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The lightcone singularity at the origin is resolved by blowing up the singular point to $CP^1$. The Lorentz group acts on the resolved lightcone and has $CP^1$ as a special orbit. Using Wigner's method of associating unitary irreducible representations of the Poincar\'e group to particle states, we find that the special orbit gives rise to new vacuum states. These vacuum states are labelled by the principal series representations of $SL(2,C)$. Some remarks are included on the applications of these results to gauge theories and asymptotically flat spacetimes., Comment: 8 pages, new material added

International audience; We consider the Schr\"odinger equation with nonlinear derivative term on $[0,+\infty)$ under Robin boundary condition at $0$. Using a virial argument, we obtain the existence of blowing up solutions and using variational techniques, we obtain stability and instability by blow up results for standing waves.

We are concerned with the existence of blowing-up solutions to the following boundary value problem \begin{document}$ -\Delta u = \lambda V(x) e^u-4\pi N {\mathit{\boldsymbol{\delta}}}_0\;\hbox{ in } \Omega, \quad u = 0 \;\hbox{ on }\partial \Omega, $\end{document} where \begin{document}$ \Omega $\end{document} is a smooth and bounded domain in \begin{document}$ \mathbb{R}^2 $\end{document} such that \begin{document}$ 0\in\Omega $\end{document} , \begin{document}$ V $\end{document} is a positive smooth potential, \begin{document}$ N $\end{document} is a positive integer and \begin{document}$ \lambda>0 $\end{document} is a small parameter. Here \begin{document}$ {\mathit{\boldsymbol{\delta}}}_0 $\end{document} defines the Dirac measure with pole at \begin{document}$ 0 $\end{document} . We assume that \begin{document}$ \Omega $\end{document} is \begin{document}$ (N+1) $\end{document} -symmetric and we find conditions on the potential \begin{document}$ V $\end{document} and the domain \begin{document}$ \Omega $\end{document} under which there exists a solution blowing up at \begin{document}$ N+1 $\end{document} points located at the vertices of a regular polygon with center \begin{document}$ 0 $\end{document} .

In the current paper, we provide a thorough investigation of the blowing up behaviour induced via diffusion of the solution of the following non local problem \begin{equation*} \left\{\begin{array}{rcl} \partial_t u &=& \Delta u - u + \displaystyle{\frac{u^p}{ \left(\mathop{\,\rlap{-}\!\!\int}\nolimits_\Omega u^r dr \right)^\gamma }}\quad\text{in}\quad \Omega \times (0,T), \\[0.2cm] \frac{ \partial u}{ \partial \nu} & = & 0 \text{ on } \Gamma = \partial \Omega \times (0,T),\\ u(0) & = & u_0, \end{array} \right. \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary $\partial \Omega;$ such problem is derived as the shadow limit of a singular Gierer-Meinhardt system, cf. \cite{KSN17, NKMI2018}. Under the Turing type condition $$ \frac{r}{p-1} < \frac{N}{2}, \gamma r \ne p-1, $$ we construct a solution which blows up in finite time and only at an interior point $x_0$ of $\Omega,$ i.e. $$ u(x_0, t) \sim (\theta^*)^{-\frac{1}{p-1}} \left[\kappa (T-t)^{-\frac{1}{p-1}} \right], $$ where $$ \theta^* := \lim_{t \to T} \left(\mathop{\,\rlap{-}\!\!\int}\nolimits_\Omega u^r dr \right)^{- \gamma} \text{ and } \kappa = (p-1)^{-\frac{1}{p-1}}. $$ More precisely, we also give a description on the final asymptotic profile at the blowup point $$ u(x,T) \sim ( \theta^* )^{-\frac{1}{p-1}} \left[ \frac{(p-1)^2}{8p} \frac{|x-x_0|^2}{ |\ln|x-x_0||} \right]^{ -\frac{1}{p-1}} \text{ as } x \to 0, $$ and thus we unveil the form of the Turing patterns occurring in that case due to driven-diffusion instability. The applied technique for the construction of the preceding blowing up solution mainly relies on the approach developed in \cite{MZnon97} and \cite{DZM3AS19}., Comment: 31 pages

A distributed binary hypothesis testing (HT) problem over a noisy (discrete and memoryless) channel studied previously by the authors is investigated from the perspective of the strong converse property. It was shown by Ahlswede and Csiszar that a strong converse holds in the above setting when the channel is rate-limited and noiseless. Motivated by this observation, we show that the strong converse continues to hold in the noisy channel setting for a special case of HT known as testing against independence (TAI), under the assumption that the channel transition matrix has non-zero elements. The proof utilizes the blowing up lemma and the recent change of measure technique of Tyagi and Watanabe as the key tools.

Graph orientations with low out-degree are one of several ways to efficiently store sparse graphs. If the graphs allow for insertion and deletion of edges, one may have to flip the orientation of some edges to prevent blowing up the maximum out-degree. We use arboricity as our sparsity measure. With an immensely simple greedy algorithm, we get parametrized trade-off bounds between out-degree and worst case number of flips, which previously only existed for amortized number of flips. We match the previous best worst-case algorithm (in O(log n) flips) for general arboricity and beat it for either constant or super-logarithmic arboricity. We also match a previous best amortized result for at least logarithmic arboricity, and give the first results with worst-case O(1) and O(sqrt(log n)) flips nearly matching degree bounds to their respective amortized solutions. Graph orientations with low out-degree are one of several ways to efficiently store sparse graphs. If the graphs allow for insertion and deletion of edges, one may have to flip the orientation of some edges to prevent blowing up the maximum out-degree. We use arboricity as our sparsity measure. With an immensely simple greedy algorithm, we get parametrized trade-off bounds between out-degree and worst case number of flips, which previously only existed for amortized number of flips. We match the previous best worst-case algorithm (in O(log n) flips) for general arboricity and beat it for either constant or super-logarithmic arboricity. We also match a previous best amortized result for at least logarithmic arboricity, and give the first results with worst-case O(1) and O(sqrt(log n)) flips nearly matching degree bounds to their respective amortized solutions.

In this paper, we explicitly construct the smooth compact base threefold for the elliptic Calabi-Yau fourfold with the largest known $h^{1,1}=303\,148$. It is generated by blowing up a smooth toric "seed" base threefold with $(E_8,E_8,E_8)$ collisions. The 4d F-theory compactification model over it has the largest geometric gauge group, $E_8^{2\,561}\times F_4^{7\,576}\times G_2^{20\,168}\times SU(2)^{30\,200}$, and the largest number of axions, $181\,820$, in the known 4d $\mathcal{N}=1$ supergravity landscape. We also prove that there are at least $1100^{15\,048}\approx 7.5\times 10^{45\,766}$ different flip and flop phases of this base threefold. Moreover, we find that many other base threefolds with large $h^{1,1}$ in the 4d F-theory landscape can be constructed in a similar way as well., v4, 32 pages

We consider a class of nonlinear elliptic problems whose prototype involves a coefficients matrix blowing up for a finite value m of the unknown u. Since datum is in $$L^1$$ , a suitable notion of renormalized solutions is introduced taking into account that u can reach the value m and the existence of such solutions is proved. We study the corresponding evolution problem as well.

We study existence of nonnegative solutions to a nonlinear parabolic boundary value problem with a general singular lower order term and a nonnegative measure as nonhomogeneous datum, of the form u t − Δ p u = h ( u ) f + μ in Ω × ( 0 , T ) , u = 0 on ∂ Ω × ( 0 , T ) , u = u 0 in Ω × { 0 } , where Ω is an open bounded subset of R N ( N ≥ 2 ), u 0 is a nonnegative integrable function, Δ p is the p -Laplace operator, μ is a nonnegative bounded Radon measure on Ω × ( 0 , T ) and f is a nonnegative function of L 1 ( Ω × ( 0 , T ) ) . The term h is a positive continuous function possibly blowing up at the origin. Furthermore, we show uniqueness of finite energy solutions in presence of a nonincreasing h .

We show a new example of blow-up behaviour for the prescribed $Q$-curvature equation in even dimension $6$ and higher, namely given a sequence $(V_k)\subset C^0(\mathbb{R}^{2n})$ suitably converging we construct {for $n\geq 3$} a sequence $(u_k)$ of radially symmetric solutions to the equation $${(-\Delta)^n u_k=V_k e^{2n u_k} \quad \text{in }\mathbb{R}^{2n},}$$ with $u_k$ blowing up at the origin \emph{and} on a sphere. We also prove sharp blow-up estimates. This is in sharp contrast with the $4$-dimensional case studied by F. Robert (J. Diff. Eq. 2006).

Following the work of Brown, we can canonically associate a family of motivic periods -- called the motivic Feynman amplitude -- to any convergent Feynman integral, viewed as a function of the kinematic variables. The motivic Galois theory of motivic Feynman amplitudes provides an organizing principle, as well as strong constraints, on the space of amplitudes in general, via Brown's "small graphs principle". This serves as motivation for explicitly computing the motivic Galois action, or, dually, the coaction of the Hopf algebra of functions on the motivic Galois group. In this paper, we study the motivic Galois coaction on the motivic Feynman amplitudes associated to one-loop Feynman graphs. We study the associated variations of mixed Hodge structures, and provide an explicit formula for the coaction on the four-edge cycle graph -- the box graph -- with non-vanishing generic kinematics, which leads to a formula for all one-loop graphs with non-vanishing generic kinematics in four-dimensional space-time. We also show how one computes the coaction in some degenerate configurations -- when defining the motive of the graph requires blowing up the underlying family of varieties -- on the example of the three-edge cycle graph., The prefactors appearing in front of the de Rham periods in the statement of Theorem 1 were corrected, some clarifications regarding the role of the Landau variety in the definition of a graph motive were added, and several typos were corrected

In the present paper initial problems for the semilinear integro-differential diffusion equation and system are considered. The analogue of Duhamel principle for the linear integro-differential diffusion equation is proved. The results on existence of local mild solutions and Fujita-type critical exponents to the semilinear integro-differential diffusion equation and system are presented., 27 pages

We consider the algebraic effective couplings for open superstring massless modes in the framework of the $A_\infty$ theory in the small Hilbert space. Focusing on quartic algebraic couplings, we reduce the effective action of the $A_\infty$ theory to the Berkovits one where we have already shown that such couplings are fully computed from contributions at the boundary of moduli space, when the massless fields under consideration are appropriately charged under an ${\cal N}\!=\!2$ $R$-symmetry. Here we offer a proof of localization which is in the small Hilbert space. We also discuss the flat directions of the obtained quartic potentials and give evidence for the existence of exactly marginal deformations in the $D3/D(-1)$ system in the framework of string field theory., 34 pages, no figures. V2: Improved presentation, typos corrected

We prove Grothendieck's Conjecture on Resolution of Singulari-ties for quasi-excellent schemes X of dimension three and of arbitrary characteristic. This applies in particular to X = SpecA, A a reduced complete Noetherian local ring of dimension three and to algebraic or arithmetical varieties of dimension three. Similarly, if F is a number field, a complete discretely valued field or more generally the quotient field of any excellent Dedekind domain O, any regular projective sur-face X/F has a proper and flat model X over O which is everywhere regular., updates and extends 'Resolution of Singularities of Arithmetical Threefolds I' posted on this website

We consider the nonlinear Schrodinger equation on $${\mathbb R}^N $$ , $$N\ge 1$$ , $$\begin{aligned} \partial _t u = i \varDelta u + \lambda | u |^\alpha u \end{aligned}$$ with $$\lambda \in {\mathbb C}$$ and $$\mathfrak {R}\lambda >0$$ , for $$H^1$$ -subcritical nonlinearities, i.e. $$\alpha >0$$ and $$(N-2) \alpha < 4$$ . Given a compact set $$K \subset {\mathbb {R}}^N $$ , we construct $$H^1$$ solutions that are defined on $$(-T,0)$$ for some $$T>0$$ , and blow up on K at $$t=0$$ . The construction is based on an appropriate ansatz. The initial ansatz is simply $$U_0(t,x) = ( \mathfrak {R}\lambda )^{- \frac{1}{\alpha }} (-\alpha t + A(x) )^{ -\frac{1}{\alpha } - i \frac{\mathfrak {I}\lambda }{\alpha \mathfrak {R}\lambda } }$$ , where $$A\ge 0$$ vanishes exactly on K, which is a solution of the ODE $$u'= \lambda | u |^\alpha u$$ . We refine this ansatz inductively, using ODE techniques. We complete the proof by energy estimates and a compactness argument. This strategy is reminiscent of Cazenave et al. (Discrete Contin Dyn Syst 39(2):1171–1183, 2019. https://doi.org/10.3934/dcds.2019050 ; Solutions blowing up on any given compact set for the energy subcritical wave equation. 2018. arXiv:1812.03949 ).

We are concerned with the mean field equation with singular data on bounded domains. By assuming a singular point to be a critical point of the 1-vortex Kirchhoff-Routh function, we prove local uniqueness and non-degeneracy of bubbling solutions blowing up at a singular point. The proof is based on sharp estimates for bubbling solutions of singular mean field equations and a suitably defined Pohozaev-type identity.

We consider the Gelfand problem on a planar domain. Under some conditions on the potential, we provide the first examples of multiplicity for blowing-up solutions at a given point in the domain. The argument is based on a refined Lyapunov-Schmidt reduction and the computation of the degree of a finite-dimensional map., 25 pages

We prove that any sufficiently differentiable space-like hypersurface of ℝ 1 + N {{\mathbb{R}}^{1+N}} coincides locally around any of its points with the blow-up surface of a finite-energy solution of the focusing nonlinear wave equation ∂ t ⁢ t ⁡ u - Δ ⁢ u = | u | p - 1 ⁢ u {\partial_{tt}u-\Delta u=|u|^{p-1}u} on ℝ × ℝ N {{\mathbb{R}}\times{\mathbb{R}}^{N}} , for any 1 ≤ N ≤ 4 {1\leq N\leq 4} and 1 < p ≤ N + 2 N - 2 {1 . We follow the strategy developed in our previous work (2018) on the construction of solutions of the nonlinear wave equation blowing up at any prescribed compact set. Here to prove blow-up on a local space-like hypersurface, we first apply a change of variable to reduce the problem to blowup on a small ball at t = 0 {t=0} for a transformed equation. The construction of an appropriate approximate solution is then combined with an energy method for the existence of a solution of the transformed problem that blows up at t = 0 {t=0} . To obtain a finite-energy solution of the original problem from trace arguments, we need to work with H 2 × H 1 {H^{2}\times H^{1}} solutions for the transformed problem.

We prove functorial weak factorization of projective birational morphisms of regular quasi-excellent schemes in characteristic 0 broadly based on the existing line of proof for varieties. From this general functorial statement we deduce factorization results for algebraic stacks, formal schemes, complex analytic germs, Berkovich analytic and rigid analytic spaces., Comment: 45 pages

We consider the focusing energy subcritical nonlinear wave equation ∂ t t u − Δ u = | u | p − 1 u in R N , N ≥ 1 . Given any compact set K ⊂ R N , we construct finite energy solutions which blow up at t = 0 exactly on K. The construction is based on an appropriate ansatz. The initial ansatz is simply U 0 ( t , x ) = κ ( t + A ( x ) ) − 2 p − 1 , where A ≥ 0 vanishes exactly on K, which is a solution of the ODE h ″ = h p . We refine this first ansatz inductively using only ODE techniques and taking advantage of the fact that (for suitably chosen A), space derivatives are negligible with respect to time derivatives. We complete the proof by an energy argument and a compactness method.

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$ and denote the regular part of the Green's function on $\Omega$ with Dirichlet boundary condition as $H(x,y)$. Assume that $q \in \Omega$ and $n\geq 5$. We prove that there exists an integer $k_0$ such that for any integer $k\geq k_0$ there exist initial data $u_0$ and smooth parameter functions $\xi(t)\to q$, $0, Comment: 60 pages; comments welcome

We consider the simplest parabolic-elliptic model of chemotaxis in the whole space in several dimensions. Criteria for the blowup of radially symmetric solutions in terms of suitable Morrey spaces norms are derived., 20 pages

We study the family of K3 surfaces of Picard rank sixteen associated with the double cover of the projective plane branched along the union of six lines, and the family of its Van Geemen-Sarti partners, i.e., K3 surfaces with special Nikulin involutions, such that quotienting by the involution and blowing up recovers the former. We prove that the family of Van Geemen-Sarti partners is a four-parameter family of K3 surfaces with $H \oplus E_7(-1) \oplus E_7(-1)$ lattice polarization. We describe explicit Weierstrass models on both families using even modular forms on the bounded symmetric domain of type $IV$. We also show that our construction provides a geometric interpretation, called geometric two-isogeny, for the F-theory/heterotic string duality in eight dimensions. As a result, we obtain novel F-theory models, dual to non-geometric heterotic string compactifications in eight dimensions with two non-vanishing Wilson line parameters., 42 pages; minor typos corrected in version 2

We prove that given a cubic fourfold $Y$ not containing any plane, the Voisin map $v: F(Y)\times F(Y) \dashrightarrow Z(Y)$ constructed in \cite{Voi}, where $F(Y)$ is the variety of lines and $Z(Y)$ is the Lehn-Lehn-Sorger-van Straten eightfold, can be resolved by blowing up the incident locus $\Gamma \subset F(Y)\times F(Y)$ endowed with the reduced scheme structure. Moreover, if $Y$ is very general, then this blowup is a relative Quot scheme over $Z(Y)$ parametrizing quotients in a heart of a Kuznetsov component of $Y.$, Comment: 16 pages, comments are welcome

In this paper we use some properties of spherical blowing-up to give an alternative and more geometric proof of Gau-Lipman Theorem about the differentiable invariance of the multiplicity of complex analytic sets. Moreover, we also provide a generalization of the Ephraim-Trotman Theorem., This is a post-peer-review, pre-copyedit version of an article published in Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas. The final authenticated version is available online at: http://doi.org/10.1007/s13398-018-0537-5

Let $$\text {Bl}_{\mathbb {P}^1} \mathbb {P}^n$$ be a Kahler manifold obtained by blowing up a complex projective space $$\mathbb {P}^n$$ along a line $$\mathbb {P}^1$$ . We prove that $$\text {Bl}_{\mathbb {P}^1} \mathbb {P}^n$$ is slope unstable with respect to any polarisation, and hence, it does not admit constant scalar curvature Kahler metrics in any rational Kahler class.

In this paper, a system of nonlinear integral equations related to combustion problems is considered. Necessary and sufficient conditions for the existence and explosion of positive solutions are given. In addition, the uniqueness of the positive solutions is shown. The main results are obtained by monotonicity methods.

In this article we extend cutting and blowing up to the nonrational symplectic toric setting. This entails the possibility of cutting and blowing up for symplectic toric manifolds and orbifolds in nonrational directions., Comment: 17 pages, 7 figures, minor changes in last version, to appear in Internat. J. Math

Blowing up a rational surface singularity in a reflexive module gives a (any) partial resolution dominated by the minimal resolution. The main theorem shows how deformations of the pair (singularity, module) relates to deformations of the corresponding pair of partial resolution and locally free strict transform, and to deformations of the underlying spaces. The results imply some recent conjectures on small resolutions and flops., v3, 26 pages. Section 2.4 on Ext and base-change is expanded. The proof of Lemma 2.9 is rewritten. Added remarks (6.5, 6.8 and 6.9), in particular relating to some added references concerning the Homological MMP. Some further additions and changes. The article is published (Adv. Math. 340 (2018), 1108--1140) with open access

In this paper we prove that if S is a smooth, irreducible, projective, rational, complex surface and D an effective, connected, reduced divisor on S, then the pair (S,D) is contractible if the log-Kodaira dimension of the pair is $-\infty$. More generally, we even prove that this contraction is possible without blowing up an assigned cluster of points on S. Using the theory of peeling, we are also able to give some information in the case D is not connected., Comment: 18 pages; comments welcome

We study systems with different diffusions (local and nonlocal), mixed boundary conditions, and reaction terms. We prove existence and uniqueness of the solutions and then analyze global existence vs blow up in finite time. For blowing up solutions, we find asymptotic bounds for the blow-up rate.

This paper deals with heat equations coupled via exponential boundary flux, where the solution is made up of $n$ components. Under certain monotone assumptions, necessary and sufficient conditions are obtained for simultaneous blow-up of at least two components for each initial datum. As for two components blowing up simultaneously, it is interesting that the representations of blow-up rates are quite different with respect to the different blow-up mechanisms and positions between the two components.