A conformal Yamabe problem with potential on the Euclidean space

We consider, in the Euclidean setting, a conformal Yamabe-type equation related to a potential generalization of the classical constant scalar curvature problem and which naturally arises in the study of Ricci solitons structures. We prove existence and nonexistence results, focusing on the radial case, under some general hypothesis on the potential.

Secrets, puzzles and mysteries of macroamylasemia

Physiological features of amylase synthesis and excretion are considered in the article, presence of other sources of amylase synthesis different from pancreas and salivary glands is emphasized. Definitions of hyperenzymemia and macroamylasemia (MAE) are given. MAE is a state characterized by presence of circulating complexes of normal serum amylase with protein or carbohydrates in blood. There are 3 types of MAE: first — classical (constant hyperamylasemia, decreased amylase level in urine, high blood concentration of macroamylase complexes); second — hyperamylasemia with slightly decreased amylase activity in urine, macroamylase/normal amylase ratio is less than in the first type; third — normal blood and urine amylase activity, low macroamylase/normal amylase ratio. Pathogenesis is explained by connection of blood amylase and acute phase protein in different inflammatory, infectious diseases, malabsorption. MAE clinical manifestations could be absent, sometimes abdominal pain is possible. Hyperamylasemia and reduced urine amylase activity are typical. MAE diagnostics means determination of macroamylase complexes in blood (chromatography, calculation of the clearance ratio of amylase and creatinine). The article presents clinical cases describing extra-pancreatic MAE in women with malignant ovarian lesions. The question of expediency of thorough diagnostic examination in asymptomatic MAE is raised, which may turn out to be a symptom of cancer. The lack of specific treatment for MAE is emphasized.

Analysis of one-dimensional models for exchange flows under strong stratification

One-dimensional models of exchange flows driven by horizontal density gradients are well known for performing poorly in situations with weak turbulent mixing. The main issue with these models is that the horizontal density gradient is usually imposed as a constant, leading to non-physically high stratification known as runaway stratification. Here, we propose two new parametrizations of the horizontal density gradient leading to one-dimensional models able to tackle strongly stratified exchange flows at high and low Schmidt number values. The models are extensively tested against results from laminar two-dimensional simulations and are shown to outperform the models using the classical constant parametrization for the horizontal density gradients. Four different flow regimes are found by exploring the parameter space defined by the gravitational Reynolds number Reg, the Schmidt number Sc, and the aspect ratio of the channel Γ. For small values of RegΓ, when diffusion dominates, all models perform well. However, as RegΓ increases, two clearly distinct regimes emerge depending on the Sc value, with an equally clear distinction of the performance of the one-dimensional models.

Option pricing in a subdiffusive constant elasticity of variance (CEV) model

In this paper, we extend the classical constant elasticity of variance (CEV) model to a subdiffusive CEV model, where the underlying CEV process is time changed by an inverse [Formula: see text]-stable subordinator. The new model can capture the subdiffusive characteristics of financial markets. We find the corresponding fractional Fokker–Planck equation governing the PDF of the new process. We also derive the analytical formula for option prices in terms of eigenfunction expansion. This method avoids the evaluation of PDF of an inverse [Formula: see text]-stable variable and also eliminates the need for numerical integration to calculate the option prices. We numerically investigate the sensitivities of the option prices to the key parameters of the newly developed model.

Debates with Small Transparent Quantum Verifiers

We study a model where two opposing provers debate over the membership status of a given string in a language, trying to convince a weak verifier whose coins are visible to all. We show that the incorporation of just two qubits to an otherwise classical constant-space verifier raises the class of debatable languages from at most NP to the collection of all Turing-decidable languages (recursive languages). When the verifier is further constrained to make the correct decision with probability 1, the corresponding class goes up from the regular languages up to at least E. We also show that the quantum model outperforms its classical counterpart when restricted to run in polynomial time, and demonstrate some non-context-free languages which have such short debates with quantum verifiers.

EXACT SOLUTIONS OF THE EINSTEIN–MAXWELL EQUATIONS IN CHARGED PERFECT FLUID SPHERES FOR THE GENERALIZED TOLMAN–OPPENHEIMER–VOLKOFF EQUATIONS

Exact solutions of the Einstein–Maxwell equations for spherically symmetric charged perfect fluid have been broadly studied so far. However, the cases with a nonzero cosmological constant are seldom focused. In the present paper, the Tolman–Oppenheimer–Volkoff (TOV) equations have been generalized from the neutral case of hydrostatic equilibrium to the charged case of hydroelectrostatic equilibrium, and base on it, for the first time we find a series of new exact solutions of Einstein–Maxwell's equations with a nonzero cosmological constant for static charged perfect fluid spheres. Moreover, two special TOV equations and two classical constant density interior solutions are also given.

Quantum expanders from any classical Cayley graph expander

We give a simple recipe for translating walks on Cayley graphs of a group G into a quantum operation on any irrep of G. Most properties of the classical walk carry over to the quantum operation: degree becomes the number of Kraus operators, the spectral gap becomes the gap of the quantum operation (viewed as a linear map on density matrices), and the quantum operation is efficient whenever the classical walk and the quantum Fourier transform on G are efficient. This means that using classical constant-degree constant-gap families of Cayley expander graphs on e.g. the symmetric group, we can construct efficient families of quantum expanders.