On a Singular Value Problem for the Fractional Laplacian on the Exterior of the Unit Ball

2000 Mathematics Subject Classification: Primary 26A33; Secondary 47G20, 31B05

Sobolev-Morrey Type Inequality for Riesz Potentials, Associated with the Laplace-Bessel Differential Operator

2000 Mathematics Subject Classification: 42B20, 42B25, 42B35

On Limiting Case of the Stein-Weiss Type Inequality for the B-Riesz Potentials

Mathematics Subject Classification: Primary 42B20, 42B25, 42B35

Singular Solutions with Exponential Growth for the (3+1)-D Wave Equation

Недю И. Попиванов, Тодор П. Попов, Рудолф Шерер - Разглеждат се четиримерни гранични задачи за нехомогенното вълново уравнение. Те са предложени от М. Протер като многомерни аналози на задачата на Дарбу в равнината. Известно е, че единственото обобщено решение може да има силна степенна особеност само в една гранична точка. Тази сингулярност е изолирана във върха на характеристичния конус и не се разпространява по конуса. Друг аспект на проблема е, че задачата не е фредхолмова, тъй като има безкрайномерно коядро. Предишни резултати сочат, че решението може да има най-много експоненциален ръст, но оставят открит въпроса дали наистина съществуват такива решения. Показваме, че отговора на този въпрос е положителен и строим обобщено решение на задачата на Протер с експоноциална особеност.

Singular Solutions of Protter’s Problem for a Class of 3-D Hyperbolic Equations

Недю Иванов Попиванов, Алексей Йорданов Николов - През 1952 г. М. Протър формулира нови гранични задачи за вълновото уравнение, които са тримерни аналози на задачите на Дарбу в равнината. Задачите са разгледани в тримерна област, ограничена от две характеристични конуса и равнина. Сега, след като са минали повече от 50 години, е добре известно, че за безброй гладки функции в дясната страна на уравнението тези задачи нямат класически решения, а обобщеното решение има силна степенна особеност във върха на характеристичния конус, която е изолирана и не се разпространява по конуса. Тук ние разглеждаме трета гранична задача за вълновото уравнение с младши членове и дясна страна във формата на тригонометричен полином. Дадена е по-нова от досега известната априорна оценка за максимално възможната особеност на решенията на тази задача. Оказва се, че при по-общото уравнение с младши членове възможната сингулярност е от същия ред като при чисто вълновото уравнение.

New Hardy-Type Inequalities with Singular Weights

2010 Mathematics Subject Classification: 26D10.

Characterization of Schrödinger Processes with Unbounded Potentials

2000 Mathematics Subject Classification: 49L20, 60J60, 93E20

Multiple Dipole Source Models for Scalp-Recorded Event-Related Potentials: Example from Complex Visual Processing in the Human Brain

2000 Mathematics Subject Classification: 62P10, 92C20

Solvability of an Infinite System of Singular Integral Equations

2000 Mathematics Subject Classification: 45G15, 26A33, 32A55, 46E15.

Existence of Solutions for a Class of Quasi-Linear Singular Integro-Differential Equations

2000 Mathematics Subject Classification: 45F15, 45G10, 46B38.

On Generalized Models and Singular Products of Distributions in Colombeau Algebra G(R)

MSC 2010: 46F30, 46F10

Dunkl-Schrödinger Equations with and without Quadratic Potentials

2000 Mathematics Subject Classification: Primary 42A38. Secondary 42B10.

Reflectionless potentials for slow whispering gallery modes in surface nanoscale axial photonic fiber resonators

We consider an optical fiber with a nanoscale variation of the effective fiber radius that supports whispering gallery modes slowly propagating along the fiber, and reveal that the radius variation can be designed to support the reflectionless propagation of these modes. We show that reflectionless modulations can realize control of the transmission amplitude and temporal delay, while enabling close packing due to the absence of cross talk, in contrast to the conventional potentials.

The association between glycated haemoglobin levels and P100 visual evoked potentials in diabetes mellitus

Diabetes mellitus (DM) is a metabolic disorder which is characterised by hyperglycaemia resulting from defects in insulin secretion, insulin action or both. The long-term specific effects of DM include the development of retinopathy, nephropathy and neuropathy. Cardiac disease, peripheral arterial and cerebrovascular disease are also known to be linked with DM. Type 1 diabetes mellitus (T1DM) accounts for approximately 10% of all individuals with DM, and insulin therapy is the only available treatment. Type 2 diabetes mellitus (T2DM) accounts for 90% of all individuals with DM. Diet, exercise, oral hypoglycaemic agents and occasionally exogenous insulin are used to manage T2DM. The diagnosis of DM is made where the glycated haemoglobin (HbA1c) percentage is greater than 6.5%. Pattern-reversal visual evoked potential (PVEP) testing is an objective means of evaluating impulse conduction along the central nervous pathways. Increased peak time of the visual P100 waveform is an expression of structural damage at the level of myelinated optic nerve fibres. This was an observational cross sectional study. The participants were grouped into two phases. Phase 1, the control group, consisted of 30 healthy non-diabetic participants. Phase 2 comprised of 104 diabetic participants of whom 52 had an HbA1c greater than 10% (poorly controlled DM) and 52 whose HbA1c was 10% and less (moderately controlled DM). The aim of this study was to firstly observe the possible association between glycated haemoglobin levels and P100 peak time of pattern-reversal visual evoked potentials (PVEPs) in DM. Secondly, to assess whether the central nervous system (CNS) and in particular visual function is affected by type and/or duration of DM. The cut-off values to define P100 peak time delay was calculated as the mean P100 peak time plus 2.5 X standard deviations as measured for the non-diabetic control group, and were 110.64 ms for the right eye. The proportion of delayed P100 peak time amounted to 38.5% for both diabetic groups, thus the poorly controlled group (HbA1c > 10%) did not pose an increased risk for delayed P100 peak time, relative to the moderately controlled group (HbA1c ≤ 10%). The P100 PVEP results for this study, do however, reflect significant delay (p < 0.001) of the DM group as compared to the non-diabetic group; thus, subclincal neuropathy of the CNS occurs in 38.5% of cases. The duration of DM and type of DM had no influence on the P100 peak time measurements.

Greenhouse gas mitigation potentials in the livestock sector

Acknowledgements This paper constitutes an output of the Belmont Forum/FACCE-JPI funded DEVIL project (NE/M021327/1). Financial support from the CGIAR Program on Climate Change, Agriculture and Food Security (CCAFS) and the EU-FP7 AnimalChange project is also recognized. P.K.T. acknowledges the support of a CSIRO McMaster Research Fellowship.