Upper tails for triangles

With $\xi$ the number of triangles in the usual (Erd\H{o}s-R\'enyi) random graph $G(m,p)$, $p>1/m$ and $\eta>0$, we show (for some $C_{\eta}>0$) \Pr(\xi> (1+\eta)\E \xi) < \exp[-C_{\eta}\min{m^2p^2\log(1/p),m^3p^3}]. This is tight up to the value of $C_{\eta}$.Comment: 10 page

Upper Tails for Cliques

With $\xi_{k}=\xi_{k}^{n,p}$ the number of copies of $K_k$ in the usual (Erd\H{o}s-R\'enyi) random graph $G(n,p)$, $p\geq n^{-2/(k-1)}$ and $\eta>0$, we show when $k>1$ \Pr(\xi_k> (1+\eta)\E \xi_k) < \exp [-\gO_{\eta,k} \min\{n^2p^{k-1}\log(1/p), n^kp^{\binom{k}{2}}\}]. This is tight up to the value of the constant in the exponent.Comment: 25 page

Evaluation of Hemodynamic Parameters as Predictors of Glaucoma Progression

Purpose. To evaluate hemodynamic parameters as possible predictors for glaucoma progression. Methods. An 18-month randomized double-masked cohort study including 30 open-angle glaucoma patients receiving fixed-combination treatment with Dorzolamide/Timolol (DTFC) or Latanoprost/Timolol (LTFC) (n = 15 per group) was performed. Intraocular pressure (IOP), arterial blood pressure (BP), ocular and diastolic perfusion pressures (OPP, DPP), color Doppler imaging, pulsatile ocular blood flow analysis, scanning laser polarimetry, and Humphrey visual field evaluations were included. Results. Both treatments showed statistically similar IOP reduction. Six patients in DTFC and 7 in LTFC group met glaucoma progression criteria. DTFC group had higher OPP, DPP, and lower vascular resistivity indices as compared to the LTFC. Progressing patients had higher nerve fiber index, lower systolic BP, OPP, DPP, higher ophthalmic and central retinal artery vascular resistance, and lower pulse volume (P < .05; t-test). Conclusions. Structural changes consistent with glaucoma progression correlate with non-IOP-dependent risk factors

Dysfunctional regulation of ocular blood flow: A risk factor for glaucoma?

Primary open angle glaucoma (OAG) is a multifactorial optic neuropathy characterized by progressive retinal ganglion cell death and associated visual field loss. OAG is an emerging disease with increasing costs and negative outcomes, yet its fundamental pathophysiology remains largely undetermined. A major treatable risk factor for glaucoma is elevated intraocular pressure (IOP). Despite the medical lowering of IOP, however, some glaucoma patients continue to experience disease progression and subsequent irreversible vision loss. The scientific community continues to accrue evidence suggesting that alterations in ocular blood flow play a prominent role in OAG disease processes. This article develops the thesis that dysfunctional regulation of ocular blood flow may contribute to glaucomatous optic neuropathy. Evidence suggests that impaired vascular autoregulation renders the optic nerve head susceptible to decreases in ocular perfusion pressure, increases in IOP, and/or increased local metabolic demands. Ischemic damage, which likely contributes to further impairment in autoregulation, results in changes to the optic nerve head consistent with glaucoma. Included in this review are discussions of conditions thought to contribute to vascular regulatory dysfunction in OAG, including atherosclerosis, vasospasm, and endothelial dysfunction

Factors Associated With Retinal Vessel Diameters in an Elderly Population: the Thessaloniki Eye Study

Purpose: To identify the factors associated with retinal vessel diameters in the population of the Thessaloniki Eye Study. Methods: Cross-sectional population-based study (age ≥ 60 years). Subjects with glaucoma, late age-related macular degeneration, and diabetic retinopathy were excluded from the analyses. Retinal vessel diameters were measured using the IVAN software, and measurements were summarized to central retinal artery equivalent (CRAE), central retinal vein equivalent (CRVE), and arteriole to venule ratio (AVR). Results: The analysis included 1614 subjects. The hypertensive group showed lower values of CRAE (P = 0.033) and AVR (P = 0.0351) compared to the normal blood pressure (BP) group. On the contrary, the group having normal BP under antihypertensive treatment did not have different values compared to the normal BP group. Diastolic BP (per mm Hg) was negatively associated with CRAE (P < 0.0001) and AVR (P < 0.0001), while systolic BP (per mm Hg) was positively associated with CRAE (P = 0.001) and AVR (P = 0.0096). Other factors significantly associated included age, sex, alcohol, smoking, cardiovascular disease history, ophthalmic medication, weight, and IOP; differences were observed in a stratified analysis based on BP medication use. Conclusions: Our study confirms previous reports about the association of age and BP with vessel diameters. The negative correlation between BP and CRAE seems to be guided by the effect of diastolic BP as higher systolic BP is independently associated with higher values of CRAE. The association of BP status with retinal vessel diameters is determined by diastolic BP status in our population. Multiple other factors are also independently associated with retinal vessel diameters

Update in intracranial pressure evaluation methods and translaminar pressure gradient role in glaucoma

Glaucoma is one of the leading causes of blindness worldwide. Historically, it has been considered an ocular disease primary caused by pathological intraocular pressure (IOP). Recently, researchers have emphasized intracranial pressure (ICP), as translaminar counter pressure against IOP may play a role in glaucoma development and progression. It remains controversial what is the best way to measure ICP in glaucoma. Currently, the ‘gold standard’ for ICP measurement is invasive measurement of the pressure in the cerebrospinal fluid via lumbar puncture or via implantation of the pressure sensor into the brains ventricle. However, the direct measurements of ICP are not without risk due to its invasiveness and potential risk of intracranial haemorrhage and infection. Therefore, invasive ICP measurements are prohibitive due to safety needs, especially in glaucoma patients. Several approaches have been proposed to estimate ICP non-invasively, including transcranial Doppler ultrasonography, tympanic membrane displacement, ophthalmodynamometry, measurement of optic nerve sheath diameter and two-depth transcranial Doppler technology. Special emphasis is put on the two-depth transcranial Doppler technology, which uses an ophthalmic artery as a natural ICP sensor. It is the only method which accurately and precisely measures absolute ICP values and may provide valuable information in glaucoma