detection Systems (BRDs) are often equipped with NaI(Tl) detectors. We demonstrate that such instruments could be provided with reasonable thermal- and fast-neutron sensitivity by means of an improved and sophisticated processing of the digitized detector signals: Fast neutrons produce nuclear recoils in the scintillation crystal. Corresponding signals are detectible and can be distinguished from that of electronic interactions by pulse-shape discrimination (PSD) techniques as used in experiments searching for weakly interacting massive particles (WIMPs). Thermal neutrons are often captured in iodine nuclei of the scintillator. The gamma-ray cascades following such captures comprise a sum energy of almost 7 MeV, and some of them involve isomeric states leading to delayed gamma emissions. Both features can be used to distinguish corresponding detector signals from responses to ambient gamma radiation. The experimental proof was adduced by offline analyses of pulse records taken with a commercial RID. An implementation of such techniques in commercial RIDs is feasible., Comment: 7 pages, 5 figures; invited paper to be presented at the 2022 SPIE Optics + Photonics Conference in San Diego, CA

This letter reports proof-of-principle demonstration of 3D printable, low-cost, and compact THz spectral splitters based on diffractive optical elements (DOEs) designed to disperse the incident collimated broadband THz radiation (0.5 THz - 0.7 THz) at a pre-specified distance. Via inverse design, we show that it is possible to design such a diffractive optic, which can split the broadband incident spectrum in any desired fashion, as is evidenced from both FDTD simulations and measured intensity profiles using a 500-750 GHz VNA. Due to its straightforward construction without the usage of any movable parts, our approach, in principle, can have various applications such as in portable, low-cost spectroscopy as well as in wireless THz communication systems as a THz demultiplexer., Comment: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible