Some inferences based on a mixture of power function and continuous logarithmic distribution

Let F=Fθ:θ∈Θ⊂R be a family of probability distributions indexed by a parameter θ and let X1,⋯,Xn be i.i.d. r.v.’s with L(X1)=Fθ∈F. Then, F is said to be reproducible if for all θ∈Θ and n∈N, there exists a sequence (αn)n≥1 and a mapping gn:Θ→Θ,θ⟼gn(θ) such that L(αn∑i=1nXi)=Fgn(θ)∈F. In this paper, we prove that a natural exponential family F is reproducible iff it possesses a variance function which is a power function of its mean. Such a result generalizes that of Bar-Lev and Enis (1986, The Annals of Statistics) who proved a similar but partial statement under the assumption that F is steep as and under rather restricted constraints on the forms of αn and gn(θ). We show that such restrictions are not required. In addition, we examine various aspects of reproducibility, both theoretically and practically, and discuss the relationship between reproducibility, convolution and infinite divisibility. We suggest new avenues for characterizing other classes of families of distributions with respect to their reproducibility and convolution properties .

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Role of Reynolds and Archimedes numbers in particle-fluid flows

Reviews in Chemical Engineering ◽

10.1515/revce-2020-0005 ◽

2020 ◽

Vol 0 (0) ◽

Cited By ~ 1

Author(s):

Haim Kalman

Keyword(s):

Flow Regime ◽

Power Function ◽

Fluid Flows ◽

Limited Range ◽

Operating Conditions ◽

Pneumatic Conveying ◽

Particulate Materials ◽

Simple Power ◽

Simple Power Function

AbstractAny scientific behavior is best represented by nondimensional numbers. However, in many cases, for pneumatic conveying systems, dimensional equations are developed and used. In some cases, many of the nondimensional equations include Reynolds (Re) and Froude (Fr) numbers; they are usually defined for a limited range of materials and operating conditions. This study demonstrates that most of the relevant flow types, whether in horizontal or vertical pipes, can be better described by Re and Archimedes (Ar) numbers. Ar can also be used in hydraulic conveying systems. This paper presents many threshold velocities that are accurately defined by Re as a simple power function of Ar. Many particulate materials are considered by Ar, thereby linking them to a common behavior. Using various threshold velocities, a flow regime chart for horizontal conveying is presented in this paper.

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Boundary Value Problems of Electroelasticity with Concentrated Singularities

Georgian Mathematical Journal ◽

10.1515/gmj.1994.459 ◽

1994 ◽

Vol 1 (5) ◽

pp. 459-467

Author(s):

T. Buchukuri ◽

D. Yanakidi

Keyword(s):

Singular Point ◽

Boundary Value Problems ◽

Power Function ◽

Boundary Value ◽

Linearly Independent

Abstract We investigate the solutions of boundary value problems of linear electroelasticity, having growth as a power function in the neighbourhood of infinity or in the neighbourhood of an isolated singular point. The number of linearly independent solutions of this type is established for homogeneous boundary value problems.

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Masking Transforms the Power Function for Loudness

The Journal of the Acoustical Society of America ◽

10.1121/1.1942916 ◽

1966 ◽

Vol 39 (6) ◽

pp. 1261-1262

Author(s):

S. S. Stevens ◽

Miguelina Guirao

Keyword(s):

Power Function

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Analytical Continuation within the Freundlich Adsorption Model. Comment on Edet, U.A.; Ifelebuegu, A.O. Kinetics, Isotherms, and Thermodynamic Modeling of the Adsorption of Phosphates from Model Wastewater Using Recycled Brick Waste. Processes 2020, 8, 665

Processes ◽

10.3390/pr9071251 ◽

2021 ◽

Vol 9 (7) ◽

pp. 1251

Author(s):

Michael Vigdorowitsch ◽

Alexander N. Pchelintsev ◽

Liudmila E. Tsygankova

Keyword(s):

Experimental Data ◽

Power Function ◽

Thermodynamic Modeling ◽

Mathematical Theory ◽

Surface Adsorption ◽

Analytical Continuation ◽

Adsorption Model ◽

Saturation Regime

Using experimental data for the adsorption of phosphates out of wastewater on waste recycled bricks, published independently in MDPI Processes before (2020), this message re-visits the mathematical theory of the Freundlich adsorption model. It demonstrates how experimental data are to be deeper treated to model the saturation regime and to bridge a chasm between those areas where the data fit the Freundlich power function and where a saturation of surface adsorption centers occurs.

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Commutative semigroups whose endomorphisms are power functions

Semigroup Forum ◽

10.1007/s00233-021-10178-x ◽

2021 ◽

Author(s):

Ryszard Mazurek

Keyword(s):

Positive Integer ◽

Power Function ◽

Cyclic Group ◽

Commutative Semigroup ◽

Commutative Monoid ◽

Power Functions ◽

Commutative Semigroups ◽

Finite Cyclic Group

AbstractFor any commutative semigroup S and positive integer m the power function $$f: S \rightarrow S$$ f : S → S defined by $$f(x) = x^m$$ f ( x ) = x m is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.

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Strength and Microscopic Damage Mechanism of Yellow Sandstone with Holes under Freezing and Thawing

Advances in Civil Engineering ◽

10.1155/2020/5921901 ◽

2020 ◽

Vol 2020 ◽

pp. 1-13 ◽

Cited By ~ 1

Author(s):

Huren Rong ◽

Jingyu Gu ◽

Miren Rong ◽

Hong Liu ◽

Jiayao Zhang ◽

...

Keyword(s):

Mechanical Properties ◽

Pore Size ◽

Power Function ◽

Damage Mechanism ◽

Uniaxial Compression Test ◽

Freezing And Thawing ◽

Freeze Thaw ◽

Thaw Cycle ◽

Freeze Thaw Cycle ◽

Microscopic Damage

In order to study the damage characteristics of the yellow sandstone containing pores under the freeze-thaw cycle, the uniaxial compression test of saturated water-stained yellow sandstones with different freeze-thaw cycles was carried out by rock servo press, the microstructure was qualitatively analyzed by Zeiss 508 stereo microscope, and the microdamage mechanism was quantitatively studied by using specific surface area and pore size analyzer. The mechanism of weakening mechanical properties of single-hole yellow sandstone was expounded from the perspective of microstructure. The results show the following. (1) The number of freeze-thaw cycles and single-pore diameter have significant effects on the strength and elastic modulus of the yellow sandstone; the more the freeze-thaw cycles and the larger the pore size, the lower the strength of the yellow sandstone. (2) The damage modes of the yellow sandstone containing pores under the freeze-thaw cycle are divided into five types, and the yellow sandstone with pores is divided into two areas: the periphery of the hole and the distance from the hole; as the number of freeze-thaw cycles increases, different regions show different microscopic damage patterns. (3) The damage degree of yellow sandstone is different with freeze-thaw cycle and pore size. Freeze-thaw not only affects the mechanical properties of yellow sandstone but also accelerates the damage process of pores. (4) The damage of the yellow sandstone by freeze-thaw is logarithmic function, and the damage of the yellow sandstone is a power function. The damage equation of the yellow sandstone with pores under the freezing and thawing is a log-power function nonlinear change law and presents a good correlation.

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Costs of transport for the scyphomedusa Stomolophus meleagris L. Agassiz

Canadian Journal of Zoology ◽

10.1139/z87-408 ◽

1987 ◽

Vol 65 (11) ◽

pp. 2690-2695 ◽

Cited By ~ 20

Author(s):

R. J. Larson

Keyword(s):

Output Power ◽

Power Function ◽

Power Output ◽

Reynolds Numbers ◽

Power Input ◽

Cost Of Transport ◽

Swimming Efficiency ◽

Wet Weight ◽

Planktonic Crustaceans ◽

Stomolophus Meleagris

The rhizostome scyphomedusa Stomolophus meleagris swims continuously at speeds up to 15 cm∙s−1. Mean velocities increased as a power function of wet weight up to 70 g but were mostly constant thereafter. Bell pulsations ranged from 1.7 to 3.6 Hz. Reynolds numbers equalled 900 – 13 000. During activity, medusae consumed 0.05 mL O2∙h−1∙g WW−1 (1.2 mL O2∙h−1∙g DW−1), at 30 °C. Rates for inactive medusae were 50% less. The estimated cost of transport ranged from 2 J∙kg−1∙m−1 at 5 g to 1 J∙kg−1∙m−1 at 1 kg. These rates are comparable to those of fishes and about 1/50th that of planktonic crustaceans. These results were unexpected in light of the typical inefficiency (power output/power input) of jet swimming. However, S. meleagris has a very low respiration rate relative to crustaceans and fish, which probably compensated for low swimming efficiency.

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