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1. On the stationarity for nonlinear optimization problems with polyhedral constraints

   https://doi.org/10.1007/s10107-023-01979-9  

   di Serafino, Daniela ; Hager, William W. ; Toraldo, Gerardo ; Viola, Marco ( May 2023 , Mathematical Programming)

   For polyhedral constrained optimization problems and a feasible point$$\textbf{x}$$$x$, it is shown that the projection of the negative gradient on the tangent cone, denoted$$\nabla _\varOmega f(\textbf{x})$$${\nabla }_{\Omega }f\left(x\right)$, has an orthogonal decomposition of the form$$\varvec{\beta }(\textbf{x}) + \varvec{\varphi }(\textbf{x})$$$\beta \left(x\right)+\phi \left(x\right)$. At a stationary point,$$\nabla _\varOmega f(\textbf{x}) = \textbf{0}$$${\nabla }_{\Omega }f\left(x\right)=0$so$$\Vert \nabla _\varOmega f(\textbf{x})\Vert $$$\Vert {\nabla }_{\Omega }f\left(x\right)\Vert$reflects the distance to a stationary point. Away from a stationary point,$$\Vert \varvec{\beta }(\textbf{x})\Vert $$$\Vert \beta \left(x\right)\Vert$and$$\Vert \varvec{\varphi }(\textbf{x})\Vert $$$\Vert \phi \left(x\right)\Vert$measure different aspects of optimality since$$\varvec{\beta }(\textbf{x})$$$\beta \left(x\right)$only vanishes when the KKT multipliers at$$\textbf{x}$$$x$have the correct sign, while$$\varvec{\varphi }(\textbf{x})$$$\phi \left(x\right)$only vanishes when$$\textbf{x}$$$x$is a stationary point in the active manifold. As an application of the theory, an active set algorithm is developed for convex quadratic programs which adapts the flow of the algorithm based on a comparison between$$\Vert \varvec{\beta }(\textbf{x})\Vert $$$\Vert \beta \left(x\right)\Vert$and$$\Vert \varvec{\varphi }(\textbf{x})\Vert $$$\Vert \phi \left(x\right)\Vert$.

    

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2. Summing $$\mu (n)$$: a faster elementary algorithm

   https://doi.org/10.1007/s40993-022-00408-8  

   Helfgott, Harald Andrés ; Thompson, Lola ( March 2023 , Research in Number Theory)

   Abstract We present a new elementary algorithm that takes $$ \textrm{time} \ \ O_\epsilon \left( x^{\frac{3}{5}} (\log x)^{\frac{8}{5}+\epsilon } \right) \ \ \textrm{and} \ \textrm{space} \ \ O\left( x^{\frac{3}{10}} (\log x)^{\frac{13}{10}} \right) $$ time O ϵ x 3 5 ( log x ) 8 5 + ϵ and space O x 3 10 ( log x ) 13 10 (measured bitwise) for computing $$M(x) = \sum _{n \le x} \mu (n),$$ M ( x ) = ∑ n ≤ x μ ( n ) , where $$\mu (n)$$ μ ( n ) is the Möbius function. This is the first improvement in the exponent of x for an elementary algorithm since 1985. We also show that it is possible to reduce space consumption to $$O(x^{1/5} (\log x)^{5/3})$$ O ( x 1 / 5 ( log x ) 5 / 3 ) by the use of (Helfgott in: Math Comput 89:333–350, 2020), at the cost of letting time rise to the order of $$x^{3/5} (\log x)^2 \log \log x$$ x 3 / 5 ( log x ) 2 log log x . 

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3. Enhanced Piezoelectric, Ferroelectric, and Electrostrictive Properties of Lead‐Free (1‐ x )BCZT‐( x )BCST Electroceramics with Energy Harvesting Capability

   https://doi.org/10.1002/smll.202300549  

   Baraskar, Bharat G. ; Kolekar, Yesappa D. ; Thombare, Balu. R. ; James, Ajit. R. ; Kambale, Rahul C. ; Ramana, C. V. ( May 2023 , Small)

   Next‐generation electronics and energy technologies can now be developed as a result of the design, discovery, and development of novel, environmental friendly lead (Pb)‐free ferroelectric materials with improved characteristics and performance. However, there have only been a few reports of such complex materials’ design with multi‐phase interfacial chemistry, which can facilitate enhanced properties and performance. In this context, herein, novel lead‐free piezoelectric materials (1‐x)Ba_0.95Ca_0.05Ti_0.95Zr_0.05O₃‐(x)Ba_0.95Ca_0.05Ti_0.95Sn_0.05O₃, are reported, which are represented as (1‐x)BCZT‐(x)BCST, with demonstrated excellent properties and energy harvesting performance. The (1‐x)BCZT‐(x)BCST materials are synthesized by high‐temperature solid‐state ceramic reaction method by varyingxin the full range (x= 0.00–1.00). In‐depth exploration research is performed on the structural, dielectric, ferroelectric, and electro‐mechanical properties of (1‐x)BCZT‐(x)BCST ceramics. The formation of perovskite structure for all ceramics without the presence of any impurity phases is confirmed by X‐ray diffraction (XRD) analyses, which also reveals that the Ca²⁺, Zr⁴⁺, and Sn⁴⁺are well dispersed within the BaTiO₃lattice. For all (1‐x)BCZT‐(x)BCST ceramics, thorough investigation of phase formation and phase‐stability using XRD, Rietveld refinement, Raman spectroscopy, high‐resolution transmission electron microscopy (HRTEM), and temperature‐dependent dielectric measurements provide conclusive evidence for the coexistence of orthorhombic + tetragonal (Amm2+P4mm) phases at room temperature. The steady transition ofAmm2crystal symmetry toP4mmcrystal symmetry with increasingxcontent is also demonstrated by Rietveld refinement data and related analyses. The phase transition temperatures, rhombohedral‐orthorhombic (T_R‐O), orthorhombic‐ tetragonal (T_O‐T), and tetragonal‐cubic (T_C), gradually shift toward lower temperature with increasingxcontent. For (1‐x)BCZT‐(x)BCST ceramics, significantly improved dielectric and ferroelectric properties are observed, including relatively high dielectric constantε_r≈ 1900–3300 (near room temperature),ε_r≈ 8800–12 900 (near Curie temperature), dielectric loss, tanδ≈ 0.01–0.02, remanent polarizationP_r≈ 9.4–14 µC cm⁻², coercive electric fieldE_c≈ 2.5–3.6 kV cm⁻¹. Further, high electric field‐induced strainS≈ 0.12–0.175%, piezoelectric charge coefficientd₃₃≈ 296–360 pC N⁻¹, converse piezoelectric coefficient ≈ 240–340 pm V⁻¹, planar electromechanical coupling coefficientk_p≈ 0.34–0.45, and electrostrictive coefficient (Q₃₃)_avg≈ 0.026–0.038 m⁴C⁻²are attained. Output performance with respect to mechanical energy demonstrates that the (0.6)BCZT‐(0.4)BCST composition (x= 0.4) displays better efficiency for generating electrical energy and, thus, the synthesized lead‐free piezoelectric (1‐x)BCZT‐(x)BCST samples are suitable for energy harvesting applications. The results and analyses point to the outcome that the (1‐x)BCZT‐(x)BCST ceramics as a potentially strong contender within the family of Pb‐free piezoelectric materials for future electronics and energy harvesting device technologies.

    

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4. Boundary regularity and stability for spaces with Ricci bounded below

   https://doi.org/10.1007/s00222-021-01092-8  

   Bruè, Elia ; Naber, Aaron ; Semola, Daniele ( May 2022 , Inventiones mathematicae)

   Abstract This paper studies the structure and stability of boundaries in noncollapsed $${{\,\mathrm{RCD}\,}}(K,N)$$ RCD ( K , N ) spaces, that is, metric-measure spaces $$(X,{\mathsf {d}},{\mathscr {H}}^N)$$ ( X , d , H N ) with Ricci curvature bounded below. Our main structural result is that the boundary $$\partial X$$ ∂ X is homeomorphic to a manifold away from a set of codimension 2, and is $$N-1$$ N - 1 rectifiable. Along the way, we show effective measure bounds on the boundary and its tubular neighborhoods. These results are new even for Gromov–Hausdorff limits $$(M_i^N,{\mathsf {d}}_{g_i},p_i) \rightarrow (X,{\mathsf {d}},p)$$ ( M i N , d g i , p i ) → ( X , d , p ) of smooth manifolds with boundary, and require new techniques beyond those needed to prove the analogous statements for the regular set, in particular when it comes to the manifold structure of the boundary $$\partial X$$ ∂ X . The key local result is an $$\varepsilon $$ ε -regularity theorem, which tells us that if a ball $$B_{2}(p)\subset X$$ B 2 ( p ) ⊂ X is sufficiently close to a half space $$B_{2}(0)\subset {\mathbb {R}}^N_+$$ B 2 ( 0 ) ⊂ R + N in the Gromov–Hausdorff sense, then $$B_1(p)$$ B 1 ( p ) is biHölder to an open set of $${\mathbb {R}}^N_+$$ R + N . In particular, $$\partial X$$ ∂ X is itself homeomorphic to $$B_1(0^{N-1})$$ B 1 ( 0 N - 1 ) near $$B_1(p)$$ B 1 ( p ) . Further, the boundary $$\partial X$$ ∂ X is $$N-1$$ N - 1 rectifiable and the boundary measure "Equation missing" is Ahlfors regular on $$B_1(p)$$ B 1 ( p ) with volume close to the Euclidean volume. Our second collection of results involve the stability of the boundary with respect to noncollapsed mGH convergence $$X_i\rightarrow X$$ X i → X . Specifically, we show a boundary volume convergence which tells us that the $$N-1$$ N - 1 Hausdorff measures on the boundaries converge "Equation missing" to the limit Hausdorff measure on $$\partial X$$ ∂ X . We will see that a consequence of this is that if the $$X_i$$ X i are boundary free then so is X . 

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5. Cohomology of line bundles on the incidence correspondence

   https://doi.org/10.1090/btran/173  

   Gao, Zhao ; Raicu, Claudiu ( January 2024 , Transactions of the American Mathematical Society, Series B)

   For a finite dimensional vector space$V$of dimension$n$, we consider the incidence correspondence (or partial flag variety)$X\subset \mathbb {P}V \times \mathbb {P}V^{\vee }$, parametrizing pairs consisting of a point and a hyperplane containing it. We completely characterize the vanishing and non-vanishing behavior of the cohomology groups of line bundles on$X$in characteristic$p>0$. If$n=3$then$X$is the full flag variety of$V$, and the characterization is contained in the thesis of Griffith from the 70s. In characteristic$0$, the cohomology groups are described for all$V$by the Borel–Weil–Bott theorem. Our strategy is to recast the problem in terms of computing cohomology of (twists of) divided powers of the cotangent sheaf on projective space, which we then study using natural truncations induced by Frobenius, along with careful estimates of Castelnuovo–Mumford regularity. When$n=3$, we recover the recursive description of characters from recent work of Linyuan Liu, while for general$n$we give character formulas for the cohomology of a restricted collection of line bundles. Our results suggest truncated Schur functions as the natural building blocks for the cohomology characters.

    

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