@article{22074, abstract = {We discuss how countable subadditivity of operators can be derived from subadditivity under mild forms of continuity, and provide examples manifesting such circumstances.}, author = {Grafakos, Loukas and Visan, Monica}, issn = {1473-7124}, journal = {Proceedings of the Royal Society of Edinburgh: Section A Mathematics}, number = {5}, pages = {1504--1517}, publisher = {Cambridge University Press}, title = {{Remarks on countable subadditivity}}, doi = {10.1017/prm.2023.77}, volume = {154}, year = {2024}, } @article{22079, abstract = {We prove that the cubic nonlinear Schrödinger equation (both focusing and defocusing) is globally well-posed in H^s(R) for any regularity s > −1/2. Well-posedness has long been known for s ≥ 0, see [55], but not previously for any s < 0. The scaling-critical value s = −1/2 is necessarily excluded here, since instantaneous norm inflation is known to occur [11, 40, 48]. We also prove (in a parallel fashion) well-posedness of the real- and complex-valued modified Korteweg–de Vries equations in H^s(R) for any s > −1/2. The best regularity achieved previously was s ≥ 1/4 (see [15, 24, 33, 39]). To overcome the failure of uniform continuity of the data-to-solution map, we employ the method of commuting flows introduced in [37]. In stark contrast with our arguments in [37], an essential ingredient in this paper is the demonstration of a local smoothing effect for both equations. Despite the nonperturbative nature of the well-posedness, the gain of derivatives matches that of the underlying linear equation. To compensate for the local nature of the smoothing estimates, we also demonstrate tightness of orbits. The proofs of both local smoothing and tightness rely on our discovery of a new one-parameter family of coercive microscopic conservation laws that remain meaningful at this low regularity. }, author = {Harrop-Griffiths, Benjamin and Killip, Rowan and Visan, Monica}, issn = {2050-5086}, journal = {Forum of Mathematics, Pi}, publisher = {Cambridge University Press}, title = {{Sharp well-posedness for the cubic NLS and mKdV in H^s(R)}}, doi = {10.1017/fmp.2024.4}, volume = {12}, year = {2024}, } @article{22123, abstract = {Frontal ablation is responsible for a large fraction of the mass loss from tidewater glaciers. The main contributors to frontal ablation are iceberg calving and submarine melting, with calving often being the largest. However, submarine melting, in addition to its direct contribution to mass loss, also promotes calving through the changes induced in the stress field at the glacier terminus, so both processes should be jointly analysed. Among the factors influencing submarine melting, the formation of a buoyant plume due to the emergence of fresh subglacial water at the glacier grounding line plays a key role. In this study we used Elmer/Ice to develop a 3D glacier dynamics model including calving and subglacial hydrology coupled with a line plume model to calculate the calving front position at every time step. We applied this model to the Hansbreen–Hansbukta glacier–fjord system in southern Spitsbergen, Svalbard, where a large set of data are available for both the glacier and the fjord from September 2008 to March 2011. We found that our 3D model reproduced the expected seasonal cycle of advance–retreat. Besides, the modelled front positions were in good agreement with the observed front positions at the central part of the calving front, with longitudinal differences, on average, below 15 m for the period from December 2009 to March 2011. But there were regions of the front, especially the eastern margin, that presented major differences.}, author = {Muñoz Hermosilla, José M and Otero, Jaime and De Andrés, Eva and Shahateet, Kaian and Navarro, Francisco and Pérez-Doña, Iván}, issn = {1994-0424}, journal = {The Cryosphere}, number = {4}, pages = {1911--1924}, publisher = {Copernicus Publications}, title = {{A 3D glacier dynamics–line plume model to estimate the frontal ablation of Hansbreen, Svalbard}}, doi = {10.5194/tc-18-1911-2024}, volume = {18}, year = {2024}, } @unpublished{20701, abstract = {A Proof of Exponentiation (PoE) allows a prover to efficiently convince a verifier that 𝑦 = 𝑥 𝑒 in some group of unknown order. PoEs are the basis for practical constructions of Verifiable Delay Functions (VDFs), which, in turn, are important for various higher-level protocols in distributed computing. In applications such as distributed consensus, many PoEs are generated regularly, motivating protocols for secure aggregation of batches of statements into a few statements to improve the efficiency for both parties. Rotem (TCC 2021) recently presented two such generic batch PoEs. In this work, we introduce two batch PoEs that outperform both proposals of Rotem and we evaluate their practicality. First, we show that the two batch PoEs of Rotem can be combined to improve the overall efficiency by at least a factor of two. Second, we revisit the work of Bellare, Garay, and Rabin (EUROCRYPT 1998) on batch verification of digital signatures and show that, under the low order assumption, their bucket test can be securely adapted to the setting of groups of unknown order. The resulting batch PoE quickly outperforms the state of the art in the expected number of group multiplications with the growing number of instances, and it decreases the cost of batching by an order of magnitude already for hundreds of thousands of instances. Importantly, it is the first batch PoE that significantly decreases both the proof size and complexity of verification. Our experimental evaluations show that even a nonoptimized implementation achieves such improvements, which would match the demands of real-life systems requiring large-scale PoE processing. Finally, even though our proof techniques are conceptually similar to Rotem, we give an improved analysis of the application of the low order assumption towards secure batching of PoE instances, resulting in a tight reduction, which is important when setting the security parameter in practice.}, author = {Hoffmann, Charlotte and Hubáček, Pavel and Ivanova, Svetlana}, booktitle = {Cryptology ePrint Archive}, publisher = {International Association for Cryptologic Research}, title = {{Practical batch proofs of exponentiation}}, year = {2024}, } @article{22154, abstract = {The inertia bound and ratio bound (also known as the Cvetković bound and Hoffman bound) are two fundamental inequalities in spectral graph theory, giving upper bounds on the independence number a(G) of a graph G in terms of spectral information about a weighted adjacency matrix of G. For both inequalities, given a graph G, one needs to make a judicious choice of weighted adjacency matrix to obtain as strong a bound as possible. While there is a well‐established theory surrounding the ratio bound, the inertia bound is much more mysterious, and its limits are rather unclear. In fact, only recently did Sinkovic find the first example of a graph for which the inertia bound is not tight (for any weighted adjacency matrix), answering a longstanding question of Godsil. We show that the inertia bound can be extremely far from tight, and in fact can significantly underperform the ratio bound: for example, one of our results is that for infinitely many n, there is an n‐vertex graph for which even the unweighted ratio bound can prove a(G)<4n^3/4, but the inertia bound is always at least n/4. In particular, these results address questions of Rooney, Sinkovic, and Wocjan–Elphick–Abiad.}, author = {Kwan, Matthew and Wigderson, Yuval}, issn = {1469-2120}, journal = {Bulletin of the London Mathematical Society}, number = {10}, pages = {3196--3208}, publisher = {Wiley}, title = {{The inertia bound is far from tight}}, doi = {10.1112/blms.13127}, volume = {56}, year = {2024}, } @article{22179, abstract = {Burr and Erd˝os in 1975 conjectured, and Chv´atal, R¨odl, Szemer´edi and Trotter later proved, that the Ramsey number of any bounded degree graph is linear in the number of vertices. In this paper, we disprove the natural directed analogue of the Burr–Erd˝os conjecture, answering a question of Buci´c, Letzter, and Sudakov. If H is an acyclic digraph, the oriented Ramsey number of H, denoted −→r1(H), is the least N such that every tournament on N vertices contains a copy of H. We show that for any Δ ≥ 2 and any sufficiently large n, there exists an acyclic digraph H with n vertices and maximum degree Δ such that −→r1(H) ≥ nΩ(Δ2/3/ log5/3 Δ). This proves that −→r1(H) is not always linear in the number of vertices for bounded-degree H. On the other hand, we show that −→r1(H) is nearly linear in the number of vertices for typical bounded-degree acyclic digraphs H, and obtain linear or nearly linear bounds for several natural families of bounded-degree acyclic digraphs. For multiple colors, we prove a quasi-polynomial upper bound −→rk(H)= 2(log n)Ok(1) for all bounded-degree acyclic digraphs H on n vertices, where −→rk(H) is the least N such that every k-edge-colored tournament on N vertices contains a monochromatic copy of H. For k ≥ 2 and n ≥ 4, we exhibit an acyclic digraph H with n vertices and maximum degree 3 such that −→rk(H) ≥ nΩ(log n/ log log n), showing that these Ramsey numbers can grow faster than any polynomial in the number of vertices.}, author = {Fox, Jacob and He, Xiaoyu and Wigderson, Yuval}, issn = {1565-8511}, journal = {Israel Journal of Mathematics}, number = {1}, pages = {1--48}, publisher = {Springer Nature}, title = {{Ramsey numbers of sparse digraphs}}, doi = {10.1007/s11856-024-2624-y}, volume = {263}, year = {2024}, } @article{22180, abstract = {Given a graph , its Ramsey number is the minimum so that every two‐coloring of contains a monochromatic copy of . It was conjectured by Conlon, Fox, and Sudakov that if one deletes a single vertex from , the Ramsey number can change by at most a constant factor. We disprove this conjecture, exhibiting an infinite family of graphs such that deleting a single vertex from each decreases the Ramsey number by a super‐constant factor. One consequence of this result is the following. There exists a family of graphs so that in any Ramsey coloring for (i.e., a coloring of a clique on vertices with no monochromatic copy of ), one of the color classes has density .}, author = {Wigderson, Yuval}, issn = {1097-0118}, journal = {Journal of Graph Theory}, number = {3}, pages = {663--675}, publisher = {Wiley}, title = {{Ramsey numbers upon vertex deletion}}, doi = {10.1002/jgt.23093}, volume = {106}, year = {2024}, } @article{22190, abstract = {We consider the set of 𝑚×𝑛 matrices with rational entries having numerator and denominator of size at most H and obtain various upper bounds on the number of such matrices of a given rank, or with a given determinant, or a given characteristic polynomial. We also consider similar questions for matrices whose entries are Egyptian fractions.}, author = {Afifurrahman, Muhammad and Kuperberg, Vivian Zieve and Ostafe, Alina and Shparlinski, Igor E.}, issn = {1435-5337}, journal = {Forum Mathematicum}, number = {4}, publisher = {De Gruyter}, title = {{Statistics of ranks, determinants and characteristic polynomials of rational matrices}}, doi = {10.1515/forum-2024-0114}, volume = {37}, year = {2024}, } @article{22197, abstract = {We study the distribution of consecutive sums of two squares in arithmetic progressions. If {En}n∈N is the sequence of sums of two squares in increasing order, we show that for any modulus q and any congruence classes a1, a2, a3 mod q which are admissible in the sense that there are solutions to x2 + y2 ≡ ai mod q, there exist infinitely many n with En+i−1 ≡ ai mod q, for i =1, 2, 3. We also show that for any r1, r2 ≥ 1, there exist infinitely many n with En+i−1 ≡ a1 mod q for 1 ≤ i ≤ r1 and En+i−1 ≡ a2 mod q for r1 +1 ≤ i ≤ r1 + r2}, author = {Kimmel, Noam and Kuperberg, Vivian Zieve}, issn = {0022-314X}, journal = {Journal of Number Theory}, pages = {135--147}, publisher = {Elsevier}, title = {{Consecutive runs of sums of two squares}}, doi = {10.1016/j.jnt.2024.05.003}, volume = {264}, year = {2024}, } @article{22209, abstract = {The curvature of elongated microscopic building blocks plays a crucial role on their self-assembly into orientationally ordered phases. While rod-like molecules form a handful of liquid crystal (LC) phases, curved or banana-shaped molecules show more than fifty phases, with fascinating physical properties, such as chirality or polarity. Despite the fundamental and technological importance of these so-called ‘banana-shaped liquid crystals’, little is known about their microscopic details at the single-molecule level. Curved colloidal liquid crystals—liquid crystals formed by curved colloidal rods—are excellent model systems to optically resolve the structure and dynamics of curved building blocks within these condensed phases. Recent advances in the synthesis of curved rod-like particles have unlocked the potential for studying—at the single-particle level—the intimate relationship between shape and phase symmetry, and even confirmed the stability of elusive LC phases. Further developments in this nascent field promise exciting findings, such as the first observation of the colloidal twist-bend nematic phase or the fabrication of functional materials with curvature-dependent properties. In this Report on Progress, we will highlight recent advances in the synthesis and assembly of curved colloidal liquid crystals and discuss the upcoming challenges and opportunities of this field.}, author = {Fernández-Rico, Carla and Dullens, Roel P A}, issn = {1361-6633}, journal = {Reports on Progress in Physics}, number = {9}, publisher = {IOP Publishing}, title = {{Liquid crystals from curved colloidal rods: Waves, twists and more}}, doi = {10.1088/1361-6633/ad627b}, volume = {87}, year = {2024}, } @article{22219, abstract = {Bicontinuous microstructures are essential to the function of diverse natural and synthetic systems. Their synthesis has been based on two approaches: arrested phase separation or self-assembly of block copolymers. The former is attractive for its chemical simplicity and the latter, for its thermodynamic robustness. Here we introduce elastic microphase separation (EMPS) as an alternative approach to make bicontinuous microstructures. Conceptually, EMPS balances the molecular-scale forces that drive demixing with large-scale elasticity to encode a thermodynamic length scale. This process features a continuous phase transition, reversible without hysteresis. Practically, EMPS is triggered by simply supersaturating an elastomeric matrix with a liquid, resulting in uniform bicontinuous materials with a well-defined microscopic length scale tuned by the matrix stiffness. The versatility of EMPS is further demonstrated by fabricating bicontinuous materials with superior mechanical properties and controlled anisotropy and microstructural gradients. Overall, EMPS presents a robust alternative for the bulk fabrication of homogeneous bicontinuous materials.}, author = {Fernández-Rico, Carla and Schreiber, Sanjay and Oudich, Hamza and Lorenz, Charlotta and Sicher, Alba and Sai, Tianqi and Bauernfeind, Viola and Heyden, Stefanie and Carrara, Pietro and Lorenzis, Laura De and Style, Robert W. and Dufresne, Eric R.}, issn = {1476-4660}, journal = {Nature Materials}, pages = {124--130}, publisher = {Springer Nature}, title = {{Elastic microphase separation produces robust bicontinuous materials}}, doi = {10.1038/s41563-023-01703-0}, volume = {23}, year = {2024}, } @unpublished{19425, abstract = {We demonstrate that periodically driven quantum rotors provide a promising and broadly applicable platform to implement multi-gap topological phases, where groups of bands can acquire topological invariants due to non-Abelian braiding of band degeneracies. By adiabatically varying the periodic kicks to the rotor we find nodal-line braiding, which causes sign flips of topological charges of band nodes and can prevent them from annihilating, indicated by non-zero values of the %non-Abelian patch Euler class. In particular, we report on the emergence of an anomalous Dirac string phase arising in the strongly driven regime, a truly out-of-equilibrium phase of the quantum rotor. This phase emanates from braiding processes involving all (quasienergy) gaps and manifests itself with edge states at zero angular momentum. Our results reveal direct applications in state-of-the-art experiments of quantum rotors, such as linear molecules driven by periodic far-off-resonant laser pulses or artificial quantum rotors in optical lattices, whose extensive versatility offers precise modification and observation of novel non-Abelian topological properties. }, author = {Karle, Volker and Lemeshko, Mikhail and Bouhon, Adrien and Slager, Robert-Jan and Ünal, F. Nur}, booktitle = {arXiv}, title = {{Anomalous multi-gap topological phases in periodically driven quantum rotors}}, doi = {10.48550/arXiv.2408.16848}, year = {2024}, } @article{19051, abstract = {This paper corrects an error in an earlier work of the author.}, author = {Browning, Timothy D}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, number = {13}, pages = {10165--10168}, publisher = {Oxford University Press}, title = {{The polynomial sieve and equal sums of like polynomials}}, doi = {10.1093/imrn/rnae066}, volume = {2024}, year = {2024}, } @unpublished{20071, abstract = {Farkas established that a system of linear inequalities has a solution if and only if we cannot obtain a contradiction by taking a linear combination of the inequalities. We state and formally prove several Farkas-like theorems over linearly ordered fields in Lean 4. Furthermore, we extend duality theory to the case when some coefficients are allowed to take "infinite values".}, author = {Dvorak, Martin and Kolmogorov, Vladimir}, booktitle = {arXiv}, keywords = {Farkas lemma, linear programming, extended reals, calculus of inductive constructions}, title = {{Duality theory in linear optimization and its extensions -- formally verified}}, doi = {10.48550/arXiv.2409.08119}, year = {2024}, } @unpublished{19545, abstract = {We prove the Eigenstate Thermalisation Hypothesis for Wigner matrices uniformly in the entire spectrum, in particular near the spectral edges, with a bound on the fluctuation that is optimal for any observable. This complements earlier works of Cipolloni et. al. (Comm. Math. Phys. 388, 2021; Forum Math., Sigma 10, 2022) and Benigni et. al. (Comm. Math. Phys. 391, 2022; arXiv: 2303.11142) that were restricted either to the bulk of the spectrum or to special observables. As a main ingredient, we prove a new multi-resolvent local law that optimally accounts for the edge scaling.}, author = {Cipolloni, Giorgio and Erdös, László and Henheik, Sven Joscha}, booktitle = {arXiv}, title = {{Eigenstate thermalisation at the edge for Wigner matrices}}, doi = {10.48550/arXiv.2309.05488}, year = {2024}, } @unpublished{19550, abstract = {We introduce a multi-band BCS free energy functional and prove that for a multi-band superconductor the effect of inter-band coupling can only increase the critical temperature, irrespective of its attractive or repulsive nature and its strength. Further, for weak coupling and weaker inter-band coupling, we prove that the dependence of the increase in critical temperature on the inter-band coupling is (1) linear, if there are two or more equally strongly superconducting bands, or (2) quadratic, if there is only one dominating band.}, author = {Henheik, Sven Joscha and Langmann, Edwin and Lauritsen, Asbjørn Bækgaard}, booktitle = {arXiv}, title = {{Multi-band superconductors have enhanced critical temperatures}}, doi = {10.48550/arXiv.2409.17297}, year = {2024}, } @unpublished{19547, abstract = {For correlated real symmetric or complex Hermitian random matrices, we prove that the local eigenvalue statistics at any cusp singularity are universal. Since the density of states typically exhibits only square root edge or cubic root cusp singularities, our result completes the proof of the Wigner-Dyson-Mehta universality conjecture in all spectral regimes for a very general class of random matrices. Previously only the bulk and the edge universality were established in this generality [arXiv:1804.07744], while cusp universality was proven only for Wigner-type matrices with independent entries [arXiv:1809.03971, arXiv:1811.04055]. As our main technical input, we prove an optimal local law at the cusp using the Zigzag strategy, a recursive tandem of the characteristic flow method and a Green function comparison argument. Moreover, our proof of the optimal local law holds uniformly in the spectrum, thus also re-establishing universality of the local eigenvalue statistics in the previously studied bulk [arXiv:1705.10661] and edge [arXiv:1804.07744] regimes.}, author = {Erdös, László and Henheik, Sven Joscha and Riabov, Volodymyr}, booktitle = {arXiv}, title = {{Cusp universality for correlated random matrices}}, doi = {10.48550/arXiv.2410.06813}, year = {2024}, } @unpublished{19551, abstract = {We introduce a notion of a \emph{local gap} for interacting many-body quantum lattice systems and prove the validity of response theory and Kubo's formula for localized perturbations in such settings. On a high level, our result shows that the usual spectral gap condition, concerning the system as a whole, is not a necessary condition for understanding local properties of the system. More precisely, we say that an equilibrium state ρ0 of a Hamiltonian H0 is locally gapped in Λgap⊂Λ, whenever the Liouvillian −i[H0,⋅] is almost invertible on local observables supported in Λgap when tested in ρ0. To put this into context, we provide other alternative notions of a local gap and discuss their relations. The validity of response theory is based on the construction of \emph{non-equilibrium almost stationary states} (NEASSs). By controlling locality properties of the NEASS construction, we show that response theory holds to any order, whenever the perturbation \(\epsilon V\) acts in a region which is further than |logϵ| away from the non-gapped region Λ∖Λgap.}, author = {Henheik, Sven Joscha and Wessel, Tom}, booktitle = {arXiv}, title = {{Response theory for locally gapped systems}}, doi = {10.48550/arXiv.2410.10809}, year = {2024}, } @article{22444, abstract = {Abstract. Hydro-pedotransfer functions (PTFs) relate easy-to-measure and readily available soil information to soil hydraulic properties (SHPs) for applications in a wide range of process-based and empirical models, thereby enabling the assessment of soil hydraulic effects on hydrological, biogeochemical, and ecological processes. At least more than 4 decades of research have been invested to derive such relationships. However, while models, methods, data storage capacity, and computational efficiency have advanced, there are fundamental concerns related to the scope and adequacy of current PTFs, particularly when applied to parameterise models used at the field scale and beyond. Most of the PTF development process has focused on refining and advancing the regression methods, while fundamental aspects have remained largely unconsidered. Most soil systems are not represented in PTFs, which have been built mostly for agricultural soils in temperate climates. Thus, existing PTFs largely ignore how parent material, vegetation, land use, and climate affect processes that shape SHPs. The PTFs used to parameterise the Richards–Richardson equation are mostly limited to predicting parameters of the van Genuchten–Mualem soil hydraulic functions, despite sufficient evidence demonstrating their shortcomings. Another fundamental issue relates to the diverging scales of derivation and application, whereby PTFs are derived based on laboratory measurements while often being applied at the field to regional scales. Scaling, modulation, and constraining strategies exist to alleviate some of these shortcomings in the mismatch between scales. These aspects are addressed here in a joint effort by the members of the International Soil Modelling Consortium (ISMC) Pedotransfer Functions Working Group with the aim of systematising PTF research and providing a roadmap guiding both PTF development and use. We close with a 10-point catalogue for funders and researchers to guide review processes and research.}, author = {Weber, Tobias Karl David and Weihermüller, Lutz and Nemes, Attila and Bechtold, Michel and Degré, Aurore and Diamantopoulos, Efstathios and Fatichi, Simone and Filipović, Vilim and Gupta, Surya and Hohenbrink, Tobias L. and Hirmas, Daniel R. and Jackisch, Conrad and de Jong van Lier, Quirijn and Koestel, John and Lehmann, Peter and Marthews, Toby R. and Minasny, Budiman and Pagel, Holger and van der Ploeg, Martine and Shojaeezadeh, Shahab Aldin and Svane, Simon Fiil and Szabó, Brigitta and Vereecken, Harry and Verhoef, Anne and Young, Michael and Zeng, Yijian and Zhang, Yonggen and Bonetti, Sara}, issn = {1607-7938}, journal = {Hydrology and Earth System Sciences}, number = {14}, pages = {3391--3433}, publisher = {Copernicus Publications}, title = {{Hydro-pedotransfer functions: A roadmap for future development}}, doi = {10.5194/hess-28-3391-2024}, volume = {28}, year = {2024}, } @article{22485, abstract = {Allometric scaling relations are widely used to link biological processes to body size in nature. Several studies have shown that such scaling laws hold also for natural ecosystems, including individual trees and forests, riverine metabolism, and river network organization. However, the derivation of scaling laws for catchment-scale water and carbon fluxes has not been achieved so far. Here, we focus on scaling relations of catchment green metabolism, defined as the set of ecohydrological and biogeochemical processes through which vegetation assemblages in catchments maintain their structure and react to the surrounding environment. By revising existing plant size–density relationships and integrating them across large-scale domains, we show that the ecohydrological fluxes occurring at the catchment scale are invariant with respect to the above-ground vegetation biomass per unit area of the basin, while they scale linearly with catchment size. We thus demonstrate that the sublinear scaling of plant metabolism results in an isometric scaling at catchment and regional scales. Deviations from such predictions are further shown to collapse onto a common distribution, thus incorporating natural fluctuations due to resource limitations into a generalized scaling theory. Results from scaling arguments are supported by hyperresolution ecohydrological simulations and remote sensing observations.}, author = {Bassani, Francesca and Fatichi, Simone and Rinaldo, Andrea and Bonetti, Sara}, issn = {1091-6490}, journal = {Proceedings of the National Academy of Sciences}, number = {42}, publisher = {National Academy of Sciences}, title = {{Toward a metabolic theory of catchments: Scaling of water and carbon fluxes with size}}, doi = {10.1073/pnas.2410736121}, volume = {121}, year = {2024}, }