Universal Formula Research Articles

Clebsch-Gordan Coefficients for Macdonald Polynomials

AbstractIn this paper we use the double affine Hecke algebra to compute the Macdonald polynomial products $$E_\ell P_m$$ E ℓ P m and $$P_\ell P_m$$ P ℓ P m for type $$SL_2$$ S L 2 and type $$GL_2$$ G L 2 Macdonald polynomials. Our method follows the ideas of Martha Yip but executes a compression to reduce the sum from $$2\cdot 3^{\ell -1}$$ 2 · 3 ℓ - 1 signed terms to $$2\ell $$ 2 ℓ positive terms. We show that our rule for $$P_\ell P_m$$ P ℓ P m is equivalent to a special case of the Pieri rule of Macdonald. Our method shows that computing $$E_\ell {\textbf {1}}_0$$ E ℓ 1 0 and $${\textbf {1}}_0 E_\ell {\textbf {1}}_0$$ 1 0 E ℓ 1 0 in terms of a special basis of the double affine Hecke algebra provides universal compressed formulas for multiplication by $$E_\ell $$ E ℓ and $$P_\ell $$ P ℓ . The formulas for a specific products $$E_\ell P_m$$ E ℓ P m and $$P_\ell P_m$$ P ℓ P m are obtained by evaluating the universal formulas at $$t^{-\frac{1}{2}}q^{-\frac{m}{2}}$$ t - 1 2 q - m 2 .

The impact of armed conflict on education and children in Jammu and Kashmir: implications for peace education

ABSTRACT The paper examines how prolonged armed conflict in Jammu and Kashmir affects both education and children. Utilizing a descriptive qualitative survey methodology, the study delves into the multifaceted experiences of the diverse educational stakeholders while analyzing the underlying factors that shape their perspectives. Empirical findings strongly suggest that while there is no universal formula for peace education in conflict contexts, a crucial element lies in establishing supportive classrooms that address the unique challenges experienced by students and teachers. The study’s implications have broad significance, offering valuable insights for policymakers and humanitarian organizations seeking a deeper understanding of the adverse childhood experiences (ACEs) faced by communities in conflict-affected areas. Implementing peace education at the school level can help alleviate the impact of conflict on students and teachers, but addressing the broader context of insecurity requires attention to political, security, social, and economic factors, with schools serving as a potential arena for specific aspects of peace education.

Formation of the national concept of transitive justice

The scientific article is devoted to researching the concept of transit justice and clarifying its significance for Ukraine. The concept of transitive justice is very relevant, which is connected with the spread of internal and interstate conflicts. It is noted that the national model of transitive justice is relatively new for domestic legal practice, therefore the understanding of its essence and the development of mechanisms is still at the initial stage. It is emphasized that transitive justice involves a set of measures aimed at overcoming the consequences of war and occupation and developing mechanisms that will prevent the recurrence of conflicts in the future. The concept of transitive justice is used by states that are transitioning from a state of conflict and occupation to a state of peace, it does not contain universal formulas for all states. Attention is focused on the fact that under modern conditions in Ukraine, the directions of transitive justice are implemented situationally, unsystematically and do not have proper legal consolidation. The international experience of the implementation of the mentioned concept, taking into account the Ukrainian realities, should be defined as useful. Recognition of the supremacy of international law and the tendency to increase the role of non-judicial mechanisms in the justice system will provide an opportunity to use non-judicial mechanisms in order to strengthen sustainable peace. Implementation of the principles of transitive justice affects the development of the legal system of Ukraine and should ensure its stabilization, creation of conditions for its modernization and sustainable development. At the same time, it is stated that it is necessary to carry out a lot of work to establish the reasons for which the war is taking place, to find out what happened directly during the armed conflict. It should be understood that the end of the war is not just the return of the occupied territories under the jurisdiction of Ukraine, but the beginning of a difficult path to resolve the consequences of the conflict and search for a compromise. It is emphasized that there is a real need to implement the model of transitive justice, as an important tool for resolving the military conflict and the main direction of state legal policy, in the conditions of Russia’s armed aggression against Ukraine.

Statistics of three-dimensional black holes from Liouville line defects

Black holes and wormholes in the gravitational path integral can be used to calculate the statistics of heavy operators. An explicit example in higher dimensions is provided by thin shells of matter. We study these solutions in 3D gravity, and reproduce the behavior of black holes and wormholes from the dual CFT using the large-c conformal bootstrap. The CFT operator that creates a thin shell black hole is a line defect, so we begin by using the bootstrap to study the statistics of line defects, both at finite c and in the holographic large-c limit. The crossing equation leads to a universal formula for the average high-energy matrix elements of the line defect in any compact, unitary 2d CFT with c > 1. The asymptotics are controlled by a line defect in Liouville CFT at the same value of the central charge. At large c, three distinct quantities are related: the statistics of line defects in holographic CFTs, the individual matrix elements of a line defect in Liouville CFT, and the on-shell action of black holes and wormholes in 3D gravity. The three calculations match for black holes, and if the statistics of the line defects are assumed to be approximately Gaussian, then a class of wormholes is also reproduced by the dual CFT.

Quantum measurement encoding for quantum metrology

Preserving the precision of the parameter of interest in the presence of environmental decoherence is an important yet challenging task in dissipative quantum sensing. In this work, we study quantum metrology when the decoherence effect is unraveled by a set of quantum measurements, dubbed quantum measurement encoding. In our case, the estimation parameter is encoded into a quantum state through a quantum measurement, unlike the parameter encoding through a unitary channel in the decoherence-free case or trace-preserving quantum channels in the case of decoherence. We identify conditions for a precision-preserving measurement encoding. These conditions can be employed to transfer metrological information from one subsystem to another through quantum measurements. Furthermore, postselected non-Hermitian sensing can also be viewed as quantum sensing with measurement encoding. When the precision-preserving conditions are violated in non-Hermitian sensing, we derive a universal formula for the loss of precision. Published by the American Physical Society 2024

A universal formula for the entanglement asymmetry of matrix product states

Abstract Symmetry breaking is a fundamental concept in understanding quantum phases of matter, studied
so far mostly through the lens of local order parameters. Recently, a new entanglement-based probe
of symmetry breaking has been introduced under the name of entanglement asymmetry, which has
been employed to investigate the mechanism of dynamical symmetry restoration. Here, we provide
a universal formula for the entanglement asymmetry of matrix product states with finite bond
dimension, valid in the large volume limit. We show that the entanglement asymmetry of any
compact – discrete or continuous – group depends only on the symmetry breaking pattern, and is
not related to any other microscopic features.

Time-Dependent Resistance of Sol–Gel HfO2 Films to In Situ High-Temperature Laser Damage

Laser damage in films under long-term high-temperature conditions is a significant concern for advancing laser applications. This study focused on HfO2 films prepared using the sol–gel method with HfCl4 as a precursor. It examined the effects of temperature on various properties of the films, including their optical properties, microstructure, surface morphology, absorption, and laser-induced damage threshold (LIDT). The prepared film demonstrated desirable characteristics at the high temperature of 423 K, such as high transmittance, low absorption, and high LIDT. As the duration of its high-temperature exposure increased, the LIDT of the films gradually decreased. An intriguing finding was that the film’s LIDT exhibited an exponential decay pattern with prolonged heating time. This observation could be attributed to the power-law increase in defects on both the internal and surface areas of the film as the duration of high-temperature exposure lengthened. Moreover, even after a 15-day heating period at 423 K, the film maintained an LIDT of 12.9 J/cm2, indicating its potential applicability in practical high-temperature environments. This study provided a general pattern and a universal formula for understanding the laser damage of sol–gel films at high temperatures over time. Furthermore, it opened possibilities for future developments of laser films suitable for extreme environments.

Topological Characterization of Coronoid Structures via Valency Based Topological Indices.

The number of atoms producing a bond with an atom a in a molecular structure (graph) is regarded as the valency of a. A number of molecular structures have been topologically characterized via a large number of valency-based topological indices, which correlate certain physio-chemical properties like boiling point, stability, strain energy, and many more chemical compounds. Our intention is to perform the topological characterization of primitive coronoid structures initially on the base of seven different valency-based topological indices. Fundamentally, our purpose is to provide the universal formula by using the topological characterization of any primitive coronoid structures with any valency-based topological index. The technique of atom-bonds partitions according to the valency of atoms besides counting principals are used to find the results. Exact values of valency-based topological index, namely the Randic index, sum connectivity index, harmonic index, atom bond connectivity index, geometric arithmetic index, forgotten index, and symmetric division index are calculated coronoid structures initially constructed with at most three benzene lyres. All these indices are valency-dependent structural invariants that have been used to elaborate several chemical properties of compounds of the structure. Moreover, we provide the general formula to compute any valency-based topological index for any type of coronoid structure constructed with any number of benzene lyres.

Peakons and compactons of the (2+1)-dimensional modified dispersive water-wave system

For a higher-dimensional nonlinear dynamical system, there exist abundant coherent excitations. The variable-separated method is a powerful approach to deriving these structures, as its solutions allow for arbitrary functions. Previous works have produced numerous results, including solitons, chaos and fractals. As the molecule structure appears, constructing the multi-soliton molecule through this technology is a meaningful work, especially considering the local peakons and compactons that were seldom discussed before. In this paper, after taking the Bäcklund transformation, the variable-separated solution for the (2+1)-dimensional modified dispersive water-wave system is first derived, which is an important physical model in describing the nonlinear and dispersive long gravity waves. As a result, the multi-peakons and multi-compactons are constructed through the derived universal formula with the aid of the variable functions p and q. These solitons include two general clusters of M × N peakons and compactons, from which the multi-soliton molecules and their interactions are presented.

Recruiting for Engagement in Health Policy.

Who participates in public and patient engagement processes, and in what capacity they participate, matters. The strategies employed to recruit participants shape the outcomes and legitimacy of engagement processes. We explore these issues through a case study of workshop recruitment. We conducted a mixed-methods study drawing on literature about existing theories of engagement, and integrated findings from the research team's own public engagement workshop in September 2022. We sought to align theoretical frameworks with practical approaches to recruiting for engagement. There are inherent trade-offs in recruitment methods. While the theory of recruitment is valuable, practical implementation is complex and highly context-dependent. Engaging existing partners and fostering relationships beyond specific events is crucial. Hybrid workshops and low-barrier honoraria promote participation; however, decisions about location and time create barriers. Finally, balancing trusting relationships with critical perspectives can create tension. Recruitment is foundational for the engagement process, and requires flexibility, responsiveness and a realistic understanding of barriers. Our study suggests that there is no universal formula for ideal participant makeup or event format. Meaningful engagement requires ongoing dialogue and constant adjustment based on practice. Policy makers can use these insights to align recruitment and engagement strategies with their goals in order to move beyond quick, technocratic fixes.

Case law of the European court of human rights on compensation for non-pecuniary damage: Certain issues

Compensation is considered a means of restoring the violated rights. Ukrainian legislation directly provides for the possibility of compensation for non-pecuniary damage, however, a number of problems arise when considering cases on this issue in national courts. Firstly, there is no legally enshrined definition of this concept in Ukraine. Secondly, there is no single methodology or model for assessing moral damage, no formula for calculation is established at the legislative level, due to which judges assign different amounts of compensation for harm caused under the same or similar circumstances. Besides, there are no clear criteria for determining the amount of damage caused, which makes it almost impossible to unambiguously calculate, for example, the extent of the mental pain or the infringement on dignity and honor in property equivalent, that is, it is in fact a conditional compensation. Reference to the case law of the ECHR on this matter may contribute to solving these problems, because its decisions are referred to by national courts. The ECHR has not developed a universal formula for calculating the amount of non-pecuniary damage, but there are certain criteria for its assessment: the nature of the offense and violated right; payments accrued based on the results of consideration of previous cases; victim’s personality; the respondent State; collective statements or mass offenses; other circumstances of the case.

The impact of constrained interacting dark energy on the bound-zone velocity profile

We numerically study the effects of constrained interacting dark energy (CIDER) on the bound-zone velocity profiles around massive dark matter halos. Analyzing the CIDER simulations performed by ref. [1] for three different cases of dark sector coupling (β = 0.03, 0.05 and 0.08) as well as for the standard ΛCDM cosmology (β = 0), we determine the mean peculiar velocity profiles in the bound zones around the friends-of-friends halos with masses larger than M cut = 3 × 1013 h-1 M ⊙ at three redshifts, z = 0, 0.5 and 1. It is found that the universal power-law formula proposed by ref. [2] originally for the ΛCDM case still describes well the bound-zone velocity profiles, V(r), even in the CIDER models. The slope of V(r), turns out to be significantly affected by the CIDER, progressively decreasing as β increases. Meanwhile, the amplitude of V(r) exhibits little dependence on β, which is ascribed to the identical Hubble parameters shared by the ΛCDM and CIDER models in the entire redshift range. Our results imply that the bound-zone velocity slope can break a degeneracy even between the ΛCDM and CIDER models with β ≤ 0.03, which the standard cosmological diagnostics fail to distinguish. We devise a simple analytic formula for the bound-zone slope as a function of β, and prove its validity at all of the three redshifts. It is concluded that the slope of the mean bound-zone peculiar velocity profile should be in principle a powerful probe of dark sector interaction.

In Search of Beauty Formula

The main category of aesthetics — beauty is the focus of the study. Our epoch is a revision of some spiritual meanings both in the sphere of human life and in science and art. The aim of this paper is to derive a universal formula of beauty through the image of Venus on the basis of a comparative analysis of works of classical art, art of postmodernity, as well as to determine the possibilities of modern neural networks. Art of the 21st century is multifaceted, neuro-art is another of its aspects, artificial intelligence and Man of Art compete here. Neural network is a program built on the principle of organisation and functioning of nerve cells of a living organism. The neural network was tasked to represent the modern ideal of beauty in the image of Venus, similar to the way painters of previous epochs acted. Can modern neural networks create? Create spiritual values? Create beauty in art? In total, more than a thousand images were generated according to our requests, which allows us to draw certain conclusions about the representation of the image of Venus by artificial intelligence. An attempt to derive a formula of beauty for each period of art development and to find a universal one is presented. The modern understanding of beauty has absorbed all historical paradigms, but on a qualitative, essential level. Beauty is a subjective sensual perception of the appearance of an object that is capable of evoking positive emotions. It is not the beauty itself that is important, but the feeling of beauty. Beauty is the dominant value of being, which contains a harmonious combination of absolute values, what makes a person a Personality.

Isotropic Kempf–Laksov flag bundles

We introduce analogs of the Kempf–Laksov desingularizations of Schubert bundles in (non-necessary Lagrangian) symplectic Grassmann bundles. In this setting, these are (possibly singular) irreducible flag bundles that are birational to Schubert bundles, and can be described as chains of zero-loci of regular sections in projectivized bundles. The orthogonal analogs are also presented. We immediately derive universal Gysin formulas for isotropic Schubert bundles from these very constructions.

Insights into the surface tension and superficial density peak of molten metals from molecular dynamics

The surface tension of molten metals plays a key role in applications such as microfluidics, welding, and additive manufacturing. These are areas where there is an increasing emphasis on micro- and nano-scale systems, for which universal and scaling relationships are needed. We report a molecular dynamics-based analysis of the size-dependence of surface tension of molten metal droplets at the nanoscale, employing the Young-Laplace law. The superficial density peak in the surface transition layer is found to persist to the nanoscale. A novel curvature radius (Rs*) based on the position of the superficial density peak is introduced to capture the effective surface of tension. This effective radius Rs* is found to reflect accurately the intrinsic surface curvature of the droplet and thereby enable an accurate and predictive universal scaling formula for the behavior of specific surface tension at the nanoscale. These results shed new light on the surface behavior of liquid metals and provide a scaling formula that can be used for further experimental and theoretical analysis.

Towards systematic evaluation of de Sitter correlators via Generalized Integration-By-Parts relations

We generalize Integration-By-Parts (IBP) and differential equations methods to de Sitter correlators related to inflation. While massive correlators in de Sitter spacetime are usually regarded as highly intricate, we find they have remarkably hidden concise structures from the perspective of IBP. We find the factorization of the IBP relations of each vertex integral family corresponding to dτi integration. Furthermore, with a smart construction of master integrals, the universal formulas for iterative reduction and d log-form differential equations of arbitrary vertex integral family are presented and proved. These formulas dominate all tree-level de Sitter correlators and play a kernel role at the loop-level as well.

Experimental and numerical investigations on tensile failure mechanism and load-bearing capacity of expansion anchors under pull-through failure mode in HPSFRC

Pull-through(PT) is a unique failure mode of torque-controlled expansion(TCE) anchor after being pulled out from concrete. No universal design formula is available for predicting tensile bearing capacity of anchor of PT mode. In normal concrete(NC), PT failure usually occurs at large embedment depths(Hef), which is uncommon in the field and received less attention than the concrete breakout failure. However, due to the improved mechanical performance of the substrate, the PT mode dominates for anchors in high-performance steel fiber reinforced concrete(HPSFRC) substrate (even at small Hef). In this study, to study the PT mode in HPSFRC, the results of 34 PT failures from previous tests, including testing scenarios with different anchor diameters(D), Hef, and fiber contents(vf), are analyzed. The cracking morphology and load-bearing capacity distribution of PT mode are elaborated. Further, finite element models(FEM) of anchor pullout behavior in HPSFRC is established, which accurately identified different failure modes. The calculated Nu by FEM agree well with the tested Nu. The difference in damage domain between different failure modes is mainly reflected in the unloading stage. Parametric analyses were conducted on pre-tightening torque(Tini), Hef, D, sleeve length(Lsleeve), anchor cone width, anchor cone length(Lcone), and vf to study their influences on Nu of PT mode. Except for Lsleeve and Lcone, which do not affect Nu, changes in other factors can cause linear or nonlinear changes in Nu. The non-linear relationship between Nu and experimental parameters is formulated. Accordingly, a new prediction model that can accurately predict Nu of PT mode for anchors in HPSFRC is developed, of which calibration factors are derived using probability functions.

Angular Deviations, Lateral Displacements, and Transversal Symmetry Breaking: An Analytical Tutorial

The study of a Gaussian laser beam interacting with an optical prism, both through reflection and transmission, provides a technical tool to examine deviations from the optical path as dictated by geometric optics principles. These deviations encompass alterations in the reflection and refraction angles, as predicted by the reflection and Snell laws, along with lateral displacements in the case of total internal reflection. The analysis of the angular distributions of both the reflected and transmitted beams allows us to understand the underlying causes of these deviations and displacements, and it aids in formulating analytic expressions that are capable of characterizing these optical phenomena. The study also extends to the examination of transverse symmetry breaking, which is a phenomenon observed in the laser beam as it traverses the oblique interface of the prism. It is essential to underscore that this analytical overview does not strive to function as an exhaustive literature review of these optical phenomena. Instead, its primary objective is to provide a comprehensive and self-referential treatment, as well as give universal analytical formulas intended to facilitate experimental validations or applications in various technological contexts.

One-loop elastic amplitudes from tree-level elasticity in 2d

In this paper we extend the study initiated in [1] to the computation of one-loop elastic amplitudes. We consider 1+1 dimensional massive bosonic Lagrangians with polynomial-like potentials and absence of inelastic processes at the tree level; starting from these assumptions we show how to write sums of one-loop diagrams as products and integrals of tree-level amplitudes. We derive in this way a universal formula for the one-loop two-to-two S-matrices in terms of tree S-matrices. We test our results on different integrable theories, such as sinh-Gordon, Bullough-Dodd and the full class of simply-laced affine Toda theories, finding perfect agreement with the bootstrapped S-matrices known in the literature. We show how Landau singularities in amplitudes are naturally captured by our universal formula while they are lost in results based on unitarity-cut methods implemented in the past [2, 3].

The observation of atmospheric HONO by wet-rotating-denuder ion chromatograph in a coastal city: Performance and influencing factors

Due to the significance of atmospheric HONO as a reservoir for radicals and the presence of substantial unknown sources of HONO, there is a pressing need for accurate and consistent measurement of its concentration. In this study, we compared the measurements obtained from the monitor for aerosols and gases in ambient air (MARGA) based on wet chemical method with those from the incoherent broadband cavity-enhanced absorption spectroscopy (IBBCEAS) based on optical method to assess the suitability of the MARGA instrument for accurate HONO detection. The diurnal patterns obtained by the two instruments are similar, with peaks at 8 a.m. and lows at 5 p.m. Over the course of the observation period, it was often observed that HONO concentrations recorded by the MARGA instrument consistently exceeded those obtained through the IBBCEAS technique, accounting for approximately 91.33% of the total observation time. Throughout the entire observation period, the R2 value between the two instruments was 0.49, indicating relatively good correlation. However, with a slope of only 0.27, it suggests poor agreement between the two instruments. Furthermore, the R2 and slopes between the two instruments vary with the seasons and day-night. The larger the quartile values of NO2, NH3, and BC, the greater the slopes of both MARGA and IBBCEAS instruments, and the higher the concentrations of NO2, NH3, and BC (indicator of semivolatile oxidizable hydrocarbons), the greater the differences between the two instruments, all indicating that NH3 may promote the reaction of NO2 with semivolatile oxidizable hydrocarbons to produce HONO. The O3 with its strong oxidizing properties may cause underestimation in the MARGA instrument by oxidizing NO2− to NO3− in the absorbing solution. It is challenging to derive a universal correction formula due to the interference of various chemical substances. Hence, MARGA should not be used for HONO research in the future.