Formula For Number Research Articles

Investigation of cellularization characteristics of hydrogen-methane-ethanol expanding spherical flame at elevated pressures

The incorporation of ethanol into hydrocarbon fuels for use is attracting increasing interest and it is necessary to investigate its inherent flame instability for better application in combustion units. The instability of hydrogen-methane-ethanol spherically expanding flame has been investigated at an initial temperature of 400 K, initial pressures of (2–4 bar), ethanol fraction of (20%, 50%, 80%), and equivalence ratios (Ф) of (0.7–1.4) using the constant volume combustion chamber (CVCC). High-speed schlieren technology was used to record flame propagation images. The effects of hydrodynamic and thermal-diffusion effect on the inherent instability of the flame were investigated. As the ethanol ratio increased, the hydrodynamic effect was enhanced. The thermal-diffusion effect was discovered to stabilize the flame surface under all conditions, as judged jointly by the effective Lewis number and the critical Lewis number. The critical conditions (critical radius and Peclet number) at the onset of unstability were evaluated, and it was found that the flames were more prone to flame instability at higher pressures. The critical Peclet number increased with the increase in the equivalence ratio when the ethanol ratio was 20%, and showed the opposite trend when the ethanol ratio was 50% and 80%. In addition, as the ethanol ratio increases, the stability of the lean mixtures flame increases, while the rich mixtures flame suffers from early onset of instability. The theoretical and experimental results were consistent, with some differences at Ф = 1.4. An empirical correlation formula for the critical Peclet number (Pec) and Markstein number (Mb) was further proposed (Pec= 18.03Mb+214.78). Finally, the Karlovitz number was used to study the instability behavior of the flame. The critical Karlovitz number (Kac) decreased with increasing Mb and the tendency of the flame to suffer from instability diminished, and the following correlation was obtained Kac=0.05635×e−0.13852Mb. Furthermore, the flame was more unstable in rich mixtures, this was consistent with the conclusion of instability derived from the critical radius.

Construction and verification of dual-feed-opening rotor spinning unit merging effect model

To explore the yarn forming mechanism of dual-feed-opening (DFO) rotor-spun yarn, this work is based on the advanced movement of the peeling point and the merging effect of conventional rotor spinning. The fiber-layers distributions in the groove at the representative stages are characterized and analyzed in the form of illustration. Then the model of merging effect is deduced by mathematical induction. This model obtains the calculation formula of the merging number and the definition of the output-section. The morphological distribution of the original fiber-layer presents a nonlinear ladder like feature. The apparent color distribution of 1:1 blue/white DFO rotor-spun yarn theoretically shows the characteristics of two colors equally spaced, and the interval length is the length of the output-section. Using digital image processing tools, the length of output-section is verified by the image processing algorithm. This research has certain guiding significance for improving the theory of DFO rotor spinning unit (DFO-RSU).

Sums of One Prime Power and Four Prime Cubes in Short Intervals

Let k ⩾ 1 be an integer. In this study, we derive an asymptotic formula for the average number of representations of integers n = p 1 k + p 2 3 + p 3 3 + p 4 3 + p 5 3 in short intervals, where p 1 , p 2 , p 3 , p 4 , p 5 are prime numbers.

Hecke nilpotency for modular forms mod 2 and an application to partition numbers

A well-known observation of Serre and Tate is that the Hecke algebra acts locally nilpotently on modular forms mod 2 on $${{\,\mathrm{\textrm{SL}}\,}}_2({{\,\mathrm{{\mathbb {Z}}}\,}})$$ . We give an algorithm for calculating the degree of Hecke nilpotency for cusp forms, and we obtain a formula for the total number of cusp forms mod 2 of any given degree of nilpotency. Using these results, we find that the degrees of Hecke nilpotency in spaces $$M_k$$ have no limiting distribution as $$k \rightarrow \infty $$ . As an application, we study the parity of the partition function using Hecke nilpotency.

Cyclic products of Szegö kernels and spin structure sums. Part I. Hyper-elliptic formulation

The summation over spin structures, which is required to implement the GSO projection in the RNS formulation of superstring theories, often presents a significant impediment to the explicit evaluation of superstring amplitudes. In this paper we discover that, for Riemann surfaces of genus two and even spin structures, a collection of novel identities leads to a dramatic simplification of the spin structure sum. Explicit formulas for an arbitrary number of vertex points are obtained in two steps. First, we show that the spin structure dependence of a cyclic product of Szegö kernels (i.e. Dirac propagators for worldsheet fermions) may be reduced to the spin structure dependence of the four-point function. Of particular importance are certain trilinear relations that we shall define and prove. In a second step, the known expressions for the genus-two even spin structure measure are used to perform the remaining spin structure sums. The dependence of the spin summand on the vertex points is reduced to simple building blocks that can already be identified from the two-point function. The hyper-elliptic formulation of genus-two Riemann surfaces is used to derive these results, and its SL(2, ℂ) covariance is employed to organize the calculations and the structure of the final formulas. The translation of these results into the language of Riemann ϑ-functions, and applications to the evaluation of higher-point string amplitudes, are relegated to subsequent companion papers.

Final Size for Epidemic Models with Asymptomatic Transmission

The final infection size is defined as the total number of individuals that become infected throughout an epidemic. Despite its importance for predicting the fraction of the population that will end infected, it does not capture which part of the infected population will present symptoms. Knowing this information is relevant because it is related to the severity of the epidemics. The objective of this work is to give a formula for the total number of symptomatic cases throughout an epidemic. Specifically, we focus on different types of structured SIR epidemic models (in which infected individuals can possibly become symptomatic before recovering), and we compute the accumulated number of symptomatic cases when time goes to infinity using a probabilistic approach. The methodology behind the strategy we follow is relatively independent of the details of the model.

Reliability Analysis on Stability of Armor Units for Foundation Mound of Composite Breakwaters

Probabilistic and deterministic analyses are implemented for the armor units of rubble foundation mound of composite breakwaters which is needed to protect the upright section against the scour of foundation mounds. By a little modification and incorporation of the previous empirical formulas that has commonly been applied to design the armor units of foundation mound, a new type formula of stability number has been suggested which is capable of taking into account slopes of foundation mounds, damage ratios of armor units, and incident wave numbers. The new proposed formula becomes mathematically identical with the previous empirical formula under the same conditions used in the developing process. Deterministic design have first been carried out to evaluate the minimum weights of armor units for several conditions associated with a typical section of composite breakwater. When the slopes of foundation mound become steepening and the incident wave numbers are increasing, the bigger armor units more than those from the previous empirical formula should be required. The opposite trends however are shown if the damage ratios is much more allowed. Meanwhile, the reliability analysis, which is one of probabilistic models, has been performed in order to quantitatively verify how the armor unit resulted from the deterministic design is stable. It has been confirmed that 1.2% of annual encounter probability of failure has been evaluated under the condition of 1% damage ratio of armor units for the design wave of 50 years return period. By additionally calculating the influence factors of the related random variables on the failure probability due to those uncertainties, it has been found that Hudson’s stability coefficient, significant wave height, and water depth above foundation mound have sequentially been given the impacts on failure regardless of the incident wave angles. Finally, sensitivity analysis has been interpreted with respect to the variations of random variables which are implicitly involved in the formula of stability number for armor units of foundation mound. Then, the probability of failure have been rapidly decreased as the water depth above foundation mound are deepening. However, it has been shown that the probability of failure have been increased according as the berm width of foundation mound are widening and wave periods become shortening.

Bounds on Intersection Number in the Join and Corona of Graphs

In this paper, we provide an upper bound for the intersection number in the join and corona of graphs. Moreover, we give formulas for the intersection number of Kn ◦G, Pn ◦G, Cn ◦G and Crn.

Projections and Angle Sums of Belt Polytopes and Permutohedra

Let Psubset mathbb {R}^n be a belt polytope, that is a polytope whose normal fan coincides with the fan of some hyperplane arrangement {mathcal {A}}. Also, let G:mathbb {R}^nrightarrow mathbb {R}^d be a linear map of full rank whose kernel is in general position with respect to the faces of P. We derive a formula for the number of j-faces of the “projected” polytope GP in terms of the j-th level characteristic polynomial of {mathcal {A}}. In particular, we show that the face numbers of GP do not depend on the linear map G provided a general position assumption is satisfied. Furthermore, we derive formulas for the sum of the conic intrinsic volumes and Grassmann angles of the tangent cones of P at all of its j-faces. We apply these results to permutohedra of types A and B, which yields closed formulas for the face numbers of projected permutohedra and the generalized angle sums of permutohedra in terms of Stirling numbers of both kinds and their B-analogues.

Stirling Numbers of Uniform Trees and Related Computational Experiments

The Stirling numbers for graphs provide a combinatorial interpretation of the number of cycle covers in a given graph. The problem of generating all cycle covers or enumerating these quantities on general graphs is computationally intractable, but recent work has shown that there exist infinite families of sparse or structured graphs for which it is possible to derive efficient enumerative formulas. In this paper, we consider the case of trees and forests of a fixed size, proposing an efficient algorithm based on matrix algebra to approximate the distribution of Stirling numbers. We also present a model application of machine learning to enumeration problems in this setting, demonstrating that standard regression techniques can be applied to this type of combinatorial structure.

A proof of the refined class number formula of Gross

In 1988, Gross proposed a conjectural congruence between Stickelberger elements and algebraic regulators, which is often referred to as the refined class number formula. In this paper, we prove this congruence.

Accurately predicting hypersonic transitional flow on cone via a symmetry approach

A new algebraic transition model is proposed based on a Structural Ensemble Dynamics (SED) theory of wall turbulence, for accurately predicting the hypersonic flow heat transfer on cone. The model defines the eddy viscosity in terms of a two-dimensional multi-regime distribution of a Stress Length (SL) function, and hence is named as SED-SL. This paper presents clear evidence of precise predictions of transition onset location and peak heat flux of a wide range of hypersonic Transitional Boundary Layers (TrBL) around straight cone at zero incidence, to an unprecedented accuracy as validated by over 70 measurements for varying five crucial influential factors (Mach number, temperature ratio, cone half angle, nose Reynolds number and noise level). The results demonstrate the universality of the postulated multi-regime similarity structure, in characterizing not only the spatial non-uniform distribution of the eddy viscosity in hypersonic TrBL on cone, but also the dependence of the transition onset location on the five influential factors. The latter yields a novel correlation formula for transition center Reynolds number which takes similar functional form as the SL function within the symmetry approach. It is concluded that the SED-SL model simulates TrBL around cone with uniformly high accuracy, and then points out to an optimistic alternative way to construct hypersonic transition model.

P-Numerical Semigroups of Generalized Fibonacci Triples

For a nonnegative integer p, we give explicit formulas for the p-Frobenius number and the p-genus of generalized Fibonacci numerical semigroups. Here, the p-numerical semigroup Sp is defined as the set of integers whose nonnegative integral linear combinations of given positive integers a1,a2,…,ak are expressed in more than p ways. When p=0, S0 with the 0-Frobenius number and the 0-genus is the original numerical semigroup with the Frobenius number and the genus. In this paper, we consider the p-numerical semigroup involving Jacobsthal polynomials, which include Fibonacci numbers as special cases. We can also deal with the Jacobsthal–Lucas polynomials, including Lucas numbers accordingly. An application on the p-Hilbert series is also provided. There are some interesting connections between Frobenius numbers and geometric and algebraic structures that exhibit symmetry properties.

A nested model with boosting and waning of immunity from Tilapia Lake Virus infection with distributed resistance to pathogens carrier-state.

This paper proposes and analyzes an immune-structured population model of tilapia subject to Tilapia Lake Virus (TiLV) disease. The model incorporates within-host dynamics, used to describe the interaction between the pathogen, the immune system and the waning of immunity. Individuals infected with a low dose acquire a low immunity level and those infected with a high dose acquire a high level of immunity. Since individuals' immune status plays an important role in the spread of infectious diseases at the population level, the within-host dynamics are connected to the between-host dynamics in the population. We define an explicit formula for the reproductive number [Formula: see text] and show that the disease-free equilibrium is locally asymptotically stable when [Formula: see text], while it is unstable when [Formula: see text]. Furthermore, we prove that an endemic equilibrium exists. We also study the influence of the initial distribution of host resistance on the spread of the disease, and find that hosts' initial resistance plays a crucial role in the disease dynamics. This suggests that the genetic selection aiming to improve hosts' initial resistance to TiLV could help fight the disease. The results also point out the crucial role played by the inoculum size. We find that the higher the initial inoculum size, the faster the dynamics of infection. Moreover, if the initial inoculum size is below a certain threshold, it may not result in an outbreak at the between-host level. Finally, the model shows that there is a strong negative correlation between heterogeneity and the probability of pathogen invasion.

The saturation number of monomial ideals

Let be the polynomial ring over a field and be the homogeneous maximal ideal of S. For an ideal , let be the minimum number k for which . In this paper, we compute the saturation number of irreducible monomial ideals and their powers. We apply this result to find the saturation number of the ordinary powers and symbolic powers of some families of monomial ideals in terms of the saturation number of irreducible components appearing in an irreducible decomposition of these ideals. Moreover, we give an explicit formula for the saturation number of monomial ideals in two variables.

The Role of Early Literacy Skills in Beginning to Read in Turkish: Longitudinal Findings from First Graders

The aim of this longitudinal study is to investigate the effects of early literacy skills on beginning to read in a transparent orthography, viz. Turkish. In line with this aim, a total of eighty-four participants were followed from preschool to the end of the first grade and the extent to which participants' early literacy skills in the preschool period predict their future reading performances was examined. All the data gathered in the study were collected individually at one-year intervals. The participants’ early literacy skills were tested by the "Test of Early Literacy (TEL)", reading comprehension skills was obtained with a text containing 162 words and 10 multiple choice questions, reading fluency performances were calculated with the formula of correct number of words per minute. Data were computed through correlation analysis, and hierarchical regression analysis. Results suggest that regarding phonological awareness, early literacy sub-skills, especially those contributing to or serving oral comprehension had positive effects on reading fluency and reading comprehension at the end of the first grade. Valuable conclusions have been drawn from this finding about the role of phonological awareness, and its contribution to reading in an agglutinative orthography such as Turkish. Obtained results are discussed in relation to the literature.

A mathematical model with numerical simulations for malaria transmission dynamics with differential susceptibility and partial immunity

Malaria is a deadly vector-borne infectious disease with high prevalence in the world’s endemic tropical and subtropical regions. Differences in individuals’ disease susceptibility may lead to their differentiation of susceptibility to infection. We formulate a mathematical model for malaria transmission dynamics that accounts for the host’s differential susceptibility, where partial immunity is acquired after infection. As customary, the explicit formula for the basic reproduction number is derived and used to determine the local stability of the model’s equilibria. An analysis of a special case with two susceptible classes shows that the model could have two endemic equilibria when the disease threshold parameter is less than unity. Numerical simulations are provided for a differential susceptibility when individuals are re-infected seven times after the initial infection. Graphical representations show that the transient transmission dynamics of the infected components are indistinguishable when there is no inflow into the susceptible classes. When there is an inflow into the various susceptible classes, the graphs of the infected component of the model are fundamentally different, showing that individuals who have been infected multiple times tend to be less infected over time. Knowledge of the inflow rate and the infection reduction rate due to prior infection in each class could be key drivers to mitigate the burden of malaria in a community.

Gaussian Quaternions Including Biperiodic Fibonacci and Lucas Numbers

In this study, we define a type of bi-periodic Fibonacci and Lucas numbers which are called bi-periodic Fibonacci and Lucas Gaussian quaternions. We also give the relationship between negabi-periodic Fibonacci and Lucas Gaussian quaternions and bi-periodic Fibonacci and Lucas Gaussian quaternions. Moreover, we obtain the Binet’s formula, generating function, d’Ocagne’s identity, Catalan’s identity, Cassini’s identity, like-Tagiuri’s identity, Honberger’s identity and some formulas for these new type numbers. Some algebraic proporties of bi-periodic Fibonacci and Lucas Gaussian quaternions which are connected between Gaussian quaternions and bi-periodic Fibonacci and Lucas numbers are investigated.

An Age of Infection Kernel, an R Formula, and Further Results for Arino–Brauer A, B Matrix Epidemic Models with Varying Populations, Waning Immunity, and Disease and Vaccination Fatalities

In this work, we first introduce a class of deterministic epidemic models with varying populations inspired by Arino et al. (2007), the parameterization of two matrices, demography, the waning of immunity, and vaccination parameters. Similar models have been focused on by Julien Arino, Fred Brauer, Odo Diekmann, and their coauthors, but mostly in the case of “closed populations” (models with varying populations have been studied in the past only in particular cases, due to the difficulty of this endeavor). Our Arino–Brauer models contain SIR–PH models of Riano (2020), which are characterized by the phase-type distribution (α→,A), modeling transitions in “disease/infectious compartments”. The A matrix is simply the Metzler/sub-generator matrix intervening in the linear system obtained by making all new infectious terms 0. The simplest way to define the probability row vector α→ is to restrict it to the case where there is only one susceptible class s, and when matrix B (given by the part of the new infection matrix, with respect to s) is of rank one, with B=bα→. For this case, the first result we obtained was an explicit formula (12) for the replacement number (not surprisingly, accounting for varying demography, waning immunity and vaccinations led to several nontrivial modifications of the Arino et al. (2007) formula). The analysis of (A,B) Arino–Brauer models is very challenging. As obtaining further general results seems very hard, we propose studying them at three levels: (A) the exact model, where only a few results are available—see Proposition 2; and (B) a “first approximation” (FA) of our model, which is related to the usually closed population model often studied in the literature. Notably, for this approximation, an associated renewal function is obtained in (7); this is related to the previous works of Breda, Diekmann, Graaf, Pugliese, Vermiglio, Champredon, Dushoff, and Earn. (C) Finally, we propose studying a second heuristic “intermediate approximation” (IA). Perhaps our main contribution is to draw attention to the importance of (A,B) Arino–Brauer models and that the FA approximation is not the only way to tackle them. As for the practical importance of our results, this is evident, once we observe that the (A,B) Arino–Brauer models include a large number of epidemic models (COVID, ILI, influenza, illnesses, etc.).

CFD Simulation of Laminar Rotating Flow Inside a Membrane Filter Vessel

In this study, computational fluid dynamics (CFD) was used to simulate a steady-state rotating flow inside a filtration system. The shape of a 3-D vessel was designed and then meshed using ANSYS design modular (DM) and workbench mesh. The laminar viscous regime was employed; it was identified using the Reynolds number formula. As a boundary layer, the zero inlet velocity and atmospheric pressure were imposed on the inlet boundary. The obtained results showed that for a rotating velocity of 50 rpm, the shear stress distribution on the membrane surface is extremely low, except for four narrow regions situated between the extremities of the impeller and the vessel wall. The pressure variation versus z axes, below the impeller, was almost constant, and it was unevenly distributed on the membrane surface, with two areas of negative pressure. The velocity values below the impeller were higher than those above. This study provides good insights that suggest designing the impeller shape under the most extreme conditions (rotating velocity, flow regime, and pressure inlet) of the membrane. The filtration process will be performed in a future study.