Counting Formula Research Articles

Platelet to Lymphocyte Ratıo in Respıratory Syncytıal Vırus Infectıon

Introduction: We aim to examine the role of Platelet to lymphocyte ratio (PLR) and Platelet to Lymphocyte and monocyte ratio (PLT/LY+MO) values in Respiratory Syncytial Virus (RSV) infection. Methodology: Complete blood counts and RSV Antigen test results of 76 patients, between the age of 0-12 were investigated retrospectively. PLR values are calculated using Platelet Count / Lymphocyte Count formula, whereas PLMR values are calculated using Platelet Count / Lymphocyte Count + Monocyte Count. Results: Out of 76, 32 patients were diagnosed as RSV infected. Lymphocyte and monocyte percentage and platelet count were significantly higher in RSV positive group. Monocyte percentage in lower respiratory tract RSV infection was significantly lower compared to upper respiratory tract RSV infection. Conclusions: Our study revealed that these blood parameters may be an important clue for the clinician for RSV infection and also play a role as a guide before advanced techniques.

Sharp Bounds for the Number of Regions of Maxout Networks and Vertices of Minkowski Sums

We present results on the number of linear regions of the functions that can be represented by artificial feedforward neural networks with maxout units. A rank- maxout unit is a function computing the maximum of linear functions. For networks with a single layer of maxout units, the linear regions correspond to the upper vertices of a Minkowski sum of polytopes. We obtain face counting formulas in terms of the intersection posets of tropical hypersurfaces or the number of upper faces of partial Minkowski sums, along with explicit sharp upper bounds for the number of regions for any input dimension, any number of units, and any ranks, in cases with and without biases. Based on these results we also obtain asymptotically sharp upper bounds for networks with multiple layers.

On atomic density of numerical semigroup algebras

A numerical semigroup S is a cofinite, additively closed subset of the nonnegative integers that contains 0. We initiate the study of atomic density, an asymptotic measure of the proportion of irreducible elements in a given ring or semigroup, for semigroup algebras. It is known that the atomic density of the polynomial ring 𝔽q[x] is zero for any finite field 𝔽q; we prove that the numerical semigroup algebra 𝔽q[S] also has atomic density zero for any numerical semigroup S. We also examine the particular algebra 𝔽2[x2,x3] in more detail, providing a bound on the rate of convergence of the atomic density as well as a counting formula for irreducible polynomials using Möbius inversion, comparable to the formula for irreducible polynomials over a finite field 𝔽q.

Congruence for Lattice Path Models with Filter Restrictions and Long Steps

We derive a path counting formula for a two-dimensional lattice path model with filter restrictions in the presence of long steps, source and target points of which are situated near the filters. This solves the problem of finding an explicit formula for multiplicities of modules in tensor product decomposition of T(1)⊗N for Uq(sl2) with divided powers, where q is a root of unity. Combinatorial treatment of this problem calls for the definition of congruence of regions in lattice path models, properties of which are explored in this paper.

Schubert Products for Permutations with Separated Descents

Abstract We say that two permutations $\pi $ and $\rho $ have separated descents at position $k$ if $\pi $ has no descents before position $k$ and $\rho $ has no descents after position $k$. We give a counting formula, in terms of reduced word tableaux, for computing the structure constants of products of Schubert polynomials indexed by permutations with separated descents, and recognize that these structure constants are certain Edelman–Greene coefficients. Our approach uses generalizations of Schützenberger’s jeu de taquin algorithm and the Edelman–Greene correspondence via bumpless pipe dreams.

Polynomial functions over dual numbers of several variables

Let [Formula: see text] be a positive integer. For a commutative ring [Formula: see text], the ring of dual numbers of [Formula: see text] variables over [Formula: see text] is the quotient ring [Formula: see text], where [Formula: see text] is the ideal generated by the set [Formula: see text]. This ring can be viewed as [Formula: see text] with [Formula: see text], where [Formula: see text] for [Formula: see text]. We investigate the polynomial functions of [Formula: see text] whenever [Formula: see text] is a finite commutative ring. We derive counting formulas for the number of polynomial functions and polynomial permutations on [Formula: see text] depending on the order of the pointwise stabilizer of the subring of constants [Formula: see text] in the group of polynomial permutations of [Formula: see text]. Further, we show that the stabilizer group of [Formula: see text] is independent of the number of variables [Formula: see text]. Moreover, we prove that a function [Formula: see text] on [Formula: see text] is a polynomial function if and only if a system of linear equations on [Formula: see text] that depends on [Formula: see text] has a solution.

Designing paths for box shapes using a Rubik’s Snake

A Rubik’s Snake is a toy that was invented over 40 years ago together with the more famous Rubik’s Cube. It starts with a straight ruler and could be twisted into many interesting shapes. One important type of shapes is a box shape. Many complicated shapes can be constructed as a combination of box shapes. Therefore it is a fundamental problem to study the box shape. We constructed a path for a general box shape and derived a counting formula. Some other properties and application were also discussed.

Riemann-type functional equations: Julia line and counting formulae

We study Riemann-type functional equations with respect to value-distribution theory and derive implications for their solutions. In particular, for a fixed complex number a≠0 and a function from the Selberg class L, we prove a Riemann–von Mangoldt formula for the number of a-points of the Δ-factor of the functional equation of L and an analog of Landau’s formula over these points. From the last formula we derive that the ordinates of these a-points are uniformly distributed modulo one. Lastly, we show the existence of the mean-value of the values of L(s) taken at these points.

Derivation of a statistical model for classical systems obeying the fractional exclusion principle.

The violation of the Pauli principle has been surmised in several models of the fractional exclusion statistics and successfully applied to several quantum systems. In this paper, a classical alternative of the exclusion statistics is studied using the maximum entropy methods. The difference between the Bose-Einstein statistics and the Maxwell-Boltzmann statistics is understood in terms of a separable quantity, namely the degree of indistinguishability. Starting from the usual Maxwell-Boltzmann microstate counting formula, a special restriction related to the degree of indistinguishability is incorporated using Lagrange multipliers to derive the probability distribution function at equilibrium under NVE conditions. It is found that the resulting probability distribution function generates real positive values within the permissible range of parameters. For a dilute system, the probability distribution function is intermediate between the Fermi-Dirac and Bose-Einstein statistics and follows the exclusion principle. Properties of various variables of this novel statistical model are studied and possible application to classical thermodynamics is discussed.

Degree counting formula for non-symmetric Toda systems of rank two

In this paper, we consider B2 and G2 Toda systems on a compact Riemann surface M. We investigate the relation between the topological property of M and the Leray-Schauder degree of the Toda systems by computing the degree jump caused by multi-bubbling phenomena of the Toda system with non-symmetric Cartan matrix. The bubbling phenomena can reduce the computations of this degree jump to the calculations of the topological degree for some mean field equation together with an additional condition, which is called the shadow system in this paper. The main purpose of this paper is to compute the topological degree of this shadow system via a suitable deformation from a “decoupled” shadow system to our shadow system, and to show the a priori bound along this deformation. The proof of a priori bounds is very subtle, which depends on the type of Lie algebra.

Structure of the nearly-degenerate manifold of lattice quasiholes on a torus

We study the nearly-degenerate quasihole manifold of the bosonic Hofstadter-Hubbard model on a torus, known to host the lattice analog of the Laughlin state at filling fraction ν=1/2. Away from ν=1/2 and in the presence of both localized and delocalized quasiholes, the ratio between the numerically calculated many-body Chern number for certain groups of states and the number of states in the relevant group turns out to be constant for this manifold, which is also manifested in the density profile as the depleted charge of localized quasiholes. Inspired by a zero-mode counting formula derivable from a generalized Pauli principle, we employ a combinatorial scheme to account for the splittings in the manifold, allowing us to interpret some groups of states as the quasihole excitations corresponding to filling fractions lower than ν=1/2. In this scheme, the many-body Chern number of subgroups appears as a simple combinatorial factor.

A characterization of GVZ groups in terms of fully ramified characters

In this paper‎, ‎we obtain a characterization of GVZ-groups in terms of commutators and monolithic quotients‎. ‎This characterization is based on counting formulas due to Gallagher‎.

Integrality structures in topological strings and quantum 2-functions

In this article, we first prove the integrality of an explicit disc counting formula obtained by Panfil and Sulkowski for a class of toric Calabi-Yau manifolds named generalized conifolds. Then, motivated by the integrality structures in open topological string theory, we introduce a mathematical notion of “quantum 2-function” which can be viewed as the quantization of the notion of “2-function” introduced by Schwarz, Vologod-sky and Walcher. Finally, we provide a basic example of quantum 2-function and discuss the quantization of 2-functions.

Counting spanning trees in a complete bipartite graph which contain a given spanning forest

AbstractIn this article, we extend Moon's classic formula for counting spanning trees in complete graphs containing a fixed spanning forest to complete bipartite graphs. Let be the bipartition of the complete bipartite graph with and . We prove that for any given spanning forest of with components , the number of spanning trees in which contain all edges in is equal to where and for .

Relationship between the prognostic nutritional index and resistant hypertension in patients with essential hypertension

ABSTRACT Objectives Immune system activation plays a role in resistant hypertension (RHTN) pathogenesis. The clinical effect of the prognostic nutritional index (PNI) on patients with RHTN remains unclear. The aim of this study investigated the possible correlation between PNI and RHTN. Methods In this cross-sectional study, we enrolled 180 adult subjects. In patients were classified into three groups according to their office and ambulatory blood pressure measurements (ABPM): RHTN (n = 60), controlled hypertension (CHTN, n = 60), and normotension-control (NT-C, n = 60). RHTN was defined as BP ≥140/90 mm Hg while taking ≥3 antihypertensive medications or BP <140/90 mm Hg while taking ≥4 medications. The PNI was calculated from the 10 x serum albumin (g/dL) + 0.005 x total lymphocyte count (/μL) formula. Results Office and ABPM were significantly higher in patients with RHTN. Patients in the RHTN (46.1 ± 5.3) had significantly lower PNI than that in the CHTN (54.9 ± 6.7) (P = .032), and PNIs of both hypertensive groups were significantly lower than the NT-C group (P = .019, for both). The ROC curve analysis performed to assess the predictive value of PNI for RHTN and using 50.9 optimal cutoff value of PNI for RHTN gave a sensitivity of 77% and a specificity of 68.5% (AUC = 0.73, 95% CI 0.69–0.96). Multivariate analysis indicated diabetes, 24-h ABPM SBP, CRP, pill burden, and PNI (<51.6) as independent predictors of RHTN. Conclusion This study showed that the level of PNI was significantly lower in patients with RHTN compared to patients with CHTN. PNI is independently related to RHTN.

The fuzzball hair removal

The gravitational description of black hole microstructure via the fuzzball proposal suggests a vast space of solutions called fuzzballs. This idea together with the 4d and 5d microscopic counting formulas necessitates that each supersymmetric 4d fuzzball admits an infinite number of hair. The hair removed fuzzballs are the true horizon degrees of freedom. We discuss how to construct/identify, count, and remove fuzzball hair.

Flux vacua: a voluminous recount

In this note we apply mathematical results for the volume of certain symmetric spaces to the problem of counting flux vacua in simple IIB Calabi--Yau compactifications. In particular we obtain estimates for the number of flux vacua including the geometric factor related to the Calabi-Yau moduli space, in the large flux limit, for the FHSV model and some closely related models. We see that these geometric factors give rise to contributions to the counting formula that are typically not of order one and might potentially affect the counting qualitatively in some cases. We also note, for simple families of Calabi-Yau moduli spaces, an interesting dependence of the moduli space volumes on the dimension of the flux space, which in turn is governed by the Betti numbers of the Calabi-Yaus.

An electron counting formula to explain and to predict hydrogenated and metallated borophenes.

A generalized electron counting rule for borophene with varying hole density explains the preference for hydrogenation in β12 borophene of hole density ⅙. Interplay between bridging and terminal BH bonds controls the strength of interaction of the 2D layer with the metal surface.

Reachability results in labelled \(t\)-ary trees

In this paper, we prove some new formulas in the enumeration of labelled \(t\)-ary trees by path lengths. We treat trees having their edges oriented from a vertex of lower label towards a vertex of higher label. Among other results, we obtain counting formulas for the number of \(t\)-ary trees on \(n\) vertices in which there are paths of length \(\ell\) starting at a root with label \(i\) and ending at a vertex, sink, leaf sink, first child, non-first child and non-leaf. For each statistic, the average number of these reachable vertices is obtained for any random \(t\)-ary tree.

Bijections for Faces of the Shi and Catalan Arrangements

In 1986, Shi derived the famous formula $(n+1)^{n-1}$ for the number of regions of the Shi arrangement, a hyperplane arrangement in ${R}^n$. There are at least two different bijective explanations of this formula, one by Pak and Stanley, another by Athanasiadis and Linusson. In 1996, Athanasiadis used the finite field method to derive a formula for the number of $k$-dimensional faces of the Shi arrangement for any $k$. Until now, the formula of Athanasiadis did not have a bijective explanation. In this paper, we extend a bijection for regions defined by Bernardi to obtain a bijection between the $k$-dimensional faces of the Shi arrangement for any $k$ and a set of decorated binary trees. Furthermore, we show how these trees can be converted to a simple set of functions of the form $f: [n-1] \to [n+1]$ together with a marked subset of $\text{Im}(f)$. This correspondence gives the first bijective proof of the formula of Athanasiadis. In the process, we also obtain a bijection and counting formula for the faces of the Catalan arrangement. All of our results generalize to both extended arrangements.