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We give a direct proof of a result of Earle, Gardiner and Lakic, that is, Kobayashias metric and TeichmA14lleras metric coincide with each other on the TeichmA14ller space of symmetric circle homeomorphisms.

\"Covering the entire sixty-plus-year history of this remote desert outpost with personal vignettes and realistic illustrations, 'Area 51' is the perfect introduction to the significant history made-- and still being made-- at this secret military base\"--Page 4 or cover.

Proving mathematical theorems at the olympiad level represents a notable milestone in human-level automated reasoning 1 – 4 , owing to their reputed difficulty among the world’s best talents in pre-university mathematics. Current machine-learning approaches, however, are not applicable to most mathematical domains owing to the high cost of translating human proofs into machine-verifiable format. The problem is even worse for geometry because of its unique translation challenges 1 , 5 , resulting in severe scarcity of training data. We propose AlphaGeometry, a theorem prover for Euclidean plane geometry that sidesteps the need for human demonstrations by synthesizing millions of theorems and proofs across different levels of complexity. AlphaGeometry is a neuro-symbolic system that uses a neural language model, trained from scratch on our large-scale synthetic data, to guide a symbolic deduction engine through infinite branching points in challenging problems. On a test set of 30 latest olympiad-level problems, AlphaGeometry solves 25, outperforming the previous best method that only solves ten problems and approaching the performance of an average International Mathematical Olympiad (IMO) gold medallist. Notably, AlphaGeometry produces human-readable proofs, solves all geometry problems in the IMO 2000 and 2015 under human expert evaluation and discovers a generalized version of a translated IMO theorem in 2004. A new neuro-symbolic theorem prover for Euclidean plane geometry trained from scratch on millions of synthesized theorems and proofs outperforms the previous best method and reaches the performance of an olympiad gold medallist.

Measurement performance evaluation of real and virtual automotive light detection and ranging (LiDAR) sensors is an active area of research. However, no commonly accepted automotive standards, metrics, or criteria exist to evaluate their measurement performance. ASTM International released the ASTM E3125-17 standard for the operational performance evaluation of 3D imaging systems commonly referred to as terrestrial laser scanners (TLS). This standard defines the specifications and static test procedures to evaluate the 3D imaging and point-to-point distance measurement performance of TLS. In this work, we have assessed the 3D imaging and point-to-point distance estimation performance of a commercial micro-electro-mechanical system (MEMS)-based automotive LiDAR sensor and its simulation model according to the test procedures defined in this standard. The static tests were performed in a laboratory environment. In addition, a subset of static tests was also performed at the proving ground in natural environmental conditions to determine the 3D imaging and point-to-point distance measurement performance of the real LiDAR sensor. In addition, real scenarios and environmental conditions were replicated in the virtual environment of a commercial software to verify the LiDAR model’s working performance. The evaluation results show that the LiDAR sensor and its simulation model under analysis pass all the tests specified in the ASTM E3125-17 standard. This standard helps to understand whether sensor measurement errors are due to internal or external influences. We have also shown that the 3D imaging and point-to-point distance estimation performance of LiDAR sensors significantly impacts the working performance of the object recognition algorithm. That is why this standard can be beneficial in validating automotive real and virtual LiDAR sensors, at least in the early stage of development. Furthermore, the simulation and real measurements show good agreement on the point cloud and object recognition levels.

   Objective:  to verify and confirm (or refute) the hypothesis about the mismatch between the means of proof and the means of criminal activity in terms of strength and diversity, which causes imbalance in the dualism of criminal activity and its proving.    Methods:  in solving the research problem, we used, first of all, dualistic and hypothetical-deductive methods, as well as methods of functional systems, comparison and other general scientific methods.    Results:  criminal activity and its proving show the dualism of two complex functional systems, which are opposite, not reducible to each other, equal, but not equipollent. The lack of equipollency is manifested in the inconsistency of the means of proof with the means of criminal activity in terms of strength and diversity. On the one hand, the available means of proof are outdated and limited; on the other hand, they are used ineffectively when forming evidentiary systems of different levels. This leads to imbalance in the dualistic system. This is confirmed by systemic problems of reality – critically low disclosure rate, violation of the rights and freedoms of participants in criminal proceedings, low level of compensation for damages from crimes, protracted procedures of multi-volume criminal cases, etc.    Scientific novelty:  for the first time, the activity of proving is presented as a complex functional system, which together with criminal activity forms a dualism. In this dualistic system, an imbalance is found, consisting in the non-equipollent relationship between the two functional systems, where the means of proving are inferior to the means of criminal activity both in strength and in diversity. In this regard, the article attempts to reveal the mechanism of proving in order to determine how the two sides in the dualistic system can be balanced.    Practical significance:  the provisions laid down in the study can in the future act as a methodological basis for improving the means of proving and increasing the effectiveness of forming evidentiary systems of different levels.

In contrast to common automated theorem proving approaches, in which the search space is a set of some formulae and what is sought is again a (goal) formula, we propose an approach based on searching for a proof (of a given length) as a whole. Namely, a proof of a formula in a fixed logical setting can be encoded as a sequence of natural numbers meeting some conditions and a suitable constraint solver can find such a sequence. The sequence can then be decoded giving a proof in the original theory language. This approach leads to several unique features, for instance, it can provide shortest proofs. In this paper, we focus on proofs in coherent logic, an expressive fragment of first-order logic, and on SAT and SMT solvers for solving sets of constraints, but the approach could be tried in other contexts as well. We implemented the proposed method and we present its features, perspectives and performances. One of the features of the implemented prover is that it can generate human understandable proofs in natural language and also machine-verifiable proofs for the interactive prover Coq.

Objective: to study issues related to the lawyer and client interaction regarding building a legal position on the case and the plea process.Methods: the article uses the universal dialectical method of cognition, as well as general (analysis, synthesis, induction, deduction) and specific scientific research methods (formal-legal method).Results: As the United States has shifted to “a system of pleas,” the role of defense attorneys has swung from trial litigator to plea negotiator. The study results indicated that defense attorneys spend a significant amount of time meeting with clients. Attorneys also cited substantial deficits in criminal defendants’ knowledge of the legal system, as well as many misconceptions regarding legal procedures. The respondents provided a diversity of responses regarding the most important advice they offer their clients with many mentioning facts related to the case resolution process, the direct and collateral consequences associated with a criminal conviction, the role of the defense attorney, and the importance of the right to silence. Further, over half of the attorneys surveyed indicated a general hesitance and 15.0% – an outright refusal to provide an explicit plea recommendation to their clients.Scientific novelty: The plea process is complex, and defense attorneys play a critical role in this process. The current work highlights how that role has potentially evolved – moving from adviser to educator and evaluator. More qualitative data collection like this is needed to illuminate how attorneys influence defendants’ decision-making rather than just whether they do and by how much. This type of research on “expanded criminal defense lawyering” often goes above and beyond typical outcomes, and instead focuses more on the process, such as plea negotiations and effective attorney-client communication, and often incorporates the defendants’ perspectives. Such studies will inform future avenues for research and how best to operationalize attorney-client communication. The research substantiates the need to reduce the burden on defense attorneys by making it easier for them to ensure that their clients are meeting the knowing and intelligent requirements for plea decisions.Practical significance: the main provisions and conclusions of the article can be used in scientific, pedagogical and law enforcement activities when considering issues related to building a legal position on the case and the plea process.

Landauerʼs principle relates entropy decrease and heat dissipation during logically irreversible processes. Most theoretical justifications of Landauerʼs principle either use thermodynamic reasoning or rely on specific models based on arguable assumptions. Here, we aim at a general and minimal setup to formulate Landauerʼs principle in precise terms. We provide a simple and rigorous proof of an improved version of the principle, which is formulated in terms of an equality rather than an inequality. The proof is based on quantum statistical mechanics concepts rather than on thermodynamic argumentation. From this equality version, we obtain explicit improvements of Landauerʼs bound that depend on the effective size of the thermal reservoir and reduce to Landauerʼs bound only for infinite-sized reservoirs.

Leo-III is an automated theorem prover for extensional type theory with Henkin semantics and choice. Reasoning with primitive equality is enabled by adapting paramodulation-based proof search to higher-order logic. The prover may cooperate with multiple external specialist reasoning systems such as first-order provers and SMT solvers. Leo-III is compatible with the TPTP/TSTP framework for input formats, reporting results and proofs, and standardized communication between reasoning systems, enabling, e.g., proof reconstruction from within proof assistants such as Isabelle/HOL. Leo-III supports reasoning in polymorphic first-order and higher-order logic, in many quantified normal modal logics, as well as in different deontic logics. Its development had initiated the ongoing extension of the TPTP infrastructure to reasoning within non-classical logics.