Informal Knowledge Sharing (KS) is vital for end-user programmers to gain expertise. To better understand how personal (self-efficacy), social (reputational gains, trust between colleagues), and software-related (codification effort) variables influence spreadsheet KS intention, we conducted a multiple regressions analysis based on survey data from spreadsheet users (n=100) in administrative and finance roles. We found that high levels of spreadsheet self-efficacy and a perception that sharing would result in reputational gains predicted higher KS intention, but individuals who found knowledge codification effortful showed lower KS intention. We also observed that regardless of occupation, users tended to report a lower sense of self-efficacy in their general spreadsheet proficiency, despite also reporting high self-efficacy in spreadsheet use for job-related contexts. Our findings suggest that acknowledging and designing for these social and personal variables can help avoid situations where experienced individuals refrain unnecessarily from sharing, with implications for spreadsheet design.

We derive in this paper Gaussian estimates for a general parabolic equation $u_{t}-\big(a(x)u_{x}\big)_x= r(x)u$ over $\mathbb{R}$. Here $a$ and $r$ are only assumed to be bounded, measurable and $\mathrm{essinf}_{\mathbb{R}} a>0$. We first consider a canonical equation $ν(x) \partial_{t}p - \partial_{x }\big( ν(x)a(x)\partial_{x}p\big)+W\partial_{x}p=0$, with $W\in \mathbb{R}$, $ν$ bounded and $\mathrm{essinf}_{\mathbb{R}} ν>0$, for which we derive Gaussian estimates for the fundamental solution: $$\forall t>0, x,y\in \mathbb{R}, \quad \displaystyle\frac{1}{Ct^{1/2}}e^{-C|T(x)-T(y)-Wt|^{2}/t} \leq P(t,x,y)\leq \frac{C}{t^{1/2}}e^{-|T(x)-T(y)-Wt|^{2}/Ct}.$$ Here, the function $T$ is a corrector, for which we are able to derive appropriate properties using one-dimensional arguments. We then show that any solution $u$ of the original equation could be divided by some generalized principal eigenfunction $ϕ_γ$ so that $p:=u/ϕ_γ$ satisfies a canonical equation. As a byproduct of our proof, we derive Nash type estimates, that is, Holder continuity in $x$, for the solutions of the canonical equation.

Data from national accounts show no effect of change in net saving or consumption, in ratio to market-value capital, on change in growth rate of market-value capital (capital acceleration). Thus it appears that capital growth and acceleration arrive without help from net saving or consumption restraint. We explore ways in which this is possible, and discuss implications for economic teaching and public policy

In this paper, we study a class of multilinear Gibbs measures with Hamiltonian given by a generalized $\mathrm{U}$-statistic and with a general base measure. Expressing the asymptotic free energy as an optimization problem over a space of functions, we obtain sufficient conditions for replica-symmetry, and provide examples to show why these conditions are also necessary. Utilizing this, we obtain weak limits for a large class of statistics of interest, which includes the \enquote{local fields/magnetization}, the Hamiltonian, the global magnetization, etc. An interesting consequence is a universal weak law for contrasts under replica symmetry, namely, $n^{-1}\sum_{i=1}^n c_i X_i\to 0$ weakly, if $\sum_{i=1}^n c_i=o(n)$. Our results yield a probabilistic interpretation for the optimizers arising out of the limiting free energy. We also prove the existence of a sharp phase transition point in terms of the temperature parameter, thereby generalizing existing results that were only known for quadratic Hamiltonians. As a by-product of our proof technique, we obtain exponential concentration bounds on local and global magnetizations, which are of independent interest.

Let $f(m)$ be the largest integer such that for every set $A = \{a_1 < \cdots < a_m\}$ of $m$ positive integers and every open interval $I$ of length $2a_m$, there exist at least $f(m)$ disjoint pairs $(a, b)$ with $a \in A$ dividing $b \in I$. Solving a problem of Erdős, we determine $f(m)$ exactly, and show $$ f(m)=\min\bigl(m,\lceil 2\sqrt{m}\,\rceil\bigr) $$ for all $m$. The proof was obtained through an AI-assisted workflow: the proof strategy was first proposed by ChatGPT, and the detailed argument was subsequently made fully rigorous and formally verified in Lean by Aristotle. The exposition and final proofs presented here are entirely human-written. [This paper solves Problem #650 on Bloom's website "Erdős problems".]

The integration of a theory regarding the dynamics of averaged magnetic moment in a turbulent setting with the concept of a self-optimized cascade loop of helical fluxes, either spontaneously or intentionally generated near the separatrix, enables the derivation of spectral laws for the floating potential and ion saturation current, which align with findings from various experiments conducted on tokamaks, stellarators, and RFX-mod reversed field pinches. The notion of distributed chaos enables a quantitative evaluation of the randomness levels of chaotic/turbulent states observed inside and outside the separatrix, linking them to self-optimized helical fluxes.

We determine the full persistence probability distribution for a non-Markovian stochastic process, motivated by first-passage questions arising in interacting spin systems and allied systems. We show that this distribution is governed by a distinguished Painlevé VI system arising from an exact Fredholm Pfaffian structure associated with the integrable sech kernel, $K_{\mathrm{sech}}=1/(2 π\cosh[(x-y)/2])$. The universal persistence exponent originally obtained by Derrida, Hakim and Pasquier is recovered as an asymptotic observable and acquires a natural geometric interpretation. In the stationary scaling regime, the persistence probability admits an exact Pfaffian decomposition into even and odd Fredholm determinants of the integrable \emph{sech} kernel. These determinants are controlled by a unique global solution of a second-order nonlinear ordinary differential equation, which is identified as a particular Painlevé VI equation. The corresponding Painlevé VI connection problem determines the persistence exponent as a limiting value at infinity. We further show that the Painlevé VI system governing persistence admits a direct geometric interpretation: the relevant solution coincides with the mean curvature of a one-parameter family of Bonnet surfaces immersed in $\mathbb R^3$. A folding transformation between such surfaces singles out the Painlevé VI equation with Manin coefficients $[0,0,0,0]$, which in particular governs the universal persistence distribution in the symmetric Ising case. In this framework, the persistence exponent is identified with the asymptotic mean curvature of the associated surface.

The Version Age of Information (VAoI) quantifies information freshness by measuring the number of versions the receiver lags behind. This paper studies VAoI minimization in an $M$-user uplink non-orthogonal multiple access (NOMA) system where users maintain single-packet buffers and transmissions are constrained by average power and information-quality constraints, modeled by a general distortion function. A fundamental trade-off arises: transmitting more bits per update improves information quality but increases power consumption, reducing transmission opportunities and increasing VAoI, while transmitting fewer bits has the opposite effect. We formulate a weighted-sum VAoI minimization problem as a convex optimization problem. However, users' power allocations are coupled through multiple-access capacity constraints per channel state, leading to exponential complexity. To address this, we develop a VAoI-agnostic stationary randomized policy that jointly optimizes scheduling, bit allocation, and power control without tracking instantaneous VAoI, and achieves a provable 2-approximation to the globally optimal average VAoI. Leveraging Lagrangian dual decomposition, we derive closed-form expressions for the scheduling probabilities and power allocations, and efficiently determine the optimal successive interference cancellation decoding order, avoiding exhaustive search Numerical results show that NOMA significantly outperforms time-division multiple access (TDMA): at high power budgets, NOMA achieves near-zero VAoI, whereas TDMA saturates at a non-zero value, consistent with the analysis. The proposed general distortion framework accommodates diverse bit-priority structures by assigning unequal importance to different bits within an update.

The numerical sign problem remains one of the central challenges in computational physics. The Worldvolume Hybrid Monte Carlo (WV-HMC) method has recently been proposed as a reliable and computationally efficient algorithm that crucially avoids the ergodicity issues inherent in Lefschetz-thimble approaches. In these proceedings, after outlining the key ideas behind WV-HMC, we present its extension to group manifolds. This provides a rigorous framework for applying WV-HMC to lattice gauge theories.

When training data are limited, data-driven models are especially vulnerable to optimization-related fluctuations from random initialization and to sampling-induced bias from insufficient training data. We address both challenges with transfer learning (TL): deep neural networks (DNNs) are first pretrained on $α$ decay half-lives and then fine-tuned on a small cluster decay dataset. The pretraining stage provides a physically informed initialization that stabilizes optimization, while transferred global decay systematics regularize the fit and reduce sensitivity to training set composition. Despite extreme data sparsity, the resulting models accurately predict cluster decay half-lives for parent nuclei from $^{221}$Fr to $^{242}$Cm. We further quantify how initialization and sample selection affect predictive accuracy and robustness, demonstrating that TL enables stable and reliable learning in the small-sample regime.

Quantum computers have the potential to perform computational tasks beyond the reach of classical machines. A prominent example is Shor's algorithm for integer factorization and discrete logarithms, which is of both fundamental importance and practical relevance to cryptography. However, due to the high overhead of quantum error correction, optimized resource estimates for cryptographically relevant instances of Shor's algorithm require millions of physical qubits. Here, by leveraging advances in high-rate quantum error-correcting codes, efficient logical instruction sets, and circuit design, we show that Shor's algorithm can be executed at cryptographically relevant scales with as few as 10,000 reconfigurable atomic qubits. Increasing the number of physical qubits improves time efficiency by enabling greater parallelism; under plausible assumptions, the runtime for discrete logarithms on the P-256 elliptic curve could be just a few days for a system with 26,000 physical qubits, while the runtime for factoring RSA-2048 integers is one to two orders of magnitude longer. Recent neutral-atom experiments have demonstrated universal fault-tolerant operations below the error-correction threshold, computation on arrays of hundreds of qubits, and trapping arrays with more than 6,000 highly coherent qubits. Although substantial engineering challenges remain, our theoretical analysis indicates that an appropriately designed neutral-atom architecture could support quantum computation at cryptographically relevant scales. More broadly, these results highlight the capability of neutral atoms for fault-tolerant quantum computing with wide-ranging scientific and technological applications.

The transition to a cloud-native 5G Service-Based Architecture (SBA) improves scalability but exposes control-plane signaling to emerging quantum threats, including Harvest-Now, Decrypt-Later (HNDL) attacks. While NIST has standardized post-quantum cryptography (PQC), practical, deployable integration in operational 5G cores remains underexplored. This work experimentally integrates NIST-standardized ML-KEM-768 and ML-DSA into an open-source 5G core (free5GC) using a sidecar proxy pattern that preserves unmodified network functions (NFs). Implemented on free5GC, we compare three deployments: (i) native HTTPS/TLS, (ii) TLS sidecar, and (iii) PQC-enabled sidecar. Measurements at the HTTP/2 request-response boundary over repeated independent runs show that PQC increases end-to-end Service-Based Interface (SBI) latency to approximately 54 ms, adding a deterministic 48-49 ms overhead relative to the classical baseline, while maintaining tightly bounded variance (IQR <= 0.2 ms, CV < 0.4%). We also quantify the impact of Certification Authority (CA) security levels, identifying certificate validation as a tunable contributor to overall delay. Overall, the results demonstrate that sidecar-based PQC insertion enables a non-disruptive and operationally predictable migration path for quantum-resilient 5G signaling.

We propose Quantum Riemannian Hamiltonian Descent (QRHD), a quantum algorithm for continuous optimization on Riemannian manifolds that extends Quantum Hamiltonian Descent (QHD) by incorporating geometric structure of the parameter space via a position-dependent metric in the kinetic term. We formulate QRHD at both operator and path integral formalisms and derive the corresponding quantum equations of motion, showing that quantum corrections appear in the action integral but they are suppressed at late times by the time-dependent dissipation factor. This implies that convergence near optimal points is controlled by the classical potential while quantum effects influence early-time dynamics. By analyzing the semiclassical equation, we estimate a lower bound on the convergence time and numerically demonstrate whether QRHD work as a quantum optimization algorithm in some examples. A quantum circuit implementation based on time-dependent Hamiltonian simulation is also discussed and the query complexity is estimated.

There are two distinct perspectives on the quantum time-of-arrival: one can ask for the probability that a particle is found at the detector at a given time, regardless of whether it was previously detected, or for the probability that the particle is detected there for the first time. In this work, we analyze the latter by constructing the time-of-arrival distribution conditioned on the particle not having been detected at earlier times -- the first-click distribution. We work within the Page and Wootters formalism, where time is treated as a quantum observable, and introduce a memory mechanism that records the outcomes of successive detection attempts separated by the detector's finite time resolution. We apply this framework to a single Gaussian wave packet and to a superposition of two overlapping wave packets. We find that conditioning on non-detection redistributes probability toward earlier arrival times, producing narrower and sharper distributions compared with the standard unconditioned case. This effect persists in the presence of quantum interference, though coarser time resolutions broaden the distribution and shift it toward later times.

We compute the radiative corrections to the process of two-pion production in association with a hard photon in $e^+ e^-$ annihilation by taking into account the non-perturbative structure of the pion in the one-loop calculation. For this purpose, we adopt the generalised vector meson dominance model to insert the pion form factor in loop integrals for the treatment of final-state radiation and initial-final state interference at next-to-leading order. We compare our predictions with the results of the naive factorised scalar QED approach for experimentally relevant observables in the measurement of the $e^+ e^- \to π^+ π^- γ$ process. The computation extends previous results obtained for the energy scan process $e^+ e^- \to π^+ π^-$ and can be used to quantify the uncertainty due to the model describing the pion-photon interaction in radiative return experiments at flavour factories.

We implement the scenario of early relaxation of the axion via a high scale confinement within $SU(5)$ grand unified theory and study an epoch of strong QCD in inflationary cosmology. We consider scenarios in which, during inflation, the $SU(5)$ is either entirely or partially in the confining phase. This generates an early potential for the axion and dilutes its energy density removing any cosmological upper bound on the decay constant. We show that a phase of strong QCD can be realized by at least two mechanisms: 1) A direct coupling between the inflaton and the gauge fields and/or 2) by restoration of the $SU(5)$ symmetry during the inflationary epoch. In the latter case, strong coupling is already achieved via the RG running of the $SU(5)$ gauge coupling. We show that the mechanism works for all known realizations of the invisible axion idea: Peccei-Quinn (PQ) type formulations in which the anomalous global symmetry is realized via additional scalars (DFSZ) or heavy fermions (KSVZ) as well as the two-form gauge axion formulation based entirely on the QCD gauge redundancy without any anomalous global symmetry. Even if the expectation value of the PQ scalar vanishes during inflation, the axion is a well defined degree of freedom represented by the phase of the fermion 't Hooft determinant. For the DFSZ case, this phase is composed out of a condensate of the ordinary quarks, amounting to an early universe version of the $η'$-meson. In all considered scenarios, the present day axion can be a viable dark matter candidate for an arbitrarily large value of the decay constant.

We study the natural action of $\mathrm{PGL}(V)$ on the Grassmannian $G=\operatorname{Gr}(2,\operatorname{Sym}^2 V^\vee)$, where $\dim V=4$ and points of $G$ are pencils of quadrics in $\mathbb{P}(V)\cong \mathbb{P}^3$. Here $\dim G=16$ while $\dim \mathrm{PGL}(V)=15$, so the generic orbit has codimension one and one expects a one-parameter family of generic orbits. We construct this family via the $j$-invariant of the discriminant binary quartic of a pencil. We then determine the codimension-one orbit closures and compute their Chow classes. The smooth codimension-one orbit closures are the reduced fibers of the $j$-map on the smooth locus, while the unique boundary divisor is the closure of the orbit of a nodal quartic complete intersection of arithmetic genus $1$ and geometric genus $0$. Every divisorial fiber of the rational $j$-map has class $12σ_1$ in $A^1(G)$. For the reduced codimension-one orbit closures one has $[\overline{O_a}]=12σ_1$ for $a\neq 0,1728,\infty$, $[\overline{O_{1728}}]=6σ_1$, $[\overline{O_0}]=4σ_1$, and $[T]=12σ_1$.

We investigate the interplay between the dimension of the space of static potentials and the geometric and topological structure of the underlying static three-manifold. A partial classification of boundaryless static manifolds is obtained in terms of this dimension. We also treat the case of static manifolds with boundary. In particular, we prove that if a compact static manifold with boundary admits a static potential whose zero set is disjoint from the boundary, then the space of static potentials is necessarily one-dimensional. These results rely on a careful analysis of the relative positions of the zero sets of linearly independent static potentials - a technique originally introduced by Miao and Tam.

In this note, we compare the two approaches to semiring schemes as topological spaces with a structure sheaf and as a functor of points. We explain and prove the following two results: (1) the topological space can be recovered from the functor of points; (2) the two notions of semiring schemes are canonically equivalent as categories.

Toxicity monitoring is essential in Phase II clinical trials to ensure participant safety. While monitoring rules are well-established for single-arm trials, two-cohort trials present unique challenges because toxicities are expected to be similar between cohorts but may still differ. Current approaches either monitor the two cohorts independently, which ignores their similarity, or pool them together as a single arm, which neglects heterogeneity between cohorts. We propose a Bayesian method based on a bivariate beta prior that provides a compromise between these two approaches. The marginal posterior distribution is derived as a mixture of beta distributions, enabling exact calculations of the proposed method's operating characteristics. Examples demonstrate that joint monitoring offers a balanced approach between the independent and pooled methods. Keywords: Toxicity; Two-cohort; Phase II clinical trial; Monitoring rules; Bivariate Beta; Exact Operating characteristics

We consider the following question of Knuth: given a directed graph $G$ and a root $r$, can the arborescences of $G$ rooted in $r$ be listed such that any two consecutive arborescences differ by only one arc? Such an ordering is called a pivot Gray code and can be formulated as a Hamiltonian path in the reconfiguration graph of the arborescences of $G$ under arc flips, also called flip graph of $G$. We give a positive answer for tournaments and explore several conditions showing that the flip graph of a directed graph may contain no Hamiltonian cycles.

Highly coherent lasers are central to modern photonics. To date, high-coherence operation has been achieved predominantly in microcavity and fiber-based platforms. More recently, free-space Brillouin-laser experiments have revealed unusually strong noise suppression whose physical origin cannot be explained by conventional continuous-medium models developed for those platforms. In conventional continuous-medium models, the optical and acoustic fields are assumed to remain continuously coupled throughout the cavity evolution, whereas in free-space implementations the coupling is confined to the nonlinear medium and interrupted by passive propagation over the rest of the round trip. To describe this interaction-propagation separation, we develop a discrete-cavity model in which the short Brillouin interaction inside the gain medium and the subsequent free-space propagation are treated as two separate stages of the round-trip evolution. This separation introduces a temporal asymmetry between optical storage and acoustic relaxation, which effectively enhances acoustic damping at the cavity level and strongly reduces pump-noise transfer to the Stokes field. If the cavity round-trip time is much longer than the interaction time in the nonlinear medium, the noise-suppression ratio scales with the ratio of the total cavity length to the nonlinear-medium length. Our discrete-cavity model further provides quantitative predictions for the lasing threshold, output power, phase-noise transfer, and fundamental linewidth, in good agreement with experiment. These results identify the discrete interaction-propagation structure as the physical origin of the unusually strong noise suppression in free-space Brillouin lasers systems.

An MPC controller uses a model of the dynamical system to plan an optimal control strategy for a finite horizon, which makes its performance intrinsically tied to the quality of the model. When faults occur, the compromised model will degrade the performance of the MPC with this impact being dependent on the designed cost function. In this paper, we aim to devise a strategy that combines active fault identification while driving the system towards the desired trajectory. The explored approaches make use of an exact formulation of the problem in terms of set-based propagation resorting to Constrained Convex Generators (CCGs) and a suboptimal version that resorts to the SVD decomposition to achieve the active fault isolation in order to adapt the model in runtime.

Beer is a phenomenal beverage. It has previously shaped the history of many peoples, states and cultures. The beauty of beer is its versatility. Starting from the original implementations that were murky or diluted, over time researchers found novel approaches to gradually develop beverages that are diverse, intense and are pleasant for the end user. Recently, the industry came up with the so-called \textit{craft beers}, that often differ from the commercial beers in production volume (due to lower capacities of the craft beer producers) and tasting profile (often having more intense unusual flavours). However, while it is often relatively easy to judge if a particular commercial beer is likely to be enjoyable, the same cannot be said about craft beers, as there are far too many styles, implementations and ingredients involved in their production. This creates a gap between the beverage producers and the consumers due to the inability of the former to judge the preferences and the consumption trends of the latter. As a response to this challenge we present a novel collaborative beverage-related data collection and analysis framework - the Distributed Beverage Analysis (DBA). The idea behind this study is to identify the common trends and support them by empirical evidence to better understand the needs of the consumers. We empirically verify DBA at the biannual \textit{Kraft Bier Fest} conducted by Vienna Kraft brewery in (you guessed it) Vienna. To showcase a need in such kind of analysis, we evaluate various large language models (LLMs) against our collaborative framework and confirm that many AI models cannot be reliably used to reason over the trends and patterns in the evolving world of craft beer.

In this paper we study representations of real numbers in a numeral system with the base $a>1$ and alphabet (digits set) $A\equiv\{0,1,...,r\}$, $a-1<r\in N$ given by \[x=\sum\limits_{n=1}^{\infty}\frac{α_n}{a^n}\equiv Δ^{r_a}_{α_1α_2...α_n...}, α_n\in A.\] Since the alphabet is redundant the numbers from the interval $[0;\frac{r}{a-1}]$ have not a single representation and can even have a continuous set of different representations. We describe the geometry (topological and metric properties) of such representations (the $r_a$-representations) in terms of cylinders defined by \[Δ^{r_a}_{c_1c_2...c_m}= \{x: x=Δ^{r_a}_{c_1c_2...c_ma_1a_2...a_n...}, a_n\in A\},\] We analyze their properties in detail, including the specific nature of overlaps. We present results on the structural, variational, topological, metric and partially fractal properties of the function defined by \[f\left(x=\sum_{n=1}^{\infty}\frac{α_n}{(r+1)^n}\right)= Δ^{r_a}_{α_1α_2...α_n...},α_n \in A.\] We prove the function is continuous at all points of the interval $[0,1]$ that have a unique representation in the classical numeral system on the base $r+1$ and prove the function is discontinuous at points of a countable everywhere dense set in $[0,1]$. Furthermore, we show that the function is nowhere monotonic and has unlimited variation. In the particular case $r=1$ and $a=\frac{1+\sqrt{5}}{2}$, we specify fractal level sets with Hausdorff--Besicovitch dimension not less than $-\log_a2$.

Trotter decomposition provides a simple approach to simulating open quantum systems by decomposing the Lindbladian into a sum of individual terms. While it is established that Trotter errors in Hamiltonian simulation depend on nested commutators of the summands, such a relationship remains poorly understood for Lindbladian dynamics. In this Letter, we derive commutator-based Trotter error bounds for Lindbladian simulation, yielding an $O(\sqrt{N})$ scaling in the number of Trotter steps for locally interacting systems on $N$ sites. When estimating observable averages, we apply Richardson extrapolation to achieve polylogarithmic precision while maintaining the commutator scaling. To bound the extrapolation remainder, we develop a general truncation bound for the Baker-Campbell-Hausdorff expansion that bypasses common convergence issues in physically relevant systems. For local Lindbladians, our results demonstrate that the Trotter-based methods outperform prior simulation techniques in system-size scaling while requiring only $O(1)$ ancillas. Numerical simulations further validate the predicted system-size and precision scaling.

Network loops are a common feature in the quiet Sun. The physical processes sustaining their energy budget is still under discussion. We rely on a multi-instrumental (Solar Orbiter/EUI, Solar Orbiter/PHI, IRIS) observation of a six hours quiet Sun region to measure the dynamics and the possible magnetic drivers of impulsive EUV emission enhancements along network loops. We report the detection of small-scale impulsive EUV emission enhancements with EUI/HRIEUV in three network loops. We selected four EUV emission enhancements to measure their plane-of-sky velocities in HRIEUV; their Doppler velocities in the line (log Si iv T = 4.8) with IRIS ; their possible relation to small-scale flux emergence and fluctuation in one of the loop footpoint. The plane-of-sky velocities of the four EUV emission enhancements have a component that seem to appear almost instantaneously along the loop (more than 220 km/s) ; and two of them had a co-temporal component with a PoS velocity of up to , starting near one of the loop footpoint. In one case, we measured with IRIS a co-temporal intensity increase in the line associated with Doppler velocities down to and up to along the line of sight. Finally, we found evidence of small-scale (8E16 Mx) mixed polarity field emergence and fluctuation near one of the loop footpoint. We concluded that the fast component on the plane-of-sky are consistent with a thermal transfer or supersonic plasma flows, while the slower component is consistent with plasma flows. A possible physical origin for these EUV emission enhancements would be magnetic reconnection driven by either photospheric motion of the loop footpoints or by the reconnection of the loop with small-scale magnetic bipoles.

The dynamics of simple two-alternative forced-choice (2AFC) decisions are well-modeled by a class of random walk models (e.g. Laming, 1968; Ratcliff, 1978; Usher & McClelland, 2001; Bogacz et al., 2006). However, in real-life, even simple decisions involve dynamically changing influence of additional information. In this work, we describe a computational theory of decision making from multiple sources of information, grounded in Bayesian inference and consistent with a simple neural network. This Contextual Diffusion Decision Model (CDDM) is a formal generalization of the Diffusion Decision Model (DDM), a popular existing model of fixed-context decision making (Ratcliff, 1978), and shares with it both a mechanistic and a probabilistic motivation. Just as the DDM is a model for a variety of simple two-alternative forced-choice (2AFC) decision making tasks, we demonstrate that the CDDM supports a variety of simple context-dependent tasks of longstanding interest in psychology, including the Flanker (Eriksen & Eriksen, 1974), AX-CPT (Servan-Schreiber et al., 1996), Stop-Signal (Logan & Cowan, 1984), Cueing (Posner, 1980), and Prospective Memory paradigms (Einstein & McDaniel, 2005). Further, we use the CDDM to perform a number of normative rational analyses exploring optimal response and memory allocation policies. Finally, we show how the use of a consistent model across tasks allows us to recover consistent qualitative data patterns in multiple tasks, using the same model parameters.

In the paper we study a class $F$ of multiparameter functions defined in terms of a polybasic $s$-adic $Q^{*}_{s}$-representation of numbers by \begin{equation*} f_a\bigl(x=Δ^{Q^{*}_s}_{α_1α_2\ldotsα_n\ldots}\bigr) = Δ^{Q^{*}s}_{|a_1-α_1|\,|a_2-α_2|\,\ldots\,|a_n-α_n|\ldots}, \end{equation*} where $(a_n)$ is the sequence of digits for $s$-adic representation of the parameter $a\in[0,1]$, and \begin{equation*} Δ^{Q^{*}_s}_{α_1α_2\ldotsα_n\ldots}= β_{α_1 1}+ \sum_{n=2}^{\infty} \left( β_{α_n n} \prod_{j=1}^{n-1} q_{α_j j} \right) \end{equation*} is the $Q^{*}_{s}$-representation of real numbers generated by a positive stochastic matrix $\|q_{ij}\|$ with $β_{α_n n}=\sum\limits_{i=0}^{α_n-1} q_{in}$. In this paper we investigate the continuity of the function $f_a$ on the sets of $Q^{*}_{s}$-binary and $Q^{*}_{s}$-unary numbers. We prove that the functions in this class are continuous on the set of numbers with a unique $Q^{*}_{s}$-representation. Furthermore, we show that except for $f_0$ and $f_1$, all functions have a countable set of discontinuities at $Q^{*}_{s}$-binary points. We classify the topological types of the value sets of $f_a$ depending on the parameter $a$. We prove that, if the value set is of Cantor type, then it is zero-dimensional. We describe the structural properties of the level sets of $f_a$ in terms of the digits of the $s$-adic representation of $a$. In particular, we establish that a level set of the function $f_a$ can be an empty set, a finite set, or a continuum. For certain values of $s$ we provide examples of fractal level sets and calculate its fractal dimensions.

This book provides an introduction to path integral methods and their application to modeling atomistic processes. The book covers both the foundational theory and recently developed simulation techniques. The text provides a self-contained resource and was originally developed for the CECAM schools on Path Integral Methods.