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### Growth estimates for exp—log functions

John Shackell
1990 Journal of symbolic computation
This also provides another proof that the dominance problem for exp-log functions is Turing-reducible to the identity problem for constant terms.  ...  Exp-log functions are those obtained from the constant 1 and the variable X by means of arithmetic operations and the function symbols exp() and logll.  ...  Firstly the identity problem for exp-log functions asks for a procedure to determine whether a given exp-log expression is identically zero.  ...

### New Insights into Modelling Bacterial Growth with Reference to the Fish Pathogen Flavobacterium psychrophilum

Christopher D. Powell, Secundino López, James France
2020 Animals
In contrast with log × exp and BAR, log × hyp can be easily manipulated mathematically allowing for simple algebraic expressions for time and microbial biomass at inflexion point, in addition to maximum  ...  The newly derived functions (log × hyp and log × exp), along with the Baranyi (BAR), simple logistic (LOG) and its modified form (MLOG) were evaluated based upon examination of residuals and measures of  ...  Using the parameter estimates from fitting log × hyp and log × exp to a given dataset, actual growth rate (Equation (9) ) can be calculated and plotted as a function of time.  ...

### Bayesian Inference of the Stochastic Gompertz Growth Model for Tumor Growth

Jayeong Paek, Ilsu Choi
2014 Communications for Statistical Applications and Methods
A stochastic Gompertz diffusion model for tumor growth is a topic of active interest as cancer is a leading cause of death in Korea.  ...  This likelihood is useful in providing a basis for the so-called continuous record or infill likelihood function and infill asymptotic.  ...  However, since Y t = log (X t /X 0 ), the probability density function of X t given X 0 = x 0 is f X t (x t |x 0 ) = 1 x t √ 2πσ 2 t exp           ( log ( x t x 0 ) − m β ( 1 − e −βt ) + σ 2  ...

### On the rate of growth of entire functions of fast growth

Daihachiro Sato
1963 Bulletin of the American Mathematical Society
The purpose of this note is to generalize the following well-known formula to give the order p and type a of an entire function/(s) = J^T-o a nZ n with M{r) =max| z |" r \f(z)\, that is [l; 2],  ...  X(3> is called the rank and i c<3> is called the title of the entire function. We call X( g > and / C( 3 > the rate of growth of the entire function of index q. Formulas for X (a) and K iq) .  ...  Consider f(x) = exp (q/p) x as a well defined solution of a simultaneous functional equation exp ltp] (f(x))=exp [tq] x (t = 0, ±1, ±2, • • • ) and for real r, define exp (r) (#) by uniform limit process  ...

### On the relation between maximum modulus, maximum term, and Taylor coefficients of an entire function

M Freund, E Görlich
1985 Journal of Approximation Theory
For entire functions of relatively slow growth, a distinct increase of precision is achieved. Our approach originates in work by Hadamard.  ...  Relations are established between the quantities in the title in form of direct estimates, instead of measuring the growth of J" by its order, type, or some generalized order.  ...  In terms of the estimate m(r) = O(A(r)) this means that we exclude functions f for which m(r) increases faster than exp{a(log r)*) for some constant a > 0.  ...

### BMOA estimates and radial growth of B_ϕ functions [article]

Peyo Stoilov, Roumyana Gesheva, Milena Racheva
2008 arXiv   pre-print
BMO estimates and the radial growth of Bloch functions have been studied by B. Korenblum . The present paper contains some natural generalizations of these results.  ...  The space BMOA is the space of functions f ∈ H 1 for which Mathematics Subject Classification: Primary 30E20, 30D50. 0 Key words and phrases: Bloch space, BMOA estimates, radial growth of functions.  ...  Korenblum for the Bloch space B are generalized for B φ . Note that the function log(1 − z) ∈ B , however log 2 (1 − z) / ∈ B.  ...

### Statistical Weight Kinetics Modeling and Estimation for Silica Nanowire Growth Catalyzed by Pd Thin Film

Qiang Huang, Li Wang, Tirthankar Dasgupta, Li Zhu, Praveen K. Sekhar, Shekhar Bhansali, Yu An
2011 IEEE Transactions on Automation Science and Engineering
Model estimation and comparison are conducted using both maximum likelihood estimation (MLE) and Bayesian approaches.  ...  We investigate six different weight kinetics models in predicting weight changes during growth.  ...  The log-likelihood function for model 5 is: L 5 (θ, σ 2 1 , σ 2 2 ) = N i=1 {I t≤t0 [− 1 2 log (σ 2 1 ) − 1 2σ 2 1 (y i − α 1 exp(α 2 /t i )) 2 ] + I t>t0 [− 1 2 log(σ 2 2 ) − 1 2σ 2 2 (y i − at i − b)  ...

### Generalized n-Laplacian: boundedness of weak solutions to the Dirichlet problem with nonlinearity in the critical growth range

Robert Černý
2014 Open Mathematics
∈ L Φ(Ω) is a non-trivial continuous function and µ ≥ 0 is a small parameter.  ...  such that the space W 01 L Φ(Ω) is embedded into an exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V(x) is a continuous potential, h  ...  Acknowledgements The author would like to thank Professors Andrea Cianchi, Pavel Drábek and Stanislav Hencl for fruitful discussions.  ...

### On Determining the Growth of Meromorphic Solutions of Algebraic Differential Equations Having Arbitrary Entire Coefficients

Steven B. Bank
1973 Nagoya mathematical journal
In this paper, we treat the problem of determining the rate of growth of meromorphic functions on the plane, which are solutions ofnthorder algebraic differential equations whose coefficients are arbitrary  ...  entire functions (i.e., equations of the form,Ω(z,y,dy/dz,• • •,dny/dzn) —0, whereΩis a polynomial iny,dy/dz,• • •,dny/dznwhose coefficients are arbitrary entire functions ofz.)  ...  ) (r + log + N(r, y 0 ) + log + N(r, Hence, in any case, the growth of a solution y Q can be estimated in terms of the counting functions N(r,y 0 ) and N(r,l/y Q ) for the poles and zeros respectively,  ...

### On the Use of the Gompertz Model to Predict Microbial Thermal Inactivation Under Isothermal and Non-Isothermal Conditions

Maria M. Gil, Fátima A. Miller, Teresa R. S. Brandão, Cristina L. M. Silva
2011 Food Engineering Reviews
The versatility of the Gompertz model in describing kinetics with different shapes, varying from a log-linear tendency till a complete sigmoidal shape, makes it attractive for predictive purposes, both  ...  In this field, the use of mathematical models that describe the microorganisms' kinetics in such conditions is an important tool for convenient design, control and optimization of efficient processes.  ...   also proposed a modification for the Gompertz equation, as follows: log N ¼ A þ C exp Àexp ÀBðt À MÞ ½ f g ð3Þ and considered: Growth rate ¼ BC e ð4Þ lag ¼ M À 1 B ð5Þ Generation time ¼ 24 Â logð2Þ  ...

### Growth/survival of Salmonella enteritidis on fresh poultry and fish stored under vacuum or modified atmosphere

G.-J.E. Nychas, C. C. Tassou
1996 Letters in Applied Microbiology
The essential oil and packing treatments had remarkable delaying effects on the growth of S. typhimurium.  ...  The structural identifiability properties of the Baranyi model were analyzed in fitting the effect of oregano and thyme essential oils on growth of Salmonella typhimurium in rainbow trout stored under  ...  These parameters can be estimated as following (Toğrul and Arslan, 2004) : n n i=1 1 = log log / log ×100 exp,i pre,i exp,i MPE N N N (8) n i=1 pre,i exp,i 1 = n log log MBE N N (9) n n i=1 1/2 exp,i  ...

### USING NONLINEAR FIXED AND MIXED MODELS WITH SWITCHING FUNCTIONS TO ALLOW FOR HORMESIS IN GROWTH OF ESCHERICHIA COLI

C. Tu, A. M. Parkhurst, L. M. Durso, R. W. Hutkins
2007 Conference on Applied Statistics in Agriculture
Classic bacterial growth curves display an increase in optical density over time. In this paper, we use the logistic function to model growth in optical density of E. coli over time.  ...  We examine several switching functions that allow for the effect of hormesis and compare the ability of nonlinear fixed and mixed models to detect the presence of hormesis.  ...  These changes can be described by logistic or log-logistic regression models. The models provide such estimates as maximum optical density, maximum rate of growth, and time to maximum rate of growth.  ...

### Growth estimates of entire functions with the help of their relative L^{∗}-types and relative L^{∗}- weak types

Sanjib Kumar Datta, Tanmay Biswas
2018 Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics
In this paper we attempt to prove some results related to the growth rates of entire functions on the basis of relative L -type and relative L -weak type of an entire function with respect to another entire  ...  function.  ...  1 inf log log M f (r) log re L(r) : An entire function for which L -order and L -lower order are the same is said to be of regular L -growth.  ...

### Nonlinear Growth Models for Modeling Oil Palm Yield Growth

Azme Khamis, Zuhaimy Ismail, Khalid Haron, Ahmad Tarmizi Mohammed
2005 Journal of Mathematics and Statistics
This study provided the basic needs of parameters estimation for nonlinear growth model such as partial derivatives of each model, determination of initial values for each parameter and statistical tests  ...  This study found that the Gompertz, logistic, log-logistic, Morgan-Mercer-Flodin and Chapman-Richard growth models have the ability for quantifying a growth phenomenon that exhibit a sigmoid pattern over  ...  The parameter estimates for the logistic, Gompertz, negative exponential, monomolecular and Morgan-Mercer-Flodin growth functions are all statistically significant at the 5% level.  ...

### Nonlinear Growth Models for Modeling Oil Palm Yield Growth

Khamiz
2005 Journal of Mathematics and Statistics
This study provided the basic needs of parameters estimation for nonlinear growth model such as partial derivatives of each model, determination of initial values for each parameter and statistical tests  ...  This study found that the Gompertz, logistic, log-logistic, Morgan-Mercer-Flodin and Chapman-Richard growth models have the ability for quantifying a growth phenomenon that exhibit a sigmoid pattern over  ...  The parameter estimates for the logistic, Gompertz, negative exponential, monomolecular and Morgan-Mercer-Flodin growth functions are all statistically significant at the 5% level.  ...
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