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The British Society for Rheumatology Biologics Registers in Ankylosing Spondylitis (BSRBR-AS) study: Protocol for a prospective cohort study of the long-term safety and quality of life outcomes of biologic treatment
Gary J. Macfarlane1,
Maxwell S. Barnish1,
Elizabeth A. Jones1,
Lesley Kay2,
Andrew Keat3,
Karen T. Meldrum1,
Ejaz Pathan4,
Roger D. Sturrock5,6,
Claudia Zabke1,
Paul McNamee7 &
Gareth T. Jones1
Axial spondyloarthropathy typically has its onset in early adulthood and can impact significantly on quality of life. In the UK, biologic anti-tumour necrosis factor therapy is recommended for patients who are unresponsive to non-steroidal anti-inflammatory drugs. There remain several unresolved issues about the long-term safety and quality of life outcomes of biologic treatment in axial spondyloarthropathy. Long-term "real-world" surveillance data are required to complement data from randomised controlled trials.
Methods/Design
We are conducting a UK-wide prospective cohort study of patients with axial spondyloarthropathy who are naïve to biologic therapy at the time of recruitment. Those about to commence anti-tumour necrosis factor biologic therapy will enter a "biologic" sub-cohort with other patients assigned to a "non-biologic" sub-cohort. The primary objective is to determine whether the use of biologic therapy is associated with an increased risk of serious infection, while secondary objectives are to assess differences in malignancy, serious comorbidity, all-cause mortality but also assess impact on specific clinical domains (physical health, mental health and quality of life) including work outcomes between biologic and non-biologic patient cohorts. Patients will be followed-up for up to 5 years. Data are obtained at baseline and at standard clinical follow-up visits – at 3, 6 and 12 months and then annually for the biologic cohort and annually for the non-biologic cohort. This study will also collect biological samples for genetic analysis.
Although biologic therapy is widely used for ankylosing spondylitis patients who are unresponsive to non-steroidal anti-inflammatory drugs, the majority of the available safety information comes from rheumatoid arthritis, where increased infection risk has consistently been shown. However, given the typical demographic differences between rheumatoid arthritis and axial spondyloarthropathy patients, it is important to develop an epidemiologically rigorous cohort of patients receiving biologic therapy to effectively evaluate outcomes with regard not only to safety but also to quantify benefits across clinical, psychosocial and work outcomes.
Clinical trial registration
This is an observational cohort study and clinical trial registration was not required or obtained
Axial spondyloarthritis (axSpA) is a chronic inflammatory condition that principally affects the spine and sacroiliac joints but is also associated with peripheral arthritis plus various extra-articular features such as enthesitis, iritis, inflammatory bowel disease and psoriasis. There are two sub-groups – radiographic axSpA, also known as ankylosing spondylitis (AS), and non-radiographic axSpA – which have been shown to have similar clinical manifestations and disease activity measures [1]. AxSpA typically has its onset early in adulthood [2] and can have a significant impact on aspects of quality of life including disability and work status [3]. Until recently, treatment has largely been limited to non-steroidal anti-inflammatory drugs (NSAIDs) and physiotherapy, while disease-modifying antirheumatic drugs (DMARDs), though effective in other inflammatory arthritides, have shown little efficacy in axSpA. The introduction of anti-tumour necrosis factor (anti-TNF) biologic therapies has been associated with significantly improved outcomes, including improvements in pain, stiffness and fatigue [4, 5]. Anti-TNF therapy was initially recommended in the United Kingdom (UK) for patients with radiographic axSpA who were unresponsive to NSAIDs and who met specific disease severity criteria [6] and the extension of this indication to non-radiographic axSpA is being considered [7].
Research has highlighted the importance of genetics in axSpA, in particular the HLA-B27 allele [2]. However, many important genetic pathways remain unknown and their relationships to the development of axSpA unconfirmed. Genome-wide genetic association studies are currently underway in international collaborations to elucidate these pathways and the roles they play in axSpA pathogenesis. Although data from randomised clinical trials (RCTs) of biologic therapy in axSpA have been highly informative, there remain a number of unresolved issues, since RCTs are conducted typically on more highly selected populations than are representative of routine clinical practice [8].
Pharmacological management in rheumatology, including the use of disease-suppressive agents, is associated with adverse side effects in a proportion of patients. In addition, longer term complications, such as malignancy, are unlikely to be detected in the relatively short follow-up period of RCTs. Immunosuppressive therapy, in particular, is considered to be a potential risk factor for both malignancy and life-threatening infection. The use of azathioprine and cyclophosphamide, for example, is associated with an increased risk of lymphoproliferative malignancies in patients with rheumatoid diseases [9–11]. Immunosuppressed patients are also at risk of serious infections such as from Mycobacterium tuberculosis, Pneumocystis carinii and fungal infections [12].
Informed prescribing of biologic agents requires knowledge of the magnitude of risk of such longer-term adverse events. To date, most of the safety information regarding anti-TNF therapy has come from rheumatoid arthritis (RA). It is problematic to extrapolate from one condition to another: RA patients generally have older age at onset than axSpA patients, have a significant burden of co-morbidity, have a longer history of medication use and have often received polypharmacy including many other immunosuppressive drugs such as corticosteroids [2, 13–15]. In terms of benefit, the impact of biologic therapy on employment maintenance and long-term quality of life is of great significance for patients and real-world data are needed in this respect. Furthermore, the ability to identify those who are most likely to benefit from a particular treatment is crucial in delivering cost-effective care. In order to provide axSpA-specific data on the long term safety and benefit of biologic therapy, we are conducting a nationwide prospective cohort study: the British Society for Rheumatology Biologics Register in Ankylosing Spondylitis (BSRBR-AS).
Cohort population
Participants are required to meet the Assessment of SpondyloArthritis international Society (ASAS) criteria for radiographic or non-radiographic axSpA, i.e they must satisfy at least one of the following: i) the modified New York criteria for AS [16]; ii) the imaging-based ASAS definition of axSpA [17]; or iii) the "clinical" ASAS definition of axSpA [17]. At the time of recruitment they are required to be naïve to biologic therapy. Individuals starting an "eligible" biologic therapy comprise the "biologic" cohort and those not starting a biologic therapy join the "non-biologic" cohort. The eligible drug list currently comprises adalimumab (Humira), etanercept (Enbrel) and certolizumab pegol (Cimzia). Participants must be aged at least 16 years and be willing to give informed consent for follow-up including access to medical records. Study exclusion criteria are: i) otherwise eligible patients who are starting a biologic therapy not on the eligible drug list and ii) inability to communicate in English. However, once recruited to the study, participants are eligible to remain in the study irrespective of the subsequent pharmacological or non-pharmacological management of their condition. Recruitment can take place through participating National Health Service (NHS) hospitals across the UK.
Study procedures
Patients are informed about BSRBR-AS prior to their clinic appointment by means of a letter from their consultant rheumatologist and provided with an information sheet. When patients attend their routine clinic appointment, they are asked whether they would like to take part in the study. Consent is taken by the consultant rheumatologist or an appropriately trained member of the site research team. In exceptional circumstances, patients may complete the consent form at home and return it by post. Patients are ordinarily given at least 24 hours to consider the Patient Information Sheet and are given the opportunity to ask questions before deciding whether to take part.
All participants are assessed at time of recruitment (baseline). Patients in the biologic cohort are followed up at 3, 6 and 12 months after their first dose of biologic therapy and then annually thereafter. Patients in the non-biologic cohort are followed up annually from study entry. This is in line with current UK clinical practice. Each follow-up involves the collection of clinical and self-report data as detailed in the section 'data collection and management'. Recruitment began in December 2012 and is currently scheduled to continue until December 2016 and follow-up will continue to the end of the study period (currently December 2017), which will provide up to 5 years follow-up for each participant. Assessment of endpoints will be achieved using a combination of physician and patient report and record linkage to routinely collected health data.
Data collection and management
Clinical data are collected by site clinicians, research nurses or appropriate delegate at routine clinical appointments. All data are entered onto the electronic Case Report Form (eCRF).
The following clinical measures are recorded:
ii)
Targeted medical history, blood pressure, weight and height
iii)
Active phase reactants (serum C reactive protein (CRP) or if not available erythrocyte sedimentation rate (ESR))
iv)
Bath Ankylosing Spondylitis Metrology Index (BASMI) [18]
v)
Extra-articular disease features
vi)
Pregnancy status of patient or partner
The following self-report measures are collected:
Current smoking status and alcohol consumption
Disease activity:
○ Bath Ankylosing Spondylitis Disease Activity Index (BASDAI) [19]
○ Ankylosing Spondylitis Disease Activity Score (ASDAS) [20]
General measures of AS:
○ Bath Ankylosing Spondylitis Global Score (BAS-G) [21]
○ Bath Ankylosing Spondylitis Functional Index (BASFI) [22]
○ Spinal pain visual analogue scale
Quality of life:
○ Short Form Health Survey (SF-12) [23]
○ Ankylosing Spondylitis Quality of Life Index (ASQoL) [24]
○ EuroQoL quality of life measures (EQ-5D-5L and EQ-VAS) [25]
Other measures including:
○ Chalder fatigue scale [26]
○ Estimation of Sleep Problems [27]
○ Hospital Anxiety and Depression Scale (HADS) [28]
○ Work Productivity and Activity Impairment Questionnaire: Specific Health Problem (WPAI: SHP) [29]
○ American College of Rheumatology (ACR) 2011 research criteria for fibromyalgia [30]
The health economics analysis will involve the construction of cost and quality of life profiles, based on EQ-5D-5L and SF-6D (derived from SF-12), for different axSpA health states. Costs will include the number of visits to NHS hospitals for inpatient treatment, day-case procedures and outpatient services associated with the condition and related side-effects, recorded from routine records, valued using standard national sources such as NHS Reference Costs [31]. Quality of life profiles will be estimated from patient self-report, and valued using published UK tariffs. The estimated cost and quality of life profiles will facilitate the development of economic models to assess the cost-effectiveness of biologic therapy versus non-biologic therapy. A unique feature of these profiles is that they will be developed from real-life registry data, and so are more likely to be generalisable to NHS patients than previously published estimates.
Pharmacovigilance and safety reporting
Clinically confirmed Serious Adverse Events related to biologic therapies on the eligible drug list are reported to the relevant pharmaceutical company.
Data analysis and statistics
The study was powered on the primary objective – an ability to determine a doubling of the risk of serious infection, i.e infection resulting in hospitalisation or death, among axSpA patients treated with biologics compared to the non-biologic cohort. Analysis of this primary endpoint will be based on comparing the risks of events over time using Cox proportional hazards regression [32], taking into account differences between groups such as potential confounders and disease modifying effects. We will control for medication switching in the analysis.
Available data from a meta-analysis of persons enrolled in trials suggest that the baseline risk of serious infection is 1.0 per 100 person years in those not randomised to biologic therapy and that there is an approximate doubling in risk in those receiving biologics [33]. Noting that adverse events will be more common in routine practice we have powered the study on a baseline risk of 1.6 cases per 100 person-years. The power calculation is taken as a time-to-event analysis and is based on an expected ratio of 35 biologic patients to every 65 non-biologic patients. Assuming we wish to be able to detect a Hazard Ratio (HR) of 2 in the biologic cohort compared to the non-biologic cohort, i.e. an event rate of 3.2 per 100 person years, the number of events required for 80 % power is 7.9/(0.65 * 0.35 * log (HR) ^ 2) = 73 events [34], which will be provided by 1184 and 2216 person-years of observation in the biologic and non-biologic cohorts respectively. Considering alternative outcomes, not necessarily related to safety, but also for example treatment response, improvements in symptoms and quality of life, power increases to >95 % for baseline hazards of 3 per 100 person-years in detecting a HR of 2. For baseline hazards of 10 per 100 person-years the study has 95 % power to detect a HR of 1.5 while the study has 80 % power to detect a HR of 1.2 for baseline hazards of 30 per 100 person-years.
Study management and oversight arrangements
The study is co-ordinated by a Study Management Group consisting of the study investigators. A study co-ordinator oversees the study and is accountable to the Chief Investigator. An International Advisory Committee advises on maximising the usefulness of the register for research purposes and on potential collaborations, harmonising opportunities for data collection with other international studies. A Data Monitoring Committee is convened by the British Society for Rheumatology (BSR). The register and data belong to the BSR and oversight of this register (and other registers of the BSR) is undertaken by the BSR Registers Committee. The committee membership comprises BSR clinical affairs staff, representatives from BSR members and from the study investigators.
This is an observational cohort study and clinical trial registration was not required or obtained. The study was peer-reviewed as part of the process of applying for funds, competitively, to the BSR. Ethical approval was obtained from the National Research Ethics Service (NRES) Committee North East – County Durham and Tees Valley (Research Ethics Committee (REC) reference 11/NE/0374). Appropriate NHS Research and Development (R&D) approvals are obtained for each site prior to the commencement of study operations in that location. The BSR is the legal sponsor for the study and delegates certain functions to the University of Aberdeen. All clinical sites will only have access to local identifiable patient information. All patient identifiable study information will be kept confidential and in compliance with the Data Protection Act (1998).
Dissemination of findings
Dissemination activities in BSRBR-AS will comply with the BSR Publication Policy. Summaries of results will be made available for investigators to disseminate within their clinical areas and reports will be produced for the sponsor, REC and NHS R&D.
This study represents one of only a small number of registers focused on biologic therapy among persons with axSpA. Such therapies have been shown to be effective in clinical trials in achieving ASAS 50 % response, in improving disease activity and function [35]. So why the need for a register? Firstly, trials are conducted on highly selected populations [8]. Specifically, trial populations tend to be younger and have fewer comorbidities than the general patient population. Trials focus on evaluating efficacy, are short-term and conducted on relatively small numbers of patients. In contrast, this register is recruiting patients with axSpA from centres throughout the UK. These include some centres with specific expertise in managing axSpA, but most are local hospitals responsible for managing all rheumatic diseases arising in patients within their local population. Secondly, eligibility for the register is determined, in terms of disease status, only in relation to having an eligible diagnosis and in being naïve to biologic therapy. Thus, patients are not excluded on the basis of multi-morbidity. Thirdly, patients will be followed up until the end of study, providing longer-term follow-up. This will then allow examination of both effectiveness, impact and safety of these therapies over the medium term.
A novel aspect of the register is the inclusion of persons who meet only the clinical arm of the ASAS criteria for axSpA, namely patients who are HLA-B27 positive and have 2 additional SpA features, but who do not meet the imaging arm of the ASAS criteria. This will allow the natural history of axSpA in this sub-group to be determined. Indeed, the lack of information on the natural history of non-radiographic axSpA was highlighted by the US Food and Drug Administration (FDA) in its ruling against widening indications for anti-TNF therapy beyond radiographically confirmed axSpA [36]. The current register is the second under the auspices of the BSR. The first, the British Society for Rheumatology Biologics Register in Rheumatoid Arthritis (BSRBR-RA), was established over 10 years ago. It uses a similar design as BSRBR-AS and has provided important evidence on safety of anti-TNF therapy in that population. For example, the addition of anti-TNF therapy to DMARDs did not alter the risk of cancer in RA patients selected for anti-TNF therapy [37]; a significantly increased risk of shingles and a doubling in risk of septic arthritis was observed in those treated with anti-TNF therapy but an observed increased risk of serious skin and soft tissue infections was not statistically significant [38, 39] and there was no excess risk of venous thromboembolism [40]. Using BSRBR-RA data, Rituximab has been shown to be a more effective option than a second anti-TNF therapy for patients who have not responded to an initial anti-TNF therapy [41].
In 2015, the first 'biosimilars' for RA, axSpA and psoriatic arthritis (PsA) were introduced to the market. A biosimilar is a biological medicine manufactured to be similar to an existing 'reference' biological medicine, with no meaningful differences in terms of safety or efficacy. Although pre-market authorisation includes comparisons against the reference medicine, this typically involves small scale trials and they are not undertaken for all indications of the reference product. There is also little evidence on the safety and efficacy of switching to biosimilars – and it is thought that many patients may have their therapy switched in order to reduce costs to the NHS. The BSR has therefore recommended strongly that all patients starting or switching to biosimilar agents should be registered with the appropriate biologics register. The current study will seek to capture such information on participating patients within BSRBR-AS, thus expanding the evidence base for these agents [42].
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Oversight of the study is provided by the BSR Registers Committee of which GJM and GTJ are members, together with investigators from BSRBR-RA, representatives from the BSR clinical affairs section and BSR independent members, currently, Alex MacGregor (University of East Anglia), Elaine Dennison (University of Southampton), Jon Packham (Keele University) and patient representatives Ailsa Bosworth and Debbie Cook. We acknowledge the contribution of the International Advisory Group members Desireé van der Heijde (Netherlands), Matthew Brown (Australia) and Walter Maksymowych (Canada). We thank Neil Basu (University of Aberdeen) for his role with regards to pharmacovigilance and the Robertson Centre for Biostatistics (University of Glasgow) for data management services. Author KTM is currently at the Tayside Clinical Trials Unit, University of Dundee.
Musculoskeletal Research Collaboration (Epidemiology Group), Institute of Applied Health Sciences, University of Aberdeen, Aberdeen, UK
Gary J. Macfarlane, Maxwell S. Barnish, Elizabeth A. Jones, Karen T. Meldrum, Claudia Zabke & Gareth T. Jones
Department of Rheumatology, Freeman Hospital, Newcastle upon Tyne, UK
Lesley Kay
Rheumatology Department, Northwick Park Hospital, London, UK
Andrew Keat
Department of Rheumatology, Aberdeen Royal Infirmary, Aberdeen, UK
Ejaz Pathan
Division of Immunology, Infection and Inflammation, University of Glasgow, Glasgow, UK
Roger D. Sturrock
Centre for Rheumatic Diseases, Glasgow Royal Infirmary, Glasgow, UK
Health Economics Research Unit, Institute of Applied Health Sciences, University of Aberdeen, Aberdeen, UK
Paul McNamee
Gary J. Macfarlane
Maxwell S. Barnish
Elizabeth A. Jones
Karen T. Meldrum
Claudia Zabke
Gareth T. Jones
Correspondence to Maxwell S. Barnish.
BSRBR-AS is funded by the BSR, which in turn receives funding from the manufacturers of the biologic therapies included in this study (currently AbbVie, Pfizer and UCB). Pharmaceutical companies providing funds to BSR do not have a role in the oversight of the study, but they do receive advance notice of publications on which they are able to comment. They do not have access to the data collected but can request analyses of the data, for which additional funds are provided. GJM chairs a Pfizer competitive grant committee for which he receives an honorarium. GJM and GTJ have received separate funding from AbbVie and Pfizer to study spondyloarthritis in the Scotland Registry for Ankylosing Spondylitis (SIRAS) study. LK has received an unrestricted educational grant from UCB. AK has received research funding from Abbvie and Pfizer as well as speaker/chairman fees and payments for attending advisory boards from Abbvie, Pfizer and UCB. The remaining authors have no competing interests.
MSB, the BSRBR-AS study analyst, wrote the first draft of the manuscript which was then revised by GJM (Chief Investigator) and comments subsequently made by all authors. The protocol of the study was originally written by GJM, EAJ, LK, AK, KTM, RDS, PM and GTJ, while EP and CZ have been involved in making revisions to the original protocol. PM designed and oversees the health economic aspects of the study. All authors approved the submission. GJM is the guarantor.
Macfarlane, G.J., Barnish, M.S., Jones, E.A. et al. The British Society for Rheumatology Biologics Registers in Ankylosing Spondylitis (BSRBR-AS) study: Protocol for a prospective cohort study of the long-term safety and quality of life outcomes of biologic treatment. BMC Musculoskelet Disord 16, 347 (2015). https://doi.org/10.1186/s12891-015-0805-x
DOI: https://doi.org/10.1186/s12891-015-0805-x
Axial spondyloarthropathy
Biologic therapy
Epidemiology of musculoskeletal disorders


October 14 Meeting Follow-Up: Publishing Paths – MWA – Montgomery Co.
Wednesday, October 14, chapter president Katherine Pickett presented on the many paths to publication. We had a great turnout, but that meant we were short on handouts! To rectify the problem, we are posting them here.
The first is the Publishing Paths Personal Assessment. The questions offer a starting point for assessing your strengths, weaknesses, and goals, which is essential in choosing the publishing path that's right for you and your book.
Click the link below to view the Publishing Paths Personal Assessment. Note that you will need Adobe Acrobat in order to access the document and may be asked to save it to your computer.
The second is the Publishing Paths Cheat Sheet. If you couldn't catch all of the tips and tidbits, this sheet will give you the highlights. For each path, Katherine has noted who this path would be ideal for and what the finances are for that path.
Click the link below to view the Publishing Paths Cheat Sheet. Note that you will need Adobe Acrobat in order to access the document and may be asked to save it to your computer.
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text:
\section{Introduction}
As two of us have argued\cite{BN2}, a particularly interesting, hydrodynamic-like picture of the evolution of the averaged characteristic polynomial (hereafter ACP) emerges if one exploits Dyson's idea \cite{D1} of introducing temporal dynamics into random matrix ensembles.
Consider for instance a Hermitian, $N\times N$ matrix whose entries perform a properly normalized, continuous
random walk in the space of complex numbers. Let this process be initiated with a matrix filled with zeros.  For this particular choice it was shown that the ACP is
equal to a certain time dependent monic Hermite polynomial in a complex variable $z$,  that satisfies a complex diffusion equation with a diffusion constant equal to $-\frac{1}{2N}$. Moreover, the ACP can be transformed into a different function (by taking its logarithmic derivative with respect to the complex variable $z$) that is governed by the viscid Burgers equation. In the large $N$ (inviscid) limit, the latter admits solutions exhibiting shocks whose positions coincide with the edges of the eigenvalue spectrum. In the finite $N$ (viscid) case one can perform an expansion around the shocks to obtain the well known Airy asymptotic behavior of the averaged characteristic polynomial. Note finally, that this analysis can be carried over to the case of diffusing Wishart matrices\cite{BNW1}.

In this paper we demonstrate the robustness of the above ideas. To this end we show that both the ACP  and the averaged inverse characteristic polynomial (AICP) satisfy the same complex diffusion equation, except that for the former the diffusion constant is equal to $-\frac{1}{2N}$, whereas for the latter it is $\frac{1}{2N}$. The new proof works regardless of the actual form of the initial condition imposed on the process and this allows us to examine two different, generic scenarios. The solution of the diffusion equation leads to simple integral representations for both the ACP and the AICP, which makes it possible to  study their asymptotic, large $N$, universal behaviors. In particular, we recover the known scaling property of the ACP and the AICP at the edge of the spectrum in terms of Airy functions. Moreover, for a process initiated with at least two distinct eigenvalues, when two edges of the spectrum meet, the Pearcey functions emerge. Thus, the present diffusion scenario provides a natural and simple way to re-derive the universal functions corresponding to the fold and cusp singularities in random matrix models.

The paper is organized as follows. We start by introducing the stochastic evolution of the studied matrix. Using the representations of the determinant and its inverse as Gaussian integrals over, respectively, Grassman or complex variables, we derive the diffusion equations for the ACP and the AICP. For the simplest scenario, in which the process is initialized with a null matrix, we crosscheck the equation for the AICP by exhibiting the equivalence of its solution with the Cauchy transform of the ACP. In the following section, we derive the corresponding Burgers equation, which we solve in the large $N$ limit with the method of complex characteristics and obtain the associated Green's function for two different generic examples. In the first one, the initial matrix is filed with zeros, whereas in the second, it has two distinct non vanishing eigenvalues. We subsequently use the saddle point method to inspect how the ACP and AICP behave in the former scenario, at the points corresponding to the edges of the probability density function for the eigenvalues, asymptotically when $N$ is increased. Next, we turn to the case of the latter initial condition. The spectrum then forms two disjoint lumps of eigenvalues that eventually collide. We perform the saddle point analysis around the time and point of this collision. Finally, we mention how the ACP and AICP can be used to reconstruct the random matrix kernel for the problem, the details are however left for the appendix.
The last section summarizes our conclusions.

\section{Diffusion of Hermitian matrices}
Let us introduce an $N\times N$ Hermitian matrix $H$ by defining its complex entries according to:
\begin{align}
H_{ij}= \left\{
	\begin{array}{l r}
	x_{ii}, & i=j, \\
	\frac{1}{\sqrt{2}} ( x_{ij}+i y_{ij} ), & i\neq j,
	\end{array}\right.
\end{align}
where $x_{ij}=x_{ji}$ and $y_{ij}=-y_{ji}$, with $x_{ij}$ and $y_{ij}$ real. Furthermore let $x_{ij}$ and $y_{ij}$ perform white noise driven, independent random walks, such that
\begin{align}
	\left \langle\delta H_{ij} \right\rangle = 0, \qquad \left \langle \left(\delta H_{ij}\right)^{2}\right\rangle=\frac{1}{N}\delta \tau
\end{align}
for any $i$ and $j$. Let $P(x_{ij},\tau)P(y_{ij},\tau)$ be the probability that the off diagonal matrix entry $H_{ij}$  will change from its initial state to $\frac{1}{\sqrt{2}} ( x_{ij}+i y_{ij} )$ after time $\tau$. Analogically, $P(x_{ii},\tau)$ is the probability of the diagonal entry $H_{ii}$ becoming equal to $x_{ii}$ at $\tau$.
The evolution of these functions is governed by the following diffusion equations:
\begin{align}
\label{entrydiff}
\frac{\partial}{\partial \tau}P(x_{ij},\tau) & = \frac{1}{2N}\frac{\partial^{2}}{\partial x_{ij}^{2}}P(x_{ij},\tau) , \nonumber \\
\frac{\partial}{\partial \tau}P(y_{ij},\tau) & = \frac{1}{2N}\frac{\partial^{2}}{\partial y_{ij}^{2}}P(y_{ij},\tau) , \quad i\ne j.
\end{align}
Moreover, the joint probability density function
\begin{align}
	 \qquad P(x,y,\tau)\equiv\prod_{k}P(x_{kk},\tau)\prod_{i<j} P(x_{ij},\tau)P(y_{ij},\tau)\label{prob}
\end{align}
satisfies  the following Smoluchowski-Fokker-Planck equation
\begin{align}
\partial_\tau P(x,y,\tau)=\mathcal{A}(x,y) P(x,y,\tau), \qquad \mathcal{A}(x,y) = \frac{1}{2N}\sum_k \frac{\partial^{2}}{\partial x_{kk}^{2}}+\frac{1}{2N}\sum_{i<j} \left ( \frac{\partial^{2}}{\partial x_{ij}^{2}}+ \frac{\partial^{2}}{\partial
y_{ij}^{2}} \right ). \label{prob1}
\end{align}
With the setting thus defined, let us proceed to the derivation of the partial differential equations governing the ACP and AICP.

\subsection{Evolution of the averaged characteristic polynomial}
Let  $\pi_{N}(z,\tau)$ be the averaged characteristic polynomial associated with the diffusing matrix $H$: $\pi_{N}(z,\tau)\equiv\langle \det\left(z-H\right)\rangle$, where the angular brackets denote the averaging over the time dependent probability density (\ref{prob}). In order to derive the partial differential equation governing the ACP, we write the determinant as a Gaussian integral over  Grassmann variables $\eta_i, \bar \eta_i$:
\begin{align}
{\rm det}~{A} = {\int {{\prod _{i,{j}}{{\rm d}{\eta }_{{i}}}}{{\rm d}{{{\overline\eta } }_{{j}}}}}}\,{\exp{\left({\overline{{\eta }_{{i}}}{A}_{{ij}}{\eta
}_{{j}}}\right)}}.\label{gras}
\end{align}
This allows us  to express the averaged characteristic polynomial in the following way:
\begin{align}
\pi_{N}(z,t) & = \int \mathcal{D}[\bar{\eta},\eta,x,y]P(x,y,\tau)\exp{\left[\bar{\eta}_i\left(z\delta_{ij}-H_{ij}\right)\eta_{j}\right]},\label{repr}
\end{align}
where the joint integration measure is defined by
\begin{align}
\mathcal{D}[\bar{\eta},\eta,x,y] & \equiv \prod _{i,j}{\rm d}\eta_{i}{\rm d}\overline{\eta}_{j} \prod_{k}{\rm d}x_{kk}\prod_{n<m}{\rm d}x_{nm}{\rm
d}{y}_{nm}.
\end{align}
The Hermicity condition ($H_{ij}=\bar{H}_{ji}$) allows us to write the argument of the exponent of (\ref{repr}) in a convenient form:
\begin{align}
T_g(\bar{\eta},\eta,x,y,z) & \equiv\sum_{r}\bar{\eta}_{r}\left(z-x_{rr}\right)\eta_{r}
-\frac{1}{\sqrt{2}}\sum_{n<m}\left[
x_{nm}\left(\bar{\eta}_{n}\eta_{m}-\eta_{n}\bar{\eta}_{m}\right)+iy_{nm}\left(\bar{\eta}_{n}\eta_{m}+\eta_{n}\bar{\eta}_{m}\right)\right].\nonumber
\end{align}
Note that the time dependence of $\pi(z,\tau)$ resides entirely in $P(x,y,\tau)$. By differentiating Eq.~(\ref{repr}) with respect to $\tau$, and using Eq.~(\ref{prob1}), one ends up with an expression where the operator $\mathcal{A}(x,y)$ acts on the joint probability density function. Integrating by parts with respect to $x_{ij}$ and $y_{ij}$, one obtains
\begin{align}
\partial_{\tau}\pi_{N}(z,\tau) & = \int \mathcal{D}[\bar{\eta},\eta,x,y]P(x,y,\tau) \mathcal{A}(x,y) \exp{\left[T_g(\bar{\eta},\eta,x,y,z)\right]}.
\end{align}
At this point, we differentiate with respect to the matrix elements (acting with $\mathcal{A}(x,y)$), exploit  some simple properties of Grassmann variables, and obtain:
\begin{align}
\partial_{\tau}\pi_{N}(z,\tau) = -\frac{1}{N} \int \mathcal{D}[\bar{\eta},\eta,x,y]P(x,y,\tau)
\sum_{i<j}\bar{\eta}_i \eta_i \bar{\eta}_j \eta_{j} \exp{\left[T_g(\bar{\eta},\eta,x,y,z)\right]}.
\end{align}
It is easily verified that this expression, when multiplied by $-2N$, matches the double differentiation with respect to $z$ of Eq.~(\ref{repr}).
We thus end up with
\begin{align}
\label{eq:acp}
\partial_{\tau}\pi_{N}(z,t)=-\frac{1}{2N}\partial_{zz}\pi_N(z,\tau).
\end{align}
This is the sought for diffusion equation for the ACP. Note that the same equation was already obtained in Ref.~\cite{BN2}, albeit for a very specific initial condition, for which $\pi_{N}(z,t)$ is a scaled Hermit polynomial. The present derivation has the advantage of being independent of the choice of initial condition.
\subsection{Evolution of the averaged inverse characteristic polynomial}
We now turn to the averaged inverse characteristic polynomial
\begin{align}
\theta_{N}(z,t)\equiv\left\langle \frac{1}{\det\left(z-H\right)}\right\rangle, \label{aicp0}
\end{align}
to which we are going to apply a similar strategy.
In this case, we use he fact that the inverse of a determinant has a well-known representation in terms of a Gaussian integral over complex variables $\xi_{i}$:
\begin{align}
	\frac{1}{{\rm det}{A}}=\int \prod _{i,j}{\rm d}\xi_{i}{\rm d}{\overline\xi}_{j}\exp{\left(-\overline{\xi}_{i}A_{ij}\xi_{j}\right)}.\label{compl}
\end{align}
As in the ACP case, we use this representation to express (\ref{aicp0}) as
\begin{align}
\label{aicpsol}
\theta_{N}(z,\tau) = \int \mathcal{D}[\bar{\xi},\xi,x,y]P(x,y,\tau)\exp{\left[\bar{\xi}_i\left(H_{ij}-z\delta_{ij}\right)\xi_{j}\right]},
\end{align}
where, again, the proper notation for the joint integration measure was introduced. Performing the differentiation with respect to $\tau$ yields:
\begin{align}
\partial_{\tau}\theta_{N}(z,\tau) & = \int \mathcal{D}[\bar{\xi},\xi,x,y]P(x,y,\tau) \mathcal{A}(x,y) \exp{\left[T_c(\bar{\xi},\xi,x,y)\right]}
,\end{align}
with
\begin{align}
T_c(\bar{\xi},\xi,x,y,z) & \equiv \sum_{r}\bar{\xi}_{r}\left(x_{rr}-z\right)\xi_{r}
+\frac{1}{\sqrt{2}}\sum_{n<m}\left [
x_{nm}\left(\bar{\xi}_{n}\xi_{m}+\xi_{n}\bar{\xi}_{m}\right)+iy_{nm}\left(\bar{\xi}_{n}\xi_{m}-\xi_{n}\bar{\xi}_{m}\right)\right ], \nonumber
\end{align}
where we have used (\ref{prob1}), the hermicity of $H$ and we have performed integrations by parts. After differentiation with respect to the matrix elements, one obtains:
\begin{align}
\partial_{\tau}\theta_{N}(z,\tau) = \frac{1}{N} \int \mathcal{D}[\bar{\xi},\xi,x,y]P(x,y,\tau)\left(
\sum_{i<j}\bar{\xi}_i \xi_i \bar{\xi}_j \xi_{j}+\frac{1}{2}\sum_{k}\bar{\xi}_k\xi_k\bar{\xi}_{k}\xi_k \right) \exp{\left[ T_c(\bar{\xi},\xi,x,y,z)\right]},
\end{align}
which, multiplied by $2N$, matches the double differentiation of Eq.~(\ref{aicpsol})  with respect to $z$.
The  final result reads
\begin{align}
\partial_{\tau}\theta_{N}(z,t)=\frac{1}{2N}\partial_{zz}\theta_N(z,t),
\label{aicp}
\end{align}
the announced diffusion equation for the AICP.
\section{The integral representation}
The main advantage of the equations derived above is that they have obvious solutions in terms of initial condition dependent integrals.
In this section we explicitly state those representations and show additionally how, for the simplest initial condition, one is a Cauchy transform of the other.
Let us also note here that these types of integrals were obtained\cite{BK2} as representations of multiple orthogonal polynomials\cite{DF1,BK1} and equivalently
as averaged characteristic polynomials of GUE matrices perturbed by a source\cite{PFOR}.

\subsection{The averaged characteristic polynomial}

One can verify by a direct calculation that the expression
\begin{align}
\label{acpsol}
\pi_N(z,\tau)= \mathcal{C} \,\tau^{-1/2} \int_{-\infty}^{\infty} \exp{\left(-N\frac{(q-iz)^{2}}{2\tau}\right)}\,\pi_N(-iq,\tau=0)\,\mathrm{d} q,
\end{align}
satisfies the complex diffusion equation (\ref{eq:acp}) governing the evolution of the averaged characteristic polynomial. The imaginary unit in the exponent and in the argument of the initial condition arises from the negative value of the diffusion constant in this equation. For finite $N$, the most general form of the initial condition is $\pi_N(z,\tau=0) = \prod_i (z-\lambda_i)$, where the $\lambda_i$'s are real due to the Hermiticity of the initial matrix $H(\tau=0)$.
Exploiting  the steepest descent method to match this with Eq. (\ref{acpsol}), one determines the constant term $\mathcal{C}$. The saddle point associated with $\tau\to 0$ is $u_0 = iz$.
 Performing the Gaussian integration around the saddle point, we obtain $\mathcal{C}=\sqrt{\frac{N}{2\pi}}$ so that Eq.~(\ref{acpsol}) reads
\begin{align}\label{integralrepres3}
\pi_N(z,\tau)= \sqrt{\frac{N}{2\pi\tau}}\int_{-\infty}^\infty \exp{\left(-N\frac{(q-iz)^{2}}{2\tau}\right)}\,\pi_{N}(-iq,\tau=0)\,\mathrm{d} q.
\end{align}

\subsection{The averaged inverse characteristic polynomial}
The integral representation of the averaged inverse characteristic polynomial arising as a solution to the partial differential equation (\ref{aicp}) is
\begin{align}
\theta_N(z,\tau)= \mathcal{C}\int_{\Gamma} ~ \exp{\left(-N\frac{(q-z)^{2}}{2\tau}\right)}\,\theta_{N}(q,\tau=0)\,\mathrm{d} q . \label{aicpsol1}
\end{align}
As in the case of the ACP, the initial condition has to be recovered. Here however, $\theta_N(z,\tau=0)$ has poles on the real axis, and the contour $\Gamma$ must avoid these poles. A first possibility is to  choose $\Gamma_+$ parallel and slightly above  the real axis. In this case the saddle point analysis for $\tau\to 0$ is performed by moving $\Gamma_+$ upward so that it crosses the saddle point $q_0=z$. Obviously, this is possible only if ${\rm Im}z>0$. If instead ${\rm Im}z<0$, we need to choose $\Gamma_-$ also  parallel to the real axis but slightly below. Imposing an integration contour that would switch from the upper to the lower half plane (and {\it vice versa}) in between the poles, would results in a function no longer being the solution of the initial problem.

In the simple case of $\pi_N(z,\tau=0) = z^N$, we can cross-check the above results using the well known\cite{FS1} Cauchy transform formula linking the ACP and the AICP.
In particular, (\ref{integralrepres3}) coincides with the integral representation of the Hermite polynomial\cite{ASKEY}:
\begin{align}\label{fs0}
	\pi_{k}(s,\tau) & = (-i)^{k} \sqrt{\frac{N}{2 \pi \tau}} \int_{-\infty}^\infty q^{k} \exp \left ( -\frac{N}{2\tau} (q-is)^2 \right )\mathrm{d} q .
\end{align}
and the aforementioned Cauchy transform formula reads:
\begin{align}
	\theta_N(z,\tau) & = \frac{1}{c_{N-1}^2} \int \frac{\mathrm{d} s}{z-s} \pi_{N-1}(s,\tau) \exp \left (-\frac{N s^2}{2\tau} \right ),
	\label{fs1}
\end{align}
where the constant $c_k^2 = \sqrt{\frac{2\pi \tau}{N}} \left ( \frac{\tau}{N} \right )^k k!$ is the normalization of the monic polynomials. Note that this is valid only for this particular, simplest initial condition. Analogical prescriptions for other cases are significantly more complicated{\color{MidnightBlue} \cite{DF1}}.

We start this calculation by plugging (\ref{fs0}) into  (\ref{fs1}). After transforming the $q^{N-1}$ term into a differentiation of the exponent with respect to $s$, followed by integrating by parts, we get the result:
\begin{align}
\theta_N(z,\tau) = \sqrt{\frac{N}{2\pi\tau}}  \int^{\infty+z}_{-\infty+z}  \frac{1}{u^N} \exp{\left(-N\frac{(u-z)^2}{2\tau}\right)} \mathrm{d} u. \label{rair}
\end{align}
This contour in turn can be deformed to the real axis and a half circle enclosing the pole at $0$ from above (${\rm Im} z>0$) or below (${\rm Im} z<0$) -
 in complete agreement with the results given before.

\section{Large $N$ spectral dynamics}

Let $\lambda_{i}$'s be the eigenvalues of the diffusing matrix $H(\tau)$.
The connection between the spectral density $\rho(\lambda)\equiv \langle \sum_i \delta(\lambda-\lambda_{i})\rangle $, in the limit of $N$ going to infinity, and the averaged characteristic polynomial, is established through the so-called Green's function, defined by:
\begin{align}\label{def:g}
G(z,\tau)\equiv\frac{1}{N}\langle{\rm Tr}\left[z-H(\tau)\right]^{-1} \rangle.
\end{align}
Note that when $N$ goes to infinity, the poles of this function merge, forming a cut on the complex plane.
The link is made with the well known Sokhotski-Plemelj formula $\rho(\lambda) = \frac{1}{\pi} \lim_{\epsilon\to 0_\pm} \textrm{Im} G(\lambda \mp i \epsilon)$ and the relation
\begin{align}
\label{eq:g}
	G(z,\tau) =  \lim_{N\to\infty}\frac{1}{N} \partial_z \ln \pi_N(z,\tau).
\end{align}
Note that $f_{N}(z, \tau)\equiv \frac{1}{N} \partial_z \ln \pi_N(z,\tau)$ is the famous Cole-Hopf transform. The diffusion equation derived for the ACP in the previous section corresponds to the following Burgers equation for $f_{N}(z, \tau)$
\begin{align}\label{eq:bev}
	\partial_\tau f_N (z,\tau) + f_N(z,\tau) \partial_z f_N(z,\tau) = -\frac{1}{2N} \partial^2_{z} f_N(z,\tau),
\end{align}
in which the ``spatial" variable $z$ is complex and the role of ``viscosity" is played by $-1/N$, a negative number.
In the large $N$ limit the viscosity vanishes, $f_{N}(z,\tau)\to G(z,\tau)$ and
\begin{align}\label{eq:beiv}
	\partial_\tau G(z,\tau) + G(z,\tau) \partial_z G(z,\tau) = 0.
\end{align}
This may be solved by determining curves, called characteristic lines, which are labeled by $\xi$, and  along which the solution is constant and equal to the initial condition, that is $G(z,\tau)=G_0(\xi)$, where $G_{0}(z)\equiv G(z,\tau=0)$.
In this particular case, these characteristic lines are defined in the $(z,\tau)$ hyperplane by
\begin{align}\label{def:char}
z=\xi+\tau G_{0}(\xi).
\end{align}
The result is an implicit equation for $G(z,\tau)$ which is solved under the condition, stemming from (\ref{def:g}), that in the limit of $|z|\to\infty$, the Green's function has to vanish.
Curves which the characteristic lines are tangent to are called envelopes or caustics. Their position, $x_c$, is given by the condition
\begin{align}\label{def:sh}
0=\frac{{\rm d}z}{{\rm d}\xi}\biggr\rvert_{\xi=\xi_c
=1+\tau G'_{0}(\xi_c).
\end{align}
Along them, the mapping between $z$ and $\xi$ ceases to be one to one, which makes the characteristic method loose its validity.

We address two types of initial conditions which are generic for this setting: $H(\tau = 0) = 0$ and $H(\tau=0) = \text{diag} (-a...,a...)$. The first one corresponds to $G_{0}=\frac{1}{z}$, while the second (for which we assume that $N$ is even) amounts to $G_{0}=\frac{1}{2(z-a)}+\frac{1}{2(z+a)}$. In the former scenario, for an infinitely large matrix, the spectrum forms a single interval throughout its evolution. In the latter, it initially occupies two separate domains of the real axis which, in turn, merge at some critical space time point.  Fig.\ref{fig:schemeevol} pictures the time evolution of the two corresponding spectral densities.
In both cases, the characteristic lines that are real at $\tau=0$, remain on the plane of ${\rm Im}z=0$ throughout the time evolution - we depict them in Fig.\ref{fig:charb}. Those that are complex, on the other hand, are symmetric under complex conjugation and never cross each other until some time, when they hit the real line and end on the cut of the Greens function. The caustics, live on the plane of ${\rm Im}( z)=0$ so long as they don't merge. Moreover, they move along the branching points of the resulting Green's function and mark, therefore, the edges of the spectra. Additionally, they constitute the positions of the shocks, curves in the $(z,\tau)$ space along which the characteristic lines have to be cut to ensure unambiguity of the solution for $G(z,\tau)$. Note finally, that if the complex characteristic lines were allowed to cross the cuts of the complex plane, they would form (in the second scenario) complex caustics evolving out of the merging point of the real ones. These are depicted by dashed lines in Fig. \ref{fig:charb}.

\begin{figure}[ht!]
  \centering
   \includegraphics[width=1.\textwidth]{collision2.pdf}
      \caption{The above figure depicts the time evolution of the large $N$ spectral density of the evolving matrices for two scenarios that differ in the imposed initial condition. The parameter $a$ was set to one.}
     \label{fig:schemeevol}
\end{figure}

\begin{figure}[ht!]
  \centering
   \includegraphics[width=0.99\textwidth]{burgers-chars.pdf}
      \caption{The thin lines are characteristics that remain real throughout their temporal evolution. They finish at the bold lines which are caustics and shocks simultaneously. The dashed bold lines are the caustics that would be formed by the strictly complex characteristics (not depicted here) if they didn't end on the branch cut.}
     \label{fig:charb}
\end{figure}

\section{Universal microscopic scaling}

In this section, we inspect  the ACP and the AICP in the vicinity of the points corresponding to the edges of the spectrum of $H(\tau)$, that is near the shocks. For the size of the matrix approaching infinity, one expects that the behaviors of the ACP and the AICP do not depend on the details of the stochastic process governing the evolution - a manifestation of the so-called microscopic universality.

As some of us have demonstrated\cite{BN2}, the asymptotic behavior of the ACP can be recovered by analyzing Eq. (\ref{eq:bev}) through an expansion of $f_N$ around the positions of the shocks. Following this approach one could recover the Airy function describing the behavior of the ACP near the propagating edge. However, this method doesn't seem to be so effective when one examines a situation when two shocks collide. As we shall now see, it is more convenient in this case to return to the diffusion equation and realize that, irrespective of the initial condition, the integral representations of the ACP and AICP have the following generic structure
\begin{align}\label{int1}
\int_{\Gamma}e^{N f(p,z,\tau)}{\rm d}p.
\end{align}
This is well suited for a steepest descent analysis in the large $N$ limit\cite{WONG}. Moreover, the saddle point condition $
\partial_{p}f(p,z,\tau)|_{p=p_i}=0 $ is equivalent to (\ref{def:char}). For the ACP, that is, we have
\begin{align}
f(p,z,\tau)=\frac{1}{N}{\rm ln}\left[\pi_0(-ip)\right]-\frac{1}{2\tau}(p-iz)^2
\end{align}
and we identify $\xi$ with $-ip_i$. For the AICP
\begin{align}
f(p,z,\tau)=\frac{1}{N}{\rm ln}\left[\theta_0(p)\right]-\frac{1}{2\tau}(p-z)^2
\end{align}
(notice that $G_{0}(p)=-\frac{1}{N}\partial_{p}{\rm ln}\left[\theta_0(p)\right]$) with $\xi$ identified as $p_i$.
The fact that the labels of the characteristics and the saddle points are connected is clearly not coincidental. The viscid Burgers equation and the diffusion equation are equivalent trough the Cole-Hopf transform. This  induces an identification between the characteristics method used to solve the inviscid limit of the former and the saddle point method applied for the large $N$ solution of the latter. Consider approaching a caustic in $(z,\tau)$ (hyper-)plane. Through the analogy pointed out above, the merging of characteristics implies the merging of two saddle points - this will be the scenario of the first example considered below. When the caustics merge, forming a cusp, three saddle points coalesce, which will be studied subsequently.

The final issue to resolve before engaging the calculations is the question of what precisely we mean by the ``vicinity" of the edges. If the width of the studied interval remains constant or shrinks too slowly, as the number of the eigenvalues grows to infinity, we will deal with an infinite number of eigenvalues and most of them won't ``feel" that they are ``close" to edge from the macroscopic point of view. On the other hand if the interval shrinks too fast, in the end there won't be any eigenvalues left inside the interval. This is a heuristic explanation of why the studied vicinity of the edge should have a span proportional to the average spacing of the eigenvalues near the shock. This quantity can be derived by inspecting the large $N$ limit of the spectral density, whose behavior is given, through the Sokhotski-Plemelj formula, by the Green's function. To study it around the spectral edge one expands $G$ around $\xi_c$:
\be
G_{0}(\xi)=G_{0}(\xi_c )+(\xi-\xi_c )G_{0}'(\xi_c )+\frac{1}{2}(\xi-\xi_c )^{2}G_{0}''(\xi_c )+\frac{1}{6}(\xi-\xi_c )^{3}G_{0}'''(\xi_c )+\ldots .
\ee
For any initial condition $G_0=(z-\xi)/\tau$ and thus we have $G_{0}'(\xi_c )=-{1}/{\tau}$, which gives
\be\label{zscale}
z-z_c =\frac{\tau}{k !}(\xi-\xi_c )^{k}G_{0}^{(k)}(\xi_c )+\ldots,
\ee
where $k$ in $G_{0}^{(k)}(\xi_c )$ indicates the power of the first after ($G_{0}'$), non-vanishing derivative of $G_{0}$ taken in $\xi_c$, for a given critical point. This leads to:
 \be\label{gwtausr}
G(z,\tau)\simeq G_{0}(\xi_c )+G_{0}'(\xi_c ) \left[\frac{k ! (z-z_c)}{\tau G_{0}^{(k)}(\xi_c )}\right]^{1/k}
\ee
Now, let $N_{\Delta}$ be the average number of eigenvalues located at an interval $\Delta$. We therefore have
\be
N_{\Delta}\sim N\int^{z_c+\Delta}_{z_c}(z-z_c)^{1/k}{\rm d}z \sim N\Delta^{1+1/k}
\ee
which for fixed $N_{\Delta}$, e.g. $N_{\Delta}=1$, implies that the average eigenvalue spacing at those points is proportional to $N^{-k/(1+k)}$.

As we shall perform the rest of the calculations using the saddle point method, let us finally show how the proper scalings arise in that framework, through the condition for the merging of the saddle points $p_i$, as was done in\cite{F1}.
Particularly, the deviation $s$ from $z_c$ has to scale with the size of the matrix in such a way that, when $N$ grows to infinity, the value of the integrand is not concentrated at separate $p_i$'s but in the single $p_c$. This is equivalent to requiring that for such an $s$, the distance between the saddle points $p_i$ is of the order of the width of the Gaussian functions arising from expanding $f(p)$ around the respective $p_i$'s in $\exp [ N f(p) ] $. In particular the condition:
\begin{align}
\label{eq:scaling}
|p_i-p_n | \sim \left[N f''(p_j)\right]^{-1/2}, \quad (i \ne n),
\end{align}
gives the relevant order of magnitude of $s$ that is $N^{\alpha}$. We therefore set $z=z_c+s=z_c+N^\alpha \eta$ and $\eta$ is of order one. Note that $\alpha$ and $\delta$ are related through $\alpha(1+k)=-k$.
This subsequently sets the scale for the distance probed by the deviation from $p_c$ and the condition $|p_i-p_c|\sim N^{\beta}$ defines the substitution $p=p_c+N^{\beta}t$. The connection between the saddle points and characteristic lines allows us to relate $\beta$ and $\alpha$ through $k$. We see that near the critical point $|p_i-p_c|\sim |\xi-\xi_{c}|$. Using Eq. (\ref{zscale}), we thus obtain $\beta=\alpha / k=-(1+\alpha)$, which can be used as a consistency check.
In the example of subsection B the merging of the saddle points happens in a particular critical time $\tau_c$ and there exists a time scale of the order of $N^\gamma$ for which, asymptotically $p_i$ are not distinguishable. This exponent is calculated by expanding the condition for the merging of saddle points around the critical value $\tau= \tau_c + N^\gamma \kappa$.

\subsection{Soft, Airy scaling}
\begin{figure}[ht!]
  \centering
    \includegraphics[scale=.54]{airy-saddle.pdf}
      \caption{In the two above graphs, the gray scale gradient portrays the value of ${\rm Re}f(p)$ (growing with the brightness), whereas the dashed lines depict the curves of constant ${\rm Im}f(p)$.
The left figure is plotted for the ACP, with $p\equiv q$, the right one for the AICP, with $p\equiv u$.  The initial condition is $H(\tau=0)=0$ and time $\tau$ is fixed to $1$ for both.
Dashed bold curves indicate contours of integration suitable for the saddle point analysis, for the AICP, we identify the black and white line with the contours $\Gamma_+$  and $\Gamma_-$ respectively.}
     \label{fig:airysaddle1}
\end{figure}
Let us start by considering the simplest initial condition, namely $\pi_N(z,\tau=0) = z^N$, for which $H(\tau=0)$ is filled with zeros:
\begin{align}
\pi_{N}(z,\tau) & = (-i)^{N} \sqrt{\frac{N}{2 \pi \tau}} \int_{-\infty}^\infty q^{N} \exp \left ( -\frac{N}{2\tau} (q-iz)^2 \right )\mathrm{d} q .
\end{align}
In this setting, the two edges of the spectrum plainly move in opposite directions along the real line (see Fig. \ref{fig:schemeevol}). We conduct a large $N$ steepest descent analysis. As a first step we set the function $f = \ln q - \frac{1}{2\tau} (q-iz)^2 $ and we compute the saddle point equation (\ref{eq:saddle}) as:
\begin{align*}
	\tau = q(q-iz).
\end{align*}
This relation has the role analogous to the hydrodynamic equation governing the characteristics as was introduced in the analysis of the Burgers equation and described above.
The positions of the saddle points are given by
$q_\pm = \frac{1}{2} \left ( i z \pm \sqrt{4\tau - z^2} \right )$.  Their merging, at $q_c=i\sqrt{\tau}$, marks the locations of the spectral edges, in particular the right one $z_c=2\sqrt{\tau}$. Similarly, in the hydrodynamic picture, a singularity emerges when reaching the critical point $z_c$, corresponding to the crossing of the characteristic lines.

The new contour, going trough $q_c$ is depicted (for $\tau=1$) on the left plot of figure \ref{fig:airysaddle1}. The contour deformation is constrained by two conditions: a) the real part of the function $f$ reaches a maximum along the contour and b) the imaginary part  must obey the condition $\textrm{Im} f = \textrm{const}$. By imposing the latter, we guarantee the steepest descent of $\textrm{Re} f$ upon integrating along the contour. These requirements fix uniquely the path marked in bold on the left plot of figure \ref{fig:airysaddle1}.

The scaling exponent $\alpha$ is equal to $-\frac{2}{3}$ and we have $\eta = (z-2\sqrt{\tau}) N^{2/3}$. Moreover, $\beta=-\frac{1}{3}$ and the change of variables in the integral
is given by $t = (q-i\sqrt{\tau})N^{1/3}$. These scaling parameters were obtained from the equation (\ref{eq:scaling}) which compares the spacing between the saddle points $q_\pm$ to the width of the Gaussian approximation. The same result can be read out from the spectral density expanded around the edge of the bulk $\rho \sim \sqrt{|z-z_c|}$. Expanding the logarithm and taking the large matrix size limit yields
\begin{align}
\pi_N \left(z=2\sqrt{\tau}+ \eta N^{-2/3},\tau\right)\approx \tau^{N/2}\frac{N^{1/6}}{\sqrt{2\pi}} \exp{\left(\frac{N}{2}+\frac{\eta N^{1/3}}{\sqrt{\tau}}\right)} \textrm{Ai} \left(\frac{\eta}{\sqrt{\tau}}\right),
\end{align}
where
\begin{align}
\label{eq:airy}
	\textrm{Ai}(x) = \int_{\Gamma_0} \mathrm{d} t \exp \left (\frac{it^3}{3} + itx \right ),
\end{align}
is the well-known Airy function. The contour $\Gamma_0$ is formed by the rays $-\infty \cdot e^{5i\pi/6}$ and $\infty \cdot e^{i \pi/6}$ emerging as $N$ goes to infinity. Along these rays the integral (\ref{eq:airy}) is convergent as can be seen by substituting $t \to t e^{i\pi/6}$.

In the case of the AICP, the initial condition used above takes the form of $\theta_N(z,\tau=0)=z^{-N}$ and (\ref{aicpsol1}) reads
\begin{align}
\theta_N(z,\tau) = \sqrt{\frac{N}{2\pi\tau}}  \int_{\Gamma_\pm} u^{-N} \exp{\left(-N\frac{(u-z)^2}{2\tau}\right)} \mathrm{d} u,
\end{align}
where the contours avoid the pole at zero from above $\Gamma_+$ (for ${\rm Im}z>0$) or below $\Gamma_-$ (for ${\rm Im}z<0$), as explained in subsection IIIB.
This time the saddle point merging for the right spectral edge occurs for $u_c=\sqrt{\tau}$. It also marks the position of the cusps of the new integration contours
(depicted for $\tau=1$ on the second plot of figure \ref{fig:airysaddle1}).  The transformation of variables is given by $\eta = (z-2 \sqrt{\tau})N^{2/3}$ and $it = (u - \sqrt{\tau})N^{1/3}$.
Notice that the complex plane of $t$ is rotated by $\pi/2$ with respect to the one of $u$. By expanding the logarithm and taking the large matrix size limit we obtain
\begin{align}
	\theta_N \left(z=2\sqrt{\tau}+\eta N^{-2/3},\tau\right) & \approx
i \tau^{-N/2}\frac{N^{1/6}}{\sqrt{2\pi}}
\exp{\left(-\frac{N}{2}-\frac{\eta N^{1/3}}{\sqrt{\tau}}\right)} \textrm{Ai} \left(e^{i\phi_{\pm}}\frac{\eta}{\sqrt{\tau}}\right),
\end{align}
the asymptotic behavior in terms of the Airy function, yet with its argument rotated by $\phi_+= -2\pi/3$, for ${\rm Im}z>0$, and by $\phi_-=2\pi/3$, for ${\rm Im}z<0$ in accordance
with previous results for static matrices\cite{AF}.

\subsection{Pearcey scaling}
\begin{figure}[ht!]
  \centering
    \includegraphics[scale=.54]{pearcey-saddle.pdf}
      \caption{In the two above graphs, the gray scale gradient portrays the value of ${\rm Re}f(p)$ (growing with the brightness), whereas the dashed lines depict the curves of constant ${\rm Im}f(p)$.
The left figure is plotted for the ACP, with $p\equiv q$, the right one for the AICP, with $p\equiv u$.  The initial condition is $H(\tau=0)={\rm diag}(1, \dots,1 , -1,\dots,-1)$ and time $\tau$ is fixed to $1$ for both.
Dashed bold curves indicate contours of integration suitable for the saddle point analysis, for the AICP, we identify the black and white line with the contours $\Gamma_+$  and $\Gamma_-$ respectively.}
     \label{fig:pearceysaddle1}
\end{figure}
To observe a collision of the edges of the spectrum, in the large $N$ limit, one has to consider a slightly different initial condition.  Let $\pi_{N}(z,\tau=0)=(z^2 - a^2)^{N/2}$, with $N$ even. This corresponds to the initial matrix eigenvalues set to $\pm a$ with equal degeneracy $N/2$. In this case the ACP takes the form:
\begin{align}
\pi_N(z,\tau)=i^N \sqrt{\frac{N}{2\pi \tau}}\int_{-\infty}^{+\infty} \mathrm{d} q \exp{\left[-\frac{N}{2\tau}(q-iz)^2 +\frac{N}{2}\log(a^2+q^2)\right]}.
\end{align}
We determine the saddle point equation:
\begin{align}
\label{eq:saddle}
	\frac{q}{q^2+a^2}- \frac{q-iz}{\tau} = 0,
\end{align}
and calculate its three solutions $q_{1,2,3}$. In this scenario the two parts of the spectra join at $z_c=0$ at $\tau_c=a^2$ and this is reflected in the saddle points merging for $q_c=0$.
The contour doesn't have to be deformed in this case as seen on the left plot of figure \ref{fig:pearceysaddle1}.
Subsequently, the leading term of the equation (\ref{eq:scaling}) relating the distance between the solutions $q_{1,2,3}$ to the Gaussian width of the saddle point approximation is used to extract the scaling $z\sim N^{-3/4}$. Further analysis of the distance between the critical saddle point $q_c$ and the points $q_{1,2,3}$ shows that the main contribution to the integral for $z\sim N^{-3/4}$ comes from $q$ of order $N^{-1/4}$ and $\tau$ of order $N^{-1/2}$. Based on this, we set $q=t N^{-1/4}, \quad \tau = a^2 + \kappa N^{-1/2}, \quad z=\eta N^{-3/4}$.
In the limit of $N \rightarrow \infty$ we expand the logarithm arising in the exponent through the initial condition and find:
\begin{align}
\pi_N\left(z=\eta N^{-3/4},\tau \approx a^2 + \kappa N^{-1/2} \right)\approx \frac{N^{1/4}}{\sqrt{2 \pi}}(ia)^{N} \textrm P\left (\frac{\kappa}{2a^2},\frac{\eta}{a} \right ),
\end{align}
where we define the Pearcey integral by:
\begin{align}
\label{pearceydef}
\textrm P(x,y) = \int_{-\infty}^{\infty} \mathrm{d} t ~ \exp \left ( -\frac{t^4}{4} + x t^2 + i t y \right ).
\end{align}
This conclusion to the analysis of the microscopic behavior of the ACP is not a surprise - the Pearcey kernel was first derived\cite{BH1} in the context of GUE matrices perturbed by a source\cite{PZJ1}, i.e. in a setting analogical to considered here, in which the sources constitute the initial condition and their critical adjustment plays the role of the critical time.

In the case of the AICP, the initial condition reads $\theta_N(z,\tau=0) = (z^2-a^2)^{-N/2}$ and the solution to the complex diffusion equation is:
\begin{align}
	\theta_N(z,\tau) = \sqrt{\frac{N}{2 \pi \tau}}\int_{\Gamma_{\pm}} \mathrm{d} u ~ (u^2-a^2)^{-N/2} \exp \left ( -\frac{N}{2 \tau} (u-z)^2 \right ),
\end{align}
where $\Gamma_\pm$ denotes contours circling the poles $u_{i} = \pm a$ from above ($\Gamma_{+}$, ${\rm Im}z>0$) or from below ($\Gamma_{-}$, ${\rm Im}z<0$).
The right plot in figure \ref{fig:pearceysaddle1} depicts the types of curves these are deformed to.
We parametrize the saddle point expansion by $\tau= a^2 + \kappa N^{-1/2}$,  $z= \eta N^{-3/4}$ and $u = e^{i\pi/4} t N^{-1/4}$.
These turn out to give:
\begin{align}
\theta_N(z=\eta N^{-3/4},\tau \approx a^2 + \kappa N^{-1/2}) \approx \frac{N^{1/4}}{\sqrt{2\pi}} (ia)^{-N}  \int_{\Gamma_{\pm}} \mathrm{d} t ~ \exp \left ( -t^4/4 - \frac{i\kappa}{2a^2}t^2 + it \frac{e^{-i\pi/4} \eta}{a} \right ).
\end{align}
We therefore obtain a Pearcey type integral along contours $\Gamma_{+}, \Gamma_{-}$ depending on the choice of sign of the imaginary part of $z$.  The former is defined by rays
with phase $\pi/2$ and $0$ whereas the later starts at $-\infty$ and after reaching zero forms a ray along a phase $-\pi/2$.

\section{Constructing the kernel}
The ACP and the AICP are the building blocks of the matrix kernel, which in turn contains, for arbitrary $N$, all the information about the matrix model.
Obtaining the kernel was the aim of many previous works\cite{BK1,DF1,BK2}. By making use of these results we can easily write down its form for the case of the studied diffusing matrix. We have
\begin{align}
\label{kernel}
	K_N(x,y,\tau) = \sum_{i=0}^{N-1} \Theta_i(x,\tau) \Pi_i(y,\tau),
\end{align}
where $\Theta_i(x,\tau)$ and $\Pi_i(y,\tau)$ are defined as follows. First, let
\begin{align}
\label{teta}
	\Theta_{\vec{m}}(x,\tau) & \equiv \theta_{|\vec{m}|}^{+}(x,\tau) - \theta_{|\vec{m}|}^{-}(x,\tau) = \sqrt{\frac{N}{2\pi \tau} }\oint_{\Gamma_0} \mathrm{d} u ~ \exp \left ( -N \frac{(u-x)^2}{2\tau} \right ) \theta_{0}(u) , \\\label{paj}
\Pi_{\vec{m}} (x,\tau) & \equiv \pi_{|\vec{m}|}(x,\tau) = \sqrt{\frac{N}{2\pi \tau} }\int_{-\infty}^\infty \mathrm{d} q ~ \exp \left ( -N \frac{(q-ix)^2}{2\tau}  \right ) \pi_{0}(-iq).
\end{align}
Here $\theta_{0}(x) = \prod_{i=1}^d (x-a_i)^{-m_i}$, $\pi_{0}(x) = \prod_{i=1}^d (x-a_i)^{m_i}$ are the initial conditions and $\vec{m}$ is a corresponding arbitrary eigenvalue multiplicity vector.
Moreover, the contour $\Gamma_0$ encircles all the sources $a_i$ clockwise and $\theta^+(z,\tau)$, $\theta^-(z,\tau)$ denote the different solutions of AICP diffusion equation valid for $\mathrm{Im} z>0$ and $\mathrm{Im} z<0$ respectively.
Finally, the functions labeled by the index $i$ in \eqref{kernel} arise through an ordering of the multiplicities according to their increasing norm  $|\vec{m}| \equiv \sum_{j=1}^d m_j$ (see \eqref{vectors}):
\begin{align}
	\Theta_i \equiv \Theta_{\vec{n}^{(i+1)}}, \qquad \Pi_i \equiv \Pi_{\vec{n}^{(i)}}, \qquad i = 0, ..., N-1.
\end{align}
The details concerning the construction of (\ref{kernel}) are delivered in Appendix A. It is shown in this Appendix that the Kernel for the Pearcey process that derives from the present construction is identical to that obtained by Brezin and Hikami\cite{BH1}:

\begin{align}
\label{bhclass}
	K_{BH}(x,y,\tau) = - \frac{N}{2\pi \tau}\oint_{\Gamma_0} \mathrm{d} u \int_{-\infty}^\infty \mathrm{d} q \frac{(-q^2-a^2)^{N/2}}{(u^2-a^2)^{N/2}} \frac{1}{u+iq} \exp \left ( -N \frac{(q-iy)^2}{2\tau} -N \frac{(u-x)^2}{2\tau}  \right ).
\end{align}

\section{Conclusions}
In this work we have studied the behavior of averaged characteristic polynomials and averaged inverse characteristic polynomials
associated with Hermitian matrices filled with entries performing Brownian motion in the space of complex numbers. A key step of our analysis was a derivation of partial differential equations governing the matrix-valued evolution. These turned out to be, independently of the initial conditions, complex heat equations with diffusion coefficients
inversely proportional to the size of the matrices, and thus provided us with integral representations of the polynomials. By using the saddle point method we were able
to examine their so-called critical microscopical behavior, which is known to be universal. In particular, the asymptotics are driven by the Airy functions, at the edges of the spectrum, and
by the Pearcey functions, when those edges meet. The first case holds for any frozen moment of time, however one can easily modify the free diffusion into an Ornstein-Uhlenbeck problem to obtain the Airy behaviour in the stationary limit at $\tau\to\infty$.

Finally we note, that such an analysis is also possible in the case of diffusing Wishart\cite{BNW2} or chiral\cite{CHIRBURG} matrices for which the Pearcey function is replaced by the Bessoid function
and an additional Bessel type behavior emerges.

\section{Acknowledgements}
MAN appreciates inspiring discussions with Nicholas Witte. PW is supported by the International PhD Projects Programme of the Foundation
for Polish Science within the European Regional Development Fund of the European Union, agreement no. MPD/2009/6 and the ETIUDA scholarship under the agreement no. UMO-2013/08/T/ST2/00105  of the National Centre of Science. MAN and JG are supported by the Grant DEC-2011/02/A/ST1/00119 of the National Centre of Science.

redpajama_set_name: RedPajamaArXiv


text:
30 seconds interview: Guy Andrews
Composer-producer Guy Andrews mines a deep love of techno, ambient and post-rock for his own nocturnal electronics.
Over the years, his submerged sounds have been released via several lauded labels including Erased Tapes and Scuba's Hotflush, with his latest home the Fabric-affiliated Houndstooth.
It's not just labels that have been quick to show their support, Guy's also won over Massive Attack, collaborating with 3D (Robert Del Naja) from the band, while Ulrich Schnauss, Max Cooper and Ben UFO are all firm fans.
Next week sees the release of, Tåke, his second album for Houndstooth. Inspired by the fjords of Norway and the snowy streets of Bergen, it's a paean to the Scandinavian hinterlands that he found himself lost in late last year.
The album arcs through pulsating techno, sinister post-rock and glistening glacial ambience, shot through with the silken voice of collaborator Alev Lenz.
Here, he tells us more about the album and his wider musical make-up…
What was the first song you developed an obsession for?
Smashing Pumpkins' Mayonnaise. It perfectly captures the Pumpkins' famous guitar distortion and had a massive impact on my sonic experimentation.
What's the first gig you went to?
A Perfect Circle at the Hammersmith Apollo, February 2004.
What's the first instrument you ever got hold of?
A Petrof upright piano when I was around four years old. It is still one of the most inviting and warm sounding pianos I have ever played.
What is your worst musical habit?
Spending too much time on audio production early on in the writing process; it's more important to focus on songwriting than producing sounds.
What's the best piece of musical advice you've ever been given?
Always keep reflecting on why you're writing the song, so you keep its inspiration and narrative at the forefront of all compositional decision-making.
Where do you discover new music?
Apart from online, I often discover my favourite new music at festivals such as Sea Change in Totnes or The Great Escape in Brighton.
What's your favourite venue?
The Barbican. It has amazing acoustics and the event programming is excellent.
Who is your current favourite band/artist?
Ben Frost, his sound design and storytelling within his composition is inspiring to listen to.
What inspirations outside of music impact your songwriting?
Exploring the outdoors and taking in new landscapes. Discovering new places influences me more than anything.
What track of yours best represents your sound?
Fjell, as it melts my main musical influences of techno and post-rock into one track.
22 September sees the release of my new eight-track album Tåke on Houndstooth, and I'm supporting Ulrich Schnauss at Islington Assembly Hall on 27 October.
Preorder the record
guy andrews
Playlist: WildKat
This week's playlist come courtesy of WildKat, a creative agency for culture and music as they celebrate a year of expansion and the host of great artist they have worked with. Dig in…
Interview: Robin Richards
'There are no boundaries and there are no limits to what you can create with this art form': composer and multi-instrumentalist Robin Richards branches out from his work with Dutch Uncles.

redpajama_set_name: RedPajamaCommonCrawl


text:
Von Hippel-Lindau disease-associated renal cell carcinoma: a call to action. November 19, 2021
FDA Approves Merck's Pembrolizumab as Adjuvant Therapy for Certain Patients With Renal Cell Carcinoma (RCC) Following Surgery November 18, 2021
Patterns of first-line targeted therapy utilization and adherence among older adults diagnosed with metastatic renal cell carcinoma. November 18, 2021
Predictive Impact of Early Changes in Serum C-Reactive Protein Levels in Nivolumab Plus Ipilimumab Therapy for Metastatic Renal Cell Carcinoma. November 18, 2021
Comparison of prognosis between patients undergoing radical nephrectomy versus partial nephrectomy for renal cell carcinoma ≤7 cm T3aN0/xM0: Survival benefit is biased toward partial nephrectomy. November 17, 2021
Kidney tumor diffusion-weighted magnetic resonance imaging derived ADC histogram parameters combined with patient characteristics and tumor volume to discriminate oncocytoma from renal cell carcinoma. November 17, 2021
Treatment-free Survival after Immune Checkpoint Inhibitor Therapy versus Targeted Therapy for Advanced Renal Cell Carcinoma: 42-Month Results of the CheckMate 214 Trial. November 16, 2021
Survival in Metastatic Renal Cell Carcinoma Patients Treated With Sunitinib With or Without Cryoablation. November 16, 2021
The The Prognosis and Immune Microenvironment Related Factor Interleukin-23 in Clear Cell Renal Cell Carcinoma: A Radiological Investigation. November 15, 2021
[Synchronous oligometastatic renal cell carcinoma-what is the role of surgery?] November 15, 2021
Positive Preliminary Data from Phase 1b Trial Evaluating Batiraxcept (AVB-500) in Combination with Cabozantinib for Treatment of Clear Cell Renal Cell Carcinoma November 12, 2021
Referral and adjuvant treatment patterns after nephrectomy in high-risk locoregional renal cell carcinoma. November 12, 2021
First-line Nivolumab plus Ipilimumab Versus Sunitinib in Patients Without Nephrectomy and With an Evaluable Primary Renal Tumor in the CheckMate 214 Trial. November 12, 2021
Association of baseline neutrophil-to-eosinophil ratio with response to nivolumab plus ipilimumab in patients with metastatic renal cell carcinoma. November 11, 2021
ROBOCOP II (ROBOtic assisted versus conventional open partial nephrectomy) randomised, controlled feasibility trial: clinical trial protocol. November 11, 2021
Treatment-free Survival after Immune Checkpoint Inhibitor Therapy versus Targeted Therapy for Advanced Renal Cell Carcinoma: 42-Month Results of the CheckMate 214 Trial November 10, 2021
The role of stereotactic body radiation therapy and its integration with systemic therapies in metastatic kidney cancer: a multicenter study on behalf of the AIRO (Italian Association of Radiotherapy and Clinical Oncology) genitourinary study group. November 10, 2021
Combination Therapy in Renal Cell Carcinoma: the Best Choice for Every Patient? November 10, 2021
Thromboembolism in Patients with Metastatic Renal Cell Carcinoma Treated with Immunotherapy. November 9, 2021
Analysis by region of outcomes for patients with advanced renal cell carcinoma treated with cabozantinib or everolimus: a sub-analysis of the METEOR study. November 9, 2021


text:
package org.citygml4j.cityjson.adapter.geometry.builder;

import com.fasterxml.jackson.databind.JsonNode;
import com.fasterxml.jackson.databind.node.ArrayNode;
import com.fasterxml.jackson.databind.node.ObjectNode;
import org.citygml4j.cityjson.adapter.Fields;
import org.citygml4j.cityjson.adapter.appearance.builder.AppearanceBuilder;
import org.citygml4j.cityjson.adapter.geometry.MultiSurfaceProvider;
import org.citygml4j.cityjson.model.geometry.GeometryType;
import org.citygml4j.cityjson.reader.CityJSONBuilderHelper;
import org.citygml4j.cityjson.util.BoundaryFilter;
import org.citygml4j.cityjson.util.lod.DefaultLodMapper;
import org.citygml4j.cityjson.util.lod.LodMapper;
import org.citygml4j.core.model.appearance.Appearance;
import org.citygml4j.core.model.core.AbstractFeature;
import org.citygml4j.core.model.core.AbstractSpace;
import org.xmlobjects.gml.model.geometry.primitives.SurfaceProperty;

import java.util.*;

public class GeometryBuilder {
    private final CityJSONBuilderHelper helper;
    private final VerticesBuilder verticesBuilder;
    private final VerticesBuilder templatesVerticesBuilder;
    private final SpaceGeometryBuilder spaceGeometryBuilder;
    private final SemanticsBuilder semanticsBuilder;
    private final MaterialBuilder materialBuilder;
    private final TextureBuilder textureBuilder;
    private final TemplateInfo templateInfo;
    private final Map<AbstractSpace, Map<Integer, MultiSurfaceProvider>> providers = new IdentityHashMap<>();

    private LodMapper lodMapper = new DefaultLodMapper();
    private boolean transformTemplateGeometries;
    private boolean assignAppearancesToImplicitGeometries;

    public GeometryBuilder(ArrayNode vertices, ObjectNode templates, AppearanceBuilder appearanceBuilder, CityJSONBuilderHelper helper) {
        this.helper = helper;

        verticesBuilder = new VerticesBuilder(vertices);
        templatesVerticesBuilder = new VerticesBuilder(helper.getOrPutArray(Fields.VERTICES_TEMPLATES, templates));

        spaceGeometryBuilder = new SpaceGeometryBuilder(this, helper);
        semanticsBuilder = new SemanticsBuilder(helper);
        materialBuilder = new MaterialBuilder(appearanceBuilder, helper);
        textureBuilder = new TextureBuilder(appearanceBuilder, helper);

        templateInfo = new TemplateInfo(helper.getOrPutArray(Fields.TEMPLATES, templates));
    }

    public VerticesBuilder getVerticesBuilder() {
        return verticesBuilder;
    }

    public VerticesBuilder getTemplatesVerticesBuilder() {
        return templatesVerticesBuilder;
    }

    public boolean isTransformTemplateGeometries() {
        return transformTemplateGeometries;
    }

    public void setTransformTemplateGeometries(boolean transformTemplateGeometries) {
        this.transformTemplateGeometries = transformTemplateGeometries;
    }

    public boolean isAssignAppearancesToImplicitGeometries() {
        return assignAppearancesToImplicitGeometries;
    }

    public void setAssignAppearancesToImplicitGeometries(boolean assignAppearancesToImplicitGeometries) {
        this.assignAppearancesToImplicitGeometries = assignAppearancesToImplicitGeometries;
    }

    public LodMapper getLodMapper() {
        return lodMapper;
    }

    public void setLodMapper(LodMapper lodMapper) {
        this.lodMapper = Objects.requireNonNull(lodMapper, "The LoD mapper must not be null.");
    }

    public List<Appearance> getGlobalAppearances() {
        return templateInfo.getAppearances();
    }

    public boolean hasGlobalAppearances() {
        return templateInfo.hasAppearances();
    }

    public void setMultiSurfaceProviders(AbstractSpace space, Map<Integer, MultiSurfaceProvider> providers) {
        if (providers != null) {
            this.providers.put(space, providers);
        }
    }

    public GeometryObject getGeometry(AbstractFeature object, JsonNode geometry, int lod, BoundaryFilter filter) {
        GeometryObject geometryObject = GeometryObject.newInstance();
        getGeometry(object, geometryObject, geometry, lod, filter, verticesBuilder);
        return geometryObject.isSetGeometry() || geometryObject.isSetImplicitGeometry() ?
                geometryObject :
                null;
    }

    void getGeometry(AbstractFeature object, GeometryObject geometryObject, JsonNode geometry, int lod, BoundaryFilter filter, VerticesBuilder verticesBuilder) {
        if (geometry != null && geometry.isObject()) {
            GeometryType type = GeometryType.fromValue(geometry.path(Fields.TYPE).asText());

            GeometryObjectBuilder builder = null;
            if (GeometryType.SURFACE_TYPES.contains(type)) {
                builder = new SurfaceGeometryBuilder(object, filter, verticesBuilder);
            } else if (GeometryType.SOLID_TYPES.contains(type)) {
                builder = new SolidGeometryBuilder(object, filter, verticesBuilder);
            } else if (type == GeometryType.MULTI_LINE_STRING) {
                builder = new CurveGeometryBuilder(object, filter, verticesBuilder);
            } else if (type == GeometryType.MULTI_POINT) {
                builder = new PointGeometryBuilder(object, filter, verticesBuilder);
            } else if (type == GeometryType.TEMPLATE_GEOMETRY) {
                builder = new TemplateGeometryBuilder(object, filter, templateInfo, this, helper);
            }

            if (builder != null) {
                builder.build(geometry, type, lod, geometryObject);

                if (geometryObject.getGeometry() != null && builder instanceof SurfaceProvider) {
                    List<SurfaceProperty> surfaces = ((SurfaceProvider) builder).getSurfaces();
                    materialBuilder.build(geometry.path(Fields.MATERIAL), surfaces, geometryObject);
                    textureBuilder.build(geometry.path(Fields.TEXTURE), surfaces, geometryObject);
                    if (object instanceof AbstractSpace) {
                        AbstractSpace space = (AbstractSpace) object;
                        JsonNode semantics = geometry.path(Fields.SEMANTICS);
                        semanticsBuilder.build(space, semantics, filter, surfaces, lod, geometryObject);
                    }
                }
            }
        }
    }

    public Map<Integer, List<GeometryObject>> getGeometries(AbstractFeature object, JsonNode geometries, BoundaryFilter filter) {
        Map<Integer, List<GeometryObject>> geometryObjects = new HashMap<>();

        Map<Double, List<JsonNode>> geometriesByLod = groupGeometriesByLod(geometries);
        if (!geometriesByLod.isEmpty()) {
            lodMapper.buildMapping(geometriesByLod.keySet());

            int maxLod = Collections.max(geometriesByLod.keySet()).intValue() + 1;
            for (int lod = 0; lod < maxLod; lod++) {
                List<JsonNode> nodes = geometriesByLod.getOrDefault(lodMapper.getMappingFor(lod), Collections.emptyList());
                for (JsonNode geometry : nodes) {
                    GeometryObject geometryObject = getGeometry(object, geometry, lod, filter);
                    if (geometryObject != null) {
                        geometryObjects.computeIfAbsent(lod, v -> new ArrayList<>()).add(geometryObject);
                    }
                }
            }
        }

        return geometryObjects;
    }

    public void addGeometries(AbstractSpace space, JsonNode geometries, BoundaryFilter filter) {
        try {
            if (geometries != null && geometries.isArray()) {
                spaceGeometryBuilder.build(space, geometries, filter, providers.getOrDefault(space, Collections.emptyMap()));
            }
        } finally {
            providers.remove(space);
        }
    }

    public Map<Double, List<JsonNode>> groupGeometriesByLod(JsonNode geometries) {
        if (!geometries.isEmpty()) {
            Map<Double, List<JsonNode>> geometriesByLod = new HashMap<>();
            for (JsonNode geometry : geometries) {
                GeometryType type = GeometryType.fromValue(geometry.path(Fields.TYPE).asText());
                JsonNode lod = type != GeometryType.TEMPLATE_GEOMETRY ?
                        geometry.path(Fields.LOD) :
                        templateInfo.getTemplate(geometry.path(Fields.TEMPLATE).asInt(-1)).path(Fields.LOD);

                if (lod.isValueNode()) {
                    geometriesByLod.computeIfAbsent(lod.asDouble(), v -> new ArrayList<>()).add(geometry);
                }
            }

            return geometriesByLod;
        }

        return Collections.emptyMap();
    }
}

redpajama_set_name: RedPajamaGithub


text:
Walker 'thrilled' to seal £45m City transfer
Kyle Walker. Pic: PA
Kyle Walker has joined Manchester City from Tottenham on a five-year deal worth an initial £45m.
Pep Guardiola's side agreed a fee with Spurs for the 27-year-old England star which could rise to £50m if the add-on clauses are met.
Walker's agency, Base Soccer, claimed that their client commanded: "a record fee for a defender and record fee for an English player" - potentially surpassing the £47.5m City paid Everton for centre-back John Stones.
Walker's arrival takes City's summer spending beyond the £100m mark after the captures of playmaker Bernardo Silva and keeper Ederson.
"I am thrilled to be signing for City," said Walker. "Pep Guardiola is one of the most respected managers in the world and I feel he can help me take my game to a new level."


text:
Q: What's the point in this xor magic defining the upper bound of a loop? I've stumbled upon this piece of code without any comments:
int delay = 5;
int maxDelay = 5 * (3 ^ 5); // << what's this xor magic?
while( Something && delay < maxDelay ) {
    Thread.Sleep( delay );
    delay *= 3;
}

What's the point in that 3 ^ 5? That's "3 xor 5" which is six, not anything interesting like maybe one third of Int32.MaxValue. Why not just write 6 instead? I guess it has to do something with the fact that the result is then multiplied by 5 and also the value is multiplied by 3 on each iteration.
Is that some well-known pattern? What does it do?

A: He meant Math.Pow(3, 5) but ^ is not the power operator (which does not exist in C#). This is a bug.
That makes sense because in the loop he is increasing delay exponentially. So the exponential upper bound makes sense.
I think this is bad style even if it worked. It is not obvious how often the loop will run. He should have used a normal for loop and computed delay freshly inside the loop instead of running it along the loop incrementally.

redpajama_set_name: RedPajamaStackExchange


This car delivers affordable transportation inside an attractive four door package with class leading standard safety features. A turbocharger further enhances performance, while also preserving fuel economy. Top features include power front seats, heated seats, high intensity discharge headlights, and leather upholstery.
$703/mo** This offer is available only to qualified customers with a credit score of or above, at participating dealers by . Advertised lease rate based on months, due at lease signing, acquisition fee, including the gross capitalized cost of . Excludes title, taxes, registration, license fees, dealer preparation and additional options. Total monthly payments equal . Cash due at signing includes capitalized cost reduction, a required security deposit of , and first month's lease payment of . charge per mile over miles per year. Early termination fees may apply. Lessee is responsible for maintenance, insurance, repair, and excess wear/tear at end of lease. Lessee will not be liable for extra charges if realized vehicle value is lower than residual value at end of lease. Lessee may incur penalties and additional charges for delinquent payments. See for complete details on this and other offers.


Experience a unique and intimate guided tour through the Waipoua Forest as you share in the culture connecting the Maori people to their land and forest. Learn firsthand from local Maori about spiritual awareness, history, genealogy and traditional song.
Meet your guide for the evening in Hokianga where you board your coach and travel to Waipoua Forest. On this journey and as you arrive at the forest you will learn about Kupe and his journey from Hawaiiki to Aotearoa. From here you will encounter the mighty Te Matua Ngahere ('The Father of the Forest') estimated to be 2,500 & 3,000 years old; 'The Four Sisters,' a graceful collection of Kauri trees; and Tane Mahuta ('The Lord of the Forest') who stands an impressive 51 metres (150ft) tall.
Your guide will greet the giant trees with spine-tingling waiata (sacred songs) and share the legends of the forest, of its gods and other spiritual inhabitants.
You will also experience the affinity that early Maori had with the forest and the deep spiritual respect they hold for the giants still growing there. Learn of the rich bird life and other fauna that inhabits the forest and witness the transitions of life and the day turning into the night.
As your continue on your tour you may encounter insects like weta, nocturnal birds, eels, fresh-water lobsters or the giant carnivorous 'kauri' snail.
At the conclusion of this unique encounter you will be returned to Hokianga.
A medium level of fitness is required.
A minimum of two people are required for the tour to proceed. There will be up to 15 people on your tour.
Wear comfortable walking shoes. Bring a warm top, and dress for the weather conditions of the day.
This experience takes place in the Waipoua Forest, 20 mins from Omapere.


Scope and Contents The collection consists of a letter written by Ambrose Stanton as Treasurer of the Indianapolis Art Association regarding the purchase of a piece by an Indianapolis artist.
Scope and Contents This collection includes typed and autographed letters from family members, friends and military officers in the United States and other countries; greeting and postal cards and telegrams; magazine and newspaper articles event programs; invoices; poems; exhibition catalogs; a Japanese Imperial Army war flag from World War II; and Japanese currency ranging from 1908-1980 regarding art and art business, family, personal, religious, and war matters.
Scope and Contents The collection is unique in that it includes his early works from his time at the John Herron Art School as well as personal journals Mr. Hopper kept listing the sales, exhibits, and loans of his artwork during a large portion of his career. The collection also includes exhibit programs and photographs of some of his later works.
Scope and Contents The collection includes numerous clippings of drawings made by cartoonist and illustrator, Franklin Booth. Most items are undated.