We introduce and study coherent polarization self-rotation (CPSR), a two-photon light-matter interaction in dense alkali-metal vapors that enables both narrowband optical spectroscopy of magnetic transitions and coherent coupling between light and collective atomic spins. Unlike conventional polarization self-rotation, CPSR requires initial spin polarization and a predominantly linearly polarized probe. It operates efficiently even in optically thick vapors with high buffer-gas pressure, rapid spin-exchange collisions, and optically-unresolved hyperfine structure. We demonstrate CPSR with near-unity contrast in rubidium and achieve an exceptionally narrow two-photon linewidth of 10 Hz in potassium. CPSR realizes a coherent interface between one optical quadrature and the long-lived collective electronic spin, offering a robust and scalable spin-light coupling in optically thick platforms. This opens new opportunities for quantum optics, including quantum-enhanced metrology in the audio-frequency band and coherent transduction between light and ultra-long-lived noble-gas spins via alkali spins.

EPR showed that two particles emitted from a source can be entangled by a shared wavefunction where two non-commuting observables (position, momentum) can be simultaneously real, leading to a contradiction with quantum mechanics (two non-commuting variables can not be simultaneously real). John Bell derived an inequality where any local hidden variables prediction is bounded and quantum mechanics can violate the inequality. Bell tests on correlated photon pairs showed a clear violation of the Bell inequality and agreement with quantum mechanics. This study revealed that the Bell inequality holds at any individual instances in a given system but it does not hold over the entire population. The Bell inequality is not an incompatibility criterion for local hidden variables vs quantum mechanics rather it is a criterion for an individual nature vs population dynamics. The nonlocality prediction given by wavefunction is a system approximation and local causality (photon polarization) provides a complete description of the Bell tests. The paper concludes with great satisfaction that the measurement of two correlated photons at a distance was a local cause (photon-filter interaction did not violate the special theory of relativity).

In this paper we theoretically discuss thermal Hall effect of magnons in insulating Néel ordered antiferromagnets at zero external magnetic field. We show that for compensated Néel order the non-zero thermal Hall effect will occur in the absence of any symmetry between the two magnetic sublattices, thus making the system ferrimagnetic. We then show that collinear Dzyaloshinskii's weak ferromagnets, in which there is a symmetry connecting the magnetic sublattices, also show magnon thermal Hall effect. The thermal Hall effect of magnons will be non-zero by a virtue of the spin-momentum splitting of the magnon spectrum due to the Dzyaloshinskii-Moriya interaction as well as second-nearest exchange interaction different in the two magnetic sublattices, both corresponding to the broken symmetries that lead to the Dzyaloshinskii's invariant. We construct a theoretical model in which an external electric field may change the symmetry of the antiferromagnetic system thus altering the thermal Hall effect of magnons.

Unconventional superconductivity typically emerges out of a strongly correlated normal state, manifesting as a highly renormalized Fermi liquid or a strange metal with $T$-linear resistivity. In Ruddlesden-Popper bilayer nickelates, superconductivity with a critical temperature $T_{\rm c}$ exceeding 80 and 40~K has been respectively realised in pressurized bulk crystals and epitaxially strained thin films. These advancements call for the characterisation of fundamental normal-state and superconducting parameters in these new materials platforms of high-$T_{\rm c}$ superconductivity. Here we report detailed magnetotransport experiments on superconducting La$_2$PrNi$_2$O$_7$ (LPNO) thin films under pulsed magnetic fields up to 64~T and access the normal-state behaviour over a wide temperature range between 1.5 and 300~K. We find that the normal state of thin-film LPNO exhibits the hallmarks of Fermi-liquid transport, including $T^2$ temperature dependence of resistivity and Hall angle, and $H^2$ magnetoresistance obeying Kohler scaling. Using the empirical Kadowaki-Woods ratio, we estimate a quasiparticle effective mass $m^*/m_e \simeq 10$, thereby revealing the highly renormalized Fermi liquid state therein. Our results demonstrate that thin-film LPNO follows the same $T_{\rm c}/T_{\rm F}$ scaling observed across a myriad of strongly correlated superconductors and establish key normal-state characteristics of strained bilayer superconducting nickelates.

In this paper, we develop a general analysis for the fixed points of the operators defining the graph splitting methods from [SIAM J. Optim., 34 (2024), pp. 1569-1594] by Bredies, Chenchene and Naldi. We particularize it to the case where the maximally monotone operators are normal cones of closed linear subspaces and provide an explicit formula for the limit points of the graph splitting schemes. We exemplify these results on some particular algorithms, unifying in this way some results previously derived as well as obtaining new ones.

This paper presents a customized second-order cone programming (SOCP) solver tailored for embedded real-time optimization, which frequently arises in modern guidance and control (G&C) applications. The solver employs a practically efficient predictor-corrector type primal-dual interior-point method (PDIPM) combined with a homogeneous embedding framework for infeasibility detection. Unlike conventional homogeneous self-dual embedding formulations, the adopted approach can directly handle quadratic cost functions without requiring problem reformulation. This capability allows the solver to directly address quadratic objective SOCP problems, while avoiding unnecessary performance degradation caused by the loss of sparsity due to problem reformulation. To support a systematic workflow, we also develop a code generation tool that analyzes the sparsity pattern of the problem to be solved and generates customized solver code using a predefined code template. The generated solver code is written in C with no external dependencies other than the standard library math.h, and it supports complete static allocation of all data. Additionally, it provides parsing information to facilitate the use of the solver by end users. Finally, benchmark and numerical experiments on an embedded platform demonstrate that the developed solver outperforms the existing solvers on problem scales typical of G&C applications.

We study a three-dimensional Boussinesq-type temperature-velocity system on a bounded smooth domain $\mathcal D\subset\mathbb R^3$, where the velocity $u^\varepsilon$ solves the Navier-Stokes equations and the temperature $θ^\varepsilon$ is driven by Dirichlet boundary noise of intensity $\sqrt{\varepsilon}$. The boundary forcing produces a stochastic convolution $Z^\varepsilon$ which is, in general, only continuous in time with values in $H^{-\frac12-δ_θ}(\mathcal D)$. To handle this roughness together with initial data $θ_0\in W^{s,6/5}(\mathcal D)$, we work in the ambient space $H^{-\frac12-δ_u}(\mathcal D)$ with $δ_u\ge \max\{δ_θ,\frac12-s\}$. Given a finite time $T>0$, for any $p>4$ and sufficiently small initial data, we prove existence and uniqueness of a mild solution $(u^\varepsilon,θ^\varepsilon)$ up to a stopping time $τ^\varepsilon\le T$ such that \[ u^\varepsilon \in W^{1,p}(0,τ^\varepsilon;H^{-\frac12-δ_u}(\mathcal D)) \cap L^p (0,τ^\varepsilon;H^{\frac32-δ_u}(\mathcal D)), \quad θ^\varepsilon \in C(0,τ^\varepsilon;H^{-\frac12-δ_u}(\mathcal D)). \] Moreover, we obtain a high-probability global existence estimate of the form $\mathbb P(τ^\varepsilon=T)\geq 1- C\varepsilon $, with $C= C( δ_θ, T)>0.$

Multi-fidelity methods that use an ensemble of models to compute a Monte Carlo estimator of the expectation of a high-fidelity model can significantly reduce computational costs compared to single-model approaches. These methods use oracle statistics, specifically the covariance between models, to optimally allocate samples to each model in the ensemble. However, in practice, the oracle statistics are estimated using additional model evaluations, whose computational cost and induced error are typically ignored. To address this issue, this paper proposes an adaptive algorithm to optimally balance the resources between oracle statistics estimation and final multi-fidelity estimator construction, leveraging ideas from multilevel best linear unbiased estimators in Schaden and Ullmann (2020) and a bandit-learning procedure in Xu et al. (2022). Under mild assumptions, we demonstrate that the multi-fidelity estimator produced by the proposed algorithm exhibits mean-squared error commensurate with that of the best linear unbiased estimator under the optimal allocation computed with oracle statistics. Our theoretical findings are supported by detailed numerical experiments, including a parametric elliptic PDE and an ice-sheet mass-change modeling problem.

Megathrusts host Earth's largest earthquakes. Understanding the physical conditions controlling their rupture dynamics is critical for assessing seismic and tsunami hazards. These earthquakes often display complex rupture dynamics, exemplified by the 2011 Tohoku-Oki earthquake, which exhibited multiple rupture episodes, depth-dependent seismic radiation, and substantial tsunamigenic slip near the trench. However, how such complexity arises from preexisting physical conditions remains uncertain. Here, we demonstrate that the observed rupture complexity of the Tohoku-Oki earthquake can spontaneously and self-consistently emerge, driven by rapid coseismic frictional restrengthening and data-informed fault heterogeneity. We use an ensemble of 3D dynamic rupture simulations to identify that mixed downdip pulse-like and updip crack-like rupture are driven by dynamic stress redistribution with episodic rupture reactivation. By featuring low fault strength compared to its dynamic stress drop, a preferred model can consistently reproduce the observed complex depth-dependent propagation speeds, multiple rupture fronts as imaged by back-projection, and large tsunamigenic slip at the trench. Our findings demonstrate that preexisting fault heterogeneity conjointly with dynamic frictional weakening and restrengthening drives seemingly unexpected megathrust rupture complexity, highlighting the need to include dynamic effects into physics-based seismic and tsunami hazard assessments of future earthquakes.

The complex phases $ϕ_d$ and $ϕ_s$, associated with the mixing between neutral $B_q^0$ and $\bar B_q^0$ mesons ($q\in\{d,s\}$), are key observables to test the Standard Model and search for contributions from new physics. They are conventionally determined from the measurements of mixing-induced CP violation in the decays $B_d^0\to J/ψK^0$, $B_s^0\to J/ψϕ$ and $B_s^0\to D_s^+D_s^-$. To reach the highest possible precision on $ϕ_d$ and $ϕ_s$, it is crucial that corrections from next-to-leading order effects - primarily associated with penguin decay topologies - are accounted for. The strategy adopted in this paper uses the $SU(3)$ flavour symmetry of QCD to relate the unknown contributions from penguin topologies to their counterparts in suitably-chosen control modes, where their effects are enhanced. Utilising new CP asymmetry measurements from LHCb on the decays $B_s^0\to D_s^+D_s^-$, $B_d^0\to D^+D^-$, $B^+\to J/ψK^+$ and $B^+\to J/ψπ^+$, as well as from Belle-II on the decay $B_d^0\to J/ψπ^0$, we present the current state-of-the-art picture on controlling the penguin contributions and extract $ϕ_d$ and $ϕ_s$ from the corresponding observables. We explore the prospects for the end of the Belle-II and HL-LHC flavour physics programmes, and demonstrate the importance of measuring the control modes with future data.

We present a new way to identify systems similar to our Local Group (LG) of galaxies in cosmological simulations. Our method uses as a new constraint the speed and direction of the LG's center of mass, which we can measure accurately from cosmic microwave background data. When we apply these criteria to different cosmological simulations and compare the results with traditional selection methods, we find statistically significant differences. Our approach produces simulated galaxy pairs where the relative M31 velocity is less radial (2 percent to 7 percent difference over the mean) and more tangential (1 percent to 3 percent difference over the mean) than in cases that do not take into account the barycenter speed. The radial change pattern appears consistently across all cosmological models we test, showing that matching the observed barycenter velocity has a measurable effect when modeling and interpreting Local Group-like systems.

Motivated by algorithmic applications, Kun, O'Brien, Pilipczuk, and Sullivan introduced the parameter linear chromatic number as a relaxation of treedepth and proved that the two parameters are polynomially related. They conjectured that treedepth could be bounded from above by twice the linear chromatic number. In this paper we investigate the properties of linear chromatic number and provide improved bounds in several graph classes.

Recently, there were works claiming that path integral quantisation of gauge theories necessarily requires relaxation of Lagrangian constraints. As has also been noted in the literature, it is of course wrong since there perfectly exist gauge field quantisations respecting the constraints, and at the same time the very idea of changing the classical theory in this way has many times appeared in other works. On the other hand, what was done in the path integral approach is fixing a gauge in terms of zero-momentum variables. We would like to show that this relaxation is what normally happens when one fixes such a gauge at the level of action principle in a Lagrangian theory. Moreover, there is an interesting analogy to be drawn. Namely, one of the ways to quantise a gauge theory is to build an extended Hamiltonian and then add new conditions by hand such as to make it a second class system. The constraints' relaxation occurs when one does the same at the level of the total Hamiltonian, i.e. a second class system with the primary constraints only.

Although QMLE is generally inconsistent, logistic regression relying on the binary choice model (BCM) with logistic errors is widely used, especially in machine learning contexts with many covariates. This paper revisits the slope consistency of QMLE for BCMs. Ruud (1983) introduced a set of conditions under which QMLE may yield a constant multiple of the slope coefficient of BCMs asymptotically. However, he did not fully establish the slope consistency of QMLE, which requires the existence of a positive multiple of the true slope that maximizes the population QMLE likelihood over an appropriately restricted parameter space. We close this gap by providing a formal proof of slope consistency under the same set of conditions for BCMs identified as in Manski (1975, 1985). Our result implies that, under suitable conditions, logistic regression yields a consistent estimate of the slope coefficient for BCMs.

We study topological boundedness of order-to-topology bounded and order-to-topology continuous operators from ordered vector spaces to topological vector spaces. The uniform boundedness principle for such operators is investigated.

This paper investigates two-player ergodic nonzero-sum stochastic differential games with McKean-Vlasov dynamics. We establish a verification theorem connecting solutions of coupled Hamilton-Jacobi-Bellman (HJB) Master equations to Nash equilibria, characterized through an auxiliary control problem defined on the measure space. A key contribution is showing that the value functions are uniquely determined (up to an additive constant) by the uniqueness of the invariant measure of the optimal state process. The theory is applied to Linear-Quadratic-Gaussian (LQG) settings, where explicit solutions to the Master equations are derived by exploiting their polynomial structure in measure variables.

The Wigner eigenfunctions of a free quantum particle propagating on a plane are derived. Two possibilities are analysed. Firstly, the particle of given energy and angular momentum is discussed. In that case, a special choice of coordinates on the symplectic space $(\mathbb{R}^{4},\,ω)$ suitable for the representation of eigenstates of the discussed particle is presented. Further, the Moyal $\star_{(\text{M})}$-product on the phase space is derived with the use of the Fedosov algorithm adapted to these coordinates on a flat phase space. Next, the eigenvalue equations for the Wigner eigenfunction are solved and the physically acceptable solutions are identified. Secondly, the particle with fixed components of the Cartesian momentum is considered. Finally, a relationship between the Wigner eigenfunction of the particle with the fixed components of the Cartesian momentum and the cross-Wigner functions of the particle with the given energy and angular momentum is found.

Using the theory of pro-p groups and relative Poincaré duality, we define a type of cobordism category well suited to arithmetic topology. We completely classify topological quantum field theories on these two-dimensional versions of our cobordism categories. This classification uses Frobenius algebras with extra operations corresponding to automorphisms of the p-adic integers. We look in more detail at the example of arithmetic Dijkgraff--Witten theory for a finite gauge p-group in this setting. This allows us to deduce formulae counting Galois extensions of local p-adic fields whose Galois groups are the given gauge group.

We establish a Murnaghan--Nakayama rule for the irreducible characters of the cyclotomic Hecke algebra $\mathscr H_{m,n}(q,u)$ on Shoji's standard elements. Combined with Shoji's determinacy result, our formula provides a direct combinatorial route to the full irreducible character table of $\mathscr H_{m,n}(q,u)$. Our construction is based on our recent multi-parameter Murnaghan--Nakayama rule for Macdonald polynomials and specializes uniformly to several previously known formulas, including those for the complex reflection group of type $G(m,1,n)$ and the Iwahori--Hecke algebras of types $A$ and $B$. In a dual framework, using the vertex operator realization of Schur functions, we also derive a complementary iterative formula for irreducible characters on upper multipartitions, which may be viewed as a dual Murnaghan--Nakayama rule. As applications, we obtain a Regev-type formula and a Lübeck--Prasad--Adin--Roichman-type formula for cyclotomic Hecke algebras, extending the corresponding formulas for the Iwahori--Hecke algebra of type $A$ and the complex reflection group, respectively. We further introduce the notion of multiple bitrace for cyclotomic Hecke algebras and give a general combinatorial formula for the multiple bitrace. As a specialization, this yields the second orthogonality relation for irreducible characters of the complex reflection group $W_{m,n}$. For practical computation, we also include in an appendix a SageMath implementation of our Murnaghan--Nakayama rule, which computes individual character values and the full character table.

We give a cohomological criterion for certain decomposition of Borel graphs, which is an analog of Dunwoody's work on accessibility of groups. As an application, we prove that a Borel graph $(X,G)$ with uniformly bounded degrees of cohomological dimension one is Lipschitz equivalent to a Borel acyclic graph on $X$. This gives a new proof of a result of Chen-Poulin-Tao-Tserunyan on Borel graphs with components quasi-isometric to trees.

Quasar pairs, a special subclass of galaxy pairs, are valuable in the investigation of quasar interaction, clustering, co-evolution between the two quasars' host galaxies, the growth of supermassive black holes, as well as the formation and evolution of galaxies. However, quasar pairs on kpc scales are observationally rare. The scarcity of available samples hindered the deeper exploration and statistics of these objects. In this work, we apply an astrometric method to systematically search for quasar candidates within a transverse distance of 100 kpc to known quasars in the Million Quasar Catalog. These candidates are \textit{Gaia} sources with proper motions and parallaxes that are consistent with zero. Visual inspection of the sample was performed to remove the contamination of crowded stellar fields and nearby galaxies. A total of 4\,112 quasar pair candidates were isolated, with a median member separation of 8.81{\arcsec}, a median \textit{Gaia} $G$-band magnitude of 20.52, and a median redshift of 1.61. Our catalog was compared with three major candidate quasar pair catalogs and identified 3\,984 new quasar pair candidates previously uncataloged in the three catalogs. Several interesting quasar pair candidates are highlighted and discussed. We also briefly discussed our quasar selection and several techniques for improving the success rate of quasar pair selection. Extensive spectroscopic follow-up campaigns are being carried out to validate their astrophysical nature.

Pendulum-like dynamics is a universal motif across many areas of physics, underlying systems ranging from classical nonlinear oscillators to superconducting qubits and cold-atom tunneling platforms. Here we present an exact frequency-domain formulation of the pendulum equation that applies uniformly across oscillatory, separatrix, and rotational regimes. The resulting spectral representation reveals a previously hidden unification: all regimes share the same analytic spectral structure and characteristic frequency scale. We discover that all regimes arise from a single universal spectral kernel, with parity selection distinguishing the periodic motions and the separatrix representing their discrete-to-continuum limit. Regime changes thus correspond to symmetry-driven reorganizations in frequency space rather than changes in the underlying spectral structure, with the stopping trajectory representing the continuum limit reached without system-size scaling. The spectral structure can be derived via a spectral discretization approach starting from the separatrix solution, without relying on the classical Jacobi elliptic formulation. Beyond providing closed-form solutions, the framework reveals a transparent spectral structure underlying a broad class of classical and quantum pendulum-like systems.

Since early 2024, highly pathogenic avian influenza virus (HPAIV) H5N1 of clade 2.3.4.4b has spilled over from wild birds to dairy cattle in the United States (U.S.), spreading to more than 1000 herds and threatening both animal and public health. Denmark's location along major migratory flyways and the lack of active HPAIV surveillance in cattle underscore the need to assess potential spillover risk from wild birds to cattle to strengthen preparedness. A quantitative spillover risk assessment model was developed to integrate data from Bird Flu Radar, eBird, and cattle density to estimate the weekly probability of HPAIV introduction from wild birds to cattle. The model was calibrated using observed U.S. spillover data and extrapolated to Denmark under the assumption of a comparable transmission rate parameter. Under the frequency-dependent model, the expected HPAIV introductions to Danish cattle via wild birds remain below 0.35 cases per week, with the highest temporal risk from December to March. High-risk areas were concentrated along the Danish coastline and near the German border. In contrast, applying a density-dependent model shifted the spatial risk toward regions with higher cattle densities, while the high-risk temporal periods remained unchanged. Overall, the results indicate a spatially and temporally variable risk of HPAIV spillover from wild birds to cattle in Denmark. The model establishes a data-driven framework to strengthen early warning and guide targeted surveillance efforts in high-risk regions.

We present an existence and uniqueness result for weak solutions of Dirichlet boundary value problems governed by a nonlocal operator in divergence form and in the presence of a datum which is assumed to belong only to $L^1(Ω)$ and to be suitably dominated. We also prove that the solution that we find converges, as $s\nearrow 1$, to a solution of the local counterpart problem, recovering the classical result as a limit case. This requires some nontrivial customized uniform estimates and representation formulas, given that the datum is only in $L^1(Ω)$ and therefore the usual regularity theory cannot be leveraged to our benefit in this framework. The limit process uses a nonlocal operator, obtained as an affine transformation of a homogeneous kernel, which recovers, in the limit as $s\nearrow 1$, every classical operator in divergence form.

Cascading failures in power systems caused by sequential tripping of components are a serious concern as they can lead to complete or partial shutdowns, disrupting vital services and causing damage and inconvenience. In prior work, we developed a new approach for identifying and preventing cascading failures in power systems. The approach uses supervisory control technique of discrete event systems (DES) by incorporating both on-line lookahead control and forcible events. In this paper, we use modular supervisory control of DES to reduce computation complexity and increase the robustness and reliability of control. Modular supervisory control allows us to predict and mitigate cascading failures in power systems more effectively. We implemented the proposed control technique on a simulation platform developed in MATLAB and applied the proposed DES controller. The calculations of modular supervisory control of DES are performed using an external tool and imported into the MATLAB platform. We conduct simulation studies for the IEEE 30-bus, 118-bus and 300-bus systems, and the results demonstrate the effectiveness of our proposed approach.

Complex and larger networks are becoming increasingly prevalent in scientific applications in various domains. Although a number of models and methods exist for such networks, cross-validation on networks remains challenging due to the unique structure of network data. In this paper, we propose a general cross-validation procedure called NETCROP (NETwork CRoss-Validation using Overlapping Partitions). The key idea is to divide the original network into multiple subnetworks with a shared overlap part, producing training sets consisting of the subnetworks and a test set with the node pairs between the subnetworks. This train-test split provides the basis for a network cross-validation procedure that can be applied on a wide range of model selection and parameter tuning problems for networks. The method is computationally efficient for large networks as it uses smaller subnetworks for the training step. We provide methodological details and theoretical guarantees for several model selection and parameter tuning tasks using NETCROP. Numerical results demonstrate that NETCROP performs accurate cross-validation on a diverse set of network model selection and parameter tuning problems. The results also indicate that NETCROP is computationally much faster while being often more accurate than the existing methods for network cross-validation.

Rydberg atom triangular arrays in an optical cavity serve as an ideal platform for understanding the interplay between geometric frustration and quantized photons. Using a large-scale quantum Monte Carlo method, we obtain a rich ground state phase diagram. Around half-filling, the infinite long-range light-matter interaction lifts the ground state degeneracy, resulting in a novel order-coexisted superradiant clock phase that completely destroys the fragile order-by-disorder phase observed in classical light fields. According to the Ginzburg-Landau theory, this replacement may result from the competition between threefold and sixfold clock terms. Similar to the spin supersolid, the clear first-order phase transition at the $Z_2$ symmetry line is attributed to the nonzero photon density, which couples to the threefold clock term. Finally, we discuss the low-energy physics in the dimer language and propose that cavity-mediated nonlocal ring exchange interactions may play a critical role in the rich physics induced by the attachment of cavity-QED. Our work opens a new arena of research on the emergent phenomena of quantum phase transitions in many-body quantum optics.

We study non-Birkhoff periodic orbits in symmetric convex planar billiards. Our main result provides a quantitative criterion for the existence of such orbits with prescribed minimal period, rotation number, and spatiotemporal symmetry. We exploit this criterion to find sufficient conditions for a symmetric billiard to possess infinitely many non-Birkhoff periodic orbits. It follows that arbitrarily small analytical perturbations of the circular billiard have non-Birkhoff periodic orbits of any rational rotation number and with arbitrarily long periods. We also generalize a known result for elliptical billiards to other $\mathbb{D}_2$-symmetric billiards. Lastly, we provide Matlab codes which can be used to numerically compute and visualize the non-Birkhoff periodic orbits whose existence we prove analytically.

The symmetric subspace of multi-qubit systems, that is, the space of states invariant under permutations, is commonly encountered in applications in the context of quantum information and communication theory. It is known that the symmetric subspace can be described in terms of irreducible representations of the group $SU(2)$, whose representation spaces form a basis of symmetric states, the so-called Dicke states. In this work, we present deformations of the symmetric subspace as deformations of this group structure, which are promoted to a quantum group $\mathcal{U}_q(\mathfrak{su}(2))$. We see that deformations of the symmetric subspace obtained in this manner correspond to local deformations of the inner product of each spin, in such a way that departure from symmetry can be encoded in a position-dependent inner product. The consequences and possible extensions of these results are also discussed.

Monolayers (MLs) of semiconducting transition metal dichalcogenides (\mbox{S-TMDs}) emit light very efficiently and display rich spin-valley physics, with gyromagnetic ($g$-) factors of about -4. Here, we investigate how these properties can be tailored by alloying. Magneto-optical spectroscopy is used to reveal the peculiar properties of excitonic complexes in Mo$_{x}$W$_{1-x}$Se$_2$ MLs with different Mo and W concentrations. We show that the alloys feature extremely high $g$-factors for neutral excitons, that change gradually with the composition up to reaching values of the order of -10 for $x \approx 0.2$. First-principles calculations corroborate the experimental findings and provide evidence that alloying in S-TMDs results in a non-trivial band structure engineering, being at the origin of the high $g$-factors. The theoretical framework also suggests a higher strain sensitivity of the alloys, making them promising candidates for tailor-made optoelectronic devices.