Analogue Of Formula Research Articles

EXCEPTIONAL SIMPLE REAL LIE ALGEBRAS AND VIA CONTACTIFICATIONS

Abstract In Cartan’s PhD thesis, there is a formula defining a certain rank 8 vector distribution in dimension 15, whose algebra of authomorphism is the split real form of the simple exceptional complex Lie algebra $\mathfrak {f}_4$ . Cartan’s formula is written in the standard Cartesian coordinates in $\mathbb {R}^{15}$ . In the present paper, we explain how to find analogous formulae for the flat models of any bracket generating distribution $\mathcal D$ whose symbol algebra $\mathfrak {n}({\mathcal D})$ is constant and 2-step graded, $\mathfrak {n}({\mathcal D})=\mathfrak {n}_{-2}\oplus \mathfrak {n}_{-1}$ . The formula is given in terms of a solution to a certain system of linear algebraic equations determined by two representations $(\rho ,\mathfrak {n}_{-1})$ and $(\tau ,\mathfrak {n}_{-2})$ of a Lie algebra $\mathfrak {n}_{00}$ contained in the $0$ th order Tanaka prolongation $\mathfrak {n}_0$ of $\mathfrak {n}({\mathcal D})$ . Numerous examples are provided, with particular emphasis put on the distributions with symmetries being real forms of simple exceptional Lie algebras $\mathfrak {f}_4$ and $\mathfrak {e}_6$ .

A Study on the Progress of Astragalus-Coptis Chinensis Medicinal Pair in the Treatment of Diabetes Mellitus

According to the relevant studies in recent years, it was found that astragalus - Coptis Chinensis drug pair and its analogous formula have a broad prospect for the treatment of type 2 diabetes mellitus (T2DM). Pharmacological studies have shown that Coptis Chinensis possesses the pharmacological activities of antimicrobial, anti-inflammatory, anti-tumor, lipid regulating, hypoglycemic, anti-arrhythmic, and antioxidant activities[1], and Astragalus possesses the effects of enhancing the function of the immune system, antimyocardial ischemia, bi-directional regulation of blood pressure, anti-tumor, anti-aging, hypoglycemia, and protection of metabolic tissues (liver and kidneys, etc.) [2] [3], the use of Astragalus and Coptis Chinensis pairs has demonstrated significant multiple effects. Specifically, this combination not only effectively reduces blood glucose levels, but also significantly enhances the body's sensitivity to insulin, thus helping to improve the phenomenon of insulin resistance and further enhance the overall health status of patients [1][2][3]. This paper analyzes and researches the research progress of astragalus-Coptis Chinensis medicinal pair and its analogous formulae in the treatment of type 2 diabetes mellitus and the improvement of its complications through the collation and summarization of the literature, in order to lay a certain theoretical research foundation for the clinical diagnosis and treatment of type 2 diabetes mellitus by astragalus-Coptis Chinensis medicinal pair and its related formulae, prevention of diabetes mellitus complications, and rational use of medicines in diabetes mellitus clinical treatment.

Scattering problems for the wave equation in 1D: D’Alembert-type representations and a reconstruction method

We derive the extension of the classical d’Alembert formula for the wave equation, which provides the analytical solution for the direct scattering problem for a medium with constant refractive index. Analogous formulae exist already in the literature, but in the current work this is derived in a natural way for general incident field, by employing results obtained via the Fokas method. This methodology is further extended to a medium with piecewise constant refractive index, providing the apparatus for the solution of the associated inverse scattering problem. Hence, we provide an exact reconstruction method which is also valid for phaseless data.

Topological impact of nanopore electrodes on the structure of the electrical double layer and the differential capacitance

The electrical double layer for three different topologies of nanopore electrodes is studied, i.e., the interior and exterior electrical double layers of planar, cylindrical and spherical nanopores immersed into a point-ions electrolyte, and not connected to a power source, are analytically attained through the linearized Poisson-Boltzmann equation. Thus, analytical formulas for the mean electrostatic potential, electrolyte's reduced concentration, and electrical field profiles, are exhibited. Their corresponding analytical expressions for their differential capacitances are presented. All the nanopores are treated as permeable, so the electrolyte outside and inside the electrodes are at the same chemical potential. Analogous analytical formulas for solid nano-electrodes are obtained as a corollary of those for nanopores. In particular, their analytical expressions for the differential capacitance here derived are shown to be consistent with the capacitive compactness proposed in the past by one of us. Numerical results of all of the above functions are analyzed as a function of the nanopores geometrical parameters and the electrolyte's temperature and molar concentration. It is found that the spherical topology, at lower temperatures, has the higher differential capacitance. It is demonstrated that for the three nanopore topologies here considered their capacitances reduce to that of a single planar electrode, in the limit of infinitely wide nanopores. The electrical double layer and mean electrostatic potential of the three topologies are in qualitatively agreement with those from the non-linearized Poisson-Boltzmann, hypernetted chain/mean-spherical approximation (HNC/MSA) equations and computer simulations results presented in the past, within the low mean electrostatic potential assumption.

The Riemann–Roch theorem for the Adams operations

We prove the classical Riemann–Roch theorems for the Adams operations ψj on K-theory: a statement with coefficients on Z[j−1], that holds for arbitrary projective morphisms, as well as another statement with integral coefficients, that is valid for closed immersions. In presence of rational coefficients, we also analyze the relation between the corresponding Riemann–Roch formula for one Adams operation and the analogous formula for the Chern character. To do so, we complete the elementary exposition of the work of Panin–Smirnov that was initiated by the first author in a previous paper. Their notion of oriented cohomology theory on algebraic varieties allows to use classical arguments to prove general and neat statements, which imply all the aforementioned results as particular cases.

Hurwitz zeta functions and Ramanujan's identity for odd zeta values

Inspired by a famous formula of Ramanujan for odd zeta values, we prove an analogous formula involving the Hurwitz zeta function. We introduce a new integral kernel related to the Hurwitz zeta function, generalizing that associated to Ramanujan's identity. We also derive several infinite families of identities analogous to Ramanujan's formula.

Synthetic geometry in hyperbolic simplices

Let $\tau$ be an $n$-simplex and let $g$ be a metric on $\tau$ with constant curvature $\kappa$. The lengths that $g$ assigns to the edges of $\tau$, along with the value of $\kappa$, uniquely determine all of the geometry of $(\tau, g)$. In this paper we focus on hyperbolic simplices ($\kappa = -1$) and develop geometric formulas which rely only on the edge lengths of $\tau$. Our main results are distance and projection formulas in hyperbolic simplices, as well as a projection formula in Euclidean simplices. We also provide analogous formulas in simplices with arbitrary constant curvature $\kappa$.

Explicit formula for the \(n\)-th derivative of a quotient

Leibniz’s rule for the \(n\)-th derivative of a product is a very well-known and extremely useful formula. This article introduces an analogous explicit formula for the \(n\)-th derivative of a quotient of two functions. Later, we use this formula to derive new partition identities and to develop expressions for some particular \(n\)-th derivatives.

Unipotent ideals for spin and exceptional groups

In the monograph [27], we define the notion of a unipotent representation of a complex reductive group. The representations we define include, as a proper subset, all special unipotent representations in the sense of [4] and form the (conjectural) building blocks of the unitary dual. In [27] we provide combinatorial formulas for the infinitesimal characters of all unipotent representations of linear classical groups. In this paper, we establish analogous formulas for spin and exceptional groups, thus completing the determination of the infinitesimal characters of all unipotent ideals. Using these formulas, we prove an old conjecture of Vogan: all unipotent ideals are maximal. For G a real reductive Lie group (not necessarily complex), we introduce the notion of a unipotent representation attached to a rigid nilpotent orbit (in the complexified Lie algebra of G). Like their complex group counterparts, these representations form the (conjectural) building blocks of the unitary dual. Using the atlas software (and the work of [2]) we show that if G is a real form of a simple group of exceptional type, all such representations are unitary.

Product formulas for hypergeometric functions over finite fields

Many product formulas are known classically for generalized hypergeometric functions over the complex numbers. In this paper, we establish some analogous formulas for generalized hypergeometric functions over finite fields.

Italian Diasporic Studies: The Then and Now

Italian Diasporic Studies: The Then and Now

The relative f-invariant and non-uniform random sofic approximations

AbstractThe f-invariant is an isomorphism invariant of free-group measure-preserving actions introduced by Lewis Bowen, who first used it to show that two finite-entropy Bernoulli shifts over a finitely generated free group can be isomorphic only if their base measures have the same Shannon entropy. Bowen also showed that the f-invariant is a variant of sofic entropy; in particular, it is the exponential growth rate of the expected number of good models over a uniform random homomorphism. In this paper we present an analogous formula for the relative f-invariant and use it to prove a formula for the exponential growth rate of the expected number of good models over a random sofic approximation which is a type of stochastic block model.

The Canny–Emiris Conjecture for the Sparse Resultant

We present a product formula for the initial parts of the sparse resultant associated with an arbitrary family of supports, generalizing a previous result by Sturmfels. This allows to compute the homogeneities and degrees of this sparse resultant, and its evaluation at systems of Laurent polynomials with smaller supports. We obtain an analogous product formula for some of the initial parts of the principal minors of the Sylvester-type square matrix associated with a mixed subdivision of a polytope. Applying these results, we prove that under suitable hypothesis, the sparse resultant can be computed as the quotient of the determinant of such a square matrix by one of its principal minors. This generalizes the classical Macaulay formula for the homogeneous resultant and confirms a conjecture of Canny and Emiris.

Quantitative determination of multi-class bioactive constituents for quality control of Yiqi Jiangzhi Granules

ObjectiveTo establish a reliable and sensitive method for evaluating quality of Yiqi Jiangzhi Granules (YQJZG). MethodsUltra performance liquid chromatography electrospray ionization tandem mass spectrometry (UPLC-ESI-MS/MS) was employed for simultaneous determination of eight marker components. Separation was performed on an AQUITY UPLC® HSS T3 column, the mobile phase consisted of acetonitrile as the organic phase and 0.1% (volume percentage) formic acid as the aqueous. Eight marker components, ginsenoside Rg1 (GRg1), ginsenoside Re (GRe), ginsenoside Rb1 (Gb1), typhaneoside (TEO), isorhamnetin-3-O-neohespeidoside (IN), hesperidin (HPD), aurantio-obtusin-6-O-β-D-glucoside (AG) and curcumin (CCM), were detected by multiple reaction monitoring (MRM) mode. The Chinese Pharmacopoeia (2020 edition) was regarded as the guidance document for this method validation. ResultsThe method showed good linearity (R2 ≥ 0.9990). The relative standard deviation (RSD) values for the instrument precision, intermediate precision and repeatability were less than 2.91%, 2.88%, and 3.54%, respectively. The average recovery varied from 91.08% to 103.89%, with RSD below 3.81%. Sample solutions were found to be stable within 24 h at 4 °C (RSD < 2.85%). Eight marker components were successfully determined from three batches of YQJZG. ConclusionThe proposed UPLC-ESI-MS/MS method was found to be simple, fast and sensitive, and can be used for the routine quality assessment of YQJZG. Simultaneously, this method may provide a new and powerful tool of quality control for other traditional Chinese medicine analogous formulae.

Odd dimensional analogue of the Euler characteristic

When compact manifolds X and Y are both even dimensional, their Euler characteristics obey the Künneth formula χ(X × Y) = χ(X)χ(Y). In terms of the Betti numbers bp(X), χ(X) = Σp(−1)pbp(X), implying that χ(X) = 0 when X is odd dimensional. We seek a linear combination of Betti numbers, called ρ, that obeys an analogous formula ρ(X × Y) = χ(X)ρ(Y) when Y is odd dimensional. The unique solution is ρ(Y) = − Σp(−1)ppbp(Y). Physical applications include: (1) ρ → (−1)mρ under a generalized mirror map in d = 2m + 1 dimensions, in analogy with χ → (−1)mχ in d = 2m; (2) ρ appears naturally in compactifications of M-theory. For example, the 4-dimensional Weyl anomaly for M-theory on X4× Y7 is given by χ(X4)ρ(Y7) = ρ(X4× Y7) and hence vanishes when Y7 is self-mirror. Since, in particular, ρ(Y × S1) = χ(Y), this is consistent with the corresponding anomaly for Type IIA on X4× Y6, given by χ(X4)χ(Y6) = χ(X4× Y6), which vanishes when Y6 is self-mirror; (3) In the partition function of p-form gauge fields, ρ appears in odd dimensions as χ does in even.

On the wavefront sets associated with theta representations

We study a conjectural formula for the maximal elements in the wavefront set associated with a theta representation of a covering group over p-adic fields. In particular, it is shown that the formula agrees with the existing work in the literature for various families of groups. We also recapitulate the results of an analogous formula in the archimedean case, which motivated the conjectural formula in the p-adic setting.

Optimal error estimates for Legendre expansions of singular functions with fractional derivatives of bounded variation

We present a new fractional Taylor formula for singular functions whose Caputo fractional derivatives are of bounded variation. It bridges and ``interpolates" the usual Taylor formulas with two consecutive integer orders. This enables us to obtain an analogous formula for the Legendre expansion coefficient of this type of singular functions, and further derive the optimal (weighted) $L^\infty$-estimates and $L^2$-estimates of the Legendre polynomial approximations. This set of results can enrich the existing theory for $p$ and $hp$ methods for singular problems, and answer some open questions posed in some recent literature.

Poisson Quantum Information

By taking a Poisson limit for a sequence of rare quantum objects, I derive simple formulas for the Uhlmann fidelity, the quantum Chernoff quantity, the relative entropy, and the Helstrom information. I also present analogous formulas in classical information theory for a Poisson model. An operator called the intensity operator emerges as the central quantity in the formalism to describe Poisson states. It behaves like a density operator but is unnormalized. The formulas in terms of the intensity operators not only resemble the general formulas in terms of the density operators, but also coincide with some existing definitions of divergences between unnormalized positive-semidefinite matrices. Furthermore, I show that the effects of certain channels on Poisson states can be described by simple maps for the intensity operators.

The relative Riemann–Hurwitz formula

For any nonconstant such that the numerator H(x, y) of is irreducible, we compute the genus of the normalization of the curve We also prove an analogous formula in arbitrary characteristic when f and g have no common wildly ramified branch points, and generalize to (possibly reducible) fiber products of nonconstant morphisms of curves and

Lifting the dual immaculate functions

We introduce a reverse variant of the dual immaculate quasisymmetric functions, mirroring the dichotomy between quasisymmetric Schur functions and Young quasisymmetric Schur functions, and establish a lift of this basis to the polynomial ring. We show that taking stable limits of these reverse dual immaculate slide polynomials produces the reverse dual immaculate quasisymmetric functions, and we establish positive formulas for expansions of these polynomials into the fundamental slide and quasi-key bases for polynomials. These formulas mirror connections between dual immaculate quasisymmetric functions, fundamental quasisymmetric functions, and Young quasisymmetric Schur functions, extending these connections from the ring of quasisymmetric functions to the full polynomial ring. We also use this lift to provide a lift of the dual immaculate quasisymmetric functions, and we establish analogous expansion formulas. We moreover use the reverse dual immaculate quasisymmetric functions to establish a connection between the dual immaculate quasisymmetric functions and the Demazure atom basis for polynomials.