Generation in singularity categories of hypersurfaces of countable representation type

The Orlov spectrum and Rouquier dimension are invariants of a triangulated category to measure how big the category is, and they have been studied actively. In this paper, we investigate the singularity category $$\textsf {D} _{\textsf {sg} }(R)$$ of a hypersurface R of countable representation type. For a thick subcategory $${\mathcal {T}}$$ of $$\textsf {D} _{\textsf {sg} }(R)$$ and a full subcategory $$\mathcal {X}$$ of $${\mathcal {T}}$$, we calculate the Rouquier dimension of $${\mathcal {T}}$$ with respect to $$\mathcal {X}$$. Furthermore, we prove that the level in $$\textsf {D} _{\textsf {sg} }(R)$$ of the residue field of R with respect to each nonzero object is at most one.     

Weak finitely regular relations and applications

In this note, we introduce the concepts of weak finitely regular relations and weak hypercontinuous posets. It is proved that when a binary relation $$\rho :X/\rightarrow Y$$ satisfies a certain condition, $$\rho $$ is weak finitely regular if and only if $$(\varphi _{\rho }(X,Y),\subseteq )$$ is a weak hypercontinuous poset. For a poset P, we get that the relation $$\not \le $$ on P is weak finitely regular if and only if P is weak hypercontinuous.     

Remarks on Rawnsley’s $$\varvec{\varepsilon }$$ε -function on the Fock–Bargmann–Hartogs domains

In this paper, we mainly study a family of unbounded non-hyperbolic domains in $$\mathbb {C}^{n+m}$$, called Fock–Bargmann–Hartogs domains $$D_{n,m}(\mu )$$ ($$\mu >0$$) which are defined as a Hartogs type domains with the fiber over each $$z\in \mathbb {C}^{n}$$ being a ball of radius $$e^{-\frac{\mu }{2} {\Vert z\Vert }^{2}}$$. The purpose of this paper is twofold. Firstly, we obtain necessary and sufficient conditions for Rawnsley’s $$\varepsilon $$-function $$\varepsilon _{(\alpha ,g)}(\widetilde{w})$$ of $$\big (D_{n,m}(\mu ), g(\mu ;\nu )\big )$$ to be a polynomial in $$\Vert \widetilde{w}\Vert ^2$$, where $$g(\mu ;\nu )$$ is a Kähler metric associated with the Kähler potential $$\nu \mu {\Vert z\Vert }^{2} -\ln (e^{-\mu {\Vert z\Vert }^{2}}-\Vert w\Vert ^2)$$. Secondly, using above results, we study the Berezin quantization on $$D_{n,m}(\mu )$$ with the metric $$\beta g(\mu ;\nu )$$$$(\beta >0)$$.     

Orders generated by character values

Let $$K:=\mathbb {Q}(G)$$ be the number field generated by the complex character values of a finite group G. Let $$\mathbb {Z}_K$$ be the ring of integers of K. In this paper we investigate the suborder $$\mathbb {Z}[G]$$ of $$\mathbb {Z}_K$$ generated by the character values of G. We prove that every prime divisor of the order of the finite abelian group $$\mathbb {Z}_K/\mathbb {Z}[G]$$ divides |G|. Moreover, if G is nilpotent, we show that the exponent of $$\mathbb {Z}_K/\mathbb {Z}[G]$$ is a proper divisor of |G| unless $$G=1$$. We conjecture that this holds for arbitrary finite groups G.     

Diophantine quadruples in $$\mathbb {Z}[i][X]$$ Z [ i ] [ X ]

Periodica Mathematica Hungarica - In this paper, we prove that every Diophantine quadruple in $$\mathbb {Z}[i][X]$$ is regular. More precisely, we prove that if $$\{a, b, c, d\}$$ is a set of four...     

Partitions of the complete hypergraph $$K_6^3$$ and a determinant-like function

Journal of Algebraic Combinatorics - In this paper, we introduce a determinant-like map $$\mathrm{det}^{S^3}$$ and study some of its properties. For this, we define a graded vector space $$\Lambda...     

A new example of an algebraic surface with canonical map of degree 16

Archiv der Mathematik - In this note, we construct a minimal surface of general type with geometric genus $$ p_g =4 $$ , self-intersection of the canonical divisor $$ K^2 = 32$$ , and irregularity...     

Sigma-subdifferential and its application to minimization problem

Positivity - In this paper, we study $$\sigma $$ -subdifferentials of $$\sigma $$ -convex functions. Two equivalent conditions for $$\sigma $$ -convexity are given. The formula for the $$\sigma $$...     

A locally convex version of Kadison’s representation theorem

Positivity - In this paper, we study the local ordered $$*$$ -vector spaces and their representations. We prove that each Archimedean local ordered $$*$$ -vector space, can be represented as a...     

A class of twisted generalized Reed–Solomon codes

Designs, Codes and Cryptography - Let $${\mathbb F}_{q}$$ be a finite field of size q and $${\mathbb F}_{q}^*$$ the set of non-zero elements of $${\mathbb F}_{q}$$ . In this paper, we study a class...     

Complete Hypersurfaces with Two Distinct Principal Curvatures in a Space Form

Results in Mathematics - In this paper, we study complete hypersurfaces M n immersed in a space form $${\mathbb{Q}_c^{n+1}}$$ , with $${c \in \{-1,0,1\}}$$ and $${n \geq 2}$$ , having two distinct...     

The Langlands parameter of a simple supercuspidal representation: symplectic groups

The Ramanujan Journal - Let $$\pi $$ be a simple supercuspidal representation of the symplectic group $${\mathrm {Sp}}_{2l}(F)$$, over a p-adic field F. In this work, we explicitly compute the...     

The number of representations of integers by 4-dimensional strongly N-modular lattices

The Ramanujan Journal - In this article, we prove that a space of cusp forms of weight 2 with level N and real character $$\chi $$ has dimension 1 if and only if $$\chi $$ is trivial and N is in...     

The $${\mathcal {F}}$$ F -transitivity and recurrence of translation semigroups on complex sectors

Semigroup Forum - In this paper, we study the $${\mathcal {F}}$$ -transitive behaviour of the translation semigroups on complex sectors, where $${\mathcal {F}}$$ is a Furstenberg family of the...     

Ideal class groups of imaginary quadratic fields

The Ramanujan Journal - Let d be a square-free positive integer and $$\mathrm{CL}(-d)$$ the ideal class group of the imaginary quadratic field $${\mathbb {Q}}(\sqrt{-d})$$ . In this paper, we show...     

Some notes on the multiplicative order of $$\alpha + \alpha ^{-1}$$α+α-1 in finite fields of characteristic two

Periodica Mathematica Hungarica - In this paper we prove some results on the possible multiplicative order of $$\alpha + \alpha ^{-1}$$ when $$\alpha $$ is a non-zero element of a finite field of...     

The universal minimal one parameter system with quasi-discrete spectrum

The universal minimal one parameter system will be characterized as the space $$\Gamma ^{\infty }$$, in which $$\Gamma$$ is the Bohr compactification of the additive group $${\mathbb {R}}$$ of real numbers. In this way, we need to show that $$\Gamma ^\infty$$ is isomorphic to the spectrum of $$W({\mathbb {R}})$$, the norm closure of the invariant algebra generated by the maps $$\exp q(t)$$, where q(t) is a real polynomial on $${\mathbb {R}}$$.     

On the Exponential Diophantine Equation $$(a^{n}-2)(b^{n}-2)=x^{2}$$

Mathematical Notes - In this paper, we deal with the equation $$(a^{n}-2)(b^{n}-2)=x^{2}$$ , $$2\leq a3$$ is odd and $$P_{k},Q_{k}$$ are the Pell and Pell Lucas numbers, respectively. We also...     

On weakening tightness to weak tightness

The weak tightness wt(X) of a space X was introduced in Carlson (Topol Appl 249:103–111, 2018) with the property $$wt(X)\le t(X)$$. We investigate several well-known results concerning t(X) and consider whether they extend to the weak tightness setting. First we give an example of a non-sequential compactum X such that $$wt(X)=\aleph _0<t(X)$$ under $$2^{\aleph _0}=2^{\aleph _1}$$. In particular, this demonstrates the celebrated Balogh’s (Proc Am Math Soc 105(3):755–764, 1989) Theorem does not hold in general if countably tight is replaced with weakly countably tight. Second, we introduce the notion of an S-free sequence and show that if X is a homogeneous compactum then $$|X|\le 2^{wt(X)\pi \chi (X)}$$. This refines a theorem of de la Vega (Topol Appl 153:2118–2123, 2006). In the case where the cardinal invariants involved are countable, this also represents a variation of a theorem of Juhász and van Mill (Proc Am Math Soc 146(1):429–437, 2018). In this connection we also show $$w(X)\le 2^{wt(X)}$$ for a homogeneous compactum. Third, we show that if X is a $$T_1$$ space, $$wt(X)\le \kappa $$, X is $$\kappa ^+$$-compact, and $$\psi (\overline{D},X)\le 2^\kappa $$ for any $$D\subseteq X$$ satisfying $$|D|\le 2^\kappa $$, then (a) $$d(X)\le 2^\kappa $$ and (b) X has at most $$2^\kappa $$-many $$G_\kappa $$-points. This is a variation of another theorem of Balogh (Topol Proc 27:9–14, 2003). Finally, we show that if X is a regular space, $$\kappa =L(X)wt(X)$$, and $$\lambda $$ is a caliber of X satisfying $$\kappa <\lambda \le \left( 2^{\kappa }\right) ^+$$, then $$d(X)\le 2^{\kappa }$$. This extends of theorem of Arhangel$$'$$skiĭ (Topol Appl 104:13–26, 2000).