Papers

A subgraph $H$ of $G$ is said to be $F$-saturated relative to $G$, if $H$ does not contain any copy of $F$, but the addition of any edge $e$ in $E(G)\backslash E(H)$ would create a copy of $F$. The minimum size of an $F$-saturated graph relative to $G$ is denoted by $sat(G,F)$. Let $K_k^n$ be the complete $k$-partite graph with $n$ vertices in each part. In this paper, we determine $sat(K_4^n,C_4)$ for all $ n \geq 2$. Moreover, we determine all extremal configurations of $sat(K_k^n,C_4)$ for all $n\ge 2$ and $k\ge 4 $.

Let $(M^n,g)$ be a complete Riemannian manifold of dimension $n\geq 5$ endowed with a critical metric of the quadratic scalar-curvature functional $$ \mathcal S^2(g)=\int_M R_g^2\,dV_g . $$ For $n\geq 10$, Catino, Mastrolia and Monticelli [J. Math. Pures Appl. 211 (2026), 103883] established that all complete noncompact critical metrics with finite energy are scalar-flat, and they conjectured that this scalar-flatness result holds for all dimensions $n\geq 5$. In this paper, we settle the conjecture by verifying its validity for the remaining dimension range $5\leq n\leq 9$.

We investigate linear lumping for parameter-dependent mass action reaction networks, distinguishing between generic and critical parameter regimes. For generic parameters – those ranging in some non-empty open subset of parameter space – we prove that exact linear lumping yields only "obvious" reductions: elimination of non-reactant species or projections along stoichiometric first integrals. This characterization extends to reaction networks with product-form kinetics, including Michaelis-Menten and Hill-type rate laws. For mass action systems we proceed to develop an algorithmic approach to identify critical parameter sets – algebraic subvarieties in parameter space where non-trivial lumpings become available. This procedure reduces the determination of lumping maps to a system of finitely many polynomial equations. It also applies to constrained lumping scenarios (which are frequently motivated by chemical considerations). We then review and extend results about proper lumpings. Finally, we discuss lumpings of a self-replicator system, and of a two-pathway enzyme mechanism, to document the viability of our methods in relevant scenarios. Our results clarify the relationship between structural (parameter-independent) and fine-tuned (parameter-dependent) reductions, with implications for approximate lumping when system parameters lie near critical values

In this paper, we study the so-called log-convergence of graphs defined by Balázs Szegedy (arXiv:1504.00858). We answer his Question 4 affirmatively: the sequence of incidence graphs of projective planes over finite fields log-converges, and the limit coincides with that of a particular random graph model.

This paper provides an isomorphism between the space of twisted gauge fields on a principal bundle $\mathcal{P}(M,H)$ and the space of standard gauge fields on a different principal bundle $\mathcal{Q}(M,G)$ associated to $\mathcal{P}$. This isomorphism extends to local fields on the base manifold, which enables the use of local twisted fields in standard gauge theories (e.g. Yang-Mills-like theories). This allows one to deal with two symmetry groups, coming from $\mathcal{P}$ and $\mathcal{Q}$, respectively. The construction makes use of a larger principal bundle $\mathcal{S}$ which has $\mathcal{P}$ and $\mathcal{Q}$ as quotient bundles. $\mathcal{S}$ is a Hilsum-Skandalis bibundle between the action groupoids $H\ltimes\mathcal{P}$ and $G\ltimes\mathcal{Q}$. The gauge structure on $\mathcal{S}$ encodes both standard and twisted gauge structures on $\mathcal{P}$. In addition, the isomorphism classes of bundles $\mathcal{S}$ up to an isomorphism covering the identity map on $\mathcal{P}$ are in 1:1 correspondence with the equivalence classes of cocycles (up to a coboundary). This paper also provides a new interpretation of (full) dressing fields as dynamic (or active) sections of a principal bundle.

We study a mean-field optimal control problem for a consensus (high-dimensional Kuramoto-type) dynamics with diffusion on the unit sphere. The control acts through a prescribed drift field and an interaction gain, and the cost functional is given to track a given target density while penalizing the control effort. At the microscopic level, we formulate the corresponding controlled $N$-particle Liouville problem and establish the existence of optimal controls. For fixed controls, we obtain a quantitative stochastic mean-field limit showing that the one-particle marginal converges to the mean-field solution with the convergence rate $\mathcal O(1/\sqrt{N})$. Finally, we show that microscopic optimal controls approximate a mean-field optimal control: any weak limit of particle-level minimizers is optimal for the mean-field problem.

The paper considers the Cauchy problem for a first-order integro-differential equation with memory in a finite-dimensional Hilbert space. The main computational difficulty of such problems is the need to store and process the solution at all previous time levels. To overcome this difficulty, an approach is used that approximates the memory kernel by a sum of exponentials, which reduces the original nonlocal problem to a local one – a system of weakly coupled evolution equations with additional ordinary differential equations for auxiliary functions. The problem is formulated in vector form on the direct sum of Hilbert spaces. Unconditional stability of two-level operator-difference schemes with weights is proved under standard restrictions. Splitting schemes are proposed and investigated by separating the local and integral operators of the problem. Possibilities for constructing similar schemes for other nonlocal problems, in particular for the equation with memory of the time derivative of the solution, are noted.

In this paper, we prove a Dolbeault geometric Langlands equivalence for $\GL_r$ and for the Langlands dual pair $\SL_r/\PGL_r$ over an open locus of the Hitchin base which strictly contains the elliptic locus. This open locus contains the points corresponding to spectral curves with at worst type $A$ singularities, without any restriction on the number of irreducible components. The Dolbeault geometric Langlands equivalence considered here is the one formulated in our previous work with Tudor Pădurariu, which links categorical Donaldson–Thomas theory with the geometric Langlands correspondence. It relates coherent sheaves on moduli stacks of semistable Higgs bundles to the limit category associated with the full moduli stack of Higgs bundles. The use of limit categories is essential beyond the elliptic locus, where the full Higgs moduli stack is no longer quasi-compact and contains infinitely many Harder–Narasimhan strata. The key step is to prove the Whittaker normalization conjecture over the locus of spectral curves with type $A$ singularities, following and extending the strategy developed in the author's proof of the $\GL_2$ case over the reduced spectral curve locus. As a consequence, we also obtain the Dolbeault geometric Langlands conjecture for $\SL_2/\PGL_2$ over the reduced spectral curve locus.

In periodically forced dynamical systems, resonance tongues are open regions of a parameter plane in which the dynamics on an invariant torus locks to a stable periodic orbit. While individual resonance tongues are well understood, the principles governing their global arrangement remain largely unexplored. We develop a topological framework, grounded in applied topology and Morse theory, whose central object is the two-dimensional resonance surface, defined as the graph of the rotation number $\rho$ over a parameter plane. Within this framework, resonance tongues appear as terraces of the resonance surface at rational values of $\rho$, and their global arrangement is determined by the singularities of this surface. Resolving the resonance surface requires the accurate computation of $\rho$, and we present an algorithm that does so efficiently and at high resolution. As a specific example, we examine a periodically forced model of vertical mixing in the North Atlantic, a process relevant to the Atlantic Meridional Overturning Circulation, and study how its resonance surface changes under variation of a third parameter. We identify six distinct resonance-tongue arrangements and show that the resonance transitions between them are due to changes in the number and type of singularities on the boundary of the resonance surface.

This paper develops a path-first theory using the signature as a universal coordinate for deterministic paths, rough paths, jump streams, and path-valued random variables. Geometricity is presented as a first-order algebraic property with second-order obstructions: a bracket for non-geometric lifts, and a covariance when averaging random paths. This framework links the shuffle identity, Marcus-Ito distinction, expected signatures, signature kernels, and free nilpotent group geometry. We offer four main contributions. (1) The Geometricity-Defect Theorem identifies quadratic covariation and coordinate covariance as the canonical failures of shuffle multiplicativity. (2) The Hopf Square proves that for pure-jump finite-variation paths, the forward Ito signature equals the iterated-sums signature, while the Marcus signature is Hoffman's exponential image of it. (3) Affine and exponential Hawkes processes are shown to admit finite-dimensional linear closures for truncated expected signatures after state-weight augmentation. For scalar Hawkes clocks, this allows explicit identification of baseline, excitation, and decay parameters. (4) An antisymmetric second-level cross-area is proved to detect two-channel Hawkes excitation direction to first order. Secondary results cover kernel-MMD decompositions, free nilpotent truncations, stable-law thresholds, heavy-tail normalizations, and a large-deviation principle. All identities and formulas are validated by a reproducibility script.

We study metric representations in finite-dimensional non-Hermitian quantum mechanics. The main purpose of this work is to emphasize that the physical description of a non-Hermitian system may be formulated in different, but isomorphic, Hilbert spaces. In particular, within the Krein-space formalism, we show that the vector space endowed with an indefinite Krein metric can be explicitly related to the standard Hilbert space through a suitable isomorphism. This observation is essential for a consistent description of non-Hermitian Hamiltonians. Physical states, metrics, and operators must be transported through the corresponding Hilbert-space isomorphism. In this way, equivalent representations of the same system can be used without changing the physical content of the theory. We illustrate these theoretical aspects by studying a two-level non-Hermitian spin model. We use the Robertson uncertainty relation as a consistency test. Apparent violations can arise when operators and states are kept fixed while the metric is changed, and therefore reflect a mismatch of representations rather than a failure of the uncertainty principle.

We present a fast and accurate potential theory-based method for the two-dimensional modified Helmholtz equation, treating the involved singular and nearly singular layer evaluations together with volume potentials within a single computational framework. The method is based on a decomposition of the free-space Green's function into a short-range local part and a smooth long-range part. The long-range contribution is evaluated efficiently using the non-uniform fast Fourier transform (NUFFT), while the local contribution is treated by asymptotic expansions. For the layer potentials, an intermediate telescoping sum over dyadic refinement levels is added, where the resulting difference kernels are smooth and rapidly decaying, allowing the dyadic levels to be evaluated without specialized quadrature rules. The volume potential is evaluated on triangular cut-cell meshes, where the mesh only enters the scheme as quadrature rule for smooth data. This makes the method robust with respect to small and distorted mesh cells, without the need for stabilization or cell-merging techniques. Numerical experiments demonstrate the expected convergence rates, high throughput of the potential evaluations, and robustness with respect to mesh quality.

Let $X$ be a smooth irreducible projective variety of dimension $n\ge 3$ over an algebraically closed field of characteristic zero, polarized by a very ample line bundle $\OO_X(1)$. Let $\E$ be an Ulrich bundle on $X$. We prove that there exists an explicitly computable integer $M\gg 0$ such that for every $m\ge M$ the global syzygy bundle $S_{\E(m)}$ is slope semistable with respect to $\OO_X(1)$. This confirms Conjecture~3.11 of Miró-Roig.

Let \((a_n)_{n\ge1}\subset\mathbb{N}\) be a lacunary sequence, \(a_{n+1}\ge q a_n\) for \(q>1\). For \(x\in\mathbb{T}\), we study the maximal empty circular gap \(G_N(x)\) of the finite orbit \(\{a_1x,\ldots,a_Nx\}\). We prove that, for Lebesgue-almost every \(x\), \[ \frac{1}{2} \le \liminf_{N\to\infty}\frac{NG_N(x)}{\log N} \le \limsup_{N\to\infty}\frac{NG_N(x)}{\log N} \le \frac{q+1}{q-1}\,. \] If, in addition, \(a_n\mid a_{n+1}\) for every \(n\), then this can be improved to \[ \lim_{N\to\infty}\frac{NG_N(x)}{\log N}=1 \] for Lebesgue-almost every \(x\).

The common factor analytic model is related to Helmholtz and Boltzmann machines, can be conceived as a linear autoencoder, or can be thought of as a single-hidden-layer generative neural network. We thus consider it a basal generative representation learner that can be used as a minimal model for studying the foundational characteristics of (deep) generative model architectures. We focus on the fundamental problem of indeterminacy in latent factor projections. This indeterminacy implies that, even when the intrinsic dimension of the latent vector is known, regularity conditions are met, and rotational indeterminacy is resolved, an inherent indefiniteness in the retrieval of causative latent sources remains: they will be uncertain, distributionally deviant, and non-unique. This can have major implications for data representation but remains an elusive issue, even to practitioners and theorists well-versed in the factor model. Moreover, this classic psychometric problem is intricately related to the modern issue of latent variable collapse in the variational autoencoder framework for deep generative modeling. Here, we assess this indeterminacy from various perspectives and show how these are mathematically and conceptually related and we discuss subsequent implications for the Psychometrics, Statistics, and Artificial Intelligence communities. We show that one has latent factor determinacy across all its facets when the feature-dimension grows to infinity. This feeds into an essentially distribution-free estimation approach in the sample case when the number of features grows very large. We conclude, as these are emergent properties at scale, that the factor model is suited for representation learning of very-high-dimensional data.

In this paper, we study regular second-order Sturm–Liouville difference equations using the discrete Prüfer transformation. By representing solutions in amplitude and phase coordinates, we analyze an exact algebraic phase system that guarantees unique, monotonic phase tracking and preserves classical oscillation properties. Using this theoretical foundation, we develop a Prüfer-based numerical shooting method to compute eigenvalues for discrete boundary value problems. To initialize the root-finding algorithm, we apply Gershgorin's theorem to the difference operator to establish mathematically guaranteed starting search intervals. Numerical experiments on classical benchmark problems demonstrate that the proposed method effectively isolates the discrete spectrum and converges to the exact continuous eigenvalues with second-order $\mathcal{O}(h^2)$ accuracy.

We consider the following derivative nonlinear Schrödinger equation with a single power-type perturbation \begin{equation*} i\partial_tu+\partial_x^2u+i|u|^2\partial_xu+b |u|^pu=0, \end{equation*} with $b\geq 0$ and $p\geq 4$. When $b=0$ or $p=4$, the equation possesses a family of two-parameter solitons; see, for instance, \cite{CoOh06,Oh14}. Moreover, the authors established the orbital stability/instability of these solitons. In \cite[Corollary]{CoOh06}, a criterion for orbital of solitons was proved. Using the explicit formula of solitons to compute the necessary quantities, the authors verify that the solitons are orbitally stable across most of their range of existence (i.e., $c^2 < 4\omega$). When $b\neq 0$ or $p>4$, an explicit formula for the soliton profile is unavailable, making it difficult to verify the criterion in \cite{CoOh06}. In this paper, we prove the soliton profile varies smoothly with respect to parameters $b$ and $p$. More precisely, we show that the solitons change slowly when $b$ is sufficiently small or $p$ is sufficiently close to $4$. Consequently, we obtain the orbital stability of solitons in these cases. In the borderline case ($c=2\sqrt{\omega}$), the soliton still depends smoothly on the parameter $b$. However, its orbital stability or instability remains an open problem. Solutions to (dNLS) fail to scatter even for small initial data, a property originally proved for $b=0$ in \cite{BaWuXu20}. We also obtain similar results regarding the existence of modified wave operators, analogous to those in \cite{HaOz94}.

For a positive integer $h$, let $R_{A,h}(n)$ denote the number of ordered representations $n=s_1+\cdots+s_h$ with all $s_i\in A$. Let \[ B=\{0\}\cup\{m\ge 1:\text{ the base-4 expansion of }m\text{ begins with }1\text{ or }2\}. \] Shallit proved that $R_{B,3}(n)$ is strictly increasing, thereby disproving a 2002 conjecture of Dombi. In this paper, by using linear bounds for $R_{B,3}(n+1)-R_{B,3}(n)$ and a convolution argument, we prove the polynomial order of $R_{B,h}(n+1)-R_{B,h}(n)$ for every integer $h\ge 3$. More precisely, for every integer $h\ge 3$, there exist constants $c_h,C_h>0$, depending only on $h$, such that \[ c_h n^{h-2}\le R_{B,h}(n+1)-R_{B,h}(n)\le C_h n^{h-2} \] for all integers $n\ge 1$. We also construct a co-infinite set $C\subset\mathbb N$ satisfying $\lim_{n\to\infty}C(n)/n=1$ such that $R_{C,h}(n)$ is strictly increasing for every integer $h\ge 3$. This answers a problem of Dombi posed in 2002. We also pose some problems for further research.

We settle the long-standing open question whether there exists a $3$-ladder of cardinality $\aleph_2$. Given a positive integer $n$, an $n$-ladder is a lower finite lattice whose elements have at most $n$ lower covers. In 1984, Ditor proved that every $n$-ladder has cardinality at most $\aleph_{n-1}$, and that this cardinal bound is sharp for $n = 1,2$. He then raised the question of whether the bound is attained for $n\ge 3$ as well. An affirmative answer is known to be consistent with $\mathsf{ZFC}$. We prove, relative to the consistency of a Mahlo cardinal, that the question is independent of $\mathsf{ZFC}$. More precisely, we show that the nonexistence of a $3$-ladder of cardinality $\aleph_2$ is equiconsistent with a Mahlo cardinal.

We study the capacity of an additive white Gaussian noise (AWGN) channel followed by a subtractive dithered uniform quantizer. Under the Schuchman conditions and with negligible overload probability, the system admits an additive-noise representation in which the effective noise is the sum of Gaussian and uniform components. Capacity bounds are derived for this model where inputs are subject to an average-power constraint as well as a peak-amplitude constraint, where the latter accounts for the limited quantizer dynamic range. Specifically, a computable lower bound is obtained based on the entropy power inequality (EPI), using the maximum-entropy input under the above constraints. Tighter numerical lower bounds are derived using discrete input constellations with finite mass points. Finally, an upper bound is obtained by exploiting the maximum-entropy property of the Gaussian distribution for a given variance. Numerical results show that, for a K-level quantizer, discrete constellations with K mass points already achieve near-optimal rates among the tested families. Moreover, our upper bound is close to the lower bounds in the moderate signal-to-noise ratio (SNR) regime; thus it provides a simple capacity approximation in this regime.

This paper investigates the asymptotic behavior of the tail of the singular product arising in the Hardy Littlewood and Bateman Horn conjectures for one dimensional systems of polynomials. A universal estimate is proved, showing that the contribution of large primes decays like the reciprocal of the logarithm, regardless of the structure of the system. For linear systems (trivial Galois group) superfast convergence is obtained. For nonlinear systems a coefficient is defined that is expressed via the average over the Galois group; in the abelian case and under the Riemann Hypothesis for Dirichlet L functions a more precise error estimate is obtained. Mixed systems containing both linear and nonlinear polynomials are also considered. Numerical experiments, presented as summary tables, confirm the theoretical conclusions. The results provide a rigorous theoretical foundation for computing singular series and refine the Bateman Horn formula.

Marstrand's projection theorem states that the Hausdorff dimension of the orthogonal projection of a Borel set in the plane onto lines is constant almost surely. This property extends to other notions of dimension, such as box and packing dimensions, but does not hold for the Assouad dimension. In this paper, we show that Marstrand's projection theorem also fails for the quasi-Assouad dimension and the Assouad spectrum, which interpolates between the upper box and quasi-Assouad dimensions. Additionally, we establish an almost sure lower bound for the Assouad spectrum of the projections using capacity-theoretic dimension profiles, and an almost sure upper bound for projections of bounded planar sets via an incidence geometry-inspired tube-counting argument. As an application, for a parametrised family of homogeneous self-similar sets, we obtain an almost sure upper bound for the Assouad spectrum which beats the trivial upper bound coming from the upper box dimension.

Here, we show that if $m\ge 5$ is fixed and odd, then there are only finitely many Carmichael numbers of the form $2^np^m+1$ for positive integers $n$ and prime $p$.

In this paper, we develop a unified approach for various operators on Lie conformal algebras. Given a quasi-twilled Lie conformal algebra $(\Ep,\Vs,\Ws)$, we introduce two dual families of operators: right deformation maps $D:\Vs\to\Ws$ and left deformation maps $B:\Ws\to\Vs$. Each family simultaneously subsumes several classical structures: modified $r$-matrices, crossed homomorphisms, derivations, and Lie conformal algebra homomorphisms in the right case, relative Rota-Baxter operators, twisted Rota-Baxter operators, Reynolds operators, and deformation maps of matched pairs in the left case. Using Voronov's derived bracket method, we construct the controlling homotopy algebras: a curved $L_\infty$-algebra governing right deformation maps and an $L_\infty$-algebra governing left deformation maps, with Maurer-Cartan elements precisely characterizing each type. We further develop the associated deformation theories via twisted $L_\infty$-algebras and define cohomology complexes for both types of deformation maps, recovering and extending the cohomologies of all classical and conformal operators already developed in the literature.

In this note, we prove that the two-dimensional Blume-Emery-Griffiths model at the triple point Ferromagnetic-Antiquadrupolar-Disordered (FAD) has a unique Gibbs measure at any temperature, thereby establishing the absence of phase transitions. The FAD point lies at the intersection of lines separating three regions of the phase diagram, and it is a singular point where the model exhibits infinitely many ground states. Our proof is based on a random-cluster type representation with configuration-dependent cluster weights and a coupling with Bernoulli site percolation with parameter $1/2$.

Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbf{Q}_p$. For a smooth representation $\pi$ of $\mathrm{GL}_2(K)$ occurring in some Hecke eigenspace of the mod $p$ cohomology of a Shimura curve, we explore different strategies (inspired by the case $K=\mathbf{Q}_p$) to attack the locality question: does $\pi$ depend only on the underlying $2$-dimensional representation $\overline{\rho}$ of ${\rm Gal}(\overline K/K)$? In particular when $[K:\mathbf{Q}_p]=2$, crucially using perfectoid geometry, we associate to $\overline{\rho}$ an infinite-dimensional mod $p$ smooth representation of $\begin{pmatrix}K^\times&K\\0&1\end{pmatrix}$ which we hope is the restriction to $\begin{pmatrix}K^\times&K\\0&1\end{pmatrix}$ of the (irreducible) supersingular subquotient of $\pi$.

In this paper, we investigate the strong algebrability and $(\alpha,\beta)$-lineability/spaceability of continuous functions with prescribed fractal dimensions. For $1< s< r< t\leq2$, we define $$H_s[0,1]=\{f\in C[0,1]:{\dim}_HG_f([0,1])=s\},$$ $$\underline{B}_r[0,1]=\{f\in C[0,1]:\underline{{\dim}}_BG_f([0,1])=r\}$$ and $$\overline{B}_t[0,1]=\{f\in C[0,1]:\overline{{\dim}}_BG_f([0,1])=t\}.$$ We prove that $H_s[0,1]\cap\underline{B}_r[0,1]\cap\overline{B}_t[0,1]$ is both strongly $\mathfrak{c}$-algebrable and spaceable. This complements recent findings of Bonilla et al. \cite{BFBS}, Esser et al. \cite{EMVVS}, and Liu et al. \cite{LZS}. We prove that for any $1<s\leq t\leq2$, $H_s[0,1]\cap\overline{B}_t[0,1]$ is $(p,\mathfrak{c})$-spaceable for $p=1,2$. We also prove that $H_s[0,1]\cap\overline{B}_t[0,1]$ is $(n,m+n)$-lineable for any $m,n\in\mathbb{N}$, thus complementing the recent work of Liu et al. \cite{LS}.

We study the leading-order fluctuation of stochastic gradient Euler-Maruyama estimators for generalized non-reversible Langevin dynamics. Under structural assumptions tailored to the small-stepsize central limit theorem and under an unbiased stochastic gradient oracle, we prove that the empirical average over a horizon of order the inverse squared stepsize satisfies a central limit theorem in the vanishing-stepsize regime. The limiting variance is characterized through the Poisson equation of the limiting full-gradient diffusion. We then rewrite this constant in an operator form that links it to the continuous-time asymptotic variance and, under standard operator-theoretic assumptions, derive a sufficient condition under which an anti-symmetric perturbation strictly reduces the leading-order fluctuation constant relative to the reversible baseline. We also identify bounded smooth predictive observables that re directly covered by the main theorem. As a separate Gaussian calculation beyond the bounded-test-function regime, we obtain closed-form formulas for quadratic Hamiltonians and linear observables. The framework covers non-reversible Langevin dynamics and augmented-state examples including Hessian-free high-resolution dynamics and a positive-definite subclass of gradient-adjusted underdamped Langevin dynamics that allow stochastic gradients. Numerical experiments on basic examples and Bayesian linear regression using synthetic data, and Bayesian logistic regression using real data support the predicted Gaussian fluctuations and show that the non-reversible schemes consistently reduce the root mean squared error (RMSE) relative to their reversible baselines.

This paper is devoted to studying the existence of normalized solutions for the following quasilinear Schrödinger equation \begin{equation*} \begin{aligned} -\Delta u-u\Delta u^2 +\lambda u=h(u) \quad\mathrm{in}\ \mathbb{R}^{3}, \end{aligned} \end{equation*} where $\lambda$ appears as a Lagrange multiplier, $h$ is a $L^2$-supercritical and Sobolev subcritical nonlinearity. The solutions correspond to critical points of the energy functional subject to the $L^2$-norm constraint $\int_{\mathbb{R}^3}|u|^2dx=a^2>0$. Taking into account the Pohozaev manifold and perturbation method, we obtain the existence of ground state normalized solutions and infinitely many normalized solutions. Moreover, our results cover several relevant existing results in \cite{LZ2023}. And in the end, we get the asymptotic properties of energy as $a$ tends to $+\infty$ and $a$ tends to $0^+$.

Let $B$ be a curve on an irrational ruled surface $X$. We prove that the logarithmic Kodaira dimension of $X-B$ equals the Iitaka dimension of $K_X+B$ and give a rough configuration of $B$ when the logarithmic Kodaira dimension of $X - B$ is less than two. Next, we study the logarithmic multicanonical system of $X-B$ when the logarithmic Kodaira dimension of $X - B$ equals one and prove that its logarithmic $m$-canonical system gives either a $\mathbb{P}^1$-fibration or an elliptic fibration if $m \geq 12$.