arXiv subjects

Martin Kolář

Publications and source records attributed to Martin Kolář.

16 recordsLinked to original sources

arXiv2026

Canonical Bochner-Kähler potentials via the Sasaki-Kähler correspondence

We introduce a new approach to the study of Bochner-Kähler manifolds, based on normal form techniques in CR geometry. As observed by Webster, Bochner-Kähler manifolds are intimately related to spherical Sasakian manifolds. Our approach utilises this relationship as well as normal forms for spherical Sasakian manifolds (known as rigid spheres). In this setting potentials of Bochner-Kähler manifolds are nothing but the defining equations of rigid spheres in rigid normal form. This leads to a canonical formula for Bochner-Kähler potentials, as well as streamlined proofs of known results on the structure and the symmetries of Bochner-Kähler manifolds. Our approach also provides a host of examples in terms of an implicit formula of the corresponding Kähler potentials.

math.DG↗

arXiv2026

Unconditional $V^0_1$-independence of a certified hitting-set principle

We show that a certified formalization of the hitting-set-existence axiom of Atserias and Tzameret, instantiated on the parity-based Nisan-Wigderson compression class of Khaniki, is independent of the two-sorted theory $V^0_1$ of $\mathrm{AC}^0$-reasoning, unconditionally: $V^0_1$ proves neither it nor its negation. The same holds for the corresponding certified dual weak pigeonhole principle, whose refutation is witnessed by a single seed that certified-computes every string of the model simultaneously. The mechanism is a bounded-arithmetic transfer of Atserias-Tzameret's reduction from hitting sets to the dual weak pigeonhole principle: the amplification half of that reduction, the sole source of its NP-oracle, is unnecessary at the native stretch of the Nisan-Wigderson map, and the compression half becomes a $V^0_1$-provable implication once circuit evaluation is replaced by its certified $Σ^B_0$ unfolding. This is, to our knowledge, the first independence result for a derandomization-flavoured existence principle at the $\mathrm{AC}^0$-reasoning level, and it makes explicit the bridge between the Khaniki Nisan-Wigderson line and the Atserias-Tzameret reverse mathematics of hitting sets.

cs.CC↗

arXiv2026

Emergent Alignment

Can Large Language Models (LLMs) discern when their own outputs are misaligned with human ethics? And can they self-correct? We endow an LLM with a conscience step that reviews its own reasoning and outputs, and we extend the training loss with an alignment component using Direct Preference Optimization (DPO) to steer the model away from non-ethical outputs. The result is an online technique to align models in a wide range of applications: training, fine-tuning, adversarial prompting, and zero-shot learning. It does not require a weaker or stronger judge, relying instead on a frozen copy of itself. In previous work, the Emergent Misalignment scenario showed a range of emergent unethical behaviors from fine-tuning the model to hack code. Instead, we empirically show how to achieve Emergent Alignment: a single high-level introspective question steers training toward an ethical model under the same code hacking scenario.

cs.AI↗

arXiv2026

On Formally Undecidable Propositions of Nondeterministic Complexity and Related Classes

The definition of \NP\ requires, for each member language~$L$, a polynomial-time checking relation~$R$ and a constant~$k$ such that $w \in L \iff \exists y\,(|y| \leq |w|^k \wedge R(w,y))$. We show that this biconditional instantiates, for each member language, Hilbert's triple: a sound, complete, decidable proof system in which truth-in-$L$ and bounded provability coincide by fiat. We show further that the polynomial-time restriction on~$R$ does not exclude Gödel's proof-checking relation, which is itself polynomial-time and fits the definition as a literal instance. Hence \NP, taken as a totality over all polynomial-time~$R$, contains languages for which the biconditional asserts a property that Gödel's First Incompleteness Theorem prohibits. The semantic definition of \NP\ is unsatisfiable, for the same reason that Hilbert's Program is.

cs.CC↗

arXiv2025

New examples of CR submanifolds for the $2$-jet determination problem

This paper addresses two questions related to mapping problems. In the first part of the paper, we discuss some recent results regarding the $2$-jet determination for biholomorphisms between smooth (weakly) pseudoconvex hypersurfaces; in light of those, we formulate a general problem and illustrate it with examples. In the second part of the paper, we provide a new example of a real submanifold of codimension $7$ in $\mathbb{C}^{10}$ that is not strictly pseudoconvex and for which its germs of CR automorphisms are determined by their $2$-jets at a given point.

math.CV↗

arXiv2025

Classification of polynomial models without 2-jet determination in $\mathbb{C}^3$

An intriguing phenomenon regarding Levi-degenerate hypersurfaces is the existence of nontrivial infinitesimal symmetries with vanishing 2-jets at a point. In this work we consider polynomial models of Levi-degenerate real hypersurfaces in $\mathbb{C}^3$ of finite Catlin multitype. Exploiting the structure of the corresponding Lie algebra, we characterize completely models without 2-jet determination, including an explicit description of their symmetry algebras.

math.CV↗

arXiv2024

Holomorphic vector fields with real integral manifolds

We classify singular holomorphic vector fields in two-dimensional complex space admitting a (Levi-nonflat) real-analytic invariant 3-fold through the singularity. In this way, we complete the classification of infinitesimal symmetries of real-analytic Levi-nonflat hypersurfaces in complex two-space. The classification of holomorphic vector fields obtained in the paper has very interesting overlaps with the recent Lombardi-Stolovitch classification theory for holomorphic vector fields at a singularity. In particular, we show that most of the resonances arising in Lombardi-Stolovitch theory do not occur under the presence of (Levi-nonflat) integral manifolds.

math.CV↗

arXiv2024

GRBAlpha and VZLUSAT-2: GRB observations with CubeSats after 3 years of operations

GRBAlpha is a 1U CubeSat launched in March 2021 to a sun-synchronous LEO at an altitude of 550 km to perform an in-orbit demonstration of a novel gamma-ray burst detector developed for CubeSats. VZLUSAT-2 followed ten months later in a similar orbit carrying as a secondary payload a pair of identical detectors as used on the first mission. These instruments detecting gamma-rays in the range of 30-900 keV consist of a 56 cm2 5 mm thin CsI(Tl) scintillator read-out by a row of multi-pixel photon counters (MPPC or SiPM). The scientific motivation is to detect gamma-ray bursts and other HE transient events and serve as a pathfinder for a larger constellation of nanosatellites that could localize these events via triangulation. At the beginning of July 2024, GRBAlpha detected 140 such transients, while VZLUSAT-2 had 83 positive detections, confirmed by larger GRB missions. Almost a hundred of them are identified as gamma-ray bursts, including extremely bright GRB 221009A and GRB 230307A, detected by both satellites. We were able to characterize the degradation of SiPMs in polar orbit and optimize the duty cycle of the detector system also by using SatNOGS radio network for downlink.

astro-ph.HE↗

arXiv2024

New examples of $2$-nondegenerate real hypersurfaces in $\mathbb{C}^N$ with arbitrary nilpotent symbols

We introduce a class of uniformly $2$-nondegenerate CR hypersurfaces in $\mathbb{C}^N$, for $N>3$, having a rank $1$ Levi kernel. The class is first of all remarkable by the fact that for every $N>3$ it forms an {\em explicit} infinite-dimensional family of everywhere $2$-nondegenerate hypersurfaces. To the best of our knowledge, this is the first such construction. Besides, the class an infinite-dimensional family of non-equivalent structures having a given constant nilpotent CR symbol for every such symbol. Using methods that are able to handle all cases with $N>5$ simultaneously, we solve the equivalence problem for the considered structures whose symbol is represented by a single Jordan block, classify their algebras of infinitesimal symmetries, and classify the locally homogeneous structures among them. We show that the remaining considered structures, which have symbols represented by a direct sum of Jordan blocks, can be constructed from the single block structures through simple linking and extension processes.

math.CV↗

arXiv2024

Models of $2$-nondegenerate CR hypersurface in $\mathbb{C}^N$

We show that every point in a uniformly $2$-nondegenerate CR hypersurface is canonically associated with a model $2$-nondegenerate structure. The $2$-nondegenerate models are basic CR invariants playing the same fundamental role as quadrics do in the Levi nondegenerate case. We characterize all $2$-nondegenerate models and show that the moduli space of such hypersurfaces in $\mathbb{C}^N$ is infinite dimensional for each $N>3$. We derive a normal form for these models' defining equations that is unique up to an action of a finite dimensional Lie group. We generalize recently introduced CR invariants termed modified symbols, and show how to compute these intrinsically defined invariants from a model's defining equation. We show that these models automatically possess infinitesimal symmetries spanning a complement to their Levi kernel and derive explicit formulas for them.

math.CV↗

arXiv2023

Force-Free and Autonomous Active Brownian Ratchets

Autonomous active Brownian ratchets rectify active Brownian particle motion solely by means of a spatially modulated but stationary activity, without external forces. We argue that such ratcheting requires at least a two-dimensional geometry. The underlying principle is similar to the ratcheting induced by steric obstacles in microswimmer baths: suitably polarized swimmers get channeled, while the others get trapped in low-activity regions until they loose direction. The maximum current is generally reached in the limit of large propulsion speeds, in which the rectification efficiency vanishes. Maximum efficiency is attained at intermediate activities and numerically found to be on the order of a few percent, for ratchets with simple wedge-shaped low-activity regions.

cond-mat.soft↗

arXiv2023

GRBAlpha: the smallest astrophysical space observatory -- Part 1: Detector design, system description and satellite operations

Aims. Since launched on 2021 March 22, the 1U-sized CubeSat GRBAlpha operates and collects scientific data on high-energy transients, making it the smallest astrophysical space observatory to date. GRBAlpha is an in-obit demonstration of a gamma-ray burst (GRB) detector concept suitably small to fit into a standard 1U volume. As it was demonstrated in a companion paper, GRBAlpha adds significant value to the scientific community with accurate characterization of bright GRBs, including the recent outstanding event of GRB 221009A. Methods. The GRB detector is a 75x75x5 mm CsI(Tl) scintillator wrapped in a reflective foil (ESR) read out by an array of SiPM detectors, multi-pixel photon counters by Hamamatsu, driven by two separate, redundant units. To further protect the scintillator block from sunlight and protect the SiPM detectors from particle radiation, we apply a multi-layer structure of Tedlar wrapping, anodized aluminium casing and a lead-alloy shielding on one edge of the assembly. The setup allows observations of gamma radiation within the energy range of 70-890 keV with an energy resolution of ~30%. Results. Here, we summarize the system design of the GRBAlpha mission, including the electronics and software components of the detector, some aspects of the platform as well as the current way of semi-autonomous operations. In addition, details are given about the raw data products and telemetry in order to encourage the community for expansion of the receiver network for our initiatives with GRBAlpha and related experiments.

astro-ph.IM↗

arXiv2022

PNPCoin: Distributed Computing on Bitcoin infrastructure

Research and applications in Machine Learning are limited by computational resources, while 1% of the world's electricity goes into calculating 34 billion billion SHA-256 hashes per second, four orders of magnitude more than the 200 petaflop power of the world's most powerful supercomputer. The work presented here describes how a simple soft fork on Bitcoin can adapt these incomparable resources to a global distributed computer. By creating an infrastructure and ledger fully compatible with blockchain technology, the hashes can be replaced with stochastic optimizations such as Deep Net training, inverse problems such as GANs, and arbitrary NP computations.

cs.DC↗

arXiv2021

On formally undecidable propositions in nondeterministic languages

Any class of languages $\mathbf{L}$ accepted in time $\mathbf{T}$ has a counterpart $\mathbf{NL}$ accepted in nondeterministic time $\mathbf{NT}$. It follows from the definition of nondeterministic languages that $\mathbf{L} \subseteq \mathbf{NL}$. This work shows that every sufficiently powerful language in $\mathbf{L}$ contains a string corresponding to Gödel's undecidable proposition, but this string is not contained in its nondeterministic counterpart. This inconsistency in the definition of nondeterministic languages shows that certain questions regarding nondeterministic time complexity equivalences are irrevocably ill-posed.

cs.FL↗

arXiv2020

Nonlinearizable CR automorphisms for polynomial models in $\mathbb C^N$

We classify polynomial models for real hypersurfaces in $\mathbb C^N$, which admit nonlinearizable infinitesimal CR automorphisms. As a consequence, this provides an optimal 1-jet determination result in the general case. Further we prove that such automorphisms arise from one common source, by pulling back via a holomorphic mapping a suitable symmetry of a hyperquadric in some complex space.

math.CV↗

arXiv2015

Technical Report: Image Captioning with Semantically Similar Images

This report presents our submission to the MS COCO Captioning Challenge 2015. The method uses Convolutional Neural Network activations as an embedding to find semantically similar images. From these images, the most typical caption is selected based on unigram frequencies. Although the method received low scores with automated evaluation metrics and in human assessed average correctness, it is competitive in the ratio of captions which pass the Turing test and which are assessed as better or equal to human captions.

cs.CV↗