Singular structure of Toda lattices and cohomology of certain compact Lie groups

We study the singularities (blow-ups) of the Toda lattice associated with a real split semisimple Lie algebra $\mathfrak g$. It turns out that the total number of blow-up points along trajectories of the Toda lattice is given by the number of points of a Chevalley group $K({\mathbb F}_q)$ related to the maximal compact subgroup $K$ of the group $\check G$ with $\check{\mathfrak g}={\rm Lie}(\check G)$ over the finite field ${\mathbb F}_q$. Here $\check{\mathfrak g}$ is the Langlands dual of ${\mathfrak g}$. The blow-ups of the Toda lattice are given by the zero set of the $\tau$-functions. For example, the blow-ups of the Toda lattice of A-type are determined by the zeros of the Schur polynomials associated with rectangular Young diagrams. Those Schur polynomials are the $\tau$-functions for the nilpotent Toda lattices. Then we conjecture that the number of blow-ups is also given by the number of real roots of those Schur polynomials for a specific variable. We also discuss the case of periodic Toda lattice in connection with the real cohomology of the flag manifold associated to an affine Kac-Moody algebra.Comment: 23 pages, 12 figures, To appear in the proceedings "Topics in Integrable Systems, Special Functions, Orthogonal Polynomials and Random Matrices: Special Volume, Journal of Computational and Applied Mathematics

The Kazhdan-Lusztig conjecture for W-algebras

The main result in this paper is the character formula for arbitrary irreducible highest weight modules of W algebras. The key ingredient is the functor provided by quantum Hamiltonian reduction, that constructs the W algebras from affine Kac-Moody algebras and in a similar fashion W modules from KM modules. Assuming certain properties of this functor, the W characters are subsequently derived from the Kazhdan-Lusztig conjecture for KM algebras. The result can be formulated in terms of a double coset of the Weyl group of the KM algebra: the Hasse diagrams give the embedding diagrams of the Verma modules and the Kazhdan-Lusztig polynomials give the multiplicities in the characters.Comment: uuencoded file, 29 pages latex, 5 figure

Surgical treatment of varicose veins caused by anterior accessory saphenous vein incompetence: long-term results of preservation of the sapheno-femoral junction

Scopul lucrării. Evaluarea rezultatelor la distanță ale tratamentului chirurgical cu prezervarea joncțiunii safeno-femurale (JSF) și a v.safena magna (VSM) la bolnavii cu maladie varicoasă cauzată de incompetența v.safene accesorii anterioare (VSAA). Materiale și metode. Cercetarea a inclus 54 pacienți cu incompetența VSAA identificată în timpul ultrasonografiei Doppler-duplex, supuși intervențiilor pe venele superficiale cu prezervarea JSF/VSM. Toți bolnavii au fost examinați clinico-imagistic pre- și postoperator. Rezultate. Mediana vârstei: 48 (IQR 35-59) ani; femei – 45 (83,3%). S-au evidențiat 2 subgrupuri: I – cu reflux în VSAA și competența VSM (n=40; 74,1%) și II – cu reflux asociat, în VSAA și segmentul supraiacent de VSM (n=14; 25,9%). Distribuția cazurilor conform clasei clinice (CEAP) în subgrupul I: C2 – 20 (50%), C3 – 18 (45%), C4 – 2 (5%); iar în subgrupul II: C2 – 1 (7,2%), C3 – 10 (71,4%), C4 – 1 (7,2%), C6 – 2 (14,2%). Miniflebectomia izolată s-a practicat la 21 (52,5%) pacienți din subgrupul I și 7 (50%) – din subgrupul II; iar asocierea miniflebectomiei cu deconectarea VSAA de la confluiența cu VSM – în 19 (47,5%) și 7 (50%), respectiv. La termenul de follow-up de 5,3 (IQR 1-7,2) ani ameliorare clinică au raportat 90% bolnavi din subgrupul I și 78,5% – din subgrupul II. Diminuarea duratei sau dispariția postoperatorie a refluxului segmentar în VSM s-a evidențiat în 64,2% cazuri. Concluzii. La bolnavii cu maladie varicoasă cauzată de incompetența VSAA miniflebectomia cu prezervarea JSF/VSM, inclusiv în prezența refluxului safenian segmentar, oferă o rată înaltă de ameliorare clinică, iar în majoritatea cazurilor – și micșorarea duratei sau chiar dispariția refluxului pre-existent în VSM.Aim of study. To evaluate long-term results of surgical treatment with preservation of sapheno-femoral junction (SFJ) and great saphenous vein (GSV) in patients with varicose veins caused by incompetence of anterior accessory saphenous vein (AASV). Materials and methods. The research included 54 patients with AASV incompetence identified during Doppler-duplex ultrasound, undergoing superficial vein surgery with SFJ/GSV preservation. All patients underwent pre- and postoperatively clinical/imaging examinations. Results. Median value of age: 48 (IQR 35-59) years; females – 45 (83.3%). Two subgroups were identified: I – with AASV reflux, but competent GSV (n=40; 74.1%) and II – with associated reflux, in AASV and overlying segment of GSV (n=14; 25.9%). Distribution of cases according to clinical class (CEAP) in subgroup I: C2 – 20 (50%), C3 – 18 (45%), C4 – 2 (5%); and in subgroup II: C2 – 1 (7.2%), C3 – 10 (71.4%), C4 – 1 (7.2%), C6 – 2 (14.2%). Isolated miniphlebectomy was performed in 21 (52.5%) patients from subgroup I, and 7 (50%) – from subgroup II; while association of miniphlebectomy with disconnection of AASV from confluence with GSV – in 19 (47.5%) and 7 (50%), respectively. At 5.3 (IQR 1-7.2) years follow-up clinical improvement was reported by 90% (subgroup I) and 78.5% (subgroup II) patients. Postoperative decreasing of duration or disappearance of segmental GSV reflux was highlighted in 64.2% cases. Conclusions. In patients with varicose veins caused by AASV incompetence, miniphlebectomy with SFJ/GSV preservation, even in presence of segmental saphenous reflux, offers a high rate of clinical improvement, and in most cases – shortening the duration or disappearance of pre-existing GSV reflux

On a family of solutions of the KP equation which also satisfy the Toda lattice hierarchy

We describe the interaction pattern in the $x$-$y$ plane for a family of soliton solutions of the Kadomtsev-Petviashvili (KP) equation, $(-4u_{t}+u_{xxx}+6uu_x)_{x}+3u_{yy}=0$. Those solutions also satisfy the finite Toda lattice hierarchy. We determine completely their asymptotic patterns for $y\to \pm\infty$, and we show that all the solutions (except the one-soliton solution) are of {\it resonant} type, consisting of arbitrary numbers of line solitons in both aymptotics; that is, arbitrary $N_-$ incoming solitons for $y\to -\infty$ interact to form arbitrary $N_+$ outgoing solitons for $y\to\infty$. We also discuss the interaction process of those solitons, and show that the resonant interaction creates a {\it web-like} structure having $(N_--1)(N_+-1)$ holes.Comment: 18 pages, 16 figures, submitted to JPA; Math. Ge

The Kazhdan-Lusztig conjecture for finite W-algebras

We study the representation theory of finite W-algebras. After introducing parabolic subalgebras to describe the structure of W-algebras, we define the Verma modules and give a conjecture for the Kac determinant. This allows us to find the completely degenerate representations of the finite W-algebras. To extract the irreducible representations we analyse the structure of singular and subsingular vectors, and find that for W-algebras, in general the maximal submodule of a Verma module is not generated by singular vectors only. Surprisingly, the role of the (sub)singular vectors can be encapsulated in terms of a `dual' analogue of the Kazhdan-Lusztig theorem for simple Lie algebras. These involve dual relative Kazhdan-Lusztig polynomials. We support our conjectures with some examples, and briefly discuss applications and the generalisation to infinite W-algebras.Comment: 11 page

Clinical outcomes of lower limbs revascularization using extra-anatomic bypasses

Scopul studiului. Evaluarea rezultatelor revascularizării extremităților inferioare prin bypass-uri cu traseul grefei diferit de cel al segmentului arterial nativ ocolit – extra-anatomice (BEA). Materiale și metode. S-au analizat datele medicale ale bolnavilor operați în Clinică prin BEA pe parcursul a 41 luni (ianuarie 2020 – mai 2023). Rezultate. Lotul a cuprins 50 pacienți, vârsta – 70 (25%-75%IQR 63-73) ani, bărbați – 40 (80%). Printre comorbidități au prevalat: hipertensiunea arterială (47/94%), insuficiența cardiacă (35/70%), boala coronariană (19/38%), diabetul zaharat (15/30%) și fibrilația atrială (14/28%). Intervenții de revascularizare în antecedente au suportat 24 (48%) pacienți, iar în 9 (18%) cazuri anterior s-au efectuat diverse amputații. BEA s-au realizat pentru ischemie cronică amenințătoare (37/74%), ischemie acută (8/16%), ocluzie postembolică (2/4%), traumă vasculară (1/2%), pseudoanevrism infectat (1/2%) și neoplasm cu implicarea vaselor magistrale (1/2%). Structura BEA: crossover femuro-femural (13/26%), femuro-tibial (13/26%), crossover ilio-femural (12/24%), ilio-femural transobturator (3/6%), profundo-tibial (3/6%), axilo-femural (2/4%), crossover ilio-femural/tibial secvențial (1/2%), ilio-femural trans-aripă iliacă (1/2%), popliteo-tibial anterior (1/2%) și femuro-popliteu proximal lateral (1/2%). Drept conduite s-au utilizat grefe sintetice (28/56%) sau autologe (22/44%). În 4 (8%) cazuri BEA s-au asociat cu angioplastii percutanate transluminale (intervenții hibrid). Rata amputațiilor majore și decesului la 30 zile postoperator – 4% și, respectiv, 12%. Pe parcursul evaluării timp de 14 (25-75%IQR 8-22) luni intervenții arteriale adiționale au necesitat 5 (10%) bolnavi; rata cumulativă de supraviețuire fără amputații constituid 82% (41/50). Concluzii. Revascularizarea extremităților inferioare prin BEA reprezintă o opțiune curativă fiabilă, asociată cu rezultate clinice satisfăcătoare chiar și la bolnavii cu patologii asociate multiple și/sau operații arteriale anterioare eșuate.Aim of study. Assessment of outcomes of lower limbs revascularization using bypasses with a graft route different from that of the bypassed native arterial segment – extra-anatomic (EAB). Materials and methods. The medical data of patients operated on using EAB during 41 months (January 2020 – May 2023) were analyzed. Results. The studied group included 50 patients, age – 70 (25%-75%IQR 63-73) years, males – 40 (80%). Among comorbidities prevailed: arterial hypertension (47/94%), heart failure (35/70%), coronary artery disease (19/38%), diabetes (15/30%) and atrial fibrillation (14/28%). Previous revascularization of affected extremity underwent 24 (48%) patients; in 9 (18%) cases various amputations were performed in past. EAB were applied for chronic limb-threatening ischemia (37/74%), acute ischemia (8/16%), postembolic occlusion (2/4%), vascular trauma (1/2%), infected pseudoaneurysm (1/2%) and neoplasm involving magistral vessels (1/2%). Types of EAB: femoro-femoral crossover (13/26%), femoro-tibial (13/26%), ilio-femoral crossover (12/24%), transobturator ilio-femoral (3/6%), profundo-tibial (3/6%), axillo-femoral (2/4%), sequential ilio-femoral/tibial crossover (1/2%), trans-iliac wing iliofemoral (1/2%), popliteo-anterior tibial (1/2%) and lateral above-knee femoro-popliteal (1/2%). Prosthetic (28/56%) or autologous (22/44%) grafts were used as conduits. In 4 (8%) patients EAB was associated with percutaneous transluminal angioplasties (hybrid interventions). The rate of major amputations and death at 30-day after surgery – 4% and 12%, respectively. During the 14 (25- 75%IQR 8-22) months follow-up additional arterial interventions were required in 5 (10%) cases; while the cumulative amputation-free survival rate was 82% (41/50). Conclusions. Lower extremity revascularization using EAB represents a reliable curative option associated with satisfactory clinical results even in patients with multiple comorbidities and/or failed previous arterial surgery

Current role of tibial artery bypass in the treatment of chronic limb threatening ischemia: a prospective study

Scopul lucrării. Deși intervențiile endovasculare reprezintă o abordare efectivă în tratamentul ischemiei cronice amenințătoare a membrelor (ICAM), unii pacienți în continuare necesită efectuarea bypass-ului chirurgical pentru salvarea extremităților. Scopul studiului a fost analiza indicațiilor, particularităților tehnice și rezultatelor bypass-urilor infrainghinale cu anastomoza distală mai jos de artera poplitee. Materiale și metode. În perioada 01.2020-04.2023 bypass-uri tibiale au fost efectuate la 44 pacienți: 84% bărbați, vârsta – 65,5±10,9 (23-86) ani. ICAM stadiul IV Fontaine a fost diagnosticată în 25 (56,8%) cazuri și 14 (31,8%) pacienți au suportat revascularizări ipsilaterale în trecut (8 – endovasculare, 6 – deschise). Conform datelor angiografiei toți bolnavii au avut stadiul III GLASS. Rezultate. Ca sursă de inflow a servit artera: femurală comună – la 14 pacienți, femurală superficială – la 22, femurală profundă – la 5, și poplitee – la 3. Anastomoza distală a fost aplicată pe: trunchiul tibioperoneal în 8 cazuri, artera tibială anterioară – în 19, artera tibială posterioară – în 12 și artera peronea – în 5. Vena safena magna reversată a fost utilizată în 39 (88,6%) cazuri (8 – contralaterală), grefa sintetică – în 3, vena in situ și vena cefalică – câte un caz. Durata medie a intervenției – 190 (120-345) minute. Indicile gleznăbraț a crescut de la 0,36±0,17 la 0,94±0,14 postoperator. Pe durata spitalizării 5 (11,3%) pacienți au dezvoltat ocluzia definitivă a bypass-ului, 4 (9%) au suportat amputație majoră și 2 (4,5%) au decedat. Supraviețuirea fără amputație la termen mediu de 12 luni a constituit 75%. Concluzii. La pacienții cu ICAM și imposibilitatea sau eșecul tratamentului endovascular bypass-urile tibiale oferă șanse acceptabile pentru salvarea extremităților.Aim of study. While endovascular interventions represent an effective approach for treatment of chronic limb threatening ischemia (CLTI) some patients still require surgical bypass for limb salvage. The aim of study was analysis of the indications, technical details and results of infrainguinal bypasses with distal anastomosis below the popliteal artery. Materials and methods. During 01.2020-04.2023 tibial bypasses were performed in 44 patients: 84% male, age 65.5±10.9 (range 23-86) years. CLTI stage IV Fontaine was diagnosed in 25 (56.8%) cases and 14 (31.8%) patients had failed previous ipsilateral revascularizations (8 – endovascular, 6 – open). Based on angiography results, all patients were classified as GLASS stage III. Results. The inflow artery was: common femoral – in 14 patients, superficial femoral – in 22, deep femoral – in 5, and popliteal – in 3. Distal anastomosis was constructed with tibioperoneal trunk in 8 cases, anterior tibial artery – in 19, posterior tibial artery – in 12 and peroneal artery – in 5. Reversed great saphenous vein was used as a conduit in 39 (88.6%) cases (8 – contralateral), synthetic graft – in 3, vein in situ – in 1 and cephalic vein – in 1. Duration of surgery was 190 (120-345) min. Ankle-brachial index increased from 0.36±0.17 to 0.94±0.14 postoperatively. During hospitalisation 5 (11.3%) patients developed definitive bypass occlusion, 4 (9%) – suffered major amputation and 2 (4.5%) died. Amputation-free survival at median follow-up of 12 months was 75%. Conclusions. In patients with CLTI and impossibility or failure of endovascular treatment tibial bypasses offer acceptable chances for limb salvage

Combination therapy with rituximab and cyclophosphamide in the treatment of anti-neutrophil cytoplasmic antibodies (ANCA) positive pulmonary hemorrhage: case report

Anti-neutrophil cytoplasmic antibody (ANCA)-associated vasculitis (AAV) with pulmonary hemorrhage is rare in childhood. Standard treatment includes corticosteroids and cyclophosphamide (CYC), which is associated with a high level of toxicity. We report a white female with ANCA positive pulmonary hemorrhage who was treated with cyclophosphamide (CYC) and rituximab (RTX) combination therapy