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Hovey and Palmieri [ The structure of the Bousfield lattice , Amer. Math. Soc., Providence, RI, 1999] proposed the retract conjecture on the Bousfield lattice of the stable homotopy category. The author, Shimomura and Tatehara [Publ. Res. Inst. Math. Sci. 50 (2014), pp. 497–513] defined the notion of monoidal posets as a generalization of the Bousfield lattice. In this paper, we prove that an analogue of the retract conjecture holds on a sublattice of monoidal posets. [ABSTRACT FROM AUTHOR]

In this paper, we prove that klt singularities are invariant under deformations if the generic fiber is Q-Gorenstein. We also obtain a similar result for slc singularities. These are generalizations of results of Esnault-Viehweg [Math. Ann. 271 (1985), 439-449] and S. Ishii [Math. Ann. 275 (1986), 139-148; Singularities (Iowa City, IA, 1986) Contemporary Mathematics, vol. 90 (American Mathematical Society, Providence, RI, 1989), 135-145]. [ABSTRACT FROM AUTHOR]

For a compact set A \subset \mathbb {R}^d and an integer k\ge 1, let us denote by \begin{equation*} A[k] = \left \{a_1+\cdots +a_k: a_1, \ldots, a_k\in A\right \}=\sum _{i=1}^k A \end{equation*} the Minkowski sum of k copies of A. A theorem of Shapley, Folkmann and Starr (1969) states that \frac {1}{k}A[k] converges to the convex hull of A in Hausdorff distance as k tends to infinity. Bobkov, Madiman and Wang [ Concentration, functional inequalities and isoperimetry , Amer. Math. Soc., Providence, RI, 2011] conjectured that the volume of \frac {1}{k}A[k] is nondecreasing in k, or in other words, in terms of the volume deficit between the convex hull of A and \frac {1}{k}A[k], this convergence is monotone. It was proved by Fradelizi, Madiman, Marsiglietti and Zvavitch [C. R. Math. Acad. Sci. Paris 354 (2016), pp. 185–189] that this conjecture holds true if d=1 but fails for any d \geq 12. In this paper we show that the conjecture is true for any star-shaped set A \subset \mathbb {R}^d for d=2 and d=3 and also for arbitrary dimensions d \ge 4 under the condition k \ge (d-1)(d-2). In addition, we investigate the conjecture for connected sets and present a counterexample to a generalization of the conjecture to the Minkowski sum of possibly distinct sets in \mathbb {R}^d, for any d \geq 7. [ABSTRACT FROM AUTHOR]

This paper provides a probabilistic proof of the comparison result for viscosity solutions of path-dependent semilinear PDEs. We consider the notion of viscosity solutions introduced in [I. Ekren, et al., Ann. Probab., 42 (2014), pp. 204-236], which considers as test functions all those smooth processes which are tangent in mean. When restricted to the Markovian case, this definition induces a larger set of test functions and reduces to the notion of stochastic viscosity solutions analyzed in [E. Bayraktar and M. Sirbu, Proc. Amer. Math. Soc., 140 (2012), pp. 3645-3654; SIAM J. Control Optim., 51 (2013), pp. 4274-4294]. Our main result takes advantage of this enlargement of the test functions and provides an easier proof of comparison. This is most remarkable in the context of the linear path-dependent heat equation. As a key ingredient for our methodology, we introduce a notion of punctual differentiation, similar to the corresponding concept in the standard viscosity solutions [L. A. Caffarelli and X. Cabre, Amer. Math. Soc. Colloq. Publ., 43, AMS, Providence, RI, 1995], and we prove that semimartingales are almost everywhere punctually differentiable. This smoothness result can be viewed as the counterpart of the Aleksandroff smoothness result for convex functions. A similar comparison result was established earlier in [I. Ekren et al., Ann. Probab., 42 (2014), pp. 204-236]. The result of this paper is more general and, more importantly, the arguments that we develop do not rely on any representation of the solution. [ABSTRACT FROM AUTHOR]

This paper provides a probabilistic proof of the comparison result for viscosity solutions of path-dependent semilinear PDEs. We consider the notion of viscosity solutions introduced in [I. Ekren, et al., Ann. Probab., 42 (2014), pp. 204-236], which considers as test functions all those smooth processes which are tangent in mean. When restricted to the Markovian case, this definition induces a larger set of test functions and reduces to the notion of stochastic viscosity solutions analyzed in [E. Bayraktar and M. Sirbu, Proc. Amer. Math. Soc., 140 (2012), pp. 3645-3654; SIAM J. Control Optim., 51 (2013), pp. 4274-4294]. Our main result takes advantage of this enlargement of the test functions and provides an easier proof of comparison. This is most remarkable in the context of the linear path-dependent heat equation. As a key ingredient for our methodology, we introduce a notion of punctual differentiation, similar to the corresponding concept in the standard viscosity solutions [L. A. Caffarelli and X. Cabre, Amer. Math. Soc. Colloq. Publ., 43, AMS, Providence, RI, 1995], and we prove that semimartingales are almost everywhere punctually differentiable. This smoothness result can be viewed as the counterpart of the Aleksandroff smoothness result for convex functions. A similar comparison result was established earlier in [I. Ekren et al., Ann. Probab., 42 (2014), pp. 204-236]. The result of this paper is more general and, more importantly, the arguments that we develop do not rely on any representation of the solution. [ABSTRACT FROM AUTHOR]

In this paper, the author studies the local existence of strong solutions and their possible blow-up in time for a quasilinear system describing the interaction of a short wave induced by an electron field with a long wave representing an extension of the motion of the director field in a nematic liquid crystal's asymptotic model introduced in [Saxton, R. A., Dynamic instability of the liquid crystal director. In: Current Progress in Hyperbolic Systems (Lindquist, W. B., ed.), Contemp. Math., Vol.100, Amer. Math. Soc., Providence, RI, 1989, pp.325–330] and [Hunter, J. K. and Saxton, R. A., Dynamics of director fields, SIAM J. Appl. Math., 51, 1991, 1498–1521] and studied in [Hunter, J. K. and Zheng, Y., On a nonlinear hyperbolic variational equation I, Arch. Rat. Mech. Anal., 129, 1995, 305–353], [Hunter, J. K. and Zheng, Y., On a nonlinear hyperbolic variational equation II, Arch. Rat. Mech. Anal., 129, 1995, 355–383] and in [Zhang, P. and Zheng, Y., On oscillation of an asymptotic equation of a nonlinear variational wave equation, Asymptotic Anal., 18, 1998, 307–327] and, more recently, in [Bressan, A., Zhang, P. and Zheng, Y., Asymptotic variational wave equations, Arch. Rat. Mech. Anal., 183, 2007, 163–185]. [ABSTRACT FROM AUTHOR]