Combinatorial invariants for nets of conics in $$\mathrm {PG}(2,q)$$

The problem of classifying linear systems of conics in projective planes dates back at least to Jordan, who classified pencils (one-dimensional systems) of conics over $${\mathbb {C}}$$ C and $$\mathbb {R}$$ R in 1906–1907. The analogous problem for finite fields $$\mathbb {F}_q$$ F q with q odd was solved by Dickson in 1908. In 1914, Wilson attempted to classify nets (two-dimensional systems) of conics over finite fields of odd characteristic, but his classification was incomplete and contained some inaccuracies. In a recent article, we completed Wilson’s classification (for q odd) of nets of rank one, namely those containing a repeated line. The aim of the present paper is to introduce and calculate certain combinatorial invariants of these nets, which we expect will be of use in various applications. Our approach is geometric in the sense that we view a net of rank one as a plane in $$\mathrm {PG}(5,q)$$ PG ( 5 , q ) , q odd, that meets the quadric Veronesean in at least one point; two such nets are then equivalent if and only if the corresponding planes belong to the same orbit under the induced action of $$\mathrm {PGL}(3,q)$$ PGL ( 3 , q ) viewed as a subgroup of $$\mathrm {PGL}(6,q)$$ PGL ( 6 , q ) . Since q is odd, the orbits of lines in $$\mathrm {PG}(5,q)$$ PG ( 5 , q ) under this action correspond to the aforementioned pencils of conics in $$\mathrm {PG}(2,q)$$ PG ( 2 , q ) . The main contribution of this paper is to determine the line-orbit distribution of a plane $$\pi $$ π corresponding to a net of rank one, namely, the number of lines in $$\pi $$ π belonging to each line orbit. It turns out that this list of invariants completely determines the orbit of $$\pi $$ π , and we will use this fact in forthcoming work to develop an efficient algorithm for calculating the orbit of a given net of rank one. As a more immediate application, we also determine the stabilisers of nets of rank one in $$\mathrm {PGL}(3,q)$$ PGL ( 3 , q ) , and hence the orbit sizes.

MULTIPLE FLAG IND-VARIETIES WITH FINITELY MANY ORBITS

Let G be one of the ind-groups GL(∞), O(∞), Sp(∞), and let P1, ..., Pℓ be an arbitrary set of ℓ splitting parabolic subgroups of G. We determine all such sets with the property that G acts with finitely many orbits on the ind-variety X1 × × Xℓ where Xi = G/Pi. In the case of a finite-dimensional classical linear algebraic group G, the analogous problem has been solved in a sequence of papers of Littelmann, Magyar–Weyman–Zelevinsky and Matsuki. An essential difference from the finite-dimensional case is that already for ℓ = 2, the condition that G acts on X1 × X2 with finitely many orbits is a rather restrictive condition on the pair P1, P2. We describe this condition explicitly. Using the description we tackle the most interesting case where ℓ = 3, and present the answer in the form of a table. For ℓ ≥ 4 there always are infinitely many G-orbits on X1 × × Xℓ.

Developing Written Mathematics Communication through Solving Analogous Problems

Developing mathematics communication, especially in writing is needed considering that communication is one of the objectives of learning mathematics which can describe students understanding so that effective strategies are needed to develop it. One of problem solving strategies that can measures written mathematics communication is solving the analogous problem. This research aims to describe the analogous problem in developing mathematics communication especially in written communication. This research was a qualitative type through a student-written test about solving target problem using analogous problems and interview. The result showed solving analogous problems gave students understanding to be able to do the target question correctly also easier than before and it effected students could communicate the correct solutions accurately, effectively and completely in writing. So it can be said that solving analogous problems develops written mathematics communication ability of student effectively.

Alienation of Drygas’ and Cauchy’s Functional Equations

Inspired by the papers [2, 10] we will study, on 2-divisible groups that need not be abelian, the alienation problem between Drygas’ and the exponential Cauchy functional equations, which is expressed by the equation f ( x + y ) + g ( x + y ) g ( x - y ) = f ( x ) f ( y ) + 2 g ( x ) + g ( y ) + g ( - y ) . f\left( {x + y} \right) + g\left( {x + y} \right)g\left( {x - y} \right) = f\left( x \right)f\left( y \right) + 2g\left( x \right) + g\left( y \right) + g\left( { - y} \right). We also consider an analogous problem for Drygas’ and the additive Cauchy functional equations as well as for Drygas’ and the logarithmic Cauchy functional equations. Interesting consequences of these results are presented.

Holomorphic one-forms, fibrations over the circle, and characteristic numbers of Kähler manifolds

We prove that the only relation imposed on the Hodge and Chern numbers of a compact Kähler manifold by the existence of a nowhere zero holomorphic one-form is the vanishing of the Hirzebruch genus. We also treat the analogous problem for nowhere zero closed one-forms on smooth manifolds.

ON THE EQUIVALENCE OF SOME CONVOLUTIONAL EQUALITIES IN SPACES OF SEQUENCES

The problem of the equivalence of two systems with $n$ convolutional equalities arose in investigation of the conditions of similarity in spaces of sequences of operators which are left inverse to the $n$-th degree of the generalized integration operator. In this paper we solve this problem. Note that we first prove the equivalence of two corresponding systems with $n$ equalities in the spaces of analytic functions, and then, using this statement, the main result of paper is obtained. Let $X$ be a vector space of sequences of complex numbers with K$\ddot{\rm o}$the normal topology from a wide class of spaces, ${\mathcal I}_{\alpha}$ be a generalized integration operator on $X$, $\ast$ be a nontrivial convolution for ${\mathcal I}_{\alpha}$ in $X$, and $(P_q)_{q=0}^{n-1}$ be a system of natural projectors with $\displaystyle x = \sum\limits_{q=0}^{n-1} P_q x$ for all $x\in X$. We established that a set $(a^{(j)})_{j=0}^{n-1}$ with $$ \max\limits_{0\le j \le n-1}\left\{\mathop{\overline{\lim}}\limits_{m\to\infty} \sqrt[m]{\left|\frac{a_{m}^{(j)}}{\alpha_m}\right|}\right\}<\infty $$ and a set $(b^{(j)})_{j=0}^{n-1}$ of elements of the space $X$ satisfy the system of equalities $$ b^{(j)}=a^{(j)}+\sum\limits_{k=0}^{n-1}({\mathcal I}_{\alpha}^{n-k-1} a^{(k)}) \ast {(P_{k}b^{(j)})}, \quad j = 0, 1, ... \, , \, n-1, $$ if and only if they satisfy the system of equalities $$ b^{(j)}=a^{(j)}+\sum\limits_{k=0}^{n-1}({\mathcal I}_{\alpha}^{n-k-1} b^{(k)}) \ast {(P_{k}a^{(j)})}, \quad j = 0, 1, ... \, , \, n-1. $$ Note that the assumption on the elements $(a^{(j)})_{j=0}^{n-1}$ of the space $X$ allows us to reduce the solution of this problem to the solution of an analogous problem in the space of functions analytic in a disc.

Future Desires, the Agony Argument, and Subjectivism about Reasons

Extant discussions of subjectivism about reasons for action have concentrated on presentist versions of the theory, on which reasons for present actions are grounded in present desires. In this article, I motivate and investigate the prospects of futurist subjectivism, on which reasons for present actions are grounded in present or future desires. Futurist subjectivism promises to answer Parfit's Agony Argument, and it is motivated by natural extensions of some of the considerations that support subjectivism in general. However, it faces a problem: because which desires one will have in the future can depend on what one does now, it must tell us which of one's possible future desires give one reasons to promote their satisfaction. I argue that the most natural solutions to this problem are unsatisfactory: they either fail to answer the Agony Argument or have unacceptable implications elsewhere. Then, I propose a more promising solution. Moreover, I argue that a closely analogous problem arises for an important class of idealizing subjectivist views and that this problem admits of a similar solution.

Statistical Normalization Methods in Interpersonal and Intertheoretic Comparisons

A major problem for interpersonal aggregation is how to compare utility across individuals; a major problem for decision-making under normative uncertainty is the formally analogous problem of how to compare choice-worthiness across theories. We introduce and study a class of methods, which we call statistical normalization methods, for making interpersonal comparisons of utility and intertheoretic comparisons of choice-worthiness. We argue against the statistical normalization methods that have been proposed in the literature. We argue, instead, in favor of normalization of variance: we claim that this is the account that most plausibly gives all individuals or theories ‘equal say’. To this end, we provide two proofs that variance normalization has desirable properties that all other normalization methods lack, though we also show how different assumptions could lead one to axiomatize alternative statistical normalization methods.

Finger Growth and Selection in a Poisson Field

Solutions are found for the growth of infinitesimally thin, two-dimensional fingers governed by Poisson’s equation in a long strip. The analytical results determine the asymptotic paths selected by the fingers which compare well with the recent numerical results of Cohen and Rothman (J Stat Phys 167:703–712, 2017) for the case of two and three fingers. The generalisation of the method to an arbitrary number of fingers is presented and further results for four finger evolution given. The relation to the analogous problem of finger growth in a Laplacian field is also discussed.

On relations between weak and strong type inequalities for modified maximal operators on non-doubling metric measure spaces

In this article, we investigate a special class of non-doubling metric measure spaces in order to describe the possible configurations of {P_{k,{\mathrm{s}}}^{{\mathrm{c}}}} , {P_{k,{\mathrm{s}}}} , {P_{k,{\mathrm{w}}}^{{\mathrm{c}}}} and {P_{k,{\mathrm{w}}}} , the sets of all {p\in[1,\infty]} for which the weak and strong type {(p,p)} inequalities hold for the centered and non-centered modified Hardy–Littlewood maximal operators {M^{{\mathrm{c}}}_{k}} and {M_{k}} , {k\geq 1} . For any fixed k we describe the necessary conditions that {P_{k,{\mathrm{s}}}^{{\mathrm{c}}}} , {P_{k,{\mathrm{s}}}} , {P_{k,{\mathrm{w}}}^{{\mathrm{c}}}} and {P_{k,{\mathrm{w}}}} must satisfy in general and illustrate each admissible configuration with a properly chosen non-doubling metric measure space. We also give some partial results related to an analogous problem stated for varying k.