scholarly journals Novel Theorems and Algorithms Relating to the Collatz Conjecture

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Michael R. Schwob ◽  
Peter Shiue ◽  
Rama Venkat
Keyword(s):  
Natural Numbers ◽  
Collatz Conjecture ◽  
Computer Scientists ◽  
The Relationship ◽  
Peak Value

Proposed in 1937, the Collatz conjecture has remained in the spotlight for mathematicians and computer scientists alike due to its simple proposal, yet intractable proof. In this paper, we propose several novel theorems, corollaries, and algorithms that explore relationships and properties between the natural numbers, their peak values, and the conjecture. These contributions primarily analyze the number of Collatz iterations it takes for a given integer to reach 1 or a number less than itself, or the relationship between a starting number and its peak value.

scholarly journals Transporting functions across ornaments

2014 ◽  
Vol 24 (2-3) ◽  
pp. 316-383 ◽  
Author(s):  
PIERRE-ÉVARISTE DAGAND ◽  
CONOR McBRIDE
Keyword(s):  
Type Theory ◽  
Structured Data ◽  
Dependent Types ◽  
Natural Numbers ◽  
Code Reuse ◽  
Domain Specific ◽  
Definition Of ◽  
The Relationship ◽  
Do So ◽  
Structural Invariants

AbstractProgramming with dependent types is a blessing and a curse. It is a blessing to be able to bake invariants into the definition of datatypes: We can finally write correct-by-construction software. However, this extreme accuracy is also a curse: A datatype is the combination of a structuring medium together with a special purpose logic. These domain-specific logics hamper any attempt to reuse code across similarly structured data. In this paper, we capitalise on the structural invariants of datatypes. To do so, we first adapt the notion of ornament to our universe of inductive families. We then show how code reuse can be achieved by ornamenting functions. Using these functional ornaments, we capture the relationship between functions such as the addition of natural numbers and the concatenation of lists. With this knowledge, we demonstrate how the implementation of the former informs the implementation of the latter: The users can ask the definition of addition to be lifted to lists and they will only be asked the details necessary to carry on adding lists rather than numbers. Our presentation is formalised in the type theory with a universe of datatypes and all our constructions have been implemented as generic programs, requiring no extension to the type theory.

Study of Earthquake Data for Cumulative Absolute Velocity and Acceleration Peak Value

2014 ◽  
Vol 539 ◽  
pp. 741-746
Author(s):  
Jie Ning Xia ◽  
Zhi Gao Chen ◽  
Jun Huang ◽  
Jiang Yang ◽  
Jian Yang ◽  
...  
Keyword(s):  
Lushan Earthquake ◽  
Absolute Velocity ◽  
Background Information ◽  
Ground Acceleration ◽  
Cumulative Absolute Velocity ◽  
Observation Systems ◽  
Recorded Data ◽  
Earthquake Observation ◽  
The Relationship ◽  
Peak Value

Characteristics of cumulative absolute velocity parameter (CAV) of Lushan earthquake is discussed and presented in this paper. Based on a brief analysis of the background information of the Lushan earthquake, the value of CAV which is calculated from the recorded data of the Lushan earthquake is compared with the commonly used value peak ground acceleration (PGA). Accordingly, the relationship between the CAV and the PGA is studied, and 3 CAV/PGA ratio charts in 3 different sub-directions are obtained. Then the linear fitting operation and the polynomial fitting operation are performed to analyze the potential discipline and characteristics thereof. The applicability of utilizing the CAV parameter in earthquake observation systems is further studied in this paper, and the CAV parameter is cooperated with the currently used value PGA to provide the work of earthquake observation and emergency response with corresponding theoretical basis.

Analysis of the Influence of Tunnel Blasting to the Neighbor Tunnel

2011 ◽  
Vol 199-200 ◽  
pp. 870-873
Author(s):  
Hai Liang Wang ◽  
Shu Cui Cong ◽  
Bi Jun Wang ◽  
Lin Sheng Liu
Keyword(s):  
Vibration Velocity ◽  
Vibration Test ◽  
Blasting Vibration ◽  
Guide Line ◽  
Analytical Results ◽  
Tunnel Blasting ◽  
The Relationship ◽  
Peak Value

According to the tunnel blasting vibration test at Kiaochow bay Cross-harbor Tunnel Guide Line Project, the regulation of the tunnel vibration velocity has been studied. Based on the analytical results, this paper finds that the change regulation of vertical, horizontal radial and tangential vibration velocity as the different distances from the work face. The tunnel vibration velocity of the rear work face is greater than the unexcavated area. The peak value of the rear work face is 2-2.5 times as large as that of the front work face, vibration velocity of the front work face attenuates gently. The paper figures out the relationship between vibration velocity and distance from sensor to the work face, which can offer a reference to similar studies.

scholarly journals On the Probabilistic Proof of the Convergence of the Collatz Conjecture

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Kamal Barghout
Keyword(s):  
Probability Distribution ◽  
Random Numbers ◽  
Natural Numbers ◽  
New Approach ◽  
Collatz Conjecture ◽  
Probabilistic Proof ◽  
Number Of Iterations

A new approach towards probabilistic proof of the convergence of the Collatz conjecture is described via identifying a sequential correlation of even natural numbers by divisions by 2 that follows a recurrent pattern of the form x,1,x,1…, where x represents divisions by 2 more than once. The sequence presents a probability of 50:50 of division by 2 more than once as opposed to division by 2 once over the even natural numbers. The sequence also gives the same 50:50 probability of consecutive Collatz even elements when counted for division by 2 more than once as opposed to division by 2 once and a ratio of 3:1. Considering Collatz function producing random numbers and over sufficient number of iterations, this probability distribution produces numbers in descending order that lead to the convergence of the Collatz function to 1, assuming that the only cycle of the function is 1-4-2-1.

Effect of water immersion on renal natriuretic peptide (urodilatin) excretion in humans

1993 ◽  
Vol 74 (6) ◽  
pp. 2881-2885 ◽  
Author(s):  
P. Norsk ◽  
C. Drummer ◽  
L. B. Johansen ◽  
R. Gerzer
Keyword(s):  
Atrial Natriuretic Peptide ◽  
Natriuretic Peptide ◽  
Tap Water ◽  
Water Immersion ◽  
Normal Subjects ◽  
Temporal Profile ◽  
And Control ◽  
The Relationship ◽  
Linear Correlations ◽  
Peak Value

We examined 1) the effect of thermoneutral (34.5 +/- 0.5 degrees C) water immersion to the neck (WI) in humans on the temporal profile of renal urodilatin [atrial natriuretic peptide- (ANP) (95–,126)] excretion and 2) the relationship between urodilatin and urinary fluid (V) and sodium (UNaV) excretion. Eight normal subjects underwent 12 h of WI, and another group of eight were studied during seated control conditions. The subjects ingested 200 ml of tap water hourly. WI induced an increase in renal urodilatin and guanosine 3′,5′-cyclic monophosphate (cGMP) excretion, V, and UNaV. After peak values were attained between the 2nd and 5th h of WI, urodilatin and cGMP excretion, V, and UNaV returned toward preimmersion and control levels. At the 12th h of WI, urodilatin and cGMP excretion and V were indistinguishable from preimmersion values but were significantly elevated compared with the control values. UNaV was maintained elevated compared with both preimmersion and control values. During WI, positive and statistically significant linear correlations could be established between V and renal urodilatin excretion in six subjects and between UNaV and urodilatin excretion in four subjects. We conclude that WI induces an increase in the rate of renal urodilatin excretion, attaining a peak value at the 3rd h followed by an attenuation toward preimmersion and control levels. Furthermore, urodilatin might participate as one of several mechanisms of the natriuresis and diuresis of WI in humans.

scholarly journals Sponge effect of aerated concrete on phosphorus adsorption-desorption from agricultural drainage water in rainfall

2020 ◽  
Vol 15 (No. 4) ◽  
pp. 220-227
Author(s):  
Jinquan Zhang ◽  
Weiguo Fu
Keyword(s):  
Drainage Water ◽  
Agricultural Drainage ◽  
Phosphorus Adsorption ◽  
P Adsorption ◽  
Initial Stage ◽  
Aerated Concrete ◽  
Initial Adsorption ◽  
Adsorption Desorption ◽  
The Relationship ◽  
Peak Value

In the initial stage of the rainfall, the nutrient element phosphorus (P) in the farmland, one of the most important factors causing agricultural non-point source pollution, flows into agriculture drainage ditches rapidly, and an instantaneous phosphorus peak value in the ditch water often occurs. Aerated concrete with high P adsorption properties was chosen as the experiment material in the laboratory to reduce the instantaneous P peak value in the drainage water in the initial stage of the rainfall. The three total P (TP) concentrations of the simulated drainage water (1.0, 2.0, and 3.0 mg/L) stood for three treatments were designed in the adsorption experiment; the same three TP concentrations of the simulated drainage water and the three TP concentrations of the simulated natural water (0.2, 0.3, and 0.4 mg/L) stood for nine treatments in the desorption experiment. The sponge effect of the aerated concrete on the P adsorption-desorption was explored by studying the dynamics of the P adsorption-desorption of the aerated concrete with an increase in the experiment’s time. The results showed the following details: (1) Both the adsorption rate and desorption rate of the aerated concrete decrease with an increase in the experiment’s time. The initial adsorption is dominant during the entire adsorption, as with the initial desorption during the entire desorption. (2) The adsorption capacity of the aerated concrete slightly decreases with the increase in the re-adsorption, whereas the desorption capacity of the aerated concrete significantly decreases with the increase in the re-desorption. Thus, the aerated concrete can be introduced into the agricultural drainage ditch to reduce the instantaneous P peak value in the drainage water in the initial stage of the rainfall, and potential further studies should explore the relationship between the different drainage water loads and the amount of the aerated concrete.

Interactions of intestinal lymph flow and secretion.

1977 ◽  
Vol 232 (1) ◽  
pp. E13 ◽  
Author(s):  
D N Granger ◽  
N A Mortillaro ◽  
A E Taylor
Keyword(s):  
Cholera Toxin ◽  
Venous Pressure ◽  
Intestinal Secretion ◽  
Venous Hypertension ◽  
Rapid Decrease ◽  
Lymph Flow ◽  
Constant Infusion ◽  
Concomitant Increase ◽  
The Relationship ◽  
Peak Value

The relationship between lymph flow and intestinal secretion was studied in an isolated, vascularly perfused cat ileal preparation in which secretion was caused by three different means: 1) plasma dilution from a constant infusion of Tyrode solution (2.5 ml/min per kg), 2) elevation of intestinal venous pressure to 30 mmHg, and 3) exposure to cholera toxin. In the plasma-dilution group, lymph flow attained a peak value of 38 times control at 60-90 min following the onset of the infusion, after which time lymph flow progressively decreased. Concomitant to the rapid decrease in lymph flow was a rapid increase in intestinal secretion (filtration secretion). A similar pattern, i.e., a rapid increase in lymph flow followed by a progressive decrease in lymph flow and concomitant increase in filtration secretion, was observed in the venous hypertension group; however, peak lymph flow (20 X control) was observed within 10 min of the pertubation. No correlation between the onset of intestinal secretion and a decline in lymph flow was observed in the cholera toxin group.

Ca2+ current and Ca2+ transients under action potential clamp in guinea pig ventricular myocytes

1991 ◽  
Vol 261 (2) ◽  
pp. C393-C397 ◽  
Author(s):  
J. Arreola ◽  
R. T. Dirksen ◽  
R. C. Shieh ◽  
D. J. Williford ◽  
S. S. Sheu
Keyword(s):  
Guinea Pig ◽  
Action Potential ◽  
Time Course ◽  
Ventricular Myocytes ◽  
Slow Component ◽  
Fast Component ◽  
Maximal Effect ◽  
The Relationship ◽  
Peak Value

Precise characterization of the magnitude and kinetics of transsarcolemmal Ca2+ influx during an action potential (AP) is essential for a complete understanding of excitation-contraction coupling in heart. Using a voltage-clamp protocol that simulated a physiological AP (AP clamp), we characterized the properties of the Ca2+ current (ICa) in guinea pig ventricular myocytes. The AP-generated ICa showed a complex time course that was different from ICa generated by a square pulse. ICa activated rapidly during the upstroke of the AP and then partially inactivated during the plateau. The fast component of ICa reached a peak value of -7.6 +/- 1.0 pA/pF at 2.40 +/- 0.30 ms after depolarization, followed by a slow component with a peak value of -2.9 +/- 0.4 pA/pF during the plateau. ICa generated by an AP was composed of both L- and T-type Ca2+ channels. T-type Ca2+ current contributed to the fast component of ICa and L-type Ca2+ current contributed to both fast and slow components of ICa. Activation of beta-adrenoceptors enhanced ICa with a maximal effect lasting throughout the entire plateau of the AP. Measurements of cytosolic Ca2+ transients using fura-2 indicated that the ICa was responsible for triggering Ca2+ release from the sarcoplasmic reticulum. The AP clamp provides a new approach for investigation of the relationship between ICa and Ca2+ transients under more physiological conditions.

Quantitative Evaluation of the Glass Forming Ability of (Cu-Zr) Based Glass Alloys with Thermodynamics Method

2011 ◽  
Vol 239-242 ◽  
pp. 1622-1625
Author(s):  
Hui Qiang Li ◽  
Long Fei Liu
Keyword(s):  
Cooling Rate ◽  
Nucleation Rate ◽  
Glass Forming Ability ◽  
Crystalline Fraction ◽  
Critical Cooling Rate ◽  
Glass Forming ◽  
Element Number ◽  
The Relationship ◽  
Peak Value ◽  
Crystalline Growth

Along nucleation → crystalline growth →crystalline fraction → critical cooling rate, the relationship between the nucleation, elements, cooling rate and the Glass Forming Ability of (Cu-Zr) based glass alloys is quantitatively studied with thermodynamics method, and a better method to evaluate the critical cooling rate of glass alloys is also proposed in this paper. The computed results show that: (1) with the increase of element number, the steady state nucleation rate drops gradually. From Cu-Zr, Cu-Zr-Al, Cu-Zr-Al-Ni, to Cu-Zr-Al-Ni-Ti, the peak value of nucleation rate decreases from 1021mol-1s-1to 1013mol-1s-1. It is also found the nucleation rate both drops with the substitution of Ni with Cu or Al with Zr; (2) with the increase of cooling rate, the nucleation rate drops sharply. When the cooling rate reaches 103K/s, the nucleation rates of Cu64Zr36, Cu54Zr42.5Al3.5, Cu55Zr40Al5and Cu30Zr55Al10Ni5drop to 109mol-1s-1, 106mol-1s-1, 107mol-1s-1and 103mol-1s-1accordingly.

scholarly journals Hardness and Conductivity of Die Forged ZYK530 Magnesium Alloy

2021 ◽  
Author(s):  
Fenghong CAO ◽  
Yaohui XU ◽  
Chang CHEN ◽  
Zhaohui QIN ◽  
Chi DENG
Keyword(s):  
Heat Treatment ◽  
Electrical Conductivity ◽  
Treatment Process ◽  
Mg Alloy ◽  
Heat Treatment Process ◽  
Optimum Heat ◽  
The Relationship ◽  
Optimum Heat Treatment ◽  
Good Agreement ◽  
Peak Value

The relationship among the microstructure, hardness and electrical conductivity of the as-forged ZYK530 Mg alloy after heat treatment was analyzed and studied using a microscope, X-Ray Diffractometer, eddy current conductivity meter, and Vickers microhardness tester, to explore optimum heat treatment process of ZYK530 Mg alloy. The results show that: with the prolongation of holding time, the electrical conductivity and microhardness show the same change trend, both of which show an oscillatory upward trend, and then decrease in an oscillatory downward trend after reaching the  peak value. There is a linear positive correlation between the conductivity and the hardness, and the fitting results of the conductivity and hardness are in good agreement with the measured results; combined with the actual production, when the heat-treatment is 480 ℃ × 8 h + 220 ℃ × 3 h, the highest hardness is 79.2 HV, the electroconductivity is 36.2%IACS, and the comprehensive performance is the best, which is the best heat treatment process.

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