scholarly journals Counting Defective Parking Functions

10.37236/816 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Peter J Cameron ◽  
Daniel Johannsen ◽  
Thomas Prellberg ◽  
Pascal Schweitzer
Keyword(s):  
Kernel Method ◽  
Rayleigh Distribution ◽  
The Other ◽  
Partial Sums ◽  
Square Root ◽  
Parking Space ◽  
Parking Functions ◽  
Parking Function ◽  
Car Park ◽  
Binomial Identity

Suppose that $m$ drivers each choose a preferred parking space in a linear car park with $n$ spaces. Each driver goes to the chosen space and parks there if it is free, and otherwise takes the first available space with a larger number (if any). If all drivers park successfully, the sequence of choices is called a parking function. In general, if $k$ drivers fail to park, we have a defective parking function of defect $k$. Let ${\rm cp}(n,m,k)$ be the number of such functions. In this paper, we establish a recurrence relation for the numbers ${\rm cp}(n,m,k)$, and express this as an equation for a three-variable generating function. We solve this equation using the kernel method, and extract the coefficients explicitly: it turns out that the cumulative totals are partial sums in Abel's binomial identity. Finally, we compute the asymptotics of ${\rm cp}(n,m,k)$. In particular, for the case $m=n$, if choices are made independently at random, the limiting distribution of the defect (the number of drivers who fail to park), scaled by the square root of $n$, is the Rayleigh distribution. On the other hand, in the case $m=\omega(n)$, the probability that all spaces are occupied tends asymptotically to one.

scholarly journals A further correspondence between $(bc,\bar{b})$-parking functions and $(bc,\bar{b})$-forests

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Heesung Shin ◽  
Jiang Zeng
Keyword(s):  
The Other ◽  
Parking Functions ◽  
Fixed Sequence ◽  
Parking Function ◽  
International Audience ◽  
Positive Integers ◽  
Appl Math

International audience For a fixed sequence of $n$ positive integers $(a,\bar{b}) := (a, b, b,\ldots, b)$, an $(a,\bar{b})$-parking function of length $n$ is a sequence $(p_1, p_2, \ldots, p_n)$ of positive integers whose nondecreasing rearrangement $q_1 \leq q_2 \leq \cdots \leq q_n$ satisfies $q_i \leq a+(i-1)b$ for any $i=1,\ldots, n$. A $(a,\bar{b})$-forest on $n$-set is a rooted vertex-colored forests on $n$-set whose roots are colored with the colors $0, 1, \ldots, a-1$ and the other vertices are colored with the colors $0, 1, \ldots, b-1$. In this paper, we construct a bijection between $(bc,\bar{b})$-parking functions of length $n$ and $(bc,\bar{b})$-forests on $n$-set with some interesting properties. As applications, we obtain a generalization of Gessel and Seo's result about $(c,\bar{1})$-parking functions [Ira M. Gessel and Seunghyun Seo, Electron. J. Combin. $\textbf{11}$(2)R27, 2004] and a refinement of Yan's identity [Catherine H. Yan, Adv. Appl. Math. $\textbf{27}$(2―3):641―670, 2001] between an inversion enumerator for $(bc,\bar{b})$-forests and a complement enumerator for $(bc,\bar{b})$-parking functions. Soit $(a,\bar{b}) := (a, b, b,\ldots, b)$ une suite d'entiers positifs. Une $(a,\bar{b})$-fonction de parking est une suite $(p_1, p_2, \ldots, p_n)$ d'entiers positives telle que son réarrangement non décroissant $q_1 \leq q_2 \leq \cdots \leq q_n$ satisfait $q_i \leq a+(i-1)b$ pour tout $i=1,\ldots, n$. Une $(a,\bar{b})$-forêt enracinée sur un $n$-ensemble est une forêt enracinée dont les racines sont colorées avec les couleurs $0, 1, \ldots, a-1$ et les autres sommets sont colorés avec les couleurs $0, 1, \ldots, b-1$. Dans cet article, on construit une bijection entre $(bc,\bar{b})$-fonctions de parking et $(bc,\bar{b})$-forêts avec des des propriétés intéressantes. Comme applications, on obtient une généralisation d'un résultat de Gessel-Seo sur $(c,\bar{1})$-fonctions de parking [Ira M. Gessel and Seunghyun Seo, Electron. J. Combin. $\textbf{11}$(2)R27, 2004] et une extension de l'identité de Yan [Catherine H. Yan, Adv. Appl. Math. $\textbf{27}$(2―3):641―670, 2001] entre l'énumérateur d'inversion de $(bc,\bar{b})$-forêts et l'énumérateur complémentaire de $(bc,\bar{b})$-fonctions de parking.

scholarly journals A Generalization of Parking Functions Allowing Backward Movement

10.37236/8948 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Alex Christensen ◽  
Pamela E. Harris ◽  
Zakiya Jones ◽  
Marissa Loving ◽  
Andrés Ramos Rodríguez ◽  
...  
Keyword(s):  
Recursive Formula ◽  
Parking Space ◽  
Parking Functions ◽  
Parking Function ◽  
Dyck Paths ◽  
Positive Integers

Classical parking functions are defined as the parking preferences for $n$ cars driving (from west to east) down a one-way street containing parking spaces labeled from $1$ to $n$ (from west to east). Cars drive down the street toward their preferred spot and park there if the spot is available. Otherwise, the car continues driving down the street and takes the first available parking space, if such a space exists. If all cars can park using this parking rule, we call the $n$-tuple containing the cars' parking preferences a parking function.   In this paper, we introduce a generalization of the parking rule allowing cars whose preferred space is taken to first proceed up to $k$ spaces west of their preferred spot to park before proceeding east if all of those $k$ spaces are occupied. We call parking preferences which allow all cars to park under this new parking rule $k$-Naples parking functions of length $n$. This generalization gives a natural interpolation between classical parking functions, the case when $k=0$, and all $n$-tuples of positive integers $1$ to $n$, the case when $k\geq n-1$. Our main result provides a recursive formula for counting $k$-Naples parking functions of length $n$. We also give a characterization for the $k=1$ case by introducing a new function that maps $1$-Naples parking functions to classical parking functions, i.e. $0$-Naples parking functions. Lastly, we present a bijection between $k$-Naples parking functions of length $n$ whose entries are in weakly decreasing order and a family of signature Dyck paths. 

scholarly journals THE PREDICTION OF PARKING SPACE AVAILABILITY

Transport ◽  
2020 ◽  
Vol 35 (5) ◽  
pp. 462-473
Author(s):  
Helena Brožová ◽  
Miroslav Růžička
Keyword(s):  
Real Time ◽  
Personal Communication ◽  
Navigation Systems ◽  
Parking Space ◽  
Transition Matrices ◽  
Communication Services ◽  
Proposed Model ◽  
Harmful Emissions ◽  
Parking Lot ◽  
Car Park

Intelligent Parking Systems (IPS) allow customers to select a car park according to their preferences, rapidly park their vehicle without searching for the available parking space (place) or even book their place in advance avoiding queues. IPS provides the possibility to reduce the wastage of fuel (energy) while finding a parking place and consequently reduce harmful emissions. Some systems interact with in-vehicle navigation systems and provide users with information in real-time such as free places available at a given parking lot (car park), the location and parking fees. Few of these systems, however, provide information on the forecasted utilisation at specific time. This paper describes results of a traffic survey carried out at the parking lot of supermarket and the proposal of the model predicting real-time parking space availability based on these surveyed data. The proposed model is formulated as the non-homogenous Markov chains that are used as a tool for the forecasting of parking space availability. The transition matrices are calculated for different time periods, which allow for and include different drivers’ behaviour and expectations. The proposed forecasting model is adequate for potential use by IPS with the support of different communication means such as the internet, navigation systems (GPS, Galileo etc.) and personal communication services (mobile-phones).

scholarly journals Constraining planetesimal stirring: how sharp are debris disc edges?

2021 ◽  
Vol 503 (4) ◽  
pp. 5100-5114
Author(s):  
Sebastian Marino
Keyword(s):  
Surface Density ◽  
Semimajor Axis ◽  
Rayleigh Distribution ◽  
Outer Edge ◽  
The Other ◽  
Dust Production ◽  
Array Data ◽  
Disc Surface ◽  
Sharp Edges ◽  
Good Agreement

ABSTRACT The dust production in debris discs by grinding collisions of planetesimals requires their orbits to be stirred. However, stirring levels remain largely unconstrained, and consequently the stirring mechanisms as well. This work shows how the sharpness of the outer edge of discs can be used to constrain the stirring levels. Namely, the sharper the edge the lower the eccentricity dispersion must be. For a Rayleigh distribution of eccentricities (e), I find that the disc surface density near the outer edge can be parametrized as tanh [(rmax  − r)/lout], where rmax  approximates the maximum semimajor axis and lout defines the edge smoothness. If the semimajor axis distribution has sharp edges erms is roughly 1.2lout/rmax  or erms = 0.77lout/rmax  if semimajor axes have diffused due to self-stirring. This model is fitted to Atacama Large Millimeter/submillimeter Array data of five wide discs: HD 107146, HD 92945, HD 206893, AU Mic, and HR 8799. The results show that HD 107146, HD 92945, and AU Mic have the sharpest outer edges, corresponding to erms values of 0.121 ± 0.05, $0.15^{+0.07}_{-0.05}$, and 0.10 ± 0.02 if their discs are self-stirred, suggesting the presence of Pluto-sized objects embedded in the disc. Although these stirring values are larger than typically assumed, the radial stirring of HD 92945 is in good agreement with its vertical stirring constrained by the disc height. HD 206893 and HR 8799, on the other hand, have smooth outer edges that are indicative of scattered discs since both systems have massive inner companions.

scholarly journals Efficacy of Fungicide Applications During and After Anthesis Against Fusarium Head Blight and Deoxynivalenol in Soft Red Winter Wheat

Plant Disease ◽  
2014 ◽  
Vol 98 (10) ◽  
pp. 1387-1397 ◽  
Author(s):  
D. L. D'Angelo ◽  
C. A. Bradley ◽  
K. A. Ames ◽  
K. T. Willyerd ◽  
L. V. Madden ◽  
...  
Keyword(s):  
Winter Wheat ◽  
Fusarium Head Blight ◽  
Field Experiments ◽  
The Other ◽  
Square Root ◽  
Head Blight ◽  
Soft Red Winter Wheat ◽  
Untreated Check ◽  
Red Winter Wheat ◽  
Days After Anthesis

Seven field experiments were conducted in Ohio and Illinois between 2011 and 2013 to evaluate postanthesis applications of prothioconazole + tebuconazole and metconazole for Fusarium head blight and deoxynivalenol (DON) control in soft red winter wheat. Treatments consisted of an untreated check and fungicide applications made at early anthesis (A), 2 (A+2), 4 (A+4), 5 (A+5), or 6 (A+6) days after anthesis. Six of the seven experiments were augmented with artificial Fusarium graminearum inoculum, and the other was naturally infected. FHB index (IND), Fusarium damaged kernels (FDK), and DON concentration of grain were quantified. All application timings led to significantly lower mean arcsine-square-root-transformed IND and FDK (arcIND and arcFDK) and log-transformed (logDON) than in the untreated check; however, arcIND, arcFDK, and logDON for the postanthesis applications were generally not significantly different from those for the anthesis applications. Relative to the check, A+2 resulted in the highest percent control for both IND and DON, 69 and 54%, respectively, followed by A+4 (62 and 52%), A+6 (62 and 48%), and A (56 and 50%). A+2 and A+6 significantly reduced IND by 30 and 14%, respectively, relative to the anthesis application. Postanthesis applications did not, however, reduce DON relative to the anthesis application. These results suggest that applications made up to 6 days following anthesis may be just as effective as, and sometimes more effective than, anthesis applications at reducing FHB and DON.

Are There Different Kinds of Rogue Waves?

2008 ◽  
Vol 130 (2) ◽  
Author(s):  
Paul C. Liu ◽  
Keith R. MacHutchon
Keyword(s):  
Rogue Wave ◽  
Rogue Waves ◽  
Rayleigh Distribution ◽  
The Other ◽  
South Indian Ocean ◽  
Gas Drilling ◽  
Wave Measurements ◽  
South Indian ◽  
Theoretical Considerations

There is clearly no immediate answer to the question posted by the title of this paper. Inasmuch as that there are not much definitively known about rogue waves and that there is still no universally accepted definition for rogue waves in the ocean, we think there might just be even more than one kind of rogue waves to contend with. While the conventional approach has generally designated waves with Hmax∕Hs greater than 2.2 as possible rogue waves, based on Rayleigh distribution considerations, there is conspicuously no provision as to how high the ratio of Hmax∕Hs can be and thus not known how high can a rogue wave be. In our analysis of wave measurements made from a gas-drilling platform in South Indian Ocean, offshore from Mossel Bay, South Africa, we found a number of cases that indicated Hmax∕Hs could be valued in the range between 4 and 10. If this were to be the case, then these records could be considered to be “uncommon” rogue waves, whereas a record of Hmax∕Hs in the range between 2 and 4 could be considered to comprise “typical” rogue waves. On the other hand, the spikes in the Hmax data could have been caused by equipment malfunction or some other phenomenon. Clearly, the question of whether or not there are different kinds of rogue waves cannot be readily answered by theoretical considerations alone and there is a crucial need for long-term wave time-series measurements for studying rogue waves.

Are There Different Kinds of Rogue Waves?

2006 ◽  
Author(s):  
Paul C. Liu ◽  
Keith R. MacHutchon
Keyword(s):  
Time Series ◽  
Rogue Waves ◽  
Rayleigh Distribution ◽  
The Other ◽  
South Indian Ocean ◽  
Gas Drilling ◽  
Wave Measurements ◽  
South Indian ◽  
Theoretical Considerations

Inasmuch as there is as yet still no universally accepted definition for rogue waves in the ocean, we think there might just be more than one kind of rogue waves to contend with. While the conventional approach has generally designated waves with Hmax/Hs greater than 2.2 as possible rogue waves, based on Rayleigh distribution considerations, there is conspicuously no provision as to how high the ratio of Hmax/Hs can be. In our analysis of wave measurements made from a gas-drilling platform in South Indian Ocean, offshore from Mossel Bay, South Africa, we found a number of cases that indicated Hmax/Hs could be valued in the range between 4 and 10. If this were to be the case these records could be considered to be “uncommon” rogue waves, whereas a record of Hmax/Hs in the range between 2 and 4 could be considered to comprise “typical” rogue waves. On the other hand the spikes in the Hmax data could have been caused by equipment malfunction or some other phenomenon. Clearly the question of whether or not there are different kinds of rogue waves can not be readily answered by theoretical considerations alone and there is a crucial need for long-term wave time series measurements for studying rogue waves.

scholarly journals FPGA-Based Synthesis of High-Speed Hybrid Carry Select Adders

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
V. Kokilavani ◽  
K. Preethi ◽  
P. Balasubramanian
Keyword(s):  
High Speed ◽  
Boolean Logic ◽  
The Other ◽  
Square Root ◽  
Verilog Hdl ◽  
Other Hand ◽  
Carry Select Adder

Carry select adder is a square-root time high-speed adder. In this paper, FPGA-based synthesis of conventional and hybrid carry select adders are described with a focus on high speed. Conventionally, carry select adders are realized using the following: (i) full adders and 2 : 1 multiplexers, (ii) full adders, binary to excess 1 code converters, and 2 : 1 multiplexers, and (iii) sharing of common Boolean logic. On the other hand, hybrid carry select adders involve a combination of carry select and carry lookahead adders with/without the use of binary to excess 1 code converters. In this work, two new hybrid carry select adders are proposed involving the carry select and section-carry based carry lookahead subadders with/without binary to excess 1 converters. Seven different carry select adders were implemented in Verilog HDL and their performances were analyzed under two scenarios, dual-operand addition and multioperand addition, where individual operands are of sizes 32 and 64-bits. In the case of dual-operand additions, the hybrid carry select adder comprising the proposed carry select and section-carry based carry lookahead configurations is the fastest. With respect to multioperand additions, the hybrid carry select adder containing the carry select and conventional carry lookahead or section-carry based carry lookahead structures produce similar optimized performance.

Cloud Based Smart Parking System

2020 ◽  
Vol 17 (4) ◽  
pp. 1578-1583
Author(s):  
V. Sathya ◽  
Bheemanadham ◽  
Surya Sai ◽  
Arpit Tharad ◽  
Ayush Kumar ◽  
...  
Keyword(s):  
Urban Communities ◽  
Traffic Congestion ◽  
Network Architecture ◽  
Performance Metrics ◽  
Parking Space ◽  
Parking Spot ◽  
Parking System ◽  
Car Park ◽  
Smart Parking ◽  
Internet Of Things Technology

Lately, the idea of brilliant urban communities has increased awesome notoriety. Parking becomes one of the tedious tasks in our day to day life. Many efforts have been done in this field to solve parking based problems like traffic congestion, limited parking space, high parking charges, and the most basic problem is to find nearby parking space. In this paper, we proposed an algorithm that increases the efficiency of the current cloud-based smart parking system and develops a network architecture based on Internet-Of-Things technology. Our proposed system helps users automatically find a free parking space at the least cost based on new performance metrics to calculate the user parking cost by considering the distance from the user to parking area and the total number of free places in each car park. A mobile application is provided that navigate the user to the free parking space based on the efficiency based on distance and cost. This cost will be utilized to offer an answer of finding an accessible parking spot upon a demand by the client and an answer of proposing another car park if the present park is full.

scholarly journals XIX. —Consideration of the objections raised against the geometrical representation of the square roots of negative quantities

1829 ◽  
Vol 119 ◽  
pp. 241-254 ◽  
Keyword(s):  
The Other ◽  
Square Root ◽  
Square Roots ◽  
The Third ◽  
Oblique Direction ◽  
Real Nature ◽  
Positive Quantity ◽  
Algebraic Operations ◽  
The One ◽  
Straight Lines

Some years ago my attention was drawn to those algebraic quantities, which are commonly called impossible roots or imaginary quantities: it appeared extraordinary, that mathematicians should be able by means of these quan­tities to pursue their investigations, both in pure and mixed mathematics, and to arrive at results which agree with the results obtained by other independent processes; and yet that the real nature of these quantities should be entirely unknown, and even their real existence denied. One thing was evident re­specting them; that they were quantities capable of undergoing algebraic operations analogous to the operations performed on what are called possible quantities, and of producing correct results: thus it was manifest, that the operations of algebra were more comprehensive than the definitions and funda­mental principles; that is, that they extended to a class of quantities, viz. those commonly called impossible roots, to which the definitions and funda­mental principles were inapplicable. It seemed probable, therefore, that there was a deficiency in the definitions and fundamental principles of algebra ; and that other definitions and fundamental principles might be discovered of a more comprehensive nature, which would extend to every class of quantities to which the operations of algebra were applicable; that is, both to possible and impossible quantities, as they are called. I was induced therefore to examine into the nature of algebraic operations, with a view, if possible, of arriving at these general definitions and fundamental principles: and I found, that, by considering algebra merely as applied to geometry, such principles and definitions might be obtained. The fundamental principles and definitions which I arrived at were these: that all straight lines drawn in a given plane from a given point, in any direction whatever, are capable of being algebra­ically represented, both in length and direction; that the addition of such lines (when estimated both in length and direction) must be performed in the same manner as composition of motion in dynamics; and that four such lines are proportionals, -both in length and direction, when they are proportionals in length, and the fourth is inclined to the third at the same angle that the second is to the first. From these principles I deduced, that, if a line drawn in any given direction be assumed as a positive quantity, and consequently its oppo­site, a negative quantity, a line drawn at right angles to the positive or nega­tive direction will be the square root of a negative quantity, and a line drawn in an oblique direction will be the sum of two quantities, the one either posi­tive or negative, and the other, the square root of a negative quantity.

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