Affine-invariant contracting-point methods for Convex Optimization

In this paper, we develop new affine-invariant algorithms for solving composite convex minimization problems with bounded domain. We present a general framework of Contracting-Point methods, which solve at each iteration an auxiliary subproblem restricting the smooth part of the objective function onto contraction of the initial domain. This framework provides us with a systematic way for developing optimization methods of different order, endowed with the global complexity bounds. We show that using an appropriate affine-invariant smoothness condition, it is possible to implement one iteration of the Contracting-Point method by one step of the pure tensor method of degree $$p \ge 1$$ p ≥ 1 . The resulting global rate of convergence in functional residual is then $${\mathcal {O}}(1 / k^p)$$ O ( 1 / k p ) , where k is the iteration counter. It is important that all constants in our bounds are affine-invariant. For $$p = 1$$ p = 1 , our scheme recovers well-known Frank–Wolfe algorithm, providing it with a new interpretation by a general perspective of tensor methods. Finally, within our framework, we present efficient implementation and total complexity analysis of the inexact second-order scheme $$(p = 2)$$ ( p = 2 ) , called Contracting Newton method. It can be seen as a proper implementation of the trust-region idea. Preliminary numerical results confirm its good practical performance both in the number of iterations, and in computational time.

Numerical Modeling of Microfluid Dynamics in Xylem Vessels of Khaya grandifoliola

Computational fluid dynamic (CFD) can be used to quantify the internal flow variables of xylem conducting vessels. This study aims to analyze through numerical simulations the xylem water ascent of African mahogany (Khaya grandifoliola) cultivated under different irrigation regimes. We determined a geometric model, defined through the variability of the anatomical structures of the species, observing characteristics of the xylem vessels such as diameter, length, number of pits, and average surface area of the pits. Then we applied numerical simulation through an Eulerian mathematical model with the discretization of volumes via CFD. Compared to other models, we observed that numerical simulation using CFD represented the xylem microstructures in a greater level of detail, contributing to the understanding of the flow of xylem vessels and the interference of its various structures. Analyzing the micrographs, we observed the non-irrigated vessels had a higher number of pits in the secondary wall thickening when compared to the irrigated treatments. This trend influenced the variability of the radial flow of the xylem vessels, causing greater fluid movement in this region and decreasing the influence of the smooth part of the wall, resulting in a lower total resistance of these vessels.

Pathogenetic Role of the Mitral Valve at Hypertrophic Cardiomyopathy

The work is devoted to the results of the study of the role of the mitral valve (MV) in the pathogenesis of hypertrophic cardiomyopathy (HCM). Purpose. To determine the role of MV in the pathogenesis of HCM. Morphological examination was performed on the operating material: 36 MV fragments, 41 specimens of the interventricular septum resected during Ferrazzi surgery, 4 autopsy cases of aortic valve stenosis (AVS). Morphological data were compared with the results of Echocardiography: 41 patients with HCM (29 men and 12 women, mean age – 39.8 ± 15.3 years), 53 patients with AVS (comparison group), 54 healthy volunteers (24 men and 30 women, mean age – 33.2 ± 8.5 years). Echocardiograms of maximal longitudinal displacement of the basal segments of the left ventricle were analyzed. The results were processed using the Statistica 6.0 application package. At HCM changes of MV in the form of atypical chords are attached, which are attached to the smooth part of the ante-rior flap, destruction of the endothelial layer on the ventricular surface of this flap, increase of the sizes of the flap and chord, expansion of the spongy and fibrosis of the compact layer. In the basal part of the interventricular septum a fibrous stain is formed on the endocardium. In the area of the myocardium adjacent to the mitral fibrosis stain, maximal hypertrophy of cardiomyocytes and interstitial fibrosis spreading from the fibrous stain are observed. Asymmetric contraction of the basal segments of the LV was reported in patients with HCM with vector-echocardiogram by reducing the longitudinal displacement of the septal, inferior and anterior segments and increasing this index for the posterior and lateral walls. This indicates the asymmetric nature of the reduction of the LV myocardium, resulting in the MV fibrous ring during systole shifting unevenly. In patients with AVS, circular myocardial hypertrophy due to stenosis does not affect MV position during systole. According to the study, morpho-functional evidence was obtained of the essential role of MV in the development of HCM, but the question remains open and needs further study.

On the Hodge, Tate and Mumford-Tate Conjectures for Fibre Products of Families of Regular Surfaces with Geometric Genus 1

The Hodge, Tate and Mumford-Tate conjectures are proved for the fibre product of two non-isotrivial 1-parameter families of regular surfaces with geometric genus 1 under some conditions on degenerated fibres, the ranks of the N\'eron - Severi groups of generic geometric fibres and representations of Hodge groups in transcendental parts of rational cohomology.Let \(\pi_i:X_i\to C\quad (i = 1, 2)\) be a projective non-isotrivial family (possibly with degeneracies) over a smooth projective curve \(C\). Assume that the discriminant loci \(\Delta_i=\{\delta\in C\,\,\vert\,\, Sing(X_{i\delta})\neq\varnothing\} \quad (i = 1, 2)\) are disjoint, \(h^{2,0}(X_{ks})=1,\quad h^{1,0}(X_{ks}) = 0\) for any smooth fibre \(X_{ks}\), and the following conditions hold:\((i)\) for any point \(\delta \in \Delta_i\) and the Picard-Lefschetz transformation \( \gamma \in GL(H^2 (X_{is}, Q)) \), associated with a smooth part \(\pi'_i: X'_i\to C\setminus\Delta_i\) of the morphism \(\pi_i\) and with a loop around the point \(\delta \in C\), we have \((\log(\gamma))^2\neq0\);\((ii)\) the variety \(X_i \, (i = 1, 2)\), the curve \(C\) and the structure morphisms \(\pi_i:X_i\to C\) are defined over a finitely generated subfield \(k \hookrightarrow C\).If for generic geometric fibres \(X_{1s}\) \, and \, \(X_{2s}\) at least one of the following conditions holds: \((a)\) \(b_2(X_{1s})- rank NS(X_{1s})\) is an odd prime number, \(\quad\,\,\) \(b_2(X_{1s})- rank NS(X_{1s})\neq b_2(X_{2s})- rank NS(X_{2s})\); \((b)\) the ring \(End_{ Hg(X_{1s})} NS_ Q(X_{1s})^\perp\) is an imaginary quadratic field, \(\quad\,\, b_2(X_{1s})- rank NS(X_{1s})\neq 4,\) \(\quad\,\, End_{ Hg(X_{2s})} NS_ Q(X_{2s})^\perp\) is a totally real field or \(\,\, b_2(X_{1s})- rank NS(X_{1s})\,>\, b_2(X_{2s})- rank NS(X_{2s})\) ; \((c)\) \([b_2(X_{1s})- rank NS(X_{1s})\neq 4, \, End_{ Hg(X_{1s})} NS_ Q(X_{1s})^\perp= Q\); \(\quad\,\,\) \(b_2(X_{1s})- rank NS(X_{1s})\neq b_2(X_{2s})- rank NS(X_{2s})\),then for the fibre product \(X_1 \times_C X_2\) the Hodge conjecture is true, for any smooth projective \(k\)-variety \(X_0\) with the condition \(X_1 \times_C X_2\) \(\widetilde{\rightarrow}\) \(X_0 \otimes_k C\) the Tate conjecture on algebraic cycles and the Mumford-Tate conjecture for cohomology of even degree are true.

Polymeric quantum mechanics and the zeros of the Riemann zeta function

We analyze the Berry–Keating model and the Sierra and Rodríguez-Laguna Hamiltonian within the polymeric quantization formalism. By using the polymer representation, we obtain for both models, the associated polymeric quantum Hamiltonians and the corresponding stationary wave functions. The self-adjointness condition provides a proper domain for the Hamiltonian operator and the energy spectrum, which turned out to be dependent on an introduced scale parameter. By performing a counting of semiclassical states, we prove that the polymer representation reproduces the smooth part of the Riemann–von Mangoldt formula, and also introduces a correction depending on the energy and the scale parameter. This may shed some light on the understanding of the fluctuation behavior of the zeros of the Riemann function from a purely quantum point of view.

Heat transfer study on a hybrid smooth and spirally corrugated tube

Corrugated tubes are widely used in a range of applications for heat transfer enhancement. The spirally corrugated tube has a better heat transfer performance than the smooth tube. In this paper, the heat transfer performance of a hybrid smooth and six-start spirally corrugated tube is studied. With a validated numerical model, the effects of the corrugation part length on the vortex in the downstream smooth tube are studied for a range of high Reynolds numbers, where the existence of the corrugation part can turn out the secondary flow and enhance heat transfer. Meanwhile, it is found that in the smooth part, the fluid flow part with whirling can reach a maximum length, even if the length of the corrugation part continuously increases. Thus a series of critical corrugation lengths can be obtained. This work can reveal the enhanced heat transfer mechanism of the hybrid smooth and spirally corrugated tube and be of interest to researchers in heat transfer issues of corrugated tubes.

Nonoverlapping Blocks Based Copy-Move Forgery Detection

In order to solve the problem of high computational complexity in block-based methods for copy-move forgery detection, we divide image into texture part and smooth part to deal with them separately. Keypoints are extracted and matched in texture regions. Instead of using all the overlapping blocks, we use nonoverlapping blocks as candidates in smooth regions. Clustering blocks with similar color into a group can be regarded as a preprocessing operation. To avoid mismatching due to misalignment, we update candidate blocks by registration before projecting them into hash space. In this way, we can reduce computational complexity and improve the accuracy of matching at the same time. Experimental results show that the proposed method achieves better performance via comparing with the state-of-the-art copy-move forgery detection algorithms and exhibits robustness against JPEG compression, rotation, and scaling.

Robust Approximation of Singularly Perturbed Delay Differential Equations by the hp Finite Element Method

We consider the finite element approximation of the solution to a singularly perturbed second order differential equation with a constant delay. The boundary value problem can be cast as a singularly perturbed transmission problem, whose solution may be decomposed into a smooth part, a boundary layer part, an interior/interface layer part and a remainder. Upon discussing the regularity of each component, we show that under the assumption of analytic input data, the hp version of the finite element method on an appropriately designed mesh yields robust exponential convergence rates. Numerical results illustrating the theory are also included.

Detection of delamination in composite laminated plates using filtered mode shapes

Filtered mode shapes are used to detect the presence, location, size and shape of the delaminations in composite laminated plates with various boundary conditions. This method is the extension of a previous study by the authors on the delamination detection in the beams using irregularities of the mode shapes. The mode shapes are filtered to separate the smooth and irregular parts. Presence and situation of delamination affects these separated parts, and these effects are used to detect the delamination. Here, two new indicators, named ‘slope of smooth part’ and ‘irregularities in the slope of smooth part’, are introduced to increase the clarity of detected damage and reduce the noisy effects. The former one is obtained by differentiating the smooth part of the mode shape and the latter by applying the filter on the slope of smooth part for another time. Using this method and the mentioned indicators, delaminations may be detected in the plates using the data of just the damaged structure. This is considered as an important advantage of the method as we do not need the intact structure data. The method is validated utilizing the numerical data for a delaminated plate model. This method lacks the ability to locate the position of delamination through the thickness and the delamination should not be too close to the edges of the plate.

ON THE GENERALIZED VOLUME CONJECTURE AND REGULATOR

In this paper, by using the regulator map of Beilinson-Deligne on a curve, we show that the quantization condition posed by Gukov is true for the SL2(ℂ) character variety of the hyperbolic knot in S3. Furthermore, we prove that the corresponding ℂ*-valued closed 1-form is a secondary characteristic class (Chern-Simons) arising from the vanishing first Chern class of the flat line bundle over the smooth part of the character variety, where the flat line bundle is the pullback of the universal Heisenberg line bundle over ℂ* × ℂ*. Based on this result, we give a reformulation of Gukov's generalized volume conjecture from a motivic perspective.