On a conjecture of Gowers and Wolf, Discrete Analysis 2022:10, 13 pp. Szemerédi's theorem asserts that for every $\delta>0$ and every positive integer $k$ there exists $n$ such that every subset $A$ ...of $\{1,2,\dots,n\}$ of size at least $\delta n$ contains an arithmetic progression of length $k$. We can very slightly reformulate the conclusion of this statement by saying that if $\phi_i$ is the linear form $(x,y)\mapsto x+(i-1)y$, then there exist $a,d$ with $d\ne 0$ such that $\phi_i(a,d)\in A$ for $i=1,2,\dots,k$. Several proofs of Szemerédi's theorem involve looking at expressions of the form $$\mathbb E_{x,d}f(x)f(x+d)\dots f(x+(k-1)d)$$ or in the alternative notation $$\mathbb E_{x,d}f(\phi_1(x,d))\dots f(\phi_k(x,d)).$$ Here $x$ and $d$ are chosen uniformly at random from a large cyclic group (or from other finite Abelian groups when related questions are studied). In particular, it is useful to find conditions that guarantee that averages of this kind are small. In particular, Gowers showed that if $\|f\|_\infty\leq 1$, then the average above is small if a certain norm of $f$, known as the $U^{k-1}$ norm, is small. This is often expressed by saying that the average is "controlled by the $U^{k-1}$ norm". The $U^k$ norms are defined as follows. First we define a "difference operator" $\partial_a$ by $\partial_af(x)=f(x)\overline{f(x-a)}$. Then the $U^k$ norm of a function $f$ is given by the formula $$\|f\|_{U^k}^{2^k}=\mathbb E_{x,a_1,\dots,a_k}\partial_{a_1}\dots\partial_{a_k}f(x).$$ It is not immediately obvious that this formula defines a norm, but this can be shown (except in the case $k=1$ when one obtains a seminorm). A rough intuitive sense of what this statement says is that if $f$ has a small $U^k$ norm, then it behaves "sufficiently quasirandomly" to guarantee that a lot of cancellation occurs in the average in question. The $U^k$ norms can be shown to increase with $k$. A key example of a function with large $U^k$ norm but small $U^j$ norm for all $j<k$ is the function $f(x)=\exp(2\pi ix^{k-1}/p)$ defined on the cyclic group of integers mod $p$. It is not hard to prove that $\partial_{a_1}\dots\partial_{a_j}f$ is identically 0 when $j<k$, making the $U^k$ norm equal to 1 (its maximum possible value given that $\|f\|_\infty=1$) and it turns out that for all smaller $j$ the average is small. In their celebrated paper Linear equations in primes(https://annals.math.princeton.edu/2010/171-3/p08), Green and Tao were interested in asymptotics for the number of configurations in the first $n$ primes. Their first aim was to obtain asymptotics for the number of arithmetic progressions of length $k$, thereby obtaining a more precise version of the Green-Tao theorem, which states (or rather easily implies) that this number tends to infinity with $n$. However, they also looked at more general configurations, which led them to consider averages of the form mentioned above for other sequences of linear forms. They showed that the proof that the average is controlled by the $U^{k-1}$ norm in the case $\phi_i(x,y)=x+(i-1)y$ can be generalized to other systems of linear forms, and gave a criterion for which $k$ is needed in order for the $U^k$ norm to control the average. Later, Gowers and Wolf observed that although the criterion identified by Green and Tao accurately identified the $k$ for which the proof (which involved repeated applications of the Cauchy-Schwarz inequality) worked, it did not appear to be best possible, in the sense that for some systems of linear forms, a smaller $k$ seemed to suffice. They conjectured that if for each $i=1,\dots,t$, $\psi_i:\mathbb{Z}^D\to\mathbb{Z}$ is a linear form, then the smallest $s$ such that the $U^s$ norm controls the corresponding average is also the smallest $s$ such that the functions $\psi_1^{s},\ldots, \psi_t^{s}$ are linearly independent. They proved this conjecture when $s\leq 2$, and also proved the whole conjecture when instead of a cyclic group one takes the group $\mathbb F_p^n$ for $p$ bounded (but also not too small -- additional complications arose in small characteristic). For this they made use of an inverse theorem for the $U^k$ norms due to Bergelson, Tao and Ziegler, which gave a description of bounded functions with large $U^k$ norms (showing that they correlate with polynomial examples). At the time of that result, the corresponding inverse theorem for functions defined on cyclic groups was still a major open problem. However, later it was solved by Green, Tao and Ziegler, and shortly after that, Green and Tao used the resulting inverse theorem to solve the conjecture of Gowers and Wolf in the cyclic groups case. Or so it seemed. Ten years later, the author of this paper noticed that their proof subtly used an assumption about the linear forms that does not always hold, which Green and Tao dubbed the "flag condition". The result is that the proof of Green and Tao works for systems of linear forms that satisfy the flag condition, and therefore covers many examples of interest, but does not work for all systems of linear forms. In this paper, the author finally proves the conjecture in full generality. The first step of the argument is to generalize Green and Tao's result in various ways for systems that do satisfy the flag condition. This results in a theorem that is invariant under dilations of the linear forms. The second step is to show that for any system of linear forms one can dilate the individual forms in such a way that they satisfy the flag condition.
The use of standardized mesh kits for repair of pelvic-organ prolapse has spread rapidly in recent years, but it is unclear whether this approach results in better outcomes than traditional ...colporrhaphy.
In this multicenter, parallel-group, randomized, controlled trial, we compared the use of a trocar-guided, transvaginal polypropylene-mesh repair kit with traditional colporrhaphy in women with prolapse of the anterior vaginal wall (cystocele). The primary outcome was a composite of the objective anatomical designation of stage 0 (no prolapse) or 1 (position of the anterior vaginal wall more than 1 cm above the hymen), according to the Pelvic Organ Prolapse Quantification system, and the subjective absence of symptoms of vaginal bulging 12 months after the surgery.
Of 389 women who were randomly assigned to a study treatment, 200 underwent prolapse repair with the transvaginal mesh kit and 189 underwent traditional colporrhaphy. At 1 year, the primary outcome was significantly more common in the women treated with transvaginal mesh repair (60.8%) than in those who underwent colporrhaphy (34.5%) (absolute difference, 26.3 percentage points; 95% confidence interval, 15.6 to 37.0). The surgery lasted longer and the rates of intraoperative hemorrhage were higher in the mesh-repair group than in the colporrhaphy group (P<0.001 for both comparisons). Rates of bladder perforation were 3.5% in the mesh-repair group and 0.5% in the colporrhaphy group (P=0.07), and the respective rates of new stress urinary incontinence after surgery were 12.3% and 6.3% (P=0.05). Surgical reintervention to correct mesh exposure during follow-up occurred in 3.2% of 186 patients in the mesh-repair group.
As compared with anterior colporrhaphy, use of a standardized, trocar-guided mesh kit for cystocele repair resulted in higher short-term rates of successful treatment but also in higher rates of surgical complications and postoperative adverse events. (Funded by the Karolinska Institutet and Ethicon; ClinicalTrials.gov number, NCT00566917.).
There is an unmet need for developing a new class of smart medical implants with novel properties and advanced functionalities. Here, the concept of “self‐aware implants” is proposed to enable the ...creation of a new generation of multifunctional metamaterial implantable devices capable of responding to their environment, empowering themselves, and self‐monitoring their condition. These functionalities are achieved via integrating nano energy harvesting and mechanical metamaterial design paradigms. Various aspects of the proposed concept are highlighted by developing proof‐of‐concept interbody spinal fusion cage implants with self‐sensing, self‐powering, and mechanical tunability features. Bench‐top testing is performed using synthetic biomimetic and human cadaver spine models to evaluate the electrical and mechanical performance of the developed patient‐specific metamaterial implants. The results show that the self‐aware cage implants can diagnose bone healing process using the voltage signals generated internally through their built‐in contact‐electrification mechanisms. The voltage and current generated by the implants under the axial compression forces of the spine models reach 9.2 V and 4.9 nA, respectively. The metamaterial implants can serve as triboelectric nanogenerators to empower low‐power electronics. The capacity of the proposed technology to revolutionize the landscape of implantable devices and to achieve better surgical outcomes is further discussed.
A new generation of patient‐specific metamaterial orthopedic implants with diagnostic capabilities is presented. The proposed metamaterial implants can respond to their environment, empower themselves, and self‐monitor their condition. Bench‐top tests are performed using synthetic biomimetic and human cadaver spine models.
Historically, humeral shaft fractures have been successfully treated with nonoperative management and functional bracing; however, various surgical options are also available. In the present study, ...we compared the outcomes of nonoperative versus operative interventions for the treatment of extra-articular humeral shaft fractures.
This study was a network meta-analysis of prospective randomized controlled trials (RCTs) in which functional bracing was compared with surgical techniques (including open reduction and internal fixation ORIF, minimally invasive plate osteosynthesis MIPO, and intramedullary nailing in both antegrade aIMN and retrograde rIMN directions) for the treatment of humeral shaft fractures. The outcomes that were assessed included time to union and the rates of nonunion, malunion, delayed union, secondary surgical intervention, iatrogenic radial nerve palsy, and infection. Mean differences and log odds ratios (ORs) were used to analyze continuous and categorical data, respectively.
Twenty-one RCTs evaluating the outcomes for 1,203 patients who had been treated with functional bracing (n = 190), ORIF (n = 479), MIPO (n = 177), aIMN (n = 312), or rIMN (n = 45) were included. Functional bracing yielded significantly higher odds of nonunion and significantly longer time to union than ORIF, MIPO, and aIMN (p < 0.05). Comparison of surgical fixation techniques demonstrated significantly faster time to union with MIPO than with ORIF (p = 0.043). Significantly higher odds of malunion were observed with functional bracing than with ORIF (p = 0.047). Significantly higher odds of delayed union were observed with aIMN than with ORIF (p = 0.036). Significantly higher odds of secondary surgical intervention were observed with functional bracing than with ORIF (p = 0.001), MIPO (p = 0.007), and aIMN (p = 0.004). However, ORIF was associated with significantly higher odds of iatrogenic radial nerve injury and superficial infection than both functional bracing and MIPO (p < 0.05).
Compared with functional bracing, most operative interventions demonstrated lower rates of reoperation. MIPO demonstrated significantly faster time to union while limiting periosteal stripping, whereas ORIF was associated with significantly higher rates of radial nerve palsy. Nonoperative management with functional bracing demonstrated higher nonunion rates than most surgical techniques, often requiring conversion to surgical fixation.
Therapeutic Level I . See Instructions for Authors for a complete description of levels of evidence.
This study compared outcomes of locked plating (LP) versus intramedullary nailing (IMN) techniques for treatment of extra-articular proximal-third tibia fractures.
Data Sources: PubMed, Ovid MEDLINE
...Studies were included if they compared LP and IMN fixation for proximal one third tibial shaft fractures without articular extension or with simple articular extension into the tibial plateau. Minimum 1 year of clinical and radiographic follow up was used.
Outcomes assessed included operative duration, postoperative knee range of motion (ROM), union outcomes (time to union, nonunion, malunion, delayed union), and incidence of postoperative complications (superficial and deep infection, secondary surgical intervention, compartment syndrome).
Separate random-effects meta-analyses were conducted for each outcome. For categorical data, relative risks were used whereas the standardized mean difference was used for continuous variables, with corresponding 95 % confidence intervals.
7 studies were included reporting the outcomes of 319 patients treated with LP and 300 treated with IMN. IMN fixation had significantly shorter time to union (p = 0.049) and lower risk for superficial infection (p = 0.028). However, LP conferred a significantly lower risk for malunion (p = 0.017) and postoperative compartment syndrome (p = 0.018).
IMN demonstrated significantly shorter time to union and lower risk of superficial infection when treating extra-articular proximal tibia fractures, while LP fixation demonstrated significantly lower risk for malunion and postoperative compartment syndrome. Although successful results can be achieved with good technique in LP and IMN fixation, a significant complication profile exists with these fractures regardless of construct choice.
Therapeutic Level III
Objective To estimate the risk for stress urinary incontinence and pelvic organ prolapse surgery related to vaginal birth or cesarean delivery. Study Design A cohort study of all women having their ...first and all subsequent deliveries by cesarean (n = 33,167), and an age-matched sample of women only having vaginal deliveries (n = 63,229) between 1973 and 1983. Hazard ratios were calculated using Cox regression models with 95% confidence intervals. Results Women only having vaginal deliveries had increased overall risks of incontinence (hazard ratio, 2.9; 95% confidence interval, 2.4–3.6) and prolapse surgery (hazard ratio, 9.2; 95% confidence interval, 7.0–12.1) compared with women only having cesarean deliveries. Conclusion Having only vaginal childbirths was associated with a significantly increased risk of stress urinary incontinence and pelvic organ prolapse surgery later in life compared with only having cesarean deliveries.
Management of Sacroiliac Joint Pain Schmidt, Gary L; Bhandutia, Amit K; Altman, Daniel T
Journal of the American Academy of Orthopaedic Surgeons,
09/2018, Letnik:
26, Številka:
17
Journal Article
Recenzirano
Sacroiliac joint (SIJ)-based pain can be difficult to diagnose definitively through physical examination and conventional radiography. A fluoroscopically guided injection into the SIJ can be both ...diagnostic and therapeutic. The initial phase of treatment involves nonsurgical modalities such as activity modification, use of a sacroiliac (SI) belt, NSAIDs, and physical therapy. Prolotherapy and radiofrequency ablation may offer a potential benefit as therapeutic modalities, although limited data support their use as a primary treatment modality. Surgical treatment is indicated for patients with a positive response to an SI injection with >75% relief, failure of nonsurgical treatment, and continued or recurrent SIJ pain. Percutaneous SI arthrodesis may be recommended as a first-line surgical treatment because of its improved safety profile compared with open arthrodesis; however, in the case of revision surgery, nonunion, and aberrant anatomy, open arthrodesis should be performed.
A tough Jewish kid from the Bronx, Dan Altman enlisted in the Army when the U.S. entered World War II. Adapting street smarts to soldiering, he became a skilled sharpshooter and attained the rank of ...sergeant in the 1st Infantry Division.On D-Day, Altman's unit was among the second wave to the assault the German defenses at Normandy. Surviving the invasion, the fighting in the lethal hedgerow country and the Hurtgen Forest, and the Battle of the Bulge, he was later assigned to gather information on Nazi atrocities for the trials at Nuremburg. Beginning with his plunge into the blood-tinged surf at Omaha Beach, his candid, often graphic memoir is presented here as told to his granddaughter.
•Repairs of the superior ramus were tested in vitro with either screw or plate fixation.•Cyclic loading with an average head, arms and trunk load of 400 N followed by cylces with 800 N of axial ...loading were applied to the whole pelvis in single leg support.•Repair with a screw provided lower translational and angular displacements across the rami fractures than plate fixation with the angular displacement statistically significantly different.
Management of the anterior component of unstable lateral compression (LC) pelvic ring injuries remains controversial. Common internal fixation options include plating and superior pubic ramus screws. These constructs have been evaluated in anterior-posterior compression (APC) fracture patterns, but no study has compared the two for unstable LC patterns, which is the purpose of this study.
A rotationally unstable LC pelvic ring injury was modeled in 10 fresh frozen cadaver specimens by creating a complete sacral fracture, disruption of posterior ligaments, and ipsilateral superior and inferior rami osteotomies. All specimens were repaired posteriorly with two fully threaded 7 mm cannulated transiliac-transsacral screws through the S1 and S2 corridors. The superior ramus was repaired with either a 3.5 mm pelvic reconstruction plate (n = 5) or a bicortical 5.5 mm cannulated retrograde superior ramus screw (n = 5). Specimens were loaded axially in single leg support for 1000 cycles at 400 N followed by an additional 3 cycles at 800 N. Displacement and angulation of the superior and inferior rami osteotomies were measured with a three-dimensional (3D) motion tracker. The two fixation methods were then compared with Mann-Whitney U-Tests.
Retrograde superior ramus screw fixation had lower average displacement and angulation than plate fixation in all categories, with the motion at the inferior ramus at 800 N of loading showing a statistically significant difference in angulation.
Although management of the anterior ring in unstable LC injuries remains controversial, indications for fixation are becoming more defined over time. In this study, the 5.5 mm cannulated retrograde superior ramus screw significantly outperformed the 3.5 mm reconstruction plate in angulation of the inferior ramus fracture at 800 N. No other significance was found, however the ramus screw demonstrated lower average displacements and angulations in all categories for both the inferior and superior ramus fractures.
Because of the established path of bacterial entry and contamination-associated mechanisms, grade 3 open orthopedic fractures represent a substantial infection risk. The Eastern Association for the ...Surgery of Trauma (EAST) guidelines recommended covering
and adding aminoglycoside gram-negative coverage. Local institutional guidelines rely on ceftriaxone for gram negative coverage and add methicillin-resistant
coverage with vancomycin.
The electronic health records of adults admitted for a grade 3 open fracture between January 1, 2016, and October 31, 2021, were retrospectively reviewed. Patients who received cefazolin and gentamicin (CZ+GM) or ceftriaxone and vancomycin (CRO+VA) as prophylaxis were included. We recorded the rate of a composite treatment failure outcome of receipt of antibiotic agents, infection-related hospitalization, or subsequent debridement for injury-site skin and soft tissue infection or osteomyelitis. The presence of acute kidney injury (AKI) was also evaluated.
There were 65 patients included in the CZ+GM group and 53 patients in the CRO+VA group. Patients in the CZ+GM group were younger (mean 42.6 compared with 50.6 years; p = 0.02). Otherwise, there were no significant differences between groups' demographics, mechanism and site of injury, timeline of care, or surgical interventions. More patients in the CZ+GM arm met the composite treatment failure outcome, but it was not statistically significant (45% vs. 32%; p = 0.2). There were similar rates of treatment failure at 30 days (21% vs. 26%; p = 0.5) and for only osteomyelitis (8% vs. 9%; p = 1).
The trend in numerically lower treatment failure rates in the CRO+VA group across outcomes provides sufficient evidence to continue the current local recommendations. Given our sample size, type 2 error may have occurred, and studies with greater power should analyze this question.
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