Mean-to-max ratio of the torsion function and honeycomb structures

In this paper we study extremal behaviors of the mean to max ratio of the $p$-torsion function with respect to the geometry of the domain. For $p$ larger than the dimension of the space $N$, we prove that the upper bound is uniformly below $1$, contrary to the case $p \in (1,N]$. For $p=+\infty$, in two dimensions, we prove that the upper bound is asymptotically attained by a disc from which is removed a network of points consisting on the vertices of a tiling of the plane with regular hexagons of vanishing size

On a reverse Kohler-Jobin inequality

We consider the shape optimization problems for the quantities $\lambda(\Omega)T^q(\Omega)$, where $\Omega$ varies among open sets of $\mathbb{R}^d$ with a prescribed Lebesgue measure. While the characterization of the infimum is completely clear, the same does not happen for the maximization in the case $q>1$. We prove that for $q$ large enough a maximizing domain exists among quasi-open sets and that the ball is optimal among {\it nearly spherical domains}

Non-neural phenotype of spinal and bulbar muscular atrophy: Results from a large cohort of Italian patients

Objective: To carry out a deep characterisation of the main androgen-responsive tissues involved in spinal and bulbar muscular atrophy (SBMA). Methods: 73 consecutive Italian patients underwent a full clinical protocol including biochemical and hormonal analyses, genitourinary examination, bone metabolism and densitometry, cardiological evaluation and muscle pathology. Results: Creatine kinase levels were slightly to markedly elevated in almost all cases (68 of the 73; 94%). 30 (41%) patients had fasting glucose above the reference limit, and many patients had total cholesterol (40; 54.7%), low-density lipoproteins cholesterol (29; 39.7%) and triglyceride (35; 48%) levels above the recommended values. Although testosterone, luteinising hormone and follicle-stimulating hormone values were generally normal, in one-third of cases we calculated an increased Androgen Sensitivity Index reflecting the presence of androgen resistance in these patients. According to the International Prostate Symptom Score (IPSS), 7/70 (10%) patients reported severe lower urinal tract symptoms (IPSS score >19), and 21/73 (30%) patients were moderately symptomatic (IPSS score from 8 to 19). In addition, 3 patients were carriers of an indwelling bladder catheter. Videourodynamic evaluation indicated that 4 of the 7 patients reporting severe urinary symptoms had an overt prostate-unrelated bladder outlet obstruction. Dual-energy X-ray absorptiometry scan data were consistent with low bone mass in 25/61 (41%) patients. Low bone mass was more frequent at the femoral than at the lumbar level. Skeletal muscle biopsy was carried out in 20 patients and myogenic changes in addition to the neurogenic atrophy were mostly observed. Conclusions: Our study provides evidence of a wide non-neural clinical phenotype in SBMA, suggesting the need for comprehensive multidisciplinary protocols for these patients. \ua9 2016 Published by the BMJ Publishing Group Limited

Left egocentric neglect in early subacute right-stroke patients is related to damage of the superior longitudinal fasciculus.

A typical consequence of stroke in the right hemisphere is unilateral spatial neglect. Distinct forms of neglect have been described, such as space-based (egocentric) and object-based (allocentric) neglect. However, the relationship between these two forms of neglect is still far from being understood, as well as their neural substrates. Here, we further explore this issue by using voxel lesion symptoms mapping (VLSM) analyses on a large sample of early subacute right-stroke patients assessed with the Apples Cancellation Test. This is a sensitive test that simultaneously measures both egocentric and allocentric neglect. Behaviourally, we found no correlation between egocentric and allocentric performance, indicating independent mechanisms supporting the two forms of neglect. This was confirmed by the VLSM analysis that pointed out a link between a damage in the superior longitudinal fasciculus and left egocentric neglect. By contrast, no association was found between brain damage and left allocentric neglect. These results indicate a higher probability to observe egocentric neglect as a consequence of white matter damages in the superior longitudinal fasciculus, while allocentric neglect appears more "globally" related to the whole lesion map. Overall, these findings on early subacute right-stroke patients highlight the role played by white matter integrity in sustaining attention-related operations within an egocentric frame of reference

Assessment of diagnostic criteria for multifocal motor neuropathy in patients included in the Italian database

Background and purposeThis study aimed to assess the diagnostic criteria, ancillary investigations and treatment response using real-life data in multifocal motor neuropathy (MMN) patients.MethodsClinical and laboratory data were collected from 110 patients enrolled in the Italian MMN database through a structured questionnaire. Twenty-six patients were excluded due to the unavailability of nerve conduction studies or the presence of clinical signs and symptoms and electrodiagnostic abnormalities inconsistent with the MMN diagnosis. Analyses were conducted on 73 patients with a confirmed MMN diagnosis and 11 patients who did not meet the diagnostic criteria.ResultsThe European Federation of Neurological Societies/Peripheral Nerve Society (EFNS/PNS) diagnostic criteria were variably applied. AUTHOR:When applying the American Association of Electrodiagnostic Medicine criteria, an additional 17% of patients fulfilled the criteria for probable/definite diagnosis whilst a further 9.5% missed the diagnosis. In 17% of the patients only compound muscle action potential amplitude, but not area, was measured and subsequently recorded in the database by the treating physician. Additional investigations, including anti-GM1 immunoglobulin M antibodies, cerebrospinal fluid analysis, nerve ultrasound and magnetic resonance imaging, supported the diagnosis in 46%-83% of the patients. Anti-GM1 immunoglobulin M antibodies and nerve ultrasound demonstrated the highest sensitivity. Additional tests were frequently performed outside the EFNS/PNS guideline recommendations.ConclusionsThis study provides insights into the real-world diagnostic and management strategies for MMN, highlighting the challenges in applying diagnostic criteria

Impact of 2021 European Academy of Neurology/Peripheral Nerve Society diagnostic criteria on diagnosis and therapy of chronic inflammatory demyelinating polyradiculoneuropathy variants

Background and purpose: there are different criteria for the diagnosis of different variants of chronic inflammatory demyelinating polyradiculoneuropathy (CIDP). The 2021 European Academy of Neurology/Peripheral Nerve Society (EAN/PNS) guidelines provide specific clinical criteria for each CIDP variant even if their therapeutical impact has not been investigated. Methods: we applied the clinical criteria for CIDP variants of the 2021 EAN/PNS guidelines to 369 patients included in the Italian CIDP database who fulfilled the 2021 EAN/PNS electrodiagnostic criteria for CIDP. Results: according to the 2021 EAN/PNS clinical criteria, 245 patients achieved a clinical diagnosis of typical CIDP or CIDP variant (66%). We identified 106 patients with typical CIDP (29%), 62 distal CIDP (17%), 28 multifocal or focal CIDP (7%), four sensory CIDP (1%), 27 sensory-predominant CIDP (7%), 10 motor CIDP (3%), and eight motor-predominant CIDP (2%). Patients with multifocal, distal, and sensory CIDP had milder impairment and symptoms. Patients with multifocal CIDP had less frequently reduced conduction velocity and prolonged F-wave latency and had lower levels of cerebrospinal fluid protein. Patients with distal CIDP more frequently had reduced distal compound muscle action potentials. Patients with motor CIDP did not improve after steroid therapy, whereas those with motor-predominant CIDP did. None of the patients with sensory CIDP responded to steroids, whereas most of those with sensory-predominant CIDP did. Conclusions: the 2021 EAN/PNS criteria for CIDP allow a better characterization of CIDP variants, permitting their distinction from typical CIDP and more appropriate treatment for patients

Shape optimization problems with competing terms

In this thesis we discuss several shape optimization problems in which the cost functionals are given by the product of two competing terms. For these problems, classical strategies based on the monotonicity of the energy with respect to the set inclusion are not applicable. The definitions and preliminaries required to present our results are collected and discussed in Chapter 1. In Chapter 2 we consider the p-torsional rigidity and the p-first eigenvalue of the Dirichlet p-Laplace operator. The goal is to find upper and lower bounds to products of suitable powers of the quantities above in various classes of domains. In Chapter 3 we further investigate the above optimization problems, restricting ourselves to the linear case. In Chapter 4 we discuss the behaviors of the mean-to-max ratio of the p-torsion function with respect to the geometry of the domain. In Chapter 5 we study a general version of the classical Cheeger inequality. Finally, in Chapter 6 we look at the optimization problems for the shape functional defined by Jq(Ω) = P(Ω)T^q(Ω) among open sets with prescribed measure

Inequalities between torsional rigidity and principal eigenvalue of the p-Laplacian

We consider the torsional rigidity and the principal eigenvalue related to the $p$-Laplace operator. The goal is to find upper and lower bounds to products of suitable powers of the quantities above in various classes of domains. The limit cases $p=1$ and $p=infty$ are also analyzed, which amount to consider the Cheeger constant of a domain and functionals involving the distance function from the boundary

Some inequalities involving perimeter and torsional rigidity

We consider shape functionals of the form $F_q(Omega)=P(Omega)T^q(Omega)$ on the class of open sets of prescribed Lebesgue measure. Here q>0 is fixed, $P(Omega)$ denotes the perimeter of $Omega$ and $T(Omega)$ is the torsional rigidity of $Omega$. The minimization and maximization of $F_q(Omega)$ is considered on various classes of admissible domains $Omega$: in the class ${cal A}_{all}$ of {it all domains}, in the class ${cal A}_{convex}$ of {it convex domains}, and in the class ${cal A}_{thin}$ of {it thin domains}

A shape optimization problem on planar sets with prescribed topology

We consider shape optimization problems involving functionals depending on perimeter, torsional rigidity and Lebesgue measure. The scaling free cost functionals are of the form $P(Omega)T^q(Omega)|Omega|^{-2q-1/2}$ and the class of admissible domains consists of two-dimensional open sets $Omega$ satisfying the topological constraints of having a prescribed number $k$ of bounded connected components of the complementary set. A relaxed procedure is needed to have a well-posed problem and we show that when q<1/2 an optimal relaxed domain exists. When q>1/2 the problem is ill-posed and for $q=1/2$ the explicit value of the infimum is provided in the cases $k=0$ and $k=1$