Solution dependence on initial conditions in differential variational inequalities

In the first part of this paper, we establish several sensitivity results of the solution x(t, ξ) to the ordinary differential equation (ODE) initial-value problem (IVP) dx/dt = f(x), x(0) =  ξ as a function of the initial value ξ for a nondifferentiable f(x). Specifically, we show that for $\Xi_T \equiv \{\,x(t,\xi^0): 0 \leq t \leq T\,\}$ , (a) if f is “B-differentiable” on $\Xi_T$ , then so is the solution operator x(t;·) at ξ⁰; (b) if f is “semismooth” on $\Xi_T$ , then so is x(t;·) at ξ⁰; (c) if f has a “linear Newton approximation” on $\Xi_T$ , then so does x(t;·) at ξ⁰; moreover, the linear Newton approximation of the solution operator can be obtained from the solution of a “linear” differential inclusion. In the second part of the paper, we apply these ODE sensitivity results to a differential variational inequality (DVI) and discuss (a) the existence, uniqueness, and Lipschitz dependence of solutions to subclasses of the DVI subject to boundary conditions, via an implicit function theorem for semismooth equations, and (b) the convergence of a “nonsmooth shooting method” for numerically computing such boundary-value solutions.     

Thresholded covering algorithms for robust and max–min optimization

In a two-stage robust covering problem, one of several possible scenarios will appear tomorrow and require to be covered, but costs are higher tomorrow than today. What should you anticipatorily buy today, so that the worst-case cost (summed over both days) is minimized? We consider the \(k\) -robust model where the possible scenarios tomorrow are given by all demand-subsets of size \(k\) . In this paper, we give the following simple and intuitive template for \(k\) -robust covering problems: having built some anticipatory solution, if there exists a single demand whose augmentation cost is larger than some threshold, augment the anticipatory solution to cover this demand as well, and repeat. We show that this template gives good approximation algorithms for \(k\) -robust versions of many standard covering problems: set cover, Steiner tree, Steiner forest, minimum-cut and multicut. Our \(k\) -robust approximation ratios nearly match the best bounds known for their deterministic counterparts. The main technical contribution lies in proving certain net-type properties for these covering problems, which are based on dual-rounding and primal–dual ideas; these properties might be of some independent interest. As a by-product of our techniques, we also get algorithms for max–min problems of the form: “given a covering problem instance, which \(k\) of the elements are costliest to cover?” For the problems mentioned above, we show that their \(k\) -max–min versions have performance guarantees similar to those for the \(k\) -robust problems.     

A symmetry problem with applications to finance

In this paper we will consider an overdetermined problem in an unknown ring-shaped domain, and we prove that the domain has to be spherical ring. We will apply this problem to an \(n\) -dimensional toy model in the mean field game theory introduced by Lasry-Lions too. We show that if in Lasry-Lions model, the additional extra data ”the boundary is a level set” is assumed, then the region, where the solution is harmonic, has to have spherical symmetry.     

A covering lemma for {K(\mathbb {R})}

The Dodd–Jensen Covering Lemma states that “if there is no inner model with a measurable cardinal, then for any uncountable set of ordinals X, there is a ${Y\in K}$ such that ${X\subseteq Y}$ and |X| = |Y|”. Assuming ZF+AD alone, we establish the following analog: If there is no inner model with an ${\mathbb {R}}$ –complete measurable cardinal, then the real core model ${K(\mathbb {R})}$ is a “very good approximation” to the universe of sets V; that is, ${K(\mathbb {R})}$ and V have exactly the same sets of reals and for any set of ordinals X with ${|{X}|\ge\Theta}$ , there is a ${Y\in K(\mathbb {R})}$ such that ${X\subseteq Y}$ and |X| = |Y|. Here ${\mathbb {R}}$ is the set of reals and ${\Theta}$ is the supremum of the ordinals which are the surjective image of ${\mathbb {R}}$ .     

Projective Hausdorff gaps

Todor?evi? (Fund Math 150(1):55–66, 1996) shows that there is no Hausdorff gap (A, B) if A is analytic. In this note we extend the result by showing that the assertion “there is no Hausdorff gap (A, B) if A is coanalytic” is equivalent to “there is no Hausdorff gap (A, B) if A is ${{\bf \it{\Sigma}}^{1}_{2}}$ ”, and equivalent to ${\forall r \; (\aleph_1^{L[r]}\,< \aleph_1)}$ . We also consider real-valued games corresponding to Hausdorff gaps, and show that ${\mathsf{AD}_\mathbb{R}}$ for pointclasses Γ implies that there are no Hausdorff gaps (A, B) if ${{\it{A}} \in {\bf \it{\Gamma}}}$ .     

Hölderabschätzungen für Ableitungen von Lösungen der Gleichung\bar \partial u = f bei streng pseudokonvexem Rand

Using the local Kerzman kernel we prove regularity of solutions of \(\bar \partial \) u=f, where f is a \(\bar \partial \) -closed (0,1)-form in a strongly pseudoconvex domain G in ?^N. If f is in H^m,∞, then the solution is in \(\tilde C^{m,\mu } \) forμ<1, that is, the m-th derivatives are in C^o,μ/2 and in addition areμ-Hölder continuous on curves “parallel” to the holomorphic tangent bundle \(\tilde T\) ?G. If f is in C^m,α with o<α<1, then the solution is in \(\tilde C^{m,1 + \mu } \) forμ<α, that is, the m-th derivatives are in C^o,(1+μ/2, and they have first derivatives “parallel” to \(\tilde T\) ?G lying in \(\tilde C^{o,\mu } \) . We derive the same results for the global solution constructed by Grauert and Lieb, and similar estimates on complex manifolds.     

Retraction closure property

We say that an algebra ${\mathcal{A}}$ has the retraction closure property (RCP) if the set of all retractions of ${\mathcal{A}}$ is closed with respect to fundamental operations of ${\mathcal{A}}$ applied pointwise. In this paper we investigate this property, both “locally” (one algebra) and “globally” (in some variety of algebras), especially emphasizing the case of groupoids. We compare the retraction closure property with the endomorphism closure property on both levels and prove that a necessary and sufficient condition for a variety V of algebras to have RCP is that V is a variety of entropic algebras that satisfy the diagonal identities.     

On Sets Defining Few Ordinary Lines

Let $P$ P be a set of $n$ n points in the plane, not all on a line. We show that if $n$ n is large then there are at least $n/2$ n / 2 ordinary lines, that is to say lines passing through exactly two points of $P$ P . This confirms, for large $n$ n , a conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for this problem, as well as all sets having fewer than $n-C$ n - C ordinary lines for some absolute constant $C$ C . We also solve, for large $n$ n , the “orchard-planting problem”, which asks for the maximum number of lines through exactly 3 points of $P$ P . Underlying these results is a structure theorem which states that if $P$ P has at most $Kn$ K n ordinary lines then all but O(K) points of $P$ P lie on a cubic curve, if $n$ n is sufficiently large depending on $K$ K .     

Towards factoring in {SL(2,\,\mathbb{F}_{2^n})}

The security of many cryptographic protocols relies on the hardness of some computational problems. Besides discrete logarithm or integer factorization, other problems are regularly proposed as potential hard problems. The factorization problem in finite groups is one of them. Given a finite group G, a set of generators generators for this group and an element ${g\in G}$ , the factorization problem asks for a “short” representation of g as a product of the generators. The problem is related to a famous conjecture of Babai on the diameter of Cayley graphs. It is also motivated by the preimage security of Cayley hash functions, a particular kind of cryptographic hash functions. The problem has been solved for a few particular generator sets, but essentially nothing is known for generic generator sets. In this paper, we make significant steps towards a solution of the factorization problem in the group ${G:=\,SL(2,\,\mathbb{F}_{2^n})}$ , a particularly interesting group for cryptographic applications. To avoid considering all generator sets separately, we first give a new reduction tool that allows focusing on some generator sets with a “nice” special structure. We then identify classes of trapdoor matrices for these special generator sets, such that the factorization of a single one of these matrices would allow efficiently factoring any element in the group. Finally, we provide a heuristic subexponential time algorithm that can compute subexponential length factorizations of any element for any pair of generators of ${SL(2,\,\mathbb{F}_{2^n})}$ . Our results do not yet completely remove the factorization problem in ${SL(2,\,\mathbb{F}_{2^n})}$ from the list of potential hard problems useful for cryptography. However, we believe that each one of our individual results is a significant step towards a polynomial time algorithm for factoring in ${SL(2,\,\mathbb{F}_{2^n})}$ .     

Capacitary estimates of solutions of semilinear parabolic equations

We prove that any positive solution of ${\partial_tu-\Delta u+u^q=0 (q > 1)}$ in ${\mathbb{R}^N \times (0, \infty)}$ with initial trace (F, 0), where F is a closed subset of ${\mathbb{R}^{N}}$ can be represented, up to two universal multiplicative constants, by a series involving the Bessel capacity ${C_{2/q, q^{\prime}}}$ . As a consequence we prove that there exists a unique positive solution of the equation with such an initial trace. We also characterize the blow-up set of u(x, t) when ${t \downarrow 0}$ , by using the “density” of F expressed in terms of the ${C_{2/q, q^{\prime}}}$ -Bessel capacity.     

The pseudo-hyperplanes and homogeneous pseudo-embeddings of the generalized quadrangles of order (3, t)

In the paper “as reported by De Bruyn (Adv Geom, to appear)”, we introduced the notions of pseudo-hyperplane and pseudo-embedding of a point-line geometry and proved that every generalized quadrangle of order (s, t), 2 ≤ s < ∞, has faithful pseudo-embeddings. The present paper focuses on generalized quadrangles of order (3, t). Using the computer algebra system GAP and invoking some theoretical relationships between pseudo-hyperplanes and pseudo-embeddings obtained in “De Bruyn (Adv Geom, to appear)”, we are able to give a complete classification of all pseudo-hyperplanes of ${\mathcal{Q}}$ . We hereby find several new examples of tight sets of generalized quadrangles, as well as a complete classification of all 2-ovoids of ${\mathcal{Q}}$ . We use the classification of the pseudo-hyperplanes of ${\mathcal{Q}}$ to obtain a list of all homogeneous pseudo-embeddings of ${\mathcal{Q}}$ .     

Symmetric 2-Structures,a Classification

We classify symmetric 2-structures ${(P, \mathfrak{G}_1, \mathfrak{G}_2, \mathfrak{K})}$ , i.e. chain structures which correspond to sharply 2-transitive permutation sets (E, Σ) satisfying the condition: “ ${(*) \, \, \forall \sigma, \tau \in \Sigma : \sigma \circ \tau^{-1} \circ \sigma \in \Sigma}$ ”. To every chain ${K \in \mathfrak{K}}$ one can associate a reflection ${\widetilde{K}}$ in K. Then (*) is equivalent to “ ${(**) \, \, \forall K \in \mathfrak{K} : \widetilde{K}(\mathfrak{K}) = \mathfrak{K}}$ ” and one can define an orthogonality “ ${\perp}$ ” for chains ${K, L \in \mathfrak{K}}$ by “ ${K \perp L \Leftrightarrow K \neq L \wedge \widetilde{K}(L) = L}$ ”. The classification is based on the cardinality of the set of chains which are orthogonal to a chain K and passing through a point p of K. For one of these classes (called point symmetric 2-structures) we proof that in each point there is a reflection and that the set of point reflections forms a regular involutory permutation set.     

A Representation of a Point Symmetric 2-Structure by a Quasi-Domain

In a symmetric 2-structure ${\Sigma =(P,\mathfrak{G}_1,\mathfrak{G}_2,\mathfrak{K})}$ we fix a chain ${E \in \mathfrak{K}}$ and define on E two binary operations “+” and “·”. Then (E,+) is a K-loop and for ${E^* := E {\setminus}\{o\}}$ , (E ^*,·) is a Bol loop. If ${\Sigma}$ is even point symmetric then (E,+ ,·) is a quasidomain and one has the set ${Aff(E,+,\cdot) := \{a^+\circ b^\bullet | a \in E, b \in E^*\}}$ of affine permutations. From Aff(E, +, ·) one can reproduce via a “chain derivation” the point symmetric 2-structure ${\Sigma}$ .     

Length of Geodesics and Quantitative Morse Theory on Loop Spaces

Let M ⁿ be a closed Riemannian manifold of diameter d. Our first main result is that for every two (not necessarily distinct) points ${p,q \in M^n}$ and every positive integer k there are at least k distinct geodesics connecting p and q of length ${\leq 4nk^2d}$ . We demonstrate that all homotopy classes of M ⁿ can be represented by spheres swept-out by “short” loops unless the length functional has “many” “deep” local minima of a “small” length on the space ${\Omega_{pq}M^n}$ of paths connecting p and q. For example, one of our results implies that for every positive integer k there are two possibilities: Either the length functional on ${\Omega_{pq} M^n}$ has k distinct non-trivial local minima with length ${\leq 2kd}$ and “depth” ${\geq 2d}$ ; or for every m every map of S ^m into ${\Omega_{pq}M^n}$ is homotopic to a map of S ^m into the subspace ${\Omega_{pq}^{4(k+2)(m+1)d}M^n}$ of ${\Omega_{pq}M^n}$ that consists of all paths of length ${\leq 4(k+2)(m+1)d}$ .     

Theta dichotomy for the genuine unramified principal series of {\widetilde{Sp}_2(F)}

Let F be a p-adic field with odd residual characteristic. This work is the continuation of a previous paper that contains some detailed computations of the doubling integral for irreducible constituents ${(\pi, \mathcal{V}_{\pi})}$ of the genuine unramified principal series of ${\widetilde{Sp}_2(F)}$ using various “good test data”. This paper aims to interpret those results in terms of the non-vanishing of local theta lifts. Assuming a technical condition on order of a particular pole for the family of doubling integrals for ${(\pi, \mathcal{V}_{\pi})}$ , we aim to determine the so-called “dichotomy sign” of ${(\pi, \mathcal{V}_{\pi})}$ .     

The stress intensity factor for non-smooth fractures in antiplane elasticity

Motivated by some questions arising in the study of quasistatic growth in brittle fracture, we investigate the asymptotic behavior of the energy of the solution u of a Neumann problem near a crack in dimension 2. We consider non smooth cracks K that are merely closed and connected. At any point of density 1/2 in K, we show that the blow-up limit of u is the usual “cracktip” function ${C\sqrt{r}\sin(\theta/2)}$ , with a well-defined coefficient (the “stress intensity factor” or SIF). The method relies on Bonnet’s monotonicity formula (Bonnet, Variational methods for discontinuous structures, pp. 93–103. Birkhäuser, Basel, 1996) together with Γ-convergence techniques.     

On the chaotic character of the stochastic heat equation, II

Consider the stochastic heat equation ${\partial_t u = (\varkappa/2)\Delta u+\sigma(u)\dot{F}}$ , where the solution u := u _t (x) is indexed by ${(t, x) \in (0, \infty) \times {\bf R}^d}$ , and ${\dot{F}}$ is a centered Gaussian noise that is white in time and has spatially-correlated coordinates. We analyze the large- ${\|x\|}$ fixed-t behavior of the solution u in different regimes, thereby study the effect of noise on the solution in various cases. Among other things, we show that if the spatial correlation function f of the noise is of Riesz type, that is ${f(x)\propto \|x\|^{-\alpha}}$ , then the “fluctuation exponents” of the solution are ${\psi}$ for the spatial variable and ${2\psi-1}$ for the time variable, where ${\psi:=2/(4-\alpha)}$ . Moreover, these exponent relations hold as long as ${\alpha \in (0, d \wedge 2)}$ ; that is precisely when Dalang’s theory [Dalang, Electron J Probab 4:(Paper no. 6):29, 1999] implies the existence of a solution to our stochastic PDE. These findings bolster earlier physical predictions [Kardar et al., Phys Rev Lett 58(20):889–892, 1985; Kardar and Zhang, Phys Rev Lett 58(20):2087–2090, 1987].     

The elliptic functions and integrals of the “{1 \mathord{\left/ {\vphantom {1 9}} \right. \kern-\nulldelimiterspace} 9}” problem

We describe the functions needed in the determination of the rate of convergence of best $L^\infty $ rational approximation to $\exp ( - x)$ on [0,∞) when the degree n of the approximation tends to ∞ (“1/9” problem).     

Reflection Spaces and Corresponding Kinematic Structures

For a reflection space (P, Γ) [introduced in Karzel and Taherian (Results Math 59:213–218, 2011)] we define the notion “Reducible Subspace”, consider two subsets of ${\Gamma, \Gamma^{+} := \{a b\,|\, a,b \in P\}}$ and ${\Gamma^{-} := \{a b c\,|\, a, b, c \in P\}}$ and the map $$ \kappa : 2^{P} \to 2^{\Gamma^+} ; X \mapsto X \cdot X := \{xy\,|\, x,y \in X\}$$ We show, for each subspace S of (P, Γ), V := κ(S) is a v-subgroup (i.e. V is a subgroup of Γ⁺ with if ${\xi = xy \in V, \xi \neq 1}$ then ${x \cdot \overline{x,y}\subseteq V}$ ) if and only if S is reducible. Our main results are stated in the items 1–5 in the introduction.     

On generalizing transitivity, persistence, and sensitivity

A subgroup property $\alpha $ is transitive in a group $G$ if $U \alpha V$ and $V \alpha G$ imply that $U \alpha G$ whenever $U \le V \le G$ , and $\alpha $ is persistent in $G$ if $U \alpha G$ implies that $U \alpha V$ whenever $U \le V \le G$ . Even though a subgroup property $\alpha $ may be neither transitive nor persistent, a given subgroup $U$ may have the property that each $\alpha $ -subgroup of $U$ is an $\alpha $ -subgroup of $G$ , or that each $\alpha $ -subgroup of $G$ in $U$ is an $\alpha $ -subgroup of $U$ . We call these subgroup properties $\alpha $ -transitivity and $\alpha $ -persistence, respectively. We introduce and develop the notions of $\alpha $ -transitivity and $\alpha $ -persistence, and we establish how the former property is related to $\alpha $ -sensitivity. In order to demonstrate how these concepts can be used, we apply the results to the cases in which $\alpha $ is replaced with “normal” and the “cover-avoidance property.” We also suggest ways in which the theory can be developed further.