A topological space is called submetrizable if it can be mapped onto a metrizable topological space by a continuous one-to-one map. In this paper we answer two questions concerning sequence-covering maps on submetrizable spaces.
A topological space is called submetrizable if it can be mapped onto a metrizable topological space by a continuous one-to-one map. In this paper we answer two questions concerning sequence-covering maps on submetrizable spaces.
For an imaginary biquadratic number field (K = mathbb Q(sqrt{-q},sqrt d)), where (q>3) is a prime congruent to (3 pmod 8), and (d) is an odd square-free integer which is not equal to q, let (K_infty) be the cyclotomic (mathbb Z_2)-extension of (K). For any integer (n geq 0), we denote by (K_n) the nth layer of (K_infty/K). We investigate the rank of the 2-class group of (K_n), then we draw the list of all number fields K such that the Galois group of the maximal unramified pro-2-extension over their cyclotomic (mathbb Z_2)-extension is metacyclic pro-2 group.
We improve and complement a result by Móricz and Siddiqi [20]. In particular, we prove that their estimate of the Nörlund means with respect to the Vilenkin system holds also without their additional condition. Moreover, we prove a similar approximation result in Lebesgue spaces for any (1leq p<infty).
Let (a_t(n)) denote the number of t-core partitions of n. In recent years, a number of congruences for (a_t(n)) have been discovered for some small t. Very recently, Fathima and Pore [4] established infinite families of congruences modulo 3 for (a_5(n)) and congruences modulo 2 for (a_7(n)). Motivated by their work, we prove some new infinite families of congruences modulo 3 for (a_5(n)) and congruences modulo 2 for (a_7(n)) by utilizing Newman's identities.
We study mappings satisfying some estimate of distortion of modulus of families of paths. Under some conditions on definition and mapped domains, we prove that these mappings are logarithmic Hölder continuous at boundary points.
We introduce two notions of fractal sumset properties. A compact set (Ksubsetmathbb{R}^d) is said to have the Hausdorff sumset property (HSP) if for any (ellinmathbb{N}_{ge 2}) there exist compact sets (K_1,K_2),..., (K_ell) such that (K_1+K_2+cdots+K_ellsubset K) and (dim_H K_i=dim_H K) for all (1le ile ell). Analogously, if we replace the Hausdorff dimension by the packing dimension in the definition of HSP, then the compact set (Ksubsetmathbb{R}^d) is said to have the packing sumset property (PSP). We show that the HSP fails for certain homogeneous self-similar sets satisfying the strong separation condition, while the PSP holds for all homogeneous self-similar sets in (mathbb{R}^d).
We prove that: I. If L is a (T_1) space, (|L|>1) and (d(L) leq kappa geq omega), then there is a submaximal dense subspace X of (L^{2^kappa}) such that (|X|=Delta(X)=kappa). II. If (mathfrak{c}leqkappa=kappa^omega<lambda) and (2^kappa=2^lambda), then there is a Tychonoff pseudocompact globally and locally connected space X such that (|X|=Delta(X)=lambda) and X is not (kappa^+)-resolvable. III. If (omega_1leqkappa<lambda) and (2^kappa=2^lambda), then there is a regular space X such that (|X|=Delta(X)=lambda), all continuous real-valued functions on X are constant (so X is connected) and X is not (kappa^+)-resolvable.
We develop the martingale theory in the framework of Lorentz amalgam spaces. Atomic decompositions for the martingale Hardy Lorentz amalgam spaces are established. As applications of atomic decompositions, the dual spaces of the martingale Hardy Lorentz amalgam spaces are characterized. Furthermore, when the stochastic basis is regular, the boundedness of the fractional integrals on martingale Hardy Lorentz amalgam spaces is proved. The results obtained here generalized the corresponding known results in martingale Hardy amalgam spaces and various classical martingale Hardy spaces.
We construct examples of bounded below, noncontractible, acyclic complexes of finitely generated projective modules over some rings S, as well as bounded above, noncontractible, acyclic complexes of injective modules. The rings S are certain rings of infinite matrices with entries in the rings of commutative polynomials or formal power series in infinitely many variables. In the world of comodules or contramodules over coalgebras over fields, similar examples exist over the cocommutative symmetric coalgebra of an infinite-dimensional vector space. A simpler, universal example of a bounded below, noncontractible, acyclic complex of free modules with one generator, communicated to the author by Canonaco, is included at the end of the paper.
This paper concerns the visibility properties of lattice points in multiple random walks on (mathbb{N}^k), where (kgeq 2) is an integer. We study two aspects of the visibility: simultaneous visibility in multiple random walkers; and that only some of these walkers are visible. Combining tools from number theory and probability theory, we prove the corresponding densities of the above two parts.
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