Electrical Networks, Lagrangian Grassmannians and Symplectic Groups

We refine the result of T. Lam \cite{L} on embedding the space $E_n$ of electrical networks on a planar graph with $n$ boundary points into the totally non-negative Grassmannian $\mathrm{Gr}_{\geq 0}(n-1,2n)$ by proving first that the image lands in $\mathrm{Gr}(n-1,V)\subset \mathrm{Gr}(n-1,2n)$ where $V\subset \mathbb{R}^{2n}$ is a certain subspace of dimension $2n-2$. The role of this reduction in the dimension of the ambient space is crucial for us. We show next that the image lands in fact inside the Lagrangian Grassmannian $\mathrm{LG}(n-1,V)\subset \mathrm{Gr}(n-1,V)$. As it is well known $\mathrm{LG}(n-1)$ can be identified with $\mathrm{Gr}(n-1,2n-2)\cap \mathbb{P} L$ where $L\subset \bigwedge^{n-1}\mathbb R^{2n-2}$ is a subspace of dimension equal to the Catalan number $C_n$, moreover it is the space of the fundamental representation of the symplectic group $Sp(2n-2)$ which corresponds to the last vertex of the Dynkin diagram. We show further that the linear relations cutting the image of $E_n$ out of $\mathrm{Gr}(n-1,2n)$ found in \cite{L} define that space $L$. This connects the combinatorial description of $E_n$ discovered in \cite{L} and representation theory of the symplectic group.Comment: Journal version, minor correction

Ground State of 2D Quantum Gravity and Spectral Density of Random Matrices

We compute the exact spectral density of random matrices in the ground state of the quantum hamiltonian corresponding to the matrix model whose double scaling limit describes pure gravity in 2D. We show that the non-perturbative effects are very large and in certain cases dominate the semi-classical WKB contribution studied in the earlier literature. The physical observables in this model are the loop averages with respect to the spectral density. We compute their exact ground-state expectation values and show that they differ significantly from the values obtained in the WKB approximation. Unlike the alternative regularizations of the nonperturbative 2D quantum gravity, based on analytic continuation of the Painlev\'e transcendent, our solution shows no pathologies.Comment: 14 pages (LaTeX) + 4 postscript figures encoded through uufiles. PUPT-1354, TAUP-2013-9

Suburbanization Processes in Sofia: Demographic, Socio-Economic and Spatial Transformation of the Agglomeration Area

The paper studies the agglomeration area of Bulgaria’s capital, Sofia. The territorial scope of the metropolitan agglomeration encompasses 7 (LAU 1) municipalities including Sofia, Bozhurishte, Elin Pelin, Gorna Malina, Kostinbrod, Slivnitsa and Svoge, and comprises 146 (LAU 2) individual settlements. The total area of Sofia agglomeration covers almost 3,550 km² (3.2% of national territory), while it had 1,359,270 inhabitants (almost 21% of national population) according to the last population census (7 September 2021). During the whole period between 2001 and 2021, Sofia agglomeration has grown in population by more than 95,000 people (7.6%). Bozhurishte, Kostinbrod and Elin Pelin, along with Sofia municipality itself, are the municipalities with the best demographic indicators within the agglomeration area. According to the typology of Webb, type 6 agglomeration settlements (migratory and natural decrease, natural > migratory) are predominant during the first subperiod (2001-2011), while type 8 (migratory increase compensates for natural decrease) has become the leading type in the second subperiod (2011-2021). Despite the increasing number of type 8 settlements, however, the population decline of Sofia city alone (19,361 people) exceeds more than twice the combined population growth of all type 8 settlements (8,929 people), and as a result – the total population number of the agglomeration declined during the second subperiod. In 2021, the GDP in Sofia agglomeration represented 43% of the national economy, while the GDP per capita in Sofia (BGN 38,891) was more than double of the national average. A significant number of the industrial sites are located in the neighbouring municipalities that fall within the agglomeration, with leading economic activities in manufacturing, transport and logistics. The distribution of FDI, however, is uneven: some 97% of them concentrated in Sofia municipality alone. Sofia agglomeration is distinguished by a relatively stable and balanced labour market, characterized by high economic activity and low unemployment rate compared to other agglomerations in the country. Together with the demographic processes, the increasing soil sealing is another evidence of the spatial expansion of suburbanization. The largest share of new soil sealing was observed in Sofia municipality itself (approximately 370 ha), followed by Elin Pelin (105 ha), Bozhurishte (46 ha) and Kostinbrod (32 ha). Based on the intensity of new constructions and population change, the so-called zone of active inbuence of the agglomeration core has been outlined. The zone is made up of settlements meeting two conditions: they have a high intensity of new constructions and they are type 8 or type 7 according to the typology of Webb in the period between 2011 and 2021. During the considered period, the northern, eastern, southern and western peripheries of the agglomeration area developed each in its own speciac way, with dieerent new construction intensity, morphological structure and inter-settlement spaces. Suburbanization processes observed in Sofia are generally comparable to those of other post-socialist European capitals in terms of historical legacies, demographic transformation, spatial dynamics, and land use shifts.

Minimal Liouville gravity correlation numbers from Douglas string equation

We continue the study of (q, p) Minimal Liouville Gravity with the help of Douglas string equation. We generalize the results of [1,2], where Lee-Yang series (2, 2s + 1) was studied, to (3, 3s + p 0) Minimal Liouville Gravity, where p 0 = 1, 2. We demonstrate that there exist such coordinates \u3c4 m,n on the space of the perturbed Minimal Liouville Gravity theories, in which the partition function of the theory is determined by the Douglas string equation. The coordinates \u3c4 m,n are related in a non-linear fashion to the natural coupling constants \u3bb m,n of the perturbations of Minimal Lioville Gravity by the physical operators O m,n . We find this relation from the requirement that the correlation numbers in Minimal Liouville Gravity must satisfy the conformal and fusion selection rules. After fixing this relation we compute three- and four-point correlation numbers when they are not zero. The results are in agreement with the direct calculations in Minimal Liouville Gravity available in the literature [3-5]. \ua9 2014 The Author(s)

Report of three novel germline CYLD mutations in unrelated patients with Brooke-Spiegler syndrome, including classic phenotype, multiple familial trichoepitheliomas and malignant transformation

Brooke-Spiegler syndrome is a rare autosomal-dominant genetic disorder characterized by multiple adnexal tumors, including cylindromas, spiradenomas, spiradenocylindromas and trichoepitheliomas. It is caused by germline CYLD mutations commonly leading to a premature stop codon. We here report on 3 novel CYLD mutations in 3 unrelated BSS patients, including the classic phenotype, multiple familial trichoepitheliomas phenotype and malignant transformation. These included c.1821_1826+1delinsCT/L607Ffs*9, c.2666A>T/p.D889V and c.2712delT/p.905Kfs*8. By extending the spectrum of CYLD mutations, better understanding of the molecular mechanisms of BSS can be gained, which might later assist in finding new treatment options

Fiducial marker placement with electromagnetic navigation bronchoscopy: a subgroup analysis of the prospective, multicenter NAVIGATE study

Fiducial markers (FMs) help direct stereotactic body radiation therapy (SBRT) and localization for surgical resection in lung cancer management. We report the safety, accuracy, and practice patterns of FM placement utilizing electromagnetic navigation bronchoscopy (ENB). Methods: NAVIGATE is a global, prospective, multicenter, observational cohort study of ENB using the superDimension™ navigation system. This prospectively collected subgroup analysis presents the patient demographics, procedural characteristics, and 1-month outcomes in patients undergoing ENB-guided FM placement. Follow up through 24 months is ongoing. Results: Two-hundred fifty-eight patients from 21 centers in the United States were included. General anesthesia was used in 68.2%. Lesion location was confirmed by radial endobronchial ultrasound in 34.5% of procedures. The median ENB procedure time was 31.0 min. Concurrent lung lesion biopsy was conducted in 82.6% (213/258) of patients. A mean of 2.2 ± 1.7 FMs (median 1.0 FMs) were placed per patient and 99.2% were accurately positioned based on subjective operator assessment. Follow-up imaging showed that 94.1% (239/254) of markers remained in place. The procedure-related pneumothorax rate was 5.4% (14/258) overall and 3.1% (8/258) grade ⩾ 2 based on the Common Terminology Criteria for Adverse Events scale. The procedure-related grade ⩾ 4 respiratory failure rate was 1.6% (4/258). There were no bronchopulmonary hemorrhages. Conclusion: ENB is an accurate and versatile tool to place FMs for SBRT and localization for surgical resection with low complication rates. The ability to perform a biopsy safely in the same procedure can also increase efficiency. The impact of practice pattern variations on therapeutic effectiveness requires further study