This paper elaborates on the implications of the relationship between the Second and Third Laws and provides a comprehensive formal and historical justification for the logical redundancy of the Nernst heat theorem. By revisiting the Nernst-Einstein debate, the underlying hypotheses that lead to the traditional view of the Third Law as an independent postulate are examined. It is argued that the historical rejection of Nernst's proof -- motivated by Einstein's insistence on the practical non-performability of cycles at absolute zero -- overlooks the fact that a universal Second Law already precludes such cycles, rendering an independent Third Law an unnecessary complexity. Ultimately, the Nernst theorem is shown to be an essential consistency regulator rather than an independent physical discovery.
The purpose of this article is to analyze the role of U.S. governmental and non-governmental institutions in supporting democratic values in Ukraine following Russia’s full-scale invasion in 2022. It examines how American assistance has contributed to preserving Ukraine’s democratic governance, sovereignty, and political freedoms during the Russian-Ukrainian armed conflict (war). The article also provides an overview of key U.S. institutions and organizations that have actively supported Ukraine’s democratic aspirations through financial aid, technical assistance, and institutional reforms.
The methodological basis of the study includes a thorough review of publications from the websites of relevant organizations and agencies, official reports and news for 2022–2024. The research also draws on analytical works from prominent U.S. think tanks, including the Brookings Institution and Carnegie Endowment for International Peace, alongside data from surveys conducted by the National Democratic Institute and Razumkov Center. Comparative analysis and case studies are used to assess the feasibility of U.S. support in areas like anti-corruption initiatives, civil society development, and local governance reforms.
The scientific novelty of the article lies in its holistic approach to understanding the diverse dimensions of U.S. support for democratic values in Ukraine. Unlike previous studies, which focus on specific aspects of U.S.–Ukraine cooperation, this research presents a comprehensive examination of how American institutions, including the U.S. Department of State, USAID, the National Endowment for Democracy (NED) etc. are making efforts to strengthen democracy in Ukraine during the war. The author concludes that U.S. assistance is important for Ukraine’s democratic foundations. However, it has not yet addressed such problems as the decline in Ukrainians’ trust in government institutions, which underscores the need for continued efforts to increase transparency, anti-corruption measures, and accountability to bring Ukraine closer to European standards in the long run.
This article explores legal personality within a hierarchical society, examining the factors influencing an individual's rights and obligations. It discusses the significance of social position, age, creditworthiness, sex, and honour in determining an individual's legal capacity. The article highlights the limited social mobility within the hierarchical structure, the importance of the age of majority, the implications of honour and reputation, the role of women in the legal system, and the impact of disabilities and non-Christian faiths. Additionally, it touches on using names and aristocratic predicates to indicate social status. The article provides a concise overview of the complex dynamics surrounding legal personality in historical contexts.
History (General) and history of Europe, History of Law
AbstractUnder the title “Consequences of decisions of International Courts and Treaty bodies on the practical independence of Prosecutors”, the Consultative Council of European Prosecutors (CCPE).
Anatol Guglielmi, Alexey Zavyalov, Oleg Zotov
et al.
After the main shock of an earthquake, a stream of aftershocks that does not subside for a long time is usually observed. Fusakichi Omori found that the frequency of aftershocks decreases hyperbolically with time. It has recently been observed that Omori's law can be viewed as a solution to a differential equation describing the evolution of aftershocks. An alternative way of describing is based on Utsu law, which states that the frequency of aftershocks decreases with time according to a power law. The presented paper is polemical. We discuss the issue of the applicability of each of the three alternative ways of describing aftershocks. The Omori law has a limited scope. The law is valid only in the so-called Omori epoch, after which the earthquake source undergoes a bifurcation. In the Omori epoch, the Utsu law is also valid, but it does not differ in this epoch from the Omori law. The general conclusion is that the existence of the Omori epoch and the phenomenon of bifurcation exclude the possibility of describing by a continuous smooth function. At the same time, the differential evolution equation is applicable both before and after the bifurcation point. Key words: earthquake, main shock, evolution equation, deactivation factor, Omori epoch, bifurcation.
We provide an alternative unified approach for proving the Pythagorean theorem (in dimension $2$ and higher), the law of sines and the law of cosines, based on the concept of shape derivative. The idea behind the proofs is very simple: we translate a triangle along a specific direction and compute the resulting change in area. Equating the change in area to zero yields the statements of the three aforementioned theorems.
Hassan Basari V. T., P. B. Krishna, Titus K. Mathew
We derive a unified expansion law for our universe from the first law of thermodynamics on the apparent horizon, where entropic evolution depicts the emergence of cosmic space. The derivation advances a general form for degrees of freedom on the surface and bulk, which provides a natural generalization for the expansion law proposed by Padmanabhan. The general expression for the surface degrees of freedom differs from the natural expectation, $ N_{sur }= 4S $ in the emergent gravity paradigm for general theories of gravity. The derivation also provides justification for the selection of Gibbons-Hawking temperature in the original expansion law and for the use of areal volume in the non-flat FRW universe. Since the unified expansion law exclusively depends on the form of entropy, the method is applicable to obtain the expansion law in any gravity theory without any additional ad hoc assumptions. From the general expansion law, we have obtained the expansion law corresponding to different theories of gravity like (n+1) Einstein, Gauss-Bonnet, Lovelock, and Horava-Lifshitz. We also obtained the expansion law for non-extensive entropy like Tsallis entropy from the unified expansion law.
Zipf's law is well known in linguistics: the frequency of a word is inversely proportional to its rank. This is a special case of a more general power law, a common phenomenon in many kinds of real-world statistical data. Here, it is shown that snooker statistics also follow such a mathematical pattern, but with varying (estimated) parameter values. Two types of rankings (prize money earned and centuries scored), and three time-frames (all-time, decade, and year) are considered. The results indicate that the power law parameter values depend on the type of ranking used as well as the time-frame considered. Furthermore, in some cases the resulting parameter values vary significantly over time, for which a plausible explanation is provided.
E-commerce has progressed within Europe before, on the basis of the pre-pandemic raise, with COVID-19 determining an explosion of its evolution. The European e-commerce market is very important, linking together more than 500 million consumers. This paper presents the results of assessing the guidelines and trends observed in the European regulatory process in the current period, when e-commerce is developing appreciably, studying the applicable legal norms and their effects on facilitating the control of remote payments. Compliance with the VAT (value-added tax) regime is addressed in the context described by the two important categories of active legal rules: the rules for consumer protection and the rules related to ensuring data protection. The investigation of the legal framework in this regard started from the presupposition that the necessary changes in regulation for facilitating e-commerce are capable to determine indirect changes in many aspects of financial rules and taxation. While analyzing the regulation of online trade, the various measures undertaken recently in respect to European sources of law and their implications for national legal framework opened additional directions of investigation. This paper points out the indirect effect of e-commerce development on the regulatory framework, both in the area of consumer protection and in line with the difficult balance between the right to information and protection of data. The fiscal effects of e-commerce payments and the challenges for the value-added tax regime are also addressed, both from a theoretical point of view and with input from the Court of Justice of the European Union’s jurisprudence. The analyzed documents show that, in this respect, the legal reform will go deeper and will continue with the rhythm of the development of the online activities.
We study the manifestly covariant and local 1-loop path integrals on $S^{d+1}$ for general massive, shift-symmetric and (partially) massless totally symmetric tensor fields of arbitrary spin $s\geq 0$ in any dimensions $d\geq 2$. After reviewing the cases of massless fields with spin $s=1,2$, we provide a detailed derivation for path integrals of massless fields of arbitrary integer spins $s\geq 1$. Following the standard procedure of Wick-rotating the negative conformal modes, we find a higher spin analog of Polchinski's phase for any integer spin $s\geq 2$. The derivations for low-spin ($s=0,1,2$) massive, shift-symmetric and partially massless fields are also carried out explicitly. Finally, we provide general prescriptions for general massive and shift-symmetric fields of arbitrary integer spins and partially massless fields of arbitrary integer spins and depths.
Godwin's law, i.e. the empirical observation that as an online discussion grows in time, the probability of a comparison with Nazis or Hitler quickly approaches unity, is one of the best-documented facts of the internet. Anticipating the quantum internet, here we show under reasonable model assumptions a polynomial quantum speedup of Godwin's law. Concretely, in quantum discussions, Hitler will be mentioned on average quadratically earlier, and we conjecture that under specific network topologies, even cubic speedups are possible. We also show that the speedup cannot be more than exponential, unless the polynomial hierarchy collapses to a certain finite level. We report on numerical experiments to simulate the appearance of the quantum Godwin law in future quantum internets; the most amazing finding of our studies is that -- unlike quantum computational speedups -- the quantum Godwin effect is not only robust against noise, but actually enhanced by decoherence. We have as yet no theoretical explanation, nor a good application, for this astonishing behaviour, which we dub quantum hyperpiesia.
Taylor's power law states that the variance function decays as a power law. It is observed for population densities of species in ecology. For random networks another power law, that is, the power law degree distribution is widely studied. In this paper the original Taylor's power law is considered for random networks. A precise mathematical proof is presented that Taylor's power law is asymptotically true for the $N$-stars network evolution model.
Charles J. Law, Dan Milisavljevic, Daniel J. Patnaude
et al.
We present a three-dimensional kinematic reconstruction of the optically-emitting, oxygen-rich ejecta of supernova remnant N132D in the Large Magellanic Cloud. Data were obtained with the 6.5 m Magellan telescope in combination with the IMACS+GISMO instrument and survey [O III] $λλ$4959,5007 line emission in a ${\sim}$3$^{\prime}~\times$ 3$^{\prime}$ region centered on N132D. The spatial and spectral resolution of our data enable detailed examination of the optical ejecta structure. The majority of N132D's optically bright oxygen ejecta are arranged in a torus-like geometry tilted approximately 28$^{\circ}$ with respect to the plane of the sky. The torus has a radius of 4.4 pc ($D_{\rm LMC}$/50 kpc), exhibits a blue-shifted radial velocity asymmetry of $-3000$ to $+2300$ km s$^{-1}$, and has a conspicuous break in its circumference. Assuming homologous expansion from the geometric center of O-rich filaments, the average expansion velocity of 1745 km s$^{-1}$ translates to an age since explosion of 2450 $\pm$ 195 yr. A faint, spatially-separated "runaway knot" (RK) with total space velocity of 3650 km s$^{-1}$ is nearly perpendicular to the torus plane and coincident with X-ray emission that is substantially enhanced in Si relative to the LMC and N132D's bulk ejecta. These kinematic and chemical signatures suggest that the RK may have had its origin deep within the progenitor star. Overall, the main shell morphology and high-velocity, Si-enriched components of N132D have remarkable similarity with that of Cassiopeia A, which was the result of a Type IIb supernova explosion. Our results underscore the need for further observations and simulations that can robustly reconcile whether the observed morphology is dominated by explosion dynamics or shaped by interaction with the environment.
In this paper, we examine the proposed first law of holographic complexity through studying different perturbations around various spacetime backgrounds. We present a general expression for the variation of the holographic complexity on arbitrary backgrounds by an explicit covariant computation. Interestingly, the general expression can be written as a function of gravitational conserved charges reminiscent of the first law of thermodynamics.
Gram's Law describes a pattern that frequently occurs in the distribution of the non-trivial zeros of the Riemann zeta function along the critical line. Whenever Gram's Law holds true, it reduces the difficulty of computing the corresponding zeta zeros. In this paper, we provide a model that estimates how often this pattern occurs. The model is based on a conjecture that relates the statistical distribution of the zeta zeros to that of the eigenvalues of random unitary matrices.