This paper develops a sensitivity analysis framework for the perturbed utility route choice (PURC) model and the accompanying stochastic traffic equilibrium model. We derive analytical sensitivity expressions for the Jacobian of the individual optimal PURC flow and equilibrium link flows with respect to link cost parameters under general assumptions. This allows us to determine the marginal change in link flows following a marginal change in link costs across the network. We show how to implement these results while exploiting the sparsity generated by the PURC model. Numerical examples illustrate the use of our method for estimating equilibrium link flows after link cost shifts, identifying critical design parameters, and quantifying uncertainty in performance predictions. Finally, we demonstrate the method in a large-scale example. The findings have implications for network design, pricing strategies, and policy analysis in transportation planning and economics, providing a bridge between theoretical models and real-world applications.

We study free Banach lattices over pre-ordered Banach spaces in the category of Banach lattices of a given convexity type. These generalise the free Banach lattices under convexity conditions over Banach spaces in the literature. Their existence is shown from the existence of free vector lattices over pre-ordered vector spaces, which are also investigated. We determine when the positive contraction from the pre-ordered Banach space into the free Banach lattice is injective or bipositive, and when it has closed range. It is a bipositive embedding with closed range if and only if the positive wedge of the space is a closed normal cone. Even for a Banach lattice it can be non-isometric. By analysing the norm of the free $p$-convex Banach lattice with convexity constant 1 over a pre-ordered Banach space, it becomes clear that it can be realised as a function lattice on the positive part of the dual unit ball. This generalises the known realisation for a free Banach lattice of that type over a Banach space. As a preparation for this analysis of the norm, characterisations of $p$-convex Banach lattices in terms of vector lattice homomorphisms into $\mathrm{L}_p(μ)$-spaces for probability measures $μ$ are given.

High-dimensional encoding of quantum information holds the potential to greatly increase the computational power of existing devices by enlarging the accessible state space for fixed register size and by reducing the number of required entangling gates. However, qudit-based quantum computation remains far less developed than conventional qubit-based approaches, in particular for photons, which represent natural multi-level information carriers that play a crucial role in the development of quantum networks. A major obstacle for realizing quantum gates between two individual photons is the restriction of direct interaction between photons in linear media. In particular, essential logic components for quantum operations such as native qudit-qudit entangling gates are still missing for optical quantum information processing. Here we address this challenge by presenting a protocol for realizing an entangling gate -- the controlled phase-flip (CPF) gate -- for two photonic qudits in arbitrary dimension. We experimentally demonstrate this protocol by realizing a four-dimensional qudit-qudit CPF gate, whose decomposition would require at least 13 two-qubit entangling gates. Our photonic qudits are encoded in orbital angular momentum (OAM) and we have developed a new active high-precision phase-locking technology to construct a high-dimensional OAM beam splitter that increases the stability of the CPF gate, resulting in a process fidelity within a range of $ [0.64 \pm 0.01, 0.82 \pm 0.01]$. Our experiment represents a significant advance for high-dimensional optical quantum information processing and has the potential for wider applications beyond optical system.

We study information aggregation in a dynamic trading model with partially informed traders. Ostrovsky [2012] showed that `separable' securities aggregate information in all equilibria, however, determining whether a security is separable requires knowing the exact information structure of agents. To remedy this problem, we allow traders to acquire signals with cost $κ$, in every period. We show that `$κ$ separable securities' characterize information aggregation and, as the cost decreases, almost all securities become $κ$ separable, irrespective of the traders' initial private information. Moreover, the switch to $κ$ separability happens not gradually but discontinuously, hence even a small decrease in costs can result in a security aggregating information. We provide a complete classification of securities in terms of how well they aggregate information, which surprisingly depends only on their payoff structure.

Primary key (PK) and foreign key (FK) constraints are widely used for query optimization. Knowledge about additional data dependencies, such as order dependencies, enables further substantial performance improvements. However, such dependencies are not maintained by database systems or are even unknown to the user. Identifying and validating relevant dependencies automatically and efficiently remains an unsolved problem. This paper presents a system that (i) recognizes dependency candidates for optimization, (ii) efficiently validates their applicability, and (iii) optimizes query plans using valid dependencies. First, we demonstrate the performance impact of optimization techniques using data dependencies additional to PKs and FKs. Using rewritten SQL queries, we empirically show that data dependencies improve performance for a wide range of analytical database systems and benchmarks. Second, we present how to integrate data dependencies into a system to use them without (i) manual declaration and maintenance or (ii) SQL rewrites. Our integrated and fully automated system matches the performance of dedicated SQL rewrites: compared to using only PKs and FKs, queries improve with geometric mean speedups of 35 % for TPC-DS and 29 % for JOB. Individual query latencies drop by more than 90 %. The dependency discovery overhead is orders of magnitude lower than the latency improvement of a single workload execution.

We seek to identify genes involved in Parkinson's Disease (PD) by combining information across different experiment types. Each experiment, taken individually, may contain too little information to distinguish some important genes from incidental ones. However, when experiments are combined using the proposed statistical framework, additional power emerges. The fundamental building block of the family of statistical models that we propose is a hierarchical three-group mixture of distributions. Each gene is modeled probabilistically as belonging to either a null group that is unassociated with PD, a deleterious group, or a beneficial group. This three-group formalism has two key features. By apportioning prior probability of group assignments with a Dirichlet distribution, the resultant posterior group probabilities automatically account for the multiplicity inherent in analyzing many genes simultaneously. By building models for experimental outcomes conditionally on the group labels, any number of data modalities may be combined in a single coherent probability model, allowing information sharing across experiment types. These two features result in parsimonious inference with few false positives, while simultaneously enhancing power to detect signals. Simulations show that our three-groups approach performs at least as well as commonly-used tools for GWAS and RNA-seq, and in some cases it performs better. We apply our proposed approach to publicly-available GWAS and RNA-seq datasets, discovering novel genes that are potential therapeutic targets.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and $F$ be $\mathbb{Q}$ or an imaginary quadratic field with certain conditions. In this article, we study the ideal class group $\mathrm{Cl}(F_E)$ of the $p$-division field $F_E:=F(E[p])$ of $E$ over $F$ for an odd prime number $p$. More precisely, we investigate the non-vanishing of the $E[p]$-component in the semi-simplification of $\mathrm{Cl}(F_E)/p\mathrm{Cl}(F_E)$ as an $\mathbb{F}_p[\mathrm{Gal}(F_E/F)]$-module when $E[p]$ is an irreducible $\mathrm{Gal}(F_E/F)$-module. When the analytic rank of $E$ over $F$ is $1$, we establish a new relationship between the non-vanishing of the $E[p]$-component and the $p$-divisibility of a certain $p$-adic analytic quantity associated with $E$. The quantity is defined by the leading coefficient of the cyclotomic $p$-adic $L$-function of $E$ when $F=\mathbb{Q}$ and by that of Bertolini--Darmon--Prasanna's anticyclotomic $p$-adic $L$-function of $E$ when $F$ is the imaginary quadratic field.

Addressing global challenges often involves stimulating the large-scale adoption of new products or behaviors. Research traditions that focus on individual decision making suggest that achieving this objective requires identifying the drivers of individual discrete adoption choices. On the other hand, computational approaches rooted in complexity science focus on maximizing the propagation of a given product or behavior throughout social networks of interconnected adopters. Here, by integrating discrete choice modeling into the complex contagion theory, we propose a method to estimate individual-level thresholds to adoption. We validate the predictive power of this approach in two choice experiments. By integrating the estimated thresholds into computational simulations, we show that state-of-the-art seeding policies for initiating large-scale behavioral change might be suboptimal if they neglect individual-level behavioral drivers, which can be corrected through the proposed experimental method.

It is shown that in a Cohen-Macaulay local ring, the generic linkage of an ideal $I$ is a deformation of the arbitrary linkage of $I$. This fact does not need $I$ to be a Cohen-Macaulay ideal. The same holds for $s$-residual intersections of $I$ when $s$ does not exceed the height of $I$ by one. Under some slight conditions on $I$, one further generalizes this principle to encompass any $s$-residual intersection.

The Messaging Layer security (MLS) and its underlying Continuous Group Key Agreement (CGKA) protocol allows a group of users to share a cryptographic secret in a dynamic manner, such that the secret is modified in member insertions and deletions. Although this flexibility makes MLS ideal for implementations in distributed environments, a number of issues need to be overcome. Particularly, the use of digital certificates for authentication in a group goes against the group members' privacy. In this work we provide an alternative method of authentication in which the solicitors, instead of revealing their identity, only need to prove possession of certain attributes, dynamically defined by the group, to become a member. Instead of digital certificates, we employ Attribute-Based Credentials accompanied with Selective Disclosure in order to reveal the minimum required amount of information and to prevent attackers from linking the activity of a user through multiple groups. We formally define a CGKA variant named Attribute-Authenticated Continuous Group Key Agreement (AA-CGKA) and provide security proofs for its properties of Requirement Integrity, Unforgeability and Unlinkability. We also provide an implementation of our AA-CGKA scheme and show that it achieves performance similar to a trivial certificate-based solution.

Cyber-physical systems are at the intersection of digital technology and engineering domains, rendering them high-value targets of sophisticated and well-funded cybersecurity threat actors. Prominent cybersecurity attacks on CPS have brought attention to the vulnerability of these systems and the inherent weaknesses of critical infrastructure reliant on them. Security modelling for CPS is an important mechanism to systematically identify and assess vulnerabilities, threats, and risks throughout system life cycles, and to ultimately ensure system resilience, safety, and reliability. This survey delves into state-of-the-art research on CPS security modelling, encompassing both threat and attack modelling. While these terms are sometimes used interchangeably, they are different concepts. This paper elaborates on the differences between threat and attack modelling, examining their implications for CPS security. We conducted a systematic search that yielded 449 papers, from which 32 were selected and categorised into three clusters: those focused on threat modelling methods, attack modelling methods, and literature reviews. Specifically, we sought to examine what security modelling methods exist today, and how they address real-world cybersecurity threats and CPS-specific attacker capabilities throughout the life cycle of CPS, which typically span longer durations compared to traditional IT systems. This paper also highlights several limitations in existing research, wherein security models adopt simplistic approaches that do not adequately consider the dynamic, multi-layer, multi-path, and multi-agent characteristics of real-world cyber-physical attacks.

Oncolytic virotherapy, utilizing genetically modified viruses to combat cancer and trigger anti-cancer immune responses, has garnered significant attention in recent years. In our previous work arXiv:2305.12386, we developed a stochastic agent-based model elucidating the spatial dynamics of infected and uninfected cells within solid tumours. Building upon this foundation, we present a novel stochastic agent-based model to describe the intricate interplay between the virus and the immune system; the agents' dynamics are coupled with a balance equation for the concentration of the chemoattractant that guides the movement of immune cells. We formally derive the continuum limit of the model and carry out a systematic quantitative comparison between this system of PDEs and the individual-based model in two spatial dimensions. Furthermore, we describe the traveling waves of the three populations, with the uninfected proliferative cells trying to escape from the infected cells while immune cells infiltrate the tumour. Simulations show a good agreement between agent-based approaches and numerical results for the continuum model. Some parameter ranges give rise to oscillations of cell number in both models, in line with the behaviour of the corresponding nonspatial model, which presents Hopf bifurcations. Nevertheless, in some situations the behaviours of the two models may differ significantly, suggesting that stochasticity plays a key role in the dynamics. Our results highlight that a too rapid immune response, before the infection is well-established, appears to decrease the efficacy of the therapy and thus some care is needed when oncolytic virotherapy is combined with immunotherapy. This further suggests the importance of clinically improving the modulation of the immune response according to the tumour's characteristics and to the immune capabilities of the patients.

Bayesian inference with empirical likelihood faces a challenge as the posterior domain is a proper subset of the original parameter space due to the convex hull constraint. We propose a regularized exponentially tilted empirical likelihood to address this issue. Our method removes the convex hull constraint using a novel regularization technique, incorporating a continuous exponential family distribution to satisfy a Kullback--Leibler divergence criterion. The regularization arises as a limiting procedure where pseudo-data are added to the formulation of exponentially tilted empirical likelihood in a structured fashion. We show that this regularized exponentially tilted empirical likelihood retains certain desirable asymptotic properties with improved finite sample performance. Simulation and data analysis demonstrate that the proposed method provides a suitable pseudo-likelihood for Bayesian inference.

A major step in the planning process of passenger railway operators is the assignment of rolling stock, i.e., train units, to the trips of the timetable. A wide variety of mathematical optimization models have been proposed to support this task, which we discuss and argue to be justified in order to deal with operational differences between railway operators, and hence different planning requirements, in the best possible way. Our investigation focuses on two commonly used models, the Composition model and the Hypergraph model, that were developed for Netherlands Railways (NS) and DB Fernverkehr AG (DB), respectively. We compare these models in a rolling stock scheduling setting similar to that of NS, which we show to be strongly NP-hard, and propose different variants of the Hypergraph model to tune the model to the NS setting. We prove that, in this setting, the linear programming bounds of both models are equally strong as long as a Hypergraph model variant is chosen that is sufficiently expressive. However, through a numerical evaluation on NS instances, we show that the Composition model is generally more compact in practice and can find optimal solutions in the shortest running time.

The Alperin weight conjecture has been reduced to simple groups by Navarro and Tiep. In this paper, we investigate the Navarro Alperin weight conjecture, which includes Galois automorphisms and group automorphisms in comparison with the original version, and give a reduction to simple groups. As an application, we prove the conjecture for the finite groups with abelian Sylow 2-subgroups.

Using a sample of 10 billion $J/ψ$ events collected with the BESIII detector, the decays $η' \to π^{+}π^{-}π^{+}π^{-}$, $η' \to π^{+}π^{-}π^{0}π^{0}$ and $η' \to 4 π^{0}$ are studied via the process $J/ψ\toγη'$. The branching fractions of $η' \to π^{+}π^{-}π^{+}π^{-}$ and $η' \to π^{+}π^{-}π^{0}$ $π^{0}$ are measured to be $( 8.56 \pm 0.25({\rm stat.}) \pm 0.23({\rm syst.}) ) \times {10^{ - 5}}$ and $(2.12 \pm 0.12({\rm stat.}) \pm 0.10({\rm syst.})) \times {10^{ - 4}}$, respectively, which are consistent with previous measurements but with improved precision. No significant $η' \to 4 π^{0}$ signal is observed, and the upper limit on the branching fraction of this decay is determined to be less than $1.24 \times {10^{-5}}$ at the $90\%$ confidence level. In addition, an amplitude analysis of $η' \to π^{+}π^{-}π^{+}π^{-}$ is performed to extract the doubly virtual isovector form factor $α$ for the first time. The measured value of $α=1.22 \pm 0.33({\rm stat.}) \pm 0.04({\rm syst.})$, is in agreement with the prediction of the VMD model.

We consider a nonequilibrium transition that leads to the formation of nonlinear steady-state structures due to the gas flow scattering on a partially penetrable obstacle. The resulting nonequilibrium steady state (NESS) corresponds to a two-domain gas structure attained at certain critical parameters. We use a simple mean-field model of the driven lattice gas with ring topology to demonstrate that this transition is accompanied by the emergence of local invariants related to a complex composed of the obstacle and its nearest gas surrounding, which we refer to as obstacle edges. These invariants are independent of the main system parameters and behave as local first integrals, at least qualitatively. As a result, the complex becomes insensitive to the noise of external driving field within the overcritical domain. The emerged invariants describe the conservation of the number of particles inside the obstacle and strong temporal synchronization or correlation of gas states at obstacle edges. Such synchronization guarantees the equality to zero of the total edge current at any time. The robustness against external drive fluctuations is shown to be accompanied by strong spatial localization of induced gas fluctuations near the domain wall separating the depleted and dense gas phases. Such a behavior can be associated with nonequilibrium protection effect and synchronization of edges. The transition rates between different NESSs are shown to be different. The relaxation rates from one NESS to another take complex and real values in the sub- and overcritical regimes, respectively. The mechanism of these transitions is governed by the generation of shock waves at the back side of the obstacle. In the subcritical regime, these solitary waves are generated sequentially many times, while only a single excitation is sufficient to rearrange the system state in the overcritical regime.

We associate a family of ideal sheaves to any Q-effective divisor on a complex manifold, called higher multiplier ideals, using the theory of mixed Hodge modules and V-filtrations. This family is indexed by two parameters, an integer indicating the Hodge level and a rational number, and these ideals admit a weight filtration. When the Hodge level is zero, they recover the usual multiplier ideals. We study the local and global properties of higher multiplier ideals systematically. In particular, we prove vanishing theorems and restriction theorems, provide criteria for the nontriviality, and introduce the center of minimal exponent (generalizing the notion of minimal log canonical center). The main idea is to exploit the global structure of the V-filtration along an effective divisor using the notion of twisted Hodge modules. As applications, we prove new cases of conjectures by Debarre, Casalaina-Martin and Grushevsky on singularities of theta divisors on principally polarized abelian varieties.

Let $k$ be a field and let $C$ be a small category. A $k$-linear representation of $C$, or a $kC$-module, is a functor from $C$ to the category of finite dimensional vector spaces over $k$. When the category $C$ is more general than a linear order, then its representation type is generally infinite and in most cases wild. Hence the task of understanding such representations in terms of their indecomposable factors becomes difficult at best, and impossible in general. This paper offers a new set of ideas designed to enable studying modules locally. Specifically, inspired by work in discrete calculus on graphs, we set the foundations for a calculus type analysis of $kC$-modules, under some restrictions on the category $C$. As a starting point, for a $kC$-module $M$ we define its gradient \emph{gradient} \(\nabla[M]\) as a virtual module in the Grothendieck group of isomorphism classes of $kC$-modules. Pushing the analogy with ordinary differential calculus and discrete calculus on graphs, we define left and right divergence via the appropriate left and right Kan extensions and two bilinear pairings on modules and study their properties, specifically with respect to adjointness relations between the gradient and the left and right divergence. The left and right divergence are shown to be rather easily computable in favourable cases. Having set the scene, we concentrate specifically on the case where the category $C$ is a finite poset. Our main result is a necessary and sufficient condition for the gradient of a module $M$ to vanish under certain hypotheses on the poset. We next investigate implications for two modules whose gradients are equal. Finally we consider the resulting left and right Laplacians, namely the compositions of the divergence with the gradient, and study an example of the relationship between the vanishing of the Laplacians and the gradient.

Quadratic Hamiltonians are important in quantum field theory and quantum statistical mechanics. Their general studies, which go back to the sixties, are relatively incomplete for the fermionic case studied here. Following Berezin, they are quadratic in the fermionic field and in this way well-defined self-adjoint operators acting on the fermionic Fock space. We analyze their diagonalization by applying a novel elliptic operator-valued differential equations studied in a companion paper. This allows for their ($\mathrm{N}$-) diagonalization under much weaker assumptions than before. Last but not least, in 1994 Bach, Lieb and Solovej defined them to be generators of strongly continuous unitary groups of Bogoliubov transformations. This is shown to be an equivalent definition, as soon as the vacuum state belongs to the domain of definition of these Hamiltonians. This second outcome is demonstrated to be reminiscent to the celebrated Shale-Stinespring condition on Bogoliubov transformations.

The model of generalized automata, introduced by Eilenberg in 1974, allows representing a regular language more concisely than conventional automata by allowing edges to be labeled not only with characters, but also strings. Giammarresi and Montalbano introduced a notion of determinism for generalized automata [STACS 1995]. While generalized deterministic automata retain many properties of conventional deterministic automata, the uniqueness of a minimal generalized deterministic automaton is lost. In the first part of the paper, we show that the lack of uniqueness can be explained by introducing a set $ \mathcal{W(A)} $ associated with a generalized automaton $ \mathcal{A} $. In this way, we derive for the first time a full Myhill-Nerode theorem for generalized automata, which contains the textbook Myhill-Nerode theorem for conventional automata as a degenerate case. In the second part of the paper, we show that the set $ \mathcal{W(A)} $ leads to applications for pattern matching and data compression. We show that a Wheeler generalized automata can be stored using $ \mathfrak{e} \log σ(1 + o(1)) + O(e) $ bits so that pattern matching queries can be solved in $ O(m \log \log σ) $ time, where $ \mathfrak{e} $ is the total length of all edge labels, $ e $ is the number of edges, $ σ$ is the size of the alphabet and $ m $ is the length of the pattern.

We study the existence of traveling waves of reaction-diffusion systems with delays in both diffusion and reaction terms of the form $\partial u(x,t)/\partial t = Δu(x,t-τ_1)+f(u(x,t),u(x,t-τ_2))$, where $τ_1,τ_2$ are positive constants. We extend the monotone iteration method to systems that satisfy typical monotone conditions by thoroughly studying the sign of the Green function associated with a linear functional differential equation. Namely, we show that for small positive $r$ the functional equation $x''(t)-ax'(t+r)-bx(t+r)=f(t)$, where $a\not=0, b>0$ has a unique bounded solution for each given bounded and continuous $f(t)$. Moreover, if $r>0$ is sufficiently small, $f(t)\ge 0$ for $t\in {\mathbb R}$, then the unique bounded solution $x_f(t)\le 0$ for all $t\in {\mathbb R}$. In the framework of the monotone iteration method that is developed based on this result, upper and lower solutions are found for Fisher-KPP and Belousov-Zhabotinski equations to show that traveling waves exist for these equations when delays are small in both diffusion and reaction terms. The obtained results appear to be new.

We construct an uncountable family of well-quasi-ordered permutation classes, each with a distinct enumeration sequence. This disproves a conjecture that all well-quasi-ordered permutation classes have algebraic generating functions, and in fact shows that many such classes lack D-finite or D-algebraic generating functions. Our construction is based on an uncountably large collection of factor-closed, well-quasi-ordered binary languages due to Pouzet.

Multi-agent systems, e.g., automobiles and UAVs (Unmanned Ariel Vehicles), rely on the precision of onboard sensors to accurately perceive their environment, which in turn depends on the precision of onboard sensors and reliable in-field calibration. This paper introduces a novel targetless camera-LiDAR extrinsic calibration approach called Multi-FEAT (Multi-Feature Edge AlignmenT). Multi-FEAT uses the cylindrical projection model to encode the 3D LiDAR point cloud into a 2D panorama and exploits diverse LiDAR feature information in panoramic images to supplement the sparse LiDAR point cloud boundaries. Furthermore, camera edges are extracted using off-the-shelf segmentation solutions. In addition, a feature-matching function is designed to optimize the calibration parameters. The performance of the proposed Multi-FEAT algorithm is evaluated using the KITTI dataset, and our approach shows more reliable results than several existing targetless calibration methods. We conclude our analysis with directions for future work.

It is argued that the new morphological and spectral information gleaned from the recently published LoFAR Two meter Sky Survey data release 2 (LoTSS-2 at 144 MHz) observations of the cluster Abell 980 (A980), in combination with its existing GMRT and VLA observations at higher frequencies, provide the much-needed evidence to strengthen the proposal that the cluster's radio emission comes mainly from two double radio sources, both produced by the brightest cluster galaxy (BCG) in two major episodes of jet activity. The two radio lobes left from the previous activity have become diffuse and developed an ultra-steep radio spectrum while rising buoyantly through the confining hot intra-cluster medium (ICM) and, concomitantly, the host galaxy has drifted to the cluster centre and entered a new active phase manifested by a coinciding younger double radio source. The new observational results and arguments presented here bolster the case that the old and young double radio sources in A980 conjointly represent a `double-double' radio galaxy whose two lobe-pairs have lost colinearity due to the (lateral) drift of their parent galaxy, making this system by far the most plausible case of a `Detached-Double-Double Radio Galaxy' (dDDRG).

For a smooth projective curve $X$ and reductive group $G$, the Whittaker functional on nilpotent sheaves on $\text{Bun}_G(X)$ is expected to correspond to global sections of coherent sheaves on the spectral side of Betti geometric Langlands. We prove that the Whittaker functional calculates the (shifted) microstalk of nilpotent sheaves at the point in the Hitchin moduli where the Kostant section intersects the global nilpotent cone. In particular, the (shifted) Whittaker functional is exact for the perverse $t$-structure and commutes with Verdier duality. Our proof is topological and depends on the intrinsic local hyperbolic symmetry of $\text{Bun}_G(X)$. It is an application of a general result relating vanishing cycles to the composition of restriction to an attracting locus followed by vanishing cycles.

We study information design in games where players choose from a continuum of actions and have continuously differentiable payoffs. We show that an information structure is optimal when the equilibrium it induces can also be implemented in a principal-agent contracting problem. Building on this result, we characterize optimal information structures in symmetric linear-quadratic games. With common values, targeted disclosure is robustly optimal across all priors. With interdependent and normally distributed values, linear disclosure is uniquely optimal. We illustrate our findings with applications in venture capital, Bayesian polarization, and price competition.

In this paper we prove lower bounds for sizes of refutations of unsatisfiable vector Subset Sum instances $\overrightarrow{a}_1 x_1 + \dots + \overrightarrow{a}_n x_n = \overrightarrow{b}$ in the proof system Res(lin$_{\mathbb{F}_q}$) where $char(\mathbb{F}_{q})\geq 5$. As a basis for the hardness criterion for such instances we choose the property of the matrix $A$ with columns $(\overrightarrow{a}_1, \ldots, \overrightarrow{a}_n)$ to be (the transpose of) the generating matrix for a good error-correcting code $C_{A} := \{x\cdot A\, |\, x \in \mathbb{F}_{q}^k\}\subset \mathbb{F}_{q}^n$ and prove the following lower bounds: 1) For a dag-like fragment of Res(lin$_{\mathbb{F}_q}$). We introduce the notion of $(s,r)$-robustness for Subset Sum instances, which in particular implies that $A$ defines an error-correcting code with the minimal distance $s\geq r$. For $(s,r)$-robust instances we prove $2^{Ω(r)}$ lower bound for sizes of refutations in a dag-like fragment of Res(lin$_{\mathbb{F}_q}$). We show that random instances are $(n / 3, Ω\left((n/(q + 1)\ln q))^{1/3}\right))$-robust and that specific examples achieving these bounds can be constructed using algebraic geometry codes. 2) For tree-like Res(lin$_{\mathbb{F}_q}$) refutations we show the size lower bound $2^{Ω({((q+1)\ln q)^{-1/3}}d^{1/5})}$ for any Subset Sum instance where $d$ is the minimal distance of $C_{A}$.

We discuss the use of comagnetometry in studying new physics that couples to fermionic spin. Modern comagnetometry is -- in absolute energy units -- the most sensitive experimental technique for measuring the energy difference between quantum states, reaching sensitivities in the $10^{-26}\,$eV range. The technique suppresses the magnetic interactions of the spins, making searches for non-standard-model interactions possible. Many implementations have been developed and optimized for various uses. New physics scenarios which can be probed with comagnetometers include: EDMs, violations of Lorentz invariance, Goldstone bosons of new high-energy symmetries, CP-violating long-range forces, and axionic dark matter. We consider the prospects for improvements in the technique, and show -- based purely on signal-to-noise ratio with existing technology -- that there is room for several orders of magnitude in further improvement. We also evaluate several sources of systematic error and instability that may limit improvements.

This paper deals with functional equations in the form of $f(x) + g(y) = h(x,y)$ where $h$ is given and $f$ and $g$ are unknown. We will show that if $h$ is a Borel measurable function associated with characterizations of the uniform or Cauchy distributions, then there is no measurable solutions of the equation. Our proof uses a characterization of the Dirac measure and it is also applicable to the arctan equation.