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In this paper we prove that the time part of the germ of an analytic family of vector fields with a Hopf bifurcation is rigid in the parameter. Time parts are associated with the temporal invariant of the analytic classification. Because the eigenvalues at zero are complex conjugate, time parts usually unfold in the hyperbolic direction, where the singular points are linearizable. We first identify the time part of a generic conformal family and prove that any weak holomorphic conjugacy between two time parts yields a biholomorphism analytic in the parameter. The existence of Fatou coordinates in both the Siegel and in the Poincare domains plays a fundamental role in the proof of this result.

We prove two inequalities for the complex orientations of a separating non-singular real algebraic curve in $${\mathbb {RP}}^2$$ of any odd degree. We also construct a separating non-singular real (i.e., invariant under the complex conjugation) pseudoholomorphic curve in $${\mathbb {CP}}^2$$ of any degree congruent to 9 mod 12 which does not satisfy one of these inequalities. Therefore the oriented isotopy type of the real locus of each of these curves is algebraically unrealizable.

The aim of this article is to apply a Floer theory to study symmetric periodic Reeb orbits. We define positive equivariant wrapped Floer homology using a (anti-)symplectic involution on a Liouville domain and investigate its algebraic properties. By a careful analysis of index iterations, we obtain a non-trivial lower bound on the minimal number of geometrically distinct symmetric periodic Reeb orbits on a certain class of real contact manifolds. This includes non-degenerate real dynamically convex starshaped hypersurfaces in $\mathbb{R}^{2n}$ which are invariant under complex conjugation. As a result, we give a partial answer to the Seifert conjecture on brake orbits in the contact setting., Comment: 42 pages, 4 figures, final version, to appear in Ergodic Theory and Dynamical Systems

A Bagnera-de Franchis variety $X = A/G$ is the quotient of an abelian variety $A$ by a free action of a finite cyclic group $G \subset Bihol(A)$, which does not contain only translations. Constructing explicit polarizations and using a method introduced by F. Catanese, we classify split Bagnera-de Franchis varieties up to complex conjugation in dimensions $\leq 4$.

In this paper, we introduce the notion of normal Jacobi operator of Codazzi type for real hypersurfaces in the complex hyperbolic quadric [Formula: see text]. The normal Jacobi operator of Codazzi type implies that the unit normal vector field [Formula: see text] becomes [Formula: see text]-principal or [Formula: see text]-isotropic. Then according to each case, we give a complete classification of Hopf real hypersurfaces in [Formula: see text] with normal Jacobi operator of Codazzi type. The result of the classification shows that no such hypersurfaces exist.

The eigenvalue hypothesis claims that any quantum Racah matrix for finite-dimensional representations of $U_q(sl_N)$ is uniquely determined by eigenvalues of the corresponding quantum $\cal{R}$-matrices. If this hypothesis turns out to be true, then it will significantly simplify the computation of Racah matrices. Also due to this hypothesis various interesting properties of colored HOMFLY-PT polynomials will be proved. In addition, it allows one to discover new symmetries of the quantum 6-j symbols, about which almost nothing is known for $N>2$, with the exception of the tetrahedral symmetries, complex conjugation and transformation $q \longleftrightarrow q^{-1}$. In this paper we prove the eigenvalue hypothesis in $U_q(sl_2)$ case and show that it is equivalent to 6-j symbol symmetries (the Regge symmetry and two argument permutations). Then we apply the eigenvalue hypothesis to inclusive Racah matrices with 3 symmetric incoming representations of $U_q(sl_N)$ and an arbitrary outcoming one. It gives us 8 new additional symmetries that are not tetrahedral ones. Finally, we apply the eigenvalue hypothesis to exclusive Racah matrices with symmetric representations and obtain 4 tetrahedral symmetries., 22 pages

In this letter, we study a few properties of Complex Conjugate Pair Sums (CCPSs) and Complex Conjugate Subspaces (CCSs). Initially, we consider an LTI system whose impulse response is one period data of CCPS. For a given input x(n), we prove that the output of this system is equivalent to computing the first order derivative of x(n). Further, with some constraints on the impulse response, the system output is also equivalent to the second order derivative. With this, we show that a fine edge detection in an image can be achieved using CCPSs as impulse response over Ramanujan Sums (RSs). Later computation of projection for CCS is studied. Here the projection matrix has a circulant structure, which makes the computation of projections easier. Finally, we prove that CCS is shift-invariant and closed under the operation of circular cross-correlation., 4 pages, 2 figures

We introduce the notion of Lie invariant structure Jacobi operators for real hypersurfaces in the complex quadric $Q^{m}=SO_{m+2}/SO_{m}SO_{2}$. The existence of invariant structure Jacobi operators implies that the unit normal vector field $N$ becomes $\mathfrak{A}$-principal or $\mathfrak{A}$-isotropic. Then, according to each case, we give a complete classification of real hypersurfaces in $Q^{m}=SO_{m+2}/SO_{m}SO_{2}$ with Lie invariant structure Jacobi operators.

We introduce a new notion of pseudo-anti commuting Ricci tensor for real hypersurfaces in the noncompact complex hyperbolic quadric $$Q^{m*}=SO_{2,m}^0/SO_2SO_m$$ and give a complete classification of these hypersurfaces.

We consider semilinear Schrodinger equations with nonlinearity that is a polynomial in the unknown function and its complex conjugate, on \begin{document}$\mathbb{R}^d$\end{document} or on the torus. Norm inflation (ill-posedness) of the associated initial value problem is proved in Sobolev spaces of negative indices. To this end, we apply the argument of Iwabuchi and Ogawa (2012), who treated quadratic nonlinearities. This method can be applied whether the spatial domain is non-periodic or periodic and whether the nonlinearity is gauge/scale-invariant or not.

We consider the Riemann–Hilbert-type boundary-value problems in the case of several unknown functions and obtain solvability conditions and index formulas for the linear conjugation problem, the Riemann–Hilbert problem, and the Riemann–Hilbert–Poincare problem on a plane cut along several regular arcs in various weighted functional classes. We also examine the general differential boundary-value problem for analytic functions and boundary-value problems with shifts and complex conjugation on a cut plane.

We describe a simple way to write down explicit derivations of octonions that form a Chevalley basis of g 2 . This uses the description of octonions as a twisted group algebra of the finite field F 8 . Generators of Gal ( F 8 / F 2 ) act on the roots as 120-degree rotations and complex conjugation acts as negation.

In this article we study the class QS2cIL of quadratic differential systems with two complex conjugate invariant lines meeting at a finite point. From the literature we know that quadratic systems with invariant lines of total multiplicity at least four or with the line at infinity filled up with singularities are integrable via the method of Darboux and hence they have no limit cycles. These could only occur if we have only the two complex lines, and the line at infinity, all simple. We first find all integrable systems in QS2cIL due to the presence of invariant lines. We next indicate a gap in the 1986 proof of Suo and Chen that systems in QS2cIL have at most one limit cycle and we give a complete proof of this result. Finally we give the topological classification of QS2cIL yielding 22 phase portraits three of which with a limit cycle.

This manuscript concerns two definite integrals that could be connected to the Bose-Einstein and the Fermi-Dirac functions in the integrands, separately, with numerators slightly modified with a difference in two expressions that contain the Fourier kernel multiplied by a polynomial and its complex conjugate. In this work, we use our contour integral method to derive these definite integrals, which are given by ∫0∞ie−imx(log(a)−ix)k−eimx(log(a)+ix)k2eαx−1dx and ∫0∞ie−imx(log(a)−ix)k−eimx(log(a)+ix)k2eαx+1dx in terms of the Lerch function. We use these two definite integrals to derive formulae by Erdéyli and Watson. We derive special cases of these integrals in terms of special functions not found in current literature. Special functions have the property of analytic continuation, which widens the range of computation of the variables involved.

Publisher Summary This chapter provides an overview on complex number. These are represented pictorially on rectangular or Cartesian axes. The horizontal (x) axis is used to represent the real axis and the vertical (y) axis is used to represent the imaginary axis. Such a diagram is called an Argand diagram. Multiplication of complex numbers is achieved by assuming all quantities involved are real and then using j2 = -1 to simplify. The complex conjugate of a complex number is obtained by changing the sign of the imaginary part. Hence, the complex conjugate of a + jb is a − jb. The product of a complex number and its complex conjugate is always a real number. Division of complex numbers is achieved by multiplying both numerator and denominator by the complex conjugate of the denominator. If two complex numbers are equal, then their real parts are equal and their imaginary parts are equal. Hence, if a + jb = c + jd, then a = c and b = d. There are several applications of complex numbers in science and engineering, in particular, in electrical alternating current theory and in mechanical vector analysis.

Fake projective planes are smooth complex surfaces of general type with Betti numbers equal to those of the usual projective plane. They come in complex conjugate pairs and have been classified as quotients of the two-dimensional ball by explicitly written arithmetic subgroups. In this paper we find equations of a projective model of a conjugate pair of fake projective planes by studying the geometry of the quotient of such surface by an order seven automorphism., Comment: This is a full version of "Research announcement: equations of a fake projective plane", arXiv:1710.04501. Key tables and some M2 and Magma code from the paper are included in separate files for convenience

The goal of this article is to give invariant necessary and sufficient conditions for a quadratic system, presented in whatever normal form, to have anyone of 17 out of the 20 phase portraits of the family of quadratic systems with two complex conjugate invariant lines intersecting at a finite real point. The systems in this family have a maximum of one limit cycle. Among the 17 phase portraits we have two with limit cycles. We also give invariant necessary and sufficient conditions for a system to have one of the three remaining phase portraits, out of which one has a limit cycle and another one a homoclinic loop. In the region $${\mathcal {R}}$$ determined by these last conditions, due to the presence of systems with a homoclinic loop, an analytic condition, the three phase portraits cannot be separated by algebraic conditions in terms of invariant polynomials. We also give the bifurcation diagram of this family, outside the region $${\mathcal {R}}$$ , in the twelve parameter space of coefficients of the systems.

A conformal map is one that preserves angles. In the case of mappings from one connected domain in \(\mathbb {C}\) to another, such a map is holomorphic, or else its complex conjugate is holomorphic.

In this paper, first, we investigate the commuting property between the normal Jacobi operator~${\bar R}_N$ and the structure Jacobi operator~$R_{\xi}$ for Hopf real hypersurfaces in the complex quadric~$Q^m = SO_{m+2}/SO_mSO_2$, $m \geq 3$, which is defined by ${\bar R}_N R_{\xi} = R_{\xi}{\bar R}_N$. Moreover, a new characterization of Hopf real hypersurfaces with $\mathfrak A$-principal singular normal vector field in the complex quadric~$Q^{m}$ is obtained. By virtue of this result, we can give a remarkable classification of Hopf real hypersurfaces in the complex quadric~$Q^{m}$ with commuting Jacobi operators., Comment: 20 pages. arXiv admin note: text overlap with arXiv:1907.04661, arXiv:1605.05316

This chapter provides a self-contained introduction to quantum groups or Hopf algebras and their associated braided categories of (co)modules in the (co)quasitriangular case and crossed modules in the case of invertible antipode. We then cover the construction of differential structures on them and on associated algebras, including the theory of the quantum Lie algebra of a bicovariant calculus and of braided-Lie algebras in the coquasitriangular case. Basic examples include q-SU2 and the associated q-sphere. The chapter ends with the notion of a bar category needed to formulate complex conjugation and *-operations in a more categorical way.

The S-transform (ST), which is an important tool in signal processing, is a conceptual principle of the Fourier transform (FT) with a Gaussian window function. It has been observed from literature study that only linearity, scaling, time-shifting and convolution theorem of ST is documented. This led to the finding of remaining properties of ST in order to establish it as complete transform technique. Along with this a new better definition of convolution theorem for ST is also presented. Therefore this paper is focused on formulation of properties such as time-reversal, time-derivatives, complex conjugate, convolution theorem, correlation theorem and Parseval’s theorem. A comparative analysis of existing convolution theorem with proposed convolution theorem is also given in this manuscript.

Conjugation spaces are topological spaces equipped with an involution such that their fixed points have the same mod $2$ cohomology (as a graded vector space, a ring, and even an unstable algebra) but with all degrees divided by two, generalizing the classical examples of complex projective spaces under complex conjugation. Spaces which are constructed from unit balls in complex Euclidean spaces are called spherical and are very well understood. Our aim is twofold. We construct "exotic" conjugation spaces and study the realization question: which spaces can be realized as real loci, i.e., fixed points of conjugation spaces. We identify obstructions and provide examples of spaces and manifolds which cannot be realized as such., Comment: 16 pages

The traditional Cohen's class time-frequency representation is extended to the linear canonical domain by using a well-established closed-form instantaneous cross-correlation function (CICF) type of linear canonical transform (LCT) free parameters embedded approach. The derived CICF type of Cohen's class (CICFCC) unifies some well-known Cohen's classes in linear canonical domains including the affine characteristic, basis function, convolution expression and instantaneous crosscorrelation function types of Cohen's classes, and can be considered as the Cohen's class's closed-form representation in linear canonical domains. A fundamental theory about the CICFCC's essential properties, such as marginal distribution, energy conservation, unique reconstruction, Moyal formula, complex conjugate symmetry, time reversal symmetry, scaling property, time shift property, frequency shift property, and LCT invariance, is then established. Possible applications are also carried out to illustrate that the CICFCC outperforms the traditional one in nonstationary signal separation and detection.

In the first part of the paper, it is proved that for 1 < p < ∞ the couple (K , K ∞ ) of coinvariant subspaces of the shift operator on the unit circle is K-closed in the couple (L p (T),L∞ (T)). This property underlies basically all problems of real interpolation for the first couple. Also, a weighted analog of the above statement is established. In the second part it is shown that, given two closed ideals I and J in a uniform algebra such that the complex conjugate of I ∩ J is not included in some of them, the sum I + J is not closed. Though the methods of study in the two parts are quite different, the topics are related by the fact that the question treated in the second part emerged during the work on the first.

Let $\unicode[STIX]{x1D70C}\in (0,\infty ]$ be a real number. In this short note, we extend a recent result of Marques and Ramirez [‘On exceptional sets: the solution of a problem posed by K. Mahler’, Bull. Aust. Math. Soc.94 (2016), 15–19] by proving that any subset of $\overline{\mathbb{Q}}\cap B(0,\unicode[STIX]{x1D70C})$, which is closed under complex conjugation and contains $0$, is the exceptional set of uncountably many analytic transcendental functions with rational coefficients and radius of convergence $\unicode[STIX]{x1D70C}$. This solves the question posed by K. Mahler completely.

In this paper, first we give a non-existence theorem for a real hypersurface M with anti-commuting shape operator S in the complex quadric Q m , that is, S ϕ + ϕ S = 0 , where ϕ denotes the structure tensor of M in Q m . By using such a non-existence theorem, we give a classification of Hopf hypersurfaces with recurrent normal Jacobi operator in the complex quadric Q m .

We consider the Schwarz problem for J-analytic functions for the case in which the Jordan basis Q of the matrix J contains complex conjugate vectors. Conditions on the matrix Q are obtained under which there exists a unique solution of the Schwarz problem in Holder classes.

We introduce the notion of Reeb invariant Ricci tensor for real hypersurfaces in the complex quadric Q m = S O m + 2 ∕ S O m S O 2 . The Reeb invariant Ricci tensor implies that the unit normal vector field N becomes A -principal or A -isotropic. Then according to each case, we give a complete classification of real hypersurfaces in Q m = S O m + 2 ∕ S O m S O 2 with Reeb invariant Ricci tensor.

First we introduce a new notion of pseudo-anti commuting Ricci tensor for real hypersurfaces in the complex quadric Q m = S O m + 2 / S O 2 S O m and give a complete classification of these hypersurfaces in the complex quadric Q m . Next as an application we give a complete classification of Ricci solitons on Hopf real hypersurfaces in the complex quadric Q m .

We prove that a totally complex algebraic number field K; having a conjugate which is not closed under complex conjugation, can be generated by a reciprocal integer, when the Galois group of its normal closure is contained in the hyperoctahedral group Bdeg(K)/2.Keywords: Reciprocal integers, unit primitive elements, reciprocal numbers

We characterize continuous, nondegenerate multiplicative maps on the algebra of upper-triangular Toeplitz matrices. As an application, we show that the continuous multiplicative isometries of (for any norm) are necessarily similarities or similarities composed with complex conjugation; for unitarily invariant norms, such isometries are precisely the unitary similarities, or unitary similarities composed with complex conjugation. We partially extend these results to algebras generated by non-derogatory matrices.

In this article, we introduce the notion of pseudo-Einstein real hypersurfaces in the complex quadric Qm=SOm+2/SO2SOm and give a complete classification of such hypersurfaces.

We review some recent results on properties of tensor product and fusion coefficients under complex conjugation of one of the factors. Some of these results have been proven, some others are conjectures awaiting a proof, one of them involving hitherto unnoticed observations on ordinary representation theory of finite simple groups of Lie type, 8 pages, 2 figures

First, we introduce the notion of shape operator of Codazzi type for real hypersurfaces in the complex quadric \({Q^m}^* = SO^{o}_{m,2}/SO_mSO_2\). Next, we give a complete proof of non-existence of real hypersurfaces in \({Q^m}^* = SO^{o}_{m,2}/SO_mSO_2\) with shape operator of Codazzi type. Motivated by this result, we give a complete classification of real hypersurfaces in \({Q^m}^* = SO^{o}_{m,2}/SO_mSO_2\) with Reeb parallel shape operator.

We investigate the statistical properties of eigenvalues of pseudo-Hermitian random matrices whose eigenvalues are real or complex conjugate. It is shown that when the spectrum splits into separated sets of real and complex conjugate eigenvalues, the real ones show characteristics of an intermediate incomplete spectrum, that is, of a so-called thinned ensemble. On the other hand, the complex ones show repulsion compatible with cubic-order repulsion of non normal matrices for the real matrices, but higher order repulsion for the complex and quaternion matrices.

We consider the standard hypergeometric differential operator $D$ regarded as an operator on the complex plane $C$ and the complex conjugate operator $\overline D$. These operators formally commute and are formally adjoint one to another with respect to an appropriate weight. We find conditions when they commute in the Nelson sense and write explicitly their joint spectral decomposition. It is determined by a two-dimensional counterpart of the Jacobi transform (synonyms: generalized Mehler--Fock transform, Olevskii transform). We also show that the inverse transform is an operator of spectral decomposition for a pair of commuting difference operators defined in terms of shifts in imaginary direction., 55p, typos were corrected

This paper introduces biquaternion eigen-decomposition theory (via Peirce decomposition) with respect to a selected quaternion with a non-zero vector part. The eigen-decomposition allows evaluation of polynomials and power series with real coefficients as functions of quaternions. This extension of analytic functions to functions of quaternions requires only standard complex function evaluation. The theory also applies to quaternion rotations. The theory uses biquaternion calculations indicated by matrix methods via the algebraic isomorphism between Hamilton’s biquaternions and appropriate $$4\times 4$$ complex matrices. The isomorphism preserves algebraic structure. In particular, the left and right biquaternion multiplication by the selected quaternion maps to left and right matrix multiplication, respectively. This unifies the representation of the left and right quaternion multiplication as a linear map into a single matrix form. This matrix, as a linear operator, acts on matrices, so that the eigenvectors have matrix form that maps into the biquaternions. Use of an alternate quaternion basis results in a similarity transform of the representation matrix, preserving eigenvalues across change of basis. The similarity transform allows simple eigenvector calculation. The matrix for the selected quaternion has two identical, complex conjugate pairs of eigenvalues. Each pair corresponds to two complex conjugate pairs of eigenvector biquaternions, an idempotent pair and a nilpotent pair. Idempotent and nilpotent eigenvectors correspond to the commuting and non-commuting parts, respectively, of quaternion multiplication.

We introduce the notion of harmonic curvature for real hypersurfaces in the complex quadric Q m = S O m + 2 / S O m S O 2 . We give a complete classification, in terms of their A -principal or their A -isotropic unit normal vector fields, of real hypersurfaces in Q m = S O m + 2 / S O m S O 2 having harmonic curvature tensor.

First we introduce the notion of parallel normal Jacobi operator for real hypersurfaces in the complex quadric . Next we give a complete classification of real hypersurfaces in the complex quadric with parallel normal Jacobi operator.

In this paper we prove that a dynamically convex starshaped hypersurface in $\mathbb{C}^2$ which is invariant under complex conjugation admits a global surface of section which is invariant under conjugation as well. We obtain this invariant global surface by embedding $\mathbb{C}^2$ into $\mathbb{CP}^2$ and applying a stretching argument to real holomorphic curves in $\mathbb{CP}^2$. The motivation for this result arises from recent progress in applying holomorphic curve techniques to gain a deeper understanding on the dynamics of the restricted three body problem., 36 pages

We introduce the notion of Killing shape operator for real hypersurfaces in the complex quadric $$Q^m = SO_{m+2}/SO_mSO_2$$ . The Killing shape operator condition implies that the unit normal vector field N becomes $$\mathfrak {A}$$ -principal or $$\mathfrak {A}$$ -isotropic. Then according to each case, we give a complete classification of Hopf real hypersurfaces in $$Q^m = SO_{m+2}/SO_mSO_2$$ with Killing shape operator.

In this article we study the action of the absolute Galois group Gal(Q/Q) on dessins d’enfants and Beauville surfaces. A foundational result in Grothendieck’s theory of dessins d’enfants is the fact that the absolute Galois group Gal(Q/Q) acts faithfully on the set of all dessins. However the question of whether this holds true when the action is restricted to the set of the, more accessible, regular dessins seems to be still an open question. In the first part of this paper we give an affirmative answer to it. In fact we prove the strongest result that the action is faithful on the set of quasiplatonic (or triangle) curves of any given hyperbolic type. Beauville surfaces are an important kind of algebraic surfaces introduced by Catanese. They are rigid surfaces of general type closely related to dessins d’enfants. Here we prove that for any σ ∈ Gal(Q/Q) different from the identity and the complex conjugation there is a Beauville surface S such that S and its Galois conjugate S have non-isomorphic fundamental groups. This in turn easily implies that the action of Gal(Q/Q) on the set of Beauville surfaces is faithful. These results were conjectured by Bauer, Catanese and Grunewald, and immediately imply that Gal(Q/Q) acts faithfully on the connected components of the moduli space of surfaces of general type, a result due to the above mentioned authors.

First we introduce the notion of Reeb invariant shape operator for real hypersurfaces in the complex quadric Q m = SO m + 2 / SO m SO 2 . Next we give a complete classification of real hypersurfaces in Q m = SO m + 2 / SO m SO 2 with invariant shape operator.

The ternary cubic equation 3 2 2 35 1 9 ) ( 5 z y x xy y x       is considered for determining its non-zero distinct integral solutions Employing the linear transformations x=u+v,y=u-v (u≠v≠0),and employing the meyhod of factorization in complex conjugates, different patterns of integral solutions to the ternary cubic equation under consideration are obtained.. In each pattern, interesting relations among the solutions, some special polygonal , pyramidal numbers and central pyramidal numbers are exhibited.

A truncation of a Haar distributed orthogonal random matrix gives rise to a matrix whose eigenvalues are either real or complex conjugate pairs, and are supported within the closed unit disk. This is also true for a product $P_m$ of $m$ independent truncated orthogonal random matrices. One of most basic questions for such asymmetric matrices is to ask for the number of real eigenvalues. In this paper, we will exploit the fact that the eigenvalues of $P_m$ form a Pfaffian point process to obtain an explicit determinant expression for the probability of finding any given number of real eigenvalues. We will see that if the truncation removes an even number of rows and columns from the original Haar distributed orthogonal matrix, then these probabilities will be rational numbers. Finally, based on exact finite formulae, we will provide conjectural expressions for the asymptotic form of the spectral density and the average number of real eigenvalues as the matrix dimension tends to infinity., 27 pages, 4 figures

We introduce the notion of normal Jacobi operator of Codazzi type for real hypersurfaces in the complex quadric \(Q^m\). The normal Jacobi operator of Codazzi type implies that the unit normal vector field N becomes \(\mathfrak {A}\)-principal or \(\mathfrak {A}\)-isotropic. Then according to each case, we give a non-existence theorem of real hypersurfaces in \(Q^m\) with normal Jacobi operator of Codazzi type.

In this paper we study the action of complex conjugation on Shimura varieties and the problem of descending these to the maximal totally real field of the reflex field. We prove the existence of such descent for many Shimura varieties whose associated adjoint group has certain factors of type A or D. This includes a large family of Shimura varieties of abelian type. Our considerations and constructions are carried out purely at the level of Shimura data and group theory., Comment: 19 pages

In this paper we introduce broccoli curves, certain plane tropical curves of genus zero related to real algebraic curves. The numbers of these broccoli curves through given points are independent of the chosen points - for arbitrary choices of the directions of the ends of the curves, possibly with higher weights, and also if some of the ends are fixed. In the toric Del Pezzo case we show that these broccoli invariants are equal to the Welschinger invariants (with real and complex conjugate point conditions), thus providing a proof of the independence of Welschinger invariants of the point conditions within tropical geometry. The general case gives rise to a tropical Caporaso-Harris formula for broccoli curves which suffices to compute all Welschinger invariants of the plane., Comment: 49 pages, minor changes to match the published version

The Dold manifold P (m,n) is obtained from the product Sm × CPn of the m-dimensional sphere and n-dimensional complex projective space by identifying (x, [z1, . . . , zn+1]) with (−x, [z1, . . . , zn+1]), where z denotes the complex conjugate of z. We answer the parallelizability question for the Dold manifolds P (m,n) and, by completing an earlier (2008) result due to Peter Novotný, we solve the vector field problem for the manifolds P (m, 1).