Reconstruction Algorithms for a Class of Restricted Ray Transforms Without Added Singularities

Let X and \(X^*\) denote a restricted ray transform along curves and a corresponding backprojection operator, respectively. Theoretical analysis of reconstruction from the data Xf is usually based on a study of the composition \(X^* D X\), where D is some local operator (usually a derivative). If \(X^*\) is chosen appropriately, then \(X^* D X\) is a Fourier integral operator (FIO) with singular symbol. The singularity of the symbol leads to the appearance of artifacts (added singularities) that can be as strong as the original (or, useful) singularities. By choosing D in a special way one can reduce the strength of added singularities, but it is impossible to get rid of them completely. In the paper we follow a similar approach, but make two changes. First, we replace D with a nonlocal operator \(\tilde{D}\) that integrates Xf along a curve in the data space. The result \(\tilde{D} Xf\) resembles the generalized Radon transform R of f. The function \(\tilde{D} Xf\) is defined on pairs \((x_0,\Theta )\in U\times S^2\), where \(U\subset {\mathbb R}^3\) is an open set containing the support of f, and \(S^2\) is the unit sphere in \({\mathbb R}^3\). Second, we replace \(X^*\) with a backprojection operator \(R^*\) that integrates with respect to \(\Theta \) over \(S^2\). It turns out that if \(\tilde{D}\) and \(R^*\) are appropriately selected, then the composition \(R^* \tilde{D} X\) is an elliptic pseudodifferential operator of order zero with principal symbol 1. Thus, we obtain an approximate reconstruction formula that recovers all the singularities correctly and does not produce artifacts. The advantage of our approach is that by inserting \(\tilde{D}\) we get access to the frequency variable \(\Theta \). In particular, we can incorporate suitable cut-offs in \(R^*\) to eliminate bad directions \(\Theta \), which lead to added singularities.     

The distortion dimension of $$\mathbb $$-rank 1 lattices

Let \(X=G/K\) be a symmetric space of noncompact type and rank \(k\ge 2\). We prove that horospheres in X are Lipschitz \((k-2)\)-connected if their centers are not contained in a proper join factor of the spherical building of X at infinity. As a consequence, the distortion dimension of an irreducible \(\mathbb {Q}\)-rank-1 lattice \(\Gamma \) in a linear, semisimple Lie group G of \(\mathbb R\)-rank k is \(k-1\). That is, given \(m< k-1\), a Lipschitz m-sphere S in (a polyhedral complex quasi-isometric to) \(\Gamma \), and a \((m+1)\)-ball B in X (or G) filling S, there is a \((m+1)\)-ball \(B'\) in \(\Gamma \) filling S such that \({{\mathrm{vol}}}B'\sim {{\mathrm{vol}}}B\). In particular, such arithmetic lattices satisfy Euclidean isoperimetric inequalities up to dimension \(k-1\).     

Generating Hyperbolic Singularities in Semitoric Systems Via Hopf Bifurcations

Let \((M,\Omega )\) be a connected symplectic 4-manifold and let \(F=(J,H) :M\rightarrow \mathbb {R}^2\) be a completely integrable system on M with only non-degenerate singularities. Assume that F does not have singularities with hyperbolic blocks and that \(p_1,\ldots ,p_n\) are the focus–focus singularities of F. For each subset \(S=\{i_1,\ldots ,i_j\}\), we will show how to modify F locally around any \(p_i, i \in S\), in order to create a new integrable system \(\widetilde{F}=(J, \widetilde{H}) :M \rightarrow \mathbb {R}^2\) such that its classical spectrum \(\widetilde{F}(M)\) contains j smooth curves of singular values corresponding to non-degenerate transversally hyperbolic singularities of \(\widetilde{F}\). Moreover the focus–focus singularities of \(\widetilde{F}\) are precisely \(p_i\), \(i \in \{1,\ldots ,n\} \setminus S\). The proof is based on Eliasson’s linearization theorem for non-degenerate singularities, and properties of the Hamiltonian Hopf bifurcation.     

Fixed point compactifications

A fixed point compactification of a locally compact noncompact group G is a faithful semigroup compactification S such that \(ap=pa=p\) for all \(p\in S\setminus G\) and \(a\in G\). Since the right translations are continuous, the remainder of a fixed point compactification is a right zero semigroup. Among all fixed point compactifications of G there is a largest one, denoted \(\theta G\). We show that if G is \(\sigma \)-compact, then \(\theta G\setminus G\) contains a copy of \(\beta \omega \setminus \omega \). In contrast, if G is not \(\sigma \)-compact, then \(\theta G\) is the one-point compactification.     

Preservation of semistability under Fourier–Mukai transforms

For a trivial elliptic fibration \(X=C \times S\) with C an elliptic curve and S a projective K3 surface of Picard rank 1, we study how various notions of stability behave under the Fourier–Mukai autoequivalence \(\Phi \) on \(D^b(X)\), where \(\Phi \) is induced by the classical Fourier–Mukai autoequivalence on \(D^b(C)\). We show that, under some restrictions on Chern classes, Gieseker semistability on coherent sheaves is preserved under \(\Phi \) when the polarisation is ‘fiber-like’. Moreover, for more general choices of Chern classes, Gieseker semistability under a ‘fiber-like’ polarisation corresponds to a notion of \(\mu _*\)-semistability defined by a ‘slope-like’ function \(\mu _*\).     

Torsion and divisibility in finitely generated commutative semirings

It is conjectured that (additive) divisibility is equivalent to (additive) idempotency in a finitely generated commutative semiring S. In this paper we extend this conjecture to weaker forms of these properties—torsion and almost-divisibility (an element \(a\in S\) is called almost-divisible in S if there is \(b\in \mathbb {N}\cdot a\) such that b is divisible in S by infinitely many primes). We show that a one-generated semiring is almost-divisible if and only if it is torsion. In the case of a free commutative semiring F(X) we characterize those elements \(f\in F(X)\) such that for every epimorphism \(\pi \) of F(X) torsion and almost-divisibility of \(\pi (f)\) are equivalent in \(\pi (F(X))\).     

A characterization of the Riemann extension in terms of harmonicity

If (M,?) is a manifold with a symmetric linear connection, then T*M can be endowed with the natural Riemann extension \(\bar g\) (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to \(\bar g\) initiated by C. L.Bejan and O.Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure \(\mathcal{P}\) on (T*M, \(\bar g\)) and prove that \(\mathcal{P}\) is harmonic (in the sense of E.Garciá-Río, L.Vanhecke and M. E.Vázquez-Abal (1997)) if and only if \(\bar g\) reduces to the classical Riemann extension introduced by E.M. Patterson and A.G. Walker (1952).     

An <Emphasis Type="Italic">A</Emphasis>-invariant subspace for bipartite distance-regular graphs with exactly two irreducible <Emphasis Type="Italic">T</Emphasis>-modules with endpoint 2, both thin

Let \(\Gamma \) denote a bipartite distance-regular graph with vertex set X, diameter \(D \ge 4\), and valency \(k \ge 3\). Let \({{\mathbb {C}}}^X\) denote the vector space over \({{\mathbb {C}}}\) consisting of column vectors with entries in \({{\mathbb {C}}}\) and rows indexed by X. For \(z \in X\), let \({{\widehat{z}}}\) denote the vector in \({{\mathbb {C}}}^X\) with a 1 in the z-coordinate, and 0 in all other coordinates. Fix a vertex x of \(\Gamma \) and let \(T = T(x)\) denote the corresponding Terwilliger algebra. Assume that up to isomorphism there exist exactly two irreducible T-modules with endpoint 2, and they both are thin. Fix \(y \in X\) such that \(\partial (x,y)=2\), where \(\partial \) denotes path-length distance. For \(0 \le i,j \le D\) define \(w_{ij}=\sum {{\widehat{z}}}\), where the sum is over all \(z \in X\) such that \(\partial (x,z)=i\) and \(\partial (y,z)=j\). We define \(W=\mathrm{span}\{w_{ij} \mid 0 \le i,j \le D\}\). In this paper we consider the space \(MW=\mathrm{span}\{mw \mid m \in M, w \in W\}\), where M is the Bose–Mesner algebra of \(\Gamma \). We observe that MW is the minimal A-invariant subspace of \({{\mathbb {C}}}^X\) which contains W, where A is the adjacency matrix of \(\Gamma \). We show that \(4D-6 \le \mathrm{dim}(MW) \le 4D-2\). We display a basis for MW for each of these five cases, and we give the action of A on these bases.     

A note on Wilson’s functional equation

Let S be a semigroup, and \(\mathbb {F}\) a field of characteristic \(\ne 2\). If the pair \(f,g:S \rightarrow \mathbb {F}\) is a solution of Wilson’s \(\mu \)-functional equation such that \(f \ne 0\), then g satisfies d’Alembert’s \(\mu \)-functional equation.     

Generic Vopěnka’s Principle,remarkable cardinals,and the weak Proper Forcing Axiom

We introduce and study the first-order Generic Vopěnka’s Principle, which states that for every definable proper class of structures \(\mathcal {C}\) of the same type, there exist \(B\ne A\) in \(\mathcal {C}\) such that B elementarily embeds into A in some set-forcing extension. We show that, for \(n\ge 1\), the Generic Vopěnka’s Principle fragment for \(\Pi _n\)-definable classes is equiconsistent with a proper class of n-remarkable cardinals. The n-remarkable cardinals hierarchy for \(n\in \omega \), which we introduce here, is a natural generic analogue for the \(C^{(n)}\)-extendible cardinals that Bagaria used to calibrate the strength of the first-order Vopěnka’s Principle in Bagaria (Arch Math Logic 51(3–4):213–240, 2012). Expanding on the theme of studying set theoretic properties which assert the existence of elementary embeddings in some set-forcing extension, we introduce and study the weak Proper Forcing Axiom, \(\mathrm{wPFA}\). The axiom \(\mathrm{wPFA}\) states that for every transitive model \(\mathcal M\) in the language of set theory with some \(\omega _1\)-many additional relations, if it is forced by a proper forcing \(\mathbb P\) that \(\mathcal M\) satisfies some \(\Sigma _1\)-property, then V has a transitive model \(\bar{\mathcal M}\), satisfying the same \(\Sigma _1\)-property, and in some set-forcing extension there is an elementary embedding from \(\bar{\mathcal M}\) into \(\mathcal M\). This is a weakening of a formulation of \(\mathrm{PFA}\) due to Claverie and Schindler (J Symb Logic 77(2):475–498, 2012), which asserts that the embedding from \(\bar{\mathcal M}\) to \(\mathcal M\) exists in V. We show that \(\mathrm{wPFA}\) is equiconsistent with a remarkable cardinal. Furthermore, the axiom \(\mathrm{wPFA}\) implies \(\mathrm{PFA}_{\aleph _2}\), the Proper Forcing Axiom for antichains of size at most \(\omega _2\), but it is consistent with \(\square _\kappa \) for all \(\kappa \ge \omega _2\), and therefore does not imply \(\mathrm{PFA}_{\aleph _3}\).     

On the Multiplication Tuples Related to Certain Reproducing Kernel Hilbert Spaces

We consider a class of subnormal operator tuples \(M_z\) consisting of multiplications by coordinate functions on a class of reproducing kernel Hilbert spaces associated with certain bounded domains \(\Omega \) in \({\mathbb {C}}^m\), with the closure \({{\bar{\Omega }}}\) of \(\Omega \) being the Taylor spectrum of \(M_z\) and the topological boundary \(\partial \Omega \) of \(\Omega \) being the Taylor essential spectrum of \(M_z\). If T is a subnormal operator tuple quasisimilar to \(M_z\), then we show that the Taylor spectrum of T is \({{\bar{\Omega }}}\) provided \({{\bar{\Omega }}}\) is polynomially convex and provided \(\Omega \) is either strictly pseudoconvex with \(C^2\) boundary or is starlike, and that the Taylor essential spectrum of T is \(\partial \Omega \) provided \(\Omega \) satisfies the Gleason property as well. This generalizes some previous work of the first-named author in the context of the unit ball and the unit polydisk. The relevant theory is then applied to the multiplication tuples on the Hardy and Bergman spaces of complex ellipsoids.     

Algebraic properties of Zappa–Szép products of semigroups and monoids

Direct, semidirect and Zappa–Szép products provide tools to decompose algebraic structures, with each being a natural generalisation of its predecessor. In this paper we examine Zappa–Szép products of monoids and semigroups and investigate generalised Greens relations \({\mathcal R}^{*},\, {\mathcal L}^{*},\, \widetilde{\mathcal {R}}_E\) and \(\widetilde{\mathcal {L}}_E\) for these Zappa–Szép products. We consider a left restriction semigroup S with semilattice of projections E and define left and right actions of S on E and E on S, respectively, to form the Zappa–Szép product \(E \bowtie S\). We further investigate properties of \(E \bowtie S\) and show that S is a retract of \(E\bowtie S\). We also find a subset T of \(E \bowtie S\) which is left restriction.     

On the Diophantine equation $$X^{2N}+2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5$$

We prove that for each prime p, positive integer \(\alpha \), and non-negative integers \(\beta \) and \(\gamma \), the Diophantine equation \(X^{2N} + 2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5\) has no solution with N, X, \(Z\in \mathbb {Z}^+\), \(N > 1\), and \(\gcd (X,Z) = 1\).     

The Terwilliger algebra of the incidence graph of the Hamming graph

Let \(\varGamma \) be a distance-semiregular graph on Y, and let \(D^Y\) be the diameter of \(\varGamma \) on Y. Let \(\varDelta \) be the halved graph of \(\varGamma \) on Y. Fix \(x \in Y\). Let T and \(T'\) be the Terwilliger algebras of \(\varGamma \) and \(\varDelta \) with respect to x, respectively. Assume, for an integer i with \(1 \le 2i \le D^Y\) and for \(y,z \in \varGamma _{2i}(x)\) with \(\partial _{\varGamma }(y,z)=2\), the numbers \(|\varGamma _{2i-1}(x) \cap \varGamma (y) \cap \varGamma (z)|\) and \(|\varGamma _{2i+1}(x) \cap \varGamma (y) \cap \varGamma (z)|\) depend only on i and do not depend on the choice of y, z. The first goal in this paper is to show the relations between T-modules of \(\varGamma \) and \(T'\)-modules of \(\varDelta \). Assume \(\varGamma \) is the incidence graph of the Hamming graph H(D, n) on the vertex set Y and the set \({\mathcal {C}}\) of all maximal cliques. Then, \(\varGamma \) satisfies above assumption and \(\varDelta \) is isomorphic to H(D, n). The second goal is to determine the irreducible T-modules of \(\varGamma \). For each irreducible T-module W, we give a basis for W the action of the adjacency matrix on this basis and we calculate the multiplicity of W.     

On some constacyclic codes over $$\mathbb {Z}_{4}\left[ u\right] /\left\langle u^{2}-1\right\rangle $$, their $$\mathbb {Z}_4$$Z4 images,and new codes

In this paper, we study \(\lambda \)-constacyclic codes over the ring \(R=\mathbb {Z}_4+u\mathbb {Z}_4\) where \(u^{2}=1\), for \(\lambda =3+2u\) and \(2+3u\). Two new Gray maps from R to \(\mathbb {Z}_4^{3}\) are defined with the goal of obtaining new linear codes over \(\mathbb {Z}_4\). The Gray images of \(\lambda \)-constacyclic codes over R are determined. We then conducted a computer search and obtained many \(\lambda \)-constacyclic codes over R whose \(\mathbb {Z}_4\)-images have better parameters than currently best-known linear codes over \(\mathbb {Z}_4\).     

Simultaneous power factorization in modules over Banach algebras

Let A be a Banach algebra with a bounded left approximate identity \(\{e_\lambda \}_{\lambda \in \Lambda }\), let \(\pi \) be a continuous representation of A on a Banach space X, and let S be a non-empty subset of X such that \(\lim _{\lambda }\pi (e_\lambda )s=s\) uniformly on S. If S is bounded, or if \(\{e_\lambda \}_{\lambda \in \Lambda }\) is commutative, then we show that there exist \(a\in A\) and maps \(x_n: S\rightarrow X\) for \(n\ge 1\) such that \(s=\pi (a^n)x_n(s)\) for all \(n\ge 1\) and \(s\in S\). The properties of \(a\in A\) and the maps \(x_n\), as produced by the constructive proof, are studied in some detail. The results generalize previous simultaneous factorization theorems as well as Allan and Sinclair’s power factorization theorem. In an ordered context, we also consider the existence of a positive factorization for a subset of the positive cone of an ordered Banach space that is a positive module over an ordered Banach algebra with a positive bounded left approximate identity. Such factorizations are not always possible. In certain cases, including those for positive modules over ordered Banach algebras of bounded functions, such positive factorizations exist, but the general picture is still unclear. Furthermore, simultaneous pointwise power factorizations for sets of bounded maps with values in a Banach module (such as sets of bounded convergent nets) are obtained. A worked example for the left regular representation of \(\mathrm {C}_0({\mathbb R})\) and unbounded S is included.     

Upper Bounds for the Paired-Domination Numbers of Graphs

A set \(S\subseteq V\) is a paired-dominating set if every vertex in \(V{\setminus } S\) has at least one neighbor in S and the subgraph induced by S contains a perfect matching. The paired-domination number of a graph G, denoted by \(\gamma _{pr}(G)\), is the minimum cardinality of a paired-dominating set of G. A conjecture of Goddard and Henning says that if G is not the Petersen graph and is a connected graph of order n with minimum degree \(\delta (G)\ge 3\), then \(\gamma _{pr}(G)\le 4n/7\). In this paper, we confirm this conjecture for k-regular graphs with \(k\ge 4\).     

Geometric Constructibility of Polygons Lying on a Circular Arc

For a positive integer n, an n-sided polygon lying on a circular arc or, shortly, an n-fan is a sequence of \(n+1\) points on a circle going counterclockwise such that the “total rotation” \(\delta \) from the first point to the last one is at most \(2\pi \). We prove that for \(n\ge 3\), the n-fan cannot be constructed with straightedge and compass in general from its central angle \(\delta \) and its central distances, which are the distances of the edges from the center of the circle. Also, we prove that for each fixed \(\delta \) in the interval \((0, 2\pi ]\) and for every \(n\ge 5\), there exists a concrete n-fan with central angle \(\delta \) that is not constructible from its central distances and \(\delta \). The present paper generalizes some earlier results published by the second author and Á. Kunos on the particular cases \(\delta =2\pi \) and \(\delta =\pi \).     

Pairs of Semisimple Lie Algebras and their Maximal Reductive Subalgebras

Let \(\mathfrak g\) be a semisimple Lie algebra over a field \(\mathbb K\), \(\text{char}\left( \mathbb{K} \right)=0\), and \(\mathfrak g_1\) a subalgebra reductive in \(\mathfrak g\). Suppose that the restriction of the Killing form B of \(\mathfrak g\) to \(\mathfrak g_1 \times \mathfrak g_1\) is nondegenerate. Consider the following statements: ( 1) For any Cartan subalgebra \(\mathfrak h_1\) of \(\mathfrak g_1\) there is a unique Cartan subalgebra \(\mathfrak h\) of \(\mathfrak g\) containing \(\mathfrak h_1\); ( 2) \(\mathfrak g_1\) is self-normalizing in \(\mathfrak g\); ( 3) The B-orthogonal \(\mathfrak p\) of \(\mathfrak g_1\) in \(\mathfrak g\) is simple as a \(\mathfrak g_1\)-module for the adjoint representation. We give some answers to this natural question: For which pairs \((\mathfrak g,\mathfrak g_1)\) do ( 1), ( 2) or ( 3) hold? We also study how \(\mathfrak p\) in general decomposes as a \(\mathfrak g_1\)-module, and when \(\mathfrak g_1\) is a maximal subalgebra of \(\mathfrak g\). In particular suppose \((\mathfrak g,\sigma )\) is a pair with \(\mathfrak g\) as above and σ its automorphism of order m. Assume that \(\mathbb K\) contains a primitive m-th root of unity. Define \(\mathfrak g_1:=\mathfrak g^{\sigma}\), the fixed point algebra for σ. We prove the following generalization of a well known result for symmetric Lie algebras, i.e., for m=2: (a) \((\mathfrak g,\mathfrak g_1)\) satisfies ( 1); (b) For m prime, \((\mathfrak g,\mathfrak g_1)\) satisfies ( 2).     

Generalizing a theorem of Gagola and Lewis characterizing nilpotent groups

Gagola and Lewis proved that a finite group G is nilpotent if and only if \(\chi (1)^2\) divides |G :  \(\mathrm{Ker}\) \(\chi |\) for all irreducible characters \(\chi \) of G. In this paper, we prove the following generalization that a finite group G is nilpotent if and only if \(\chi (1)^2\) divides |G :  \(\mathrm{Ker}\) \(\chi |\) for all monolithic characters \(\chi \) of G.