Minimal Degree Research Articles

Kernels of minimal characters of solvable groups

Abstract Let G be a finite solvable group. We prove that if $\chi\in{{\operatorname{Irr}}}(G)$ has odd degree and $\chi(1)$ is the minimal degree of the nonlinear irreducible characters of G, then $G/\operatorname{Ker}\chi$ is nilpotent-by-abelian.

Snap-through and indirect reduced-order modelling

Snap-through is a nonlinear jump phenomenon that is increasingly being designed into engineering structures. The rich and varied dynamics of snap-through present a significant bottleneck in the design process as costly dynamic simulations are required to ensure predictable behaviour. The purpose of this research is to investigate the use of a recent projection-based, reduced-order modelling technique, namely, implicit condensation and expansion with inertial compensation, to reproduce the snap-through behaviours found in bistable finite element systems in analogous models with minimal degrees of freedom. In this work, the free response of a simple mass-spring oscillator is used to differentiate different types of snap-through. This provides a framework for discussing the dynamics of a family of buckled beams. The ability of the reduced-order models to faithfully reproduce these dynamics is investigated by testing the existence and stability of predicted periodic orbits. It is shown that the minimum projection basis required to capture snap-through accurately can be found through considering the minimum energy required for different types of snap-through. For bistable beams, this can be further simplified to just requiring an eigenvalue analysis at the most unstable equilibrium.

ENHANCED ALGORITHM FOR COMPUTING CAYLEY’S FIRST HYPERDETERMINANT

Combinatorial hyperdeterminant DET – is the homogeneous polynomial in the entries of a hypermatrix of even number of indices, which is also a unique SL-invariant of minimal degree. It was first studied by Cayley in the middle of 19-th century. Given its fundamental nature, the computation of this polynomial is an important task. For fixed d and a cubical hypermatrix X of length n Barvinok introduced an algorithm of computing hyperdeterminant in 0(2nd nd-1) . Since the problem of deciding whether for the given hypermatrix X the hyperdeterminant DET(X) is equal to zero is NP-hard, it is essential to develop efficient algorithm for computing hyperdeterminant, as the size of problem grows exponentially. We provide enhanced algorithm of computing hyperdeterminant that requires 0(2 n(d-1) nd-1) arithmetic operations.

The Bourbaki degree of a plane projective curve

Bourbaki sequences and Bourbaki ideals have been studied by several authors since its inception sixty years ago circa. Generic Bourbaki sequences have been thoroughly examined by the senior author with B. Ulrich and W. Vasconcelos, but due to their nature, no numerical invariant was immediately available. Recently, J. Herzog, S. Kumashiro, and D. Stamate introduced the Bourbaki number in the category of graded modules as the shifted degree of a Bourbaki ideal corresponding to submodules generated in degree at least the maximal degree of a minimal generator of the given module. The present work introduces the Bourbaki degree as the algebraic multiplicity of a Bourbaki ideal corresponding to choices of minimal generators of minimal degree. The main intent is a study of plane curve singularities via this new numerical invariant. Accordingly, quite naturally, the focus is on the case where the standing graded module is the first syzygy module of the gradient ideal of a reduced form f ∈ k [ x , y , z ] f\in k[x,y,z] – i.e., the main component of the module of logarithmic derivations of the corresponding curve. The overall goal of this project is to allow for a facet of classification of projective plane curves based on the behavior of this new numerical invariant, with emphasis on results about its lower and upper bounds.

On the relation of character codegrees and the minimal faithful quasi-permutation representation degree of p-groups

For a finite group G, we denote by c ( G ) , the minimal degree of a faithful representation of G by quasi-permutation matrices over C . For an irreducible character χ of G, the codegree of χ is defined as cod ( χ ) = | G / ker ( χ ) | / χ ( 1 ) . In this article, we establish equality between c ( G ) and a Q ≥ 0 -sum of codegrees of some irreducible characters of a non-abelian p-group G of odd order. We also study the relation between c ( G ) and irreducible character codegrees for various classes of non-abelian p-groups, such as, p-groups with cyclic center, maximal class p-groups, GVZ p-groups, and others.

Cones between the cones of positive semidefinite forms and sums of squares

Abstract For n, d ∈ ℕ, the cone 𝓟 n+1,2d of positive semidefinite real forms in n + 1 variables of degree 2d contains the subcone Σ n+1,2d of those representable as finite sums of squares of real forms. Hilbert [11] proved that these cones coincide exactly in the Hilbert cases (n + 1, 2d) with n + 1 = 2 or 2d = 2 or (n + 1, 2d) = (3, 4). In this paper, we induce a filtration of intermediate cones between Σ n+1,2d and 𝓟 n+1,2d via the Gram matrix approach in [4] on a filtration of irreducible projective varieties V k−n ⊊ … ⊊ Vn ⊊ … ⊊ V 0 containing the Veronese variety. Here, k is the dimension of the vector space of real forms in n + 1 variables of degree d. By showing that V 0, …, V n (and V n+1 when n = 2) are varieties of minimal degree, we demonstrate that the corresponding intermediate cones coincide with Σ n+1,2d . We moreover prove that, in the non-Hilbert cases of (n + 1)-ary quartics for n ≥ 3 and (n + 1)-ary sextics for n ≥ 2, all the remaining cone inclusions are strict.

CRISPR-Cas9 and Cas12a target site richness reflects genomic diversity in natural populations of Anopheles gambiae and Aedes aegypti mosquitoes

Due to limitations in conventional disease vector control strategies including the rise of insecticide resistance in natural populations of mosquitoes, genetic control strategies using CRISPR gene drive systems have been under serious consideration. The identification of CRISPR target sites in mosquito populations is a key aspect for developing efficient genetic vector control strategies. While genome-wide Cas9 target sites have been explored in mosquitoes, a precise evaluation of target sites focused on coding sequence (CDS) is lacking. Additionally, target site polymorphisms have not been characterized for other nucleases such as Cas12a, which require a different DNA recognition site (PAM) and would expand the accessibility of mosquito genomes for genetic engineering. We undertook a comprehensive analysis of potential target sites for both Cas9 and Cas12a nucleases within the genomes of natural populations of Anopheles gambiae and Aedes aegypti from multiple continents. We demonstrate that using two nucleases increases the number of targets per gene. Also, we identified differences in nucleotide diversity between North American and African Aedes populations, impacting the abundance of good target sites with a minimal degree of polymorphisms that can affect the binding of gRNA. Lastly, we screened for gRNAs targeting sex-determination genes that could be widely applicable for developing field genetic control strategies. Overall, this work highlights the utility of employing both Cas9 and Cas12a nucleases and underscores the importance of designing universal genetic strategies adaptable to diverse mosquito populations.

Meaningful changes in motor function in Duchenne muscular dystrophy (DMD): A multi-center study.

Evaluations of treatment efficacy in Duchenne muscular dystrophy (DMD), a rare genetic disease that results in progressive muscle wasting, require an understanding of the 'meaningfulness' of changes in functional measures. We estimated the minimal detectable change (MDC) for selected motor function measures in ambulatory DMD, i.e., the minimal degree of measured change needed to be confident that true underlying change has occurred rather than transient variation or measurement error. MDC estimates were compared across multiple data sources, representing >1000 DMD patients in clinical trials and real-world clinical practice settings. Included patients were ambulatory, aged ≥4 to <18 years and receiving steroids. Minimal clinically important differences (MCIDs) for worsening were also estimated. Estimated MDC thresholds for >80% confidence in true change were 2.8 units for the North Star Ambulatory Assessment (NSAA) total score, 1.3 seconds for the 4-stair climb (4SC) completion time, 0.36 stairs/second for 4SC velocity and 36.3 meters for the 6-minute walk distance (6MWD). MDC estimates were similar across clinical trial and real-world data sources, and tended to be slightly larger than MCIDs for these measures. The identified thresholds can be used to inform endpoint definitions, or as benchmarks for monitoring individual changes in motor function in ambulatory DMD.

Exposure to polyhexamethyleneguanidine phosphate in early life dampens pulmonary damage compared to adult mice

Polyhexamethyleneguanidine phosphate (PHMG-P) is a biocide of guanidine family that can cause a fatal lung damage if exposed directly to the lungs. No reports exist regarding the toxicity of PHMG-P in neonatal animals. Therefore, this study aimed to determine PHMG-P toxicity in neonatal and 8-week-old mice after they were intranasally instilled with 1.5 mg/kg, 3 mg/kg, and 4.5 mg/kg PHMG-P. PHMG-P lung exposure resulted in more severe pulmonary toxicity in adult mice than in newborn mice. In the high-dose group of newborn mice, a minimal degree of inflammatory cell infiltration and fibrosis in the lung were detected, whereas more severe pathological lesions including granulomatous inflammation, fibrosis, and degeneration of the bronchiolar epithelium were observed in adult mice. At day 4, C–C motif chemokine ligand 2 (CCL2), a potent chemokine for monocytes, was upregulated but recovered to normal levels at day 15 in newborn mice. However, increased CCL2 and IL-6 levels were sustained at day 15 in adult mice. When comparing the differentially expressed genes of newborn and adult mice through RNA-seq analysis, there were expression changes in several genes associated with inflammation in neonates that were similar or different from those in adults. Although no significant lung damage occurred in newborns, growth inhibition was observed which was not reversed until the end of the experiment. Further research is needed to determine how growth inhibition from neonatal exposure to PHMG-P affects adolescent and young adult health.

Some remarks on the Kp,1$\mathcal {K}_{p,1}$ theorem

AbstractLet be a non‐degenerate projective irreducible variety of dimension , degree , and codimension over an algebraically closed field of characteristic 0. Let be the th graded Betti number of . Green proved the celebrating ‐theorem about the vanishing of for high values for and potential examples of nonvanishing graded Betti numbers. Later, Nagel–Pitteloud and Brodmann–Schenzel classified varieties with nonvanishing . It is clear that when there is an ‐dimensional variety of minimal degree containing , however, this is not always the case as seen in the example of the triple Veronese surface in .In this paper, we completely classify varieties with nonvanishing such that does not lie on an ‐dimensional variety of minimal degree. They are exactly cones over smooth del Pezzo varieties, whose Picard number is .

Spheroid models to elaborate the broken symmetry and equivalent volume of molecules in crystalline phase.

Dense packing of particles has provided powerful models to elaborate the important structural features of matter in various systems such as liquid, glassy, and crystalline phases. The simplest sphere packing models can represent and capture salient properties of the building blocks for covalent, metallic, and ionic crystals; it, however, becomes insufficient to reflect the broken symmetry of the commonly anisotropic molecules in molecular crystals. Here, we develop spheroid models with a minimal degree of anisotropy, which serve as a simple geometrical representation for a rich spectrum of molecules-including both isotropic and anisotropic, convex and concave ones-in crystalline phases. Our models are determined via an inverse packing approach: Given a molecular crystal, an optimal spheroid model is constructed using a contact diagram, which depicts the packing relationship between neighboring molecules within the crystal. The spheroid models are capable of accurately capturing the broken symmetry and characterizing the equivalent volume of molecules in the crystalline phases. Moreover, our model retrieves such molecular information from low-quality x-ray diffraction data with poorly resolved structures, and by using soft spheroids, it can also describe the packing behavior in cocrystals.

Natural metric-affine inflation

We consider here natural inflation in the low energy (two-derivative) metric-affine theory containing only the minimal degrees of freedom in the inflationary sector, i.e. the massless graviton and the pseudo-Nambu-Goldstone boson (PNGB). This theory contains the Ricci-like and parity-odd Holst invariants together with non-minimal couplings between the PNGB and the above-mentioned invariants. The Palatini and Einstein-Cartan realizations of natural inflation are particular cases of our construction. Explicit models of this type featuring non-minimal couplings are shown to emerge from the microscopic dynamics of a QCD-like theory with an either sub-Planckian or trans-Planckian confining scale and that is renormalizable on Minkowski spacetime. Moreover, for these models, we find regions of the parameter space where the inflationary predictions agree with the most recent observations at the 2σ level. We find that in order to enter the 1σ region it is necessary (and sufficient) to have a finite value of the Barbero-Immirzi parameter and a sizable non-minimal coupling between the inflaton and the Holst invariant (with sign opposite to the Barbero-Immirzi parameter). Indeed, in this case the potential of the canonically normalized inflaton develops a plateau as shown analytically.

Soft-landing dynamics of a type of four-legged space lander

Analyzing the landing dynamics response is essential for enhancing the success rate of surface explorations on celestial bodies like Mars, the moon, and asteroids. This paper develops a comprehensive dynamics model for a space lander equipped with four sets of buffer legs. Each set comprises a primary strut, two secondary struts, and a footpad. The compression force of the buffer struts is determined through interpolation of real experimental data, while the contact force between the footpad and the ground is calculated using Archimedes law for granular media. The Lagrange equation of the second kind is employed to construct the dynamical model for the lander with minimal degrees of freedom. The proposed dynamics model is validated through experiments involving touchdowns of a single legged lander and a four legged lander. Moreover, key physical quantities such as the buffer force, energy absorption and body acceleration are computed for both 1-2-1 and 2-2 landing configurations. The results indicate that the buffer strut is capable of absorbing approximately half of the impact energy, leading to a 50-fold reduction in the shock acceleration of the main body in both landing configurations. However, it is observed that the maximum compression of the buffer strut is greater in the 1-2-1 landing configuration than in the 2-2 landing configuration. Furthermore, these simulations can be completed within seconds, enabling extensive simulations for the design and optimization of soft landing processes.

On the parameterized complexity of non-hereditary relaxations of clique

We investigate the parameterized complexity of several problems formalizing cluster identification in graphs. In other words, we ask whether a graph contains a large enough and sufficiently connected subgraph. We study here three relaxations of Clique: s-Club and s-Clique, in which the relaxation is focused on the distances in respectively the cluster and the original graph, and γ-Complete Subgraph in which the relaxation is made on the minimal degree in the cluster. As these three problems are known to be NP-hard, we study here their parameterized complexities. We prove that s-Club and s-Clique are NP-hard even restricted to graphs of degeneracy ≤3 whenever s≥3, and to graphs of degeneracy ≤2 whenever s≥5, which is a strictly stronger result than its W[1]-hardness parameterized by the degeneracy. Concerning γ-Complete Subgraph, we prove that it is W[1]-hard parameterized both by the degeneracy, implying the W[1]-hardness parameterized by the number of vertices in the γ-complete-subgraph, and by the number of elements outside the γ-complete subgraph.

Forestier disease: A possible cause of decannulation failure

Diffuse Idiopathic Skeletal Hyperostosis (DISH) [1] also known as Forestier disease, is a noninflammatory, systemic skeletal disease of unknown aetiology. DISH is a common but underdiagnosed disease that is usually observed in elderly people and has been reported to affect up to 10% of patients over 65 years of age [2]. The diagnosis is primarily radiological characterised by the ossification of the anterior longitudinal ligament, with osteophytes formation along the spinal column of at least four contiguous vertebral bodies, a minimal degree of degenerative disc disease and absence of apophyseal joint ankylosis and sacroiliac joint fusion, erosions, or sclerosis [3]. DISH is usually asymptomatic but may compress the posterior wall of the aero digestive tract and lead to dysphagia, globus, hoarseness, stridor, dyspnea and neurological problems. Although dysphagia is not uncommon among the presenting symptoms of DISH, dysphonia and stridor are rarely reported. Cervical hyperostosis in patients with DISH is responsible for 17%-28% incidence of dysphagia, and about 8% of patients failed to respond to conservative treatment and surgical excision through an anterior cervical approach was required [3]. Many studies have been conducted on the surgical approach [4] and anaesthetic evaluation of patients with DISH [5]. Less known are the difficulties in weaning from the tracheostomy tube in these patients, especially when the diagnosis of DISH is unknown.

Comparison Regarding the Carbon Footprint of Various Sustainable Seismic Consolidation Solutions for Romanian Orthodox Churches

In Romania, there are numerous Orthodox churches, many of which are historical monuments of great cultural value that have suffered multiple degradations over time due to various natural or man-made reasons. In a context that is currently increasingly focused on environmental protection, we aim to analyse the carbon footprint of several different consolidation proposals to an Orthodox church with structural deteriorations (and more) and the equivalent impact if a similar building were erected with new materials. The research is proposed to be a stepping stone for determining the sustainability of interventions for orthodox churches, as the existing literature is scarce when it comes to the emissions of these churches and there is no norm to prevent unsustainable interventions. The Orthodox Church “Sfintii Voievozi”, the subject of the analysis, is in the city of Tg. Jiu, Gorj County. The construction was documented to be between 1748 and 1764 and is a historical monument listed in the LMI GJ-II-m-A-09189 registry. The architectural solutions for the church and the structural elements that comprise the load-bearing system are presented. A detailed investigation was conducted to determine structural and non-structural degradations, specifying the main causes that have produced them. With regard to consolidation solutions, two options are presented and compared in this paper: Alternative I—minimal intervention and Alternative II—maximal intervention, both of which are reversible. The carbon footprint calculation was carried out for both options, determining the associated material consumption, and compared to the carbon footprint for the case of a new construction. In conclusion, the consolidation methods with a minimal degree of intervention is recommended as the “most environmentally friendly”, considering carbon emissions when comparing the options.

On rationally integrable planar dual multibilliards and piecewise smooth projective billiards

The billiard flow in a planar domain Ω acts on the tangent bundle TR2|Ω as geodesic flow with reflections from the boundary. It has the trivial first integral: squared modulus of the velocity. Bolotin’s conjecture, now a joint theorem of Bialy, Mironov and the author, deals with those billiards whose flow admits an additional integral that is polynomial in the velocity and whose restriction to the unit tangent bundle is non-constant. It states that (1) if the boundary of such a billiard is C 2-smooth, nonlinear and connected, then it is a conic; (2) if it is piecewise C 2-smooth and contains a nonlinear arc, then it consists of arcs of conics from a confocal pencil and segments of ‘admissible lines’ for the pencil; (3) the minimal degree of the additional integral is either 2, or 4. In 1997 Sergei Tabachnikov introduced projective billiards: planar curves equipped with a transversal line field, which defines a reflection acting on oriented lines and the projective billiard flow. They are common generalization of billiards on surfaces of constant curvature, but in general may have no canonical conserved quantity. In a previous paper the author classified those C 4-smooth connected nonlinear planar projective billiards whose flow admits a non-constant integral that is a rational 0-homogeneous function of the velocity (with coefficients depending on the position): these billiards are called rationally 0-homogeneously integrable. It was shown that: (1) the underlying curve is a conic; (2) the minimal degree of integral is equal to two, if the billiard is defined by a dual pencil of conics; (3) otherwise it can be arbitrary even number. In the present paper we classify piecewise C4-smooth rationally 0-homogeneously integrable projective billiards. Unexpectedly, we show that such a billiard associated to a dual pencil of conics may have integral of minimal degree 2, 4, or 12. For the proof of main results we prove dual results for the so-called dual multibilliards introduced in the present paper.

A $ \prod _{2}^{0}$ SINGLETON OF MINIMAL ARITHMETIC DEGREE

Abstract In the study of the arithmetic degrees the $\omega \text {-REA}$ sets play a role analogous to the role the r.e. degrees play in the study of the Turing degrees. However, much less is known about the arithmetic degrees and the role of the $\omega \text {-REA}$ sets in that structure than about the Turing degrees. Indeed, even basic questions such as the existence of an $\omega \text {-REA}$ set of minimal arithmetic degree are open. This paper makes progress on this question by demonstrating that some promising approaches inspired by the analogy with the r.e. sets fail to show that no $\omega \text {-REA}$ set is arithmetically minimal. Finally, it constructs a $\prod ^0_{2}$ singleton of minimal arithmetic degree. Not only is this a result of considerable interest in its own right, constructions of $\prod ^0_{2}$ singletons often pave the way for constructions of $\omega \text {-REA}$ sets with similar properties. Along the way, a number of interesting results relating arithmetic reducibility and rates of growth are established.

Analysis of Graft Detachments and Re-Bubblings After 450 Descemet Membrane Endothelial Keratoplasty Procedures.

To objectify the indication for re-bubbling by analyzing graft detachments (GDs) after Descemet membrane endothelial keratoplasty. In this retrospective monocentric observational study, re-bubbling cases of 450 Descemet membrane endothelial keratoplasties and the percentage of the residual gas filling (RGF) in the anterior chamber on the first postoperative day were collected. The number/location/extent of GDs and the corneal thickness above GDs were analyzed using anterior segment optical coherence tomography. From a total of 450 grafts, 384 (85.3%) had at least a minimal degree GD. One hundred twenty-two of 450 grafts (27.1%) underwent at least 1 re-bubbling. The mean RGF was significantly lower in eyes with GD (67.7 ± 12.6%) than in eyes without GD (74.2 ± 11.3%). GDs occurred most frequently in the inferotemporal quadrant (46.0%). GDs were significantly more likely to require a re-bubbling when the central parts of the graft were affected (94.0% vs. 35.7%). The number of detachments per graft was directly proportional to the re-bubbling rate. The GDs which required a re-bubbling were on average 56 μm higher and 461 μm wider than the untreated ones. The cornea above the GDs that needed a re-bubbling was significantly thicker than above the untreated GDs (mean 988 ± 102 μm vs. 951 ± 99 μm). The RGF seems to be a major influencing factor for graft attachment. The most susceptible location of the GD is inferotemporal. The main factors that need to be investigated to decide if a re-bubbling is required are the number of detachments per graft, their dimensions, whether the central portions of the graft are involved, and the corneal thickness above GDs.

Weak polynomial identities of small degree for the Weyl algebra

In this paper we investigate weak polynomial identities for the Weyl algebra $\mathsf{A}_1$ over an infinite field of arbitrary characteristic. Namely, we describe weak polynomial identities of the minimal degree, which is three, and of degrees 4 and 5. We also describe weak polynomial identities is two variables.