Remarks on Ramanujam-Kawamata-Viehweg Vanishing Theorem

In this article we prove a general result on a nef vector bundle E on a projective manifold X of dimension n depending on the vector space H^n,n(X,E): It is also shown that H^n,n(X,E) = 0 for an indecomposable nef rank 2 vector bundles E on some specific type of n dimensional projective manifold X. The same vanishing shown to hold for indecomposable nef and big rank 2 vector bundles on any variety with trivial canonical bundle.     

Free <Emphasis Type="Italic">n</Emphasis>-Tuple Semigroups

The present paper is devoted to the study of n-tuple semigroups. A free n-tuple semigroup of arbitrary rank is constructed and, as a consequence, singly generated free n-tuple semigroups are characterized. Moreover, examples of n-tuple semigroups are presented, the independence of the n-tuple semigroup axioms is proved, and it is shown that the natural semigroups of the constructed free n-tuple semigroup are isomorphic and the automorphism group of this n-tuple semigroup is isomorphic to a symmetric group.     

<Emphasis Type="Italic">C</Emphasis>*-simplicity of <Emphasis Type="Italic">n</Emphasis>-periodic products

The C*-simplicity of n-periodic products is proved for a large class of groups. In particular, the n-periodic products of any finite or cyclic groups (including the free Burnside groups) are C*-simple. Continuum-many nonisomorphic 3-generated nonsimple C*-simple groups are constructed in each of which the identity xⁿ = 1 holds, where n ≥ 1003 is any odd number. The problem of the existence of C*-simple groups without free subgroups of rank 2 was posed by de la Harpe in 2007.     

Slow entropy for some smooth flows on surfaces

We study slow entropy in some classes of smooth mixing flows on surfaces. The flows we study can be represented as special flows over irrational rotations and under roof functions which are C² everywhere except one point (singularity). If the singularity is logarithmic asymmetric (Arnol’d flows), we show that in the scale a_n(t) = n(log n)^t slow entropy equals 1 (the speed of orbit growth is n log n) for a.e. irrational α. If the singularity is of power type (x^?γ, γ ∈ (0, 1)) (Kochergin flows), we show that in the scale a_n(t) = n^t slow entropy equals 1 + γ for a.e. α.We show moreover that for local rank one flows, slow entropy equals 0 in the n(log n)^t scale and is at most 1 for scale n^t. As a consequence we get that a.e. Arnol’d and a.e Kochergin flow is never of local rank one.     

Rational realization of the minimum ranks of nonnegative sign pattern matrices

A sign pattern matrix (or nonnegative sign pattern matrix) is a matrix whose entries are from the set {+,?, 0} ({+, 0}, respectively). The minimum rank (or rational minimum rank) of a sign pattern matrix A is the minimum of the ranks of the matrices (rational matrices, respectively) whose entries have signs equal to the corresponding entries of A. Using a correspondence between sign patterns with minimum rank r ≥ 2 and point-hyperplane configurations in R^r?1 and Steinitz’s theorem on the rational realizability of 3-polytopes, it is shown that for every nonnegative sign pattern of minimum rank at most 4, the minimum rank and the rational minimum rank are equal. But there are nonnegative sign patterns with minimum rank 5 whose rational minimum rank is greater than 5. It is established that every d-polytope determines a nonnegative sign pattern with minimum rank d + 1 that has a (d + 1) × (d + 1) triangular submatrix with all diagonal entries positive. It is also shown that there are at most min{3m, 3n} zero entries in any condensed nonnegative m × n sign pattern of minimum rank 3. Some bounds on the entries of some integer matrices achieving the minimum ranks of nonnegative sign patterns with minimum rank 3 or 4 are established.     

The normalizers of free subgroups in free burnside groups of odd period <Emphasis Type="Italic">n ≥</Emphasis> 1003

Let B(m, n) be a free periodic group of arbitrary rank m with period n. In this paper, we prove that for all odd numbers n ≥ 1003 the normalizer of any nontrivial subgroup N of the group B(m, n) coincides with N if the subgroup N is free in the variety of all n-periodic groups. From this, there follows a positive answer for all prime numbers n > 997 to the following problem set by S. I. Adian in the Kourovka Notebook: is it true that none of the proper normal subgroups of the group B(m, n) of prime period n > 665 is a free periodic group? The obtained result also strengthens a similar result of A. Yu. Ol’shanskii by reducing the boundary of exponent n from n > 10⁷⁸ to n ≥ 1003. For primes 665 < n ≤ 997, the mentioned question is still open.     

A family of non-isomorphism results

We calculate the local groups of germs associated with the higher dimensional R. Thompson groups nV. For a given \({n\in N\cup\left\{\omega\right\}}\) , these groups of germs are free abelian groups of rank r, for r ≤ n (there are some groups of germs associated with nV with rank precisely k for each index 1 ≤ k ≤ n). By Rubin’s theorem, any conjectured isomorphism between higher dimensional R. Thompson groups induces an isomorphism between associated groups of germs. Thus, if m ≠ n the groups mV and nV cannot be isomorphic. This answers a question of Brin.     

Economical separability in free groups

Consider the rank n free group F _n with basis X. Bogopol’ski? conjectured in [1, Problem 15.35] that each element w ∈ F _n of length |w| ≥ 2 with respect to X can be separated by a subgroup H ≤ F _n of index at most C log |w| with some constant C. We prove this conjecture for all w outside the commutant of F _n , as well as the separability by a subgroup of index at most |w|/2 + 2 in general.     

On a Special Case of the Herbert Stahl Theorem

The BMV conjecture states that for n ×  n Hermitian matrices A and B the function f_A,B(t) = trace e^tA+B is exponentially convex. Recently the BMV conjecture was proved by Herbert Stahl. The proof of Herbert Stahl is based on ingenious considerations related to Riemann surfaces of algebraic functions. In the present paper we give a purely “matrix” proof of the BMV conjecture for the special case rank A = 1. This proof is based on the Lie product formula for the exponential of the sum of two matrices and does not require complex analysis.     

Full residual finiteness growths of nilpotent groups

Full residual finiteness growth of a finitely generated group G measures how efficiently word metric n-balls of G inject into finite quotients of G. We initiate a study of this growth over the class of nilpotent groups. When the last term of the lower central series of G has finite index in the center of G we show that the growth is precisely n^b, where b is the product of the nilpotency class and dimension of G. In the general case, we give a method for finding an upper bound of the form n^b where b is a natural number determined by what we call a terraced filtration of G. Finally, we characterize nilpotent groups for which the word growth and full residual finiteness growth coincide.     

On the pronormality of subgroups of odd index in finite simple symplectic groups

A subgroup H of a group G is pronormal if the subgroups H and H ^g are conjugate in 〈H,H ^g 〉 for every g ∈ G. It was conjectured in [1] that a subgroup of a finite simple group having odd index is always pronormal. Recently the authors [2] verified this conjecture for all finite simple groups other than PSL _n (q), PSU _n (q), E ₆(q), ² E ₆(q), where in all cases q is odd and n is not a power of 2, and P Sp_2n (q), where q ≡ ±3 (mod 8). However in [3] the authors proved that when q ≡ ±3 (mod 8) and n ≡ 0 (mod 3), the simple symplectic group P Sp²ⁿ (q) has a nonpronormal subgroup of odd index, thereby refuted the conjecture on pronormality of subgroups of odd index in finite simple groups.The natural extension of this conjecture is the problem of classifying finite nonabelian simple groups in which every subgroup of odd index is pronormal. In this paper we continue to study this problem for the simple symplectic groups P Sp_2n (q) with q ≡ ±3 (mod 8) (if the last condition is not satisfied, then subgroups of odd index are pronormal). We prove that whenever n is not of the form 2 ^m or 2 ^m (2^2k +1), this group has a nonpronormal subgroup of odd index. If n = 2 ^m , then we show that all subgroups of P Sp_2n (q) of odd index are pronormal. The question of pronormality of subgroups of odd index in P Sp_2n (q) is still open when n = 2 ^m (2^2k + 1) and q ≡ ±3 (mod 8).     

On finite alperin <Emphasis Type="Italic">p</Emphasis>-groups with homocyclic commutator subgroup

We study metabelian Alperin groups, i.e., metabelian groups in which every 2-generated subgroup has a cyclic commutator subgroup. It is known that, if the minimum number d(G) of generators of a finite Alperin p-group G is n ≥ 3, then d(G′) ≤ C _n ² for p≠ 3 and d(G′) ≤ C _n ² + C _n ³ for p = 3. The first section of the paper deals with finite Alperin p-groups G with p≠ 3 and d(G) = n ≥ 3 that have a homocyclic commutator subgroup of rank C _n ² . In addition, a corollary is deduced for infinite Alperin p-groups. In the second section, we prove that, if G is a finite Alperin 3-group with homocyclic commutator subgroup G- of rank C _n ² + C _n ³ , then G″ is an elementary abelian group.     

Boundary Crossing Probabilities for General Exponential Families

We consider parametric exponential families of dimension K on the real line. We study a variant of boundary crossing probabilities coming from the multi-armed bandit literature, in the case when the real-valued distributions form an exponential family of dimension K. Formally, our result is a concentration inequality that bounds the probability that B ^ψ (θ? _n , θ*) ≥ f(t/n)/n, where θ* is the parameter of an unknown target distribution, θ? _n is the empirical parameter estimate built from n observations, ψ is the log-partition function of the exponential family and B ^ψ is the corresponding Bregman divergence. From the perspective of stochastic multi-armed bandits, we pay special attention to the case when the boundary function f is logarithmic, as it is enables to analyze the regret of the state-of-the-art KL-ucb and KL-ucb+ strategies, whose analysis was left open in such generality. Indeed, previous results only hold for the case when K = 1, while we provide results for arbitrary finite dimension K, thus considerably extending the existing results. Perhaps surprisingly, we highlight that the proof techniques to achieve these strong results already existed three decades ago in the work of T. L. Lai, and were apparently forgotten in the bandit community. We provide a modern rewriting of these beautiful techniques that we believe are useful beyond the application to stochastic multi-armed bandits.     

Elliptic function of level 4

The article is devoted to the theory of elliptic functions of level n. An elliptic function of level n determines a Hirzebruch genus called an elliptic genus of level n. Elliptic functions of level n are also of interest because they are solutions of the Hirzebruch functional equations. The elliptic function of level 2 is the Jacobi elliptic sine function, which determines the famous Ochanine–Witten genus. It is the exponential of the universal formal group of the form F(u, v) = (u² ? v²)/(uB(v) ? vB(u)), B(0) = 1. The elliptic function of level 3 is the exponential of the universal formal group of the form F(u, v) = (u²A(v) ? v²A(u))/(uA(v)² ? vA(u)²), A(0) = 1, A″(0) = 0. In the present study we show that the elliptic function of level 4 is the exponential of the universal formal group of the form F(u, v) = (u²A(v) ? v²A(u))/(uB(v) ? vB(u)), where A(0) = B(0) = 1 and for B′(0) = A″(0) = 0, A′(0) = A₁, and B″(0) = 2B₂ the following relation holds: (2B(u) + 3A₁u)² = 4A(u)³ ? (3A₁² ? 8B₂)u²A(u)². To prove this result, we express the elliptic function of level 4 in terms of the Weierstrass elliptic functions.     

Congruent elliptic curves with non-trivial Shafarevich-Tate groups

We study 2-primary parts ⅢX(E~((n))/Q)[2~∞] of Shafarevich-Tate groups of congruent elliptic curves E~((n)): y~2= x~3-n~2x, n ∈Q~×/Q~(×2). Previous results focused on finding sufficient conditions for ⅢX(E~((n))/Q)[2~∞]trivial or isomorphic to(Z/2Z)~2. Our first result gives necessary and sufficient conditions such that the 2-primary part of the Shafarevich-Tate group of E~((n))is isomorphic to(Z/2Z)～2 and the Mordell-Weil rank of E~((n)) is zero,provided that all prime divisors of n are congruent to 1 modulo 4. Our second result provides sufficient conditions for ⅢX(E~((n))/Q)[2~∞]■(Z/2Z)~(2k), where k≥2.     

The Index of a 1-Form on a Real Quotient Singularity

Let G be a finite Abelian group acting (linearly) on space ?ⁿ and, therefore, on its complexification ?ⁿ, and let W be the real part of the quotient ?ⁿ/G (in the general case, W ≠ ?ⁿ/G). The index of an analytic 1-form on the space W is expressed in terms of the signature of the residue bilinear form on the G-invariant part of the quotient of the space of germs of n-forms on (?ⁿ, 0) by the subspace of forms divisible by the 1-form under consideration.     

Rates of decay in the classical Katznelson-Tzafriri theorem

The Katznelson-Tzafriri Theorem states that, given a power-bounded operator T, ∥Tⁿ(I ? T)∥ → 0 as n → ∞ if and only if the spectrum σ(T) of T intersects the unit circle T in at most the point 1. This paper investigates the rate at which decay takes place when σ(T) ∩ T = {1}. The results obtained lead, in particular, to both upper and lower bounds on this rate of decay in terms of the growth of the resolvent operator R(e^iθ, T) as θ → 0. In the special case of polynomial resolvent growth, these bounds are then shown to be optimal for general Banach spaces but not in the Hilbert space case.     

Exponential growth rates of free and amalgamated products

We prove that there is a gap between \(\sqrt 2 and\left( {1 + \sqrt 5 } \right)/2\) for the exponential growth rate of nontrivial free products. For amalgamated products G = A_*CB with ([A: C] ? 1)([B: C] ? 1) ≥ 2, we show that an exponential growth rate lower than \(\sqrt 2 \) can be achieved. Indeed, there are infinitely many amalgamated products for which the exponential growth rate is equal to ψ ≈ 1.325, where ψ is the unique positive root of the polynomial z³?z?1. One of these groups is \(PGL\left( {2,\mathbb{Z}} \right) \cong \left( {{C_2} \times {C_2}} \right){*_{{C_2}}}{D_6}\). However, under some natural conditions the lower bound can be put up to \(\sqrt 2 \). This answers two questions by Avinoam Mann [The growth of free products, Journal of Algebra 326, no. 1 (2011), 208–217]. We also prove that ψ is a lower bound for the minimal growth rates of a large class of Coxeter groups, including cofinite non-cocompact planar hyperbolic groups, which strengthens a result obtained earlier by William Floyd, who considered only standard Coxeter generators.     

Minimal self-joinings of infinite mixing actions of rank 1

We prove that measure-preserving actions of rank 1 of the groups ? ⁿ and ? ⁿ on a Lebesgue space with a σ-finite measure have minimal self-joinings.     

Invariable generation of prosoluble groups

A group G is invariably generated by a subset S of G if G = 〈s^g(s) | s ∈ S〉 for each choice of g(s) ∈ G, s ∈ S. Answering two questions posed by Kantor, Lubotzky and Shalev in [8], we prove that the free prosoluble group of rank d ≥ 2 cannot be invariably generated by a finite set of elements, while the free solvable profinite group of rank d and derived length l is invariably generated by precisely l(d ? 1) + 1 elements.