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# What is the intuition behind why a rank deficient matrix does not have an inverse?

Suppose that we have a $$p$$ dimensional square matrix $$A$$ whose rank is less than $$p$$. We know that such a matrix cannot have an inverse and there are several different ways to prove that the $$A$$ does not have an inverse.

However, I am struggling to obtain an intuition behind why the inverse does not exist. I considered the following ideas to generate an intuition but failed to do so.

1. The matrix $$A$$ can be viewed as a set of transformations such as scaling, translation, rotation etc. Thus, when we apply $$A$$ to a vector in $$p$$ dimensions it always maps the $$p$$ dimensional vector to a vector in a subspace spanned by $$A$$ if it is less than full rank. Lack of an inverse implies that we cannot reverse the transformations. Why not?

2. The columns of $$A$$ span only a subspace of $$R^p$$ if it is less than full rank. Thus, a transformation such as $$A y$$ takes a vector from $$R^p$$ to a vector that always belongs to that subspace. The above viewpoint did not help in obtaining an intuition either.

Is there a way to obtain an intuition as to why a rank deficient matrix does not have an inverse?

• I think point 2 will give you the intuition if you consider the action on a basis. (The image of a basis is a basis of the image.) – saulspatz Jan 17 at 14:57
• To elaborate on the above comment, if $A$ does not have full rank, then a nontrivial subspace gets sent to the zero vector. What happens if we try to invert that? In other words, how can we find the inverse image of the zero vector if lots of vectors are mapped to it (the answer: we can't!). – OldGodzilla Jan 17 at 15:00
• It hasn't an inverse but it has a pseudo-inverse, a very useful notion. – Jean Marie Jan 25 at 15:44

"Invertible" when talking about linear transformations means "reversible". In other words, a linear transformation (and the corresponding matrix in a given basis) is invertible iff it is possible, given an output, to figure out exactly what the input was.

A rank deficient linear transformation will collapse at least one dimension, meaning each output could be the result of any of a number of different inputs. Specifically, it will have a non-trivial kernel, so multiple different inputs result in the output $$\vec 0$$.

The row reduced echelon form of your matrix will have some rows of zero at the bottom , which is not invertible.

An invertible matrix when reduced to its row reduced echelon form becomes the identity matrix.

This stems from the fact that a mapping $$f:V\to V$$ cannot possess any left inverse if it is not injective and it cannot possess any right inverse if it is not surjective:

• if $$f$$ is not injective, i.e. if $$f(u)=f(v)$$ for some $$u\ne v$$, then $$f$$ cannot have any left inverse, otherwise we would have $$u=(f^{-1}\circ f)(u)=f^{-1}(f(u))=f^{-1}(f(v))=(f^{-1}\circ f)(v)=v$$, which is a contradiction;
• if $$f$$ is not surjective, i.e. if there is some member $$w$$ of $$V$$ that lies outside $$f(V)$$, then $$f$$ cannot possess any right inverse, otherwise we would have $$w=f(f^{-1}(w))\in f(V)$$, which is a contradiction.

Now, if a square matrix $$A$$ is rank deficient, its columns are linearly dependent. Therefore $$Au=0$$ for some nonzero vector $$u$$. In other words, $$A$$ maps both $$u$$ and $$0$$ to $$0$$. Hence $$A$$ has not any left inverse, because the mapping $$x\mapsto Ax$$ is not injective. (If you invert back, what should $$A^{-1}0$$ be? $$u$$ or $$0$$?)

Also, as $$A$$ is rank deficient, its column space $$A$$ is a proper subspace of the ambient space. Hence $$A$$ has not any right inverse, because the mapping $$x\mapsto Ax$$ is not surjective. (If $$w$$ lies outside the column space of $$A$$ and it has an inverse image, then $$w$$ itself is the image of its own inverse image, hence $$w$$ also lies inside the column space of $$A$$. How paradoxical!)

The way i learned is was like this: You should see the Det function as sending a matrix (of dim=n),to the oriented n-dim volume it's column vector's span. if this is zero, this will be a n-1 dim hyper-surface. Thus losing surjectivity into its image, we know injectivity and surjectivity are equivalent for finite dim square matrices. So it can't be invertible.

• All functions are surjective onto their image. Onto their codomains, on the other hand... – Arthur Jan 17 at 15:17
• yes ofcourse, i meant onto it's n dim vector space – Aylon Pinto Jan 17 at 15:21


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Let $A$ be a sorted array containing $n$ distinct integers, such that, for all $1 \leq i<j \leq n$, we have $A[i]<A[j]$. Note that the integers stored in the array $A$ are not necessarily from the set $\{1, \ldots, n\}$. Design an algorithm that outputs an index $i \in\{1, \ldots, n\}$ such that $A[i]=i$, if such an $i$ exists, and outputs $-1$ otherwise. The worst case running time of the algorithm should be asymptotically better than $O(n)$. Prove the correctness of your algorithm and state its asymptotic time complexity.

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Despite accumulating evidence of cardiorenal benefits from sodium–glucose cotransporter 2 (SGLT2) inhibitors, prescription of agents in this drug class may be limited by concerns regarding adverse effects and interdisciplinary care coordination. To investigate these potential barriers, we performed a cross-sectional study of SGLT2 inhibitor prescriptions in 2017 in 3,779 adults with type 2 diabetes and proteinuric chronic kidney disease from a nationwide database. Only 173 (5%) of these patients received an SGLT2 inhibitor in 2017. Younger age, renin-angiotensin-aldosterone system inhibitor prescription, and higher estimated glomerular filtration rate were associated with SGLT2 inhibitor prescription. Primary care providers were responsible for the majority of the prescriptions. Continued efforts should be made to track and improve SGLT2 inhibitor use in indicated populations.

Patients with type 2 diabetes and chronic kidney disease (CKD) have been waiting for innovation for decades. Despite hope for novel agents including aliskiren (1), bardoxolone (2), and other treatments (35), no new medications were found to slow the progression of CKD in patients with type 2 diabetes since the angiotensin receptor blockers (ARBs) in 2001 (6,7). However, sodium–glucose cotransporter 2 (SGLT2) inhibitors, originally developed as medications to enhance glycemic control in type 2 diabetes, have been shown to decrease major adverse cardiovascular events and the incidence and progression of CKD in multiple randomized controlled trials (811). In fact, the CREDENCE (Canagliflozin and Renal Events in Diabetes with Established Nephropathy Clinical Evaluation) trial showed a 32% lower risk of end-stage kidney disease (ESKD) in patients with proteinuric CKD randomized to canagliflozin versus placebo (hazard ratio 0.68, 95% CI 0.54–0.86, P = 0.002) (10). The American Diabetes Association (ADA) has released new recommendations that all patients with type 2 diabetes and diabetic kidney disease whose estimated glomerular filtration rate (eGFR) is ≥30 mL/min/1.73 m2, and particularly those with albuminuria >300 mg/day, should be considered for treatment with an SGLT2 inhibitor (based on A-level evidence) (12).

Despite the impressive effects of these medications, clinicians may be reluctant to prescribe them for several reasons. In addition to being relatively new and unfamiliar, SGLT2 inhibitors face another barrier: uncertainty regarding which medical specialty “owns” them. Because these medications have effects on glycemic control, blood pressure, cardiovascular events, and progression of CKD, prescribers may wonder whether endocrinologists, nephrologists, cardiologists, or primary care providers (PCPs) should be the practitioners responsible for prescribing them. Other barriers to SGLT2 inhibitor initiation may include reported adverse effects (i.e., increased risk of amputations, urinary tract infections [UTIs] and genital mycotic infections, diabetic ketoacidosis [DKA], and fractures) and the costs of these medications, which are not yet available in generic formulations. These barriers may aggregate and prevent eligible patients from enjoying important health benefits, including lower risks of needing dialysis or having a cardiovascular event.

In this study, we sought to estimate the proportion of eligible patients with type 2 diabetes and proteinuric CKD who were treated with SGLT2 inhibitors in 2017. We further aimed to investigate patient characteristics associated with SGLT2 inhibitor prescription such as eGFR or history of peripheral arterial disease (PAD), UTIs, fractures, or DKA. We also examined which medical specialties typically prescribed SGLT2 inhibitors, with the hypothesis that most prescriptions were written by endocrinologists. We hypothesized that the majority of eligible patients did not receive SGLT2 inhibitors and that SGLT2 inhibitor prescription was less common among older patients with lower eGFRs and a history of the comorbidities selected.

The Clinformatics Data Mart, OptumInsight Life Sciences dataset (OptumInsight, Eden Prairie, MN), is a single-payer, closed data system that consists of patient-level administrative and demographic information, including but not limited to type of insurance plan, age, sex, eligibility, income, medical and prescription claims, laboratory values, and unique identifiers for linking patients. We used fully de-identified data from this large claims and integrated database that included employed and commercially insured patients in the United States. This study was approved by the institutional review board of Stanford University and was conducted in accordance with the Declaration of Helsinki guidelines.

We first identified patients with at least one serum creatinine value and one urine albumin-to-creatinine ratio (UACR) in 2016. We then selected the patients with diabetes defined by at least one A1C value ≥6.5% between 1 January 2016 and 1 January 2017. We next required proteinuric CKD, defined using laboratory data from 2016 as the combination of a UACR ≥300 mg/g creatinine (Cr) and the latest eGFR ≥30 and <90 mL/min/1.73 m2 determined by the four-variable Chronic Kidney Disease Epidemiology Collaboration formula (13). Patients without race or ethnicity identified were excluded because this variable is required for the eGFR calculation. We chose this definition of CKD because it mirrors the inclusion criteria for the CREDENCE trial (10) and defines the group for whom ADA recommends consideration of SGLT2 inhibitors (12). We excluded patients with an International Classification of Diseases, 10th edition (ICD-10), code for type 1 diabetes (E10). Our final cohort included only adults (age ≥18 years on 1 January 2017) with type 2 diabetes and proteinuric CKD who had continuous enrollment from 1 January 2016 or earlier until at least 1 January 2017 (Figure 1).

FIGURE 1

Cohort assembly of adult patients with type 2 diabetes and CKD in the Clinformatics Data Mart, OptumInsight Life Sciences dataset, in 2017.

FIGURE 1

Cohort assembly of adult patients with type 2 diabetes and CKD in the Clinformatics Data Mart, OptumInsight Life Sciences dataset, in 2017.

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### SGLT2 Inhibitor Use

We defined SGLT2 inhibitor use for 2017 as the presence of at least one prescription for an SGLT2 inhibitor with either the prescription date or the prescription date + the number of days prescribed falling between 1 January 2017 and 1 January 2018. The prescribing provider’s specialty was determined by taxonomy code. Prescribing providers without a taxonomy code or with a nonspecific taxonomy code were classified as specialty unknown. Patients with SGLT2 inhibitor prescriptions from both specialty providers and PCPs were classified as specialty-prescribed.

### Covariate Definitions

We defined the following comorbidities by ICD-9/ICD-10 codes (Supplementary Appendix A) recorded before 1 January 2017: hypertension, ischemic heart disease, heart failure, cerebrovascular disease, PAD, amputation, UTI, genital mycotic infection, fracture, osteoporosis, and DKA. Inpatient and outpatient codes were included. We obtained the other sociodemographic variables from the Optum SES Member file (SES data file, v. 7.0). Co-prescription of an ACE inhibitor or ARB in 2017 was measured in the same method described for SGLT2 inhibitors above.

### Statistical Analysis

We report binary variables as proportions and continuous variables as median (25th–75th percentiles). We used univariate logistic regression to evaluate the association of patient characteristics (all Table 1 variables included) with receipt of an SGLT2 inhibitor. Data were missing only for the income variable (14% missing). Analysis of the other variables by income category suggested that these data were not missing completely at random; patients with missing income information were younger with higher UACRs (Supplementary Table 1). However, it is reasonable to assume that missingness is related to the measured characteristics and not to itself or to unmeasured factors (missing at random assumption). Therefore, we performed multiple imputation, generating 15 datasets with imputed income category values, with pooling of the results of logistic regression from each dataset. We performed multivariable logistic regression including only Table 1 variables for which univariate P values were <0.2. We considered two-sided P values <0.05 as significant. In a sensitivity analysis to check for dropout from the database during 2017, we evaluated the percentage receiving SGLT2 inhibitors in the subset of patients with continuous enrollment for all of 2017. We performed all statistical analyses using SAS, v. 9.4, software (SAS Institute, Cary, NC).

TABLE 1

Demographic and Clinical Characteristics of Patients

All patients (n = 3,779)eGFR 30–44 mL/min/1.73 m2 (n = 1,000)eGFR 45–59 mL/min/1.73 m2 (n = 1,111)eGFR 60–89 mL/min/1.73 m2 (n = 1,668)
Age, years 72 (67–78) 74 (69–81) 74 (67–79) 70 (64–76)
Female sex 43 48 39 42
Race/ethnicity
White 43 44 45 41
Hispanic 37 36 36 37
Black 14 15 13 14
Asian
Income
<$40,000 33 35 32 31$40,000–49,999
$50,000–59,999$60,000–74,999 11 10 11 10
$75,000–99,999 11 10 13 11 ≥$100,000 15 12 14 16
Unknown 14 15 13 13
Comorbidities
Hypertension 97 98 97 96
Ischemic heart disease 44 50 47 38
Heart failure 30 38 32 23
Cerebrovascular disease 30 34 31 27
Amputation
Fracture 13 14 13 12
Osteoporosis 11 13 10
DKA
UTI 33 36 32 32
Genital mycotic infection
ACE inhibitor/ARB use 77 73 79 79
Laboratory data
eGFR, mL/min/1.73 m2 57 (44–72) 38 (34–42) 52 (49–56) 74 (66–82)
UACR, mg/g Cr 753 (449–1,437) 916 (494–1,924) 754 (448–1,407) 671 (430–1,256)
A1C, % 8.0 (7.1–9.5) 7.9 (7.0–9.2) 8.0 (7.1–9.4) 8.1 (7.2–9.7)
All patients (n = 3,779)eGFR 30–44 mL/min/1.73 m2 (n = 1,000)eGFR 45–59 mL/min/1.73 m2 (n = 1,111)eGFR 60–89 mL/min/1.73 m2 (n = 1,668)
Age, years 72 (67–78) 74 (69–81) 74 (67–79) 70 (64–76)
Female sex 43 48 39 42
Race/ethnicity
White 43 44 45 41
Hispanic 37 36 36 37
Black 14 15 13 14
Asian
Income
<$40,000 33 35 32 31$40,000–49,999
$50,000–59,999$60,000–74,999 11 10 11 10
$75,000–99,999 11 10 13 11 ≥$100,000 15 12 14 16
Unknown 14 15 13 13
Comorbidities
Hypertension 97 98 97 96
Ischemic heart disease 44 50 47 38
Heart failure 30 38 32 23
Cerebrovascular disease 30 34 31 27
Amputation
Fracture 13 14 13 12
Osteoporosis 11 13 10
DKA
UTI 33 36 32 32
Genital mycotic infection
ACE inhibitor/ARB use 77 73 79 79
Laboratory data
eGFR, mL/min/1.73 m2 57 (44–72) 38 (34–42) 52 (49–56) 74 (66–82)
UACR, mg/g Cr 753 (449–1,437) 916 (494–1,924) 754 (448–1,407) 671 (430–1,256)
A1C, % 8.0 (7.1–9.5) 7.9 (7.0–9.2) 8.0 (7.1–9.4) 8.1 (7.2–9.7)

Data on categorical variables are proportions; data on continuous variables are median (25th–75th percentiles).

We identified 3,779 adult patients with type 2 diabetes and proteinuric CKD. They were elderly (median age 72 years) with a median eGFR of 57 mL/min/1.73 m2, median UACR of 753 mg/g Cr, and median A1C of 8.0% (Table 1). Almost all of the patients (97%) had hypertension, and 77% had a prescription for an ACE inhibitor or ARB. At least one SGLT2 inhibitor was prescribed for 5% of the patients in the cohort (173 of 3,779). This percentage was unchanged in our sensitivity analysis assessing for dropout during the 2017 measurement period (5%; 156 of 3,402). Among patients with an eGFR of 30–44 mL/min/1.73 m2, only 2% were prescribed an SGLT2 inhibitor.

Younger age, ACE inhibitor or ARB prescription, higher A1C, higher eGFR, and lower UACR were associated with SGLT2 inhibitor prescription in multivariable logistic regression (all P <0.05) (Table 2). The odds of SGLT2 inhibitor prescription were 76% higher in patients with than in those without an ACE inhibitor or ARB prescription. Patients with a history of UTI were less likely to be prescribed an SGLT2 inhibitor, but the results were not significant in multivariable analysis (adjusted odds ratio [OR] 0.69, 95% CI 0.47–1.03, P = 0.07).

TABLE 2

Factors Associated With SGLT2 Inhibitor Prescription

Age (per 5 years) 0.74 0.69–0.79 0.77 0.72–0.84 <0.001
Female sex 0.60 0.43–0.83 0.78 0.55–1.10 0.16
Heart failure 0.73 0.51–1.04 1.01 0.69–1.47 0.97
Cerebrovascular disease 0.73 0.51–1.04 1.08 0.74–1.57 0.70
Fracture 0.64 0.38–1.10 0.83 0.48–1.43 0.50
UTI 0.52 0.36–0.75 0.69 0.47–1.03 0.07
ACE inhibitor/ARB 1.95 1.25–3.04 1.76 1.12–2.77 0.01
eGFR (per 10 mL/min/1.73 m2) 1.32 1.21–1.45 1.21 1.09–1.33 <0.001
UACR (per 100 mg/g Cr) 0.99 0.97–1.00 0.98 0.97–1.00 0.02
A1C 1.16 1.09–1.24 1.09 1.01–1.17 0.03
Age (per 5 years) 0.74 0.69–0.79 0.77 0.72–0.84 <0.001
Female sex 0.60 0.43–0.83 0.78 0.55–1.10 0.16
Heart failure 0.73 0.51–1.04 1.01 0.69–1.47 0.97
Cerebrovascular disease 0.73 0.51–1.04 1.08 0.74–1.57 0.70
Fracture 0.64 0.38–1.10 0.83 0.48–1.43 0.50
UTI 0.52 0.36–0.75 0.69 0.47–1.03 0.07
ACE inhibitor/ARB 1.95 1.25–3.04 1.76 1.12–2.77 0.01
eGFR (per 10 mL/min/1.73 m2) 1.32 1.21–1.45 1.21 1.09–1.33 <0.001
UACR (per 100 mg/g Cr) 0.99 0.97–1.00 0.98 0.97–1.00 0.02
A1C 1.16 1.09–1.24 1.09 1.01–1.17 0.03

Unadjusted ORs are shown for Table 1 variables with univariate P values <0.2. Those variables were included in the multivariable model, and those with adjusted ORs with P values <0.05 are shown in bold.

Cardiologists did not prescribe any SGLT2 inhibitors in this study, and only two patients received SGLT2 inhibitor prescriptions from nephrologists (not shown). Only four patients received SGLT2 inhibitor prescriptions from multiple providers (i.e., from both an endocrinologist and a PCP). The majority of the SGLT2 inhibitor prescriptions identified were written by PCPs (59% of patients) (Table 3). Endocrinologists prescribed most of the remainder (25% of patients). Compared with patients prescribed SGLT2 inhibitors by endocrinologists, patients prescribed SGLT2 inhibitors by other providers had a higher prevalence of male sex, heart failure, fracture, osteoporosis, DKA, and UTI.

TABLE 3

Characteristics of Patients Prescribed SGLT2 Inhibitors, Stratified by Prescribing Specialty

Endocrinology (n = 43, 25%)Primary Care (n = 102, 59%)Unknown* (n = 26, 15%)
Age, years 64 (57–70) 67 (59–71) 64 (58–72)
Female sex 40 31
Race/ethnicity
White 53 37 46
Hispanic 26 46 38
Black 12 12 15
Asian
Income
<$40,000 26 33 42$40,000–49,999
$50,000–59,999 12$60,000–74,999 14
$75,000–99,999 12 12 12 ≥$100,000 21 11 31
Unknown 19 15
Comorbidities
Hypertension 100 97 96
Ischemic heart disease 40 39 42
Heart failure 14 24 42
Cerebrovascular disease 28 23 27
Amputation
Fracture 12
Osteoporosis 11
DKA
UTI 21 38
Genital mycotic infection
ACE inhibitor/ARB use 88 83% 96
eGFR, mL/min/1.73 m2
30–44 14
45–59 26 26 31
60–89 65 60 62
UACR, mg/g Cr 548 (399–1,203) 637 (429–1,148) 893 (532–1,431)
A1C, % 8.8 (7.9–10.1) 8.8 (7.8–9.8) 9.1 (8.0–11.0)
Endocrinology (n = 43, 25%)Primary Care (n = 102, 59%)Unknown* (n = 26, 15%)
Age, years 64 (57–70) 67 (59–71) 64 (58–72)
Female sex 40 31
Race/ethnicity
White 53 37 46
Hispanic 26 46 38
Black 12 12 15
Asian
Income
<$40,000 26 33 42$40,000–49,999
$50,000–59,999 12$60,000–74,999 14
$75,000–99,999 12 12 12 ≥$100,000 21 11 31
Unknown 19 15
Comorbidities
Hypertension 100 97 96
Ischemic heart disease 40 39 42
Heart failure 14 24 42
Cerebrovascular disease 28 23 27
Amputation
Fracture 12
Osteoporosis 11
DKA
UTI 21 38
Genital mycotic infection
ACE inhibitor/ARB use 88 83% 96
eGFR, mL/min/1.73 m2
30–44 14
45–59 26 26 31
60–89 65 60 62
UACR, mg/g Cr 548 (399–1,203) 637 (429–1,148) 893 (532–1,431)
A1C, % 8.8 (7.9–10.1) 8.8 (7.8–9.8) 9.1 (8.0–11.0)

Data on categorical variables are proportions; data on continuous variables are medians (25th–75th percentiles). *The Unknown category includes prescribers missing taxonomy codes and prescribers with nonspecific taxonomy codes.

In this study of patients with type 2 diabetes and CKD from a nationwide cohort, we found that only 5% of patients for whom the ADA particularly recommends SGLT2 inhibitors received them in 2017. However, this finding was expected given the results of a 2017 study of patients in the National Cardiovascular Disease Registry, in which only 5% of patients who met the eligibility criteria for the EMPA-REG OUTCOME (BI 10773 [Empagliflozin] Cardiovascular Outcome Event Trial in Type 2 Diabetes Mellitus Patients) trial were receiving SGLT2 inhibitor therapy (14).

The low prescription rate we observed may be in part because of a lack of recognition of the benefits of SGLT2 inhibitors, so it is important to consider the dates of our study in the context of the evidence available to prescribers at the time. We evaluated SGLT2 inhibitor prescriptions during 2017. Canagliflozin was approved by the U.S. Food and Drug Administration (FDA) in 2013, followed by dapagliflozin and empagliflozin in 2014. The results of the EMPA-REG OUTCOME trial (8) were published in 2015; those from the CANVAS (Canagliflozin Cardiovascular Assessment Study) research program (9) were published during our measurement period in 2017, and the CREDENCE trial (10) had been enrolling since 2014, but its results were not published until 2019. Analyses of EMPA-REG OUTCOME in 2016 (15) and the CANVAS program in 2017 (9) had both shown improvement in secondary kidney outcomes with SGLT2 inhibitors. In 2016, the FDA granted empagliflozin a second indication, apart from glycemic control, for the reduction of risk of cardiovascular death.

Thus, during our data collection period, SGLT2 inhibitors had been available in the United States for more than 4 years. Benefits with regard to cardiovascular mortality and CKD incidence and progression had been observed in two large, randomized controlled trials (8,9), and a trial evaluating canagliflozin in more advanced CKD (10) had been enrolling for more than 3 years without being discontinued for safety concerns. However, the evidence on the benefits of SGLT2 inhibitors had not yet been incorporated into clinical practice guidelines, so many health care providers may have been unaware of them. Despite the FDA granting empagliflozin an indication for cardiovascular risk reduction regardless of glycemic control in 2016, we found no association between cardiovascular morbidities and SGLT2 inhibitor prescription.

Finally, SGLT2 inhibitor prescription was limited for patients with an eGFR of 30–44 mL/min/1.73 m2, likely because of drug labeling requirements and consequent issues with insurance approval. Although the FDA-approved prescribing information at the time recommended against using these drugs in patients with an eGFR <45 mL/min/1.73 m2, the CREDENCE trial initiated canagliflozin in patients with an eGFR as low as 30 mL/min/1.73 m2 and did not stop the drug for eGFR until patients reached ESKD (16). Updated FDA-approved prescribing information (17) lowering the threshold to 30 mL/min/1.73 m2 will likely increase SGLT2 inhibitor prescription to patients with an eGFR of 30–44 mL/min/1.73 m2.

In addition to the costs of and limited experience with newer medications, as well as the need for prior insurance authorization that involves additional resources, fear of adverse side effects may also have contributed to the low rate of SGLT2 inhibitor prescription in eligible patients. Concerns regarding specific adverse events have been raised, most notably amputation (18), but also genital mycotic infections, UTIs, euglycemic DKA, and osteoporotic fractures (19). However, we did not detect an association between SGLT2 inhibitor prescription and patient history of these conditions, although UTI history was associated with 31% lower odds of SGLT2 inhibitor prescription, which did not reach statistical significance with multivariable adjustment, perhaps because of the low number of patients. We did find that older age was associated with decreased odds of SGLT2 inhibitor prescription, possibly indicating increased provider concerns about adverse effects in the more vulnerable elderly population.

Prescription of an ACE inhibitor or ARB was associated with SGLT2 inhibitor prescription, perhaps reflecting provider recognition of proteinuric CKD and intensity of care. Although ACE inhibitor or ARB therapy is indicated in proteinuric CKD and 97% of our cohort had hypertension, 23% of patients in our cohort were not on an ACE inhibitor or ARB concurrently. Patients with higher A1C were also more likely to receive SGLT2 inhibitors, which is expected for a class of drugs initially developed and marketed to lower A1C.

The task of bringing these powerful medications to patients with CKD who need them may be complicated by inter-specialty culture and already complex medical regimens. Contrary to our a priori hypothesis that most SGLT2 inhibitor prescriptions would come from endocrinologists, most patients received their SGLT2 inhibitors from a PCP. Despite an FDA-approved indication to lower cardiovascular risk regardless of glycemic control, few SGLT2 inhibitor prescriptions were written by cardiologists or nephrologists. Analysis of 2013–2017 data from a single health care system in Boston found that only 5% of SGLT2 inhibitor prescriptions were written by cardiologists, compared with endocrinologists (40%) and PCPs (23%) (20). The benefits of SGLT2 inhibitors on the risks of cardiovascular death and ESKD place these medications within the purviews of cardiology and nephrology, and we will likely see more SGLT2 inhibitor use in these specialties in the future.

Our study has several strengths, most notably the real-world data on usage of SGLT2 inhibitors in a contemporary nationwide cohort. The claims and integrated database allowed us to use both laboratory and diagnosis codes to construct our cohort and to capture both prescription and prescriber information. Limitations of our study include the single laboratory values (serum creatinine, urine albumin) rather than trends or averages for inclusion. UACR data were only available for a small proportion of patients, although this circumstance is consistent with other large clinical datasets (e.g., a finding that UACR values were available for <7% of patients in a dataset from five health care organizations [21]). Because new cohorts are added to the OptumInsight dataset each year, we were unable to perform trend analyses over time. Finally, patients may have died or changed insurance providers in 2017, although our sensitivity analysis indicated that dropout did not affect the proportion of patients prescribed SGLT2 inhibitors.

Using data from a national cohort in 2017, we found very low utilization of SGLT2 inhibitors among patients with type 2 diabetes and proteinuric CKD, a group with high cardiovascular risk and high rates of progression to ESKD (22). These findings manifested despite ample evidence from randomized clinical trials demonstrating the cardiorenal benefits of SGLT2 inhibitors. The early termination of the CREDENCE trial for efficacy and publication of its results, which occurred after the time period of our study, should focus PCPs, endocrinologists, nephrologists, and cardiologists on using this class of agents in patients with type 2 diabetes and proteinuric CKD. Continued efforts should be made to track how the proportion of eligible patients receiving SGLT2 inhibitors changes as knowledge of their benefits is disseminated and long-term data on their safety and effectiveness in clinical practice accumulate.

### Funding

This work was supported by the National Institutes of Health grant numbers 5T32DK007357 (I.E.M.), K24DK085446 (G.M.C. and M.E.M.-R.), and 5K01DK110221 (J.J.R.).

### Author Contributions

I.E.M. participated in the study design and data analysis and wrote the manuscript. J.H. participated in data collection and analysis. M.E.M.-R. participated in the study design, data collection, and data analysis. G.M.C. participated in data analysis. J.J.R. participated in the study design and data analysis. All authors edited the manuscript and approved the final version for submission. I.E.M. is the guarantor of this work and, as such, had full access to all of the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis.

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Parving
H-H
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text:
# Step-by-step Solution

Go!
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=
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csc

asin
acos
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sinh
cosh
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csch

asinh
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## Step-by-step explanation

Problem to solve:

$\int t e^{\left(5t+\pi \right)}dt$

Learn how to solve integrals of exponential functions problems step by step online.

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$

Learn how to solve integrals of exponential functions problems step by step online. Compute the integral int(t*2.718281828459045^(5*t+3.141592653589793))dt. Use the integration by parts theorem to calculate the integral \int te^{\left(5t+\pi \right)}dt, using the following formula. First, identify u and calculate du. Now, identify dv and calculate v. Solve the integral.

$\frac{1}{5}e^{\left(5t+\pi \right)}t-\frac{1}{25}e^{\left(5t+\pi \right)}+C_0$

### Problem Analysis

$\int t e^{\left(5t+\pi \right)}dt$

### Main topic:

Integrals of Exponential Functions

~ 0.11 seconds


text:
A New Family of q-analogue of Genocchi Numbers and Polynomials of Higher Order

• Journal title : Kyungpook mathematical journal
• Volume 54, Issue 1,  2014, pp.131-141
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2014.54.1.131
Title & Authors
A New Family of q-analogue of Genocchi Numbers and Polynomials of Higher Order
Araci, Serkan; Acikgoz, Mehmet; Seo, Jong Jin;

Abstract
In the present paper, we introduce the new generalization of q-Genocchi polynomials and numbers of higher order. Also, we give some interesting identities. Finally, by applying q-Mellin transformation to the generating function for q-Genocchi polynomials of higher order put we define novel q-Hurwitz-Zeta type function which is an interpolation for this polynomials at negative integers.
Keywords
Genocchi numbers and polynomials;q-Genocchi numbers and polynomials of higher order;q-Mellin transformation;q-Hurwitz-zeta function;q-Gamma function;q-Exponential function;
Language
English
Cited by
1.
Bounds for q-integrals of ψ r + 1 r + 1 ${}_{r+1}\psi_{r+1}$ with applications, Journal of Inequalities and Applications, 2015, 2015, 1
References
1.
S. Araci, Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus, Applied Mathematics and Computation, 233(2014), 599-607.

2.
M. Acikgoz, S. Araci and I. N. Cangul, A note on the modified q-Bernstein polynomials for functions of several variables and their reflections on q-Volkenborn integration, Applied Mathematics and Computation, 218(3)(2011), 707-712.

3.
S. Araci, M. Acikgoz, E. Sen, On the extended Kim's p-adic q-deformed fermionic integrals in the p-adic integer ring, Journal of Number Theory, 133(2013), 3348-3361.

4.
S. Araci, M. Acikgoz and K. H. Park, A note on the q-analogue of Kim's p-adic log gamma type functions associated with q-extension of Genocchi and Euler numbers with weight ${\alpha}$, Bull. Korean Math. Soc., 50(2)(2013), 583-588.

5.
S. Araci, D. Erdal and J. J. Seo, A study on the fermionic p-adic q-integral representation on $\mathbb{Z}_p$ associated with weighted q-Bernstein and q-Genocchi polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 649248, 10 pages.

6.
S. Araci, M. Acikgoz and A. Gursul, Analytic continuation of weighted q-Genocchi numbers and polynomials, Commun. Korean Math. Soc., 28(3)(2013), 457-462.

7.
S. Araci and M. Acikgoz, A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials, Advanced Studies in Contemporary Mathematics, 22(3)(2012), 399-406.

8.
S. Araci, M. Acikgoz, H. Jolany, J. J. Seo, A unified generating function of the q-Genocchi polynomials with their interpolation functions, Proc. Jangjeon Math. Soc.,15(2)(2012), 227-233.

9.
S. Araci, J. J. Seo and D. Erdal, New construction weighted (h, q)-Genocchi numbers and polynomials related to Zeta-type functions, Discrete Dynamics in Nature and Society, 2011(2011), Article ID 487490, 7 pages.

10.
K. Brahim, and R. Guanes, Some applications of the q-Mellin Transform, Tamsui Oxford Journal of Mathematical Sciences, 26(3)(2010), 335-343.

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Y. He and C.Wang, Some formulae of products of the Apostol-Bernoulli and Apostol-Euler polynomials, Discrete Dynamics in Nature and Society, vol. 2012, Article ID 927953, 11 pages, 2012.

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F. H. Jackson, On q-definite integrals, The Quarterly Journal of Pure and Applied Mathematics, 41(1910), 193-2036.

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L. C. Jang, The q-analogue of twisted Lerch type Euler Zeta functions, Bull. Korean Math. Soc., 47(6)(2010), 1181-1188.

14.
H. Jolany and H. Sharifi, Some results for Apostol-Genocchi Polynomials of higher order, Bulletin of the Malaysian Mathematical Sciences Society, 36(2)(2013).

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V. G. Kac and P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, 2002.

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T. Kim, q-Generalized Euler numbers and polynomials, Russ. J. Math. Phys., 13(3)(2006), pp. 293-308

17.
T. Kim, Some identities on the q-Euler polynomials of higher order and q-stirling numbers by the fermionic p-adic integral on $\mathbb{Z}_p$, Russian J. Math. Phys., 16(2009), 484-491.

18.
T. Kim, On the q-extension of Euler and Genocchi numbers, J. Math. Anal. Appl., 326(2007), 1458-1465.

19.
T. Kim, On the analogs of Euler numbers and polynomials associated with p-adic q-integral on $\mathbb{Z}_p$ at q = -1, J. Math. Anal. Appl., 331 (2007), 779-792.

20.
T. Kim, Some Identities on the integral representation of the product of several q-Bernstein-type polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 634675, 11 pages.

21.
T. Kim, S. H. Lee, H. H. Han and C. S. Ryoo, On the values of the weighted q-Zeta and L-functions, Discrete Dynamics in Nature and Society, Volume 2011, Article ID 476381, 7 pp.

22.
T. Kim, Euler numbers and polynomials associated with Zeta functions, Abstract and Applied Analysis, Volume 2008, Article ID 581582, 11 pages.

23.
T. Kim, Power series and asymptotic series associated with the q-analog of the twovariable p-adic L-function, Russian Journal of Mathematical Physics, 12(2)(2005), 186-196.

24.
T. Kim, On the q-extension of Euler and Genocchi numbers, J. Math. Anal. Appl., 326(2007), 1458-1465.

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26.
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text:
Is there a name for this identity-like object?

I was grading for a linear algebra class just now and someone remarked that since $U + U = U$, then $U$ would be an identity for the set of subspaces of some vector space with binary operation subspace addition. Obviously I corrected their mistake, since it isn't true that this fixed $U$ has the property $U + V = V$ for all subspaces $V$ of this vector space (except perhaps in trivial cases), but it made me curious whether or not there is a name for this type of object since they have exhibited a nontrivial instance of the existence of such an element.

I suppose it would only be meaningful in a semigroup or algebraic object with less structure than a semigroup, since the standard proof that the identity element is unique in, say, a group, would probably show that any element like this would have to be the identity.

So, my question is, is there a name for an element of a semigroup (or an algebraic object with less structure if even a semigroup in general is too restrictive), let's call the element $x$ and the operation of the semigroup +, with the property that $x + x = x$.

• Looks kind of like idempotence – G Tony Jacobs Sep 16 '17 at 13:42
• When thinking about linear operators, $P^2 = P$ means $P$ is a projection. – Ethan Bolker Sep 16 '17 at 13:42
• Ah! I was not at all thinking in terms of functions. I would think idempotence would be it! – Christian Sep 16 '17 at 13:47
• You might be interested in the semiring of ideals associated to any ring. The set of ideals becomes a semiring using normal ideal addition and multiplication. It has idempotent addition. The multiplication becomes idempotent too if you use a von Neumann regular ring to start with. – rschwieb Sep 18 '17 at 11:08

Idempotents behave like identities in the following sense: if $m$ is an idempotent element of a semigroup $S$, you can consider the subsemigroup of elements $a$ satisfying $ma = am = a$. This is also a semigroup, even a monoid, and (by construction) has identity element $m$; in fact by construction it's the largest subsemigroup of $S$ on which $m$ acts as the identity.
When applied to an element $U$ of the lattice of subspaces of a vector space $V$, with monoid operation given by intersection, this construction produces the lattice of subspaces of $U$. With the monoid operation given by sum, this construction produces the lattice of subspaces of $V$ containing $U$, which can naturally be identified with the lattice of subspaces of $V/U$ by the isomorphism theorems.


text:
## The free rigid body dynamics: generalized versus classic    [PDF]

Razvan M. Tudoran
In this paper we analyze the normal forms of a general quadratic Hamiltonian system defined on the dual of the Lie algebra $\mathfrak{o}(K)$ of real $K$ - skew - symmetric matrices, where $K$ is an arbitrary $3\times 3$ real symmetric matrix. A consequence of the main results is that any first-order autonomous three-dimensional differential equation possessing two independent quadratic constants of motion which admits a positive/negative definite linear combination, is affinely equivalent to the classical "relaxed" free rigid body dynamics with linear controls.
View original: http://arxiv.org/abs/1102.2725


text:
## Dumbser, Michael

Compute Distance To:
 Author ID: dumbser.michael Published as: Dumbser, Michael; Dumbser, M. External Links: ORCID
 Documents Indexed: 143 Publications since 2003 6 Contributions as Editor Co-Authors: 114 Co-Authors with 142 Joint Publications 2,007 Co-Co-Authors
all top 5

### Co-Authors

 7 single-authored 23 Munz, Claus-Dieter 20 Boscheri, Walter 16 Toro, Eleuterio F. 14 Balsara, Dinshaw S. 13 Tavelli, Maurizio 10 Zanotti, Olindo 9 Romenskii, Evgenii I. 8 Busto, Saray 8 Castro, Manuel J. 8 Gaburro, Elena 8 Loubère, Raphaël 8 Peshkov, Ilya M. 7 Fambri, Francesco 6 Hidalgo, Arturo 6 Ioriatti, Matteo 5 Gassner, Gregor J. 4 Armanini, Aronne 4 Casulli, Vincenzo 4 Chiocchetti, Simone 4 Titarev, Vladimir A. 3 Avesani, Diego 3 Bader, Michael 3 Ferrari, Angela 3 Filimon, Alexander 3 Gavrilyuk, Sergey L. 3 Iben, Uwe 3 Käser, Martin 3 Righetti, Maurizio 3 Roller, Sabine 3 Schwartzkopff, Thomas 2 Bassi, Caterina 2 Bellin, Alberto 2 Després, Bruno 2 Diot, Steven 2 Escalante, Cipriano 2 Fortenbach, Roland 2 Goetz, Claus R. 2 Hindenlang, Florian J. 2 Parés Madroñal, Carlos 2 Qiu, Jianxian 2 Rannabauer, Leonhard 2 Semplice, Matteo 2 Shu, Chi-Wang 2 Taube, Arne 2 Utyuzhnikov, Sergey V. 2 Utzmann, Jens 2 Vázquez-Cendón, María Elena 2 Weinzierl, Tobias 1 Abgrall, Rémi 1 Altmann, Christoph 1 Ariunaa, Uuriintsetseg 1 Audit, Edouard 1 Babucke, Andreas 1 Bankole, Adeleke O. 1 Bermúdez, Alfredo 1 Bonaventura, Luca 1 Castro-Díaz, Manuel Jesús 1 Castro, Cristóbal E. 1 Charrier, Dominic Etienne 1 Chiogna, Gabriele 1 Demidenko, Gennadiĭ Vladimirovich 1 Du, Huijing 1 Enaux, Cedric 1 Facchini, M. 1 Favrie, Nicolas 1 Fernández, E. Guerrero 1 Ferrín, Jose Luis 1 Fraccarollo, Luigi 1 Furci, Isabella 1 Gabriel, Alice-Agnes 1 Garain, Sudip K. 1 Gressier, Jérémie 1 Gurski, Katharine F. 1 Iske, Armin 1 Ivanova, Kseniya 1 Kamm, James R. 1 Kemm, Friedemann 1 Leibinger, Julia 1 Maire, Pierre-Henri 1 Mazza, Mariarosa 1 Meyer, Chad 1 Montecinos, Gino I. 1 Moraga, Nelson O. 1 Morales de Luna, Tomás 1 Moschetta, Jean-Marc 1 Nguyen, Nguyen Thanh 1 Nkonga, Boniface 1 Reinarz, Anne 1 Reshetova, Galina Vitalevna 1 Río-Martín, Laura 1 Rohde, Christian 1 Romeo, F. L. 1 Rossi, Giulia 1 Rumpf, Tobias 1 Rung, Thomas 1 Saavedra, Laura 1 Salmon, Stéphanie 1 Sarantuya, Tsedendamba 1 Saurel, Richard 1 Schneider, Rudolf ...and 14 more Co-Authors
all top 5

### Serials

 44 Journal of Computational Physics 24 Computers and Fluids 11 Journal of Scientific Computing 8 Computer Methods in Applied Mechanics and Engineering 8 Applied Mathematics and Computation 7 Applied Numerical Mathematics 6 Communications in Computational Physics 2 SIAM Journal on Scientific Computing 2 European Series in Applied and Industrial Mathematics (ESAIM): Proceedings 2 Comptes Rendus. Mécanique. Académie des Sciences, Paris 2 Oberwolfach Reports 1 Computers & Mathematics with Applications 1 Computer Physics Communications 1 International Journal for Numerical Methods in Fluids 1 Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV) 1 Journal of Fluid Mechanics 1 Journal of Mathematical Biology 1 Zhurnal Vychislitel’noĭ Matematiki i Matematicheskoĭ Fiziki 1 International Journal for Numerical Methods in Engineering 1 Physics of Fluids 1 Numerical Linear Algebra with Applications 1 ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik 1 International Journal of Numerical Modelling 1 International Journal of Applied Mathematics and Computer Science 1 CMES. Computer Modeling in Engineering & Sciences 1 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 1 PAMM. Proceedings in Applied Mathematics and Mechanics 1 International Journal for Numerical Methods in Biomedical Engineering 1 Izvestiya Irkutskogo Gosudarstvennogo Universiteta. Seriya Matematika 1 Axioms 1 Philosophical Transactions of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences 1 Communications on Applied Mathematics and Computation
all top 5

### Fields

 117 Fluid mechanics (76-XX) 102 Numerical analysis (65-XX) 35 Partial differential equations (35-XX) 15 Mechanics of deformable solids (74-XX) 6 General and overarching topics; collections (00-XX) 5 Geophysics (86-XX) 4 Astronomy and astrophysics (85-XX) 3 Classical thermodynamics, heat transfer (80-XX) 3 Biology and other natural sciences (92-XX) 2 Optics, electromagnetic theory (78-XX) 2 Relativity and gravitational theory (83-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Statistical mechanics, structure of matter (82-XX)

### Citations contained in zbMATH Open

122 Publications have been cited 3,684 times in 1,242 Documents Cited by Year
A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes. Zbl 1147.65075
Dumbser, Michael; Balsara, Dinshaw S.; Toro, Eleuterio F.; Munz, Claus-Dieter
2008
Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. Zbl 1110.65077
Dumbser, Michael; Käser, Martin
2007
Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. Zbl 1124.65074
Dumbser, Michael; Käser, Martin; Titarev, Vladimir A.; Toro, Eleuterio F.
2007
Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. Zbl 1142.65070
Dumbser, Michael; Enaux, Cedric; Toro, Eleuterio F.
2008
A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws. Zbl 1349.65448
Dumbser, Michael; Zanotti, Olindo; Loubère, Raphaël; Diot, Steven
2014
Arbitrary high order $$P_{N}P_{M}$$ schemes on unstructured meshes for the compressible Navier-Stokes equations. Zbl 1242.76161
Dumbser, Michael
2010
Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics. Zbl 1275.76169
Balsara, Dinshaw S.; Rumpf, Tobias; Dumbser, Michael; Munz, Claus-Dieter
2009
A simple extension of the Osher Riemann solver to non-conservative hyperbolic systems. Zbl 1220.65110
Dumbser, Michael; Toro, Eleuterio F.
2011
ADER schemes on unstructured meshes for nonconservative hyperbolic systems: applications to geophysical flows. Zbl 1177.76222
Dumbser, Michael; Castro, Manuel; Parés, Carlos; Toro, Eleuterio F.
2009
Runge-Kutta discontinuous Galerkin method using WENO limiters II: Unstructured meshes. Zbl 1157.65453
Zhu, Jun; Qiu, Jianxian; Shu, Chi-Wang; Dumbser, Michael
2008
A new 3D parallel SPH scheme for free surface flows. Zbl 1242.76270
Ferrari, Angela; Dumbser, Michael; Toro, Eleuterio F.; Armanini, Aronne
2009
Dumbser, Michael; Zanotti, Olindo; Hidalgo, Arturo; Balsara, Dinshaw S.
2013
On universal Osher-type schemes for general nonlinear hyperbolic conservation laws. Zbl 1373.76125
Dumbser, Michael; Toro, Eleuterio F.
2011
Building blocks for arbitrary high order discontinuous Galerkin schemes. Zbl 1115.65100
Dumbser, Michael; Munz, Claus-Dieter
2006
Efficient implementation of ADER schemes for Euler and magnetohydrodynamical flows on structured meshes – speed comparisons with Runge-Kutta methods. Zbl 1291.76237
Balsara, Dinshaw S.; Meyer, Chad; Dumbser, Michael; Du, Huijing; Xu, Zhiliang
2013
A direct arbitrary-Lagrangian-Eulerian ADER-WENO finite volume scheme on unstructured tetrahedral meshes for conservative and non-conservative hyperbolic systems in 3D. Zbl 1349.76310
Boscheri, Walter; Dumbser, Michael
2014
FORCE schemes on unstructured meshes. II: Non-conservative hyperbolic systems. Zbl 1227.76043
Dumbser, Michael; Hidalgo, Arturo; Castro, Manuel; Parés, Carlos; Toro, Eleuterio F.
2010
Very high order $$P_NP_M$$ schemes on unstructured meshes for the resistive relativistic MHD equations. Zbl 1261.76028
Dumbser, Michael; Zanotti, Olindo
2009
Multidimensional HLLC Riemann solver for unstructured meshes – with application to Euler and MHD flows. Zbl 1349.76426
Balsara, Dinshaw S.; Dumbser, Michael; Abgrall, Remi
2014
High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids. Zbl 1349.76324
Dumbser, Michael; Peshkov, Ilya; Romenski, Evgeniy; Zanotti, Olindo
2016
The discontinuous Galerkin method with Lax–Wendroff type time discretizations. Zbl 1093.76038
Qiu, Jianxian; Dumbser, Michael; Shu, Chi-Wang
2005
A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems. Zbl 1349.76603
Dumbser, Michael; Balsara, Dinshaw S.
2016
Hidalgo, Arturo; Dumbser, Michael
2011
A new family of high order unstructured MOOD and ADER finite volume schemes for multidimensional systems of hyperbolic conservation laws. Zbl 1373.76137
Loubère, Raphaël; Dumbser, Michael; Diot, Steven
2014
Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting. Zbl 1390.76381
Zanotti, Olindo; Fambri, Francesco; Dumbser, Michael; Hidalgo, Arturo
2015
Fast high order ADER schemes for linear hyperbolic equations. Zbl 1052.65078
Schwartzkopff, Thomas; Dumbser, Michael; Munz, Claus-Dieter
2004
A simple robust and accurate a posteriori sub-cell finite volume limiter for the discontinuous Galerkin method on unstructured meshes. Zbl 1349.65447
Dumbser, Michael; Loubère, Raphaël
2016
A sub-cell based indicator for troubled zones in RKDG schemes and a novel class of hybrid RKDG+HWENO schemes. Zbl 1124.65072
Balsara, Dinshaw S.; Altmann, Christoph; Munz, Claus-Dieter; Dumbser, Michael
2007
FORCE schemes on unstructured meshes. I: Conservative hyperbolic systems. Zbl 1168.65377
Toro, Eleuterio F.; Hidalgo, Arturo; Dumbser, Michael
2008
A matrix stability analysis of the carbuncle phenomenon. Zbl 1079.76607
Dumbser, Michael; Moschetta, Jean-Marc; Gressier, Jérémie
2004
High-order unstructured Lagrangian one-step WENO finite volume schemes for non-conservative hyperbolic systems: applications to compressible multi-phase flows. Zbl 1290.76081
Dumbser, Michael; Boscheri, Walter
2013
Arbitrary-Lagrangian-Eulerian one-step WENO finite volume schemes on unstructured triangular meshes. Zbl 1388.65075
Boscheri, Walter; Dumbser, Michael
2013
High order space-time adaptive ADER-WENO finite volume schemes for non-conservative hyperbolic systems. Zbl 1295.65088
Dumbser, Michael; Hidalgo, Arturo; Zanotti, Olindo
2014
Lagrangian ADER-WENO finite volume schemes on unstructured triangular meshes based on genuinely multidimensional HLL Riemann solvers. Zbl 1349.76309
Boscheri, Walter; Balsara, Dinshaw S.; Dumbser, Michael
2014
Divergence-free MHD on unstructured meshes using high order finite volume schemes based on multidimensional Riemann solvers. Zbl 1351.76092
Balsara, Dinshaw S.; Dumbser, Michael
2015
Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes. Zbl 1378.76044
Boscheri, Walter; Dumbser, Michael
2017
Multidimensional Riemann problem with self-similar internal structure. Part II: Application to hyperbolic conservation laws on unstructured meshes. Zbl 1351.76091
Balsara, Dinshaw S.; Dumbser, Michael
2015
A high order semi-implicit discontinuous Galerkin method for the two dimensional shallow water equations on staggered unstructured meshes. Zbl 1298.76120
Tavelli, Maurizio; Dumbser, Michael
2014
Central weighted ENO schemes for hyperbolic conservation laws on fixed and moving unstructured meshes. Zbl 1377.65115
Dumbser, Michael; Boscheri, Walter; Semplice, Matteo; Russo, Giovanni
2017
A staggered semi-implicit spectral discontinuous Galerkin scheme for the shallow water equations. Zbl 1366.76050
Dumbser, Michael; Casulli, Vincenzo
2013
A pressure-based semi-implicit space-time discontinuous Galerkin method on staggered unstructured meshes for the solution of the compressible Navier-Stokes equations at all Mach numbers. Zbl 1376.76028
Tavelli, Maurizio; Dumbser, Michael
2017
A staggered space-time discontinuous Galerkin method for the three-dimensional incompressible Navier-Stokes equations on unstructured tetrahedral meshes. Zbl 1349.76271
Tavelli, Maurizio; Dumbser, Michael
2016
ADER discontinuous Galerkin schemes for aeroacoustics. Zbl 1107.76044
Dumbser, Michael; Munz, Claus-Dieter
2005
A new class of moving-least-squares WENO-SPH schemes. Zbl 1349.76661
Avesani, Diego; Dumbser, Michael; Bellin, Alberto
2014
Explicit one-step time discretizations for discontinuous Galerkin and finite volume schemes based on local predictors. Zbl 1220.65122
Gassner, Gregor; Dumbser, Michael; Hindenlang, Florian; Munz, Claus-Dieter
2011
A high-order discontinuous Galerkin method with time-accurate local time stepping for the Maxwell equations. Zbl 1156.78012
Taube, Arne; Dumbser, Michael; Munz, Claus-Dieter; Schneider, Rudolf
2009
Comparison of solvers for the generalized Riemann problem for hyperbolic systems with source terms. Zbl 1284.35268
Montecinos, G.; Castro, C. E.; Dumbser, M.; Toro, E. F.
2012
A staggered semi-implicit discontinuous Galerkin method for the two dimensional incompressible Navier-Stokes equations. Zbl 1338.76068
Tavelli, Maurizio; Dumbser, Michael
2014
Direct arbitrary-Lagrangian-Eulerian ADER-MOOD finite volume schemes for multidimensional hyperbolic conservation laws. Zbl 1349.76312
Boscheri, Walter; Loubère, Raphaël; Dumbser, Michael
2015
Arbitrary high-order discontinuous Galerkin schemes for the magnetohydrodynamic equations. Zbl 1176.76075
Taube, Arne; Dumbser, Michael; Balsara, Dinshaw S.; Munz, Claus-Dieter
2007
Arbitrary high order discontinuous Galerkin schemes. Zbl 1210.65165
Dumbser, M.; Munz, C.-D.
2005
A conservative, weakly nonlinear semi-implicit finite volume scheme for the compressible Navier-Stokes equations with general equation of state. Zbl 1410.76220
Dumbser, Michael; Casulli, Vincenzo
2016
On arbitrary-Lagrangian-Eulerian one-step WENO schemes for stiff hyperbolic balance laws. Zbl 1373.76126
Dumbser, Michael; Uuriintsetseg, Ariunaa; Zanotti, Olindo
2013
A staggered space-time discontinuous Galerkin method for the incompressible Navier-Stokes equations on two-dimensional triangular meshes. Zbl 1390.76360
Tavelli, Maurizio; Dumbser, Michael
2015
Cell centered direct arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes for nonlinear hyperelasticity. Zbl 1390.76399
Boscheri, Walter; Dumbser, Michael; Loubère, Raphaël
2016
Efficient implementation of high order unstructured WENO schemes for cavitating flows. Zbl 1290.76098
Dumbser, Michael; Iben, Uwe; Munz, Claus-Dieter
2013
Construction and comparison of parallel implicit kinetic solvers in three spatial dimensions. Zbl 1349.76741
Titarev, Vladimir; Dumbser, Michael; Utyuzhnikov, Sergey
2014
Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes with time-accurate local time stepping for hyperbolic conservation laws. Zbl 1423.76296
Dumbser, Michael
2014
Linearized acoustic perturbation equations for low Mach number flow with variable density and temperature. Zbl 1261.76049
Munz, Claus-Dieter; Dumbser, Michael; Roller, Sabine
2007
A diffuse interface method for complex three-dimensional free surface flows. Zbl 1286.76099
Dumbser, Michael
2013
Semi-implicit numerical modeling of axially symmetric flows in compliant arterial systems. Zbl 1242.92019
Casulli, Vincenzo; Dumbser, Michael; Toro, Eleuterio F.
2012
Spectral semi-implicit and space-time discontinuous Galerkin methods for the incompressible Navier-Stokes equations on staggered Cartesian grids. Zbl 1432.76157
Fambri, Francesco; Dumbser, Michael
2016
An efficient semi-implicit finite volume method for axially symmetric compressible flows in compliant tubes. Zbl 1326.76070
Dumbser, Michael; Iben, Uwe; Ioriatti, Matteo
2015
High order cell-centered Lagrangian-type finite volume schemes with time-accurate local time stepping on unstructured triangular meshes. Zbl 1349.76311
Boscheri, Walter; Dumbser, Michael; Zanotti, Olindo
2015
Direct arbitrary-Lagrangian-Eulerian finite volume schemes on moving nonconforming unstructured meshes. Zbl 1390.76433
Gaburro, Elena; Dumbser, Michael; Castro, Manuel J.
2017
A second-order cell-centered Lagrangian ADER-MOOD finite volume scheme on multidimensional unstructured meshes for hydrodynamics. Zbl 1381.76212
Boscheri, Walter; Dumbser, Michael; Loubère, Raphaël; Maire, Pierre-Henri
2018
Three-dimensional flow evolution after a dam break. Zbl 1205.76228
Ferrari, A.; Fraccarollo, L.; Dumbser, M.; Toro, E. F.; Armanini, A.
2010
An efficient quadrature-free formulation for high order arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes on unstructured meshes. Zbl 1338.65219
Boscheri, Walter; Dumbser, Michael
2016
A simple two-phase method for the simulation of complex free surface flows. Zbl 1225.76210
Dumbser, Michael
2011
A space-time discontinuous Galerkin method for Boussinesq-type equations. Zbl 1410.76167
Dumbser, M.; Facchini, M.
2016
Efficient high order accurate staggered semi-implicit discontinuous Galerkin methods for natural convection problems. Zbl 07157294
Busto, S.; Tavelli, M.; Boscheri, W.; Dumbser, M.
2020
High-order unstructured one-step $$P_{N}P_{M}$$ schemes for the viscous and resistive MHD equations. Zbl 1231.76345
Dumbser, M.; Balsara, D. S.
2009
A high order special relativistic hydrodynamic and magnetohydrodynamic code with space-time adaptive mesh refinement. Zbl 1344.76058
Zanotti, Olindo; Dumbser, Michael
2015
High order accurate direct arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes on moving curvilinear unstructured meshes. Zbl 1390.76398
Boscheri, Walter; Dumbser, Michael
2016
Efficient implementation of ADER discontinuous Galerkin schemes for a scalable hyperbolic PDE engine. Zbl 1434.65179
Dumbser, Michael; Fambri, Francesco; Tavelli, Maurizio; Bader, Michael; Weinzierl, Tobias
2018
Staggered discontinuous Galerkin methods for the incompressible Navier-Stokes equations: spectral analysis and computational results. Zbl 06986989
Dumbser, M.; Fambri, F.; Furci, I.; Mazza, M.; Serra-Capizzano, S.; Tavelli, M.
2018
A well balanced diffuse interface method for complex nonhydrostatic free surface flows. Zbl 1410.76224
Gaburro, Elena; Castro, Manuel J.; Dumbser, Michael
2018
A two-dimensional Riemann solver with self-similar sub-structure - alternative formulation based on least squares projection. Zbl 1349.76157
Balsara, Dinshaw S.; Vides, Jeaniffer; Gurski, Katharine; Nkonga, Boniface; Dumbser, Michael; Garain, Sudip; Audit, Edouard
2016
A simple diffuse interface approach for compressible flows around moving solids of arbitrary shape based on a reduced Baer-Nunziato model. Zbl 07200216
Kemm, Friedemann; Gaburro, Elena; Thein, Ferdinand; Dumbser, Michael
2020
A hyperbolic reformulation of the Serre-Green-Naghdi model for general bottom topographies. Zbl 07335348
Bassi, C.; Bonaventura, L.; Busto, S.; Dumbser, M.
2020
A novel solver for the generalized Riemann problem based on a simplified LeFloch-Raviart expansion and a local space-time discontinuous Galerkin formulation. Zbl 1368.65155
Goetz, Claus R.; Dumbser, Michael
2016
Theoretical and numerical comparison of hyperelastic and hypoelastic formulations for Eulerian non-linear elastoplasticity. Zbl 1452.74022
Peshkov, Ilya; Boscheri, Walter; Loubère, Raphaël; Romenski, Evgeniy; Dumbser, Michael
2019
An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution with high order discontinuous Galerkin schemes. Zbl 1452.65188
Escalante, C.; Dumbser, M.; Castro, M. J.
2019
On high order ADER discontinuous Galerkin schemes for first order hyperbolic reformulations of nonlinear dispersive systems. Zbl 1465.76060
Busto, Saray; Dumbser, Michael; Escalante, Cipriano; Favrie, Nicolas; Gavrilyuk, Sergey
2021
A new stable version of the SPH method in Lagrangian coordinates. Zbl 1364.76175
Ferrari, A.; Dumbser, M.; Toro, E. F.; Armanini, A.
2008
A family of HLL-type solvers for the generalized Riemann problem. Zbl 1410.76228
Goetz, Claus R.; Balsara, Dinshaw S.; Dumbser, Michael
2018
High resolution methods for scalar transport problems in compliant systems of arteries. Zbl 1302.76134
Tavelli, M.; Dumbser, Michael; Casulli, V.
2013
A novel numerical method of high-order accuracy for flow in unsaturated porous media. Zbl 1242.76188
Zambra, C. E.; Dumbser, M.; Toro, E. F.; Moraga, N. O.
2012
Arbitrary high order accurate space-time discontinuous Galerkin finite element schemes on staggered unstructured meshes for linear elasticity. Zbl 1406.74647
Tavelli, Maurizio; Dumbser, Michael
2018
On source terms and boundary conditions using arbitrary high order discontinuous Galerkin schemes. Zbl 1159.35305
Dumbser, Michael; Munz, Claus-Dieter
2007
Semi-implicit discontinuous Galerkin methods for the incompressible Navier-Stokes equations on adaptive staggered Cartesian grids. Zbl 1439.76066
Fambri, Francesco; Dumbser, Michael
2017
ADER-DG with a-posteriori finite-volume limiting to simulate tsunamis in a parallel adaptive mesh refinement framework. Zbl 1410.86006
Rannabauer, Leonhard; Dumbser, Michael; Bader, Michael
2018
A unified hyperbolic formulation for viscous fluids and elastoplastic solids. Zbl 1417.35191
Dumbser, Michael; Peshkov, Ilya; Romenski, Evgeniy
2018
Semi-implicit staggered discontinuous Galerkin schemes for axially symmetric viscous compressible flows in elastic tubes. Zbl 1390.76322
Ioriatti, Matteo; Dumbser, Michael
2018
On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows. Zbl 07384714
Busto, Saray; Dumbser, Michael; Gavrilyuk, Sergey; Ivanova, Kseniya
2021
A path-conservative Osher-type scheme for axially symmetric compressible flows in flexible visco-elastic tubes. Zbl 1381.76219
Leibinger, Julia; Dumbser, Michael; Iben, Uwe; Wayand, Isabell
2016
Calculation of low Mach number acoustics: a comparison of MPV, EIF and linearized Euler equations. Zbl 1130.76072
Roller, Sabine; Schwartzkopff, Thomas; Fortenbach, Roland; Dumbser, Michael; Munz, Claus-Dieter
2005
A posteriori sub-cell finite volume limiting of staggered semi-implicit discontinuous Galerkin schemes for the shallow water equations. Zbl 1403.76046
Ioriatti, Matteo; Dumbser, Michael
2019
Implicit multiblock method for solving a kinetic equation on unstructured meshes. Zbl 1299.76222
Dumbser, M.; Titarev, V. A.; Utyuzhnikov, S. V.
2013
A path-conservative finite volume scheme for compressible multi-phase flows with surface tension. Zbl 1410.76463
Nguyen, Nguyen T.; Dumbser, Michael
2015
On high order ADER discontinuous Galerkin schemes for first order hyperbolic reformulations of nonlinear dispersive systems. Zbl 1465.76060
Busto, Saray; Dumbser, Michael; Escalante, Cipriano; Favrie, Nicolas; Gavrilyuk, Sergey
2021
On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows. Zbl 07384714
Busto, Saray; Dumbser, Michael; Gavrilyuk, Sergey; Ivanova, Kseniya
2021
A semi-implicit hybrid finite volume/finite element scheme for all Mach number flows on staggered unstructured meshes. Zbl 07423581
Busto, S.; Río-Martín, L.; Vázquez-Cendón, M. E.; Dumbser, M.
2021
Arbitrary high order central non-oscillatory schemes on mixed-element unstructured meshes. Zbl 07426260
Tsoutsanis, Panagiotis; Dumbser, Michael
2021
A posteriori subcell finite volume limiter for general $$P_NP_M$$ schemes: applications from gasdynamics to relativistic magnetohydrodynamics. Zbl 1475.65091
Gaburro, Elena; Dumbser, Michael
2021
An alternative SPH formulation: ADER-WENO-SPH. Zbl 07415027
Avesani, Diego; Dumbser, Michael; Vacondio, Renato; Righetti, Maurizio
2021
Efficient high order accurate staggered semi-implicit discontinuous Galerkin methods for natural convection problems. Zbl 07157294
Busto, S.; Tavelli, M.; Boscheri, W.; Dumbser, M.
2020
A simple diffuse interface approach for compressible flows around moving solids of arbitrary shape based on a reduced Baer-Nunziato model. Zbl 07200216
Kemm, Friedemann; Gaburro, Elena; Thein, Ferdinand; Dumbser, Michael
2020
A hyperbolic reformulation of the Serre-Green-Naghdi model for general bottom topographies. Zbl 07335348
Bassi, C.; Bonaventura, L.; Busto, S.; Dumbser, M.
2020
High order ADER-DG schemes for the simulation of linear seismic waves induced by nonlinear dispersive free-surface water waves. Zbl 1447.74042
Bassi, C.; Busto, S.; Dumbser, M.
2020
A new continuum model for general relativistic viscous heat-conducting media. Zbl 1462.82032
Romenski, E.; Peshkov, I.; Dumbser, M.; Fambri, F.
2020
On GLM curl cleaning for a first order reduction of the CCZ4 formulation of the Einstein field equations. Zbl 1453.85002
Dumbser, Michael; Fambri, Francesco; Gaburro, Elena; Reinarz, Anne
2020
Modeling wavefields in saturated elastic porous media based on thermodynamically compatible system theory for two-phase solid-fluid mixtures. Zbl 07211859
Romenski, Evgeniy; Reshetova, Galina; Peshkov, Ilya; Dumbser, Michael
2020
Theoretical and numerical comparison of hyperelastic and hypoelastic formulations for Eulerian non-linear elastoplasticity. Zbl 1452.74022
Peshkov, Ilya; Boscheri, Walter; Loubère, Raphaël; Romenski, Evgeniy; Dumbser, Michael
2019
An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution with high order discontinuous Galerkin schemes. Zbl 1452.65188
Escalante, C.; Dumbser, M.; Castro, M. J.
2019
A posteriori sub-cell finite volume limiting of staggered semi-implicit discontinuous Galerkin schemes for the shallow water equations. Zbl 1403.76046
Ioriatti, Matteo; Dumbser, Michael
2019
A simple diffuse interface approach on adaptive Cartesian grids for the linear elastic wave equations with complex topography. Zbl 1452.74113
Tavelli, Maurizio; Dumbser, Michael; Charrier, Dominic Etienne; Rannabauer, Leonhard; Weinzierl, Tobias; Bader, Michael
2019
Central WENO subcell finite volume limiters for ADER discontinuous Galerkin schemes on fixed and moving unstructured meshes. Zbl 1474.65307
Boscheri, Walter; Semplice, Matteo; Dumbser, Michael
2019
FORCE schemes on moving unstructured meshes for hyperbolic systems. Zbl 1442.65191
Boscheri, Walter; Dumbser, Michael; Righetti, Maurizio
2019
A second-order cell-centered Lagrangian ADER-MOOD finite volume scheme on multidimensional unstructured meshes for hydrodynamics. Zbl 1381.76212
Boscheri, Walter; Dumbser, Michael; Loubère, Raphaël; Maire, Pierre-Henri
2018
Efficient implementation of ADER discontinuous Galerkin schemes for a scalable hyperbolic PDE engine. Zbl 1434.65179
Dumbser, Michael; Fambri, Francesco; Tavelli, Maurizio; Bader, Michael; Weinzierl, Tobias
2018
Staggered discontinuous Galerkin methods for the incompressible Navier-Stokes equations: spectral analysis and computational results. Zbl 06986989
Dumbser, M.; Fambri, F.; Furci, I.; Mazza, M.; Serra-Capizzano, S.; Tavelli, M.
2018
A well balanced diffuse interface method for complex nonhydrostatic free surface flows. Zbl 1410.76224
Gaburro, Elena; Castro, Manuel J.; Dumbser, Michael
2018
A family of HLL-type solvers for the generalized Riemann problem. Zbl 1410.76228
Goetz, Claus R.; Balsara, Dinshaw S.; Dumbser, Michael
2018
Arbitrary high order accurate space-time discontinuous Galerkin finite element schemes on staggered unstructured meshes for linear elasticity. Zbl 1406.74647
Tavelli, Maurizio; Dumbser, Michael
2018
ADER-DG with a-posteriori finite-volume limiting to simulate tsunamis in a parallel adaptive mesh refinement framework. Zbl 1410.86006
Rannabauer, Leonhard; Dumbser, Michael; Bader, Michael
2018
A unified hyperbolic formulation for viscous fluids and elastoplastic solids. Zbl 1417.35191
Dumbser, Michael; Peshkov, Ilya; Romenski, Evgeniy
2018
Semi-implicit staggered discontinuous Galerkin schemes for axially symmetric viscous compressible flows in elastic tubes. Zbl 1390.76322
Ioriatti, Matteo; Dumbser, Michael
2018
Reprint of: “Direct arbitrary-Lagrangian-Eulerian finite volume schemes on moving nonconforming unstructured meshes”. Zbl 1391.76406
Gaburro, Elena; Dumbser, Michael; Castro, Manuel J.
2018
Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes. Zbl 1378.76044
Boscheri, Walter; Dumbser, Michael
2017
Central weighted ENO schemes for hyperbolic conservation laws on fixed and moving unstructured meshes. Zbl 1377.65115
Dumbser, Michael; Boscheri, Walter; Semplice, Matteo; Russo, Giovanni
2017
A pressure-based semi-implicit space-time discontinuous Galerkin method on staggered unstructured meshes for the solution of the compressible Navier-Stokes equations at all Mach numbers. Zbl 1376.76028
Tavelli, Maurizio; Dumbser, Michael
2017
Direct arbitrary-Lagrangian-Eulerian finite volume schemes on moving nonconforming unstructured meshes. Zbl 1390.76433
Gaburro, Elena; Dumbser, Michael; Castro, Manuel J.
2017
Semi-implicit discontinuous Galerkin methods for the incompressible Navier-Stokes equations on adaptive staggered Cartesian grids. Zbl 1439.76066
Fambri, Francesco; Dumbser, Michael
2017
A well-balanced path conservative SPH scheme for nonconservative hyperbolic systems with applications to shallow water and multi-phase flows. Zbl 1390.76760
Rossi, Giulia; Dumbser, Michael; Armanini, Aronne
2017
An alternative smooth particle hydrodynamics formulation to simulate chemotaxis in porous media. Zbl 1360.76210
Avesani, Diego; Dumbser, Michael; Chiogna, Gabriele; Bellin, Alberto
2017
High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids. Zbl 1349.76324
Dumbser, Michael; Peshkov, Ilya; Romenski, Evgeniy; Zanotti, Olindo
2016
A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems. Zbl 1349.76603
Dumbser, Michael; Balsara, Dinshaw S.
2016
A simple robust and accurate a posteriori sub-cell finite volume limiter for the discontinuous Galerkin method on unstructured meshes. Zbl 1349.65447
Dumbser, Michael; Loubère, Raphaël
2016
A staggered space-time discontinuous Galerkin method for the three-dimensional incompressible Navier-Stokes equations on unstructured tetrahedral meshes. Zbl 1349.76271
Tavelli, Maurizio; Dumbser, Michael
2016
A conservative, weakly nonlinear semi-implicit finite volume scheme for the compressible Navier-Stokes equations with general equation of state. Zbl 1410.76220
Dumbser, Michael; Casulli, Vincenzo
2016
Cell centered direct arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes for nonlinear hyperelasticity. Zbl 1390.76399
Boscheri, Walter; Dumbser, Michael; Loubère, Raphaël
2016
Spectral semi-implicit and space-time discontinuous Galerkin methods for the incompressible Navier-Stokes equations on staggered Cartesian grids. Zbl 1432.76157
Fambri, Francesco; Dumbser, Michael
2016
An efficient quadrature-free formulation for high order arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes on unstructured meshes. Zbl 1338.65219
Boscheri, Walter; Dumbser, Michael
2016
A space-time discontinuous Galerkin method for Boussinesq-type equations. Zbl 1410.76167
Dumbser, M.; Facchini, M.
2016
High order accurate direct arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes on moving curvilinear unstructured meshes. Zbl 1390.76398
Boscheri, Walter; Dumbser, Michael
2016
A two-dimensional Riemann solver with self-similar sub-structure - alternative formulation based on least squares projection. Zbl 1349.76157
Balsara, Dinshaw S.; Vides, Jeaniffer; Gurski, Katharine; Nkonga, Boniface; Dumbser, Michael; Garain, Sudip; Audit, Edouard
2016
A novel solver for the generalized Riemann problem based on a simplified LeFloch-Raviart expansion and a local space-time discontinuous Galerkin formulation. Zbl 1368.65155
Goetz, Claus R.; Dumbser, Michael
2016
A path-conservative Osher-type scheme for axially symmetric compressible flows in flexible visco-elastic tubes. Zbl 1381.76219
Leibinger, Julia; Dumbser, Michael; Iben, Uwe; Wayand, Isabell
2016
Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting. Zbl 1390.76381
Zanotti, Olindo; Fambri, Francesco; Dumbser, Michael; Hidalgo, Arturo
2015
Divergence-free MHD on unstructured meshes using high order finite volume schemes based on multidimensional Riemann solvers. Zbl 1351.76092
Balsara, Dinshaw S.; Dumbser, Michael
2015
Multidimensional Riemann problem with self-similar internal structure. Part II: Application to hyperbolic conservation laws on unstructured meshes. Zbl 1351.76091
Balsara, Dinshaw S.; Dumbser, Michael
2015
Direct arbitrary-Lagrangian-Eulerian ADER-MOOD finite volume schemes for multidimensional hyperbolic conservation laws. Zbl 1349.76312
Boscheri, Walter; Loubère, Raphaël; Dumbser, Michael
2015
A staggered space-time discontinuous Galerkin method for the incompressible Navier-Stokes equations on two-dimensional triangular meshes. Zbl 1390.76360
Tavelli, Maurizio; Dumbser, Michael
2015
An efficient semi-implicit finite volume method for axially symmetric compressible flows in compliant tubes. Zbl 1326.76070
Dumbser, Michael; Iben, Uwe; Ioriatti, Matteo
2015
High order cell-centered Lagrangian-type finite volume schemes with time-accurate local time stepping on unstructured triangular meshes. Zbl 1349.76311
Boscheri, Walter; Dumbser, Michael; Zanotti, Olindo
2015
A high order special relativistic hydrodynamic and magnetohydrodynamic code with space-time adaptive mesh refinement. Zbl 1344.76058
Zanotti, Olindo; Dumbser, Michael
2015
A path-conservative finite volume scheme for compressible multi-phase flows with surface tension. Zbl 1410.76463
Nguyen, Nguyen T.; Dumbser, Michael
2015
A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws. Zbl 1349.65448
Dumbser, Michael; Zanotti, Olindo; Loubère, Raphaël; Diot, Steven
2014
A direct arbitrary-Lagrangian-Eulerian ADER-WENO finite volume scheme on unstructured tetrahedral meshes for conservative and non-conservative hyperbolic systems in 3D. Zbl 1349.76310
Boscheri, Walter; Dumbser, Michael
2014
Multidimensional HLLC Riemann solver for unstructured meshes – with application to Euler and MHD flows. Zbl 1349.76426
Balsara, Dinshaw S.; Dumbser, Michael; Abgrall, Remi
2014
A new family of high order unstructured MOOD and ADER finite volume schemes for multidimensional systems of hyperbolic conservation laws. Zbl 1373.76137
Loubère, Raphaël; Dumbser, Michael; Diot, Steven
2014
High order space-time adaptive ADER-WENO finite volume schemes for non-conservative hyperbolic systems. Zbl 1295.65088
Dumbser, Michael; Hidalgo, Arturo; Zanotti, Olindo
2014
Lagrangian ADER-WENO finite volume schemes on unstructured triangular meshes based on genuinely multidimensional HLL Riemann solvers. Zbl 1349.76309
Boscheri, Walter; Balsara, Dinshaw S.; Dumbser, Michael
2014
A high order semi-implicit discontinuous Galerkin method for the two dimensional shallow water equations on staggered unstructured meshes. Zbl 1298.76120
Tavelli, Maurizio; Dumbser, Michael
2014
A new class of moving-least-squares WENO-SPH schemes. Zbl 1349.76661
Avesani, Diego; Dumbser, Michael; Bellin, Alberto
2014
A staggered semi-implicit discontinuous Galerkin method for the two dimensional incompressible Navier-Stokes equations. Zbl 1338.76068
Tavelli, Maurizio; Dumbser, Michael
2014
Construction and comparison of parallel implicit kinetic solvers in three spatial dimensions. Zbl 1349.76741
Titarev, Vladimir; Dumbser, Michael; Utyuzhnikov, Sergey
2014
Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes with time-accurate local time stepping for hyperbolic conservation laws. Zbl 1423.76296
Dumbser, Michael
2014
Dumbser, Michael; Zanotti, Olindo; Hidalgo, Arturo; Balsara, Dinshaw S.
2013
Efficient implementation of ADER schemes for Euler and magnetohydrodynamical flows on structured meshes – speed comparisons with Runge-Kutta methods. Zbl 1291.76237
Balsara, Dinshaw S.; Meyer, Chad; Dumbser, Michael; Du, Huijing; Xu, Zhiliang
2013
High-order unstructured Lagrangian one-step WENO finite volume schemes for non-conservative hyperbolic systems: applications to compressible multi-phase flows. Zbl 1290.76081
Dumbser, Michael; Boscheri, Walter
2013
Arbitrary-Lagrangian-Eulerian one-step WENO finite volume schemes on unstructured triangular meshes. Zbl 1388.65075
Boscheri, Walter; Dumbser, Michael
2013
A staggered semi-implicit spectral discontinuous Galerkin scheme for the shallow water equations. Zbl 1366.76050
Dumbser, Michael; Casulli, Vincenzo
2013
On arbitrary-Lagrangian-Eulerian one-step WENO schemes for stiff hyperbolic balance laws. Zbl 1373.76126
Dumbser, Michael; Uuriintsetseg, Ariunaa; Zanotti, Olindo
2013
Efficient implementation of high order unstructured WENO schemes for cavitating flows. Zbl 1290.76098
Dumbser, Michael; Iben, Uwe; Munz, Claus-Dieter
2013
A diffuse interface method for complex three-dimensional free surface flows. Zbl 1286.76099
Dumbser, Michael
2013
High resolution methods for scalar transport problems in compliant systems of arteries. Zbl 1302.76134
Tavelli, M.; Dumbser, Michael; Casulli, V.
2013
Implicit multiblock method for solving a kinetic equation on unstructured meshes. Zbl 1299.76222
Dumbser, M.; Titarev, V. A.; Utyuzhnikov, S. V.
2013
High-order finite volume schemes based on defect corrections. Zbl 1275.65052
Filimon, Alexander; Dumbser, Michael; Munz, Claus-Dieter
2013
A semi-implicit scheme for 3D free surface flows with high-order velocity reconstruction on unstructured Voronoi meshes. Zbl 1455.76104
Boscheri, Walter; Dumbser, M.; Righetti, M.
2013
Comparison of solvers for the generalized Riemann problem for hyperbolic systems with source terms. Zbl 1284.35268
Montecinos, G.; Castro, C. E.; Dumbser, M.; Toro, E. F.
2012
Semi-implicit numerical modeling of axially symmetric flows in compliant arterial systems. Zbl 1242.92019
Casulli, Vincenzo; Dumbser, Michael; Toro, Eleuterio F.
2012
A novel numerical method of high-order accuracy for flow in unsaturated porous media. Zbl 1242.76188
Zambra, C. E.; Dumbser, M.; Toro, E. F.; Moraga, N. O.
2012
Discontinuous Galerkin schemes with defect corrections based on reconstruction. Zbl 1365.76114
Filimon, Alexander; Dumbser, Michael; Munz, Claus-Dieter
2012
A simple extension of the Osher Riemann solver to non-conservative hyperbolic systems. Zbl 1220.65110
Dumbser, Michael; Toro, Eleuterio F.
2011
On universal Osher-type schemes for general nonlinear hyperbolic conservation laws. Zbl 1373.76125
Dumbser, Michael; Toro, Eleuterio F.
2011
Hidalgo, Arturo; Dumbser, Michael
2011
Explicit one-step time discretizations for discontinuous Galerkin and finite volume schemes based on local predictors. Zbl 1220.65122
Gassner, Gregor; Dumbser, Michael; Hindenlang, Florian; Munz, Claus-Dieter
2011
A simple two-phase method for the simulation of complex free surface flows. Zbl 1225.76210
Dumbser, Michael
2011
$$P_N P_M$$ schemes on unstructured meshes for time-dependent partial differential equations. Zbl 1358.76039
Dumbser, Michael
2011
Arbitrary high order $$P_{N}P_{M}$$ schemes on unstructured meshes for the compressible Navier-Stokes equations. Zbl 1242.76161
Dumbser, Michael
2010
FORCE schemes on unstructured meshes. II: Non-conservative hyperbolic systems. Zbl 1227.76043
Dumbser, Michael; Hidalgo, Arturo; Castro, Manuel; Parés, Carlos; Toro, Eleuterio F.
2010
Three-dimensional flow evolution after a dam break. Zbl 1205.76228
Ferrari, A.; Fraccarollo, L.; Dumbser, M.; Toro, E. F.; Armanini, A.
2010
Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics. Zbl 1275.76169
Balsara, Dinshaw S.; Rumpf, Tobias; Dumbser, Michael; Munz, Claus-Dieter
2009
ADER schemes on unstructured meshes for nonconservative hyperbolic systems: applications to geophysical flows. Zbl 1177.76222
Dumbser, Michael; Castro, Manuel; Parés, Carlos; Toro, Eleuterio F.
2009
A new 3D parallel SPH scheme for free surface flows. Zbl 1242.76270
Ferrari, Angela; Dumbser, Michael; Toro, Eleuterio F.; Armanini, Aronne
2009
Very high order $$P_NP_M$$ schemes on unstructured meshes for the resistive relativistic MHD equations. Zbl 1261.76028
Dumbser, Michael; Zanotti, Olindo
2009
A high-order discontinuous Galerkin method with time-accurate local time stepping for the Maxwell equations. Zbl 1156.78012
Taube, Arne; Dumbser, Michael; Munz, Claus-Dieter; Schneider, Rudolf
2009
High-order unstructured one-step $$P_{N}P_{M}$$ schemes for the viscous and resistive MHD equations. Zbl 1231.76345
Dumbser, M.; Balsara, D. S.
2009
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text:
# Is this calculation assuming that the neutrino is massless?

This is a calculation that I don't understand in Griffiths Intro to Elementary Particles page 105 second edition.

This the decay of a pion into a moun and a neutrino.

From Conservation of energy and momentum,

the momentum four vector

$$p_{neutrino} = p_{pion} - p_{muon}$$

implies

$$p_{neutrino}^2 = p_{pion}^2 + p_{muon}^2 - 2 p_{pion} \cdot p_{muon}$$

and

$$p_{neutrino}^2=0 :p_{pion}^2= m_{pion}^2 c^2 , p_{muon}^2 = m_{muon}^2 c^2$$

Ok. Why is $$p_{neutrino}^2=0$$?

Is it just because he's taking the mass of the neutrino to be 0? But it isn't 0. It's very small.

## 1 Answer

In his "Formulas and Constants" table on XIII, Griffiths says, under the Lepton table, "Neutrino masses are extremely small, and for most purposes can be taken to be zero; for details see Chapter 11."


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# Lab 06: Balmer Series

Steve Koch 04:01, 21 December 2010 (EST):Missing standard error of the mean and comparison of discrepancy with this SEM.

### Summary

The Balmer Series is of one of a set of six different series describing the spectral line emissions of the hydrogen atom. The Balmer series is calculated using the Balmer formula which was discovered by Johann Balmer in 1885. The visible spectrum of light from hydrogen displays four wavelengths, 410.2 nm, 434.1 nm, 486.1 nm, and 656.3 nm. These correspond to electron transitioning from n = 6,5,4,3 to n = 2 respectively, where n is the principal quantum number of the electron. These are the first four wavelengths in the Balmer Series.

In our lab we used a Constant Deviation Spectrometer to observe and measure the spectral lines of mercury, hydrogen, and deuterium. Deuterium is an isotope of hydrogen that has a proton and a neutron in its nucleus. We measured the wavelengths of light emitted from a mercury tube and compared them against known values of the spectral lines for mercury to get a correction for systematic error caused by the instrument. Then we measured the wavelengths of the first four lines of the Balmer series for hydrogen, and the first three lines of the Balmer series for deuterium. We were unable to locate the fourth line for deuterium. We used our corrected wavelength measurements to determine the Rydberg constant.

Part of the Deuterium Emission Spectrum as seen through the Spectrometer.
Courtesy of Emran Qassem.

### Analysis

The Rydberg constant can also be found with the following equation when the wavelengths and corresponding transitions are known:

$\frac{1}{\lambda }=R(\frac{1}{2^2}-\frac{1}{n^2})$

where n=3, 4, 5, 6 in the Balmer Series.

Solving for R gives:

$R=\frac{4n^2}{\lambda(n^2-4)}\,\!$

Calculating R for each wavelength for the first three terms in the Balmer series for hydrogen and deuterium, and then averaging the results for hydrogen and deuterium separately gives:

RHydrogen = (RHalpha + RHbeta + RHgamma) / 3 = (10953407.40062 + 11017462.67835 + 11001027.49597) / 3 = 10990630.85831m − 1
RDeuterium = (RDalpha + RDbeta + RDgamma) / 3 = (10970091.26506 + 10989086.46351 + 11014513.82487) / 3 = 10991230.51781m − 1

The known values for RHydrogen and RDeuterium are found by using the following equation:

$R_M = \frac{R_\infty}{1+m_e/M},$

where me is the rest mass of the electron, and M is the mass of the atomic nucleus.

These values are:

$R_{Hydrogen}= 10967758.3406 m^{-1}\ \$
$R_{Deuterium}=10970746.1986 m^{-1} \ \$

The calculated percent errors are given by:

$\% error=\frac{R_{accepted}-R_{measured}}{R_{accepted}}$

These values are:

${Hydrogen}\approx0.21%\,\!$
${Deuterium}\approx0.19%\,\!$

### Conclusion

We were able to gather some good data since we have a low percent error. I think the change factor I used was off and it didn't compensated for the systematic error from the spectrometer very well. I would like to generate a new one with more than just the readings and known values for mercury. In the future I think this lab should include a few more elements so that the differences in the Rydberg constant for small nuclei become apparent. That was very interesting to me when I discussed it with Katie, the TA.


text:
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# Create table in Latex, then save it as image in vector graphics format

I have created a table in Latex, and I would now like to save the table as an image using a vector graphics format such as EPS. Is there a way I can do this?

• Compile only your table with no page number and run latex+dvips to obtain a ps file. Or run pdflatex to obtain a pdf file. Configure the margin to reduce white space. – Sigur Oct 12 '14 at 1:23


text:
Question

# For the identification of beta $\beta$-naphthol using dye test, it is necessary to use:A. dichloromethane solution of $\beta$-naphtholB. acidic solution of $\beta$-naphtholC. neutral solution of$\beta$-naphtholD. alkaline solution of $\beta$-naphthol

Hint: We know that azo dye test is used for identification of aromatic primary amines. Primary amine is the amine which is in the form of ${\rm{R}} - {\rm{N}}{{\rm{H}}_{\rm{2}}}$ (R is any alkyl group).

The aromatic primary amine can be confirmed by azo dye test. In this test, primary aromatic amine like aniline reacts with nitrous acid (produced by reaction of sodium nitrite with hydrochloric acid) at zero to five degree Celsius to form diazonium salt or benzene diazonium chloride. The reaction of formed benzene diazonium chloride with $\beta$-naphthol gives a scarlet red dye and this die is sparingly soluble in water. The reaction can be shown as follows:

Let’s come to the question. We have to identify the type of solution of $\beta$-naphthol to be used in dye test. If the solution of $\beta$-naphthol is alkaline, -OH convert to ${{\rm{O}}^ - }$ which reacts with azo compound (benzene diazonium chloride) to form azo dye. The chemical reaction is shown below.

Hence, the correct option is D, that is, an alkaline solution.

Additional Information Another test of amine is carbylamine test. In this test aromatic as well as aliphatic primary amine is heated with chloroform to result in carbylamine. The reaction is,

${\rm{R}} - {\rm{N}}{{\rm{H}}_{\rm{2}}} + {\rm{CHC}}{{\rm{l}}_{\rm{3}}} + 3{\rm{KOH}} \to {\rm{RNC}}\left( {{\rm{Carbylamine}}} \right) + 3{\rm{KCl + 3}}{{\rm{H}}_2}{\rm{O}}$

Note: It is to be noted that aromatic primary can be distinguished from secondary and tertiary amine by azo dye test and aliphatic or aromatic primary amine can be differentiated from secondary and tertiary amine by Carbylamine test.