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Let $\mathsf{s}(n)$ denote the sum of binary digits of an integer $n \geq 0$. In the recent years there has been interest in the behavior of the differences $\mathsf{s}(n+t)-\mathsf{s}(n)$, where $t \geq 0$ is an integer. In particular, Spiegelhofer and Wallner showed that for $t$ whose binary expansion contains sufficiently many blocks of $\mathtt{1}$s the inequality $\mathsf{s}(n+t) -\mathsf{s}(n) \geq 0$ holds for $n$ belonging to a set of asymptotic density $>1/2$, partially answering a question by Cusick. Furthermore, for such $t$ the values $\mathsf{s}(n+t) - \mathsf{s}(n)$ are approximately normally distributed. In this paper we consider a natural generalization to the family of block-counting functions $N^w$, giving the number of occurrences of a block of binary digits $w$ in the binary expansion. Our main result show that for any $w$ of length at least $2$ the distribution of the differences $N^w(n+t) - N^w(n)$ is close to a Gaussian when $t$ contains many blocks of $\mathtt{1}$s in its binary expansion. This extends an earlier result by the author and Spiegelhofer for $w=\mathtt{11}$., Comment: 36 pages

Let $(u_{n})_{n\in \mathbb{N}}$ be a $k$-automatic sequence with values in the set $\{0, 1\}$. In the paper, we consider properties of sequences $(f(n))_{n\in\mathbb{N}}$ governed by the recurrence relations of the form $f(n)=af(n-u_{n}-1)+bf(n-u_{n}-2)$. One of our main results states that under mild assumptions on the sequence $(u_{n})_{n\in \mathbb{N}}$, the corresponding set of quotients $\cal{V}(f):=\{f(n+1)/f(n):\;n\in\mathbb{N}\}$ is finite and $k$-automatic. In particular, this property holds in the case when $u_{n}=T_{n}$, where $(T_{n})_{n\in\mathbb{N}}$ is the famous Prouhet-Thue-Morse sequence. We also study the cardinality of $\cal{V}(f)$ in the case when $(u_{n})_{n\in \mathbb{N}}$ contains arbitrarily long blocks of zeros or is ultimately periodic., Comment: 20 pages, 3 figures

DEtection TRansformer (DETR) has emerged as a promising architecture for object detection, offering an end-to-end prediction pipeline. In practice, however, DETR generates hundreds of predictions that far outnumber the actual number of objects present in an image. This raises the question: can we trust and use all of these predictions? Addressing this concern, we present empirical evidence highlighting how different predictions within the same image play distinct roles, resulting in varying reliability levels across those predictions. More specifically, while multiple predictions are often made for a single object, our findings show that most often one such prediction is well-calibrated, and the others are poorly calibrated. Based on these insights, we demonstrate identifying a reliable subset of DETR's predictions is crucial for accurately assessing the reliability of the model at both object and image levels. Building on this viewpoint, we first tackle the shortcomings of widely used performance and calibration metrics, such as average precision and various forms of expected calibration error. Specifically, they are inadequate for determining which subset of DETR's predictions should be trusted and utilized. In response, we present Object-level Calibration Error (OCE), which is capable of assessing the calibration quality both across different models and among various configurations within a specific model. As a final contribution, we introduce a post hoc Uncertainty Quantification (UQ) framework that predicts the accuracy of the model on a per-image basis. By contrasting the average confidence scores of positive (i.e., likely to be matched) and negative predictions determined by OCE, the framework assesses the reliability of the DETR model for each test image.

Let $s(n)$ denote the number of ones in the binary expansion of the nonnegative integer $n$. How does $s$ behave under addition of a constant $t$? In order to study the differences \[s(n+t)-s(n),\] for all $n\ge0$, we consider the associated characteristic function $\gamma_t$. Our main theorem is a structural result on the decomposition of $\gamma_t$ into a sum of \emph{components}. We also study in detail the case that $t$ contains at most two blocks of consecutive $1$s. The results in this paper are motivated by \emph{Cusick's conjecture} on the sum-of-digits function. This conjecture is concerned with the \emph{central tendency} of the corresponding probability distributions, and is still unsolved., Comment: 31 pages

A novel, computationally designed, class of triangular-shape organic molecules with an inverted singlet-triplet (IST) energy gap is investigated with the aid of ab initio methods of electronic structure theory. The considered molecular systems have a form of cyclic oligomers and their common feature is electronic conjugation localized along the molecular rim. Analysis of vertical transition energies from the electronic ground state, as well as from the lowest excited singlet and triplet states of selected molecules, is conducted. The results underscore the significance of optimizing excited-state geometries in theoretical models to accurately describe the optoelectronic properties of the IST molecules, particularly in relation to their applications in OLEDs., Comment: 22 pages, 12 figures, 2 tables

In toxicology research, experiments are often conducted to determine the effect of toxicant exposure on the behavior of mice, where mice are randomized to receive the toxicant or not. In particular, in fixed interval experiments, one provides a mouse reinforcers (e.g., a food pellet), contingent upon some action taken by the mouse (e.g., a press of a lever), but the reinforcers are only provided after fixed time intervals. Often, to analyze fixed interval experiments, one specifies and then estimates the conditional state-action distribution (e.g., using an ANOVA). This existing approach, which in the reinforcement learning framework would be called modeling the mouse's "behavioral policy," is sensitive to misspecification. It is likely that any model for the behavioral policy is misspecified; a mapping from a mouse's exposure to their actions can be highly complex. In this work, we avoid specifying the behavioral policy by instead learning the mouse's reward function. Specifying a reward function is as challenging as specifying a behavioral policy, but we propose a novel approach that incorporates knowledge of the optimal behavior, which is often known to the experimenter, to avoid specifying the reward function itself. In particular, we define the reward as a divergence of the mouse's actions from optimality, where the representations of the action and optimality can be arbitrarily complex. The parameters of the reward function then serve as a measure of the mouse's tolerance for divergence from optimality, which is a novel summary of the impact of the exposure. The parameter itself is scalar, and the proposed objective function is differentiable, allowing us to benefit from typical results on consistency of parametric estimators while making very few assumptions., Comment: 28 pages, 4 figures

Exceptional points (EPs) -- singularities in the parameter space of non-Hermitian systems where two nearby eigenmodes coalesce -- feature unique properties with applications such as sensitivity enhancement and chiral emission. Existing realizations of EP lasers operate with static populations in the gain medium. By analyzing the full-wave Maxwell--Bloch equations, here we show that in a laser operating sufficiently close to an EP, the nonlinear gain will spontaneously induce a multi-spectral multi-modal instability above a pump threshold, which initiates an oscillating population inversion and generates a frequency comb. The efficiency of comb generation is enhanced by both the spectral degeneracy and the spatial coalescence of modes near an EP. Such an ``EP comb'' has a widely tunable repetition rate, self-starts without external modulators or a continuous-wave pump, and can be realized with an ultra-compact footprint. We develop an exact solution of the Maxwell--Bloch equations with an oscillating inversion, describing all spatiotemporal properties of the EP comb as a limit cycle. We numerically illustrate this phenomenon in a 5-\textmu m-long gain-loss coupled AlGaAs cavity and adjust the EP comb repetition rate from 20 to 27~GHz. This work provides a rigorous spatiotemporal description of the rich laser behaviors that arise from the interplay between the non-Hermiticity, nonlinearity, and dynamics of a gain medium.

The binary sum-of-digits function $\mathsf{s}$ returns the number of ones in the binary expansion of a nonnegative integer. Cusick's Hamming weight conjecture states that, for all integers $t\geq 0$, the set of nonnegative integers $n$ such that $\mathsf{s}(n+t)\geq \mathsf{s}(n)$ has asymptotic density strictly larger than $1/2$. We are concerned with the block-additive function $\mathsf{r}$ returning the number of (overlapping) occurrences of the block $\mathtt{11}$ in the binary expansion of $n$. The main result of this paper is a central limit-type theorem for the difference $\mathsf{r}(n+t)-\mathsf{r}(n)$: the corresponding probability function is uniformly close to a Gaussian, where the uniform error tends to $0$ as the number of blocks of ones in the binary expansion of $t$ tends to $\infty$., Comment: 19 pages

End-to-end learning has emerged as a major paradigm for developing autonomous systems. Unfortunately, with its performance and convenience comes an even greater challenge of safety assurance. A key factor of this challenge is the absence of the notion of a low-dimensional and interpretable dynamical state, around which traditional assurance methods revolve. Focusing on the online safety prediction problem, this paper proposes a configurable family of learning pipelines based on generative world models, which do not require low-dimensional states. To implement these pipelines, we overcome the challenges of learning safety-informed latent representations and missing safety labels under prediction-induced distribution shift. These pipelines come with statistical calibration guarantees on their safety chance predictions based on conformal prediction. We perform an extensive evaluation of the proposed learning pipelines on two case studies of image-controlled systems: a racing car and a cartpole., Comment: This is supplementary material to the paper: How Safe Am I Given What I See? Calibrated Prediction of Safety Chances for Image-Controlled Autonomy in 6th Annual Learning for Dynamics & Control Conference (L4DC 2024)

Let $m$ be a positive integer and $b_{m}(n)$ be the number of partitions of $n$ with parts being powers of 2, where each part can take $m$ colors. We show that if $m=2^{k}-1$, then there exists the natural density of integers $n$ such that $b_{m}(n)$ can not be represented as a sum of three squares and it is equal to $1/12$ for $k=1, 2$ and $1/6$ for $k\geq 3$. In particular, for $m=1$ the equation $b_{1}(n)=x^2+y^2+z^2$ has a solution in integers if and only if $n$ is not of the form $2^{2k+2}(8s+2t_{s}+3)+i$ for $i=0, 1$ and $k, s$ are non-negative integers, and where $t_{n}$ is the $n$th term in the Prouhet-Thue-Morse sequence. A similar characterization is obtained for the solutions in $n$ of the equation $b_{2^k-1}(n)=x^2+y^2+z^2$., Comment: 24 pages, 4 figures

When irradiated with violet light, hexaazatrinaphthylene (HATN) extracts a hydrogen atom from an alcohol forming a long-living hydrogenated species. The apparent kinetic isotope effect for fluorescence decay time in deuterated methanol (1.56) indicates that the lowest singlet excited state of the molecule is a precursor for intermolecular hydrogen transfer. The photochemical hydrogenation occurs in several alcohols (methanol, ethanol, isopropanol) but not in water. Hydrogenated HATN can be detected optically by an absorption band at 1.78 eV as well as with EPR and NMR techniques. Mass spectroscopy of photoproducts reveal di-hydrogenated HATN structures along with methoxylated and methylated HATN molecules which are generated through the reaction with methoxy radicals (remnants from alcohol splitting). Experimental findings are consistent with the theoretical results which predicted that for the excited state of the HATN-solvent molecular complex, there exists a barrierless hydrogen transfer from methanol but a small barrier for the similar oxidation of water., Comment: 14 pages, 7 figures, 2 schemes, 1 table

We consider bounded Hankel operators $H_{\psi}$ acting on the Hardy space $H^2$ to $L^2\ominus H^2$ and obtain results on the Schmidt subspaces $E^+_s(H_\psi)$ of such operators defined as the kernels of $ H_{\psi}^{\ast}H_{\psi}-s^2I$ where $s>0$. These spaces have been recently studied in \cite{GP} and \cite{GP1} in the context of anti-linear Hankel operators. We also discuss the range of the Hankel operators with symbols being the complex conjugates of functions in the unit ball of $H^{\infty}$.

Let $e_v(n)$ denote the number of occurrences of a fixed pattern $v$ in the binary expansion of $n \in \mathbb{N}$. In this paper we study monochromatic arithmetic progressions in the class of binary words $(e_v(n) \bmod{2})_{n \geq 0}$, which includes the famous Thue--Morse word $\mathbf{t}$ and Rudin--Shapiro word $\mathbf{r}$. We prove that the length of a monochromatic arithmetic progression of difference $d \geq 3$ starting at $0$ in $\mathbf{r}$ is at most $(d+3)/2$, with equality for infinitely many $d$. Moreover, we compute the maximal length of a monochromatic arithmetic progression in $\mathbf{r}$ of difference $2^k-1$ and $2^k+1$. For a general pattern $v$ we provide an upper bound on the length of a monochromatic arithmetic progression of any difference $d$. We also prove other miscellaneous results and offer a number of related problems and conjectures., Comment: 31 pages, 3 figures. Slightly changed the title and abstract, added references and additional comments in Section 1, merged Sections 3 and 4 into one and rewrote the proofs in there

In this note we investigate the solutions of certain meta-Fibonacci recurrences of the form $f(n)=f(n-f(n-1))+f(n-2)$ for various sets of initial conditions. In the case when $f(n)=1$ for $n\leq 1$, we prove that the resulting integer sequence is closely related to the function counting binary partitions of a certain type., Comment: 18 pages

In this perspective, we discuss a novel water-splitting scenario, namely the direct oxidation of water molecules by organic photooxidants in hydrogen-bonded chromophore-water complexes. In comparison with the established scenario of semiconductor-based water splitting, the distance of electron transfer processes is thereby reduced from mesoscopic scales to the {\AA}ngstr{\o}m scale and the time scale is reduced from milliseconds to femtoseconds, which suppresses competing loss processes. The concept is illustrated by computational studies for the heptazine-H2O complex. The excited-state landscape of this complex has been characterized with ab initio electronic-structure methods and the proton-coupled electron-transfer dynamics has been explored with nonadiabatic dynamics simulations. A unique feature of the heptazine chromophore is the existence of a low-lying and exceptionally long-lived 1{\pi}{\pi}* state in which a substantial part of the photon energy can be stored for hundreds of nanoseconds and is available for the oxidation of water molecules. The calculations reveal that the absorption spectra and the photochemical functionalities of heptazine chromophores can be systematically tailored by chemical substitution. The options of harvesting hydrogen and the problems posed by the high reactivity of OH radicals are discussed., Comment: 24 pages, 2 tables, 8 figures

Progress in high-performance computing demands significant advances in memory technology. Among novel memory technologies that promise efficient device operation on a sub-ns timescale, resistance switching between charge ordered phases of the 1T-TaS$_2$ has shown to be potentially useful for the development of high-speed, energy efficient non-volatile memory device. While ultrafast switching was previously reported with optical pulses, determination of the intrinsic speed limits of actual devices that are triggered by electrical pulses is technically challenging and hitherto still largely unexplored. A new optoelectronic laboratory-on-a-chip, designed for measurements of ultrafast memory switching, enables an accurate measurement of the electrical switching parameters with 100 fs temporal resolution. A photoconductive response is used for ultrashort electrical pulse generation, while its propagation along a coplanar transmission line is detected using electro-optical sampling using a purpose-grown highly-resistive electro-optic (Cd,Mn)Te crystal substrate. By combining the transmission line and the 1T-TaS$_2$ device in a single optoelectronic circuit a non-volatile resistance switching with a single 1.9 ps electrical pulse is demonstrated, with an extremely small switching energy density per unit area E$_A$ = 9.4 fJ/$\mu$m$^2$. The experiments demonstrate ultrafast, energy-efficient circuits utilizing switching between non-volatile charge-ordered states offers a new technological platform for cryogenic memory devices., Comment: 13 pages, 4 figures

Let $b \geq 2$ be an integer base with prime factors $p_1, \ldots, p_s$. In this paper we study sequences of "$b$-adic valuations" and last nonzero digits in $b$-adic expansions of the values $f(n) = (f_1(n), \ldots, f_s(n))$, where each $f_i$ is a $p_i$-adic analytic function. We give a complete classification concerning $k$-regularity of these sequences, which generalizes a result for $b$ prime obtained by Shu and Yao. As an application, we strengthen a theorem by Murru and Sanna on $b$-adic valuations of Lucas sequences of the first kind. Moreover, we derive a method to determine precisely which terms of these sequences can be represented by certain ternary quadratic forms.

Two chromophores derived from heptazine, HAP-3MF and HAP-3TPA, were synthesized and tested as emitters in light-emitting diodes (OLEDs) by Adachi and coworkers. Both emitters were shown to exhibit quantum efficiencies which exceed the theoretical maximum of conventional fluorescent OLEDs. The enhanced emission efficiency was explained by the mechanism of thermally activated delayed fluorescence (TADF). In the present work, the electronic excitation energies and essential features of the topography of the excited-state potential-energy surfaces of HAP-3MF and HAP-3TPA have been investigated with a wave-function based ab initio method (ADC(2)). It is found that HAP-3MF is an inverted singlet-triplet (IST) system, that is, the energies of the S1and T1states are robustly inverted in violation of Hund's multiplicity rule. Notably, HAP-3MF presumably is the first IST emitter which was implemented in an OLED device. In HAP-3TPA, on the other hand, the vertical excitation energies of the S1 and T1 states are essentially degenerate. The excited states exhibit vibrational stabilization energies of similar magnitude along different relaxation coordinates, resulting in adiabatic excitation energies which also are nearly degenerate. HAP-3TPA is found to be a chromophore at the borderline of TADF and IST systems. The spectroscopic data reported by Adachi and coworkers forHAP-3MF and HAP-3TPA are analysed in the light of these computational results., Comment: 24 pages, 7 figures, 4 tables

There were two objectives. The first was to determine if a real time graphing activity will influence the interpretation of the subject of kinematic graphs, as measured by the TUGK. The second was to determine if attitude, as measured by CLASS, has a connection with graphical interpretation. The graphing activity did not affect graphical interpretation; this might be due to the short time spent on the treatment. However, there was a connection between attitude and graphical interpretation. The subjects of this study were non science majors attending a medium sized state owned university.

We present a conspectus of recent joint spectroscopic and computational studies which provided novel insight into the photochemistry of hydrogen-bonded complexes of the heptazine (Hz) chromophore with hydroxylic substrate molecules (water and phenol). It was found that a functionalized derivative of Hz, tri-anisole-heptazine (TAHz), can photooxidize water and phenol in a homogeneous photochemical reaction. This allows the exploration of the basic mechanisms of the proton-coupled electron-transfer (PCET) process involved in the water photooxidation reaction in well-defined complexes of chemically tunable molecular chromophores with chemically tunable substrate molecules. The unique properties of the excited electronic states of the Hz molecule and derivatives thereof are highlighted. The potential energy landscape relevant for the PCET reaction has been characterized by judicious computational studies. These data provided the basis for the demonstration of rational laser control of PCET reactions in TAHz-phenol complexes by pump-push-probe spectroscopy, which sheds light on the branching mechanisms occurring by the interaction of nonreactive locally excited states of the chromophore with reactive intermolecular charge-transfer states. Extrapolating from these results, we propose a general scenario which unravels the complex photoinduced water-splitting reaction into simple sequential light-driven one-electron redox reactions followed by simple dark radical-radical recombination reactions., Comment: 33 pages, 5 tables, 17 figures

Let $b$ be a nonextreme function in the unit ball of $H^{\infty}$ on the unit disk $\mathbb{D} $ and let $a$ be an outer $H^{\infty}$ function such that $|a|^2+|b|^2=1$ almost everywhere on $\partial \mathbb{D}$. The sufficient and necessary conditions for the orthogonal complement of $\mathcal{M}(a)$ in $\mathcal{H}(b)$ be finite dimensional has been given by D. Sarason . Here we describe this space explicitly.

We apply the theory of de Branges-Rovnyak spaces to describe kernels of some Toeplitz operators on the classical Hardy space $H^2$. In particular, we discuss the kernels of the operators $T_{\bar f/ f}$ and $T_{\bar I\bar f/ f}$, where $f$ is an outer function in $H^2$ and $I$ is inner such that $I(0)=0$. We also obtain results on de Branges-Rovnyak spaces generated by nonextreme functions.

We show that polynomials associated with automatic sequences satisfy a certain recurrence relation when evaluated at a root of unity, which generalizes a result of Brillhart, Lomont and Morton on the Rudin--Shapiro polynomials. We study the minimal order of such a relation and the integrality of its coefficients., Comment: 19 pages, 4 figures; typos corrected

Here we consider the balayage operator in the setting of $H^p$ spaces and its Bergman space version (B-balayage) introduced by H. Wulan, J. Yang and K. Zhu \cite{WYZ}, and extend some known results on these operators.