The best way to gain credibility and trust with patients is to share engaging, human stories about other successful patients.
When done correctly, case studies serve as a powerful marketing tool, establishing your authority as an expert and answering questions about therapy, through the patient's voice.
They create the right psychological framework for patients (this therapist is going to help me because she helped others) and forge an emotional bond between your practice and your patients. The readers can relate to someone else's situation, as if it were their own.
This helps the clinic achieve the desired outcome – the patient returns for more services and invests in cash paying services and related products.
They recommend you to family and friends through verbal and social media communication. They are more willing to pay a premium for services and products, and the clinic enjoys a positive reputation.
Everybody wins, because trust and loyalty is established. This trust is more important than all the marketing in the world, but it takes work to earn it.
Trust in companies and businesses is on the decline, and that's why it's so important to use case studies to gain, and to build that trust.
According to an article in the Washington post, the rise of 'virtual space' and text messaging has made us less trusting.
Trust equals loyalty, but trust is getting more difficult to earn (and maintain).
That's where case studies can help demonstrate expertise, commitment and success. Marketing expert, Steve Rayson, noted that case studies establish credibility and provide evidence. They provide a learning experience for the reader and unique content that competitors can't easily copy.
Highly specific case studies provide more details, are more specific, and a tactic that's surpassed only by an in-person interaction as a method of social proof.
In other ways, the best way to motivate and influence a patient is in person.
The next best way is through case studies.
A case study encompasses 400-500 words, while testimonials range from 50-100 words. Readers want details about symptoms and results to compare with their own situation and case studies are tailor made to accomplish that.
In an interview with author Bob Bly, copywriter Heather Sloan explains that everyone loves a story and the greatest value of a case study is that it evokes emotion within the reader. Memorable stories remain with individuals longer than a clinical dissertation or pitch about the clinic's value.
When the clinician's audience sees the words case study, they perceive it as more real and assign a higher value to the information. It includes quotes from actual people.
Testimonials are an important part of physical therapy marketing, but the way in which they're written often appears like an advertisement to potential patients, turning them off.
One of the biggest patient deterrents is uncertainty and it's critical that practitioners understand this and find ways to dispel it. People don't really care about the process. They just want to know the results.
Can physical therapy really help?
New patient decisions are based on emotion and the level of risk versus benefits, combined with countless other qualifiers ranging from practical to financial concerns. During each encounter, either on line or in a clinical setting, patients are unconsciously collecting reasons to continue with the practitioner or go elsewhere.
The more evidence clinicians can provide when marketing physical therapy, the more fear and uncertainty is removed from the potential patient.
Individuals generally avoid taking risks, especially when they have no experience from which to draw a conclusion.
Potential patients are more likely to become actual patients after reading cases studies, but the key is to get them to actually read a blog post, newsletter or download information.
Headlines that create curiosity and interest draw readers in, but physical therapy marketing requires that the headline also be descriptive and the content results-oriented.
A few statistics and percentages may be helpful, but readers want the human touch in a case study. They want to know how the patient felt, how their life was limited and the circumstances that led to the need for physical therapy. Readers want to be able to relate to the patient. They want specifics and practitioners can provide that without disclosing any confidential information.
A case study should focus on the patient and be a real account of what happened, not a veiled advertisement.
The audience wants to feel as if they could have been the patient that was treated. Explain why the specific treatment was chosen and maximize impact with images. An image breaks up a long piece of copy, adds interest, and can convey complex concepts to further understanding.
Many marketers advocate a photo of the patient, but it can have a myriad of legal ramifications involving patient confidentiality and privacy.
Clinicians who choose to ask a patient for the use of their image must ensure the individual understands how it will be used and obtain their permission in writing. It is also important to consult an attorney, and the state practice act before you use a patients picture or full name. To be on the safe side, de-identify the patient as much as possible, using first name, initial of last name, and avoiding the use of pictures.
Just because a case study can be told doesn't mean it should. The most effective case studies provide readers with a positive or beneficial end result. Case studies can be used as a powerful tool not only because they demonstrate how patients can be helped but they should also reinforce the brand.
Content strategist Amanda Farmer has worked extensively with case studies for Fortune 500 businesses. She says an effective case study should provide a quick summary, the challenge faced, the solution and results, and a clear conclusion that tells readers the customer's plans for the future or what they can do now that they couldn't do before.
The reader that has followed throughout the entire case study will be interested enough or envious of the patient's result and want it for their own.
It's the perfect opportunity to insert a call to action by making an appointment, opt-in for a newsletter, or download information.
The Internet has changed the way consumers and patients locate the products and services they need. Patients used to rely almost exclusively on word-of-mouth recommendations from friends and family when they sought medical expertise. The Internet has expanded that circle of referral sources through a variety of social media platforms, websites and blogs.
According to Pardot senior content strategist, Jenna Hanington, today's individuals are increasingly engaged in research through reviews, testimonials, videos and case studies.
Modern patients want to see the human aspect behind brands. Clinicians that aren't afraid to share that information are doing themselves a huge favor toward creating trust that translates into conversions.
A valuable tool for managing and implementing a physical therapy business plan is the In Touch EMR™ integrated software. It tracks and monitors multiple advertising efforts and campaigns, provides analytics and metrics, and offers an easy way to distribute newsletters. The software is capable of communicating with established patients and opt-ins in a variety of ways that fits patients' busy, mobile lifestyles.
A physical therapy practice should use case studies to answer questions, and make it a no-brainer for the patient to say 'yes' to physical therapy. Building trust through case studies results in increased trust and credibility that can be transitioned into more patient visits.
Click here to register for Private Practice Summit 2019 and to learn more.
You now have the opportunity to join the ranks of the most successful and highest earning physical therapists on the planet when you attend the 2019 Private Practice Summit Event on October 11, 12 and 13 in Las Vegas, NV.
You'll have access to every tool, resource, and system available to help you reach your goals.
If you want to learn more about how to create case studies and take advantage of other trends, I go into a lot more detail during the event.
This is ideal for owners, clinicians, front desk staff, managers, and billing staff.
Click below to learn more about the educational sessions, schedule and hotel information and secure your fast action price tickets.
This is a must-attend event to take your practice to the next level, and to learn the things they don't (but should) teach us in PT school. The conference is ideal owners, clinicians, front desk staff, managers, and billers in physical therapy private practice.
Click here to learn more about the educational sessions, schedule and hotel information and to secure your tickets.


What makes us diffrent? Most of the diagnostic interfaces available at an affordable price are only Chinese counterfeits products. Poor quality, you can't count on updates and you don't have contact with the technical support. What more - most of them don't have necessary approvals so that is a reason you can get unpleasantness during the inspection in the workshop.
Below You will find the search engine of MaxiEcu functions. Check which brands and functe purchase please make sure to what extent does MaxiEcu run a chosen brand.
Check out what extent does MaxiEcu run a chosen brand. Remember, that we are still working to improve our software - You can be sure that we will add new brands and develop possibilities.
The price is the basic price with a USB interface. Wireless interfaces are more expensive, after selecting the interface the current price of the set will appear.
What interface are you choosing?


Hello! I'm Sarah–a recovering teacher, workaholic turned work-at-home/stay-at-home mom to twins, and wife to the most incredible woman I've ever met.
I have a bachelor's and master's in education and 8 years of experience teaching middle school English. My planned and organized world turned upside down when our twin boys were born. I never imagined myself as a stay-at-home mom; I never even thought I would marry, much less have kids. When I had kids, I believed I would be a better mom if I worked outside the home. Turns out I was wrong. Taking care of my family became the most important thing to me and I grew tired of giving the best of myself to a career that didn't value me. Professional turned wife and mom, that's my story.
I've wanted to have a blog for as long as I remember, but I have always lacked the self-confidence to do it.
Between the best cheering section ever (my wife) and the desire to share our parenting story, I have finally found the motivation to begin. I knew I wanted to be a teacher when I was a kid, well before I ever started school. I have believed for years that teaching is a gift and a calling and always viewed my classroom as a mission field. Honestly, I wholeheartedly believed that I could help make the world a better place if I was a teacher. Maybe I did, maybe I didn't, but I gave it my best. I gave all of myself, and now I feel at peace about this transition to something else, whatever it may be.
I believe that I accomplished whatever I was called to accomplish and my new calling is to make the world better by raising good kids who are equipped to fight for justice and love.
So now I'm here at The Same Sunset–a name inspired by a quote from S.E. Hinton's novel The Outsiders–my favorite novel to teach. "Maybe the two different worlds we live in weren't so different. We saw the same sunset," says Ponyboy Curtis. It is my reminder that even though my life and goals have changed drastically, the sunset hasn't. And even though you and I may be totally different, we still see the same sunset–so maybe you can find some comfort in my stories–or a laugh, some inspiration, or help.
Does this feel like a scattered mess?
That's my life now. Mom Brain is REAL and somewhat debilitating. I'm still working through it. In the meantime, I am coping by continuing to develop my book obsession (Harry Potter), upping my crafting/scrapbooking game (I just upgraded my Cricut Expression to the Cricut Maker!) and learning more about Anglican traditions. The wife and I were both raised in conservative Protestant traditions, but have since joined Episcopal/Lutheran church. The transition has been wonderful.
Follow me on social media and you can see just what a huge mess it is!
Is self-doubt preventing you from following a new path? Click here to read my blog post about how I overcame self-doubt and wound up at The Same Sunset. Then download my 28 Day Follow Your Dreams Inspirational Journal to start your own journey of confidently chasing your dreams.


text:
The year 1927 in television involved some significant events.
Below is a list of television-related events during 1927.

Global television events

Births

References

redpajama_set_name: RedPajamaWikipedia


"The right of the foolish to be guided and governed by the wise, is the first right of man, beside which all the others pale in comparison..."
Who gets to decide who is wise? It is a foolish question. It is pellucid, to any with understanding, who is wiser - he who knows what to do in an emergency IS the government. Khuenhelt-Ledhin used to tell the story about a sinking ship, in which a stampede to the lifeboats is stopped by a passenger with a gun; this passenger demands that the men do their duty, that women and children be allowed first, and then that an orderly evacuation follow. It cannot be said that this is democratic; nor can it be said to be peaceful or "normative"; nor can it be said that it is voluntary. And yet it is, in a sense, voluntary. The man with the gun might have happened to be a violent, selfish criminal. Indeed, in such situations, this may be the first element to show its head. Much as in the story of Wyatt Earp, the West faces an increasing lawlessness in which it is unclear what "law and order" any longer mean. In these situations, even the most peaceful, milk-drinking, kindliest old gentleman may well pull forth an old rusty Birmingham horse pistol, and hoarsely demand of the rising element of chaos or criminals that they "be so good as to retire, & that with promptitude". This is not democracy, but it is good government. Those who resist the anarchists, the opportunists, and the tyrants - look to these. It matters not what form of government they impose; it will be doubly, triply just - just because they oppose the wrong; just because they are good men; just because they are brave. They may well be said to be Jefferson's natural aristoi, of whom Romans 13 was written in truth, or Calvin's "lesser magistrates" - men who do not fear death enough to bow before its Incarnation. Pray for these men, and support them. Let the devil back his gift to us of the magical power of 51%.


text:
Need For A Comprehensive Traffic Simulation Model In Indian Context
Download & View Need For A Comprehensive Traffic Simulation Model In Indian Context as PDF for free.
Need for a Comprehensive Traffic Simulation Model in Indian Context M. Sreekumar, Post Graduate Student Akhilesh Kumar Maurya, Assistant Professor Department of Civil Engineering, Indian Institute of Technology Guwahati, Assam- 781039. Email: [email protected]
In developing countries like India, the rapid growth of cities has resulted an extensive increase in the road network. The numbers of motorized vehicles have increased manifolds, especially during the peak hours. Congestion problems in cities simultaneously alarm both the users as well as the decision makers. Controlling, suggesting remedial measures, making changes to the existing traffic system in urban India is becoming more and more complex day after day. Traffic Engineering is hence coming to the limelight in India to understand and analyze the traffic stream behaviour for an efficient design of traffic facilities. Unfortunately, the large body of literature that exists in the area of traffic flow theory cannot be used as is in countries like India due to the following reasons. Firstly, traffic here includes innumerable classes of vehicles having different operating characteristics. Secondly, drivers in India do not follow lanes i.e., the traffic streams do not have lane discipline. Therefore any study on Indian traffic scenario will be meaningful only if it can incorporate both vehicular heterogeneity and "no lane discipline" conditions. To analyze the traffic stream behaviour which is very complex and stochastic in nature, one need to understand how vehicles interact each other not only along the longitudinal but also along the lateral direction of the lane and also with the existing traffic facilities? So while modeling the heterogeneous traffic conditions prevailing in most of the developing countries, the system have to be viewed microscopically. The complexity of stream behaviour and the difficulty that exists with performing experiments make computer simulation an important analysis tool in traffic engineering. Secondly, through simulation, it is easier to model individual vehicle actions in different situations than to understand the system dynamics directly. By making use of different traffic simulation models, one can simulate large scale real-world situations in great detail. This paper is being written, identifying the necessity of a comprehensive microscopic traffic simulation model which can simulate Indian traffic scenario in a realistic manner. A brief review of various categories of models consisting of longitudinal control models, Lateral control models and combined lateral and longitudinal control models is included in this paper. It is felt that the quality of any model lies in the level of realism present in the traffic stream simulated using that simulation model. A few recent models are concisely explained to address the fact that the assumptions made in them are far from Indian traffic conditions. 1.2
A PRECIS OF MICROSCOPIC TRAFFIC SIMULATION MODELS
A realistic model of driving or driver behaviour must be a comprehensive model which models both lateral control (steering control) and longitudinal control (speed control) under the impact of both roadway and traffic features [1]. According to this aspect of control, microscopic models can be categorized into (i) longitudinal control models, (ii) lateral control models, and (iii) comprehensive models and discussed in present section.
Longitudinal Control Models
Longitudinal control models assume that the driver behaviour is influenced only by leading vehicles traveling in the same path or lane. This assumption is largely true where the road characteristics are reasonably same for long distances, vehicles have well demarcated paths and vehicles do not generally cross these demarcations. The process of achieving the longitudinal control is referred to as car-following in transportation literature. The car-following behaviour is basically, a human perception-reaction process, where driver of the following vehicle (FV) attempts to reach a stable condition (by accelerating or decelerating) and then maintain it while following a leading vehicle (LV). In this process of reaching a stable situation and maintaining it, the driver of following vehicle continuously takes corrective actions by accelerating or decelerating [1]. Research on car-following theory started dates back to the mid 20th century. From then, a lot of studies have been done in order to develop ever more accurate models on car-following. In the early 1950's, car-following behaviour was modeled using a stimulus-response concept. The research team of General Motors (GM) Company produced five generations of car-following models which were all based on the analogy that the response of the following driver (acceleration or deceleration) is a function of the sensitivity of this driver and the stimulus. This stimulus is generally the relative speed between the two vehicles and the sensitivity term is affected by the distance headway between the two vehicles and the speed of the following vehicle. Some models suggested a linear increase of minimum safe distance headway with increasing speed. Subsequently safety distance models were formulated based on the driving tendency to keep a safe following-distance. The model proposed by Gipps relates the sped of FV to parameters like distance between LV and FV, desired speed of FV, current speeds of FV and LV, maximum acceleration of FV, maximum decelerations of LV and FV through a set of two equations [2]. Later psycho-physical or action point models were developed based on the idea that a driver can be in different driving modes like free driving, approaching, following and braking. The driver behaviour in car-following mode was also modeled using certain rule-based models by specifying some set of rules on what to do under different driving scenarios [3]. Although the cellular automata models which can be considered as the advanced series of rule-based models are computationally efficient and can simulate large traffic streams, the simulated streams often do not replicate many of the observed microscopic properties of real-world streams. There is many more car following models available in the literature, in addition to the above summary. Each model has certain advantages and disadvantages. A detailed review of these models can be found in [4]. A few traffic flow models which seem to be more relevant in Indian context are briefed in the next section of this paper. 1.2.2
Lateral Control Models
Most of the works on lateral control models relates to how different roadway objects and obstacles initiate a lateral shift in driver's path. These studies reinforce the fact that objects other than those on the travel path of a vehicle also have impact on the driver's actions. The steering control in lane maintenance was also studied under very restrictive conditions. The intent of this study was to see how precisely humans can steer the vehicles. None of these studies, however, actually try to model what path a driver would like to use in the presence of various static and dynamic obstacles [1]. 1.2.3
Comprehensive Models As discussed above, the final goal of microscopic modeling is to be able to devise a comprehensive model which models both lateral and longitudinal control of vehicles as a comprehensive whole. There are comprehensive models developed based on a force field (or potential field) analogy of the driving
environment which consist of goals (where the driver wants to be) and obstacles (dynamic as well as static).The force field models perform well when a simulated driver's or a simulated platoon's behaviour is evaluated. But, the computational inefficiency of these models prevents their use in simulating large traffic streams [5]. The CARSIM simulation model, developed for the simulation of stop-and-go traffic had incorporated a collision a collision algorithm and a minimum separation constraint for car-following [6]. Another simulation model, INTELSIM was also developed in which the reaction times of drivers were not restricted by simulation-time steps [7]. Since all these studies have been conducted under relatively homogeneous traffic conditions, these models cannot find their application in heterogeneous traffic streams. Now a days, traffic simulation packages like CORSIM and VISSIM are frequently used as tools for analyzing traffic. VISSIM is a microscopic, time-step and behaviour based simulation model developed to analyze the full range of functionally classified roadways and public transportation operations. Since all these are commercial software packages, sufficient changes to the internal parameters satisfying customized user requirements cannot be made easily. 1.3
OTHER TRAFFIC FLOW MODELS RELEVANT TO INDIAN SCENARIO
Arasan and Koshy developed a simulation model suitable for replicating heterogeneous traffic flow, based on the interval scanning technique with fixed increment time advance [8]. It has been found that the method of treating the entire road space as a single unit, for the purpose of simulation, and representing the different types of vehicles as rectangular blocks on the surface, is appropriate for simulating highly heterogeneous traffic flow. This model adopted sequential system to update the positions of vehicles. Dey et al. have developed simulation program for traffic flow modelling on a two-lane road. Random placement for vehicles is assigned within a lane [9]. Moreover, some clearance is also incorporated from edge and shoulder of a roadway. Logical overtaking is implemented using a set of well defined rules to imitate continuous decision making of drivers. This study is not limited only to speed-flow relationship, while model is applied to estimate the capacity of the road, to check the variation of PCU (passenger car unit) values for mix traffic, and effect of traffic mix on capacity. A modified car-following model has been developed by Gunay, with particular reference to weak discipline of lane based driving [10]. The movement of the following vehicle was formulated as a function of the off-centre effects of the leader(s). This incorporation of lateral friction offers a potential breakthrough in the fields of car-following theory and microscopic simulation of traffic flow. Using a stopping-distance car following approach, Gunay pointed out the effect of the travel path width on the following vehicle, and the reduced following distance with increased lateral separation between the leader and the follower. The model also incorporated the concept of Maximum Escape Speed (MES) to the conventional Gipp's model. But the model had no reference on vehicular heterogeneity to find their applications in Indian context. Further attempts were also made to modify the widely used Gipp's car-following model to find its use in heterogeneous traffic conditions. The model stated in [11] had incorporated the vehicle-type dependent parameters to the conventional model. The choice of model and their parameters can be estimated only after elaborate field studies addressing driver and vehicle heterogeneity. Even though this approach attempted to model explicitly the vehicle-type dependent behaviour in traffic simulation, the effect of vehicle-type on lane changing and non-lane based movement of vehicles needs to be considered in order to replicate traffic flow in a greater detail.
CUTSiM which was developed as a comprehensive microscopic model, is found to be more relevant in Indian perspective since it can be used for simulating the traffic stream with and without lane discipline. A description about this model is also briefed in the next section of this paper. From the review of various microscopic simulation models presented in this paper, it is very clear that a full-fledged simulation model need to be developed which can simulate Indian traffic conditions in a realistic way. 1.4
CUTSiM – COMPREHENSIVE UNINTERRUPTED TRAFFIC SIMULATION MODEL
As discussed earlier, the ultimate goal of modeling traffic flow must be to evolve a comprehensive model of driver behaviour which can handle both longitudinal and lateral interactions and also simple enough to be used to simulate large traffic streams. CUTSiM (Comprehensive Unidirectional Traffic Simulation Model) developed by Maurya (2007), find its use as a suitable program package to simulate heterogeneous, uninterrupted traffic streams with or without lane discipline i.e. similar to Indian traffic streams. Like any microscopic models, the description of traffic stream behaviour presented in this model is also a discussion on drivers and their behaviour. The model concentrates on studying and modeling unidirectional and uninterrupted traffic streams and also, the program code was written in C language. It was developed as a discrete-event traffic simulation model, using interval scanning technique with fixedincrement time advance. The road space is discretized into cells with specific dimensions l and w. Time is discretized into small time-steps, ∆t at which the system is updated. This model consists of two modules; one of the modules defines the actions of the vehicle aimed at controlling its lateral position on the road while the other module defines the actions of the vehicle aimed at controlling its speed. Both the modules are discussed in next two subsections. 1.4.1
Lateral Control Module
The lateral control module describes how a driver chooses a suitable steering angle to maintain his/her vehicle on the "best path." The best path is assumed to be the one which allows a driver to move safely and at satisfactory speeds given the driving environment at that time. It is hypothesized here that in order to choose the best path the driver evaluates the goodness of each available path within its vicinity based on (i) the maximum distance headway available on the path, (ii) the difficulty in maneuvering towards that path, (iii) the obstacles present in and around the path,(iv) whether the path ends up (within a short time) leading the vehicle to the edge of the road, and (v) whether the path crosses the expected path of faster moving vehicles coming from behind. Simple functions are defined to capture the impact of these factors on goodness of path. All these individual goodness values are multiplied to determine the overall goodness of a path. The path which has the maximum goodness among the available paths (within the vicinity) at any instant of time is assumed as the best path for the movement of that vehicle that point of time. Also, all the goodness values based on different criteria mentioned above have equal weight in deciding the best path based on the maximum goodness. So, all these five goodness values vary within the equal range (0 and 1). 1.4.2
Longitudinal Control Module
The longitudinal control module defines the actions of the driver (in terms of acceleration or deceleration) which he/she employs to control the speed or alternatively the longitudinal position of the vehicle. It is postulated that the driver's behaviour can be classified in two types: (i) when the driver behaviour is not impeded by the actions of other vehicles, and (ii) when the driver behaviour is impacted by the actions of the leading vehicle on the chosen path. The first type is referred to as "free-flow" behaviour while the other is referred to as "car-following" behaviour. In the free-flow mode, actions of the driver is assumed to depend on the difference between his/her desired speed (speed which one wants to maintain if
not impeded by other vehicles for a given road condition) and the current speed. Actions of the driver in carfollowing mode is assumed to depend on the anticipated relative speed, relative acceleration and the deviation of available distance headway from stable distance headway (distance headway at which a driver feels safe while moving at a given speed). The distance headway at which a driver switches from one mode of behaviour to another is referred to as the threshold distance headway. CUTSiM utilizes both modules at every instant of time for each vehicle to simulate large traffic streams. It employs a parallel update strategy and incorporates the driver reaction time explicitly. Vehicles are initially generated over the entire road space of vehicle generation stretch with desired lateral and longitudinal clearances between generated vehicles. CUTSiM can be used to simulate single lane traffic streams as well as traffic streams on wide roads with and without lane discipline. In fact, it has been shown that lane behaviour can be obtained as a special case of "no lane behaviour" traffic streams. 1.5
CUTSiM is used to simulate single lane traffic streams as well as traffic streams on wide roads with and without lane discipline. Further traffic streams on roads of varying widths and surface conditions with a variety of vehicle mix have been simulated. These simulated streams have been studied for their microscopic and macroscopic properties. Some of the study results are briefed in this paper. Results from these studies and their comparisons with appropriate observations from real-world traffic streams have been used to validate CUTSiM. 1.5.1
Microscopic Analysis
The various microscopic properties studied include (i) stability behaviour of CUTSiM in carfollowing situations, (ii) acceleration noise, and (iii) speed distribution. CUTSiM, when used to simulate single lane homogeneous traffic flow with cars, only the longitudinal control module is operational and this situation can be used to analyze car-following behaviour. It is observed that simulated stream shows local stability, closing-in and shying-away behaviour but detailed results are not provided due to space restriction. A five vehicles platoon is simulated to study asymptotic stability. All the vehicles are placed at 32.31 m (106 ft) initial distance headways and 18.29 m/s (60 ft/s) initial speed. leading vehicle (LV) moves at constant speed till 1 s then it accelerated its speed from 18.29 m/s to 24.38 m/s (80 ft/s) in 4 s then decelerate from 24.38 m/s to 12.19 m/s (40 ft/s) in 8 s and again accelerated to attain its initial speed in 4 s. Distance headway variation between other following vehicle due to this perturbation of LV is presented in Fig. 1(a). It can be seen from figure that perturbation gets damped as it propagated upstream side (i.e. i increases) which shows the simulated stream is the asymptotically stable. The above mentioned study establishes that CUTSiM behaves realistically from the perspective of car-following. Although acceleration noise is an average quantity, it is still discussed here because it relates to driver actions in terms of acceleration at every time-step. In a way, this value indicates the driving pattern of drivers. Fig. 1(b) shows the acceleration noise of different vehicle types at different density levels in simulated streams on two lane wide road without lane discipline. It is observed that acceleration noise initially increases with density and later it decreases with further increase in density. This is similar to as observed by other researchers [14]. The observed speed distributions from Delhi-Gurgaon highway (NH8) is compared with the simulated speed distributions in order to study how well CUTSiM replicates a real-world traffic stream on wide roads with no lane discipline. The comparison is shown in Fig. 2(a). The traffic on Delhi-Gurgaon
Figure 1(a) Study of asymptotic stability of single lane traffic simulated using CUTSiM and (b) Acceleration noise obtained from CUTSiM simulated heterogeneous traffic stream on two lane wide roads without lane discipline.
highway is unidirectional, uninterrupted and mainly consists of passenger cars (≈70%) and motorized two wheelers (≈30%). The road is 14.6 m wide. The observed speed distribution is found to be reasonably symmetric around the mean speed of 64.16 km/hr. A similar traffic condition has been simulated using CUTSiM and the speed distribution of simulated traffic is also distributed symmetrically around a mean speed of 64.92 km/hr. The observed and simulated data shows a good match with an RMS error of 0.016 and also satisfies the KS-test at 5% significance level. Speed-flow comparison of observed and simulated stream is also compared and result is presented in macroscopic analysis subsection. 1.5.2
Macroscopic Analysis
The macroscopic features include pair wise relations between speed, flow and density. These macroscopic properties of single lane and two lane traffic flow are evaluated using the flow simulated using CUTSiM. The simulated stream is assumed to have 100% passenger-cars. Idealized speed-density and speed-flow plots presented in Fig. 3(a) and 3(b) respectively shows the behaviour as expected in real-world traffic streams. Table 1 shows the capacity values obtained and those suggested in Highway Capacity Manual for both the above cases. It can be observed from the table that unlike the multilane highway, doubling the road width does not result in doubling of capacity of road. This is due to increase lateral interaction between vehicles in case of no lane discipline traffic. For validation purpose, a comparison of speed-flow relation observed at Delhi-Gurgaon highway and that obtained from a traffic stream simulated using CUTSiM (for similar road conditions and vehicle mix) is also made. Fig. 2(b) shows a good match in observed and simulated speed-flow relationship and the
Figure 2(a) Speed distributions and (b) Speed-flow relationships obtained from simulated and observed traffic at
Delhi-Gurgaon highway
Figure 3(a). Speed-density and (b) Speed-flow relationship obtained from simulated single and two lane road traffic stream using CUTSiM TABLE 1. OBTAINED AND SUGGESTED CAPACITY VALUES Road width (m)
Capacity obtained using CUTSiM in passenger-cars/hr/lane
Capacity suggested in Highway Capacity Manual in passenger-cars/hr/lane
2200 NA NA NA
capacity values are also found to be a closer match with an RMS error value of 1.76 and it also satisfies the KS-test at 5% significance level. CUTSiM can even model lane behaviour as a special case of the more generic "no lane discipline" behaviour which proves CUTSiM as truly comprehensive model to simulate all kinds of traffic streams. The way CUTSiM converted to behaviour with lane discipline is by simply suggesting that a driver shifts to a new path only if the new path is "substantially" better than the present otherwise continue in his/her current path even if its "goodness" value is not the best. Note that in the "no lane discipline" case the driver would have moved to the path with maximum goodness even if the gain in goodness over current path was marginal. CUTSiM is evaluated for various microscopic and macroscopic properties of lane disciplined traffic streams like speed-flow relationship, speed distribution, acceleration noise, time headway distribution, lane change frequency, etc. However these results are not presented here due to space restrictions. CUTSiM can simulate large traffic streams in reasonable time (i.e. it can simulate 4000 vehicles in less than half of the real time) on desktop computers (with dual core 2.2 GHz processor and 32 GB RAM).
As explained before, this study is primarily motivated by the necessity of a comprehensive simulation model which can simulate Indian traffic conditions in a realistic way. Keeping in view of this, various models relevant to Indian context have been discussed in the previous sections. It can very well observed from the discussion on these models that even though there are models which can incorporate vehicular heterogeneity, they cannot be applied directly to simulate Indian traffic scenario. This is due to the fact that the drivers here do not follow any lane discipline. Therefore in the present study, the methodology of a comprehensive model which is found closer to replicate Indian traffic conditions was briefly explained and
the results were analyzed both microscopically and macroscopically. The results of above studies show that CUTSiM is able to simulate Indian traffic streams with and without lane discipline on various kinds of roads with different vehicle mixes reasonably realistically. It has been seen that CUTSiM can simulate large streams very realistically with limited computation power. Further, a large scale validation with field data is required for its wider acceptability. However, a full-fledged comprehensive simulation model needs to be developed considering the complex behaviours such as vehicular interactions in bidirectional streams and their non-lane based movement. With such step by step improvements, simulation models can be expected to get refined, that in future, there will be models that can study Indian traffic conditions in an appreciable way. REFERENCES [1] A. K. Maurya,"Development of a comprehensive microscopic model for simulation of large uninterrupted traffic streams without lane discipline," PhD Thesis, Indian Institute of Technology, Kanpur, 2007. [2] P. G. Gipps, "A behavioural car-following model for computer simulation," Transportation Research Part B, vol. 15(2), pp. 105-111, 1981. [3] P. Chakroborty and S. Kikuchi, "Evaluation of the general motors based car-following models and a proposed fuzzy inference model," Transportation Research C, vol. 7(4), pp. 209-235, 1999. [4] M. Brackstone and M. McDonald, "Car-following: A historical review," Transportation Research F, vol. 2(4), pp. 181-196, 1999. [5] S. Gupta, P. Chakroborty and A. Mukharjee, "Microscopic simulation of vehicular traffic on congested roads," Proceedings of International Symposium on industrial robotic system – 98, Bangalore, 1998. [6] R. F. Benekohal and J. Treiterer, "CARSIM: Car-following model for simulation of traffic in normal and stop-and-go conditions,"Transportation Research Record, vol. 1194, Transportation Research Board, National Research Council, Washington, DC, pp. 99-111, 1988. [7] M. F. Aycin and R. F. Benekohal, "Comparison of car-following models for simulation," Transportation Research Record, vol. 1678, pp. 116-127, 2000. [8] P. P. Dey, S. Chandra and S. Gangopadhyay, "Simulation of mixed traffic flow on two-lane roads," Journal of Transportation Engineering, ASCE, vol. 134(9), pp. 361-369, 2008. [9] B. Gunay, "Car-following theory with lateral discomfort", Transportation Research Part B, vol. 41, pp. 722-735, 2007. [10] K. V. R. Ravishankar and Tom. V. Mathew, "Vehicle-type dependent car-following model for heterogeneous traaffic conditions," Journal of Transportation Engineering, ASCE, in press. [11] A. K. Maurya and P. Chakroborty, "CASIM: A realistic CA based traffic flow model," Proceedings of 11th World Conference on Transport Research, Berkeley, California, USA, 2007. [12] P. Chakraborty and A. Das, "Principles of Transportation Engineering," Prentice Hall of India Private Limited, New Delhi, 2003. [13] A. D. May, "Traffic Flow Fundamentals," Prentice Hall, Englewood Cliffs, New Jersey, 1990. [14] T. Winzer, "Measurement of Acceleration Noise and Discussion of the Energy Model Development by Drew", Transportation Research A, 15 (6), pp. 437-443, 1981.
A Queuing Model For Road Traffic Simulation
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redpajama_set_name: RedPajamaCommonCrawl


text:
6 Best Delta 8 Social Media Accounts to Follow in 2022
General November 29, 2022
CBD oil, CBD Products, Delta 8 Social Media, Delta-8
You may be familiar with delta-8, a cannabinoid extracted from legal hemp plants. It has been a flood of hemp-derived talent creating a fantastic new business, with fascinating brands appearing per day and a symphony of well-known CBD brands battling for the Delta 8 mainstream. Not just us but others are using CBD. The CBD sector has grown extraordinarily over the past few years. It has been made feasible by several studies proving the advantages of CBD and the broad legalization of the substance.
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Celebrities that promote and use CBD & THC
Alessandra Ambrosio, Victoria's Secret supermodel and parent to two children, devoted 17 years to walk the runways for the well-known lingerie company. She has appeared in a lot of Hollywood productions.
Given her active lifestyle, Alessandra constantly concerns about the importance of getting proper rest and sleep. Alessandra stated that utilizing CBD oil has immensely facilitated her in achieving the eight hours of beauty sleep she truly deserves while in an interview with an American magazine.
Kim Kardashian, a recent multi-millionaire turned entrepreneur and mother of four children, is well-known for more than just her phenomenally successful cosmetics line. She has only lately begun utilizing CBD cosmetics, including face creams. Kim states that she uses CBD gummies routinely to help her sleep well at night.
Kim is one of the celebrities with the largest social media followings and one of the most well-liked. On her massive 214 million-follower Instagram feed, Kim has often been seen promoting CBD products. Just a few weeks before Psalm West was born, she held a CBD-themed baby shower to raise awareness of the advantages of CBD among her family and friends.
The man with a terrific record for his voice-over roles has long supported marijuana legalization. Freeman got involved in a car accident in 2008 that almost cost him his life. Jaws of life had to be used to release him. Freeman had fibromyalgia following several operations to heal his shoulder. The medical disorder fibromyalgia affects the nervous system and results in physical pain and emotional distress.
Freeman claimed that marijuana is the one thing that helps with the fibromyalgia agony in his left arm. When it comes to children who experience seizures, marijuana is proven to remarkably minimize them, allowing these youngsters to lead healthy lives. Another well-known supporter of marijuana legalization says that since marijuana is less adverse to use than alcohol, it can't be a toxic substance.
Morgan acknowledges that he consumes CBD and THC products and that he takes cannabis for both therapeutic and recreational purposes.
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Beautiful American actress and director Olivia Wilde recently shared her love of CBD with the media. Olivia discussed her introduction to CBD with the New York Times and Vanity Fair. She was introduced to CBS by a friend, and it ultimately helped her manage the chronic pain she had been dealing with for six months. Olivia Wilde thinks CBD offers a healthier approach to the prescription painkillers we often use. She claims that due to misconceptions about cannabis, CBD has often been criticized.
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The 47-year-old has been open and honest about her healing process, including how postpartum depression led to her looking into holistic health treatments. She said I love this topic (cannabis). It can alter old thought patterns and transform people's lives. We could be on edge about something scientifically far more significant than we think.
We've reached the end of our list now. Many more well-known celebrities, not included in the names on our list, also use and support cannabis. Given that so many celebrities have publicly acknowledged using CBD, its use is likely to increase in the coming years. Of course, anyone can use CBD products, but with its expanding legality and research into its possible benefits, we could only anticipate CBD's popularity to keep rising.


text:
New York teams that can make it to March Madness
James Hastings 11 months ago No Comments
With March Madness just around the corner, 68 college basketball teams would be up against each other in one match eliminator format. Going on in the NCAA Division-1 tournament, only a few teams have the potential of claiming the title and consolidating their reputation in the circuit.
Teams hoping to directly qualify for March Madness would need to win their conference tournament or get a winning bid from the NCAA Tournament Selection and Organization Committee.
Like every year this year, too, teams would be gearing up for thrilling matches and pre-season qualifiers in conference tournaments. The exciting action would begin from the first week of March, with the final to be played on 4th April 2022.
According to sources, till now, 32 teams have gained systematic entry in March Madness by winning their respective conference tournaments.
According to recent sources and standings, the top 16 seeds include Gonzaga, Auburn, Arizona, Kansas, Baylor, Kentucky, Purdue, Duke, Villanova, Texas Tech, Tennessee, Illinois, Wisconsin, UCLA, Providence, and Texas.
Talking about New York teams that can make it to March Madness include Syracuse and the other 3 teams which are capable of going through March Madness 2022. Though the upcoming Selection Sunday would be an amazing clash of rivals to find a spot in March Madness.
Every state across the US has invested thousands of dollars in upgrading its college sports program structure.
Teams capable of making It through the March Madness from New York in NCAA Division-1
A list of teams representing New York can be prepared, but only a few have the caliber and instincts to perform on such an important stage and make it through the Selection Sunday. Continue reading to find out about the possible teams that can make it through and how one can bet on their selective odds and outcomes.
If you are betting on Iona or any of the teams of New York, just ensure what Caesars sportsbook NY is offering. With Caesars sportsbook, one can make instant online payments, betting on their favorite college basketball team.
Moreover, every online user is entitled to decent signup or welcome bonus. If you are playing regularly at Caesars, there's a chance you might get frequent promotional offers and rewards. Mentioned below are NCAA Division 1 team from New York that can make it through March Madness this year.
Syracuse University- Syracuse Orange Men's Basketball
Cornell University – Cornell Big Red Men's Basketball
Siena College – Siena Saints
Iona College- Iona Gaels
St John's University- St John's University- St John's Red Storm Men's Basketball
For years Syracuse has claimed the throne of being the best NCAA Division-1 College in New York. With seasoned players averaging around 8 game points per match, Syracuse would be tough to beat in Selection Sunday event. Syracuse has a consistent track record playing, representing the state of New York in the NCAA Division 1 tournament.
Apart from their loss against Pittsburgh, they have managed to win big games and handle pressure. If you are betting on pre-match odds, then it should definitely be Syracuse University boys. In the upcoming days, Syracuse Orange would be up against 7 Duke Blue Devils, UNC, AND Miami, FL.
Cornell University- Cornell Big Red Men's Basketball
Cornell has also demonstrated great game skills over the years, managing to take the game against Kansas to the wire, keeping everyone interested in the final outcomes.
They are well reputed and well placed in the NCAA Division-1 above St John's. Cornell also won against Alabama, and since then, they are on their merry way of winning and taking challenges to the wire. On 6th March, Cornell Big Red would be up against Columbia, and it would certainly be a thriller of a game.
Siena College- Siena Saints
Siena Saints have been successful in winning their last 10 games on the trot. Siena has a well-gelled pool of college players who have the best ability to perform under pressure and secure big wins.
Siena Saints are looking to make an impact and make it through to March Madness starting in a couple of weeks. Despite winning against Cornell (32 -37), they couldn't suppress Cornell in NCAA rankings as they weren't able to win their conference game.
With the likes of Quinn Slazinski, Dylan van Eyck, Trey James, Elijah Joiner, Nelly Junior Joseph, Ryan Myers, Walter Clayton Jr., and Osborn Shema in the lineup, Iona Gaels is looking for a formidable lineup. From warming the 9th spot, they have done considerably well to rise at 3rd. Iona Gaels recently managed to suppress Baylor to 58 points, depicting their shear defense which is almost the best in MAAC.
St John's University- St John's Red Storm Men's Basketball
St John's University Red Storm Men's Basketball team would be backing on the availability of Julian Champagnie, Aaron Wheeler, Posh Alexander, Joel Soriano, Tareq Coburn, and other main players in their lineup.
St John's has considerably done well in Big East Conference. In the 2019-2020 season, St John's Red Storm was ranked 9th in NCAA Division 1 with a total of 1871 wins. St John's have has shown consistency in the last 3 games. St John's have made their way through the top 5 NCAA Division 1 from New York State.
Final word!
Be it any team from New York or Texas; it would be coming out all guns blazing to make most of the opportunity and make it through to the lateral stages of March Madness.
Teams from New York are backed with decent players and readily polished coaching staff that can outsmart opponents in the league. Be it Syracuse Orange men's basketball team or Cornell Big Red men's basketball team, it would be an exciting clash between the top teams coming from various states of the US.


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text:
\section{Introduction}
Classical inverse and implicit function theorems have attracted many researchers because of their applications in mathematics.  These theorems  are stated for the  class of $C^k$ mappings, and there have been researches for non-smooth mappings and global expansions. We recall some typical results of them.

\vspace*{8pt} \noindent
F. H. Clarke (1976 - \cite{C1}) presented local inverse function theorem for Lipschitz mappings.
Let $ f: \mathbb{R}^n \rightarrow  \mathbb{R}^n $ be a Lipschitz mapping in a neighborhood of $x_0 \in \mathbb{R}^n$. If the generalized Jacobian $ \partial f(x_0)$ at $ x_0 $  is of maximal rank (see Def. \ref{dn2}, \ref{dn3} below), then there exist neighborhoods $ U $ and $ V $ of $ x_0 $ and $ f(x_0) $, respectively, and a Lipschitz function $g: V  \rightarrow  \mathbb{R}^n$ such that

(a) $g(f(u)) = u $ for all $ u \in U$,

(b) $f(g (v)) = v $ for all $ v \in V$.

\vspace * {8pt} \noindent
More general, M. S. Gowda (2004 -\cite{Go}) considered inverse and implicit function theorems for the class of H - differentiable mappings. For the global case,
J. Hadamard (1906 -\cite{H}) presented global diffeomorphism conditions for $ C^1$ mapping class. Generalizing the results of J. Hadamard, P. J. Rabier (1997 -\cite{R}) demonstrated the results for $C^1$ mapping class  on smooth manifolds. O. Gut\'{u} and J. A. Jaramillo (2007 -\cite{Gu-J}) demonstrated global invertible conditions for the class of quasi-isometric mappings between complete metric spaces. Recently, T. Fukui, K. Kurdyka, and L. Paunescu (2010 -\cite{F-K-P}) demonstrated some global inverse function theorems for the class of tame continuous mappings.

\vspace*{8pt} Most of the results on the inverse and implicit function theorems only show the existence of neighborhoods $ U $ and $ V $ to ensure $ f: U \rightarrow V $ is invertible. The quantitative assessments for the subjects in the results were not considered.
It is necessary to use the quantitative assessments for these theorems in several different fields such as: number theory, optimization, theory of measurement, assessment of complex algorithms, ...

\vspace*{8pt} Up to now, in  general case, the problem of quantitative assessment for the classical inverse and implicit function theorems  is unresolved. For the case $ n\leq 2$, P. Henrici (1988 - \cite{He}) gave a quantitative form for analytic function (one variable). Recently, D. Cohen (2005 - \cite{Y}) gave a different proof for the case of analytic functions.  Under the result,  a quantitative form for the theorems in the case of analytic function with two variables was given.

\vspace*{8pt} In this paper, we present a quantitative form for the Clarke inverse function theorem, where $U$, $V$ and the Lipschitz constant of inverse mapping are evaluated quantitatively by  $\partial f (x_0)$. Moreover, we also give a quantitative form for Lipschitz implicit function theorem and prove  that the class of Lipschitz mappings satisfying Clarke's theorem are open: If $f$ is perturbed by a mapping $h$ with  the Lipschitz constant small enough, then the mapping $f + h$ is locally invertible.

The remaining of the paper is organized as follows.
In Section 2 we introduce necessary concepts and results.
Section 3 presents the main results and examples.

\section{Preliminaries}
\subsection{Perturbations and the Inverse}
We give here some definitions, notations and results that will be used later.
\begin{itemize}
\item[] Let $\mathbf{M}_{m\times n}$ denote the vector space of real $m \times n$ matrices,
\item[] $\|x\| =(|x_1|^2 + \cdots + |x_n|^2)^{\frac{1}{2}}$, ~where $x\in \mathbb{R}^n$,\\
$\mathbf{B}^n$ denotes the unit ball in $ \mathbb{R}^n$, $\mathbf{B}_r^n$ denotes the ball of radius $r$, centered at $0\in \mathbb{R}^n$, $\mathbf{B}_r^n(x_0)$ denotes the ball of radius $r$, centered at $x_0\in \mathbb{R}^n$, and $\mathbf{S}^{n-1}$ denote the unit sphere in $\mathbb{R}^n$,
\item[] $\|A\| = \max_{\|x\|=1}\|Ax\|, ~$where $A \in \mathbf{M}_{m\times n}$,
\item[] $\|A\|_F = \ds\left(\sum_{i=1}^m\sum_{j=1}^n\|a_{ij}^2\|\right)^\frac{1}{2},$ where $A$ is a $m\times n$ matrix,
    \item[] If $ A \in \mathbf{M}_{n\times n}$ is an invertible matrix, then
\[\|A^{-1}\|=\frac{1}{\min_{\|x\|=1}\|Ax\|}.\]
\end{itemize}
\noindent Matrix norms have some of the following properties:
\begin{itemize}
\item[(i)] $\|AB\| \leq \|A\|\|B\|$, $\|AB\|_F \leq \|A\|_F\|B\|_F$.
\item[(ii)] $\|A\| \leq \|A\|_F \leq \sqrt{n}\|A\|$, where $A$ is a $m\times n$ matrix.
\item[(iii)] For all $ A \in \mathbf{M}_{m\times n}$ and $ x \in \mathbb{R}^n$, we have $\|Ax\| \leq \|A\| \|x\|$.
\end{itemize}
We topologize $\mathbf{M}_{m\times n}$ with the norm $\|\cdot\|$, and $\mathcal{B}_{m\times n}$ denote the unit ball in $\mathbf{M}_{m\times n}$.
\begin{lem}\label{bd1}
 If $F \in \mathbf{M}_{n\times n}$ and $\|F\|< 1$, then $I - F$ is nonsingular and
  \[(I - F)^{-1} = \sum_{k=0}^\infty F^k\]
 with
 \[\|(I - F)^{-1}\| \leq \frac{1}{1 - \|F\|}.\]
 \end{lem}
\begin{proof} See \cite[Lemma 2.3.3]{G-L}.
\end{proof}
\noindent Based on Lemma \ref{bd1}, we have the following theorem.
\begin{thm}\label{dl1}
Let $A, E \in \mathbf{M}_{n\times n}$. If $A$ is nonsingular and $r = \|A^{-1}E\| < 1$, then $A + E$ is nonsingular and $\|(A + E)^{-1} - A^{-1}\| \leq \|E\|\|A^{-1}\|^2/(1-r)$.
\end{thm}
\begin{proof} See \cite[Theorem 2.3.4]{G-L}.
\end{proof}
\subsection{Generalized Jacobians}
\begin{defin}\label{dn1}
A mapping $ f: \mathbb{R}^m  \rightarrow  \mathbb{R}^n $ is called \textbf{Lipschitz} in a neighborhood of a point $ x_0 $ in $ \mathbb{R}^n $  if there exist a constant $ K $ such that for all $x$ and $y$ near $ x_0 $, we have
\begin{equation}\label{ct1}
\|f(x) - f(y)\|\leq K\|x-y\|.
\end{equation}
\noindent
 If $K \geq 1$ and
 \[\frac{1}{K}\|x-y\| \leq \|f(x) - f(y)\| \leq K \|x-y\|,\]
then $f$ is called \textbf{bi-Lipschitz} or $K$-bi-Lipschitz.
 \end{defin}
 \begin{thm}[Rademacher] \label{dlr} If $f: \mathbb{R}^m \rightarrow \mathbb{R}^n$ is Lipschitz, then $f$ is almost everywhere differentiable.
\end{thm}
\begin{proof}See \cite[Theorem 3.1.6]{F}.
\end{proof}
\noindent The usual $n \times m$ Jacobian matrix of partial derivatives of $f$ at $x$, when it exists, is denoted by $Jf(x)$. By Rademacher's theorem, we have the following definition.
\begin{defin}[F. H. Clarke - \textrm{\cite{C1}, \cite{C2}}]\label{dn2}  The \textbf{Generalized Jacobian} of $f$ at $x_0$, denoted by $\partial f(x_0)$, is the convex hull of all matrices $M$ of the form
\[M = \lim_{i\rightarrow \infty}Jf(x_i),\]
where $x_i$ converges to $x_0$ and $f$ is differentiable at $x_i$ for each $i$.\\
When $f: \mathbb{R}^m \rightarrow \mathbb{R}$, $\partial f(x_0)$ is called the generalized gradient of $f$ at $x_0$.
\end{defin}
\begin{defin} \label{dn3} $\partial f(x_0)$ is said to be of \textbf{maximal rank} if every $M$ in $\partial f(x_0)$ is of maximal rank.
\end{defin}
\begin{rem}
From (\ref{ct1}), $ \partial f(x) $ is bounded in the neighborhood of $x_0$.
\end{rem}
\begin{prop}[\cite{C1}]\label{md1}
$\partial f(x_0)$ is a nonempty compact convex subset of $\mathbf{M}_{n\times m}$.
\end{prop}
\begin{lem}[\cite{C1}] \label{bd22}Let $\varepsilon$ be a positive number. Then for all $x$ sufficiently near $x_0$,
\[\partial f(x) \subset \partial f(x_0) + \varepsilon \mathcal{B}_{n \times m}.\]
\end{lem}
\subsection{Topology of Lipschitz mappings}
Let $f: \mathbb{R}^m \rightarrow \mathbb{R}^n$. Then the Lipschitz constant of $f$ is defined by
\[\textrm{L}(f) = \sup \left\{\frac{\|f(x)-f(y)\|}{\|x-y\|}, ~ x \neq y\right\}.\]
Note that $f$ is Lipschitz if and only if $\textrm{L}(f) < \infty$.\\
Set
$$\textrm{Lip}(\mathbb{R}^m, \mathbb{R}^n) = \left\{f: \textrm{L}(f) < +\infty \right\}.$$
For $f, g \in \textrm{Lip}(\mathbb{R}^m,  \mathbb{R}^n)$ and $\alpha \in \mathbb{R}$, we have the following properties:
\begin{itemize}
\item[(i)] $f+g, \alpha f \in \textrm{Lip}(\mathbb{R}^m,  \mathbb{R}^n)$,
\item[(ii)] $\textrm{L}(f) \geq 0$,
\item[(iii)] $\textrm{L}(f+g) \leq \textrm{L}(f) + \textrm{L}(g)$,
\item[(iv)] $\textrm{L}(\alpha f) = \alpha \textrm{L}(f)$,
\item[(v)] $\textrm{L}(f) = 0 \Leftrightarrow f = \textrm{constant}$.
\end{itemize}

\vspace*{8pt}\noindent
By (v), for $x_0 \in \mathbb{R}^m$, set
\[\textrm{Lip}_{x_0}(\mathbb{R}^m,  \mathbb{R}^n) = \left\{f: f ~\textrm{is Lipschitz  and}~ f(x_0) = 0 \right\}.\]
Then
\[L(f) = 0 \Leftrightarrow f \equiv 0, ~~ \textrm{for all}~ f \in \textrm{Lip}_{x_0}(\mathbb{R}^m,  \mathbb{R}^n).\]
Thus $\textrm{Lip}_{x_0} (\mathbb{R}^m, \mathbb{R}^n) $ is a normed vector space with the norm $\textrm{L}(\cdot)$.
\section{Results - Examples}
Applying the results of F. H. Clarke (\cite{C1}), perturbation matrix, the properties of Lipschitz mappings and differentiable mappings, in Theorem \ref{dl2} we present a quantitative form of the Lipschitz inverse function theorem of F. H. Clarke, in Theorem \ref{dl4} we give a quantitative form of the Lipschitz implicit function theorem, openness of the class of Lipschitz mappings satisfying Clarke's inverse function theorem is proved in Theorem \ref{dl3} and Corollary  \ref{hq31}.
\begin{thm}[c.f.   {\cite[Theorem 1]{C1}}] \label{dl2}
Let $f: \mathbb{R}^n \rightarrow\mathbb{R}^n$ be a Lipschitz mapping with Lipschitz constant $K$. If $~\partial f(x_0)$ is of maximal rank, set
\[\delta = \frac{1}{2}\inf_{M_0\in \partial f(x_0)}\frac{1}{\|M_0^{-1}\|},\]
$r$ be chosen so that $f$ satisfies Lipschitz condition (\ref{ct1})  and
$$\partial f(x) \subset \partial f(x_0) + \delta \mathcal{B}_{n\times n},~~ \textrm{when}~~x \in \mathbf{B}_r^n(x_0),$$ then there exist neighborhoods $U$ and $V$ of  $x_0$ and $f(x_0)$, respectively, and a Lipschitz mapping $~g: V \rightarrow \mathbb{R}^n$ such that

(a) $g(f(u)) = u $ for every $u \in U$,

(b) $f(g(v)) = v $ for every $v \in V$,\\
where,
\[U=\mathbf{B}^n_{\frac{r\delta}{2}.\frac{1}{K}}(x_0), ~~V= \mathbf{B}^n_{\frac{r\delta}{2}}(f(x_0) ),~~\textrm{and}~~L(g) = \ds\frac{1}{\delta}.\]
\end{thm}
First we have:
\begin{lem}[c.f. {\cite[Lemma 3]{C1}}] \label{bd2}
Let $ f: \mathbb{R}^n \rightarrow \mathbb{R}^n $ be a Lipschitz mapping, $\partial f(x_0)$ has maximal rank, set
\[\delta = \frac{1}{2} \inf_{M_0 \in \partial f(x_0)} \frac{1}{\|M_0^{-1} \|},\]
$r$ be chosen so that in $ \mathbf{B}_r^n(x_0)$ $ f $ satisfies the Lipschitz condition (\ref{ct1}) and $ \partial f(x) \subset \partial f(x_0) + \delta \mathcal{B}_{n \times n}$. Then, for every unit vector $v$ in $ \mathbb{R}^n$, there exists a unit vector $w$ in $ \mathbb{R}^n $ such that, whenever $ x$ lies in $x_0 + r \mathbf{B}^n$ and $M$ belongs to $ \partial f(x) $,
\begin{equation}\label{ct4}w.(Mv) \geq \delta.
\end{equation}
\end{lem}
\begin{proof}
By Proposition \ref{md1} and $ \partial f (x_0) $ is of maximal rank, the subset $ \partial f(x_0) \mathbf{S}^{n-1}$ of $ \mathbb{R}^n$ is compact and not containing $0$. For $M_0 \in \partial f(x_0)$, we have
\[\min_{\|x\|=1}\|M_0x\| = \frac{1}{\|M_0^{-1}\|}.\]
Set
\[\delta = \frac{1}{2}\inf_{M_0\in \partial f(x_0)}\frac{1}{\|M_0^{-1}\|},\]
we get $ \partial f(x_0) \mathbf{S}^{n-1}$ distances $2 \delta $ from $0$. \\
If $M \in G = \partial f(x_0) + \varepsilon \mathcal{B}_{n\times n}$, then
\[\min_{\|x\|=1}\|Mx\| \geq \min_{\|x\|=1}\|M_0x\| - \varepsilon.\]
Choosing
\[\varepsilon = \delta = \frac{1}{2}\inf_{M_0\in \partial f(x_0)}\frac{1}{\|M_0^{-1}\|},\]
we get $G \mathbf{S}^{n-1}$ distances at least $\delta$ from $0$. By Lemma \ref{bd22}, there exists a positive number $r$, such that
\begin{equation} \label{ct5}
x \in x_0 + r\mathbf{B}^n \Rightarrow \partial f(x) \subset G.
\end{equation}
Let $r$ be chosen so that $f$ satisfies (\ref{ct1}) on $x_0 + r \mathbf{B}^n$. \\
Thus, let any unit vector $v$ be given, apply the above results, the convex set $Gv$ distances at least $\delta$ from $0$. By the usual separation theorem for convex sets, there exists a unit vector $w$ such that
\[w.(\gamma v) \geq \delta,\]
for every $\gamma \in G$.
Hence, applying (\ref{ct5}) we obtain (\ref {ct4}).
\end{proof}
\begin{proof}[Proof of  Theorem \ref{bd1}]
Using the proof of  \cite[Theorem 1]{C1}  to replace \cite[Lemma 3]{C1}  by  the preceding lemma.\\
\textbf{Estimation of the neighborhood $U$, $V$ and the Lipschitz constant $L(g)$}: \\
According to the proof of \cite[Theorem 1]{C1}, we have
\[L(g) = \ds\frac{1}{\delta},\]
\[V = f(x_0) + (r\delta/2)\mathbf{B}^n=\mathbf{B}^n_{\frac{r\delta}{2}}\left(f(x_0)\right),\]
and choose $U$ being an arbitrary neighborhood of $x_0 $ and satisfying $f(U) \subset V$. Then for all $ x \in U $ we have
\[\|f(x) - f(x_0)\| \leq K\|x-x_0\|\leq \frac{r\delta}{2}.\]
Hence,
\[\|x-x_0\|\leq \frac{r\delta}{2}\frac{1}{K}.\]
So
\[U = \mathbf{B}^n_{\frac{r\delta}{2}\frac{1}{K}}\left(x_0\right).\]
\end{proof}
\begin{rem} When $f$ is $C^1$, $\partial f(x_0)$ reduces to $ Jf(x_0) $, and $g$ is in the class $C^1$. Thus we get the quantitative form of the classical inverse function theorem.
\end{rem}
\begin{rem}If $F: A \rightarrow \mathbb{R}^n$ be a Lipschitz mapping in a neighborhood of $(x_0, y_0)$, $A = U \times V$ be a open subset of $\mathbb{R}^m \times \mathbb{R}^n$, then the generalized Jacobian of $F$ at $(x_0, y_0)$ satisfies
\[\partial F(x_0, y_0) \subset \left\{\left(
                                  \begin{array}{cc}
                                    M_1 & M_2 \\
                                  \end{array}
                                \right): M_1 \in \partial_1 F(x_0, y_0), M_2 \in \partial_2 F(x_0, y_0)\right\},\]
where $\partial_1 F(x_0, y_0) ~\textrm{and}~  \partial_2 F(x_0, y_0)$ are generalized Jacobians of $F(\cdot, y_0): U \rightarrow \mathbb{R}^n  ~\textrm{and}~ F(x_0, \cdot): U \rightarrow \mathbb{R}^n$ at $(x_0, y_0)$, respectively.
\end{rem}
\begin{thm} \label{dl4}
Let $F: A \rightarrow \mathbb{R}^n$ be a Lipschitz mapping in a neighborhood of $(x_0, y_0)$ with Lipschitz constant $K$, $A = U \times V$ be a open subset of $\mathbb{R}^m \times \mathbb{R}^n$. Suppose that $\partial_2F(x_0, y_0)$ is of maximal rank, $F(x_0, y_0) = 0$, set
\[\delta = \frac{1}{2} \inf_{M_2 \in \partial_2 F(x_0, y_0)}\frac{1}{(m+(1+mK^2)n\| M_2^{-1}\|^2)^\frac{1}{2}},\]
 $r$ be chosen so that  $F$ satisfies Lipschitz condition (\ref{ct1}) and $$ \partial F (x, y) \subset \partial F(x_0, y_0) + \delta \mathcal{B}_{n\times(m+n)}, ~\textrm{when}~ (x, y) \in  \mathbf{B}_r^{m+n}((x_0, y_0)).$$
Then there exist a Lipschitz mapping $g: U_0 \rightarrow V$ defined in a neighborhood $U_0 \subset \mathbb{R}^m$ of $x_0 $ such that $g(x_0) = y_0$ and
\[F(x, g (x)) = 0,\]
for all $ x \in U_0$. \\
Moreover,
\[U_0 = \mathbf{B}^m_{\frac{r  \delta}{2} \frac{1}{K +1} }\left (x_0 \right ), ~\textrm{and}~~ L(g) \leq  \sup_{M_2 \in \partial_2 F(x_0, y_0)} K \|M_2^{-1}\|.\]
 \end{thm}

 \begin{proof}\textit{ }\\
\textbf{ 1.} Set $f(x, y) = (x, F (x, y)), ~ \textrm{for all}~ (x, y) \in U \times V$.
 Since $F$ is Lipschitz, according to Radamacher's theorem, there exists the generalized Jacobian of $F$ at $(x_0, y_0)$.
In a neighborhood of $(x_0, y_0)$, $F$ is almost everywhere differentiable, therefore existing generalized Jacobian of $f$ at $(x_0, y_0)$ and
\[\partial f(x_0, y_0) \subset \left\{\left(\begin{array}{cc}
                                       I_m & 0 \\
                                       M_1 & M_2
                                     \end{array}
\right): \ \ M_1 \in \partial_1 F(x_0, y_0), \ \ M_2 \in \partial_2 F(x_0, y_0) \right\}.\]
Then $\partial f(x_0, y_0)$ is of maximal rank, because $\partial_2 F(x_0, y_0)$ is of maximal rank.

\vspace*{8pt} \noindent
\textbf{2.} For $M = \left(\begin{array}{cc}
                                       I_m & 0 \\
                                       M_1 & M_2
                                     \end{array}
\right) \in \partial f(x_0, y_0)$,
\[M^{-1} = \left(
             \begin{array}{cc}
               I_m & 0 \\
               -M_2^{-1}M_1 & M_2^{-1} \\
             \end{array}
           \right).
\]
Therefore, we have
\[\ds\begin{array}{rcl}
    \|M^{-1}\| \leq \|M^{-1}\|_F & = & \left(m + \|M_2^{-1}M_1\|_F^2 + \|M_2^{-1}\|_F^2\right)^\frac{1}{2} \\
     & \leq & \left(m + \|M_2^{-1}\|_F^2( \|M_1\|_F^2 + 1)\right)^\frac{1}{2} \\
     & \leq & \left(m + \|M_2^{-1}\|_F^2( m\|M_1\|^2 + 1)\right)^\frac{1}{2} \\
     & \leq & \left(m + (1 + mK^2) \|M_2^{-1}\|_F^2\right)^\frac{1}{2}.
  \end{array}
\]
Thus
\[\frac{1}{\|M^{-1}\|} \geq \frac{1}{\left(m + (1 + mK^2)\|M_2^{-1}\|_F^2\right)^\frac{1}{2}}\geq \frac{1}{\left(m + (1 + mK^2)n\|M_2^{-1}\|^2\right)^\frac{1}{2}}.\]
Set
\[\Delta = \frac{1}{2}\inf_{M\in \partial f(x_0, y_0)}\frac{1}{\|M^{-1}\|},\]
we get
\begin{equation}\label{ct341}\Delta \geq \delta.
\end{equation}

\vspace*{8pt} \noindent
\textbf{3.} According to the theorem, we have
$$ \partial F (x, y) \subset \partial F(x_0, y_0) + \delta \mathcal{B}_{n\times (m+n)}, ~\textrm{when}~ (x, y) \in  \mathbf{B}^{m+n}_r((x_0, y_0)),$$
by (\ref{ct341}), we get
$$ \partial F (x, y) \subset \partial F(x_0, y_0) + \Delta  \mathcal{B}_{n\times (m+n)}, ~\textrm{when}~ (x, y) \in  \mathbf{B}^{m+n}_r((x_0, y_0)).$$
Therefore, we can chose $r$ so that $f$ satisfies Lipschitz condition (\ref{ct1}) and
\[\partial f(x, y) \subset \partial f(x_0, y_0) + \Delta \mathcal{B}_{(m+n)\times(m+n)}, ~\textrm{when}~ (x, y) \in  \mathbf{B}^{m+n}_r((x_0, y_0)).\]

\vspace*{8pt} \noindent
\textbf{4.} Since $F$ is a Lipschitz mapping with coefficient $K$, $f$ is Lipschitz with coefficient $K +1$. Applying Theorem \ref{dl2}, $f$ is locally invertible and
\[f^{-1}(x, z) = (x, h(x, z)),\]
with $h$ is a Lipschitz mapping. Define
\[g(x) = h(x, 0).\]
Then $g$ is Lipschitz and
\[(x, F(x, g (x))) = f(x, g(x)) = f(x, h(x, 0)) = f (f ^{-1}(x, 0 )) = (x, 0).\]
This indicates the existence of $g$ satisfying the requirements of the theorem.

\vspace*{8pt} \noindent
\textbf{5.} Estimation of the neighborhood $U_0 $ of $ x_0 $ and $L(g)$:\\
Applying Theorem \ref{dl2},  we obtain
\[(x, h(x, z)) \in U= \mathbf{B}^{m+n}_{\frac{r \Delta}{2} \frac{1}{k +1}} \left((x_0, y_0)\right).\]
Thus
\[(x, g(x)) \in U'= \mathbf{B}^{m+n}_{\frac{r \Delta}{2} \frac{1}{k +1}}\left((x_0, 0)\right).\]
Hence, by (\ref{ct341}), the theorem is satisfied for all $(x, g(x)) \in U'' = \mathbf{B}^{m+n}_{\frac{r \delta}{2} \frac{1}{k +1}}\left((x_0, 0)\right).$
Projecting $U''$ onto the space $ \mathbb{R}^m$, we get
\[U_0 = \mathbf{B}^m_{\frac{r  \delta}{2} \frac{1}{k +1}} \left (x_0\right ).\]
Moreover, applying the formula of implicit function derivative,
\[Dg = -\left( \frac{\partial F}{\partial y}\right)^{-1} \frac{\partial F}{\partial x}, ~~\textrm{whenever}~~ \left( \frac{\partial F}{\partial y}\right)^{-1}~~\textrm{exist}. \]
Therefore, we get
\[L(g) \leq  \sup_{M_2 \in \partial_2 F(x_0, y_0)} K \|M_2^{-1}\|.\]
 \end{proof}

\begin{rem} When $F$ is $C^1$, $\partial_2 F(x_0, y_0)$ reduces to $ J_2F(x_0, y_0) $, and $g$ is in the class $C^1$. Thus we get the quantitative form of the classical implicit function theorem.
\end{rem}
\begin{exa}
For $m = 1, n=2$, consider $F(x, y, z) = (2x+|y|+3y, 2x+|z|+3z)$ in $\mathbf{B}^3\left((0, 0, 0)\right)$. Then
\[\|F(x, y, z) - F(x', y', z')\| \leq \sqrt{24}\|(x, y, z) - (x', y', z')\|.\]
Thus $F$ is Lipschitz with Lipschitz constant $K = \sqrt{24}$.\\
We have
\[J_2F(x_i, y_i, z_i) = \left(
                   \begin{array}{cc}
                     \frac{|y_i|}{y_i}+3 & 0 \\
                     0 &  \frac{|z_i|}{z_i}+3  \\
                   \end{array}
                 \right),~~ (x_i, y_i, z_i) ~\textrm{near}~(0, 0, 0).
\]
Hence,
\[\partial_2F(0, 0, 0)=\left\{\left(
                            \begin{array}{cc}
                              s+3 & 0 \\
                              0 & t+3 \\
                            \end{array}
                          \right): -1 \leq s \leq 1, -1 \leq t \leq 1
\right\},\]
and $\partial_2F(0, 0, 0)$ is of maximal rank.\\
Let $M_2 \in \partial_2F(0, 0, 0)$. Then there exist $M_2^{-1}$ defined by
\[M_2^{-1} = \left(
                            \begin{array}{cc}
                              \frac{1}{s+3} & 0 \\
                              0 &  \frac{1}{t+3} \\
                            \end{array}
                          \right).\]
So
\[\delta = \frac{1}{2} \inf_{M_2 \in \partial_2 F(x_0, y_0)}\frac{1}{(m+(1+mK^2)n\| M_2^{-1}\|^2)^\frac{1}{2}} = \frac{1}{\sqrt{54}}.\]
By the preceding theorem, there exist a Lipschitz mapping $g$ such that $g(0) = (0, 0)$ and
\[F(x, g(x)) = (0, 0), ~~\textrm{for all}~x \in U_0.\]
Moreover, for $(x, y, z)$ near $(0, 0, 0)$, we have
\[JF(x, y, z) = \left(
                   \begin{array}{ccc}
                    2& \frac{|y|}{y}+3 & 0 \\
                    2& 0 &  \frac{|z|}{z}+3  \\
                   \end{array}
                 \right).\]
                 We can chose $r = 1$, and then
               \[ \partial F(x, y, z) \subset \partial F(0, 0, 0) + \frac{1}{\sqrt{54}} \mathcal{B}_{2\times 3}, ~\textrm{every}~ (x, y, z) \in  \mathbf{B}^3_r((0,0, 0)).\]
Hence, we obtain
\[U_0 = \mathbf{B}^1_{\frac{1}{6\sqrt{6}}\cdot \frac{1}{1+2\sqrt{6}}}\left(0 \right), ~\textrm{and}~ L(g) \leq \sqrt{6}.\]
\end{exa}
\begin{thm}\label{dl3}
Let $f_0: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a Lipschitz mapping in the neighborhood of $x_0$ so that $\partial f_0(x_0)$ is of maximal rank and satisfies
\[K \| x-y \| \leq \| f_0(x)-f_0(y) \| \leq K' \| x-y \|.\]
Let $f = f_0 + h$, with $ h: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is a Lipschitz mapping with Lipschitz constant L so that
\[L < K.\]
Set
\[\delta = \frac{1}{2} \inf_{M_0 \in \partial f(x_0)} \frac{1}{\| M_0^{-1}\|}.\]
Suppose that  $~r$ was chosen so that  $f$ satisfies Lipschitz condition (\ref{ct1}) and $\partial f(x) \subset \partial f(x_0) + \delta \mathcal{B}_{n\times n}$ in $\mathbf{B}^n_r(x_0)$.
Then there exist neighborhoods $U$ and $V$ of $x_0$ and $f(x_0)$, respectively, and a Lipschitz mapping $g: V \rightarrow \mathbb{R}^n$ such that

\vspace*{8pt}
(a) $g(f(u)) = u$ for every $u \in U$,

\vspace {8pt}
(b) $f(g(v)) = v $ for every $v \in V$.\\
Moreover, $U, V$ and $L(g)$ are determined by
\[U = \mathbf{B}^n_{\frac{r \delta}{2} \frac{1}{K '+ L} }\left(x_0\right ),~~~~ V = \mathbf{B}^n_{\frac{r \delta}{2}} \left (f(x_0)\right), ~~\textrm{and}~~L(g) = \ds\frac{1}{\delta}.\]
\end{thm}
\begin{proof}\textit{ }\\
\textit{Remark}: $f$ is differentiable at $x_i$ if and only if $f$ can be approximated in a neighborhood of $x_i$ by the affine mapping
\[T(x) = f(x_i) + Jf(x_i)(x-x_i).\]
Hence,
\[K\|x-x_i\| \leq \|f(x) - f(x_i)\| \approx \|Jf(x_i)(x-x_i)\| \leq K'\|x-x_i\|.\]
From which we obtain
\begin{equation}\label{ct2}K \leq \inf_{y\neq 0}\frac{\|Jf(x_i)y\|}{\|y\|}, \forall Jf(x_i) \Rightarrow K \leq \inf_{Jf(x_i)}\left(\min_{\|x\|=1}\|Jf(x_i)x\|\right),
\end{equation}
\begin{equation}\begin{array}{ccl}
                  \sup_{y\neq 0}\frac{\|Jf(x_i)y\|}{\|y\|}\leq K', \forall Jf(x_i) & \Rightarrow & \sup_{Jf(x_i)}\left(\max_{\|x\|=1}\|Jf(x_i)x\| \right)\leq K' \\
                  \\
                  & \Rightarrow & \sup_{Jf(x_i)}\|Jf(x_i)\| \leq K'.
                \end{array}
    \label{ct3}
\end{equation}

\vspace*{8pt}
\noindent
From the remarks above we will prove that $f$ is a Lipschitz mapping and  $\partial f(x_0)$ is of maximal rank. \\
We have
\[\begin{array}{rcl}
    \|f(x)-f(y)\| & = & \|(f_0(x)-f_0(y))+(h(x)-h(y))\| \\
     & \leq & \|(f_0(x)-f_0(y))\|+\|(h(x)-h(y))\| \\
     & \leq & (K'+L)\|x-y\|.\\
     \end{array}
\]
So $f$ is Lipschitzian with Lipschitz constant $K'+L$.

\vspace*{8pt}
\noindent
According to Rademacher's theorem, $f$ is almost everywhere differentiable near $x_0$, so that the general Jacobian $ \partial f(x_0)$ exits. Moreover,
\[\partial f(x_0) \subset \partial f_0(x_0) + \partial h(x_0).\]
Indeed, let $E$ be the set of points where $f_0$ or $h$ is fail to be differentiable. Then every $M \in \partial f(x_0), M$ has the form
\[M = \lim_{i\rightarrow \infty}J(f_0+h)(x_0+h_i), h_i \rightarrow 0 ~\textrm{when}~i \rightarrow \infty,\]
here the sequence $ \{x_0 + h_i \} $ lies  in the complement of $E$, and admits a subsequence $\{x_0 + h_{n_i}\}$ such that $Jf_0(x_0 + h_{n_i})$ and $Jh(x_0 + h_{n_i})$ both exist and converge. Hence
 \[M= \lim_{i\rightarrow \infty}Jf_0(x_0+h_{n_i}) + \lim_{i\rightarrow \infty}Jh(x_0+h_{n_i}) = M_0+H,\]
where $M_0 \in \partial f_0(x_0), H \in \partial h(x_0)$.

\vspace*{8pt}
\noindent
Next, we prove $ \partial f(x_0) $ is of maximal rank: \\
Using (\ref{ct2}) and (\ref{ct3}) we have
\[\ds\begin{array}{rcrcl}
    L<K & \Rightarrow & \sup_{H\in \partial h(x_0)}\|H\| & < & \inf_{M_0\in\partial f_0(x_0)}\left(\min_{\|x\|=1}\|M_0x\|\right) \\
     & \Rightarrow & \sup_{H\in \partial h(x_0)}\|H\| & < &\inf_{M_0\in\partial f_0(x_0)} \left( \frac{1}{\frac{1}{\min_{\|x\|=1}\|M_0x\|}}\right) \\
          & \Rightarrow & \sup_{H\in \partial h(x_0)}\|H\| & < &\frac{1}{ \sup_{M_0\in\partial f_0(x_0)} \left(\frac{1}{\min_{\|x\|=1}\|M_0x\|}\right)} \\
    & \Rightarrow & \sup_{H\in \partial h(x_0)}\|H\| & < & \frac{1}{\sup_{M_0\in\partial f_0(x_0)}\|M_0^{-1}\|},
     \end{array}
\]
so we get
\[\sup_{H\in \partial h(x_0), M_0\in \partial f_0(x_0)}\|M_0^{-1} H\| < 1. \]
According to Theorem \ref{dl1}, $M_0 + H$ is of maximal rank for all $H\in \partial h(x_0), M_0\in \partial f_0(x_0)$.\\
According to the proof above,  if $M \in \partial f(x_0)$ then $M = M_0 + H$, with $M_0\in \partial f_0(x_0), H\in \partial h(x_0)$. Hence, $M$ is of maximal rank for every $M\in \partial f(x_0)$.\\
So $\partial f(x_0)$ is of maximal rank.

\vspace*{8pt}
\noindent
Thus $f$ is Lipschitz and $ \partial f(x_0)$ is of maximal rank, applying Theorem \ref{dl2}, we get the results of the theorem.

\vspace*{8pt}
\noindent
Moreover, according to the proof of Theorem \ref{dl2}, we have
\[L(g) = \frac{1}{\delta}, \ \ \ \ V = f(x_0) + (r\delta/2)\mathbf{B}^n=\mathbf{B}^n_{\frac{r\delta}{2}}\left(f(x_0)\right),\]
and choose $U$ being an arbitrary neighborhood of $x_0 $ and satisfying $f(U) \subset V$. Then for all $ x \in U $ we have
\[\|f(x) - f(x_0)\| \leq (K'+L)\|x-x_0\|\leq \frac{r\delta}{2}.\]
Hence,
\[\|x-x_0\|\leq \frac{r\delta}{2}\frac{1}{K'+L}.\]
So
\[U = \mathbf{B}^n_{\frac{r\delta}{2}\frac{1}{K'+L}}\left(x_0\right).\]
\end{proof}
\begin{rem}If $ L> K $, then $f = f_0 + h $ may not satisfy local invertible condition.
\end{rem}
\begin{cor}\label{hq31} The class of Lipschitz mappings satisfying Clarke's inverse function theorem  is open in the space
$\textrm{Lip}_{x_0}(\mathbb{R}^m,  \mathbb{R}^n)$.
\end{cor}
\begin{exa}
For $n=2$, consider $f_0(x, y) = (|x|+2x, |y|+2y)$ in $\mathbf{B}^2\left((0, 0)\right)$. We have
\[Jf_0(x_i, y_i) = \left(
                   \begin{array}{cc}
                     \frac{|x_i|}{x_i}+2 & 0 \\
                     0 &  \frac{|y_i|}{y_i}+2  \\
                   \end{array}
                 \right),~~ (x_i, y_i) ~\textrm{near}~(0, 0).
\]
Thus
\[\partial f_0(0, 0)=\left\{\left(
                            \begin{array}{cc}
                              s+2 & 0 \\
                              0 & t+2 \\
                            \end{array}
                          \right): -1 \leq s \leq 1, -1 \leq t \leq 1
\right\},\]
and $\partial f_0(0, 0)$ is of maximal rank.\\
We have
\[\begin{array}{ccl}
    \|f_0(x, y) - f_0(x', y')\| & = & \|(|x|+2x, |y|+2y)-(|x'|+2x', |y'|+2y')\| \\
    & = & \|((|x|-|x'|)+2(x-x'), (|y|-|y'|)+2(y-y'))\|.
  \end{array}
\]
Hence,
\[\|(x, y) - (x', y')\| \leq \|f_0(x, y) - f_0(x', y')\| \leq 3\|(x, y) - (x', y')\|. \]
Thus, $f_0$ is Lipschitz with Lipschitz constants $K = 1, K' = 3$.

\vspace*{8pt}
\noindent \textbf{1. Case $L<K$.}
We find $h(x, y) = (\frac{1}{2}|x|, \frac{1}{2}|y|)$ is Lipschitz with Lipschitz constant $L = \frac{1}{2}$ satisfying
\[L < K.\]
Set $f = f_0+h$, we have
\[f(x, y) =( \frac{3}{2}|x|+2|x|, \frac{3}{2}|y|+2y),\]
and $f$ is Lipschitz in $\mathbf{B}^2\left((0, 0)\right)$ with Lipschitz constant $\frac{7}{2}.$\\
Then,
\[Jf(x_i, y_i) = \left(
                   \begin{array}{cc}
                    \frac{3}{2} \frac{|x_i|}{x_i}+2 & 0 \\
                     0 &  \frac{3}{2}\frac{|y_i|}{y_i}+2  \\
                   \end{array}
                 \right),~~ (x_i, y_i) ~\textrm{near}~(0, 0),
\]
\[\partial f(0, 0)=\left\{\left(
                            \begin{array}{cc}
                               s+2 & 0 \\
                              0 &  t+2 \\
                            \end{array}
                          \right): -\frac{3}{2} \leq s \leq \frac{3}{2}, -\frac{3}{2} \leq t \leq \frac{3}{2}
\right\},\]
and $\partial f(0, 0)$ is of maximal rank.\\
So $f = f_0+h$ is locally invertible.\\
We have
\[\delta = \frac{1}{2} \inf_{M_0 \in \partial f(x_0)} \frac{1}{\| M_0^{-1}\|} = 1.\]
Moreover,
\[Jf(x, y) = \left(
                   \begin{array}{cc}
                    \frac{3}{2} \frac{|x|}{x}+2 & 0 \\
                     0 &  \frac{3}{2}\frac{|y|}{y}+2  \\
                   \end{array}
                 \right),~~ (x, y) \in \mathbf{B}^2_r\left((0, 0)\right), ~ r \leq 1.
\]
We can chose $r = 1$, and then
\[\partial f(x, y) \subset \partial f(0, 0) + \mathcal{B}_{2\times 2}, ~~\textrm{when} ~~(x, y) \in \mathbf{B}^2_r\left((0, 0)\right).\]
Hence, we obtain
\[U = \mathbf{B}^2_{\frac{1}{7}} \left((0, 0)\right ),~~~~ V = \mathbf{B}^2_{\frac{1}{2}}\left ((0, 0)\right), ~~\textrm{and}~~L(g) = 1.\]

\vspace*{8pt}
\noindent \textbf{2. Case $L>K$.}
We find $h(x, y) = (\frac{3}{2}|x|, \frac{3}{2}|y|)$ is Lipschitz with Lipschitz constant $L = \frac{3}{2}>K$.
Set $f = f_0+h$, we have
\[f(x, y) =( \frac{5}{2}|x|+2x, \frac{5}{2}|y|+2y),\]
and $f$ is Lipschitz with Lipschitz constant $\frac{9}{2}.$\\
Then,
\[Jf(x_i, y_i) = \left(
                   \begin{array}{cc}
                    \frac{5}{2} \frac{|x_i|}{x_i}+2 & 0 \\
                     0 &  \frac{5}{2}\frac{|y_i|}{y_i}+2  \\
                   \end{array}
                 \right),~~ (x_i, y_i) ~\textrm{near}~(0, 0),
\]
\[\partial f(0, 0)=\left\{\left(
                            \begin{array}{cc}
                              s+2 & 0 \\
                              0 & t +2\\
                            \end{array}
                          \right): -\frac{5}{2} \leq s \leq \frac{5}{2}, -\frac{5}{2} \leq t \leq \frac{5}{2}
\right\},\]
and $\partial f(0, 0)$ is not of maximal rank.
\end{exa}
\noindent {\large{\textbf{Acknowledgements.}}}

\vspace*{8pt}\noindent
The author wishes to thank Professor Ta Le Loi for the suggestion of writing down this paper.
This research is supported  by Vietnam's
National Foundation for Science and Technology Development
(NAFOSTED).

redpajama_set_name: RedPajamaArXiv