Terms Of Formula Research Articles

Weighted Bergman kernels for nearly holomorphic functions on bounded symmetric domains

We identify the standard weighted Bergman kernels of spaces of nearly holomorphic functions, in the sense of Shimura, on bounded symmetric domains. This also yields a description of the analogous kernels for spaces of “invariantly-polyanalytic” functions — a generalization of the ordinary polyanalytic functions on the ball which seems to be the most appropriate one from the point of view of holomorphic invariance. In both cases, the kernels turn out to be given by certain spherical functions, or equivalently Heckman-Opdam hypergeometric functions, and a conjecture relating some of these to a Faraut-Koranyi hypergeometric function is formulated based on the study of low rank situations. Finally, analogous results are established also for compact Hermitian symmetric spaces, where explicit formulas in terms of multivariable Jacobi polynomials are given.

Beyond islands: a free probabilistic approach

We give a free probabilistic proposal to compute the fine-grained radiation entropy for an arbitrary bulk radiation state, in the context of the Penington-Shenker-Stanford-Yang (PSSY) model where the gravitational path integral can be implemented with full control. We observe that the replica trick gravitational path integral is combinatorially matching the free multiplicative convolution between the spectra of the gravitational sector and the matter sector respectively. The convolution formula computes the radiation entropy accurately even in cases when the island formula fails to apply. It also helps to justify this gravitational replica trick as a soluble Hausdorff moment problem. We then work out how the free convolution formula can be evaluated using free harmonic analysis, which also gives a new free probabilistic treatment of resolving the separable sample covariance matrix spectrum.The free convolution formula suggests that the quantum information encoded in competing quantum extremal surfaces can be modelled as free random variables in a finite von Neumann algebra. Using the close tie between free probability and random matrix theory, we show that the PSSY model can be described as a random matrix model that is essentially a generalization of Page’s model. It is then manifest that the island formula is only applicable when the convolution factorizes in regimes characterized by the one-shot entropies. We further show that the convolution formula can be reorganized to a generalized entropy formula in terms of the relative entropy.

Arithmetic properties of the Herglotz-Zagier-Novikov function

In this article, we undertake the study of the function F(x;u,v), which we refer to as the Herglotz-Zagier-Novikov function. This function appears in Novikov's work on the Kronecker limit formula, which was motivated by Zagier's paper where he obtained the Kronecker limit formula in terms of the Herglotz function F(x). Two, three, and six-term functional equations satisfied by F(x;u,v) are exhibited. These are cohomological relations coming from the action of an involution and SL2(Z) on C×D12 (the unit circle D1). We also provide the special values of F(x;u,v) at rational arguments of x. Importantly, F(x;u,v) serves as a unified generalization of three other interesting functions, namely F(x), J(x), and T(x), which also appear in various Kronecker limit formulas and are previously studied by Cohen, Herglotz, Muzaffar and Williams, and Radchenko and Zagier. Consequently, our study not only reveals the numerous elegant properties of F(x;u,v) but also helps us to further develop the theories of functions related to its special cases such as J(x) and T(x).

The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces

The generalized cluster complex was introduced by Fomin and Reading, as a natural extension of the Fomin-Zelevinsky cluster complex coming from finite type cluster algebras. In this work, to each face of this complex we associate a parabolic conjugacy class of the underlying finite Coxeter group. We show that the refined enumeration of faces (respectively, positive faces) according to this data gives an explicit formula in terms of the corresponding characteristic polynomial (equivalently, in terms of Orlik-Solomon exponents). This characteristic polynomial originally comes from the theory of hyperplane arrangements, but it is conveniently defined via the parabolic Burnside ring. This makes a connection with the theory of parking spaces: our results eventually rely on some enumeration of chains of noncrossing partitions that were obtained in this context. The precise relations between the formulas counting faces and the one counting chains of noncrossing partitions are combinatorial reciprocities, generalizing the one between Narayana and Kirkman numbers.

Simulating capture efficiency of pitfall traps based on sampling strategy and the movement of ground‐dwelling arthropods

Abstract Pitfall traps are frequently used to capture ground‐dwelling arthropods, particularly beetles, ants and spiders. The capture efficiency of a pitfall trapping system strongly depends on the number and opening size of traps, how traps are distributed over the sampling area (spatial arrangement) and the movement characteristics of arthropods. We use numerical simulations for a single species to analyse the trap count patterns that emerge from these variables. Arthropod movement of individuals is modelled as correlated random walks, with multiple traps placed over an area, and catches are simulated as individual interaction with traps. We consider four different types of spatial arrangements of traps across a homogeneous landscape: grid (i.e. rectangular array), transect, nested‐cross and randomised. We contextualise our results by considering the locomotion of Pterostichus melanarius, a highly active carabid beetle often serving as a biocontrol agent for the suppression of pest insects and weeds. By simulating the trapping of randomly moving ground‐dwelling arthropods, we show that there is an optimal inter‐trap separation distance (trap spacing) that maximises captures, that can be expressed using exact formulae in terms of trap opening sizes, sampling area and trap number. Moreover, for the grid and nested‐cross arrangements, larger trap spacing to maximise spatial coverage over the whole sampling area is suboptimal. Also, we find that over a large sampling area, there is a hierarchical order for spatial arrangements in relation to capture efficiency: grid, randomised, transect, followed by the nested‐cross. However, over smaller sampling areas, this order is changed as the rate at which trap counts accumulate with trap number varies across arrangements—eventually saturating at different levels. In terms of movement effects, capture efficiency is maximised over a narrow diffusive range and does not depend strongly on the type of spatial arrangement—indicating an approximate optimal mode of arthropod activity, i.e. rate of spread. Our approach simultaneously considers several important experimental design aspects of pitfall trapping providing a basis to optimise and adapt sampling protocols to other types of traps to better reflect their various purposes, such as monitoring, conservation or pest management.

Formulasi dan Evaluasi Stabilitas Fisik Sediaan Lip Balm Minyak Bekatul (Rice Bran Oil)

Utilization of rice bran into oil through the extraction process. The content of rice bran can be used as a moisturizer in cosmetic preparation, especially lip balm. Lip balm is a cosmetic preparation with main components such as waxes, fats, and oils from natural or synthesized extracts to prevent lip dryness by increasing humidity and protecting it from adverse environmental effects. This research aims to determine the physical stability of lip balm preparations containing bran oil and the effect of bran oil concentration on the hedonic test results of lip balm preparations containing bran oil. This experimental research method focuses on the manufacture of lip balm preparations, the evaluation of lip balm preparations including organoleptic examination, homogeneity, melting point test, pH test, and spread-ability test during 28 days of storage. The results showed that all formulas of rice bran oil lip balm preparations met the requirements and the hedonic testing of rice bran oil lip balm preparations revealed that F1 (5% rice bran oil) was the panelist’s most preferred formula in terms of aroma, F2 (7.5% rice bran oil) was the most preferred formula in terms of texture, while F3 (10% rice bran oil) was the most preferred formula by panelists in terms of color.
 

Organoleptic test Smoothies Ambon Banana fruit (Musa Acuminata) and Cantaloupe (Cucumis Melo var. cantaloupe)

Bananas and cantaloupe are two fruits that can be found in Indonesia and are very easy to get at affordable prices and can even be grown alone. These two fruits are a good combination if processed in new ways, one of which is in the form of processed smoothies. This study aims to determine consumer assessment of the organoleptic quality of beverages smoothies Ambon bananas and cantaloupe. The method used in this study was to perform an organoleptic test on the panelists using the Hedonic Test by giving 3 drink samples smoothies Ambon bananas and cantaloupe with variations in the amount of different fruit extracts to untrained panelists. The criteria tested included color, aroma, taste, texture and overall assessment so that the best formulation was produced from several treatments. The scoring scores used were 1=Very Disliked, 2=Disliked, 3=Liked, and 4=Very Liked. The Results, F1 was the panelist’s most preferred formula in terms of color, F2 was the panelist’s most preferred formula in terms of aroma, F1 and F3 were the panelists’ most preferred formula in terms of taste, F3 was the panelist’s most preferred formula in terms of texture, F3 was the best formula the most preferred overall by the panelists. Overall, in terms of texture and taste, the panelists preferred the treatment of smoothies Ambon bananas and cantaloupe 100 ml with a ratio of 1:1:1 or F3 between Ambon bananas, cantaloupe and low-fat milk. The taste of the resulting drink is neither bitter nor astringent. In terms of color, panelists preferred F1 while for aroma, panelists preferred F2 because the aroma of bananas was stronger.

Comparing the Accuracy of the Kane, Barrett Universal II, Hill-Radial Basis Function, Emmetropia Verifying Optical, and Ladas Super Formula Intraocular Lens Power Calculation Formulas.

To assess the accuracy of five new-generation intraocular lens (IOL) power formulas: Barrett Universal II (BUII), Emmetropia Verifying Optical (EVO) Formula, Hill-Radial Basis Function (Hill-RBF), Kane Formula, and Ladas Super Formula (LSF). This is a retrospective single-surgeon study from a refractive clinic and clinical research center in Draper, UT, USA. The primary outcome measures were mean absolute error (MAE) and median absolute error (MedAE). Secondary outcome measures were the standard deviation (SD) of each formula's refractive prediction errors (RPE) and the percentage of eyes within ±0.50D. Refractive predictions were compared to the postoperative spherical equivalent to determine the RPE for each formula. RPEs were optimized, and MAE, MedAE, SD of the AME, and percent of eyes achieving RPEs within the specified ranges of ±0.125 D, ±0.25 D, ±0.50 D, ±0.75 D, ±1.0 D were calculated. Subgroup analysis between different axial lengths was attempted but yielded insufficient statistical power to draw meaningful conclusions. A total of 103 eyes of 103 patients were included in our study after applying inclusion and exclusion criteria to 606 eyes from 2019 to 2021. Formulas ranked in ascending order by MAE were Kane, EVO, BUII, Hill-RBF, and LSF. The ascending rankings of MedAE were Kane, BUII, Hill-RBF, EVO, Ladas. Kane had a significantly lower MAE than Hill-RBF (p<0.001). EVO had the lowest SD of AMEs and the highest percentage of eyes within ±0.50 D. According to heteroscedastic testing, EVO also had a statistically significant lower SD than Hill-RBF. Kane was the most accurate formula in terms of MAE and MedAE. EVO and BUII achieved marginally higher MAEs than Kane, suggesting these three formulas are comparable in performance. With the exception EVO and Hill-RBF, the heteroscedastic test yielded no significant differences in SD between the formulas. Although there were multiple statistically significant differences between the formulas in terms of MAE, MedAE, and SD, these differences may not be appreciable clinically. Lastly, there were no statistically significant differences in the percent of eyes with RPEs within ±0.50 D, suggesting similar clinical performance between formulas.

Sign changes in statistics for plane partitions

Recent work of Cesana, Craig and the third author shows that the trace of plane partitions is asymptotically equidistributed in residue classes mod b. Applying a technique of the first two authors and Garnowski, we prove asymptotic formulas for the secondary terms in this equidistribution, which are controlled by certain complex numbers generated by a twisted MacMahon-type product. We further carry out a similar analysis for a statistic related to plane overpartitions.

Antioxidant and Anti-Propionibacterium acnes Activities of Citronella Oil and Clove Oil, and their Formulation Into Emulgel

Propionibacterium acnes is a common acne-causing bacteria in adolescents and adults. The combination of citronella oil and clove oil has the potential to inhibit the growth of Propionibacterium acnes. Both essential oils are also known to have antioxidant effects. The combination of the two essential oils is expected to have better activity. The purpose of this research was to determine the activity of the Fractional Inhibitory Concentration (FIC) index against Propionibacterium acnes, antioxidant activity, and formulation in emulgel. Antibacterial assay of combination citronella oil and clove oil using the FIC index. Antioxidant assay by DPPH (2,2-diphenyl-1-picrihydrazil) method. The emulgels formulation uses Viscolam AT 100P as the base of the emulgels. The results showed that the combination of citronella and clove oil an antibacterial activity with an FIC index is 2 (indifferent), antioxidant citronella oil had an IC50 of a moderate category (185.04 µg/mL) and clove oil in a very strong category (23.48 µg/mL) so that it could be formulated in an emulgel with good characteristics. The most preferred formula in terms of color, scent, consistency, and texture is a formula with a concentration of 5% viscolam AT 100P.

Dissolvable shower gel tablets with enhanced skin benefits.

The excessive use of plastic packaging, even though it is possible to recycle it, is one of the main causes of global warming. In this study, dissolvable shower gel tablets for multiple uses have been developed in order to reduce the use of fresh plastic packaging. The appropriate ratio of two surfactants, cocoyl glutamic acid (CGA) and sodium coco sulphate (SCS), was optimized using design of experiments. Additionally, skin hydration of the emollient either omega oil or glycerine was taken into account when determining its concentration. After that, powdered shower gel formulations were made and tested for their properties, including cleaning power, and foamability. On 30 humanvolunteers, the effects of reconstituted shower gel on skin redness, cleaning effectiveness and overall satisfaction were evaluated. The study found that, when cleaning power and foam height were taken into account, the proper surfactant ratio was 7.5:0 (SCS:CGA). The 5% glycerine shower gel formula demonstrated significantly greater skin hydration than other formulas. The in vivo study's findings demonstrated that there was no statistically significant difference between selected formulas (5% glycerine and 2.5% omega oil) in terms of cleaning ability. Comparing both formulas to the control, neither showed any skin redness. Additionally, it was discovered that the developed products were much more effective at cleaning and easier to use when washing for the volunteers than regular liquid soap. Overall satisfaction and the moisturizing feel were not significantly different among all products. The formula with 7.5% SCS and 5% glycerine is said to be the best one for both cleaning ability and moisturizing effect. These findings suggest that dissolvable shower gel tablets with enhanced skin benefits could offer a promising innovation in the personal care industry.

An Efficient Solver for ASP(Q)

AbstractAnswer Set Programming with Quantifiers ASP(Q) extends Answer Set Programming (ASP) to allow for declarative and modular modeling of problems from the entire polynomial hierarchy. The first implementation of ASP(Q), called QASP, was based on a translation to Quantified Boolean Formulae (QBF) with the aim of exploiting the well-developed and mature QBF-solving technology. However, the implementation of the QBF encoding employed in qasp is very general and might produce formulas that are hard to evaluate for existing QBF solvers because of the large number of symbols and subclauses. In this paper, we present a new implementation that builds on the ideas of QASP and features both a more efficient encoding procedure and new optimized encodings of ASP(Q) programs in QBF. The new encodings produce smaller formulas (in terms of the number of quantifiers, variables, and clauses) and result in a more efficient evaluation process. An algorithm selection strategy automatically combines several QBF-solving back-ends to further increase performance. An experimental analysis, conducted on known benchmarks, shows that the new system outperforms QASP.

Tax curvature

In a Mirrleesian environment, a monopsonist sets hourly wages and individuals choose how many hours to work. Labor market outcomes do not only depend on the level and slope of the income tax function, but also on its curvature. A more concave tax schedule raises the elasticity of labor supply, which boosts wages. Consequently, optimal marginal tax rates for low-skilled workers are declining in income. I derive an optimal tax formula in terms of sufficient statistics that accounts for the impact of tax curvature on labor market outcomes. Numerical simulations suggest that optimal marginal tax rates decline rapidly at low incomes.

Bulk-boundary thermodynamic equivalence: a topology viewpoint

Setting the cosmological constant to be dynamical, we study the bulk and boundary thermodynamics of charged Anti-de Sitter black holes. We develop mass/energy formulas in terms of thermodynamic state functions for the extended thermodynamics, mixed thermodynamics, and boundary conformal field theory thermodynamics. We employ the residue method to study the topological properties of the phase transitions. Our analysis reveals that the bulk and boundary thermodynamics are topologically equivalent for both criticalities and first-order phase transitions in the canonical ensembles, as well as for the Hawking-Page(-like) phase transitions in the grand canonical ensembles. Additionally, those three kinds of phase transitions are shown to be distinguished by their unique topological charges. Our results exemplify the gravity-gauge duality in terms of topology.

Predicting the climate impact of aviation for en-route emissions: the algorithmic climate change function submodel ACCF 1.0 of EMAC 2.53

Abstract. Using climate-optimized flight trajectories is one essential measure to reduce aviation's climate impact. Detailed knowledge of temporal and spatial climate sensitivity for aviation emissions in the atmosphere is required to realize such a climate mitigation measure. The algorithmic Climate Change Functions (aCCFs) represent the basis for such purposes. This paper presents the first version of the Algorithmic Climate Change Function submodel (ACCF 1.0) within the European Centre HAMburg general circulation model (ECHAM) and Modular Earth Submodel System (MESSy) Atmospheric Chemistry (EMAC) model framework. In the ACCF 1.0, we implement a set of aCCFs (version 1.0) to estimate the average temperature response over 20 years (ATR20) resulting from aviation CO2 emissions and non-CO2 impacts, such as NOx emissions (via ozone production and methane destruction), water vapour emissions, and contrail cirrus. While the aCCF concept has been introduced in previous research, here, we publish a consistent set of aCCF formulas in terms of fuel scenario, metric, and efficacy for the first time. In particular, this paper elaborates on contrail aCCF development, which has not been published before. ACCF 1.0 uses the simulated atmospheric conditions at the emission location as input to calculate the ATR20 per unit of fuel burned, per NOx emitted, or per flown kilometre. In this research, we perform quality checks of the ACCF 1.0 outputs in two aspects. Firstly, we compare climatological values calculated by ACCF 1.0 to previous studies. The comparison confirms that in the Northern Hemisphere between 150–300 hPa altitude (flight corridor), the vertical and latitudinal structure of NOx-induced ozone and H2O effects are well represented by the ACCF model output. The NOx-induced methane effects increase towards lower altitudes and higher latitudes, which behaves differently from the existing literature. For contrail cirrus, the climatological pattern of the ACCF model output corresponds with the literature, except that contrail-cirrus aCCF generates values at low altitudes near polar regions, which is caused by the conditions set up for contrail formation. Secondly, we evaluate the reduction of NOx-induced ozone effects through trajectory optimization, employing the tagging chemistry approach (contribution approach to tag species according to their emission categories and to inherit these tags to other species during the subsequent chemical reactions). The simulation results show that climate-optimized trajectories reduce the radiative forcing contribution from aviation NOx-induced ozone compared to cost-optimized trajectories. Finally, we couple the ACCF 1.0 to the air traffic simulation submodel AirTraf version 2.0 and demonstrate the variability of the flight trajectories when the efficacy of individual effects is considered. Based on the 1 d simulation results of a subset of European flights, the total ATR20 of the climate-optimized flights is significantly lower (roughly 50 % less) than that of the cost-optimized flights, with the most considerable contribution from contrail cirrus. The CO2 contribution observed in this study is low compared with the non-CO2 effects, which requires further diagnosis.

Inverse scattering transform for the two‐component Gerdjikov–Ivanov equation with nonzero boundary conditions: Dark‐dark solitons

AbstractThe two‐component Gerdjikov–Ivanov equation with nonzero boundary conditions is studied by the inverse scattering transform. A fundamental set of analytic eigenfunctions is obtained with the aid of the associated adjoint problem. Three symmetry conditions are discussed to curb the scattering data. The behavior of the Jost functions and the scattering matrix at the branch points is discussed. The inverse scattering problem is formulated by a matrix Riemann–Hilbert problem. The trace formula in terms of the scattering data and the so‐called asymptotic phase difference for the potential are obtained. The solitons classification is described in detail. When the discrete eigenvalues lie on the circle, the dark‐dark soliton is obtained for the first time in this work. And the discrete eigenvalues off the circle generate the dark‐bright, bright‐bright, breather‐breather, M(‐type)‐W(‐type) solitons, and their interactions.

Inverse Problem for a Planar Conductivity Inclusion

This paper concerns the inverse problem of determining a planar conductivity inclusion. Our aim is to analytically recover from the generalized polarization tensors (GPTs), which can be obtained from exterior measurements, a homogeneous inclusion with arbitrary constant conductivity. The primary outcome of recovering a homogeneous inclusion is an inversion formula in terms of the GPTs for conformal mapping coefficients associated with the inclusion. To prove the formula, we establish matrix factorizations for the GPTs.

Fertilitopes

We introduce tools from discrete convexity theory and polyhedral geometry into the theory of West’s stack-sorting map s. Associated to each permutation pi is a particular set mathcal V(pi ) of integer compositions that appears in a formula for the fertility of pi , which is defined to be |s^{-1}(pi )|. These compositions also feature prominently in more general formulas involving families of colored binary plane trees called troupes and in a formula that converts from free to classical cumulants in noncommutative probability theory. We show that mathcal V(pi ) is a transversal discrete polymatroid when it is nonempty. We define the fertilitope of pi to be the convex hull of mathcal V(pi ), and we prove a surprisingly simple characterization of fertilitopes as nestohedra arising from full binary plane trees. Using known facts about nestohedra, we provide a procedure for describing the structure of the fertilitope of pi directly from pi using Bousquet-Mélou’s notion of the canonical tree of pi . As a byproduct, we obtain a new combinatorial cumulant conversion formula in terms of generalizations of canonical trees that we call quasicanonical trees. We also apply our results on fertilitopes to study combinatorial properties of the stack-sorting map. In particular, we show that the set of fertility numbers has density 1, and we determine all infertility numbers of size at most 126. Finally, we reformulate the conjecture that sum _{sigma in s^{-1}(pi )}x^{textrm{des}(sigma )+1} is always real-rooted in terms of nestohedra, and we propose natural ways in which this new version of the conjecture could be extended.

Trigonometric Sums and Riemann Zeta Function

ABSTRACTSeveral trigonometric sums are examined by means of partial fraction decompositions. Their generating functions will be evaluated in closed forms and the related asymptotic formulae in terms of Riemann zeta function will be established.

Reduction formula for cross-spectral purity ofnonstationary light fields.

We examine cross-spectral purity of random, nonstationary (pulsed), scalar light fields with arbitrary spectral bandwidth. In particular, we derive a reduction formula in terms of time-integrated coherence functions, which ensures cross-spectral purity of interfering fields having identical normalized spectra. We further introduce fields that are cross-spectrally pure in either a global or local sense. Our analysis is based on an ideal field superposition realizable with all-reflective wavefront-shearing interferometers. Such devices avoid certain problems related to Young's interferometer, which is the framework customarily employed in assessing cross-spectral purity. We show that any partially coherent beam can be transformed into a locally cross-spectrally pure beam whose cross-spectral density is specular. On the other hand, lack of space-frequency (and space-time) coupling ensures cross-spectral purity in the global sense, i.e.,across an entire transverse plane, regardless of the spectral bandwidth or the temporalshape of the pulses.