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Given a field K, a rational function \phi \in K(x), and a point b \in \mathbb {P}^1(K), we study the extension K(\phi ^{-\infty }(b)) generated by the union over n of all solutions to \phi ^n(x) = b, where \phi ^n is the nth iterate of \phi. We ask when a finite extension of K(\phi ^{-\infty }(b)) can contain all m-power roots of unity for some m \geq 2, and prove that this occurs for several families of rational functions. A motivating application is to understand the higher ramification filtration when K is a finite extension of \mathbb {Q}_p and p divides the degree of \phi, especially when \phi is post-critically finite (PCF). We show that all higher ramification groups are infinite for new families of iterated extensions, for example those given by bicritical rational functions with periodic critical points. We also give new examples of iterated extensions with subextensions satisfying an even stronger ramification-theoretic condition called arithmetic profiniteness. We conjecture that every iterated extension arising from a PCF map should have a subextension with this stronger property, which would give a dynamical analogue of Sen's theorem for PCF maps. [ABSTRACT FROM AUTHOR]

We describe all possible coactions of finite groups (equivalently, all group gradings) on two-dimensional Artin-Schelter regular algebras. We give necessary and sufficient conditions for the associated Auslander map to be an isomorphism, and determine precisely when the invariant ring for the coaction is Artin-Schelter regular. The proofs of our results are combinatorial and exploit the structure of the McKay quiver associated to the coaction. [ABSTRACT FROM AUTHOR]

For a finite set \mathcal {F} of polynomials from {\mathbf {F}_{p}[\overline x]} (p is a fixed prime) containing all polynomials x^2 - x, a Nullstellensatz proof of the unsolvability of the system \begin{equation*} f = 0, \text { all } f \in \mathcal {F} \end{equation*} in \mathbf {F}_{p} is an {\mathbf {F}_{p}[\overline x]}-linear combination \sum _{f \in \mathcal {F}} \ h_f \cdot f that equals to 1 in {\mathbf {F}_{p}[\overline x]}. The measure of complexity of such a proof is its degree: \max _f deg(h_f f). We study the problem to establish degree lower bounds for some extended NS proof systems: these systems prove the unsolvability of \mathcal {F} (in \mathbf {F}_{p}) by proving the unsolvability of a bigger set \mathcal {F}\cup \mathcal {E}, where the set \mathcal {E}\subseteq {\mathbf {F}_{p}[\overline x, \overline r]} contains all polynomials r^p - r and satisfies the following soundness condition: [ABSTRACT FROM AUTHOR]

On any complete three dimensional Riemannian manifold with a pole and non-negative Ricci curvature, we show that the asymptotic scaling invariant integral of scalar curvature is equal to a term determined by the asymptotic volume ratio of this Riemannian manifold. [ABSTRACT FROM AUTHOR]

We give asymptotically optimal constructions in generalized Ramsey theory using results about conflict-free hypergraph matchings. For example, we present an edge-coloring of K_{n,n} with 2n/3 + o(n) colors such that each 4-cycle receives at least three colors on its edges. This answers a question of Axenovich, Füredi and the second author [J. Combin. Theory Ser B 79 (2000), pp. 66–86]. We also exhibit an edge-coloring of K_n with 5n/6+o(n) colors that assigns each copy of K_4 at least five colors. This gives an alternative very short solution to an old question of Erdős and Gyárfás [Combinatorica 17 (1997), pp. 459–467] that was recently answered by Bennett, Cushman, Dudek, and Prałat [J. Combin. Theory Ser. B 169 (2024), pp. 253–297] by analyzing a colored modification of the triangle removal process. [ABSTRACT FROM AUTHOR]

We show that there are no distance one surgeries on non-null-homologous knots in M that yield -M (M with opposite orientation) if M is a 3-manifold obtained by a Dehn surgery on a knot K in S^{3}, such that the order of its first homology is divisible by 9 but is not divisible by 27. As an application, we show several knots, including the (2,9) torus knot, do not have chirally cosmetic bandings. This simplifies the proof of a result first proven by Yang that the (2,k) torus knot (k>1) has a chirally cosmetic banding if and only if k=5. [ABSTRACT FROM AUTHOR]

Let \mathbb {K} be a number field with ring of integers \mathfrak {O} and let \mathcal {G} be a Chevalley group scheme not of type \mathtt {E}_8, \mathtt {F}_4 or \mathtt {G}_2. We use the theory of Tits buildings and a result of Tóth on Steinberg modules to prove that H^{{vcd}}(\mathcal {G}(\mathfrak {O}); \mathbb {Q}) = 0 if \mathfrak {O} is Euclidean. [ABSTRACT FROM AUTHOR]

We prove a strong dichotomy result for countably-infinite oriented graphs; that is, we prove that for all countably-infinite oriented graphs G, either (i) there is a countably-infinite tournament K such that G\not \subseteq K, or (ii) every countably-infinite tournament contains a spanning copy of G. Furthermore, we are able to give a concise characterization of such oriented graphs. Our characterization becomes even simpler in the case of transitive acyclic oriented graphs (i.e. strict partial orders). For uncountable oriented graphs, we are able to extend the dichotomy result mentioned above to all regular cardinals \kappa; however, we are only able to provide a concise characterization in the case when \kappa =\aleph _1. [ABSTRACT FROM AUTHOR]

Starting from billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d-dimensional Euclidean space, we introduce a new type of separable integer partition classes, called type B. We study the numbers of basis partitions with d parts and relate them to the Fibonacci sequence and its natural generalizations. Remarkably, the generating series of basis partitions can be related to the quiver generating series of symmetric quivers corresponding to the framed unknot via knots-quivers correspondence, and to the count of Schröder paths. [ABSTRACT FROM AUTHOR]

We study the continuous extension of discrete shift translations on one-dimensional quantum lattice systems. We explore a specific example provided by a quasi-free {C}^{\ast }-flow on the one-dimensional fermion lattice system, comparing it with its original discrete shift-translations. Specifically, we demonstrate its explicit dynamical formula, which reveals violations of causality and locality. Furthermore, we prove that this quasi-free {C}^{\ast }-flow on the one-dimensional fermion lattice system cannot be extended to the one-dimensional quantum spin lattice system via the Jordan-Wigner transformation. [ABSTRACT FROM AUTHOR]

We prove that for any countable directed graph E with Condition (K), the associated graph C^*-algebra C^*(E) has nuclear dimension at most 2. Furthermore, we provide a sufficient condition producing an upper bound of 1. [ABSTRACT FROM AUTHOR]

We consider the existence of invariant manifolds to evolution equations u'(t)=Au(t), A:D(A)\subset \mathbb {X}\to \mathbb {X} near its equilibrium A(0)=0 under the assumption that its proto-derivative \partial A(x) exists and is continuous in x\in D(A) in the sense of Yosida distance. Yosida distance between two (unbounded) linear operators U and V in a Banach space \mathbb {X} is defined as d_Y(U,V)≔\limsup _{\mu \to +\infty } \| U_\mu -V_\mu \|, where U_\mu and V_\mu are the Yosida approximations of U and V, respectively. We show that the above-mentioned equation has local stable and unstable invariant manifolds near an exponentially dichotomous equilibrium if the proto-derivative of \partial A is continuous in the sense of Yosida distance. The Yosida distance approach allows us to generalize the well-known results with possible applications to larger classes of partial differential equations and functional differential equations. The obtained results seem to be new. [ABSTRACT FROM AUTHOR]

We introduce the Gelfand-Kirillov base for a numerical semigroup ring over the prime field of characteristic p, where p is prime, and show its existence for the semigroup ring of the ordinary cusp by establishing a growth recurrence with respect to Frobenius. [ABSTRACT FROM AUTHOR]

We consider associative algebras with an action by derivations by some finite dimensional and semisimple Lie algebra. We prove that if a differential variety has almost polynomial growth, then it is generated by one of the algebras UT_2(W_\lambda) or End(W_\mu) for some integral dominant weight \lambda,\mu with \mu \neq 0. In the special case L=\mathfrak {sl}_2 we prove that this is a sufficient condition too. [ABSTRACT FROM AUTHOR]

Consider a unimodular random graph, or just a finitely generated Cayley graph. When does its cycle space have an invariant random generating set of cycles such that every edge is contained in finitely many of the cycles? Generating the free Loop O(1) model as a factor of iid is closely connected to having such a generating set for FK-Ising cluster. We show that geodesic cycles do not always form such a generating set, by showing for a parameter regime of the FK-Ising model on the lamplighter group the origin is contained in infinitely many geodesic cycles. This answers a question by Angel, Ray and Spinka. Then we take a look at how the property of having an invariant locally finite generating set for the cycle space is preserved by Bernoulli percolation, and apply it to the problem of factor of iid presentations of the free Loop O(1) model. [ABSTRACT FROM AUTHOR]

We consider word complexity and topological entropy for random substitution subshifts. In contrast to previous work, we do not assume that the underlying random substitution is compatible. We show that the subshift of a primitive random substitution has zero topological entropy if and only if it can be obtained as the subshift of a deterministic substitution, answering in the affirmative an open question of Rust and Spindeler [Indag. Math. (N.S.) 29 (2018), pp. 1131–1155]. For constant length primitive random substitutions, we develop a systematic approach to calculating the topological entropy of the associated subshift. Further, we prove lower and upper bounds that hold even without primitivity. For subshifts of non-primitive random substitutions, we show that the complexity function can exhibit features not possible in the deterministic or primitive random setting, such as intermediate growth, and provide a partial classification of the permissible complexity functions for subshifts of constant length random substitutions. [ABSTRACT FROM AUTHOR]

We obtain partial affirmative answers to the question of whether the isomorphism of the unitary groups of two C*-algebras, either as topological groups or as discrete groups, implies the isomorphism of the C*-algebras as real C*-algebras. [ABSTRACT FROM AUTHOR]

In this note, we provide an adaptation of the Kohler-Jobin rearrangement technique to the setting of the Gauss space. As a result, we prove the Gaussian analogue of the Kohler-Jobin resolution of a conjecture of Pólya-Szegö: when the Gaussian torsional rigidity of a domain is fixed, the Gaussian principal frequency is minimized for the half-space. At the core of this rearrangement technique is the idea of considering a "modified" torsional rigidity, with respect to a given function, and rearranging its layers to half-spaces, in a particular way; the Rayleigh quotient decreases with this procedure. We emphasize that the analogy of the Gaussian case with the Lebesgue case is not to be expected here, as in addition to some soft symmetrization ideas, the argument relies on the properties of some special functions; the fact that this analogy does hold is somewhat of a miracle. [ABSTRACT FROM AUTHOR]

We describe an explicit formula of the canonical pairing on the twisted de Rham cohomology associated with the category of local matrix factorizations and by characterizing its relation to Saito's higher residue pairings we reprove the conjecture of Shklyarov [Adv. Math. 292 (2016), pp. 181–209]. [ABSTRACT FROM AUTHOR]

This manuscript studies a no-flux initial-boundary value problem for a four-component chemotaxis system that has been proposed as a model for the response of cytotoxic T-lymphocytes to a solid tumor. In contrast to classical Keller-Segel type situations focusing on two-component interplay of chemotaxing populations with a signal directly secreted by themselves, the presently considered system accounts for a certain indirect mechanism of attractant evolution. Despite the presence of a zero-order exciting nonlinearity of quadratic type that forms a core mathematical feature of the model, the manuscript asserts the global existence of classical solutions for initial data of arbitrary size in three-dimensional domains. [ABSTRACT FROM AUTHOR]

We consider the problem of estimating the eigenvalues and the integral of the corresponding eigenfunctions, associated to the Newtonian potential operator, defined in a bounded domain \Omega \subset \mathbb {R}^{d}, where d=2,3, in terms of the maximum radius of \Omega. We first provide these estimations in the particular case of a ball and a disc. Then we extend them to general shapes using a, derived, monotonicity property of the eigenvalues of the Newtonian operator. The derivation of the lower bounds is quite tedious for the 2D-Logarithmic potential operator. Such upper/lower bounds appear naturally while estimating the electric/acoustic fields propagating in \mathbb {R}^{d} in the presence of small scaled and highly heterogeneous particles. [ABSTRACT FROM AUTHOR]

We introduce a natural generalization of the notion of strongly approximately transitive (SAT) states for actions of locally compact quantum groups. In the case of discrete quantum groups of Kac type, we show that the existence of unique stationary SAT states entails rigidity results concerning injective extensions of quantum group von Neumann algebras. [ABSTRACT FROM AUTHOR]

We study pairs of non-constant maps between two integral schemes of finite type over two (possibly different) fields of positive characteristic. When the target is quasi-affine, Tamagawa showed that the two maps are equal up to a power of Frobenius if and only if they induce the same homomorphism on their étale fundamental groups. We extend Tamagawa's result by adding a purely topological criterion for maps to agree up to a power of Frobenius. [ABSTRACT FROM AUTHOR]

We establish a Pythagorean theorem for the absolute values of the blocks of a partitioned matrix. This leads to a series of remarkable operator inequalities. For instance, if the matrix \mathbb {A} is partitioned into three blocks A,B,C, then \begin{gather*} |\mathbb {A}|^3 \ge U|A|^3U^* + V|B|^3V^*+ W|C|^3W^*,\\ \sqrt {3} |\mathbb {A}| \ge U|A|U^* + V|B|V^*+ W|C|W^*, \end{gather*} for some isometries U,V,W, and \begin{equation*} \mu _4^2(\mathbb {A}) \le \mu _3^2(A) +\mu _2^2(B) + \mu _1^2(C) \end{equation*} where \mu _j stands for the j-th singular value. Our theorem may be used to extend a result by Bhatia and Kittaneh for the Schatten p-norms and to give a singular value version of Cauchy's Interlacing Theorem. [ABSTRACT FROM AUTHOR]

Let G be a \pi-separable group with a Hall \pi-subgroup H or order n. For x\in H let \lambda (x) be the number of Hall \pi-subgroups of G containing x. We show that \prod _{d\mid n}\prod _{x\in H}\lambda (x^{d})^{\frac {n}{d}\mu (d)}=1, where \mu is the Möbius function. This generalizes fixed point formulas for coprime actions by Brauer, Wielandt and Navarro–Rizo. We further investigate an additive version of this formula. [ABSTRACT FROM AUTHOR]

Let \operatorname {Diff}_{\partial }(D^{n}) be the topological group of diffeomorphisms of D^{n} which agree with the identity near the boundary. In this short note, we compute the fundamental group \pi _1 \operatorname {Diff}_{\partial }(D^{4k}) for k\geq 3. [ABSTRACT FROM AUTHOR]

Consider a symmetric space G/H with simple Lie group G. We demonstrate that when G/H is not irreducible, it is necessarily even dimensional and noncompact. Furthermore, the subgroup H is also both noncompact and non-semisimple. Additionally, we establish that the only G-invariant connection on G/H is the canonical connection. On the other hand, we show that if G/H has an odd dimension, it must be irreducible, and the subgroup H must be semisimple. Finally, we present an explicit example, and we show that there exists no other torsion-free G-invariant connection on a symmetric space G/H with semisimple Lie group G which has the same curvature as the canonical one. [ABSTRACT FROM AUTHOR]

In this work we examine the existence of positive classical solutions of \begin{equation*} \begin {cases} -\Delta u +u = |x|^\alpha u^{p-1} & \text { in } B_1, \\ u>0 & \text { in } B_1, \\ \partial _\nu u= 0 & \text { on } \partial B_1, \end{cases} \end{equation*} where p>1, \alpha >0 and B_1 is the unit ball in {\mathbb {R}}^N where N \ge 4 and is even. Of particular interest is the existence of nonradial position classical solutions. We show that under suitable conditions on p,\alpha and N there are positive classical nonradial solutions. Our approach is to utilize a variational approach on suitable convex cones. [ABSTRACT FROM AUTHOR]

In this note, we relate the basepoint-freeness threshold of a polarized abelian variety, introduced by Jiang and Pareschi, with k-jet very ampleness. Then, we derive several applications of this fact, including a criterion for the k-very ampleness of Kummer varieties. [ABSTRACT FROM AUTHOR]

We provide necessary and sufficient conditions for operator-valued functions on arbitrary sets associated with a collection of test functions to have factorizations in several situations. [ABSTRACT FROM AUTHOR]

We prove that a virtually periodic object in an abelian category gives rise to a non-vanishing result on certain Hom groups in the singularity category. Consequently, for any artin algebra with infinite global dimension, its singularity category has no silting subcategory, and the associated differential graded Leavitt algebra has a non-vanishing cohomology in each degree. We verify the Singular Presilting Conjecture for singularly-minimal algebras and ultimately-closed algebras. We obtain a trichotomy on the Hom-finiteness of the cohomologies of differential graded Leavitt algebras. [ABSTRACT FROM AUTHOR]

In this note, we prove a family of sharp weighed inequalities which involve weighted k-th mean curvature integral and two distinct quermassintegrals for closed hypersurfaces in \mathbb {R}^n. This inequality generalizes the corresponding result of Wei and Zhou [Bull. Lond. Math. Soc. 55 (2023), pp. 263–281] where their proof is based on earlier results of Kwong-Miao [Pacific J. Math. 267 (2014), pp. 417–422; Commun. Contemp. Math. 17 (2015), p. 1550014]. Here we present a proof which does not rely on Kwong-Miao's results. [ABSTRACT FROM AUTHOR]

In this article, we show the necessary and sufficient conditions for the inequality \begin{equation*} \|u\|_{L_t^qL_x^r}\lesssim \|u\|_{X^{s,b}}, \end{equation*} where \|u\|_{X^{s,b}}≔\|\hat {u}(\tau,\xi)\langle \xi \rangle ^s\langle \tau +|\xi |^2\rangle ^b \|_{L_{\tau,\xi }^2}. These estimates are also referred to as Strichartz estimates related to Schrödinger equation. We also give a new proof of the maximal function estimates for solutions to Schrödinger and Airy equations. [ABSTRACT FROM AUTHOR]

Let \Sigma be a \sigma-algebra of subsets of a set \Omega and X be a Banach space, and let B(\Sigma,X) stand for the Banach space of all X-valued totally \Sigma-measurable functions on \Omega, equipped with the sup-norm. We study nuclear operators T:B(\Sigma,X)\rightarrow Y between Banach spaces B(\Sigma,X) and Y in terms of their representing measures m:\Sigma \rightarrow \mathcal {N}(X,Y), where \mathcal {N}(X,Y) stands for the Banach space of all nuclear operators U:X\rightarrow Y, equipped with the nuclear norm. We establish the relationship between nuclearity of a bounded linear operator T:B(\Sigma,X)\rightarrow Y and nuclearity of its conjugate operator. [ABSTRACT FROM AUTHOR]

There is a long-standing problem, posed by A.T.-M. Lau [ Fixed point theory and its applications , Academic Press, New York-London, 1976, pp. 121–129], whether left amenability is sufficient to ensure the existence of a common fixed point for every jointly weak^{\ast } continuous nonexpansive semigroup action on a nonempty weak^{\ast } compact convex set in a dual Banach space. In this note we discuss the current status of this problem and give a partial solution in the case of weak^{\ast } compact convex sets with the Radon–Nikodým property. [ABSTRACT FROM AUTHOR]

We consider periodic Jacobi operators and obtain upper and lower estimates on the sizes of the spectral bands. Our proofs are based on estimates on the logarithmic capacities and connections between the Chebyshev polynomials and logarithmic capacity of compact subsets of the real line. [ABSTRACT FROM AUTHOR]

We prove the strong Lefschetz property for Artinian Gorenstein algebras generated by the relative invariants of prehomogeneous vector spaces of commutative parabolic type. [ABSTRACT FROM AUTHOR]

We study the existence and uniqueness of strict solution for a general class of abstract explicit neutral equations with state-dependent delay. Some examples concerning explicit partial neutral differential equations with state dependent delay are presented. [ABSTRACT FROM AUTHOR]

We identify natural conditions for a countable group acting on a countable tree which imply that the orbit equivalence relation of the induced action on the Gromov boundary is Borel hyperfinite. Examples of this condition include acylindrical actions. We also identify a natural weakening of the aforementioned conditions that implies measure hyperfiniteness of the boundary action. We then document examples of group actions on trees whose boundary action is not hyperfinite. [ABSTRACT FROM AUTHOR]

We give sufficient conditions for an expansive partially hyperbolic diffeomorphism with one-dimensional center to be (topologically) Anosov. [ABSTRACT FROM AUTHOR]

Let V be a vertex algebra of countable dimension, G a subgroup of AutV of finite order, V^{G} the fixed point subalgebra of V under the action of G, and \mathscr {S} a finite G-stable set of inequivalent irreducible twisted weak V-modules associated with possibly different automorphisms in G. We show a Schur–Weyl type duality for the actions of \mathscr {A}_{\alpha }(G,\mathscr {S}) and V^G on the direct sum of twisted weak V-modules in \mathscr {S} where \mathscr {A}_{\alpha }(G,\mathscr {S}) is a finite dimensional semisimple associative algebra associated with G,\mathscr {S}, and a 2-cocycle \alpha naturally determined by the G-action on \mathscr {S}. It follows as a natural consequence of the result that for any g\in G every irreducible g-twisted weak V-module is a completely reducible weak V^G-module. [ABSTRACT FROM AUTHOR]

We consider a nonstationary random walk on a compact metrizable abelian group. Under a classical strict aperiodicity assumption we establish a weak-* convergence to the Haar measure, Ergodic Theorem and Large Deviation Type Estimate. [ABSTRACT FROM AUTHOR]

In this note, we establish a vanishing result for telescopically localized topological restriction homology TR. More precisely, we prove that T(k)-local TR vanishes on connective L_n^{p,f}-acyclic \mathbb {E}_1-rings for every 1 \leq k \leq n and deduce consequences for connective Morava K-theory and the Thom spectra y(n). The proof relies on the relationship between TR and the spectrum of curves on K-theory together with fact that algebraic K-theory preserves infinite products of additive \infty-categories which was recently established by Córdova Fedeli [ Topological Hochschild homology of adic rings , Ph.D. thesis, University of Copenhagen, 2023]. [ABSTRACT FROM AUTHOR]

Let \alpha be a non-zero algebraic number. Let K be the Galois closure of \mathbb {Q}(\alpha) with Galois group G and \bar {\mathbb {Q}} be the algebraic closure of \mathbb {Q}. In this article, among the other results, we prove the following. If f\in \bar {\mathbb {Q}}[G] is a non-zero element of the group ring \bar {\mathbb {Q}}[G] and \alpha is a given algebraic number such that f(\alpha ^n) is a non-zero algebraic integer for infinitely many natural numbers n, then \alpha is an algebraic integer. This result generalises the result of Polya [Rend. Circ Mat. Palermo, 40 (1915), pp. 1–16], Corvaja and Zannier [Acta Math. 193 (2004), pp. 175–191] and Philippon and Rath [J. Number Theory 219 (2021), pp. 198–211]. We also prove the analogue of this result for rational functions with algebraic coefficients. Inspired by a result of B. de Smit [J. Number Theory 45 (1993), pp. 112–116], we prove a finite version of the Polya type result for a binary recurrence sequences of non-zero algebraic numbers. In order to prove these results, we apply the techniques of Corvaja and Zannier along with the results of Kulkarni et al. [Trans. Amer. Math. Soc. 371 (2019), pp. 3787–3804], which are applications of the Schmidt subspace theorem. [ABSTRACT FROM AUTHOR]

A new highly symmetrical model of the compact Lie algebra \mathfrak {g}^c_2 is provided as a twisted ring group for the group \mathbb {Z}_2^3 and the ring \mathbb {R}\oplus \mathbb {R}. The model is self-contained and can be used without previous knowledge on roots, derivations on octonions or cross products. In particular, it provides an orthogonal basis with integer structure constants, consisting entirely of semisimple elements, which is a generalization of the Pauli matrices in \mathfrak {su}(2) and of the Gell-Mann matrices in \mathfrak {su}(3). As a bonus, the split Lie algebra \mathfrak {g}_2 is also seen as a twisted ring group. [ABSTRACT FROM AUTHOR]

We study the BMO-type functional \kappa _{\varepsilon }(f,\mathbb {R}^n), which can be used to characterize bounded variation functions f\in \mathrm {BV}(\mathbb {R}^n). The \Gamma-limit of this functional, taken with respect to L^1_{\mathrm {loc}}-convergence, is known to be \tfrac 14 |Df|(\mathbb {R}^n). We show that the \Gamma-limit with respect to L^{\infty }_{\mathrm {loc}}-convergence is \[ \tfrac 14 |D^a f|(\mathbb {R}^n)+\tfrac 14 |D^c f|(\mathbb {R}^n)+\tfrac 12 |D^j f|(\mathbb {R}^n), \] which agrees with the "pointwise" limit in the case of special functions of bounded varation. [ABSTRACT FROM AUTHOR]

We show that every proper, dense ideal in a C^{*}-algebra is contained in a prime ideal. It follows that a subset generates a C^{*}-algebra as a not necessarily closed ideal if and only if it is not contained in any prime ideal. This allows us to transfer Lie theory results from prime rings to C^{*}-algebras. For example, if a C^{*}-algebra A is generated by its commutator subspace [A,A] as a ring, then [[A,A],[A,A]] = [A,A]. Further, given Lie ideals K and L in A, then [K,L] generates A as a not necessarily closed ideal if and only if [K,K] and [L,L] do, and moreover this implies that [K,L]=[A,A]. We also discover new properties of the subspace generated by square-zero elements and relate it to the commutator subspace of a C^{*}-algebra. [ABSTRACT FROM AUTHOR]

Using elementary probabilistic methods, in particular a variant of the Weak Law of Large Numbers related to the Bernoulli distribution, we prove that for every infinite compact spaces K and L the product K\times L admits a sequence \langle \mu _n\colon n\in \mathbb {N}\rangle of normalized signed measures with finite supports which converges to 0 with respect to the weak* topology of the dual Banach space C(K\times L)^*. Our approach is completely constructive—the measures \mu _n are defined by an explicit simple formula. We also show that this result generalizes the classical theorem of Cembranos [Proc. Amer. Math. Soc. 91 (1984), pp. 556–558] and Freniche [Math. Ann. 267 (1984), pp. 479–486] which states that for every infinite compact spaces K and L the Banach space C(K\times L) contains a complemented copy of the space c_0. [ABSTRACT FROM AUTHOR]

We establish a Liouville type result for stable solutions for a wide class of second order semilinear elliptic equations in \mathbb {R}^{n} with sign-changing nonlinearity f. Under the hypothesis that the equation does not have any nonconstant one dimensional stable solution, and a further nondegeneracy condition of f at its zero points, we show that in any dimension, stable solutions of the equation must be constant. This partially answers a question raised by Dancer. [ABSTRACT FROM AUTHOR]

We prove that order convergence on a Boolean algebra turns it into a compact convergence space if and only if this Boolean algebra is complete and atomic. We also show that on an Archimedean vector lattice, order intervals are compact with respect to order convergence if and only the vector lattice is complete and atomic. Additionally we provide a direct proof of the fact that uo convergence on an Archimedean vector lattice is induced by a topology if and only if the vector lattice is atomic. [ABSTRACT FROM AUTHOR]