Journal articles on the topic 'Morse theory; Donaldson's theorem'

This paper contains two remarks about the application of the [Formula: see text]-invariant in Heegaard Floer homology and Donaldson's diagonalization theorem to knot theory. The first is the equivalence of two obstructions they give to a 2-bridge knot being smoothly slice. The second carries out a suggestion by Stefan Friedl to replace the use of Heegaard Floer homology by Donaldson's theorem in the proof of the main result of [J. E. Greene, Lattices, graphs, and Conway mutation, Invent. Math. 192(3) (2013) 717–750] concerning Conway mutation of alternating links.

Abstract We show that Hamiltonian monodromy of an integrable two degrees of freedom system with a global circle action can be computed by applying Morse theory to the Hamiltonian of the system. Our proof is based on Takens’s index theorem, which specifies how the energy-h Chern number changes when h passes a non-degenerate critical value, and a choice of admissible cycles in Fomenko–Zieschang theory. Connections of our result to some of the existing approaches to monodromy are discussed.

We define a new one-form HA based on the second fundamental tensor [Formula: see text], the Gauss–Bonnet–Chern form can be novelly expressed with this one-form. Using the ϕ-mapping theory we find that the Gauss–Bonnet–Chern density can be expressed in terms of the δ-function δ(ϕ) and the relationship between the Gauss–Bonnet–Chern theorem and Hopf–Poincaré theorem is given straightforwardly. The topological current of the Gauss–Bonnet–Chern theorem and its topological structure are discussed in details. At last, the Morse theory formula of the Euler characteristic is generalized.

AbstractWe show that the main theorem of Morse theory holds for a large class of functions on singular spaces. The function must satisfy certain conditions extending the usual requirements on a manifold that Condition C holds and the gradient flow around the critical sets is well-behaved, and the singular space must satisfy a local deformation retract condition. We then show that these conditions are satisfied when the function is the norm-square of a moment map on an affine variety, and that the homotopy equivalence from this theorem is equivariant with respect to the associated Hamiltonian group action. An important special case of these results is that the main theorem of Morse theory holds for the norm square of a moment map on the space of representations of a finite quiver with relations.

In the study of smooth functions on manifolds, min-max theory provides a mechanism for identifying critical values of a function. We introduce a discretized version of this theory associated to a discrete Morse function on a (regular) cell complex. As applications we prove a discrete version of the mountain pass lemma and give an alternate proof of a discrete Lusternik–Schnirelmann theorem.

We give a construction of an eigenstate for a non-critical level of the Hamiltonian function, and investigate the contribution of Morse critical points to the spectral decomposition. We compare the rigorous result with the series obtained by a perturbation theory. As an example the relation to the spectral asymptotics is discussed.

AbstractWe study a Dirichlet-type boundary value problem for a pseudodifferential equation driven by the fractional Laplacian, proving the existence of three non-zero solutions. When the reaction term is sublinear at infinity, we apply the second deformation theorem and spectral theory. When the reaction term is superlinear at infinity, we apply the mountain pass theorem and Morse theory.

In this paper we construct a Universal chain complex, counting zeros of closed 1-forms on a manifold. The Universal complex is a refinement of the well known Novikov complex; it relates the homotopy type of the manifold, after a suitable noncommutative localization, with the numbers of zeros of different indices which may have closed 1-forms within a given cohomology class. The main theorem of the paper generalizes the result of a joint paper with A. Ranicki, which treats the special case of closed 1-forms having integral cohomology classes. The present paper also describes a number of new inequalities, giving topological lower bounds on the minimum number of zeros of closed 1-forms. In particular, such estimates are provided by the homology of flat line bundles with monodromy described by complex numbers, which are not Dirichlet units.

The first and second homology groups, H1 and H2, are computed for configuration spaces of framed three-dimensional point particles with annihilation included, when up to two particles and an antiparticle are present, the types of frames considered being S2 and SO(3). Whereas a recent calculation for two-dimensional particles used the Mayer–Vietoris sequence, in the present work Morse theory is used. By constructing a potential function none of whose critical indices is less than four, we find that (for coefficients in an arbitrary field K) the homology groups H1 and H2 reduce to those of the frame space, S2 or SO(3) as the case may be. In the case of SO(3) frames this result implies that H1 (with coefficients in ℤ2) is generated by the cycle corresponding to a 2π rotation of the frame. (This same cycle is homologous to the exchange loop: the spin-statistics correlation.) It also implies that H2 is trivial, which means that there does not exist a topologically nontrivial Wess–Zumino term for SO(3) frames [in contrast to the two-dimensional case, where SO(2) frames do possess such a term]. In the case of S2 frames (with coefficients in ℝ), we conclude H2=ℝ, the generator being in effect the frame space itself. This implies that for S2 frames there does exist a Wess–Zumino term, as indeed is needed for the possibility of half-integer spin and the corresponding Fermi statistics. Taken together, these results for H1 and H2 imply that our configuration space “admits spin 1/2” for either choice of frame, meaning that the spin-statistics theorem previously proved for this space is not vacuous.

We study a smooth analogue of jumping curves of a holomorphic vector bundle, and use Yang–Mills theory over S2 to show that any non-trivial, smooth Hermitian vector bundle E over a smooth simply connected manifold, must have such curves. This is used to give new examples complex manifolds for which a non-trivial holomorphic vector bundle must have jumping curves in the classical sense (when c1(E) is zero). We also use this to give a new proof of a theorem of Gromov on the norm of curvature of unitary connections, and make the theorem slightly sharper. Lastly we define a sequence of new non-trivial integer invariants of smooth manifolds, connected to this theory of smooth jumping curves, and make some computations of these invariants. Our methods include an application of the recently developed Morse–Bott chain complex for the Yang–Mills functional over S2.

The level of distribution of a complex-valued sequence $b$ measures the quality of distribution of $b$ along sparse arithmetic progressions $nd+a$. We prove that the Thue–Morse sequence has level of distribution $1$, which is essentially best possible. More precisely, this sequence gives one of the first nontrivial examples of a sequence satisfying a Bombieri–Vinogradov-type theorem for each exponent $\theta <1$. This result improves on the level of distribution $2/3$ obtained by Müllner and the author. As an application of our method, we show that the subsequence of the Thue–Morse sequence indexed by $\lfloor n^c\rfloor$, where $1 < c < 2$, is simply normal. This result improves on the range $1 < c < 3/2$ obtained by Müllner and the author and closes the gap that appeared when Mauduit and Rivat proved (in particular) that the Thue–Morse sequence along the squares is simply normal.

The multiplicity of classical solutions for impulsive fractional differential equations has been studied by many scholars. Using Morse theory, Brezis and Nirenberg’s Linking Theorem, and Clark theorem, we aim to solve this kind of problems. By this way, we obtain the existence of at least three classical solutions and k distinct pairs of classical solutions. Finally, an example is presented to illustrate the feasibility of the main results in this paper.

AbstractWe consider a nonlinear periodic problem driven by the scalar p-Laplacian and with a reaction term which exhibits a (p – 1)-superlinear growth near ±∞ but need not satisfy the Ambrosetti-Rabinowitz condition. Combining critical point theory with Morse theory we prove an existence theorem. Then, using variational methods together with truncation techniques, we prove a multiplicity theorem establishing the existence of at least five non-trivial solutions, with precise sign information for all of them (two positive solutions, two negative solutions and a nodal (sign changing) solution).

In this article, Conley’s connection matrix theory and a spectral sequence analysis of a filtered Morse chain complex $(C,{\rm\Delta})$ are used to study global continuation results for flows on surfaces. The briefly described unfoldings of Lyapunov graphs have been proved to be a well-suited combinatorial tool to keep track of continuations. The novelty herein is a global dynamical cancellation theorem inferred from the differentials of the spectral sequence $(E^{r},d^{r})$. The local version of this theorem relates differentials $d^{r}$ of the $r$th page $E^{r}$ to Smale’s theorem on cancellation of critical points.

This paper gives two results that show that the dynamics of a time-periodic Lagrangian system on a hyperbolic manifold are at least as complicated as the geodesic flow of a hyperbolic metric. Given a hyperbolic geodesic in the Poincaré ball, Theorem A asserts that there are minimizers of the lift of the Lagrangian system that are a bounded distance away and have a variety of approximate speeds. Theorem B gives the existence of a collection of compact invariant sets of the Euler–Lagrange flow that are semiconjugate to the geodesic flow of a hyperbolic metric. These results can be viewed as a generalization of the Aubry–Mather theory of twist maps and the Hedlund–Morse–Gromov theory of minimal geodesics on closed surfaces and hyperbolic manifolds.

AbstractFor a real valued periodic smooth function u on R, n ≥ 0, one defines the osculating polynomial φs (of order 2n + 1) at a point s ∈ R to be the unique trigonometric polynomial of degree n, whose value and first 2n derivatives at s coincide with those of u at s. We will say that a point s is a clean maximal flex (resp. clean minimal flex) of the function u on S1 if and only if φs ≥ u (resp. φs ≤ u) and the preimage (φ - u)-1(0) is connected. We prove that any smooth periodic function u has at least n + 1 clean maximal flexes of order 2n + 1 and at least n + 1 clean minimal flexes of order 2n + 1. The assertion is clearly reminiscent of Morse theory and generalizes the classical four vertex theorem for convex plane curves.

An anticommuting analogue of Brownian motion, corresponding to fermionic quantum mechanics, is developed, and combined with classical Brownian motion to give a generalised Feynman-Kac-Itô formula for paths in geometric supermanifolds. This formula is applied to give a rigorous version of the proofs of the Atiyah-Singer index theorem based on supersymmetric quantum mechanics. After a discussion of the BFV approach to the quantization of theories with symmetry, it is shown how the quantization of the topological particle leads to the supersymmetric model introduced by Witten in his study of Morse theory.

This paper studies the existence of multiple solutions of the second-order difference boundary value problemΔ2u(n−1)+V′(u(n))=0,n∈ℤ(1,T),u(0)=0=u(T+1). By applying Morse theory, critical groups, and the mountain pass theorem, we prove that the previous equation has at least three nontrivial solutions when the problem is resonant at the eigenvalueλk (k≥2)of linear difference problemΔ2u(n−1)+λu(n)=0,n∈ℤ(1,T),u(0)=0=u(T+1)near infinity and the trivial solution of the first equation is a local minimizer under some assumptions onV.

Abstract We call a continuous self-map that reveals itself through a discrete set of point-value pairs a sampled dynamical system. Capturing the available information with chain maps on Delaunay complexes, we use persistent homology to quantify the evidence of recurrent behavior. We establish a sampling theorem to recover the eigenspaces of the endomorphism on homology induced by the self-map. Using a combinatorial gradient flow arising from the discrete Morse theory for Čech and Delaunay complexes, we construct a chain map to transform the problem from the natural but expensive Čech complexes to the computationally efficient Delaunay triangulations. The fast chain map algorithm has applications beyond dynamical systems.

AbstractWe define and study jet bundles in the geometric orbifold category. We show that the usual arguments from the compact and the logarithmic settings do not all extend to this more general framework. This is illustrated by simple examples of orbifold pairs of general type that do not admit any global jet differential, even if some of these examples satisfy the Green–Griffiths–Lang conjecture. This contrasts with an important result of Demailly (Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture, Pure Appl. Math. Q. 7 (2011), 1165–1207) proving that compact varieties of general type always admit jet differentials. We illustrate the usefulness of the study of orbifold jets by establishing the hyperbolicity of some orbifold surfaces, that cannot be derived from the current techniques in Nevanlinna theory. We also conjecture that Demailly's theorem should hold for orbifold pairs with smooth boundary divisors under a certain natural multiplicity condition, and provide some evidence towards it.

This is a sequel to the papers Oh and Wang (Real and Complex Submanifolds, Springer Proceedings in Mathematics and Statistics 106 (2014), 43–63, eds. by Y.-J. Suh and et al. for ICM-2014 satellite conference, Daejeon, Korea, August 2014; arXiv:1212.4817; Analysis of contact Cauchy–Riemann maps I: a priori$C^{k}$estimates and asymptotic convergence, submitted, preprint, 2012, arXiv:1212.5186v3). In Oh and Wang (Real and Complex Submanifolds, Springer Proceedings in Mathematics and Statistics 106 (2014), 43–63, eds. by Y.-J. Suh and et al. for ICM-2014 satellite conference, Daejeon, Korea, August 2014; arXiv:1212.4817), the authors introduced a canonical affine connection on $M$ associated to the contact triad $(M,\unicode[STIX]{x1D706},J)$. In Oh and Wang (Analysis of contact Cauchy–Riemann maps I: a priori$C^{k}$estimates and asymptotic convergence, submitted, preprint, 2012, arXiv:1212.5186v3), they used the connection to establish a priori$W^{k,p}$-coercive estimates for maps $w:\dot{\unicode[STIX]{x1D6F4}}\rightarrow M$ satisfying $\overline{\unicode[STIX]{x2202}}^{\unicode[STIX]{x1D70B}}w=0$, $d(w^{\ast }\unicode[STIX]{x1D706}\circ j)=0$without involving symplectization. We call such a pair $(w,j)$ a contact instanton. In this paper, we first prove a canonical neighborhood theorem of the locus $Q$ foliated by closed Reeb orbits of a Morse–Bott contact form. Then using a general framework of the three-interval method, we establish exponential decay estimates for contact instantons $(w,j)$ of the triad $(M,\unicode[STIX]{x1D706},J)$, with $\unicode[STIX]{x1D706}$ a Morse–Bott contact form and $J$ a CR-almost complex structure adapted to $Q$, under the condition that the asymptotic charge of $(w,j)$ at the associated puncture vanishes.We also apply the three-interval method to the symplectization case and provide an alternative approach via tensorial calculations to exponential decay estimates in the Morse–Bott case for the pseudoholomorphic curves on the symplectization of contact manifolds. This was previously established by Bourgeois (A Morse–Bott approach to contact homology, Ph.D. dissertation, Stanford University, 2002) (resp. by Bao (On J-holomorphic curves in almost complex manifolds with asymptotically cylindrical ends, Pacific J. Math. 278(2) (2015), 291–324)), by using special coordinates, for the cylindrical (resp. for the asymptotically cylindrical) ends. The exponential decay result for the Morse–Bott case is an essential ingredient in the setup of the moduli space of pseudoholomorphic curves which plays a central role in contact homology and symplectic field theory (SFT).

In 1970, Kenneth Kunen showed that there is no non-trivial elementary embedding of the universe V into itself [2] using the axiom of choice. Kunen remarked in his paper that the result can be formalized in Morse-Kelley theory of sets and classes. In this paper, we will work within ZF, Zermelo-Fraenkel axioms, and deal with embeddings definable with a formula and a parameter.In ZF, a “class” is usually synonymous with “property”, that is a class definable with a parameter, C = {x: φ(x,p)}, where φ is a formula in the language [∈}. Using this convention, let j be a class. Then “j is an elementary embedding of V into V” is not a single statement but a schema of statements “j preserves ψ” for each formula ψ. We prove that this schema is expressible in the language {∈} by a single formula:Lemma. An embedding j: V → V is elementary iff j preservesψ.Here ψ(α, ψ, a) is the property “a is an ordinal, φ is a formula and Vα.”The lemma is of course a schema of lemmas, one for each formula denning j and for each ψ to be preserved.Using this we prove our theorem in ZF (again, a schema of theorems.):Theorem 1.1. There is no nontrivial definable elementary embedding j: V → V.Many symbols and their definitions follow those used by Drake's book [1]. The formula Sat expresses the satisfaction relation . The formula Fmla(u) expresses that u is the Gödel-set for a formula.

The results of data mining can be used to predict the physical health status of sports athletes and college sports students and provide physical fitness warnings, so that students can pay attention to physical health status and adjust their physical exercise status. Discrete Morse theory, as a powerful optimization theory, plays a big role in algorithm optimization. This paper combines data mining and discrete Morse theory to propose a grid clustering algorithm based on discrete Morse theory. Moreover, according to the theorem that the cell complex reaches the optimum when it has the smallest possible critical point, this study applies the concept of critical points in the discrete Morse theory to optimize the grid clustering process to obtain clustering results. In addition, this study uses the improved C4.5 algorithm to analyze the physical fitness assessment results and obtains a valuable analysis of the physical fitness assessment results.

We obtain a simple and complete characterisation of which matchings on the Tait graph of a knot diagram induce a discrete Morse function (dMf) on the two sphere, extending a construction due to Cohen. We show these dMfs are in bijection with certain rooted spanning forests in the Tait graph. We use this to count the number of such dMfs with a closed formula involving the graph Laplacian. We then simultaneously generalise Kauffman's Clock Theorem and Kenyon-Propp-Wilson's correspondence in two different directions; we first prove that the image of the correspondence induces a bijection on perfect dMfs, then we show that all perfect matchings, subject to an admissibility condition, are related by a finite sequence of click and clock moves. Finally, we study and compare the matching and discrete Morse complexes associated to the Tait graph, in terms of partial Kauffman states, and provide some computations.

AbstractGiven a Lorentzian manifold (M, g), a geodesic γ in M and a timelike Jacobi field γ along γ, we introduce a special class of instants along γ that we call γ- pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the γ-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jacobi field γ is obtained as the restriction of a globally defined timelike Killing vector field.

We consider the finite binary words $Z(n)$, $n \in {\Bbb N}$, defined by the following self-similar process: $Z(0):=0$, $Z(1):=01$, and $Z(n+1):=Z(n)\cdot\overline{Z(n-1)}$, where the dot $\cdot$ denotes word concatenation, and $\overline{w}$ the word obtained from $w$ by exchanging the zeros and the ones. Denote by $Z(\infty)=01110100 \dots$ the limiting word of this process, and by $z(n)$ the $n$'th bit of this word. This sequence $z$ is an analogue of the Thue-Morse sequence. We show that a theorem of Bacher and Chapman relating the latter to a "Sierpiński matrix" has a natural analogue involving $z$. The semi-infinite self-similar matrix which plays the role of the Sierpiński matrix here is the zeta matrix of the poset of finite subsets of ${\Bbb N}$ without two consecutive elements, ordered by inclusion. We observe that this zeta matrix is nothing but the exponential of the incidence matrix of the Hasse diagram of this poset. We prove that the corresponding Möbius matrix has a simple expression in terms of the zeta matrix and the sequence $z$.

AbstractWe consider a parametric Neumann problem with an indefinite and unbounded potential. Using a combination of critical point theory with truncation and comparison techniques, with Morse theory and with invariance arguments for a suitable negative gradient flow, we prove two multiplicity theorems for certain values of the parameter. In the first theorem we produce three solutions and in the second five. For all solutions we provide sign information. Our work improves significantly results existing in the literature.

AbstractWe show the mapping class group, $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan–Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.

AbstractWe present a rigidity theorem for the action of the mapping class group $$\pi _0({\mathrm{Diff}}(M))$$ π 0 ( Diff ( M ) ) on the space $$\mathcal {R}^+(M)$$ R + ( M ) of metrics of positive scalar curvature for high dimensional manifolds M. This result is applicable to a great number of cases, for example to simply connected 6-manifolds and high dimensional spheres. Our proof is fairly direct, using results from parametrised Morse theory, the 2-index theorem and computations on certain metrics on the sphere. We also give a non-triviality criterion and a classification of the action for simply connected 7-dimensional $${\mathrm{Spin}}$$ Spin -manifolds.

AbstractFor the Newtonian (gravitational) n-body problem in the Euclidean d-dimensional space, the simplest possible solutions are provided by those rigid motions (homographic solutions) in which each body moves along a Keplerian orbit and the configuration of the n-body is a (constant up to rotations and scalings) central configuration. For $$d\le 3$$ d ≤ 3 , the only possible homographic motions are those given by central configurations. For $$d \ge 4$$ d ≥ 4 instead, new possibilities arise due to the higher complexity of the orthogonal group $$\mathrm {O}(d)$$ O ( d ) , as observed by Albouy and Chenciner (Invent Math 131(1):151–184, 1998). For instance, in $$\mathbb {R}^4$$ R 4 it is possible to rotate in two mutually orthogonal planes with different angular velocities. This produces a new balance between gravitational forces and centrifugal forces providing new periodic and quasi-periodic motions. So, for $$d\ge 4$$ d ≥ 4 there is a wider class of S-balanced configurations (containing the central ones) providing simple solutions of the n-body problem, which can be characterized as well through critical point theory. In this paper, we first provide a lower bound on the number of balanced (non-central) configurations in $$\mathbb {R}^d$$ R d , for arbitrary $$d\ge 4$$ d ≥ 4 , and establish a version of the $$45^\circ $$ 45 ∘ -theorem for balanced configurations, thus answering some of the questions raised in Moeckel (Central configurations, 2014). Also, a careful study of the asymptotics of the coefficients of the Poincaré polynomial of the collision free configuration sphere will enable us to derive some rather unexpected qualitative consequences on the count of S-balanced configurations. In the last part of the paper, we focus on the case $$d=4$$ d = 4 and provide a lower bound on the number of periodic and quasi-periodic motions of the gravitational n-body problem which improves a previous celebrated result of McCord (Ergodic Theory Dyn Syst 16:1059–1070, 1996).

AbstractIn this paper, we study the existence of multiple periodic solutions for the following fractional equation:(-\Delta)^{s}u+F^{\prime}(u)=0,\qquad u(x)=u(x+T)\quad x\in\mathbb{R}.For an even double-well potential, we establish more and more periodic solutions for a large period T. Without the evenness of F we give the existence of two periodic solutions of the problem. We make use of variational arguments, in particular Clark’s theorem and Morse theory.

AbstractWe study here the relative cohomology and the Gauss-Manin connections associated to an isolated singularity of a function on a manifold with boundary, i.e. with a fixed hyperplane section. We prove several relative analogs of classical theorems obtained mainly by E. Brieskorn and B. Malgrange, concerning the properties of the Gauss-Manin connection as well as its relations with the Picard-Lefschetz monodromy and the asymptotics of integrals of holomorphic forms along the vanishing cycles. Finally, we give an application in isochore deformation theory, i.e. the deformation theory of boundary singularities with respect to a volume form. In particular, we prove the relative analog of J. Vey's isochore Morse lemma, J .-P. Fran~oise's generalisation on the local normal forms of volume forms with respect to the boundary singularity-preserving diffeomorphisms, as well as M. D. Garay's theorem on the isochore version of Mather's versa! unfolding theorem.

AbstractWe consider a nonlinear parametric Robin problem driven by the p-Laplacian. We assume that the reaction exhibits a concave term near the origin. First we prove a multiplicity theorem producing three solutions with sign information (positive, negative and nodal) without imposing any growth condition near ±∞ on the reaction. Then, for problems with subcritical reaction, we produce two more solutions of constant sign, for a total of five solutions. For the semilinear problem (that is, for p = 2), we generate a sixth solution but without any sign information. Our approach is variational, coupled with truncation, perturbation and comparison techniques and with Morse theory.

We prove that a sequence satisfying a certain symmetry property is $2$-regular in the sense of Allouche and Shallit, i.e., the $\mathbb{Z}$-module generated by its $2$-kernel is finitely generated. We apply this theorem to develop a general approach for studying the $\ell$-abelian complexity of $2$-automatic sequences. In particular, we prove that the period-doubling word and the Thue-Morse word have $2$-abelian complexity sequences that are $2$-regular. Along the way, we also prove that the $2$-block codings of these two words have $1$-abelian complexity sequences that are $2$-regular.

The classical van der Waerden Theorem says that for every every finite set $S$ of natural numbers and every $k$-coloring of the natural numbers, there is a monochromatic set of the form $aS+b$ for some $a>0$ and $b\geq 0$. I.e., monochromatism is obtained by a dilation followed by a translation. We investigate the effect of reversing the order of dilation and translation. $S$ has the variant van der Waerden property for $k$ colors if for every $k$-coloring there is a monochromatic set of the form $a(S+b)$ for some $a>0$ and $b\geq 0$. On the positive side it is shown that every two-element set has the variant van der Waerden property for every $k$. Also, for every finite $S$ and $k$ there is an $n$ such that $nS$ has the variant van der Waerden property for $k$ colors. This extends the classical van der Waerden Theorem. On the negative side it is shown that if $S$ has at least three elements, the variant van der Waerden property fails for a sufficiently large $k$. The counterexamples to the variant van der Waerden property are constructed by specifying colorings as Thue-Morse sequences.

AbstractWe study mathematical programs with switching constraints (for short, MPSC) from the topological perspective. Two basic theorems from Morse theory are proved. Outside the W-stationary point set, continuous deformation of lower level sets can be performed. However, when passing a W-stationary level, the topology of the lower level set changes via the attachment of a w-dimensional cell. The dimension w equals the W-index of the nondegenerate W-stationary point. The W-index depends on both the number of negative eigenvalues of the restricted Lagrangian’s Hessian and the number of bi-active switching constraints. As a consequence, we show the mountain pass theorem for MPSC. Additionally, we address the question if the assumption on the nondegeneracy of W-stationary points is too restrictive in the context of MPSC. It turns out that all W-stationary points are generically nondegenerate. Besides, we examine the gap between nondegeneracy and strong stability of W-stationary points. A complete characterization of strong stability for W-stationary points by means of first and second order information of the MPSC defining functions under linear independence constraint qualification is provided. In particular, no bi-active Lagrange multipliers of a strongly stable W-stationary point can vanish.

This issue of M/C Journal is devoted to all things cute – Internet animals and stuffed toys, cartoon characters and branded bears. In what follows our nine contributors scrutinise a diverse range of media objects, discussing everything from the economics of Grumpy Cat and the aesthetics of Furbys to Reddit’s intellectual property dramas and the ethics of kitten memes. The articles range across diverse sites, from China to Canada, and equally diverse disciplines, including cultural studies, evolutionary economics, media anthropology, film studies and socio-legal studies. But they share a common aim of tracing out the connections between degraded media forms and wider questions of culture, identity, economy and power. Our contributors tell riveting stories about these connections, inviting us to see the most familiar visual culture in a new way. We are not the first to take cute media seriously as a site of cultural politics, and as an industry in its own right. Cultural theory has a long, antagonistic relationship with the kitsch and the disposable. From the Frankfurt School’s withering critique of cultural commodification to revisionist feminist accounts that emphasise the importance of the everyday, critics have been conducting sporadic incursions into this space for the better part of a century. The rise of cultural studies, a discipline committed to analysing “the scrap of ordinary or banal existence” (Morris and Frow xviii), has naturally provided a convincing intellectual rationale for such research, and has inspired an impressive array of studies on such things as Victorian-era postcards (Milne), Disney films (Forgacs), Hallmark cards (West, Jaffe) and stock photography (Frosh). A parallel strand of literary theory considers the diverse registers of aesthetic experience that characterize cute content (Brown, Harris). Sianne Ngai has written elegantly on this topic, noting that “while the avant-garde is conventionally imagined as sharp and pointy, as hard- or cutting-edge, cute objects have no edge to speak of, usually being soft, round, and deeply associated with the infantile and the feminine” (814). Other scholars trace the historical evolution of cute aesthetics and commodities. Cultural historians have documented the emergence of consumer markets for children and how these have shaped what we think of as cute (Cross). Others have considered the history of domestic animal imagery and its symptomatic relationship with social anxieties around Darwinism, animal rights, and pet keeping (Morse and Danahay, Ritvo). And of course, Japanese popular culture – with its distinctive mobilization of cute aesthetics – has attracted its own rich literature in anthropology and area studies (Allison, Kinsella). The current issue of M/C Journal extends these lines of research while also pushing the conversation in some new directions. Specifically, we are interested in the collision between cute aesthetics, understood as a persistent strand of mass culture, and contemporary digital media. What might the existing tradition of “cute theory” mean in an Internet economy where user-generated content sites and social media have massively expanded the semiotic space of “cute” – and the commercial possibilities this entails? As the heir to a specific mode of degraded populism, the Internet cat video may be to the present what the sitcom, the paperback novel, or the Madonna video was to an earlier moment of cultural analysis. Millions of people worldwide start their days with kittens on Roombas. Global animal brands, such as Maru and Grumpy Cat, are appearing, along with new talent agencies for celebrity pets. Online portal I Can Haz Cheezburger has received millions of dollars in venture capital funding, becoming a diversified media business (and then a dotcom bubble). YouTube channels, Twitter hashtags and blog rolls form an infrastructure across which a vast amount of cute-themed user-generated content, as well as an increasing amount of commercially produced and branded material, now circulates. All this reminds us of the oft-quoted truism that the Internet is “made of kittens”, and that it’s “kittens all the way down”. Digitization of cute culture leads to some unusual tweaks in the taste hierarchies explored in the aforementioned scholarship. Cute content now functions variously as an affective transaction, a form of fandom, and as a subcultural discourse. In some corners of the Internet it is also being re-imagined as something contemporary, self-reflexive and flecked with irony. The example of 4Chan and LOLcats, a jocular, masculinist remix of the feminized genre of pet photography, is particularly striking here. How might the topic of cute look if we moving away from the old dialectics of mass culture critique vs. defense and instead foreground some of these more counter-intuitive aspects, taking seriously the enormous scale and vibrancy of the various “cute” content production systems – from children’s television to greeting cards to CuteOverload.com – and their structural integration into current media, marketing and lifestyle industries? Several articles in this issue adopt this approach, investigating the undergirding economic and regulatory structures of cute culture. Jason Potts provides a novel economic explanation for why there are so many animals on the Internet, using a little-known economic theory (the Alchian-Allen theorem) to explain the abundance of cat videos on YouTube. James Meese explores the complex copyright politics of pet images on Reddit, showing how this online community – which is the original source of much of the Internet’s animal gifs, jpegs and videos – has developed its own procedures for regulating animal image “piracy”. These articles imaginatively connect the soft stuff of cute content with the hard stuff of intellectual property and supply-and-demand dynamics. Another line of questioning investigates the political and bio-political work involved in everyday investments in cute culture. Seen from this perspective, cute is an affect that connects ground-level consumer subjectivity with various economic and political projects. Carolyn Stevens’ essay offers an absorbing analysis of the Japanese cute character Rilakkuma (“Relaxed Bear”), a wildly popular cartoon bear that is typically depicted lying on the couch and eating sweets. She explores what this representation means in the context of a stagnant Japanese economy, when the idea of idleness is taking on a new shade of meaning due to rising under-employment and precarity. Sharalyn Sanders considers a fascinating recent case of cute-powered activism in Canada, when animal rights activists used a multimedia stunt – a cat, Tuxedo Stan, running for mayor of Halifax, Canada – to highlight the unfortunate situation of stray and feral felines in the municipality. Sanders offers a rich analysis of this unusual political campaign and the moral questions it provokes. Elaine Laforteza considers another fascinating collision of the cute and the political: the case of Lil’ Bub, an American cat with a rare genetic condition that results in a perpetually kitten-like facial expression. During 2011 Lil’ Bub became an online phenomenon of the first order. Laforteza uses this event, and the controversies that brewed around it, as an entry point for a fascinating discussion of the “cute-ification” of disability. These case studies remind us once more of the political stakes of representation and viral communication, topics taken up by other contributors in their articles. Radha O’Meara’s “Do Cats Know They Rule YouTube? How Cat Videos Disguise Surveillance as Unselfconscious Play” provides a wide-ranging textual analysis of pet videos, focusing on the subtle narrative structures and viewer positioning that are so central to the pleasures of this genre. O’Meara explains how the “cute” experience is linked to the frisson of surveillance, and escape from surveillance. She also explains the aesthetic differences that distinguish online dog videos from cat videos, showing how particular ideas about animals are hardwired into the apparently spontaneous form of amateur content production. Gabriele de Seta investigates the linguistics of cute in his nuanced examination of how a new word – meng – entered popular discourse amongst Mandarin Chinese Internet users. de Seta draws our attention to the specificities of cute as a concept, and how the very notion of cuteness undergoes a series of translations and reconfigurations as it travels across cultures and contexts. As the term meng supplants existing Mandarin terms for cute such as ke’ai, debates around how the new word should be used are common. De Seta shows us how deploying these specific linguistic terms for cuteness involve a range of linguistic and aesthetic judgments. In short, what exactly is cute and in what context? Other contributors offer much-needed cultural analyses of the relationship between cute aesthetics, celebrity and user-generated culture. Catherine Caudwell looks at the once-popular Furby toy brand its treatment in online fan fiction. She notes that these forms of online creative practice offer a range of “imaginative and speculative” critiques of cuteness. Caudwell – like de Seta – reminds us that “cuteness is an unstable aesthetic that is culturally contingent and very much tied to behaviour”, an affect that can encompass friendliness, helplessness, monstrosity and strangeness. Jonathon Hutchinson’s article explores “petworking”, the phenomenon of social media-enabled celebrity pets (and pet owners). Using the famous example of Boo, a “highly networked” celebrity Pomeranian, Hutchinson offers a careful account of how cute is constructed, with intermediaries (owners and, in some cases, agents) negotiating a series of careful interactions between pet fans and the pet itself. Hutchinson argues if we wish to understand the popularity of cute content, the “strategic efforts” of these intermediaries must be taken into account. Each of our contributors has a unique story to tell about the aesthetics of commodity culture. The objects they analyse may be cute and furry, but the critical arguments offered here have very sharp teeth. We hope you enjoy the issue.Acknowledgments Thanks to Axel Bruns at M/C Journal for his support, to our hard-working peer reviewers for their insightful and valuable comments, and to the Swinburne Institute for Social Research for the small grant that made this issue possible. ReferencesAllison, Anne. “Cuteness as Japan’s Millenial Product.” Pikachu’s Global Adventure: The Rise and Fall of Pokemon. Ed. Joseph Tobin. Durham: Duke University Press, 2004. 34-48. Brown, Laura. Homeless Dogs and Melancholy Apes: Humans and Other Animals in the Modern Literary Imagination. Ithaca: Cornell University Press, 2010. Cross, Gary. The Cute and the Cool: Wondrous Innocence and Modern American Children's Culture. Oxford: Oxford University Press, 2004. Forgacs, David. 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