Anderson’s Orthogonality Catastrophe for One-Dimensional Systems

The overlap, \({\mathcal{D}_N}\) , between the ground state of N free fermions and the ground state of N fermions in an external potential in one spatial dimension is given by a generalized Gram determinant. An upper bound is \({\mathcal{D}_N\leq\exp(-\mathcal{I}_N)}\) with the so-called Anderson integral \({\mathcal{I}_N}\) . We prove, provided the external potential satisfies some conditions, that in the thermodynamic limit \({\mathcal{I}_N = \gamma\ln N + O(1)}\) as \({N\to\infty}\) . The coefficient γ > 0 is given in terms of the transmission coefficient of the one-particle scattering matrix. We obtain a similar lower bound on \({\mathcal{D}_N}\) concluding that \({\tilde{C} N^{-\tilde{\gamma}} \leq \mathcal{D}_N \leq CN^{-\gamma}}\) with constants C, \({\tilde{C}}\) , and \({\tilde{\gamma}}\) . In particular, \({\mathcal{D}_N\to 0}\) as \({N\to\infty}\) which is known as Anderson’s orthogonality catastrophe.     

Solid Extensions of the Cesàro Operator on ℓ p and c 0

We introduce and study the largest Banach lattice (for the coordinate-wise order) which is a solid subspace of \({\mathbb{C}^\mathbb{N}}\) and to which the classical Cesàro operator \({\mathcal{C}\colon\ell^p \to \ell^p}\) (a positive operator) can be continuously extended while still maintaining its values in ? ^p . Properties of this optimal Banach lattice \({[\mathcal{C}, \ell^p]_s}\) are presented. In addition, all continuous convolution operators of \({[\mathcal{C}, \ell^p]_s}\) into itself are identified and the spectrum of \({\mathcal{C}\colon[\mathcal{C}, \ell^p]_s \to[\mathcal{C}, \ell^p]_s}\) is determined. A similar investigation is undertaken for the Cesàro operator \({\mathcal{C}\colon c_0\to c_0}\) .     

Erdős-Ginzburg-Ziv theorem for finite commutative semigroups

Let \(\mathcal{S}\) be a finite additively written commutative semigroup, and let \(\exp(\mathcal{S})\) be its exponent which is defined as the least common multiple of all periods of the elements in \(\mathcal{S}\) . For every sequence T of elements in \(\mathcal{S}\) (repetition allowed), let \(\sigma(T) \in\mathcal{S}\) denote the sum of all terms of T. Define the Davenport constant \(\mathsf{D}(\mathcal{S})\) of \(\mathcal{S}\) to be the least positive integer d such that every sequence T over \(\mathcal{S}\) of length at least d contains a proper subsequence T′ with σ(T′)=σ(T), and define \(\mathsf{E}(\mathcal{S})\) to be the least positive integer ? such that every sequence T over \(\mathcal{S}\) of length at least ? contains a subsequence T′ with \(|T|-|T'|= \lceil\frac{|\mathcal{S}|}{\exp(\mathcal{S})} \rceil \exp(\mathcal{S})\) and σ(T′)=σ(T). When \(\mathcal{S}\) is a finite abelian group, it is well known that \(\lceil\frac{|\mathcal{S}|}{\exp(\mathcal{S})} \rceil\exp (\mathcal{S})=|\mathcal{S}|\) and \(\mathsf{E}(\mathcal{S})=\mathsf{D}(\mathcal{S})+|\mathcal{S}|-1\) . In this paper we investigate whether \(\mathsf{E}(\mathcal{S})\leq \mathsf{D}(\mathcal{S})+ \lceil\frac{|\mathcal{S}|}{\exp(\mathcal {S})} \rceil \exp(\mathcal{S})-1\) holds true for all finite commutative semigroups \(\mathcal{S}\) . We provide a positive answer to the question above for some classes of finite commutative semigroups, including group-free semigroups, elementary semigroups, and archimedean semigroups with certain constraints.     

Hall Polynomials and Composition Algebra of Representation Finite Algebras

Let \(\mathcal{A}\) be a representation finite algebra over finite field k such that the indecomposable \(\mathcal{A}\) -modules are determined by their dimension vectors and for each \(M, L \in ind(\mathcal{A})\) and \(N\in mod(\mathcal{A})\) , either \(F^{M}_{N L}=0\) or \(F^{M}_{L N}=0\) . We show that \(\mathcal{A}\) has Hall polynomials and the rational extension of its Ringel–Hall algebra equals the rational extension of its composition algebra. This result extend and unify some known results about Hall polynomials. As a consequence we show that if \(\mathcal{A}\) is a representation finite simply-connected algebra, or finite dimensional k-algebra such that there are no short cycles in \(mod(\mathcal{A})\) , or representation finite cluster tilted algebra, then \(\mathcal{A}\) has Hall polynomials and \(\mathcal{H}(\mathcal{A})\otimes_\mathbb{Z}Q=\mathcal{C}(\mathcal{A})\otimes_\mathbb{Z}Q\) .     

On the Relaxed Colouring Game and the Unilateral Colouring Game

In the paper we introduce the new game—the unilateral \({\mathcal{P}}\) -colouring game which can be used as a tool to study the r-colouring game and the (r, d)-relaxed colouring game. Let be given a graph G, an additive hereditary property \({\mathcal {P}}\) and a set C of r colours. In the unilateral \({\mathcal {P}}\) -colouring game similarly as in the r-colouring game, two players, Alice and Bob, colour the uncoloured vertices of the graph G, but in the unilateral \({\mathcal {P}}\) -colouring game Bob is more powerful than Alice. Alice starts the game, the players play alternately, but Bob can miss his move. Bob can colour the vertex with an arbitrary colour from C, while Alice must colour the vertex with a colour from C in such a way that she cannot create a monochromatic minimal forbidden subgraph for the property \({\mathcal {P}}\) . If after |V(G)| moves the graph G is coloured, then Alice wins the game, otherwise Bob wins. The \({\mathcal {P}}\) -unilateral game chromatic number, denoted by \({\chi_{ug}^\mathcal {P}(G)}\) , is the least number r for which Alice has a winning strategy for the unilateral \({\mathcal {P}}\) -colouring game with r colours on G. We prove that the \({\mathcal {P}}\) -unilateral game chromatic number is monotone and is the upper bound for the game chromatic number and the relaxed game chromatic number. We give the winning strategy for Alice to play the unilateral \({\mathcal {P}}\) -colouring game. Moreover, for k ≥  2 we define a class of graphs \({\mathcal {H}_k =\{G|{\rm every \;block \;of\;}G \; {\rm has \;at \;most}\; k \;{\rm vertices}\}}\) . The class \({\mathcal {H}_k }\) contains, e.g., forests, Husimi trees, line graphs of forests, cactus graphs. Let \({\mathcal {S}_d}\) be the class of graphs with maximum degree at most d. We find the upper bound for the \({\mathcal {S}_2}\) -unilateral game chromatic number for graphs from \({\mathcal {H}_3}\) and we study the \({\mathcal {S}_d}\) -unilateral game chromatic number for graphs from \({\mathcal {H}_4}\) for \({d \in \{2,3\}}\) . As the conclusion from these results we obtain the result for the d-relaxed game chromatic number: if \({G \in \mathcal {H}_k}\) , then \({\chi_g^{(d)}(G) \leq k + 2-d}\) , for \({k \in \{3, 4\}}\) and \({d \in \{0, \ldots, k-1\}}\) . This generalizes a known result for trees.     

\mathcal{D}-modules with finite support are semi-simple

Let \((R, \frak{m}, k_{R})\) be a regular local k-algebra satisfying the weak Jacobian criterion, and such that k _R /k is an algebraic field extension. Let \(\mathcal{D}_{R}\) be the ring of k-linear differential operators of R. We give an explicit decomposition of the \(\mathcal{D}_{R}\) -module \(\mathcal{D}_{R}/\mathcal{D}_{R} \frak{m}_{R}^{n+1}\) as a direct sum of simple modules, all isomorphic to \(\mathcal{D}_{R}/\mathcal{D}_{R} \frak{m}\) , where certain “Pochhammer” differential operators are used to describe generators of the simple components.     

On Graphlike k-Dissimilarity Vectors

Let \({\mathcal{G} = (G, w)}\) be a positive-weighted simple finite connected graph, that is, let G be a simple finite connected graph endowed with a function w from the set of edges of G to the set of positive real numbers. For any subgraph \({G^\prime}\) of G, we define \({w(G^\prime)}\) to be the sum of the weights of the edges of \({G^\prime}\) . For any i ₁, . . . , i _k vertices of G, let \({D_{\{i_1,..., i_k\}} (\mathcal{G})}\) be the minimum of the weights of the subgraphs of G connecting i ₁, . . . , i _k . The \({D_{\{i_1,..., i_k\}}(\mathcal{G})}\) are called k-weights of \({\mathcal{G}}\) . Given a family of positive real numbers parametrized by the k-subsets of {1, . . . , n}, \({{\{D_I\}_{I} \in { \{1,...,n\} \choose k}}}\) , we can wonder when there exist a weighted graph \({\mathcal{G}}\) (or a weighted tree) and an n-subset {1, . . . , n} of the set of its vertices such that \({D_I (\mathcal{G}) = D_I}\) for any \({I} \in { \{1,...,n\} \choose k}\) . In this paper we study this problem in the case k =  n?1.     

Approximating minimum-cost edge-covers of crossing biset-families

Part of this paper appeared in the preliminary version [16]. An ordered pair ? = (S, S ₊) of subsets of a groundset V is called a biset if S ? S+; (V S ⁺;V S) is the co-biset of ?. Two bisets \(\hat X,\hat Y\) intersect if ^X X ∩ Y ≠ \(\not 0\) and cross if both X ∩ Y \(\not 0\) and X ⁺ ∪ Y ⁺ ≠= V. The intersection and the union of two bisets \(\hat X,\hat Y\) are defined by \(\hat X \cap \hat Y = (X \cap Y,X^ + \cap Y^ + )\) and \(\hat X \cup \hat Y = (X \cup Y,X^ + \cup Y^ + )\) . A biset-family \(\mathcal{F}\) is crossing (intersecting) if \(\hat X \cap \hat Y,\hat X \cup \hat Y \in \mathcal{F}\) for any \(\hat X,\hat Y \in \mathcal{F}\) that cross (intersect). A directed edge covers a biset ? if it goes from S to V S ⁺. We consider the problem of covering a crossing biset-family \(\mathcal{F}\) by a minimum-cost set of directed edges. While for intersecting \(\mathcal{F}\) , a standard primal-dual algorithm computes an optimal solution, the approximability of the case of crossing \(\mathcal{F}\) is not yet understood, as it includes several NP-hard problems, for which a poly-logarithmic approximation was discovered only recently or is not known. Let us say that a biset-family \(\mathcal{F}\) is k-regular if \(\hat X \cap \hat Y,\hat X \cup \hat Y \in \mathcal{F}\) for any \(\hat X,\hat Y \in \mathcal{F}\) with |V (X∪Y)≥k+1 that intersect. In this paper we obtain an O(log |V|)-approximation algorithm for arbitrary crossing \(\mathcal{F}\) if in addition both \(\mathcal{F}\) and the family of co-bisets of \(\mathcal{F}\) are k-regular, our ratios are: \(O\left( {\log \frac{{|V|}} {{|V| - k}}} \right) \) if |S ⁺ \ S| = k for all \(\hat S \in \mathcal{F}\) , and \(O\left( {\frac{{|V|}} {{|V| - k}}\log \frac{{|V|}} {{|V| - k}}} \right) \) if |S ⁺ \ S| = k for all \(\hat S \in \mathcal{F}\) . Using these generic algorithms, we derive for some network design problems the following approximation ratios: \(O\left( {\log k \cdot \log \tfrac{n} {{n - k}}} \right) \) for k-Connected Subgraph, and O(logk) \(\min \{ \tfrac{n} {{n - k}}\log \tfrac{n} {{n - k}},\log k\} \) for Subset k-Connected Subgraph when all edges with positive cost have their endnodes in the subset.     

New approximations for the cone of copositive matrices and its dual

We provide convergent hierarchies for the convex cone $\mathcal{C }$ of copositive matrices and its dual $\mathcal{C }^*$ , the cone of completely positive matrices. In both cases the corresponding hierarchy consists of nested spectrahedra and provide outer (resp. inner) approximations for $\mathcal{C }$ (resp. for its dual $\mathcal{C }^*$ ), thus complementing previous inner (resp. outer) approximations for $\mathcal{C }$ (for $\mathcal{C }^*$ ). In particular, both inner and outer approximations have a very simple interpretation. Finally, extension to $\mathcal{K }$ -copositivity and $\mathcal{K }$ -complete positivity for a closed convex cone $\mathcal{K }$ , is straightforward.     

Generalized Complex Spherical Harmonics, Frame Functions, and Gleason Theorem

Consider a finite dimensional complex Hilbert space ${\mathcal{H}}$ , with ${dim(\mathcal{H}) \geq 3}$ , define ${\mathbb{S}(\mathcal{H}):= \{x\in \mathcal{H} \:|\: \|x\|=1\}}$ , and let ${\nu_\mathcal{H}}$ be the unique regular Borel positive measure invariant under the action of the unitary operators in ${\mathcal{H}}$ , with ${\nu_\mathcal{H}(\mathbb{S}(\mathcal{H}))=1}$ . We prove that if a complex frame function ${f : \mathbb{S}(\mathcal{H})\to \mathbb{C}}$ satisfies ${f \in \mathbb{L}^2(\mathbb{S}(\mathcal{H}), \nu_\mathcal{H})}$ , then it verifies Gleason’s statement: there is a unique linear operator ${A: \mathcal{H} \to \mathcal{H}}$ such that ${f(u) = \langle u| A u\rangle}$ for every ${u \in \mathbb{S}(\mathcal{H}).\,A}$ is Hermitean when f is real. No boundedness requirement is thus assumed on f a priori.     

Hypercentralizing generalized skew derivations on left ideals in prime rings

Let $\mathcal{R }$ be a prime ring of characteristic different from $2, \mathcal{Q }_r$ the right Martindale quotient ring of $\mathcal{R }, \mathcal{C }$ the extended centroid of $\mathcal{R }, \mathcal{I }$ a nonzero left ideal of $\mathcal{R }, F$ a nonzero generalized skew derivation of $\mathcal{R }$ with associated automorphism $\alpha $ , and $n,k \ge 1$ be fixed integers. If $[F(r^n),r^n]_k=0$ for all $r \in \mathcal{I }$ , then there exists $\lambda \in \mathcal{C }$ such that $F(x)=\lambda x$ , for all $x\in \mathcal{I }$ . More precisely one of the following holds: (1) $\alpha $ is an $X$ -inner automorphism of $\mathcal{R }$ and there exist $b,c \in \mathcal{Q }_r$ and $q$ invertible element of $\mathcal{Q }_r$ , such that $F(x)=bx-qxq^{-1}c$ , for all $x\in \mathcal{Q }_r$ . Moreover there exists $\gamma \in \mathcal{C }$ such that $\mathcal{I }(q^{-1}c-\gamma )=(0)$ and $b-\gamma q \in \mathcal{C }$ ; (2) $\alpha $ is an $X$ -outer automorphism of $\mathcal{R }$ and there exist $c \in \mathcal{Q }_r, \lambda \in \mathcal{C }$ , such that $F(x)=\lambda x-\alpha (x)c$ , for all $x\in \mathcal{Q }_r$ , with $\alpha (\mathcal{I })c=0$ .     

Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space

Let \(M\) and \(N\) be two connected smooth manifolds, where \(M\) is compact and oriented and \(N\) is Riemannian. Let \(\mathcal {E}\) be the Fréchet manifold of all embeddings of \(M\) in \(N\) , endowed with the canonical weak Riemannian metric. Let \(\sim \) be the equivalence relation on \(\mathcal {E}\) defined by \(f\sim g\) if and only if \(f=g\circ \phi \) for some orientation preserving diffeomorphism \(\phi \) of \(M\) . The Fréchet manifold \(\mathcal {S}= \mathcal {E}/_{\sim }\) of equivalence classes, which may be thought of as the set of submanifolds of \(N\) diffeomorphic to \(M\) and is called the nonlinear Grassmannian (or Chow manifold) of \(N\) of type \(M\) , inherits from \( \mathcal {E}\) a weak Riemannian structure. We consider the following particular case: \(N\) is a compact irreducible symmetric space and \(M\) is a reflective submanifold of \(N\) (that is, a connected component of the set of fixed points of an involutive isometry of \( N\) ). Let \(\mathcal {C}\) be the set of submanifolds of \(N\) which are congruent to \(M\) . We prove that the natural inclusion of \(\mathcal {C}\) in \(\mathcal {S}\) is totally geodesic.     

A Jost–Pais-Type Reduction of Fredholm Determinants and Some Applications

We study the analog of semi-separable integral kernels in \({\mathcal {H}}\) of the type $$ K(x, x') = \left\{\begin{array}{ll} F_1(x) G_1(x'), \quad& a < x' < x < b,\\ F_2 (x)G_2(x'), \quad& a < x < x' < b,\end{array}\right.$$ where \({-\infty \leqslant a < b \leqslant \infty}\) , and for a.e. \({x \in (a, b)}\) , \({F_j (x) \in \mathcal{B}_2(\mathcal{H}_j, \mathcal{H})}\) and \({G_j(x) \in \mathcal {B}_2(\mathcal {H},\mathcal {H}_j)}\) such that F _j (·) and G _j (·) are uniformly measurable, and $$\begin{array}{ll} || F_j ( \cdot) ||_{\mathcal {B}_2(\mathcal {H}_j,\mathcal {H})} \in L^2((a, b)), ||G_j (\cdot)||_{\mathcal {B}_2(\mathcal {H},\mathcal {H}_j)} \in L^2((a, b)), \quad j=1,2, \end{array}$$ with \({\mathcal {H}}\) and \({\mathcal {H}_j}\) , j = 1, 2, complex, separable Hilbert spaces. Assuming that K(·, ·) generates a trace class operator K in \({L^2((a, b);\mathcal {H})}\) , we derive the analog of the Jost–Pais reduction theory that succeeds in proving that the Fredholm determinant \({{\rm det}_{L^2((a,b);\mathcal{H})}}\) (I ? α K), \({\alpha \in \mathbb{C}}\) , naturally reduces to appropriate Fredholm determinants in the Hilbert spaces \({\mathcal{H}}\) (and \({\mathcal{H}_1 \oplus \mathcal{H}_2}\) ). Explicit applications of this reduction theory to Schrödinger operators with suitable bounded operator-valued potentials are made. In addition, we provide an alternative approach to a fundamental trace formula first established by Pushnitski which leads to a Fredholm index computation of a certain model operator.     

Irregular and Singular Loci of Commuting Varieties

Let $ {\user1{\mathcal{C}}} $ be the commuting variety of the Lie algebra $ \mathfrak{g} $ of a connected noncommutative reductive algebraic group G over an algebraically closed field of characteristic zero. Let $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ be the singular locus of $ {\user1{\mathcal{C}}} $ and let $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ be the locus of points whose G-stabilizers have dimension > rk G. We prove that: (a) $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ is a nonempty subset of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ ; (b) $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{irr}}}} = 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ where the maximum is taken over all simple ideals $ \mathfrak{a} $ of $ \mathfrak{g} $ and $ l{\left( \mathfrak{a} \right)} $ is the “lacety” of $ \mathfrak{a} $ ; and (c) if $ \mathfrak{t} $ is a Cartan subalgebra of $ \mathfrak{g} $ and $ \alpha \in \mathfrak{t}^{*} $ root of $ \mathfrak{g} $ with respect to $ \mathfrak{t} $ , then $ \overline{{G{\left( {{\text{Ker}}\,\alpha \times {\text{Ker }}\alpha } \right)}}} $ is an irreducible component of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ of codimension 4 in $ {\user1{\mathcal{C}}} $ . This yields the bound $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ and, in particular, $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 2 $ . The latter may be regarded as an evidence in favor of the known longstanding conjecture that $ {\user1{\mathcal{C}}} $ is always normal. We also prove that the algebraic variety $ {\user1{\mathcal{C}}} $ is rational.     

Classical limit of the tensor product of holomorphic discrete series representations

For three coadjoint orbits \(\mathcal {O}_1, \mathcal {O}_2\) and \(\mathcal {O}_3\) in \(\mathfrak {g}^*\) , the Corwin–Greenleaf function \(n(\mathcal {O}_1 \times \mathcal {O}_2, \mathcal {O}_3)\) is given by the number of \(G\) -orbits in \(\{(\lambda , \mu ) \in \mathcal {O}_1 \times \mathcal {O}_2 \, : \, \lambda + \mu \in \mathcal {O}_3 \}\) under the diagonal action. In the case where \(G\) is a simple Lie group of Hermitian type, we give an explicit formula of \(n(\mathcal {O}_1 \times \mathcal {O}_2, \mathcal {O}_3)\) for coadjoint orbits \(\mathcal {O}_1\) and \(\mathcal {O}_2\) that meet \(\left( [\mathfrak {k}, \mathfrak {k}] + \mathfrak {p}\right) ^{\perp }\) , and show that the formula is regarded as the ‘classical limit’ of a special case of Kobayashi’s multiplicity-free theorem (Progr. Math. 2007) in the branching law to symmetric pairs.     

Some results on metric n-Lie algebras

We study the structure of a metric n-Lie algebra G over the complex field C. Let G = SR be the Levi decomposition, where R is the radical of G and S is a strong semisimple subalgebra of G. Denote by m(G) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R ⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R ⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to m(G) + 1. The centralizer of R in G is equal to the sum of all minimal ideals; it is the direct sum of R ⊥ and the center of G. Finally, G has no strong semisimple ideals if and only if R⊥■R.     

On additive decompositions of the set of primitive roots modulo p

It is conjectured that the set ${\mathcal {G}}$ of the primitive roots modulo p has no decomposition (modulo p) of the form ${\mathcal {G}= \mathcal {A} +\mathcal {B}}$ with ${|\mathcal {A}|\ge 2}$ , ${|\mathcal {B} |\ge 2}$ . This conjecture seems to be beyond reach but it is shown that if such a decomposition of ${\mathcal {G}}$ exists at all, then ${|\mathcal {A} |}$ , ${|\mathcal {B} |}$ must be around p ^1/2, and then this result is applied to show that ${\mathcal {G}}$ has no decomposition of the form ${\mathcal {G} =\mathcal {A} + \mathcal {B} + \mathcal {C}}$ with ${|\mathcal {A} |\ge 2}$ , ${|\mathcal {B} |\ge 2}$ , ${|\mathcal {C} |\ge 2}$ .     

A Geometric Uncertainty Principle with an Application to Pleijel’s Estimate

Let \({\Omega \subset \mathbb{R}^2}\) be an open, bounded domain and \({\Omega = \bigcup_{i = 1}^{N} \Omega_{i}}\) be a partition. Denote the Fraenkel asymmetry by \({0 \leq \mathcal{A}(\Omega_i) \leq 2}\) and write $$D(\Omega_i) := \frac{|\Omega_{i}| - {\rm min}_{1 \leq j \leq N}{|\Omega_{j}|}}{|\Omega_{i}|}$$ with \({0 \leq D(\Omega_{i}) \leq 1}\) . For N sufficiently large depending only on \({\Omega}\) , there is an uncertainty principle $$\left(\sum_{i=1}^{N}{\frac{|\Omega_{i}|}{|\Omega|}{\mathcal{A}}(\Omega_i)}\right) + \left(\sum_{i=1}^{N}{\frac{|\Omega_i|}{|\Omega|}D(\Omega_i)}\right) \geq \frac{1}{60000}.$$ The statement remains true in dimensions \({n \geq 3}\) for some constant \({c_{n} > 0}\) . As an application, we give an (unspecified) improvement of Pleijel’s estimate on the number of nodal domains of a Laplacian eigenfunction and an improved inequality for a spectral partition problem.     

On the Fučik spectrum of non-local elliptic operators

In this article, we study the Fu?ik spectrum of the fractional Laplace operator which is defined as the set of all \({(\alpha, \beta)\in \mathbb{R}^2}\) such that $$\quad \left.\begin{array}{ll}\quad (-\Delta)^s u = \alpha u^{+} - \beta u^{-} \quad {\rm in}\;\Omega \\ \quad \quad \quad u = 0 \quad \quad \quad \qquad {\rm in}\; \mathbb{R}^n{\setminus}\Omega.\end{array}\right\}$$ has a non-trivial solution u, where \({\Omega}\) is a bounded domain in \({\mathbb{R}^n}\) with Lipschitz boundary, n > 2s, \({s \in (0, 1)}\) . The existence of a first nontrivial curve \({\mathcal{C}}\) of this spectrum, some properties of this curve \({\mathcal{C}}\) , e.g. Lipschitz continuous, strictly decreasing and asymptotic behavior are studied in this article. A variational characterization of second eigenvalue of the fractional eigenvalue problem is also obtained. At the end, we study a nonresonance problem with respect to the Fu?ik spectrum.     

Descent in Locally Presentable Categories

Let \(\mathbb {V}=(VV, \otimes , I)\) be a symmetric monoidal category such that \(\mathcal {V}\) is locally presentable and that all functors \(V\otimes - : \mathcal {V} \rightarrow \mathcal {V}\) for \(V \in \mathcal {V}\) preserve reflexive coequalizers and directed colimits. It is proved that any pure morphism of commutative ??-monoids is an effective descent morphism with respect to the indexed category given by commutative ??-monoids and modules over them. As a by-product, we prove that pure morphisms in a locally presentable category are effective for codescent.