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The electric polarizability and the spread of the total position tensors are used to characterize the metallic vs insulator nature of large (finite) systems. Finite clusters are usually treated within the open boundary condition formalism. This introduces border effects, which prevent a fast convergence to the thermodynamic limit and can be eliminated within the formalism of periodic boundary conditions. Recently, we introduced an original approach to periodic boundary conditions, named Clifford boundary conditions. It considers a finite fragment extracted from a periodic system and the modification of its topology into that of a Clifford torus. The quantity representing the position is modified in order to fulfill the system periodicity. In this work, we apply the formalism of Clifford boundary conditions to the case of carbon nanotubes, whose treatment results in a particularly simple zigzag geometry. Indeed, we demonstrate that at the Hückel level, these nanotubes, either finite or periodic, are formally equivalent to a collection of non-interacting dimerized linear chains, thus simplifying their treatment. This equivalence is used to describe some nanotube properties as the sum of the contributions of the independent chains and to identify the origin of peculiar behaviors (such as conductivity). Indeed, if the number of hexagons along the circumference is a multiple of three, a metallic behavior is found, namely a divergence of both the (per electron) polarizability and total position spread of at least one linear chain. These results are in agreement with those in the literature from tight-binding calculations. [ABSTRACT FROM AUTHOR]

Let S be a smooth del Pezzo surface that is defined over a field K and splits over a Galois extension L. Let G be either the split reductive group given by the root system of S L in Pic S L , or a form of it containing the Néron–Severi torus. Let G be the G -torsor over S L obtained by extension of structure group from a universal torsor T over S L. We prove that G does not descend to S unless T does. This is in contrast to a result of Friedman and Morgan that such G always descend to singular del Pezzo surfaces over C from their desingularizations. [ABSTRACT FROM AUTHOR]

We analyse the dynamics of an infinitesimal particle around an elongated body, which is modelled as a homogeneous fixed straight segment centred at the origin. We assume that the length of the segment is small compared with the distance to the particle. After a Lie–Deprit normalization, we end up with a Hamiltonian that has not only the mean anomaly but also the argument of the perigee relegated to terms or third order or higher. We employ invariant and reduction theories to reduce the artificial symmetries associated with the Kepler flow and the central action of the angular momentum. Analysing the relative equilibria in the first and second reduced spaces allows us to determine the existence of near-polar circular periodic orbits and KAM tori. [ABSTRACT FROM AUTHOR]

Over the years, polynuclear cyclic or torus complexes have attracted increasing interest due to their unique metal topologies and properties. However, the isolation of polynuclear cyclic organometallic complexes is extremely challenging due to their inherent reactivity, which stems from the labile and reactive metal‐carbon bonds. In this study, the pyrazine ligand undergoes a radical‐radical cross‐coupling reaction leading to the formation of a decanuclear [(Cp*)20Dy10(L1)10] ⋅ 12(C7H8) (1; where L1 = anion of 2‐prop‐2‐enyl‐2H‐pyrazine; Cp* = pentamethylcyclopentadienyl) complex, where all DyIII metal centres are bridged by the anionic L1 ligand. Amongst the family of polynuclear Ln organometallic complexes bearing CpR2Lnx units (CpR = substituted cyclopentadienyl), 1 features the highest nuclearity obtained to date. In‐depth computational studies were conducted to elucidate the proposed reaction mechanism and formation of L1, while probing of the magnetic properties of 1, revealed slow magnetic relaxation upon application of a static dc field. [ABSTRACT FROM AUTHOR]

We consider an H(2)-move between two torus links of type (2, 2n), T2n. We give some necessary conditions for two links T2n and T2m which are related by a single H(2)-move. In particular, we show that T2n is obtained by a single H(2)-move from the Hopf link if and only if n = ±1, ± 3, and T2n is obtained by a single H(2)-move from the trivial 2-component link if and only if n = 0, ± 2, [ABSTRACT FROM AUTHOR]

Magnetic flux ropes (MFRs) at the center of interplanetary coronal mass ejections (ICMEs) are often characterized as simplistic cylindrical or toroidal tubes with field lines that twist around the cylinder or torus axis. Recent multipoint observations suggest that the overall geometry of these large-scale structures may be significantly more complex. As such, contemporary modeling approaches are likely insufficient to properly understand the global structure of any ICME. In an attempt to rectify this issue, we have developed a novel flux rope modeling approach that allows for the description of arbitrary distortions of the flux rope cross section or deformation of the magnetic axis. The resulting distorted MFR model is a fully analytic model that can be used to describe a complex geometry and is numerically efficient enough to be used for event reconstructions. To demonstrate the usefulness of our approach, we focus on a specific implementation of our model and apply it to an ICME event that was observed in situ on 2023 April 23 at the L1 point by the Wind spacecraft and also by the STEREO-A spacecraft, which was 10.°2 further east and 0.°9 south in heliographic coordinates. We demonstrate that our model can accurately reconstruct each observation individually and also gives a fair reconstruction of both events simultaneously using a multipoint reconstruction algorithm, which results in a geometry that is inconsistent with a cylindrical or toroidal approximation. [ABSTRACT FROM AUTHOR]

A toric braid is a braid whose closure is a torus link in ℝ3. Manturov [A combinatorial representation of links by quasitoric braids, European J. Combin. 23(2) (2002) 207–212] generalized toric braids that is called quasitoric braids and showed that the subset of quasitoric braids in the classical braid group is a subgroup of the braid group. We call this subgroup the quasitoric braid group. In this paper, we give two minimal generating sets for the quasitoric braid group and determine its abelianization. The minimalities of these two generating sets are obtained from a lower bound by the number of generators for the abelianization. [ABSTRACT FROM AUTHOR]

We show that certain smooth tori with group \mathbb {Z} in S^4 have exteriors with standard equivariant intersection forms, and so are topologically unknotted. These include the turned 1-twist-spun tori in the 4-sphere constructed by Boyle, the union of the genus one Seifert surface of Cochran and Davis that has no slice derivative with a ribbon disc, and tori with precisely four critical points whose middle level set is a 2-component link with vanishing Alexander polynomial. This gives evidence towards the conjecture that all \mathbb {Z}-surfaces in S^4 are topologically unknotted, which is open for genus one and two. It is unclear whether these tori are smoothly unknotted, except for tori with four critical points whose middle level set is a split link. The double cover of S^4 branched along any of these surfaces is a potentially exotic copy of S^2 \times S^2, and, in the case of turned twisted tori, we show they cannot be distinguished from S^2 \times S^2 using Seiberg–Witten invariants. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity of brake orbits on certain symplectic manifold. We give a criterion to find brake orbits for even Hamiltonian on the cotangent space of T n by the methods of the Maslov-index theory and a critical point theorem formulated by Bartsch and Wang in [1]. More specifically, if H is even and satisfies certain growth conditions, one can find more brake orbits on the cotangent space of T n . [ABSTRACT FROM AUTHOR]

We give a recursive construction of the categorified Young symmetrizer introduced by Abel and Hogancamp (Sel Math (NS) 23:1739–1801, 2017) corresponding to the single-column partition. As a consequence, we obtain new expressions for the uncolored y-ified HOMFLYPT homology of positive torus links and the y-ified column-colored HOMFLYPT homology of positive torus knots. In the latter case, we compare with the row-colored homology of positive torus knots computed by Hogancamp and Mellit (Torus link homology, 2019), verifying the mirror symmetry conjectures of Gukov and Stošić (Geom Topol Monogr 18:309–367, 2012) and Gorsky et al. (Fund Math 243:209–299, 2018) in this case. [ABSTRACT FROM AUTHOR]

Let 퐺 be a finite simple group of Lie type and let 푇 be a maximal torus of 퐺. It is well known that if the defining field of 퐺 is large enough, then the normaliser of 푇 in 퐺 is equal to the algebraic normaliser N ⁢ (G , T) . We identify explicitly all the cases when N G ⁢ (T) is not equal to N ⁢ (G , T) . [ABSTRACT FROM AUTHOR]

We study coloured invariants of torus knots T (p , p ′) (where p , p ′ are coprime positive integers). When the colouring Lie algebra is simply-laced, and when p , p ′ ≥ h ∨ , we use the representation theory of the corresponding principal affine algebras to understand the trailing monomials of the coloured invariants. In these cases, we show that the appropriate limits of the renormalized invariants are equal to the characters of certain algebra modules (up to some factors); this result on limits rests on a purely Lie-algebraic conjecture on asymptotic weight multiplicities which we verify in some examples. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the effect of motor suspension parameters, specifically damping and stiffness, on the bifurcations of a bogie system. A motor bogie model with a nonlinear smooth equivalent conicity function is established. The study includes qualitative analyses of the stability and the Hopf bifurcation of the equilibrium, with the running speed as the single parameter. Furthermore, the generalised Hopf bifurcation and Hopf-Hopf bifurcation of the equilibrium, based on two motor suspension parameters, are also analysed. The paper delves into the codimension-1 and codimension-2 bifurcations of the limit cycles generated by the Hopf bifurcation, including Neimark–Sacker bifurcation, fold bifurcation, cusp bifurcation, 1:3 resonance, and 1:4 resonance. Analytical investigations reveal that the cusp bifurcation alters the type of fold bifurcation (subcritical or supercritical), and the fold bifurcation influences the stability and bifurcation direction of the limit cycle. Additionally, resonance occurs between the motor and the bogie frame. Since the subcritical (supercritical) Neimark–Sacker bifurcation produces an unstable (a stable) torus, the resonance points associated with the subcritical Neimark–Sacker bifurcation will lead to the instability of the motor bogie. The bifurcation analysis on the motor suspension parameters in this paper offers a theoretical reference for enhancing the stability of the motor bogie. [ABSTRACT FROM AUTHOR]

We present an example of a monotone two-parameter family of vector fields on a torus whose bifurcation diagram we demonstrate to be in the class of 'simplest' diagrams proposed by Baesens and MacKay (2018 Nonlinearity 31 2928–81). This shows that the proposed class is realisable. [ABSTRACT FROM AUTHOR]

The article targets binomial ideals in quantum tori and quantum affine spaces. First, noncommutative analogs of known results for commutative (Laurent) polynomial rings are obtained, including the following: Under the assumption of an algebraically closed base field, it is proved that primitive ideals are binomial, as are radicals of binomial ideals and prime ideals minimal over binomial ideals. In the case of a quantum torus T q , the results are strongest: In this situation, the binomial ideals are parametrized by characters on sublattices of the free abelian group whose group algebra is the center of T q ; the sublattice-character pairs corresponding to primitive ideals as well as to radicals and minimal primes of binomial ideals are determined. As for occurrences of binomial ideals in quantum algebras: It is shown that cocycle-twisted group algebras of finitely generated abelian groups are quotients of quantum tori modulo binomial ideals. Another appearance is as follows: Cocycle-twisted semigroup algebras of finitely generated commutative monoids, as well as quantum affine toric varieties, are quotients of quantum affine spaces modulo certain types of binomial ideals. [ABSTRACT FROM AUTHOR]

We show that the exterior algebra Λ R [ α 1 , ⋯ , α n ] , which is the cohomology of the torus T = (S 1) n , and the polynomial ring R [ t 1 , ... , t n ] , which is the cohomology of the classifying space B (S 1) n = (C P ∞) n , are S n -equivariantly log-concave. We do so by explicitly giving the S n -representation maps on the appropriate sequences of tensor products of polynomials or exterior powers and proving that these maps satisfy the hard Lefschetz theorem. Furthermore, we prove that the whole Kähler package, including the Poincaré duality, the hard Lefschetz theorem, and the Hodge–Riemann bilinear relations, holds on the corresponding sequences in an equivariant setting. [ABSTRACT FROM AUTHOR]

Over function fields of p -adic curves, we construct stably rational varieties in the form of homogeneous spaces of SL n with semisimple simply connected stabilizers and we show that strong approximation away from a non-empty set of places fails for such varieties. The construction combines the Lichtenbaum duality and the degree 3 cohomological invariants of the stabilizers. We then establish a reciprocity obstruction which accounts for this failure of strong approximation. We show that this reciprocity obstruction to strong approximation is the only one for counterexamples we constructed, and also for classifying varieties of tori. We also show that this reciprocity obstruction to strong approximation is compatible with known results for tori. At the end, we explain how a similar point of view shows that the reciprocity obstruction to weak approximation is the only one for classifying varieties of tori over p -adic function fields. [ABSTRACT FROM AUTHOR]

The infrastructure to support human societies' safety, health, and utility supply can not be negotiated, and to have basic reliability in monitoring tasks, plural or dual surveillance, rooted in biologic senses, is needed. To fit the contemporary wireless communication, MIMO (multi-in multi-out) being incorporated with suggested network prototypes, aimed especially on the naturally bipartite Honeycomb Tori (HT) for cellular communications, to prevent information loss, interference, unexpected changes caused by such as clogged water. The mutually independent Hamiltonian Path (MIHP) property can be used for parallel processing and supporting cipher coding to offer efficiency, integrity, and privacy. Isomorphism between Honeycomb Tori and Generalized Honeycomb Tori (GHT) is utilized; mathematical HT(m) m ≥ 2 can be presented as GHT (m, 6m, 3m). Setting rational configurations, incremental scalability qualifying, it is proved that GHT (m, n, n/2) "even m, n ≥ 12" and "odd m > 1, n ≥ 10," which naturally include full Honeycomb Tori, can have a property dual MIHP. [ABSTRACT FROM AUTHOR]

We prove that the knot Floer complex of a fibered knot detects whether the monodromy of its fibration is right-veering. In particular, this leads to a purely knot Floer-theoretic characterization of tight contact structures, by the work of Honda–Kazez–Matić. Our proof makes use of the relationship between the Heegaard Floer homology of mapping tori and the symplectic Floer homology of area-preserving surface diffeomorphisms. We describe applications of this work to Dehn surgeries and taut foliations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the average geodesic distance on Sierpiński torus networks. We construct Sierpiński torus networks based on the classic Sierpiński carpet in an iterative way. By applying finite patterns on integrals, we deduce the exact value of the average geodesic distance of the Sierpiński torus. Furthermore, the asymptotic formula for the average geodesic distance of the corresponding networks can be obtained. [ABSTRACT FROM AUTHOR]

Let |$Z\subset \operatorname{Fl}(n)$| be the closure of a generic torus orbit in the full flag variety. Anderson–Tymoczko express the cohomology class of |$Z$| as a sum of classes of Richardson varieties. Harada–Horiguchi–Masuda–Park give a decomposition of the permutohedron, the moment map image of |$Z$|⁠ , into subpolytopes corresponding to the summands of the Anderson–Tymoczko formula. We construct an explicit toric degeneration inside |$\operatorname{Fl}(n)$| of |$Z$| into Richardson varieties, whose moment map images coincide with the HHMP decomposition, thereby obtaining a new proof of the Anderson–Tymoczko formula. [ABSTRACT FROM AUTHOR]

We solve the equivariant generalized Nash problem for any non-rational normal variety with torus action of complexity one. Namely, we give an explicit combinatorial description of the Nash order on the set of equivariant divisorial valuations on any such variety. Using this description, we positively solve the classical Nash problem in this setting, showing that every essential valuation is a Nash valuation. We also describe terminal valuations and use our results to answer negatively a question of de Fernex and Docampo by constructing examples of Nash valuations which are neither minimal nor terminal, thus illustrating a striking new feature of the class of singularities under consideration. [ABSTRACT FROM AUTHOR]

The Spherical Tokamak for Energy Production (STEP) environment will include magnetic, thermal, mechanical and environmental loads far greater than those seen in the Joint European Torus campaigns of the past decade or currently contemplated for ITER. Greater still are the neutron peak dose rates of 10−6 displacements per atom, per second, which in-vessel materials in STEP are anticipated to be exposed to. Reduced activation and high-fluence resilience therefore dominate the materials strategy to support the STEP Programme. The latter covers the full life cycle from downselected compositions and new microstructural developments to irradiation-informed modelling and end-of-life strategies. This article discusses how the materials downselection is oriented in plant power trade-off space, outlines the development of an advanced ferritic-martensitic structural steel, describes the 'Design by Fundamentals' mesoscale modelling approach and reports some of the waste mitigation routes intended to make STEP operations as sustainable as possible. This article is part of the theme issue 'Delivering Fusion Energy – The Spherical Tokamak for Energy Production (STEP)'. [ABSTRACT FROM AUTHOR]

The aim of the present study was to analyze palatal marginal alveolar exostosis (PMAE) and palatal torus (PT). Cone‐beam computed tomography (CBCT) of the maxilla in multiplanar sections and volumetric renderings were used to assess this. PT and PMAE were classified according to location and morphology. Height, width, length, and thickness of the overlying mucosa were determined. The prevalence of PT and PMAE was assessed according to sex and age group. The correlation between the occurrence of PMAE and PT was also evaluated. A total of 385 CBCT scans were examined. PT was found in 38.70% of the sample and located more frequently in the middle third of the maxilla (52.35%) with a flat shape (42.95%). PMAE was found in 54.80% of the sample, bilaterally in 56.40% of the cases, and located more frequently in the molar region (62.42%) in the form of small nodules (36.97%). The mucosa covering PMAE was generally thicker than that over PT. The use of CBCT for the identification of PT and PMAE in vivo showed high frequencies of both conditions. The occurrence of PMAE was independent of the presence of PT. [ABSTRACT FROM AUTHOR]

In this paper, the entropy of discrete Heisenberg group actions is considered. Let α be a discrete Heisenberg group action on a compact metric space X. Two types of entropies, h ~ (α) and h(α) are introduced, in which h ~ (α) is defined in Ruelle's way and h(α) is defined via the natural extension of α. It is shown that when X is the torus and α is induced by integer matrices then h ~ (α) is zero and h(α) can be expressed via the eigenvalues of the matrices. [ABSTRACT FROM AUTHOR]

We will consider a thermostatic system, Sprott B, that is a generalization of the well-known one-parameter Sprott A system. Sprott B contains an explicit periodic solution for all positive values of the parameter a. As for Sprott A, we find dissipative KAM tori associated with time-reversal symmetry and canards in dissipative systems. The exact periodic solution is characterized by an infinite number of instability intervals of the parameter. The investigation of the dynamics in these intervals shows the presence of families of stable and unstable periodic solutions, tori, and strange attractors. For large values of the control parameter a , we find non-hyperbolic slow manifolds producing violent vibrations. We discuss a generalization of the Sprott B system with related dynamics. [ABSTRACT FROM AUTHOR]

This paper deals with the study of torsional energy (total squared torsion) at infinitesimal bending of curves and knots in three dimensional Euclidean space. During bending, the curve is subject to change, and its properties are changed. The effect that deformation has on the curve is measured by variations. Here, we observe the infinitesimal bending of the second order and variations of the first and the second order that occur in this occasion. The subjects of study are curves and knots, in particular torus knots. We analyze various examples both analytically and graphically, using our own calculation and visualization software tool. [ABSTRACT FROM AUTHOR]

In this paper, we extend the theory of planar pseudo knots to the theories of annular and toroidal pseudo knots. Pseudo knots are defined as equivalence classes under Reidemeister-like moves of knot diagrams characterized by crossings with undefined over/under information. In the theories of annular and toroidal pseudo knots, we introduce their respective lifts to the solid and the thickened torus. Then, we interlink these theories by representing annular and toroidal pseudo knots as planar O -mixed and H -mixed pseudo links. We also explore the inclusion relations between planar, annular and toroidal pseudo knots, as well as of O -mixed and H -mixed pseudo links. Finally, we extend the planar weighted resolution set to annular and toroidal pseudo knots, defining new invariants for classifying pseudo knots and links in the solid and in the thickened torus. [ABSTRACT FROM AUTHOR]

We formulate and prove a version of the arithmetic Siegel–Weil formula for (zero dimensional) Shimura varieties attached to tori, equipped with some additional data. More precisely, we define a family of "special" divisors in terms of Green functions at archimedean and non-archimedean places and prove that their degrees coincide with the Fourier coefficients of the central derivative of an Eisenstein series. The proof relies on the usual Siegel–Weil formula to provide a direct link between both sides of the identity, and in some sense, offers a more conceptual point of view on prior results in the literature. [ABSTRACT FROM AUTHOR]

Cavity optomechanical systems have received widespread attentions because they provide a novel platform for metrology, sensing, hybrid systems and quantum information processing. Their nonlinear dynamics has rich physics and plays an important role in the application scenarios. Previous works devoted to this subject have usually focused on the self-induced oscillation and chaos, whereas other parts of the rich nonlinear-dynamics picture are almost uncharted waters. In this study, we fill this gap and report the first experimental observation of limit-torus attractor, whose dynamics exhibits a torus-like trajectory in phase space. Moreover, we investigate the sharp decrease of oscillating amplitude along the up scanning transmission spectrum, referred to as catastrophe point, for the first time. The location of catastrophe point is independent of the pump power and the coupling distance. Our findings enrich the nonlinear dynamics in optomechanical systems, and open up new ways towards exploiting these systems as versatile building blocks in various applications including communication, quantum information processing, sensing and metrology. [ABSTRACT FROM AUTHOR]

Establishing the (non)existence of a spectral gap above the ground state in the thermodynamic limit is one of the fundamental steps for characterizing the topological phase of a quantum lattice model. This is particularly challenging when a model is expected to have low-lying edge excitations, but nevertheless a positive bulk gap. We review the bulk gap strategy introduced in [S. Warzel and A. Young, The spectral gap of a fractional quantum Hall system on a thin torus, J. Math. Phys.63 (2022) 041901; S. Warzel and A. Young, A bulk spectral gap in the presence of edge states for a truncated pseudopotential, Ann. Henri Poincaré24 (2023) 133–178], while studying truncated Haldane pseudopotentials. This approach is able to avoid low-lying edge modes by separating the ground states and edge states into different invariant subspaces before applying spectral gap bounding techniques. The approach is stated in a general context, and we reformulate specific spectral gap methods in an invariant subspace context to illustrate the necessary conditions for combining them with the bulk gap strategy. We then review its application to a truncation of the 1/3-filled Haldane pseudopotential in the cylinder geometry. [ABSTRACT FROM AUTHOR]

We show that there are no distance one surgeries on non-null-homologous knots in M that yield -M (M with opposite orientation) if M is a 3-manifold obtained by a Dehn surgery on a knot K in S^{3}, such that the order of its first homology is divisible by 9 but is not divisible by 27. As an application, we show several knots, including the (2,9) torus knot, do not have chirally cosmetic bandings. This simplifies the proof of a result first proven by Yang that the (2,k) torus knot (k>1) has a chirally cosmetic banding if and only if k=5. [ABSTRACT FROM AUTHOR]

The classical Hard Lefschetz theorem (HLT), Hodge-Riemann bilinear relation theorem (HRR) and Lefschetz decomposition theorem (LD) are stated for a power of a Kähler class on a compact Kähler manifold. These theorems are not true for an arbitrary class, even if it contains a smooth strictly positive representative. Dinh-Nguyên proved the mixed HLT, HRR and LD for a product of arbitrary Kähler classes. Instead of products, they asked whether determinants of Griffiths positive k\times k matrices with (1,1)-form entries in \mathbb {C}^n satisfy these theorems in the linear case. This paper answered their question positively when k=2 and n=2,3. Moreover, assume that the matrix only has diagonalized entries, for k=2 and n\geqslant 4, the determinant satisfies HLT for bidegrees (n-2,0), (n-3,1), (1,n-3) and (0,n-2). In particular, for k=2 and n=4,5 with this extra assumption, the determinant satisfies HRR, HLT and LD. Two applications: First, a Griffiths positive 2\times 2 matrix with (1,1)-form entries, if all entries are \mathbb {C}-linear combinations of the diagonal entries, then its determinant also satisfies these theorems. Second, on a complex torus of dimension \leqslant 5, the determinant of a Griffiths positive 2\times 2 matrix with diagonalized entries satisfies these theorems. [ABSTRACT FROM AUTHOR]

We study the triangular θ -mean of the partial sums of f ∈ L p (T q 2) and prove the following noncommutative weak and strong type maximal inequalities: ‖ (σ n Δ , θ (f)) n ≥ 1 ‖ Λ 1 , ∞ (T q 2 , ℓ ∞) ≤ c θ ‖ f ‖ L 1 (T q 2) , p = 1 and σ n Δ , θ (f) n ≥ 1 L p (T q 2 , ℓ ∞) ≤ c p , θ ‖ f ‖ L p (T q 2) , 1 < p < ∞ , where T q 2 is a 2-dimensional quantum torus. As a consequence, we obtain the bilateral almost uniform convergence of σ n Δ , θ (f) provided f ∈ L p (T q 2). [ABSTRACT FROM AUTHOR]

We study algebraic cycles in the moduli space of PGL2-shtukas, arising from the diagonal torus. Our main result shows that their intersection pairing with the Heegner--Drinfeld cycle is the product of the r-th central derivative of an automorphic L-function (π, s) and Waldspurger's toric period integral. When L(π, 1/2) ≠ 0, this gives a new geometric interpretation for the Taylor series expansion. When L(π, 1/2) 0, the pairing vanishes, suggesting higher order analogues of the vanishing of cusps in the modular Jacobian, as well as other new phenomena. Our proof sheds new light on the algebraic correspondence introduced by Yun and Zhang, which is the geometric incarnation of "differentiating the L-function". We realize it as the Lie algebra action of e + ƒ ∈ sl2 on (ℚℓ²)⊗2d. The comparison of relative trace formulas needed to prove our formula is then a consequence of Schur--Weyl duality. [ABSTRACT FROM AUTHOR]

This paper mainly concerns the KAM persistence of the mapping $\mathscr {F}:\mathbb {T}^{n}\times E\rightarrow \mathbb {T}^{n}\times \mathbb {R}^{n}$ with intersection property, where $E\subset \mathbb {R}^{n}$ is a connected closed bounded domain with interior points. By assuming that the frequency mapping satisfies certain topological degree condition and weak convexity condition, we prove some Moser-type results about the invariant torus of mapping $\mathscr {F}$ with frequency-preserving under small perturbations. To our knowledge, this is the first approach to Moser's theorem with frequency-preserving. Moreover, given perturbed mappings over $\mathbb {T}^n$ , it is shown that such persistence still holds when the frequency mapping and perturbations are only continuous about parameter beyond Lipschitz or even Hölder type. We also touch the parameter without dimension limitation problem under such settings. [ABSTRACT FROM AUTHOR]

In this paper we investigate the pair difference cordial labeling behaviour of some product related graphs. [ABSTRACT FROM AUTHOR]

Motivated by the L-space conjecture, we investigate various notions of order-detection of slopes on knot manifolds. These notions are designed to characterise when rational homology 3-spheres, obtained by gluing compact manifolds along torus boundary components, have leftorderable fundamental groups and when a Dehn filling of a knot manifold has a left-orderable fundamental group. Our developments parallel the results by Hanselman et al. (2020) in the case of Heegaard Floer slope detection and by Boyer et al. (2021) in the case of foliation slope detection, leading to several conjectured structure theorems that connect relative Heegaard Floer homology and the boundary behaviour of co-oriented taut foliations with the set of left-orders supported by the fundamental group of a 3-manifold. The dynamics of the actions of 3-manifold groups on the real line play a key role in our constructions and proofs. Our analysis leads to conjectured dynamical constraints on such actions in the case where the underlying manifold is Floer simple. [ABSTRACT FROM AUTHOR]

We study the law of random self-similar series defined above an irrational rotation on the Circle. This provides a natural class of continuous singular non-Rajchman measures. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, all biderivations of solvable Lie algebras of maximal rank g=Q⊕N are characterized. Namely, we consider a solvable Lie algebra of the form g=Q⊕N , where Q is the maximal torus subalgebra of g, N is the nilradical of g and dimQ=dimN/N2 .In the case dimQ=N we characterize the form of skew-symmetric and symmetric biderivations of g . As applications, the forms of linear commuting (skew-commuting) maps and the commutative post-Lie algebra structures on g are given. [ABSTRACT FROM AUTHOR]

We consider a space–time SI epidemic model with infection age dependent infectivity and non-local infections constructed on a grid of the torus T d = [ 0 , 1) d , where the individuals may migrate from node to node. The migration processes in either of the two states are assumed to be Markovian. We establish a functional law of large numbers by letting the initial approximate number of individuals on each node, N, to go to infinity and the mesh size of the grid, ε , to go to zero jointly. The limit is a system of parabolic PDE/integral equations. The constraint on the speed of convergence of the parameters N and ε is that N ε d → ∞ as (N , ε) → (+ ∞ , 0) . [ABSTRACT FROM AUTHOR]

One way to characterize an orbit of a dynamical system is through its frequency content. Using a "spectral bifurcation diagram", it has been shown earlier how the frequency content changes when a system undergoes a period-doubling cascade. In this paper, we extend the scope of this technique to obtain newer insights into various bifurcations like pitchfork bifurcation, border-collision bifurcation, period-adding cascade, and various torus bifurcations. We show that applying this method can enrich our understanding of bifurcations by providing vital information about generating or annihilating frequency components in a bifurcation. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem for the stochastically perturbed high-dimensional modified Euler-Poincaré system (MEP2) on the torus T d , d ≥ 1. We first establish a local well-posedness framework in the sense of Hadamard for the MEP2 driven by general nonlinear multiplicative noises. Then two kinds of global existence and uniqueness results are demonstrated: One indicates that the MEP2 perturbed by nonlocal-type random noises with proper intensity admits a unique large global strong solution; The other one infers that, if the initial data is sufficiently small, then the MEP2 perturbed by linear multiplicative noise has a unique global solution with high probability. In the case of one dimension, we find that the stochastic MEP2 will break down in finite time when the initial data meets appropriate shape condition. [ABSTRACT FROM AUTHOR]

The Kuratowski graphs K 3 , 3 and K 5 characterize planarity. Counting distinct 2-cell embeddings of these two graphs on orientable surfaces was previously done by Mull (1999) and Mull et al. (2008), using Burnside's Lemma and automorphism groups of K 3 , 3 and K 5 , without actually constructing the embeddings. We obtain all 2-cell embeddings of these graphs on the double torus, using a constructive approach. This shows that there is a unique non-orientable 2-cell embedding of K 3 , 3 , and 14 orientable and 17 non-orientable 2-cell embeddings of K 5 on the double torus, which are explicitly obtained using an algorithmic procedure of expanding from minors. Therefore we confirm the numbers of embeddings obtained by Mull (1999) and Mull et al. (2008). As a consequence, several new polygonal representations of the double torus are presented. Rotation systems for the one-face embeddings of K 5 on the triple torus are also found, using exhaustive search. [ABSTRACT FROM AUTHOR]

Our goal in this paper is to initiate the rigorous investigation of wave turbulence and derivation of wave kinetic equations (WKEs) for water waves models. This problem has received intense attention in recent years in the context of semilinear models, such as Schrödinger equations or multidimensional KdV‐type equations. However, our situation here is different since the water waves equations are quasilinear and solutions cannot be constructed by iteration of the Duhamel formula due to unavoidable derivative loss. This is the first of two papers in which we design a new strategy to address this issue. We investigate solutions of the gravity water waves system in two dimensions. In the irrotational case, this system can be reduced to an evolution equation on the one‐dimensional interface, which is a large torus TR${\mathbb {T}}_R$ of size R≥1$R\ge 1$. Our first main result is a deterministic energy inequality, which provides control of (possibly large) Sobolev norms of solutions for long times, under the condition that a certain L∞$L^\infty$‐type norm is small. This energy inequality is of “quintic” type: if the L∞$L^\infty$ norm is O(ε)$O(\varepsilon)$, then the increment of the high‐order energies is controlled for times of the order ε−3$\varepsilon ^{-3}$, consistent with the approximate quartic integrability of the system. In the second paper in this sequence, we will show how to use this energy estimate and a propagation of randomness argument to prove a probabilistic regularity result up to times of the order ε−3R3/2−${\varepsilon }^{-3} R^{3/2-}$, in a suitable scaling regime relating ε→0${\varepsilon }\rightarrow 0$ and R→∞$R \rightarrow \infty$. For our second main result, we combine the quintic energy inequality with a bootstrap argument using a suitable Z$Z$‐norm of Strichartz‐type to prove that deterministic solutions with Sobolev data of size ε≪1${\varepsilon }\ll 1$ are regular for times of the order ε−3min(ε−3,R3/4)${\varepsilon }^{-3} \min \big ({\varepsilon }^{-3}, R^{3/4} \big)$. In particular, on the real line, solutions exist for times of order O(ε−6)$O(\varepsilon ^{-6})$. This improves substantially on all the earlier extended lifespan results for 2D gravity water waves with small Sobolev data. [ABSTRACT FROM AUTHOR]

The optimal L4-Strichartz estimate for the Schrödinger equation on the two-dimensional rational torus T² is proved, which improves an estimate of Bourgain. A new method based on incidence geometry is used. The approach yields a stronger L4 bound on a logarithmic time scale, which implies global existence of solutions to the cubic (mass-critical) nonlinear Schrödinger equation in Hs (T²) for any s > 0 and data that are small in the critical norm. [ABSTRACT FROM AUTHOR]

For a connected reductive group G over a nonarchimedean local field F of positive characteristic, GenestierLafforgue and Fargues-Scholze have attached a semisimple parameter Lss to each irreducible representation π. Our first result shows that the Genestier-Lafforgue parameter of a tempered π can be uniquely refined to a tempered L-parameter L(π), thus giving the unique local Langlands correspondence which is compatible with the GenestierLafforgue construction. Our second result establishes ramification properties of Lss (π) for unramified G and supercuspidal π constructed by induction from an open compact (modulo center) subgroup. If Lss is pure in an appropriate sense, we show that Lss (π) is ramified (unless G is a torus). If the inducing subgroup is sufficiently small in a precise sense, we show Lss is wildly ramified. The proofs are via global arguments, involving the construction of Poincaré series with strict control on ramification when the base curve is P1 and a simple application of Deligne’s Weil II. [ABSTRACT FROM AUTHOR]

We derive rigorously the reduced dynamical law for quantized vortex dynamics of the complex Ginzburg-Landau equation on the torus when the core size of vortex ε → 0. The reduced dynamical law of the complex Ginzburg-Landau equation is governed by a mixed flow of gradient flow and Hamiltonian flow which are both driven by a renormalized energy on the torus. Finally, some first integrals and analytic solutions of the reduced dynamical law are discussed. [ABSTRACT FROM AUTHOR]

The equations of plane oscillations of a satellite in an elliptical orbit are considered. For the numerical detection of periodic solutions, a combination of the Poincaré section method and the previously proposed approach based on an analogue of the principle of contraction mappings is used. A number of classes of periodic solutions are numerically identified, and necessary conditions for their stability are studied. These motions are given special attention, since, in the general case, they are difficult to study analytically. [ABSTRACT FROM AUTHOR]