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What is the neurobiological basis of our ability to create complex messages with language? Results from multiple methodologies have converged on a set of brain regions as relevant for this general process, but the computational details of these areas remain to be characterized. The left anterior temporal lobe (LATL) has been a consistent node within this network, with results suggesting that although it rather systematically shows increased activation for semantically complex structured stimuli, this effect does not extend to number phrases such as ‘three books.’ In the present work we used magnetoencephalography to investigate whether numbers in general are an invalid input to the combinatory operations housed in the LATL or whether the lack of LATL engagement for stimuli such as ‘three books’ is due to the quantificational nature of such phrases. As a relevant test case, we employed complex number terms such as ‘twenty-three’, where one number term is not a quantifier of the other but rather, the two terms form a type of complex concept. In a number naming paradigm, participants viewed rows of numbers and depending on task instruction, named them as complex number terms (‘twenty-three’), numerical quantifications (‘two threes’), adjectival modifications (‘blue threes’) or non-combinatory lists (e.g., ‘two, three’). While quantificational phrases failed to engage the LATL as compared to non-combinatory controls, both complex number terms and adjectival modifications elicited a reliable activity increase in the LATL. Our results show that while the LATL does not participate in the enumeration of tokens within a set, exemplified by the quantificational phrases, it does support conceptual combination, including the composition of complex number concepts. [ABSTRACT FROM AUTHOR]

The units of a number field k act naturally on the real vector space k ⊗ Q R , and so on open subsets of (k ⊗ Q R) ⁎ that are stable under the units. A Shintani domain for this action consists of a finite number of polyhedral cones, all having generators in k , whose union is a fundamental domain. Aside from the trivial case of imaginary quadratic fields, no practical method for computing Shintani domains for totally complex number fields has been published. Here we give a quick way to compute Shintani domains for totally complex quartic number fields. To construct these domains we exploit the existence of a forward attractor and backward repeller set for this action. We prove that the number of polyhedral cones needed for the Shintani domain is bounded by an absolute constant, a fact previously known only for cubic or quadratic fields. We also show that our algorithm for finding a Shintani domain runs in polynomial time and we give a table describing the results of running the algorithm on more than 168,000 fields. [ABSTRACT FROM AUTHOR]

Any complex-analytic vector bundle E admits naturally defined homotheties ϕ α , α ∈ C ⁎ , i.e. ϕ α is the multiplication of a local section by a complex number α. We investigate the question when such automorphisms can be lifted to a non-split supermanifold corresponding to E. Further, we compute the automorphism supergroup of a Π-symmetric super-Grassmannian Π Gr n , k , and, using Galois cohomology, we classify the real structures on Π Gr n , k and compute the corresponding supermanifolds of real points. [ABSTRACT FROM AUTHOR]

In this paper, we associate the TKK algebra G ˆ (J) with vertex algebras through twisted modules. Firstly, we prove that for any complex number ℓ , the category of restricted G ˆ (J) -modules of level ℓ is canonically isomorphic to the category of σ -twisted V C g ˆ (ℓ , 0) -modules, where V C g ˆ (ℓ , 0) is a vertex algebra arising from the toroidal Lie algebra of type C 2 and σ is an isomorphism of V C g ˆ (ℓ , 0) induced from the involution of this toroidal Lie algebra. Secondly, we prove that for any nonnegative integer ℓ , the integrable restricted G ˆ (J) -modules of level ℓ are exactly the σ -twisted modules for the quotient vertex algebra L C g ˆ (ℓ , 0) of V C g ˆ (ℓ , 0). Finally, we classify the irreducible 1 2 N -graded σ -twisted L C g ˆ (ℓ , 0) -modules. [ABSTRACT FROM AUTHOR]

This paper is to study what we call twisted regular representations for vertex operator algebras. Let V be a vertex operator algebra, let σ 1 , σ 2 be commuting finite-order automorphisms of V and let σ = (σ 1 σ 2) − 1. Among the main results, for any σ -twisted V -module W and any nonzero complex number z , we construct a weak σ 1 ⊗ σ 2 -twisted V ⊗ V -module D σ 1 , σ 2 (z) (W) inside W ⁎. Let W 1 , W 2 be σ 1 -twisted, σ 2 -twisted V -modules, respectively. We show that P (z) -intertwining maps from W 1 ⊗ W 2 to W ⁎ are the same as homomorphisms of weak σ 1 ⊗ σ 2 -twisted V ⊗ V -modules from W 1 ⊗ W 2 into D σ 1 , σ 2 (z) (W). We also show that a P (z) -intertwining map from W 1 ⊗ W 2 to W ⁎ is equivalent to an intertwining operator of type ( W ′ W 1 W 2 ) , which is a twisted version of a result of Huang and Lepowsky. Finally, we show that for each τ -twisted V -module M with τ any finite-order automorphism of V , the coefficients of the q -graded trace function lie in D τ , τ − 1 (− 1) (V) and generate a τ ⊗ τ − 1 -twisted V ⊗ V -submodule isomorphic to M ⊗ M ′. [ABSTRACT FROM AUTHOR]

Comparisons of evolutionary features between human disease and non-disease genes have a wide implication to understand the genetic basis of human disease genes. However, it has not yet been resolved whether disease genes evolve at slower or faster rate than the non-disease genes. To resolve this controversy, here we integrated human disease genes from several databases and compared their protein evolutionary rates with non-disease genes in both housekeeping and tissue-specific group. We noticed that in tissue specific group, disease genes evolve significantly at a slower rate than non-disease genes. However, we found no significant difference in evolutionary rates between disease and non-disease genes in housekeeping group. Tissue specific disease genes have a higher protein complex number, elevated gene expression level and are also associated with conserve biological processes. Finally, our regression analysis suggested that protein complex number followed by protein multifunctionality independently modulates the evolutionary rate of human disease genes. [ABSTRACT FROM AUTHOR]

Complex number words (e.g., “twenty two”) are formed by merging together several simple number words (e.g., “twenty” and “two”). In the present study, we explored the neural correlates of this operation and investigated to what extent it engages brain areas involved processing numerical quantity and linguistic syntactic structure. Participants speaking two typologically distinct languages, French and Chinese, were required to read aloud sequences of simple number words while their cerebral activity was recorded by functional magnetic resonance imaging. Each number word could either be merged with the previous ones (e.g., ‘twenty three’) or not (e.g., ‘three twenty’), thus forming four levels ranging from lists of number words to complex numerals. When a number word could be merged with the preceding ones, it was named faster than when it could not. Neuroimaging results showed that the number of merges correlated with activation in the left inferior frontal gyrus and in the left inferior parietal lobule. Consistent findings across Chinese and French participants suggest that these regions serve as the neural bases for forming complex number words in different languages. [ABSTRACT FROM AUTHOR]

Let sl N ˆ (C q) be the core of extended affine Lie algebra of type A N − 1 coordinated by the rational quantum 2-torus C q. In this paper, we first prove that for any complex number ℓ , the category of restricted sl N ˆ (C q) -modules of level ℓ is canonically isomorphic to the category of twisted modules for the vertex algebra V C g ˆ (ℓ , 0) arising from a conformal Lie algebra C g , where C g ˆ is isomorphic to a toroidal Lie algebra. Then we prove that for any nonnegative integer ℓ , the integrable restricted sl N ˆ (C q) -modules of level ℓ are exactly the twisted modules for the quotient vertex algebra L C g ˆ (ℓ , 0) of V C g ˆ (ℓ , 0). Finally, we classify irreducible graded twisted L C g ˆ (ℓ , 0) -modules. [ABSTRACT FROM AUTHOR]

In this paper, we associate vertex algebras and their two different kinds of module categories with the unitary Lie algebra u ˆ N ( C Γ ˜ ) for N ≥ 2 being a positive integer and Γ ˜ = { q n | n ∈ Z } , where the nonzero complex number q is not a root of unity. It is proved that for any complex number ℓ , the category of restricted u ˆ N ( C Γ ˜ ) -modules of level ℓ is canonically isomorphic to the category of quasi modules for certain vertex algebra. And we also prove that the category of restricted u ˆ N ( C Γ ˜ ) -modules of level ℓ is isomorphic to the category of Γ -equivariant ϕ -coordinated quasi modules for the same vertex algebra, where Γ is an automorphism group of this vertex algebra. [ABSTRACT FROM AUTHOR]

We consider a generalized Mathieu series where the summands of the classical Mathieu series are multiplied by powers of a complex number. The Mellin transform of this series can be expressed by the polylogarithm or the Hurwitz zeta function. From this we derive a full asymptotic expansion, generalizing known expansions for alternating Mathieu series. Another asymptotic regime for trigonometric Mathieu series is also considered, to first order, by applying known results on the asymptotic behavior of trigonometric series. [ABSTRACT FROM AUTHOR]

We introduce a family of rational functions R N (a , d 0 , d 1 , ... , d N) with the following property. Let d 0 , d 1 , ... , d N be equal respectively to the value of some function f (s) and the values of its first N derivatives calculated at a certain complex number a lying not too far from a zero ρ of this function. It is expected that the value of R N (a , d 0 , d 1 , ... , d N) is very close to ρ. We demonstrate this phenomenon on several numerical instances with the Riemann zeta function in the role of f (s). For example, for N = 10 and a = 0.6 + 14 i we have | R 10 (a , d 0 , d 1 , ... , d 10) − ρ 1 | < 10 − 18 where ρ 1 = 0.5 + 14.13... i is the first non-trivial zeta zero. Also we define rational functions R N , n (a , d 0 , d 1 , ... , d N) which (under the same assumptions) have values which are very close to n − ρ , that is, to the terms from the Dirichlet series for the zeta function calculated at its zero. In the case when a is between two consecutive zeros, say ρ l and ρ l + 1 , functions R N , n (a , d 0 , d 1 , ... , d N) approximate neither of n − ρ l no n − ρ l + 1 ; nevertheless, they allow us to approximate the sum n − ρ l + n − ρ l + 1 and the product n − ρ l n − ρ l + 1 and hence to calculate both n − ρ l and n − ρ l + 1 by solving corresponding quadratic equation. [ABSTRACT FROM AUTHOR]

We study quantum current algebra A (R ‾) associated with the rational R -matrix of gl N and we give explicit formulae for the elements of its center at the critical level. Due to Etingof–Kazhdan's construction, the level c vacuum module V c (R ‾) for the algebra A (R ‾) possesses a quantum vertex algebra structure for any complex number c. We prove that any module for the quantum vertex algebra V c (R ‾) is naturally equipped with a structure of restricted A (R ‾) -module of level c and vice versa. [ABSTRACT FROM AUTHOR]

Abstract Let g be any untwisted affine Kac–Moody algebra, μ any fixed complex number, and g ˜ (μ) the corresponding toroidal extended affine Lie algebra of nullity two. For any k -tuple λ = (λ 1 , ⋯ , λ k) of weights of g , and k -tuple a = (a 1 , ⋯ , a k) of distinct non-zero complex numbers, we construct a class of modules V ˜ (λ , a) for the extended affine Lie algebra g ˜ (μ). We prove that the g ˜ (μ) -module V ˜ (λ , a) is completely reducible. We also prove that the g ˜ (μ) -module V ˜ (λ , a) is integrable when all weights λ i in λ are dominant. Thus, we obtain a new class of irreducible integrable weight modules for the toroidal extended affine Lie algebra g ˜ (μ). [ABSTRACT FROM AUTHOR]

• Using the structural dynamic modification theory, the general damped system is considered a perturbation of the un-damped system, where the modification is defined by the damping matrix. C • The complexity of the mode shapes depends on the level of orthogonality of the normal mode shapes with respect to damping matrix. C. • The modal parameters of general damped systems (poles, complex modal masses, length of mode shapes) can be expressed as linear combination of the corresponding un-damped modal parameters. • If the mass density of the system is constant, the modal mass is always equal to the product between the total mass of the structure and the magnitude of the length squared, but with a phase to introduce a complex number. In this paper the modal mass is considered in the case of general damping where it is well-known that the modal mass becomes a complex valued quantity. The aim of this paper is to present a definition of the complex modal mass, which leads to an easier physical understanding of this modal parameter. It is demonstrated that based on the structural modification theory, the complex modal mass can be expressed as a linear combination of the un-damped modal masses, where a complex coefficient matrix can be defined that depends on the general damping matrix and on the mode shapes of the damped and the un-damped system. An important finding is that an apparent mass can be defined so that the modal mass is always equal to the product between the apparent mass and the length squared. The paper includes a 2-DOF example that illustrates the influence of the amount of general damping on modal quantities that are normally considered non-sensitive to damping. [ABSTRACT FROM AUTHOR]

Abstract It is well known that for two univariate polynomials over the complex number field the number of their common roots is equal to the order of their resultant. In this paper, we show that this fundamental relationship still holds for the tropical polynomials under suitable adaptation of the notion of order, if the roots are simple and non-zero. [ABSTRACT FROM AUTHOR]

Text For any complex number c , define the divisor function σ c : N → C by σ c ( n ) = ∑ d | n d c . Let σ c ( N ) ‾ denote the topological closure of the range of σ c . Extending previous work of the current author and Sanna, we prove that σ c ( N ) ‾ has nonempty interior and has finitely many connected components if ℜ ( c ) ≤ 0 and c ≠ 0 . We end with some open problems. Video For a video summary of this paper, please visit https://youtu.be/-Zic04j6sRg . [ABSTRACT FROM AUTHOR]

We classify finite irreducible conformal modules over a class of infinite Lie conformal algebras B ( p ) of Block type, where p is a nonzero complex number. In particular, we obtain that a finite irreducible conformal module over B ( p ) may be a nontrivial extension of a finite conformal module over Vir if p = − 1 , where Vir is a Virasoro conformal subalgebra of B ( p ) . As a byproduct, we also obtain the classification of finite irreducible conformal modules over a series of finite Lie conformal algebras b ( n ) for n ≥ 1 . [ABSTRACT FROM AUTHOR]

This paper studies controllability of the Poisson equation on the unit disk in C subject to the third boundary condition when the control is imposed on the boundary. We use complex analytic methods to prove existence and uniqueness of the control when the parameter λ is a nonzero complex number but not a negative integer (not an eigenvalue). Otherwise, due to multiplicity of solutions to the underlying problem, when λ is a negative integer, controllability could only be obtained if proper additional conditions on the boundary are imposed. [ABSTRACT FROM AUTHOR]

The main purposes of this paper are to accomplish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear system of reaction diffusion equations u t = u x x + α [ β H ( u − θ ) − u ] − w , w t = ε ( u − γ w ) and to establish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear scalar reaction diffusion equation u t = u x x + α [ β H ( u − θ ) − u ] , under different conditions on the model constants. To establish the bifurcation for the system, we will study the existence and instability of a standing pulse solution if 0 < 2 ( 1 + α γ ) θ < α β γ ; the existence and stability of two standing wave fronts if 2 ( 1 + α γ ) θ = α β γ and γ 2 ε > 1 ; the existence and instability of two standing wave fronts if 2 ( 1 + α γ ) θ = α β γ and 0 < γ 2 ε < 1 ; the existence and instability of an upside down standing pulse solution if 0 < ( 1 + α γ ) θ < α β γ < 2 ( 1 + α γ ) θ . To establish the bifurcation for the scalar equation, we will study the existence and stability of a traveling wave front as well as the existence and instability of a standing pulse solution if 0 < 2 θ < β ; the existence and stability of two standing wave fronts if 2 θ = β ; the existence and stability of a traveling wave front as well as the existence and instability of an upside down standing pulse solution if 0 < θ < β < 2 θ . By the way, we will also study the existence and stability of a traveling wave back of the nonlinear scalar reaction diffusion equation u t = u x x + α [ β H ( u − θ ) − u ] − w 0 , where w 0 = α ( β − 2 θ ) > 0 is a positive constant, if 0 < 2 θ < β . To achieve the main goals, we will make complete use of the special structures of the model equations and we will construct Evans functions and apply them to study the eigenvalues and eigenfunctions of several eigenvalue problems associated with several linear differential operators. It turns out that a complex number λ 0 is an eigenvalue of the linear differential operator, if and only if λ 0 is a zero of the Evans function. The stability, instability and bifurcations of the nonlinear waves follow from the zeros of the Evans functions. A very important motivation to study the existence, stability, instability and bifurcations of the nonlinear waves is to study the existence and stability/instability of infinitely many fast/slow multiple traveling pulse solutions of the nonlinear system of reaction diffusion equations. The existence and stability of infinitely many fast multiple traveling pulse solutions are of great interests in mathematical neuroscience. [ABSTRACT FROM AUTHOR]

We deal with the following functional equation: f ( x y ) + λ f ( x ) f ( y ) = ϕ ( x , y ) , where λ is a complex number and f and ϕ are complex mappings defined on a semigroup. Moreover, we assume an addition formula of trigonometric type for ϕ . Our research is motivated by some earlier results related to a problem posed by S. Butler in 2003. [ABSTRACT FROM AUTHOR]

Let J be the set of all infinite complex continued fractions with partial numerators equal to one and partial denominators b 1 , b 2 , … , where each b k is a complex number of the form m + n i with m being a positive integer and n being any integer. In this paper we give an algorithm to determine lower bounds for the Hausdorff dimension of J . We show that the Hausdorff dimension of J is greater than 1.825, which is the best known lower bound, to the best of our knowledge. The ideas used to obtain this improved bound is different from those used by other authors and could be applied to estimate Hausdorff dimensions of other sets as well. [ABSTRACT FROM AUTHOR]

Let H ( C ) be the set of entire functions endowed with the topology of local uniform convergence. Fix a sequence of non-zero complex numbers ( λ n ) , | λ n | → + ∞ , which satisfies the following property: for every M > 0 there exists a subsequence ( μ n ) of ( λ n ) such that (i) | μ n + 1 | − | μ n | > M for every n = 1 , 2 , … and (ii) ∑ n = 1 + ∞ 1 | μ n | = + ∞ . We prove that there exists a residual set G ⊂ H ( C ) such that for every f ∈ G and every non-zero complex number a the set { f ( z + λ n a ) : n = 1 , 2 , … } is dense in H ( C ) . This answers in the affirmative Question 1 in [29] and it also provides an extension of a theorem due to Costakis and Sambarino in [32] . [ABSTRACT FROM AUTHOR]

This paper is devoted to studying an interpolation problem on the circle, which can be considered an intermediate problem between Lagrange and Hermite interpolation. The difference as well as the novelty is that we prescribe Lagrange values at the 2 n roots of a complex number with modulus one and we prescribe values for the first derivative only on half of the nodes. We obtain two types of expressions for the interpolation polynomials: the barycentric expressions and another one given in terms of an orthogonal basis of the corresponding subspace of Laurent polynomials. These expressions are very suitable for numerical computation. Moreover, we give sufficient conditions in order to obtain convergence in case of continuous functions and we obtain the rate of convergence for smooth functions. Finally we present some numerical experiments to highlight the results obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study complex Jacobi matrices obtained by the Christoffel and Geronimus transformations at a nonreal complex number, including the properties of the corresponding sequences of orthogonal polynomials. We also present some invariant and semi-invariant properties of Jacobi matrices under such transformations. For instance, we show that a Nevai class is invariant under the transformations in question, which is not true in general, and that the ratio asymptotic still holds outside the spectrum of the corresponding symmetric complex Jacobi matrix but the spectrum could include one extra point. In principal, these transformations can be iterated and, for example, we demonstrate how Geronimus transformations can lead to R I I -recurrence relations, which in turn are related to orthogonal rational functions and pencils of Jacobi matrices. [ABSTRACT FROM AUTHOR]

We define and study representation categories based on Deligne categories Rep ( G L t ) , Rep ( O t ) , Rep ( S p 2 t ) , where t is any (non-integer) complex number. Namely, we define complex rank analogs of the parabolic category O and the representation categories of real reductive Lie groups and supergroups, affine Lie algebras, and Yangians. We develop a framework and language for studying these categories, prove basic results about them, and outline a number of directions of further research. [ABSTRACT FROM AUTHOR]

Recently, M. Abért and T. Hubai studied the following problem. The chromatic measure of a finite simple graph is defined to be the uniform distribution on its chromatic roots. Abért and Hubai proved that for a Benjamini–Schramm convergent sequence of finite graphs, the chromatic measures converge in holomorphic moments. They also showed that the normalized logarithm of the chromatic polynomial converges to a harmonic real function outside a bounded disc. In this paper we generalize their work to a wide class of graph polynomials, namely, multiplicative graph polynomials of bounded exponential type. A special case of our results is that for any fixed complex number v 0 the measures arising from the Tutte polynomial Z G n ( z , v 0 ) converge in holomorphic moments if the sequence ( G n ) of finite graphs is Benjamini–Schramm convergent. This answers a question of Abért and Hubai in the affirmative. Even in the original case of the chromatic polynomial, our proof is considerably simpler. [ABSTRACT FROM AUTHOR]

A polynomial is called unimodular if each of its coefficients is a complex number of modulus 1. A polynomial P of the form P ( z ) = ∑ j = 0 n a j z j is called conjugate reciprocal if a n − j = a ‾ j , a j ∈ C for each j = 0 , 1 , … , n . Let ∂ D be the unit circle of the complex plane. We prove that there is an absolute constant ε > 0 such that max z ∈ ∂ D ⁡ | f ( z ) | ≥ ( 1 + ε ) 4 / 3 m 1 / 2 , for every conjugate reciprocal unimodular polynomial of degree m . We also prove that there is an absolute constant ε > 0 such that M q ( f ′ ) ≤ exp ⁡ ( ε ( q − 2 ) / q ) 1 / 3 m 3 / 2 , 1 ≤ q < 2 , and M q ( f ′ ) ≥ exp ⁡ ( ε ( q − 2 ) / q ) 1 / 3 m 3 / 2 , 2 < q , for every conjugate reciprocal unimodular polynomial of degree m , where M q ( g ) = ( 1 2 π ∫ 0 2 π | g ( e i t ) | q d t ) 1 / q , q > 0 . [ABSTRACT FROM AUTHOR]

Let η ≠ − 1 be a non-zero complex number, and let ϕ be a not necessarily linear bijection between two von Neumann algebras, one of which has no central abelian projections, satisfying ϕ ( I ) = I and preserving the Jordan triple η -⁎-product. It is showed that ϕ is a linear ⁎ -isomorphism if η is not real and ϕ is the sum of a linear ⁎ -isomorphism and a conjugate linear ⁎ -isomorphism if η is real. [ABSTRACT FROM AUTHOR]

This study considers three stochastic transforms for jump diffusion processes: the stochastic Laplace transform, stochastic Fourier transform, and stochastic wavelet transform. First, we introduce the stochastic Laplace transform for processes adapted by Brownian filtration as a solution for complex number valued backward stochastic differential equations (BSDEs). This transform can be explained well by the Girsanov theorem, which also allows us to define the stochastic Laplace transform for jump diffusion processes directly. Based on this perspective, we give natural definitions of the stochastic Fourier transform and stochastic wavelet transform. The advantages of these stochastic transforms are all related to the uniqueness of the processes. Compared with the classical transforms, the newly introduced parameters guarantee the uniqueness of the stochastic transforms for the adapted processes, while they also agree with the corresponding parameters in the classical transforms, which can represent the frequency property of the processes. In addition, these three stochastic transforms can also be regarded as the solutions of related BSDEs with jumps. [ABSTRACT FROM AUTHOR]

Using the basic fact that any formal power series over the real or complex number field can always be expressed in terms of given polynomials {p n (t)}, where p n (t) is of degree n, we extend the ordinary Riordan array (resp. Riordan group) to a generalized Riordan array (resp. generalized Riordan group) associated with {p n (t)}. As new application of the latter, a rather general Vandermonde-type convolution formula and certain of its particular forms are presented. The construction of the Abel type identities using the generalized Riordan arrays is also discussed. [ABSTRACT FROM AUTHOR]

Abstract: We say that a complex number λ is an extended eigenvalue of a bounded linear operator T on a Hilbert space if there exists a nonzero bounded linear operator X acting on , called extended eigenvector associated to λ, and satisfying the equation . In this paper we describe the sets of extended eigenvalues and extended eigenvectors for the product of a positive and a self-adjoint operator which are both injective. We also treat the case of normal operators. [Copyright &y& Elsevier]

We prove Feigin-Frenkel type dualities between subregular W -algebras of type A , B and principal W -superalgebras of type sl (1 | n) , osp (2 | 2 n). The type A case proves a conjecture of Feigin and Semikhatov. Let (g 1 , g 2) = (sl n + 1 , sl (1 | n + 1)) or (so 2 n + 1 , osp (2 | 2 n)) and let r be the lacity of g 1. Let k be a complex number and ℓ defined by r (k + h 1 ∨) (ℓ + h 2 ∨) = 1 with h i ∨ the dual Coxeter numbers of the g i. Our first main result is that the Heisenberg cosets C k (g 1) and C ℓ (g 2) of these W -algebras at these dual levels are isomorphic, i.e. C k (g 1) ≃ C ℓ (g 2) for generic k. We determine the generic levels and furthermore establish analogous results for the cosets of the simple quotients of the W -algebras. Our second result is a novel Kazama-Suzuki type coset construction: We show that a diagonal Heisenberg coset of the subregular W -algebra at level k times the lattice vertex superalgebra V Z is the principal W -superalgebra at the dual level ℓ. Conversely a diagonal Heisenberg coset of the principal W -superalgebra at level ℓ times the lattice vertex superalgebra V − 1 Z is the subregular W -algebra at the dual level k. Again this is proven for the universal W -algebras as well as for the simple quotients. We show that a consequence of the Kazama-Suzuki type construction is that the simple principal W -superalgebra and its Heisenberg coset at level ℓ are rational and/or C 2 -cofinite if the same is true for the simple subregular W -algebra at dual level ℓ. This gives many new C 2 -cofiniteness and rationality results. [ABSTRACT FROM AUTHOR]

Let X be a complex Banach space and denote by B (X) the algebra of all linear bounded operators on X. For T ∈ B (X) , denote by r (T) its spectral radius. We prove that if φ : B (X) → B (X) is a linear and surjective map such that i) given any A , B ∈ B (X) , there exists a positive real number M A , B such that r (φ (A) + λ φ (B)) ≤ M A , B r (A + λ B) for all complex numbers λ satisfying | λ | ≤ 1 ; ii) r (φ (T)) = 0 implies r (T) = 0 , then there exist a linear functional f : B (X) → C , a nonzero complex number c and either a bounded invertible linear map U : X → X such that φ (T) = c U T U − 1 + f (T) I (T ∈ B (X)) , or a bounded invertible linear map U : X ⁎ → X such that φ (T) = c U T ⁎ U − 1 + f (T) I (T ∈ B (X)). Also, the linear functional f satisfies the property that given any A , B ∈ B (X) there exists N A , B ≥ 0 such that | f (A) + λ f (B) | ≤ N A , B r (A + λ B) for all complex numbers λ satisfying | λ | ≤ 1. [ABSTRACT FROM AUTHOR]

Suppose X is an N × n complex matrix whose entries are centered, independent, and identically distributed random variables with variance 1 ∕ n and whose fourth moment is of order O (n − 2). Suppose A is a deterministic matrix whose smallest and largest singular values are bounded below and above respectively, and z ≠ 0 is a complex number. First we consider the matrix X A X ∗ − z , and obtain asymptotic probability bounds for its smallest singular value when N and n diverge to infinity and N ∕ n → γ , 0 < γ < ∞. Then we consider the special case where A = J = 1 i − j = 1 mod n is a circulant matrix. Using the above result, we show that the limit spectral distribution of X J X ∗ exists when N ∕ n → γ , 0 < γ < ∞ and describe the limit explicitly. Assuming that X represents a ℂ N -valued time series which is observed over a time window of length n , the matrix X J X ∗ represents the one-step sample autocovariance matrix of this time series. A whiteness test against an MA correlation model for this time series is introduced based on the above limit result. Numerical simulations show the excellent performance of this test. [ABSTRACT FROM AUTHOR]

On the Fréchet space of entire functions H (C) , we show that every nonscalar continuous linear operator L : H (C) → H (C) which commutes with differentiation has a hypercyclic vector f (z) in the form of the infinite product of linear polynomials: f (z) = ∏ j = 1 ∞ (1 − z a j ) , where each a j is a nonzero complex number. [ABSTRACT FROM AUTHOR]