Изопериметрические и функциональные неравенства

We establish lower estimates for an integral functional$$\int\limits_\Omega f(u(x), \nabla u(x)) \, dx ,$$where $\Omega$ -- a bounded domain in $\mathbb{R}^n \; (n \geqslant 2)$, an integrand $f(t,p) \, (t \in [0, \infty),\; p \in \mathbb{R}^n)$ -- a function that is $B$-measurable with respect to a variable $t$ and is convex and even in the variable $p$, $\nabla u(x)$ -- a gradient (in the sense of Sobolev) of the function $u \colon \Omega \rightarrow \mathbb{R}$. In the first and the second sections we utilize properties of permutations of differentiable functions and an isoperimetric inequality $H^{n-1}( \partial A) \geqslant \lambda(m_n A)$, that connects $(n-1)$-dimensional Hausdorff measure $H^{n-1}(\partial A )$ of relative boundary $\partial A$ of the set $A \subset \Omega$ with its $n$-dimensional Lebesgue measure $m_n A$. The integrand $f$ is assumed to be isotropic, i.e. $f(t,p) = f(t,q)$ if $|p| = |q|$.Applications of the established results to multidimensional variational problems are outlined. For functions $u$ that vanish on the boundary of the domain $\Omega$, the assumption of the isotropy of the integrand $f$ can be omitted. In this case, an important role is played by the Steiner and Schwartz symmetrization operations of the integrand $f$ and of the function $u$. The corresponding variants of the lower estimates are discussed in the third section. What is fundamentally new here is that the symmetrization operation is applied not only to the function $u$, but also to the integrand $f$. The geometric basis of the results of the third section is the Brunn-Minkowski inequality, as well as the symmetrization properties of the algebraic sum of sets.Устанавливаются оценки снизу интегрального функционала$$\int\limits_\Omega f(u(x), \nabla u(x)) \, dx ,$$где $\Omega$ -- ограниченная область в пространстве $\mathbb{R}^n \; (n \geqslant 2)$, интегрант $f(t,p) \, (t \in [0, \infty),\; p \in \mathbb{R}^n)$ -- функция, $B$-измеримая по переменному $t$ и выпуклая и четная по переменному $p$, $\nabla u(x)$ -- градиент (в смысле Соболева) функции $u \colon \Omega \rightarrow \mathbb{R}$. В первом и втором разделах существенную роль играют свойства перестановок дифференцируемых функций, а также изопериметрическое неравенство вида $H^{n-1}( \partial A) \geqslant \lambda(m_n A)$, связывающее $(n-1)$-мерную меру Хаусдорфа $H^{n-1}(\partial A )$ относительной границы $\partial A$ множества $A \subset \Omega$ с его $n$-мерной мерой Лебега $m_n A$. Интегрант $f$ при этом предполагается изотропным, т.е. $f(t,p) = f(t,q)$, если $|p| = |q|$. Намечены приложения установленных результатов к многомерным вариационным задачам.Для функций $u$, обращающихся в нуль на границе области $\Omega$, предположение об изотропности $f$ можно опустить. В этом случае существенную роль начинают играть операции симметризации по Штейнеру и Шварцу интегранта $f$ и функции $u$. Соответствующие варианты оценок снизу обсуждаются в третьем пункте. Принципиально новым здесь является то, что операция симметризации применяется не только к функции $u$, но и к интегранту $f$. Геометрическую основу результатов третьего пункта составляют неравенство Брунна--Минковского, а также свойства симметризации алгебраической суммы множеств

T-functions revisited: New criteria for bijectivity/transitivity

The paper presents new criteria for bijectivity/transitivity of T-functions and fast knapsack-like algorithm of evaluation of a T-function. Our approach is based on non-Archimedean ergodic theory: Both the criteria and algorithm use van der Put series to represent 1-Lipschitz $p$-adic functions and to study measure-preservation/ergodicity of these

Syndromic Assessment of Degenerative Disorders of the Lumbar Spine in Elderly Patients

Background. The choice of the method and options for surgical treatment of degenerative pathology of the lumbar spine is difficult due to the lack of clear clinical and radiological criteria for diagnosis and a direct correlation between the severity of the radiological manifestations of the disease and clinical symptoms. The aim of this study was to analyze the clinical and neurological characteristics of elderly patients with degenerative disorders of the lumbar spine and to identify the dominant clinical and radiologic syndromes. Methods. Сohort of 1013 patients were operated using MIS technologies (decompression alone, TLIF, LLIF, ALIF) in the period 2013–2017 (367 male/646 female). The age range is 60-89 years (mean 66 years). The criteria for identifying the leading syndromes: leg pain/back pain with a threshold value of 5 points according to VAS, X-ray criteria for clinical instability by A.A. White and M.M. Panjabi (value 5 points), Cobb angle 10°, markers of sagittal imbalance: Index Barrey (II and III), PT increase above target values, L4-S1 and LL deficiency. Results. Symptoms of compression were identified in 97% of patients. Radiculopathy syndrome was detected in 665 (66%) patients with mean leg pain 7 points, neurogenic intermittent claudication — in 319 (31%) patients. Degenerative spondylolisthesis according to radiological criteria was detected in 428 (42%) patients. Degenerative scoliotic deformity had 91 (9%) patients. In accordance with the proposed criteria, the dominant compression syndrome was determined in 624 patients (62%), clinical instability syndrome — in 338 (33%), deformity syndrome with sagittal imbalance — in 51 (5%). Conclusion. Syndromic assessment of clinical, neurological and radiological manifestations of degenerative disorders provides the possibility of identifying the dominant syndrome requiring operative surgical treatment and a differentiated approach to choosing the optimal surgical option

The use of LLIF technology in adult patients with degenerative scoliosis: retrospective cohort analysis and literature review

Introduction Incidence of adult degenerative scoliosis (ADS) among individuals over 50 years old reaches 68%. Surgical interventions aimed at correcting the spinal deformity in patients of the older age group are accompanied by a high risk of complications. The use of LLIF is associated with lower complications as compared with open anterior or posterior fusion. Materials and methods Seventy-one patients with ADS (13 men, 58 women) were operated at the Federal Neurosurgical Center. Their average age was 60.4/60 (average/median) [55;64.5] (1: 3 quartile) years. The follow-up was from 12 to 18 months. X-ray study, SCT, MRI of the lumbar spine were used. Questionnaire surveys were conducted using the visual analog pain scale (VAS), Oswestry Disability Index (ODI) and the Short Form-36 (SF-36). Deformity correction was estimated in the frontal plane with Cobb’s method. Scoliosis was classified according to SRS-Schwab classification. Parameters of sagittal balance were estimated: PI (Pelvic incidence), SS (Sacral slope), PT (Pelvic tilt), LL (Lumbar lordosis). SVA, PT and PILL (PI minus LL) were defined adjusted for the age. Results Back pain according to VAS relieved from 6.1/6 [4;8] to 2.2/2 [2;3] points (p < 0.001) and was statistically significant at 12 months after the surgery. Leg pain according to VAS decreased from 5.4/5 [4;8] to 2.1/2 [1;3] points (p < 0.001) and was statistically significant at 12 months after the surgery. Functional adaptation according to ODI improved from 51.2/52.2 [38.6;64.1] to 31.8/33.3 [26.1;35.9] (p < 0.001). According to SF36, PH before the surgery was 25.7/24.3 [21.8;28.9] on average and at 12 months after the surgery - 38.7/38.7 [35.4;41.2] (p < 0.001). SF-36 MH before surgery was 27.1/26.3 [21.8;31.4] on average and 12 months later – 41.3/40.6 [36.5;43.7] (p < 0.001). PT before the surgery was 23.3/22° [17.5;28], 12 months later it was 17.9/17° [15;20] (p < 0.001). PI-LL was 11.5/10 ° [4;17.5], 12 months later – 8.4/8 ° [5.5;11.5] (p = 0.11). Transient paresis of femur flexors on the ipsilateral side was observed in five (7 %) cases; transient hyposthesia on the anterior thigh surface occurred in eight (11.2 %) cases. There were two cases of medial malposition (0.4 %) of pedicle screws (474 screws), pseudoarthrosis at two levels (1.2 %) (Grade 4 Bridwell) out of 166 levels performed, and seven (4.2 %) cases of damage to cortical endplates. Conclusion Restoration of local sagittal balance in ADS patients by short-segment fixation using LLIF technology leads to a statistically significant improvement in the quality of life and increases functional adaptation. Few early and late postoperative complications, less intraoperative blood loss and shorter hospital stay make LLIF in combination with MIS transpedicular fixation a method of choice in determining the surgical tactics for ADS in elderly and old age patients

Unitary and Non-Unitary Matrices as a Source of Different Bases of Operators Acting on Hilbert Spaces

Columns of d^2 x N matrices are shown to create different sets of N operators acting on $d$-dimensional Hilbert space. This construction corresponds to a formalism of the star-product of operator symbols. The known bases are shown to be partial cases of generic formulas derived by using d^2 x N matrices as a source for constructing arbitrary bases. The known examples of the SIC-POVM, MUBs, and the phase-space description of qubit states are considered from the viewpoint of the developed unified approach. Star-product schemes are classified with respect to associated d^2 x N matrices. In particular, unitary matrices correspond to self-dual schemes. Such self-dual star-product schemes are shown to be determined by dequantizers which do not form POVM.Comment: 12 pages, 1 figure, 1 table, to appear in Journal of Russian Laser Researc

Molecular Interpretation of ACTH-β-Endorphin Coaggregation: Relevance to Secretory Granule Biogenesis

Peptide/protein hormones could be stored as non-toxic amyloid-like structures in pituitary secretory granules. ACTH and β-endorphin are two of the important peptide hormones that get co-stored in the pituitary secretory granules. Here, we study molecular interactions between ACTH and β-endorphin and their colocalization in the form of amyloid aggregates. Although ACTH is known to be a part of ACTH-β-endorphin aggregate, ACTH alone cannot aggregate into amyloid under various plausible conditions. Using all atom molecular dynamics simulation we investigate the early molecular interaction events in the ACTH-β-endorphin system, β-endorphin-only system and ACTH-only system. We find that β-endorphin and ACTH formed an interacting unit, whereas negligible interactions were observed between ACTH molecules in ACTH-only system. Our data suggest that ACTH is not only involved in interaction with β-endorphin but also enhances the stability of mixed oligomers of the entire system

Measurement-Induced State Transitions in a Superconducting Qubit: Within the Rotating Wave Approximation

Superconducting qubits typically use a dispersive readout scheme, where a resonator is coupled to a qubit such that its frequency is qubit-state dependent. Measurement is performed by driving the resonator, where the transmitted resonator field yields information about the resonator frequency and thus the qubit state. Ideally, we could use arbitrarily strong resonator drives to achieve a target signal-to-noise ratio in the shortest possible time. However, experiments have shown that when the average resonator photon number exceeds a certain threshold, the qubit is excited out of its computational subspace, which we refer to as a measurement-induced state transition. These transitions degrade readout fidelity, and constitute leakage which precludes further operation of the qubit in, for example, error correction. Here we study these transitions using a transmon qubit by experimentally measuring their dependence on qubit frequency, average photon number, and qubit state, in the regime where the resonator frequency is lower than the qubit frequency. We observe signatures of resonant transitions between levels in the coupled qubit-resonator system that exhibit noisy behavior when measured repeatedly in time. We provide a semi-classical model of these transitions based on the rotating wave approximation and use it to predict the onset of state transitions in our experiments. Our results suggest the transmon is excited to levels near the top of its cosine potential following a state transition, where the charge dispersion of higher transmon levels explains the observed noisy behavior of state transitions. Moreover, occupation in these higher energy levels poses a major challenge for fast qubit reset

Overcoming leakage in scalable quantum error correction

Leakage of quantum information out of computational states into higher energy states represents a major challenge in the pursuit of quantum error correction (QEC). In a QEC circuit, leakage builds over time and spreads through multi-qubit interactions. This leads to correlated errors that degrade the exponential suppression of logical error with scale, challenging the feasibility of QEC as a path towards fault-tolerant quantum computation. Here, we demonstrate the execution of a distance-3 surface code and distance-21 bit-flip code on a Sycamore quantum processor where leakage is removed from all qubits in each cycle. This shortens the lifetime of leakage and curtails its ability to spread and induce correlated errors. We report a ten-fold reduction in steady-state leakage population on the data qubits encoding the logical state and an average leakage population of less than $1 \times 10^{-3}$ throughout the entire device. The leakage removal process itself efficiently returns leakage population back to the computational basis, and adding it to a code circuit prevents leakage from inducing correlated error across cycles, restoring a fundamental assumption of QEC. With this demonstration that leakage can be contained, we resolve a key challenge for practical QEC at scale.Comment: Main text: 7 pages, 5 figure

Non-Abelian braiding of graph vertices in a superconducting processor

Indistinguishability of particles is a fundamental principle of quantum mechanics. For all elementary and quasiparticles observed to date - including fermions, bosons, and Abelian anyons - this principle guarantees that the braiding of identical particles leaves the system unchanged. However, in two spatial dimensions, an intriguing possibility exists: braiding of non-Abelian anyons causes rotations in a space of topologically degenerate wavefunctions. Hence, it can change the observables of the system without violating the principle of indistinguishability. Despite the well developed mathematical description of non-Abelian anyons and numerous theoretical proposals, the experimental observation of their exchange statistics has remained elusive for decades. Controllable many-body quantum states generated on quantum processors offer another path for exploring these fundamental phenomena. While efforts on conventional solid-state platforms typically involve Hamiltonian dynamics of quasi-particles, superconducting quantum processors allow for directly manipulating the many-body wavefunction via unitary gates. Building on predictions that stabilizer codes can host projective non-Abelian Ising anyons, we implement a generalized stabilizer code and unitary protocol to create and braid them. This allows us to experimentally verify the fusion rules of the anyons and braid them to realize their statistics. We then study the prospect of employing the anyons for quantum computation and utilize braiding to create an entangled state of anyons encoding three logical qubits. Our work provides new insights about non-Abelian braiding and - through the future inclusion of error correction to achieve topological protection - could open a path toward fault-tolerant quantum computing