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Based on the algebraic independence of Drinfeld logarithms at algebraic points proved by Chang and Papanikolas, we show the four exponentials conjecture of Drinfeld modules, by following the typical approach in the classical transcendental number theory. [ABSTRACT FROM AUTHOR]

In this paper, we give a general criterion for elimination of imaginaries using an abstract independence relation. We also study germs of definable functions at certain well-behaved invariant types. Finally we apply these results to prove the elimination of imaginaries in bounded pseudo p-adically closed fields. [ABSTRACT FROM AUTHOR]

We study correlations of multiplicative functions taken along deterministic sequences and sequences that satisfy certain linear independence assumptions. The results obtained extend recent results of Tao and Teräväinen and results of the author. Our approach is to use tools from ergodic theory in order to effectively exploit feedback from analytic number theory. The results on deterministic sequences crucially use structural properties of measure preserving systems associated with bounded multiplicative functions that were recently obtained by the author and Host. The results on independent sequences depend on multiple ergodic theorems obtained using the theory of characteristic factors and qualitative equidistribution results on nilmanifolds. [ABSTRACT FROM AUTHOR]

Let pn be a random, degree n polynomial whose roots are chosen independently according to the probability measure μ on the complex plane. For a deterministic point ξ lying outside the support of μ, we show that almost surely the polynomial qn(z) := pn(z)(z − ξ) has a critical point at distance O(1/n) from ξ. In other words, conditioning the random polynomials pn to have a root at ξ almost surely forces a critical point near ξ. More generally, we prove an analogous result for the critical points of qn(z) := pn(z)(z − ξ1) · · · (z−ξk), where ξ1, . . ., ξk are deterministic. In addition, when k = o(n), we show that the empirical distribution constructed from the critical points of qn converges to μ in probability as the degree tends to infinity, extending a recent result of Kabluchko. [ABSTRACT FROM AUTHOR]

We construct several new spaces of quantum sequences and their quantum families of maps in the sense of Sołtan. The noncommutative distributional symmetries associated with these quantum maps are noncommutative versions of spreadability and partial exchangeability. Then, we study simple relations between these symmetries. We will focus on studying two kinds of noncommutative distributional symmetries: monotone spreadability and boolean spreadability. We provide an example of a spreadable sequence of random variables for which the usual unilateral shift is an unbounded map. As a result, it is natural to study bilateral sequences of random objects, which are indexed by integers, rather than unilateral sequences. At the end of the paper, we will show Ryll-Nardzewski type theorems for monotone independence and boolean independence: Roughly speaking, an infinite bilateral sequence of random variables is monotonically (boolean) spreadable if and only if the variables are identically distributed and monotone (boolean) with respect to the conditional expectation onto its tail algebra. For an infinite sequence of non-commutative random variables, boolean spreadability is equivalent to boolean exchangeability. [ABSTRACT FROM AUTHOR]

We study correlations in the multispecies TASEP on a ring. Results on the correlation of two adjacent points prove two conjectures by Thomas Lam on (a) the limiting direction of a reduced random walk in Ãn-1 and (b) the asymptotic shape of a random integer partition with no hooks of length n, a so called n-core. We further investigate two-point correlations far apart and three-point nearest neighbour correlations and prove explicit formulas in almost all cases. These results can be seen as a finite strengthening of correlations in the TASEP speed process by Amir, Angel and Valk'o. We also give conjectures for certain higher order nearest neighbour correlations. We find an unexplained independence property (provably for two points, conjecturally for more points) between points that are closer in position than in value that deserves more study. [ABSTRACT FROM AUTHOR]

The well-known "necklace splitting theorem" of Alon (1987) asserts that every k-colored necklace can be fairly split into q parts using at most t cuts, provided k(q - 1) = t. In a joint paper with Alon et al. (2009) we studied a kind of opposite question. Namely, for which values of k and t is there a measurable k-coloring of the real line such that no interval has a fair splitting into 2 parts with at most t cuts? We proved that k > t + 2 is a sufficient condition (while k > t is necessary). We generalize this result to Euclidean spaces of arbitrary dimension d, and to arbitrary number of parts q. We prove that if k(q -1) > t+d+q -1, then there is a measurable k-coloring of Rd such that no axis-aligned cube has a fair q-splitting using at most t axis-aligned hyperplane cuts. Our bound is of the same order as a necessary condition k(q - 1) > t implied by Alon's 1987 work. Moreover, for d = 1, q = 2 we get exactly the result of the 2009 work. Additionally, we prove that if a stronger inequality k(q-1) > dt+d+q-1 is satisfied, then there is a measurable k-coloring of ℝd with no axis-aligned cube having a fair q-splitting using at most t arbitrary hyperplane cuts. The proofs are based on the topological Baire category theorem and use algebraic independence over suitably chosen fields. [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n ≥ 3. Ore's classical theorem states that if d(x) + d(y) ≥ n for each pair of nonadjacent vertices x, y ∈ V (G), then G is Hamiltonian (Amer. Math. Monthly 67 (1960), 55). In 1984, Fan proved that if G is 2-connected and max{d(x), d(y)} ≥ n/2 for each pair of vertices x, y with distance 2, then G is Hamiltonian. Fan's result is a significant improvement to Ore's theorem, and the condition stated is called Fan's condition. Then in 1987, Benhocine and Wojda showed that if G is a 2-connected graph with independence number a(G) ≤ n/2, and max{d(x), d(y)} ≥ n-1/2 for each pair of vertices x, y with distance 2, then G is Hamiltonian with some exceptions: either G is Hamiltonian or G belongs to one of two classes of well characterized graphs. In 2007, Li et al. removed the independence restriction, but they also reversed the Fan type bigger degree lower bound condition to the stronger Ore type degree sum requirement. In our present work, we drop the independence restriction while keeping Benhocine and Wojda's relaxed Fan type condition to prove that if G is 2- connected and max{d(x), d(y)} ≥ n-1/2 for each pair of vertices x, y with distance 2, then either G is Hamiltonian or G is among certain classes of well characterized graphs. This extends previous results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, by establishing uniform lower bounds for the Poisson kernel and (-δ)-12 on the Heisenberg type group H(2n,m) with m ≽ 2, which follow from the various properties of Bessel functions and Legendre functions, we prove that there exists a constantA > 0 such that, for all f ε L¹(H(2n,m)) and all n,m ε N* satisfying 4 ≼ m² ≪ log n, we have ‖MKf‖L1,∞ ≼ An‖f‖1, where MK denotes the centered Hardy-Littlewood maximal function defined by the Kor'anyi norm. For the centered Hardy-Littlewood maximal function MCC defined by the Carnot-Carath'edory distance, we prove ‖MCCf‖L1,∞ ≼ A(m)n‖f‖1 holds for some constant A(m) independent of n. [ABSTRACT FROM AUTHOR]

It is a striking and elegant fact (proved independently by Furstenberg and S'arközy) that in any subset of the natural numbers of positive upper density there necessarily exist two distinct elements whose difference is given by a perfect square. In this article we present a new and simple proof of this result by adapting an argument originally developed by Croot and Sisask to give a new proof of Roth's theorem. [ABSTRACT FROM AUTHOR]

For a family F (a collection of subsets of Z+), the notion of Findependence is defined both for topological dynamics (t.d.s.) and measurable dynamics (m.d.s.). It is shown that there is no non-trivial {syndetic}- independent m.d.s.; an m.d.s. is {positive-density}-independent if and only if it has completely positive entropy; and an m.d.s. is weakly mixing if and only if it is {IP}-independent. For a t.d.s. it is proved that there is no non-trivial minimal {syndetic}-independent system; a t.d.s. is weakly mixing if and only if it is {IP}-independent. Moreover, a non-trivial proximal topological K system is constructed, and a topological proof of the fact that minimal topological K implies strong mixing is presented. [ABSTRACT FROM AUTHOR]

Let K be an imaginary quadratic field and f an integral ideal. Denote by Cl(f) the ray class group of f. For every non-trivial character χ of Cl(f), we show that L(1, χ)/π is transcendental. If f = f̄, then complex conjugation acts on the character group of Cl(f). Denoting by Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. the orbits of the group of characters, we show that the values L(1, χ) as χ ranges over elements of Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. are linearly independent over Q̄. We give applications of this result to the study of transcendental values of Petersson inner products and certain special values of Artin L-series attached to dihedral extensions. [ABSTRACT FROM AUTHOR]

We answer a number of open problems in frame theory concerning the decomposition of frames into linearly independent and/or spanning sets. We prove that Parseval frames with norms bounded away from 1 can be decomposed into a number of sets whose complements are spanning, where the number of these sets only depends on the norm bound. Further, we prove a stronger result for Parseval frames whose norms are uniformly small, which shows that in addition to the spanning property, the sets can be chosen to be independent and the complement of each set can contain a number of disjoint, spanning sets. [ABSTRACT FROM AUTHOR]

We prove that ω-categorical rings with NIP are nilpotent-by-finite and that ω-categorical groups with NIP and fsg are nilpotent-by-finite, too. We give an easy proof that each infinite, ω-categorical p-group with NIP has an infinite, definable abelian subgroup. Assuming additionally that the group in question is characteristically simple and has a non-algebraic type which is generically stable over θ, we show that the group is abelian. Moreover, we prove that in any group with at least one strongly regular type all non-central elements are conjugated, and we conclude that assuming in addition ω-categoricity, such a group must be abelian. [ABSTRACT FROM AUTHOR]

Let $ K$. It is proved that each automorphism $ \sigma \in \operatorname{Aut}_K(\mathcal D(P_n))$ $ \mathcal D(P_n)$ $ P_n= K[x_1, \ldots, x_n]$uniquely determined by the elements $ \sigma (x_1), \ldots ,\sigma (x_n)$ $ \operatorname{Frob}(\mathcal D(P_n))$ $ \mathcal D(P_n)$, to the ring $ \mathcal D(P_n)$ $ \operatorname{Aut}_K(\mathcal D(P_n) ) \cdot \mathcal{F}$ $ \mathcal{F}$ to $ \mathcal D(P_n)$ $ \mathcal{F}$countably many independent parameters). [ABSTRACT FROM AUTHOR]

We show that for decaying solutions of the Ablowitz-Ladik system, the leading asymptotic term is time independent. In addition, two arbitrary bounded solutions of the Ablowitz-Ladik system which are asymptotically close at the initial time stay close. All results are also derived for the associated hierarchy. [ABSTRACT FROM AUTHOR]

Inspired by several recent papers on the edge ideal of a graph $G$, we study the equivalent notion of the independence complex of $G$. Using the tool of vertex decomposability from geometric combinatorics, we show that $5$-chordal graphs with no chordless $4$-cycles are shellable and sequentially Cohen-Macaulay. We use this result to characterize the obstructions to shellability in flag complexes, extending work of Billera, Myers, and Wachs. We also show how vertex decomposability may be used to show that certain graph constructions preserve shellability. [ABSTRACT FROM AUTHOR]

We give sufficient conditions under which solutions of discretized in space second-order parabolic and elliptic equations, perhaps degenerate, admit estimates of the first derivatives in the space variables independent of the mesh size. [ABSTRACT FROM AUTHOR]

This paper presents a nonstandard local approach to Richardson extrapolation, when it is used to increase the accuracy of the standard finite element approximation of solutions of second order elliptic boundary value problems in $mathbb R^N$, $N ge 2$. The main feature of the approach is that it does not rely on a traditional asymptotic error expansion, but rather depends on a more easily proved weaker a priori estimate, derived in [19], called an asymptotic error expansion inequality. In order to use this inequality to verify that the Richardson procedure works at a point, we require a local condition which links the different subspaces used for extrapolation. Roughly speaking, this condition says that the subspaces are similar about a point, i.e., any one of them can be made to locally coincide with another by a simple scaling of the independent variable about that point. Examples of finite element subspaces that occur in practice and satisfy this condition are given. [ABSTRACT FROM AUTHOR]

We investigate the algebraic independence of some derivatives of certain multiplicative arithmetical functions over the field $mathbb {C}$ of complex numbers. [ABSTRACT FROM AUTHOR]

We discuss the past and future of set theory, axiom systems and independence results. We deal in particular with cardinal arithmetic. [ABSTRACT FROM AUTHOR]