arXiv subjects

Rauan Akylzhanov

Publications and source records attributed to Rauan Akylzhanov.

11 recordsLinked to original sources

arXiv2026

KazByte: Adapting Qwen models to Kazakh via Byte-level Adapter

Large language models fragment Kazakh text into many more tokens than equivalent English text, because their tokenizers were built for high-resource languages. This tokenizer tax inflates compute, shortens the effective context window, and weakens the model's grip on Kazakh morphology. We propose to bypass the tokenizer entirely by feeding raw bytes through a small adapter that learns to speak the internal language of a frozen Qwen2.5-7B. Once the adapter is trained, we freeze it and fine-tune only the attention layers of Qwen on Kazakh text. Our central hypothesis is that this two-stage process -- first teach the interface, then adapt the model -- should match or exceed the accuracy of the original Qwen2.5-7B on standard Kazakh benchmarks. This report describes the ByteKaz architecture and training protocol. Empirical validation is ongoing; this version stakes the design and hypotheses for the record.

cs.CL↗

arXiv2026

Hormander-Mikhlin type theorem on non-commutative spaces

In this paper, we introduce a Fourier-type formalism on non-commutative spaces. As a result, we obtain two versions of Hormander-Mikhlin Lp-multiplier theorem: on locally compact Kac groups and on semi-finite von Neumann algebras, respectively. In the simplest case our result coincides with a sharp version of the classical Hormander Lp-multiplier theorem, which was obtained by Grafakos and Slavikova in [11]. Finally, we present some applications to the evolution equation in non-commutative setting.

math.OA↗

arXiv2022

Norms of certain functions of a distinguished Laplacian on the $ax+b$ groups

The aim of this paper is to find new estimates for the norms of functions of the (minus) Laplace operator $\cal L$ on the `$ax+b$' groups. The central part is devoted to spectrally localized wave propagators, that is, functions of the type $ψ(\sqrt{\cal L})\exp(it \sqrt{\cal L})$, with $ψ\in C_0(\mathbb{R})$. We show that for $t\to+\infty$, the convolution kernel $k_t$ of this operator satisfies $$ \|k_t\|_1\asymp t, \qquad \|k_t\|_\infty\asymp 1, $$ so that the upper estimates of D. Müller and C. Thiele (Studia Math., 2007) are sharp. As a necessary component, we recall the Plancherel density of $\cal L$ and spend certain time presenting and comparing different approaches to its calculation. Using its explicit form, we estimate uniform norms of several functions of the shifted Laplace-Beltrami operator $\tildeΔ$, closely related to $\cal L$. The functions include in particular $\exp(-t\tildeΔ^γ)$, $t>0,γ>0$, and $(\tildeΔ-z)^s$, with complex $z,s$.

math.FA↗

arXiv2019

Re-expansions on compact Lie groups

In this paper we consider the re-expansion problems on compact Lie groups. First, we establish weighted versions of classical re-expansion results in the setting of multi-dimensional tori. A natural extension of the classical re-expansion problem to general compact Lie groups can be formulated as follows: given a function on the maximal torus of a compact Lie group, what conditions on its (toroidal) Fourier coefficients are sufficient in order to have that the group Fourier coefficients of its central extension are summable. We derive the necessary and sufficient conditions for the above property to hold in terms of the root system of the group. Consequently, we show how this problem leads to the re-expansions of even/odd functions on compact Lie groups, giving a necessary and sufficient condition in terms of the discrete Hilbert transform and the root system. In the model case of the group SU(2) a simple sufficient condition is given.

math.CA↗

arXiv2018

Contractions of group representations via geometric quantisation

We propose a general framework to contract unitary dual of Lie groups via holomorphic quantization of their co-adjoint orbits. The sufficient condition for the contractability of a representation is expressed via cocycles on coadjoint orbits. This condition is checked explicitly for the contraction of ${\mathrm SU}_2$ into $\mathbb{H}$. The main tool is the geometric quantization. We construct two types of contractions that can be implemented on every matrix Lie group with diagonal contraction matrix.

math.RT↗

arXiv2017

Smooth dense subalgebras and Fourier multipliers on compact quantum groups

We define and study dense Frechet subalgebras of compact quantum groups consisting of elements rapidly decreasing with respect to an unbounded sequence of real numbers. Further, this sequence can be viewed as the eigenvalues of a Dirac-like operator and we characterize the boundedness of its commutators in terms of the eigenvalues. Grotendieck's theory of topological tensor products immediately yields a Schwartz kernel theorem for linear operators on compact quantum groups and allows us to introduce a natural class of pseudo-differential operators on compact quantum groups. As a by-product, we develop elements of the distribution theory and corresponding Fourier analysis. We give applications of our construction to obtain sufficient conditions for $L^p-L^q$ boundedness of coinvariant linear operators. We provide necessary and sufficient conditions for algebraic differential calculi on Hopf subalgebras of compact quantum groups to extend to the proposed smooth structure. We check explicitly that these conditions hold true on the quantum $SU(2)$ for both its 3-dimensional and 4-dimensional calculi.

math.OA↗

arXiv2017

$L^p$-$L^q$ multipliers on locally compact groups

In this paper we discuss the $L^p$-$L^q$ boundedness of both spectral and Fourier multipliers on general locally compact separable unimodular groups $G$ for the range $1<p\leq q<\infty$. We prove a Lizorkin type multiplier theorem for $1<p\leq q<\infty$, and then refine it as a Hörmander type multiplier theorem for $1<p\leq 2\leq q<\infty$. In the process, we establish versions of Paley and Hausdorff-Young-Paley inequalities on general locally compact separable unimodular groups. As a consequence of the Hörmander type multiplier theorem we derive a spectral multiplier theorem on general locally compact separable unimodular groups. We then apply it to obtain embedding theorems as well as time-asymptotics for the $L^p$-$L^q$ norms of the heat kernels for general positive unbounded invariant operators on $G$. We illustrate the obtained results for sub-Laplacians on compact Lie groups and on the Heisenberg group. We show that our results imply the known results for $L^p$-$L^q$ multipliers such as Hörmander's Fourier multiplier theorem on $\mathbb{R}^{n}$ or known results for Fourier multipliers on compact Lie groups. The new approach developed in this paper relies on the analysis in the group von Neumann algebra for the derivation of the desired multiplier theorems.

math.RT↗

arXiv2016

Fourier multipliers and group von Neumann algebras

In this paper we establish the $L^p$-$L^q$ boundedness of Fourier multipliers on locally compact separable unimodular groups for the range of indices $1<p\leq 2 \leq q<\infty$. Our approach is based on the operator algebras techniques. The result depends on a version of the Hausdorff-Young-Paley inequality that we establish on general locally compact separable unimodular groups. In particular, the obtained result implies the corresponding Hörmander's Fourier multiplier theorem on $\mathbb{R}^{n}$ and the corresponding known results for Fourier multipliers on compact Lie groups.

math.OA↗

arXiv2016

Hardy-Littlewood inequalities and Fourier multipliers on SU(2)

In this paper we prove a noncommutative version of Hardy-Littlewood inequalities relating a function and its Fourier coefficients on the group $SU(2)$. As a consequence, we use it to obtain lower bounds for the $L^p-L^q$ norms of Fourier multipliers on the group $SU(2)$, for $1 < p \leq 2 \leq q < 1$. In addition, we give upper bounds of a similar form, analogous to the known results on the torus, but now in the noncommutative setting of $SU(2)$.

math.FA↗

arXiv2015

Hardy-Littlewood, Hausdorff-Young-Paley inequalities, and Lp-Lq Fourier multipliers on compact homogeneous manifolds

In this paper we prove new inequalities describing the relationship between the "size" of a function on a compact homogeneous manifold and the "size" of its Fourier coefficients. These inequalities can be viewed as noncommutative versions of the Hardy-Littlewood inequalities obtained by Hardy and Littlewood on the circle. For the example case of the group SU(2) we show that the obtained Hardy-Littlewood inequalities are sharp, yielding a criterion for a function to be in $L^p$ on SU(2) in terms of its Fourier coefficients. We also establish Paley and Hausdorff-Young-Paley inequalities on general compact homogeneous manifolds. The latter is applied to obtain conditions for the $L^p$-$L^q$ boundedness of Fourier multipliers for $1<p\leq 2\leq q<\infty$ on compact homogeneous manifolds as well as the $L^p$-$L^q$ boundedness of general (non-invariant) operators on compact Lie groups. We also record an abstract version of the Marcinkiewicz interpolation theorem on totally ordered discrete sets, to be used in the proofs with different Plancherel measures on the unitary duals.

math.FA↗

arXiv2015

Net spaces on lattices, Hardy-Littlewood type inequalities, and their converses

We introduce abstract net spaces on directed sets and prove their embedding and interpolation properties. Typical examples of interest are lattices of irreducible unitary representations of compact Lie groups and of class I representations with respect to a subgroup. As an application, we prove Hardy-Littlewood type inequalities and their converses on compact Lie groups and on compact homogeneous manifolds.

math.FA↗