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In this paper, we establish a framework for the analysis of linear parabolic equations on conical surfaces and use them to study the conical Ricci flow. In particular, we prove the long time existence of the conical Ricci flow for general cone angle and show that this solution has the optimal regularity, namely, the time derivatives of the conformal factor are bounded and for each fixed time, the conformal factor has an explicit asymptotic expansion near the cone points. [ABSTRACT FROM AUTHOR]

Based on the endpoint Strichartz estimates for the fourth order Schr¨odinger equation with potentials for n ≥ 5 by [Feng, H., Soffer, A., Yao, X.: Decay estimates and Strichartz estimates of the fourth-order Schr¨odinger operator. J. Funct. Anal., 274, 605-658 (2018)], in this paper, the authors further derive Strichartz type estimates with gain of derivatives similar to the one in [Pausader, B.: The cubic fourth-order Schr¨odinger equation. J. Funct. Anal., 256, 2473-2517 (2009)]. As their applications, we combine the classical Morawetz estimate and the interaction Morawetz estimate to establish scattering theory in the energy space for the defocusing fourth order NLS with potentials and pure power nonlinearity 1 + 8 n < p < 1 + 8 n-4 in dimensions n ≥ 7. [ABSTRACT FROM AUTHOR]

Continuing our previous work (arXiv:1509.07981v1), we derive another global gradient estimate for positive functions, particularly for positive solutions to the heat equation on finite or locally finite graphs. In general, the gradient estimate in the present paper is independent of our previous one. As applications, it can be used to get an upper bound and a lower bound of the heat kernel on locally finite graphs. These global gradient estimates can be compared with the Li-Yau inequality on graphs contributed by Bauer et al. [ J. Differential Geom., 99, 359-409 (2015)]. In many topics, such as eigenvalue estimate and heat kernel estimate (not including the Liouville type theorems), replacing the Li-Yau inequality by the global gradient estimate, we can get similar results. [ABSTRACT FROM AUTHOR]

In this paper, the structure of Jordan higher derivable maps on triangular algebras by commutative zero products is given. As an application, the form of Jordan higher derivable maps of nest algebras by commutative zero products is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider the scattering for the nonlinear Schrödinger equation with small, smooth, and localized data. In particular, we prove that the solution of the quadratic nonlinear Schrödinger equation with nonlinear term | u| involving some derivatives in two dimension exists globally and scatters. It is worth to note that there exist blow-up solutions of these equations without derivatives. Moreover, for radial data, we prove that for the equation with p-order nonlinearity with derivatives, the similar results hold for $p \geqslant \tfrac{{2d + 3}} {{2d - 1}} $ and d ≥ 2, which is lower than the Strauss exponents. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the growth of meromorphic functions concerning Picard values with a radially distributed value. This generalizes a classic result due to Hayman [Hayman, W. K.: Picard values of meromorphic functions and their derivatives. Ann. of Math., 70, 9–42 (1959)]. [ABSTRACT FROM AUTHOR]

In the recent work, we have developed a decay framework in general Lp critical spaces and established optimal time-decay estimates for barotropic compressible Navier-Stokes equations. Those decay rates of Lq-Lr type of the solution and its derivatives are available in the critical regularity framework, which were exactly firstly observed by Matsumura & Nishida, and subsequently generalized by Ponce for solutions with high Sobolev regularity. We would like to mention that our approach is likely to be effective for other hyperbolic/parabolic systems that are encountered in fluid mechanics or mathematical physics. In this paper, a new observation is involved in the high frequency, which enables us to improve decay exponents for the high frequencies of solutions. [ABSTRACT FROM AUTHOR]

In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-covering constant problem, Hadwiger's covering conjecture and Borsuk's partition conjecture. They are fundamental and fascinating problems about the same objects. However, up to now, both the methodology and the technique applied to them are essentially different. Therefore, a common foundation for them has been much expected. By treating problems of these types as functionals defined on the spaces of n-dimensional convex bodies, this paper tries to create such a foundation. In particular, supderivatives for these functionals will be studied. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Heisenberg-Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined. [ABSTRACT FROM AUTHOR]

In the paper under review, we consider the generation of fractional resolvent families by abstract differential operators. Our results can be simply incorporated in the study of corresponding abstract time-fractional equations with Caputo fractional derivatives. [ABSTRACT FROM AUTHOR]

This paper focuses on two cases of two-dimensional wave equations with fractal boundaries. The first case is the equation with classical derivative. The formal solution is obtained. And a definition of the solution is given. Then we prove that under certain conditions, the solution is a kind of fractal function, which is continuous, differentiable nowhere in its domain. Next, for specific given initial position and 3 different initial velocities, the graphs of solutions are sketched. By computing the box dimensions of boundaries of cross-sections for solution surfaces, we evaluate the range of box dimension of the vibrating membrane. The second case is the equation with p-type derivative. The corresponding solution is shown and numerical example is given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a type of approximation operators of neural networks with sigmodal functions on compact intervals, and obtain the pointwise and uniform estimates of the approximation. To improve the approximation rate, we further introduce a type of combinations of neural networks. Moreover, we show that the derivatives of functions can also be simultaneously approximated by the derivatives of the combinations. We also apply our method to construct approximation operators of neural networks with sigmodal functions on infinite intervals. [ABSTRACT FROM AUTHOR]

Let $$\mathcal{A}$$ and $$\mathcal{B}$$ be unital rings, and $$\mathcal{M}$$ be an $$\left( {\mathcal{A},\mathcal{B}} \right)$$-bimodule, which is faithful as a left $$\mathcal{A}$$-module and also as a right $$\mathcal{B}$$-module. Let $$\mathcal{U} = Tri\left( {\mathcal{A},\mathcal{M},\mathcal{B}} \right)$$ be the triangular algebra. In this paper, we give some different characterizations of Lie higher derivations on $$\mathcal{U}$$. [ABSTRACT FROM AUTHOR]

By using the stable t-structure induced by an adjoint pair, we extend several results concerning recollements to upper (resp. lower) recollements. These include the fundamental results of Parshall and Scott on comparisons of recollements, Wiedemann's result on the global dimension and Happel's result on the finitistic dimension, occurring in a recollement ( D ( A′), D ( A), D ( A″)) of bounded derived categories of Artin algebras. We introduce and describe a triangle expansion of a triangulated category and illustrate it by examples. [ABSTRACT FROM AUTHOR]

This work focuses on the second type of generalized Feigenbaum's equation where φ( x) is C-increasing function on [0, 1] and satisfies that φ(0) = 0,0 < φ′( x) < 1 ( x ∈ [0, 1]). Using constructive method, we discuss the existence of C-single-valley solutions whose derivatives are not equal to 0 on origin of the above equation. [ABSTRACT FROM AUTHOR]

In this note, we obtain an asymptotic estimate for the time derivative of the Φ-entropy in terms of the lower bound of the Bakry-Emery Γ curvature. In the cases of hyperbolic space and the Heisenberg group (more generally, the nilpotent Lie group of rank two), we show that the time derivative of the Φ-entropy is non-increasing and concave in time t, also we get a sharp asymptotic bound for the time derivative of the entropy in these cases. [ABSTRACT FROM AUTHOR]

Let 0 ≤ α < n, Ω be a rough kernel, and let A have derivatives of order m−1 in $$C\dot BMO^{q,\mu _2 }$$ with m ≥ 2. We consider a class of generalized commutators TΩ,αA of Cohen-Gosselin type, and obtain the boundedness of TΩ,αA from the central Morrey spaces $$\dot E^{p,\mu _1 }$$ to $$\dot E^{r,\lambda }$$ for λ = µ 1 + µ 2 + α/n and 1/ r = 1/ p + 1/ q − α/n. [ABSTRACT FROM AUTHOR]

Using forward-backward stochastic calculus, we prove convex concentration inequalities for some additive functionals of the solution of stochastic differential equations with jumps admitting an invariant probability measure. As a consequence, transportation-information inequalities are obtained and bounds on option prices for interest rate derivatives are given as an application. [ABSTRACT FROM AUTHOR]

Let f be a nonconstant meromorphic function in the plane and h be a nonconstant elliptic function. We show that if all zeros of f are multiple except finitely many and T( r, h) = o{ T( r, f)} as r→ ∞, then f′ = h has infinitely many solutions (including poles). [ABSTRACT FROM AUTHOR]

This contribution gives results on the action of the Laplace-Beltrami derivative on sufficiently smooth kernels on the sphere, those defined by absolutely and uniformly expansions generated by a family of at least continuous functions. Among other things, the results show that convenient Laplace-Beltrami derivatives of positive definite kernels on the sphere are positive definite too. We also include similar results on the action of the Laplace-Beltrami derivative on condensed spherical harmonic expansions. [ABSTRACT FROM AUTHOR]