Analogue Of Formula Research Articles

Tail asymptotics of the Brownian signature

The signature of a path \gamma is a sequence whose n-th term is the order-n iterated integrals of \gamma. It arises from solving multidimensional linear differential equations driven by \gamma. We are interested in relating the path properties of \gamma with its signature. If \gamma is C1, then an elegant formula of Hambly and Lyons relates the length of \gamma to the tail asymptotics of the signature. We show an analogous formula for the multidimensional Brownian motion,with the quadratic variation playing a similar role to the length. In the proof, we study the hyperbolic development of Brownian motion and also obtain a new subadditive estimate for the asymptotic of signature, which may be of independent interest. As a corollary, we strengthen the existing uniqueness results for the signatures of Brownian motion.

The Varchenko determinant for oriented matroids

We generalize the Varchenko matrix of a hyperplane arrangement to oriented matroids. We show that the celebrated determinant formula for the Varchenko matrix, first proved by Varchenko, generalizes to oriented matroids. It follows that the determinant only depends on the matroid underlying the oriented matroid and analogous formulas hold for closed supertopes in oriented matroids. We follow a proof strategy for the original Varchenko formula first suggested by Denham and Hanlon. Besides several technical lemmas this strategy also requires a topological result on supertopes which is of independent interest. We show that a supertope considered as a subposet of the tope poset has a contractible order complex.

Heights of hypersurfaces in toric varieties

For a cycle of codimension 1 in a toric variety, its degree with respect to a nef toric divisor can be understood in terms of the mixed volume of the polytopes associated to the divisor and to the cycle. We prove here that an analogous combinatorial formula holds in the arithmetic setting: the global height of a 1-codimensional cycle with respect to a toric divisor equipped with a semipositive toric metric can be expressed in terms of mixed integrals of the $v$-adic roof functions associated to the metric and the Legendre-Fenchel dual of the $v$-adic Ronkin function of the Laurent polynomial of the cycle.

A note on discrete mixed volume and Hodge–Deligne numbers

Generalizing the famous Bernstein–Kushnirenko theorem, Khovanskĭi proved in 1978 a combinatorial formula for the arithmetic genus of the compactification of a generic complete intersection associated to a family of lattice polytopes. Recently, an analogous combinatorial formula, called the discrete mixed volume, was introduced by Bihan and shown to be nonnegative. By making a footnote of Khovanskĭi in his paper explicit, we interpret this invariant as the (motivic) arithmetic genus of the non-compact generic complete intersection associated to the family of lattice polytopes.

Lattice Points and Simultaneous Core Partitions

We apply lattice point techniques to the study of simultaneous core partitions. Our central observation is that for $a$ and $b$ relatively prime, the abacus construction identifies the set of simultaneous $(a,b)$-core partitions with lattice points in a rational simplex. We apply this result in two main ways: using Ehrhart theory, we reprove Anderson's theorem that there are $(a+b-1)!/a!b!$ simultaneous $(a,b)$-cores; and using Euler-Maclaurin theory we prove Armstrong's conjecture that the average size of an $(a,b)$-core is $(a+b+1)(a-1)(b-1)/24$. Our methods also give new derivations of analogous formulas for the number and average size of self-conjugate $(a,b)$-cores.

Prediction of papillomavirus infection by after therapy of women and men from family couples

An important role in carcinogenesis in both women and men is played by the immune system. With a high established classifi cation accuracy by the method of discriminant analysis was revealed that in women from married couples with HPV VKR the fi nal function allows to determine the prognosis of the eff ectiveness of the elimination of the virus after the treatment as a result of substituting the value of the functional reserve of neutrophils of cervical mucus in the function, while the fi nal result is compared with the values of centroids 1.163 and –0.263, i. e. at a higher value of the last HPV, stimulation is not detected (p <0.002). In men from married couples, when substituting in the analogous formula the functions of the absolute neutrophil viability, the functional reserve of the neutrophils of the ejaculate, and the viral load, the centroid values fi t into the values 11,700 and –1,003 and the HPV of stimulation is not detected at a higher value of the fi nal result (p <0.001).

The Chevalley-Weil Formula for Orbifold Curves

In the 1930s Chevalley and Weil gave a formula for decomposing the canonical representation on the space of differential forms of the Galois group of a ramified Galois cover of Riemann surfaces. In this article we prove an analogous Chevalley-Weil formula for ramified Galois covers of orbifold curves. We then specialize the formula to the case when the base orbifold curve is the (reduced) modular orbifold. As an application of this latter formula we decompose the canonical representations of modular curves of full, prime level and of Fermat curves of arbitrary exponent.

Velocity formulae between entropy and hitting time for Markov chains

In the absence of acceleration, the velocity formula gives “distance travelled equals speed multiplied by time”. For a broad class of Markov chains such as circulant Markov chains or random walk on complete graphs, we prove a probabilistic analogue of the velocity formula between entropy and hitting time, where distance is the entropy of the Markov trajectories from state i to state j in the sense of Ekroot and Cover (1993), speed is the classical entropy rate of the chain, and the time variable is the expected hitting time between i and j. This motivates us to define new entropic counterparts of various hitting time parameters such as average hitting time or commute time, and prove analogous velocity formulae and estimates between these quantities.

Bijections for planar maps with boundaries

We present bijections for planar maps with boundaries. In particular, we obtain bijections for triangulations and quadrangulations of the sphere with boundaries of prescribed lengths. For triangulations we recover the beautiful factorized formula obtained by Krikun using a (technically involved) generating function approach. The analogous formula for quadrangulations is new. We also obtain a far-reaching generalization for other face-degrees. In fact, all the known enumerative formulas for maps with boundaries are proved bijectively in the present article (and several new formulas are obtained).Our method is to show that maps with boundaries can be endowed with certain “canonical” orientations, making them amenable to the master bijection approach we developed in previous articles. As an application of our enumerative formulas, we note that they provide an exact solution of the dimer model on rooted triangulations and quadrangulations.

Novel Spreadsheet Direct Method for Optimal Control Problems

We devise a simple yet highly effective technique for solving general optimal control problems in Excel spreadsheets. The technique exploits Excel’s native nonlinear programming (NLP) Solver Command, in conjunction with two calculus worksheet functions, namely, an initial value problem solver and a discrete data integrator, in a direct solution paradigm adapted to the spreadsheet. The technique is tested on several highly nonlinear constrained multivariable control problems with remarkable results in terms of reliability, consistency with pseudo-spectral reported answers, and computing times in the order of seconds. The technique requires no more than defining a few analogous formulas to the problem mathematical equations using basic spreadsheet operations, and no programming skills are needed. It introduces an alternative, simpler tool for solving optimal control problems in social and natural science disciplines.

Cilia Function as a Target for Respiratory Diseases and TCM

Cilia in the lungs have a primary role in many respiratory diseases. Disruption of ciliated epithelial cell functions results in excess mucus in the airways, termed watery phlegm according to the Traditional Chinese Medicine Theory. Traditional Chinese Medicine prescription to treat respiratory diseases caused by watery phlegm attenuate cilia shortening, aid the clearance of ciliated epithelial cells and reduce excess mucus in the airways. Further studies are required to determine the role and mechanism of Traditional Chinese Medicine decoction and its analogous formulae and this might help develop new drugs for the treatment of respiratory diseases.

Short proofs in extrema of spectrally one sided Lévy processes

We provide short and simple proofs of the continuous time ballot theorem for processes with cyclically interchangeable increments and Kendall’s identity for spectrally positive Levy processes. We obtain the later result as a direct consequence of the former. The ballot theorem is extended to processes having possible negative jumps. Then we prove through straightforward arguments based on the law of bridges and Kendall’s identity, Theorem 2.4 in [20] which gives an expression for the law of the supremum of spectrally positive Levy processes. An analogous formula is obtained for the supremum of spectrally negative Levy processes.

ГРАВІТАЦІЙНЕ МІКРОЛІНЗУВАННЯ ЕЛІПТИЧНОГО ДЖЕРЕЛА ПОБЛИЗУ КАУСТИКИ-СКЛАДКИ

We consider the amplification factor for the luminosity of an extended source near the fold caustic of the gravitational lens. It is assumed that the source has elliptical shape, and the brightness distribution along the radial directions is Gaussian. During the microlensing event the total brightness of all microimages is observed, which changes when the source moves relative to the caustic. The main contribution to the variable component is given by the so-called critical images that arise/disappear at the intersection of the caustic by the source. In the present paper we obtained an analogous formula for elliptical Gaussian source. The formula involves a dependence on the coordinates of the source centre, its geometric dimensions, and its orientation relative to the caustic. We show that in the linear caustic approximation the amplification of the circular and elliptical sources is described by the same (rescaled) formula. However, in the next approximations the differences are significant. We compare analytical calculations of the amplification curves for different orientations of an elliptical source and for a circular source with the same luminosity for the model example.

Formulation, Evaluation and Antioxidant activity of Caffeine Fast Melt Tablets

The aim of this study was to investigate the effect of various Prosolv® co-processed excipients in the manufacture of different grades of fast melt tablets (FMTs). Caffeine was chosen as a model drug. Caffeine FMTs were prepared by direct compression using Prosolv® based excipients;Prosolv® SMCC 50, Prosolv® SMCC 90,Prosolv® SMCC HD90, Prosolv® ODT, Prosolv®Easytab Nutra,and Prosolv®Easytab SP. Caffeine FMTs were evaluated for weight, thickness, hardness, friability, wetting time (WT), in-vitro disintegration time, and in-vitro dissolutionstudies. The selected Caffeine FMT was compared with oral commercial tablets (Stay Awake Caffeine Alertness Aid®, Equate, USA) for their antioxidant activities and their impact on reproductive organs of male rats. All Caffeine FMTs were within the pharmacopeial limits for weight, and content. All formulae had acceptable friability percentage (<1%) except Prosolv® SMCC HD based formula. Prosolv® Easytab Nutra based formula showed superiority over Prosolv® Easytab SP and Prosolv® ODT regarding WT and DT. Furthermore, Prosolv® Easytab Nutra based formula gave comparablefunctionality to its analogous formulae based Prosolv® SMCC 50 and Prosolv® SMCC 90 with superior advantages regarding simplicity of manufacture and the potential minimizing of undesirable issuesas segregation by reducing the number of steps during production of tablets. The data concluded that the Prosolv® co-processed excipients are promising carriers for Caffeine FMT designed formodulating male reproductive hormone reproductive values and oxidative stress with having greaterbenefits of dosage form which, in turn, increase compliance.

Some formulas for the restricted r-Lah numbers

The r-Lah numbers, which we denote here by `(r)(n, k), enumerate partitions of an (n+r)-element set into k+r contents-ordered blocks in which the smallest r elements belong to distinct blocks. In this paper, we consider a restricted version `(r) m (n, k) of the r-Lah numbers in which block cardinalities are at most m. We establish several combinatorial identities for `(r) m (n, k) and obtain as limiting cases for large m analogous formulas for `(r)(n, k). Some of these formulas correspond to previously established results for `(r)(n, k), while others are apparently new also in the r-Lah case. Some generating function formulas are derived as a consequence and we conclude by considering a polynomial generalization of `(r) m (n, k) which arises as a joint distribution for two statistics defined on restricted r-Lah distributions. Keywords: restricted Lah numbers, polynomial generalization, r-Lah numbers, combinatorial identities MSC: 11B73, 05A19, 05A18

Research progress on chemical constituents, pharmacological mechanism and clinical application of Guizhi decoction

Guizhi Decoction was one of the most commonly used traditional Chinese Medicine which possesses the effects of "jie-ji-fa-biao, regulating Ying and Wei". It was mainly used to treat mind-cold due to exogenous evils such as fever, headache, sweating, hate the wind, et al. Modern studies indicated that the chemical constituents of Guizhi decoction mainly include phenylpropanoid, monoterpenes, organic acids, flavonoids, triterpenoid saponins and so on. Pharmacological experimental studies had shown that Guizhi decoction could play a big role in dual-directional regulation on sweat gland, body temperature, immune function, gastrointestinal peristalsis, and blood pressure, and could also play the role of anti-inflammatory, antibacterial, antiviral, anti-allergic, analgesic, hypoglycemic, and cardiovascular protection. Many diseases such as internal, external, gynecological and pediatric diseases were treated in the clinical by using Guizhi decoction and its analogous formulae involving circulatory, immune, urinary, reproductive, endocrine, digestive, nervous and other systems. This article reviews the latest research progress of Guizhi decoction from three aspects: chemical constituents, pharmacological mechanism and clinical application. It will provide reference for further research and development of Guizhi decoction.

Preparation and Characterization of Dual-Template Molecularly Imprinted Membrane with High Flux Based on Blending the Inorganic Nanoparticles

In this study, the dual-template molecularly imprinted composite membrane (D-MIM) was synthesized using ferulic acid and cinnamic acid as the mixed templates, α-methacrylic acid as functional monomer, ethylene glycol dimethacrylate as crosslinker, 2,2-azobisisobutyronitrile as photoinitiator inducing polymerization reaction, furthermore blended nanoparticles so as to improve mechanical strength and pure water flux of membrane. This paper has researched that the effect of variety of nanoparticles and its quantity, the quantity of photoinitiator, total template–functional monomer (T/F) ratio on performance of prepared membrane. The study has founded optimum parameters of synthesis: nanoparticles 18 nm TiO2 of 1 wt%, photoinitiator of 0.3 g, T/F ratio is 1:3. Under the above conditions, D-MIM would present a high separation efficiency of template molecules (ferulic acid and cinnamic acid) and the analogues (caffeic acid and syringic acid) with the permeation rate of about 100%. However, the permeation rate of glucose which has an analogous formula weight and extremely different structure is about 60%. From the foregoing, it showed that the D-MIM has specific selectively to the template molecules and its analogues, but the substances which have obvious distinctions in structure would present low permeation rate. It can be seen that D-MIM can identify the same kind of substance with specificity. Therefore, it can be used in the enrichment and separation of some active ingredients of traditional Chinese medicine, the detection of toxic substances in food and so on. Finally, the scanning electron microscope and FT-IR spectrometer has been used to visualize the structure morphology of the prepared membrane and to quantity the surface characteristics.

Trace formulas for perturbations of operators with Hilbert-Schmidt resolvents

Trace formulas for self-adjoint perturbations \(V\) of self-adjoint operators \(H\) such that \(V\) is in Schatten class were obtained in the works of L.S. Koplienko, M.G. Krein, and the joint paper of D. Potapov, A. Skripka and F. Sukochev. In this article, we obtain an analogous trace formula under the assumptions that the perturbation \(V\) is bounded and the resolvent of \(H\) belongs to Hilbert-Schmidt class.

Master formulas for the dressed scalar propagator in a constant field

The worldline formalism has previously been used for deriving compact master formulas for the one-loop N-photon amplitudes in both scalar and spinor QED, and in the vacuum as well as in a constant external field. For scalar QED, there is also an analogous master formula for the propagator dressed with N photons in the vacuum. Here, we extend this master formula to include a constant field. The two-photon case is worked out explicitly, yielding an integral representation for the Compton scattering cross section in the field suitable for numerical integration in the full range of electric and magnetic field strengths.

Conformal Nets II: Conformal Blocks

Conformal nets provide a mathematical formalism for conformal field theory. Associated to a conformal net with finite index, we give a construction of the ‘bundle of conformal blocks’, a representation of the mapping class groupoid of closed topological surfaces into the category of finite-dimensional projective Hilbert spaces. We also construct infinite-dimensional spaces of conformal blocks for topological surfaces with smooth boundary. We prove that the conformal blocks satisfy a factorization formula for gluing surfaces along circles, and an analogous formula for gluing surfaces along intervals. We use this interval factorization property to give a new proof of the modularity of the category of representations of a conformal net.