Smoothing Scheme Research Articles

RECIPROCITY SHEAVES AND THEIR RAMIFICATION FILTRATIONS

AbstractWe define a motivic conductor for any presheaf with transfersFusing the categorical framework developed for the theory of motives with modulus by Kahn, Miyazaki, Saito and Yamazaki. IfFis a reciprocity sheaf, this conductor yields an increasing and exhaustive filtration on$F(L)$, whereLis any henselian discrete valuation field of geometric type over the perfect ground field. We show that ifFis a smooth group scheme, then the motivic conductor extends the Rosenlicht–Serre conductor; ifFassigns toXthe group of finite characters on the abelianised étale fundamental group ofX, then the motivic conductor agrees with the Artin conductor defined by Kato and Matsuda; and ifFassigns toXthe group of integrable rank$1$connections (in characteristic$0$), then it agrees with the irregularity. We also show that this machinery gives rise to a conductor for torsors under finite flat group schemes over the base field, which we believe to be new. We introduce a general notion of conductors on presheaves with transfers and show that on a reciprocity sheaf, the motivic conductor is minimal and any conductor which is defined only for henselian discrete valuation fields of geometric type withperfectresidue field can be uniquely extended to all such fields without any restriction on the residue field. For example, the Kato–Matsuda Artin conductor is characterised as the canonical extension of the classical Artin conductor defined in the case of a perfect residue field.

Characteristic cycle and wild ramification for nearby cycles of étale sheaves

Abstract In this article, we give a bound for the wild ramification of the monodromy action on the nearby cycles complex of a locally constant étale sheaf on the generic fiber of a smooth scheme over an equal characteristic trait in terms of Abbes and Saito’s logarithmic ramification filtration. This provides a positive answer to the main conjecture in [24] for smooth morphisms in equal characteristic. We also study the ramification along vertical divisors of étale sheaves on relative curves and abelian schemes over a trait.

An entropy-stable Smooth Particle Hydrodynamics algorithm for large strain thermo-elasticity

This paper presents a novel Smooth Particle Hydrodynamics computational framework for the simulation of large strain fast solid dynamics in thermo-elasticity. The formulation is based on the Total Lagrangian description of a system of first order conservation laws written in terms of the linear momentum, the triplet of deformation measures (also known as minors of the deformation gradient tensor) and the total energy of the system, extending thus the previous work carried out by some of the authors in the context of isothermal elasticity and elasto-plasticity (Lee et al., 2016; Lee et al., 2017; Lee et al., 2019). To ensure the stability (i.e. hyperbolicity) of the formulation from the continuum point of view, the internal energy density is expressed as a polyconvex combination of the triplet of deformation measures and the entropy density. Moreover, and to guarantee stability from the spatial discretisation point of view, consistently derived Riemann-based numerical dissipation is carefully introduced where local numerical entropy production is demonstrated via a novel technique in terms of the time rate of the so-called ballistic free energy of the system. For completeness, an alternative and equally competitive formulation (in the case of smooth solutions), expressed in terms of the entropy density, is also implemented and compared. A series of numerical examples is presented in order to assess the applicability and robustness of the proposed formulations, where the Smooth Particle Hydrodynamics scheme is benchmarked against an alternative in-house Finite Volume Vertex Centred implementation.

Semi-orthogonal decompositions of GIT quotient stacks

If G is a reductive group acting on a linearized smooth scheme X then we show that under suitable standard conditions the derived category \({{{\mathcal {D}}}}(X^{ss}{/}G)\) of the corresponding GIT quotient stack \(X^{ss}{/}G\) has a semi-orthogonal decomposition consisting of derived categories of coherent sheaves of rings on \(X^{ss}{/\!\!/}G\) which are locally of finite global dimension. One of the components of the decomposition is a certain non-commutative resolution of \(X^{ss}{/\!\!/}G\) constructed earlier by the authors. As a concrete example we obtain in the case of odd Pfaffians a semi-orthogonal decomposition of the corresponding quotient stack in which all the parts are certain specific non-commutative crepant resolutions of Pfaffians of lower or equal rank which had also been constructed earlier by the authors. In particular this semi-orthogonal decomposition cannot be refined further since its parts are Calabi–Yau. The results in this paper complement results by Halpern–Leistner, Ballard–Favero–Katzarkov and Donovan–Segal that assert the existence of a semi-orthogonal decomposition of \({{{\mathcal {D}}}}(X/G)\) in which one of the parts is \({{{\mathcal {D}}}}(X^{ss}/G)\).

Robust Finger Vein Recognition Based on Deep CNN with Spatial Attention and Bias Field Correction

As one of the biometric information based authentication technologies, finger vein recognition has received increasing attention due to its safety and convenience. However, it is still a challenging task to design an efficient and robust finger vein recognition system because of the low quality of the finger vein images, lack of sufficient number of training samples with image-level annotated information and no pixel-level finger vein texture labels in the public available finger vein databases. In this paper, we propose a novel CNN-based finger vein recognition approach with bias field correction, spatial attention mechanism and a multistage transfer learning strategy to cope with the difficulties mentioned above. In the proposed method, the bias field correction module is to remove the unbalanced bias field of the original images by using a two-dimensional polynomial fitting algorithm, the spatial attention module is to enhance the informative vein texture regions while suppressing the other less informative regions, and the multistage transfer learning strategy is to solve the problem caused by insufficient training for CNN-based model due to lack of labeled training samples in the public finger vein databases. Moreover, several measures, including a label smoothing scheme and data augmentation, are exploited to improve the performance of the proposed method. Extensive experiments have been conducted in the work on three public databases, and the results show that the proposed approach outperforms the existing state-of-the-art methods.

Lower-Estimates on the Hochschild (Co)Homological Dimension of Commutative Algebras and Applications to Smooth Affine Schemes and Quasi-Free Algebras

The Hochschild cohomological dimension of any commutative k-algebra is lower-bounded by the least-upper bound of the flat-dimension difference and its global dimension. Our result is used to show that for a smooth affine scheme X satisfying Pointcaré duality, there must exist a vector bundle with section M and suitable n which the module of algebraic differential n-forms Ωn(X,M). Further restricting the notion of smoothness, we use our result to show that most k-algebras fail to be smooth in the quasi-free sense. This consequence, extends the currently known results, which are restricted to the case where k=C.

An asymptotic vanishing theorem for the cohomology of thickenings

Let X be a closed equidimensional local complete intersection subscheme of a smooth projective scheme Y over a field, and let \(X_t\) denote the t-th thickening of X in Y. Fix an ample line bundle \(\mathcal {O}_Y(1)\) on Y. We prove the following asymptotic formulation of the Kodaira vanishing theorem: there exists an integer c, such that for all integers \(t \geqslant 1\), the cohomology group \(H^k(X_t,\mathcal {O}_{X_t}(j))\) vanishes for \(k < \dim X\) and \(j < -ct\). Note that there are no restrictions on the characteristic of the field, or on the singular locus of X. We also construct examples illustrating that a linear bound is indeed the best possible, and that the constant c is unbounded, even in a fixed dimension.

Effective optical smoothing scheme to suppress laser plasma instabilities by time-dependent polarization rotation via pulse chirping.

In this paper, we propose a novel effective optical smoothing scheme to suppress laser plasma instabilities (LPIs) by time-dependent polarization rotation (TPR) on a picosecond timescale. The polarization rotation with time-dependent frequency is generated by the superposition of chirped light pulses with dynamic frequency shift and counter-rotating circular polarization. Compared to light without polarization rotation or pulse chirping, such superposed light with TPR has a broader spectrum and lower temporal coherence. Using the one-dimensional fluid laser-plasma-instability code (FLAME) and PIC simulation, TPR is demonstrated working well in suppressing parametric backscattering, which provides an effective approach to suppress LPIs. In the meantime, a significant improvement of irradiation uniformity of the chirped pulses is achieved by the introduction of proper spatial phase modulation and grating dispersion.

Optimized dynamic interference smoothing via asynchronous phasemodulation of SSD

In dynamic interference smoothing (DIS) scheme, the speckles of focal spots are swept in multi-directions within several picoseconds, leading to the improvement of irradiation uniformity. In this work, the theoretical limitation of the smoothing performance of DIS scheme is revealed and then an optimized dynamic interference smoothing (ODIS) scheme is proposed. The ODIS scheme is implemented by applying asynchronous sinusoidal phase modulations induced by electro-optic modulators with time-delayed voltage signals to the beamlets in a laser quad. Results indicate that the irradiation uniformity of the ODIS scheme is further improved than that of DIS scheme owing to the asynchronous spatiotemporal phases. In addition, the polarization of the ODIS scheme also rotates on picosecond timescale, which may be potential to mitigate the parametric backscattering. Moreover, though the ODIS scheme has a relatively high requirement for temporal synchronization of beamlets, it shows a wide tolerance for the spatial wavefront distortion.

On the decomposition of the De Rham complex on formal schemes

We show that, for a pseudo-proper smooth noetherian formal scheme $\mathfrak{X}$ over a positive characteristic $p$ field, its truncated De Rham complex up to the characteristic $p$ is decomposable. Moreover, if the dimension of $\mathfrak{X}$ is exactly $p$, then the full De Rham complex is decomposable. Along the way we establish the Cartier isomorphism associated to a smooth morphism of positive characteristic noetherian formal schemes.

Uniform boundedness for Brauer groups of forms in positive characteristic

Let $k$ be a finitely generated field of characteristic $p>0$ and $X$ a smooth and proper scheme over $k$. Recent works of Cadoret, Hui and Tamagawa show that, if $X$ satisfies the $\ell$-adic Tate conjecture for divisors for every prime $\ell\neq p$, the Galois invariant subgroup $Br(X_{\overline k})[p']^{\pi_1(k)}$ of the prime-to-$p$ torsion of the geometric Brauer group of $X$ is finite. The main result of this note is that, for every integer $d\geq 1$, there exists a constant $C:=C(X,d)$ such that for every finite field extension $k \subseteq k'$ with $[k':k]\leq d$ and every $(\overline k/k')$-form $Y$ of $X$ one has $|(Br(Y\times_{k'}\overline k)[p']^{\pi_1(k')}|\leq C$. The theorem is a consequence of general results on forms of compatible systems of $\pi_1(k)$-representations and it extends to positive characteristic a recent result of Orr and Skorobogatov in characteristic zero.

On a torsion analogue of the weight-monodromy conjecture

We formulate and study a torsion analogue of the weight-monodromy conjecture for a proper smooth scheme over a non-archimedean local field. We prove it for proper smooth schemes over equal characteristic non-archimedean local fields, abelian varieties, surfaces, varieties uniformized by Drinfeld upper half spaces, and set-theoretic complete intersections in projective smooth toric varieties. In the equal characteristic case, our methods rely on an ultraproduct variant of Weil II established by Cadoret.

Filtering and Smoothing of Hidden Monotonic Trends and Application to Fouling Detection

Abstract In this paper, we present a filtering and smoothing scheme for process variables characterized by a hidden monotonic trend. The proposed method models the transition probability distribution of the hidden monotonic trend as a closed skew normal distribution and the observed data is assumed to have a Gaussian noise added onto this monotonic trend. The objective is to extract the monotonic trend given the noisy observations. The proposed method has advantages in process monitoring applications involving processes driven by a monotonic trend where vanilla Kalman filter may not be the apt option. The proposed method has been verified on an industrial dataset of a hot lime softener process to detect the fouling buildup.

Equivariant Hodge theory and noncommutative geometry

We develop a version of Hodge theory for a large class of smooth formally proper quotient stacks $X/G$ analogous to Hodge theory for smooth projective schemes. We show that the noncommutative Hodge-de Rham sequence for the category of equivariant coherent sheaves degenerates. This spectral sequence converges to the periodic cyclic homology, which we canonically identify with the topological equivariant K-theory of $X$ with respect to a maximal compact subgroup of $G$, equipping the latter with a canonical pure Hodge structure. We also establish Hodge-de Rham degeneration for categories of matrix factorizations for a large class of equivariant Landau-Ginzburg models.

Macaulayfication of Noetherian schemes

To reduce to resolving Cohen–Macaulay singularities, Faltings initiated the program of “Macaulayfying” a given Noetherian scheme X. For a wide class of X, Kawasaki built the sought-for Cohen–Macaulay modifications, with a crucial drawback that his blowups did not preserve the locus CM(X)⊂X, where X is already Cohen–Macaulay. We extend Kawasaki’s methods to show that every quasi-excellent, Noetherian scheme X has a Cohen–Macaulay X˜ with a proper map X˜→X that is an isomorphism over CM(X). This completes Faltings’s program, reduces the conjectural resolution of singularities to the Cohen–Macaulay case, and implies that every proper, smooth scheme over a number field has a proper, flat, Cohen–Macaulay model over the ring of integers.

Characteristic cycles and the conductor of direct image

We prove the functoriality for a proper push-forward of the characteristic cycles of constructible complexes by morphisms of smooth projective schemes over a perfect field, under the assumption that the direct image of the singular support has the dimension at most that of the target of the morphism. The functoriality is deduced from a conductor formula which is a special case for morphisms to curves. The conductor formula in the constant coefficient case gives the geometric case of a formula conjectured by Bloch.

Cartier modules and cyclotomic spectra

We construct and study a t t -structure on p p -typical cyclotomic spectra and explain how to recover crystalline cohomology of smooth schemes over perfect fields using this t t -structure. Our main tool is a new approach to p p -typical cyclotomic spectra via objects we call p p -typical topological Cartier modules. Using these, we prove that the heart of the cyclotomic t t -structure is the full subcategory of derived V V -complete objects in the abelian category of p p -typical Cartier modules.

Linear smoothed finite element method for quasi-incompressible hyperelastic media

This work presents a linear smoothing scheme over high-order triangular elements within the framework of the cell-based strain smoothed finite element method for two-dimensional nonlinear problems. The main idea behind the proposed linear smoothing scheme is that it unlike the classical SFEM, it does not require the subdivision of the finite element cells into smoothing sub-cell. The other features of the classical SFEM are retained, such as: it does not require an explicit form of the derivatives of the basis functions, all the computations are done in the physical space, and the results are less sensitive to mesh distortion. A series of benchmark tests are done to demonstrate the validity and the stability of the proposed scheme. The validity and accuracy are confirmed by comparing the obtained numerical results with the standard FEM using quadratic triangular element and the exact solutions.

Morphisms of Rational Motivic Homotopy Types

We investigate several interrelated foundational questions pertaining to the study of motivic dga’s of Dan-Cohen and Schlank (Rational motivic path spaces and Kim’s relative unipotent section conjecture. arXiv:1703.10776 ) and Iwanari (Motivic rational homotopy type. arXiv:1707.04070 ). In particular, we note that morphisms of motivic dga’s can reasonably be thought of as a nonabelian analog of motivic cohomology. Just as abelian motivic cohomology is a homotopy group of a spectrum coming from K-theory, the space of morphisms of motivic dga’s is a certain limit of such spectra; we give an explicit formula for this limit—a possible first step towards explicit computations or dimension bounds. We also consider commutative comonoids in Chow motives, which we call “motivic Chow coalgebras”. We discuss the relationship between motivic Chow coalgebras and motivic dga’s of smooth proper schemes. As a small first application of our results, we show that among schemes which are finite etale over a number field, morphisms of associated motivic dga’s are no different than morphisms of schemes. This may be regarded as a small consequence of a plausible generalization of Kim’s relative unipotent section conjecture, hence as an ounce of evidence for the latter.

Elimination of Transient Current Mutation and Voltage Spike for Buck–Boost Current-Fed Isolated DC–DC Converter

The bidirectional buck-boost current-fed isolated dc-dc converter is an emerging topology and has the advantages of low current stress, high efficiency, and wide power/voltage conversion range. However, it suffers from the inductor current mutation and transient voltage spike when the converter is directly switched between the buck and boost modes in the actual operation. By analyzing in detail the process of directly switching the two operation modes, it reals the reason of causing these transient problems is the duty ratio, and shoot-through duty ratio is not correctly set when the operation mode is switched. In order to solve this problem, a smooth switching scheme between the buck/boost operation modes is proposed by setting a special relationship between the closed-loop controller output of the inductor current and the two control variables based on the system model. The detailed simulation and experimental results verify the theoretical analysis and the proposed solutions.