In this paper, we consider singular systems of linear forms over global function fields of class number one and give an upper bound for the Hausdorff dimension of the set of singular systems of linear forms by constructing an appropriate Margulis height function on the space of lattices over global function fields.

We study the joint distribution of values of a pair consisting of a quadratic form and a linear form over the set of integral vectors, a problem initiated by Dani and Margulis [Orbit closures of generic unipotent flows on homogeneous spaces of SL3(R). Math. Ann. 286 (1990), 101-128]. In the spirit of the celebrated theorem of Eskin, Margulis and Mozes on the quantitative version of the Oppenheim conjecture, we show that if n >= 5, then under the assumptions that for every (alpha, beta) is an element of R-2 \ {(0, 0)}, the form alpha q + beta l(2) is irrational and that the signature of the restriction of q to the kernel of l is (p, n - 1 - p), where 3 <= p <= n - 2, the number of vectors v is an element of Z(n) for which IIvII < T, a < q(v) < b and c < l(v) is asymptotically C(q, 1) (d - c)(b - a)Tn-3 as T -> infinity where C(q, 1) only depends on q and l. The density of the set of joint values of (q, 1) under the same assumptions is shown by Gorodnik [Oppenheim conjecture for pairs consisting of a linear form and a quadratic form. Trans. Amer. Math. Soc. (2004), 4447-4463].

For given & varepsilon;>0 and b is an element of R-m, we say that a real mxn matrix A is & varepsilon;-badly approximable for the target b if lim inf(q is an element of Zn,& Vert;q & Vert;->infinity)& Vert;q & Vert;(n)< Aq-b >(m)>=& varepsilon;, where <& sdot;> denotes the distance from the nearest integral vector. In this article, we obtain upper bounds for the Hausdorff dimensions of the set of & varepsilon;-badly approximable matrices for fixed target b and the set of & varepsilon;-badly approximable targets for fixed matrix A. Moreover, we give a Diophantine condition of A equivalent to the full Hausdorff dimension of the set of & varepsilon;-badly approximable targets for fixed A. The upper bounds are established by effectivizing entropy rigidity in homogeneous dynamics, which is of independent interest. For the A-fixed case, our method also works for the weighted setting where the supremum norms are replaced by certain weighted quasinorms.

We study inhomogeneous Diophantine approximation over the completion Kv of a function field K (over a finite field) for a discrete valuation v, with affine algebra Rv. We obtain an effective upper bound for the Hausdorff dimension of the set �BadA(⠂) = & UTheta; E Kvm : � lim inf ⠄q ⠄n ⠄Aq - & UTheta; - p ⠄m ⠂ ⠂ , (p,q)ERvm xRvn,IIqII-& iota;oo of ⠂-badly approximable targets & UTheta; E Kvm given A E Mm,n(Kv), using an effective version of entropy rigidity in homogeneous dynamics for some diagonal action on the space of Rv-grids. We characterize matrices A for which BadA(⠂) has full Hausdorff dimension for some ⠂ > 0 by a Diophantine condition of singularity on average. Our methods work for the approximation using weighted ultrametric distances.& COPY; 2023 Elsevier Inc. All rights reserved.

Tinnitus can be defined as the conscious perception of phantom sounds in the absence of corresponding external auditory signals. Tinnitus can develop in the setting of sudden sensorineural hearing loss (SSNHL), but the underlying mechanism is largely unknown. Using electroencephalography, we investigated differences in afferent node capacity between 15 SSNHL patients without tinnitus (NT) and 30 SSNHL patients with tinnitus (T). Where the T group showed increased afferent node capacity in regions constituting a "triple brain network" [default mode network (DMN), central executive network (CEN), and salience network (SN)], the NT group showed increased information flow in regions implicated in temporal auditory processing and noise-canceling pathways. Our results demonstrate that when all components of the triple network are activated due to sudden-onset auditory deprivation, tinnitus ensues. By contrast, auditory processing-associated and tinnitus-suppressing networks are highly activated in the NT group, to overcome the activation of the triple network and effectively suppress the generation of tinnitus.

BackgroundTopological data analysis (TDA) can generate patient-patient similarity networks by analyzing large, complex data and derive new insights that may not be possible with standard statistics.ObjectivesThe purpose of this paper was to discover novel phenotypes of chronic primary mitral regurgitation (MR) patients and to analyze their clinical implications using network analysis of echocardiographic data.MethodsPatients with chronic moderate to severe primary MR were prospectively enrolled from 11 Asian tertiary hospitals (n = 850; mean age 56.9 ± 14.2 years, 57.9% men). We performed TDA to generate network models using 14 demographic and echocardiographic variables. The patients were grouped by phenotypes in the network, and the prognosis was compared by groups.ResultsThe network model by TDA revealed 3 distinct phenogroups. Group A was the youngest with fewer comorbidities but increased left ventricular (LV) end-systolic volume, representing compensatory LV dilation commonly seen in chronic primary MR. Group B was the oldest with high blood pressure and a predominant diastolic dysfunction but relatively preserved LV size, an unnoticed phenotype in chronic primary MR. Group C showed advanced LV remodeling with impaired systolic, diastolic function, and LV dilation, indicating advanced chronic primary MR. During follow-up (median 3.5 years), 60 patients received surgery for symptomatic MR or died of cardiovascular causes. Kaplan-Meier curves demonstrated that although group C had the worst clinical outcome (P < 0.001), group B, characterized by diastolic dysfunction, had an event-free survival comparable to group A despite preserved LV chamber size. The grouping information by the network model was an independent predictor for the composite of MR surgery or cardiovascular death (adjusted HR: 1.918; 95% CI: 1.257-2.927; P = 0.003).ConclusionsThe patient-patient similarity network by TDA visualized diverse remodeling patterns in chronic primary MR and revealed distinct phenotypes not emphasized currently. Importantly, diastolic dysfunction deserves equal attention when understanding the clinical presentation of chronic primary MR.

. For given ǫ ą 0 and b P R m , we say that a real m ˆ n matrix A is ǫ -badly approximable for the target b if lim inf q P Z n , } q }Ñ8 } q } n x Aq ´ b y m ě ǫ, where x¨y denotes the distance from the nearest integral vector. In this article, we obtain upper bounds for the Hausdorff dimensions of the set of ǫ -badly approximable matrices for fixed target b and the set of ǫ -badly approximable targets for fixed matrix A . Moreover, we give a Diophantine condition of A equivalent to the full Hausdorff dimension of the set of ǫ -badly approximable targets for fixed A . The upper bounds are established by effectivizing entropy rigidity in homogeneous dynamics, which is of independent interest. For the A -fixed case, our method also works for the weighted setting where the supremum norms are replaced by certain weighted quasinorms.

Consider the heat kernel $p(t,x,y)$ on the universal cover $X$ of a Riemannian manifold $M$ of negative curvature. We show the local limit theorem for $p$ : $$\lim_{t \to \infty} t^{3/2}e^{\lambda_0 t} p(t,x,y)=C(x,y),$$ where $\lambda_0$ is the bottom of the spectrum of the geometric Laplacian and $C(x,y)$ is a positive function which depends on $x, y \in X$. We also show that the $\lambda_0$-Martin boundary of $X$ is equal to its topological boundary. The Martin decomposition of $C(x,y)$ gives a family of measures $\{\mu^{\lambda_0}_x \}$ on $\partial \widetilde{M}$. We show that $\{\mu^{\lambda_0}_x \}$ is the unique family minimizing the energy or the Rayleigh quotient of Mohsen. We use the uniform Harnack inequality on the boundary $\partial X$ and the uniform three-mixing of the geodesic flow on the unit tangent bundle $SM$ for suitable Gibbs-Margulis measures.

Let $\alpha $ be an irrational real number. We show that the set of $\varepsilon $-badly approximable numbers $$\begin{equation*} \textrm{Bad}^\varepsilon (\alpha):= \Big\{x\in [0,1]\,: \, \liminf_{|q| \to \infty} |q| \cdot \| q\alpha -x \| \geq \varepsilon \Big\} \end{equation*}$$has full Hausdorff dimension for some positive $\varepsilon $ if and only if $\alpha $ is singular on average. The condition is equivalent to the average $\frac{1}{k} \sum _{i=1, \cdots , k} \log a_i$ of the logarithms of the partial quotients $a_i$ of $\alpha $ going to infinity with $k$. We also consider one-sided approximation, obtain a stronger result when $a_i$ tends to infinity, and establish a partial result in higher dimensions.

Let X be a locally finite Gromov hyperbolic graph whose Gromov boundary consists of infinitely many points and with a cocompact isometric action of a discrete group. We show the uniform Ancona inequality for the Brownian motion which implies that the lambda-Martin boundary coincides with the Gromov boundary for any lambda is an element of [0, lambda(0)], in particular at the bottom of the spectrum lambda(0).

Along with phantom pain, tinnitus, a phantom auditory perception occurring in the absence of an external acoustic stimulus, is one of the most representative phantom perceptions that develops in subjects with decreased peripheral sensory input. Although tinnitus is closely associated with peripheral hearing loss (HL), it remains unclear why only some individuals with HL develop tinnitus. In this study, we investigated the differences between 65 HL with tinnitus (HL-T) and 104 HL with no tinnitus (HL-NT) using a resting-state electroencephalography data-based volume entropy model of the brain network, by comparing the afferent node capacities, that quantify the contribution of each node to the spread of information, of all Brodmann areas. While the HL-T group showed increased information flow in areas involved in Bayesian inference (the left orbitofrontal cortex, the left subgenual anterior cingulate cortex, and the left ventrolateral prefrontal cortex) and auditory memory storage (the right hippocampus/parahippocampus), the HL-NT group showed increased afferent node capacity in hub areas of the default mode network (DMN; the right posterior cingulate cortex and the right medial temporal gyrus). These results suggest that the balance of activity between the Bayesian inferential network (updating missing auditory information by retrieving auditory memories from the hippocampus/parahippocampus) and DMN (maintaining the "silent status quo") determines whether phantom auditory perception occurs in a brain with decreased peripheral auditory input.

In this article, we prove an extreme value theorem on the limit distribution of geodesics in a geometrically finite quotient of $\Gamma\backslash\mathcal{T}$ a locally finite tree. Main examples of such graphs are quotients of a Bruhat-Tits tree $\mathcal{T}$ by non-cocompact discrete subgroups $\Gamma$ of $PGL(2,\mathbf{K})$ of a positive characteristic local field $\mathbf{K}$. We investigate, for a given time $T$, the measure of the set of $\Gamma$-equivalent geodesic classes which stay up to time $T$ the region of distance $d$ at most $N$ depending on $T$ from a fixed compact subset $D$ of $\Gamma\backslash\mathcal{T}$. Namely, for Bowen-Margulis measure $\mu$ on the space $\Gamma\backslash\mathcal{GT}$ of geodesics and the critical exponent $\delta$ of $\Gamma$, we show that there exists a constant $C$ depending on $\Gamma$ and $D$ such that $$\lim_{T\to\infty}\mu\left(\left\{[l]\in\Gamma\backslash\mathcal{GT}\colon \underset{0\le t \le T}{\textrm{max}}d(D,l(t))\le N+y\right\}\right)=e^{-q^y/e^{2\delta y}}$$ with $$N=\log_{e^{2\delta/q}}\left(\frac{T(e^{2\delta-q)}}{2e^{2\delta}-C(e^{2\delta}-q)}\right).$$

OBJECTIVES This study sought to identify distinct patient groups and their association with outcome based on the patient similarity network using quantitative coronary plaque characteristics from coronary computed tomography angiography (CTA). BACKGROUND Coronary CTA can noninvasively assess coronary plaques quantitatively. METHODS Patients who underwent 2 coronary CTAs at a minimum of 24 months' interval were analyzed (n = 1,264). A similarity Mapper network of patients was built by topological data analysis (TDA) based on the whole-heart quantitative coronary plaque analysis on coronary CTA to identify distinct patient groups and their association with outcome. RESULTS Three distinct patient groups were identified by TDA, and the patient similarity network by TDA showed a dosed loop, demonstrating a continuous trend of coronary plaque progression. Group A had the least coronary plaque amount (median 12.4 mm(3) [interquartile range (IQR): 0.0 to 39.6 mm(3)]) in the entire coronary tree. Group B had a moderate coronary plaque amount (31.7 mm(3) [IQR: 0.0 to 127.4 mm(3)]) with relative enrichment of fibrofatty and necrotic core (32.6% [IQR: 16.7% to 46.2%] and 2.7% [IQR: 0.1% to 6.9%] of the total plaque, respectively) components. Group C had the largest coronary plaque amount (187.0 mm(3) [IQR: 96.7 to 306.4 mm(3)]) and was enriched for dense calcium component (46.8% [IQR: 32.0% to 63.7%] of the total plaque). At follow-up, total plaque volume, fibrous, and dense calcium volumes increased in all groups, but the proportion of fibrofatty component decreased in groups B and C, whereas the necrotic core portion decreased in only group B (all p< 0.05). Group B showed a higher acute coronary syndrome incidence than other groups (0.3% vs. 2.6% vs. 0.6%; p= 0.009) but both group B and C had a higher revascularization incidence than group A (3.1% vs. 15.5% vs. 17.8%; p < 0.001). Incorporating group information from TDA demonstrated increase of model fitness for predicting acute coronary syndrome or revascularization compared with that incorporating clinical risk factors, percentage diameter stenosis, and high-risk plaque features. CONCLUSIONS The TDA of quantitative whole-heart coronary plaque characteristics on coronary CTA identified distinct patient groups with different plaque dynamics and clinical outcomes. (Progression of AtheRosclerotic PlAque Determined by Computed TomoGraphic Angiography Imaging [PARADIGM]; NCT02803411) (C) 2021 by the American College of Cardiology Foundation.

Attention-deficit hyperactivity disorder (ADHD) is a complex brain development disorder characterized by hyperactivity/impulsivity and inattention. A major hypothesis of ADHD is a lag of maturation, which is supported mainly by anatomical studies evaluating cortical thickness. Here, we analyzed changes of topological characteristics of whole-brain metabolic connectivity in twelve SHR rats selected as ADHD-model rats by confirming behavior abnormalities using the marble burying test, open field test, and delay discounting task and 12 Wistar Kyoto rats as the control group, across development from 4 weeks old (childhood) and 6 weeks old (entry of puberty). A topological approach based on graph filtrations revealed a lag in the strengthening of limbic-cortical/subcortical connections in ADHD-model rats. This in turn related to impaired modularization of memory and reward-motivation associated regions. Using mathematical network analysis techniques such as single linkage hierarchical clustering and volume entropy, we observed left-lateralized connectivity in the ADHD-model rats at 6 weeks old. Our findings supported the maturational delay of metabolic connectivity in the SHR model of ADHD, and also suggested the possibility of impaired and compensative reconfiguration of information flow over the brain network.

Quasi-Sturmian words, which are infinite words with factor complexity eventually n + c share many properties with Sturmian words. In this article, we study the quasi-Sturmian colorings on regular trees. There are two different types, bounded and unbounded, of quasi-Sturmian colorings. We obtain an induction algorithm similar to Sturmian colorings. We distinguish them by the recurrence function.

Brain regions send and receive information through neuronal connections in an efficient way. However, when these brain networks are organized in an abnormal way, a symptom-generating pathologic network may be generated. In this regard, based on the fact that only some of subjects with hearing loss develop tinnitus, we conjecture that subjects with hearing loss and tinnitus (HLT) may be different from subjects with hearing loss without tinnitus (HL-NT) with regard to the cortical network property. This assumption prompted us to conduct the current study comparing the HL-T group and the HL-NT group with regard to volume entropy and inflow using a relatively large quantitative electroencephalography (qEEG) data. We use volume entropy and the weights on vertices and edges, as well as other type of entropylike invariants in modeling the brain networks of various populations including tinnitus and hearing loss using EEG. Volume entropy of a metric graph, a global measure of information, measures the exponential growth rate of the number of network paths. On the other hand, weight vectors of nodes and edges, which are local measures of information, represent the stationary distribution of information propagation in brain networks.

Factor complexity b(n) (phi) for a vertex coloring phi of a regular tree is the number of classes of n-balls up to color-preserving automorphisms. Sturmian colorings are colorings of minimal unbounded factor complexity b(n) (phi) = n + 2. In this article, we prove an induction algorithm for Sturmian colorings using colored balls in a way analogous to the continued fraction algorithm for Sturmian words. Furthermore, we characterize Sturmian colorings in terms of the data appearing in the induction algorithm.

We show that for almost any vector $v$ in $\mathbb{R}^n$, for any $\epsilon>0$ there exists $\delta>0$ such that the dimension of the set of vectors $w$ satisfying $\liminf_{k\to\infty} k^{1/n} \ge \epsilon$ (where $ $ denotes the distance from the nearest integer), is bounded above by $n-\delta$. This result is obtained as a corollary of a discussion in homogeneous dynamics and the main tool in the proof is a relative version of the principle of uniqueness of measures with maximal entropy.