Narrow Results

Lie conformal algebras ${𝒲}(a,b)$ are the semi-direct sums of Virasoro Lie conformal algebra and its nontrivial conformal modules of rank one. In this paper, we first give a complete classification of all finite nontrivial irreducible conformal modules of ${𝒲}(a,b)$. It is shown that all such modules are of rank one. Moreover, with a similar method, all finite nontrivial irreducible conformal modules of Schrödinger–Virasoro type Lie conformal algebras $\mathrm{\text{TSV}}(a,b)$ and $\mathrm{\text{TSV}}(c)$ are characterized.

We study a local system associated with a system ${ℱ}$ of hypergeometric differential equations in two variables of rank $9$ with seven parameters $a_1,a_2,a_3$ and $b_1,b_2,b_3,b_4$. We modify the fundamental system of solutions to ${ℱ}$ given in [A system of hypergeometric differential equations in two variables of rank 9, Internat. J. Math. 28 (2017), 1750015, 34 pp] so that it is valid even in cases where $b_1,b_2,b_3,b_4$ satisfy some integral conditions. By using this fundamental system, we show the irreducibility of the monodromy representation of ${ℱ}$ under some conditions on the parameters. We characterize the fundamental group of the base space of this local system as the group generated by three loops with four relations among them.

For each integer $n\ge 2$, we identify new infinite families of monogenic trinomials $f(x)=x^n+A{x}^m+B$ with non-squarefree discriminant, many of which have small Galois group. Moreover, in certain situations when $A=B\ge 2$ with fixed $n$ and $m$, we produce asymptotics on the number of such trinomials with $A\le X$.

We show that there are infinitely many two component links in S³ whose complements have weakly reducible and irreducible non-minimal genus Heegaard splittings, yet the construction given in the theorem of Casson and Gordon does not produce an essential closed surface. The situation for manifolds with a single boundary component is still unresolved though we obtain partial results regarding manifolds with a non-minimal genus weakly reducible and irreducible Heegaard splitting.

Three disjoint rays in ℝ³ form Borromean rays provided their union is knotted, but the union of any two components is unknotted. We construct infinitely many Borromean rays, uncountably many of which are pairwise inequivalent. We obtain uncountably many Borromean hyperplanes.

We prove that if K⊂ S³ is either: (I) a link with an essential n-string arcbody decomposition, where at least one arcspace has incompressible boundary, or a knot with an essential n-string tangle decomposition, where (II) each tangle has no parallel strings, or (III) one tangle space is not a handlebody and K is not cabled, then any nontrivial surgery on every component of K produces irreducible manifolds in all cases (with some exceptional surgeries in case (I)) and, in particular, Haken manifolds in cases (I) and (III). Moreover, if K is hyperbolic in (III) and at least one tangle space has incompressible boundary, then all nontrivial surgeries on K are also hyperbolic; this last result is also established for type (I) decompositions under some constraints.

We report a rare case of irreducible chronic palmar dislocation of the distal radioulnar joint (DRUJ). This case showed that the dislocated ulnar head was impacted to the palmar cortex of the radius probably due to the dynamic force of the pronator quadratus muscle. Re-attachment of the ulnar styloid and partial resection of the ulnar head were necessary to make the reduction of the DRUJ possible. The continuity of the radioulnar ligament to the ulnar head was restored and the stability of DRUJ was maintained after reduction.

Irreducible volar subluxation should be considered when assessing a patient with flexion deformity of the proximal interphalangeal finger joint (PIPJ). Primary assessment requires careful examination of the collateral ligaments and extensor tendon. Preoperative imaging such as ultrasound and MRI can help identify the interposed structures and plan the subsequent operation. Although rare, irreducible volar subluxation due to radial collateral ligament interposition is an important entity to be aware of. Prompt and appropriate management can prevent joint stiffness and loss of function.

The generator of a Gaussian quantum Markov semigroup on the algebra of bounded operator on a $d$-mode Fock space is represented in a generalized GKLS form with an operator $G$ quadratic in creation and annihilation operators and Kraus operators $L_1,\dots ,L_m$ linear in creation and annihilation operators. Kraus operators, commutators $[G,L_{\ell }]$ and iterated commutators $[G,[G,L_{\ell }]],\dots$ up to the order $2d-m$, as linear combinations of creation and annihilation operators determine a vector in ${ℂ}^{2d}$. We show that a Gaussian quantum Markov semigroup is irreducible if such vectors generate ${ℂ}^{2d}$, under the technical condition that the domains of $G$ and the number operator coincide. Conversely, we show that this condition is also necessary if the linear space generated by Kraus operators and their iterated commutator with $G$ is fully non-commutative.

In this paper we study prime preradicals, irreducible preradicals, ∧-prime preradicals, prime submodules and diuniform modules. We study some relations between these concepts, using the lattice structure of preradicals developed in previous papers. In particular, we give a characterization of prime preradicals using an operator named the relative annihilator. We also characterize prime submodules by means of prime preradicals. We give some characterizations of rings that have certain conditions on prime radicals and on irreducible preradicals, such as left local left V-rings, as well as 1-spr rings, which we introduce.

Let $C_n$ denote the cyclic group of order $n$, and let Hol($C_n$) denote the holomorph of $C_n$. In this paper, for any odd integer $m\ge 3$, we find necessary and sufficient conditions on an integer $A$, with $\left|A\right|\ge 3$, such that ${{ℱ}}_{m,A}(x)=x^{2m}+A{x}^m+1$ is irreducible over $\mathbb{Q}$. When $m=q\ge 3$ is prime and ${{ℱ}}_{q,A}(x)$ is irreducible, we show that the Galois group over $\mathbb{Q}$ of ${{ℱ}}_{q,A}(x)$ is isomorphic to either Hol($C_q$) or Hol($C_{2q}$), depending on whether there exists $y\in \mathbb{Z}$ such that $A^2-4=q{y}^2$. Finally, we prove that there exist infinitely many positive integers $A$ such that ${{ℱ}}_{q,A}(x)$ is irreducible over $\mathbb{Q}$ and that $\{1,𝜃,{𝜃}^2,\dots ,{𝜃}^{2q-1}\}$ is a basis for the ring of integers of $K=\mathbb{Q}(𝜃)$, where ${{ℱ}}_{q,A}(𝜃)=0$.

Jędrzejewicz showed that a polynomial map over a field of characteristic zero is invertible, if and only if the corresponding endomorphism maps irreducible polynomials to irreducible polynomials. Furthermore, he showed that a polynomial map over a field of characteristic zero is a Keller map, if and only if the corresponding endomorphism maps irreducible polynomials to square-free polynomials. We show that the latter endomorphism maps other square-free polynomials to square-free polynomials as well.

In connection with the above classification of invertible polynomial maps and the Jacobian Conjecture, we study irreducibility properties of several types of Keller maps, to each of which the Jacobian Conjecture can be reduced. Herewith, we generalize the result of Bakalarski that the components of cubic homogeneous Keller maps with a symmetric Jacobian matrix (over ℂ and hence any field of characteristic zero) are irreducible.

Furthermore, we show that the Jacobian Conjecture can even be reduced to any of these types with the extra condition that each affinely linear combination of the components of the polynomial map is irreducible. This is somewhat similar to reducing the planar Jacobian Conjecture to the so-called (planar) weak Jacobian Conjecture by Kaliman.

In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k · 2ⁿ + 1 is composite for all integers n ≥ 0. Variations of this problem, using polynomials with integer coefficients, and considering reducibility over the rationals, have been investigated by several authors. In particular, if S is the set of all positive integers d for which there exists a polynomial f(x) ∈ ℤ[x], with f(1) ≠ -d, such that f(x)xⁿ + d is reducible over the rationals for all integers n ≥ 0, then what are the elements of S? Interest in this problem stems partially from the fact that if S contains an odd integer, then a question of Erdös and Selfridge concerning the existence of an odd covering of the integers would be resolved. Filaseta has shown that S contains all positive integers d ≡ 0 (mod 4), and until now, nothing else was known about the elements of S. In this paper, we show that S contains infinitely many positive integers d ≡ 6 (mod 12). We also consider the corresponding problem over 𝔽_p, and in that situation, we show 1 ∈ S for all primes p.

In this paper we investigate the factorization of trinomials of the form xⁿ + cx^n-1 + d ∈ ℤ[x]. We then use these results about trinomials to prove results about the factorization of polynomials of the form xⁿ + c(x^n-1 +⋯+ x + 1) ∈ ℤ[x].

Given relatively prime polynomials f(x) and g(x) in ℤ[x] with non-zero constant terms, we show that for n greater than an explicitly determined bound depending on f(x) and g(x), if the polynomial f(x)xⁿ + g(x) is non-reciprocal, then its non-cyclotomic part is irreducible except for some explicit cases where a known factorization of f(x)xⁿ + g(x) can easily be described. Prior work of a similar nature is discussed which shows under similar circumstances the non-reciprocal part off(x)xⁿ + g(x) is irreducible. The current paper establishes and makes use of a result which shows that a reciprocal polynomial f(x) with a root off the unit circle must have a root bounded away from the unit circle by an explicitly given function of the degree of f(x), the leading coefficient a of f(x) and the discriminant of f(x). Notably in this result, a need not be 1.

In 1908, Schur raised the question of the irreducibility over ℚ of polynomials of the form f(x) = (x - a₁)(x - a₂)⋯(x - a_n) + 1, where the a_i are distinct integers. Since then, many authors have addressed variations and generalizations of this question. In this article, we investigate the analogous question when replacing the linear polynomials with cyclotomic polynomials and allowing the constant perturbation of the product to be any integer d ∉ {-1, 0}. One interesting consequence of our investigations is that we are able to construct, for any positive integer N, an infinite set S of cyclotomic polynomials such that 1 plus the product of any k (not necessarily distinct) polynomials from S, where k ≢ 0(mod 2^N+1), is reducible over ℚ.

Ideal flow network is a strongly connected network with flow, where the flows are in steady state and conserved. The matrix of ideal flow is premagic, where vector, the sum of rows, is equal to the transposed vector containing the sum of columns. The premagic property guarantees the flow conservation in all nodes. The scaling factor as the sum of node probabilities of all nodes is equal to the total flow of an ideal flow network. The same scaling factor can also be applied to create the identical ideal flow network, which has from the same transition probability matrix. Perturbation analysis of the elements of the stationary node probability vector shows an insight that the limiting distribution or the stationary distribution is also the flow-equilibrium distribution. The process is reversible that the Markov probability matrix can be obtained from the invariant state distribution through linear algebra of ideal flow matrix. Finally, we show that recursive transformation $F_k\to F_{k+1}$ to represent $k$-vertices path-tracing also preserved the properties of ideal flow, which is irreducible and premagic.

We describe 4 cases of irreducible volar rotatory subluxation of the proximal interphalangeal (PIP) joint of the finger that required open reduction. All of the patients had radiographically proven (in lateral-view radiographs) volar rotatory subluxation of the PIP joint, without fracture. The causes of irreducibility were interposition of the lateral band about the condyle of the middle phalanx in 2 cases, interposition of the collateral ligament in 1 case, and scarring of the injured central slip in 1 case. Rupture of the collateral ligament of one side was found in all cases. Acceptable results were provided with all cases after restoration of the collateral ligaments and the damaged parts. Accurate early diagnosis by careful physical examination and obtaining true lateral radiographs of the PIP joint is important.