Elliptic Form Research Articles

Extraction of newly soliton wave structure of generalized stochastic NLSE with standard Brownian motion, quintuple power law of nonlinearity and nonlinear chromatic dispersion

Stochastic optical solitons are a fascinating phenomenon in nonlinear optics where soliton-like behavior emerges in systems affected by stochastic noise. This study investigates the influence of Brownian motion on wave propagation in optical fibers. The propagation is modeled using a stochastic nonlinear Schrödinger equation incorporating quintuple power-law nonlinearity and nonlinear chromatic dispersion. To explore this, the improved modified extended tanh (IMET) scheme, leveraging the extended Riccati equation, is employed. This technique facilitates the extraction of various stochastic solutions, including bright, dark, and singular solitons. Furthermore, solutions in the shapes of exponential, singular periodic, and Weierstrass elliptic forms are investigated. The study looks at how the strength of noise impacts various solutions, and Matlab software is used to create 2D and 3D graphs that show the results. It has been noted that when noise intensity rises, signal level falls and surface flattens.

Construction of modular differential equations by using Rankin-Cohen brackets

In this paper, we construct higher-order modular differential equations for elliptic modular forms, holomorphic Jacobi forms, and skew-holomorphic Jacobi forms by using the Rankin-Cohen brackets extended to the Eisenstein series of weight two. Rankin-Cohen brackets are typical tools in the construction of modular forms by differential operators. We introduce the extended Rankin-Cohen brackets for holomorphic or skew-holomorphic Jacobi forms paired with elliptic modular forms. They provide a unified method of construction of higher-order modular differential equations for the modular forms above. In particular, with these extended Rankin-Cohen brackets, we can reproduce the modular differential equations of previous studies.

Wave nature of Rosensweig instability

The explicit analytical solution of Rosensweig instability spikes’ shapes obtained by Navier–Stokes (NS) equation in diverse magnetic field H vertical to the flat free surface of ferrofluids are systematically studied experimentally and theoretically. After carefully analyzing and solving the NS equation in elliptic form, the force balanced surface equations of spikes in Rosensweig instability are expressed as cosine wave in perturbated magnetic field and hyperbolic tangent in large magnetic field, whose results both reveal the wave-like nature of Rosensweig instability. The results of hyperbolic tangent form are perfectly fitted to the experimental results in this paper, which indicates that the analytical solution is basically correct. Using the forementioned theoretical results, the total energy of the spike distribution pattern is calculated. By analyzing the energy components under different magnetic field intensities H, the hexagon–square transition of Rosensweig instability is systematically discussed and explained in an explicit way.

Structure of fine Selmer groups over -extensions

AbstractThis paper is concerned with the study of the fine Selmer group of an abelian variety over a $\mathbb{Z}_{p}$ -extension which is not necessarily cyclotomic. It has been conjectured that these fine Selmer groups are always torsion over $\mathbb{Z}_{p}[[ \Gamma ]]$ , where $\Gamma$ is the Galois group of the $\mathbb{Z}_{p}$ -extension in question. In this paper, we shall provide several strong evidences towards this conjecture. Namely, we show that the conjectural torsionness is consistent with the pseudo-nullity conjecture of Coates–Sujatha. We also show that if the conjecture is known for the cyclotomic $\mathbb{Z}_{p}$ -extension, then it holds for almost all $\mathbb{Z}_{p}$ -extensions. We then carry out a similar study for the fine Selmer group of an elliptic modular form. When the modular forms are ordinary and come from a Hida family, we relate the torsionness of the fine Selmer groups of the specialization. This latter result allows us to show that the conjectural torsionness in certain cases is consistent with the growth number conjecture of Mazur. Finally, we end with some speculations on the torsionness of fine Selmer groups over an arbitrary p-adic Lie extension.

On approximation of the Dirichlet problem for divergence form operators by Robin problems

We show that, under natural assumptions, solutions of Dirichlet problems for uniformly elliptic divergence form operators can be approximated pointwise by solutions of some versions of Robin problems. The proof is based on the stochastic representation of solutions and properties of reflected diffusions corresponding to divergence form operators.

L2 error estimates for a family of cubic finite volume methods on triangular meshes

This paper considers the error estimates of a family of Lagrange cubic finite volume methods for solving two dimensional elliptic boundary value problems over triangular meshes. By introducing a novel map of finite volume method, we propose a family of cubic finite volume method schemes with four independent map coefficients. It is detailedly proved that the proposed finite volume method schemes is exactly equivalent to the existing traditional finite volume method schemes of the same boundary value problems. A geometric constraint requirement on the triangular meshes of primary partitions is imposed to guarantee the uniform ellipticity of discrete elliptic bilinear forms resulted from the proposed schemes. The geometric requirements of the primary partitions could be greatly attenuated when the map coefficients are appropriately designed. Once the uniform ellipticity of the discrete elliptic bilinear forms is established, the proposed schemes have the unique solvability, and the optimal H1-norm error estimates without any restriction on dual elements of dual partitions of the finite volume methods, which is consistent with the results of literatures. With the help of the introduced map, the proposed schemes have the optimal L2-norm error estimates with a mesh restriction for the dual elements of their dual partitions significantly weaker than the existing theoretical results, in which the Aubin-Nitsche duality argument and the orthogonal conditions based on the dual partitions are employed. Finally, we validate our theoretical results by numerical experiments.

Jacobi-Eisenstein series over number fields

For any given totally real number field K, we compute the Fourier developments of the Jacobi Eisenstein series over K at the cusp at infinity. As main application we prove, for any K with class number 1, that the L-series of the Jacobi Eisenstein series of weight k≥3 for indices with rank and modified level 1 coincide with the L-series of the Eisenstein series of weight 2k−2 on the full Hilbert modular group of K. Moreover, under this correspondence the Fourier coefficients of the Jacobi Eisenstein series are related to the twisted L-series of the Hilbert Eisenstein series at the critical point by a Waldspurger type identity. This is a first step in the proof that Skoruppa's and Zagier's lifting from Jacobi forms over Q to elliptic modular forms holds true over arbitrary totally real number fields too.

A Compact Wideband (22–44GHz) Printed 2×4 MIMO Array Antenna with High Gain for 26/28/38GHz Millimeter-Wave 5G Applications

In this work, a novel multiple input multiple output (MIMO) array antenna system with a large bandwidth and high gain has been simulated, analyzed, fabricated, and measured. The proposed antenna is structured in [Formula: see text] patch configuration along with a cross shaped ground plane loaded with four square and one circular shaped defect. The projected antenna occupies a total size of [Formula: see text]. Several slots in an elliptic form have been added to the patches to achieve the required results in terms of wide bandwidth and high gain. The MIMO antenna array is fabricated and experimentally tested to confirm the simulation results. The suggested MIMO array antenna offers an impedance bandwidth of 22[Formula: see text]GHz covering 22–44[Formula: see text]GHz wide range of frequencies with a high peak gain of 17[Formula: see text]dBi at 38[Formula: see text]GHz. The designed MIMO antenna offers superior diversity performance and it supports several 5G NR bands n257/n258/n259/n260/n261 in the mm-wave spectrum. The suggested MIMO antenna supports 5G application bands that are deployed in UK, USA, China, Europe, Canada, India, and Europe.

Offers in Saudi EFL talks: A focus on the learners’ pragmatic competence in interactions

The current study was driven by the most recent research trend of interlanguage pragmatics, which regards communication as a complex phenomenon and considers a learner's ability to produce social actions in extended discourse. It investigated the interactional structure of offers in natural conversations by nine Saudi learners of English (SEFL) when interacting with a British English native speaker. Conversations were recorded during a dinner attended by three to four friends. Comparable sets of data were collected from native speakers of Saudi Arabic (SA) and British English (BE). Both quantitative and qualitative analysis revealed that the three cultural groups in this study did not get involved in intricate negotiations of offers during the conversation among friends; however, this was not necessarily to the same degree. The SEFL learners' interactional tools were not native-like. They showed more limited interactional resources and a pattern of direct offers by using elliptic forms or conveying the offer nonverbally. A possible pragmatic transfer was observed in their preference for employing nonverbal offers and invoking God. However, they were like the BE speakers in their insistence, which was often made with strategies that consider imposition on the addressee and usually closed in two adjacency pairs.

Eigenform product identities of genus two Siegel modular forms of general congruence level

Given two eigenforms, it is a natural question to ask if the product of the eigenforms is again an eigenform. In the case of elliptic modular forms, this was answered in the full level case by Duke and Ghate and in the general level case by Johnson. In this paper, we consider the case of genus two Siegel modular forms with general congruence level.

Overconvergent Hilbert modular forms via perfectoid modular varieties

We give a new construction of p-adic overconvergent Hilbert modular forms by using Scholze’s perfectoid Shimura varieties at infinite level and the Hodge–Tate period map. The definition is analytic, closely resembling that of complex Hilbert modular forms as holomorphic functions satisfying a transformation property under congruence subgroups. As a special case, we first revisit the case of elliptic modular forms, extending recent work of Chojecki, Hansen and Johansson. We then construct sheaves of geometric Hilbert modular forms, as well as subsheaves of integral modular forms, and vary our definitions in p-adic families. We show that the resulting spaces are isomorphic as Hecke modules to earlier constructions of Andreatta, Iovita and Pilloni. Finally, we give a new direct construction of sheaves of arithmetic Hilbert modular forms, and compare this to the construction via descent from the geometric case.

On the rank-part of the Mazur–Tate refined conjecture for higher weight modular forms

Under some assumptions, we prove the rank-part of the Mazur–Tate refined conjecture of BSD type. More concretely, we prove that the rank of the Selmer group of an elliptic modular form is less than or equal to the order of zeros of Mazur–Tate elements, or modular elements, which are elements in certain group rings constructed from special values of the associated L-function. Our main result is regarded as a generalization of our previous work on elliptic curves.

Modular forms via invariant theory

We discuss two simple but useful observations that allow the construction of modular forms from given ones using invariant theory. The first one deals with elliptic modular forms and their derivatives, and generalizes the Rankin-Cohen bracket, while the second one deals with vector-valued modular forms of genus greater than 1.

Dynamic Properties of Non-Autonomous Femtosecond Waves Modeled by the Generalized Derivative NLSE with Variable Coefficients

The primary purpose of this study is to analyze non-autonomous femtosecond waves with various geometrical configurations correlated to the generalized derivative nonlinear Shrödinger equation (NLSE) with variable coefficients. Numerous academic publications, especially in nonlinear optics, material science, semiconductor, chemical engineering, and many other fields, have looked into this model since it is closer to real-world situations and has more complex wave structures than models with constant coefficients. It can serve as a reflection for the slowly altering inhomogeneities, non-uniformities, and forces acting on boundaries. New complex wave solutions in two different categories are proposed: implicit and elliptic (or periodic or hyperbolic) forms are obtained for this model via the unified method. Indeed, the innovative wave solutions that were achieved and reported here are helpful for investigating optical communication applications as well as the transmission characteristics of light pulses.

Time Regularity for Local Weak Solutions of the Heat Equation on Local Dirichlet Spaces

We study the time regularity of local weak solutions of the heat equation in the context of local regular symmetric Dirichlet spaces. Under two basic and rather minimal assumptions, namely, the existence of certain cut-off functions and a very weak L2 Gaussian type upper bound for the heat semigroup, we prove that the time derivatives of a local weak solution of the heat equation are themselves local weak solutions. This applies, for instance, to local weak solutions of parabolic equations with uniformly elliptic symmetric divergence form second order operators with measurable coefficients. We describe some applications to the structure of ancient local weak solutions of such equations which generalize recent results of Colding and Minicozzi (Duke Math. J., 170(18), 4171–4182 2021) and Zhang (Proc. Amer. Math. Soc., 148(4), 1665–1670 2020).

Exact double averages of twisted L-values

Consider central L-values of even weight elliptic or Hilbert modular forms f twisted by ideal class characters $$\chi $$ of an imaginary quadratic extension K. Fixing $$\chi $$ , and assuming K is inert at each prime dividing the level, one knows simple exact formulas for averages over newforms f of squarefree levels satisfying a parity condition on the number of prime factors. These averages stabilize when the level is large with respect to K (the “stable range”). In weight 2, we obtain exact formulas for a simultaneous average over both f and $$\chi $$ . We allow for non-squarefree levels with any number of prime factors, and ramification or splitting of K above the level. Under elementary conditions on the level, these double averages are “stable” in all ranges. Two consequences are generalizations of the aforementioned stable (single) averages and effective results on nonvanishing of central L-values.

A stabilized [formula omitted] domain decomposition finite element method for time harmonic Maxwell’s equations

One way of improving the behavior of finite element schemes for classical, time-dependent Maxwell’s equations is to render their hyperbolic character to elliptic form. This paper is devoted to the study of a stabilized linear, domain decomposition, finite element method for the time harmonic Maxwell’s equations, in a dual form, obtained through the Laplace transformation in time. The model problem is for the particular case of the dielectric permittivity function which is assumed to be constant in a boundary neighborhood. The discrete problem is coercive in a symmetrized norm, equivalent to the discrete norm of the model problem. This yields discrete stability, which together with continuity guarantees the well-posedness of the discrete problem, cf Arnold et al. (2002) [3], Di Pietro and Ern (2012) [45]. The convergence is addressed both in a priori and a posteriori settings. In the a priori error estimates we confirm the theoretical convergence of the scheme in a L2-based, gradient dependent, triple norm. The order of convergence is O(h) in weighted Sobolev space Hw2(Ω), and hence optimal. Here, the weight w≔w(ɛ,s) where ɛ is the dielectric permittivity function and s is the Laplace transformation variable. We also derive, similar, optimal a posteriori error estimates controlled by a certain, weighted, norm of the residuals of the computed solution over the domain and at the boundary (involving the relevant jump terns) and hence independent of the unknown exact solution. The a posteriori approach is used, e.g. in constructing adaptive algorithms for the computational purposes, which is the subject of a forthcoming paper. Finally, through implementing several numerical examples, we validate the robustness of the proposed scheme.

Integral period relations and congruences

Under relatively mild and natural conditions, we establish an integral period relations for the (real or imaginary) quadratic base change of an elliptic cusp form. This answers a conjecture of Hida regarding the {\it congruence number} controlling the congruences between this base change and other eigenforms which are not base change. As a corollary, we establish the Bloch-Kato conjecture for adjoint modular Galois representations twisted by an even quadratic character. In the odd case, we formulate a conjecture linking the degree two topological period attached to the base change Bianchi modular form, the cotangent complex of the corresponding Hecke algebra and the archimedean regulator attached to some Beilinson-Flach element.

On the Petersson norm of the Hilbert-Siegel cusp form under the Ikeda lift

In this paper, we show an arithmeticity of the ratio of the Petersson norm of the Hilbert-Siegel cusp form determined by the Ikeda lift from a Hilbert cusp form to that of the Hilbert cusp form. This is a generalization of the algebraicity of the ratio of the Petersson norm of the Siegel cusp form under the Ikeda lift of an elliptic cusp form to that of the elliptic cusp form due to Y. Choie and W. Kohnen.

Carleson estimates for the Green function on domains with lower dimensional boundaries

In the present paper we consider an elliptic divergence form operator in Rn∖Rd with d<n−1 and prove that its Green function is almost affine, in the sense that the normalized difference between the Green function with a sufficiently far away pole and a suitable affine function at every scale satisfies a Carleson measure estimate. The coefficients of the operator can be very oscillatory, and only need to satisfy some condition similar to the traditional quadratic Carleson condition.