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Daniel Strzelecki

Publications and source records attributed to Daniel Strzelecki.

10 recordsLinked to original sources
arXiv2026

An Index Theorem in Relative K-Theory for First-Order Systems

Motivated by bifurcation of branches of homoclinic orbits of dynamical systems, we consider families of first-order equations on the real line and introduce a generalisation of previous index theorems by Pejsachowicz, and by Hu and Portaluri. The main novelties of our approach firstly concern the analytical setting, where we lift the common assumption that the equations are asymptotically hyperbolic. Secondly, we consider general compact parameter spaces instead of a single parameter, which results in a remarkably simple index formula in relative $K$-theory.

math.DS↗
arXiv2025

Nonlinear scalar field equations with a critical Hardy potential

We study the existence of solutions for the nonlinear scalar field equation $$-\Delta u - \frac{(N-2)^2}{4|x|^2} u = g(u), \quad \mbox{in } \mathbb{R}^N \setminus \{0\},$$ where the potential $-\frac{(N-2)^2}{4|x|^2}$ is the critical Hardy potential and $N \geq 3$. The nonlinearity $g$ is continuous and satisfies general subcritical growth assumptions of the Berestycki-Lions type. The problem is approached using variational methods within a non-standard functional setting. The natural energy functional associated with the equation is defined on the space $X^1(\mathbb{R}^N)$, which is the completion of $H^1(\mathbb{R}^N)$ with respect to the norm induced by the quadratic part of the functional. We establish the existence of a nontrivial solution $u_0 \in X^1(\mathbb{R}^N)$ that satisfies the Poho\v{z}aev constraint $\mathcal{M}$ and minimizes the energy functional on $\mathcal{M}$. Furthermore, assuming $g$ is odd, we prove the existence of at least one non-radial solution.

math.AP↗
arXiv2025

A Lichnerowicz equation in the Einstein-scalar field theory on non-CMC closed manifolds

In the paper, we prove the existence of a positive and essentially bounded solution to a Lichnerowicz equation in the Einstein-scalar field theory on a closed manifold with non-constant mean curvature. In particular, the non-constant mean curvature gives rise to supercritical terms in the equation, on top of singular ones. We employ a recent fixed-point argument, which involves sub- and supersolutions. Additionally, we provide several conditions on the coefficients in the equation that prevent the existence of positive classical solutions.

math.AP↗
arXiv2025

Putnam-like dataset summary: LLMs as mathematical competition contestants

In this paper we summarize the results of the Putnam-like benchmark published by Google DeepMind. This dataset consists of 96 original problems in the spirit of the Putnam Competition and 576 solutions generated by LLMs. We analyze the performance of models on this set of problems to verify their ability to solve problems from mathematical contests. We find that top models, particularly Gemini 2.5 Pro, achieve high scores, demonstrating strong mathematical reasoning capabilities, although their performance was lower on problems from the 2024 Putnam competition. The analysis highlights distinct behavioral patterns among models, including bimodal scoring distributions and challenges in providing fully rigorous justifications.

cs.LG↗
arXiv2024

Multiplicity of critical orbits to nonlinear, strongly indefinite functionals with sign-changing nonlinear part

We show an abstract critical point theorem about existence of infinitely many critical orbits to strongly indefinite functionals with sign-changing nonlinear part defined on a dislocation space with a discrete group action. We apply the abstract result to a Schr\"odinger equation $$ -\Delta u + V(x) u = f(u) - \lambda g(u) $$ with $0$ in the spectral gap of the Schr\"odinger operator $-\Delta + V(x)$, that appears in nonlinear optics, as well as to the equations with singular potentials arising from the study of cylindrically symmetric, electromagnetic waves to the system of Maxwell equations.

math.AP↗
arXiv2024

Note on homoclinic solutions to nonautonomous Hamiltonian systems with sign-changing nonlinear part

In the paper, we utilize the recent variational, abstract theorem to show the existence of homoclinic solutions to the Hamiltonian system $$ \dot{z} = J D_z H(z, t), \quad t \in \mathbb{R}, $$ where the Hamiltonian $H : \mathbb{R}^{2N} \times \mathbb{R} \rightarrow \mathbb{R}$ is of the form $$ H(z, t) = \frac12 Az \cdot z + \Gamma(t) \left( F(z) - \lambda G(z) \right) $$ for some symmetric matrix $A$.

math.CA↗
arXiv2019

Periodic solutions of symmetric Hamiltonian systems

This paper is devoted to the study of periodic solutions of Hamiltonian system $\dot z(t)=J \nabla H(z(t))$, where $H$ is symmetric under an action of a compact Lie group. We are looking for periodic solutions in a nearby of non-isolated critical points of $H$ which form orbits of the group action. We prove Lyapunov-type theorem for symmetric Hamiltonian systems.

math.CA↗
arXiv2017

Symmetric Liapunov center theorem for minimal orbit

Using the techniques of equivariant bifurcation theory we prove the existence of non-stationary periodic solutions of $\Gamma$-symmetric systems $\ddot q(t)=-\nabla U(q(t))$ in any neighborhood of an isolated orbit of minima $\Gamma(q_0)$ of the potential $U$. We show the strength of our result by proving the existence of new families of periodic orbits in the Lennard-Jones two- and three-body problems and in the Schwarzschild three-body problem.

math.CA↗
arXiv2016

Symmetric Liapunov center theorem

In this article, using an infinite-dimensional equivariant Conley index, we prove a generalization of the profitable Liapunov center theorem for symmetric potentials. Consider the system $\ddot{q}= -\nabla U(q),$ where $U(q)$ is a $\Gamma$-symmetric potential, where $\Gamma$ is a compact Lie group acting linearly on $\mathbb{R}^n$. If the system possess a non-degenerate orbit of stationary solutions $\Gamma(q_0)$ with trivial isotropy group, such that there exists at least one positive eigenvalue of the Hessian $\nabla^2 U(q_0)$, then in any neighbourhood of orbit $\Gamma(q_0)$ there is a periodic orbit of solutions of the system.

math.CA↗