Circuit complexity in interacting QFTs and RG flows

We consider circuit complexity in certain interacting scalar quantum field theories, mainly focusing on the $\phi^4$ theory. We work out the circuit complexity for evolving from a nearly Gaussian unentangled reference state to the entangled ground state of the theory. Our approach uses Nielsen's geometric method, which translates into working out the geodesic equation arising from a certain cost functional. We present a general method, making use of integral transforms, to do the required lattice sums analytically and give explicit expressions for the $d=2,3$ cases. Our method enables a study of circuit complexity in the epsilon expansion for the Wilson-Fisher fixed point. We find that with increasing dimensionality the circuit depth increases in the presence of the $\phi^4$ interaction eventually causing the perturbative calculation to breakdown. We discuss how circuit complexity relates with the renormalization group.Comment: 50 pages, 2 figures; references updated; version to appear in JHE

Biomimetic route to hybrid nano-Composite scaffold for tissue engineering

Hydroxyapatite-poly(vinyl) alcohol-protein composites have been prepared by a biomimetic route at ambient conditions, aged for a fortnight at 30±2°C and given a shape in the form of blocks by thermal cycling. The structural characterizations reveal a good control over the morphology mainly the size and shape of the particles. Initial mechanical studies are very encouraging. Three biocompatibility tests, i.e., hemocompatibility, cell adhesion, and toxicity have been done from Shree Chitra Tirunal, Trivandrum and the results qualify their standards. Samples are being sent for more biocompatibility tests. Optimization of the blocks in terms of hydroxyapatite and polymer composition w.r.t the applications and its affect on the mechanical strength have been initiated. Rapid prototyping and a β-tricalcium – hydroxyapatite combination in composites are in the offing

Biomimetic patterning of polymer hydrogels with hydroxyapatite nanoparticles

We report here an in situ process to produce nano-composite polymer hydrogels having surfaces patterned with hydroxyapatite (HA) nanoparticles (100 nm). Poly (vinyl alcohol) (PVA) has been used as a hydrogel forming medium. A three step process, comprising precipitation of HA nanoparticles in presence of PVA molecules and freeze thawing of obtained PVA-HA emulsion, followed by critical point drying, has been devised to produce three dimensional nanocomposite hydrogels. Interaction of Ca2+ with oxygen atoms of PVA and the hydrogen bonding characteristic of the polymer have been exploited to have controlled size distribution of HA in a continuous and macroporous network of PVA. A systematic variation in the polymer concentration could be correlated with microstructural features of the hydrogel

Cores of partitions in rectangles

For a positive integer $t \geq 2$, the $t$-core of a partition plays an important role in modular representation theory and combinatorics. We initiate the study of $t$-cores of partitions contained in an $s \times r$ rectangle. Our main results are as follows. We first give a simple formula for the number of partitions in the rectangle which are themselves $t$-cores and compute its asymptotics for large $r,s$. We then prove that the number of partitions inside the rectangle whose $t$-cores are a fixed partition $\rho$ is given by a product of binomial coefficients. Finally, we use this formula to compute the distribution of the $t$-core of a uniformly random partition inside the rectangle extending our previous work on all partitions of a fixed integer $n$ (Ann. Appl. Prob. 2023). In particular, we show that in the limit as $r,s \to \infty$ maintaining a fixed aspect ratio, we again obtain a Gamma distribution with the same shape parameter $\alpha = (t-1)/2$ and scale parameter $\beta$ that depends on the aspect ratio.Comment: 16 pages, 1 figure, improved exposition, references adde

The size of tt-cores and hook lengths of random cells in random partitions

Fix $t \geq 2$. We first give an asymptotic formula for certain sums of the number of $t$-cores. We then use this result to compute the distribution of the size of the $t$-core of a uniformly random partition of an integer $n$. We show that this converges weakly to a gamma distribution after dividing by $\sqrt{n}$. As a consequence, we find that the size of the $t$-core is of the order of $\sqrt{n}$ in expectation. We then apply this result to show that the probability that $t$ divides the hook length of a uniformly random cell in a uniformly random partition equals $1/t$ in the limit. Finally, we extend this result to all modulo classes of $t$ using abacus representations for cores and quotients.Comment: 28 pages, 3 figures, significant revisions. Several minor errors fixed and results stated in a more concise manner. From v1, Sections 2.4 and 5.2 deleted and Corollary 5.6 is stated as Lemma 5.16 in this version, thanks to a suggestion of D. Grinber

On the fractal nature of Penrose tiling

An earliest preoccupation of man has been to find ways of partitioning infinite space into regions having a finite number of distinct shapes and yielding beautiful patterns called tiling. Archaeological edifices, everyday objects of use like baskets, carpets, textiles, etc. and many biological systems such as beehives, onion peels and spider webs also exhibit a variety of tiling. Escher’s classical paintings have not only given a new dimension to the artistic value of tiling but also aroused the curiosity of mathematicians. The generation of aperiodic tiling with five-fold rotational symmetry by Penrose in 1974 and the more recent production of decorated pentagonal tiles by Rosemary Grazebrook have heightened the interest in the subject among artists, engineers, biologists, crystall ographers and mathematicians1–5. In spite of its long history, the subject of tiling is still evolving. In this communication, we propose a novel algorithm for the growth of a Penrose tiling and relate it to the equally fascinating subject of fractal geometry pioneered by Mandelbrot6. The algorithm resembles those for generation of fractal objects such as Koch’s recursion curve, Peano curve, etc. and enables consideration of the tiling as cluster growth as well. Thus it clearly demonstrates the dual nature of a Penrose tiling as a natural and a nonrandom fractal

Open tibial fracture with severe soft tissue injury and bone loss managed with ipsilateral fibular transport and its complications: a case report

Massive segmental bone defects of tibia present as a challenging task to manage specially when associated with extensive soft tissue injury. A 30 year old male presented to Paras HMRI hospital, Patna, post road traffic accident with Gustilo Anderson 3B comminuted open tibia shaft fracture and with an external fixator in situ with a grossly inflamed and infected wound. Initially patient was managed with serial wound debridement and skin grafting was done early to obtain adequate soft tissue coverage. The patient then underwent application of Ilizarov external fixator with plan of one level fibular osteotomy for ipsilateral fibular transport. With good outcome of the procedure clinically and radiologically, Ilizarov fixator was removed after time duration of about 1.3 years and limb was immobilized in plaster of Paris (POP) cast which was removed after 8 weeks. Within 1 month of removal of POP cast the patient presented to hospital again with complaints of pain and instability when his leg was run over by his child’s bicycle while playing. Diagnosed as fracture of proximal (transported) fibula he was managed then with locking plates; one of which was used as an internal fixator and the other as external fixator which was outside the body and acted as a support to the operated limb. After about 1 year the external locking plate was removed and patient was able to bear weight on his extremities. Despite various modalities to treat massive tibial gap, fibular transport procedure with Ilizarov external fixator seems to be the most viable option

Synthesis of organized inorganic crystal assemblies

Organized crystalline assemblies of cobalt salt and g-iron oxide have been produced by in situ matrixmediated biomimetic route. The process makes use of an organized supramolecular matrix and produces cobalt chloride crystals with characteristic morphology of coccolith of alga and nacreous structure of Pinctada martensii. Crystals of g-iron oxide have been produced with typical morphology of aragonite spherulites in regenerated shell of Pomaceae paludosa

Lymphoma of frontotemporal region with massive bone destruction and intracranial and intraorbital extension

Primary non-Hodgkin\u2032s lymphoma with unilateral proptosis and diffuse involvement of the cra\uacnial vault and brain parenchyma is extremely rare. A 50-year-old woman developed a progressively increasing proptosis of her right eye, associated with a subcutaneous mass over the right frontotemporal region over the last 5 months. CT scan showed a high-density contrast-enhancing lesion with wide involvement of the cranium and intracranial and intraorbital extension. We performed a wedge biopsy for further analysis. Histological examination revealed that the tumor was non-Hodgkin\u2032s lymphoma. There was no evidence of systemic involvement. The patient received radiotherapy and was doing well at 18 months\u2032 follow-up. Primary malignant lymphoma involving the orbit and cranial vault is a rare malignancy, and treatment remains to be defined