We prove that over an algebraically closed field $\mathbb{K}$ of characteristic different from $2$, the group algebra $R=\mathbb{K} D_\infty$ of the infinite dihedral group $D_\infty$ has exactly six conjugacy classes of involutions (equivalently, of idempotents). This allows us to recover the fact that $R$ admits exactly four non-isomorphic indecomposable projective modules of the form $eR$ where $e$ is an idempotent, a result that was first established by Berman and Buz\'asi.

We generalize the notion of a root system by relaxing the conditions that ensure that it is invariant under reflections and study the resulting structures, which we call generalized root systems (GRSs for short). Since both Kostant root systems and root systems of Lie superalgebras are examples of GRSs, studying GRSs provides a uniform axiomatic approach to studying both of them. GRSs inherit many of the properties of root systems. In particular, every GRS defines a crystallographic hyperplane arrangement. We believe that GRSs provide an intrinsic counterpart to finite Weyl groupoids and crystallographic hyperplane arrangements, extending the relationship between finite Weyl groupoids and crystallographic hyperplane arrangements established by Cuntz. An important difference between GRSs and root systems is that GRSs may lack a (large enough) Weyl group. In order to compensate for this, we introduce the notion of a virtual reflection, building on a construction of Penkov and Serganova in the context of root systems of Lie superalgebras. The most significant new feature of GRSs is that, along with subsystems, one can define quotient GRSs. Both Kostant root systems and root systems of Lie superalgebras are equivalent to quotients of root systems and all root systems are isomorphic to quotients of simply-laced root systems. We classify all rank 2 GRSs and show that they are equivalent to quotients of root systems. Finally, we discuss in detail quotients of root systems. In particular we provide all isomorphisms and equivalences among them. Our results on quotient of root systems provide a different point of view on flag manifolds, reproving results of Alekseevsky and Graev.

We use quiver representation theory and King's notion of stable representations to study the (finite dimensional) representations of free products of semi-simple associative K-algebras over an algebraically closed field K. We obtain numerical criteria giving necessary conditions for a module over such a free product to be simple and prove that these conditions are sufficient for modules in general position. We also prove that modules in general position are always semisimple, although non-semisimple modules always exist when the algebra is infinite dimensional. With essentially a single exception, all module categories are strictly wild; moreover these categories admit no bounds on the dimensions of simple modules. We use quiver moduli spaces to derive a closed formula for the number of parameters needed to describe all simple modules in a given dimension. As a special case, we apply our results to free products of finite groups, and recover the representation theory of the projective modular group PSL2(Z) (strictly wild) and of the infinite dihedral group (tame).

Current breast cancer diagnosis and treatment protocols achieve survival rates of>80% at 10 years.However, long-term cancer survivors have>4-fold higher incidence of cardiovascular complications, including heart failure.Early potentially reversible changes that may underlie downstream irreversible loss of mechanical function are unknown.Therefore, we aimed to test the feasibility of a noninvasive cardiac phenotyping protocol in patients undergoing treatment for breast cancer with doxorubicin, without or with immune checkpoint inhibitors, including multiparametric proton cardiac magnetic resonance(CMR) imaging and novel metabolic carbon13MR. A comprehensive set of studies included conventional 1HCMR,as well as metabolic 13CCMR using hyperpolarized(HP)[1-13C] pyruvate, following an oral glucose load. 13CCMR with HP[1-13C] pyruvate allows the detection of downstream metabolites of injected pyruvate within the myocardium: bicarbonate from mitochondrial oxidation and lactate from anaerobic cytosolic metabolism.HP[1-13C]pyruvate was prepared using dynamic nuclear polarization in a commercial polarizer.Imaging and spectroscopy(MRS) of hyperpolarized pyruvate was conducted using a 13C transmit/receive Helmholtz loop-pair coil on a 3Tscanner.13C data were acquired from a short-axis slice, ECG-triggered in end-systole. ProtonCMR images were analyzed using Circle, and 13CCMR images were analyzed using MATLAB. The left ventricular mid-myocardium and blood pool were segmented for metabolite quantitation. Baseline cardiac studies were performed in a consecutive series of 25 women treated for breast cancer. Of these 14(56%) were Caucasian, 3(12%) were Hispanic, 5(20%) Black, and 3(12%) Asian, with ages ranging 46±11 years.Patients received standard-of-care treatment plans based on the tumor type: 11 patients were treated with dose-dense anthracycline and cyclophosphamide, and 14 also received pembrolizumab. The comprehensive baseline phenotyping protocol is planned to be repeated after standard of care therapy in all patients. Baseline data are presented in Table1, with values reported as mean±standard deviation. Noninvasive cardiac phenotyping, including dynamic detection of metabolic substrate utilization is feasible,without showing significant adverse effects,and is well tolerated in a diverse group of patients.Completion of this protocol post-oncological therapy may allow to detect early changes in cardiac function due to chemo-and immunotherapy in susceptible patients.

In this paper, we study the existence and classification problems of left-symmetric superalgebras on special linear Lie superalgebras sl(m|n) with m≠n. The main three results of this paper are: (i) a complete classification of the left-symmetric superalgebras on sl(2|1), (ii) sl(m|1) does not admit left-symmetric superalgebras for m≥3, and (iii) sl(m+1|m) admits a left-symmetric superalgebra for every m≥1. To prove these results we combine previous results on the existence and classification of left-symmetric algebras on the Lie algebras glm with a detailed analysis of small representations of the Lie superalgebras sl(m|1). We also conjecture that sl(m|n) admits left-symmetric superalgebras if and only if m=n+1.

The main result of this paper is a recursive description of all decompositions Δ^+ = Φ_1 ⊔Φ_2 ⊔…⊔Φ_k of the positive roots Δ^+ of an arbitrary root system Δ into a disjoint union of inversion sets. Such decompositions play a central role in geometric invariant theory (GIT) in connection with studying the Littlewood-Richardson cone and related problems. This work can be considered as a continuation of the work of Dewji, Dimitrov, McCabe, Roth, Wehlau, and Wilson in which similar questions were studied for root systems of type 𝔸. Their methods relied on properties of permutations and are not transferable to an arbitrary root system. In order to develop a type-independent approach, we go beyond root systems and consider quotient root systems (QRSs for short). We study subsets of positive roots in an arbitrary QRS R. We prove that every Φ⊆ R^+ can be represented in a canonical way as an inflation and develop methods to study recursively properties of such subsets. We extend the notion of an inversion to subsets of any QRS, i.e., beyond the case where a Weyl group is associated with R. If Φ⊆ R^+ is an inversion set, we introduce a graph G(Φ) and endow the set Comp(Φ) of connected components of G(Φ) with a partial addition. The resulting monoid-like structure (Comp(Φ),+) is a further generalization of root systems beyond QRSs. We study in detail the properties of (Comp(Φ),+) and their applications to studying the properties of Φ. In particular, we investigate the relationship between Φ being primitive and Φ being irreducible. Apart from describing recursively all decompositions of Δ^+ into the disjoint union of inversion sets, we provide applications to GIT and derive enumerative results which may be of independent interest.

Chemical exchange saturation transfer (CEST) MRI has gained recognition as a valuable addition to the molecular imaging and quantitative biomarker arsenal, especially for characterization of brain tumors. There is also increasing interest in the use of CEST-MRI for applications beyond the brain. However, its translation to body oncology applications lags behind those in neuro-oncology. The slower migration of CEST-MRI to non-neurologic applications reflects the technical challenges inherent to imaging of the torso. In this review, we discuss the application of CEST-MRI to oncologic conditions of the breast and torso (i.e., body imaging), emphasizing the challenges and potential solutions to address them. While data are still limited, reported studies suggest that CEST signal is associated with important histology markers such as tumor grade, receptor status, and proliferation index, some of which are often associated with prognosis and response to therapy. However, further technical development is still needed to make CEST a reliable clinical application for body imaging and establish its role as a predictive and prognostic biomarker.

Hydrogen peroxide (H2O2) is a type of reactive oxygen species that regulates essential biological processes. Despite the central role of H2O2 in pathophysiological states, available molecular probes for assessing H2O2 in vivo are still limited. This work develops hyperpolarized 15N-boronobenzyl-4-cyanopyridinium (15N-BBCP) as a rationally designed molecular probe for detecting H2O2. The 15N-BBCP demonstrated favorable physicochemical and biochemical properties for H2O2 detection and dynamic nuclear polarization, allowing noninvasive detection of H2O2. In particular, 15N-BBCP and the products possessed long spin-lattice relaxation times and spectrally resolvable 15N chemical shift differences. The performance of hyperpolarized 15N-BBCP was demonstrated both in vitro and in vivo with time-resolved 15N-MRS. This study highlights a promising approach to designing a reaction-based 15N-labeled molecular imaging agent for detecting oxidative stress in vivo.

We use quivers and their representations to bring new perspectives on the subregular J-ring JC of a Coxeter system (W,S), a subring of Lusztig's J-ring. We prove that JC is isomorphic to a suitable quotient of the path algebra of the double quiver of (W,S). Up to Morita equivalence, such quotients include the group algebras of all free products of finite cyclic groups. We then use quiver representations to study the category mod-AK of finite dimensional right modules of the algebra AK=K⊗ZJC over an algebraically closed field K of characteristic zero. Our results include classifications of Coxeter systems for which mod-AK is semisimple, has finitely many simple modules up to isomorphism, or has a bound on the dimensions of simple modules.

Objective: Considering the reported elevation of ω-6/ω-3 fatty acid ratios in breast neoplasms, one particularly important application of 13C MRS could be in more fully understanding the breast lipidome's relationship to breast cancer incidence. However, the low natural abundance and gyromagnetic ratio of the 13C isotope lead to detection sensitivity challenges. Previous 13C MRS studies have relied on the use of small surface coils with limited field-of-view and shallow penetration depths to achieve adequate signal-to-noise ratio (SNR), and the use of receive array coils is still mostly unexplored. Methods: This work presents a unilateral breast 16-channel 13C array coil and interfacing hardware designed to retain the surface sensitivity of a single small loop coil while improving penetration depth and extending the field-of-view over the entire breast at 7T. The coil was characterized through bench measurements and phantom 13C spectroscopy experiments. Results: Bench measurements showed receive coil matching better than -17 dB and average preamplifier decoupling of 16.2 dB with no evident peak splitting. Phantom MRS studies show better than a three-fold increase in average SNR over the entirety of the breast region compared to volume coil reception alone as well as an ability for individual array elements to be used for coarse metabolite localization without the use of single-voxel or spectroscopic imaging methods. Conclusion: Our current study has shown the benefits of the array. Future in vivo lipidomics studies can be pursued. Significance: Development of the 16-channel breast array coil opens possibilities of in vivo lipidomics studies to elucidate the link between breast cancer incidence and lipid metabolics.

Purpose shimming is an important method for mitigating B1 inhomogeneity in high‐field MRI. Using independent power amplifiers for each transmit (Tx) element is the preferred method for B1 shimming but comes with a high cost. Conversely, the simplest approach to control a Tx array is by using coaxial cables of varying length in the Tx chain, but this approach is cumbersome and impractical for dynamic shimming. In this article, a system is described that enables dynamic, phase‐only, eight‐channel steering on a 7T MR scanner with only two power amplifiers.MethodsPower dividers were utilized to first split the existing two‐channel Tx signal into eight channels. Digitally controlled phase shifters on each channel were designed to provide independent phase shifts with a resolution of 22.5° (from 0°, 22.5° … 337.5°). To validate the system, an eight‐channel body dipole array was simulated and constructed for bench and 7T imaging and evaluation.ResultsThe phase conjugate steering method was employed at three different spatial positions in simulation, bench measurements, and scanner measurements—all with matching results. At the desired points, regions with homogenous were generated, indicating good Tx steering to the selected region.ConclusionThe described system can be used as a simple retrofit to existing hardware to provide phase control while avoiding the need to manually switch cables and without requiring independent power amplifiers for each channel, thus demonstrating the ability to perform dynamic shimming with increased degrees of freedom but without significantly increased hardware cost.

We study the eigenspace decomposition of a basic classical Lie superalgebra under the adjoint action of a toral subalgebra, thus extending results of Kostant. In recognition of Kostant's contribution we refer to the eigenfunctions appearing in the decomposition as Kostant roots. We then prove that Kostant root systems inherit the main properties of classical root systems. Our approach is combinatorial in nature and utilizes certain graphs naturally associated with Kostant root systems. In particular, we reprove Kostant's results without making use of the Killing form.

Purpose: H-1 MRS provides a noninvasive tool for identifying mutations in isocitrate dehydrogenase (IDH). Quantification of the prominent 2-hydroxyglutarate (2HG) resonance at 2.25 ppm is often confounded by the lipid resonance at the same frequency in tumors with elevated lipids. We propose a new spectral fitting approach to separate these overlapped signals, therefore, improving 2HG evaluation. Methods: TE 97 ms PRESS was acquired at 3T from 42 glioma patients. New lipid basis sets were created, in which the small lipid 2.25-ppm signal strength was preset with reference to the lipid signal at 0.9 ppm, incorporating published fat relaxation data. LCModel fitting using the new lipid bases (Fitting method 2) was conducted along with fitting using the LCModel built-in lipid basis set (Fitting method 1), in which the lipid 2.25-ppm signal is assessed with reference to the lipid 1.3-ppm signal. In-house basis spectra of low-molecular-weight metabolites were used in both fitting methods. Results: Fitting method 2 showed marked improvement in identifying IDH mutational status compared with Fitting method 1. 2HG estimates from Fitting method 2 were overall smaller than those from Fitting method 1, which was because of differential assignment of the signal at 2.25 ppm to lipids. In receiver operating characteristic analysis, Fitting method 2 provided a complete distinction between IDH mutation and wild-type whereas Fitting method 1 did not. Conclusion: The data suggest that H-1 MR spectral fitting using the new lipid basis set provides a robust fitting strategy that improves 2HG evaluation in brain tumors with elevated lipids.

Purpose The recently introduced inhomogeneous magnetization transfer (ihMT) method has predominantly been applied for imaging the central nervous system. Future applications of ihMT, such as in peripheral nerves and muscles, will involve imaging in the vicinity of adipose tissues. This work aims to systematically investigate the partial volume effect of fat on the ihMT signal and to propose an efficient fat-separation method that does not interfere with ihMT measurements. Methods First, the influence of fat on ihMT signal was studied using simulations. Next, the ihMT sequence was combined with a multi-echo Dixon acquisition for fat separation. The sequence was tested in 9 healthy volunteers using a 3T human scanner. The ihMT ratio (ihMTR) values were calculated in regions of interest in the brain and the spinal cord using standard acquisition (no fat saturation), water-only, in-phase, and out-of-phase reconstructions. The values obtained were compared with a standard fat suppression method, spectral presaturation with inversion recovery. Results Simulations showed variations in the ihMTR values in the presence of fat, depending on the TEs used. The IhMTR values in the brain and spinal cord derived from the water-only ihMT multi-echo Dixon images were in good agreement with values from the unsuppressed sequence. The ihMT-spectral presaturation with inversion recovery combination resulted in 24%-35% lower ihMTR values compared with the standard non-fat-suppressed acquisition. Conclusion The presence of fat within a voxel affects the ihMTR calculations. The IhMT multi-echo Dixon method does not compromise the observable ihMT effect and can potentially be used to remove fat influence in ihMT.

OBJECTIVE:Most MRI scanners are equipped to receive signals from 1H array coils but few support multi-channel reception for other nuclei. Using receive arrays can provide significant SNR benefits, usually exploited to enable accelerated imaging, but the extension of these arrays to non-1H nuclei has received less attention because of the relative lack of broadband array receivers. Non-1H nuclei often have low sensitivity and stand to benefit greatly from the increase in SNR that arrays can provide. This paper presents a cost-effective approach for adapting standard 1H multi-channel array receivers for use with other nuclei - in this case, 13C.METHODS:A frequency translation system has been developed that uses active mixers residing at the magnet bore to convert the received signal from a non-1H array to the 1H frequency for reception by the host system receiver.RESULTS:This system has been demonstrated at 4.7T and 7T while preserving SNR and isolation. 1H decoupling, particularly important for 13C detection, can be straightforwardly accommodated.CONCLUSION:Frequency translation can convert 1H-only multi-channel receivers for use with other nuclei while maintaining SNR and channel isolation while still enabling 1H decoupling.SIGNIFICANCE:This work allows existing multi-channel MRI receivers to be adapted to receive signals from nuclei other than 1H, allowing for the use of receive arrays for in vivo multi-nuclear NMR.

We evaluated an alternative diffusion-weighted imaging (DWI) acquisition for prostate magnetic resonance imaging of men with pelvic hardware, using radial k -space sampling (MultiVane [MV]), short-tau inversion-recovery (STIR) fat suppression, and split acquisition of turbo spin-echo signals. The optimized STIR-MV-DWI reduced metal-associated artifacts and image distortion, and aided in visualization of the prostate and lesions. The STIR-MV-DWI can be a valuable adjunct in prostate magnetic resonance imaging of men with pelvic hardware, among whom the conventional echo-planar DWI is compromised.

BACKGROUND AND PURPOSE:Spiral MR imaging has several advantages compared with Cartesian MR imaging that can be leveraged for added clinical value. A multicenter multireader study was designed to compare spiral with standard-of-care Cartesian postcontrast structural brain MR imaging on the basis of relative performance in 10 metrics of image quality, artifact prevalence, and diagnostic benefit.MATERIALS AND METHODS:Seven clinical sites acquired 88 total subjects. For each subject, sites acquired 2 postcontrast MR imaging scans: a spiral 2D T1 spin-echo, and 1 of 4 routine Cartesian 2D T1 spin-echo/TSE scans (fully sampled spin-echo at 3T, 1.5T, partial Fourier, TSE). The spiral acquisition matched the Cartesian scan for scan time, geometry, and contrast. Nine neuroradiologists independently reviewed each subject, with the matching pair of spiral and Cartesian scans compared side-by-side, and scored on 10 image-quality metrics (5-point Likert scale) focused on intracranial assessment. The Wilcoxon signed rank test evaluated relative performance of spiral versus Cartesian, while the Kruskal-Wallis test assessed interprotocol differences.RESULTS:Spiral was superior to Cartesian in 7 of 10 metrics (flow artifact mitigation, SNR, GM/WM contrast, image sharpness, lesion conspicuity, preference for diagnosing abnormal enhancement, and overall intracranial image quality), comparable in 1 of 10 metrics (motion artifacts), and inferior in 2 of 10 metrics (susceptibility artifacts, overall extracranial image quality) related to magnetic susceptibility (P < .05). Interprotocol comparison confirmed relatively higher SNR and GM/WM contrast for partial Fourier and TSE protocol groups, respectively (P < .05).CONCLUSIONS:Spiral 2D T1 spin-echo for routine structural brain MR imaging is feasible in the clinic with conventional scanners and was preferred by neuroradiologists for overall postcontrast intracranial evaluation.

PurposeThis work describes the construction and evaluation of a bilateral 32‐channel receive array for breast imaging at 7T.MethodsThe receive array consisted of 32 receive coils, placed on two 3D‐printed hemispherical formers. Each side of the receive array consisted of 16 receive loops, each loop having a corresponding detachable board with match/tune capacitors, active detuning circuitry, and a balun. Coil performance was evaluated on homogeneous canola oil phantoms using a Philips Achieva 7T system. Array coil performance was compared with a bilateral forced current excitation volume coil in transmit/receive mode and with a previously reported 16‐channel unilateral coil with a similar design.ResultsThe 32‐channel array had an increase in average SNR throughout both phantoms by a factor of five as compared with the volume coil, with SNR increases up to 10 times along the periphery and three times in the center. Noise measurements showed low interelement noise correlation (average: 5.4%; maximum: 16.8%). Geometry factor maps were acquired for various acceleration factors and showed mean geometry factors <1.2, for combined acceleration factors of up to six.ConclusionsThe improvements achieved demonstrate the clear potential for use in dynamic contrast‐enhanced or diffusion‐weighted MR studies, while maintaining diagnostically relevant spatial and temporal resolutions.

Background. High-grade gliomas likely remodel the metabolic machinery to meet the increased demands for amino acids and nucleotides during rapid cell proliferation. Glycine, a non-essential amino acid and intermediate of nucleotide biosynthesis, may increase with proliferation. Non-invasive measurement of glycine by magnetic resonance spectroscopy (MRS) was evaluated as an imaging biomarker for assessment of tumor aggressiveness. Methods. We measured glycine, 2-hydroxyglutarate (2HG), and other tumor-related metabolites in 35 glioma patients using an MRS sequence tailored for co-detection of glycine and 2HG in gadolinium-enhancing and non-enhancing tumor regions on 3T MRI. Glycine and 2HG concentrations as measured by MRS were correlated with tumor cell proliferation (MIB-1 labeling index), expression of mitochondria' serine hydroxymethyltransferase (SHMT2), and glycine decarboxylase (GLDC) enzymes, and patient overall survival. Results. Elevated glycine was strongly associated with presence of gadolinium enhancement, indicating more rapidly proliferative disease. Glycine concentration was positively correlated with MIB-1, and levels higher than 2.5 mM showed significant association with shorter patient survival, irrespective of isocitrate dehydrogenase status. Concentration of 2HG did not correlate with MIB-1 index. A high glycine/2HG concentration ratio, >2.5, was strongly associated with shorter survival (P< 0.0001). GLDC and SHMT2 expression were detectable in all tumors with glycine concentration, demonstrating an inverse correlation with GLDC. Conclusions. The data suggest that aggressive gliomas reprogram glycine-mediated one-carbon metabolism to meet the biosynthetic demands for rapid cell proliferation. MRS evaluation of glycine provides a non-invasive metabolic imaging biomarker that is predictive of tumor progression and clinical outcome.

In this note we show that when G is a classical semi-simple algebraic group, B ⊂ G a Borel subgroup, and X = G/B, then the structure coefficients of the Belkale–Kumar product ⨀0 on H*(X, Z) are all either 0 or 1.