Papers

Let $N\geq 3$, $2<q<p<2^*$, and let $V\in L^\infty(\mathbb{R}^N)$ be real-valued and $\mathbb{Z}^N$-periodic. We assume that $0$ belongs to a finite spectral gap of the periodic Schrödinger operator $L=-\Delta+V$. We prove that there exists $\lambda_\infty>0$ such that, for every $\lambda\geq\lambda_\infty$, the competing-power equation $$ -\Delta u+V(x)u=|u|^{p-2}u-\lambda |u|^{q-2}u \quad\text{in }\mathbb{R}^N $$ has a nontrivial solution $u_\lambda\in H^1(\mathbb{R}^N)$. Moreover, $$ \|u_\lambda\|_{H^1(\mathbb{R}^N)}+\|u_\lambda\|_{L^\infty(\mathbb{R}^N)} \leq C\lambda^{-1/(q-2)}. $$ We also show that, after the natural rescaling and lattice translations, a sequence of such solutions converges weakly to a nontrivial solution of the pure defocusing equation $Lv=-|v|^{q-2}v$.

Let $(K^h,v^h)$ be a Henselization of a valued field $(K,v)$, $\mu$ any extension of $v$ to $K[x]$, and $\mu^h$ the canonical extension of $\mu$ and $v^h$ to $K^h[x].$ In this paper, we study two new invariants associated with a Mac Lane-Vaquié chain (MLV) of $\mu$, namely, the defect of each augmentation step and the regularity of an MLV chain of $\mu$. This, in turn, extends the notion of regularity of a complete sequence of abstract key polynomials for $\mu$ to the regularity of $\mu$ itself. We also prove that the regularity of $\mu$ is a sufficient condition for the equality of the depths of $\mu$ and $\mu^h$. Furthermore, it is a necessary and sufficient condition for the equality of the defect and relative gap of each augmentation step of an MLV chain of $\mu.$ Finally, we show that if $\mu$ has finite degree, then the regularity of $\mu$ is equivalent to $ \operatorname{deg}(\mu)=\operatorname{deg}(\mu^h)$.

Let $G$ be a finite group. In this article, we develop some results of equivariant persistent homotopy theory for persistent $G$-spaces. We introduce the notions of $G$-stable and $G$-homotopy invariant distances and define the $G$-homotopy interleaving distance, an equivariant analogue $d^G_{HI}$ of the homotopy interleaving distance of Blumberg and Lesnick. We prove that this distance $d^G_{HI}$ is $G$-stable and $G$-homotopy invariant and establish its universality by showing that it dominates any such distance. We further prove a version of equivariant persistent Whitehead theorem relating interleavings of equivariant persistent homotopy groups to $G$-homotopy interleavings of persistent $G$-spaces. We also prove an equivariant persistent nerve lemma for equivariant good covers of persistent $G$-spaces. As a consequence, we obtain an equivariant weak law of large numbers for filtrations, providing a homotopy-theoretic consistency result for random filtrations equipped with finite group of symmetries.

In characteristic $p =3$, we establish a concrete relationship between the torsion rank of the elementary abelian group of global sections of quasi-elliptic surfaces over the projective line ${\mathbf P}^1$ and the $p$-rank of certain hyperelliptic curves, thereby explaining a mysterious relationship between these two invariants previously observed by I. Dolgachev. As an application, we determine the torsion ranks of the Mordell–Weil groups of a class of quasi-elliptic surfaces.

We introduce a geometric structure – a conformal Killing–Yano Ricci soliton (CKY–RS) – that couples conformal Ricci soliton (CRS) geometry to conformal Killing–Yano (CKY) 2–forms. The soliton field of the CRS geometry is given by the divergence of the CKY 2–form. We introduce a conserved CKY–Cotton current and derive a compatibility identity relating the Cotton tensor, the CRS obstruction tensor, and the CKY 2–form. In 4–dimensional Lorentzian signature, we show that, under non-degeneracy and closedness assumptions on the CKY form, a CKY–RS structure forces the conformal representative to be locally Kerr–NUT–(A)dS. For a closed non-degenerate CKY on a Kerr–NUT–(A)dS background, the conformal deformation is necessarily trivial. For Einstein backgrounds of arbitrary dimension and signature, the conformal factor satisfies an eigenvalue equation and an Obata–type Hessian equation. If the background is also compact or a CKY orbit is periodic, the conformal factor is an invariant of the CKY–flow and we obtain simple spectral obstructions to non-trivial CKY–RS structures. From the Hessian equation, we obtain obstruction and classification results for the non-trivial conformal sector, including product/Brinkmann geometries and a Weyl–aligned branch. Finally, we give explicit constructions for static spherically symmetric geometries and BTZ backgrounds, including a CKY–RS realization with a time-dependent conformally flat representative. These results provide a geometric framework for studying CRS with hidden symmetry structure, with potential applications to exact geometries in general relativity.

We study the arithmetic of the elliptic curves \[ A_m:y^2=x(x-m)(x+4m) \] attached to the $3/5$ congruent number problem. A difference of ternary representation numbers controls the relevant central $L$-values. For $p\equiv11\pmod{40}$ the ordinary Cassels pairing degenerates; we construct an explicit $4$-cover and show that the next Cassels–Tate pairing is governed by the normalized representation defect, equivalently by a factorial character, a Pell symbol, and a class number congruence. For composite parameters we compute the ordinary Cassels matrices of four twists and, in the two prime case, a degree $1024$ governing field for their joint distribution. The same higher descent extends to larger radicals: a second rational pushout determines the full $\Lambda'$ row of the next pairing. We also determine two explicit Selmer towers with four dimensional ordinary radical, and all finite $2$-power Selmer groups when the ordinary radical is one dimensional.

This paper concerns continuous sonic-supersonic flows in a two-dimensional infinitely long divergent nozzle with straight solid walls. For a given inlet that is a small perturbation of an arc centered at the vertex of the nozzle, we establish the global existence and uniqueness of a continuous sonic-supersonic potential flow which is close to the radially symmetric sonic-supersonic flow. The flow is sonic at the inlet, with its velocity along the normal direction, and becomes supersonic in the nozzle. Such a sonic-supersonic model is governed by a quasilinear nonstrictly hyperbolic equation with degeneracy at the inlet, and the degeneracy is weak in the sense that there exist exactly two distinct characteristics from each sonic point, except for those at the walls, pointing into the supersonic region. One of the crucial ingredients of the analysis is to obtain the precise asymptotic behavior near the sonic inlet by the method of characteristics, which is essential for establishing the local existence of continuous sonic-supersonic flows. Another one is to extend the local solution globally and derive uniform estimates in the supersonic region.

We study the mean square of the Euler–Zagier double zeta-function when the second variable moves vertically. Mean-square formulas in and beyond the region of absolute convergence were previously obtained by Matsumoto and Tsumura and by Ikeda, Matsuoka and Nagata. In particular, the latter authors determined the leading terms in the following three boundary cases: \[ \zeta_2\left(1+a,\frac12+it\right),\qquad \zeta_2\left(1+ib,\frac12+it\right),\qquad \zeta_2\left(1-a,\frac12+a+it\right), \] where $a\in\mathbb C$ with $\Re a>0$ and $b\in\mathbb R$ are fixed. The mean squares in the first and third cases, as well as in the second case when $b\neq0$, have leading terms of order $T\log T$. When $b=0$, the second case reduces to the corner $(1,1/2)$, where the leading term is of order $T(\log T)^3$. In the present paper, we refine these results by obtaining asymptotic formulas with remainder $o(T)$. In the case $b\neq0$, an additional oscillatory term of order $T$ occurs. We also show that, at the parameter points $(s_1,\sigma_2)=(k,\ell/2)$ with integers $k\geq1$ and $\ell\geq2$, the constant multiplying $T$ in the mean-square formula is a linear combination of multiple zeta values. At the boundary points $(k,1/2)$ with integers $k\geq1$, the harmonic finite parts of the corresponding divergent sums are expressed in terms of harmonic regularized multiple zeta values.

In this paper, we focus on the boundedness and compactness of the Toeplitz operators and the Hankel operators on the $\mathcal{D}$-weighted Bergman spaces $A_{\omega}^p$ ($1\leq p<\infty$). In particular, the boundedness of Toeplitz operators with $\mathrm{BMO}_\omega^p$-symbols on $A_{\omega}^p$ is characterized in terms of the Berezin-type transform. Several sufficient conditions for the Toeplitz operators $T_f^\omega$ with locally integrable symbols to be bounded (resp. compact) for each $1\leq p<\infty$ are presented. Moreover, sufficient and necessary conditions for the boundedness of the Toeplitz operator $T_{\bar{f}}^\omega:A_\omega^1\rightarrow A_\omega^1$ and the Hankel operator $H_{\bar{f}}^\omega:A_\omega^1\rightarrow L_\omega^1$ with co-analytic symbol are established.

Let $M_1$, $\cdots$, $M_\ell$ be complete manifolds of the same dimension, where $2\le \ell\in\mathbb{N}$. Suppose that each $M_i$ satisfies two side Gaussian bounds. If some of these manifolds are parabolic and there exists a constant $1\le n_i\le 2$ such that for some $x_i\in M_i$ with $1 \le r \le R < \infty$, $$c_i\left(\frac{R}{r}\right)^{n_i}\le \frac{Vol_{M_i}(B(x_i,R))}{Vol_{M_i}(B(x_i,r))}\le C_i\left(\frac{R}{r}\right)^{n_i},$$ by assuming that all the manifolds satisfy the relative connectedness of the annuli ($RCA$) condition introduced by Grigor'yan and Saloff-Coste, we show that the Riesz transform $\nabla \mathcal L^{-1/2}$ is bounded on $L^q(M)$ for each $1<q<2$ on the gluing manifold $M=M_1 \# M_2 \# \cdots \# M_\ell$.

In this paper we prove in full detail the isomorphism between the solvable group $\mathcal{S}_{2,2+p}$, metric equivalent to the Calabi Vesentini symmetric space $\mathrm{SO(2,2+p)/SO(2)\times SO(2+p)}$, and the solvable group $\mathcal{S}_{\mathrm{L}(-1,p)}$ supporting the Kähler metric of the homogeneous non symmetric special manifold L$(-1,p)$. From an Alekseevskyan point of view the two spaces simply correspond to two different quadratic forms on the same solvable Lie algebra that we present and compare in detail. Furthermore considering the full group of isometries $\mathrm{Iso}_{\mathrm{L}(-1,p)}$ of L$(-1,p)$ we show that $\mathrm{Iso_{L(-1,p)}}\subset \mathrm{SO}(2,2+p)$ is a non-semisimple subgroup of the simple isometry group of Calabi-Vesentini manifolds. Altogether the Special Kähler manifold L$(-1,p)$ can be seen as a coset manifold $\mathrm{Iso_{L(-1,p)}}/\mathrm{H}$ where $\mathrm{H}=\mathrm{U(1)}_L\times \mathrm{SO(p)}$, the generator of $\mathrm{U(1)}_L$ being in $\so(2,2+p)$, yet not in the canonical $\so(2)\oplus\so(2+p)$ subalgebra. The action of $\mathrm{Iso}_{\mathrm{L}(-1,p)}$ on the solvable coordinates of the CV manifold can be constructed directly. This identification provides a valuable tool for Cartan Neural Networks, introducing additional non linear transformations in every map from one layer to the next one of an $r=2$ Neural Network based on the CV Tits Satake universality class; in perspective, this new tool increases expressivity.

We say that a graph is resistance nonnegative or RN if it admits a positive edge-weight that yields nonnegative resistance curvature in the sense of Devriendt and Lambiotte. Analogously, a graph may be resistance positive or RP; we say a graph is strictly RN if it is RN but not RP. In this paper, we show that every $2$-connected strictly RN graph is bipartite with parts whose sizes differ by one, demonstrating that there are no $1$-tough strictly RN graphs. As a consequence, we prove that every $1$-tough RN graph is also RP. Lastly, we quantify the exact toughness values an RN graph can attain below $1$.

In this paper, we apply the resonance method to exhibit large values at the central point of the family of quadratic Hecke $L$-functions with prime-related moduli of the Gaussian field under the generalized Riemann hypothesis.

Consider an $N\times N$ random band matrix $H$ with bandwidth $W$ with centered real Gaussian entries. We prove that, under the (almost) sharp assumption $W^2\ll N/\log N$, the bulk local statistics of $H$ are asymptotically a Poisson point process. Combined with the bulk universality results from arXiv:2501.01718 and arXiv:2506.06441, this establishes the Poisson-RMT transition of the local statistics of random band matrices, and complements the results about the localization-delocalization transition of the bulk eigenvectors previously established in arXiv:2501.01718 and arXiv:2506.06441, and arXiv:2508.05802. The key technical inputs are: (1) the fractional moment estimates from arXiv:2508.05802; (2) a novel estimate for the density of states, obtained from the local laws in arXiv:2501.01718 and arXiv:2506.06441.

Periodic resource-block (RB) scanning leaves most of the current wideband channel unobserved and mixes measurements of different ages. We develop an online canonical polyadic tracker with proximal block updates (CP-PBCD) that reconstructs the full channel after each narrow-RB acquisition. Age-weighted finite histories, frequency and temporal regularization, and bounded component management maintain an adaptive low-rank representation. Two warm-started conjugate-gradient iterations and parallel shifted frequency solves provide a fixed per-frame update budget. Training-user spatial projection strengthens noise suppression at low pilot SNR. Experiments cover ten test users, three speeds, four SNRs, and complete 1000-frame trajectories with 1-ms RB acquisition. Direct first-RB CP-PBCD achieves lower mean normalized mean squared error (NMSE) than all six baselines in all twelve conditions. At 3.6 km/h and 20 dB, it improves on periodic physical refitting by 3.91 dB and runs 35.1 times faster, averaging 0.717 ms per online update. Optional dispersed startup pilots improve early acquisition, while a controlled initialization study yields a last-100-frame NMSE range of 0.427 dB across pilot budgets. These results demonstrate accurate current-channel reconstruction through incremental updates with submillisecond average computation.

We are concerned with the following partial differential equations arising from urban crime modeling: \begin{equation} \label{main} \begin{cases} u_t = \Delta u - \chi \nabla \cdot \left( u \dfrac{\nabla v}{v} \right) - uv + B_1 + ru - \mu u^2, \\[4pt] v_t = \Delta v - v + uv + B_2, \end{cases} \end{equation} under no-flux boundary conditions in a smoothly bounded domain $\Omega \subset \mathbb{R}^n$ with $n \geq 2$, where $\chi$, $r$, and $\mu$ are positive constants. It is shown in this paper that if the nonnegative source terms $B_1$ and $B_2$ are sufficiently regular and \begin{equation*} \mu > \frac{3}{n} + \frac{1}{n}\left( \chi n + \frac{n(\chi(n-1)-2)^2}{4(n-1)} \right)^{\frac{n+1}{n}} \cdot \left( \frac{(2n+\sqrt{n})^2}{2n-1} \right)^{\frac{1}{n}} + n^{12n+3}, \end{equation*} then the system \eqref{main} with suitably regular initial data possesses a global classical solution. Moreover, under the assumption that \begin{equation*} \label{cond.B2} \inf_{t>0} \int_\Omega B_2(\cdot,t) > 0, \end{equation*} the solutions are uniformly bounded in time.

Changing solvent inertia changes the kinetic part of a molecular Hamiltonian and hence its flow in phase space, without requiring a change in the potential energy surface (PES). We study how this affects product probability in a solute-solvent model with two degrees of freedom. Initial conditions are chosen at the same total energy for each solvent mass. For this family, the deterministic outcome changes from reaching products to returning to reactants at a threshold mass. The initial condition at that mass belongs to the stable manifold of the transition-state periodic orbit. Weak Gaussian forcing of the solvent momentum replaces the abrupt change in outcome by a continuous change in product probability over a mass interval proportional to the square root of the noise strength. The deterministic threshold and a linear response calculation predict the position and width of this probability change without fitting the stochastic simulations. Monte Carlo results at three noise strengths agree with the prediction within sampling uncertainty. The calculation connects solvent inertia to product probability through the geometry of the full Hamiltonian flow, rather than through a change in the PES.

Delsarte (1973) conjectured that there are no nontrivial perfect codes in the Johnson scheme. In this paper, we prove that there are no nontrivial $e$-perfect codes in the Johnson scheme for $e\in\{1,2,4,9,10,12,16\}$. This paper complements "Perfect Codes in the Johnson Scheme Hardly Exist", thus proving Delsarte's conjecture completely.

Global channel state information (CSI) acquisition is essential for cooperative precoding in distributed multiple-input multiple-output (DMIMO) systems, but uploading full instantaneous CSI from all distributed antennas creates heavy fronthaul overhead. This paper proposes a fronthaul-efficient acquisition framework based on graph neural network (GNN) reconstruction and task-driven antenna selection. Each transmission and reception point (TRP) uploads only selected antenna CSI, while the centralized unit (CU) reconstructs the full global CSI from partial observations. A universal mask-conditioned GNN is trained with random upload masks, used to evaluate antenna subsets under a fronthaul budget, and then fine-tuned for the selected deployment mask. Simulation results show improved CSI reconstruction accuracy with lower fronthaul and pilot overhead.

We investigate an inverse coefficient problem for an augmented Keller–Segel model arising in the description of phototactic and chemotactic population dynamics. The spatiotemporal evolution of the population density is governed by an advection-diffusion equation in which both the diffusion coefficient and the advection coefficient depend on space and time. Due to the underlying intracellular adaptation mechanism, even in a time-independent external environment, the diffusion coefficient \(D(x,t)\) and the scalar advection coefficient \(K(x,t)\) remain time-dependent and relax exponentially toward their respective steady-state profiles. Importantly, their relaxation is governed by the same rate, determined by the intracellular adaptation dynamics. From internal measurements of the density, we establish conditional Lipschitz stability for recovering either coefficient when the other is known, and derive cross-sensitivity estimates quantifying the compensation between diffusion and advection perturbations that produce the same density data. We further identify a finite-time separation between the transient and steady regimes: the full dynamics admit an exponentially accurate frozen-coefficient approximation at late times. When the steady-state coefficients are known, we also prove unique identification of the relaxation rate from the long-time behavior of the solution. Finally, we derive an explicit linearization of the forward map and, motivated by the temporal separation above, develop a two-stage gradient-based reconstruction strategy. Numerical experiments demonstrate the feasibility of the proposed method.

We show that almost every graph admits a partition of its vertex set into three parts such that no two adjacent vertices have the same number of neighbors in each of the three parts. Equivalently, for $G\sim G(n,1/2)$, $\chi_m(G)\le3$ with high probability, improving the previously known bound of five. Here $\chi_m(G)$ denotes the multiset chromatic number of $G$, the minimum number of parts in a vertex partition whose neighbor-count vectors distinguish every pair of adjacent vertices. In fact, the three-part bound holds for every fixed $0.185<p<0.509$. More generally, for every fixed $p\in(0,1)$, $G\sim G(n,p)$ satisfies $\chi_m(G)\le4$ with high probability. These results are obtained by converting the unresolved edges of a carefully chosen initial partition into hyperplanes of a Boolean cube and applying the Linial–Radhakrishnan theory of essential covers. We also determine how $\chi_m(G)$ grows when the graph is polynomially close to complete. For every fixed $\beta\in(0,1)$ and $G\sim G\!\left(n,1-n^{-(1-\beta)}\right)$, with high probability $\frac{2}{\beta}\le \chi_m(G)\le \left\lfloor\frac{2}{\beta}\right\rfloor+5$. Thus $\chi_m(G)=2/\beta+O(1)$. The lower bound is spectral, while the upper bound follows from multinomial anti-concentration and the Lov'asz Local Lemma.

A new approach is developed to study the convergence of parametric finite element approximations to the mean curvature flow of closed surfaces in three-dimensional space. In this approach, the error analysis is conducted by comparing the numerical solution to a dynamic Ritz projection of the mean curvature flow introduced in this paper, rather than an interpolation of the mean curvature flow, as commonly used in the literature. The errors associated with the dynamic Ritz projection in approximating the mean curvature flow are established in the $L^2$ and $W^{1,p}$ norms. Leveraging these results, optimal-order convergence of parametric finite element methods for mean curvature flow of closed surfaces in the $L^\infty(0,T;L^2)$ norm is proved, including the convergence of parametric finite element methods with piecewise linear finite elements.

Let $N$ be a compact Riemannian manifold whose universal covering supports a convex function. We prove that it admits no quasi-harmonic sphere. No global growth condition is imposed on the convex function.

We study the instability of the naked singularities arising in the spherically symmetric self-similar collapsing of the Einstein–Euler system under gravitational perturbations. We show that small $C^{1,\alpha}$ perturbations (without any symmetries) of the initial conformal metric lead to trapped surface formation. One key point is that the speed of sound is strictly slower the the speed of light, so the gravitational radiation can become sufficiently concentrated to form a trapped surface before the fluid develops any singularity.

Let $G_n=\sum_{k=0}^n4^k\binom{2n-2k}{n-k}^{2}\binom{2k}{k}$. Zhi-Hong Sun conjectured that, for primes $p\equiv3\pmod4$, positive odd integers $m$, and $r\ge2$, the two-step congruence $G_{(mp^r-1)/2}\equiv p^2G_{(mp^{r-2}-1)/2}\pmod {p^{2r-1}}$ holds. We prove the stronger valuation statement $G_{(p^2M-1)/2}-p^2G_{(M-1)/2}\in p^{2v_p(M)+3}\mathbb Z_p$ for every positive odd $M$. The proof converts the sum to a terminating ${}_3F_2$, constructs a cancelled digit-transfer operator, and identifies a two-dimensional analytic quotient of its cubic difference operator. The quotient operator has characteristic polynomial $X^2-p^2$; its second trace is evaluated through the Gross–Koblitz formula and Greene's finite-field Dixon identity. A logarithmic-loss Green inverse converts this spectral identity into an actual integral analytic primitive. The exceptional prime $3$ requires a finite exact PARI/GP certificate, while the infinite tail is bounded symbolically. Thus the computation is finite, reproducible, and separated from the uniform part of the proof.

We introduce the coproduct of ideals and the notion of coprime ideal, extending the Heyting-algebra perspective on ideal lattices to arbitrary commutative rings. The resulting coprime spectrum is a topological space, defines a covariant functor on CF-morphisms of commutative rings, i.e. morphisms that extend ideals and are coprime faithful, classifies fields among integral domains, and in dual rings is homeomorphic to the prime spectrum. This offers an elementary, lattice-theoretic counterpart to the geometric study of nilpotents.

Conformal prediction intervals are used to construct marginally well-calibrated regions around point predictions. Recent work has focused on extending these ideas to conformal predictive distributions, whose marginal probability integral transform (PIT) follows a Unif[0, 1] distribution. In this work, we investigate how to recover predictive densities from conformal predictive distributions. We extend the theory of conformal predictive distributions by proving asymptotic marginal validity for a tail-corrected version of conformal predictive distributions. We propose a method called quantile matching, which preserves an upper bound on the deviation of the marginal PIT from uniformity. Furthermore, we show that the distribution induced by quantile matching is equivalent to the crisp version of conformal predictive distributions when the number of quantiles equals the size of the calibration set. For the recovery of conformal densities, we construct a fidelity-constrained bandwidth optimization for kernel smoothing that preserves asymptotic marginal validity and has a closed-form solution. We apply and compare our proposed methodology on a large simulated real estate transactions dataset based on the Hierarchical Trend Model.

We study the behavior of generalized Shalika models under the local theta correspondence for type II pairs of equal rank. This generalizes a result of Gan on the relation between Shalika models and linear models.

We develop an effective radial framework for analyzing new field-modified potential function of a charged spinless particle in collinear electric and magnetic fields. Using the weak- magnetic field approximation, we neglect the diamagnetic term and other angular projections under an explicit smallness condition $\ll \Delta E$, where $E$ is the electric field, and the replacement of the angular factor by $C=\cos\theta$ in the spherical coordinates. The resulting equation is therefore a field-aligned radial model whose validity is limited to angularly localized states. Within this setting, we apply the indirect mode (or search mode) of Tridiagonal representation approach in the Laguerre and Jacobi bases. In the Laguerre basis, the tridiagonal constraint generates linear-plus-inverse and linear-plus-quadratic radial potentials, and the coefficient recurrences are connected with Meixner–Pollaczek and Meixner polynomial type structures, therefore yielding the analytic spectra and expansion wavefunctions. In the Jacobi basis, the mapping of the radial half-line to a finite interval produces a singular confining effective potential and a three-term recurrence that is treated through a truncated generalized eigenvalue problem. The formulation provides analytically and numerically tractable benchmark models for field-modified potential functions, while keeping the assumptions, parameter restrictions, endpoint behavior, and convergence tests explicit. In the Laguerre basis the resultant spectra are obtained in closed form and are independent of the orientation and of the electric field, which enter through the shape of the effective central potential, whereas in the Jacobi basis the spectrum is obtained numerically from a new three-term recurrence; in all cases the magnetic field enters through a linear Zeeman shift.

This paper introduces a novel formulation and an associated space-time finite element method for simulating solutions to the nonlinear Schrödinger equation. A major advantage of the proposed algorithm is its intrinsic ability to preserve the conservation of mass, energy, and momentum at the discrete level. This is proved for the numerical solutions determined by the fully discrete implicit scheme. An effective iterative scheme is proposed for solving the nonlinear system based on an equivalent formulation which suggests using Newton's iteration for the solution and no iteration for the Lagrange multipliers in the nonlinear system. Extensive numerical examples are provided to demonstrate the high-order convergence and effectiveness of the proposed algorithm in conserving mass, energy, and momentum in the simulation of one-dimensional Ma-solitons and bi-solitons, as well as of two-dimensional solitons governed by the nonlinear Schrödinger equation. The numerical results show that the mass-, energy- and momentum-conserving method designed in this paper also significantly reduces the errors of the numerical solutions in long-time simulations compared with methods which do not conserve these quantities.