Papers

We modify the approach to the arithmetical form of the large sieve by relying on the Parseval identity rather than on an approximate Bessel inequality and as a consequence, improve on the weighted large sieve inequality. We compute in passing the $L^2$-norm of the Beurling-Selberg majorant of the characteristic function of an interval.

We study the structure of compact Vaisman and l.c.K. manifolds with nonnegative bisectional curvature in this paper. Our first main result (cf. Theorem 1.3) is a uniformization theorem for compact Vaisman manifolds, extending Mok's uniformization theorem in the Kähler setting (cf. [Mok88]). Our second result (cf. Theorem 1.6) gives a structural trichotomy for compact l.c.K. manifolds with nonnegative Chern bisectional curvature: the underlying complex manifold admits a Kähler metric, admits a Vaisman metric, or its universal cover is conformally equivalent to the product of an incomplete Kähler manifold and a positive-dimensional complex Euclidean space.

We modify the approach to the arithmetical form of the large sieve by relying on the Parseval identity rather than on an approximate Bessel inequality and as a consequence, we prove that sifted sets are either small or the attached Fourier polynomial is very regular at the origin. We also discuss the optimality of this approach and shall show in passing that $N\int_{-\delta}^\delta|S(\beta)|^2d\beta\ge(1-2/\sqrt{\delta N})|S(0)|^2$ for any trigonometric polynomial of length~$N$.

Deep neural networks approximate functions by composing affine maps with nonlinear activations, but how composition itself creates approximation power is not yet fully understood. We investigate a fundamental mechanism: geometrically weighted sums of iterates of a single scalar generator function. This mechanism underpins the classical tent-map construction of the function \(x - x^2\) and related recursive representations used by Yarotsky, W. E, et al., to analyze the approximation powers of deep neural networks. First, we establish a rigidity theorem: for continuous piecewise linear generators with a finite number of segments, any \(C^3\) function that can be represented in this way is at most quadratic. For non-affine quadratic functions, the geometric factor is at least $1/4$. This result both reveals limitations of the tent-map approach and complements existing methods based on hierarchical bases and recursive polynomial constructions. Second, using an exact remainder identity as guidance, we construct a smooth generator whose iterates yield doubly exponential error decay in total depth for square approximation and, through multiplication modules, for each fixed polynomial. For power series with absolutely summable coefficients on \([-1,1]^d\), distributing depth according to monomial degree yields a uniform approximation error of order \(O(e^{-cL^{1/d}})\) on each interior cube. These findings demonstrate how generator dynamics and remainder estimates govern depth allocation and approximation rates of deep neural networks.

Using operator-valued subordination, we derive a formula for the $2$-norm distance on reflexive lattices generated by double triangles of projections under freeness assumptions on their graph coefficients, and provide explicit evaluations in both the scalar and free-projection cases.

In this paper we extend the description of spatial topological chaos for bistable lattice equations in a small diffusion regime. We generalize the Keener's proof and show that stationary solutions are topologically conjugate to biinfinite sequences on a three-symbol alphabet. We provide a priori estimates for the values of stationary solutions and use them to show that there are exactly two monotone stationary fronts (up to a translation). Finally, we apply the localization of all stationary solutions to characterize their stability and get the existence of unique stable and unstable monotone fronts (up to a translation).

We develop Hachimori's study on the unramified Iwasawa modules using extensions with restricted $p$-ramification. Let $K$ be an algebraic number field, and $p$ a prime number which splits into two distinct primes $\mathfrak{p}$, $\mathfrak{p}'$ in $K$. Assume that $K$ and $p$ satisfies several (somewhat strict) conditions. Let $K_\infty /K$ be the cyclotomic $\mathbb{Z}_p$-extension. In the present paper, we give a method to study the structure of the unramified Iwasawa module $X (K_\infty)$ by using abelian $p$-extensions unramified outside $\mathfrak{p}$. We give a sufficient condition for $|X (K_\infty)|$ to be finite in terms of such extensions. We also give a similar criterion for $X (K_\infty)$ to be finitely generated over $\mathbb{Z}_p$. In the latter part of the present paper, we consider the case where $k = \mathbb{Q} (\sqrt{q})$ with an odd prime number $q$ and apply our results to the cyclotomic $\mathbb{Z}_2$-extension $k_\infty /k$. We give a necessary and sufficient condition for $X (k_\infty)$ to be cyclic over $\mathbb{Z}_2$, which is different from the former results given by either Mouhib-Movahhedi or Mizusawa-Mouhib. We also give several sufficient conditions for the validity of Greenberg's conjecture.

The Ehrhart series of the $n$th Birkhoff polytope is $\sum_{r\geq0}H_n(r)z^r$, where $H_n(r)$ counts nonnegative integer $n\times n$ matrices whose row and column sums all equal $r$. We present an exact method for computing this series using constant terms and finite fields. A root filter expresses $H_n(r)$ as a weighted sum of the values $h_r(M)^n$, where $h_r$ is the complete homogeneous symmetric polynomial and $M$ ranges over multisets of $m$th roots of unity with $m=r+1$. Constant-term cancellation reduces the evaluation of $h_r(M)$ to a sum over repeated elements $a$ of $M$. For a particular $a$ of multiplicity $\mu_a$, the computation uses a generalized Todd coefficient of degree $\mu_a-2$. Sums of $h_r(M)^n$ over selected multiplicity classes are handled using symmetric function techniques. Together with the remaining individual evaluations, this gives $O_n(m^{n-5}+m^4)$ field operations for each admissible prime and fixed $n\geq5$. An explicit bound and the Chinese remainder theorem recover the integer counts, and Ehrhart symmetry determines the full series. The same method applies to the World Cup problem, which counts the same matrices with diagonal entries required to be $0$. We prove correctness and compute complete series for both families through order $13$. The Birkhoff series for orders $10$–$13$ and the World Cup series for orders $9$–$13$ are tabulated in the appendices.

In this paper, we investigate the elastic inverse scattering problem in a two-dimensional cavity with an impedance boundary condition. We prove that the scattered fields inside the cavity satisfy a reciprocity relation, and prove that both the shape of cavity and the impedance parameter on the boundary can be uniquely determined from the scattered fields measured on an interior curve. Numerically, we first employ the linear sampling method to reconstruct the shape, and then recover the impedance parameter by combining potential theory with the least squares method. Several numerical experiments are presented to demonstrate the feasibility and effectiveness of our methods.

We study the strong-shear limit of bistable traveling fronts in infinite cylinders with periodic transverse boundary conditions. We prove that, for every sufficiently regular periodic shear profile, the front speed normalized by the flow amplitude converges as the amplitude tends to infinity. After suitable longitudinal rescaling and translations, the corresponding profiles converge along subsequences to full fronts of the limiting degenerate equation. Furthermore, under a Hörmander-type non-degeneracy condition on the shear, the limiting front is proved to be regular and unique up to translation, and the whole normalized family converges uniformly. A main difficulty is that the sign-changing bistable reaction allows the limiting transition to split through intermediate transverse equilibria. We rule out this possibility by exploiting the instability of such equilibria together with suitable regularization and comparison arguments. Finally, in contrast with the combustion case, where every nonconstant mean-zero shear yields a positive limiting speed, we construct an example showing that, for a fixed bistable reaction, smooth mean-zero shears can produce negative, zero, or positive limiting speeds. In particular, the example shows that a shear which accelerates propagation at small amplitudes may reverse the propagation direction when its amplitude becomes large.

The multirate partial differential equation (PDE) technique is considered as one of the most powerful methods for the simulation of radio frequency (RF) circuits. The method reformulates the underlying ordinary differential algebraic equation (DAE) by a PDE. The DAE solution is obtained along a characteristic curve of the PDE. Numerical solution techniques comprise (envelope modulated) Harmonic Balance (HB), multistep finite difference (FD) and spline/wavelet techniques. All techniques have disadvantages when mixed lumped/distributed devices occur and the waveforms exhibit pulse shaped behavior. Furthermore, communication systems are modeled and simulated by the Equivalent Complex Baseband (ECB) approach. Signals and impulse responses are described by their baseband counterparts, employing the Hilbert transformation. A coupled system/circuit simulation requires therefore novel numerical techniques for the solution of integro-differential equations.

We prove that the variety generated by the max-plus algebra on the nonnegative integers has a unique limit subvariety, namely the variety generated by the six-element semiring constructed by Shao, Ren and Gao. More generally, we characterize hereditary finite basedness in a finitely defined variety of commutative additively idempotent semirings containing the max-plus variety. Its hereditarily finitely based subvarieties form a finite lattice and admit equational bases involving at most ten variables. We also prove that every proper subvariety of the max-plus variety is locally finite. A finite basis theorem for finite semirings with a multiplicative identity establishes finite basedness of all finite truncations. Finally, we determine the finite basis properties of two families of subvarieties defined by power identities and of subvarieties relatively defined by arbitrary families of identities making individual additive subterms greatest elements.

Cancer progression often requires multiple driver mutations, but the same drivers may be acquired in different orders. How these pathways jointly shape tumor clonal structure remains unclear. We develop a multitype branching-process model in which malignant transformation requires two driver mutations, distinguishing malignant cells by mutation order and the independent transformation event that founded their clone. Under a successive exponential approximation, we establish point-process limits for pathway-specific clone sizes and derive a closed-form expression for the limiting expected Simpson's index of the combined malignant population. When both mutation orders yield malignant cells with the same net growth rate, the index decomposes into effective pathway weights, determined by mutation rates and birth-death dynamics at preceding stages, and within-pathway concentration terms, determined by intermediate-to-malignant growth-rate ratios. A driver's effect on heterogeneity thus depends critically on when it is acquired. A strong driver acquired early expands the intermediate lineage and increases the supply of independent malignant founders, whereas the same driver acquired last strengthens the growth and age advantage of early-founded malignant clones. Under additive fitness effects, these opposing mechanisms can produce a non-monotone relationship between selective advantage and clonal concentration. Threshold-like non-additive fitness effects can generate highly concentrated malignant populations, while order-dependent terminal fitness causes the faster-growing pathway to dominate asymptotically. These results show how mutation order, mutational accessibility, selection, and epistasis jointly determine lineage-level intratumor heterogeneity.

The tensor product multilevel method (TPML) approximates high-dimensional functions from scattered data on general bounded domains by combining Smolyak's sparse grid construction with the kernel-based multilevel interpolation, coupling the kernel scale to the fill distance at each level so that condition numbers stay uniformly bounded. Its original formulation depended only implicitly on the target function, which precluded numerical evaluation. Using a recent nodal representation of the low-dimensional multilevel operators, we derive an explicit, computable representation of the high-dimensional interpolant. On this basis we prove error estimates in mixed-regularity Sobolev and continuity norms, extending the existing $ L_2 $-theory, and we characterise the full range of anisotropy weights that attain the optimal rate against the number of degrees of freedom. Three explicit weight strategies follow, equilibrating the accuracy, the degrees of freedom, or the cost-benefit ratio. Numerical experiments in up to ten dimensions illustrate the theory.

We analyse Schrödinger operators with accretive potentials in weighted spaces. We find conditions on potentials and weights for which the Dirichlet realisation, introduced by generalised form methods, has non-empty resolvent set. We establish a domain and graph norm separation property, as well as sufficient conditions for the compactness and Schatten class of the resolvent. Moreover, we investigate the relation between discrete spectra and eigenfunctions of operators in standard and weighted spaces. As applications we extend results on the completeness of eigensystems of operators with accretive potentials from standard to weighted spaces and analyse operator matrices exhibiting a Schur dominance property, in particular, related to a wave equation with strong accretive damping.

We solve the finite basis problem for all subvarieties of $\mathsf{V}(S^0_7)$. By considering their intersections with $\mathsf{V}(S_7)$, we reduce the problem to six intervals. We show that $\mathsf{V}(S_7^0)$ has exactly fifteen finitely based subvarieties and that every other subvariety is nonfinitely based. Moreover, $\mathsf{V}(S_c(abc))$ and $\mathsf{V}(SR_6)$ are the only limit subvarieties of $\mathsf{V}(S_7^0)$.

Let $[a_1(x),a_2(x),\ldots,a_n(x),\ldots]$ be the continued fraction expansion of irrational $x\in[0,1)$, and let $q_n(x)$ be the denominator of the $n$-th convergent. In this paper, we study how the growth rate of $a_{n+1}(x)$ on a prescribed logarithmic size interacts with its upper growth rate relative to $q_n(x)$. For $\psi:\mathbb N\to\mathbb{R}_{\ge0}$ satisfying $\psi(n)\to \infty$ and $\alpha, \beta\in [0, \infty]$, define the joint level set \[F_{\alpha,\beta}:=\Big\{x\in [0,1)\colon \liminf_{n\to\infty}\frac{\log (a_{n+1}(x))}{\psi(n)}=\alpha,\ \limsup_{n\to\infty}\frac{\log (a_{n+1}(x))}{\log q_n(x)}=\beta\Big\}. \] We determine the Hausdorff dimension of $F_{\alpha,\beta}$ for all values of $ \alpha$ and $\beta$. Our results is related to several earlier results on the metric theory of continued fractions, including those of Bugeaud [Math. Ann. {327} (2003)] on irrationality exponents, Wang–Wu [Adv. Math. {218} (2008)] on the growth of partial quotients, and Song–Tan–Zhang [Nonlinearity {37} (2024)] on the joint distribution of convergence and irrationality exponents.

We construct the moduli space of marked polystable $\mathrm{SL}_r(\mathbb C)$-Higgs bundles via analytic methods. The resulting space is normal and Hausdorff, and its local models are products of a Teichmüller neighbourhood with a quotient of a quadratic cone. These models preserve the curve parameter and identify the stabilizers. We prove agreement with the complex and unitary gauge-quotient topologies and establish the coarse moduli property for semistable families over reduced analytic bases. The mapping class group and Higgs scaling act holomorphically. Besides, on the stable locus, the moduli spaces that we consider agree with the joint moduli constructed by Collier, Toulisse and Wentworth.

Finite monodromy provides a bridge between group representations and algebraic solutions of differential equations. We study this connection for the Katz-Long-Moody construction, which transforms representations of the semidirect product of a free group and a braid group into new representations of the same group and is related to Knizhnik-Zamolodchikov (KZ)-type equations. For a fixed finite-image input, we classify the parameter values for which the resulting representations have finite image, both on the semidirect product and on its free-group and braid-group subgroups. In particular, finiteness of the braid-group image is independent of the admissible parameter. These results give necessary and sufficient conditions for all solutions of the corresponding regular-singular KZ-type equations to be algebraic. On restriction to the free group, they also characterize finite monodromy and algebraicity of all solutions of the associated Fuchsian systems, connecting the classification to classical questions about algebraic hypergeometric functions.

We study harmonic mappings between the source domain in the $\alpha$-Grushin plane and the target domain in the $\beta$-Grushin plane for $\alpha$ not necessarily equal to $\beta$ and $\alpha\in [0,1)$. Our harmonic mappings arise as a Euler–Lagrange system of equations related to the Dirichlet energy and are studied in weak and strong forms. The key results enclose the second order regularity of harmonic mappings, the Bochner identity, the weak Harnack estimates for coordinate functions of a harmonic mapping, as well as the Caccioppoli estimates and the Liouville type theorem. Furthermore, we discuss several examples of (Grushin) harmonic mappings and, in particular, relate them to holomorphic mappings in the plane.

We study the cyclic steepest descent method (CSD) for strictly convex quadratic minimization. CSD repeats, for j consecutive iterations, the exact steepest-descent step size computed at the beginning of each cycle. The only rigorous result for the real algorithm has so far been restricted to two dimensions, where the cycle-starting gradient sequence is known to converge doubly exponentially. For the simplified ("simple") model obtained by discarding bounded logarithmic terms, Dai and Fletcher predicted that CSD is superlinear whenever the number n of distinct eigenvalues is below twice the cycle length m, and linear otherwise. Let A be symmetric positive definite with q distinct eigenvalues, and suppose 2j > q. We prove the superlinear side of this threshold for the real algorithm in arbitrary dimensions, and in fact obtain a faster, doubly exponential, decay. Except for a Lebesgue-null set of initial points, for every kappa below an explicit positive threshold kappa_0, the cycle-starting gradient satisfies ||g_k|| <= exp(-C exp(kappa k)), and the full gradient and iterate errors satisfy analogous bounds with exponent kappa/j after m iterations. This is the first rigorous proof for the real CSD in arbitrary dimensions, including repeated eigenvalues, and confirms the threshold 2j > q (the simple-model prediction n < 2m); the doubly exponential rate is faster than the superlinear one predicted by the simple model. The proof combines arbitrary-reference ratio coordinates, inverse-image volume contraction, thin-band estimates, and a per-component Borel-Cantelli argument.

We study various discretisation schemes for the regular Sturm–Liouville operator on a bounded interval and for the Laplace operator on an arbitrary planar domain, in both cases subject to zero Dirichlet boundary conditions. The objective is to identify a discretisation scheme such that the corresponding discretised operator—a matrix of size $N \times N$—produces $N$ eigenvalues that approximate as closely as possible the first $N$ eigenvalues of the corresponding operator at the continuous level. By means of numerical experiments we examine several conventional discretisation schemes, and we corroborate the known fact that the conventional discretisations fail to achieve the objective, with the failure attributable to the poor approximation of high-index eigenvalues, and, as a result, to the non-uniform spectrum approximation. In contrast, the newly proposed discrete sine transform based discretisation scheme is designed in such a way that it replicates the asymptotic behaviour of high-index eigenvalues, thereby providing the sought uniform spectrum approximation. Of equal significance is the fact that the proposed discrete sine transform based scheme can be, unlike many Fourier transform based methods, applied to non-rectangular domains.

Let $\tau(G)$ denote the maximum number of edge-disjoint spanning trees in a connected graph $G$ of order $n$, and let $\rho_D(G)$ denote its distance spectral radius. For an integer $k\ge2$, Fan, He and Zhao [Discrete Appl. Math. 376 (2025) 31–40] obtained a sharp distance spectral radius condition for $\tau(G)\ge k$ when $n\ge2k+6$. In this paper, we fill the gap $2k\le n\le2k+5$ and thus complete the result for all $n\ge2k$. The extremal graph given by Fan, He and Zhao remains valid for $n\ge2k+2$, while we determine the unique extremal graph for each of the orders $n=2k$ and $n=2k+1$. We further obtain sharp distance spectral radius conditions and characterize all extremal graphs under the minimum degree condition $\delta(G)\ge k$ for all $n\ge2k$. Finally, for graphs with the stronger minimum degree condition $\delta(G)\ge6k-4$ and order $n\ge2\delta(G)+2$, we obtain a sharp distance spectral radius condition ensuring $\tau(G)\ge k$ and determine the unique extremal graph.

A functional central limit theorem for locally stationary time series taking values in a separable Banach space $B$ is established. The result does not require type-2 or cotype assumptions and therefore covers spaces central to functional data analysis, including $C([0,1])$ and $L^p([0,1])$. Under moment conditions, summable physical dependence coefficients, and a bracketing entropy condition controlling the infinite-dimensional tails, the centered and rescaled partial sum process converges weakly in $D([0,1],B)$. The limit is a centered $B$-valued Gaussian process whose covariance is given by the integral of the local long-run covariance, interpreted as an element of the projective tensor product. We also obtain a stochastic integral representation with respect to a cylindrical Brownian motion, connecting the Banach-space limit to the familiar scalar locally stationary structure. As an application, we derive a self-normalized CUSUM procedure for detecting changes in the mean of linear projections of Banach-valued observations, yielding a pivotal asymptotic null distribution. The finite-sample behavior is illustrated through Monte Carlo experiments and exploratory applications to EEG recordings and daily temperature curves. Examples based on the Faber-Schauder system in $C([0,1])$ and on a $p$-Laplacian model in $W^{1,p}_0([0,1])$ demonstrate how the entropy condition can be verified.

Estimating two-sample comparison functionals within heterogeneous groups is challenging when outcomes are observed only for a small labelled subset while covariates are widely available. We develop semi-supervised estimation of group-specific two-sample comparison functionals and their prespecified aggregates. The proposed estimator combines single-index conditional imputation from labelled cross-arm pairs with averaging over all available covariate pairs. To handle index-score densities that may vanish at attainable endpoints, we use centred stabilization, with a smooth stabilizer in the profile criterion and a shrinking lower truncation of the kernel denominator (the hard floor) in the point estimator. The outer average retains all covariate pairs. Under a correctly specified single-index conditional-mean model and the stated regularity conditions, we establish consistency and asymptotic normality for bounded comparison functionals. A four-role original-observation influence representation accounts for sample overlap and identifies the variance component that unlabelled covariates can reduce relative to supervised estimation. We further establish conditional validity of observation-level multinomial perturbation resampling with complete refitting and shared labelled counts, distinguishing ideal variance consistency from finite-replicate variance estimation. Simulations show small bias and lower mean squared error than supervised estimation across the point-estimation settings, together with near-nominal coverage in the perturbation experiment. A descriptive NHANES 2011-2018 illustration comparing serum creatinine by hypertension history across age groups yields point estimates similar to their supervised counterparts and smaller semi-supervised standard errors for all reported comparisons.

Nearest-neighbor graphs are discrete objects whose membership may change under small perturbations of the underlying coordinates. We establish an explicit stability guarantee for finite labeled configurations in an arbitrary metric space. If the gap between the kth and (k+1)st distances is positive, simultaneous per-label perturbations smaller than one quarter of that gap preserve the directed k-nearest-neighbor membership. The factor four is sharp under the stated uniform displacement assumptions, and every fixed deterministic construction based solely on the labeled neighbor family is consequently invariant. Under an interval-valid Lipschitz bound for labeled information-space trajectories, the same result yields a certified lower bound on the first possible rewiring time, with a refinement for label-specific motion bounds. We apply the finite theorem to frozen standardized Taylor-Green future-information coordinates [d_B, log A_B] for 585 particles at three observed time strata. Outward-rounded interval arithmetic certified a sufficient perturbation radius of approximately 1.023 x 10^-6, and a separately implemented checker within the same research workflow verified the rank and distance-margin calculations. An outcome-blind audit then examined whether the construction supported a numerical continuous-time horizon. Because the complete information map involved discrete clustering and boundary reconstruction and lacked an analytic derivative bound, validated dense-time bound, or certified modulus of continuity, the application was correctly classified as DISCRETE_ONLY with STOP_NO_INTERVAL_VALID_BOUND. Thus finite local structural invariance is proved and numerically certified for the frozen configuration, while temporal specialization and predictive generalization remain separate questions requiring additional evidence.

Memory constraints in artificial intelligence motivate accurate function approximation with fewer parameters. We study looping, which repeatedly composes one update function with shared parameters; each output becomes the next input. A looped Transformer, for example, reuses one block, whereas its conventional untied counterpart uses separately parameterized blocks. We compare their parameter requirements for a given worst-case approximation accuracy, or equivalently, their approximation accuracy under a common budget limiting distinct trainable coefficients. We then ask whether this representational parsimony improves statistical accuracy. For general likelihood models, we establish an upper squared Hellinger risk bound for looped sieve maximum likelihood and a minimax lower bound for the jointly tuned untied family. Further loop iterations improve the approximation bound without adding parameters, while increasing computation and the fitted-class complexity bound. For targets of known Hölder smoothness, looped residual feedforward networks and post-layer-normalized Transformers attain the minimax polynomial rate up to logarithmic factors with a fixed number of bounded real parameters. At sufficiently large fixed budgets, looped worst-case risk vanishes while optimal worst-case untied risk remains bounded away from zero. The loop-to-untied risk ratio also tends to zero under specified growing-budget conditions. Regression, binary response, and energy-based generative models illustrate the theory.

Nonlinear activations can create equivariant interactions between irreducible representations that linear maps cannot. We use the Gaussian degree decomposition to extend ordinary polynomial degree to such nonlinear maps, and prove that for a fixed coordinatewise equivariant layer each degree factors into a polynomial determined by the linear maps and a scalar determined by the activation. This separates three distinct obstructions, coming from symmetry, coordinates, and activation.

This paper is concerned with stochastic Hamiltonian systems driven by Lévy processes. By employing the lent particle method developed by Bouleau and Denis in the framework of Malliavin calculus with jumps, we establish an explicit Bismut–Elworthy–Li type derivative formula for the associated Markov semigroup, as well as corresponding gradient estimates. The main novelty lies in extending the known results for Brownian-motion-driven degenerate systems to the nonlocal setting with jumps, where the presence of a pure jump noise and the degeneracy of the coefficients pose substantial difficulties. Our approach provides a systematic treatment of such nonlocal degenerate operators and fills a gap in the literature on derivative formulae for jump-type stochastic Hamiltonian systems.

Let $1 \leq q < p$ and let $\lambda_1$ be the first eigenvalue of the $p$-Laplacian in a bounded domain $\Omega$. We study the energy functional $$ E_\lambda(u)=\frac{1}{p}\left(\int_\Omega|\nabla u|^p\,dx -\lambda\int_\Omega|u|^p\,dx\right)-\mathcal{F}(u), \quad u\in W_0^{1,p}(\Omega), $$ where $\mathcal{F}$ is positively $q$-homogeneous and vanishes along the first eigenspace. Assuming a suitable relation between $\mathcal{F}$ and the $\kappa$-th power of the principal part of $E_{\lambda_1}$ near this eigenspace, we describe the behavior of $E_{\lambda_1}$ according to the relations $p\kappa<q$, $p\kappa=q$, or $p\kappa>q$. In particular, the functional is unbounded from below in the first case, and has a negative infimum in the last case. We then study how sufficiently small $q$-homogeneous perturbations of $\mathcal{F}$ influence the geometry of $E_\lambda$. In this way, we describe assumptions guaranteeing the existence of three critical points in a left neighborhood of $\lambda_1$ and two critical points in a right neighborhood of $\lambda_1$, which indicates an $S$-shaped structure of the solution set. The results are applied to double-phase functionals and the nonlinear Fredholm alternative.