Classical public-key cryptography standards rely on the Rivest-Shamir-Adleman (RSA) encryption protocol. The security of this protocol is based on the exponential computational complexity of the most efficient classical algorithms for factoring large semiprime numbers into their two prime components. Here, we address RSA factorization building on Schnorr's mathematical framework where factorization translates into a combinatorial optimization problem. We solve the optimization task via tensor network methods, a quantum-inspired classical numerical technique. This tensor network Schnorr's sieving algorithm displays numerical evidence of polynomial scaling of resources with the bit-length of the semiprime. We factorize RSA numbers up to 100 bits and assess how computational resources scale through numerical simulations up to 130 bits, encoding the optimization problem in quantum systems with up to 256 qubits. Only the high-order polynomial scaling of the required resources limits the factorization of larger numbers. Although these results do not currently undermine the security of the present communication infrastructure, they strongly highlight the urgency of implementing post-quantum cryptography or quantum key distribution.

Understanding the large-scale structure of the universe remains a fundamental challenge in cosmology, with computational simulations providing critical insights into non-linear structure growth. Particularly, computational simulations critical information about the non-linear growth processes behind the observed large-scale structures. Inspired by the similarly porous structure of polymer membranes prepared using phase-inversion, this work presents a novel thermodynamic approach to cosmic structure formation. A numerical framework is presented, based on the Cahn-Hilliard model of spinodal decomposition for a binary mixture treating the universe as a two-component fluid of matter and dark-energy. The dimensionless Cahn-Hilliard equation is solved using finite-element methods, with parameters calibrated to Planck 2018 cosmology. The simulation evolves an initially homogeneous matter distribution through 500 timesteps, corresponding to 35 million years of cosmological evolution. The simulated matter distribution exhibits quantitative agreement with observational surveys across multiple metrics. Void fraction evolves to 0.416 at z ~ 0.65, right at the edge of domain of applicability of Lambda-CDM model. Filamentarity reaches 0.42, comparable to Millennium Simulation results. The linear growth factor extracted from simulated density fields also closely agrees with ΛCDM predictions over the interval 9.300 < t < 9.335 Gyr. This work establishes spinodal decomposition as a viable thermodynamic framework for cosmic structure formation, offering a computationally efficient alternative to traditional N-body methods while reproducing key quantitative observables. The approach opens new avenues for exploring matter-dark energy interactions and may prove valuable for next-generation survey analysis.

Characterizing the dimensionality of the superconducting state in infinite-layer (IL) nickelates is essential for understanding its nature. Most studies have addressed this by examining the anisotropy of the upper critical fields. However, the dominance of Pauli paramagnetic effects over orbital effects complicates the interpretation of these experiments in terms of dimensionality. Here, we approach the question from a different perspective by mapping the vortex phase diagram. We show that superconducting Pr0.8Sr0.2NiO2 thin films with low disorder exhibit a vortex liquid-to-glass transition of a quasi-two-dimensional (2D) nature. In contrast, increasing disorder drives a crossover into a pure 2D state. This demonstrates that pure bidimensionality is an extrinsic property, resulting from the decoupling of NiO2 planes due to enhanced disorder. Our findings establish disorder as a key control parameter of superconductivity in IL nickelates and suggest that it resides within the NiO2 planes, providing two fundamental insights for understanding these materials.

One of the main open questions in the field of luminous ($L_{\rm bol}>10^{47}\,\rm erg\,s^{-1}$) quasars (QSOs) at $z \gtrsim 6$ is the rapid formation ($< 1\,$Gyr) of their supermassive black holes (SMBHs). For this work we analysed the relation between the X-ray properties and other properties describing the physics and growth of both the accretion disc and the SMBH in QSOs at the Epoch of Reionization (EoR). The sample consists of 21 $z>6$ QSOs, which includes 16 sources from the rapidly grown QSOs from the HYPERION sample and five other luminous QSOs with available high-quality archival X-ray data. We discovered a strong and statistically significant ($>3σ$) relation between the X-ray continuum photon index ($Γ$) and the $\rm C\,IV$ disc wind velocity ($v_{\rm C\,IV}$) in $z>6$ luminous QSOs, whereby the higher the $v_{\rm C\,IV}$, the steeper the $Γ$. This relation suggests a link between the disc-corona configuration and the kinematics of disc winds. Furthermore, we find evidence at $>2-3σ$ level that $Γ$ and $v_{\rm C\,IV}$ are correlated to the growth rate history of the SMBH. Although additional data are needed to confirm it, this result may suggest that, in luminous $z>6$ QSOs, the SMBH predominantly grows via fast accretion rather than via initial high seed BH mass.

We advance the theory of parametric bootstrap in constructing highly efficient empirical best (EB) prediction intervals of small area means. The coverage error of such a prediction interval is of the order $O(m^{-3/2})$, where $m$ is the number of small areas to be pooled using a linear mixed normal model. In the context of an area level model where the random effects follow a non-normal known distribution except possibly for unknown hyperparameters, we analytically show that the order of coverage error of empirical best linear (EBL) prediction interval remains the same even if we relax the normality of the random effects by the existence of pivot for a suitably standardized random effects when hyperpameters are known. Recognizing the challenge of showing existence of a pivot, we develop a simple moment-based method to claim non-existence of pivot. We show that existing parametric bootstrap EBL prediction interval fails to achieve the desired order of the coverage error, i.e. $O(m^{-3/2})$, in absence of a pivot. We obtain a surprising result that the order $O(m^{-1})$ term is always positive under certain conditions indicating possible overcoverage of the existing parametric bootstrap EBL prediction interval. In general, we analytically show for the first time that the coverage problem can be corrected by adopting a suitably devised double parametric bootstrap. Our Monte Carlo simulations show that our proposed single bootstrap method performs reasonably well when compared to rival methods.

This paper presents a revolutionary approach to the characterization, forecast, and control of collective systems. Collective systems are an ensemble of conservatively interacting entities. The evolution of the entities are determined by symmetries of the entities. Collective systems take many different forms. A plasma is a collective of charged particles, a fluid is a collective of molecules, a elementary field is a collective of elementary particles, and a cosmos is a collective of celestial bodies. Our new theory builds on the canonical transformation approach to dynamics. This approach recognizes that the symmetry leads to the conservation of a real function, that is the infinitesimal generator of a Lie group. The finite generator of the canonical transformation is derived from the infinitesimal generator by the solution of the Hamilton-Jacobi equation. This generating function is also known as the action, the entropy, and the logarithmic likelihood. The new theory generalizes this generating function to the generating functional of the collective field. Finally, this paper derives the formula for the Mayer Cluster Expansion, or the S-matrix expansion of the generating functional. We call it the Heisenberg Scattering Transformation (HST). Practically, this is a localized Fourier Transformation, whose principal components give the singularity spectrums, that is the solution to the Renormalization Group Equations. Limitations on the measurement of the system (that is the Born Rule and the Heisenberg Uncertainty Principle) lead to quantization of the stochastic probabilities of the collective field. How different collective systems couple together to form systems-of-systems is formalized. The details of a practical implementation of the HST will be presented.

We study a self-similar fragmentation process with dislocation measure $ν$ and self-similarity index $α> 0$. Let $e^{-m_t}$ denote the size of the largest fragment at time $t \geq 0$. For dislocation measures satisfying a regularity condition of the form $ν(1 - s_1 > δ) = δ^{-θ} \ell(1/δ)$ with $θ\in [0,1)$ and slowly varying $\ell$, we prove almost sure convergence \[ \lim_{t \to \infty} (m_t - g(t)) = 0, \] where $g(t) = (\log t - (1 - θ) \log \log t + f(t))/α$, and $f(t) = o(\log \log t)$ is a lower order correction that can be described explicitly in terms of $\ell$ and $θ$. Our results sharpen substantially the best prior result on general self-similar fragmentation processes, due to Bertoin, which states that $m_t = (1+o(1)) \log (t)/α$.

Motivated by applications to spaces of embeddings and automorphisms of manifolds, we consider a tower of $\infty$-categories of truncated right-modules over a unital $\infty$-operad $\mathcal{O}$. We study monoidality and naturality properties of this tower, identify its layers, describe the difference between the towers as $\mathcal{O}$ varies, and generalise these results to the level of Morita $(\infty,2)$-categories. Applied to the ${\rm BO}(d)$-framed $E_d$-operad, this extends Goodwillie-Weiss' embedding calculus and its layer identification to the level of bordism categories. Applied to other variants of the $E_d$-operad, it yields new versions of embedding calculus, such as one for topological embeddings, based on ${\rm BTop}(d)$, or one similar to Boavida de Brito-Weiss' configuration categories, based on ${\rm BAut}(E_d)$. In addition, we prove a delooping result in the context of embedding calculus, establish a convergence result for topological embedding calculus, improve upon the smooth convergence result of Goodwillie, Klein, and Weiss, and deduce an Alexander trick for homology 4-spheres.

Identifying conservation laws is central to every subfield of physics, as they illuminate the underlying symmetries and fundamental principles. A prime example can be found in quantum optics: The conservation of orbital angular momentum (OAM) during spontaneous parametric down-conversion (SPDC) enables the generation of a photon pair with entangled OAM. In this article, we report on the first study of OAM conservation in SPDC pumped by single photons. Our results present the first implementation of cascaded down-conversion without waveguides, setting the stage for experiments on the direct generation of multi-photon high-dimensional entanglement using all degrees of freedom of light.

Let $ Λ(s) := Γ(s+1)\, (1-2^{1-s}) \, ζ(s) $, and denote its set of zeros by $ Z_Λ:= Z_ζ\cup Z_\mathrm{p} $, where $ Z_ζ$ consists of the nontrivial zeros of $ ζ(s) $ and $ Z_\mathrm{p} $ those of the prefactor $ ( 1-2^{1-s} ) $, with $ s \neq 1 $. We introduce a non-symmetric operator $ R $ on $ L^2([0,\infty)) $ with spectrum \[ σ(R) = \left\{ i\left(1/2- λ\right) \mid λ\in Z_Λ\right\} \, . \] Assuming the simplicity of all nontrivial Riemann zeros, we construct the compression $ R_{Z_ζ} $ of $ R $ to the spectral subspace associated with $ Z_ζ$, and show that $ R_{Z_ζ} $ is intertwined with its adjoint by a positive semidefinite operator $ W $; i.e., $ W \, R_{Z_ζ} = R_{Z_ζ}^\dagger \, W $ with $ W \ge 0 $. The positivity of $ W $, viewed as an operator-theoretic form of (Bombieri's refinement of) Weil's positivity criterion, enforces $ \Re(ρ)=1/2 $ for all $ ρ\in Z_ζ$, in accordance with the Riemann Hypothesis. Under the same positivity condition, the intertwining relation yields a self-adjoint operator whose spectrum coincides with the set $ \{ \Im(ρ) \mid ρ\in Z_ζ\} $. We further extend the framework to accommodate higher-order nontrivial Riemann zeros, should they exist, and to cover any Mellin-transformable $ L $-function satisfying a functional equation.

We consider the continuous superposition of operators of the form \[ \iint_{[0, 1]\times (1, N)} (-Δ)_p^s \,u\,dμ(s,p), \] where $μ$ denotes a signed measure over the set $[0, 1]\times (1, N)$, joined to a nonlinearity satisfying a proper subcritical growth. The novelty of the paper relies in the fact that, differently from the existing literature, the superposition occurs in both $s$ and $p$. Here we introduce a new framework which is so broad to include, for example, the scenarios of the finite sum of different (in both $s$ and $p$) Laplacians, or of a fractional $p$-Laplacian plus a $p$-Laplacian, or even combinations involving some fractional Laplacians with the "wrong" sign. The development of this new setting comes with two applications, which are related to the Weierstrass Theorem and a Mountain Pass technique. The results obtained contribute to the existing literature with several specific cases of interest which are entirely new.

For a dust-like self-similar set (generated by IFSs with the strong separation condition), Elekes, Keleti and Máthé found an invariant, called `algebraic dependence number', by considering its generating IFSs and isometry invariant self-similar measures. We find an intrinsic quantitative characterisation of this number: it is the dimension over $\mathbb{Q}$ of the vector space generated by the logarithms of all the common ratios of infinite geometric sequences in the gap length set, minus 1. With this concept, we present a lower bound on the cardinality of generating IFS (with or without separation conditions) in terms of the gap lengths of a dust-like set. We also establish analogous result for dust-like graph-directed attractors on complete metric spaces. This is a new application of the ratio analysis method and the gap sequence.

There is a fast-growing literature on estimating optimal treatment rules directly by maximizing the expected outcome. In biomedical studies and operations applications, censored survival outcome is frequently observed, in which case the truncated mean survival time and survival probability are of great interest. In this paper, we propose two robust criteria for learning optimal treatment rules with censored survival outcomes; the former one targets an optimal treatment rule maximizing the truncated mean survival time, where the cutoff is specified by a given quantile such as median; the latter one targets an optimal treatment rule maximizing buffered survival probabilities, where the predetermined threshold is adjusted to account for the truncated mean survival time. We develop a sampling-based difference-of-convex algorithm for learning the proposed optimal treatment rules, and provide theoretical justifications for them. In simulation studies, our estimators show improved performance compared to existing methods. We also demonstrate the proposed method using AIDS clinical trial data.

We prove that the equational theory of the positive calculus of relations with transitive closure (PCoR*) is EXPSPACE-complete. Here, PCoR* terms consist of the following standard operators on binary relations: identity, empty, universality, union, intersection, composition, converse, and reflexive transitive closure (so, PCoR* terms subsume Kleene algebra and allegory terms as fragments). Additionally, we show that the equational theory of PCoR* extended with tests and nominals (in hybrid logic) is still EXPSPACE-complete; moreover, it is PSPACE-complete for its intersection-free fragment. To this end, we design derivatives on graphs by extending derivatives on words for regular expressions. The derivatives give a finite automata construction on path decompositions, like those on words. Because the equational theory has a linearly bounded pathwidth model property, we can decide the equational theory of PCoR* using these automata.

Boolean satisfiability (SAT) is a propositional logic problem of determining whether an assignment of variables satisfies a Boolean formula. Many combinatorial optimization problems can be formulated in Boolean SAT logic -- either as k-SAT decision problems or Max k-SAT optimization problems, with conflict-driven (CDCL) solvers being the most prominent. Despite their ability to handle large instances, CDCL-based solvers have fundamental scalability limitations. In light of this, we propose recently-developed quadratic unconstrained binary optimization (QUBO) solvers as an alternative platform for 3-SAT problems. To utilize them, we implement a 2-step [3-SAT]-[Max 2-SAT]-[QUBO] conversion procedure and present a rigorous proof to explicitly calculate the number of both satisfied and violated clauses of the original 3-SAT instance from the transformed Max 2-SAT formulation. We then demonstrate, through numerical simulations on several benchmark instances, that digital QUBO solvers can achieve state-of-the-art accuracy on 78-variable 3-SAT benchmark problems. Our work facilitates the broader use of quantum annealers on noisy intermediate-scale quantum (NISQ) devices, as well as their quantum-inspired digital counterparts, for solving 3-SAT problems.

We study abelian surfaces defined over finite fields which do not contain any possibly singular curve of genus less than or equal to $3$. Firstly, we complete and expand the characterisation of isogeny classes of abelian surfaces with no curves of genus up to $2$ initiated by the first author \emph{et al.~}in previous work. Secondly, we show that, for simple abelian surfaces, containing a curve of genus $3$ is equivalent to admitting a polarisation of degree $4$. Thanks to this result, we can use existing algorithms to check which isomorphism classes in the isogeny classes containing no genus $2$ curves have a polarisation of degree $4$. Thirdly, we characterise isogeny classes of abelian surfaces with no curves of genus $\leq 2$, containing no abelian surface with a polarisation of degree $4$. Finally, we describe the absolutely irreducible genus $3$ curves lying on abelian surfaces containing no curves of genus less than or equal to $2$.

We present an interpretable emulator for the linear matter power spectrum (MPS) in the standard cosmological model $Λ$CDM, constructed via a physics-informed symbolic regression framework. By combining domain knowledge with a machine learning technique known as genetic algorithms, we explore the space of analytic expressions to derive closed-form, smooth, physically motivated approximations of the MPS that match the accuracy of standard broadband reconstruction methodologies such as the Savitzky-Golay filter. Building upon this baseline, we incorporate transparent oscillatory corrections informed by the physics of baryon acoustic oscillations (BAO). The resulting expression delivers mean sub-percent fractional errors across a broad range of scales ($k \in [10^{-5}, 1.5]~h\,\mathrm{Mpc}^{-1}$) with an average deviation of $\sim 0.4\%$ when tested against spectra computed with a Boltzmann solver. Moreover, a comparable level of fractional deviation is maintained on smaller scales when the GA-derived formulation is used as input to the nonlinear emulator halofit. To illustrate the versatility of the framework beyond $Λ$CDM, we apply it to a representative $f(R)$ gravity model. Rather than training a general modified-gravity emulator, we compute the corresponding linear spectra with a Boltzmann solver and fit a parametric deformation of the $Λ$CDM smoothed component. This procedure achieves average errors at the 1.5-1.8\% level and captures the leading modulation of the MPS induced by modified gravity, enabling a controlled study of its impact on the BAO scale. Our results provide compact, accurate, and physically motivated fitting functions for the linear MPS in both standard and MG cosmologies, offering a fast and transparent alternative to existing emulators for parameter inference and theoretical modeling in large-scale structure analyses.

Let $f$ be a real arithmetic function and let $g:[1,\infty[\to{\mathbb R}$ be a smooth function. We describe two emblematic instances in which saddle-point estimates may be used to evaluate the frequency, on the set of integers $n\leqslant x$, of the event $\{f(n)\leqslant g(n)\}$ from those relevant to the event $\{f(n)\leqslant y\}$. The first example revisits Dickman's historical contribution to the theory of friable integers. The second is concerned with the distribution of the squarefree kernel of an integer.

We show that under mild assumptions, the Fano surfaces of lines on smooth cubic threefolds are the only smooth subvarieties of abelian varieties whose Tannaka group for the convolution of perverse sheaves is an exceptional simple group. This in particular leads to a considerable strengthening of our previous work on the Shafarevich conjecture. A key idea is to control the Hodge decomposition on cohomology by a cocharacter of the Tannaka group of Hodge modules, and to play this off against an improvement of the Hodge number estimates for irregular varieties by Lazarsfeld-Popa and Lombardi.

We study ``V-shaped'' solutions to the KPZ equation, those having opposite asymptotic slopes $θ$ and $-θ$, with $θ>0$, at positive and negative infinity, respectively. Answering a question of Janjigian, Rassoul-Agha, and Seppäläinen, we show that the spatial increments of V-shaped solutions cannot be statistically stationary in time. This completes the classification of statistically time-stationary spatial increments for the KPZ equation by ruling out the last case left by those authors. To show that these V-shaped time-stationary measures do not exist, we study the location of the corresponding ``viscous shock,'' which, roughly speaking, is the location of the bottom of the V. We describe the limiting rescaled fluctuations, and in particular show that the fluctuations of the shock location are not tight, for both stationary and flat initial data. We also show that if the KPZ equation is started with V-shaped initial data, then the long-time limits of the time-averaged laws of the spatial increments of the solution are mixtures of the laws of the spatial increments of $x\mapsto B(x)+θx$ and $x\mapsto B(x)-θx$, where $B$ is a standard two-sided Brownian motion.

Polar dielectrics with low crystal symmetry and sharp phonon resonances can support hyperbolic shear polaritons - highly confined surface modes with frequency-dependent optical axes and asymmetric dissipation features. So far, these modes have been observed only in bulk natural materials at mid-infrared frequencies, with properties limited by available crystal geometries and phonon resonance strength. Here we introduce hyperbolic shear metasurfaces: ultrathin engineered surfaces supporting hyperbolic surface modes with symmetry-tailored axial dispersion and loss redistribution that can maximally enhance light-matter interactions. By engineering effective shear phenomena in these engineered surfaces, we demonstrate geometry-controlled, ultra-confined, low-loss hyperbolic surface waves with broadband Purcell enhancements, applicable across a broad range of the electromagnetic spectrum.

Synchrotron radiation-based X-ray microtomography is uniquely suited for post mortem three-dimensional visualization of organs such as the mouse brain. Tomographic imaging of the entire mouse brain with isotropic cellular resolution requires an extended field-of-view and produces datasets of multiple terabytes in size. These data must be processed and made accessible to domain experts who may have only limited image processing knowledge. We present extended-field X-ray microtomography with 0.65 $μ$m voxel size covering an entire mouse brain. The 4,495 projections from 8 x 8 offset acquisitions were stitched to reconstruct a volume of 15,000$^3$ voxels. The microtomography volume was non-rigidly registered to the Allen Mouse Brain Common Coordinate Framework v3 based on a combination of image intensity and landmark pairs. The data were transformed block-wise and stored in a public repository with a hierarchical format for navigation and overlay with anatomical annotations in online viewers such as Neuroglancer or siibra-explorer. This study demonstrates X-ray imaging and data processing for a full mouse brain, augmenting current atlases by improving resolution in the third dimension by an order of magnitude. The 3.3-teravoxel dataset is publicly available and easily accessible for domain experts via browser-based viewers.

We present estimates for the covering numbers of the unit ball of Reproducing Kernel Hilbert Spaces (RKHSs) of functions on $M^d$ a d-dimensional compact two-point homogeneous space. The RKHS is generated by a continuous zonal/isotropic positive definite kernel. We employ the representation in terms of the Schoenberg/Fourier series expansion for continuous isotropic positive definite kernels, given in terms of a family of orthogonal polynomials on $M^d$. The bounds we present carry accurate information about the asymptotic constants depending on the dimension of the manifold and the decay or growth rate of the coefficients of the kernel. The results we present extend the estimates previously known for continuous isotropic positive definite kernels on the d-dimensional unit sphere. We present the weak asymptotic equivalence for the order of the growth of covering numbers associated to kernels on $M^d$ with a convergent geometric sequence of coefficients. We apply our estimates in order to present a bound for the covering numbers of the spherical Gaussian kernel, and to present bounds for formal examples on $M^d$.

Case-I interval-censored (current status) data from multistate systems are often encountered in biomedical and epidemiological studies. In this article, we focus on the problem of estimating state entry distribution and occupation probabilities, contingent on a preceding state occupation. This endeavor is particularly complex owing to the inherent challenge of the unavailability of directly observed counts of individuals at risk of transitioning from a state, due to severe interval censoring. We propose two nonparametric approaches, one using the fractional at-risk set approach recently adopted in the right-censoring framework and the other a new estimator based on the ratio of marginal state occupation probabilities. Both estimation approaches utilize innovative applications of concepts from the competing risks paradigm. The finite-sample behavior of the proposed estimators is studied via extensive simulation studies where we show that the estimators based on severely censored current status data have good performance when compared with those based on complete data. We demonstrate the application of the two methods to analyze data from patients diagnosed with breast cancer.

For $R>0$, we give a rigorous probabilistic construction on the cylinder $\mathbb{R} \times (\mathbb{R}/(2πR\mathbb{Z}))$ of the (massless) Sinh-Gordon model. In particular we define the $n$-point correlation functions of the model and show that these exhibit a scaling relation with respect to $R$. The construction, which relies on the massless Gaussian Free Field, is based on the spectral analysis of a quantum operator associated to the model. Using the theory of Gaussian multiplicative chaos, we prove that this operator has discrete spectrum and a strictly positive ground state.

The relevance of shallow-depth quantum circuits has recently increased, mainly due to their applicability to near-term devices. In this context, one of the main goals of quantum circuit complexity is to find problems that can be solved by shallow quantum circuits but require more computational resources classically. Our first contribution in this work is to prove new separations between classical and quantum constant-depth circuits. Firstly, we show a separation between constant-depth quantum circuits with quantum advice $\mathsf{QNC}^0/\mathsf{qpoly}$, and $\mathsf{AC}^0[p]$, which is the class of classical constant-depth circuits with unbounded-fan in and $\mathsf{MOD}_{p}$ gates. Additionally, we show a separation between $\mathsf{QAC}^0$, the circuit class containing Toffoli gates with unbounded control, and $\mathsf{AC}^0[p]$, when $\mathsf{QAC}^0$ is augmented with additional mid-circuit measurements and classical fanout. This establishes the first such separation for a shallow-depth quantum class that does not involve quantum fanout gates, while relying solely on finite quantum gate sets. Equivalently, this yields a separation between $\mathsf{AC}^0[p]$ and $[\mathsf{QNC}^0, \mathsf{AC}^0]^2$, i.e., shallow quantum circuits interleaved with simple classical computation. Secondly, we consider $\mathsf{QNC}^0$ circuits with infinite-size gate sets. We show that these circuits, along with quantum prime modular gates or classical prime modular gates in combination with classical fanout, can implement threshold gates, showing that $\mathsf{QNC}^0[p]=\mathsf{QTC}^0$. Finally, we also show that in the infinite-size gate set case, these quantum circuit classes for higher-dimensional Hilbert spaces do not offer any advantage to standard qubit implementations.

We discuss the existence theory of a nonlinear problem of nonlocal type subject to Neumann boundary conditions. Differently from the existing literature, the elliptic operator under consideration is obtained as a superposition of operators of mixed order. The setting that we introduce is very general and comprises, for instance, the sum of two fractional Laplacians, or of a fractional Laplacian and a Laplacian, as particular cases (the situation in which there are infinitely many operators, and even a continuous distribution of operators, can be considered as well). New bits of functional analysis are introduced to deal with this problem. An eigenvalue analysis divides the existence theory into two streams, one related to a Mountain Pass method, the other to a Linking technique.

We consider multi-path routing of entanglement in quantum networks, where a pre-prepared multipartite entangled 2D cluster state serves as a resource to perform different tasks on demand. We show how to achieve parallel connections between multiple, freely chosen groups of parties by performing appropriate local measurements among diagonal paths, which preserves the entanglement structure of the remaining state. We demonstrate how to route multiple Bell-states along parallel lines via crossings, turns and fade-in/-outs, analogously to a data bus. The results apply to networks at any scale.

This paper sets out to explore the modulational (or Benjamin-Feir) instability of a monochromatic wave propagating in the presence of damping such as that induced by sea-ice on the ocean surface. The fundamental wave motion is modelled using the spatial Zakharov equation, to which either uniform or non-uniform (frequency dependent) damping is added. By means of mode truncation the spatial analogue of the classical Benjamin-Feir instability can be studied analytically using dynamical systems techniques. The formulation readily yields the free surface envelope, giving insight into the physical implications of damping on the modulational instability. The evolution of an initially unstable mode is also studied numerically by integrating the damped, spatial Zakharov equation, in order to complement the analytical theory. This sheds light on the effects of damping on spectral broadening arising from this instability.

We present DMaC, a novel distributed, finite-time algorithm that guarantees max-consensus in directed networks with unreliable communication links experiencing packet drops. Unlike existing methods, DMaC ensures all nodes compute the exact maximum state under arbitrary packet loss patterns. It incorporates a fully distributed termination mechanism, enabling nodes to autonomously determine whether convergence has occurred. Our algorithm leverages narrowband error-free feedback channels to acknowledge successful (single-bit) transmissions with minimal communication overhead. We analyze our algorithm's operation, and we provide a convergence proof establishing explicit bounds on the required time steps. We validate its correctness in a wireless sensor network for environmental monitoring, and finally, we compare against existing approaches highlighting our algorithm's operational advantages.