scholarly journals Some Remarks on the Self-Exponential Function: Minimum Value, Inverse Function, and Indefinite Integral

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
J. L. González-Santander ◽  
G. Martín
Keyword(s):  
Closed Form ◽  
Exponential Function ◽  
Real Function ◽  
Inverse Function ◽  
Real Variable ◽  
The Self ◽  
Series Expansions ◽  
Minimum Value ◽  
Indefinite Integral

Considering the function xx as a real function of real variable, what is its minimum value? Surprisingly, the minimum value is reached for a negative value of x. Furthermore, considering the function fβx=x-βx, β∈R and x>0, two different expressions in closed form for the inverse function fβ-1 can be obtained. Also, two different series expansions for the indefinite integral of fβ and fβ-1 are derived. The latter does not seem to be found in the literature.

Convergence in the intermediate representation

1953 ◽  
Vol 49 (4) ◽  
pp. 642-649
Author(s):  
J. Hamilton
Keyword(s):  
The Self ◽  
Series Expansions ◽  
Intermediate Representation ◽  
Self Energy

ABSTRACTIt is shown that for the quantized φ3 field the series expansions for intermediate representation field operators are convergent for summation over all graphs except those containing self-energy and vertex parts. The method used does not permit an investigation of the self-energy and vertex parts.

Explicit formulae for stationary distributions of stress release processes

2000 ◽  
Vol 37 (2) ◽  
pp. 315-321 ◽  
Author(s):  
K. Borovkov ◽  
D. Vere-Jones
Keyword(s):  
Closed Form ◽  
Stationary Distribution ◽  
Exponential Function ◽  
Markov Models ◽  
Risk Process ◽  
Stress Release ◽  
Stationary Distributions ◽  
Explicit Formulae ◽  
Tectonic Forces ◽  
Release Processes

Stress release processes are special Markov models attempting to describe the behaviour of stress and occurrence of earthquakes in seismic zones. The stress is built up linearly by tectonic forces and released spontaneously when earthquakes occur. Assuming that the risk is an exponential function of the stress, we derive closed form expressions for the stationary distribution of such processes, the moments of the risk, and the autocovariance function of the reciprocal risk process.

On the derivation of discontinuous functions

1939 ◽  
Vol 35 (3) ◽  
pp. 373-381
Author(s):  
D. R. Dickinson
Keyword(s):  
Real Function ◽  
Limit Point ◽  
Real Variable ◽  
Positive Direction ◽  
The Real ◽  
Discontinuous Functions ◽  
Set Of Points

Let f(x) be a real function of the real variable x, let P be any point lying on the graph of f(x) and let l be a ray from P making an angle θ (− π < θ ≤ π) with the positive direction of the x-axis. We say that θ is a derivate direction of f(x) at the point P if the ray l meets the graph of f(x) in a set of points having a limit point at P.

Study of Self-Locking Mechanism of Belt Friction

2007 ◽  
Author(s):  
Keiji Imado
Keyword(s):  
Contact Angle ◽  
Coefficient Of Friction ◽  
Exponential Function ◽  
Frictional Force ◽  
Cylindrical Surface ◽  
Loading Condition ◽  
The Self ◽  
Locking Mechanism ◽  
The Coefficient Of Friction ◽  
Severe Loading

It is well know that the belt friction is expressed in an exponential function of a product μ and θ, the coefficient of friction and the angle of contact between the flexible belt and the cylindrical surface respectively. So the frictional force increases greatly with an increment of contact angle θ. Using this property, many kinds of buckles were developed to fasten belt. But the locking condition of belt is not obtained from the equation unless θ is of infinity. Their locking conditions were not clarified theoretically. In practice, the product of μθ is usually less than θ, so that the exponent of the product μθ is not so large. Then some slippage may occur in case of severe loading condition. This study is focusing on a self-locking mechanism of a simple buckle developed for flat belt. The belt in the buckle is partially wound again over the belt. According to the equation derived, the fraction of the tight side belt tension to the loose side belt tension is significantly affected by the angle of double-layered segment. With an increment of angle of doublelayered segment, the fraction increases to infinity, which means the occurrence of belt locking. The locking condition is determined by the geometry of the buckle and the coefficient of frictions. The frictional force is automatically generated by the tension of belt so that the self-locking mechanism is realized in the buckle. The equation derived was confirmed by the experiments.

scholarly journals On the self-length of two-dimensional Banach spaces

1996 ◽  
Vol 53 (1) ◽  
pp. 101-107 ◽  
Author(s):  
B. Chalmers ◽  
C. Franchetti ◽  
M. Giaquinta
Keyword(s):  
Banach Space ◽  
Euclidean Space ◽  
Banach Spaces ◽  
Real Banach Space ◽  
The Self ◽  
Two Dimensional ◽  
Minimum Value

The aim of this paper is to prove the following result: if X is a 2-dimensional symmetric real Banach space, then its self-length is greater than or equal to 2π. Moreover, the minimum value 2π is uniquely attained (up to isometries) by euclidean space.

Criterion for Preventing Self-Loosening of Preloaded Cap Screws Under Transverse Cyclic Excitation

2011 ◽  
Vol 133 (4) ◽  
Author(s):  
Xianjie Yang ◽  
Sayed A. Nassar ◽  
Zhijun Wu
Keyword(s):  
Closed Form ◽  
Test Procedure ◽  
Closed Form Solution ◽  
Form Solution ◽  
The Self ◽  
Experimental Setup ◽  
Transverse Loading ◽  
System Parameters ◽  
Transverse Excitation ◽  
Key Variables

In this paper, a novel criterion is developed for preventing the self-loosening of preloaded threaded cap screws under cyclic transverse loading. For a known cyclic excitation, the system parameters are used in the formulation of a closed form solution for the minimum fastener preload required for preventing self-loosening. The effect of several key variables is investigated; this includes bearing and thread friction coefficients, cap screw grip length, thread pitch, material, and cyclic amplitude of the transverse excitation. An experimental setup and test procedure are established. Comparison between the experimental and analytical clamp load variation results shows that the proposed criterion can accurately predict the requirements for preventing self-loosening.

scholarly journals Thermal properties [compressibility and Moelwyn-Hughes Parameter] of NaCl crystals under high pressure

BIBECHANA ◽  
1970 ◽  
Vol 7 ◽  
pp. 49-53
Author(s):  
AK Khan
Keyword(s):  
High Pressure ◽  
Thermal Properties ◽  
Potential Function ◽  
Power Function ◽  
Exponential Function ◽  
Ionic Crystal ◽  
Inverse Function ◽  
Potential Functions ◽  
Inverse Power ◽  
Ionic Potential

Many inter ionic potential functions have been proposed since the Born and Lande's inverse function. An exponential function for the repulsive component of the potential has been proposed by Born and Lande', none of the potential function proposed is successful. A new interionic function proposed by Jha and Thakur to study the properties of ionic crystal under high pressure. Its repulsive component includes both an inverse power function due to Born and Lande' and an exponential function due to Born and Mayer. Jha has used it to study the thermal properties of NaCl and CsCl crystal under high pressure up to 100 Kilo bar. Keywords: NaCl crystals; Moelwyn-Hughes parameter; Isothermal compressibilityDOI: 10.3126/bibechana.v7i0.4045BIBECHANA 7 (2011) 49-53

scholarly journals Development of the Theory of the Functions of Real Variables in the First Decades of the Twentieth Century

2020 ◽  
Author(s):  
Loredana Biacino
Keyword(s):  
Twentieth Century ◽  
Calculus Of Variations ◽  
Real Function ◽  
Measure Theory ◽  
Real Variable ◽  
Italian Mathematicians ◽  
Lebesgue Theorem

In (Biacino 2018) the evolution of the concept of a real function of a real variable at the beginning of the twentieth century is outlined, reporting the discussions and the polemics, in which some young French mathematicians of those years as Baire, Borel and Lebesgue were involved, about what had to be considered a genuine real function. In this paper a technical survey of the arising function and measure theory is given with a particular regard to the contribution of the Italian mathematicians Vitali, Beppo Levi, Fubini, Severini, Tonelli etc … and also with the purpose of exposing the intermediate steps before the final formulation of Radom-Nicodym-Lebesgue Theorem and the Italian method of calculus of variations.

scholarly journals Compressibility and Moelwyn-Hughes Parameter of NaI crystals under high pressure, as the function of thermal properties

BIBECHANA ◽  
2012 ◽  
Vol 9 ◽  
pp. 88-91
Author(s):  
Arun Kumar Khan
Keyword(s):  
High Pressure ◽  
Thermal Properties ◽  
Potential Function ◽  
Power Function ◽  
Exponential Function ◽  
Ionic Crystal ◽  
Inverse Function ◽  
Inverse Power ◽  
Ionic Potential

The credit for the commencement of inter ionic potential function goes to Born and Lande, one of whose milestone has been their expression into inverse function. Similarly, Born and Mayer also proposed the exponential function of the potential for the repulsive component. However none of them could get the desired success. Subsequently, many more prominent scientists devoted their effort towards these thermal properties of the crystals, among which the endeavour of Jha and Thakur cannot be overlooked, who proposed a new interionic function to observe the properties of ionic crystal under high pressure. During its formulation they included both an inverse power function due to Born and Lande and an exponential function due to to Born and Mayer. Appreciably, Jha and Khan used it for studying the hyped thermal properties of NAI and CsI crystal under high pressure up to 100 Kilo bar. DOI: http://dx.doi.org/10.3126/bibechana.v9i0.7180 BIBECHANA 9 (2013) 88-91

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