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In this paper we investigate the rigidity property of a wave component coupled in a wave-Klein-Gordon system. We prove that when the radiation field of the wave component vanishes at the null infinity, the initial data of this component also vanish, therefore there is no wave in the whole spacetime. [ABSTRACT FROM AUTHOR]

The main purpose of this research was to determine the relationship between family functioning and aggression among Spanish adolescents. A sample of 1,196 adolescents between 14 and 18 years old (50 per cent female) were selected from twenty‐three educational centres, ten university degrees and eighteen specific juvenile facilities. The Buss and Perry Aggression Questionnaire (BPAQ), and the Family Adaptability and Cohesion Evaluation Scale (FACES IV Package) were administered. In regard to results, adolescent hostility was the dimension of aggression most strongly related to poor family functioning. The family functioning risk factors for aggression were disengagement, chaos and rigidity, while family communication was an essential protective factor. Enmeshment did not correlate significantly with aggression but became an aggression risk factor in late adolescence. The family functioning variable most related to adolescent aggression in multi‐problematic families was disengagement, while it was the absence of rigidity among adolescent offenders. The results are discussed considering the cultural context, concluding the importance of Spanish parents combining affection and leadership, and enhancing positive family communication. [ABSTRACT FROM AUTHOR]

In the new construction of a seventeen-storey building, a provisional prop of fourteen-meter height, horizontally braced on two intermediate levels, has been used. Despite the fact that structural logic suggests that it can be cut without having any added safety precautions, the structure of the building, made up of cores and reinforced concrete slabs working spatially, indicates that certain mechanisms be introduced so that, in the event that different and worse behaviors than expected are detected, the process can be stopped and the consequences of the new situation observed can be analyzed. For this purpose, two pairs of four metallic cantilevers were introduced at mid-height with four hydraulic pistons. In addition, the best position of a series of strain gauges as well as transducers were analyzed. At first, a load test was carried out to check that the brackets worked correctly. Once this step was verified, the abutment was cut, and the results were read. The results of stresses and deformations were compared with those expected, always being satisfactory. Finally, four provisional profiles were placed in case after a few days the structure suddenly gave way. The research focuses on obtaining an efficient control system and achieving total security throughout the process, with the comparison of the results strictly necessary for this case. Few resources were used so as not to make the work excessively expensive. We have found important divergences, on the side of security, between calculation and reality. We have also considered that the construction process has an impact on the final results. In the same way, the rigidity of the temporal abutment must be considered before the calculation. All these factors have generated a lower-than-expected deformation in an 8 m cantilever. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold. First, we show that the results in a companion paper on product rigidity for maps $ f: \Omega_1 \times \Omega_2 \subset \mathbb R^n \times \mathbb R^n \to \mathbb R^n \times \mathbb R^n $ in the Sobolev space $ W^{1, p} $ are sharp with respect to $ p $. Specifically, we show that for all $ n \ge 2 $ and all $ p < 2 $ there exist maps $ f \in W^{1, p} $ such that the weak differential $ Df $ is invertible almost everywhere and preserves or reverses the product structure almost everywhere, but $ f $ is not of the form $ f(x, y) = (f_1(x), f_2(y)) $ or = $ f(x, y) = (f_2(y), f_1(x)) $. Secondly, we develop a general toolbox to study $ W^{1, p} $ solutions of differential inclusions $ Du \in K $ for unbounded sets $ K $. As an illustration we give short proofs of results by Astala et al. on optimal $ W^{1, p} $ regularity for solutions of elliptic equations with measurable coefficients and results by Colombo and Tione on irregular solutions of the $ p $-Laplace equation. [ABSTRACT FROM AUTHOR]

In this paper, we prove a Poincaré-type inequality for any set of finite perimeter which is stable with respect to the free energy among volume-preserving perturbation, provided that the Hausdorff dimension of its singular set is at most n - 3 . With this inequality, we classify all volume-constraint local energy-minimizing sets in a unit ball, a half-space or a wedge-shaped domain. In particular, we prove that the relative boundary of any energy-minimizing set is smooth. [ABSTRACT FROM AUTHOR]

In this paper, we study rigidity problems between Lyapunov exponents along periodic orbits and geometric structures. More specifically, we prove that for a surface M without focal points, if the value of the Lyapunov exponents is constant over all periodic orbits, then M is the flat 2-torus or a surface of constant negative curvature. We obtain the same result for the case of Anosov geodesic flow for surface, which generalizes C. Butler's result [5] in dimension two. Using completely different techniques, we also prove an extension of [5] to the finite volume case, where the value of the Lyapunov exponents along all periodic orbits is constant, being the maximum or minimum possible. [ABSTRACT FROM AUTHOR]

Abstract Background Progressive encephalomyelitis with rigidity and myoclonus (PERM) is a rare and life-threatening autoimmune disease of the central nervous system. So far, only ten cases of PERM have been reported in children worldwide, including the one in this study. Case presentation We report a case of an 11-year-old boy with PERM with an initial presentation of abdominal pain, skin itching, dysuria, urinary retention, truncal and limb rigidity, spasms of the trunk and limbs during sleep, deep and peripheral sensory disturbances, and dysphagia. A tissue-based assay using peripheral blood was positive, demonstrated by fluorescent staining of mouse cerebellar sections. He showed gradual and persistent clinical improvement after immunotherapy with intravenous immunoglobulin, steroids, plasmapheresis and rituximab. Conclusions We summarized the diagnosis and treatment of a patient with PERM and performed a literature review of pediatric PERM to raise awareness among pediatric neurologists. A better comprehension of this disease is required to improve its early diagnosis, treatment, and prognosis.

Objectives Methods Results Conclusions Inflexibility of thought and behaviour is a transdiagnostic feature of many neuropsychiatric disorders and presents several empirical measurement challenges. Here, we developed and validated the Flexibility in Daily Life scale (FIDL); a novel, self‐report questionnaire, which captures expressions of cognitive and behavioural flexibility in daily life and is sensitive to natural shifts in these processes across the adult lifespan.The FIDL was developed using a deductive scale development approach, which aimed to capture common themes within the flexibility literature and across diagnoses (e.g. insistence on sameness, preference for routines). Following multidisciplinary consensus, an initial 37‐item questionnaire was submitted for validation in an online sample of 295 healthy adult participants (19–78 years).Exploratory factor analysis produced a revised 21‐item version comprising five factors, labelled: Repetition, Switching, Predictability/Control, Routine, and Thoughts/Beliefs. Internal consistency reliability was good‐to‐strong for the total FIDL score and moderate‐to‐strong for individual subscales. Convergent validity was established between the FIDL and an existing measure of cognitive flexibility. Critically, the FIDL total score evinced a U‐shaped relationship with age, whereby flexibility was lower at the younger and older tails of the lifespan and greater in middle age. The same U‐shaped trajectory emerged for the Repetition, Routine, and Thoughts/Beliefs factors.Overall, the FIDL is a valid and reliable multidimensional measure of flexibility, which upholds a clearly defined factor structure and good psychometric properties. It promises to be a valuable clinical and research tool to assess the natural fluctuations in flexibility across the lifespan and departures thereof. [ABSTRACT FROM AUTHOR]

Deep brain stimulation (DBS) of the internal segment of the globus pallidus (GPi) can markedly reduce muscle rigidity in people with Parkinson's disease (PD); however, the mechanisms mediating this effect are poorly understood. Computational modeling of DBS provides a method to estimate the relative contributions of neural pathway activations to changes in outcomes. In this study, we generated subject-specific biophysical models of GPi DBS (derived from individual 7-T MRI), including pallidal efferent, putamenal efferent, and internal capsule pathways, to investigate how activation of neural pathways contributed to changes in forearm rigidity in PD. Ten individuals (17 arms) were tested off medication under four conditions: off stimulation, on clinically optimized stimulation, and on stimulation specifically targeting the dorsal GPi or ventral GPi. Quantitative measures of forearm rigidity, with and without a contralateral activation maneuver, were obtained with a robotic manipulandum. Clinically optimized GPi DBS settings significantly reduced forearm rigidity (P < 0.001), which aligned with GPi efferent fiber activation. The model demonstrated that GPi efferent axons could be activated at any location along the GPi dorsal-ventral axis. These results provide evidence that rigidity reduction produced by GPi DBS is mediated by preferential activation of GPi efferents to the thalamus, likely leading to a reduction in excitability of the muscle stretch reflex via overdriving pallidofugal output. NEW & NOTEWORTHY: Subject-specific computational models of pallidal deep brain stimulation, in conjunction with quantitative measures of forearm rigidity, were used to examine the neural pathways mediating stimulation-induced changes in rigidity in people with Parkinson's disease. The model uniquely included internal, efferent and adjacent pathways of the basal ganglia. The results demonstrate that reductions in rigidity evoked by deep brain stimulation were principally mediated by the activation of globus pallidus internus efferent pathways. [ABSTRACT FROM AUTHOR]

Let $ (G, X) $ (G , X) be a G-system, which means that X is a compact Hausdorff space and G is an infinite topological group continuously acting on X, and let μ be a G-invariant measure of $ (G, X) $ (G , X). In this paper, we introduce the concepts of rigidity, uniform rigidity and μ-Ω-equicontinuity of $ (G,X) $ (G , X) with respect to an infinite sequence Ω of G and the notions of μ-Ω-equicontinuity and μ-Ω-mean-equicontinuity of a function $ f\in L^2(\mu) $ f ∈ L 2 (μ) with respect to an infinite sequence Ω of G. Then we give some equivalent conditions for $ f\in L^2(\mu) $ f ∈ L 2 (μ) and $ (G,X) $ (G , X) to be rigid, respectively. In addition, if G is commutative and X satisfies the first axiom of countability, we present some equivalent conditions for $ (G,X) $ (G , X) to be uniformly rigid. [ABSTRACT FROM AUTHOR]

We consider a class of recent rigidity results in a convex cone \Sigma \subseteq \mathbb {R}^N. These include overdetermined Serrin-type problems for a mixed boundary value problem relative to \Sigma, Alexandrov's soap bubble-type results relative to \Sigma, and Heintze-Karcher's inequality relative to \Sigma. Each rigidity result is obtained here by means of a single integral identity and holds true under weak integral overdeterminations in possibly non-smooth cones. Optimal quantitative stability estimates are obtained in terms of an L^2-pseudodistance. In particular, the optimal stability estimate for Heintze-Karcher's inequality is new even in the classical case \Sigma = \mathbb {R}^N. Stability bounds in terms of the Hausdorff distance are also provided. Several new results are established and exploited, including a new Poincaré-type inequality for vector fields whose normal components vanish on a portion of the boundary and an explicit (possibly weighted) trace theory – relative to the cone \Sigma – for harmonic functions satisfying a homogeneous Neumann condition on the portion of the boundary contained in \partial \Sigma. We also introduce new notions of uniform interior and exterior sphere conditions relative to the cone \Sigma \subseteq \mathbb {R}^N, which allow to obtain (via barrier arguments) uniform lower and upper bounds for the gradient in the mixed boundary value-setting. In the particular case \Sigma =\mathbb {R}^N, these conditions return the classical uniform interior and exterior sphere conditions (together with the associated classical gradient bounds of the Dirichlet setting). [ABSTRACT FROM AUTHOR]

We prove a rigidity criterion for period maps of admissible variations of graded-polarizable mixed Hodge structure, and establish rigidity in a number of cases, including families of quasi-projective curves, projective curves with ordinary double points, the complement of the canonical curve in families of Kynev-Todorov surfaces, period maps attached to the fundamental groups of smooth varieties and normal functions. [ABSTRACT FROM AUTHOR]

In this paper, we prove a rigidity result for the product metric on the product of spheres $S^1\times S^{n-1}$. [ABSTRACT FROM AUTHOR]

We introduce the notion of pullback along a measurable cocycle and we use it to extend the Borel invariant studied by Bucher, Burger and Iozzi to the world of measurable cocycles. The Borel invariant is constant along cohomology classes and has bounded absolute value. This allows to define maximal cocycles. We conclude by proving that maximal cocycles are actually trivializable to the restriction of the irreducible representation. [ABSTRACT FROM AUTHOR]

Parkinson's disease (PD) is a neurological condition affecting millions, marked by mobility issues and characterized by motor and non-motor symptoms, including tremors, bradykinesia, postural instability, and rigidity. Diagnosis often relies on subjective assessments such as the Movement Disorder Society Unified PD Rating Scale (MDS-UPDRS). This study focuses on validating a wrist orthosis designed to quantify rigidity in PD patients objectively. Developed at the Center for Innovation and Technological Evaluation in Health (NIATS), the orthosis integrates a Faulhaber linear motor (LM 2070-080-11) and microcontroller (MCLM 3006 S RS). Calibration experiments, including varied mass assessments, established the orthosis's reliability. Results indicated a newly calculated force constant of 14.28 N/A, 18.49% higher than the manufacturer's value, with a strong Pearson correlation coefficient (0.9997189). The orthosis detected masses ranging from 39.07 to 812.64 grams without yielding. Angular displacement calibration, utilizing a GP10 goniometer and Myosystem-Br1 software, demonstrated linearity, supported by Pearson coefficients of 0.9995091 and 0.995259. These findings underscore the orthosis's potential as a reliable tool for measuring rigidity in PD patients, promising advancements in physiotherapy and disease monitoring. [ABSTRACT FROM AUTHOR]

Motivated by Guo–Luo’s generalized circle packings on surfaces with boundary [Guo and Luo, Geom. Topol. 13(3) (2009) 1265–1312], we introduce the generalized Thurston’s sphere packings on 3-dimensional manifolds with boundary. Then we investigate the rigidity of the generalized Thurston’s sphere packings. We prove that the generalized Thurston’s sphere packings are locally determined by the combinatorial scalar curvatures. We further prove the infinitesimal rigidity that the generalized Thurston’s sphere packings cannot be deformed while keeping the combinatorial Ricci curvatures fixed. [ABSTRACT FROM AUTHOR]

The goal of this paper is to investigate complete noncompact warped product gradient Ricci solitons. Nonexistence results, estimates for the warping function and for its gradient are proven. When the soliton is steady or expanding these nonexistence results generalize to a broader context certain estimates and rigidity obtained when studying warped product Einstein manifolds. When the soliton is shrinking, it is presented as a nonexistence theorem with no counterpart in the Einstein case, which is proved using properties of the first eigenvalue of a weighted Laplacian. [ABSTRACT FROM AUTHOR]

In this paper, we consider self-similar solutions of the r -mean curvature flow into the (n + 1) -dimensional Euclidean space. By employing some general maximum principles as the main tool, we characterize some self-similar solutions of the r -mean curvature flow under some suitable geometric constraints. [ABSTRACT FROM AUTHOR]

Video head impulse tests (video-HITs) are commonly used for vestibular evaluation; however, the results can be contaminated by various artifacts, including technical errors, recording problems, and participant factors. Although video-HITs can be used in patients with Parkinson's disease (PD), the effect of neck rigidity has not been systematically investigated. This study aimed to investigate the effect of neck rigidity on video-HIT results in patients with PD. We prospectively recruited 140 consecutive patients with PD (mean age ± standard deviation = 68 ± 10 years, 69 men) between September 2021 and April 2024 at Korea University Medical Center. The video-HIT results were compared with those of 19 age- and sex-matched healthy participants. Neck rigidity was stratified as a subdomain of the Movement Disorder Society-Unified Parkinson's Disease Rating Scale motor part (MDS-UPDRS-III). In 59 patients, the vestibulo-ocular reflex (VOR) gain was overestimated in at least one canal plane (58/140, 41%), mostly in the anterior canal (AC, n = 44), followed by the horizontal (HC, n = 15) and posterior canals (PC, n = 7). VOR gain overestimation was also observed in patients with no (18/58, 35%), subtle (20/58, 34%), or mild (17/58, 29%) neck rigidity. Multivariable logistic regression analysis showed that VOR overestimation was positively associated with neck rigidity (odds ratio [OR] [95% confidence interval] = 1.51 [1.01–2.25], p = 0.043). The head velocities of patients decreased during head impulses for the AC (p = 0.033 for the right AC; p = 0.014 for the left AC), whereas eye velocities were similar to those of healthy participants. Our findings suggest that neck rigidity may be a confounder that can contaminate video-HIT results. Thus, the results of video-HITs, especially for the AC, should be interpreted with the context of head velocity during head impulses in patients with neck rigidity. [ABSTRACT FROM AUTHOR]

Autism is a neurological disability characterised by a number of 'deficits' in multiple areas of functioning and everyday life. Or is it? Damian Milton's theory of the 'double empathy problem' attempts to empirically posit that the socialisation and communication difficulties present in autistic people are due to bidirectional differences in ways of communication between autistic and non-autistic people. This Current Issues piece extends said theory to encompass the restricted and repetitive behaviours that are commonly observed in autistic people, arguing that such behaviours may also be present in non-autistic people, albeit in a less stereotypical, more socially acceptable way. Because autism is characterised in part by the presence of restricted and repetitive behaviours, this piece fundamentally challenges the manner in which autism spectrum conditions are diagnosed in individuals and seen far and wide, ultimately promoting the view of cognitive-behavioural difference in autistic people instead of 'deficit'. [ABSTRACT FROM AUTHOR]

We consider a Serrin’s type problem in convex cones in the Euclidean space and motivated by recent rigidity results we study the quantitative stability issue for this problem. In particular, we prove both sharp Lipschitz estimates for an L 2 -pseudodistance and estimates in terms of the Hausdorff distance. [ABSTRACT FROM AUTHOR]

We prove that under restrictions on the fiber, any fibered partially hyperbolic system over a nilmanifold is leaf conjugate to a smooth model that is isometric on the fibers and descends to a hyperbolic nilmanifold automorphism on the base. One ingredient is a result of independent interest generalizing a result of Hiraide: an Anosov homeomorphism of a nilmanifold is topologically conjugate to a hyperbolic nilmanifold automorphism. [ABSTRACT FROM AUTHOR]

Methyl methacrylate-acrylonitrile-butadiene-styrene (MABS) was obtained by blending the rubber phase with matrix resin. Matching the refractive index was crucial for transmittance in MABS resins. Additionally, the superimposed effect of particle size and the interfacial area further determined the MABS resin transparency. The MABS resin with a rubber particle size of 201nm had a transmission of 88.9% when the refractive indexes of the two phases were close. By utilizing a bimodal distribution of toughened particles in the phase domain structure, the rubber particle size could be altered while maintaining control over the interfacial area. It resulted in an increase in the transmittance of the MABS to 90.1%. The analysis of mechanical properties and morphology shows that the phase structure with a bimodal distribution of toughened particles is also beneficial to the impact and tensile strength of MABS resin. When toughening particles with a size of 73nm and 304nm were mixed in a mass ratio of 60:40, the impact strength was 145 J/m, the tensile strength was 47MPa, and the elongation at break was 20%. And the study investigated the impact of rubber content on the properties of transparent MABS resins. [ABSTRACT FROM AUTHOR]

Autonomic synchrony plays an important role in work team performance where coordinated actions are required on the part of the team members. The present study examined the connection between nine psychological variables that represent types of elasticity-rigidity, which are closely related to adaptability and autonomic synchrony, within teams playing a computer game that involved dynamic decision making. Elasticity-rigidity variables were first identified as part of the dynamics that transpire between workload and performance. They are used here to determine why some individuals within teams synchronize with teammates more strongly than others. The driver-empath model of group synchrony produces a single metric of synchrony (SE) within a team of three or more members. Driver scores, which are produced from the algorithm, indicate each person's total influence on the other group members. Empath scores, which are also produced from the SE algorithm, indicate a person's total receptivity to all other group members. It was found that coping flexibility, monitoring, emotional intelligence, and solving anagrams significantly predicted empath scores in the earlier part of the session. Anxiety and monitoring significantly predicted empath scores in the later part of the session. There were no significant correlations between driver scores and elasticity-rigidity variables. [ABSTRACT FROM AUTHOR]

Background: Progressive encephalomyelitis with rigidity and myoclonus (PERM) is a rare and life-threatening autoimmune disease of the central nervous system. So far, only ten cases of PERM have been reported in children worldwide, including the one in this study. Case presentation: We report a case of an 11-year-old boy with PERM with an initial presentation of abdominal pain, skin itching, dysuria, urinary retention, truncal and limb rigidity, spasms of the trunk and limbs during sleep, deep and peripheral sensory disturbances, and dysphagia. A tissue-based assay using peripheral blood was positive, demonstrated by fluorescent staining of mouse cerebellar sections. He showed gradual and persistent clinical improvement after immunotherapy with intravenous immunoglobulin, steroids, plasmapheresis and rituximab. Conclusions: We summarized the diagnosis and treatment of a patient with PERM and performed a literature review of pediatric PERM to raise awareness among pediatric neurologists. A better comprehension of this disease is required to improve its early diagnosis, treatment, and prognosis. [ABSTRACT FROM AUTHOR]

The analysis focuses on how the physical parameters of the attachment–workpiece and tool–chuck systems affect the stability when milling aviation materials that are hard to machine. Methods of stabilizing the milling process in terms of vibration are outlined, as well as means of increasing the productivity without damage to the equipment. Examples are given of stable and unstable machining of titanium alloy by end mills and its influence on the cutting process. [ABSTRACT FROM AUTHOR]

In this paper, we deal with Serrin-type problems in Riemannian manifolds. First, we obtain a Heintze-Karcher inequality and a Soap Bubble result, with its respective rigidity, when the ambient space has a Ricci tensor bounded below. After, we approach a Serrin problem in bounded domains of manifolds endowed with a closed conformal vector field. Our primary tool, in this case, is a new Pohozaev identity, which depends on the scalar curvature of the manifold. Applications involve Einstein and constant scalar curvature spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study centrally symmetric Birkhoff billiard tables. We introduce a closed invariant set $\mathcal {M}_{\mathcal {B}}$ consisting of locally maximizing orbits of the billiard map lying inside the region $\mathcal {B}$ bounded by two invariant curves of $4$ -periodic orbits. We give an effective bound from above on the measure of this invariant set in terms of the isoperimetric defect of the curve. The equality case occurs if and only if the curve is a circle. [ABSTRACT FROM AUTHOR]

The paper considers a design process of an aerodynamic rudder, which structure comprises the skin of constant thickness, a load-bearing structure and a trimmed nose that plays the role of an anti-flutter balancer. The aim of the work is to set and solve the design problem of a rational structural and technological solution of the rudder that meets the requirements of strength, rigidity, aeroelastic stability and minimum mass. To solve this problem, a design algorithm for the rudder, using topological and parametric optimization, is proposed. The main parameters of the design area and the trimmed nose required for topological optimization are determined. The ANSYS Workbench software package was used for the finite element analysis and topological optimization. Based on the results of optimization, post-processing was carried out. A structural and technological solution, that combines structural layouts with constant and variable width of the trimmed nose, was proposed. An analysis of the stress-strain state was carried out, and it was found that the designed structure meets the strength requirements for the given design case. A scheme for solving the parametric optimization problem of the rudder under the condition of aeroelastic stability is proposed. Within the framework of solving this problem, a flutter study was conducted, using a multi-mode model, which makes it possible to study the rudder and body-rudder flutter forms of an unmanned aerial vehicle (UAV) equipped with aerodynamic rudders. The results of the flutter study for the design mode of the UAV flight are obtained in the form of dependencies of the critical flutter velocity and frequency on the average width of the trimmed nose. The analysis of these dependencies allowed us to derive the optimal values of the trimmed nose parameters from the minimum weight condition for two rudder configurations: with a constant and variable width of the trimmed nose.

Deer mice (Peromyscus maniculatus bairdii), a wildtype species native to North America, have been investigated for their spontaneous compulsive-like behaviour. The repetitive and persistence nature of three unique compulsive-like phenotypes in deer mice, i.e., high stereotypy (HS), large nesting behaviour (LNB) and high marble burying (HMB), are characterized by behavioural and cognitive rigidity. In this narrative review, we summarize key advances in the model's application to study obsessive-compulsive disorder (OCD), emphasizing how it may be used to investigate neurobiological and neurocognitive aspects of rigidity. Indeed, deer mice provide the field with a unique naturalistic and spontaneous model system of behavioural and cognitive rigidity that is useful for investigating the psychobiological mechanisms that underpin a range of compulsive-like phenotypes. Throughout the review, we highlight new opportunities for future research.

BackgroundSevere muscle stiffness (SMS) in dogs with hypercortisolism (HC) is uncommon.ObjectivesTo evaluate signalment, presentation, treatments, and long-term outcomes of dogs with concurrent HC and SMS.AnimalsThirty-seven dogs.MethodsMedical records of dogs with HC and concurrent SMS were recruited from 10 institutions. Clinical information, test results, therapeutic responses, and survival times were reviewed.ResultsAll 37 dogs with HC and SMS had pituitary-dependent hypercortisolism (PDH); 36/37 weighed

We show that, for points along the moment curve, the bar-and-joint rigidity matroid and the hyperconnectivity matroid coincide, and that both coincide with the \(C^{d-2}_{d-1}\)-cofactor rigidity of points along any (non-degenerate) conic in the plane. For hyperconnectivity in dimension two, having the points in the moment curve is no loss of generality. We also show that, restricted to bipartite graphs, the bar-and-joint rigidity matroid is freer than the hyperconnectivity matroid.Mathematics Subject Classifications: 52C25, 52B40Keywords: Rigidity, hyperconnectivity, moment curve, cofactor rigidity

Parkinson's disease (PD) is a progressive neurological disorder that is typically characterized by a range of motor dysfunctions, and its impact extends beyond physical abnormalities into emotional well-being and cognitive symptoms. The loss of dopaminergic neurons in the substantia nigra pars compacta (SNc) leads to an array of dysfunctions in the functioning of the basal ganglia (BG) circuitry that manifests into PD. While active research is being carried out to find the root cause of SNc cell death, various therapeutic techniques are used to manage the symptoms of PD. The most common approach in managing the symptoms is replenishing the lost dopamine in the form of taking dopaminergic medications such as levodopa, despite its long-term complications. Another commonly used intervention for PD is deep brain stimulation (DBS). DBS is most commonly used when levodopa medication efficacy is reduced, and, in combination with levodopa medication, it helps reduce the required dosage of medication, prolonging the therapeutic effect. DBS is also a first choice option when motor complications such as dyskinesia emerge as a side effect of medication. Several studies have also reported that though DBS is found to be effective in suppressing severe motor symptoms such as tremors and rigidity, it has an adverse effect on cognitive capabilities. Henceforth, it is important to understand the exact mechanism of DBS in alleviating motor symptoms. A computational model of DBS stimulation for motor symptoms will offer great insights into understanding the mechanisms underlying DBS, and, along this line, in our current study, we modeled a cortico-basal ganglia circuitry of arm reaching, where we simulated healthy control (HC) and PD symptoms as well as the DBS effect on PD tremor and bradykinesia. Our modeling results reveal that PD tremors are more correlated with the theta band, while bradykinesia is more correlated with the beta band of the frequency spectrum of the local field potential (LFP) of the subthalamic nucleus (STN) neurons. With a DBS current of 220 pA, 130 Hz, and a 100 microsecond pulse-width, we could found the maximum therapeutic effect for the pathological dynamics simulated using our model using a set of parameter values. However, the exact DBS characteristics vary from patient to patient, and this can be further studied by exploring the model parameter space. This model can be extended to study different DBS targets and accommodate cognitive dynamics in the future to study the impact of DBS on cognitive symptoms and thereby optimize the parameters to produce optimal performance effects across modalities. Combining DBS with rehabilitation is another frontier where DBS can reduce symptoms such as tremors and rigidity, enabling patients to participate in their therapy. With DBS providing instant relief to patients, a combination of DBS and rehabilitation can enhance neural plasticity. One of the key motivations behind combining DBS with rehabilitation is to expect comparable results in motor performance even with milder DBS currents. [ABSTRACT FROM AUTHOR]

We consider the problem of finding an inductive construction, based on vertex splitting, of triangulated spheres with a fixed number of additional edges (braces). We show that for any positive integer b there is such an inductive construction of triangulations with b braces, having finitely many base graphs. In particular we establish a bound for the maximum size of a base graph with b braces that is linear in b. In the case that b = 1 or 2 we determine the list of base graphs explicitly. Using these results we show that doubly braced triangulations are (generically) minimally rigid in two distinct geometric contexts arising from a hypercylinder in R 4 and a class of mixed norms on R 3 . [ABSTRACT FROM AUTHOR]